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University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 19 1997
Please answer six (6) of the seven (7) questions using complete sentences and well thought out responses
You will be graded on the correctness logic and clarity of your answer All questions are equally
weighted There are four (4) pages to this exam
I The Pep Pill Company producer of pep pills sleeping pills and tranquilizers is considering
marketing an inexpensive pill call Vita-Power which will supply the user with the minimum adult
requirement of vitamins A and B A bottle of I00 Vita-Power tablets requires a minimum of 10 units of
vitamin A and 40 units of Vitamin B
After considerable research the Pep Pill people found that a unit of calf-liver extract contaiiis 14 of
a unit of vitamin A while a unit of soybean oil contains I unit of vitamin A Similarly they found that a
unit of calf-liver extract contains 2 units of vitamin B while a unit of soybean oil contains 112 of a unit of
vitamin B The Pep Pill Company also notes that a unit of calf-liver extract costs $200 and a unit of
soybeanmiddotmiddotoil costs $300 The firm wants to minimize the cost of supplying the minimum requirements of
vitamins A and B per bottle of Vita-Power tablets
a) Set up the primal linear programming problem corresponding to the above information
Provide an economic interpretation of the problem and carefully defme all variables
b) Graph the constraints of the problem and identify the area of feasible solutions
Write down the dual linear programming problem corresponding to the primal problem you
developed in part a) Provide an economic interpretation of the problem and carefully
define all variables
2) Much of your microeconomic theory has been developed in terms of production functions cost
functions and profit functions
a) Give a precise mathematical and economic definition of the cost function and profit
function Be sure to carefully define all the terms and symbols used
b) What kind of functional form would you specify for these three functions and why
c) Carefully explain how you would go about estimating such functions including the type of
estimator that you would use
d) It is sometimes said that All economics questions eventually reduce to questions of
demand and supply Do your answers to a) b) and c) above relate to demand and supply
issues Why or why not
62 1bull
d)
ii)
3) The United States is a large importer of beef With this in mind answer the following questions
a) Is the import (excess) demand for beef more or less elastic than domestic demand Explain
why
b) In general what is the relationship between domestic demand and import demand
Support your explanation with a graph and some mathematics
c) List the three factors which are the most important determinants of the elasticity of import
demand and discuss each briefly
4) Define the following terms in words and with some mathematics or a graph being careful to define
any notation that you use
a) Giffen good
b) Engel curve
c) Cross-price elasticity ci demanl
Consumers surplus
e) lndiect utility functirn
5) The following questions all deal with basic issues in econometrics
a) What is heteroskedasticity Draw a picture of a heteroskedastic error term
b) What statistical problems if any does heteroskedasticity create for the ordinary least
squares parameter estimates
c) Consider the linear model Y = 30 + 3 X1 + e where E[e] = 0 and Var[e1] 1i = 12 n Transform this model so that the new error term say e is homoskedastic
d) Suppose you are estimating the parameters of a multiple linear regression model with four
independent variables Explain carefully how you would go about testing the null
hypothesis that the last two independent variables can be omitted from the model Note that
this question is independent of the previous ones
e) Consider the following set of residuals
i) -23-10 -2 3 - 1 1
iii) 551-11
Which set of residuals can be ordinary least squares residuals Explain why Note that
this question is independent of the previous ones too
2
6) You have learned that if a monopolist can perfectly price discriminate (ie can first-ltlegree price
discriminate) then it will make the maximum profits possible and in the process capture all of the
consumers surplus Therefore it might appear that all consumers are worse off under this economic
arrangement This question is designed to probe this issue more deeply Consider the following graph
which should look quite familiaf to you by now
price
D
quantity Ym
The points A 8 and C represent three (3) different consumers For example point A on the demand
cuivemiddot represents a consumer who is willing to pay at most the price pA for a unit of the good that is point
A on the demand curve represents a consumer who has a reservation price p for a unit of the good
Points B and C can be interpreted similarly
a) Would a consumer represented by point A on the market demand curve be better off if the
monopolist charged only the simple (ie nondiscriminating) monopoly price Pm or if the
monopolist practiced perfect price discrimination Explain clearly
b) Would a consumer represented by point B on the market demand curve be better off if the
monopolist charged only the simple (ie nondiscriminating) monopoly price Pm or if the
monopolist practiced perfect price discrimination Explain clearly
3
c) Would a consumer represented by point C on the market demand curve be better off if the
monopolist charged only the simple (ie nondiscriminating) monopoly price Pmbull or if the
monopolist practiced perfect price discrimination Explain clearly
7) Assume that you own a house on which you make a mortgage payment of $200month that
moving costs are negligible and that apartments provide a set of housing services that are equivalent to
that provided by homes
a) If you can rent your house out for $250month and live in an apartment for $220month
should you move into the apartment and rent out your house Explain using some
numbers
b) If you can rent your house out for $250month and live in an apartment for $220month
but your mortgage payment is now $220month should you move into the apartment and
rent out your house Explain using some numbers
c) If you can rent your house out for $250month and live in an apartment for $220month
but your mortgage payment is now $1000month should you move into the apartment and
rent out your house Explain using some numbers
d) What do your answers to this sequence of questions reveal about the importance of the
mortgage payment for this problem
e) In general what must hold in this problem for renting out your house and renting an
apartment to be the correct response
4
(035) (095)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 26 1998
There are two sections to this exam Each section consists of four questions Follow the instructions for each section carefully Use complete sentences and well thought out responses when writing down your answers Note that if we cannot read your writing then we will assume that your answer is wrong All questions are equally weighted There are five pages to this exam
Section 1 This section consists of questions 1 ) 2) 3) and 4) Answer question 1 ) and then select two of the remaining three questions to answer Do not answer all four questions
1) Below is a highly simplified model of wine supply and demand Selected t-ratios are below each
estimated coefficient
Two-Stage Least-Squares Estimates of Deinand Q = -101 +026PWHAT
(210)
R = 055 DW=066 Two-Stage Least Squares Estimates of Supply
Q =-101+ l22PWHAT- 222 WR + 340PGRAPE + 123Qt-l(110) (056)
R2 =096 DW=l95
Auxiliary First Stage Regression PWHAT = 201 + 133WR + 440PGRAPE + 023Q_
R2 =0995 The variables are defined as follows
Q - Quantity of wine supplied and demanded
PWHAT- Price of wine from auxiliary regression
WR- Wage rate in wineries
PGRAPE - Price of wine grapes
There are (at least) 11 things wrong with this model and its estimates Can you find them List them by
number and indicate the consequence of each problem
middot 2) You have been asked to estimate a linear econometric model say Y =30 + 3 X + Jl Assume that
Jl = PJl_ + e (I)
2rsquowhere e is a normally distributed random variable with a mean of 0 and variance of u bull The above
model (1 ) is often termed autocorrelation
a) How do you test for autocorrelation
There are two kinds of autocorrelation- pure and impure The Durbin-Watson statistic can be indicative
of either kind
b) Show the consequences of both kinds of autocorrelation and how to correct for each
3) Consider the following primal (P) linear programming problem
max R = Ox + 8x2 XJxl
st tX1 + x2 5125
2x1 + 3x2 5 300 (P)
x 5 90
Xp X2 0
Answer the questions below being sure to explain your steps and reasoning
a) Graph the constraints of (P) and identify the primal feasible solution set ie the set of all
admissible solutions for (P)
b) Identify all the extreme points of the primal feasible solution set
c) Find the optimal solution to (P)
d) Determine the slope of the isorevenue curves sketch a few of them and verify your
answer in part c)
e) Write down the dual (D) linear programming problem to (P)
4) An individual investor has $70000 to divide among several investments The investments
available to this person are municipal bonds with an 85 annual return certificates of deposit with a 5
annual return treasury bills with a 65 annual return and a growth stock fund with a 13 annual return
The investments are all re-evaluated after one year ie the time frame for the investor is one year Each
investment has a different perceived risk to the investor so it is optimal to diversify
The following guidelines have been established for diversifying investments bull No more than 20 of the total investment should be in the growth stock fund
bull The amount invested in certificates of deposit should not exceed the amount invested in the other three
investment alternatives bull At least 30 of the investment should be in treasury bills and certificates of deposit
2
bull More should be invested in certificates of deposit and treasury bills than in municipal bonds and the
growth stock fund by a ratio of 12 to 10 The investor wants to know how much to invest in each investment alternative in order to maximize the
total return from all investments All $70000 is to be spent in this investment plan
Formulate the linear programming problem for this investor Defme all the variables constraints and
equations in the formulation
Section 2 This section consists of questions 5) 6) 7) and 8) Answer any three of these four questions Do not answer all four questions
5) Which of the following describes an externality and which does not Explain your reasoning
clearly in each case
a) A policy of restricted coffee exports in Brazil causes the US price of coffee to rise which
in turn causes the price of tea to increase
b) An advertising blimp distracts a motorist who then hits a telephone pole and damages the
car
c) In a homogeneous product Cournot duopoly an increase in the output of firm 1 lowers the
market price received by firm 2 for its product (due to the law of demand) ceteris paribus thereby reducing the profit of firm 2
6) Consider a competitive industry with a large number of identical firms each of which has a simple
cost curve of the form
C(q)= l +1 qgtO 0 q =O
where q 0 is the output of the firm Think of these costs in a long-run context ie if the firm produces
nothing it incurs no costs but any q gt 0 incurs the 10 set up cost Suppose industry demand is given by
Q0 =52-P
where Pis the market price of the good produced by the firms
a) Derive a supply curve for one of the identical fmns
b) Derive the industry supply curve given that there are n firms
c) What is the long-run equilibrium price in this industry Explain your answer and show any
calculations
3
e)
a)
d)
d) How many firms will populate this industry in long-run equilibrium Explain your
answer
f)
Given your solution for n from part d) assume demand shifts to Q0 =53- P What is
industry price and output in the short-run equilibrium
Given your result in part e) explain in words what will happen as the industry adjusts to
long-run equilibrium
7) The following is a demand equation for residential electricity consumption in the US estimated by
H Houthakker (Energy Jouma1980)
ln E = 0586 + 1388lnY-1118 ln PE + 0199lnH + 0483lnC +0566LnP0
where E is per capita electricity consumption in KWH per year Y is disposable personal income per
capita PE is the price per KWH of electricity H is heating degree days (deviation over the year of
temperature below 65deg) C is cooling degree days (deviation over the year of temperature above 75deg) and
P 0 is the price of natural gas per BTU
Based on this model what are the own price cross price and income elasticities of
demand for electricity in the US
b) A deregulation plan for the electricity industry is currently being implemented Suppose
electricity prices rise as a result of deregulation Based on Houthakkers equation will
consumers spend more less or the same on electricity following deregulation Explain
c) Many poor families live in apartments with electric heating and cooling A concern of
policy makers is that high energy costs make it difficult for poor families to adequately heat
or cool their homes Consider a typical poor family whose demand for electricity is given
by Houthakkers equation The family pays $010 per KWH and has per capita income of
$3000 ($12000 for a family of four) Given normal values for H C and P 0 the family
consumers on average 6000 KWH per year Energy experts have determined that 8000
KWH are needed to provide satisfactory heating and cooling in the familys climatic region
Consider two programs to eliminate the deficiency (a) a cash payments plan (b) an energy
subsidy program under which the family will get a rebate from the government for a
specific percentage of its monthly electric bill
(i) How large must the cash payment be to accomplish the plans goal
(ii) What percentage subsidy will achieve the programs goal
lllustrate the two plans using indifference curves for electricity and all other goods and the
budget constraint Begin with the situation where E = 6000 and show how either program
can attain the goal
A third possible program involves coupons that can be spent only on electricity Will e)
coupons entitling the household to buy 2000 KWH achieve the goal Explain
4
8 ) Consider a monopolist who sells in two markets The inverse demands and cost function for the
monopolist are as follows
I = 80-SQ market I
R =180- 20Q2 market2
C (Q)=50 + 20(Q + Q2) cost function
a) Find the firms profit maximizing price and output in each market assuming it is free to set
whatever price it wishes in each market Show your work
b) Assuming absence of any externalities what is the socially optimal output in each market
Explain your answer and show any calculations
c) Define the term deadweight loss and then calculate the deadweight loss from monopoly
power in each market
d) TrueIFalse Explain The firm in this problem is a natural monopolist
e) The following statement is true The firm in this problem is practicing price
discrimination Explain carefully and precisely what this statement means in terms a
noneconomist could understand
f) Suppose the government passed a law preventing the firm from practicing price
discrimination What price and output would the firm then set in market I Market 2
Provide a precise answer if you can showing all work but dont get bogged down At a
minimum explain in words how if at all this law would affect the firm
5
University of California Davi Jepartment of Agricultural and Resource bconomics
MS Comprehensive Exam September 151998
There are two sections to this exam Each section consists of four questions Follow the instructions for each section carefully Use complete sentences and well thought out responses when writing down your answers Note that if we cannot read your writing then we will assume that your answer is wrong All questions are equally weighted There are five pages to this exam
Section 1 This section consists of questions 1 ) 2) 3) and 4) Answer question 1 ) and then select two of the remaining three questions to answer Do not answer all four questions
I) Below is a highly simplified model of wine supply and demand Selected t-ratios are below each estimated coefficient
Two-Stage Least-Squares Estimates of Supply
Q = -101 + 026PWHAT(210) (220)
R2 =035 DW=041 n=21
Two-Stage Least Squares Estimates of Demand
Q = IOI -122PWHAT + 222NCOME +340CPIALL + 0753Q_1(23S) (110) (056) (400)
R2 =096 DW=05 Durbins h- statistic= 083 n=21
Auxiliary First Stage Regression PWHAT = 201+ 133NCOME + 440CPIALL + 023Q_1
R2 = 067 DW= 187
Regression On Residuals From Demand Equation
Jl = 05+155NCOME +055CPIALL (620) (ll)
R2=072 The variables are defmed as follows
Q - Quantity of wine supplied and demanded PWHAT - Price of wine from auxiliary regression INCOME- Per capita personal income CPIALL - Consumer price index of all items
Examine the above model for econometric problems In particular you should look for
I autocorrelation 2 heteroscedasticity
2)
3 omitted variables
4 multicollinearity
5 identification
6 simultaneous equation bias
For each of the above possible problems explain why you think there is or is not a problem Explain
what should be done to remedy the problem
You have been asked to estimate a linear econometric model say Y = 30 + 31X + Jl Assume that
var(Jl) = a2X (I)
where Jl is a normally distributed random variable with a mean of 0 and a variance given in Eq () The
above model (I) is termed heteroscedasticity
a) Clearly explain the problems that heteroscedasticity creates
b) How might you go about detecting the presence of heteroscedasticity in the above linear
model
middotc) How might you correct for the heteroscedasticity given that it occurs in the form ofEq (1)
in the above linear model Be specific and give details
3) Answer the following jive TrueFalseExplain questions concerning a general linear programming
(LP) problem Please note that the explanation is worth the bulk of the credit
a) An optimal solution to aLP problem uses up all of the limited resources available
b) A point on the boundary of the feasible set (or region) satisfies all of the constraints of the
LP problem
c) The optimal solution of an LP problem always occurs at an extreme point of the feasible
set
d) If the first COilStraint of aLP problem evaluated at given value xo of the decision vector
has a zero value for its corresponding slack variable then the point xo lies on the boundary
of the feasible region
e) If the shadow value (or dual variable) corresponding to a given constraint is positive than
that constraint is binding
4) PROTRAC INC produces two lines of heavy equipment One of the product lines termed
earthmoving equipment is essentially for construction applications while the other line termed forestry
2
equipment is destined for Ute timber industry The largest member or ihe earthmoving line (the E-9) and
the largest member of the forestry equipment line (the F-9) are produced in the same departments and with
the same equipment Using economic forecasts for the next month PROTRACs marketing manager has
judged that during that period it will be possible to sell as many E-9s and F-9s as the flrm cam produce
Management must now recommend a production target for the next month
Making this decision requires that the following major facts be considered
bull PROTRACs net revenue is $5000 on each E-9 that is sold and $4000 on each F-9
bull Each product is put through PROTRACs machining operations in department A and department B
bull For next months production department A hasl50 hours available and department B has 160 hours
available Each E-9 uses 10 hours of machining in department A and 20 hours of machining in
department B Each F-9 uses 15 hours of machining in department A and 10 hours of machining in
department B
bull In order for management to honor an agreement with the union the total labor hours used in next
months testing of finished products cannot fall more than 10 below an arbitrated goal of 150 hours
The testing is performed in a third department and has nothing to do with the activities in departments
A and B Each E-9 is given 30 hours of testing and each F-9 is given I0 hours of testing
bull In order to maintain the current market position senior management has decreed the operating policy
that it is necessary to build at least one F-9 for every three E-9 produced
bullmiddot A major dealer has ordered a total of at least five E-9s and F-9s in any combination for the next
month and so at least that many must be produced
Given the above considerations managements problem is to decide how many E-9s and many F-9s to
produce next month so as to maximize net revenue
Formulate the linear programming problem for PROTRAC INC Deflne all the variables constraints
and equations in the formulation
Section 2 This section consists of questions 5 ) 6) 7) and 8) Answer any three of these four questions Do
not answer all four questions
5) A simple monopolist faces the linear inverse market demand curve given by p(y) a - by where
a gt 0 b gt 0 and y is the output of the monopolist In addition the monopolist faces the total cost
function given by c(y) Fe where k gt 0 Finally the government pays the monopolist a subsidy of
s gt 0 dollars per unit of output produced in order to encourage the monopolist to produce more output and
sell it at a lower price
a) Set up the proflt maximization problem facing the monopolist
3
--------
b) Find the monopolists profit maximizing level of output say y = ym(abks) as well as its profit maximizing price say p = pm(abk s) How do you know you have found the profit maximizing solution
c) Will an increase in the subsidy achieve its desired goal of increasing the monopolists output and lowering its price Show your work
d) Provide an economic interpretation of the above comparative statics e) How large would the subsidy have to be in order to get the monopolist to produce the
Pareto efficient level of output and charge the Pareto efficient price
6) The following questions pertain to a production setting characterized by the usual neoclassical production function of the form Q = f(xlx2x ) where Q is output and (x1x2x ) are inputs
a) Define compare and contrast the following terms from production economics i) Decreasing returns to a variable input and ii) decreasing returns to scale (Decreasing returns to scale is also called
diseconomies of scale) b) What economic forces cause decreasing returns to a variable input and what economic
forces might cause decreasing returns to scale c) Given the production function as specified show mathematically how you would measure
the returns to the variable input x1 and the returns to scale d) Suppose you were studying the question of whether small farms could compete effectively
against large farms in the market for the product Q Would either or both of the concepts of returns to a variable input and returns to scale be valuable in your analysis Explain
7) The following are demand and supply curves that apply to a competitive industry
Q0 = 250- SOP Qs = ()P
a) Derive the market equilibrium price and quantity b) Is the equilibrium you derived in (a) a short-run equilibrium a long-run equilibrium or
could it be both Explain your answer in detail demonstrating your knowledge of behavior in competitive markets and the distinction in these markets between the short and long run
c) Politicians are floating a proposal to place a per unit tax of $ 100 on this commodity Producers in lobbying against the tax argue that they will have no choice but to pass the
full tax on to consumers and therefore the tax if enacted will only harm consumers Evaluate the producers argument concerning the tax Use either algebra (preferred tool) or
4
--------------
------------------------------ X
bull
a)
a graph to support your argument If you use a graph label it carefully using the demand and supply equations Q0 = 250- 50P and Qs = ( )P
8) The following graph depicts a consumers Marshallian demand curve for the commodity X The market price of X is given as
p
Demand Curve
Given identify the consumers surplus on the graph and give an intuitive definition of consumers surplus
b) Assuming X is a normal good for this consumer draw the consumers Hicksian demand curve given price
c) Explain the difference between Hicksian and Marshallian demand curves
d) Suppose price increases to P_ Use Hicksian demands to identify the consumers compensating variation and equivalent variation due to the price change Label carefully
5
University of California Davis Department of Agricultural and Resource Economics
MS Exam
July 2 1999
Directions You have three hours to write this exam Please write legibly Answers that cannot be read are regarded as incorrect answers This exam consists of two parts In part I quantitative methods you must answer question 1 Then choose to answer either question 2 or question 3 In part II microeconomics answer three of the four questions You should thus answer five questions in total two from part I and three from part II Do not under any circumstances answer more than five questions
Part I Answer 1 estion I and either Question 2 01 luestion 3
l You are constructing a si mple model of cotton su pply for a short-term project you are working on and h ave formulated the following conceptual model
where y5 i s bales of cotton ps i s the price paid to growers w1 is the labor wage rate w2 i s the price of water WJ i s pesticide price and DO is degree-days a measure of the growi ng conditions during the season
(a) A nother variable you might include in thi s speci ficati on is annual rainfall (R) whose coefficient i f included in the model would be a6 While you are sure th at all th e other va ri ables in the model are part of the correct speci fication you are not sure about annual rainfall As an applied econometrician explai n the tradeoffs made to the properties of you r estimates i f you include or exclude thi s vari able from the specification
(b) One must always be concerned with measurement error i nempirical models Explain th e effects on the properties of your estimates of (a) measurement error in th e dependent variable vs (b) measurement error in one or more explanatory vari ables
(c) You esti mate th e model wi th R included and the results are
yS = -6354 + 8727ps - 0 579lw1 - 7907w2 - 09045w3 + 769000 + 0045R ( -0 79) (2 36) ( -177) (-234) ( -190) (157) (045)
R2 = 0819 n =56
where th e Students-t statistics are gi ven in parentheses and the following sums of squared errors resu lt
Total SSY=660 9 Model SSR=5412 Errors SSE= l l 97
In answering the following part s justify your answers with quantitati ve test results using th e attached tables Quali tative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 0 5 level
(ii) Is the overall model significant Th at i s i s the hypothesis th at a1 = a2 = a3 = a4 = as = a6 = 0 rejected
(iii) The correlation between the price of water (w2) and the price of pesticides (w3) i s zero Do these two vari ables have the same effect on cotton supply i e are their coeffi cients equ al
2 Suppose you have the following econometric model
where y = yieldacre x1 = wateracre x2 = (wateracre)2 x3 = fertilizeracre
(a) For each of the following questions set up the appropriate hypothesis test and provide a brief explanation
(i) Does water have any influence on yieldacre
(ii) Assuming an unconstrained (interior) solution is wateracre = 300 significantly different from the amount that yields the maximum yield per unit water
(b) Suppose you work in a governmental agency and that your boss is not trained in economics or statistics and thus she does not know what e N(O all) means Explain-
in words the meaning and significance of this expression
3
a Given the regional farm modeling problem
Max subject to
px- cx Ax b
where x is an n x I vector of production activities p and c are vectors of prices and variable cost coefficients respectively b is an m x 1 vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Comment briefly on the connection between the dual problem and the Dual Complementary Slackness condition
(iii) If you had to calculate the derived demand for a single input b describe the type of function you would obtain from the problem
b Now use the information that the product prices p are determined according to the following demand relationships
p =a + Dx
Assume that the individual farmers act as perfect competitors
(i) Show how you would reformulate your problem and give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit )
(ii) Derive the dual to the problem in part b(i) showing your steps clearly
Part II Answer of the 4 questions in this Section _
I For some goods consumption is subject to more than the usual single constraint on money income time can also be a constrai nt Consider a consumer with utility f unction u(x) defined on goods x = (x1 xn) subject to th e two constraints T 2 tx and M 2 px where T is the total amount of time avai lable and M is th e total amount of income avai lable and t = (t1 tn) and p = (p o pn ) are the time and money prices per unit of consumption of each good
(a) Write th e primal utility max i mi zation problem for th is consumer What interpretation do you attach to th e sh adow values on each constraint
(b) F rom your answer to part (a) ex plain how the consumer s choices reveal her money value of time Develop an ex pression f or the marginal money value of time from th is pri mal problem
(c) Under what conditions would the problem revert to the standard consumer choice problem subject to a single money budget constraint
(d) Suppose th at the shadow value on the money budget constraint is zero W rite the dual expenditure mi nimization problem for thi s case Provide a brief interpretation in words of thi s dual ex penditure f unction Also provide an interpretation for the sh adow value of the constraint for th is problem
2 Assume t hat t he US wheat mar ket can be considered in isolation and is charact erized by
yS = sops - 20 wl
y0 = 60 - 12 5p0 +2M
where
and
ys is quantity of wheat suppl ied in millions of bushels per year p5 is the price recei ved by suppliers in dollars per bushel y0 is quantity demanded (in millions of bushels) by consumers p0 i s the price paid by consumers also in dollars per bushel w1 is the price of a composite input to wheat product ion M is consumer income in t housands of dollars
For purposes of this analysis consumer income i s M = 45 and the composite input price is w1 = 5
(a) What are t he equilibrium f ree-mar ket price and quant it y of wheat
(b) Is t he supply of wheat at this point price-elastic or price-inelast i c Ex plain your reasomng
(c) W h at is t he t otal net economic sur plus f rom production of wheat and how middotis it distr ibuted bet ween suppliers and demanders
(d) A var iet y of schemes are used in agricult ural markets to enhance producer welf are C onsider a pr ice support of $6 per bushel Explain t he effects of such a policy on the dist ri bution of welfare and economic efficiency Support your answer with quant itat ive est imates of t hese effect s
jgtrice
3 Suppose a monopoly fum faces the following linear demand curve P = a- bQ where P is price and Q is quantity produced and sold The monopolists cost function is
(1) c(Q) = cQ for Q Q c(Q) qo for Q gt Q
where Q is a capacity constraint
a Solve mathematically for the monopolists profit maxumzmg and output as functions of the parameters a b and c Show your work Also illustrate yourresults graphically labeling carefully all relevant information
b Suppose the function P(Q) from part (a) is the demand in the domestic market and that the monopolist can also sell unlimited quantities into the export market at price P E gt c Assuming that quantities exported cannot be resold into the domestic market show both mathematically and on your graph from part (a) how much the firm will sell into the domestic market and into the export market
c Consider now a potential entrant into this market For simplicity ignore the existence of the export market To be able to produce in this market an entrant must incur a one time set up or entry cost of E gt 0 It can then produce according to the cost function in (1) Describe in words andor diagrams how you would model this entrants decision problem You should consider the role of the incumbent monopolist in framing your answer
4 Consider a food processing firm that produces processed product Q from a raw agricultural product Rand a composite marketing input L which we will call labor
a Write a production function and draw some rough sketches of isoquants for each of the following possible production scenarios
1 There is no substitution between R and L in producing Q
ii The production relationship is Cobb-Douglas
b Suppose the true production relationship is Cobb-Douglas and that this firm produces Q exclusively for a large retail grocery chain It has a contract for 1999 to deliver Q units of product to the retailer Assume the firm is a perfect competitor in the markets for its inputs Set up the firms optimization problem given this information and derive the first-order conditions for the problem Defme any notation or terminology you need to introduce Also illustrate the optimization problem with a graph
c Discuss the firms demand function for R given the conditions of the problem in part (b) Specifically what variables determine how much R the firm will utilize
d Re-express the problem in part (b) and take first-order conditions assuming now that the firm is the only buyer of the farm input R which is produced according to the following inverse supply function
w = a + 3R a3 gt 0
OJ 04
J ---------
Amiddot __ __
09
0QgtI
4
2257
Jl59
412
14JK
446 -1564
4649
41161
29
0
495
4974
04711
164 9-9
267 2967
4170
llll
99
09X7
76
714
H
middotmiddot 74(
4791
4978
49119
2454 2764
1051
1 115
1441
IIIOK
2157
1577
U74
1517
5
4545
46JJ
4767
4974
4990
middot-
92()
J 7J7 717
2447 ll 4
middot55
4 144
19
24
1699
677
676 SlATISliCI1 tA9LFS
lA Rlf lgtl Areas under the standardized normal distribution
Example
Pr (0 z 196) = 04750
Pr (z 196) = 05 - 04750 = 0025
TABLE 01 Percentage points of lhe t distribution
Example
Pr (r gt 2086) = Oo25
Pr (r gt 1725) = 005 for df = 20
Pr(ltl gt 1725) = 010 0 I 715
01$ 010 middot-00
001 0 00010015010 002 0010 0002ozo
J 07H 127116I 11216 14 61657 IIUI1nm
22 279925IXH6 6965UK16 4 lP 00 01 01 1 0765 I 6X 251 lIK2 4541 51141 102 JJ
4 074 I 15J 21J1 176 461 moo 00411no
Ol 0middot1111
(Xl)lt) 11119 059UIO 0161 t tiiCW 029 0279 1650675 0751 s 1476 2015 511910727 25710596 0616 4010517 115 7
071 II 1064 IIOJ 1141 52111106 bull 1440 1941 l7H7 0791 UKl 0171 091ll 09411
149914KU 1517
17tMl 1406 299K J7XS7 0 711 1415 lH95 2165116111179 1217 1155 129l
1591 161M 1664
1915 1950 19115 W19
10611144 1179 K u 706 lW7 11601772 451111196
212 071) 421172054 219() 122420XX UHl 111 2262 21i21 1251)-9
0700 J72 1112 222K 2764 11692422
1021 214 25492-IK6 102157 2X92291
2794 2121 21152 4025II 0697 1361 1796 22111 2711lt lit
2611 27041gt1 25XU ll7K 1106 1111 190 0695 1156 17H1 21 79 26111 lOSS
ll40 11111 1910
2650 2M9 J65 l119 IJ 0694 1350 1771 2160 1lt5211112311lt6 21 lJH 164
145 1761 21J5 2614 1621 14 069214115 55-1 2977 7117 461 150111 0 1411
l711I 2602 2 94717011 770 790 Kill lX0 15 0691 1141 175 729 21 lfgt65 16116 JHHH
I I
11
1641
1144012
J997 2921907 JOISJ9-U 96 WKO 1 6 0690 bull IJJ7 1 746 2120 2511 l6M6 925 JK(fl 0619 11114011 4147 4162 4177
4194066 41H l6462567 2gt19H411 17 1740 21W40994(19 406 068H UJO 174 210142164192
I
4279 4292 6102X7K4265 25524251 -11117 4222 4441 068H U211 1 729 209J 253944294 112 494 J406 44111 579lK61LJ5 4157
20 0687 1325 I 715 21186 252K 14545115 4515 4515 455 5524474 44114 4495lh 4J 2 0616 121 1711 1010 251 H l517211 I 4571 45112 4591 4599 4616 4625
4554 505J67H A6K6 4691 4699 4706 22 06K6 I Jl l 1717 2014 l5HK 1XI94656 4664 4671I K 4641 41151K074756 4761 17144744 4750 2J 06KS 1319 250020694716 412 47 111 9 470 4719
41101 41112 4M17 14924791 AKOJ20 06K5 IJ UI 2797 1 4671711 20644772 47711 4710 47KK
1
I 4K21 1 4504Kl0 4104 41UH 41141 41146 4150 4H54 4H57 2l 06114 1316 1708 2060 24115 271741116 J4JS1706 20 56 2479 27794K6H 4171 4X75 4H7H 4HIH 4Hol4 4HH7 middot 4K90 26 0614 IJIS41164
J906 4911 06M4 14212471 27714913 4916 170 4909 27 L14 10524196 489R 4901 49044191 06114925 4927 4929 4931 492 4914 4916 140K2467 27612M U13 1701 204849111 4920 4922
4941 4941 4945 4946 494K 4949 4951 4951 IJII JJ9629 068) 2462 2756204S49111 494025
UJO 1697 2042 2457 17504QSS 4964 lJK S4961 JO 06KJ4960 4961 4961
496 J951J4956 49576
7 10 06KI 10749 1 6K4497 241J i7044970 4971 49724967 496K 1021J966 60 0679 129611 4981
4979 4979 49SO 16714977 2 190 26604976 4977Igrave 12122000 4975
l Jfi) 1090
49864914 4914 4985 498S 4986 120 0677 11M9 21671651 19KO 2J58
49111 49H9 49M9 4990 _ 1 1810674 164S 1960 1 126 1576 49H I 49K2 4982 498)
49X7 49K7 49117 49111
N This tahlc gives the area in the riahtmiddothand tail or the distribution tic bull t 0) But since the normal di5tribumiddot Nolt 111e smaller probability shown at the head of each column is the area in one tail the 1in is symmetrical about r- Othe area in the ldtmiddothlnd tail is the same u the area in the corresponding right-hand larser probability is the area in both tails bullbullil rltgtr eumplc PI- 196 S z S 01-04750 TMrdore -196 S t S 196)-2(047S0)-09S Sourr From E S Pearwn and H 0 Hartley eds bull Blomtuila TabftJot Slulislidans vol I
ld ed table 12 Cbullmbridgc University Prns New York 1966 Reproduced by perminion of the middot
editors and trustees of Biomeria
bull bull bull bull 10 II 30 110 100 500
(
I
n
I)
4]1
7 71
4 20
116
1 H4
4 1
1 111
4 1
1
U4
1 75
192
1 gt6
44
2 54
244
)07
142middot
2 2
41J
144
U1
199
245
l-14
2 1
14)
161
190
1 4 1
25J
)70
137
195
2)7
U 5
190
229
1 59
245
U 7
I 91
1 15
96
142
194
1 94
l2
IJ1
117
174
104
1 0
190
U4
109
284
Ill 2 71 2JJ
1 77
194 175
i-1
174
149
157
I y I 7
1 94 1 gt9
194
1 9)
174 159
145 1 4
195
119
179 147
682 AIPFNHIX ll
TARLE Ol l lpper percentage poinls or the F distribution (continut-d)
I I
STATISTICAl TA8LfS 681
df for dl lor llllmtntor N
100 N
I 16 111 I 12 I y I 2ltI 10 1 10 111 1 1 1 1 111 I Y 1 41 1 4 1 1-10 I 19 1 19 UHI middotW 1171411 147 145
I 176 IJl 170 IUI 167 165 I M 161 1 bull1 101 1 97 191 1 90 11111 1116Hi 4 l)
2 15 212
Ill 1117 201 I I I M4 I IIII I I II
2 911 1 II 1 1 1gt 67 251 11 an z llbull 2 1J ll I 26 1 1 11 1 2
191 lOS 2Kl 1 2 55 146 2 -m 24 2 lU 226 2 2l
US1 i9 l-IS S72 4112 4 1 1 199
I 1 I J I 110 I Y 12Y I 211 I M I 7 1 7 I 6 1 15 1 11 U9 UK IJK 1middot17 U6
17M I 71 1 101 167 1 162 161 I I S1 n6 I 51 IU1115 I ll 1 4 1 1401 19 1 47 146 1 44
I 98
2111 191 1 1111 1 19 210 1 042B 9 I H6 1 gt4 1 MO 119 1 1 1 1 1) 1 1 1 211 2111 2JO 22S 21 2111JOI 2SI 242 2 16 11114 16 140
2M9 2 66 UK 2 40 111 1 1 1 2-1 2 1 Ul167 middot 1 16 126 117 109 ill2 H I 472 4 12 01
114 I 12 111 1 10 129 I 211 UK 16 I 26 126 I 25 I 2 I 42 1 4 1 19 us I H 17 l l6 2 1 111 146 us 1 16 I 1611 165 1 61 I 5Y UK us IS-1 I S1 I S l I t HIIll 11111 186 2 II 1 2 1 11 2 1111 11111 Y I 151 107 199 190 1-JIS 1112 1111) 1 16 1 1S I 71 I 7 1 211 222 2111 215Ull OS 219U7 198 1 14 2 t7 2111 2511 lSO 242 1 16 2 225 22J 219 216 2 IJ UH l09 )02 196HI 129SSl 41gt4 41401
IJJ I I 1 10 129 1 18 117 127 126 1 1gt 125 124 1 24 136 usI 41 141 140 119 UK1 111 146 145 1 69 1 66 16 159 I S1 156 I 51 152 I SO 149 1 19 Ill 1114 1 812 11 1 1 229 216 1116 190 187 100
H14 196 1 9 1 1 81 1 112 119 1 71 I 71 1 11 1 69 167 165 0)219 2 15 1 1 1 JN 295 1 1 1 2242)6 229 V5 1 1 144 IS 210 2 26 1 19 1 1 1 2 1 1 109 U6 U l
0 1 407 I 75 )51 U6 123 312 101
Ul l ltJ 129 Ull 127 1 26 126 IH 1 2-1 1 2-1 12J t n l iM I 45 1 44 1 41 U9 Ull IJ6 us IB
172 167 1 64 ltll I H 15-1 Ul I SU I 1 lt7 1 6 10 10IIIS 182 179 171I 10 11111 1 22K 114 20S 1911 19) 188
2111 1 91 I H4 179 1 16 1 14 110 I M 1 IM I 62 O)1 41 lB 227 221 216 213OS l 2 291 169
270 2-17 210 2 2 1 IJ I I 207 2 Ul ] 0 1 147 J0 1 1 7 )07 298 2910 1 1 56 5 19 451 40
110 I 126 125 124 Ut 12 Ul 121 119 1 19 25IJ6 us 1 14 IJJ Ul U l140 11914425
1117 1113 179 1 66 161 157 1 )4 I l l 14K 141 142 1-11 119 I 111
176 173 1 71IllI 21)9 1 00 192 IM4 179 1611 1 64 I us I 5J U l 100I 21Oi
01
1R4 261 2 14 225 218 2 12 1 08 252 217 229 220 2 1 1 1116 202 1 94 I 92 I IM1 01 664 II 181 lSI 129 1 12 199 2119 280 27)511S
127 125 1 14 122 121 1 11l 119 1 1 7 1 1 1 1 16 1 1 5 1 1 5 I J J 112 I l l IJO 119 119us 1 41 1 4 1 UK 160 154 Ul 1411 144 141 140 116 I 15 I J I ll I 29 10 I
111 168 1 66117IK7 1821 19 2 19 211S 104 111-1 175 110 165 159 156 15] 1 4H 147 144 I-II 1 9 0)
125 2 17 210 104 199 195 1921 1 5 1 16 2 S 110 212 20 194 11111 1 04 17 I 71 I 61 160 til2S6 1SO 1 ll 295 lKl 272 261 165 ) 14
124 1221 16 129 128 127 1 2 1 1 19 1 111 1 17 116 1 14 1 1 1 1 1 1 I l l 1 10
IJJ IJI 1 10 140 119 141 117 132 111 126 1 24 1 2 1 119 10 120I us1 60I gtI 182 111 168 16S 1622 1)10 us
166 161 155 150 146 14 117 115 1 12 1211 125 I Hl 217 109 101 191 IKOl 219 201 I 176 1 101 156 UJ I 1 41 I IX
6 liS I 1701 4 J9 195 14H 296 2 79 266 256 247 240 middot 234
Ul 1 2 1 l iD 1 1 11 116 114 1 12 I l l 1 10 1119 I 0 25 111 IJ9 1111 I J6 I a I J2 I) I 129 128 127 126 US
152 146 1 41 1 111 1 1-1 I I 1211 I 24 122 uu 1 1 7 Ia Ill 2 I I 197 IKII IKO 1 75 170 166 161 160 157
172 162 IS7 IS2 146 141 119 I J2 129 U6 1 22 119 m198 19J I 88 I 114 180OS H9 1 04 265 242 226 2 14 2lt16
01 676 411 lllll l41 J I I 289 271 I 91 1119 I 79 169 I 61 1511 148 144 119 1 11 1 11 01
260 2SO 241 2 14 227
122 1 10 1 16 11-1 1 1 1 1 12 1119 Uli I 07 l tiO 124 124I 12 I J9 IJ7 US I J 111 129 1211 127 125
I II 142 118 114 uo 126 124 1 1 11 1 1 1 1 1 1 I OK Hill 10
i lS 25
IS72 JO 160 us172 167 16J10 2 71 2011 194 U2 1 14 1 12 1 1 1 1 1 1 I IIII 167 152 146 179l 11-1 HXI 260 2J7 221 210 201 1811 181liSmiddot
21KIll 6 6 4bl 3-JII J l2 l 02 2110 264 2 51 241 232 22S 1 10 U6 I ll I S I OtlIIIK104 U9 152 0 1
Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
62 1bull
d)
ii)
3) The United States is a large importer of beef With this in mind answer the following questions
a) Is the import (excess) demand for beef more or less elastic than domestic demand Explain
why
b) In general what is the relationship between domestic demand and import demand
Support your explanation with a graph and some mathematics
c) List the three factors which are the most important determinants of the elasticity of import
demand and discuss each briefly
4) Define the following terms in words and with some mathematics or a graph being careful to define
any notation that you use
a) Giffen good
b) Engel curve
c) Cross-price elasticity ci demanl
Consumers surplus
e) lndiect utility functirn
5) The following questions all deal with basic issues in econometrics
a) What is heteroskedasticity Draw a picture of a heteroskedastic error term
b) What statistical problems if any does heteroskedasticity create for the ordinary least
squares parameter estimates
c) Consider the linear model Y = 30 + 3 X1 + e where E[e] = 0 and Var[e1] 1i = 12 n Transform this model so that the new error term say e is homoskedastic
d) Suppose you are estimating the parameters of a multiple linear regression model with four
independent variables Explain carefully how you would go about testing the null
hypothesis that the last two independent variables can be omitted from the model Note that
this question is independent of the previous ones
e) Consider the following set of residuals
i) -23-10 -2 3 - 1 1
iii) 551-11
Which set of residuals can be ordinary least squares residuals Explain why Note that
this question is independent of the previous ones too
2
6) You have learned that if a monopolist can perfectly price discriminate (ie can first-ltlegree price
discriminate) then it will make the maximum profits possible and in the process capture all of the
consumers surplus Therefore it might appear that all consumers are worse off under this economic
arrangement This question is designed to probe this issue more deeply Consider the following graph
which should look quite familiaf to you by now
price
D
quantity Ym
The points A 8 and C represent three (3) different consumers For example point A on the demand
cuivemiddot represents a consumer who is willing to pay at most the price pA for a unit of the good that is point
A on the demand curve represents a consumer who has a reservation price p for a unit of the good
Points B and C can be interpreted similarly
a) Would a consumer represented by point A on the market demand curve be better off if the
monopolist charged only the simple (ie nondiscriminating) monopoly price Pm or if the
monopolist practiced perfect price discrimination Explain clearly
b) Would a consumer represented by point B on the market demand curve be better off if the
monopolist charged only the simple (ie nondiscriminating) monopoly price Pm or if the
monopolist practiced perfect price discrimination Explain clearly
3
c) Would a consumer represented by point C on the market demand curve be better off if the
monopolist charged only the simple (ie nondiscriminating) monopoly price Pmbull or if the
monopolist practiced perfect price discrimination Explain clearly
7) Assume that you own a house on which you make a mortgage payment of $200month that
moving costs are negligible and that apartments provide a set of housing services that are equivalent to
that provided by homes
a) If you can rent your house out for $250month and live in an apartment for $220month
should you move into the apartment and rent out your house Explain using some
numbers
b) If you can rent your house out for $250month and live in an apartment for $220month
but your mortgage payment is now $220month should you move into the apartment and
rent out your house Explain using some numbers
c) If you can rent your house out for $250month and live in an apartment for $220month
but your mortgage payment is now $1000month should you move into the apartment and
rent out your house Explain using some numbers
d) What do your answers to this sequence of questions reveal about the importance of the
mortgage payment for this problem
e) In general what must hold in this problem for renting out your house and renting an
apartment to be the correct response
4
(035) (095)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 26 1998
There are two sections to this exam Each section consists of four questions Follow the instructions for each section carefully Use complete sentences and well thought out responses when writing down your answers Note that if we cannot read your writing then we will assume that your answer is wrong All questions are equally weighted There are five pages to this exam
Section 1 This section consists of questions 1 ) 2) 3) and 4) Answer question 1 ) and then select two of the remaining three questions to answer Do not answer all four questions
1) Below is a highly simplified model of wine supply and demand Selected t-ratios are below each
estimated coefficient
Two-Stage Least-Squares Estimates of Deinand Q = -101 +026PWHAT
(210)
R = 055 DW=066 Two-Stage Least Squares Estimates of Supply
Q =-101+ l22PWHAT- 222 WR + 340PGRAPE + 123Qt-l(110) (056)
R2 =096 DW=l95
Auxiliary First Stage Regression PWHAT = 201 + 133WR + 440PGRAPE + 023Q_
R2 =0995 The variables are defined as follows
Q - Quantity of wine supplied and demanded
PWHAT- Price of wine from auxiliary regression
WR- Wage rate in wineries
PGRAPE - Price of wine grapes
There are (at least) 11 things wrong with this model and its estimates Can you find them List them by
number and indicate the consequence of each problem
middot 2) You have been asked to estimate a linear econometric model say Y =30 + 3 X + Jl Assume that
Jl = PJl_ + e (I)
2rsquowhere e is a normally distributed random variable with a mean of 0 and variance of u bull The above
model (1 ) is often termed autocorrelation
a) How do you test for autocorrelation
There are two kinds of autocorrelation- pure and impure The Durbin-Watson statistic can be indicative
of either kind
b) Show the consequences of both kinds of autocorrelation and how to correct for each
3) Consider the following primal (P) linear programming problem
max R = Ox + 8x2 XJxl
st tX1 + x2 5125
2x1 + 3x2 5 300 (P)
x 5 90
Xp X2 0
Answer the questions below being sure to explain your steps and reasoning
a) Graph the constraints of (P) and identify the primal feasible solution set ie the set of all
admissible solutions for (P)
b) Identify all the extreme points of the primal feasible solution set
c) Find the optimal solution to (P)
d) Determine the slope of the isorevenue curves sketch a few of them and verify your
answer in part c)
e) Write down the dual (D) linear programming problem to (P)
4) An individual investor has $70000 to divide among several investments The investments
available to this person are municipal bonds with an 85 annual return certificates of deposit with a 5
annual return treasury bills with a 65 annual return and a growth stock fund with a 13 annual return
The investments are all re-evaluated after one year ie the time frame for the investor is one year Each
investment has a different perceived risk to the investor so it is optimal to diversify
The following guidelines have been established for diversifying investments bull No more than 20 of the total investment should be in the growth stock fund
bull The amount invested in certificates of deposit should not exceed the amount invested in the other three
investment alternatives bull At least 30 of the investment should be in treasury bills and certificates of deposit
2
bull More should be invested in certificates of deposit and treasury bills than in municipal bonds and the
growth stock fund by a ratio of 12 to 10 The investor wants to know how much to invest in each investment alternative in order to maximize the
total return from all investments All $70000 is to be spent in this investment plan
Formulate the linear programming problem for this investor Defme all the variables constraints and
equations in the formulation
Section 2 This section consists of questions 5) 6) 7) and 8) Answer any three of these four questions Do not answer all four questions
5) Which of the following describes an externality and which does not Explain your reasoning
clearly in each case
a) A policy of restricted coffee exports in Brazil causes the US price of coffee to rise which
in turn causes the price of tea to increase
b) An advertising blimp distracts a motorist who then hits a telephone pole and damages the
car
c) In a homogeneous product Cournot duopoly an increase in the output of firm 1 lowers the
market price received by firm 2 for its product (due to the law of demand) ceteris paribus thereby reducing the profit of firm 2
6) Consider a competitive industry with a large number of identical firms each of which has a simple
cost curve of the form
C(q)= l +1 qgtO 0 q =O
where q 0 is the output of the firm Think of these costs in a long-run context ie if the firm produces
nothing it incurs no costs but any q gt 0 incurs the 10 set up cost Suppose industry demand is given by
Q0 =52-P
where Pis the market price of the good produced by the firms
a) Derive a supply curve for one of the identical fmns
b) Derive the industry supply curve given that there are n firms
c) What is the long-run equilibrium price in this industry Explain your answer and show any
calculations
3
e)
a)
d)
d) How many firms will populate this industry in long-run equilibrium Explain your
answer
f)
Given your solution for n from part d) assume demand shifts to Q0 =53- P What is
industry price and output in the short-run equilibrium
Given your result in part e) explain in words what will happen as the industry adjusts to
long-run equilibrium
7) The following is a demand equation for residential electricity consumption in the US estimated by
H Houthakker (Energy Jouma1980)
ln E = 0586 + 1388lnY-1118 ln PE + 0199lnH + 0483lnC +0566LnP0
where E is per capita electricity consumption in KWH per year Y is disposable personal income per
capita PE is the price per KWH of electricity H is heating degree days (deviation over the year of
temperature below 65deg) C is cooling degree days (deviation over the year of temperature above 75deg) and
P 0 is the price of natural gas per BTU
Based on this model what are the own price cross price and income elasticities of
demand for electricity in the US
b) A deregulation plan for the electricity industry is currently being implemented Suppose
electricity prices rise as a result of deregulation Based on Houthakkers equation will
consumers spend more less or the same on electricity following deregulation Explain
c) Many poor families live in apartments with electric heating and cooling A concern of
policy makers is that high energy costs make it difficult for poor families to adequately heat
or cool their homes Consider a typical poor family whose demand for electricity is given
by Houthakkers equation The family pays $010 per KWH and has per capita income of
$3000 ($12000 for a family of four) Given normal values for H C and P 0 the family
consumers on average 6000 KWH per year Energy experts have determined that 8000
KWH are needed to provide satisfactory heating and cooling in the familys climatic region
Consider two programs to eliminate the deficiency (a) a cash payments plan (b) an energy
subsidy program under which the family will get a rebate from the government for a
specific percentage of its monthly electric bill
(i) How large must the cash payment be to accomplish the plans goal
(ii) What percentage subsidy will achieve the programs goal
lllustrate the two plans using indifference curves for electricity and all other goods and the
budget constraint Begin with the situation where E = 6000 and show how either program
can attain the goal
A third possible program involves coupons that can be spent only on electricity Will e)
coupons entitling the household to buy 2000 KWH achieve the goal Explain
4
8 ) Consider a monopolist who sells in two markets The inverse demands and cost function for the
monopolist are as follows
I = 80-SQ market I
R =180- 20Q2 market2
C (Q)=50 + 20(Q + Q2) cost function
a) Find the firms profit maximizing price and output in each market assuming it is free to set
whatever price it wishes in each market Show your work
b) Assuming absence of any externalities what is the socially optimal output in each market
Explain your answer and show any calculations
c) Define the term deadweight loss and then calculate the deadweight loss from monopoly
power in each market
d) TrueIFalse Explain The firm in this problem is a natural monopolist
e) The following statement is true The firm in this problem is practicing price
discrimination Explain carefully and precisely what this statement means in terms a
noneconomist could understand
f) Suppose the government passed a law preventing the firm from practicing price
discrimination What price and output would the firm then set in market I Market 2
Provide a precise answer if you can showing all work but dont get bogged down At a
minimum explain in words how if at all this law would affect the firm
5
University of California Davi Jepartment of Agricultural and Resource bconomics
MS Comprehensive Exam September 151998
There are two sections to this exam Each section consists of four questions Follow the instructions for each section carefully Use complete sentences and well thought out responses when writing down your answers Note that if we cannot read your writing then we will assume that your answer is wrong All questions are equally weighted There are five pages to this exam
Section 1 This section consists of questions 1 ) 2) 3) and 4) Answer question 1 ) and then select two of the remaining three questions to answer Do not answer all four questions
I) Below is a highly simplified model of wine supply and demand Selected t-ratios are below each estimated coefficient
Two-Stage Least-Squares Estimates of Supply
Q = -101 + 026PWHAT(210) (220)
R2 =035 DW=041 n=21
Two-Stage Least Squares Estimates of Demand
Q = IOI -122PWHAT + 222NCOME +340CPIALL + 0753Q_1(23S) (110) (056) (400)
R2 =096 DW=05 Durbins h- statistic= 083 n=21
Auxiliary First Stage Regression PWHAT = 201+ 133NCOME + 440CPIALL + 023Q_1
R2 = 067 DW= 187
Regression On Residuals From Demand Equation
Jl = 05+155NCOME +055CPIALL (620) (ll)
R2=072 The variables are defmed as follows
Q - Quantity of wine supplied and demanded PWHAT - Price of wine from auxiliary regression INCOME- Per capita personal income CPIALL - Consumer price index of all items
Examine the above model for econometric problems In particular you should look for
I autocorrelation 2 heteroscedasticity
2)
3 omitted variables
4 multicollinearity
5 identification
6 simultaneous equation bias
For each of the above possible problems explain why you think there is or is not a problem Explain
what should be done to remedy the problem
You have been asked to estimate a linear econometric model say Y = 30 + 31X + Jl Assume that
var(Jl) = a2X (I)
where Jl is a normally distributed random variable with a mean of 0 and a variance given in Eq () The
above model (I) is termed heteroscedasticity
a) Clearly explain the problems that heteroscedasticity creates
b) How might you go about detecting the presence of heteroscedasticity in the above linear
model
middotc) How might you correct for the heteroscedasticity given that it occurs in the form ofEq (1)
in the above linear model Be specific and give details
3) Answer the following jive TrueFalseExplain questions concerning a general linear programming
(LP) problem Please note that the explanation is worth the bulk of the credit
a) An optimal solution to aLP problem uses up all of the limited resources available
b) A point on the boundary of the feasible set (or region) satisfies all of the constraints of the
LP problem
c) The optimal solution of an LP problem always occurs at an extreme point of the feasible
set
d) If the first COilStraint of aLP problem evaluated at given value xo of the decision vector
has a zero value for its corresponding slack variable then the point xo lies on the boundary
of the feasible region
e) If the shadow value (or dual variable) corresponding to a given constraint is positive than
that constraint is binding
4) PROTRAC INC produces two lines of heavy equipment One of the product lines termed
earthmoving equipment is essentially for construction applications while the other line termed forestry
2
equipment is destined for Ute timber industry The largest member or ihe earthmoving line (the E-9) and
the largest member of the forestry equipment line (the F-9) are produced in the same departments and with
the same equipment Using economic forecasts for the next month PROTRACs marketing manager has
judged that during that period it will be possible to sell as many E-9s and F-9s as the flrm cam produce
Management must now recommend a production target for the next month
Making this decision requires that the following major facts be considered
bull PROTRACs net revenue is $5000 on each E-9 that is sold and $4000 on each F-9
bull Each product is put through PROTRACs machining operations in department A and department B
bull For next months production department A hasl50 hours available and department B has 160 hours
available Each E-9 uses 10 hours of machining in department A and 20 hours of machining in
department B Each F-9 uses 15 hours of machining in department A and 10 hours of machining in
department B
bull In order for management to honor an agreement with the union the total labor hours used in next
months testing of finished products cannot fall more than 10 below an arbitrated goal of 150 hours
The testing is performed in a third department and has nothing to do with the activities in departments
A and B Each E-9 is given 30 hours of testing and each F-9 is given I0 hours of testing
bull In order to maintain the current market position senior management has decreed the operating policy
that it is necessary to build at least one F-9 for every three E-9 produced
bullmiddot A major dealer has ordered a total of at least five E-9s and F-9s in any combination for the next
month and so at least that many must be produced
Given the above considerations managements problem is to decide how many E-9s and many F-9s to
produce next month so as to maximize net revenue
Formulate the linear programming problem for PROTRAC INC Deflne all the variables constraints
and equations in the formulation
Section 2 This section consists of questions 5 ) 6) 7) and 8) Answer any three of these four questions Do
not answer all four questions
5) A simple monopolist faces the linear inverse market demand curve given by p(y) a - by where
a gt 0 b gt 0 and y is the output of the monopolist In addition the monopolist faces the total cost
function given by c(y) Fe where k gt 0 Finally the government pays the monopolist a subsidy of
s gt 0 dollars per unit of output produced in order to encourage the monopolist to produce more output and
sell it at a lower price
a) Set up the proflt maximization problem facing the monopolist
3
--------
b) Find the monopolists profit maximizing level of output say y = ym(abks) as well as its profit maximizing price say p = pm(abk s) How do you know you have found the profit maximizing solution
c) Will an increase in the subsidy achieve its desired goal of increasing the monopolists output and lowering its price Show your work
d) Provide an economic interpretation of the above comparative statics e) How large would the subsidy have to be in order to get the monopolist to produce the
Pareto efficient level of output and charge the Pareto efficient price
6) The following questions pertain to a production setting characterized by the usual neoclassical production function of the form Q = f(xlx2x ) where Q is output and (x1x2x ) are inputs
a) Define compare and contrast the following terms from production economics i) Decreasing returns to a variable input and ii) decreasing returns to scale (Decreasing returns to scale is also called
diseconomies of scale) b) What economic forces cause decreasing returns to a variable input and what economic
forces might cause decreasing returns to scale c) Given the production function as specified show mathematically how you would measure
the returns to the variable input x1 and the returns to scale d) Suppose you were studying the question of whether small farms could compete effectively
against large farms in the market for the product Q Would either or both of the concepts of returns to a variable input and returns to scale be valuable in your analysis Explain
7) The following are demand and supply curves that apply to a competitive industry
Q0 = 250- SOP Qs = ()P
a) Derive the market equilibrium price and quantity b) Is the equilibrium you derived in (a) a short-run equilibrium a long-run equilibrium or
could it be both Explain your answer in detail demonstrating your knowledge of behavior in competitive markets and the distinction in these markets between the short and long run
c) Politicians are floating a proposal to place a per unit tax of $ 100 on this commodity Producers in lobbying against the tax argue that they will have no choice but to pass the
full tax on to consumers and therefore the tax if enacted will only harm consumers Evaluate the producers argument concerning the tax Use either algebra (preferred tool) or
4
--------------
------------------------------ X
bull
a)
a graph to support your argument If you use a graph label it carefully using the demand and supply equations Q0 = 250- 50P and Qs = ( )P
8) The following graph depicts a consumers Marshallian demand curve for the commodity X The market price of X is given as
p
Demand Curve
Given identify the consumers surplus on the graph and give an intuitive definition of consumers surplus
b) Assuming X is a normal good for this consumer draw the consumers Hicksian demand curve given price
c) Explain the difference between Hicksian and Marshallian demand curves
d) Suppose price increases to P_ Use Hicksian demands to identify the consumers compensating variation and equivalent variation due to the price change Label carefully
5
University of California Davis Department of Agricultural and Resource Economics
MS Exam
July 2 1999
Directions You have three hours to write this exam Please write legibly Answers that cannot be read are regarded as incorrect answers This exam consists of two parts In part I quantitative methods you must answer question 1 Then choose to answer either question 2 or question 3 In part II microeconomics answer three of the four questions You should thus answer five questions in total two from part I and three from part II Do not under any circumstances answer more than five questions
Part I Answer 1 estion I and either Question 2 01 luestion 3
l You are constructing a si mple model of cotton su pply for a short-term project you are working on and h ave formulated the following conceptual model
where y5 i s bales of cotton ps i s the price paid to growers w1 is the labor wage rate w2 i s the price of water WJ i s pesticide price and DO is degree-days a measure of the growi ng conditions during the season
(a) A nother variable you might include in thi s speci ficati on is annual rainfall (R) whose coefficient i f included in the model would be a6 While you are sure th at all th e other va ri ables in the model are part of the correct speci fication you are not sure about annual rainfall As an applied econometrician explai n the tradeoffs made to the properties of you r estimates i f you include or exclude thi s vari able from the specification
(b) One must always be concerned with measurement error i nempirical models Explain th e effects on the properties of your estimates of (a) measurement error in th e dependent variable vs (b) measurement error in one or more explanatory vari ables
(c) You esti mate th e model wi th R included and the results are
yS = -6354 + 8727ps - 0 579lw1 - 7907w2 - 09045w3 + 769000 + 0045R ( -0 79) (2 36) ( -177) (-234) ( -190) (157) (045)
R2 = 0819 n =56
where th e Students-t statistics are gi ven in parentheses and the following sums of squared errors resu lt
Total SSY=660 9 Model SSR=5412 Errors SSE= l l 97
In answering the following part s justify your answers with quantitati ve test results using th e attached tables Quali tative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 0 5 level
(ii) Is the overall model significant Th at i s i s the hypothesis th at a1 = a2 = a3 = a4 = as = a6 = 0 rejected
(iii) The correlation between the price of water (w2) and the price of pesticides (w3) i s zero Do these two vari ables have the same effect on cotton supply i e are their coeffi cients equ al
2 Suppose you have the following econometric model
where y = yieldacre x1 = wateracre x2 = (wateracre)2 x3 = fertilizeracre
(a) For each of the following questions set up the appropriate hypothesis test and provide a brief explanation
(i) Does water have any influence on yieldacre
(ii) Assuming an unconstrained (interior) solution is wateracre = 300 significantly different from the amount that yields the maximum yield per unit water
(b) Suppose you work in a governmental agency and that your boss is not trained in economics or statistics and thus she does not know what e N(O all) means Explain-
in words the meaning and significance of this expression
3
a Given the regional farm modeling problem
Max subject to
px- cx Ax b
where x is an n x I vector of production activities p and c are vectors of prices and variable cost coefficients respectively b is an m x 1 vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Comment briefly on the connection between the dual problem and the Dual Complementary Slackness condition
(iii) If you had to calculate the derived demand for a single input b describe the type of function you would obtain from the problem
b Now use the information that the product prices p are determined according to the following demand relationships
p =a + Dx
Assume that the individual farmers act as perfect competitors
(i) Show how you would reformulate your problem and give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit )
(ii) Derive the dual to the problem in part b(i) showing your steps clearly
Part II Answer of the 4 questions in this Section _
I For some goods consumption is subject to more than the usual single constraint on money income time can also be a constrai nt Consider a consumer with utility f unction u(x) defined on goods x = (x1 xn) subject to th e two constraints T 2 tx and M 2 px where T is the total amount of time avai lable and M is th e total amount of income avai lable and t = (t1 tn) and p = (p o pn ) are the time and money prices per unit of consumption of each good
(a) Write th e primal utility max i mi zation problem for th is consumer What interpretation do you attach to th e sh adow values on each constraint
(b) F rom your answer to part (a) ex plain how the consumer s choices reveal her money value of time Develop an ex pression f or the marginal money value of time from th is pri mal problem
(c) Under what conditions would the problem revert to the standard consumer choice problem subject to a single money budget constraint
(d) Suppose th at the shadow value on the money budget constraint is zero W rite the dual expenditure mi nimization problem for thi s case Provide a brief interpretation in words of thi s dual ex penditure f unction Also provide an interpretation for the sh adow value of the constraint for th is problem
2 Assume t hat t he US wheat mar ket can be considered in isolation and is charact erized by
yS = sops - 20 wl
y0 = 60 - 12 5p0 +2M
where
and
ys is quantity of wheat suppl ied in millions of bushels per year p5 is the price recei ved by suppliers in dollars per bushel y0 is quantity demanded (in millions of bushels) by consumers p0 i s the price paid by consumers also in dollars per bushel w1 is the price of a composite input to wheat product ion M is consumer income in t housands of dollars
For purposes of this analysis consumer income i s M = 45 and the composite input price is w1 = 5
(a) What are t he equilibrium f ree-mar ket price and quant it y of wheat
(b) Is t he supply of wheat at this point price-elastic or price-inelast i c Ex plain your reasomng
(c) W h at is t he t otal net economic sur plus f rom production of wheat and how middotis it distr ibuted bet ween suppliers and demanders
(d) A var iet y of schemes are used in agricult ural markets to enhance producer welf are C onsider a pr ice support of $6 per bushel Explain t he effects of such a policy on the dist ri bution of welfare and economic efficiency Support your answer with quant itat ive est imates of t hese effect s
jgtrice
3 Suppose a monopoly fum faces the following linear demand curve P = a- bQ where P is price and Q is quantity produced and sold The monopolists cost function is
(1) c(Q) = cQ for Q Q c(Q) qo for Q gt Q
where Q is a capacity constraint
a Solve mathematically for the monopolists profit maxumzmg and output as functions of the parameters a b and c Show your work Also illustrate yourresults graphically labeling carefully all relevant information
b Suppose the function P(Q) from part (a) is the demand in the domestic market and that the monopolist can also sell unlimited quantities into the export market at price P E gt c Assuming that quantities exported cannot be resold into the domestic market show both mathematically and on your graph from part (a) how much the firm will sell into the domestic market and into the export market
c Consider now a potential entrant into this market For simplicity ignore the existence of the export market To be able to produce in this market an entrant must incur a one time set up or entry cost of E gt 0 It can then produce according to the cost function in (1) Describe in words andor diagrams how you would model this entrants decision problem You should consider the role of the incumbent monopolist in framing your answer
4 Consider a food processing firm that produces processed product Q from a raw agricultural product Rand a composite marketing input L which we will call labor
a Write a production function and draw some rough sketches of isoquants for each of the following possible production scenarios
1 There is no substitution between R and L in producing Q
ii The production relationship is Cobb-Douglas
b Suppose the true production relationship is Cobb-Douglas and that this firm produces Q exclusively for a large retail grocery chain It has a contract for 1999 to deliver Q units of product to the retailer Assume the firm is a perfect competitor in the markets for its inputs Set up the firms optimization problem given this information and derive the first-order conditions for the problem Defme any notation or terminology you need to introduce Also illustrate the optimization problem with a graph
c Discuss the firms demand function for R given the conditions of the problem in part (b) Specifically what variables determine how much R the firm will utilize
d Re-express the problem in part (b) and take first-order conditions assuming now that the firm is the only buyer of the farm input R which is produced according to the following inverse supply function
w = a + 3R a3 gt 0
OJ 04
J ---------
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09
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4
2257
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412
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4649
41161
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4974
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164 9-9
267 2967
4170
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76
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4791
4978
49119
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677
676 SlATISliCI1 tA9LFS
lA Rlf lgtl Areas under the standardized normal distribution
Example
Pr (0 z 196) = 04750
Pr (z 196) = 05 - 04750 = 0025
TABLE 01 Percentage points of lhe t distribution
Example
Pr (r gt 2086) = Oo25
Pr (r gt 1725) = 005 for df = 20
Pr(ltl gt 1725) = 010 0 I 715
01$ 010 middot-00
001 0 00010015010 002 0010 0002ozo
J 07H 127116I 11216 14 61657 IIUI1nm
22 279925IXH6 6965UK16 4 lP 00 01 01 1 0765 I 6X 251 lIK2 4541 51141 102 JJ
4 074 I 15J 21J1 176 461 moo 00411no
Ol 0middot1111
(Xl)lt) 11119 059UIO 0161 t tiiCW 029 0279 1650675 0751 s 1476 2015 511910727 25710596 0616 4010517 115 7
071 II 1064 IIOJ 1141 52111106 bull 1440 1941 l7H7 0791 UKl 0171 091ll 09411
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1591 161M 1664
1915 1950 19115 W19
10611144 1179 K u 706 lW7 11601772 451111196
212 071) 421172054 219() 122420XX UHl 111 2262 21i21 1251)-9
0700 J72 1112 222K 2764 11692422
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2794 2121 21152 4025II 0697 1361 1796 22111 2711lt lit
2611 27041gt1 25XU ll7K 1106 1111 190 0695 1156 17H1 21 79 26111 lOSS
ll40 11111 1910
2650 2M9 J65 l119 IJ 0694 1350 1771 2160 1lt5211112311lt6 21 lJH 164
145 1761 21J5 2614 1621 14 069214115 55-1 2977 7117 461 150111 0 1411
l711I 2602 2 94717011 770 790 Kill lX0 15 0691 1141 175 729 21 lfgt65 16116 JHHH
I I
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4194066 41H l6462567 2gt19H411 17 1740 21W40994(19 406 068H UJO 174 210142164192
I
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4554 505J67H A6K6 4691 4699 4706 22 06K6 I Jl l 1717 2014 l5HK 1XI94656 4664 4671I K 4641 41151K074756 4761 17144744 4750 2J 06KS 1319 250020694716 412 47 111 9 470 4719
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1
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J906 4911 06M4 14212471 27714913 4916 170 4909 27 L14 10524196 489R 4901 49044191 06114925 4927 4929 4931 492 4914 4916 140K2467 27612M U13 1701 204849111 4920 4922
4941 4941 4945 4946 494K 4949 4951 4951 IJII JJ9629 068) 2462 2756204S49111 494025
UJO 1697 2042 2457 17504QSS 4964 lJK S4961 JO 06KJ4960 4961 4961
496 J951J4956 49576
7 10 06KI 10749 1 6K4497 241J i7044970 4971 49724967 496K 1021J966 60 0679 129611 4981
4979 4979 49SO 16714977 2 190 26604976 4977Igrave 12122000 4975
l Jfi) 1090
49864914 4914 4985 498S 4986 120 0677 11M9 21671651 19KO 2J58
49111 49H9 49M9 4990 _ 1 1810674 164S 1960 1 126 1576 49H I 49K2 4982 498)
49X7 49K7 49117 49111
N This tahlc gives the area in the riahtmiddothand tail or the distribution tic bull t 0) But since the normal di5tribumiddot Nolt 111e smaller probability shown at the head of each column is the area in one tail the 1in is symmetrical about r- Othe area in the ldtmiddothlnd tail is the same u the area in the corresponding right-hand larser probability is the area in both tails bullbullil rltgtr eumplc PI- 196 S z S 01-04750 TMrdore -196 S t S 196)-2(047S0)-09S Sourr From E S Pearwn and H 0 Hartley eds bull Blomtuila TabftJot Slulislidans vol I
ld ed table 12 Cbullmbridgc University Prns New York 1966 Reproduced by perminion of the middot
editors and trustees of Biomeria
bull bull bull bull 10 II 30 110 100 500
(
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7 71
4 20
116
1 H4
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1 75
192
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44
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244
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144
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199
245
l-14
2 1
14)
161
190
1 4 1
25J
)70
137
195
2)7
U 5
190
229
1 59
245
U 7
I 91
1 15
96
142
194
1 94
l2
IJ1
117
174
104
1 0
190
U4
109
284
Ill 2 71 2JJ
1 77
194 175
i-1
174
149
157
I y I 7
1 94 1 gt9
194
1 9)
174 159
145 1 4
195
119
179 147
682 AIPFNHIX ll
TARLE Ol l lpper percentage poinls or the F distribution (continut-d)
I I
STATISTICAl TA8LfS 681
df for dl lor llllmtntor N
100 N
I 16 111 I 12 I y I 2ltI 10 1 10 111 1 1 1 1 111 I Y 1 41 1 4 1 1-10 I 19 1 19 UHI middotW 1171411 147 145
I 176 IJl 170 IUI 167 165 I M 161 1 bull1 101 1 97 191 1 90 11111 1116Hi 4 l)
2 15 212
Ill 1117 201 I I I M4 I IIII I I II
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191 lOS 2Kl 1 2 55 146 2 -m 24 2 lU 226 2 2l
US1 i9 l-IS S72 4112 4 1 1 199
I 1 I J I 110 I Y 12Y I 211 I M I 7 1 7 I 6 1 15 1 11 U9 UK IJK 1middot17 U6
17M I 71 1 101 167 1 162 161 I I S1 n6 I 51 IU1115 I ll 1 4 1 1401 19 1 47 146 1 44
I 98
2111 191 1 1111 1 19 210 1 042B 9 I H6 1 gt4 1 MO 119 1 1 1 1 1) 1 1 1 211 2111 2JO 22S 21 2111JOI 2SI 242 2 16 11114 16 140
2M9 2 66 UK 2 40 111 1 1 1 2-1 2 1 Ul167 middot 1 16 126 117 109 ill2 H I 472 4 12 01
114 I 12 111 1 10 129 I 211 UK 16 I 26 126 I 25 I 2 I 42 1 4 1 19 us I H 17 l l6 2 1 111 146 us 1 16 I 1611 165 1 61 I 5Y UK us IS-1 I S1 I S l I t HIIll 11111 186 2 II 1 2 1 11 2 1111 11111 Y I 151 107 199 190 1-JIS 1112 1111) 1 16 1 1S I 71 I 7 1 211 222 2111 215Ull OS 219U7 198 1 14 2 t7 2111 2511 lSO 242 1 16 2 225 22J 219 216 2 IJ UH l09 )02 196HI 129SSl 41gt4 41401
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0 1 407 I 75 )51 U6 123 312 101
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172 167 1 64 ltll I H 15-1 Ul I SU I 1 lt7 1 6 10 10IIIS 182 179 171I 10 11111 1 22K 114 20S 1911 19) 188
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270 2-17 210 2 2 1 IJ I I 207 2 Ul ] 0 1 147 J0 1 1 7 )07 298 2910 1 1 56 5 19 451 40
110 I 126 125 124 Ut 12 Ul 121 119 1 19 25IJ6 us 1 14 IJJ Ul U l140 11914425
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125 2 17 210 104 199 195 1921 1 5 1 16 2 S 110 212 20 194 11111 1 04 17 I 71 I 61 160 til2S6 1SO 1 ll 295 lKl 272 261 165 ) 14
124 1221 16 129 128 127 1 2 1 1 19 1 111 1 17 116 1 14 1 1 1 1 1 1 I l l 1 10
IJJ IJI 1 10 140 119 141 117 132 111 126 1 24 1 2 1 119 10 120I us1 60I gtI 182 111 168 16S 1622 1)10 us
166 161 155 150 146 14 117 115 1 12 1211 125 I Hl 217 109 101 191 IKOl 219 201 I 176 1 101 156 UJ I 1 41 I IX
6 liS I 1701 4 J9 195 14H 296 2 79 266 256 247 240 middot 234
Ul 1 2 1 l iD 1 1 11 116 114 1 12 I l l 1 10 1119 I 0 25 111 IJ9 1111 I J6 I a I J2 I) I 129 128 127 126 US
152 146 1 41 1 111 1 1-1 I I 1211 I 24 122 uu 1 1 7 Ia Ill 2 I I 197 IKII IKO 1 75 170 166 161 160 157
172 162 IS7 IS2 146 141 119 I J2 129 U6 1 22 119 m198 19J I 88 I 114 180OS H9 1 04 265 242 226 2 14 2lt16
01 676 411 lllll l41 J I I 289 271 I 91 1119 I 79 169 I 61 1511 148 144 119 1 11 1 11 01
260 2SO 241 2 14 227
122 1 10 1 16 11-1 1 1 1 1 12 1119 Uli I 07 l tiO 124 124I 12 I J9 IJ7 US I J 111 129 1211 127 125
I II 142 118 114 uo 126 124 1 1 11 1 1 1 1 1 1 I OK Hill 10
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Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
6) You have learned that if a monopolist can perfectly price discriminate (ie can first-ltlegree price
discriminate) then it will make the maximum profits possible and in the process capture all of the
consumers surplus Therefore it might appear that all consumers are worse off under this economic
arrangement This question is designed to probe this issue more deeply Consider the following graph
which should look quite familiaf to you by now
price
D
quantity Ym
The points A 8 and C represent three (3) different consumers For example point A on the demand
cuivemiddot represents a consumer who is willing to pay at most the price pA for a unit of the good that is point
A on the demand curve represents a consumer who has a reservation price p for a unit of the good
Points B and C can be interpreted similarly
a) Would a consumer represented by point A on the market demand curve be better off if the
monopolist charged only the simple (ie nondiscriminating) monopoly price Pm or if the
monopolist practiced perfect price discrimination Explain clearly
b) Would a consumer represented by point B on the market demand curve be better off if the
monopolist charged only the simple (ie nondiscriminating) monopoly price Pm or if the
monopolist practiced perfect price discrimination Explain clearly
3
c) Would a consumer represented by point C on the market demand curve be better off if the
monopolist charged only the simple (ie nondiscriminating) monopoly price Pmbull or if the
monopolist practiced perfect price discrimination Explain clearly
7) Assume that you own a house on which you make a mortgage payment of $200month that
moving costs are negligible and that apartments provide a set of housing services that are equivalent to
that provided by homes
a) If you can rent your house out for $250month and live in an apartment for $220month
should you move into the apartment and rent out your house Explain using some
numbers
b) If you can rent your house out for $250month and live in an apartment for $220month
but your mortgage payment is now $220month should you move into the apartment and
rent out your house Explain using some numbers
c) If you can rent your house out for $250month and live in an apartment for $220month
but your mortgage payment is now $1000month should you move into the apartment and
rent out your house Explain using some numbers
d) What do your answers to this sequence of questions reveal about the importance of the
mortgage payment for this problem
e) In general what must hold in this problem for renting out your house and renting an
apartment to be the correct response
4
(035) (095)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 26 1998
There are two sections to this exam Each section consists of four questions Follow the instructions for each section carefully Use complete sentences and well thought out responses when writing down your answers Note that if we cannot read your writing then we will assume that your answer is wrong All questions are equally weighted There are five pages to this exam
Section 1 This section consists of questions 1 ) 2) 3) and 4) Answer question 1 ) and then select two of the remaining three questions to answer Do not answer all four questions
1) Below is a highly simplified model of wine supply and demand Selected t-ratios are below each
estimated coefficient
Two-Stage Least-Squares Estimates of Deinand Q = -101 +026PWHAT
(210)
R = 055 DW=066 Two-Stage Least Squares Estimates of Supply
Q =-101+ l22PWHAT- 222 WR + 340PGRAPE + 123Qt-l(110) (056)
R2 =096 DW=l95
Auxiliary First Stage Regression PWHAT = 201 + 133WR + 440PGRAPE + 023Q_
R2 =0995 The variables are defined as follows
Q - Quantity of wine supplied and demanded
PWHAT- Price of wine from auxiliary regression
WR- Wage rate in wineries
PGRAPE - Price of wine grapes
There are (at least) 11 things wrong with this model and its estimates Can you find them List them by
number and indicate the consequence of each problem
middot 2) You have been asked to estimate a linear econometric model say Y =30 + 3 X + Jl Assume that
Jl = PJl_ + e (I)
2rsquowhere e is a normally distributed random variable with a mean of 0 and variance of u bull The above
model (1 ) is often termed autocorrelation
a) How do you test for autocorrelation
There are two kinds of autocorrelation- pure and impure The Durbin-Watson statistic can be indicative
of either kind
b) Show the consequences of both kinds of autocorrelation and how to correct for each
3) Consider the following primal (P) linear programming problem
max R = Ox + 8x2 XJxl
st tX1 + x2 5125
2x1 + 3x2 5 300 (P)
x 5 90
Xp X2 0
Answer the questions below being sure to explain your steps and reasoning
a) Graph the constraints of (P) and identify the primal feasible solution set ie the set of all
admissible solutions for (P)
b) Identify all the extreme points of the primal feasible solution set
c) Find the optimal solution to (P)
d) Determine the slope of the isorevenue curves sketch a few of them and verify your
answer in part c)
e) Write down the dual (D) linear programming problem to (P)
4) An individual investor has $70000 to divide among several investments The investments
available to this person are municipal bonds with an 85 annual return certificates of deposit with a 5
annual return treasury bills with a 65 annual return and a growth stock fund with a 13 annual return
The investments are all re-evaluated after one year ie the time frame for the investor is one year Each
investment has a different perceived risk to the investor so it is optimal to diversify
The following guidelines have been established for diversifying investments bull No more than 20 of the total investment should be in the growth stock fund
bull The amount invested in certificates of deposit should not exceed the amount invested in the other three
investment alternatives bull At least 30 of the investment should be in treasury bills and certificates of deposit
2
bull More should be invested in certificates of deposit and treasury bills than in municipal bonds and the
growth stock fund by a ratio of 12 to 10 The investor wants to know how much to invest in each investment alternative in order to maximize the
total return from all investments All $70000 is to be spent in this investment plan
Formulate the linear programming problem for this investor Defme all the variables constraints and
equations in the formulation
Section 2 This section consists of questions 5) 6) 7) and 8) Answer any three of these four questions Do not answer all four questions
5) Which of the following describes an externality and which does not Explain your reasoning
clearly in each case
a) A policy of restricted coffee exports in Brazil causes the US price of coffee to rise which
in turn causes the price of tea to increase
b) An advertising blimp distracts a motorist who then hits a telephone pole and damages the
car
c) In a homogeneous product Cournot duopoly an increase in the output of firm 1 lowers the
market price received by firm 2 for its product (due to the law of demand) ceteris paribus thereby reducing the profit of firm 2
6) Consider a competitive industry with a large number of identical firms each of which has a simple
cost curve of the form
C(q)= l +1 qgtO 0 q =O
where q 0 is the output of the firm Think of these costs in a long-run context ie if the firm produces
nothing it incurs no costs but any q gt 0 incurs the 10 set up cost Suppose industry demand is given by
Q0 =52-P
where Pis the market price of the good produced by the firms
a) Derive a supply curve for one of the identical fmns
b) Derive the industry supply curve given that there are n firms
c) What is the long-run equilibrium price in this industry Explain your answer and show any
calculations
3
e)
a)
d)
d) How many firms will populate this industry in long-run equilibrium Explain your
answer
f)
Given your solution for n from part d) assume demand shifts to Q0 =53- P What is
industry price and output in the short-run equilibrium
Given your result in part e) explain in words what will happen as the industry adjusts to
long-run equilibrium
7) The following is a demand equation for residential electricity consumption in the US estimated by
H Houthakker (Energy Jouma1980)
ln E = 0586 + 1388lnY-1118 ln PE + 0199lnH + 0483lnC +0566LnP0
where E is per capita electricity consumption in KWH per year Y is disposable personal income per
capita PE is the price per KWH of electricity H is heating degree days (deviation over the year of
temperature below 65deg) C is cooling degree days (deviation over the year of temperature above 75deg) and
P 0 is the price of natural gas per BTU
Based on this model what are the own price cross price and income elasticities of
demand for electricity in the US
b) A deregulation plan for the electricity industry is currently being implemented Suppose
electricity prices rise as a result of deregulation Based on Houthakkers equation will
consumers spend more less or the same on electricity following deregulation Explain
c) Many poor families live in apartments with electric heating and cooling A concern of
policy makers is that high energy costs make it difficult for poor families to adequately heat
or cool their homes Consider a typical poor family whose demand for electricity is given
by Houthakkers equation The family pays $010 per KWH and has per capita income of
$3000 ($12000 for a family of four) Given normal values for H C and P 0 the family
consumers on average 6000 KWH per year Energy experts have determined that 8000
KWH are needed to provide satisfactory heating and cooling in the familys climatic region
Consider two programs to eliminate the deficiency (a) a cash payments plan (b) an energy
subsidy program under which the family will get a rebate from the government for a
specific percentage of its monthly electric bill
(i) How large must the cash payment be to accomplish the plans goal
(ii) What percentage subsidy will achieve the programs goal
lllustrate the two plans using indifference curves for electricity and all other goods and the
budget constraint Begin with the situation where E = 6000 and show how either program
can attain the goal
A third possible program involves coupons that can be spent only on electricity Will e)
coupons entitling the household to buy 2000 KWH achieve the goal Explain
4
8 ) Consider a monopolist who sells in two markets The inverse demands and cost function for the
monopolist are as follows
I = 80-SQ market I
R =180- 20Q2 market2
C (Q)=50 + 20(Q + Q2) cost function
a) Find the firms profit maximizing price and output in each market assuming it is free to set
whatever price it wishes in each market Show your work
b) Assuming absence of any externalities what is the socially optimal output in each market
Explain your answer and show any calculations
c) Define the term deadweight loss and then calculate the deadweight loss from monopoly
power in each market
d) TrueIFalse Explain The firm in this problem is a natural monopolist
e) The following statement is true The firm in this problem is practicing price
discrimination Explain carefully and precisely what this statement means in terms a
noneconomist could understand
f) Suppose the government passed a law preventing the firm from practicing price
discrimination What price and output would the firm then set in market I Market 2
Provide a precise answer if you can showing all work but dont get bogged down At a
minimum explain in words how if at all this law would affect the firm
5
University of California Davi Jepartment of Agricultural and Resource bconomics
MS Comprehensive Exam September 151998
There are two sections to this exam Each section consists of four questions Follow the instructions for each section carefully Use complete sentences and well thought out responses when writing down your answers Note that if we cannot read your writing then we will assume that your answer is wrong All questions are equally weighted There are five pages to this exam
Section 1 This section consists of questions 1 ) 2) 3) and 4) Answer question 1 ) and then select two of the remaining three questions to answer Do not answer all four questions
I) Below is a highly simplified model of wine supply and demand Selected t-ratios are below each estimated coefficient
Two-Stage Least-Squares Estimates of Supply
Q = -101 + 026PWHAT(210) (220)
R2 =035 DW=041 n=21
Two-Stage Least Squares Estimates of Demand
Q = IOI -122PWHAT + 222NCOME +340CPIALL + 0753Q_1(23S) (110) (056) (400)
R2 =096 DW=05 Durbins h- statistic= 083 n=21
Auxiliary First Stage Regression PWHAT = 201+ 133NCOME + 440CPIALL + 023Q_1
R2 = 067 DW= 187
Regression On Residuals From Demand Equation
Jl = 05+155NCOME +055CPIALL (620) (ll)
R2=072 The variables are defmed as follows
Q - Quantity of wine supplied and demanded PWHAT - Price of wine from auxiliary regression INCOME- Per capita personal income CPIALL - Consumer price index of all items
Examine the above model for econometric problems In particular you should look for
I autocorrelation 2 heteroscedasticity
2)
3 omitted variables
4 multicollinearity
5 identification
6 simultaneous equation bias
For each of the above possible problems explain why you think there is or is not a problem Explain
what should be done to remedy the problem
You have been asked to estimate a linear econometric model say Y = 30 + 31X + Jl Assume that
var(Jl) = a2X (I)
where Jl is a normally distributed random variable with a mean of 0 and a variance given in Eq () The
above model (I) is termed heteroscedasticity
a) Clearly explain the problems that heteroscedasticity creates
b) How might you go about detecting the presence of heteroscedasticity in the above linear
model
middotc) How might you correct for the heteroscedasticity given that it occurs in the form ofEq (1)
in the above linear model Be specific and give details
3) Answer the following jive TrueFalseExplain questions concerning a general linear programming
(LP) problem Please note that the explanation is worth the bulk of the credit
a) An optimal solution to aLP problem uses up all of the limited resources available
b) A point on the boundary of the feasible set (or region) satisfies all of the constraints of the
LP problem
c) The optimal solution of an LP problem always occurs at an extreme point of the feasible
set
d) If the first COilStraint of aLP problem evaluated at given value xo of the decision vector
has a zero value for its corresponding slack variable then the point xo lies on the boundary
of the feasible region
e) If the shadow value (or dual variable) corresponding to a given constraint is positive than
that constraint is binding
4) PROTRAC INC produces two lines of heavy equipment One of the product lines termed
earthmoving equipment is essentially for construction applications while the other line termed forestry
2
equipment is destined for Ute timber industry The largest member or ihe earthmoving line (the E-9) and
the largest member of the forestry equipment line (the F-9) are produced in the same departments and with
the same equipment Using economic forecasts for the next month PROTRACs marketing manager has
judged that during that period it will be possible to sell as many E-9s and F-9s as the flrm cam produce
Management must now recommend a production target for the next month
Making this decision requires that the following major facts be considered
bull PROTRACs net revenue is $5000 on each E-9 that is sold and $4000 on each F-9
bull Each product is put through PROTRACs machining operations in department A and department B
bull For next months production department A hasl50 hours available and department B has 160 hours
available Each E-9 uses 10 hours of machining in department A and 20 hours of machining in
department B Each F-9 uses 15 hours of machining in department A and 10 hours of machining in
department B
bull In order for management to honor an agreement with the union the total labor hours used in next
months testing of finished products cannot fall more than 10 below an arbitrated goal of 150 hours
The testing is performed in a third department and has nothing to do with the activities in departments
A and B Each E-9 is given 30 hours of testing and each F-9 is given I0 hours of testing
bull In order to maintain the current market position senior management has decreed the operating policy
that it is necessary to build at least one F-9 for every three E-9 produced
bullmiddot A major dealer has ordered a total of at least five E-9s and F-9s in any combination for the next
month and so at least that many must be produced
Given the above considerations managements problem is to decide how many E-9s and many F-9s to
produce next month so as to maximize net revenue
Formulate the linear programming problem for PROTRAC INC Deflne all the variables constraints
and equations in the formulation
Section 2 This section consists of questions 5 ) 6) 7) and 8) Answer any three of these four questions Do
not answer all four questions
5) A simple monopolist faces the linear inverse market demand curve given by p(y) a - by where
a gt 0 b gt 0 and y is the output of the monopolist In addition the monopolist faces the total cost
function given by c(y) Fe where k gt 0 Finally the government pays the monopolist a subsidy of
s gt 0 dollars per unit of output produced in order to encourage the monopolist to produce more output and
sell it at a lower price
a) Set up the proflt maximization problem facing the monopolist
3
--------
b) Find the monopolists profit maximizing level of output say y = ym(abks) as well as its profit maximizing price say p = pm(abk s) How do you know you have found the profit maximizing solution
c) Will an increase in the subsidy achieve its desired goal of increasing the monopolists output and lowering its price Show your work
d) Provide an economic interpretation of the above comparative statics e) How large would the subsidy have to be in order to get the monopolist to produce the
Pareto efficient level of output and charge the Pareto efficient price
6) The following questions pertain to a production setting characterized by the usual neoclassical production function of the form Q = f(xlx2x ) where Q is output and (x1x2x ) are inputs
a) Define compare and contrast the following terms from production economics i) Decreasing returns to a variable input and ii) decreasing returns to scale (Decreasing returns to scale is also called
diseconomies of scale) b) What economic forces cause decreasing returns to a variable input and what economic
forces might cause decreasing returns to scale c) Given the production function as specified show mathematically how you would measure
the returns to the variable input x1 and the returns to scale d) Suppose you were studying the question of whether small farms could compete effectively
against large farms in the market for the product Q Would either or both of the concepts of returns to a variable input and returns to scale be valuable in your analysis Explain
7) The following are demand and supply curves that apply to a competitive industry
Q0 = 250- SOP Qs = ()P
a) Derive the market equilibrium price and quantity b) Is the equilibrium you derived in (a) a short-run equilibrium a long-run equilibrium or
could it be both Explain your answer in detail demonstrating your knowledge of behavior in competitive markets and the distinction in these markets between the short and long run
c) Politicians are floating a proposal to place a per unit tax of $ 100 on this commodity Producers in lobbying against the tax argue that they will have no choice but to pass the
full tax on to consumers and therefore the tax if enacted will only harm consumers Evaluate the producers argument concerning the tax Use either algebra (preferred tool) or
4
--------------
------------------------------ X
bull
a)
a graph to support your argument If you use a graph label it carefully using the demand and supply equations Q0 = 250- 50P and Qs = ( )P
8) The following graph depicts a consumers Marshallian demand curve for the commodity X The market price of X is given as
p
Demand Curve
Given identify the consumers surplus on the graph and give an intuitive definition of consumers surplus
b) Assuming X is a normal good for this consumer draw the consumers Hicksian demand curve given price
c) Explain the difference between Hicksian and Marshallian demand curves
d) Suppose price increases to P_ Use Hicksian demands to identify the consumers compensating variation and equivalent variation due to the price change Label carefully
5
University of California Davis Department of Agricultural and Resource Economics
MS Exam
July 2 1999
Directions You have three hours to write this exam Please write legibly Answers that cannot be read are regarded as incorrect answers This exam consists of two parts In part I quantitative methods you must answer question 1 Then choose to answer either question 2 or question 3 In part II microeconomics answer three of the four questions You should thus answer five questions in total two from part I and three from part II Do not under any circumstances answer more than five questions
Part I Answer 1 estion I and either Question 2 01 luestion 3
l You are constructing a si mple model of cotton su pply for a short-term project you are working on and h ave formulated the following conceptual model
where y5 i s bales of cotton ps i s the price paid to growers w1 is the labor wage rate w2 i s the price of water WJ i s pesticide price and DO is degree-days a measure of the growi ng conditions during the season
(a) A nother variable you might include in thi s speci ficati on is annual rainfall (R) whose coefficient i f included in the model would be a6 While you are sure th at all th e other va ri ables in the model are part of the correct speci fication you are not sure about annual rainfall As an applied econometrician explai n the tradeoffs made to the properties of you r estimates i f you include or exclude thi s vari able from the specification
(b) One must always be concerned with measurement error i nempirical models Explain th e effects on the properties of your estimates of (a) measurement error in th e dependent variable vs (b) measurement error in one or more explanatory vari ables
(c) You esti mate th e model wi th R included and the results are
yS = -6354 + 8727ps - 0 579lw1 - 7907w2 - 09045w3 + 769000 + 0045R ( -0 79) (2 36) ( -177) (-234) ( -190) (157) (045)
R2 = 0819 n =56
where th e Students-t statistics are gi ven in parentheses and the following sums of squared errors resu lt
Total SSY=660 9 Model SSR=5412 Errors SSE= l l 97
In answering the following part s justify your answers with quantitati ve test results using th e attached tables Quali tative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 0 5 level
(ii) Is the overall model significant Th at i s i s the hypothesis th at a1 = a2 = a3 = a4 = as = a6 = 0 rejected
(iii) The correlation between the price of water (w2) and the price of pesticides (w3) i s zero Do these two vari ables have the same effect on cotton supply i e are their coeffi cients equ al
2 Suppose you have the following econometric model
where y = yieldacre x1 = wateracre x2 = (wateracre)2 x3 = fertilizeracre
(a) For each of the following questions set up the appropriate hypothesis test and provide a brief explanation
(i) Does water have any influence on yieldacre
(ii) Assuming an unconstrained (interior) solution is wateracre = 300 significantly different from the amount that yields the maximum yield per unit water
(b) Suppose you work in a governmental agency and that your boss is not trained in economics or statistics and thus she does not know what e N(O all) means Explain-
in words the meaning and significance of this expression
3
a Given the regional farm modeling problem
Max subject to
px- cx Ax b
where x is an n x I vector of production activities p and c are vectors of prices and variable cost coefficients respectively b is an m x 1 vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Comment briefly on the connection between the dual problem and the Dual Complementary Slackness condition
(iii) If you had to calculate the derived demand for a single input b describe the type of function you would obtain from the problem
b Now use the information that the product prices p are determined according to the following demand relationships
p =a + Dx
Assume that the individual farmers act as perfect competitors
(i) Show how you would reformulate your problem and give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit )
(ii) Derive the dual to the problem in part b(i) showing your steps clearly
Part II Answer of the 4 questions in this Section _
I For some goods consumption is subject to more than the usual single constraint on money income time can also be a constrai nt Consider a consumer with utility f unction u(x) defined on goods x = (x1 xn) subject to th e two constraints T 2 tx and M 2 px where T is the total amount of time avai lable and M is th e total amount of income avai lable and t = (t1 tn) and p = (p o pn ) are the time and money prices per unit of consumption of each good
(a) Write th e primal utility max i mi zation problem for th is consumer What interpretation do you attach to th e sh adow values on each constraint
(b) F rom your answer to part (a) ex plain how the consumer s choices reveal her money value of time Develop an ex pression f or the marginal money value of time from th is pri mal problem
(c) Under what conditions would the problem revert to the standard consumer choice problem subject to a single money budget constraint
(d) Suppose th at the shadow value on the money budget constraint is zero W rite the dual expenditure mi nimization problem for thi s case Provide a brief interpretation in words of thi s dual ex penditure f unction Also provide an interpretation for the sh adow value of the constraint for th is problem
2 Assume t hat t he US wheat mar ket can be considered in isolation and is charact erized by
yS = sops - 20 wl
y0 = 60 - 12 5p0 +2M
where
and
ys is quantity of wheat suppl ied in millions of bushels per year p5 is the price recei ved by suppliers in dollars per bushel y0 is quantity demanded (in millions of bushels) by consumers p0 i s the price paid by consumers also in dollars per bushel w1 is the price of a composite input to wheat product ion M is consumer income in t housands of dollars
For purposes of this analysis consumer income i s M = 45 and the composite input price is w1 = 5
(a) What are t he equilibrium f ree-mar ket price and quant it y of wheat
(b) Is t he supply of wheat at this point price-elastic or price-inelast i c Ex plain your reasomng
(c) W h at is t he t otal net economic sur plus f rom production of wheat and how middotis it distr ibuted bet ween suppliers and demanders
(d) A var iet y of schemes are used in agricult ural markets to enhance producer welf are C onsider a pr ice support of $6 per bushel Explain t he effects of such a policy on the dist ri bution of welfare and economic efficiency Support your answer with quant itat ive est imates of t hese effect s
jgtrice
3 Suppose a monopoly fum faces the following linear demand curve P = a- bQ where P is price and Q is quantity produced and sold The monopolists cost function is
(1) c(Q) = cQ for Q Q c(Q) qo for Q gt Q
where Q is a capacity constraint
a Solve mathematically for the monopolists profit maxumzmg and output as functions of the parameters a b and c Show your work Also illustrate yourresults graphically labeling carefully all relevant information
b Suppose the function P(Q) from part (a) is the demand in the domestic market and that the monopolist can also sell unlimited quantities into the export market at price P E gt c Assuming that quantities exported cannot be resold into the domestic market show both mathematically and on your graph from part (a) how much the firm will sell into the domestic market and into the export market
c Consider now a potential entrant into this market For simplicity ignore the existence of the export market To be able to produce in this market an entrant must incur a one time set up or entry cost of E gt 0 It can then produce according to the cost function in (1) Describe in words andor diagrams how you would model this entrants decision problem You should consider the role of the incumbent monopolist in framing your answer
4 Consider a food processing firm that produces processed product Q from a raw agricultural product Rand a composite marketing input L which we will call labor
a Write a production function and draw some rough sketches of isoquants for each of the following possible production scenarios
1 There is no substitution between R and L in producing Q
ii The production relationship is Cobb-Douglas
b Suppose the true production relationship is Cobb-Douglas and that this firm produces Q exclusively for a large retail grocery chain It has a contract for 1999 to deliver Q units of product to the retailer Assume the firm is a perfect competitor in the markets for its inputs Set up the firms optimization problem given this information and derive the first-order conditions for the problem Defme any notation or terminology you need to introduce Also illustrate the optimization problem with a graph
c Discuss the firms demand function for R given the conditions of the problem in part (b) Specifically what variables determine how much R the firm will utilize
d Re-express the problem in part (b) and take first-order conditions assuming now that the firm is the only buyer of the farm input R which is produced according to the following inverse supply function
w = a + 3R a3 gt 0
OJ 04
J ---------
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09
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4
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412
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Example
Pr (0 z 196) = 04750
Pr (z 196) = 05 - 04750 = 0025
TABLE 01 Percentage points of lhe t distribution
Example
Pr (r gt 2086) = Oo25
Pr (r gt 1725) = 005 for df = 20
Pr(ltl gt 1725) = 010 0 I 715
01$ 010 middot-00
001 0 00010015010 002 0010 0002ozo
J 07H 127116I 11216 14 61657 IIUI1nm
22 279925IXH6 6965UK16 4 lP 00 01 01 1 0765 I 6X 251 lIK2 4541 51141 102 JJ
4 074 I 15J 21J1 176 461 moo 00411no
Ol 0middot1111
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071 II 1064 IIOJ 1141 52111106 bull 1440 1941 l7H7 0791 UKl 0171 091ll 09411
149914KU 1517
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1591 161M 1664
1915 1950 19115 W19
10611144 1179 K u 706 lW7 11601772 451111196
212 071) 421172054 219() 122420XX UHl 111 2262 21i21 1251)-9
0700 J72 1112 222K 2764 11692422
1021 214 25492-IK6 102157 2X92291
2794 2121 21152 4025II 0697 1361 1796 22111 2711lt lit
2611 27041gt1 25XU ll7K 1106 1111 190 0695 1156 17H1 21 79 26111 lOSS
ll40 11111 1910
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145 1761 21J5 2614 1621 14 069214115 55-1 2977 7117 461 150111 0 1411
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l Jfi) 1090
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49111 49H9 49M9 4990 _ 1 1810674 164S 1960 1 126 1576 49H I 49K2 4982 498)
49X7 49K7 49117 49111
N This tahlc gives the area in the riahtmiddothand tail or the distribution tic bull t 0) But since the normal di5tribumiddot Nolt 111e smaller probability shown at the head of each column is the area in one tail the 1in is symmetrical about r- Othe area in the ldtmiddothlnd tail is the same u the area in the corresponding right-hand larser probability is the area in both tails bullbullil rltgtr eumplc PI- 196 S z S 01-04750 TMrdore -196 S t S 196)-2(047S0)-09S Sourr From E S Pearwn and H 0 Hartley eds bull Blomtuila TabftJot Slulislidans vol I
ld ed table 12 Cbullmbridgc University Prns New York 1966 Reproduced by perminion of the middot
editors and trustees of Biomeria
bull bull bull bull 10 II 30 110 100 500
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4 20
116
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199
245
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190
1 4 1
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137
195
2)7
U 5
190
229
1 59
245
U 7
I 91
1 15
96
142
194
1 94
l2
IJ1
117
174
104
1 0
190
U4
109
284
Ill 2 71 2JJ
1 77
194 175
i-1
174
149
157
I y I 7
1 94 1 gt9
194
1 9)
174 159
145 1 4
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682 AIPFNHIX ll
TARLE Ol l lpper percentage poinls or the F distribution (continut-d)
I I
STATISTICAl TA8LfS 681
df for dl lor llllmtntor N
100 N
I 16 111 I 12 I y I 2ltI 10 1 10 111 1 1 1 1 111 I Y 1 41 1 4 1 1-10 I 19 1 19 UHI middotW 1171411 147 145
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2 15 212
Ill 1117 201 I I I M4 I IIII I I II
2 911 1 II 1 1 1gt 67 251 11 an z llbull 2 1J ll I 26 1 1 11 1 2
191 lOS 2Kl 1 2 55 146 2 -m 24 2 lU 226 2 2l
US1 i9 l-IS S72 4112 4 1 1 199
I 1 I J I 110 I Y 12Y I 211 I M I 7 1 7 I 6 1 15 1 11 U9 UK IJK 1middot17 U6
17M I 71 1 101 167 1 162 161 I I S1 n6 I 51 IU1115 I ll 1 4 1 1401 19 1 47 146 1 44
I 98
2111 191 1 1111 1 19 210 1 042B 9 I H6 1 gt4 1 MO 119 1 1 1 1 1) 1 1 1 211 2111 2JO 22S 21 2111JOI 2SI 242 2 16 11114 16 140
2M9 2 66 UK 2 40 111 1 1 1 2-1 2 1 Ul167 middot 1 16 126 117 109 ill2 H I 472 4 12 01
114 I 12 111 1 10 129 I 211 UK 16 I 26 126 I 25 I 2 I 42 1 4 1 19 us I H 17 l l6 2 1 111 146 us 1 16 I 1611 165 1 61 I 5Y UK us IS-1 I S1 I S l I t HIIll 11111 186 2 II 1 2 1 11 2 1111 11111 Y I 151 107 199 190 1-JIS 1112 1111) 1 16 1 1S I 71 I 7 1 211 222 2111 215Ull OS 219U7 198 1 14 2 t7 2111 2511 lSO 242 1 16 2 225 22J 219 216 2 IJ UH l09 )02 196HI 129SSl 41gt4 41401
IJJ I I 1 10 129 1 18 117 127 126 1 1gt 125 124 1 24 136 usI 41 141 140 119 UK1 111 146 145 1 69 1 66 16 159 I S1 156 I 51 152 I SO 149 1 19 Ill 1114 1 812 11 1 1 229 216 1116 190 187 100
H14 196 1 9 1 1 81 1 112 119 1 71 I 71 1 11 1 69 167 165 0)219 2 15 1 1 1 JN 295 1 1 1 2242)6 229 V5 1 1 144 IS 210 2 26 1 19 1 1 1 2 1 1 109 U6 U l
0 1 407 I 75 )51 U6 123 312 101
Ul l ltJ 129 Ull 127 1 26 126 IH 1 2-1 1 2-1 12J t n l iM I 45 1 44 1 41 U9 Ull IJ6 us IB
172 167 1 64 ltll I H 15-1 Ul I SU I 1 lt7 1 6 10 10IIIS 182 179 171I 10 11111 1 22K 114 20S 1911 19) 188
2111 1 91 I H4 179 1 16 1 14 110 I M 1 IM I 62 O)1 41 lB 227 221 216 213OS l 2 291 169
270 2-17 210 2 2 1 IJ I I 207 2 Ul ] 0 1 147 J0 1 1 7 )07 298 2910 1 1 56 5 19 451 40
110 I 126 125 124 Ut 12 Ul 121 119 1 19 25IJ6 us 1 14 IJJ Ul U l140 11914425
1117 1113 179 1 66 161 157 1 )4 I l l 14K 141 142 1-11 119 I 111
176 173 1 71IllI 21)9 1 00 192 IM4 179 1611 1 64 I us I 5J U l 100I 21Oi
01
1R4 261 2 14 225 218 2 12 1 08 252 217 229 220 2 1 1 1116 202 1 94 I 92 I IM1 01 664 II 181 lSI 129 1 12 199 2119 280 27)511S
127 125 1 14 122 121 1 11l 119 1 1 7 1 1 1 1 16 1 1 5 1 1 5 I J J 112 I l l IJO 119 119us 1 41 1 4 1 UK 160 154 Ul 1411 144 141 140 116 I 15 I J I ll I 29 10 I
111 168 1 66117IK7 1821 19 2 19 211S 104 111-1 175 110 165 159 156 15] 1 4H 147 144 I-II 1 9 0)
125 2 17 210 104 199 195 1921 1 5 1 16 2 S 110 212 20 194 11111 1 04 17 I 71 I 61 160 til2S6 1SO 1 ll 295 lKl 272 261 165 ) 14
124 1221 16 129 128 127 1 2 1 1 19 1 111 1 17 116 1 14 1 1 1 1 1 1 I l l 1 10
IJJ IJI 1 10 140 119 141 117 132 111 126 1 24 1 2 1 119 10 120I us1 60I gtI 182 111 168 16S 1622 1)10 us
166 161 155 150 146 14 117 115 1 12 1211 125 I Hl 217 109 101 191 IKOl 219 201 I 176 1 101 156 UJ I 1 41 I IX
6 liS I 1701 4 J9 195 14H 296 2 79 266 256 247 240 middot 234
Ul 1 2 1 l iD 1 1 11 116 114 1 12 I l l 1 10 1119 I 0 25 111 IJ9 1111 I J6 I a I J2 I) I 129 128 127 126 US
152 146 1 41 1 111 1 1-1 I I 1211 I 24 122 uu 1 1 7 Ia Ill 2 I I 197 IKII IKO 1 75 170 166 161 160 157
172 162 IS7 IS2 146 141 119 I J2 129 U6 1 22 119 m198 19J I 88 I 114 180OS H9 1 04 265 242 226 2 14 2lt16
01 676 411 lllll l41 J I I 289 271 I 91 1119 I 79 169 I 61 1511 148 144 119 1 11 1 11 01
260 2SO 241 2 14 227
122 1 10 1 16 11-1 1 1 1 1 12 1119 Uli I 07 l tiO 124 124I 12 I J9 IJ7 US I J 111 129 1211 127 125
I II 142 118 114 uo 126 124 1 1 11 1 1 1 1 1 1 I OK Hill 10
i lS 25
IS72 JO 160 us172 167 16J10 2 71 2011 194 U2 1 14 1 12 1 1 1 1 1 1 I IIII 167 152 146 179l 11-1 HXI 260 2J7 221 210 201 1811 181liSmiddot
21KIll 6 6 4bl 3-JII J l2 l 02 2110 264 2 51 241 232 22S 1 10 U6 I ll I S I OtlIIIK104 U9 152 0 1
Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
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University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
c) Would a consumer represented by point C on the market demand curve be better off if the
monopolist charged only the simple (ie nondiscriminating) monopoly price Pmbull or if the
monopolist practiced perfect price discrimination Explain clearly
7) Assume that you own a house on which you make a mortgage payment of $200month that
moving costs are negligible and that apartments provide a set of housing services that are equivalent to
that provided by homes
a) If you can rent your house out for $250month and live in an apartment for $220month
should you move into the apartment and rent out your house Explain using some
numbers
b) If you can rent your house out for $250month and live in an apartment for $220month
but your mortgage payment is now $220month should you move into the apartment and
rent out your house Explain using some numbers
c) If you can rent your house out for $250month and live in an apartment for $220month
but your mortgage payment is now $1000month should you move into the apartment and
rent out your house Explain using some numbers
d) What do your answers to this sequence of questions reveal about the importance of the
mortgage payment for this problem
e) In general what must hold in this problem for renting out your house and renting an
apartment to be the correct response
4
(035) (095)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 26 1998
There are two sections to this exam Each section consists of four questions Follow the instructions for each section carefully Use complete sentences and well thought out responses when writing down your answers Note that if we cannot read your writing then we will assume that your answer is wrong All questions are equally weighted There are five pages to this exam
Section 1 This section consists of questions 1 ) 2) 3) and 4) Answer question 1 ) and then select two of the remaining three questions to answer Do not answer all four questions
1) Below is a highly simplified model of wine supply and demand Selected t-ratios are below each
estimated coefficient
Two-Stage Least-Squares Estimates of Deinand Q = -101 +026PWHAT
(210)
R = 055 DW=066 Two-Stage Least Squares Estimates of Supply
Q =-101+ l22PWHAT- 222 WR + 340PGRAPE + 123Qt-l(110) (056)
R2 =096 DW=l95
Auxiliary First Stage Regression PWHAT = 201 + 133WR + 440PGRAPE + 023Q_
R2 =0995 The variables are defined as follows
Q - Quantity of wine supplied and demanded
PWHAT- Price of wine from auxiliary regression
WR- Wage rate in wineries
PGRAPE - Price of wine grapes
There are (at least) 11 things wrong with this model and its estimates Can you find them List them by
number and indicate the consequence of each problem
middot 2) You have been asked to estimate a linear econometric model say Y =30 + 3 X + Jl Assume that
Jl = PJl_ + e (I)
2rsquowhere e is a normally distributed random variable with a mean of 0 and variance of u bull The above
model (1 ) is often termed autocorrelation
a) How do you test for autocorrelation
There are two kinds of autocorrelation- pure and impure The Durbin-Watson statistic can be indicative
of either kind
b) Show the consequences of both kinds of autocorrelation and how to correct for each
3) Consider the following primal (P) linear programming problem
max R = Ox + 8x2 XJxl
st tX1 + x2 5125
2x1 + 3x2 5 300 (P)
x 5 90
Xp X2 0
Answer the questions below being sure to explain your steps and reasoning
a) Graph the constraints of (P) and identify the primal feasible solution set ie the set of all
admissible solutions for (P)
b) Identify all the extreme points of the primal feasible solution set
c) Find the optimal solution to (P)
d) Determine the slope of the isorevenue curves sketch a few of them and verify your
answer in part c)
e) Write down the dual (D) linear programming problem to (P)
4) An individual investor has $70000 to divide among several investments The investments
available to this person are municipal bonds with an 85 annual return certificates of deposit with a 5
annual return treasury bills with a 65 annual return and a growth stock fund with a 13 annual return
The investments are all re-evaluated after one year ie the time frame for the investor is one year Each
investment has a different perceived risk to the investor so it is optimal to diversify
The following guidelines have been established for diversifying investments bull No more than 20 of the total investment should be in the growth stock fund
bull The amount invested in certificates of deposit should not exceed the amount invested in the other three
investment alternatives bull At least 30 of the investment should be in treasury bills and certificates of deposit
2
bull More should be invested in certificates of deposit and treasury bills than in municipal bonds and the
growth stock fund by a ratio of 12 to 10 The investor wants to know how much to invest in each investment alternative in order to maximize the
total return from all investments All $70000 is to be spent in this investment plan
Formulate the linear programming problem for this investor Defme all the variables constraints and
equations in the formulation
Section 2 This section consists of questions 5) 6) 7) and 8) Answer any three of these four questions Do not answer all four questions
5) Which of the following describes an externality and which does not Explain your reasoning
clearly in each case
a) A policy of restricted coffee exports in Brazil causes the US price of coffee to rise which
in turn causes the price of tea to increase
b) An advertising blimp distracts a motorist who then hits a telephone pole and damages the
car
c) In a homogeneous product Cournot duopoly an increase in the output of firm 1 lowers the
market price received by firm 2 for its product (due to the law of demand) ceteris paribus thereby reducing the profit of firm 2
6) Consider a competitive industry with a large number of identical firms each of which has a simple
cost curve of the form
C(q)= l +1 qgtO 0 q =O
where q 0 is the output of the firm Think of these costs in a long-run context ie if the firm produces
nothing it incurs no costs but any q gt 0 incurs the 10 set up cost Suppose industry demand is given by
Q0 =52-P
where Pis the market price of the good produced by the firms
a) Derive a supply curve for one of the identical fmns
b) Derive the industry supply curve given that there are n firms
c) What is the long-run equilibrium price in this industry Explain your answer and show any
calculations
3
e)
a)
d)
d) How many firms will populate this industry in long-run equilibrium Explain your
answer
f)
Given your solution for n from part d) assume demand shifts to Q0 =53- P What is
industry price and output in the short-run equilibrium
Given your result in part e) explain in words what will happen as the industry adjusts to
long-run equilibrium
7) The following is a demand equation for residential electricity consumption in the US estimated by
H Houthakker (Energy Jouma1980)
ln E = 0586 + 1388lnY-1118 ln PE + 0199lnH + 0483lnC +0566LnP0
where E is per capita electricity consumption in KWH per year Y is disposable personal income per
capita PE is the price per KWH of electricity H is heating degree days (deviation over the year of
temperature below 65deg) C is cooling degree days (deviation over the year of temperature above 75deg) and
P 0 is the price of natural gas per BTU
Based on this model what are the own price cross price and income elasticities of
demand for electricity in the US
b) A deregulation plan for the electricity industry is currently being implemented Suppose
electricity prices rise as a result of deregulation Based on Houthakkers equation will
consumers spend more less or the same on electricity following deregulation Explain
c) Many poor families live in apartments with electric heating and cooling A concern of
policy makers is that high energy costs make it difficult for poor families to adequately heat
or cool their homes Consider a typical poor family whose demand for electricity is given
by Houthakkers equation The family pays $010 per KWH and has per capita income of
$3000 ($12000 for a family of four) Given normal values for H C and P 0 the family
consumers on average 6000 KWH per year Energy experts have determined that 8000
KWH are needed to provide satisfactory heating and cooling in the familys climatic region
Consider two programs to eliminate the deficiency (a) a cash payments plan (b) an energy
subsidy program under which the family will get a rebate from the government for a
specific percentage of its monthly electric bill
(i) How large must the cash payment be to accomplish the plans goal
(ii) What percentage subsidy will achieve the programs goal
lllustrate the two plans using indifference curves for electricity and all other goods and the
budget constraint Begin with the situation where E = 6000 and show how either program
can attain the goal
A third possible program involves coupons that can be spent only on electricity Will e)
coupons entitling the household to buy 2000 KWH achieve the goal Explain
4
8 ) Consider a monopolist who sells in two markets The inverse demands and cost function for the
monopolist are as follows
I = 80-SQ market I
R =180- 20Q2 market2
C (Q)=50 + 20(Q + Q2) cost function
a) Find the firms profit maximizing price and output in each market assuming it is free to set
whatever price it wishes in each market Show your work
b) Assuming absence of any externalities what is the socially optimal output in each market
Explain your answer and show any calculations
c) Define the term deadweight loss and then calculate the deadweight loss from monopoly
power in each market
d) TrueIFalse Explain The firm in this problem is a natural monopolist
e) The following statement is true The firm in this problem is practicing price
discrimination Explain carefully and precisely what this statement means in terms a
noneconomist could understand
f) Suppose the government passed a law preventing the firm from practicing price
discrimination What price and output would the firm then set in market I Market 2
Provide a precise answer if you can showing all work but dont get bogged down At a
minimum explain in words how if at all this law would affect the firm
5
University of California Davi Jepartment of Agricultural and Resource bconomics
MS Comprehensive Exam September 151998
There are two sections to this exam Each section consists of four questions Follow the instructions for each section carefully Use complete sentences and well thought out responses when writing down your answers Note that if we cannot read your writing then we will assume that your answer is wrong All questions are equally weighted There are five pages to this exam
Section 1 This section consists of questions 1 ) 2) 3) and 4) Answer question 1 ) and then select two of the remaining three questions to answer Do not answer all four questions
I) Below is a highly simplified model of wine supply and demand Selected t-ratios are below each estimated coefficient
Two-Stage Least-Squares Estimates of Supply
Q = -101 + 026PWHAT(210) (220)
R2 =035 DW=041 n=21
Two-Stage Least Squares Estimates of Demand
Q = IOI -122PWHAT + 222NCOME +340CPIALL + 0753Q_1(23S) (110) (056) (400)
R2 =096 DW=05 Durbins h- statistic= 083 n=21
Auxiliary First Stage Regression PWHAT = 201+ 133NCOME + 440CPIALL + 023Q_1
R2 = 067 DW= 187
Regression On Residuals From Demand Equation
Jl = 05+155NCOME +055CPIALL (620) (ll)
R2=072 The variables are defmed as follows
Q - Quantity of wine supplied and demanded PWHAT - Price of wine from auxiliary regression INCOME- Per capita personal income CPIALL - Consumer price index of all items
Examine the above model for econometric problems In particular you should look for
I autocorrelation 2 heteroscedasticity
2)
3 omitted variables
4 multicollinearity
5 identification
6 simultaneous equation bias
For each of the above possible problems explain why you think there is or is not a problem Explain
what should be done to remedy the problem
You have been asked to estimate a linear econometric model say Y = 30 + 31X + Jl Assume that
var(Jl) = a2X (I)
where Jl is a normally distributed random variable with a mean of 0 and a variance given in Eq () The
above model (I) is termed heteroscedasticity
a) Clearly explain the problems that heteroscedasticity creates
b) How might you go about detecting the presence of heteroscedasticity in the above linear
model
middotc) How might you correct for the heteroscedasticity given that it occurs in the form ofEq (1)
in the above linear model Be specific and give details
3) Answer the following jive TrueFalseExplain questions concerning a general linear programming
(LP) problem Please note that the explanation is worth the bulk of the credit
a) An optimal solution to aLP problem uses up all of the limited resources available
b) A point on the boundary of the feasible set (or region) satisfies all of the constraints of the
LP problem
c) The optimal solution of an LP problem always occurs at an extreme point of the feasible
set
d) If the first COilStraint of aLP problem evaluated at given value xo of the decision vector
has a zero value for its corresponding slack variable then the point xo lies on the boundary
of the feasible region
e) If the shadow value (or dual variable) corresponding to a given constraint is positive than
that constraint is binding
4) PROTRAC INC produces two lines of heavy equipment One of the product lines termed
earthmoving equipment is essentially for construction applications while the other line termed forestry
2
equipment is destined for Ute timber industry The largest member or ihe earthmoving line (the E-9) and
the largest member of the forestry equipment line (the F-9) are produced in the same departments and with
the same equipment Using economic forecasts for the next month PROTRACs marketing manager has
judged that during that period it will be possible to sell as many E-9s and F-9s as the flrm cam produce
Management must now recommend a production target for the next month
Making this decision requires that the following major facts be considered
bull PROTRACs net revenue is $5000 on each E-9 that is sold and $4000 on each F-9
bull Each product is put through PROTRACs machining operations in department A and department B
bull For next months production department A hasl50 hours available and department B has 160 hours
available Each E-9 uses 10 hours of machining in department A and 20 hours of machining in
department B Each F-9 uses 15 hours of machining in department A and 10 hours of machining in
department B
bull In order for management to honor an agreement with the union the total labor hours used in next
months testing of finished products cannot fall more than 10 below an arbitrated goal of 150 hours
The testing is performed in a third department and has nothing to do with the activities in departments
A and B Each E-9 is given 30 hours of testing and each F-9 is given I0 hours of testing
bull In order to maintain the current market position senior management has decreed the operating policy
that it is necessary to build at least one F-9 for every three E-9 produced
bullmiddot A major dealer has ordered a total of at least five E-9s and F-9s in any combination for the next
month and so at least that many must be produced
Given the above considerations managements problem is to decide how many E-9s and many F-9s to
produce next month so as to maximize net revenue
Formulate the linear programming problem for PROTRAC INC Deflne all the variables constraints
and equations in the formulation
Section 2 This section consists of questions 5 ) 6) 7) and 8) Answer any three of these four questions Do
not answer all four questions
5) A simple monopolist faces the linear inverse market demand curve given by p(y) a - by where
a gt 0 b gt 0 and y is the output of the monopolist In addition the monopolist faces the total cost
function given by c(y) Fe where k gt 0 Finally the government pays the monopolist a subsidy of
s gt 0 dollars per unit of output produced in order to encourage the monopolist to produce more output and
sell it at a lower price
a) Set up the proflt maximization problem facing the monopolist
3
--------
b) Find the monopolists profit maximizing level of output say y = ym(abks) as well as its profit maximizing price say p = pm(abk s) How do you know you have found the profit maximizing solution
c) Will an increase in the subsidy achieve its desired goal of increasing the monopolists output and lowering its price Show your work
d) Provide an economic interpretation of the above comparative statics e) How large would the subsidy have to be in order to get the monopolist to produce the
Pareto efficient level of output and charge the Pareto efficient price
6) The following questions pertain to a production setting characterized by the usual neoclassical production function of the form Q = f(xlx2x ) where Q is output and (x1x2x ) are inputs
a) Define compare and contrast the following terms from production economics i) Decreasing returns to a variable input and ii) decreasing returns to scale (Decreasing returns to scale is also called
diseconomies of scale) b) What economic forces cause decreasing returns to a variable input and what economic
forces might cause decreasing returns to scale c) Given the production function as specified show mathematically how you would measure
the returns to the variable input x1 and the returns to scale d) Suppose you were studying the question of whether small farms could compete effectively
against large farms in the market for the product Q Would either or both of the concepts of returns to a variable input and returns to scale be valuable in your analysis Explain
7) The following are demand and supply curves that apply to a competitive industry
Q0 = 250- SOP Qs = ()P
a) Derive the market equilibrium price and quantity b) Is the equilibrium you derived in (a) a short-run equilibrium a long-run equilibrium or
could it be both Explain your answer in detail demonstrating your knowledge of behavior in competitive markets and the distinction in these markets between the short and long run
c) Politicians are floating a proposal to place a per unit tax of $ 100 on this commodity Producers in lobbying against the tax argue that they will have no choice but to pass the
full tax on to consumers and therefore the tax if enacted will only harm consumers Evaluate the producers argument concerning the tax Use either algebra (preferred tool) or
4
--------------
------------------------------ X
bull
a)
a graph to support your argument If you use a graph label it carefully using the demand and supply equations Q0 = 250- 50P and Qs = ( )P
8) The following graph depicts a consumers Marshallian demand curve for the commodity X The market price of X is given as
p
Demand Curve
Given identify the consumers surplus on the graph and give an intuitive definition of consumers surplus
b) Assuming X is a normal good for this consumer draw the consumers Hicksian demand curve given price
c) Explain the difference between Hicksian and Marshallian demand curves
d) Suppose price increases to P_ Use Hicksian demands to identify the consumers compensating variation and equivalent variation due to the price change Label carefully
5
University of California Davis Department of Agricultural and Resource Economics
MS Exam
July 2 1999
Directions You have three hours to write this exam Please write legibly Answers that cannot be read are regarded as incorrect answers This exam consists of two parts In part I quantitative methods you must answer question 1 Then choose to answer either question 2 or question 3 In part II microeconomics answer three of the four questions You should thus answer five questions in total two from part I and three from part II Do not under any circumstances answer more than five questions
Part I Answer 1 estion I and either Question 2 01 luestion 3
l You are constructing a si mple model of cotton su pply for a short-term project you are working on and h ave formulated the following conceptual model
where y5 i s bales of cotton ps i s the price paid to growers w1 is the labor wage rate w2 i s the price of water WJ i s pesticide price and DO is degree-days a measure of the growi ng conditions during the season
(a) A nother variable you might include in thi s speci ficati on is annual rainfall (R) whose coefficient i f included in the model would be a6 While you are sure th at all th e other va ri ables in the model are part of the correct speci fication you are not sure about annual rainfall As an applied econometrician explai n the tradeoffs made to the properties of you r estimates i f you include or exclude thi s vari able from the specification
(b) One must always be concerned with measurement error i nempirical models Explain th e effects on the properties of your estimates of (a) measurement error in th e dependent variable vs (b) measurement error in one or more explanatory vari ables
(c) You esti mate th e model wi th R included and the results are
yS = -6354 + 8727ps - 0 579lw1 - 7907w2 - 09045w3 + 769000 + 0045R ( -0 79) (2 36) ( -177) (-234) ( -190) (157) (045)
R2 = 0819 n =56
where th e Students-t statistics are gi ven in parentheses and the following sums of squared errors resu lt
Total SSY=660 9 Model SSR=5412 Errors SSE= l l 97
In answering the following part s justify your answers with quantitati ve test results using th e attached tables Quali tative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 0 5 level
(ii) Is the overall model significant Th at i s i s the hypothesis th at a1 = a2 = a3 = a4 = as = a6 = 0 rejected
(iii) The correlation between the price of water (w2) and the price of pesticides (w3) i s zero Do these two vari ables have the same effect on cotton supply i e are their coeffi cients equ al
2 Suppose you have the following econometric model
where y = yieldacre x1 = wateracre x2 = (wateracre)2 x3 = fertilizeracre
(a) For each of the following questions set up the appropriate hypothesis test and provide a brief explanation
(i) Does water have any influence on yieldacre
(ii) Assuming an unconstrained (interior) solution is wateracre = 300 significantly different from the amount that yields the maximum yield per unit water
(b) Suppose you work in a governmental agency and that your boss is not trained in economics or statistics and thus she does not know what e N(O all) means Explain-
in words the meaning and significance of this expression
3
a Given the regional farm modeling problem
Max subject to
px- cx Ax b
where x is an n x I vector of production activities p and c are vectors of prices and variable cost coefficients respectively b is an m x 1 vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Comment briefly on the connection between the dual problem and the Dual Complementary Slackness condition
(iii) If you had to calculate the derived demand for a single input b describe the type of function you would obtain from the problem
b Now use the information that the product prices p are determined according to the following demand relationships
p =a + Dx
Assume that the individual farmers act as perfect competitors
(i) Show how you would reformulate your problem and give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit )
(ii) Derive the dual to the problem in part b(i) showing your steps clearly
Part II Answer of the 4 questions in this Section _
I For some goods consumption is subject to more than the usual single constraint on money income time can also be a constrai nt Consider a consumer with utility f unction u(x) defined on goods x = (x1 xn) subject to th e two constraints T 2 tx and M 2 px where T is the total amount of time avai lable and M is th e total amount of income avai lable and t = (t1 tn) and p = (p o pn ) are the time and money prices per unit of consumption of each good
(a) Write th e primal utility max i mi zation problem for th is consumer What interpretation do you attach to th e sh adow values on each constraint
(b) F rom your answer to part (a) ex plain how the consumer s choices reveal her money value of time Develop an ex pression f or the marginal money value of time from th is pri mal problem
(c) Under what conditions would the problem revert to the standard consumer choice problem subject to a single money budget constraint
(d) Suppose th at the shadow value on the money budget constraint is zero W rite the dual expenditure mi nimization problem for thi s case Provide a brief interpretation in words of thi s dual ex penditure f unction Also provide an interpretation for the sh adow value of the constraint for th is problem
2 Assume t hat t he US wheat mar ket can be considered in isolation and is charact erized by
yS = sops - 20 wl
y0 = 60 - 12 5p0 +2M
where
and
ys is quantity of wheat suppl ied in millions of bushels per year p5 is the price recei ved by suppliers in dollars per bushel y0 is quantity demanded (in millions of bushels) by consumers p0 i s the price paid by consumers also in dollars per bushel w1 is the price of a composite input to wheat product ion M is consumer income in t housands of dollars
For purposes of this analysis consumer income i s M = 45 and the composite input price is w1 = 5
(a) What are t he equilibrium f ree-mar ket price and quant it y of wheat
(b) Is t he supply of wheat at this point price-elastic or price-inelast i c Ex plain your reasomng
(c) W h at is t he t otal net economic sur plus f rom production of wheat and how middotis it distr ibuted bet ween suppliers and demanders
(d) A var iet y of schemes are used in agricult ural markets to enhance producer welf are C onsider a pr ice support of $6 per bushel Explain t he effects of such a policy on the dist ri bution of welfare and economic efficiency Support your answer with quant itat ive est imates of t hese effect s
jgtrice
3 Suppose a monopoly fum faces the following linear demand curve P = a- bQ where P is price and Q is quantity produced and sold The monopolists cost function is
(1) c(Q) = cQ for Q Q c(Q) qo for Q gt Q
where Q is a capacity constraint
a Solve mathematically for the monopolists profit maxumzmg and output as functions of the parameters a b and c Show your work Also illustrate yourresults graphically labeling carefully all relevant information
b Suppose the function P(Q) from part (a) is the demand in the domestic market and that the monopolist can also sell unlimited quantities into the export market at price P E gt c Assuming that quantities exported cannot be resold into the domestic market show both mathematically and on your graph from part (a) how much the firm will sell into the domestic market and into the export market
c Consider now a potential entrant into this market For simplicity ignore the existence of the export market To be able to produce in this market an entrant must incur a one time set up or entry cost of E gt 0 It can then produce according to the cost function in (1) Describe in words andor diagrams how you would model this entrants decision problem You should consider the role of the incumbent monopolist in framing your answer
4 Consider a food processing firm that produces processed product Q from a raw agricultural product Rand a composite marketing input L which we will call labor
a Write a production function and draw some rough sketches of isoquants for each of the following possible production scenarios
1 There is no substitution between R and L in producing Q
ii The production relationship is Cobb-Douglas
b Suppose the true production relationship is Cobb-Douglas and that this firm produces Q exclusively for a large retail grocery chain It has a contract for 1999 to deliver Q units of product to the retailer Assume the firm is a perfect competitor in the markets for its inputs Set up the firms optimization problem given this information and derive the first-order conditions for the problem Defme any notation or terminology you need to introduce Also illustrate the optimization problem with a graph
c Discuss the firms demand function for R given the conditions of the problem in part (b) Specifically what variables determine how much R the firm will utilize
d Re-express the problem in part (b) and take first-order conditions assuming now that the firm is the only buyer of the farm input R which is produced according to the following inverse supply function
w = a + 3R a3 gt 0
OJ 04
J ---------
Amiddot __ __
09
0QgtI
4
2257
Jl59
412
14JK
446 -1564
4649
41161
29
0
495
4974
04711
164 9-9
267 2967
4170
llll
99
09X7
76
714
H
middotmiddot 74(
4791
4978
49119
2454 2764
1051
1 115
1441
IIIOK
2157
1577
U74
1517
5
4545
46JJ
4767
4974
4990
middot-
92()
J 7J7 717
2447 ll 4
middot55
4 144
19
24
1699
677
676 SlATISliCI1 tA9LFS
lA Rlf lgtl Areas under the standardized normal distribution
Example
Pr (0 z 196) = 04750
Pr (z 196) = 05 - 04750 = 0025
TABLE 01 Percentage points of lhe t distribution
Example
Pr (r gt 2086) = Oo25
Pr (r gt 1725) = 005 for df = 20
Pr(ltl gt 1725) = 010 0 I 715
01$ 010 middot-00
001 0 00010015010 002 0010 0002ozo
J 07H 127116I 11216 14 61657 IIUI1nm
22 279925IXH6 6965UK16 4 lP 00 01 01 1 0765 I 6X 251 lIK2 4541 51141 102 JJ
4 074 I 15J 21J1 176 461 moo 00411no
Ol 0middot1111
(Xl)lt) 11119 059UIO 0161 t tiiCW 029 0279 1650675 0751 s 1476 2015 511910727 25710596 0616 4010517 115 7
071 II 1064 IIOJ 1141 52111106 bull 1440 1941 l7H7 0791 UKl 0171 091ll 09411
149914KU 1517
17tMl 1406 299K J7XS7 0 711 1415 lH95 2165116111179 1217 1155 129l
1591 161M 1664
1915 1950 19115 W19
10611144 1179 K u 706 lW7 11601772 451111196
212 071) 421172054 219() 122420XX UHl 111 2262 21i21 1251)-9
0700 J72 1112 222K 2764 11692422
1021 214 25492-IK6 102157 2X92291
2794 2121 21152 4025II 0697 1361 1796 22111 2711lt lit
2611 27041gt1 25XU ll7K 1106 1111 190 0695 1156 17H1 21 79 26111 lOSS
ll40 11111 1910
2650 2M9 J65 l119 IJ 0694 1350 1771 2160 1lt5211112311lt6 21 lJH 164
145 1761 21J5 2614 1621 14 069214115 55-1 2977 7117 461 150111 0 1411
l711I 2602 2 94717011 770 790 Kill lX0 15 0691 1141 175 729 21 lfgt65 16116 JHHH
I I
11
1641
1144012
J997 2921907 JOISJ9-U 96 WKO 1 6 0690 bull IJJ7 1 746 2120 2511 l6M6 925 JK(fl 0619 11114011 4147 4162 4177
4194066 41H l6462567 2gt19H411 17 1740 21W40994(19 406 068H UJO 174 210142164192
I
4279 4292 6102X7K4265 25524251 -11117 4222 4441 068H U211 1 729 209J 253944294 112 494 J406 44111 579lK61LJ5 4157
20 0687 1325 I 715 21186 252K 14545115 4515 4515 455 5524474 44114 4495lh 4J 2 0616 121 1711 1010 251 H l517211 I 4571 45112 4591 4599 4616 4625
4554 505J67H A6K6 4691 4699 4706 22 06K6 I Jl l 1717 2014 l5HK 1XI94656 4664 4671I K 4641 41151K074756 4761 17144744 4750 2J 06KS 1319 250020694716 412 47 111 9 470 4719
41101 41112 4M17 14924791 AKOJ20 06K5 IJ UI 2797 1 4671711 20644772 47711 4710 47KK
1
I 4K21 1 4504Kl0 4104 41UH 41141 41146 4150 4H54 4H57 2l 06114 1316 1708 2060 24115 271741116 J4JS1706 20 56 2479 27794K6H 4171 4X75 4H7H 4HIH 4Hol4 4HH7 middot 4K90 26 0614 IJIS41164
J906 4911 06M4 14212471 27714913 4916 170 4909 27 L14 10524196 489R 4901 49044191 06114925 4927 4929 4931 492 4914 4916 140K2467 27612M U13 1701 204849111 4920 4922
4941 4941 4945 4946 494K 4949 4951 4951 IJII JJ9629 068) 2462 2756204S49111 494025
UJO 1697 2042 2457 17504QSS 4964 lJK S4961 JO 06KJ4960 4961 4961
496 J951J4956 49576
7 10 06KI 10749 1 6K4497 241J i7044970 4971 49724967 496K 1021J966 60 0679 129611 4981
4979 4979 49SO 16714977 2 190 26604976 4977Igrave 12122000 4975
l Jfi) 1090
49864914 4914 4985 498S 4986 120 0677 11M9 21671651 19KO 2J58
49111 49H9 49M9 4990 _ 1 1810674 164S 1960 1 126 1576 49H I 49K2 4982 498)
49X7 49K7 49117 49111
N This tahlc gives the area in the riahtmiddothand tail or the distribution tic bull t 0) But since the normal di5tribumiddot Nolt 111e smaller probability shown at the head of each column is the area in one tail the 1in is symmetrical about r- Othe area in the ldtmiddothlnd tail is the same u the area in the corresponding right-hand larser probability is the area in both tails bullbullil rltgtr eumplc PI- 196 S z S 01-04750 TMrdore -196 S t S 196)-2(047S0)-09S Sourr From E S Pearwn and H 0 Hartley eds bull Blomtuila TabftJot Slulislidans vol I
ld ed table 12 Cbullmbridgc University Prns New York 1966 Reproduced by perminion of the middot
editors and trustees of Biomeria
bull bull bull bull 10 II 30 110 100 500
(
I
n
I)
4]1
7 71
4 20
116
1 H4
4 1
1 111
4 1
1
U4
1 75
192
1 gt6
44
2 54
244
)07
142middot
2 2
41J
144
U1
199
245
l-14
2 1
14)
161
190
1 4 1
25J
)70
137
195
2)7
U 5
190
229
1 59
245
U 7
I 91
1 15
96
142
194
1 94
l2
IJ1
117
174
104
1 0
190
U4
109
284
Ill 2 71 2JJ
1 77
194 175
i-1
174
149
157
I y I 7
1 94 1 gt9
194
1 9)
174 159
145 1 4
195
119
179 147
682 AIPFNHIX ll
TARLE Ol l lpper percentage poinls or the F distribution (continut-d)
I I
STATISTICAl TA8LfS 681
df for dl lor llllmtntor N
100 N
I 16 111 I 12 I y I 2ltI 10 1 10 111 1 1 1 1 111 I Y 1 41 1 4 1 1-10 I 19 1 19 UHI middotW 1171411 147 145
I 176 IJl 170 IUI 167 165 I M 161 1 bull1 101 1 97 191 1 90 11111 1116Hi 4 l)
2 15 212
Ill 1117 201 I I I M4 I IIII I I II
2 911 1 II 1 1 1gt 67 251 11 an z llbull 2 1J ll I 26 1 1 11 1 2
191 lOS 2Kl 1 2 55 146 2 -m 24 2 lU 226 2 2l
US1 i9 l-IS S72 4112 4 1 1 199
I 1 I J I 110 I Y 12Y I 211 I M I 7 1 7 I 6 1 15 1 11 U9 UK IJK 1middot17 U6
17M I 71 1 101 167 1 162 161 I I S1 n6 I 51 IU1115 I ll 1 4 1 1401 19 1 47 146 1 44
I 98
2111 191 1 1111 1 19 210 1 042B 9 I H6 1 gt4 1 MO 119 1 1 1 1 1) 1 1 1 211 2111 2JO 22S 21 2111JOI 2SI 242 2 16 11114 16 140
2M9 2 66 UK 2 40 111 1 1 1 2-1 2 1 Ul167 middot 1 16 126 117 109 ill2 H I 472 4 12 01
114 I 12 111 1 10 129 I 211 UK 16 I 26 126 I 25 I 2 I 42 1 4 1 19 us I H 17 l l6 2 1 111 146 us 1 16 I 1611 165 1 61 I 5Y UK us IS-1 I S1 I S l I t HIIll 11111 186 2 II 1 2 1 11 2 1111 11111 Y I 151 107 199 190 1-JIS 1112 1111) 1 16 1 1S I 71 I 7 1 211 222 2111 215Ull OS 219U7 198 1 14 2 t7 2111 2511 lSO 242 1 16 2 225 22J 219 216 2 IJ UH l09 )02 196HI 129SSl 41gt4 41401
IJJ I I 1 10 129 1 18 117 127 126 1 1gt 125 124 1 24 136 usI 41 141 140 119 UK1 111 146 145 1 69 1 66 16 159 I S1 156 I 51 152 I SO 149 1 19 Ill 1114 1 812 11 1 1 229 216 1116 190 187 100
H14 196 1 9 1 1 81 1 112 119 1 71 I 71 1 11 1 69 167 165 0)219 2 15 1 1 1 JN 295 1 1 1 2242)6 229 V5 1 1 144 IS 210 2 26 1 19 1 1 1 2 1 1 109 U6 U l
0 1 407 I 75 )51 U6 123 312 101
Ul l ltJ 129 Ull 127 1 26 126 IH 1 2-1 1 2-1 12J t n l iM I 45 1 44 1 41 U9 Ull IJ6 us IB
172 167 1 64 ltll I H 15-1 Ul I SU I 1 lt7 1 6 10 10IIIS 182 179 171I 10 11111 1 22K 114 20S 1911 19) 188
2111 1 91 I H4 179 1 16 1 14 110 I M 1 IM I 62 O)1 41 lB 227 221 216 213OS l 2 291 169
270 2-17 210 2 2 1 IJ I I 207 2 Ul ] 0 1 147 J0 1 1 7 )07 298 2910 1 1 56 5 19 451 40
110 I 126 125 124 Ut 12 Ul 121 119 1 19 25IJ6 us 1 14 IJJ Ul U l140 11914425
1117 1113 179 1 66 161 157 1 )4 I l l 14K 141 142 1-11 119 I 111
176 173 1 71IllI 21)9 1 00 192 IM4 179 1611 1 64 I us I 5J U l 100I 21Oi
01
1R4 261 2 14 225 218 2 12 1 08 252 217 229 220 2 1 1 1116 202 1 94 I 92 I IM1 01 664 II 181 lSI 129 1 12 199 2119 280 27)511S
127 125 1 14 122 121 1 11l 119 1 1 7 1 1 1 1 16 1 1 5 1 1 5 I J J 112 I l l IJO 119 119us 1 41 1 4 1 UK 160 154 Ul 1411 144 141 140 116 I 15 I J I ll I 29 10 I
111 168 1 66117IK7 1821 19 2 19 211S 104 111-1 175 110 165 159 156 15] 1 4H 147 144 I-II 1 9 0)
125 2 17 210 104 199 195 1921 1 5 1 16 2 S 110 212 20 194 11111 1 04 17 I 71 I 61 160 til2S6 1SO 1 ll 295 lKl 272 261 165 ) 14
124 1221 16 129 128 127 1 2 1 1 19 1 111 1 17 116 1 14 1 1 1 1 1 1 I l l 1 10
IJJ IJI 1 10 140 119 141 117 132 111 126 1 24 1 2 1 119 10 120I us1 60I gtI 182 111 168 16S 1622 1)10 us
166 161 155 150 146 14 117 115 1 12 1211 125 I Hl 217 109 101 191 IKOl 219 201 I 176 1 101 156 UJ I 1 41 I IX
6 liS I 1701 4 J9 195 14H 296 2 79 266 256 247 240 middot 234
Ul 1 2 1 l iD 1 1 11 116 114 1 12 I l l 1 10 1119 I 0 25 111 IJ9 1111 I J6 I a I J2 I) I 129 128 127 126 US
152 146 1 41 1 111 1 1-1 I I 1211 I 24 122 uu 1 1 7 Ia Ill 2 I I 197 IKII IKO 1 75 170 166 161 160 157
172 162 IS7 IS2 146 141 119 I J2 129 U6 1 22 119 m198 19J I 88 I 114 180OS H9 1 04 265 242 226 2 14 2lt16
01 676 411 lllll l41 J I I 289 271 I 91 1119 I 79 169 I 61 1511 148 144 119 1 11 1 11 01
260 2SO 241 2 14 227
122 1 10 1 16 11-1 1 1 1 1 12 1119 Uli I 07 l tiO 124 124I 12 I J9 IJ7 US I J 111 129 1211 127 125
I II 142 118 114 uo 126 124 1 1 11 1 1 1 1 1 1 I OK Hill 10
i lS 25
IS72 JO 160 us172 167 16J10 2 71 2011 194 U2 1 14 1 12 1 1 1 1 1 1 I IIII 167 152 146 179l 11-1 HXI 260 2J7 221 210 201 1811 181liSmiddot
21KIll 6 6 4bl 3-JII J l2 l 02 2110 264 2 51 241 232 22S 1 10 U6 I ll I S I OtlIIIK104 U9 152 0 1
Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
(035) (095)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 26 1998
There are two sections to this exam Each section consists of four questions Follow the instructions for each section carefully Use complete sentences and well thought out responses when writing down your answers Note that if we cannot read your writing then we will assume that your answer is wrong All questions are equally weighted There are five pages to this exam
Section 1 This section consists of questions 1 ) 2) 3) and 4) Answer question 1 ) and then select two of the remaining three questions to answer Do not answer all four questions
1) Below is a highly simplified model of wine supply and demand Selected t-ratios are below each
estimated coefficient
Two-Stage Least-Squares Estimates of Deinand Q = -101 +026PWHAT
(210)
R = 055 DW=066 Two-Stage Least Squares Estimates of Supply
Q =-101+ l22PWHAT- 222 WR + 340PGRAPE + 123Qt-l(110) (056)
R2 =096 DW=l95
Auxiliary First Stage Regression PWHAT = 201 + 133WR + 440PGRAPE + 023Q_
R2 =0995 The variables are defined as follows
Q - Quantity of wine supplied and demanded
PWHAT- Price of wine from auxiliary regression
WR- Wage rate in wineries
PGRAPE - Price of wine grapes
There are (at least) 11 things wrong with this model and its estimates Can you find them List them by
number and indicate the consequence of each problem
middot 2) You have been asked to estimate a linear econometric model say Y =30 + 3 X + Jl Assume that
Jl = PJl_ + e (I)
2rsquowhere e is a normally distributed random variable with a mean of 0 and variance of u bull The above
model (1 ) is often termed autocorrelation
a) How do you test for autocorrelation
There are two kinds of autocorrelation- pure and impure The Durbin-Watson statistic can be indicative
of either kind
b) Show the consequences of both kinds of autocorrelation and how to correct for each
3) Consider the following primal (P) linear programming problem
max R = Ox + 8x2 XJxl
st tX1 + x2 5125
2x1 + 3x2 5 300 (P)
x 5 90
Xp X2 0
Answer the questions below being sure to explain your steps and reasoning
a) Graph the constraints of (P) and identify the primal feasible solution set ie the set of all
admissible solutions for (P)
b) Identify all the extreme points of the primal feasible solution set
c) Find the optimal solution to (P)
d) Determine the slope of the isorevenue curves sketch a few of them and verify your
answer in part c)
e) Write down the dual (D) linear programming problem to (P)
4) An individual investor has $70000 to divide among several investments The investments
available to this person are municipal bonds with an 85 annual return certificates of deposit with a 5
annual return treasury bills with a 65 annual return and a growth stock fund with a 13 annual return
The investments are all re-evaluated after one year ie the time frame for the investor is one year Each
investment has a different perceived risk to the investor so it is optimal to diversify
The following guidelines have been established for diversifying investments bull No more than 20 of the total investment should be in the growth stock fund
bull The amount invested in certificates of deposit should not exceed the amount invested in the other three
investment alternatives bull At least 30 of the investment should be in treasury bills and certificates of deposit
2
bull More should be invested in certificates of deposit and treasury bills than in municipal bonds and the
growth stock fund by a ratio of 12 to 10 The investor wants to know how much to invest in each investment alternative in order to maximize the
total return from all investments All $70000 is to be spent in this investment plan
Formulate the linear programming problem for this investor Defme all the variables constraints and
equations in the formulation
Section 2 This section consists of questions 5) 6) 7) and 8) Answer any three of these four questions Do not answer all four questions
5) Which of the following describes an externality and which does not Explain your reasoning
clearly in each case
a) A policy of restricted coffee exports in Brazil causes the US price of coffee to rise which
in turn causes the price of tea to increase
b) An advertising blimp distracts a motorist who then hits a telephone pole and damages the
car
c) In a homogeneous product Cournot duopoly an increase in the output of firm 1 lowers the
market price received by firm 2 for its product (due to the law of demand) ceteris paribus thereby reducing the profit of firm 2
6) Consider a competitive industry with a large number of identical firms each of which has a simple
cost curve of the form
C(q)= l +1 qgtO 0 q =O
where q 0 is the output of the firm Think of these costs in a long-run context ie if the firm produces
nothing it incurs no costs but any q gt 0 incurs the 10 set up cost Suppose industry demand is given by
Q0 =52-P
where Pis the market price of the good produced by the firms
a) Derive a supply curve for one of the identical fmns
b) Derive the industry supply curve given that there are n firms
c) What is the long-run equilibrium price in this industry Explain your answer and show any
calculations
3
e)
a)
d)
d) How many firms will populate this industry in long-run equilibrium Explain your
answer
f)
Given your solution for n from part d) assume demand shifts to Q0 =53- P What is
industry price and output in the short-run equilibrium
Given your result in part e) explain in words what will happen as the industry adjusts to
long-run equilibrium
7) The following is a demand equation for residential electricity consumption in the US estimated by
H Houthakker (Energy Jouma1980)
ln E = 0586 + 1388lnY-1118 ln PE + 0199lnH + 0483lnC +0566LnP0
where E is per capita electricity consumption in KWH per year Y is disposable personal income per
capita PE is the price per KWH of electricity H is heating degree days (deviation over the year of
temperature below 65deg) C is cooling degree days (deviation over the year of temperature above 75deg) and
P 0 is the price of natural gas per BTU
Based on this model what are the own price cross price and income elasticities of
demand for electricity in the US
b) A deregulation plan for the electricity industry is currently being implemented Suppose
electricity prices rise as a result of deregulation Based on Houthakkers equation will
consumers spend more less or the same on electricity following deregulation Explain
c) Many poor families live in apartments with electric heating and cooling A concern of
policy makers is that high energy costs make it difficult for poor families to adequately heat
or cool their homes Consider a typical poor family whose demand for electricity is given
by Houthakkers equation The family pays $010 per KWH and has per capita income of
$3000 ($12000 for a family of four) Given normal values for H C and P 0 the family
consumers on average 6000 KWH per year Energy experts have determined that 8000
KWH are needed to provide satisfactory heating and cooling in the familys climatic region
Consider two programs to eliminate the deficiency (a) a cash payments plan (b) an energy
subsidy program under which the family will get a rebate from the government for a
specific percentage of its monthly electric bill
(i) How large must the cash payment be to accomplish the plans goal
(ii) What percentage subsidy will achieve the programs goal
lllustrate the two plans using indifference curves for electricity and all other goods and the
budget constraint Begin with the situation where E = 6000 and show how either program
can attain the goal
A third possible program involves coupons that can be spent only on electricity Will e)
coupons entitling the household to buy 2000 KWH achieve the goal Explain
4
8 ) Consider a monopolist who sells in two markets The inverse demands and cost function for the
monopolist are as follows
I = 80-SQ market I
R =180- 20Q2 market2
C (Q)=50 + 20(Q + Q2) cost function
a) Find the firms profit maximizing price and output in each market assuming it is free to set
whatever price it wishes in each market Show your work
b) Assuming absence of any externalities what is the socially optimal output in each market
Explain your answer and show any calculations
c) Define the term deadweight loss and then calculate the deadweight loss from monopoly
power in each market
d) TrueIFalse Explain The firm in this problem is a natural monopolist
e) The following statement is true The firm in this problem is practicing price
discrimination Explain carefully and precisely what this statement means in terms a
noneconomist could understand
f) Suppose the government passed a law preventing the firm from practicing price
discrimination What price and output would the firm then set in market I Market 2
Provide a precise answer if you can showing all work but dont get bogged down At a
minimum explain in words how if at all this law would affect the firm
5
University of California Davi Jepartment of Agricultural and Resource bconomics
MS Comprehensive Exam September 151998
There are two sections to this exam Each section consists of four questions Follow the instructions for each section carefully Use complete sentences and well thought out responses when writing down your answers Note that if we cannot read your writing then we will assume that your answer is wrong All questions are equally weighted There are five pages to this exam
Section 1 This section consists of questions 1 ) 2) 3) and 4) Answer question 1 ) and then select two of the remaining three questions to answer Do not answer all four questions
I) Below is a highly simplified model of wine supply and demand Selected t-ratios are below each estimated coefficient
Two-Stage Least-Squares Estimates of Supply
Q = -101 + 026PWHAT(210) (220)
R2 =035 DW=041 n=21
Two-Stage Least Squares Estimates of Demand
Q = IOI -122PWHAT + 222NCOME +340CPIALL + 0753Q_1(23S) (110) (056) (400)
R2 =096 DW=05 Durbins h- statistic= 083 n=21
Auxiliary First Stage Regression PWHAT = 201+ 133NCOME + 440CPIALL + 023Q_1
R2 = 067 DW= 187
Regression On Residuals From Demand Equation
Jl = 05+155NCOME +055CPIALL (620) (ll)
R2=072 The variables are defmed as follows
Q - Quantity of wine supplied and demanded PWHAT - Price of wine from auxiliary regression INCOME- Per capita personal income CPIALL - Consumer price index of all items
Examine the above model for econometric problems In particular you should look for
I autocorrelation 2 heteroscedasticity
2)
3 omitted variables
4 multicollinearity
5 identification
6 simultaneous equation bias
For each of the above possible problems explain why you think there is or is not a problem Explain
what should be done to remedy the problem
You have been asked to estimate a linear econometric model say Y = 30 + 31X + Jl Assume that
var(Jl) = a2X (I)
where Jl is a normally distributed random variable with a mean of 0 and a variance given in Eq () The
above model (I) is termed heteroscedasticity
a) Clearly explain the problems that heteroscedasticity creates
b) How might you go about detecting the presence of heteroscedasticity in the above linear
model
middotc) How might you correct for the heteroscedasticity given that it occurs in the form ofEq (1)
in the above linear model Be specific and give details
3) Answer the following jive TrueFalseExplain questions concerning a general linear programming
(LP) problem Please note that the explanation is worth the bulk of the credit
a) An optimal solution to aLP problem uses up all of the limited resources available
b) A point on the boundary of the feasible set (or region) satisfies all of the constraints of the
LP problem
c) The optimal solution of an LP problem always occurs at an extreme point of the feasible
set
d) If the first COilStraint of aLP problem evaluated at given value xo of the decision vector
has a zero value for its corresponding slack variable then the point xo lies on the boundary
of the feasible region
e) If the shadow value (or dual variable) corresponding to a given constraint is positive than
that constraint is binding
4) PROTRAC INC produces two lines of heavy equipment One of the product lines termed
earthmoving equipment is essentially for construction applications while the other line termed forestry
2
equipment is destined for Ute timber industry The largest member or ihe earthmoving line (the E-9) and
the largest member of the forestry equipment line (the F-9) are produced in the same departments and with
the same equipment Using economic forecasts for the next month PROTRACs marketing manager has
judged that during that period it will be possible to sell as many E-9s and F-9s as the flrm cam produce
Management must now recommend a production target for the next month
Making this decision requires that the following major facts be considered
bull PROTRACs net revenue is $5000 on each E-9 that is sold and $4000 on each F-9
bull Each product is put through PROTRACs machining operations in department A and department B
bull For next months production department A hasl50 hours available and department B has 160 hours
available Each E-9 uses 10 hours of machining in department A and 20 hours of machining in
department B Each F-9 uses 15 hours of machining in department A and 10 hours of machining in
department B
bull In order for management to honor an agreement with the union the total labor hours used in next
months testing of finished products cannot fall more than 10 below an arbitrated goal of 150 hours
The testing is performed in a third department and has nothing to do with the activities in departments
A and B Each E-9 is given 30 hours of testing and each F-9 is given I0 hours of testing
bull In order to maintain the current market position senior management has decreed the operating policy
that it is necessary to build at least one F-9 for every three E-9 produced
bullmiddot A major dealer has ordered a total of at least five E-9s and F-9s in any combination for the next
month and so at least that many must be produced
Given the above considerations managements problem is to decide how many E-9s and many F-9s to
produce next month so as to maximize net revenue
Formulate the linear programming problem for PROTRAC INC Deflne all the variables constraints
and equations in the formulation
Section 2 This section consists of questions 5 ) 6) 7) and 8) Answer any three of these four questions Do
not answer all four questions
5) A simple monopolist faces the linear inverse market demand curve given by p(y) a - by where
a gt 0 b gt 0 and y is the output of the monopolist In addition the monopolist faces the total cost
function given by c(y) Fe where k gt 0 Finally the government pays the monopolist a subsidy of
s gt 0 dollars per unit of output produced in order to encourage the monopolist to produce more output and
sell it at a lower price
a) Set up the proflt maximization problem facing the monopolist
3
--------
b) Find the monopolists profit maximizing level of output say y = ym(abks) as well as its profit maximizing price say p = pm(abk s) How do you know you have found the profit maximizing solution
c) Will an increase in the subsidy achieve its desired goal of increasing the monopolists output and lowering its price Show your work
d) Provide an economic interpretation of the above comparative statics e) How large would the subsidy have to be in order to get the monopolist to produce the
Pareto efficient level of output and charge the Pareto efficient price
6) The following questions pertain to a production setting characterized by the usual neoclassical production function of the form Q = f(xlx2x ) where Q is output and (x1x2x ) are inputs
a) Define compare and contrast the following terms from production economics i) Decreasing returns to a variable input and ii) decreasing returns to scale (Decreasing returns to scale is also called
diseconomies of scale) b) What economic forces cause decreasing returns to a variable input and what economic
forces might cause decreasing returns to scale c) Given the production function as specified show mathematically how you would measure
the returns to the variable input x1 and the returns to scale d) Suppose you were studying the question of whether small farms could compete effectively
against large farms in the market for the product Q Would either or both of the concepts of returns to a variable input and returns to scale be valuable in your analysis Explain
7) The following are demand and supply curves that apply to a competitive industry
Q0 = 250- SOP Qs = ()P
a) Derive the market equilibrium price and quantity b) Is the equilibrium you derived in (a) a short-run equilibrium a long-run equilibrium or
could it be both Explain your answer in detail demonstrating your knowledge of behavior in competitive markets and the distinction in these markets between the short and long run
c) Politicians are floating a proposal to place a per unit tax of $ 100 on this commodity Producers in lobbying against the tax argue that they will have no choice but to pass the
full tax on to consumers and therefore the tax if enacted will only harm consumers Evaluate the producers argument concerning the tax Use either algebra (preferred tool) or
4
--------------
------------------------------ X
bull
a)
a graph to support your argument If you use a graph label it carefully using the demand and supply equations Q0 = 250- 50P and Qs = ( )P
8) The following graph depicts a consumers Marshallian demand curve for the commodity X The market price of X is given as
p
Demand Curve
Given identify the consumers surplus on the graph and give an intuitive definition of consumers surplus
b) Assuming X is a normal good for this consumer draw the consumers Hicksian demand curve given price
c) Explain the difference between Hicksian and Marshallian demand curves
d) Suppose price increases to P_ Use Hicksian demands to identify the consumers compensating variation and equivalent variation due to the price change Label carefully
5
University of California Davis Department of Agricultural and Resource Economics
MS Exam
July 2 1999
Directions You have three hours to write this exam Please write legibly Answers that cannot be read are regarded as incorrect answers This exam consists of two parts In part I quantitative methods you must answer question 1 Then choose to answer either question 2 or question 3 In part II microeconomics answer three of the four questions You should thus answer five questions in total two from part I and three from part II Do not under any circumstances answer more than five questions
Part I Answer 1 estion I and either Question 2 01 luestion 3
l You are constructing a si mple model of cotton su pply for a short-term project you are working on and h ave formulated the following conceptual model
where y5 i s bales of cotton ps i s the price paid to growers w1 is the labor wage rate w2 i s the price of water WJ i s pesticide price and DO is degree-days a measure of the growi ng conditions during the season
(a) A nother variable you might include in thi s speci ficati on is annual rainfall (R) whose coefficient i f included in the model would be a6 While you are sure th at all th e other va ri ables in the model are part of the correct speci fication you are not sure about annual rainfall As an applied econometrician explai n the tradeoffs made to the properties of you r estimates i f you include or exclude thi s vari able from the specification
(b) One must always be concerned with measurement error i nempirical models Explain th e effects on the properties of your estimates of (a) measurement error in th e dependent variable vs (b) measurement error in one or more explanatory vari ables
(c) You esti mate th e model wi th R included and the results are
yS = -6354 + 8727ps - 0 579lw1 - 7907w2 - 09045w3 + 769000 + 0045R ( -0 79) (2 36) ( -177) (-234) ( -190) (157) (045)
R2 = 0819 n =56
where th e Students-t statistics are gi ven in parentheses and the following sums of squared errors resu lt
Total SSY=660 9 Model SSR=5412 Errors SSE= l l 97
In answering the following part s justify your answers with quantitati ve test results using th e attached tables Quali tative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 0 5 level
(ii) Is the overall model significant Th at i s i s the hypothesis th at a1 = a2 = a3 = a4 = as = a6 = 0 rejected
(iii) The correlation between the price of water (w2) and the price of pesticides (w3) i s zero Do these two vari ables have the same effect on cotton supply i e are their coeffi cients equ al
2 Suppose you have the following econometric model
where y = yieldacre x1 = wateracre x2 = (wateracre)2 x3 = fertilizeracre
(a) For each of the following questions set up the appropriate hypothesis test and provide a brief explanation
(i) Does water have any influence on yieldacre
(ii) Assuming an unconstrained (interior) solution is wateracre = 300 significantly different from the amount that yields the maximum yield per unit water
(b) Suppose you work in a governmental agency and that your boss is not trained in economics or statistics and thus she does not know what e N(O all) means Explain-
in words the meaning and significance of this expression
3
a Given the regional farm modeling problem
Max subject to
px- cx Ax b
where x is an n x I vector of production activities p and c are vectors of prices and variable cost coefficients respectively b is an m x 1 vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Comment briefly on the connection between the dual problem and the Dual Complementary Slackness condition
(iii) If you had to calculate the derived demand for a single input b describe the type of function you would obtain from the problem
b Now use the information that the product prices p are determined according to the following demand relationships
p =a + Dx
Assume that the individual farmers act as perfect competitors
(i) Show how you would reformulate your problem and give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit )
(ii) Derive the dual to the problem in part b(i) showing your steps clearly
Part II Answer of the 4 questions in this Section _
I For some goods consumption is subject to more than the usual single constraint on money income time can also be a constrai nt Consider a consumer with utility f unction u(x) defined on goods x = (x1 xn) subject to th e two constraints T 2 tx and M 2 px where T is the total amount of time avai lable and M is th e total amount of income avai lable and t = (t1 tn) and p = (p o pn ) are the time and money prices per unit of consumption of each good
(a) Write th e primal utility max i mi zation problem for th is consumer What interpretation do you attach to th e sh adow values on each constraint
(b) F rom your answer to part (a) ex plain how the consumer s choices reveal her money value of time Develop an ex pression f or the marginal money value of time from th is pri mal problem
(c) Under what conditions would the problem revert to the standard consumer choice problem subject to a single money budget constraint
(d) Suppose th at the shadow value on the money budget constraint is zero W rite the dual expenditure mi nimization problem for thi s case Provide a brief interpretation in words of thi s dual ex penditure f unction Also provide an interpretation for the sh adow value of the constraint for th is problem
2 Assume t hat t he US wheat mar ket can be considered in isolation and is charact erized by
yS = sops - 20 wl
y0 = 60 - 12 5p0 +2M
where
and
ys is quantity of wheat suppl ied in millions of bushels per year p5 is the price recei ved by suppliers in dollars per bushel y0 is quantity demanded (in millions of bushels) by consumers p0 i s the price paid by consumers also in dollars per bushel w1 is the price of a composite input to wheat product ion M is consumer income in t housands of dollars
For purposes of this analysis consumer income i s M = 45 and the composite input price is w1 = 5
(a) What are t he equilibrium f ree-mar ket price and quant it y of wheat
(b) Is t he supply of wheat at this point price-elastic or price-inelast i c Ex plain your reasomng
(c) W h at is t he t otal net economic sur plus f rom production of wheat and how middotis it distr ibuted bet ween suppliers and demanders
(d) A var iet y of schemes are used in agricult ural markets to enhance producer welf are C onsider a pr ice support of $6 per bushel Explain t he effects of such a policy on the dist ri bution of welfare and economic efficiency Support your answer with quant itat ive est imates of t hese effect s
jgtrice
3 Suppose a monopoly fum faces the following linear demand curve P = a- bQ where P is price and Q is quantity produced and sold The monopolists cost function is
(1) c(Q) = cQ for Q Q c(Q) qo for Q gt Q
where Q is a capacity constraint
a Solve mathematically for the monopolists profit maxumzmg and output as functions of the parameters a b and c Show your work Also illustrate yourresults graphically labeling carefully all relevant information
b Suppose the function P(Q) from part (a) is the demand in the domestic market and that the monopolist can also sell unlimited quantities into the export market at price P E gt c Assuming that quantities exported cannot be resold into the domestic market show both mathematically and on your graph from part (a) how much the firm will sell into the domestic market and into the export market
c Consider now a potential entrant into this market For simplicity ignore the existence of the export market To be able to produce in this market an entrant must incur a one time set up or entry cost of E gt 0 It can then produce according to the cost function in (1) Describe in words andor diagrams how you would model this entrants decision problem You should consider the role of the incumbent monopolist in framing your answer
4 Consider a food processing firm that produces processed product Q from a raw agricultural product Rand a composite marketing input L which we will call labor
a Write a production function and draw some rough sketches of isoquants for each of the following possible production scenarios
1 There is no substitution between R and L in producing Q
ii The production relationship is Cobb-Douglas
b Suppose the true production relationship is Cobb-Douglas and that this firm produces Q exclusively for a large retail grocery chain It has a contract for 1999 to deliver Q units of product to the retailer Assume the firm is a perfect competitor in the markets for its inputs Set up the firms optimization problem given this information and derive the first-order conditions for the problem Defme any notation or terminology you need to introduce Also illustrate the optimization problem with a graph
c Discuss the firms demand function for R given the conditions of the problem in part (b) Specifically what variables determine how much R the firm will utilize
d Re-express the problem in part (b) and take first-order conditions assuming now that the firm is the only buyer of the farm input R which is produced according to the following inverse supply function
w = a + 3R a3 gt 0
OJ 04
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lA Rlf lgtl Areas under the standardized normal distribution
Example
Pr (0 z 196) = 04750
Pr (z 196) = 05 - 04750 = 0025
TABLE 01 Percentage points of lhe t distribution
Example
Pr (r gt 2086) = Oo25
Pr (r gt 1725) = 005 for df = 20
Pr(ltl gt 1725) = 010 0 I 715
01$ 010 middot-00
001 0 00010015010 002 0010 0002ozo
J 07H 127116I 11216 14 61657 IIUI1nm
22 279925IXH6 6965UK16 4 lP 00 01 01 1 0765 I 6X 251 lIK2 4541 51141 102 JJ
4 074 I 15J 21J1 176 461 moo 00411no
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071 II 1064 IIOJ 1141 52111106 bull 1440 1941 l7H7 0791 UKl 0171 091ll 09411
149914KU 1517
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1591 161M 1664
1915 1950 19115 W19
10611144 1179 K u 706 lW7 11601772 451111196
212 071) 421172054 219() 122420XX UHl 111 2262 21i21 1251)-9
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ll40 11111 1910
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145 1761 21J5 2614 1621 14 069214115 55-1 2977 7117 461 150111 0 1411
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4979 4979 49SO 16714977 2 190 26604976 4977Igrave 12122000 4975
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49864914 4914 4985 498S 4986 120 0677 11M9 21671651 19KO 2J58
49111 49H9 49M9 4990 _ 1 1810674 164S 1960 1 126 1576 49H I 49K2 4982 498)
49X7 49K7 49117 49111
N This tahlc gives the area in the riahtmiddothand tail or the distribution tic bull t 0) But since the normal di5tribumiddot Nolt 111e smaller probability shown at the head of each column is the area in one tail the 1in is symmetrical about r- Othe area in the ldtmiddothlnd tail is the same u the area in the corresponding right-hand larser probability is the area in both tails bullbullil rltgtr eumplc PI- 196 S z S 01-04750 TMrdore -196 S t S 196)-2(047S0)-09S Sourr From E S Pearwn and H 0 Hartley eds bull Blomtuila TabftJot Slulislidans vol I
ld ed table 12 Cbullmbridgc University Prns New York 1966 Reproduced by perminion of the middot
editors and trustees of Biomeria
bull bull bull bull 10 II 30 110 100 500
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4 20
116
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192
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199
245
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161
190
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137
195
2)7
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190
229
1 59
245
U 7
I 91
1 15
96
142
194
1 94
l2
IJ1
117
174
104
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190
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109
284
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1 77
194 175
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174
149
157
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174 159
145 1 4
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682 AIPFNHIX ll
TARLE Ol l lpper percentage poinls or the F distribution (continut-d)
I I
STATISTICAl TA8LfS 681
df for dl lor llllmtntor N
100 N
I 16 111 I 12 I y I 2ltI 10 1 10 111 1 1 1 1 111 I Y 1 41 1 4 1 1-10 I 19 1 19 UHI middotW 1171411 147 145
I 176 IJl 170 IUI 167 165 I M 161 1 bull1 101 1 97 191 1 90 11111 1116Hi 4 l)
2 15 212
Ill 1117 201 I I I M4 I IIII I I II
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191 lOS 2Kl 1 2 55 146 2 -m 24 2 lU 226 2 2l
US1 i9 l-IS S72 4112 4 1 1 199
I 1 I J I 110 I Y 12Y I 211 I M I 7 1 7 I 6 1 15 1 11 U9 UK IJK 1middot17 U6
17M I 71 1 101 167 1 162 161 I I S1 n6 I 51 IU1115 I ll 1 4 1 1401 19 1 47 146 1 44
I 98
2111 191 1 1111 1 19 210 1 042B 9 I H6 1 gt4 1 MO 119 1 1 1 1 1) 1 1 1 211 2111 2JO 22S 21 2111JOI 2SI 242 2 16 11114 16 140
2M9 2 66 UK 2 40 111 1 1 1 2-1 2 1 Ul167 middot 1 16 126 117 109 ill2 H I 472 4 12 01
114 I 12 111 1 10 129 I 211 UK 16 I 26 126 I 25 I 2 I 42 1 4 1 19 us I H 17 l l6 2 1 111 146 us 1 16 I 1611 165 1 61 I 5Y UK us IS-1 I S1 I S l I t HIIll 11111 186 2 II 1 2 1 11 2 1111 11111 Y I 151 107 199 190 1-JIS 1112 1111) 1 16 1 1S I 71 I 7 1 211 222 2111 215Ull OS 219U7 198 1 14 2 t7 2111 2511 lSO 242 1 16 2 225 22J 219 216 2 IJ UH l09 )02 196HI 129SSl 41gt4 41401
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H14 196 1 9 1 1 81 1 112 119 1 71 I 71 1 11 1 69 167 165 0)219 2 15 1 1 1 JN 295 1 1 1 2242)6 229 V5 1 1 144 IS 210 2 26 1 19 1 1 1 2 1 1 109 U6 U l
0 1 407 I 75 )51 U6 123 312 101
Ul l ltJ 129 Ull 127 1 26 126 IH 1 2-1 1 2-1 12J t n l iM I 45 1 44 1 41 U9 Ull IJ6 us IB
172 167 1 64 ltll I H 15-1 Ul I SU I 1 lt7 1 6 10 10IIIS 182 179 171I 10 11111 1 22K 114 20S 1911 19) 188
2111 1 91 I H4 179 1 16 1 14 110 I M 1 IM I 62 O)1 41 lB 227 221 216 213OS l 2 291 169
270 2-17 210 2 2 1 IJ I I 207 2 Ul ] 0 1 147 J0 1 1 7 )07 298 2910 1 1 56 5 19 451 40
110 I 126 125 124 Ut 12 Ul 121 119 1 19 25IJ6 us 1 14 IJJ Ul U l140 11914425
1117 1113 179 1 66 161 157 1 )4 I l l 14K 141 142 1-11 119 I 111
176 173 1 71IllI 21)9 1 00 192 IM4 179 1611 1 64 I us I 5J U l 100I 21Oi
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1R4 261 2 14 225 218 2 12 1 08 252 217 229 220 2 1 1 1116 202 1 94 I 92 I IM1 01 664 II 181 lSI 129 1 12 199 2119 280 27)511S
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111 168 1 66117IK7 1821 19 2 19 211S 104 111-1 175 110 165 159 156 15] 1 4H 147 144 I-II 1 9 0)
125 2 17 210 104 199 195 1921 1 5 1 16 2 S 110 212 20 194 11111 1 04 17 I 71 I 61 160 til2S6 1SO 1 ll 295 lKl 272 261 165 ) 14
124 1221 16 129 128 127 1 2 1 1 19 1 111 1 17 116 1 14 1 1 1 1 1 1 I l l 1 10
IJJ IJI 1 10 140 119 141 117 132 111 126 1 24 1 2 1 119 10 120I us1 60I gtI 182 111 168 16S 1622 1)10 us
166 161 155 150 146 14 117 115 1 12 1211 125 I Hl 217 109 101 191 IKOl 219 201 I 176 1 101 156 UJ I 1 41 I IX
6 liS I 1701 4 J9 195 14H 296 2 79 266 256 247 240 middot 234
Ul 1 2 1 l iD 1 1 11 116 114 1 12 I l l 1 10 1119 I 0 25 111 IJ9 1111 I J6 I a I J2 I) I 129 128 127 126 US
152 146 1 41 1 111 1 1-1 I I 1211 I 24 122 uu 1 1 7 Ia Ill 2 I I 197 IKII IKO 1 75 170 166 161 160 157
172 162 IS7 IS2 146 141 119 I J2 129 U6 1 22 119 m198 19J I 88 I 114 180OS H9 1 04 265 242 226 2 14 2lt16
01 676 411 lllll l41 J I I 289 271 I 91 1119 I 79 169 I 61 1511 148 144 119 1 11 1 11 01
260 2SO 241 2 14 227
122 1 10 1 16 11-1 1 1 1 1 12 1119 Uli I 07 l tiO 124 124I 12 I J9 IJ7 US I J 111 129 1211 127 125
I II 142 118 114 uo 126 124 1 1 11 1 1 1 1 1 1 I OK Hill 10
i lS 25
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Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
2rsquowhere e is a normally distributed random variable with a mean of 0 and variance of u bull The above
model (1 ) is often termed autocorrelation
a) How do you test for autocorrelation
There are two kinds of autocorrelation- pure and impure The Durbin-Watson statistic can be indicative
of either kind
b) Show the consequences of both kinds of autocorrelation and how to correct for each
3) Consider the following primal (P) linear programming problem
max R = Ox + 8x2 XJxl
st tX1 + x2 5125
2x1 + 3x2 5 300 (P)
x 5 90
Xp X2 0
Answer the questions below being sure to explain your steps and reasoning
a) Graph the constraints of (P) and identify the primal feasible solution set ie the set of all
admissible solutions for (P)
b) Identify all the extreme points of the primal feasible solution set
c) Find the optimal solution to (P)
d) Determine the slope of the isorevenue curves sketch a few of them and verify your
answer in part c)
e) Write down the dual (D) linear programming problem to (P)
4) An individual investor has $70000 to divide among several investments The investments
available to this person are municipal bonds with an 85 annual return certificates of deposit with a 5
annual return treasury bills with a 65 annual return and a growth stock fund with a 13 annual return
The investments are all re-evaluated after one year ie the time frame for the investor is one year Each
investment has a different perceived risk to the investor so it is optimal to diversify
The following guidelines have been established for diversifying investments bull No more than 20 of the total investment should be in the growth stock fund
bull The amount invested in certificates of deposit should not exceed the amount invested in the other three
investment alternatives bull At least 30 of the investment should be in treasury bills and certificates of deposit
2
bull More should be invested in certificates of deposit and treasury bills than in municipal bonds and the
growth stock fund by a ratio of 12 to 10 The investor wants to know how much to invest in each investment alternative in order to maximize the
total return from all investments All $70000 is to be spent in this investment plan
Formulate the linear programming problem for this investor Defme all the variables constraints and
equations in the formulation
Section 2 This section consists of questions 5) 6) 7) and 8) Answer any three of these four questions Do not answer all four questions
5) Which of the following describes an externality and which does not Explain your reasoning
clearly in each case
a) A policy of restricted coffee exports in Brazil causes the US price of coffee to rise which
in turn causes the price of tea to increase
b) An advertising blimp distracts a motorist who then hits a telephone pole and damages the
car
c) In a homogeneous product Cournot duopoly an increase in the output of firm 1 lowers the
market price received by firm 2 for its product (due to the law of demand) ceteris paribus thereby reducing the profit of firm 2
6) Consider a competitive industry with a large number of identical firms each of which has a simple
cost curve of the form
C(q)= l +1 qgtO 0 q =O
where q 0 is the output of the firm Think of these costs in a long-run context ie if the firm produces
nothing it incurs no costs but any q gt 0 incurs the 10 set up cost Suppose industry demand is given by
Q0 =52-P
where Pis the market price of the good produced by the firms
a) Derive a supply curve for one of the identical fmns
b) Derive the industry supply curve given that there are n firms
c) What is the long-run equilibrium price in this industry Explain your answer and show any
calculations
3
e)
a)
d)
d) How many firms will populate this industry in long-run equilibrium Explain your
answer
f)
Given your solution for n from part d) assume demand shifts to Q0 =53- P What is
industry price and output in the short-run equilibrium
Given your result in part e) explain in words what will happen as the industry adjusts to
long-run equilibrium
7) The following is a demand equation for residential electricity consumption in the US estimated by
H Houthakker (Energy Jouma1980)
ln E = 0586 + 1388lnY-1118 ln PE + 0199lnH + 0483lnC +0566LnP0
where E is per capita electricity consumption in KWH per year Y is disposable personal income per
capita PE is the price per KWH of electricity H is heating degree days (deviation over the year of
temperature below 65deg) C is cooling degree days (deviation over the year of temperature above 75deg) and
P 0 is the price of natural gas per BTU
Based on this model what are the own price cross price and income elasticities of
demand for electricity in the US
b) A deregulation plan for the electricity industry is currently being implemented Suppose
electricity prices rise as a result of deregulation Based on Houthakkers equation will
consumers spend more less or the same on electricity following deregulation Explain
c) Many poor families live in apartments with electric heating and cooling A concern of
policy makers is that high energy costs make it difficult for poor families to adequately heat
or cool their homes Consider a typical poor family whose demand for electricity is given
by Houthakkers equation The family pays $010 per KWH and has per capita income of
$3000 ($12000 for a family of four) Given normal values for H C and P 0 the family
consumers on average 6000 KWH per year Energy experts have determined that 8000
KWH are needed to provide satisfactory heating and cooling in the familys climatic region
Consider two programs to eliminate the deficiency (a) a cash payments plan (b) an energy
subsidy program under which the family will get a rebate from the government for a
specific percentage of its monthly electric bill
(i) How large must the cash payment be to accomplish the plans goal
(ii) What percentage subsidy will achieve the programs goal
lllustrate the two plans using indifference curves for electricity and all other goods and the
budget constraint Begin with the situation where E = 6000 and show how either program
can attain the goal
A third possible program involves coupons that can be spent only on electricity Will e)
coupons entitling the household to buy 2000 KWH achieve the goal Explain
4
8 ) Consider a monopolist who sells in two markets The inverse demands and cost function for the
monopolist are as follows
I = 80-SQ market I
R =180- 20Q2 market2
C (Q)=50 + 20(Q + Q2) cost function
a) Find the firms profit maximizing price and output in each market assuming it is free to set
whatever price it wishes in each market Show your work
b) Assuming absence of any externalities what is the socially optimal output in each market
Explain your answer and show any calculations
c) Define the term deadweight loss and then calculate the deadweight loss from monopoly
power in each market
d) TrueIFalse Explain The firm in this problem is a natural monopolist
e) The following statement is true The firm in this problem is practicing price
discrimination Explain carefully and precisely what this statement means in terms a
noneconomist could understand
f) Suppose the government passed a law preventing the firm from practicing price
discrimination What price and output would the firm then set in market I Market 2
Provide a precise answer if you can showing all work but dont get bogged down At a
minimum explain in words how if at all this law would affect the firm
5
University of California Davi Jepartment of Agricultural and Resource bconomics
MS Comprehensive Exam September 151998
There are two sections to this exam Each section consists of four questions Follow the instructions for each section carefully Use complete sentences and well thought out responses when writing down your answers Note that if we cannot read your writing then we will assume that your answer is wrong All questions are equally weighted There are five pages to this exam
Section 1 This section consists of questions 1 ) 2) 3) and 4) Answer question 1 ) and then select two of the remaining three questions to answer Do not answer all four questions
I) Below is a highly simplified model of wine supply and demand Selected t-ratios are below each estimated coefficient
Two-Stage Least-Squares Estimates of Supply
Q = -101 + 026PWHAT(210) (220)
R2 =035 DW=041 n=21
Two-Stage Least Squares Estimates of Demand
Q = IOI -122PWHAT + 222NCOME +340CPIALL + 0753Q_1(23S) (110) (056) (400)
R2 =096 DW=05 Durbins h- statistic= 083 n=21
Auxiliary First Stage Regression PWHAT = 201+ 133NCOME + 440CPIALL + 023Q_1
R2 = 067 DW= 187
Regression On Residuals From Demand Equation
Jl = 05+155NCOME +055CPIALL (620) (ll)
R2=072 The variables are defmed as follows
Q - Quantity of wine supplied and demanded PWHAT - Price of wine from auxiliary regression INCOME- Per capita personal income CPIALL - Consumer price index of all items
Examine the above model for econometric problems In particular you should look for
I autocorrelation 2 heteroscedasticity
2)
3 omitted variables
4 multicollinearity
5 identification
6 simultaneous equation bias
For each of the above possible problems explain why you think there is or is not a problem Explain
what should be done to remedy the problem
You have been asked to estimate a linear econometric model say Y = 30 + 31X + Jl Assume that
var(Jl) = a2X (I)
where Jl is a normally distributed random variable with a mean of 0 and a variance given in Eq () The
above model (I) is termed heteroscedasticity
a) Clearly explain the problems that heteroscedasticity creates
b) How might you go about detecting the presence of heteroscedasticity in the above linear
model
middotc) How might you correct for the heteroscedasticity given that it occurs in the form ofEq (1)
in the above linear model Be specific and give details
3) Answer the following jive TrueFalseExplain questions concerning a general linear programming
(LP) problem Please note that the explanation is worth the bulk of the credit
a) An optimal solution to aLP problem uses up all of the limited resources available
b) A point on the boundary of the feasible set (or region) satisfies all of the constraints of the
LP problem
c) The optimal solution of an LP problem always occurs at an extreme point of the feasible
set
d) If the first COilStraint of aLP problem evaluated at given value xo of the decision vector
has a zero value for its corresponding slack variable then the point xo lies on the boundary
of the feasible region
e) If the shadow value (or dual variable) corresponding to a given constraint is positive than
that constraint is binding
4) PROTRAC INC produces two lines of heavy equipment One of the product lines termed
earthmoving equipment is essentially for construction applications while the other line termed forestry
2
equipment is destined for Ute timber industry The largest member or ihe earthmoving line (the E-9) and
the largest member of the forestry equipment line (the F-9) are produced in the same departments and with
the same equipment Using economic forecasts for the next month PROTRACs marketing manager has
judged that during that period it will be possible to sell as many E-9s and F-9s as the flrm cam produce
Management must now recommend a production target for the next month
Making this decision requires that the following major facts be considered
bull PROTRACs net revenue is $5000 on each E-9 that is sold and $4000 on each F-9
bull Each product is put through PROTRACs machining operations in department A and department B
bull For next months production department A hasl50 hours available and department B has 160 hours
available Each E-9 uses 10 hours of machining in department A and 20 hours of machining in
department B Each F-9 uses 15 hours of machining in department A and 10 hours of machining in
department B
bull In order for management to honor an agreement with the union the total labor hours used in next
months testing of finished products cannot fall more than 10 below an arbitrated goal of 150 hours
The testing is performed in a third department and has nothing to do with the activities in departments
A and B Each E-9 is given 30 hours of testing and each F-9 is given I0 hours of testing
bull In order to maintain the current market position senior management has decreed the operating policy
that it is necessary to build at least one F-9 for every three E-9 produced
bullmiddot A major dealer has ordered a total of at least five E-9s and F-9s in any combination for the next
month and so at least that many must be produced
Given the above considerations managements problem is to decide how many E-9s and many F-9s to
produce next month so as to maximize net revenue
Formulate the linear programming problem for PROTRAC INC Deflne all the variables constraints
and equations in the formulation
Section 2 This section consists of questions 5 ) 6) 7) and 8) Answer any three of these four questions Do
not answer all four questions
5) A simple monopolist faces the linear inverse market demand curve given by p(y) a - by where
a gt 0 b gt 0 and y is the output of the monopolist In addition the monopolist faces the total cost
function given by c(y) Fe where k gt 0 Finally the government pays the monopolist a subsidy of
s gt 0 dollars per unit of output produced in order to encourage the monopolist to produce more output and
sell it at a lower price
a) Set up the proflt maximization problem facing the monopolist
3
--------
b) Find the monopolists profit maximizing level of output say y = ym(abks) as well as its profit maximizing price say p = pm(abk s) How do you know you have found the profit maximizing solution
c) Will an increase in the subsidy achieve its desired goal of increasing the monopolists output and lowering its price Show your work
d) Provide an economic interpretation of the above comparative statics e) How large would the subsidy have to be in order to get the monopolist to produce the
Pareto efficient level of output and charge the Pareto efficient price
6) The following questions pertain to a production setting characterized by the usual neoclassical production function of the form Q = f(xlx2x ) where Q is output and (x1x2x ) are inputs
a) Define compare and contrast the following terms from production economics i) Decreasing returns to a variable input and ii) decreasing returns to scale (Decreasing returns to scale is also called
diseconomies of scale) b) What economic forces cause decreasing returns to a variable input and what economic
forces might cause decreasing returns to scale c) Given the production function as specified show mathematically how you would measure
the returns to the variable input x1 and the returns to scale d) Suppose you were studying the question of whether small farms could compete effectively
against large farms in the market for the product Q Would either or both of the concepts of returns to a variable input and returns to scale be valuable in your analysis Explain
7) The following are demand and supply curves that apply to a competitive industry
Q0 = 250- SOP Qs = ()P
a) Derive the market equilibrium price and quantity b) Is the equilibrium you derived in (a) a short-run equilibrium a long-run equilibrium or
could it be both Explain your answer in detail demonstrating your knowledge of behavior in competitive markets and the distinction in these markets between the short and long run
c) Politicians are floating a proposal to place a per unit tax of $ 100 on this commodity Producers in lobbying against the tax argue that they will have no choice but to pass the
full tax on to consumers and therefore the tax if enacted will only harm consumers Evaluate the producers argument concerning the tax Use either algebra (preferred tool) or
4
--------------
------------------------------ X
bull
a)
a graph to support your argument If you use a graph label it carefully using the demand and supply equations Q0 = 250- 50P and Qs = ( )P
8) The following graph depicts a consumers Marshallian demand curve for the commodity X The market price of X is given as
p
Demand Curve
Given identify the consumers surplus on the graph and give an intuitive definition of consumers surplus
b) Assuming X is a normal good for this consumer draw the consumers Hicksian demand curve given price
c) Explain the difference between Hicksian and Marshallian demand curves
d) Suppose price increases to P_ Use Hicksian demands to identify the consumers compensating variation and equivalent variation due to the price change Label carefully
5
University of California Davis Department of Agricultural and Resource Economics
MS Exam
July 2 1999
Directions You have three hours to write this exam Please write legibly Answers that cannot be read are regarded as incorrect answers This exam consists of two parts In part I quantitative methods you must answer question 1 Then choose to answer either question 2 or question 3 In part II microeconomics answer three of the four questions You should thus answer five questions in total two from part I and three from part II Do not under any circumstances answer more than five questions
Part I Answer 1 estion I and either Question 2 01 luestion 3
l You are constructing a si mple model of cotton su pply for a short-term project you are working on and h ave formulated the following conceptual model
where y5 i s bales of cotton ps i s the price paid to growers w1 is the labor wage rate w2 i s the price of water WJ i s pesticide price and DO is degree-days a measure of the growi ng conditions during the season
(a) A nother variable you might include in thi s speci ficati on is annual rainfall (R) whose coefficient i f included in the model would be a6 While you are sure th at all th e other va ri ables in the model are part of the correct speci fication you are not sure about annual rainfall As an applied econometrician explai n the tradeoffs made to the properties of you r estimates i f you include or exclude thi s vari able from the specification
(b) One must always be concerned with measurement error i nempirical models Explain th e effects on the properties of your estimates of (a) measurement error in th e dependent variable vs (b) measurement error in one or more explanatory vari ables
(c) You esti mate th e model wi th R included and the results are
yS = -6354 + 8727ps - 0 579lw1 - 7907w2 - 09045w3 + 769000 + 0045R ( -0 79) (2 36) ( -177) (-234) ( -190) (157) (045)
R2 = 0819 n =56
where th e Students-t statistics are gi ven in parentheses and the following sums of squared errors resu lt
Total SSY=660 9 Model SSR=5412 Errors SSE= l l 97
In answering the following part s justify your answers with quantitati ve test results using th e attached tables Quali tative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 0 5 level
(ii) Is the overall model significant Th at i s i s the hypothesis th at a1 = a2 = a3 = a4 = as = a6 = 0 rejected
(iii) The correlation between the price of water (w2) and the price of pesticides (w3) i s zero Do these two vari ables have the same effect on cotton supply i e are their coeffi cients equ al
2 Suppose you have the following econometric model
where y = yieldacre x1 = wateracre x2 = (wateracre)2 x3 = fertilizeracre
(a) For each of the following questions set up the appropriate hypothesis test and provide a brief explanation
(i) Does water have any influence on yieldacre
(ii) Assuming an unconstrained (interior) solution is wateracre = 300 significantly different from the amount that yields the maximum yield per unit water
(b) Suppose you work in a governmental agency and that your boss is not trained in economics or statistics and thus she does not know what e N(O all) means Explain-
in words the meaning and significance of this expression
3
a Given the regional farm modeling problem
Max subject to
px- cx Ax b
where x is an n x I vector of production activities p and c are vectors of prices and variable cost coefficients respectively b is an m x 1 vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Comment briefly on the connection between the dual problem and the Dual Complementary Slackness condition
(iii) If you had to calculate the derived demand for a single input b describe the type of function you would obtain from the problem
b Now use the information that the product prices p are determined according to the following demand relationships
p =a + Dx
Assume that the individual farmers act as perfect competitors
(i) Show how you would reformulate your problem and give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit )
(ii) Derive the dual to the problem in part b(i) showing your steps clearly
Part II Answer of the 4 questions in this Section _
I For some goods consumption is subject to more than the usual single constraint on money income time can also be a constrai nt Consider a consumer with utility f unction u(x) defined on goods x = (x1 xn) subject to th e two constraints T 2 tx and M 2 px where T is the total amount of time avai lable and M is th e total amount of income avai lable and t = (t1 tn) and p = (p o pn ) are the time and money prices per unit of consumption of each good
(a) Write th e primal utility max i mi zation problem for th is consumer What interpretation do you attach to th e sh adow values on each constraint
(b) F rom your answer to part (a) ex plain how the consumer s choices reveal her money value of time Develop an ex pression f or the marginal money value of time from th is pri mal problem
(c) Under what conditions would the problem revert to the standard consumer choice problem subject to a single money budget constraint
(d) Suppose th at the shadow value on the money budget constraint is zero W rite the dual expenditure mi nimization problem for thi s case Provide a brief interpretation in words of thi s dual ex penditure f unction Also provide an interpretation for the sh adow value of the constraint for th is problem
2 Assume t hat t he US wheat mar ket can be considered in isolation and is charact erized by
yS = sops - 20 wl
y0 = 60 - 12 5p0 +2M
where
and
ys is quantity of wheat suppl ied in millions of bushels per year p5 is the price recei ved by suppliers in dollars per bushel y0 is quantity demanded (in millions of bushels) by consumers p0 i s the price paid by consumers also in dollars per bushel w1 is the price of a composite input to wheat product ion M is consumer income in t housands of dollars
For purposes of this analysis consumer income i s M = 45 and the composite input price is w1 = 5
(a) What are t he equilibrium f ree-mar ket price and quant it y of wheat
(b) Is t he supply of wheat at this point price-elastic or price-inelast i c Ex plain your reasomng
(c) W h at is t he t otal net economic sur plus f rom production of wheat and how middotis it distr ibuted bet ween suppliers and demanders
(d) A var iet y of schemes are used in agricult ural markets to enhance producer welf are C onsider a pr ice support of $6 per bushel Explain t he effects of such a policy on the dist ri bution of welfare and economic efficiency Support your answer with quant itat ive est imates of t hese effect s
jgtrice
3 Suppose a monopoly fum faces the following linear demand curve P = a- bQ where P is price and Q is quantity produced and sold The monopolists cost function is
(1) c(Q) = cQ for Q Q c(Q) qo for Q gt Q
where Q is a capacity constraint
a Solve mathematically for the monopolists profit maxumzmg and output as functions of the parameters a b and c Show your work Also illustrate yourresults graphically labeling carefully all relevant information
b Suppose the function P(Q) from part (a) is the demand in the domestic market and that the monopolist can also sell unlimited quantities into the export market at price P E gt c Assuming that quantities exported cannot be resold into the domestic market show both mathematically and on your graph from part (a) how much the firm will sell into the domestic market and into the export market
c Consider now a potential entrant into this market For simplicity ignore the existence of the export market To be able to produce in this market an entrant must incur a one time set up or entry cost of E gt 0 It can then produce according to the cost function in (1) Describe in words andor diagrams how you would model this entrants decision problem You should consider the role of the incumbent monopolist in framing your answer
4 Consider a food processing firm that produces processed product Q from a raw agricultural product Rand a composite marketing input L which we will call labor
a Write a production function and draw some rough sketches of isoquants for each of the following possible production scenarios
1 There is no substitution between R and L in producing Q
ii The production relationship is Cobb-Douglas
b Suppose the true production relationship is Cobb-Douglas and that this firm produces Q exclusively for a large retail grocery chain It has a contract for 1999 to deliver Q units of product to the retailer Assume the firm is a perfect competitor in the markets for its inputs Set up the firms optimization problem given this information and derive the first-order conditions for the problem Defme any notation or terminology you need to introduce Also illustrate the optimization problem with a graph
c Discuss the firms demand function for R given the conditions of the problem in part (b) Specifically what variables determine how much R the firm will utilize
d Re-express the problem in part (b) and take first-order conditions assuming now that the firm is the only buyer of the farm input R which is produced according to the following inverse supply function
w = a + 3R a3 gt 0
OJ 04
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09
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4
2257
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412
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4649
41161
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495
4974
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164 9-9
267 2967
4170
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76
714
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4791
4978
49119
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1051
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677
676 SlATISliCI1 tA9LFS
lA Rlf lgtl Areas under the standardized normal distribution
Example
Pr (0 z 196) = 04750
Pr (z 196) = 05 - 04750 = 0025
TABLE 01 Percentage points of lhe t distribution
Example
Pr (r gt 2086) = Oo25
Pr (r gt 1725) = 005 for df = 20
Pr(ltl gt 1725) = 010 0 I 715
01$ 010 middot-00
001 0 00010015010 002 0010 0002ozo
J 07H 127116I 11216 14 61657 IIUI1nm
22 279925IXH6 6965UK16 4 lP 00 01 01 1 0765 I 6X 251 lIK2 4541 51141 102 JJ
4 074 I 15J 21J1 176 461 moo 00411no
Ol 0middot1111
(Xl)lt) 11119 059UIO 0161 t tiiCW 029 0279 1650675 0751 s 1476 2015 511910727 25710596 0616 4010517 115 7
071 II 1064 IIOJ 1141 52111106 bull 1440 1941 l7H7 0791 UKl 0171 091ll 09411
149914KU 1517
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1591 161M 1664
1915 1950 19115 W19
10611144 1179 K u 706 lW7 11601772 451111196
212 071) 421172054 219() 122420XX UHl 111 2262 21i21 1251)-9
0700 J72 1112 222K 2764 11692422
1021 214 25492-IK6 102157 2X92291
2794 2121 21152 4025II 0697 1361 1796 22111 2711lt lit
2611 27041gt1 25XU ll7K 1106 1111 190 0695 1156 17H1 21 79 26111 lOSS
ll40 11111 1910
2650 2M9 J65 l119 IJ 0694 1350 1771 2160 1lt5211112311lt6 21 lJH 164
145 1761 21J5 2614 1621 14 069214115 55-1 2977 7117 461 150111 0 1411
l711I 2602 2 94717011 770 790 Kill lX0 15 0691 1141 175 729 21 lfgt65 16116 JHHH
I I
11
1641
1144012
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4194066 41H l6462567 2gt19H411 17 1740 21W40994(19 406 068H UJO 174 210142164192
I
4279 4292 6102X7K4265 25524251 -11117 4222 4441 068H U211 1 729 209J 253944294 112 494 J406 44111 579lK61LJ5 4157
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4554 505J67H A6K6 4691 4699 4706 22 06K6 I Jl l 1717 2014 l5HK 1XI94656 4664 4671I K 4641 41151K074756 4761 17144744 4750 2J 06KS 1319 250020694716 412 47 111 9 470 4719
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1
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J906 4911 06M4 14212471 27714913 4916 170 4909 27 L14 10524196 489R 4901 49044191 06114925 4927 4929 4931 492 4914 4916 140K2467 27612M U13 1701 204849111 4920 4922
4941 4941 4945 4946 494K 4949 4951 4951 IJII JJ9629 068) 2462 2756204S49111 494025
UJO 1697 2042 2457 17504QSS 4964 lJK S4961 JO 06KJ4960 4961 4961
496 J951J4956 49576
7 10 06KI 10749 1 6K4497 241J i7044970 4971 49724967 496K 1021J966 60 0679 129611 4981
4979 4979 49SO 16714977 2 190 26604976 4977Igrave 12122000 4975
l Jfi) 1090
49864914 4914 4985 498S 4986 120 0677 11M9 21671651 19KO 2J58
49111 49H9 49M9 4990 _ 1 1810674 164S 1960 1 126 1576 49H I 49K2 4982 498)
49X7 49K7 49117 49111
N This tahlc gives the area in the riahtmiddothand tail or the distribution tic bull t 0) But since the normal di5tribumiddot Nolt 111e smaller probability shown at the head of each column is the area in one tail the 1in is symmetrical about r- Othe area in the ldtmiddothlnd tail is the same u the area in the corresponding right-hand larser probability is the area in both tails bullbullil rltgtr eumplc PI- 196 S z S 01-04750 TMrdore -196 S t S 196)-2(047S0)-09S Sourr From E S Pearwn and H 0 Hartley eds bull Blomtuila TabftJot Slulislidans vol I
ld ed table 12 Cbullmbridgc University Prns New York 1966 Reproduced by perminion of the middot
editors and trustees of Biomeria
bull bull bull bull 10 II 30 110 100 500
(
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7 71
4 20
116
1 H4
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4 1
1
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1 75
192
1 gt6
44
2 54
244
)07
142middot
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144
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199
245
l-14
2 1
14)
161
190
1 4 1
25J
)70
137
195
2)7
U 5
190
229
1 59
245
U 7
I 91
1 15
96
142
194
1 94
l2
IJ1
117
174
104
1 0
190
U4
109
284
Ill 2 71 2JJ
1 77
194 175
i-1
174
149
157
I y I 7
1 94 1 gt9
194
1 9)
174 159
145 1 4
195
119
179 147
682 AIPFNHIX ll
TARLE Ol l lpper percentage poinls or the F distribution (continut-d)
I I
STATISTICAl TA8LfS 681
df for dl lor llllmtntor N
100 N
I 16 111 I 12 I y I 2ltI 10 1 10 111 1 1 1 1 111 I Y 1 41 1 4 1 1-10 I 19 1 19 UHI middotW 1171411 147 145
I 176 IJl 170 IUI 167 165 I M 161 1 bull1 101 1 97 191 1 90 11111 1116Hi 4 l)
2 15 212
Ill 1117 201 I I I M4 I IIII I I II
2 911 1 II 1 1 1gt 67 251 11 an z llbull 2 1J ll I 26 1 1 11 1 2
191 lOS 2Kl 1 2 55 146 2 -m 24 2 lU 226 2 2l
US1 i9 l-IS S72 4112 4 1 1 199
I 1 I J I 110 I Y 12Y I 211 I M I 7 1 7 I 6 1 15 1 11 U9 UK IJK 1middot17 U6
17M I 71 1 101 167 1 162 161 I I S1 n6 I 51 IU1115 I ll 1 4 1 1401 19 1 47 146 1 44
I 98
2111 191 1 1111 1 19 210 1 042B 9 I H6 1 gt4 1 MO 119 1 1 1 1 1) 1 1 1 211 2111 2JO 22S 21 2111JOI 2SI 242 2 16 11114 16 140
2M9 2 66 UK 2 40 111 1 1 1 2-1 2 1 Ul167 middot 1 16 126 117 109 ill2 H I 472 4 12 01
114 I 12 111 1 10 129 I 211 UK 16 I 26 126 I 25 I 2 I 42 1 4 1 19 us I H 17 l l6 2 1 111 146 us 1 16 I 1611 165 1 61 I 5Y UK us IS-1 I S1 I S l I t HIIll 11111 186 2 II 1 2 1 11 2 1111 11111 Y I 151 107 199 190 1-JIS 1112 1111) 1 16 1 1S I 71 I 7 1 211 222 2111 215Ull OS 219U7 198 1 14 2 t7 2111 2511 lSO 242 1 16 2 225 22J 219 216 2 IJ UH l09 )02 196HI 129SSl 41gt4 41401
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172 167 1 64 ltll I H 15-1 Ul I SU I 1 lt7 1 6 10 10IIIS 182 179 171I 10 11111 1 22K 114 20S 1911 19) 188
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110 I 126 125 124 Ut 12 Ul 121 119 1 19 25IJ6 us 1 14 IJJ Ul U l140 11914425
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124 1221 16 129 128 127 1 2 1 1 19 1 111 1 17 116 1 14 1 1 1 1 1 1 I l l 1 10
IJJ IJI 1 10 140 119 141 117 132 111 126 1 24 1 2 1 119 10 120I us1 60I gtI 182 111 168 16S 1622 1)10 us
166 161 155 150 146 14 117 115 1 12 1211 125 I Hl 217 109 101 191 IKOl 219 201 I 176 1 101 156 UJ I 1 41 I IX
6 liS I 1701 4 J9 195 14H 296 2 79 266 256 247 240 middot 234
Ul 1 2 1 l iD 1 1 11 116 114 1 12 I l l 1 10 1119 I 0 25 111 IJ9 1111 I J6 I a I J2 I) I 129 128 127 126 US
152 146 1 41 1 111 1 1-1 I I 1211 I 24 122 uu 1 1 7 Ia Ill 2 I I 197 IKII IKO 1 75 170 166 161 160 157
172 162 IS7 IS2 146 141 119 I J2 129 U6 1 22 119 m198 19J I 88 I 114 180OS H9 1 04 265 242 226 2 14 2lt16
01 676 411 lllll l41 J I I 289 271 I 91 1119 I 79 169 I 61 1511 148 144 119 1 11 1 11 01
260 2SO 241 2 14 227
122 1 10 1 16 11-1 1 1 1 1 12 1119 Uli I 07 l tiO 124 124I 12 I J9 IJ7 US I J 111 129 1211 127 125
I II 142 118 114 uo 126 124 1 1 11 1 1 1 1 1 1 I OK Hill 10
i lS 25
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Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
bull More should be invested in certificates of deposit and treasury bills than in municipal bonds and the
growth stock fund by a ratio of 12 to 10 The investor wants to know how much to invest in each investment alternative in order to maximize the
total return from all investments All $70000 is to be spent in this investment plan
Formulate the linear programming problem for this investor Defme all the variables constraints and
equations in the formulation
Section 2 This section consists of questions 5) 6) 7) and 8) Answer any three of these four questions Do not answer all four questions
5) Which of the following describes an externality and which does not Explain your reasoning
clearly in each case
a) A policy of restricted coffee exports in Brazil causes the US price of coffee to rise which
in turn causes the price of tea to increase
b) An advertising blimp distracts a motorist who then hits a telephone pole and damages the
car
c) In a homogeneous product Cournot duopoly an increase in the output of firm 1 lowers the
market price received by firm 2 for its product (due to the law of demand) ceteris paribus thereby reducing the profit of firm 2
6) Consider a competitive industry with a large number of identical firms each of which has a simple
cost curve of the form
C(q)= l +1 qgtO 0 q =O
where q 0 is the output of the firm Think of these costs in a long-run context ie if the firm produces
nothing it incurs no costs but any q gt 0 incurs the 10 set up cost Suppose industry demand is given by
Q0 =52-P
where Pis the market price of the good produced by the firms
a) Derive a supply curve for one of the identical fmns
b) Derive the industry supply curve given that there are n firms
c) What is the long-run equilibrium price in this industry Explain your answer and show any
calculations
3
e)
a)
d)
d) How many firms will populate this industry in long-run equilibrium Explain your
answer
f)
Given your solution for n from part d) assume demand shifts to Q0 =53- P What is
industry price and output in the short-run equilibrium
Given your result in part e) explain in words what will happen as the industry adjusts to
long-run equilibrium
7) The following is a demand equation for residential electricity consumption in the US estimated by
H Houthakker (Energy Jouma1980)
ln E = 0586 + 1388lnY-1118 ln PE + 0199lnH + 0483lnC +0566LnP0
where E is per capita electricity consumption in KWH per year Y is disposable personal income per
capita PE is the price per KWH of electricity H is heating degree days (deviation over the year of
temperature below 65deg) C is cooling degree days (deviation over the year of temperature above 75deg) and
P 0 is the price of natural gas per BTU
Based on this model what are the own price cross price and income elasticities of
demand for electricity in the US
b) A deregulation plan for the electricity industry is currently being implemented Suppose
electricity prices rise as a result of deregulation Based on Houthakkers equation will
consumers spend more less or the same on electricity following deregulation Explain
c) Many poor families live in apartments with electric heating and cooling A concern of
policy makers is that high energy costs make it difficult for poor families to adequately heat
or cool their homes Consider a typical poor family whose demand for electricity is given
by Houthakkers equation The family pays $010 per KWH and has per capita income of
$3000 ($12000 for a family of four) Given normal values for H C and P 0 the family
consumers on average 6000 KWH per year Energy experts have determined that 8000
KWH are needed to provide satisfactory heating and cooling in the familys climatic region
Consider two programs to eliminate the deficiency (a) a cash payments plan (b) an energy
subsidy program under which the family will get a rebate from the government for a
specific percentage of its monthly electric bill
(i) How large must the cash payment be to accomplish the plans goal
(ii) What percentage subsidy will achieve the programs goal
lllustrate the two plans using indifference curves for electricity and all other goods and the
budget constraint Begin with the situation where E = 6000 and show how either program
can attain the goal
A third possible program involves coupons that can be spent only on electricity Will e)
coupons entitling the household to buy 2000 KWH achieve the goal Explain
4
8 ) Consider a monopolist who sells in two markets The inverse demands and cost function for the
monopolist are as follows
I = 80-SQ market I
R =180- 20Q2 market2
C (Q)=50 + 20(Q + Q2) cost function
a) Find the firms profit maximizing price and output in each market assuming it is free to set
whatever price it wishes in each market Show your work
b) Assuming absence of any externalities what is the socially optimal output in each market
Explain your answer and show any calculations
c) Define the term deadweight loss and then calculate the deadweight loss from monopoly
power in each market
d) TrueIFalse Explain The firm in this problem is a natural monopolist
e) The following statement is true The firm in this problem is practicing price
discrimination Explain carefully and precisely what this statement means in terms a
noneconomist could understand
f) Suppose the government passed a law preventing the firm from practicing price
discrimination What price and output would the firm then set in market I Market 2
Provide a precise answer if you can showing all work but dont get bogged down At a
minimum explain in words how if at all this law would affect the firm
5
University of California Davi Jepartment of Agricultural and Resource bconomics
MS Comprehensive Exam September 151998
There are two sections to this exam Each section consists of four questions Follow the instructions for each section carefully Use complete sentences and well thought out responses when writing down your answers Note that if we cannot read your writing then we will assume that your answer is wrong All questions are equally weighted There are five pages to this exam
Section 1 This section consists of questions 1 ) 2) 3) and 4) Answer question 1 ) and then select two of the remaining three questions to answer Do not answer all four questions
I) Below is a highly simplified model of wine supply and demand Selected t-ratios are below each estimated coefficient
Two-Stage Least-Squares Estimates of Supply
Q = -101 + 026PWHAT(210) (220)
R2 =035 DW=041 n=21
Two-Stage Least Squares Estimates of Demand
Q = IOI -122PWHAT + 222NCOME +340CPIALL + 0753Q_1(23S) (110) (056) (400)
R2 =096 DW=05 Durbins h- statistic= 083 n=21
Auxiliary First Stage Regression PWHAT = 201+ 133NCOME + 440CPIALL + 023Q_1
R2 = 067 DW= 187
Regression On Residuals From Demand Equation
Jl = 05+155NCOME +055CPIALL (620) (ll)
R2=072 The variables are defmed as follows
Q - Quantity of wine supplied and demanded PWHAT - Price of wine from auxiliary regression INCOME- Per capita personal income CPIALL - Consumer price index of all items
Examine the above model for econometric problems In particular you should look for
I autocorrelation 2 heteroscedasticity
2)
3 omitted variables
4 multicollinearity
5 identification
6 simultaneous equation bias
For each of the above possible problems explain why you think there is or is not a problem Explain
what should be done to remedy the problem
You have been asked to estimate a linear econometric model say Y = 30 + 31X + Jl Assume that
var(Jl) = a2X (I)
where Jl is a normally distributed random variable with a mean of 0 and a variance given in Eq () The
above model (I) is termed heteroscedasticity
a) Clearly explain the problems that heteroscedasticity creates
b) How might you go about detecting the presence of heteroscedasticity in the above linear
model
middotc) How might you correct for the heteroscedasticity given that it occurs in the form ofEq (1)
in the above linear model Be specific and give details
3) Answer the following jive TrueFalseExplain questions concerning a general linear programming
(LP) problem Please note that the explanation is worth the bulk of the credit
a) An optimal solution to aLP problem uses up all of the limited resources available
b) A point on the boundary of the feasible set (or region) satisfies all of the constraints of the
LP problem
c) The optimal solution of an LP problem always occurs at an extreme point of the feasible
set
d) If the first COilStraint of aLP problem evaluated at given value xo of the decision vector
has a zero value for its corresponding slack variable then the point xo lies on the boundary
of the feasible region
e) If the shadow value (or dual variable) corresponding to a given constraint is positive than
that constraint is binding
4) PROTRAC INC produces two lines of heavy equipment One of the product lines termed
earthmoving equipment is essentially for construction applications while the other line termed forestry
2
equipment is destined for Ute timber industry The largest member or ihe earthmoving line (the E-9) and
the largest member of the forestry equipment line (the F-9) are produced in the same departments and with
the same equipment Using economic forecasts for the next month PROTRACs marketing manager has
judged that during that period it will be possible to sell as many E-9s and F-9s as the flrm cam produce
Management must now recommend a production target for the next month
Making this decision requires that the following major facts be considered
bull PROTRACs net revenue is $5000 on each E-9 that is sold and $4000 on each F-9
bull Each product is put through PROTRACs machining operations in department A and department B
bull For next months production department A hasl50 hours available and department B has 160 hours
available Each E-9 uses 10 hours of machining in department A and 20 hours of machining in
department B Each F-9 uses 15 hours of machining in department A and 10 hours of machining in
department B
bull In order for management to honor an agreement with the union the total labor hours used in next
months testing of finished products cannot fall more than 10 below an arbitrated goal of 150 hours
The testing is performed in a third department and has nothing to do with the activities in departments
A and B Each E-9 is given 30 hours of testing and each F-9 is given I0 hours of testing
bull In order to maintain the current market position senior management has decreed the operating policy
that it is necessary to build at least one F-9 for every three E-9 produced
bullmiddot A major dealer has ordered a total of at least five E-9s and F-9s in any combination for the next
month and so at least that many must be produced
Given the above considerations managements problem is to decide how many E-9s and many F-9s to
produce next month so as to maximize net revenue
Formulate the linear programming problem for PROTRAC INC Deflne all the variables constraints
and equations in the formulation
Section 2 This section consists of questions 5 ) 6) 7) and 8) Answer any three of these four questions Do
not answer all four questions
5) A simple monopolist faces the linear inverse market demand curve given by p(y) a - by where
a gt 0 b gt 0 and y is the output of the monopolist In addition the monopolist faces the total cost
function given by c(y) Fe where k gt 0 Finally the government pays the monopolist a subsidy of
s gt 0 dollars per unit of output produced in order to encourage the monopolist to produce more output and
sell it at a lower price
a) Set up the proflt maximization problem facing the monopolist
3
--------
b) Find the monopolists profit maximizing level of output say y = ym(abks) as well as its profit maximizing price say p = pm(abk s) How do you know you have found the profit maximizing solution
c) Will an increase in the subsidy achieve its desired goal of increasing the monopolists output and lowering its price Show your work
d) Provide an economic interpretation of the above comparative statics e) How large would the subsidy have to be in order to get the monopolist to produce the
Pareto efficient level of output and charge the Pareto efficient price
6) The following questions pertain to a production setting characterized by the usual neoclassical production function of the form Q = f(xlx2x ) where Q is output and (x1x2x ) are inputs
a) Define compare and contrast the following terms from production economics i) Decreasing returns to a variable input and ii) decreasing returns to scale (Decreasing returns to scale is also called
diseconomies of scale) b) What economic forces cause decreasing returns to a variable input and what economic
forces might cause decreasing returns to scale c) Given the production function as specified show mathematically how you would measure
the returns to the variable input x1 and the returns to scale d) Suppose you were studying the question of whether small farms could compete effectively
against large farms in the market for the product Q Would either or both of the concepts of returns to a variable input and returns to scale be valuable in your analysis Explain
7) The following are demand and supply curves that apply to a competitive industry
Q0 = 250- SOP Qs = ()P
a) Derive the market equilibrium price and quantity b) Is the equilibrium you derived in (a) a short-run equilibrium a long-run equilibrium or
could it be both Explain your answer in detail demonstrating your knowledge of behavior in competitive markets and the distinction in these markets between the short and long run
c) Politicians are floating a proposal to place a per unit tax of $ 100 on this commodity Producers in lobbying against the tax argue that they will have no choice but to pass the
full tax on to consumers and therefore the tax if enacted will only harm consumers Evaluate the producers argument concerning the tax Use either algebra (preferred tool) or
4
--------------
------------------------------ X
bull
a)
a graph to support your argument If you use a graph label it carefully using the demand and supply equations Q0 = 250- 50P and Qs = ( )P
8) The following graph depicts a consumers Marshallian demand curve for the commodity X The market price of X is given as
p
Demand Curve
Given identify the consumers surplus on the graph and give an intuitive definition of consumers surplus
b) Assuming X is a normal good for this consumer draw the consumers Hicksian demand curve given price
c) Explain the difference between Hicksian and Marshallian demand curves
d) Suppose price increases to P_ Use Hicksian demands to identify the consumers compensating variation and equivalent variation due to the price change Label carefully
5
University of California Davis Department of Agricultural and Resource Economics
MS Exam
July 2 1999
Directions You have three hours to write this exam Please write legibly Answers that cannot be read are regarded as incorrect answers This exam consists of two parts In part I quantitative methods you must answer question 1 Then choose to answer either question 2 or question 3 In part II microeconomics answer three of the four questions You should thus answer five questions in total two from part I and three from part II Do not under any circumstances answer more than five questions
Part I Answer 1 estion I and either Question 2 01 luestion 3
l You are constructing a si mple model of cotton su pply for a short-term project you are working on and h ave formulated the following conceptual model
where y5 i s bales of cotton ps i s the price paid to growers w1 is the labor wage rate w2 i s the price of water WJ i s pesticide price and DO is degree-days a measure of the growi ng conditions during the season
(a) A nother variable you might include in thi s speci ficati on is annual rainfall (R) whose coefficient i f included in the model would be a6 While you are sure th at all th e other va ri ables in the model are part of the correct speci fication you are not sure about annual rainfall As an applied econometrician explai n the tradeoffs made to the properties of you r estimates i f you include or exclude thi s vari able from the specification
(b) One must always be concerned with measurement error i nempirical models Explain th e effects on the properties of your estimates of (a) measurement error in th e dependent variable vs (b) measurement error in one or more explanatory vari ables
(c) You esti mate th e model wi th R included and the results are
yS = -6354 + 8727ps - 0 579lw1 - 7907w2 - 09045w3 + 769000 + 0045R ( -0 79) (2 36) ( -177) (-234) ( -190) (157) (045)
R2 = 0819 n =56
where th e Students-t statistics are gi ven in parentheses and the following sums of squared errors resu lt
Total SSY=660 9 Model SSR=5412 Errors SSE= l l 97
In answering the following part s justify your answers with quantitati ve test results using th e attached tables Quali tative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 0 5 level
(ii) Is the overall model significant Th at i s i s the hypothesis th at a1 = a2 = a3 = a4 = as = a6 = 0 rejected
(iii) The correlation between the price of water (w2) and the price of pesticides (w3) i s zero Do these two vari ables have the same effect on cotton supply i e are their coeffi cients equ al
2 Suppose you have the following econometric model
where y = yieldacre x1 = wateracre x2 = (wateracre)2 x3 = fertilizeracre
(a) For each of the following questions set up the appropriate hypothesis test and provide a brief explanation
(i) Does water have any influence on yieldacre
(ii) Assuming an unconstrained (interior) solution is wateracre = 300 significantly different from the amount that yields the maximum yield per unit water
(b) Suppose you work in a governmental agency and that your boss is not trained in economics or statistics and thus she does not know what e N(O all) means Explain-
in words the meaning and significance of this expression
3
a Given the regional farm modeling problem
Max subject to
px- cx Ax b
where x is an n x I vector of production activities p and c are vectors of prices and variable cost coefficients respectively b is an m x 1 vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Comment briefly on the connection between the dual problem and the Dual Complementary Slackness condition
(iii) If you had to calculate the derived demand for a single input b describe the type of function you would obtain from the problem
b Now use the information that the product prices p are determined according to the following demand relationships
p =a + Dx
Assume that the individual farmers act as perfect competitors
(i) Show how you would reformulate your problem and give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit )
(ii) Derive the dual to the problem in part b(i) showing your steps clearly
Part II Answer of the 4 questions in this Section _
I For some goods consumption is subject to more than the usual single constraint on money income time can also be a constrai nt Consider a consumer with utility f unction u(x) defined on goods x = (x1 xn) subject to th e two constraints T 2 tx and M 2 px where T is the total amount of time avai lable and M is th e total amount of income avai lable and t = (t1 tn) and p = (p o pn ) are the time and money prices per unit of consumption of each good
(a) Write th e primal utility max i mi zation problem for th is consumer What interpretation do you attach to th e sh adow values on each constraint
(b) F rom your answer to part (a) ex plain how the consumer s choices reveal her money value of time Develop an ex pression f or the marginal money value of time from th is pri mal problem
(c) Under what conditions would the problem revert to the standard consumer choice problem subject to a single money budget constraint
(d) Suppose th at the shadow value on the money budget constraint is zero W rite the dual expenditure mi nimization problem for thi s case Provide a brief interpretation in words of thi s dual ex penditure f unction Also provide an interpretation for the sh adow value of the constraint for th is problem
2 Assume t hat t he US wheat mar ket can be considered in isolation and is charact erized by
yS = sops - 20 wl
y0 = 60 - 12 5p0 +2M
where
and
ys is quantity of wheat suppl ied in millions of bushels per year p5 is the price recei ved by suppliers in dollars per bushel y0 is quantity demanded (in millions of bushels) by consumers p0 i s the price paid by consumers also in dollars per bushel w1 is the price of a composite input to wheat product ion M is consumer income in t housands of dollars
For purposes of this analysis consumer income i s M = 45 and the composite input price is w1 = 5
(a) What are t he equilibrium f ree-mar ket price and quant it y of wheat
(b) Is t he supply of wheat at this point price-elastic or price-inelast i c Ex plain your reasomng
(c) W h at is t he t otal net economic sur plus f rom production of wheat and how middotis it distr ibuted bet ween suppliers and demanders
(d) A var iet y of schemes are used in agricult ural markets to enhance producer welf are C onsider a pr ice support of $6 per bushel Explain t he effects of such a policy on the dist ri bution of welfare and economic efficiency Support your answer with quant itat ive est imates of t hese effect s
jgtrice
3 Suppose a monopoly fum faces the following linear demand curve P = a- bQ where P is price and Q is quantity produced and sold The monopolists cost function is
(1) c(Q) = cQ for Q Q c(Q) qo for Q gt Q
where Q is a capacity constraint
a Solve mathematically for the monopolists profit maxumzmg and output as functions of the parameters a b and c Show your work Also illustrate yourresults graphically labeling carefully all relevant information
b Suppose the function P(Q) from part (a) is the demand in the domestic market and that the monopolist can also sell unlimited quantities into the export market at price P E gt c Assuming that quantities exported cannot be resold into the domestic market show both mathematically and on your graph from part (a) how much the firm will sell into the domestic market and into the export market
c Consider now a potential entrant into this market For simplicity ignore the existence of the export market To be able to produce in this market an entrant must incur a one time set up or entry cost of E gt 0 It can then produce according to the cost function in (1) Describe in words andor diagrams how you would model this entrants decision problem You should consider the role of the incumbent monopolist in framing your answer
4 Consider a food processing firm that produces processed product Q from a raw agricultural product Rand a composite marketing input L which we will call labor
a Write a production function and draw some rough sketches of isoquants for each of the following possible production scenarios
1 There is no substitution between R and L in producing Q
ii The production relationship is Cobb-Douglas
b Suppose the true production relationship is Cobb-Douglas and that this firm produces Q exclusively for a large retail grocery chain It has a contract for 1999 to deliver Q units of product to the retailer Assume the firm is a perfect competitor in the markets for its inputs Set up the firms optimization problem given this information and derive the first-order conditions for the problem Defme any notation or terminology you need to introduce Also illustrate the optimization problem with a graph
c Discuss the firms demand function for R given the conditions of the problem in part (b) Specifically what variables determine how much R the firm will utilize
d Re-express the problem in part (b) and take first-order conditions assuming now that the firm is the only buyer of the farm input R which is produced according to the following inverse supply function
w = a + 3R a3 gt 0
OJ 04
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Example
Pr (0 z 196) = 04750
Pr (z 196) = 05 - 04750 = 0025
TABLE 01 Percentage points of lhe t distribution
Example
Pr (r gt 2086) = Oo25
Pr (r gt 1725) = 005 for df = 20
Pr(ltl gt 1725) = 010 0 I 715
01$ 010 middot-00
001 0 00010015010 002 0010 0002ozo
J 07H 127116I 11216 14 61657 IIUI1nm
22 279925IXH6 6965UK16 4 lP 00 01 01 1 0765 I 6X 251 lIK2 4541 51141 102 JJ
4 074 I 15J 21J1 176 461 moo 00411no
Ol 0middot1111
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071 II 1064 IIOJ 1141 52111106 bull 1440 1941 l7H7 0791 UKl 0171 091ll 09411
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1591 161M 1664
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10611144 1179 K u 706 lW7 11601772 451111196
212 071) 421172054 219() 122420XX UHl 111 2262 21i21 1251)-9
0700 J72 1112 222K 2764 11692422
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2794 2121 21152 4025II 0697 1361 1796 22111 2711lt lit
2611 27041gt1 25XU ll7K 1106 1111 190 0695 1156 17H1 21 79 26111 lOSS
ll40 11111 1910
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145 1761 21J5 2614 1621 14 069214115 55-1 2977 7117 461 150111 0 1411
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l Jfi) 1090
49864914 4914 4985 498S 4986 120 0677 11M9 21671651 19KO 2J58
49111 49H9 49M9 4990 _ 1 1810674 164S 1960 1 126 1576 49H I 49K2 4982 498)
49X7 49K7 49117 49111
N This tahlc gives the area in the riahtmiddothand tail or the distribution tic bull t 0) But since the normal di5tribumiddot Nolt 111e smaller probability shown at the head of each column is the area in one tail the 1in is symmetrical about r- Othe area in the ldtmiddothlnd tail is the same u the area in the corresponding right-hand larser probability is the area in both tails bullbullil rltgtr eumplc PI- 196 S z S 01-04750 TMrdore -196 S t S 196)-2(047S0)-09S Sourr From E S Pearwn and H 0 Hartley eds bull Blomtuila TabftJot Slulislidans vol I
ld ed table 12 Cbullmbridgc University Prns New York 1966 Reproduced by perminion of the middot
editors and trustees of Biomeria
bull bull bull bull 10 II 30 110 100 500
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229
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245
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194
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117
174
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109
284
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149
157
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TARLE Ol l lpper percentage poinls or the F distribution (continut-d)
I I
STATISTICAl TA8LfS 681
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I 16 111 I 12 I y I 2ltI 10 1 10 111 1 1 1 1 111 I Y 1 41 1 4 1 1-10 I 19 1 19 UHI middotW 1171411 147 145
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1R4 261 2 14 225 218 2 12 1 08 252 217 229 220 2 1 1 1116 202 1 94 I 92 I IM1 01 664 II 181 lSI 129 1 12 199 2119 280 27)511S
127 125 1 14 122 121 1 11l 119 1 1 7 1 1 1 1 16 1 1 5 1 1 5 I J J 112 I l l IJO 119 119us 1 41 1 4 1 UK 160 154 Ul 1411 144 141 140 116 I 15 I J I ll I 29 10 I
111 168 1 66117IK7 1821 19 2 19 211S 104 111-1 175 110 165 159 156 15] 1 4H 147 144 I-II 1 9 0)
125 2 17 210 104 199 195 1921 1 5 1 16 2 S 110 212 20 194 11111 1 04 17 I 71 I 61 160 til2S6 1SO 1 ll 295 lKl 272 261 165 ) 14
124 1221 16 129 128 127 1 2 1 1 19 1 111 1 17 116 1 14 1 1 1 1 1 1 I l l 1 10
IJJ IJI 1 10 140 119 141 117 132 111 126 1 24 1 2 1 119 10 120I us1 60I gtI 182 111 168 16S 1622 1)10 us
166 161 155 150 146 14 117 115 1 12 1211 125 I Hl 217 109 101 191 IKOl 219 201 I 176 1 101 156 UJ I 1 41 I IX
6 liS I 1701 4 J9 195 14H 296 2 79 266 256 247 240 middot 234
Ul 1 2 1 l iD 1 1 11 116 114 1 12 I l l 1 10 1119 I 0 25 111 IJ9 1111 I J6 I a I J2 I) I 129 128 127 126 US
152 146 1 41 1 111 1 1-1 I I 1211 I 24 122 uu 1 1 7 Ia Ill 2 I I 197 IKII IKO 1 75 170 166 161 160 157
172 162 IS7 IS2 146 141 119 I J2 129 U6 1 22 119 m198 19J I 88 I 114 180OS H9 1 04 265 242 226 2 14 2lt16
01 676 411 lllll l41 J I I 289 271 I 91 1119 I 79 169 I 61 1511 148 144 119 1 11 1 11 01
260 2SO 241 2 14 227
122 1 10 1 16 11-1 1 1 1 1 12 1119 Uli I 07 l tiO 124 124I 12 I J9 IJ7 US I J 111 129 1211 127 125
I II 142 118 114 uo 126 124 1 1 11 1 1 1 1 1 1 I OK Hill 10
i lS 25
IS72 JO 160 us172 167 16J10 2 71 2011 194 U2 1 14 1 12 1 1 1 1 1 1 I IIII 167 152 146 179l 11-1 HXI 260 2J7 221 210 201 1811 181liSmiddot
21KIll 6 6 4bl 3-JII J l2 l 02 2110 264 2 51 241 232 22S 1 10 U6 I ll I S I OtlIIIK104 U9 152 0 1
Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
e)
a)
d)
d) How many firms will populate this industry in long-run equilibrium Explain your
answer
f)
Given your solution for n from part d) assume demand shifts to Q0 =53- P What is
industry price and output in the short-run equilibrium
Given your result in part e) explain in words what will happen as the industry adjusts to
long-run equilibrium
7) The following is a demand equation for residential electricity consumption in the US estimated by
H Houthakker (Energy Jouma1980)
ln E = 0586 + 1388lnY-1118 ln PE + 0199lnH + 0483lnC +0566LnP0
where E is per capita electricity consumption in KWH per year Y is disposable personal income per
capita PE is the price per KWH of electricity H is heating degree days (deviation over the year of
temperature below 65deg) C is cooling degree days (deviation over the year of temperature above 75deg) and
P 0 is the price of natural gas per BTU
Based on this model what are the own price cross price and income elasticities of
demand for electricity in the US
b) A deregulation plan for the electricity industry is currently being implemented Suppose
electricity prices rise as a result of deregulation Based on Houthakkers equation will
consumers spend more less or the same on electricity following deregulation Explain
c) Many poor families live in apartments with electric heating and cooling A concern of
policy makers is that high energy costs make it difficult for poor families to adequately heat
or cool their homes Consider a typical poor family whose demand for electricity is given
by Houthakkers equation The family pays $010 per KWH and has per capita income of
$3000 ($12000 for a family of four) Given normal values for H C and P 0 the family
consumers on average 6000 KWH per year Energy experts have determined that 8000
KWH are needed to provide satisfactory heating and cooling in the familys climatic region
Consider two programs to eliminate the deficiency (a) a cash payments plan (b) an energy
subsidy program under which the family will get a rebate from the government for a
specific percentage of its monthly electric bill
(i) How large must the cash payment be to accomplish the plans goal
(ii) What percentage subsidy will achieve the programs goal
lllustrate the two plans using indifference curves for electricity and all other goods and the
budget constraint Begin with the situation where E = 6000 and show how either program
can attain the goal
A third possible program involves coupons that can be spent only on electricity Will e)
coupons entitling the household to buy 2000 KWH achieve the goal Explain
4
8 ) Consider a monopolist who sells in two markets The inverse demands and cost function for the
monopolist are as follows
I = 80-SQ market I
R =180- 20Q2 market2
C (Q)=50 + 20(Q + Q2) cost function
a) Find the firms profit maximizing price and output in each market assuming it is free to set
whatever price it wishes in each market Show your work
b) Assuming absence of any externalities what is the socially optimal output in each market
Explain your answer and show any calculations
c) Define the term deadweight loss and then calculate the deadweight loss from monopoly
power in each market
d) TrueIFalse Explain The firm in this problem is a natural monopolist
e) The following statement is true The firm in this problem is practicing price
discrimination Explain carefully and precisely what this statement means in terms a
noneconomist could understand
f) Suppose the government passed a law preventing the firm from practicing price
discrimination What price and output would the firm then set in market I Market 2
Provide a precise answer if you can showing all work but dont get bogged down At a
minimum explain in words how if at all this law would affect the firm
5
University of California Davi Jepartment of Agricultural and Resource bconomics
MS Comprehensive Exam September 151998
There are two sections to this exam Each section consists of four questions Follow the instructions for each section carefully Use complete sentences and well thought out responses when writing down your answers Note that if we cannot read your writing then we will assume that your answer is wrong All questions are equally weighted There are five pages to this exam
Section 1 This section consists of questions 1 ) 2) 3) and 4) Answer question 1 ) and then select two of the remaining three questions to answer Do not answer all four questions
I) Below is a highly simplified model of wine supply and demand Selected t-ratios are below each estimated coefficient
Two-Stage Least-Squares Estimates of Supply
Q = -101 + 026PWHAT(210) (220)
R2 =035 DW=041 n=21
Two-Stage Least Squares Estimates of Demand
Q = IOI -122PWHAT + 222NCOME +340CPIALL + 0753Q_1(23S) (110) (056) (400)
R2 =096 DW=05 Durbins h- statistic= 083 n=21
Auxiliary First Stage Regression PWHAT = 201+ 133NCOME + 440CPIALL + 023Q_1
R2 = 067 DW= 187
Regression On Residuals From Demand Equation
Jl = 05+155NCOME +055CPIALL (620) (ll)
R2=072 The variables are defmed as follows
Q - Quantity of wine supplied and demanded PWHAT - Price of wine from auxiliary regression INCOME- Per capita personal income CPIALL - Consumer price index of all items
Examine the above model for econometric problems In particular you should look for
I autocorrelation 2 heteroscedasticity
2)
3 omitted variables
4 multicollinearity
5 identification
6 simultaneous equation bias
For each of the above possible problems explain why you think there is or is not a problem Explain
what should be done to remedy the problem
You have been asked to estimate a linear econometric model say Y = 30 + 31X + Jl Assume that
var(Jl) = a2X (I)
where Jl is a normally distributed random variable with a mean of 0 and a variance given in Eq () The
above model (I) is termed heteroscedasticity
a) Clearly explain the problems that heteroscedasticity creates
b) How might you go about detecting the presence of heteroscedasticity in the above linear
model
middotc) How might you correct for the heteroscedasticity given that it occurs in the form ofEq (1)
in the above linear model Be specific and give details
3) Answer the following jive TrueFalseExplain questions concerning a general linear programming
(LP) problem Please note that the explanation is worth the bulk of the credit
a) An optimal solution to aLP problem uses up all of the limited resources available
b) A point on the boundary of the feasible set (or region) satisfies all of the constraints of the
LP problem
c) The optimal solution of an LP problem always occurs at an extreme point of the feasible
set
d) If the first COilStraint of aLP problem evaluated at given value xo of the decision vector
has a zero value for its corresponding slack variable then the point xo lies on the boundary
of the feasible region
e) If the shadow value (or dual variable) corresponding to a given constraint is positive than
that constraint is binding
4) PROTRAC INC produces two lines of heavy equipment One of the product lines termed
earthmoving equipment is essentially for construction applications while the other line termed forestry
2
equipment is destined for Ute timber industry The largest member or ihe earthmoving line (the E-9) and
the largest member of the forestry equipment line (the F-9) are produced in the same departments and with
the same equipment Using economic forecasts for the next month PROTRACs marketing manager has
judged that during that period it will be possible to sell as many E-9s and F-9s as the flrm cam produce
Management must now recommend a production target for the next month
Making this decision requires that the following major facts be considered
bull PROTRACs net revenue is $5000 on each E-9 that is sold and $4000 on each F-9
bull Each product is put through PROTRACs machining operations in department A and department B
bull For next months production department A hasl50 hours available and department B has 160 hours
available Each E-9 uses 10 hours of machining in department A and 20 hours of machining in
department B Each F-9 uses 15 hours of machining in department A and 10 hours of machining in
department B
bull In order for management to honor an agreement with the union the total labor hours used in next
months testing of finished products cannot fall more than 10 below an arbitrated goal of 150 hours
The testing is performed in a third department and has nothing to do with the activities in departments
A and B Each E-9 is given 30 hours of testing and each F-9 is given I0 hours of testing
bull In order to maintain the current market position senior management has decreed the operating policy
that it is necessary to build at least one F-9 for every three E-9 produced
bullmiddot A major dealer has ordered a total of at least five E-9s and F-9s in any combination for the next
month and so at least that many must be produced
Given the above considerations managements problem is to decide how many E-9s and many F-9s to
produce next month so as to maximize net revenue
Formulate the linear programming problem for PROTRAC INC Deflne all the variables constraints
and equations in the formulation
Section 2 This section consists of questions 5 ) 6) 7) and 8) Answer any three of these four questions Do
not answer all four questions
5) A simple monopolist faces the linear inverse market demand curve given by p(y) a - by where
a gt 0 b gt 0 and y is the output of the monopolist In addition the monopolist faces the total cost
function given by c(y) Fe where k gt 0 Finally the government pays the monopolist a subsidy of
s gt 0 dollars per unit of output produced in order to encourage the monopolist to produce more output and
sell it at a lower price
a) Set up the proflt maximization problem facing the monopolist
3
--------
b) Find the monopolists profit maximizing level of output say y = ym(abks) as well as its profit maximizing price say p = pm(abk s) How do you know you have found the profit maximizing solution
c) Will an increase in the subsidy achieve its desired goal of increasing the monopolists output and lowering its price Show your work
d) Provide an economic interpretation of the above comparative statics e) How large would the subsidy have to be in order to get the monopolist to produce the
Pareto efficient level of output and charge the Pareto efficient price
6) The following questions pertain to a production setting characterized by the usual neoclassical production function of the form Q = f(xlx2x ) where Q is output and (x1x2x ) are inputs
a) Define compare and contrast the following terms from production economics i) Decreasing returns to a variable input and ii) decreasing returns to scale (Decreasing returns to scale is also called
diseconomies of scale) b) What economic forces cause decreasing returns to a variable input and what economic
forces might cause decreasing returns to scale c) Given the production function as specified show mathematically how you would measure
the returns to the variable input x1 and the returns to scale d) Suppose you were studying the question of whether small farms could compete effectively
against large farms in the market for the product Q Would either or both of the concepts of returns to a variable input and returns to scale be valuable in your analysis Explain
7) The following are demand and supply curves that apply to a competitive industry
Q0 = 250- SOP Qs = ()P
a) Derive the market equilibrium price and quantity b) Is the equilibrium you derived in (a) a short-run equilibrium a long-run equilibrium or
could it be both Explain your answer in detail demonstrating your knowledge of behavior in competitive markets and the distinction in these markets between the short and long run
c) Politicians are floating a proposal to place a per unit tax of $ 100 on this commodity Producers in lobbying against the tax argue that they will have no choice but to pass the
full tax on to consumers and therefore the tax if enacted will only harm consumers Evaluate the producers argument concerning the tax Use either algebra (preferred tool) or
4
--------------
------------------------------ X
bull
a)
a graph to support your argument If you use a graph label it carefully using the demand and supply equations Q0 = 250- 50P and Qs = ( )P
8) The following graph depicts a consumers Marshallian demand curve for the commodity X The market price of X is given as
p
Demand Curve
Given identify the consumers surplus on the graph and give an intuitive definition of consumers surplus
b) Assuming X is a normal good for this consumer draw the consumers Hicksian demand curve given price
c) Explain the difference between Hicksian and Marshallian demand curves
d) Suppose price increases to P_ Use Hicksian demands to identify the consumers compensating variation and equivalent variation due to the price change Label carefully
5
University of California Davis Department of Agricultural and Resource Economics
MS Exam
July 2 1999
Directions You have three hours to write this exam Please write legibly Answers that cannot be read are regarded as incorrect answers This exam consists of two parts In part I quantitative methods you must answer question 1 Then choose to answer either question 2 or question 3 In part II microeconomics answer three of the four questions You should thus answer five questions in total two from part I and three from part II Do not under any circumstances answer more than five questions
Part I Answer 1 estion I and either Question 2 01 luestion 3
l You are constructing a si mple model of cotton su pply for a short-term project you are working on and h ave formulated the following conceptual model
where y5 i s bales of cotton ps i s the price paid to growers w1 is the labor wage rate w2 i s the price of water WJ i s pesticide price and DO is degree-days a measure of the growi ng conditions during the season
(a) A nother variable you might include in thi s speci ficati on is annual rainfall (R) whose coefficient i f included in the model would be a6 While you are sure th at all th e other va ri ables in the model are part of the correct speci fication you are not sure about annual rainfall As an applied econometrician explai n the tradeoffs made to the properties of you r estimates i f you include or exclude thi s vari able from the specification
(b) One must always be concerned with measurement error i nempirical models Explain th e effects on the properties of your estimates of (a) measurement error in th e dependent variable vs (b) measurement error in one or more explanatory vari ables
(c) You esti mate th e model wi th R included and the results are
yS = -6354 + 8727ps - 0 579lw1 - 7907w2 - 09045w3 + 769000 + 0045R ( -0 79) (2 36) ( -177) (-234) ( -190) (157) (045)
R2 = 0819 n =56
where th e Students-t statistics are gi ven in parentheses and the following sums of squared errors resu lt
Total SSY=660 9 Model SSR=5412 Errors SSE= l l 97
In answering the following part s justify your answers with quantitati ve test results using th e attached tables Quali tative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 0 5 level
(ii) Is the overall model significant Th at i s i s the hypothesis th at a1 = a2 = a3 = a4 = as = a6 = 0 rejected
(iii) The correlation between the price of water (w2) and the price of pesticides (w3) i s zero Do these two vari ables have the same effect on cotton supply i e are their coeffi cients equ al
2 Suppose you have the following econometric model
where y = yieldacre x1 = wateracre x2 = (wateracre)2 x3 = fertilizeracre
(a) For each of the following questions set up the appropriate hypothesis test and provide a brief explanation
(i) Does water have any influence on yieldacre
(ii) Assuming an unconstrained (interior) solution is wateracre = 300 significantly different from the amount that yields the maximum yield per unit water
(b) Suppose you work in a governmental agency and that your boss is not trained in economics or statistics and thus she does not know what e N(O all) means Explain-
in words the meaning and significance of this expression
3
a Given the regional farm modeling problem
Max subject to
px- cx Ax b
where x is an n x I vector of production activities p and c are vectors of prices and variable cost coefficients respectively b is an m x 1 vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Comment briefly on the connection between the dual problem and the Dual Complementary Slackness condition
(iii) If you had to calculate the derived demand for a single input b describe the type of function you would obtain from the problem
b Now use the information that the product prices p are determined according to the following demand relationships
p =a + Dx
Assume that the individual farmers act as perfect competitors
(i) Show how you would reformulate your problem and give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit )
(ii) Derive the dual to the problem in part b(i) showing your steps clearly
Part II Answer of the 4 questions in this Section _
I For some goods consumption is subject to more than the usual single constraint on money income time can also be a constrai nt Consider a consumer with utility f unction u(x) defined on goods x = (x1 xn) subject to th e two constraints T 2 tx and M 2 px where T is the total amount of time avai lable and M is th e total amount of income avai lable and t = (t1 tn) and p = (p o pn ) are the time and money prices per unit of consumption of each good
(a) Write th e primal utility max i mi zation problem for th is consumer What interpretation do you attach to th e sh adow values on each constraint
(b) F rom your answer to part (a) ex plain how the consumer s choices reveal her money value of time Develop an ex pression f or the marginal money value of time from th is pri mal problem
(c) Under what conditions would the problem revert to the standard consumer choice problem subject to a single money budget constraint
(d) Suppose th at the shadow value on the money budget constraint is zero W rite the dual expenditure mi nimization problem for thi s case Provide a brief interpretation in words of thi s dual ex penditure f unction Also provide an interpretation for the sh adow value of the constraint for th is problem
2 Assume t hat t he US wheat mar ket can be considered in isolation and is charact erized by
yS = sops - 20 wl
y0 = 60 - 12 5p0 +2M
where
and
ys is quantity of wheat suppl ied in millions of bushels per year p5 is the price recei ved by suppliers in dollars per bushel y0 is quantity demanded (in millions of bushels) by consumers p0 i s the price paid by consumers also in dollars per bushel w1 is the price of a composite input to wheat product ion M is consumer income in t housands of dollars
For purposes of this analysis consumer income i s M = 45 and the composite input price is w1 = 5
(a) What are t he equilibrium f ree-mar ket price and quant it y of wheat
(b) Is t he supply of wheat at this point price-elastic or price-inelast i c Ex plain your reasomng
(c) W h at is t he t otal net economic sur plus f rom production of wheat and how middotis it distr ibuted bet ween suppliers and demanders
(d) A var iet y of schemes are used in agricult ural markets to enhance producer welf are C onsider a pr ice support of $6 per bushel Explain t he effects of such a policy on the dist ri bution of welfare and economic efficiency Support your answer with quant itat ive est imates of t hese effect s
jgtrice
3 Suppose a monopoly fum faces the following linear demand curve P = a- bQ where P is price and Q is quantity produced and sold The monopolists cost function is
(1) c(Q) = cQ for Q Q c(Q) qo for Q gt Q
where Q is a capacity constraint
a Solve mathematically for the monopolists profit maxumzmg and output as functions of the parameters a b and c Show your work Also illustrate yourresults graphically labeling carefully all relevant information
b Suppose the function P(Q) from part (a) is the demand in the domestic market and that the monopolist can also sell unlimited quantities into the export market at price P E gt c Assuming that quantities exported cannot be resold into the domestic market show both mathematically and on your graph from part (a) how much the firm will sell into the domestic market and into the export market
c Consider now a potential entrant into this market For simplicity ignore the existence of the export market To be able to produce in this market an entrant must incur a one time set up or entry cost of E gt 0 It can then produce according to the cost function in (1) Describe in words andor diagrams how you would model this entrants decision problem You should consider the role of the incumbent monopolist in framing your answer
4 Consider a food processing firm that produces processed product Q from a raw agricultural product Rand a composite marketing input L which we will call labor
a Write a production function and draw some rough sketches of isoquants for each of the following possible production scenarios
1 There is no substitution between R and L in producing Q
ii The production relationship is Cobb-Douglas
b Suppose the true production relationship is Cobb-Douglas and that this firm produces Q exclusively for a large retail grocery chain It has a contract for 1999 to deliver Q units of product to the retailer Assume the firm is a perfect competitor in the markets for its inputs Set up the firms optimization problem given this information and derive the first-order conditions for the problem Defme any notation or terminology you need to introduce Also illustrate the optimization problem with a graph
c Discuss the firms demand function for R given the conditions of the problem in part (b) Specifically what variables determine how much R the firm will utilize
d Re-express the problem in part (b) and take first-order conditions assuming now that the firm is the only buyer of the farm input R which is produced according to the following inverse supply function
w = a + 3R a3 gt 0
OJ 04
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Pr (0 z 196) = 04750
Pr (z 196) = 05 - 04750 = 0025
TABLE 01 Percentage points of lhe t distribution
Example
Pr (r gt 2086) = Oo25
Pr (r gt 1725) = 005 for df = 20
Pr(ltl gt 1725) = 010 0 I 715
01$ 010 middot-00
001 0 00010015010 002 0010 0002ozo
J 07H 127116I 11216 14 61657 IIUI1nm
22 279925IXH6 6965UK16 4 lP 00 01 01 1 0765 I 6X 251 lIK2 4541 51141 102 JJ
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N This tahlc gives the area in the riahtmiddothand tail or the distribution tic bull t 0) But since the normal di5tribumiddot Nolt 111e smaller probability shown at the head of each column is the area in one tail the 1in is symmetrical about r- Othe area in the ldtmiddothlnd tail is the same u the area in the corresponding right-hand larser probability is the area in both tails bullbullil rltgtr eumplc PI- 196 S z S 01-04750 TMrdore -196 S t S 196)-2(047S0)-09S Sourr From E S Pearwn and H 0 Hartley eds bull Blomtuila TabftJot Slulislidans vol I
ld ed table 12 Cbullmbridgc University Prns New York 1966 Reproduced by perminion of the middot
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195
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190
229
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245
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96
142
194
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117
174
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190
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109
284
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149
157
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I I
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17M I 71 1 101 167 1 162 161 I I S1 n6 I 51 IU1115 I ll 1 4 1 1401 19 1 47 146 1 44
I 98
2111 191 1 1111 1 19 210 1 042B 9 I H6 1 gt4 1 MO 119 1 1 1 1 1) 1 1 1 211 2111 2JO 22S 21 2111JOI 2SI 242 2 16 11114 16 140
2M9 2 66 UK 2 40 111 1 1 1 2-1 2 1 Ul167 middot 1 16 126 117 109 ill2 H I 472 4 12 01
114 I 12 111 1 10 129 I 211 UK 16 I 26 126 I 25 I 2 I 42 1 4 1 19 us I H 17 l l6 2 1 111 146 us 1 16 I 1611 165 1 61 I 5Y UK us IS-1 I S1 I S l I t HIIll 11111 186 2 II 1 2 1 11 2 1111 11111 Y I 151 107 199 190 1-JIS 1112 1111) 1 16 1 1S I 71 I 7 1 211 222 2111 215Ull OS 219U7 198 1 14 2 t7 2111 2511 lSO 242 1 16 2 225 22J 219 216 2 IJ UH l09 )02 196HI 129SSl 41gt4 41401
IJJ I I 1 10 129 1 18 117 127 126 1 1gt 125 124 1 24 136 usI 41 141 140 119 UK1 111 146 145 1 69 1 66 16 159 I S1 156 I 51 152 I SO 149 1 19 Ill 1114 1 812 11 1 1 229 216 1116 190 187 100
H14 196 1 9 1 1 81 1 112 119 1 71 I 71 1 11 1 69 167 165 0)219 2 15 1 1 1 JN 295 1 1 1 2242)6 229 V5 1 1 144 IS 210 2 26 1 19 1 1 1 2 1 1 109 U6 U l
0 1 407 I 75 )51 U6 123 312 101
Ul l ltJ 129 Ull 127 1 26 126 IH 1 2-1 1 2-1 12J t n l iM I 45 1 44 1 41 U9 Ull IJ6 us IB
172 167 1 64 ltll I H 15-1 Ul I SU I 1 lt7 1 6 10 10IIIS 182 179 171I 10 11111 1 22K 114 20S 1911 19) 188
2111 1 91 I H4 179 1 16 1 14 110 I M 1 IM I 62 O)1 41 lB 227 221 216 213OS l 2 291 169
270 2-17 210 2 2 1 IJ I I 207 2 Ul ] 0 1 147 J0 1 1 7 )07 298 2910 1 1 56 5 19 451 40
110 I 126 125 124 Ut 12 Ul 121 119 1 19 25IJ6 us 1 14 IJJ Ul U l140 11914425
1117 1113 179 1 66 161 157 1 )4 I l l 14K 141 142 1-11 119 I 111
176 173 1 71IllI 21)9 1 00 192 IM4 179 1611 1 64 I us I 5J U l 100I 21Oi
01
1R4 261 2 14 225 218 2 12 1 08 252 217 229 220 2 1 1 1116 202 1 94 I 92 I IM1 01 664 II 181 lSI 129 1 12 199 2119 280 27)511S
127 125 1 14 122 121 1 11l 119 1 1 7 1 1 1 1 16 1 1 5 1 1 5 I J J 112 I l l IJO 119 119us 1 41 1 4 1 UK 160 154 Ul 1411 144 141 140 116 I 15 I J I ll I 29 10 I
111 168 1 66117IK7 1821 19 2 19 211S 104 111-1 175 110 165 159 156 15] 1 4H 147 144 I-II 1 9 0)
125 2 17 210 104 199 195 1921 1 5 1 16 2 S 110 212 20 194 11111 1 04 17 I 71 I 61 160 til2S6 1SO 1 ll 295 lKl 272 261 165 ) 14
124 1221 16 129 128 127 1 2 1 1 19 1 111 1 17 116 1 14 1 1 1 1 1 1 I l l 1 10
IJJ IJI 1 10 140 119 141 117 132 111 126 1 24 1 2 1 119 10 120I us1 60I gtI 182 111 168 16S 1622 1)10 us
166 161 155 150 146 14 117 115 1 12 1211 125 I Hl 217 109 101 191 IKOl 219 201 I 176 1 101 156 UJ I 1 41 I IX
6 liS I 1701 4 J9 195 14H 296 2 79 266 256 247 240 middot 234
Ul 1 2 1 l iD 1 1 11 116 114 1 12 I l l 1 10 1119 I 0 25 111 IJ9 1111 I J6 I a I J2 I) I 129 128 127 126 US
152 146 1 41 1 111 1 1-1 I I 1211 I 24 122 uu 1 1 7 Ia Ill 2 I I 197 IKII IKO 1 75 170 166 161 160 157
172 162 IS7 IS2 146 141 119 I J2 129 U6 1 22 119 m198 19J I 88 I 114 180OS H9 1 04 265 242 226 2 14 2lt16
01 676 411 lllll l41 J I I 289 271 I 91 1119 I 79 169 I 61 1511 148 144 119 1 11 1 11 01
260 2SO 241 2 14 227
122 1 10 1 16 11-1 1 1 1 1 12 1119 Uli I 07 l tiO 124 124I 12 I J9 IJ7 US I J 111 129 1211 127 125
I II 142 118 114 uo 126 124 1 1 11 1 1 1 1 1 1 I OK Hill 10
i lS 25
IS72 JO 160 us172 167 16J10 2 71 2011 194 U2 1 14 1 12 1 1 1 1 1 1 I IIII 167 152 146 179l 11-1 HXI 260 2J7 221 210 201 1811 181liSmiddot
21KIll 6 6 4bl 3-JII J l2 l 02 2110 264 2 51 241 232 22S 1 10 U6 I ll I S I OtlIIIK104 U9 152 0 1
Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
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Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
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EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
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-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
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a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
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a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
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bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
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Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
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Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
8 ) Consider a monopolist who sells in two markets The inverse demands and cost function for the
monopolist are as follows
I = 80-SQ market I
R =180- 20Q2 market2
C (Q)=50 + 20(Q + Q2) cost function
a) Find the firms profit maximizing price and output in each market assuming it is free to set
whatever price it wishes in each market Show your work
b) Assuming absence of any externalities what is the socially optimal output in each market
Explain your answer and show any calculations
c) Define the term deadweight loss and then calculate the deadweight loss from monopoly
power in each market
d) TrueIFalse Explain The firm in this problem is a natural monopolist
e) The following statement is true The firm in this problem is practicing price
discrimination Explain carefully and precisely what this statement means in terms a
noneconomist could understand
f) Suppose the government passed a law preventing the firm from practicing price
discrimination What price and output would the firm then set in market I Market 2
Provide a precise answer if you can showing all work but dont get bogged down At a
minimum explain in words how if at all this law would affect the firm
5
University of California Davi Jepartment of Agricultural and Resource bconomics
MS Comprehensive Exam September 151998
There are two sections to this exam Each section consists of four questions Follow the instructions for each section carefully Use complete sentences and well thought out responses when writing down your answers Note that if we cannot read your writing then we will assume that your answer is wrong All questions are equally weighted There are five pages to this exam
Section 1 This section consists of questions 1 ) 2) 3) and 4) Answer question 1 ) and then select two of the remaining three questions to answer Do not answer all four questions
I) Below is a highly simplified model of wine supply and demand Selected t-ratios are below each estimated coefficient
Two-Stage Least-Squares Estimates of Supply
Q = -101 + 026PWHAT(210) (220)
R2 =035 DW=041 n=21
Two-Stage Least Squares Estimates of Demand
Q = IOI -122PWHAT + 222NCOME +340CPIALL + 0753Q_1(23S) (110) (056) (400)
R2 =096 DW=05 Durbins h- statistic= 083 n=21
Auxiliary First Stage Regression PWHAT = 201+ 133NCOME + 440CPIALL + 023Q_1
R2 = 067 DW= 187
Regression On Residuals From Demand Equation
Jl = 05+155NCOME +055CPIALL (620) (ll)
R2=072 The variables are defmed as follows
Q - Quantity of wine supplied and demanded PWHAT - Price of wine from auxiliary regression INCOME- Per capita personal income CPIALL - Consumer price index of all items
Examine the above model for econometric problems In particular you should look for
I autocorrelation 2 heteroscedasticity
2)
3 omitted variables
4 multicollinearity
5 identification
6 simultaneous equation bias
For each of the above possible problems explain why you think there is or is not a problem Explain
what should be done to remedy the problem
You have been asked to estimate a linear econometric model say Y = 30 + 31X + Jl Assume that
var(Jl) = a2X (I)
where Jl is a normally distributed random variable with a mean of 0 and a variance given in Eq () The
above model (I) is termed heteroscedasticity
a) Clearly explain the problems that heteroscedasticity creates
b) How might you go about detecting the presence of heteroscedasticity in the above linear
model
middotc) How might you correct for the heteroscedasticity given that it occurs in the form ofEq (1)
in the above linear model Be specific and give details
3) Answer the following jive TrueFalseExplain questions concerning a general linear programming
(LP) problem Please note that the explanation is worth the bulk of the credit
a) An optimal solution to aLP problem uses up all of the limited resources available
b) A point on the boundary of the feasible set (or region) satisfies all of the constraints of the
LP problem
c) The optimal solution of an LP problem always occurs at an extreme point of the feasible
set
d) If the first COilStraint of aLP problem evaluated at given value xo of the decision vector
has a zero value for its corresponding slack variable then the point xo lies on the boundary
of the feasible region
e) If the shadow value (or dual variable) corresponding to a given constraint is positive than
that constraint is binding
4) PROTRAC INC produces two lines of heavy equipment One of the product lines termed
earthmoving equipment is essentially for construction applications while the other line termed forestry
2
equipment is destined for Ute timber industry The largest member or ihe earthmoving line (the E-9) and
the largest member of the forestry equipment line (the F-9) are produced in the same departments and with
the same equipment Using economic forecasts for the next month PROTRACs marketing manager has
judged that during that period it will be possible to sell as many E-9s and F-9s as the flrm cam produce
Management must now recommend a production target for the next month
Making this decision requires that the following major facts be considered
bull PROTRACs net revenue is $5000 on each E-9 that is sold and $4000 on each F-9
bull Each product is put through PROTRACs machining operations in department A and department B
bull For next months production department A hasl50 hours available and department B has 160 hours
available Each E-9 uses 10 hours of machining in department A and 20 hours of machining in
department B Each F-9 uses 15 hours of machining in department A and 10 hours of machining in
department B
bull In order for management to honor an agreement with the union the total labor hours used in next
months testing of finished products cannot fall more than 10 below an arbitrated goal of 150 hours
The testing is performed in a third department and has nothing to do with the activities in departments
A and B Each E-9 is given 30 hours of testing and each F-9 is given I0 hours of testing
bull In order to maintain the current market position senior management has decreed the operating policy
that it is necessary to build at least one F-9 for every three E-9 produced
bullmiddot A major dealer has ordered a total of at least five E-9s and F-9s in any combination for the next
month and so at least that many must be produced
Given the above considerations managements problem is to decide how many E-9s and many F-9s to
produce next month so as to maximize net revenue
Formulate the linear programming problem for PROTRAC INC Deflne all the variables constraints
and equations in the formulation
Section 2 This section consists of questions 5 ) 6) 7) and 8) Answer any three of these four questions Do
not answer all four questions
5) A simple monopolist faces the linear inverse market demand curve given by p(y) a - by where
a gt 0 b gt 0 and y is the output of the monopolist In addition the monopolist faces the total cost
function given by c(y) Fe where k gt 0 Finally the government pays the monopolist a subsidy of
s gt 0 dollars per unit of output produced in order to encourage the monopolist to produce more output and
sell it at a lower price
a) Set up the proflt maximization problem facing the monopolist
3
--------
b) Find the monopolists profit maximizing level of output say y = ym(abks) as well as its profit maximizing price say p = pm(abk s) How do you know you have found the profit maximizing solution
c) Will an increase in the subsidy achieve its desired goal of increasing the monopolists output and lowering its price Show your work
d) Provide an economic interpretation of the above comparative statics e) How large would the subsidy have to be in order to get the monopolist to produce the
Pareto efficient level of output and charge the Pareto efficient price
6) The following questions pertain to a production setting characterized by the usual neoclassical production function of the form Q = f(xlx2x ) where Q is output and (x1x2x ) are inputs
a) Define compare and contrast the following terms from production economics i) Decreasing returns to a variable input and ii) decreasing returns to scale (Decreasing returns to scale is also called
diseconomies of scale) b) What economic forces cause decreasing returns to a variable input and what economic
forces might cause decreasing returns to scale c) Given the production function as specified show mathematically how you would measure
the returns to the variable input x1 and the returns to scale d) Suppose you were studying the question of whether small farms could compete effectively
against large farms in the market for the product Q Would either or both of the concepts of returns to a variable input and returns to scale be valuable in your analysis Explain
7) The following are demand and supply curves that apply to a competitive industry
Q0 = 250- SOP Qs = ()P
a) Derive the market equilibrium price and quantity b) Is the equilibrium you derived in (a) a short-run equilibrium a long-run equilibrium or
could it be both Explain your answer in detail demonstrating your knowledge of behavior in competitive markets and the distinction in these markets between the short and long run
c) Politicians are floating a proposal to place a per unit tax of $ 100 on this commodity Producers in lobbying against the tax argue that they will have no choice but to pass the
full tax on to consumers and therefore the tax if enacted will only harm consumers Evaluate the producers argument concerning the tax Use either algebra (preferred tool) or
4
--------------
------------------------------ X
bull
a)
a graph to support your argument If you use a graph label it carefully using the demand and supply equations Q0 = 250- 50P and Qs = ( )P
8) The following graph depicts a consumers Marshallian demand curve for the commodity X The market price of X is given as
p
Demand Curve
Given identify the consumers surplus on the graph and give an intuitive definition of consumers surplus
b) Assuming X is a normal good for this consumer draw the consumers Hicksian demand curve given price
c) Explain the difference between Hicksian and Marshallian demand curves
d) Suppose price increases to P_ Use Hicksian demands to identify the consumers compensating variation and equivalent variation due to the price change Label carefully
5
University of California Davis Department of Agricultural and Resource Economics
MS Exam
July 2 1999
Directions You have three hours to write this exam Please write legibly Answers that cannot be read are regarded as incorrect answers This exam consists of two parts In part I quantitative methods you must answer question 1 Then choose to answer either question 2 or question 3 In part II microeconomics answer three of the four questions You should thus answer five questions in total two from part I and three from part II Do not under any circumstances answer more than five questions
Part I Answer 1 estion I and either Question 2 01 luestion 3
l You are constructing a si mple model of cotton su pply for a short-term project you are working on and h ave formulated the following conceptual model
where y5 i s bales of cotton ps i s the price paid to growers w1 is the labor wage rate w2 i s the price of water WJ i s pesticide price and DO is degree-days a measure of the growi ng conditions during the season
(a) A nother variable you might include in thi s speci ficati on is annual rainfall (R) whose coefficient i f included in the model would be a6 While you are sure th at all th e other va ri ables in the model are part of the correct speci fication you are not sure about annual rainfall As an applied econometrician explai n the tradeoffs made to the properties of you r estimates i f you include or exclude thi s vari able from the specification
(b) One must always be concerned with measurement error i nempirical models Explain th e effects on the properties of your estimates of (a) measurement error in th e dependent variable vs (b) measurement error in one or more explanatory vari ables
(c) You esti mate th e model wi th R included and the results are
yS = -6354 + 8727ps - 0 579lw1 - 7907w2 - 09045w3 + 769000 + 0045R ( -0 79) (2 36) ( -177) (-234) ( -190) (157) (045)
R2 = 0819 n =56
where th e Students-t statistics are gi ven in parentheses and the following sums of squared errors resu lt
Total SSY=660 9 Model SSR=5412 Errors SSE= l l 97
In answering the following part s justify your answers with quantitati ve test results using th e attached tables Quali tative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 0 5 level
(ii) Is the overall model significant Th at i s i s the hypothesis th at a1 = a2 = a3 = a4 = as = a6 = 0 rejected
(iii) The correlation between the price of water (w2) and the price of pesticides (w3) i s zero Do these two vari ables have the same effect on cotton supply i e are their coeffi cients equ al
2 Suppose you have the following econometric model
where y = yieldacre x1 = wateracre x2 = (wateracre)2 x3 = fertilizeracre
(a) For each of the following questions set up the appropriate hypothesis test and provide a brief explanation
(i) Does water have any influence on yieldacre
(ii) Assuming an unconstrained (interior) solution is wateracre = 300 significantly different from the amount that yields the maximum yield per unit water
(b) Suppose you work in a governmental agency and that your boss is not trained in economics or statistics and thus she does not know what e N(O all) means Explain-
in words the meaning and significance of this expression
3
a Given the regional farm modeling problem
Max subject to
px- cx Ax b
where x is an n x I vector of production activities p and c are vectors of prices and variable cost coefficients respectively b is an m x 1 vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Comment briefly on the connection between the dual problem and the Dual Complementary Slackness condition
(iii) If you had to calculate the derived demand for a single input b describe the type of function you would obtain from the problem
b Now use the information that the product prices p are determined according to the following demand relationships
p =a + Dx
Assume that the individual farmers act as perfect competitors
(i) Show how you would reformulate your problem and give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit )
(ii) Derive the dual to the problem in part b(i) showing your steps clearly
Part II Answer of the 4 questions in this Section _
I For some goods consumption is subject to more than the usual single constraint on money income time can also be a constrai nt Consider a consumer with utility f unction u(x) defined on goods x = (x1 xn) subject to th e two constraints T 2 tx and M 2 px where T is the total amount of time avai lable and M is th e total amount of income avai lable and t = (t1 tn) and p = (p o pn ) are the time and money prices per unit of consumption of each good
(a) Write th e primal utility max i mi zation problem for th is consumer What interpretation do you attach to th e sh adow values on each constraint
(b) F rom your answer to part (a) ex plain how the consumer s choices reveal her money value of time Develop an ex pression f or the marginal money value of time from th is pri mal problem
(c) Under what conditions would the problem revert to the standard consumer choice problem subject to a single money budget constraint
(d) Suppose th at the shadow value on the money budget constraint is zero W rite the dual expenditure mi nimization problem for thi s case Provide a brief interpretation in words of thi s dual ex penditure f unction Also provide an interpretation for the sh adow value of the constraint for th is problem
2 Assume t hat t he US wheat mar ket can be considered in isolation and is charact erized by
yS = sops - 20 wl
y0 = 60 - 12 5p0 +2M
where
and
ys is quantity of wheat suppl ied in millions of bushels per year p5 is the price recei ved by suppliers in dollars per bushel y0 is quantity demanded (in millions of bushels) by consumers p0 i s the price paid by consumers also in dollars per bushel w1 is the price of a composite input to wheat product ion M is consumer income in t housands of dollars
For purposes of this analysis consumer income i s M = 45 and the composite input price is w1 = 5
(a) What are t he equilibrium f ree-mar ket price and quant it y of wheat
(b) Is t he supply of wheat at this point price-elastic or price-inelast i c Ex plain your reasomng
(c) W h at is t he t otal net economic sur plus f rom production of wheat and how middotis it distr ibuted bet ween suppliers and demanders
(d) A var iet y of schemes are used in agricult ural markets to enhance producer welf are C onsider a pr ice support of $6 per bushel Explain t he effects of such a policy on the dist ri bution of welfare and economic efficiency Support your answer with quant itat ive est imates of t hese effect s
jgtrice
3 Suppose a monopoly fum faces the following linear demand curve P = a- bQ where P is price and Q is quantity produced and sold The monopolists cost function is
(1) c(Q) = cQ for Q Q c(Q) qo for Q gt Q
where Q is a capacity constraint
a Solve mathematically for the monopolists profit maxumzmg and output as functions of the parameters a b and c Show your work Also illustrate yourresults graphically labeling carefully all relevant information
b Suppose the function P(Q) from part (a) is the demand in the domestic market and that the monopolist can also sell unlimited quantities into the export market at price P E gt c Assuming that quantities exported cannot be resold into the domestic market show both mathematically and on your graph from part (a) how much the firm will sell into the domestic market and into the export market
c Consider now a potential entrant into this market For simplicity ignore the existence of the export market To be able to produce in this market an entrant must incur a one time set up or entry cost of E gt 0 It can then produce according to the cost function in (1) Describe in words andor diagrams how you would model this entrants decision problem You should consider the role of the incumbent monopolist in framing your answer
4 Consider a food processing firm that produces processed product Q from a raw agricultural product Rand a composite marketing input L which we will call labor
a Write a production function and draw some rough sketches of isoquants for each of the following possible production scenarios
1 There is no substitution between R and L in producing Q
ii The production relationship is Cobb-Douglas
b Suppose the true production relationship is Cobb-Douglas and that this firm produces Q exclusively for a large retail grocery chain It has a contract for 1999 to deliver Q units of product to the retailer Assume the firm is a perfect competitor in the markets for its inputs Set up the firms optimization problem given this information and derive the first-order conditions for the problem Defme any notation or terminology you need to introduce Also illustrate the optimization problem with a graph
c Discuss the firms demand function for R given the conditions of the problem in part (b) Specifically what variables determine how much R the firm will utilize
d Re-express the problem in part (b) and take first-order conditions assuming now that the firm is the only buyer of the farm input R which is produced according to the following inverse supply function
w = a + 3R a3 gt 0
OJ 04
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TABLE 01 Percentage points of lhe t distribution
Example
Pr (r gt 2086) = Oo25
Pr (r gt 1725) = 005 for df = 20
Pr(ltl gt 1725) = 010 0 I 715
01$ 010 middot-00
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N This tahlc gives the area in the riahtmiddothand tail or the distribution tic bull t 0) But since the normal di5tribumiddot Nolt 111e smaller probability shown at the head of each column is the area in one tail the 1in is symmetrical about r- Othe area in the ldtmiddothlnd tail is the same u the area in the corresponding right-hand larser probability is the area in both tails bullbullil rltgtr eumplc PI- 196 S z S 01-04750 TMrdore -196 S t S 196)-2(047S0)-09S Sourr From E S Pearwn and H 0 Hartley eds bull Blomtuila TabftJot Slulislidans vol I
ld ed table 12 Cbullmbridgc University Prns New York 1966 Reproduced by perminion of the middot
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I I
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US1 i9 l-IS S72 4112 4 1 1 199
I 1 I J I 110 I Y 12Y I 211 I M I 7 1 7 I 6 1 15 1 11 U9 UK IJK 1middot17 U6
17M I 71 1 101 167 1 162 161 I I S1 n6 I 51 IU1115 I ll 1 4 1 1401 19 1 47 146 1 44
I 98
2111 191 1 1111 1 19 210 1 042B 9 I H6 1 gt4 1 MO 119 1 1 1 1 1) 1 1 1 211 2111 2JO 22S 21 2111JOI 2SI 242 2 16 11114 16 140
2M9 2 66 UK 2 40 111 1 1 1 2-1 2 1 Ul167 middot 1 16 126 117 109 ill2 H I 472 4 12 01
114 I 12 111 1 10 129 I 211 UK 16 I 26 126 I 25 I 2 I 42 1 4 1 19 us I H 17 l l6 2 1 111 146 us 1 16 I 1611 165 1 61 I 5Y UK us IS-1 I S1 I S l I t HIIll 11111 186 2 II 1 2 1 11 2 1111 11111 Y I 151 107 199 190 1-JIS 1112 1111) 1 16 1 1S I 71 I 7 1 211 222 2111 215Ull OS 219U7 198 1 14 2 t7 2111 2511 lSO 242 1 16 2 225 22J 219 216 2 IJ UH l09 )02 196HI 129SSl 41gt4 41401
IJJ I I 1 10 129 1 18 117 127 126 1 1gt 125 124 1 24 136 usI 41 141 140 119 UK1 111 146 145 1 69 1 66 16 159 I S1 156 I 51 152 I SO 149 1 19 Ill 1114 1 812 11 1 1 229 216 1116 190 187 100
H14 196 1 9 1 1 81 1 112 119 1 71 I 71 1 11 1 69 167 165 0)219 2 15 1 1 1 JN 295 1 1 1 2242)6 229 V5 1 1 144 IS 210 2 26 1 19 1 1 1 2 1 1 109 U6 U l
0 1 407 I 75 )51 U6 123 312 101
Ul l ltJ 129 Ull 127 1 26 126 IH 1 2-1 1 2-1 12J t n l iM I 45 1 44 1 41 U9 Ull IJ6 us IB
172 167 1 64 ltll I H 15-1 Ul I SU I 1 lt7 1 6 10 10IIIS 182 179 171I 10 11111 1 22K 114 20S 1911 19) 188
2111 1 91 I H4 179 1 16 1 14 110 I M 1 IM I 62 O)1 41 lB 227 221 216 213OS l 2 291 169
270 2-17 210 2 2 1 IJ I I 207 2 Ul ] 0 1 147 J0 1 1 7 )07 298 2910 1 1 56 5 19 451 40
110 I 126 125 124 Ut 12 Ul 121 119 1 19 25IJ6 us 1 14 IJJ Ul U l140 11914425
1117 1113 179 1 66 161 157 1 )4 I l l 14K 141 142 1-11 119 I 111
176 173 1 71IllI 21)9 1 00 192 IM4 179 1611 1 64 I us I 5J U l 100I 21Oi
01
1R4 261 2 14 225 218 2 12 1 08 252 217 229 220 2 1 1 1116 202 1 94 I 92 I IM1 01 664 II 181 lSI 129 1 12 199 2119 280 27)511S
127 125 1 14 122 121 1 11l 119 1 1 7 1 1 1 1 16 1 1 5 1 1 5 I J J 112 I l l IJO 119 119us 1 41 1 4 1 UK 160 154 Ul 1411 144 141 140 116 I 15 I J I ll I 29 10 I
111 168 1 66117IK7 1821 19 2 19 211S 104 111-1 175 110 165 159 156 15] 1 4H 147 144 I-II 1 9 0)
125 2 17 210 104 199 195 1921 1 5 1 16 2 S 110 212 20 194 11111 1 04 17 I 71 I 61 160 til2S6 1SO 1 ll 295 lKl 272 261 165 ) 14
124 1221 16 129 128 127 1 2 1 1 19 1 111 1 17 116 1 14 1 1 1 1 1 1 I l l 1 10
IJJ IJI 1 10 140 119 141 117 132 111 126 1 24 1 2 1 119 10 120I us1 60I gtI 182 111 168 16S 1622 1)10 us
166 161 155 150 146 14 117 115 1 12 1211 125 I Hl 217 109 101 191 IKOl 219 201 I 176 1 101 156 UJ I 1 41 I IX
6 liS I 1701 4 J9 195 14H 296 2 79 266 256 247 240 middot 234
Ul 1 2 1 l iD 1 1 11 116 114 1 12 I l l 1 10 1119 I 0 25 111 IJ9 1111 I J6 I a I J2 I) I 129 128 127 126 US
152 146 1 41 1 111 1 1-1 I I 1211 I 24 122 uu 1 1 7 Ia Ill 2 I I 197 IKII IKO 1 75 170 166 161 160 157
172 162 IS7 IS2 146 141 119 I J2 129 U6 1 22 119 m198 19J I 88 I 114 180OS H9 1 04 265 242 226 2 14 2lt16
01 676 411 lllll l41 J I I 289 271 I 91 1119 I 79 169 I 61 1511 148 144 119 1 11 1 11 01
260 2SO 241 2 14 227
122 1 10 1 16 11-1 1 1 1 1 12 1119 Uli I 07 l tiO 124 124I 12 I J9 IJ7 US I J 111 129 1211 127 125
I II 142 118 114 uo 126 124 1 1 11 1 1 1 1 1 1 I OK Hill 10
i lS 25
IS72 JO 160 us172 167 16J10 2 71 2011 194 U2 1 14 1 12 1 1 1 1 1 1 I IIII 167 152 146 179l 11-1 HXI 260 2J7 221 210 201 1811 181liSmiddot
21KIll 6 6 4bl 3-JII J l2 l 02 2110 264 2 51 241 232 22S 1 10 U6 I ll I S I OtlIIIK104 U9 152 0 1
Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
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University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
University of California Davi Jepartment of Agricultural and Resource bconomics
MS Comprehensive Exam September 151998
There are two sections to this exam Each section consists of four questions Follow the instructions for each section carefully Use complete sentences and well thought out responses when writing down your answers Note that if we cannot read your writing then we will assume that your answer is wrong All questions are equally weighted There are five pages to this exam
Section 1 This section consists of questions 1 ) 2) 3) and 4) Answer question 1 ) and then select two of the remaining three questions to answer Do not answer all four questions
I) Below is a highly simplified model of wine supply and demand Selected t-ratios are below each estimated coefficient
Two-Stage Least-Squares Estimates of Supply
Q = -101 + 026PWHAT(210) (220)
R2 =035 DW=041 n=21
Two-Stage Least Squares Estimates of Demand
Q = IOI -122PWHAT + 222NCOME +340CPIALL + 0753Q_1(23S) (110) (056) (400)
R2 =096 DW=05 Durbins h- statistic= 083 n=21
Auxiliary First Stage Regression PWHAT = 201+ 133NCOME + 440CPIALL + 023Q_1
R2 = 067 DW= 187
Regression On Residuals From Demand Equation
Jl = 05+155NCOME +055CPIALL (620) (ll)
R2=072 The variables are defmed as follows
Q - Quantity of wine supplied and demanded PWHAT - Price of wine from auxiliary regression INCOME- Per capita personal income CPIALL - Consumer price index of all items
Examine the above model for econometric problems In particular you should look for
I autocorrelation 2 heteroscedasticity
2)
3 omitted variables
4 multicollinearity
5 identification
6 simultaneous equation bias
For each of the above possible problems explain why you think there is or is not a problem Explain
what should be done to remedy the problem
You have been asked to estimate a linear econometric model say Y = 30 + 31X + Jl Assume that
var(Jl) = a2X (I)
where Jl is a normally distributed random variable with a mean of 0 and a variance given in Eq () The
above model (I) is termed heteroscedasticity
a) Clearly explain the problems that heteroscedasticity creates
b) How might you go about detecting the presence of heteroscedasticity in the above linear
model
middotc) How might you correct for the heteroscedasticity given that it occurs in the form ofEq (1)
in the above linear model Be specific and give details
3) Answer the following jive TrueFalseExplain questions concerning a general linear programming
(LP) problem Please note that the explanation is worth the bulk of the credit
a) An optimal solution to aLP problem uses up all of the limited resources available
b) A point on the boundary of the feasible set (or region) satisfies all of the constraints of the
LP problem
c) The optimal solution of an LP problem always occurs at an extreme point of the feasible
set
d) If the first COilStraint of aLP problem evaluated at given value xo of the decision vector
has a zero value for its corresponding slack variable then the point xo lies on the boundary
of the feasible region
e) If the shadow value (or dual variable) corresponding to a given constraint is positive than
that constraint is binding
4) PROTRAC INC produces two lines of heavy equipment One of the product lines termed
earthmoving equipment is essentially for construction applications while the other line termed forestry
2
equipment is destined for Ute timber industry The largest member or ihe earthmoving line (the E-9) and
the largest member of the forestry equipment line (the F-9) are produced in the same departments and with
the same equipment Using economic forecasts for the next month PROTRACs marketing manager has
judged that during that period it will be possible to sell as many E-9s and F-9s as the flrm cam produce
Management must now recommend a production target for the next month
Making this decision requires that the following major facts be considered
bull PROTRACs net revenue is $5000 on each E-9 that is sold and $4000 on each F-9
bull Each product is put through PROTRACs machining operations in department A and department B
bull For next months production department A hasl50 hours available and department B has 160 hours
available Each E-9 uses 10 hours of machining in department A and 20 hours of machining in
department B Each F-9 uses 15 hours of machining in department A and 10 hours of machining in
department B
bull In order for management to honor an agreement with the union the total labor hours used in next
months testing of finished products cannot fall more than 10 below an arbitrated goal of 150 hours
The testing is performed in a third department and has nothing to do with the activities in departments
A and B Each E-9 is given 30 hours of testing and each F-9 is given I0 hours of testing
bull In order to maintain the current market position senior management has decreed the operating policy
that it is necessary to build at least one F-9 for every three E-9 produced
bullmiddot A major dealer has ordered a total of at least five E-9s and F-9s in any combination for the next
month and so at least that many must be produced
Given the above considerations managements problem is to decide how many E-9s and many F-9s to
produce next month so as to maximize net revenue
Formulate the linear programming problem for PROTRAC INC Deflne all the variables constraints
and equations in the formulation
Section 2 This section consists of questions 5 ) 6) 7) and 8) Answer any three of these four questions Do
not answer all four questions
5) A simple monopolist faces the linear inverse market demand curve given by p(y) a - by where
a gt 0 b gt 0 and y is the output of the monopolist In addition the monopolist faces the total cost
function given by c(y) Fe where k gt 0 Finally the government pays the monopolist a subsidy of
s gt 0 dollars per unit of output produced in order to encourage the monopolist to produce more output and
sell it at a lower price
a) Set up the proflt maximization problem facing the monopolist
3
--------
b) Find the monopolists profit maximizing level of output say y = ym(abks) as well as its profit maximizing price say p = pm(abk s) How do you know you have found the profit maximizing solution
c) Will an increase in the subsidy achieve its desired goal of increasing the monopolists output and lowering its price Show your work
d) Provide an economic interpretation of the above comparative statics e) How large would the subsidy have to be in order to get the monopolist to produce the
Pareto efficient level of output and charge the Pareto efficient price
6) The following questions pertain to a production setting characterized by the usual neoclassical production function of the form Q = f(xlx2x ) where Q is output and (x1x2x ) are inputs
a) Define compare and contrast the following terms from production economics i) Decreasing returns to a variable input and ii) decreasing returns to scale (Decreasing returns to scale is also called
diseconomies of scale) b) What economic forces cause decreasing returns to a variable input and what economic
forces might cause decreasing returns to scale c) Given the production function as specified show mathematically how you would measure
the returns to the variable input x1 and the returns to scale d) Suppose you were studying the question of whether small farms could compete effectively
against large farms in the market for the product Q Would either or both of the concepts of returns to a variable input and returns to scale be valuable in your analysis Explain
7) The following are demand and supply curves that apply to a competitive industry
Q0 = 250- SOP Qs = ()P
a) Derive the market equilibrium price and quantity b) Is the equilibrium you derived in (a) a short-run equilibrium a long-run equilibrium or
could it be both Explain your answer in detail demonstrating your knowledge of behavior in competitive markets and the distinction in these markets between the short and long run
c) Politicians are floating a proposal to place a per unit tax of $ 100 on this commodity Producers in lobbying against the tax argue that they will have no choice but to pass the
full tax on to consumers and therefore the tax if enacted will only harm consumers Evaluate the producers argument concerning the tax Use either algebra (preferred tool) or
4
--------------
------------------------------ X
bull
a)
a graph to support your argument If you use a graph label it carefully using the demand and supply equations Q0 = 250- 50P and Qs = ( )P
8) The following graph depicts a consumers Marshallian demand curve for the commodity X The market price of X is given as
p
Demand Curve
Given identify the consumers surplus on the graph and give an intuitive definition of consumers surplus
b) Assuming X is a normal good for this consumer draw the consumers Hicksian demand curve given price
c) Explain the difference between Hicksian and Marshallian demand curves
d) Suppose price increases to P_ Use Hicksian demands to identify the consumers compensating variation and equivalent variation due to the price change Label carefully
5
University of California Davis Department of Agricultural and Resource Economics
MS Exam
July 2 1999
Directions You have three hours to write this exam Please write legibly Answers that cannot be read are regarded as incorrect answers This exam consists of two parts In part I quantitative methods you must answer question 1 Then choose to answer either question 2 or question 3 In part II microeconomics answer three of the four questions You should thus answer five questions in total two from part I and three from part II Do not under any circumstances answer more than five questions
Part I Answer 1 estion I and either Question 2 01 luestion 3
l You are constructing a si mple model of cotton su pply for a short-term project you are working on and h ave formulated the following conceptual model
where y5 i s bales of cotton ps i s the price paid to growers w1 is the labor wage rate w2 i s the price of water WJ i s pesticide price and DO is degree-days a measure of the growi ng conditions during the season
(a) A nother variable you might include in thi s speci ficati on is annual rainfall (R) whose coefficient i f included in the model would be a6 While you are sure th at all th e other va ri ables in the model are part of the correct speci fication you are not sure about annual rainfall As an applied econometrician explai n the tradeoffs made to the properties of you r estimates i f you include or exclude thi s vari able from the specification
(b) One must always be concerned with measurement error i nempirical models Explain th e effects on the properties of your estimates of (a) measurement error in th e dependent variable vs (b) measurement error in one or more explanatory vari ables
(c) You esti mate th e model wi th R included and the results are
yS = -6354 + 8727ps - 0 579lw1 - 7907w2 - 09045w3 + 769000 + 0045R ( -0 79) (2 36) ( -177) (-234) ( -190) (157) (045)
R2 = 0819 n =56
where th e Students-t statistics are gi ven in parentheses and the following sums of squared errors resu lt
Total SSY=660 9 Model SSR=5412 Errors SSE= l l 97
In answering the following part s justify your answers with quantitati ve test results using th e attached tables Quali tative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 0 5 level
(ii) Is the overall model significant Th at i s i s the hypothesis th at a1 = a2 = a3 = a4 = as = a6 = 0 rejected
(iii) The correlation between the price of water (w2) and the price of pesticides (w3) i s zero Do these two vari ables have the same effect on cotton supply i e are their coeffi cients equ al
2 Suppose you have the following econometric model
where y = yieldacre x1 = wateracre x2 = (wateracre)2 x3 = fertilizeracre
(a) For each of the following questions set up the appropriate hypothesis test and provide a brief explanation
(i) Does water have any influence on yieldacre
(ii) Assuming an unconstrained (interior) solution is wateracre = 300 significantly different from the amount that yields the maximum yield per unit water
(b) Suppose you work in a governmental agency and that your boss is not trained in economics or statistics and thus she does not know what e N(O all) means Explain-
in words the meaning and significance of this expression
3
a Given the regional farm modeling problem
Max subject to
px- cx Ax b
where x is an n x I vector of production activities p and c are vectors of prices and variable cost coefficients respectively b is an m x 1 vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Comment briefly on the connection between the dual problem and the Dual Complementary Slackness condition
(iii) If you had to calculate the derived demand for a single input b describe the type of function you would obtain from the problem
b Now use the information that the product prices p are determined according to the following demand relationships
p =a + Dx
Assume that the individual farmers act as perfect competitors
(i) Show how you would reformulate your problem and give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit )
(ii) Derive the dual to the problem in part b(i) showing your steps clearly
Part II Answer of the 4 questions in this Section _
I For some goods consumption is subject to more than the usual single constraint on money income time can also be a constrai nt Consider a consumer with utility f unction u(x) defined on goods x = (x1 xn) subject to th e two constraints T 2 tx and M 2 px where T is the total amount of time avai lable and M is th e total amount of income avai lable and t = (t1 tn) and p = (p o pn ) are the time and money prices per unit of consumption of each good
(a) Write th e primal utility max i mi zation problem for th is consumer What interpretation do you attach to th e sh adow values on each constraint
(b) F rom your answer to part (a) ex plain how the consumer s choices reveal her money value of time Develop an ex pression f or the marginal money value of time from th is pri mal problem
(c) Under what conditions would the problem revert to the standard consumer choice problem subject to a single money budget constraint
(d) Suppose th at the shadow value on the money budget constraint is zero W rite the dual expenditure mi nimization problem for thi s case Provide a brief interpretation in words of thi s dual ex penditure f unction Also provide an interpretation for the sh adow value of the constraint for th is problem
2 Assume t hat t he US wheat mar ket can be considered in isolation and is charact erized by
yS = sops - 20 wl
y0 = 60 - 12 5p0 +2M
where
and
ys is quantity of wheat suppl ied in millions of bushels per year p5 is the price recei ved by suppliers in dollars per bushel y0 is quantity demanded (in millions of bushels) by consumers p0 i s the price paid by consumers also in dollars per bushel w1 is the price of a composite input to wheat product ion M is consumer income in t housands of dollars
For purposes of this analysis consumer income i s M = 45 and the composite input price is w1 = 5
(a) What are t he equilibrium f ree-mar ket price and quant it y of wheat
(b) Is t he supply of wheat at this point price-elastic or price-inelast i c Ex plain your reasomng
(c) W h at is t he t otal net economic sur plus f rom production of wheat and how middotis it distr ibuted bet ween suppliers and demanders
(d) A var iet y of schemes are used in agricult ural markets to enhance producer welf are C onsider a pr ice support of $6 per bushel Explain t he effects of such a policy on the dist ri bution of welfare and economic efficiency Support your answer with quant itat ive est imates of t hese effect s
jgtrice
3 Suppose a monopoly fum faces the following linear demand curve P = a- bQ where P is price and Q is quantity produced and sold The monopolists cost function is
(1) c(Q) = cQ for Q Q c(Q) qo for Q gt Q
where Q is a capacity constraint
a Solve mathematically for the monopolists profit maxumzmg and output as functions of the parameters a b and c Show your work Also illustrate yourresults graphically labeling carefully all relevant information
b Suppose the function P(Q) from part (a) is the demand in the domestic market and that the monopolist can also sell unlimited quantities into the export market at price P E gt c Assuming that quantities exported cannot be resold into the domestic market show both mathematically and on your graph from part (a) how much the firm will sell into the domestic market and into the export market
c Consider now a potential entrant into this market For simplicity ignore the existence of the export market To be able to produce in this market an entrant must incur a one time set up or entry cost of E gt 0 It can then produce according to the cost function in (1) Describe in words andor diagrams how you would model this entrants decision problem You should consider the role of the incumbent monopolist in framing your answer
4 Consider a food processing firm that produces processed product Q from a raw agricultural product Rand a composite marketing input L which we will call labor
a Write a production function and draw some rough sketches of isoquants for each of the following possible production scenarios
1 There is no substitution between R and L in producing Q
ii The production relationship is Cobb-Douglas
b Suppose the true production relationship is Cobb-Douglas and that this firm produces Q exclusively for a large retail grocery chain It has a contract for 1999 to deliver Q units of product to the retailer Assume the firm is a perfect competitor in the markets for its inputs Set up the firms optimization problem given this information and derive the first-order conditions for the problem Defme any notation or terminology you need to introduce Also illustrate the optimization problem with a graph
c Discuss the firms demand function for R given the conditions of the problem in part (b) Specifically what variables determine how much R the firm will utilize
d Re-express the problem in part (b) and take first-order conditions assuming now that the firm is the only buyer of the farm input R which is produced according to the following inverse supply function
w = a + 3R a3 gt 0
OJ 04
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TABLE 01 Percentage points of lhe t distribution
Example
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Pr(ltl gt 1725) = 010 0 I 715
01$ 010 middot-00
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N This tahlc gives the area in the riahtmiddothand tail or the distribution tic bull t 0) But since the normal di5tribumiddot Nolt 111e smaller probability shown at the head of each column is the area in one tail the 1in is symmetrical about r- Othe area in the ldtmiddothlnd tail is the same u the area in the corresponding right-hand larser probability is the area in both tails bullbullil rltgtr eumplc PI- 196 S z S 01-04750 TMrdore -196 S t S 196)-2(047S0)-09S Sourr From E S Pearwn and H 0 Hartley eds bull Blomtuila TabftJot Slulislidans vol I
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2111 191 1 1111 1 19 210 1 042B 9 I H6 1 gt4 1 MO 119 1 1 1 1 1) 1 1 1 211 2111 2JO 22S 21 2111JOI 2SI 242 2 16 11114 16 140
2M9 2 66 UK 2 40 111 1 1 1 2-1 2 1 Ul167 middot 1 16 126 117 109 ill2 H I 472 4 12 01
114 I 12 111 1 10 129 I 211 UK 16 I 26 126 I 25 I 2 I 42 1 4 1 19 us I H 17 l l6 2 1 111 146 us 1 16 I 1611 165 1 61 I 5Y UK us IS-1 I S1 I S l I t HIIll 11111 186 2 II 1 2 1 11 2 1111 11111 Y I 151 107 199 190 1-JIS 1112 1111) 1 16 1 1S I 71 I 7 1 211 222 2111 215Ull OS 219U7 198 1 14 2 t7 2111 2511 lSO 242 1 16 2 225 22J 219 216 2 IJ UH l09 )02 196HI 129SSl 41gt4 41401
IJJ I I 1 10 129 1 18 117 127 126 1 1gt 125 124 1 24 136 usI 41 141 140 119 UK1 111 146 145 1 69 1 66 16 159 I S1 156 I 51 152 I SO 149 1 19 Ill 1114 1 812 11 1 1 229 216 1116 190 187 100
H14 196 1 9 1 1 81 1 112 119 1 71 I 71 1 11 1 69 167 165 0)219 2 15 1 1 1 JN 295 1 1 1 2242)6 229 V5 1 1 144 IS 210 2 26 1 19 1 1 1 2 1 1 109 U6 U l
0 1 407 I 75 )51 U6 123 312 101
Ul l ltJ 129 Ull 127 1 26 126 IH 1 2-1 1 2-1 12J t n l iM I 45 1 44 1 41 U9 Ull IJ6 us IB
172 167 1 64 ltll I H 15-1 Ul I SU I 1 lt7 1 6 10 10IIIS 182 179 171I 10 11111 1 22K 114 20S 1911 19) 188
2111 1 91 I H4 179 1 16 1 14 110 I M 1 IM I 62 O)1 41 lB 227 221 216 213OS l 2 291 169
270 2-17 210 2 2 1 IJ I I 207 2 Ul ] 0 1 147 J0 1 1 7 )07 298 2910 1 1 56 5 19 451 40
110 I 126 125 124 Ut 12 Ul 121 119 1 19 25IJ6 us 1 14 IJJ Ul U l140 11914425
1117 1113 179 1 66 161 157 1 )4 I l l 14K 141 142 1-11 119 I 111
176 173 1 71IllI 21)9 1 00 192 IM4 179 1611 1 64 I us I 5J U l 100I 21Oi
01
1R4 261 2 14 225 218 2 12 1 08 252 217 229 220 2 1 1 1116 202 1 94 I 92 I IM1 01 664 II 181 lSI 129 1 12 199 2119 280 27)511S
127 125 1 14 122 121 1 11l 119 1 1 7 1 1 1 1 16 1 1 5 1 1 5 I J J 112 I l l IJO 119 119us 1 41 1 4 1 UK 160 154 Ul 1411 144 141 140 116 I 15 I J I ll I 29 10 I
111 168 1 66117IK7 1821 19 2 19 211S 104 111-1 175 110 165 159 156 15] 1 4H 147 144 I-II 1 9 0)
125 2 17 210 104 199 195 1921 1 5 1 16 2 S 110 212 20 194 11111 1 04 17 I 71 I 61 160 til2S6 1SO 1 ll 295 lKl 272 261 165 ) 14
124 1221 16 129 128 127 1 2 1 1 19 1 111 1 17 116 1 14 1 1 1 1 1 1 I l l 1 10
IJJ IJI 1 10 140 119 141 117 132 111 126 1 24 1 2 1 119 10 120I us1 60I gtI 182 111 168 16S 1622 1)10 us
166 161 155 150 146 14 117 115 1 12 1211 125 I Hl 217 109 101 191 IKOl 219 201 I 176 1 101 156 UJ I 1 41 I IX
6 liS I 1701 4 J9 195 14H 296 2 79 266 256 247 240 middot 234
Ul 1 2 1 l iD 1 1 11 116 114 1 12 I l l 1 10 1119 I 0 25 111 IJ9 1111 I J6 I a I J2 I) I 129 128 127 126 US
152 146 1 41 1 111 1 1-1 I I 1211 I 24 122 uu 1 1 7 Ia Ill 2 I I 197 IKII IKO 1 75 170 166 161 160 157
172 162 IS7 IS2 146 141 119 I J2 129 U6 1 22 119 m198 19J I 88 I 114 180OS H9 1 04 265 242 226 2 14 2lt16
01 676 411 lllll l41 J I I 289 271 I 91 1119 I 79 169 I 61 1511 148 144 119 1 11 1 11 01
260 2SO 241 2 14 227
122 1 10 1 16 11-1 1 1 1 1 12 1119 Uli I 07 l tiO 124 124I 12 I J9 IJ7 US I J 111 129 1211 127 125
I II 142 118 114 uo 126 124 1 1 11 1 1 1 1 1 1 I OK Hill 10
i lS 25
IS72 JO 160 us172 167 16J10 2 71 2011 194 U2 1 14 1 12 1 1 1 1 1 1 I IIII 167 152 146 179l 11-1 HXI 260 2J7 221 210 201 1811 181liSmiddot
21KIll 6 6 4bl 3-JII J l2 l 02 2110 264 2 51 241 232 22S 1 10 U6 I ll I S I OtlIIIK104 U9 152 0 1
Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
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Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
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EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
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a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
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a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
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bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
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Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
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Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
2)
3 omitted variables
4 multicollinearity
5 identification
6 simultaneous equation bias
For each of the above possible problems explain why you think there is or is not a problem Explain
what should be done to remedy the problem
You have been asked to estimate a linear econometric model say Y = 30 + 31X + Jl Assume that
var(Jl) = a2X (I)
where Jl is a normally distributed random variable with a mean of 0 and a variance given in Eq () The
above model (I) is termed heteroscedasticity
a) Clearly explain the problems that heteroscedasticity creates
b) How might you go about detecting the presence of heteroscedasticity in the above linear
model
middotc) How might you correct for the heteroscedasticity given that it occurs in the form ofEq (1)
in the above linear model Be specific and give details
3) Answer the following jive TrueFalseExplain questions concerning a general linear programming
(LP) problem Please note that the explanation is worth the bulk of the credit
a) An optimal solution to aLP problem uses up all of the limited resources available
b) A point on the boundary of the feasible set (or region) satisfies all of the constraints of the
LP problem
c) The optimal solution of an LP problem always occurs at an extreme point of the feasible
set
d) If the first COilStraint of aLP problem evaluated at given value xo of the decision vector
has a zero value for its corresponding slack variable then the point xo lies on the boundary
of the feasible region
e) If the shadow value (or dual variable) corresponding to a given constraint is positive than
that constraint is binding
4) PROTRAC INC produces two lines of heavy equipment One of the product lines termed
earthmoving equipment is essentially for construction applications while the other line termed forestry
2
equipment is destined for Ute timber industry The largest member or ihe earthmoving line (the E-9) and
the largest member of the forestry equipment line (the F-9) are produced in the same departments and with
the same equipment Using economic forecasts for the next month PROTRACs marketing manager has
judged that during that period it will be possible to sell as many E-9s and F-9s as the flrm cam produce
Management must now recommend a production target for the next month
Making this decision requires that the following major facts be considered
bull PROTRACs net revenue is $5000 on each E-9 that is sold and $4000 on each F-9
bull Each product is put through PROTRACs machining operations in department A and department B
bull For next months production department A hasl50 hours available and department B has 160 hours
available Each E-9 uses 10 hours of machining in department A and 20 hours of machining in
department B Each F-9 uses 15 hours of machining in department A and 10 hours of machining in
department B
bull In order for management to honor an agreement with the union the total labor hours used in next
months testing of finished products cannot fall more than 10 below an arbitrated goal of 150 hours
The testing is performed in a third department and has nothing to do with the activities in departments
A and B Each E-9 is given 30 hours of testing and each F-9 is given I0 hours of testing
bull In order to maintain the current market position senior management has decreed the operating policy
that it is necessary to build at least one F-9 for every three E-9 produced
bullmiddot A major dealer has ordered a total of at least five E-9s and F-9s in any combination for the next
month and so at least that many must be produced
Given the above considerations managements problem is to decide how many E-9s and many F-9s to
produce next month so as to maximize net revenue
Formulate the linear programming problem for PROTRAC INC Deflne all the variables constraints
and equations in the formulation
Section 2 This section consists of questions 5 ) 6) 7) and 8) Answer any three of these four questions Do
not answer all four questions
5) A simple monopolist faces the linear inverse market demand curve given by p(y) a - by where
a gt 0 b gt 0 and y is the output of the monopolist In addition the monopolist faces the total cost
function given by c(y) Fe where k gt 0 Finally the government pays the monopolist a subsidy of
s gt 0 dollars per unit of output produced in order to encourage the monopolist to produce more output and
sell it at a lower price
a) Set up the proflt maximization problem facing the monopolist
3
--------
b) Find the monopolists profit maximizing level of output say y = ym(abks) as well as its profit maximizing price say p = pm(abk s) How do you know you have found the profit maximizing solution
c) Will an increase in the subsidy achieve its desired goal of increasing the monopolists output and lowering its price Show your work
d) Provide an economic interpretation of the above comparative statics e) How large would the subsidy have to be in order to get the monopolist to produce the
Pareto efficient level of output and charge the Pareto efficient price
6) The following questions pertain to a production setting characterized by the usual neoclassical production function of the form Q = f(xlx2x ) where Q is output and (x1x2x ) are inputs
a) Define compare and contrast the following terms from production economics i) Decreasing returns to a variable input and ii) decreasing returns to scale (Decreasing returns to scale is also called
diseconomies of scale) b) What economic forces cause decreasing returns to a variable input and what economic
forces might cause decreasing returns to scale c) Given the production function as specified show mathematically how you would measure
the returns to the variable input x1 and the returns to scale d) Suppose you were studying the question of whether small farms could compete effectively
against large farms in the market for the product Q Would either or both of the concepts of returns to a variable input and returns to scale be valuable in your analysis Explain
7) The following are demand and supply curves that apply to a competitive industry
Q0 = 250- SOP Qs = ()P
a) Derive the market equilibrium price and quantity b) Is the equilibrium you derived in (a) a short-run equilibrium a long-run equilibrium or
could it be both Explain your answer in detail demonstrating your knowledge of behavior in competitive markets and the distinction in these markets between the short and long run
c) Politicians are floating a proposal to place a per unit tax of $ 100 on this commodity Producers in lobbying against the tax argue that they will have no choice but to pass the
full tax on to consumers and therefore the tax if enacted will only harm consumers Evaluate the producers argument concerning the tax Use either algebra (preferred tool) or
4
--------------
------------------------------ X
bull
a)
a graph to support your argument If you use a graph label it carefully using the demand and supply equations Q0 = 250- 50P and Qs = ( )P
8) The following graph depicts a consumers Marshallian demand curve for the commodity X The market price of X is given as
p
Demand Curve
Given identify the consumers surplus on the graph and give an intuitive definition of consumers surplus
b) Assuming X is a normal good for this consumer draw the consumers Hicksian demand curve given price
c) Explain the difference between Hicksian and Marshallian demand curves
d) Suppose price increases to P_ Use Hicksian demands to identify the consumers compensating variation and equivalent variation due to the price change Label carefully
5
University of California Davis Department of Agricultural and Resource Economics
MS Exam
July 2 1999
Directions You have three hours to write this exam Please write legibly Answers that cannot be read are regarded as incorrect answers This exam consists of two parts In part I quantitative methods you must answer question 1 Then choose to answer either question 2 or question 3 In part II microeconomics answer three of the four questions You should thus answer five questions in total two from part I and three from part II Do not under any circumstances answer more than five questions
Part I Answer 1 estion I and either Question 2 01 luestion 3
l You are constructing a si mple model of cotton su pply for a short-term project you are working on and h ave formulated the following conceptual model
where y5 i s bales of cotton ps i s the price paid to growers w1 is the labor wage rate w2 i s the price of water WJ i s pesticide price and DO is degree-days a measure of the growi ng conditions during the season
(a) A nother variable you might include in thi s speci ficati on is annual rainfall (R) whose coefficient i f included in the model would be a6 While you are sure th at all th e other va ri ables in the model are part of the correct speci fication you are not sure about annual rainfall As an applied econometrician explai n the tradeoffs made to the properties of you r estimates i f you include or exclude thi s vari able from the specification
(b) One must always be concerned with measurement error i nempirical models Explain th e effects on the properties of your estimates of (a) measurement error in th e dependent variable vs (b) measurement error in one or more explanatory vari ables
(c) You esti mate th e model wi th R included and the results are
yS = -6354 + 8727ps - 0 579lw1 - 7907w2 - 09045w3 + 769000 + 0045R ( -0 79) (2 36) ( -177) (-234) ( -190) (157) (045)
R2 = 0819 n =56
where th e Students-t statistics are gi ven in parentheses and the following sums of squared errors resu lt
Total SSY=660 9 Model SSR=5412 Errors SSE= l l 97
In answering the following part s justify your answers with quantitati ve test results using th e attached tables Quali tative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 0 5 level
(ii) Is the overall model significant Th at i s i s the hypothesis th at a1 = a2 = a3 = a4 = as = a6 = 0 rejected
(iii) The correlation between the price of water (w2) and the price of pesticides (w3) i s zero Do these two vari ables have the same effect on cotton supply i e are their coeffi cients equ al
2 Suppose you have the following econometric model
where y = yieldacre x1 = wateracre x2 = (wateracre)2 x3 = fertilizeracre
(a) For each of the following questions set up the appropriate hypothesis test and provide a brief explanation
(i) Does water have any influence on yieldacre
(ii) Assuming an unconstrained (interior) solution is wateracre = 300 significantly different from the amount that yields the maximum yield per unit water
(b) Suppose you work in a governmental agency and that your boss is not trained in economics or statistics and thus she does not know what e N(O all) means Explain-
in words the meaning and significance of this expression
3
a Given the regional farm modeling problem
Max subject to
px- cx Ax b
where x is an n x I vector of production activities p and c are vectors of prices and variable cost coefficients respectively b is an m x 1 vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Comment briefly on the connection between the dual problem and the Dual Complementary Slackness condition
(iii) If you had to calculate the derived demand for a single input b describe the type of function you would obtain from the problem
b Now use the information that the product prices p are determined according to the following demand relationships
p =a + Dx
Assume that the individual farmers act as perfect competitors
(i) Show how you would reformulate your problem and give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit )
(ii) Derive the dual to the problem in part b(i) showing your steps clearly
Part II Answer of the 4 questions in this Section _
I For some goods consumption is subject to more than the usual single constraint on money income time can also be a constrai nt Consider a consumer with utility f unction u(x) defined on goods x = (x1 xn) subject to th e two constraints T 2 tx and M 2 px where T is the total amount of time avai lable and M is th e total amount of income avai lable and t = (t1 tn) and p = (p o pn ) are the time and money prices per unit of consumption of each good
(a) Write th e primal utility max i mi zation problem for th is consumer What interpretation do you attach to th e sh adow values on each constraint
(b) F rom your answer to part (a) ex plain how the consumer s choices reveal her money value of time Develop an ex pression f or the marginal money value of time from th is pri mal problem
(c) Under what conditions would the problem revert to the standard consumer choice problem subject to a single money budget constraint
(d) Suppose th at the shadow value on the money budget constraint is zero W rite the dual expenditure mi nimization problem for thi s case Provide a brief interpretation in words of thi s dual ex penditure f unction Also provide an interpretation for the sh adow value of the constraint for th is problem
2 Assume t hat t he US wheat mar ket can be considered in isolation and is charact erized by
yS = sops - 20 wl
y0 = 60 - 12 5p0 +2M
where
and
ys is quantity of wheat suppl ied in millions of bushels per year p5 is the price recei ved by suppliers in dollars per bushel y0 is quantity demanded (in millions of bushels) by consumers p0 i s the price paid by consumers also in dollars per bushel w1 is the price of a composite input to wheat product ion M is consumer income in t housands of dollars
For purposes of this analysis consumer income i s M = 45 and the composite input price is w1 = 5
(a) What are t he equilibrium f ree-mar ket price and quant it y of wheat
(b) Is t he supply of wheat at this point price-elastic or price-inelast i c Ex plain your reasomng
(c) W h at is t he t otal net economic sur plus f rom production of wheat and how middotis it distr ibuted bet ween suppliers and demanders
(d) A var iet y of schemes are used in agricult ural markets to enhance producer welf are C onsider a pr ice support of $6 per bushel Explain t he effects of such a policy on the dist ri bution of welfare and economic efficiency Support your answer with quant itat ive est imates of t hese effect s
jgtrice
3 Suppose a monopoly fum faces the following linear demand curve P = a- bQ where P is price and Q is quantity produced and sold The monopolists cost function is
(1) c(Q) = cQ for Q Q c(Q) qo for Q gt Q
where Q is a capacity constraint
a Solve mathematically for the monopolists profit maxumzmg and output as functions of the parameters a b and c Show your work Also illustrate yourresults graphically labeling carefully all relevant information
b Suppose the function P(Q) from part (a) is the demand in the domestic market and that the monopolist can also sell unlimited quantities into the export market at price P E gt c Assuming that quantities exported cannot be resold into the domestic market show both mathematically and on your graph from part (a) how much the firm will sell into the domestic market and into the export market
c Consider now a potential entrant into this market For simplicity ignore the existence of the export market To be able to produce in this market an entrant must incur a one time set up or entry cost of E gt 0 It can then produce according to the cost function in (1) Describe in words andor diagrams how you would model this entrants decision problem You should consider the role of the incumbent monopolist in framing your answer
4 Consider a food processing firm that produces processed product Q from a raw agricultural product Rand a composite marketing input L which we will call labor
a Write a production function and draw some rough sketches of isoquants for each of the following possible production scenarios
1 There is no substitution between R and L in producing Q
ii The production relationship is Cobb-Douglas
b Suppose the true production relationship is Cobb-Douglas and that this firm produces Q exclusively for a large retail grocery chain It has a contract for 1999 to deliver Q units of product to the retailer Assume the firm is a perfect competitor in the markets for its inputs Set up the firms optimization problem given this information and derive the first-order conditions for the problem Defme any notation or terminology you need to introduce Also illustrate the optimization problem with a graph
c Discuss the firms demand function for R given the conditions of the problem in part (b) Specifically what variables determine how much R the firm will utilize
d Re-express the problem in part (b) and take first-order conditions assuming now that the firm is the only buyer of the farm input R which is produced according to the following inverse supply function
w = a + 3R a3 gt 0
OJ 04
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Example
Pr (0 z 196) = 04750
Pr (z 196) = 05 - 04750 = 0025
TABLE 01 Percentage points of lhe t distribution
Example
Pr (r gt 2086) = Oo25
Pr (r gt 1725) = 005 for df = 20
Pr(ltl gt 1725) = 010 0 I 715
01$ 010 middot-00
001 0 00010015010 002 0010 0002ozo
J 07H 127116I 11216 14 61657 IIUI1nm
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N This tahlc gives the area in the riahtmiddothand tail or the distribution tic bull t 0) But since the normal di5tribumiddot Nolt 111e smaller probability shown at the head of each column is the area in one tail the 1in is symmetrical about r- Othe area in the ldtmiddothlnd tail is the same u the area in the corresponding right-hand larser probability is the area in both tails bullbullil rltgtr eumplc PI- 196 S z S 01-04750 TMrdore -196 S t S 196)-2(047S0)-09S Sourr From E S Pearwn and H 0 Hartley eds bull Blomtuila TabftJot Slulislidans vol I
ld ed table 12 Cbullmbridgc University Prns New York 1966 Reproduced by perminion of the middot
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I I
STATISTICAl TA8LfS 681
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I 16 111 I 12 I y I 2ltI 10 1 10 111 1 1 1 1 111 I Y 1 41 1 4 1 1-10 I 19 1 19 UHI middotW 1171411 147 145
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176 173 1 71IllI 21)9 1 00 192 IM4 179 1611 1 64 I us I 5J U l 100I 21Oi
01
1R4 261 2 14 225 218 2 12 1 08 252 217 229 220 2 1 1 1116 202 1 94 I 92 I IM1 01 664 II 181 lSI 129 1 12 199 2119 280 27)511S
127 125 1 14 122 121 1 11l 119 1 1 7 1 1 1 1 16 1 1 5 1 1 5 I J J 112 I l l IJO 119 119us 1 41 1 4 1 UK 160 154 Ul 1411 144 141 140 116 I 15 I J I ll I 29 10 I
111 168 1 66117IK7 1821 19 2 19 211S 104 111-1 175 110 165 159 156 15] 1 4H 147 144 I-II 1 9 0)
125 2 17 210 104 199 195 1921 1 5 1 16 2 S 110 212 20 194 11111 1 04 17 I 71 I 61 160 til2S6 1SO 1 ll 295 lKl 272 261 165 ) 14
124 1221 16 129 128 127 1 2 1 1 19 1 111 1 17 116 1 14 1 1 1 1 1 1 I l l 1 10
IJJ IJI 1 10 140 119 141 117 132 111 126 1 24 1 2 1 119 10 120I us1 60I gtI 182 111 168 16S 1622 1)10 us
166 161 155 150 146 14 117 115 1 12 1211 125 I Hl 217 109 101 191 IKOl 219 201 I 176 1 101 156 UJ I 1 41 I IX
6 liS I 1701 4 J9 195 14H 296 2 79 266 256 247 240 middot 234
Ul 1 2 1 l iD 1 1 11 116 114 1 12 I l l 1 10 1119 I 0 25 111 IJ9 1111 I J6 I a I J2 I) I 129 128 127 126 US
152 146 1 41 1 111 1 1-1 I I 1211 I 24 122 uu 1 1 7 Ia Ill 2 I I 197 IKII IKO 1 75 170 166 161 160 157
172 162 IS7 IS2 146 141 119 I J2 129 U6 1 22 119 m198 19J I 88 I 114 180OS H9 1 04 265 242 226 2 14 2lt16
01 676 411 lllll l41 J I I 289 271 I 91 1119 I 79 169 I 61 1511 148 144 119 1 11 1 11 01
260 2SO 241 2 14 227
122 1 10 1 16 11-1 1 1 1 1 12 1119 Uli I 07 l tiO 124 124I 12 I J9 IJ7 US I J 111 129 1211 127 125
I II 142 118 114 uo 126 124 1 1 11 1 1 1 1 1 1 I OK Hill 10
i lS 25
IS72 JO 160 us172 167 16J10 2 71 2011 194 U2 1 14 1 12 1 1 1 1 1 1 I IIII 167 152 146 179l 11-1 HXI 260 2J7 221 210 201 1811 181liSmiddot
21KIll 6 6 4bl 3-JII J l2 l 02 2110 264 2 51 241 232 22S 1 10 U6 I ll I S I OtlIIIK104 U9 152 0 1
Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
equipment is destined for Ute timber industry The largest member or ihe earthmoving line (the E-9) and
the largest member of the forestry equipment line (the F-9) are produced in the same departments and with
the same equipment Using economic forecasts for the next month PROTRACs marketing manager has
judged that during that period it will be possible to sell as many E-9s and F-9s as the flrm cam produce
Management must now recommend a production target for the next month
Making this decision requires that the following major facts be considered
bull PROTRACs net revenue is $5000 on each E-9 that is sold and $4000 on each F-9
bull Each product is put through PROTRACs machining operations in department A and department B
bull For next months production department A hasl50 hours available and department B has 160 hours
available Each E-9 uses 10 hours of machining in department A and 20 hours of machining in
department B Each F-9 uses 15 hours of machining in department A and 10 hours of machining in
department B
bull In order for management to honor an agreement with the union the total labor hours used in next
months testing of finished products cannot fall more than 10 below an arbitrated goal of 150 hours
The testing is performed in a third department and has nothing to do with the activities in departments
A and B Each E-9 is given 30 hours of testing and each F-9 is given I0 hours of testing
bull In order to maintain the current market position senior management has decreed the operating policy
that it is necessary to build at least one F-9 for every three E-9 produced
bullmiddot A major dealer has ordered a total of at least five E-9s and F-9s in any combination for the next
month and so at least that many must be produced
Given the above considerations managements problem is to decide how many E-9s and many F-9s to
produce next month so as to maximize net revenue
Formulate the linear programming problem for PROTRAC INC Deflne all the variables constraints
and equations in the formulation
Section 2 This section consists of questions 5 ) 6) 7) and 8) Answer any three of these four questions Do
not answer all four questions
5) A simple monopolist faces the linear inverse market demand curve given by p(y) a - by where
a gt 0 b gt 0 and y is the output of the monopolist In addition the monopolist faces the total cost
function given by c(y) Fe where k gt 0 Finally the government pays the monopolist a subsidy of
s gt 0 dollars per unit of output produced in order to encourage the monopolist to produce more output and
sell it at a lower price
a) Set up the proflt maximization problem facing the monopolist
3
--------
b) Find the monopolists profit maximizing level of output say y = ym(abks) as well as its profit maximizing price say p = pm(abk s) How do you know you have found the profit maximizing solution
c) Will an increase in the subsidy achieve its desired goal of increasing the monopolists output and lowering its price Show your work
d) Provide an economic interpretation of the above comparative statics e) How large would the subsidy have to be in order to get the monopolist to produce the
Pareto efficient level of output and charge the Pareto efficient price
6) The following questions pertain to a production setting characterized by the usual neoclassical production function of the form Q = f(xlx2x ) where Q is output and (x1x2x ) are inputs
a) Define compare and contrast the following terms from production economics i) Decreasing returns to a variable input and ii) decreasing returns to scale (Decreasing returns to scale is also called
diseconomies of scale) b) What economic forces cause decreasing returns to a variable input and what economic
forces might cause decreasing returns to scale c) Given the production function as specified show mathematically how you would measure
the returns to the variable input x1 and the returns to scale d) Suppose you were studying the question of whether small farms could compete effectively
against large farms in the market for the product Q Would either or both of the concepts of returns to a variable input and returns to scale be valuable in your analysis Explain
7) The following are demand and supply curves that apply to a competitive industry
Q0 = 250- SOP Qs = ()P
a) Derive the market equilibrium price and quantity b) Is the equilibrium you derived in (a) a short-run equilibrium a long-run equilibrium or
could it be both Explain your answer in detail demonstrating your knowledge of behavior in competitive markets and the distinction in these markets between the short and long run
c) Politicians are floating a proposal to place a per unit tax of $ 100 on this commodity Producers in lobbying against the tax argue that they will have no choice but to pass the
full tax on to consumers and therefore the tax if enacted will only harm consumers Evaluate the producers argument concerning the tax Use either algebra (preferred tool) or
4
--------------
------------------------------ X
bull
a)
a graph to support your argument If you use a graph label it carefully using the demand and supply equations Q0 = 250- 50P and Qs = ( )P
8) The following graph depicts a consumers Marshallian demand curve for the commodity X The market price of X is given as
p
Demand Curve
Given identify the consumers surplus on the graph and give an intuitive definition of consumers surplus
b) Assuming X is a normal good for this consumer draw the consumers Hicksian demand curve given price
c) Explain the difference between Hicksian and Marshallian demand curves
d) Suppose price increases to P_ Use Hicksian demands to identify the consumers compensating variation and equivalent variation due to the price change Label carefully
5
University of California Davis Department of Agricultural and Resource Economics
MS Exam
July 2 1999
Directions You have three hours to write this exam Please write legibly Answers that cannot be read are regarded as incorrect answers This exam consists of two parts In part I quantitative methods you must answer question 1 Then choose to answer either question 2 or question 3 In part II microeconomics answer three of the four questions You should thus answer five questions in total two from part I and three from part II Do not under any circumstances answer more than five questions
Part I Answer 1 estion I and either Question 2 01 luestion 3
l You are constructing a si mple model of cotton su pply for a short-term project you are working on and h ave formulated the following conceptual model
where y5 i s bales of cotton ps i s the price paid to growers w1 is the labor wage rate w2 i s the price of water WJ i s pesticide price and DO is degree-days a measure of the growi ng conditions during the season
(a) A nother variable you might include in thi s speci ficati on is annual rainfall (R) whose coefficient i f included in the model would be a6 While you are sure th at all th e other va ri ables in the model are part of the correct speci fication you are not sure about annual rainfall As an applied econometrician explai n the tradeoffs made to the properties of you r estimates i f you include or exclude thi s vari able from the specification
(b) One must always be concerned with measurement error i nempirical models Explain th e effects on the properties of your estimates of (a) measurement error in th e dependent variable vs (b) measurement error in one or more explanatory vari ables
(c) You esti mate th e model wi th R included and the results are
yS = -6354 + 8727ps - 0 579lw1 - 7907w2 - 09045w3 + 769000 + 0045R ( -0 79) (2 36) ( -177) (-234) ( -190) (157) (045)
R2 = 0819 n =56
where th e Students-t statistics are gi ven in parentheses and the following sums of squared errors resu lt
Total SSY=660 9 Model SSR=5412 Errors SSE= l l 97
In answering the following part s justify your answers with quantitati ve test results using th e attached tables Quali tative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 0 5 level
(ii) Is the overall model significant Th at i s i s the hypothesis th at a1 = a2 = a3 = a4 = as = a6 = 0 rejected
(iii) The correlation between the price of water (w2) and the price of pesticides (w3) i s zero Do these two vari ables have the same effect on cotton supply i e are their coeffi cients equ al
2 Suppose you have the following econometric model
where y = yieldacre x1 = wateracre x2 = (wateracre)2 x3 = fertilizeracre
(a) For each of the following questions set up the appropriate hypothesis test and provide a brief explanation
(i) Does water have any influence on yieldacre
(ii) Assuming an unconstrained (interior) solution is wateracre = 300 significantly different from the amount that yields the maximum yield per unit water
(b) Suppose you work in a governmental agency and that your boss is not trained in economics or statistics and thus she does not know what e N(O all) means Explain-
in words the meaning and significance of this expression
3
a Given the regional farm modeling problem
Max subject to
px- cx Ax b
where x is an n x I vector of production activities p and c are vectors of prices and variable cost coefficients respectively b is an m x 1 vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Comment briefly on the connection between the dual problem and the Dual Complementary Slackness condition
(iii) If you had to calculate the derived demand for a single input b describe the type of function you would obtain from the problem
b Now use the information that the product prices p are determined according to the following demand relationships
p =a + Dx
Assume that the individual farmers act as perfect competitors
(i) Show how you would reformulate your problem and give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit )
(ii) Derive the dual to the problem in part b(i) showing your steps clearly
Part II Answer of the 4 questions in this Section _
I For some goods consumption is subject to more than the usual single constraint on money income time can also be a constrai nt Consider a consumer with utility f unction u(x) defined on goods x = (x1 xn) subject to th e two constraints T 2 tx and M 2 px where T is the total amount of time avai lable and M is th e total amount of income avai lable and t = (t1 tn) and p = (p o pn ) are the time and money prices per unit of consumption of each good
(a) Write th e primal utility max i mi zation problem for th is consumer What interpretation do you attach to th e sh adow values on each constraint
(b) F rom your answer to part (a) ex plain how the consumer s choices reveal her money value of time Develop an ex pression f or the marginal money value of time from th is pri mal problem
(c) Under what conditions would the problem revert to the standard consumer choice problem subject to a single money budget constraint
(d) Suppose th at the shadow value on the money budget constraint is zero W rite the dual expenditure mi nimization problem for thi s case Provide a brief interpretation in words of thi s dual ex penditure f unction Also provide an interpretation for the sh adow value of the constraint for th is problem
2 Assume t hat t he US wheat mar ket can be considered in isolation and is charact erized by
yS = sops - 20 wl
y0 = 60 - 12 5p0 +2M
where
and
ys is quantity of wheat suppl ied in millions of bushels per year p5 is the price recei ved by suppliers in dollars per bushel y0 is quantity demanded (in millions of bushels) by consumers p0 i s the price paid by consumers also in dollars per bushel w1 is the price of a composite input to wheat product ion M is consumer income in t housands of dollars
For purposes of this analysis consumer income i s M = 45 and the composite input price is w1 = 5
(a) What are t he equilibrium f ree-mar ket price and quant it y of wheat
(b) Is t he supply of wheat at this point price-elastic or price-inelast i c Ex plain your reasomng
(c) W h at is t he t otal net economic sur plus f rom production of wheat and how middotis it distr ibuted bet ween suppliers and demanders
(d) A var iet y of schemes are used in agricult ural markets to enhance producer welf are C onsider a pr ice support of $6 per bushel Explain t he effects of such a policy on the dist ri bution of welfare and economic efficiency Support your answer with quant itat ive est imates of t hese effect s
jgtrice
3 Suppose a monopoly fum faces the following linear demand curve P = a- bQ where P is price and Q is quantity produced and sold The monopolists cost function is
(1) c(Q) = cQ for Q Q c(Q) qo for Q gt Q
where Q is a capacity constraint
a Solve mathematically for the monopolists profit maxumzmg and output as functions of the parameters a b and c Show your work Also illustrate yourresults graphically labeling carefully all relevant information
b Suppose the function P(Q) from part (a) is the demand in the domestic market and that the monopolist can also sell unlimited quantities into the export market at price P E gt c Assuming that quantities exported cannot be resold into the domestic market show both mathematically and on your graph from part (a) how much the firm will sell into the domestic market and into the export market
c Consider now a potential entrant into this market For simplicity ignore the existence of the export market To be able to produce in this market an entrant must incur a one time set up or entry cost of E gt 0 It can then produce according to the cost function in (1) Describe in words andor diagrams how you would model this entrants decision problem You should consider the role of the incumbent monopolist in framing your answer
4 Consider a food processing firm that produces processed product Q from a raw agricultural product Rand a composite marketing input L which we will call labor
a Write a production function and draw some rough sketches of isoquants for each of the following possible production scenarios
1 There is no substitution between R and L in producing Q
ii The production relationship is Cobb-Douglas
b Suppose the true production relationship is Cobb-Douglas and that this firm produces Q exclusively for a large retail grocery chain It has a contract for 1999 to deliver Q units of product to the retailer Assume the firm is a perfect competitor in the markets for its inputs Set up the firms optimization problem given this information and derive the first-order conditions for the problem Defme any notation or terminology you need to introduce Also illustrate the optimization problem with a graph
c Discuss the firms demand function for R given the conditions of the problem in part (b) Specifically what variables determine how much R the firm will utilize
d Re-express the problem in part (b) and take first-order conditions assuming now that the firm is the only buyer of the farm input R which is produced according to the following inverse supply function
w = a + 3R a3 gt 0
OJ 04
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Example
Pr (0 z 196) = 04750
Pr (z 196) = 05 - 04750 = 0025
TABLE 01 Percentage points of lhe t distribution
Example
Pr (r gt 2086) = Oo25
Pr (r gt 1725) = 005 for df = 20
Pr(ltl gt 1725) = 010 0 I 715
01$ 010 middot-00
001 0 00010015010 002 0010 0002ozo
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22 279925IXH6 6965UK16 4 lP 00 01 01 1 0765 I 6X 251 lIK2 4541 51141 102 JJ
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49111 49H9 49M9 4990 _ 1 1810674 164S 1960 1 126 1576 49H I 49K2 4982 498)
49X7 49K7 49117 49111
N This tahlc gives the area in the riahtmiddothand tail or the distribution tic bull t 0) But since the normal di5tribumiddot Nolt 111e smaller probability shown at the head of each column is the area in one tail the 1in is symmetrical about r- Othe area in the ldtmiddothlnd tail is the same u the area in the corresponding right-hand larser probability is the area in both tails bullbullil rltgtr eumplc PI- 196 S z S 01-04750 TMrdore -196 S t S 196)-2(047S0)-09S Sourr From E S Pearwn and H 0 Hartley eds bull Blomtuila TabftJot Slulislidans vol I
ld ed table 12 Cbullmbridgc University Prns New York 1966 Reproduced by perminion of the middot
editors and trustees of Biomeria
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TARLE Ol l lpper percentage poinls or the F distribution (continut-d)
I I
STATISTICAl TA8LfS 681
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100 N
I 16 111 I 12 I y I 2ltI 10 1 10 111 1 1 1 1 111 I Y 1 41 1 4 1 1-10 I 19 1 19 UHI middotW 1171411 147 145
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Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
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a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
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b) Find the monopolists profit maximizing level of output say y = ym(abks) as well as its profit maximizing price say p = pm(abk s) How do you know you have found the profit maximizing solution
c) Will an increase in the subsidy achieve its desired goal of increasing the monopolists output and lowering its price Show your work
d) Provide an economic interpretation of the above comparative statics e) How large would the subsidy have to be in order to get the monopolist to produce the
Pareto efficient level of output and charge the Pareto efficient price
6) The following questions pertain to a production setting characterized by the usual neoclassical production function of the form Q = f(xlx2x ) where Q is output and (x1x2x ) are inputs
a) Define compare and contrast the following terms from production economics i) Decreasing returns to a variable input and ii) decreasing returns to scale (Decreasing returns to scale is also called
diseconomies of scale) b) What economic forces cause decreasing returns to a variable input and what economic
forces might cause decreasing returns to scale c) Given the production function as specified show mathematically how you would measure
the returns to the variable input x1 and the returns to scale d) Suppose you were studying the question of whether small farms could compete effectively
against large farms in the market for the product Q Would either or both of the concepts of returns to a variable input and returns to scale be valuable in your analysis Explain
7) The following are demand and supply curves that apply to a competitive industry
Q0 = 250- SOP Qs = ()P
a) Derive the market equilibrium price and quantity b) Is the equilibrium you derived in (a) a short-run equilibrium a long-run equilibrium or
could it be both Explain your answer in detail demonstrating your knowledge of behavior in competitive markets and the distinction in these markets between the short and long run
c) Politicians are floating a proposal to place a per unit tax of $ 100 on this commodity Producers in lobbying against the tax argue that they will have no choice but to pass the
full tax on to consumers and therefore the tax if enacted will only harm consumers Evaluate the producers argument concerning the tax Use either algebra (preferred tool) or
4
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a)
a graph to support your argument If you use a graph label it carefully using the demand and supply equations Q0 = 250- 50P and Qs = ( )P
8) The following graph depicts a consumers Marshallian demand curve for the commodity X The market price of X is given as
p
Demand Curve
Given identify the consumers surplus on the graph and give an intuitive definition of consumers surplus
b) Assuming X is a normal good for this consumer draw the consumers Hicksian demand curve given price
c) Explain the difference between Hicksian and Marshallian demand curves
d) Suppose price increases to P_ Use Hicksian demands to identify the consumers compensating variation and equivalent variation due to the price change Label carefully
5
University of California Davis Department of Agricultural and Resource Economics
MS Exam
July 2 1999
Directions You have three hours to write this exam Please write legibly Answers that cannot be read are regarded as incorrect answers This exam consists of two parts In part I quantitative methods you must answer question 1 Then choose to answer either question 2 or question 3 In part II microeconomics answer three of the four questions You should thus answer five questions in total two from part I and three from part II Do not under any circumstances answer more than five questions
Part I Answer 1 estion I and either Question 2 01 luestion 3
l You are constructing a si mple model of cotton su pply for a short-term project you are working on and h ave formulated the following conceptual model
where y5 i s bales of cotton ps i s the price paid to growers w1 is the labor wage rate w2 i s the price of water WJ i s pesticide price and DO is degree-days a measure of the growi ng conditions during the season
(a) A nother variable you might include in thi s speci ficati on is annual rainfall (R) whose coefficient i f included in the model would be a6 While you are sure th at all th e other va ri ables in the model are part of the correct speci fication you are not sure about annual rainfall As an applied econometrician explai n the tradeoffs made to the properties of you r estimates i f you include or exclude thi s vari able from the specification
(b) One must always be concerned with measurement error i nempirical models Explain th e effects on the properties of your estimates of (a) measurement error in th e dependent variable vs (b) measurement error in one or more explanatory vari ables
(c) You esti mate th e model wi th R included and the results are
yS = -6354 + 8727ps - 0 579lw1 - 7907w2 - 09045w3 + 769000 + 0045R ( -0 79) (2 36) ( -177) (-234) ( -190) (157) (045)
R2 = 0819 n =56
where th e Students-t statistics are gi ven in parentheses and the following sums of squared errors resu lt
Total SSY=660 9 Model SSR=5412 Errors SSE= l l 97
In answering the following part s justify your answers with quantitati ve test results using th e attached tables Quali tative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 0 5 level
(ii) Is the overall model significant Th at i s i s the hypothesis th at a1 = a2 = a3 = a4 = as = a6 = 0 rejected
(iii) The correlation between the price of water (w2) and the price of pesticides (w3) i s zero Do these two vari ables have the same effect on cotton supply i e are their coeffi cients equ al
2 Suppose you have the following econometric model
where y = yieldacre x1 = wateracre x2 = (wateracre)2 x3 = fertilizeracre
(a) For each of the following questions set up the appropriate hypothesis test and provide a brief explanation
(i) Does water have any influence on yieldacre
(ii) Assuming an unconstrained (interior) solution is wateracre = 300 significantly different from the amount that yields the maximum yield per unit water
(b) Suppose you work in a governmental agency and that your boss is not trained in economics or statistics and thus she does not know what e N(O all) means Explain-
in words the meaning and significance of this expression
3
a Given the regional farm modeling problem
Max subject to
px- cx Ax b
where x is an n x I vector of production activities p and c are vectors of prices and variable cost coefficients respectively b is an m x 1 vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Comment briefly on the connection between the dual problem and the Dual Complementary Slackness condition
(iii) If you had to calculate the derived demand for a single input b describe the type of function you would obtain from the problem
b Now use the information that the product prices p are determined according to the following demand relationships
p =a + Dx
Assume that the individual farmers act as perfect competitors
(i) Show how you would reformulate your problem and give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit )
(ii) Derive the dual to the problem in part b(i) showing your steps clearly
Part II Answer of the 4 questions in this Section _
I For some goods consumption is subject to more than the usual single constraint on money income time can also be a constrai nt Consider a consumer with utility f unction u(x) defined on goods x = (x1 xn) subject to th e two constraints T 2 tx and M 2 px where T is the total amount of time avai lable and M is th e total amount of income avai lable and t = (t1 tn) and p = (p o pn ) are the time and money prices per unit of consumption of each good
(a) Write th e primal utility max i mi zation problem for th is consumer What interpretation do you attach to th e sh adow values on each constraint
(b) F rom your answer to part (a) ex plain how the consumer s choices reveal her money value of time Develop an ex pression f or the marginal money value of time from th is pri mal problem
(c) Under what conditions would the problem revert to the standard consumer choice problem subject to a single money budget constraint
(d) Suppose th at the shadow value on the money budget constraint is zero W rite the dual expenditure mi nimization problem for thi s case Provide a brief interpretation in words of thi s dual ex penditure f unction Also provide an interpretation for the sh adow value of the constraint for th is problem
2 Assume t hat t he US wheat mar ket can be considered in isolation and is charact erized by
yS = sops - 20 wl
y0 = 60 - 12 5p0 +2M
where
and
ys is quantity of wheat suppl ied in millions of bushels per year p5 is the price recei ved by suppliers in dollars per bushel y0 is quantity demanded (in millions of bushels) by consumers p0 i s the price paid by consumers also in dollars per bushel w1 is the price of a composite input to wheat product ion M is consumer income in t housands of dollars
For purposes of this analysis consumer income i s M = 45 and the composite input price is w1 = 5
(a) What are t he equilibrium f ree-mar ket price and quant it y of wheat
(b) Is t he supply of wheat at this point price-elastic or price-inelast i c Ex plain your reasomng
(c) W h at is t he t otal net economic sur plus f rom production of wheat and how middotis it distr ibuted bet ween suppliers and demanders
(d) A var iet y of schemes are used in agricult ural markets to enhance producer welf are C onsider a pr ice support of $6 per bushel Explain t he effects of such a policy on the dist ri bution of welfare and economic efficiency Support your answer with quant itat ive est imates of t hese effect s
jgtrice
3 Suppose a monopoly fum faces the following linear demand curve P = a- bQ where P is price and Q is quantity produced and sold The monopolists cost function is
(1) c(Q) = cQ for Q Q c(Q) qo for Q gt Q
where Q is a capacity constraint
a Solve mathematically for the monopolists profit maxumzmg and output as functions of the parameters a b and c Show your work Also illustrate yourresults graphically labeling carefully all relevant information
b Suppose the function P(Q) from part (a) is the demand in the domestic market and that the monopolist can also sell unlimited quantities into the export market at price P E gt c Assuming that quantities exported cannot be resold into the domestic market show both mathematically and on your graph from part (a) how much the firm will sell into the domestic market and into the export market
c Consider now a potential entrant into this market For simplicity ignore the existence of the export market To be able to produce in this market an entrant must incur a one time set up or entry cost of E gt 0 It can then produce according to the cost function in (1) Describe in words andor diagrams how you would model this entrants decision problem You should consider the role of the incumbent monopolist in framing your answer
4 Consider a food processing firm that produces processed product Q from a raw agricultural product Rand a composite marketing input L which we will call labor
a Write a production function and draw some rough sketches of isoquants for each of the following possible production scenarios
1 There is no substitution between R and L in producing Q
ii The production relationship is Cobb-Douglas
b Suppose the true production relationship is Cobb-Douglas and that this firm produces Q exclusively for a large retail grocery chain It has a contract for 1999 to deliver Q units of product to the retailer Assume the firm is a perfect competitor in the markets for its inputs Set up the firms optimization problem given this information and derive the first-order conditions for the problem Defme any notation or terminology you need to introduce Also illustrate the optimization problem with a graph
c Discuss the firms demand function for R given the conditions of the problem in part (b) Specifically what variables determine how much R the firm will utilize
d Re-express the problem in part (b) and take first-order conditions assuming now that the firm is the only buyer of the farm input R which is produced according to the following inverse supply function
w = a + 3R a3 gt 0
OJ 04
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Example
Pr (0 z 196) = 04750
Pr (z 196) = 05 - 04750 = 0025
TABLE 01 Percentage points of lhe t distribution
Example
Pr (r gt 2086) = Oo25
Pr (r gt 1725) = 005 for df = 20
Pr(ltl gt 1725) = 010 0 I 715
01$ 010 middot-00
001 0 00010015010 002 0010 0002ozo
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22 279925IXH6 6965UK16 4 lP 00 01 01 1 0765 I 6X 251 lIK2 4541 51141 102 JJ
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1591 161M 1664
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ll40 11111 1910
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l Jfi) 1090
49864914 4914 4985 498S 4986 120 0677 11M9 21671651 19KO 2J58
49111 49H9 49M9 4990 _ 1 1810674 164S 1960 1 126 1576 49H I 49K2 4982 498)
49X7 49K7 49117 49111
N This tahlc gives the area in the riahtmiddothand tail or the distribution tic bull t 0) But since the normal di5tribumiddot Nolt 111e smaller probability shown at the head of each column is the area in one tail the 1in is symmetrical about r- Othe area in the ldtmiddothlnd tail is the same u the area in the corresponding right-hand larser probability is the area in both tails bullbullil rltgtr eumplc PI- 196 S z S 01-04750 TMrdore -196 S t S 196)-2(047S0)-09S Sourr From E S Pearwn and H 0 Hartley eds bull Blomtuila TabftJot Slulislidans vol I
ld ed table 12 Cbullmbridgc University Prns New York 1966 Reproduced by perminion of the middot
editors and trustees of Biomeria
bull bull bull bull 10 II 30 110 100 500
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4 20
116
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192
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244
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199
245
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161
190
1 4 1
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137
195
2)7
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190
229
1 59
245
U 7
I 91
1 15
96
142
194
1 94
l2
IJ1
117
174
104
1 0
190
U4
109
284
Ill 2 71 2JJ
1 77
194 175
i-1
174
149
157
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1 94 1 gt9
194
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174 159
145 1 4
195
119
179 147
682 AIPFNHIX ll
TARLE Ol l lpper percentage poinls or the F distribution (continut-d)
I I
STATISTICAl TA8LfS 681
df for dl lor llllmtntor N
100 N
I 16 111 I 12 I y I 2ltI 10 1 10 111 1 1 1 1 111 I Y 1 41 1 4 1 1-10 I 19 1 19 UHI middotW 1171411 147 145
I 176 IJl 170 IUI 167 165 I M 161 1 bull1 101 1 97 191 1 90 11111 1116Hi 4 l)
2 15 212
Ill 1117 201 I I I M4 I IIII I I II
2 911 1 II 1 1 1gt 67 251 11 an z llbull 2 1J ll I 26 1 1 11 1 2
191 lOS 2Kl 1 2 55 146 2 -m 24 2 lU 226 2 2l
US1 i9 l-IS S72 4112 4 1 1 199
I 1 I J I 110 I Y 12Y I 211 I M I 7 1 7 I 6 1 15 1 11 U9 UK IJK 1middot17 U6
17M I 71 1 101 167 1 162 161 I I S1 n6 I 51 IU1115 I ll 1 4 1 1401 19 1 47 146 1 44
I 98
2111 191 1 1111 1 19 210 1 042B 9 I H6 1 gt4 1 MO 119 1 1 1 1 1) 1 1 1 211 2111 2JO 22S 21 2111JOI 2SI 242 2 16 11114 16 140
2M9 2 66 UK 2 40 111 1 1 1 2-1 2 1 Ul167 middot 1 16 126 117 109 ill2 H I 472 4 12 01
114 I 12 111 1 10 129 I 211 UK 16 I 26 126 I 25 I 2 I 42 1 4 1 19 us I H 17 l l6 2 1 111 146 us 1 16 I 1611 165 1 61 I 5Y UK us IS-1 I S1 I S l I t HIIll 11111 186 2 II 1 2 1 11 2 1111 11111 Y I 151 107 199 190 1-JIS 1112 1111) 1 16 1 1S I 71 I 7 1 211 222 2111 215Ull OS 219U7 198 1 14 2 t7 2111 2511 lSO 242 1 16 2 225 22J 219 216 2 IJ UH l09 )02 196HI 129SSl 41gt4 41401
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0 1 407 I 75 )51 U6 123 312 101
Ul l ltJ 129 Ull 127 1 26 126 IH 1 2-1 1 2-1 12J t n l iM I 45 1 44 1 41 U9 Ull IJ6 us IB
172 167 1 64 ltll I H 15-1 Ul I SU I 1 lt7 1 6 10 10IIIS 182 179 171I 10 11111 1 22K 114 20S 1911 19) 188
2111 1 91 I H4 179 1 16 1 14 110 I M 1 IM I 62 O)1 41 lB 227 221 216 213OS l 2 291 169
270 2-17 210 2 2 1 IJ I I 207 2 Ul ] 0 1 147 J0 1 1 7 )07 298 2910 1 1 56 5 19 451 40
110 I 126 125 124 Ut 12 Ul 121 119 1 19 25IJ6 us 1 14 IJJ Ul U l140 11914425
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1R4 261 2 14 225 218 2 12 1 08 252 217 229 220 2 1 1 1116 202 1 94 I 92 I IM1 01 664 II 181 lSI 129 1 12 199 2119 280 27)511S
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111 168 1 66117IK7 1821 19 2 19 211S 104 111-1 175 110 165 159 156 15] 1 4H 147 144 I-II 1 9 0)
125 2 17 210 104 199 195 1921 1 5 1 16 2 S 110 212 20 194 11111 1 04 17 I 71 I 61 160 til2S6 1SO 1 ll 295 lKl 272 261 165 ) 14
124 1221 16 129 128 127 1 2 1 1 19 1 111 1 17 116 1 14 1 1 1 1 1 1 I l l 1 10
IJJ IJI 1 10 140 119 141 117 132 111 126 1 24 1 2 1 119 10 120I us1 60I gtI 182 111 168 16S 1622 1)10 us
166 161 155 150 146 14 117 115 1 12 1211 125 I Hl 217 109 101 191 IKOl 219 201 I 176 1 101 156 UJ I 1 41 I IX
6 liS I 1701 4 J9 195 14H 296 2 79 266 256 247 240 middot 234
Ul 1 2 1 l iD 1 1 11 116 114 1 12 I l l 1 10 1119 I 0 25 111 IJ9 1111 I J6 I a I J2 I) I 129 128 127 126 US
152 146 1 41 1 111 1 1-1 I I 1211 I 24 122 uu 1 1 7 Ia Ill 2 I I 197 IKII IKO 1 75 170 166 161 160 157
172 162 IS7 IS2 146 141 119 I J2 129 U6 1 22 119 m198 19J I 88 I 114 180OS H9 1 04 265 242 226 2 14 2lt16
01 676 411 lllll l41 J I I 289 271 I 91 1119 I 79 169 I 61 1511 148 144 119 1 11 1 11 01
260 2SO 241 2 14 227
122 1 10 1 16 11-1 1 1 1 1 12 1119 Uli I 07 l tiO 124 124I 12 I J9 IJ7 US I J 111 129 1211 127 125
I II 142 118 114 uo 126 124 1 1 11 1 1 1 1 1 1 I OK Hill 10
i lS 25
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Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
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Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
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Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
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Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
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d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
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University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
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a)
a graph to support your argument If you use a graph label it carefully using the demand and supply equations Q0 = 250- 50P and Qs = ( )P
8) The following graph depicts a consumers Marshallian demand curve for the commodity X The market price of X is given as
p
Demand Curve
Given identify the consumers surplus on the graph and give an intuitive definition of consumers surplus
b) Assuming X is a normal good for this consumer draw the consumers Hicksian demand curve given price
c) Explain the difference between Hicksian and Marshallian demand curves
d) Suppose price increases to P_ Use Hicksian demands to identify the consumers compensating variation and equivalent variation due to the price change Label carefully
5
University of California Davis Department of Agricultural and Resource Economics
MS Exam
July 2 1999
Directions You have three hours to write this exam Please write legibly Answers that cannot be read are regarded as incorrect answers This exam consists of two parts In part I quantitative methods you must answer question 1 Then choose to answer either question 2 or question 3 In part II microeconomics answer three of the four questions You should thus answer five questions in total two from part I and three from part II Do not under any circumstances answer more than five questions
Part I Answer 1 estion I and either Question 2 01 luestion 3
l You are constructing a si mple model of cotton su pply for a short-term project you are working on and h ave formulated the following conceptual model
where y5 i s bales of cotton ps i s the price paid to growers w1 is the labor wage rate w2 i s the price of water WJ i s pesticide price and DO is degree-days a measure of the growi ng conditions during the season
(a) A nother variable you might include in thi s speci ficati on is annual rainfall (R) whose coefficient i f included in the model would be a6 While you are sure th at all th e other va ri ables in the model are part of the correct speci fication you are not sure about annual rainfall As an applied econometrician explai n the tradeoffs made to the properties of you r estimates i f you include or exclude thi s vari able from the specification
(b) One must always be concerned with measurement error i nempirical models Explain th e effects on the properties of your estimates of (a) measurement error in th e dependent variable vs (b) measurement error in one or more explanatory vari ables
(c) You esti mate th e model wi th R included and the results are
yS = -6354 + 8727ps - 0 579lw1 - 7907w2 - 09045w3 + 769000 + 0045R ( -0 79) (2 36) ( -177) (-234) ( -190) (157) (045)
R2 = 0819 n =56
where th e Students-t statistics are gi ven in parentheses and the following sums of squared errors resu lt
Total SSY=660 9 Model SSR=5412 Errors SSE= l l 97
In answering the following part s justify your answers with quantitati ve test results using th e attached tables Quali tative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 0 5 level
(ii) Is the overall model significant Th at i s i s the hypothesis th at a1 = a2 = a3 = a4 = as = a6 = 0 rejected
(iii) The correlation between the price of water (w2) and the price of pesticides (w3) i s zero Do these two vari ables have the same effect on cotton supply i e are their coeffi cients equ al
2 Suppose you have the following econometric model
where y = yieldacre x1 = wateracre x2 = (wateracre)2 x3 = fertilizeracre
(a) For each of the following questions set up the appropriate hypothesis test and provide a brief explanation
(i) Does water have any influence on yieldacre
(ii) Assuming an unconstrained (interior) solution is wateracre = 300 significantly different from the amount that yields the maximum yield per unit water
(b) Suppose you work in a governmental agency and that your boss is not trained in economics or statistics and thus she does not know what e N(O all) means Explain-
in words the meaning and significance of this expression
3
a Given the regional farm modeling problem
Max subject to
px- cx Ax b
where x is an n x I vector of production activities p and c are vectors of prices and variable cost coefficients respectively b is an m x 1 vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Comment briefly on the connection between the dual problem and the Dual Complementary Slackness condition
(iii) If you had to calculate the derived demand for a single input b describe the type of function you would obtain from the problem
b Now use the information that the product prices p are determined according to the following demand relationships
p =a + Dx
Assume that the individual farmers act as perfect competitors
(i) Show how you would reformulate your problem and give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit )
(ii) Derive the dual to the problem in part b(i) showing your steps clearly
Part II Answer of the 4 questions in this Section _
I For some goods consumption is subject to more than the usual single constraint on money income time can also be a constrai nt Consider a consumer with utility f unction u(x) defined on goods x = (x1 xn) subject to th e two constraints T 2 tx and M 2 px where T is the total amount of time avai lable and M is th e total amount of income avai lable and t = (t1 tn) and p = (p o pn ) are the time and money prices per unit of consumption of each good
(a) Write th e primal utility max i mi zation problem for th is consumer What interpretation do you attach to th e sh adow values on each constraint
(b) F rom your answer to part (a) ex plain how the consumer s choices reveal her money value of time Develop an ex pression f or the marginal money value of time from th is pri mal problem
(c) Under what conditions would the problem revert to the standard consumer choice problem subject to a single money budget constraint
(d) Suppose th at the shadow value on the money budget constraint is zero W rite the dual expenditure mi nimization problem for thi s case Provide a brief interpretation in words of thi s dual ex penditure f unction Also provide an interpretation for the sh adow value of the constraint for th is problem
2 Assume t hat t he US wheat mar ket can be considered in isolation and is charact erized by
yS = sops - 20 wl
y0 = 60 - 12 5p0 +2M
where
and
ys is quantity of wheat suppl ied in millions of bushels per year p5 is the price recei ved by suppliers in dollars per bushel y0 is quantity demanded (in millions of bushels) by consumers p0 i s the price paid by consumers also in dollars per bushel w1 is the price of a composite input to wheat product ion M is consumer income in t housands of dollars
For purposes of this analysis consumer income i s M = 45 and the composite input price is w1 = 5
(a) What are t he equilibrium f ree-mar ket price and quant it y of wheat
(b) Is t he supply of wheat at this point price-elastic or price-inelast i c Ex plain your reasomng
(c) W h at is t he t otal net economic sur plus f rom production of wheat and how middotis it distr ibuted bet ween suppliers and demanders
(d) A var iet y of schemes are used in agricult ural markets to enhance producer welf are C onsider a pr ice support of $6 per bushel Explain t he effects of such a policy on the dist ri bution of welfare and economic efficiency Support your answer with quant itat ive est imates of t hese effect s
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3 Suppose a monopoly fum faces the following linear demand curve P = a- bQ where P is price and Q is quantity produced and sold The monopolists cost function is
(1) c(Q) = cQ for Q Q c(Q) qo for Q gt Q
where Q is a capacity constraint
a Solve mathematically for the monopolists profit maxumzmg and output as functions of the parameters a b and c Show your work Also illustrate yourresults graphically labeling carefully all relevant information
b Suppose the function P(Q) from part (a) is the demand in the domestic market and that the monopolist can also sell unlimited quantities into the export market at price P E gt c Assuming that quantities exported cannot be resold into the domestic market show both mathematically and on your graph from part (a) how much the firm will sell into the domestic market and into the export market
c Consider now a potential entrant into this market For simplicity ignore the existence of the export market To be able to produce in this market an entrant must incur a one time set up or entry cost of E gt 0 It can then produce according to the cost function in (1) Describe in words andor diagrams how you would model this entrants decision problem You should consider the role of the incumbent monopolist in framing your answer
4 Consider a food processing firm that produces processed product Q from a raw agricultural product Rand a composite marketing input L which we will call labor
a Write a production function and draw some rough sketches of isoquants for each of the following possible production scenarios
1 There is no substitution between R and L in producing Q
ii The production relationship is Cobb-Douglas
b Suppose the true production relationship is Cobb-Douglas and that this firm produces Q exclusively for a large retail grocery chain It has a contract for 1999 to deliver Q units of product to the retailer Assume the firm is a perfect competitor in the markets for its inputs Set up the firms optimization problem given this information and derive the first-order conditions for the problem Defme any notation or terminology you need to introduce Also illustrate the optimization problem with a graph
c Discuss the firms demand function for R given the conditions of the problem in part (b) Specifically what variables determine how much R the firm will utilize
d Re-express the problem in part (b) and take first-order conditions assuming now that the firm is the only buyer of the farm input R which is produced according to the following inverse supply function
w = a + 3R a3 gt 0
OJ 04
J ---------
Amiddot __ __
09
0QgtI
4
2257
Jl59
412
14JK
446 -1564
4649
41161
29
0
495
4974
04711
164 9-9
267 2967
4170
llll
99
09X7
76
714
H
middotmiddot 74(
4791
4978
49119
2454 2764
1051
1 115
1441
IIIOK
2157
1577
U74
1517
5
4545
46JJ
4767
4974
4990
middot-
92()
J 7J7 717
2447 ll 4
middot55
4 144
19
24
1699
677
676 SlATISliCI1 tA9LFS
lA Rlf lgtl Areas under the standardized normal distribution
Example
Pr (0 z 196) = 04750
Pr (z 196) = 05 - 04750 = 0025
TABLE 01 Percentage points of lhe t distribution
Example
Pr (r gt 2086) = Oo25
Pr (r gt 1725) = 005 for df = 20
Pr(ltl gt 1725) = 010 0 I 715
01$ 010 middot-00
001 0 00010015010 002 0010 0002ozo
J 07H 127116I 11216 14 61657 IIUI1nm
22 279925IXH6 6965UK16 4 lP 00 01 01 1 0765 I 6X 251 lIK2 4541 51141 102 JJ
4 074 I 15J 21J1 176 461 moo 00411no
Ol 0middot1111
(Xl)lt) 11119 059UIO 0161 t tiiCW 029 0279 1650675 0751 s 1476 2015 511910727 25710596 0616 4010517 115 7
071 II 1064 IIOJ 1141 52111106 bull 1440 1941 l7H7 0791 UKl 0171 091ll 09411
149914KU 1517
17tMl 1406 299K J7XS7 0 711 1415 lH95 2165116111179 1217 1155 129l
1591 161M 1664
1915 1950 19115 W19
10611144 1179 K u 706 lW7 11601772 451111196
212 071) 421172054 219() 122420XX UHl 111 2262 21i21 1251)-9
0700 J72 1112 222K 2764 11692422
1021 214 25492-IK6 102157 2X92291
2794 2121 21152 4025II 0697 1361 1796 22111 2711lt lit
2611 27041gt1 25XU ll7K 1106 1111 190 0695 1156 17H1 21 79 26111 lOSS
ll40 11111 1910
2650 2M9 J65 l119 IJ 0694 1350 1771 2160 1lt5211112311lt6 21 lJH 164
145 1761 21J5 2614 1621 14 069214115 55-1 2977 7117 461 150111 0 1411
l711I 2602 2 94717011 770 790 Kill lX0 15 0691 1141 175 729 21 lfgt65 16116 JHHH
I I
11
1641
1144012
J997 2921907 JOISJ9-U 96 WKO 1 6 0690 bull IJJ7 1 746 2120 2511 l6M6 925 JK(fl 0619 11114011 4147 4162 4177
4194066 41H l6462567 2gt19H411 17 1740 21W40994(19 406 068H UJO 174 210142164192
I
4279 4292 6102X7K4265 25524251 -11117 4222 4441 068H U211 1 729 209J 253944294 112 494 J406 44111 579lK61LJ5 4157
20 0687 1325 I 715 21186 252K 14545115 4515 4515 455 5524474 44114 4495lh 4J 2 0616 121 1711 1010 251 H l517211 I 4571 45112 4591 4599 4616 4625
4554 505J67H A6K6 4691 4699 4706 22 06K6 I Jl l 1717 2014 l5HK 1XI94656 4664 4671I K 4641 41151K074756 4761 17144744 4750 2J 06KS 1319 250020694716 412 47 111 9 470 4719
41101 41112 4M17 14924791 AKOJ20 06K5 IJ UI 2797 1 4671711 20644772 47711 4710 47KK
1
I 4K21 1 4504Kl0 4104 41UH 41141 41146 4150 4H54 4H57 2l 06114 1316 1708 2060 24115 271741116 J4JS1706 20 56 2479 27794K6H 4171 4X75 4H7H 4HIH 4Hol4 4HH7 middot 4K90 26 0614 IJIS41164
J906 4911 06M4 14212471 27714913 4916 170 4909 27 L14 10524196 489R 4901 49044191 06114925 4927 4929 4931 492 4914 4916 140K2467 27612M U13 1701 204849111 4920 4922
4941 4941 4945 4946 494K 4949 4951 4951 IJII JJ9629 068) 2462 2756204S49111 494025
UJO 1697 2042 2457 17504QSS 4964 lJK S4961 JO 06KJ4960 4961 4961
496 J951J4956 49576
7 10 06KI 10749 1 6K4497 241J i7044970 4971 49724967 496K 1021J966 60 0679 129611 4981
4979 4979 49SO 16714977 2 190 26604976 4977Igrave 12122000 4975
l Jfi) 1090
49864914 4914 4985 498S 4986 120 0677 11M9 21671651 19KO 2J58
49111 49H9 49M9 4990 _ 1 1810674 164S 1960 1 126 1576 49H I 49K2 4982 498)
49X7 49K7 49117 49111
N This tahlc gives the area in the riahtmiddothand tail or the distribution tic bull t 0) But since the normal di5tribumiddot Nolt 111e smaller probability shown at the head of each column is the area in one tail the 1in is symmetrical about r- Othe area in the ldtmiddothlnd tail is the same u the area in the corresponding right-hand larser probability is the area in both tails bullbullil rltgtr eumplc PI- 196 S z S 01-04750 TMrdore -196 S t S 196)-2(047S0)-09S Sourr From E S Pearwn and H 0 Hartley eds bull Blomtuila TabftJot Slulislidans vol I
ld ed table 12 Cbullmbridgc University Prns New York 1966 Reproduced by perminion of the middot
editors and trustees of Biomeria
bull bull bull bull 10 II 30 110 100 500
(
I
n
I)
4]1
7 71
4 20
116
1 H4
4 1
1 111
4 1
1
U4
1 75
192
1 gt6
44
2 54
244
)07
142middot
2 2
41J
144
U1
199
245
l-14
2 1
14)
161
190
1 4 1
25J
)70
137
195
2)7
U 5
190
229
1 59
245
U 7
I 91
1 15
96
142
194
1 94
l2
IJ1
117
174
104
1 0
190
U4
109
284
Ill 2 71 2JJ
1 77
194 175
i-1
174
149
157
I y I 7
1 94 1 gt9
194
1 9)
174 159
145 1 4
195
119
179 147
682 AIPFNHIX ll
TARLE Ol l lpper percentage poinls or the F distribution (continut-d)
I I
STATISTICAl TA8LfS 681
df for dl lor llllmtntor N
100 N
I 16 111 I 12 I y I 2ltI 10 1 10 111 1 1 1 1 111 I Y 1 41 1 4 1 1-10 I 19 1 19 UHI middotW 1171411 147 145
I 176 IJl 170 IUI 167 165 I M 161 1 bull1 101 1 97 191 1 90 11111 1116Hi 4 l)
2 15 212
Ill 1117 201 I I I M4 I IIII I I II
2 911 1 II 1 1 1gt 67 251 11 an z llbull 2 1J ll I 26 1 1 11 1 2
191 lOS 2Kl 1 2 55 146 2 -m 24 2 lU 226 2 2l
US1 i9 l-IS S72 4112 4 1 1 199
I 1 I J I 110 I Y 12Y I 211 I M I 7 1 7 I 6 1 15 1 11 U9 UK IJK 1middot17 U6
17M I 71 1 101 167 1 162 161 I I S1 n6 I 51 IU1115 I ll 1 4 1 1401 19 1 47 146 1 44
I 98
2111 191 1 1111 1 19 210 1 042B 9 I H6 1 gt4 1 MO 119 1 1 1 1 1) 1 1 1 211 2111 2JO 22S 21 2111JOI 2SI 242 2 16 11114 16 140
2M9 2 66 UK 2 40 111 1 1 1 2-1 2 1 Ul167 middot 1 16 126 117 109 ill2 H I 472 4 12 01
114 I 12 111 1 10 129 I 211 UK 16 I 26 126 I 25 I 2 I 42 1 4 1 19 us I H 17 l l6 2 1 111 146 us 1 16 I 1611 165 1 61 I 5Y UK us IS-1 I S1 I S l I t HIIll 11111 186 2 II 1 2 1 11 2 1111 11111 Y I 151 107 199 190 1-JIS 1112 1111) 1 16 1 1S I 71 I 7 1 211 222 2111 215Ull OS 219U7 198 1 14 2 t7 2111 2511 lSO 242 1 16 2 225 22J 219 216 2 IJ UH l09 )02 196HI 129SSl 41gt4 41401
IJJ I I 1 10 129 1 18 117 127 126 1 1gt 125 124 1 24 136 usI 41 141 140 119 UK1 111 146 145 1 69 1 66 16 159 I S1 156 I 51 152 I SO 149 1 19 Ill 1114 1 812 11 1 1 229 216 1116 190 187 100
H14 196 1 9 1 1 81 1 112 119 1 71 I 71 1 11 1 69 167 165 0)219 2 15 1 1 1 JN 295 1 1 1 2242)6 229 V5 1 1 144 IS 210 2 26 1 19 1 1 1 2 1 1 109 U6 U l
0 1 407 I 75 )51 U6 123 312 101
Ul l ltJ 129 Ull 127 1 26 126 IH 1 2-1 1 2-1 12J t n l iM I 45 1 44 1 41 U9 Ull IJ6 us IB
172 167 1 64 ltll I H 15-1 Ul I SU I 1 lt7 1 6 10 10IIIS 182 179 171I 10 11111 1 22K 114 20S 1911 19) 188
2111 1 91 I H4 179 1 16 1 14 110 I M 1 IM I 62 O)1 41 lB 227 221 216 213OS l 2 291 169
270 2-17 210 2 2 1 IJ I I 207 2 Ul ] 0 1 147 J0 1 1 7 )07 298 2910 1 1 56 5 19 451 40
110 I 126 125 124 Ut 12 Ul 121 119 1 19 25IJ6 us 1 14 IJJ Ul U l140 11914425
1117 1113 179 1 66 161 157 1 )4 I l l 14K 141 142 1-11 119 I 111
176 173 1 71IllI 21)9 1 00 192 IM4 179 1611 1 64 I us I 5J U l 100I 21Oi
01
1R4 261 2 14 225 218 2 12 1 08 252 217 229 220 2 1 1 1116 202 1 94 I 92 I IM1 01 664 II 181 lSI 129 1 12 199 2119 280 27)511S
127 125 1 14 122 121 1 11l 119 1 1 7 1 1 1 1 16 1 1 5 1 1 5 I J J 112 I l l IJO 119 119us 1 41 1 4 1 UK 160 154 Ul 1411 144 141 140 116 I 15 I J I ll I 29 10 I
111 168 1 66117IK7 1821 19 2 19 211S 104 111-1 175 110 165 159 156 15] 1 4H 147 144 I-II 1 9 0)
125 2 17 210 104 199 195 1921 1 5 1 16 2 S 110 212 20 194 11111 1 04 17 I 71 I 61 160 til2S6 1SO 1 ll 295 lKl 272 261 165 ) 14
124 1221 16 129 128 127 1 2 1 1 19 1 111 1 17 116 1 14 1 1 1 1 1 1 I l l 1 10
IJJ IJI 1 10 140 119 141 117 132 111 126 1 24 1 2 1 119 10 120I us1 60I gtI 182 111 168 16S 1622 1)10 us
166 161 155 150 146 14 117 115 1 12 1211 125 I Hl 217 109 101 191 IKOl 219 201 I 176 1 101 156 UJ I 1 41 I IX
6 liS I 1701 4 J9 195 14H 296 2 79 266 256 247 240 middot 234
Ul 1 2 1 l iD 1 1 11 116 114 1 12 I l l 1 10 1119 I 0 25 111 IJ9 1111 I J6 I a I J2 I) I 129 128 127 126 US
152 146 1 41 1 111 1 1-1 I I 1211 I 24 122 uu 1 1 7 Ia Ill 2 I I 197 IKII IKO 1 75 170 166 161 160 157
172 162 IS7 IS2 146 141 119 I J2 129 U6 1 22 119 m198 19J I 88 I 114 180OS H9 1 04 265 242 226 2 14 2lt16
01 676 411 lllll l41 J I I 289 271 I 91 1119 I 79 169 I 61 1511 148 144 119 1 11 1 11 01
260 2SO 241 2 14 227
122 1 10 1 16 11-1 1 1 1 1 12 1119 Uli I 07 l tiO 124 124I 12 I J9 IJ7 US I J 111 129 1211 127 125
I II 142 118 114 uo 126 124 1 1 11 1 1 1 1 1 1 I OK Hill 10
i lS 25
IS72 JO 160 us172 167 16J10 2 71 2011 194 U2 1 14 1 12 1 1 1 1 1 1 I IIII 167 152 146 179l 11-1 HXI 260 2J7 221 210 201 1811 181liSmiddot
21KIll 6 6 4bl 3-JII J l2 l 02 2110 264 2 51 241 232 22S 1 10 U6 I ll I S I OtlIIIK104 U9 152 0 1
Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
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d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
University of California Davis Department of Agricultural and Resource Economics
MS Exam
July 2 1999
Directions You have three hours to write this exam Please write legibly Answers that cannot be read are regarded as incorrect answers This exam consists of two parts In part I quantitative methods you must answer question 1 Then choose to answer either question 2 or question 3 In part II microeconomics answer three of the four questions You should thus answer five questions in total two from part I and three from part II Do not under any circumstances answer more than five questions
Part I Answer 1 estion I and either Question 2 01 luestion 3
l You are constructing a si mple model of cotton su pply for a short-term project you are working on and h ave formulated the following conceptual model
where y5 i s bales of cotton ps i s the price paid to growers w1 is the labor wage rate w2 i s the price of water WJ i s pesticide price and DO is degree-days a measure of the growi ng conditions during the season
(a) A nother variable you might include in thi s speci ficati on is annual rainfall (R) whose coefficient i f included in the model would be a6 While you are sure th at all th e other va ri ables in the model are part of the correct speci fication you are not sure about annual rainfall As an applied econometrician explai n the tradeoffs made to the properties of you r estimates i f you include or exclude thi s vari able from the specification
(b) One must always be concerned with measurement error i nempirical models Explain th e effects on the properties of your estimates of (a) measurement error in th e dependent variable vs (b) measurement error in one or more explanatory vari ables
(c) You esti mate th e model wi th R included and the results are
yS = -6354 + 8727ps - 0 579lw1 - 7907w2 - 09045w3 + 769000 + 0045R ( -0 79) (2 36) ( -177) (-234) ( -190) (157) (045)
R2 = 0819 n =56
where th e Students-t statistics are gi ven in parentheses and the following sums of squared errors resu lt
Total SSY=660 9 Model SSR=5412 Errors SSE= l l 97
In answering the following part s justify your answers with quantitati ve test results using th e attached tables Quali tative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 0 5 level
(ii) Is the overall model significant Th at i s i s the hypothesis th at a1 = a2 = a3 = a4 = as = a6 = 0 rejected
(iii) The correlation between the price of water (w2) and the price of pesticides (w3) i s zero Do these two vari ables have the same effect on cotton supply i e are their coeffi cients equ al
2 Suppose you have the following econometric model
where y = yieldacre x1 = wateracre x2 = (wateracre)2 x3 = fertilizeracre
(a) For each of the following questions set up the appropriate hypothesis test and provide a brief explanation
(i) Does water have any influence on yieldacre
(ii) Assuming an unconstrained (interior) solution is wateracre = 300 significantly different from the amount that yields the maximum yield per unit water
(b) Suppose you work in a governmental agency and that your boss is not trained in economics or statistics and thus she does not know what e N(O all) means Explain-
in words the meaning and significance of this expression
3
a Given the regional farm modeling problem
Max subject to
px- cx Ax b
where x is an n x I vector of production activities p and c are vectors of prices and variable cost coefficients respectively b is an m x 1 vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Comment briefly on the connection between the dual problem and the Dual Complementary Slackness condition
(iii) If you had to calculate the derived demand for a single input b describe the type of function you would obtain from the problem
b Now use the information that the product prices p are determined according to the following demand relationships
p =a + Dx
Assume that the individual farmers act as perfect competitors
(i) Show how you would reformulate your problem and give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit )
(ii) Derive the dual to the problem in part b(i) showing your steps clearly
Part II Answer of the 4 questions in this Section _
I For some goods consumption is subject to more than the usual single constraint on money income time can also be a constrai nt Consider a consumer with utility f unction u(x) defined on goods x = (x1 xn) subject to th e two constraints T 2 tx and M 2 px where T is the total amount of time avai lable and M is th e total amount of income avai lable and t = (t1 tn) and p = (p o pn ) are the time and money prices per unit of consumption of each good
(a) Write th e primal utility max i mi zation problem for th is consumer What interpretation do you attach to th e sh adow values on each constraint
(b) F rom your answer to part (a) ex plain how the consumer s choices reveal her money value of time Develop an ex pression f or the marginal money value of time from th is pri mal problem
(c) Under what conditions would the problem revert to the standard consumer choice problem subject to a single money budget constraint
(d) Suppose th at the shadow value on the money budget constraint is zero W rite the dual expenditure mi nimization problem for thi s case Provide a brief interpretation in words of thi s dual ex penditure f unction Also provide an interpretation for the sh adow value of the constraint for th is problem
2 Assume t hat t he US wheat mar ket can be considered in isolation and is charact erized by
yS = sops - 20 wl
y0 = 60 - 12 5p0 +2M
where
and
ys is quantity of wheat suppl ied in millions of bushels per year p5 is the price recei ved by suppliers in dollars per bushel y0 is quantity demanded (in millions of bushels) by consumers p0 i s the price paid by consumers also in dollars per bushel w1 is the price of a composite input to wheat product ion M is consumer income in t housands of dollars
For purposes of this analysis consumer income i s M = 45 and the composite input price is w1 = 5
(a) What are t he equilibrium f ree-mar ket price and quant it y of wheat
(b) Is t he supply of wheat at this point price-elastic or price-inelast i c Ex plain your reasomng
(c) W h at is t he t otal net economic sur plus f rom production of wheat and how middotis it distr ibuted bet ween suppliers and demanders
(d) A var iet y of schemes are used in agricult ural markets to enhance producer welf are C onsider a pr ice support of $6 per bushel Explain t he effects of such a policy on the dist ri bution of welfare and economic efficiency Support your answer with quant itat ive est imates of t hese effect s
jgtrice
3 Suppose a monopoly fum faces the following linear demand curve P = a- bQ where P is price and Q is quantity produced and sold The monopolists cost function is
(1) c(Q) = cQ for Q Q c(Q) qo for Q gt Q
where Q is a capacity constraint
a Solve mathematically for the monopolists profit maxumzmg and output as functions of the parameters a b and c Show your work Also illustrate yourresults graphically labeling carefully all relevant information
b Suppose the function P(Q) from part (a) is the demand in the domestic market and that the monopolist can also sell unlimited quantities into the export market at price P E gt c Assuming that quantities exported cannot be resold into the domestic market show both mathematically and on your graph from part (a) how much the firm will sell into the domestic market and into the export market
c Consider now a potential entrant into this market For simplicity ignore the existence of the export market To be able to produce in this market an entrant must incur a one time set up or entry cost of E gt 0 It can then produce according to the cost function in (1) Describe in words andor diagrams how you would model this entrants decision problem You should consider the role of the incumbent monopolist in framing your answer
4 Consider a food processing firm that produces processed product Q from a raw agricultural product Rand a composite marketing input L which we will call labor
a Write a production function and draw some rough sketches of isoquants for each of the following possible production scenarios
1 There is no substitution between R and L in producing Q
ii The production relationship is Cobb-Douglas
b Suppose the true production relationship is Cobb-Douglas and that this firm produces Q exclusively for a large retail grocery chain It has a contract for 1999 to deliver Q units of product to the retailer Assume the firm is a perfect competitor in the markets for its inputs Set up the firms optimization problem given this information and derive the first-order conditions for the problem Defme any notation or terminology you need to introduce Also illustrate the optimization problem with a graph
c Discuss the firms demand function for R given the conditions of the problem in part (b) Specifically what variables determine how much R the firm will utilize
d Re-express the problem in part (b) and take first-order conditions assuming now that the firm is the only buyer of the farm input R which is produced according to the following inverse supply function
w = a + 3R a3 gt 0
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191 lOS 2Kl 1 2 55 146 2 -m 24 2 lU 226 2 2l
US1 i9 l-IS S72 4112 4 1 1 199
I 1 I J I 110 I Y 12Y I 211 I M I 7 1 7 I 6 1 15 1 11 U9 UK IJK 1middot17 U6
17M I 71 1 101 167 1 162 161 I I S1 n6 I 51 IU1115 I ll 1 4 1 1401 19 1 47 146 1 44
I 98
2111 191 1 1111 1 19 210 1 042B 9 I H6 1 gt4 1 MO 119 1 1 1 1 1) 1 1 1 211 2111 2JO 22S 21 2111JOI 2SI 242 2 16 11114 16 140
2M9 2 66 UK 2 40 111 1 1 1 2-1 2 1 Ul167 middot 1 16 126 117 109 ill2 H I 472 4 12 01
114 I 12 111 1 10 129 I 211 UK 16 I 26 126 I 25 I 2 I 42 1 4 1 19 us I H 17 l l6 2 1 111 146 us 1 16 I 1611 165 1 61 I 5Y UK us IS-1 I S1 I S l I t HIIll 11111 186 2 II 1 2 1 11 2 1111 11111 Y I 151 107 199 190 1-JIS 1112 1111) 1 16 1 1S I 71 I 7 1 211 222 2111 215Ull OS 219U7 198 1 14 2 t7 2111 2511 lSO 242 1 16 2 225 22J 219 216 2 IJ UH l09 )02 196HI 129SSl 41gt4 41401
IJJ I I 1 10 129 1 18 117 127 126 1 1gt 125 124 1 24 136 usI 41 141 140 119 UK1 111 146 145 1 69 1 66 16 159 I S1 156 I 51 152 I SO 149 1 19 Ill 1114 1 812 11 1 1 229 216 1116 190 187 100
H14 196 1 9 1 1 81 1 112 119 1 71 I 71 1 11 1 69 167 165 0)219 2 15 1 1 1 JN 295 1 1 1 2242)6 229 V5 1 1 144 IS 210 2 26 1 19 1 1 1 2 1 1 109 U6 U l
0 1 407 I 75 )51 U6 123 312 101
Ul l ltJ 129 Ull 127 1 26 126 IH 1 2-1 1 2-1 12J t n l iM I 45 1 44 1 41 U9 Ull IJ6 us IB
172 167 1 64 ltll I H 15-1 Ul I SU I 1 lt7 1 6 10 10IIIS 182 179 171I 10 11111 1 22K 114 20S 1911 19) 188
2111 1 91 I H4 179 1 16 1 14 110 I M 1 IM I 62 O)1 41 lB 227 221 216 213OS l 2 291 169
270 2-17 210 2 2 1 IJ I I 207 2 Ul ] 0 1 147 J0 1 1 7 )07 298 2910 1 1 56 5 19 451 40
110 I 126 125 124 Ut 12 Ul 121 119 1 19 25IJ6 us 1 14 IJJ Ul U l140 11914425
1117 1113 179 1 66 161 157 1 )4 I l l 14K 141 142 1-11 119 I 111
176 173 1 71IllI 21)9 1 00 192 IM4 179 1611 1 64 I us I 5J U l 100I 21Oi
01
1R4 261 2 14 225 218 2 12 1 08 252 217 229 220 2 1 1 1116 202 1 94 I 92 I IM1 01 664 II 181 lSI 129 1 12 199 2119 280 27)511S
127 125 1 14 122 121 1 11l 119 1 1 7 1 1 1 1 16 1 1 5 1 1 5 I J J 112 I l l IJO 119 119us 1 41 1 4 1 UK 160 154 Ul 1411 144 141 140 116 I 15 I J I ll I 29 10 I
111 168 1 66117IK7 1821 19 2 19 211S 104 111-1 175 110 165 159 156 15] 1 4H 147 144 I-II 1 9 0)
125 2 17 210 104 199 195 1921 1 5 1 16 2 S 110 212 20 194 11111 1 04 17 I 71 I 61 160 til2S6 1SO 1 ll 295 lKl 272 261 165 ) 14
124 1221 16 129 128 127 1 2 1 1 19 1 111 1 17 116 1 14 1 1 1 1 1 1 I l l 1 10
IJJ IJI 1 10 140 119 141 117 132 111 126 1 24 1 2 1 119 10 120I us1 60I gtI 182 111 168 16S 1622 1)10 us
166 161 155 150 146 14 117 115 1 12 1211 125 I Hl 217 109 101 191 IKOl 219 201 I 176 1 101 156 UJ I 1 41 I IX
6 liS I 1701 4 J9 195 14H 296 2 79 266 256 247 240 middot 234
Ul 1 2 1 l iD 1 1 11 116 114 1 12 I l l 1 10 1119 I 0 25 111 IJ9 1111 I J6 I a I J2 I) I 129 128 127 126 US
152 146 1 41 1 111 1 1-1 I I 1211 I 24 122 uu 1 1 7 Ia Ill 2 I I 197 IKII IKO 1 75 170 166 161 160 157
172 162 IS7 IS2 146 141 119 I J2 129 U6 1 22 119 m198 19J I 88 I 114 180OS H9 1 04 265 242 226 2 14 2lt16
01 676 411 lllll l41 J I I 289 271 I 91 1119 I 79 169 I 61 1511 148 144 119 1 11 1 11 01
260 2SO 241 2 14 227
122 1 10 1 16 11-1 1 1 1 1 12 1119 Uli I 07 l tiO 124 124I 12 I J9 IJ7 US I J 111 129 1211 127 125
I II 142 118 114 uo 126 124 1 1 11 1 1 1 1 1 1 I OK Hill 10
i lS 25
IS72 JO 160 us172 167 16J10 2 71 2011 194 U2 1 14 1 12 1 1 1 1 1 1 I IIII 167 152 146 179l 11-1 HXI 260 2J7 221 210 201 1811 181liSmiddot
21KIll 6 6 4bl 3-JII J l2 l 02 2110 264 2 51 241 232 22S 1 10 U6 I ll I S I OtlIIIK104 U9 152 0 1
Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
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University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
Part I Answer 1 estion I and either Question 2 01 luestion 3
l You are constructing a si mple model of cotton su pply for a short-term project you are working on and h ave formulated the following conceptual model
where y5 i s bales of cotton ps i s the price paid to growers w1 is the labor wage rate w2 i s the price of water WJ i s pesticide price and DO is degree-days a measure of the growi ng conditions during the season
(a) A nother variable you might include in thi s speci ficati on is annual rainfall (R) whose coefficient i f included in the model would be a6 While you are sure th at all th e other va ri ables in the model are part of the correct speci fication you are not sure about annual rainfall As an applied econometrician explai n the tradeoffs made to the properties of you r estimates i f you include or exclude thi s vari able from the specification
(b) One must always be concerned with measurement error i nempirical models Explain th e effects on the properties of your estimates of (a) measurement error in th e dependent variable vs (b) measurement error in one or more explanatory vari ables
(c) You esti mate th e model wi th R included and the results are
yS = -6354 + 8727ps - 0 579lw1 - 7907w2 - 09045w3 + 769000 + 0045R ( -0 79) (2 36) ( -177) (-234) ( -190) (157) (045)
R2 = 0819 n =56
where th e Students-t statistics are gi ven in parentheses and the following sums of squared errors resu lt
Total SSY=660 9 Model SSR=5412 Errors SSE= l l 97
In answering the following part s justify your answers with quantitati ve test results using th e attached tables Quali tative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 0 5 level
(ii) Is the overall model significant Th at i s i s the hypothesis th at a1 = a2 = a3 = a4 = as = a6 = 0 rejected
(iii) The correlation between the price of water (w2) and the price of pesticides (w3) i s zero Do these two vari ables have the same effect on cotton supply i e are their coeffi cients equ al
2 Suppose you have the following econometric model
where y = yieldacre x1 = wateracre x2 = (wateracre)2 x3 = fertilizeracre
(a) For each of the following questions set up the appropriate hypothesis test and provide a brief explanation
(i) Does water have any influence on yieldacre
(ii) Assuming an unconstrained (interior) solution is wateracre = 300 significantly different from the amount that yields the maximum yield per unit water
(b) Suppose you work in a governmental agency and that your boss is not trained in economics or statistics and thus she does not know what e N(O all) means Explain-
in words the meaning and significance of this expression
3
a Given the regional farm modeling problem
Max subject to
px- cx Ax b
where x is an n x I vector of production activities p and c are vectors of prices and variable cost coefficients respectively b is an m x 1 vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Comment briefly on the connection between the dual problem and the Dual Complementary Slackness condition
(iii) If you had to calculate the derived demand for a single input b describe the type of function you would obtain from the problem
b Now use the information that the product prices p are determined according to the following demand relationships
p =a + Dx
Assume that the individual farmers act as perfect competitors
(i) Show how you would reformulate your problem and give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit )
(ii) Derive the dual to the problem in part b(i) showing your steps clearly
Part II Answer of the 4 questions in this Section _
I For some goods consumption is subject to more than the usual single constraint on money income time can also be a constrai nt Consider a consumer with utility f unction u(x) defined on goods x = (x1 xn) subject to th e two constraints T 2 tx and M 2 px where T is the total amount of time avai lable and M is th e total amount of income avai lable and t = (t1 tn) and p = (p o pn ) are the time and money prices per unit of consumption of each good
(a) Write th e primal utility max i mi zation problem for th is consumer What interpretation do you attach to th e sh adow values on each constraint
(b) F rom your answer to part (a) ex plain how the consumer s choices reveal her money value of time Develop an ex pression f or the marginal money value of time from th is pri mal problem
(c) Under what conditions would the problem revert to the standard consumer choice problem subject to a single money budget constraint
(d) Suppose th at the shadow value on the money budget constraint is zero W rite the dual expenditure mi nimization problem for thi s case Provide a brief interpretation in words of thi s dual ex penditure f unction Also provide an interpretation for the sh adow value of the constraint for th is problem
2 Assume t hat t he US wheat mar ket can be considered in isolation and is charact erized by
yS = sops - 20 wl
y0 = 60 - 12 5p0 +2M
where
and
ys is quantity of wheat suppl ied in millions of bushels per year p5 is the price recei ved by suppliers in dollars per bushel y0 is quantity demanded (in millions of bushels) by consumers p0 i s the price paid by consumers also in dollars per bushel w1 is the price of a composite input to wheat product ion M is consumer income in t housands of dollars
For purposes of this analysis consumer income i s M = 45 and the composite input price is w1 = 5
(a) What are t he equilibrium f ree-mar ket price and quant it y of wheat
(b) Is t he supply of wheat at this point price-elastic or price-inelast i c Ex plain your reasomng
(c) W h at is t he t otal net economic sur plus f rom production of wheat and how middotis it distr ibuted bet ween suppliers and demanders
(d) A var iet y of schemes are used in agricult ural markets to enhance producer welf are C onsider a pr ice support of $6 per bushel Explain t he effects of such a policy on the dist ri bution of welfare and economic efficiency Support your answer with quant itat ive est imates of t hese effect s
jgtrice
3 Suppose a monopoly fum faces the following linear demand curve P = a- bQ where P is price and Q is quantity produced and sold The monopolists cost function is
(1) c(Q) = cQ for Q Q c(Q) qo for Q gt Q
where Q is a capacity constraint
a Solve mathematically for the monopolists profit maxumzmg and output as functions of the parameters a b and c Show your work Also illustrate yourresults graphically labeling carefully all relevant information
b Suppose the function P(Q) from part (a) is the demand in the domestic market and that the monopolist can also sell unlimited quantities into the export market at price P E gt c Assuming that quantities exported cannot be resold into the domestic market show both mathematically and on your graph from part (a) how much the firm will sell into the domestic market and into the export market
c Consider now a potential entrant into this market For simplicity ignore the existence of the export market To be able to produce in this market an entrant must incur a one time set up or entry cost of E gt 0 It can then produce according to the cost function in (1) Describe in words andor diagrams how you would model this entrants decision problem You should consider the role of the incumbent monopolist in framing your answer
4 Consider a food processing firm that produces processed product Q from a raw agricultural product Rand a composite marketing input L which we will call labor
a Write a production function and draw some rough sketches of isoquants for each of the following possible production scenarios
1 There is no substitution between R and L in producing Q
ii The production relationship is Cobb-Douglas
b Suppose the true production relationship is Cobb-Douglas and that this firm produces Q exclusively for a large retail grocery chain It has a contract for 1999 to deliver Q units of product to the retailer Assume the firm is a perfect competitor in the markets for its inputs Set up the firms optimization problem given this information and derive the first-order conditions for the problem Defme any notation or terminology you need to introduce Also illustrate the optimization problem with a graph
c Discuss the firms demand function for R given the conditions of the problem in part (b) Specifically what variables determine how much R the firm will utilize
d Re-express the problem in part (b) and take first-order conditions assuming now that the firm is the only buyer of the farm input R which is produced according to the following inverse supply function
w = a + 3R a3 gt 0
OJ 04
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09
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lA Rlf lgtl Areas under the standardized normal distribution
Example
Pr (0 z 196) = 04750
Pr (z 196) = 05 - 04750 = 0025
TABLE 01 Percentage points of lhe t distribution
Example
Pr (r gt 2086) = Oo25
Pr (r gt 1725) = 005 for df = 20
Pr(ltl gt 1725) = 010 0 I 715
01$ 010 middot-00
001 0 00010015010 002 0010 0002ozo
J 07H 127116I 11216 14 61657 IIUI1nm
22 279925IXH6 6965UK16 4 lP 00 01 01 1 0765 I 6X 251 lIK2 4541 51141 102 JJ
4 074 I 15J 21J1 176 461 moo 00411no
Ol 0middot1111
(Xl)lt) 11119 059UIO 0161 t tiiCW 029 0279 1650675 0751 s 1476 2015 511910727 25710596 0616 4010517 115 7
071 II 1064 IIOJ 1141 52111106 bull 1440 1941 l7H7 0791 UKl 0171 091ll 09411
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1591 161M 1664
1915 1950 19115 W19
10611144 1179 K u 706 lW7 11601772 451111196
212 071) 421172054 219() 122420XX UHl 111 2262 21i21 1251)-9
0700 J72 1112 222K 2764 11692422
1021 214 25492-IK6 102157 2X92291
2794 2121 21152 4025II 0697 1361 1796 22111 2711lt lit
2611 27041gt1 25XU ll7K 1106 1111 190 0695 1156 17H1 21 79 26111 lOSS
ll40 11111 1910
2650 2M9 J65 l119 IJ 0694 1350 1771 2160 1lt5211112311lt6 21 lJH 164
145 1761 21J5 2614 1621 14 069214115 55-1 2977 7117 461 150111 0 1411
l711I 2602 2 94717011 770 790 Kill lX0 15 0691 1141 175 729 21 lfgt65 16116 JHHH
I I
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4194066 41H l6462567 2gt19H411 17 1740 21W40994(19 406 068H UJO 174 210142164192
I
4279 4292 6102X7K4265 25524251 -11117 4222 4441 068H U211 1 729 209J 253944294 112 494 J406 44111 579lK61LJ5 4157
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4554 505J67H A6K6 4691 4699 4706 22 06K6 I Jl l 1717 2014 l5HK 1XI94656 4664 4671I K 4641 41151K074756 4761 17144744 4750 2J 06KS 1319 250020694716 412 47 111 9 470 4719
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1
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J906 4911 06M4 14212471 27714913 4916 170 4909 27 L14 10524196 489R 4901 49044191 06114925 4927 4929 4931 492 4914 4916 140K2467 27612M U13 1701 204849111 4920 4922
4941 4941 4945 4946 494K 4949 4951 4951 IJII JJ9629 068) 2462 2756204S49111 494025
UJO 1697 2042 2457 17504QSS 4964 lJK S4961 JO 06KJ4960 4961 4961
496 J951J4956 49576
7 10 06KI 10749 1 6K4497 241J i7044970 4971 49724967 496K 1021J966 60 0679 129611 4981
4979 4979 49SO 16714977 2 190 26604976 4977Igrave 12122000 4975
l Jfi) 1090
49864914 4914 4985 498S 4986 120 0677 11M9 21671651 19KO 2J58
49111 49H9 49M9 4990 _ 1 1810674 164S 1960 1 126 1576 49H I 49K2 4982 498)
49X7 49K7 49117 49111
N This tahlc gives the area in the riahtmiddothand tail or the distribution tic bull t 0) But since the normal di5tribumiddot Nolt 111e smaller probability shown at the head of each column is the area in one tail the 1in is symmetrical about r- Othe area in the ldtmiddothlnd tail is the same u the area in the corresponding right-hand larser probability is the area in both tails bullbullil rltgtr eumplc PI- 196 S z S 01-04750 TMrdore -196 S t S 196)-2(047S0)-09S Sourr From E S Pearwn and H 0 Hartley eds bull Blomtuila TabftJot Slulislidans vol I
ld ed table 12 Cbullmbridgc University Prns New York 1966 Reproduced by perminion of the middot
editors and trustees of Biomeria
bull bull bull bull 10 II 30 110 100 500
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7 71
4 20
116
1 H4
4 1
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1
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1 75
192
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44
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244
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199
245
l-14
2 1
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161
190
1 4 1
25J
)70
137
195
2)7
U 5
190
229
1 59
245
U 7
I 91
1 15
96
142
194
1 94
l2
IJ1
117
174
104
1 0
190
U4
109
284
Ill 2 71 2JJ
1 77
194 175
i-1
174
149
157
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1 94 1 gt9
194
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174 159
145 1 4
195
119
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682 AIPFNHIX ll
TARLE Ol l lpper percentage poinls or the F distribution (continut-d)
I I
STATISTICAl TA8LfS 681
df for dl lor llllmtntor N
100 N
I 16 111 I 12 I y I 2ltI 10 1 10 111 1 1 1 1 111 I Y 1 41 1 4 1 1-10 I 19 1 19 UHI middotW 1171411 147 145
I 176 IJl 170 IUI 167 165 I M 161 1 bull1 101 1 97 191 1 90 11111 1116Hi 4 l)
2 15 212
Ill 1117 201 I I I M4 I IIII I I II
2 911 1 II 1 1 1gt 67 251 11 an z llbull 2 1J ll I 26 1 1 11 1 2
191 lOS 2Kl 1 2 55 146 2 -m 24 2 lU 226 2 2l
US1 i9 l-IS S72 4112 4 1 1 199
I 1 I J I 110 I Y 12Y I 211 I M I 7 1 7 I 6 1 15 1 11 U9 UK IJK 1middot17 U6
17M I 71 1 101 167 1 162 161 I I S1 n6 I 51 IU1115 I ll 1 4 1 1401 19 1 47 146 1 44
I 98
2111 191 1 1111 1 19 210 1 042B 9 I H6 1 gt4 1 MO 119 1 1 1 1 1) 1 1 1 211 2111 2JO 22S 21 2111JOI 2SI 242 2 16 11114 16 140
2M9 2 66 UK 2 40 111 1 1 1 2-1 2 1 Ul167 middot 1 16 126 117 109 ill2 H I 472 4 12 01
114 I 12 111 1 10 129 I 211 UK 16 I 26 126 I 25 I 2 I 42 1 4 1 19 us I H 17 l l6 2 1 111 146 us 1 16 I 1611 165 1 61 I 5Y UK us IS-1 I S1 I S l I t HIIll 11111 186 2 II 1 2 1 11 2 1111 11111 Y I 151 107 199 190 1-JIS 1112 1111) 1 16 1 1S I 71 I 7 1 211 222 2111 215Ull OS 219U7 198 1 14 2 t7 2111 2511 lSO 242 1 16 2 225 22J 219 216 2 IJ UH l09 )02 196HI 129SSl 41gt4 41401
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172 167 1 64 ltll I H 15-1 Ul I SU I 1 lt7 1 6 10 10IIIS 182 179 171I 10 11111 1 22K 114 20S 1911 19) 188
2111 1 91 I H4 179 1 16 1 14 110 I M 1 IM I 62 O)1 41 lB 227 221 216 213OS l 2 291 169
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110 I 126 125 124 Ut 12 Ul 121 119 1 19 25IJ6 us 1 14 IJJ Ul U l140 11914425
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125 2 17 210 104 199 195 1921 1 5 1 16 2 S 110 212 20 194 11111 1 04 17 I 71 I 61 160 til2S6 1SO 1 ll 295 lKl 272 261 165 ) 14
124 1221 16 129 128 127 1 2 1 1 19 1 111 1 17 116 1 14 1 1 1 1 1 1 I l l 1 10
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166 161 155 150 146 14 117 115 1 12 1211 125 I Hl 217 109 101 191 IKOl 219 201 I 176 1 101 156 UJ I 1 41 I IX
6 liS I 1701 4 J9 195 14H 296 2 79 266 256 247 240 middot 234
Ul 1 2 1 l iD 1 1 11 116 114 1 12 I l l 1 10 1119 I 0 25 111 IJ9 1111 I J6 I a I J2 I) I 129 128 127 126 US
152 146 1 41 1 111 1 1-1 I I 1211 I 24 122 uu 1 1 7 Ia Ill 2 I I 197 IKII IKO 1 75 170 166 161 160 157
172 162 IS7 IS2 146 141 119 I J2 129 U6 1 22 119 m198 19J I 88 I 114 180OS H9 1 04 265 242 226 2 14 2lt16
01 676 411 lllll l41 J I I 289 271 I 91 1119 I 79 169 I 61 1511 148 144 119 1 11 1 11 01
260 2SO 241 2 14 227
122 1 10 1 16 11-1 1 1 1 1 12 1119 Uli I 07 l tiO 124 124I 12 I J9 IJ7 US I J 111 129 1211 127 125
I II 142 118 114 uo 126 124 1 1 11 1 1 1 1 1 1 I OK Hill 10
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Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
2 Suppose you have the following econometric model
where y = yieldacre x1 = wateracre x2 = (wateracre)2 x3 = fertilizeracre
(a) For each of the following questions set up the appropriate hypothesis test and provide a brief explanation
(i) Does water have any influence on yieldacre
(ii) Assuming an unconstrained (interior) solution is wateracre = 300 significantly different from the amount that yields the maximum yield per unit water
(b) Suppose you work in a governmental agency and that your boss is not trained in economics or statistics and thus she does not know what e N(O all) means Explain-
in words the meaning and significance of this expression
3
a Given the regional farm modeling problem
Max subject to
px- cx Ax b
where x is an n x I vector of production activities p and c are vectors of prices and variable cost coefficients respectively b is an m x 1 vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Comment briefly on the connection between the dual problem and the Dual Complementary Slackness condition
(iii) If you had to calculate the derived demand for a single input b describe the type of function you would obtain from the problem
b Now use the information that the product prices p are determined according to the following demand relationships
p =a + Dx
Assume that the individual farmers act as perfect competitors
(i) Show how you would reformulate your problem and give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit )
(ii) Derive the dual to the problem in part b(i) showing your steps clearly
Part II Answer of the 4 questions in this Section _
I For some goods consumption is subject to more than the usual single constraint on money income time can also be a constrai nt Consider a consumer with utility f unction u(x) defined on goods x = (x1 xn) subject to th e two constraints T 2 tx and M 2 px where T is the total amount of time avai lable and M is th e total amount of income avai lable and t = (t1 tn) and p = (p o pn ) are the time and money prices per unit of consumption of each good
(a) Write th e primal utility max i mi zation problem for th is consumer What interpretation do you attach to th e sh adow values on each constraint
(b) F rom your answer to part (a) ex plain how the consumer s choices reveal her money value of time Develop an ex pression f or the marginal money value of time from th is pri mal problem
(c) Under what conditions would the problem revert to the standard consumer choice problem subject to a single money budget constraint
(d) Suppose th at the shadow value on the money budget constraint is zero W rite the dual expenditure mi nimization problem for thi s case Provide a brief interpretation in words of thi s dual ex penditure f unction Also provide an interpretation for the sh adow value of the constraint for th is problem
2 Assume t hat t he US wheat mar ket can be considered in isolation and is charact erized by
yS = sops - 20 wl
y0 = 60 - 12 5p0 +2M
where
and
ys is quantity of wheat suppl ied in millions of bushels per year p5 is the price recei ved by suppliers in dollars per bushel y0 is quantity demanded (in millions of bushels) by consumers p0 i s the price paid by consumers also in dollars per bushel w1 is the price of a composite input to wheat product ion M is consumer income in t housands of dollars
For purposes of this analysis consumer income i s M = 45 and the composite input price is w1 = 5
(a) What are t he equilibrium f ree-mar ket price and quant it y of wheat
(b) Is t he supply of wheat at this point price-elastic or price-inelast i c Ex plain your reasomng
(c) W h at is t he t otal net economic sur plus f rom production of wheat and how middotis it distr ibuted bet ween suppliers and demanders
(d) A var iet y of schemes are used in agricult ural markets to enhance producer welf are C onsider a pr ice support of $6 per bushel Explain t he effects of such a policy on the dist ri bution of welfare and economic efficiency Support your answer with quant itat ive est imates of t hese effect s
jgtrice
3 Suppose a monopoly fum faces the following linear demand curve P = a- bQ where P is price and Q is quantity produced and sold The monopolists cost function is
(1) c(Q) = cQ for Q Q c(Q) qo for Q gt Q
where Q is a capacity constraint
a Solve mathematically for the monopolists profit maxumzmg and output as functions of the parameters a b and c Show your work Also illustrate yourresults graphically labeling carefully all relevant information
b Suppose the function P(Q) from part (a) is the demand in the domestic market and that the monopolist can also sell unlimited quantities into the export market at price P E gt c Assuming that quantities exported cannot be resold into the domestic market show both mathematically and on your graph from part (a) how much the firm will sell into the domestic market and into the export market
c Consider now a potential entrant into this market For simplicity ignore the existence of the export market To be able to produce in this market an entrant must incur a one time set up or entry cost of E gt 0 It can then produce according to the cost function in (1) Describe in words andor diagrams how you would model this entrants decision problem You should consider the role of the incumbent monopolist in framing your answer
4 Consider a food processing firm that produces processed product Q from a raw agricultural product Rand a composite marketing input L which we will call labor
a Write a production function and draw some rough sketches of isoquants for each of the following possible production scenarios
1 There is no substitution between R and L in producing Q
ii The production relationship is Cobb-Douglas
b Suppose the true production relationship is Cobb-Douglas and that this firm produces Q exclusively for a large retail grocery chain It has a contract for 1999 to deliver Q units of product to the retailer Assume the firm is a perfect competitor in the markets for its inputs Set up the firms optimization problem given this information and derive the first-order conditions for the problem Defme any notation or terminology you need to introduce Also illustrate the optimization problem with a graph
c Discuss the firms demand function for R given the conditions of the problem in part (b) Specifically what variables determine how much R the firm will utilize
d Re-express the problem in part (b) and take first-order conditions assuming now that the firm is the only buyer of the farm input R which is produced according to the following inverse supply function
w = a + 3R a3 gt 0
OJ 04
J ---------
Amiddot __ __
09
0QgtI
4
2257
Jl59
412
14JK
446 -1564
4649
41161
29
0
495
4974
04711
164 9-9
267 2967
4170
llll
99
09X7
76
714
H
middotmiddot 74(
4791
4978
49119
2454 2764
1051
1 115
1441
IIIOK
2157
1577
U74
1517
5
4545
46JJ
4767
4974
4990
middot-
92()
J 7J7 717
2447 ll 4
middot55
4 144
19
24
1699
677
676 SlATISliCI1 tA9LFS
lA Rlf lgtl Areas under the standardized normal distribution
Example
Pr (0 z 196) = 04750
Pr (z 196) = 05 - 04750 = 0025
TABLE 01 Percentage points of lhe t distribution
Example
Pr (r gt 2086) = Oo25
Pr (r gt 1725) = 005 for df = 20
Pr(ltl gt 1725) = 010 0 I 715
01$ 010 middot-00
001 0 00010015010 002 0010 0002ozo
J 07H 127116I 11216 14 61657 IIUI1nm
22 279925IXH6 6965UK16 4 lP 00 01 01 1 0765 I 6X 251 lIK2 4541 51141 102 JJ
4 074 I 15J 21J1 176 461 moo 00411no
Ol 0middot1111
(Xl)lt) 11119 059UIO 0161 t tiiCW 029 0279 1650675 0751 s 1476 2015 511910727 25710596 0616 4010517 115 7
071 II 1064 IIOJ 1141 52111106 bull 1440 1941 l7H7 0791 UKl 0171 091ll 09411
149914KU 1517
17tMl 1406 299K J7XS7 0 711 1415 lH95 2165116111179 1217 1155 129l
1591 161M 1664
1915 1950 19115 W19
10611144 1179 K u 706 lW7 11601772 451111196
212 071) 421172054 219() 122420XX UHl 111 2262 21i21 1251)-9
0700 J72 1112 222K 2764 11692422
1021 214 25492-IK6 102157 2X92291
2794 2121 21152 4025II 0697 1361 1796 22111 2711lt lit
2611 27041gt1 25XU ll7K 1106 1111 190 0695 1156 17H1 21 79 26111 lOSS
ll40 11111 1910
2650 2M9 J65 l119 IJ 0694 1350 1771 2160 1lt5211112311lt6 21 lJH 164
145 1761 21J5 2614 1621 14 069214115 55-1 2977 7117 461 150111 0 1411
l711I 2602 2 94717011 770 790 Kill lX0 15 0691 1141 175 729 21 lfgt65 16116 JHHH
I I
11
1641
1144012
J997 2921907 JOISJ9-U 96 WKO 1 6 0690 bull IJJ7 1 746 2120 2511 l6M6 925 JK(fl 0619 11114011 4147 4162 4177
4194066 41H l6462567 2gt19H411 17 1740 21W40994(19 406 068H UJO 174 210142164192
I
4279 4292 6102X7K4265 25524251 -11117 4222 4441 068H U211 1 729 209J 253944294 112 494 J406 44111 579lK61LJ5 4157
20 0687 1325 I 715 21186 252K 14545115 4515 4515 455 5524474 44114 4495lh 4J 2 0616 121 1711 1010 251 H l517211 I 4571 45112 4591 4599 4616 4625
4554 505J67H A6K6 4691 4699 4706 22 06K6 I Jl l 1717 2014 l5HK 1XI94656 4664 4671I K 4641 41151K074756 4761 17144744 4750 2J 06KS 1319 250020694716 412 47 111 9 470 4719
41101 41112 4M17 14924791 AKOJ20 06K5 IJ UI 2797 1 4671711 20644772 47711 4710 47KK
1
I 4K21 1 4504Kl0 4104 41UH 41141 41146 4150 4H54 4H57 2l 06114 1316 1708 2060 24115 271741116 J4JS1706 20 56 2479 27794K6H 4171 4X75 4H7H 4HIH 4Hol4 4HH7 middot 4K90 26 0614 IJIS41164
J906 4911 06M4 14212471 27714913 4916 170 4909 27 L14 10524196 489R 4901 49044191 06114925 4927 4929 4931 492 4914 4916 140K2467 27612M U13 1701 204849111 4920 4922
4941 4941 4945 4946 494K 4949 4951 4951 IJII JJ9629 068) 2462 2756204S49111 494025
UJO 1697 2042 2457 17504QSS 4964 lJK S4961 JO 06KJ4960 4961 4961
496 J951J4956 49576
7 10 06KI 10749 1 6K4497 241J i7044970 4971 49724967 496K 1021J966 60 0679 129611 4981
4979 4979 49SO 16714977 2 190 26604976 4977Igrave 12122000 4975
l Jfi) 1090
49864914 4914 4985 498S 4986 120 0677 11M9 21671651 19KO 2J58
49111 49H9 49M9 4990 _ 1 1810674 164S 1960 1 126 1576 49H I 49K2 4982 498)
49X7 49K7 49117 49111
N This tahlc gives the area in the riahtmiddothand tail or the distribution tic bull t 0) But since the normal di5tribumiddot Nolt 111e smaller probability shown at the head of each column is the area in one tail the 1in is symmetrical about r- Othe area in the ldtmiddothlnd tail is the same u the area in the corresponding right-hand larser probability is the area in both tails bullbullil rltgtr eumplc PI- 196 S z S 01-04750 TMrdore -196 S t S 196)-2(047S0)-09S Sourr From E S Pearwn and H 0 Hartley eds bull Blomtuila TabftJot Slulislidans vol I
ld ed table 12 Cbullmbridgc University Prns New York 1966 Reproduced by perminion of the middot
editors and trustees of Biomeria
bull bull bull bull 10 II 30 110 100 500
(
I
n
I)
4]1
7 71
4 20
116
1 H4
4 1
1 111
4 1
1
U4
1 75
192
1 gt6
44
2 54
244
)07
142middot
2 2
41J
144
U1
199
245
l-14
2 1
14)
161
190
1 4 1
25J
)70
137
195
2)7
U 5
190
229
1 59
245
U 7
I 91
1 15
96
142
194
1 94
l2
IJ1
117
174
104
1 0
190
U4
109
284
Ill 2 71 2JJ
1 77
194 175
i-1
174
149
157
I y I 7
1 94 1 gt9
194
1 9)
174 159
145 1 4
195
119
179 147
682 AIPFNHIX ll
TARLE Ol l lpper percentage poinls or the F distribution (continut-d)
I I
STATISTICAl TA8LfS 681
df for dl lor llllmtntor N
100 N
I 16 111 I 12 I y I 2ltI 10 1 10 111 1 1 1 1 111 I Y 1 41 1 4 1 1-10 I 19 1 19 UHI middotW 1171411 147 145
I 176 IJl 170 IUI 167 165 I M 161 1 bull1 101 1 97 191 1 90 11111 1116Hi 4 l)
2 15 212
Ill 1117 201 I I I M4 I IIII I I II
2 911 1 II 1 1 1gt 67 251 11 an z llbull 2 1J ll I 26 1 1 11 1 2
191 lOS 2Kl 1 2 55 146 2 -m 24 2 lU 226 2 2l
US1 i9 l-IS S72 4112 4 1 1 199
I 1 I J I 110 I Y 12Y I 211 I M I 7 1 7 I 6 1 15 1 11 U9 UK IJK 1middot17 U6
17M I 71 1 101 167 1 162 161 I I S1 n6 I 51 IU1115 I ll 1 4 1 1401 19 1 47 146 1 44
I 98
2111 191 1 1111 1 19 210 1 042B 9 I H6 1 gt4 1 MO 119 1 1 1 1 1) 1 1 1 211 2111 2JO 22S 21 2111JOI 2SI 242 2 16 11114 16 140
2M9 2 66 UK 2 40 111 1 1 1 2-1 2 1 Ul167 middot 1 16 126 117 109 ill2 H I 472 4 12 01
114 I 12 111 1 10 129 I 211 UK 16 I 26 126 I 25 I 2 I 42 1 4 1 19 us I H 17 l l6 2 1 111 146 us 1 16 I 1611 165 1 61 I 5Y UK us IS-1 I S1 I S l I t HIIll 11111 186 2 II 1 2 1 11 2 1111 11111 Y I 151 107 199 190 1-JIS 1112 1111) 1 16 1 1S I 71 I 7 1 211 222 2111 215Ull OS 219U7 198 1 14 2 t7 2111 2511 lSO 242 1 16 2 225 22J 219 216 2 IJ UH l09 )02 196HI 129SSl 41gt4 41401
IJJ I I 1 10 129 1 18 117 127 126 1 1gt 125 124 1 24 136 usI 41 141 140 119 UK1 111 146 145 1 69 1 66 16 159 I S1 156 I 51 152 I SO 149 1 19 Ill 1114 1 812 11 1 1 229 216 1116 190 187 100
H14 196 1 9 1 1 81 1 112 119 1 71 I 71 1 11 1 69 167 165 0)219 2 15 1 1 1 JN 295 1 1 1 2242)6 229 V5 1 1 144 IS 210 2 26 1 19 1 1 1 2 1 1 109 U6 U l
0 1 407 I 75 )51 U6 123 312 101
Ul l ltJ 129 Ull 127 1 26 126 IH 1 2-1 1 2-1 12J t n l iM I 45 1 44 1 41 U9 Ull IJ6 us IB
172 167 1 64 ltll I H 15-1 Ul I SU I 1 lt7 1 6 10 10IIIS 182 179 171I 10 11111 1 22K 114 20S 1911 19) 188
2111 1 91 I H4 179 1 16 1 14 110 I M 1 IM I 62 O)1 41 lB 227 221 216 213OS l 2 291 169
270 2-17 210 2 2 1 IJ I I 207 2 Ul ] 0 1 147 J0 1 1 7 )07 298 2910 1 1 56 5 19 451 40
110 I 126 125 124 Ut 12 Ul 121 119 1 19 25IJ6 us 1 14 IJJ Ul U l140 11914425
1117 1113 179 1 66 161 157 1 )4 I l l 14K 141 142 1-11 119 I 111
176 173 1 71IllI 21)9 1 00 192 IM4 179 1611 1 64 I us I 5J U l 100I 21Oi
01
1R4 261 2 14 225 218 2 12 1 08 252 217 229 220 2 1 1 1116 202 1 94 I 92 I IM1 01 664 II 181 lSI 129 1 12 199 2119 280 27)511S
127 125 1 14 122 121 1 11l 119 1 1 7 1 1 1 1 16 1 1 5 1 1 5 I J J 112 I l l IJO 119 119us 1 41 1 4 1 UK 160 154 Ul 1411 144 141 140 116 I 15 I J I ll I 29 10 I
111 168 1 66117IK7 1821 19 2 19 211S 104 111-1 175 110 165 159 156 15] 1 4H 147 144 I-II 1 9 0)
125 2 17 210 104 199 195 1921 1 5 1 16 2 S 110 212 20 194 11111 1 04 17 I 71 I 61 160 til2S6 1SO 1 ll 295 lKl 272 261 165 ) 14
124 1221 16 129 128 127 1 2 1 1 19 1 111 1 17 116 1 14 1 1 1 1 1 1 I l l 1 10
IJJ IJI 1 10 140 119 141 117 132 111 126 1 24 1 2 1 119 10 120I us1 60I gtI 182 111 168 16S 1622 1)10 us
166 161 155 150 146 14 117 115 1 12 1211 125 I Hl 217 109 101 191 IKOl 219 201 I 176 1 101 156 UJ I 1 41 I IX
6 liS I 1701 4 J9 195 14H 296 2 79 266 256 247 240 middot 234
Ul 1 2 1 l iD 1 1 11 116 114 1 12 I l l 1 10 1119 I 0 25 111 IJ9 1111 I J6 I a I J2 I) I 129 128 127 126 US
152 146 1 41 1 111 1 1-1 I I 1211 I 24 122 uu 1 1 7 Ia Ill 2 I I 197 IKII IKO 1 75 170 166 161 160 157
172 162 IS7 IS2 146 141 119 I J2 129 U6 1 22 119 m198 19J I 88 I 114 180OS H9 1 04 265 242 226 2 14 2lt16
01 676 411 lllll l41 J I I 289 271 I 91 1119 I 79 169 I 61 1511 148 144 119 1 11 1 11 01
260 2SO 241 2 14 227
122 1 10 1 16 11-1 1 1 1 1 12 1119 Uli I 07 l tiO 124 124I 12 I J9 IJ7 US I J 111 129 1211 127 125
I II 142 118 114 uo 126 124 1 1 11 1 1 1 1 1 1 I OK Hill 10
i lS 25
IS72 JO 160 us172 167 16J10 2 71 2011 194 U2 1 14 1 12 1 1 1 1 1 1 I IIII 167 152 146 179l 11-1 HXI 260 2J7 221 210 201 1811 181liSmiddot
21KIll 6 6 4bl 3-JII J l2 l 02 2110 264 2 51 241 232 22S 1 10 U6 I ll I S I OtlIIIK104 U9 152 0 1
Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
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d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
3
a Given the regional farm modeling problem
Max subject to
px- cx Ax b
where x is an n x I vector of production activities p and c are vectors of prices and variable cost coefficients respectively b is an m x 1 vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Comment briefly on the connection between the dual problem and the Dual Complementary Slackness condition
(iii) If you had to calculate the derived demand for a single input b describe the type of function you would obtain from the problem
b Now use the information that the product prices p are determined according to the following demand relationships
p =a + Dx
Assume that the individual farmers act as perfect competitors
(i) Show how you would reformulate your problem and give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit )
(ii) Derive the dual to the problem in part b(i) showing your steps clearly
Part II Answer of the 4 questions in this Section _
I For some goods consumption is subject to more than the usual single constraint on money income time can also be a constrai nt Consider a consumer with utility f unction u(x) defined on goods x = (x1 xn) subject to th e two constraints T 2 tx and M 2 px where T is the total amount of time avai lable and M is th e total amount of income avai lable and t = (t1 tn) and p = (p o pn ) are the time and money prices per unit of consumption of each good
(a) Write th e primal utility max i mi zation problem for th is consumer What interpretation do you attach to th e sh adow values on each constraint
(b) F rom your answer to part (a) ex plain how the consumer s choices reveal her money value of time Develop an ex pression f or the marginal money value of time from th is pri mal problem
(c) Under what conditions would the problem revert to the standard consumer choice problem subject to a single money budget constraint
(d) Suppose th at the shadow value on the money budget constraint is zero W rite the dual expenditure mi nimization problem for thi s case Provide a brief interpretation in words of thi s dual ex penditure f unction Also provide an interpretation for the sh adow value of the constraint for th is problem
2 Assume t hat t he US wheat mar ket can be considered in isolation and is charact erized by
yS = sops - 20 wl
y0 = 60 - 12 5p0 +2M
where
and
ys is quantity of wheat suppl ied in millions of bushels per year p5 is the price recei ved by suppliers in dollars per bushel y0 is quantity demanded (in millions of bushels) by consumers p0 i s the price paid by consumers also in dollars per bushel w1 is the price of a composite input to wheat product ion M is consumer income in t housands of dollars
For purposes of this analysis consumer income i s M = 45 and the composite input price is w1 = 5
(a) What are t he equilibrium f ree-mar ket price and quant it y of wheat
(b) Is t he supply of wheat at this point price-elastic or price-inelast i c Ex plain your reasomng
(c) W h at is t he t otal net economic sur plus f rom production of wheat and how middotis it distr ibuted bet ween suppliers and demanders
(d) A var iet y of schemes are used in agricult ural markets to enhance producer welf are C onsider a pr ice support of $6 per bushel Explain t he effects of such a policy on the dist ri bution of welfare and economic efficiency Support your answer with quant itat ive est imates of t hese effect s
jgtrice
3 Suppose a monopoly fum faces the following linear demand curve P = a- bQ where P is price and Q is quantity produced and sold The monopolists cost function is
(1) c(Q) = cQ for Q Q c(Q) qo for Q gt Q
where Q is a capacity constraint
a Solve mathematically for the monopolists profit maxumzmg and output as functions of the parameters a b and c Show your work Also illustrate yourresults graphically labeling carefully all relevant information
b Suppose the function P(Q) from part (a) is the demand in the domestic market and that the monopolist can also sell unlimited quantities into the export market at price P E gt c Assuming that quantities exported cannot be resold into the domestic market show both mathematically and on your graph from part (a) how much the firm will sell into the domestic market and into the export market
c Consider now a potential entrant into this market For simplicity ignore the existence of the export market To be able to produce in this market an entrant must incur a one time set up or entry cost of E gt 0 It can then produce according to the cost function in (1) Describe in words andor diagrams how you would model this entrants decision problem You should consider the role of the incumbent monopolist in framing your answer
4 Consider a food processing firm that produces processed product Q from a raw agricultural product Rand a composite marketing input L which we will call labor
a Write a production function and draw some rough sketches of isoquants for each of the following possible production scenarios
1 There is no substitution between R and L in producing Q
ii The production relationship is Cobb-Douglas
b Suppose the true production relationship is Cobb-Douglas and that this firm produces Q exclusively for a large retail grocery chain It has a contract for 1999 to deliver Q units of product to the retailer Assume the firm is a perfect competitor in the markets for its inputs Set up the firms optimization problem given this information and derive the first-order conditions for the problem Defme any notation or terminology you need to introduce Also illustrate the optimization problem with a graph
c Discuss the firms demand function for R given the conditions of the problem in part (b) Specifically what variables determine how much R the firm will utilize
d Re-express the problem in part (b) and take first-order conditions assuming now that the firm is the only buyer of the farm input R which is produced according to the following inverse supply function
w = a + 3R a3 gt 0
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127 125 1 14 122 121 1 11l 119 1 1 7 1 1 1 1 16 1 1 5 1 1 5 I J J 112 I l l IJO 119 119us 1 41 1 4 1 UK 160 154 Ul 1411 144 141 140 116 I 15 I J I ll I 29 10 I
111 168 1 66117IK7 1821 19 2 19 211S 104 111-1 175 110 165 159 156 15] 1 4H 147 144 I-II 1 9 0)
125 2 17 210 104 199 195 1921 1 5 1 16 2 S 110 212 20 194 11111 1 04 17 I 71 I 61 160 til2S6 1SO 1 ll 295 lKl 272 261 165 ) 14
124 1221 16 129 128 127 1 2 1 1 19 1 111 1 17 116 1 14 1 1 1 1 1 1 I l l 1 10
IJJ IJI 1 10 140 119 141 117 132 111 126 1 24 1 2 1 119 10 120I us1 60I gtI 182 111 168 16S 1622 1)10 us
166 161 155 150 146 14 117 115 1 12 1211 125 I Hl 217 109 101 191 IKOl 219 201 I 176 1 101 156 UJ I 1 41 I IX
6 liS I 1701 4 J9 195 14H 296 2 79 266 256 247 240 middot 234
Ul 1 2 1 l iD 1 1 11 116 114 1 12 I l l 1 10 1119 I 0 25 111 IJ9 1111 I J6 I a I J2 I) I 129 128 127 126 US
152 146 1 41 1 111 1 1-1 I I 1211 I 24 122 uu 1 1 7 Ia Ill 2 I I 197 IKII IKO 1 75 170 166 161 160 157
172 162 IS7 IS2 146 141 119 I J2 129 U6 1 22 119 m198 19J I 88 I 114 180OS H9 1 04 265 242 226 2 14 2lt16
01 676 411 lllll l41 J I I 289 271 I 91 1119 I 79 169 I 61 1511 148 144 119 1 11 1 11 01
260 2SO 241 2 14 227
122 1 10 1 16 11-1 1 1 1 1 12 1119 Uli I 07 l tiO 124 124I 12 I J9 IJ7 US I J 111 129 1211 127 125
I II 142 118 114 uo 126 124 1 1 11 1 1 1 1 1 1 I OK Hill 10
i lS 25
IS72 JO 160 us172 167 16J10 2 71 2011 194 U2 1 14 1 12 1 1 1 1 1 1 I IIII 167 152 146 179l 11-1 HXI 260 2J7 221 210 201 1811 181liSmiddot
21KIll 6 6 4bl 3-JII J l2 l 02 2110 264 2 51 241 232 22S 1 10 U6 I ll I S I OtlIIIK104 U9 152 0 1
Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
Part II Answer of the 4 questions in this Section _
I For some goods consumption is subject to more than the usual single constraint on money income time can also be a constrai nt Consider a consumer with utility f unction u(x) defined on goods x = (x1 xn) subject to th e two constraints T 2 tx and M 2 px where T is the total amount of time avai lable and M is th e total amount of income avai lable and t = (t1 tn) and p = (p o pn ) are the time and money prices per unit of consumption of each good
(a) Write th e primal utility max i mi zation problem for th is consumer What interpretation do you attach to th e sh adow values on each constraint
(b) F rom your answer to part (a) ex plain how the consumer s choices reveal her money value of time Develop an ex pression f or the marginal money value of time from th is pri mal problem
(c) Under what conditions would the problem revert to the standard consumer choice problem subject to a single money budget constraint
(d) Suppose th at the shadow value on the money budget constraint is zero W rite the dual expenditure mi nimization problem for thi s case Provide a brief interpretation in words of thi s dual ex penditure f unction Also provide an interpretation for the sh adow value of the constraint for th is problem
2 Assume t hat t he US wheat mar ket can be considered in isolation and is charact erized by
yS = sops - 20 wl
y0 = 60 - 12 5p0 +2M
where
and
ys is quantity of wheat suppl ied in millions of bushels per year p5 is the price recei ved by suppliers in dollars per bushel y0 is quantity demanded (in millions of bushels) by consumers p0 i s the price paid by consumers also in dollars per bushel w1 is the price of a composite input to wheat product ion M is consumer income in t housands of dollars
For purposes of this analysis consumer income i s M = 45 and the composite input price is w1 = 5
(a) What are t he equilibrium f ree-mar ket price and quant it y of wheat
(b) Is t he supply of wheat at this point price-elastic or price-inelast i c Ex plain your reasomng
(c) W h at is t he t otal net economic sur plus f rom production of wheat and how middotis it distr ibuted bet ween suppliers and demanders
(d) A var iet y of schemes are used in agricult ural markets to enhance producer welf are C onsider a pr ice support of $6 per bushel Explain t he effects of such a policy on the dist ri bution of welfare and economic efficiency Support your answer with quant itat ive est imates of t hese effect s
jgtrice
3 Suppose a monopoly fum faces the following linear demand curve P = a- bQ where P is price and Q is quantity produced and sold The monopolists cost function is
(1) c(Q) = cQ for Q Q c(Q) qo for Q gt Q
where Q is a capacity constraint
a Solve mathematically for the monopolists profit maxumzmg and output as functions of the parameters a b and c Show your work Also illustrate yourresults graphically labeling carefully all relevant information
b Suppose the function P(Q) from part (a) is the demand in the domestic market and that the monopolist can also sell unlimited quantities into the export market at price P E gt c Assuming that quantities exported cannot be resold into the domestic market show both mathematically and on your graph from part (a) how much the firm will sell into the domestic market and into the export market
c Consider now a potential entrant into this market For simplicity ignore the existence of the export market To be able to produce in this market an entrant must incur a one time set up or entry cost of E gt 0 It can then produce according to the cost function in (1) Describe in words andor diagrams how you would model this entrants decision problem You should consider the role of the incumbent monopolist in framing your answer
4 Consider a food processing firm that produces processed product Q from a raw agricultural product Rand a composite marketing input L which we will call labor
a Write a production function and draw some rough sketches of isoquants for each of the following possible production scenarios
1 There is no substitution between R and L in producing Q
ii The production relationship is Cobb-Douglas
b Suppose the true production relationship is Cobb-Douglas and that this firm produces Q exclusively for a large retail grocery chain It has a contract for 1999 to deliver Q units of product to the retailer Assume the firm is a perfect competitor in the markets for its inputs Set up the firms optimization problem given this information and derive the first-order conditions for the problem Defme any notation or terminology you need to introduce Also illustrate the optimization problem with a graph
c Discuss the firms demand function for R given the conditions of the problem in part (b) Specifically what variables determine how much R the firm will utilize
d Re-express the problem in part (b) and take first-order conditions assuming now that the firm is the only buyer of the farm input R which is produced according to the following inverse supply function
w = a + 3R a3 gt 0
OJ 04
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09
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4
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412
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164 9-9
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4791
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676 SlATISliCI1 tA9LFS
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Example
Pr (0 z 196) = 04750
Pr (z 196) = 05 - 04750 = 0025
TABLE 01 Percentage points of lhe t distribution
Example
Pr (r gt 2086) = Oo25
Pr (r gt 1725) = 005 for df = 20
Pr(ltl gt 1725) = 010 0 I 715
01$ 010 middot-00
001 0 00010015010 002 0010 0002ozo
J 07H 127116I 11216 14 61657 IIUI1nm
22 279925IXH6 6965UK16 4 lP 00 01 01 1 0765 I 6X 251 lIK2 4541 51141 102 JJ
4 074 I 15J 21J1 176 461 moo 00411no
Ol 0middot1111
(Xl)lt) 11119 059UIO 0161 t tiiCW 029 0279 1650675 0751 s 1476 2015 511910727 25710596 0616 4010517 115 7
071 II 1064 IIOJ 1141 52111106 bull 1440 1941 l7H7 0791 UKl 0171 091ll 09411
149914KU 1517
17tMl 1406 299K J7XS7 0 711 1415 lH95 2165116111179 1217 1155 129l
1591 161M 1664
1915 1950 19115 W19
10611144 1179 K u 706 lW7 11601772 451111196
212 071) 421172054 219() 122420XX UHl 111 2262 21i21 1251)-9
0700 J72 1112 222K 2764 11692422
1021 214 25492-IK6 102157 2X92291
2794 2121 21152 4025II 0697 1361 1796 22111 2711lt lit
2611 27041gt1 25XU ll7K 1106 1111 190 0695 1156 17H1 21 79 26111 lOSS
ll40 11111 1910
2650 2M9 J65 l119 IJ 0694 1350 1771 2160 1lt5211112311lt6 21 lJH 164
145 1761 21J5 2614 1621 14 069214115 55-1 2977 7117 461 150111 0 1411
l711I 2602 2 94717011 770 790 Kill lX0 15 0691 1141 175 729 21 lfgt65 16116 JHHH
I I
11
1641
1144012
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4194066 41H l6462567 2gt19H411 17 1740 21W40994(19 406 068H UJO 174 210142164192
I
4279 4292 6102X7K4265 25524251 -11117 4222 4441 068H U211 1 729 209J 253944294 112 494 J406 44111 579lK61LJ5 4157
20 0687 1325 I 715 21186 252K 14545115 4515 4515 455 5524474 44114 4495lh 4J 2 0616 121 1711 1010 251 H l517211 I 4571 45112 4591 4599 4616 4625
4554 505J67H A6K6 4691 4699 4706 22 06K6 I Jl l 1717 2014 l5HK 1XI94656 4664 4671I K 4641 41151K074756 4761 17144744 4750 2J 06KS 1319 250020694716 412 47 111 9 470 4719
41101 41112 4M17 14924791 AKOJ20 06K5 IJ UI 2797 1 4671711 20644772 47711 4710 47KK
1
I 4K21 1 4504Kl0 4104 41UH 41141 41146 4150 4H54 4H57 2l 06114 1316 1708 2060 24115 271741116 J4JS1706 20 56 2479 27794K6H 4171 4X75 4H7H 4HIH 4Hol4 4HH7 middot 4K90 26 0614 IJIS41164
J906 4911 06M4 14212471 27714913 4916 170 4909 27 L14 10524196 489R 4901 49044191 06114925 4927 4929 4931 492 4914 4916 140K2467 27612M U13 1701 204849111 4920 4922
4941 4941 4945 4946 494K 4949 4951 4951 IJII JJ9629 068) 2462 2756204S49111 494025
UJO 1697 2042 2457 17504QSS 4964 lJK S4961 JO 06KJ4960 4961 4961
496 J951J4956 49576
7 10 06KI 10749 1 6K4497 241J i7044970 4971 49724967 496K 1021J966 60 0679 129611 4981
4979 4979 49SO 16714977 2 190 26604976 4977Igrave 12122000 4975
l Jfi) 1090
49864914 4914 4985 498S 4986 120 0677 11M9 21671651 19KO 2J58
49111 49H9 49M9 4990 _ 1 1810674 164S 1960 1 126 1576 49H I 49K2 4982 498)
49X7 49K7 49117 49111
N This tahlc gives the area in the riahtmiddothand tail or the distribution tic bull t 0) But since the normal di5tribumiddot Nolt 111e smaller probability shown at the head of each column is the area in one tail the 1in is symmetrical about r- Othe area in the ldtmiddothlnd tail is the same u the area in the corresponding right-hand larser probability is the area in both tails bullbullil rltgtr eumplc PI- 196 S z S 01-04750 TMrdore -196 S t S 196)-2(047S0)-09S Sourr From E S Pearwn and H 0 Hartley eds bull Blomtuila TabftJot Slulislidans vol I
ld ed table 12 Cbullmbridgc University Prns New York 1966 Reproduced by perminion of the middot
editors and trustees of Biomeria
bull bull bull bull 10 II 30 110 100 500
(
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7 71
4 20
116
1 H4
4 1
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1 75
192
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44
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244
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199
245
l-14
2 1
14)
161
190
1 4 1
25J
)70
137
195
2)7
U 5
190
229
1 59
245
U 7
I 91
1 15
96
142
194
1 94
l2
IJ1
117
174
104
1 0
190
U4
109
284
Ill 2 71 2JJ
1 77
194 175
i-1
174
149
157
I y I 7
1 94 1 gt9
194
1 9)
174 159
145 1 4
195
119
179 147
682 AIPFNHIX ll
TARLE Ol l lpper percentage poinls or the F distribution (continut-d)
I I
STATISTICAl TA8LfS 681
df for dl lor llllmtntor N
100 N
I 16 111 I 12 I y I 2ltI 10 1 10 111 1 1 1 1 111 I Y 1 41 1 4 1 1-10 I 19 1 19 UHI middotW 1171411 147 145
I 176 IJl 170 IUI 167 165 I M 161 1 bull1 101 1 97 191 1 90 11111 1116Hi 4 l)
2 15 212
Ill 1117 201 I I I M4 I IIII I I II
2 911 1 II 1 1 1gt 67 251 11 an z llbull 2 1J ll I 26 1 1 11 1 2
191 lOS 2Kl 1 2 55 146 2 -m 24 2 lU 226 2 2l
US1 i9 l-IS S72 4112 4 1 1 199
I 1 I J I 110 I Y 12Y I 211 I M I 7 1 7 I 6 1 15 1 11 U9 UK IJK 1middot17 U6
17M I 71 1 101 167 1 162 161 I I S1 n6 I 51 IU1115 I ll 1 4 1 1401 19 1 47 146 1 44
I 98
2111 191 1 1111 1 19 210 1 042B 9 I H6 1 gt4 1 MO 119 1 1 1 1 1) 1 1 1 211 2111 2JO 22S 21 2111JOI 2SI 242 2 16 11114 16 140
2M9 2 66 UK 2 40 111 1 1 1 2-1 2 1 Ul167 middot 1 16 126 117 109 ill2 H I 472 4 12 01
114 I 12 111 1 10 129 I 211 UK 16 I 26 126 I 25 I 2 I 42 1 4 1 19 us I H 17 l l6 2 1 111 146 us 1 16 I 1611 165 1 61 I 5Y UK us IS-1 I S1 I S l I t HIIll 11111 186 2 II 1 2 1 11 2 1111 11111 Y I 151 107 199 190 1-JIS 1112 1111) 1 16 1 1S I 71 I 7 1 211 222 2111 215Ull OS 219U7 198 1 14 2 t7 2111 2511 lSO 242 1 16 2 225 22J 219 216 2 IJ UH l09 )02 196HI 129SSl 41gt4 41401
IJJ I I 1 10 129 1 18 117 127 126 1 1gt 125 124 1 24 136 usI 41 141 140 119 UK1 111 146 145 1 69 1 66 16 159 I S1 156 I 51 152 I SO 149 1 19 Ill 1114 1 812 11 1 1 229 216 1116 190 187 100
H14 196 1 9 1 1 81 1 112 119 1 71 I 71 1 11 1 69 167 165 0)219 2 15 1 1 1 JN 295 1 1 1 2242)6 229 V5 1 1 144 IS 210 2 26 1 19 1 1 1 2 1 1 109 U6 U l
0 1 407 I 75 )51 U6 123 312 101
Ul l ltJ 129 Ull 127 1 26 126 IH 1 2-1 1 2-1 12J t n l iM I 45 1 44 1 41 U9 Ull IJ6 us IB
172 167 1 64 ltll I H 15-1 Ul I SU I 1 lt7 1 6 10 10IIIS 182 179 171I 10 11111 1 22K 114 20S 1911 19) 188
2111 1 91 I H4 179 1 16 1 14 110 I M 1 IM I 62 O)1 41 lB 227 221 216 213OS l 2 291 169
270 2-17 210 2 2 1 IJ I I 207 2 Ul ] 0 1 147 J0 1 1 7 )07 298 2910 1 1 56 5 19 451 40
110 I 126 125 124 Ut 12 Ul 121 119 1 19 25IJ6 us 1 14 IJJ Ul U l140 11914425
1117 1113 179 1 66 161 157 1 )4 I l l 14K 141 142 1-11 119 I 111
176 173 1 71IllI 21)9 1 00 192 IM4 179 1611 1 64 I us I 5J U l 100I 21Oi
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111 168 1 66117IK7 1821 19 2 19 211S 104 111-1 175 110 165 159 156 15] 1 4H 147 144 I-II 1 9 0)
125 2 17 210 104 199 195 1921 1 5 1 16 2 S 110 212 20 194 11111 1 04 17 I 71 I 61 160 til2S6 1SO 1 ll 295 lKl 272 261 165 ) 14
124 1221 16 129 128 127 1 2 1 1 19 1 111 1 17 116 1 14 1 1 1 1 1 1 I l l 1 10
IJJ IJI 1 10 140 119 141 117 132 111 126 1 24 1 2 1 119 10 120I us1 60I gtI 182 111 168 16S 1622 1)10 us
166 161 155 150 146 14 117 115 1 12 1211 125 I Hl 217 109 101 191 IKOl 219 201 I 176 1 101 156 UJ I 1 41 I IX
6 liS I 1701 4 J9 195 14H 296 2 79 266 256 247 240 middot 234
Ul 1 2 1 l iD 1 1 11 116 114 1 12 I l l 1 10 1119 I 0 25 111 IJ9 1111 I J6 I a I J2 I) I 129 128 127 126 US
152 146 1 41 1 111 1 1-1 I I 1211 I 24 122 uu 1 1 7 Ia Ill 2 I I 197 IKII IKO 1 75 170 166 161 160 157
172 162 IS7 IS2 146 141 119 I J2 129 U6 1 22 119 m198 19J I 88 I 114 180OS H9 1 04 265 242 226 2 14 2lt16
01 676 411 lllll l41 J I I 289 271 I 91 1119 I 79 169 I 61 1511 148 144 119 1 11 1 11 01
260 2SO 241 2 14 227
122 1 10 1 16 11-1 1 1 1 1 12 1119 Uli I 07 l tiO 124 124I 12 I J9 IJ7 US I J 111 129 1211 127 125
I II 142 118 114 uo 126 124 1 1 11 1 1 1 1 1 1 I OK Hill 10
i lS 25
IS72 JO 160 us172 167 16J10 2 71 2011 194 U2 1 14 1 12 1 1 1 1 1 1 I IIII 167 152 146 179l 11-1 HXI 260 2J7 221 210 201 1811 181liSmiddot
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Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
2 Assume t hat t he US wheat mar ket can be considered in isolation and is charact erized by
yS = sops - 20 wl
y0 = 60 - 12 5p0 +2M
where
and
ys is quantity of wheat suppl ied in millions of bushels per year p5 is the price recei ved by suppliers in dollars per bushel y0 is quantity demanded (in millions of bushels) by consumers p0 i s the price paid by consumers also in dollars per bushel w1 is the price of a composite input to wheat product ion M is consumer income in t housands of dollars
For purposes of this analysis consumer income i s M = 45 and the composite input price is w1 = 5
(a) What are t he equilibrium f ree-mar ket price and quant it y of wheat
(b) Is t he supply of wheat at this point price-elastic or price-inelast i c Ex plain your reasomng
(c) W h at is t he t otal net economic sur plus f rom production of wheat and how middotis it distr ibuted bet ween suppliers and demanders
(d) A var iet y of schemes are used in agricult ural markets to enhance producer welf are C onsider a pr ice support of $6 per bushel Explain t he effects of such a policy on the dist ri bution of welfare and economic efficiency Support your answer with quant itat ive est imates of t hese effect s
jgtrice
3 Suppose a monopoly fum faces the following linear demand curve P = a- bQ where P is price and Q is quantity produced and sold The monopolists cost function is
(1) c(Q) = cQ for Q Q c(Q) qo for Q gt Q
where Q is a capacity constraint
a Solve mathematically for the monopolists profit maxumzmg and output as functions of the parameters a b and c Show your work Also illustrate yourresults graphically labeling carefully all relevant information
b Suppose the function P(Q) from part (a) is the demand in the domestic market and that the monopolist can also sell unlimited quantities into the export market at price P E gt c Assuming that quantities exported cannot be resold into the domestic market show both mathematically and on your graph from part (a) how much the firm will sell into the domestic market and into the export market
c Consider now a potential entrant into this market For simplicity ignore the existence of the export market To be able to produce in this market an entrant must incur a one time set up or entry cost of E gt 0 It can then produce according to the cost function in (1) Describe in words andor diagrams how you would model this entrants decision problem You should consider the role of the incumbent monopolist in framing your answer
4 Consider a food processing firm that produces processed product Q from a raw agricultural product Rand a composite marketing input L which we will call labor
a Write a production function and draw some rough sketches of isoquants for each of the following possible production scenarios
1 There is no substitution between R and L in producing Q
ii The production relationship is Cobb-Douglas
b Suppose the true production relationship is Cobb-Douglas and that this firm produces Q exclusively for a large retail grocery chain It has a contract for 1999 to deliver Q units of product to the retailer Assume the firm is a perfect competitor in the markets for its inputs Set up the firms optimization problem given this information and derive the first-order conditions for the problem Defme any notation or terminology you need to introduce Also illustrate the optimization problem with a graph
c Discuss the firms demand function for R given the conditions of the problem in part (b) Specifically what variables determine how much R the firm will utilize
d Re-express the problem in part (b) and take first-order conditions assuming now that the firm is the only buyer of the farm input R which is produced according to the following inverse supply function
w = a + 3R a3 gt 0
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df for dl lor llllmtntor N
100 N
I 16 111 I 12 I y I 2ltI 10 1 10 111 1 1 1 1 111 I Y 1 41 1 4 1 1-10 I 19 1 19 UHI middotW 1171411 147 145
I 176 IJl 170 IUI 167 165 I M 161 1 bull1 101 1 97 191 1 90 11111 1116Hi 4 l)
2 15 212
Ill 1117 201 I I I M4 I IIII I I II
2 911 1 II 1 1 1gt 67 251 11 an z llbull 2 1J ll I 26 1 1 11 1 2
191 lOS 2Kl 1 2 55 146 2 -m 24 2 lU 226 2 2l
US1 i9 l-IS S72 4112 4 1 1 199
I 1 I J I 110 I Y 12Y I 211 I M I 7 1 7 I 6 1 15 1 11 U9 UK IJK 1middot17 U6
17M I 71 1 101 167 1 162 161 I I S1 n6 I 51 IU1115 I ll 1 4 1 1401 19 1 47 146 1 44
I 98
2111 191 1 1111 1 19 210 1 042B 9 I H6 1 gt4 1 MO 119 1 1 1 1 1) 1 1 1 211 2111 2JO 22S 21 2111JOI 2SI 242 2 16 11114 16 140
2M9 2 66 UK 2 40 111 1 1 1 2-1 2 1 Ul167 middot 1 16 126 117 109 ill2 H I 472 4 12 01
114 I 12 111 1 10 129 I 211 UK 16 I 26 126 I 25 I 2 I 42 1 4 1 19 us I H 17 l l6 2 1 111 146 us 1 16 I 1611 165 1 61 I 5Y UK us IS-1 I S1 I S l I t HIIll 11111 186 2 II 1 2 1 11 2 1111 11111 Y I 151 107 199 190 1-JIS 1112 1111) 1 16 1 1S I 71 I 7 1 211 222 2111 215Ull OS 219U7 198 1 14 2 t7 2111 2511 lSO 242 1 16 2 225 22J 219 216 2 IJ UH l09 )02 196HI 129SSl 41gt4 41401
IJJ I I 1 10 129 1 18 117 127 126 1 1gt 125 124 1 24 136 usI 41 141 140 119 UK1 111 146 145 1 69 1 66 16 159 I S1 156 I 51 152 I SO 149 1 19 Ill 1114 1 812 11 1 1 229 216 1116 190 187 100
H14 196 1 9 1 1 81 1 112 119 1 71 I 71 1 11 1 69 167 165 0)219 2 15 1 1 1 JN 295 1 1 1 2242)6 229 V5 1 1 144 IS 210 2 26 1 19 1 1 1 2 1 1 109 U6 U l
0 1 407 I 75 )51 U6 123 312 101
Ul l ltJ 129 Ull 127 1 26 126 IH 1 2-1 1 2-1 12J t n l iM I 45 1 44 1 41 U9 Ull IJ6 us IB
172 167 1 64 ltll I H 15-1 Ul I SU I 1 lt7 1 6 10 10IIIS 182 179 171I 10 11111 1 22K 114 20S 1911 19) 188
2111 1 91 I H4 179 1 16 1 14 110 I M 1 IM I 62 O)1 41 lB 227 221 216 213OS l 2 291 169
270 2-17 210 2 2 1 IJ I I 207 2 Ul ] 0 1 147 J0 1 1 7 )07 298 2910 1 1 56 5 19 451 40
110 I 126 125 124 Ut 12 Ul 121 119 1 19 25IJ6 us 1 14 IJJ Ul U l140 11914425
1117 1113 179 1 66 161 157 1 )4 I l l 14K 141 142 1-11 119 I 111
176 173 1 71IllI 21)9 1 00 192 IM4 179 1611 1 64 I us I 5J U l 100I 21Oi
01
1R4 261 2 14 225 218 2 12 1 08 252 217 229 220 2 1 1 1116 202 1 94 I 92 I IM1 01 664 II 181 lSI 129 1 12 199 2119 280 27)511S
127 125 1 14 122 121 1 11l 119 1 1 7 1 1 1 1 16 1 1 5 1 1 5 I J J 112 I l l IJO 119 119us 1 41 1 4 1 UK 160 154 Ul 1411 144 141 140 116 I 15 I J I ll I 29 10 I
111 168 1 66117IK7 1821 19 2 19 211S 104 111-1 175 110 165 159 156 15] 1 4H 147 144 I-II 1 9 0)
125 2 17 210 104 199 195 1921 1 5 1 16 2 S 110 212 20 194 11111 1 04 17 I 71 I 61 160 til2S6 1SO 1 ll 295 lKl 272 261 165 ) 14
124 1221 16 129 128 127 1 2 1 1 19 1 111 1 17 116 1 14 1 1 1 1 1 1 I l l 1 10
IJJ IJI 1 10 140 119 141 117 132 111 126 1 24 1 2 1 119 10 120I us1 60I gtI 182 111 168 16S 1622 1)10 us
166 161 155 150 146 14 117 115 1 12 1211 125 I Hl 217 109 101 191 IKOl 219 201 I 176 1 101 156 UJ I 1 41 I IX
6 liS I 1701 4 J9 195 14H 296 2 79 266 256 247 240 middot 234
Ul 1 2 1 l iD 1 1 11 116 114 1 12 I l l 1 10 1119 I 0 25 111 IJ9 1111 I J6 I a I J2 I) I 129 128 127 126 US
152 146 1 41 1 111 1 1-1 I I 1211 I 24 122 uu 1 1 7 Ia Ill 2 I I 197 IKII IKO 1 75 170 166 161 160 157
172 162 IS7 IS2 146 141 119 I J2 129 U6 1 22 119 m198 19J I 88 I 114 180OS H9 1 04 265 242 226 2 14 2lt16
01 676 411 lllll l41 J I I 289 271 I 91 1119 I 79 169 I 61 1511 148 144 119 1 11 1 11 01
260 2SO 241 2 14 227
122 1 10 1 16 11-1 1 1 1 1 12 1119 Uli I 07 l tiO 124 124I 12 I J9 IJ7 US I J 111 129 1211 127 125
I II 142 118 114 uo 126 124 1 1 11 1 1 1 1 1 1 I OK Hill 10
i lS 25
IS72 JO 160 us172 167 16J10 2 71 2011 194 U2 1 14 1 12 1 1 1 1 1 1 I IIII 167 152 146 179l 11-1 HXI 260 2J7 221 210 201 1811 181liSmiddot
21KIll 6 6 4bl 3-JII J l2 l 02 2110 264 2 51 241 232 22S 1 10 U6 I ll I S I OtlIIIK104 U9 152 0 1
Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
jgtrice
3 Suppose a monopoly fum faces the following linear demand curve P = a- bQ where P is price and Q is quantity produced and sold The monopolists cost function is
(1) c(Q) = cQ for Q Q c(Q) qo for Q gt Q
where Q is a capacity constraint
a Solve mathematically for the monopolists profit maxumzmg and output as functions of the parameters a b and c Show your work Also illustrate yourresults graphically labeling carefully all relevant information
b Suppose the function P(Q) from part (a) is the demand in the domestic market and that the monopolist can also sell unlimited quantities into the export market at price P E gt c Assuming that quantities exported cannot be resold into the domestic market show both mathematically and on your graph from part (a) how much the firm will sell into the domestic market and into the export market
c Consider now a potential entrant into this market For simplicity ignore the existence of the export market To be able to produce in this market an entrant must incur a one time set up or entry cost of E gt 0 It can then produce according to the cost function in (1) Describe in words andor diagrams how you would model this entrants decision problem You should consider the role of the incumbent monopolist in framing your answer
4 Consider a food processing firm that produces processed product Q from a raw agricultural product Rand a composite marketing input L which we will call labor
a Write a production function and draw some rough sketches of isoquants for each of the following possible production scenarios
1 There is no substitution between R and L in producing Q
ii The production relationship is Cobb-Douglas
b Suppose the true production relationship is Cobb-Douglas and that this firm produces Q exclusively for a large retail grocery chain It has a contract for 1999 to deliver Q units of product to the retailer Assume the firm is a perfect competitor in the markets for its inputs Set up the firms optimization problem given this information and derive the first-order conditions for the problem Defme any notation or terminology you need to introduce Also illustrate the optimization problem with a graph
c Discuss the firms demand function for R given the conditions of the problem in part (b) Specifically what variables determine how much R the firm will utilize
d Re-express the problem in part (b) and take first-order conditions assuming now that the firm is the only buyer of the farm input R which is produced according to the following inverse supply function
w = a + 3R a3 gt 0
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TABLE 01 Percentage points of lhe t distribution
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Pr (r gt 1725) = 005 for df = 20
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l Jfi) 1090
49864914 4914 4985 498S 4986 120 0677 11M9 21671651 19KO 2J58
49111 49H9 49M9 4990 _ 1 1810674 164S 1960 1 126 1576 49H I 49K2 4982 498)
49X7 49K7 49117 49111
N This tahlc gives the area in the riahtmiddothand tail or the distribution tic bull t 0) But since the normal di5tribumiddot Nolt 111e smaller probability shown at the head of each column is the area in one tail the 1in is symmetrical about r- Othe area in the ldtmiddothlnd tail is the same u the area in the corresponding right-hand larser probability is the area in both tails bullbullil rltgtr eumplc PI- 196 S z S 01-04750 TMrdore -196 S t S 196)-2(047S0)-09S Sourr From E S Pearwn and H 0 Hartley eds bull Blomtuila TabftJot Slulislidans vol I
ld ed table 12 Cbullmbridgc University Prns New York 1966 Reproduced by perminion of the middot
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TARLE Ol l lpper percentage poinls or the F distribution (continut-d)
I I
STATISTICAl TA8LfS 681
df for dl lor llllmtntor N
100 N
I 16 111 I 12 I y I 2ltI 10 1 10 111 1 1 1 1 111 I Y 1 41 1 4 1 1-10 I 19 1 19 UHI middotW 1171411 147 145
I 176 IJl 170 IUI 167 165 I M 161 1 bull1 101 1 97 191 1 90 11111 1116Hi 4 l)
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Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
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Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
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d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
4 Consider a food processing firm that produces processed product Q from a raw agricultural product Rand a composite marketing input L which we will call labor
a Write a production function and draw some rough sketches of isoquants for each of the following possible production scenarios
1 There is no substitution between R and L in producing Q
ii The production relationship is Cobb-Douglas
b Suppose the true production relationship is Cobb-Douglas and that this firm produces Q exclusively for a large retail grocery chain It has a contract for 1999 to deliver Q units of product to the retailer Assume the firm is a perfect competitor in the markets for its inputs Set up the firms optimization problem given this information and derive the first-order conditions for the problem Defme any notation or terminology you need to introduce Also illustrate the optimization problem with a graph
c Discuss the firms demand function for R given the conditions of the problem in part (b) Specifically what variables determine how much R the firm will utilize
d Re-express the problem in part (b) and take first-order conditions assuming now that the firm is the only buyer of the farm input R which is produced according to the following inverse supply function
w = a + 3R a3 gt 0
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I 98
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2M9 2 66 UK 2 40 111 1 1 1 2-1 2 1 Ul167 middot 1 16 126 117 109 ill2 H I 472 4 12 01
114 I 12 111 1 10 129 I 211 UK 16 I 26 126 I 25 I 2 I 42 1 4 1 19 us I H 17 l l6 2 1 111 146 us 1 16 I 1611 165 1 61 I 5Y UK us IS-1 I S1 I S l I t HIIll 11111 186 2 II 1 2 1 11 2 1111 11111 Y I 151 107 199 190 1-JIS 1112 1111) 1 16 1 1S I 71 I 7 1 211 222 2111 215Ull OS 219U7 198 1 14 2 t7 2111 2511 lSO 242 1 16 2 225 22J 219 216 2 IJ UH l09 )02 196HI 129SSl 41gt4 41401
IJJ I I 1 10 129 1 18 117 127 126 1 1gt 125 124 1 24 136 usI 41 141 140 119 UK1 111 146 145 1 69 1 66 16 159 I S1 156 I 51 152 I SO 149 1 19 Ill 1114 1 812 11 1 1 229 216 1116 190 187 100
H14 196 1 9 1 1 81 1 112 119 1 71 I 71 1 11 1 69 167 165 0)219 2 15 1 1 1 JN 295 1 1 1 2242)6 229 V5 1 1 144 IS 210 2 26 1 19 1 1 1 2 1 1 109 U6 U l
0 1 407 I 75 )51 U6 123 312 101
Ul l ltJ 129 Ull 127 1 26 126 IH 1 2-1 1 2-1 12J t n l iM I 45 1 44 1 41 U9 Ull IJ6 us IB
172 167 1 64 ltll I H 15-1 Ul I SU I 1 lt7 1 6 10 10IIIS 182 179 171I 10 11111 1 22K 114 20S 1911 19) 188
2111 1 91 I H4 179 1 16 1 14 110 I M 1 IM I 62 O)1 41 lB 227 221 216 213OS l 2 291 169
270 2-17 210 2 2 1 IJ I I 207 2 Ul ] 0 1 147 J0 1 1 7 )07 298 2910 1 1 56 5 19 451 40
110 I 126 125 124 Ut 12 Ul 121 119 1 19 25IJ6 us 1 14 IJJ Ul U l140 11914425
1117 1113 179 1 66 161 157 1 )4 I l l 14K 141 142 1-11 119 I 111
176 173 1 71IllI 21)9 1 00 192 IM4 179 1611 1 64 I us I 5J U l 100I 21Oi
01
1R4 261 2 14 225 218 2 12 1 08 252 217 229 220 2 1 1 1116 202 1 94 I 92 I IM1 01 664 II 181 lSI 129 1 12 199 2119 280 27)511S
127 125 1 14 122 121 1 11l 119 1 1 7 1 1 1 1 16 1 1 5 1 1 5 I J J 112 I l l IJO 119 119us 1 41 1 4 1 UK 160 154 Ul 1411 144 141 140 116 I 15 I J I ll I 29 10 I
111 168 1 66117IK7 1821 19 2 19 211S 104 111-1 175 110 165 159 156 15] 1 4H 147 144 I-II 1 9 0)
125 2 17 210 104 199 195 1921 1 5 1 16 2 S 110 212 20 194 11111 1 04 17 I 71 I 61 160 til2S6 1SO 1 ll 295 lKl 272 261 165 ) 14
124 1221 16 129 128 127 1 2 1 1 19 1 111 1 17 116 1 14 1 1 1 1 1 1 I l l 1 10
IJJ IJI 1 10 140 119 141 117 132 111 126 1 24 1 2 1 119 10 120I us1 60I gtI 182 111 168 16S 1622 1)10 us
166 161 155 150 146 14 117 115 1 12 1211 125 I Hl 217 109 101 191 IKOl 219 201 I 176 1 101 156 UJ I 1 41 I IX
6 liS I 1701 4 J9 195 14H 296 2 79 266 256 247 240 middot 234
Ul 1 2 1 l iD 1 1 11 116 114 1 12 I l l 1 10 1119 I 0 25 111 IJ9 1111 I J6 I a I J2 I) I 129 128 127 126 US
152 146 1 41 1 111 1 1-1 I I 1211 I 24 122 uu 1 1 7 Ia Ill 2 I I 197 IKII IKO 1 75 170 166 161 160 157
172 162 IS7 IS2 146 141 119 I J2 129 U6 1 22 119 m198 19J I 88 I 114 180OS H9 1 04 265 242 226 2 14 2lt16
01 676 411 lllll l41 J I I 289 271 I 91 1119 I 79 169 I 61 1511 148 144 119 1 11 1 11 01
260 2SO 241 2 14 227
122 1 10 1 16 11-1 1 1 1 1 12 1119 Uli I 07 l tiO 124 124I 12 I J9 IJ7 US I J 111 129 1211 127 125
I II 142 118 114 uo 126 124 1 1 11 1 1 1 1 1 1 I OK Hill 10
i lS 25
IS72 JO 160 us172 167 16J10 2 71 2011 194 U2 1 14 1 12 1 1 1 1 1 1 I IIII 167 152 146 179l 11-1 HXI 260 2J7 221 210 201 1811 181liSmiddot
21KIll 6 6 4bl 3-JII J l2 l 02 2110 264 2 51 241 232 22S 1 10 U6 I ll I S I OtlIIIK104 U9 152 0 1
Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
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Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
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EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
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a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
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a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
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bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
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Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
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Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
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Pr (0 z 196) = 04750
Pr (z 196) = 05 - 04750 = 0025
TABLE 01 Percentage points of lhe t distribution
Example
Pr (r gt 2086) = Oo25
Pr (r gt 1725) = 005 for df = 20
Pr(ltl gt 1725) = 010 0 I 715
01$ 010 middot-00
001 0 00010015010 002 0010 0002ozo
J 07H 127116I 11216 14 61657 IIUI1nm
22 279925IXH6 6965UK16 4 lP 00 01 01 1 0765 I 6X 251 lIK2 4541 51141 102 JJ
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071 II 1064 IIOJ 1141 52111106 bull 1440 1941 l7H7 0791 UKl 0171 091ll 09411
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1591 161M 1664
1915 1950 19115 W19
10611144 1179 K u 706 lW7 11601772 451111196
212 071) 421172054 219() 122420XX UHl 111 2262 21i21 1251)-9
0700 J72 1112 222K 2764 11692422
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2794 2121 21152 4025II 0697 1361 1796 22111 2711lt lit
2611 27041gt1 25XU ll7K 1106 1111 190 0695 1156 17H1 21 79 26111 lOSS
ll40 11111 1910
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145 1761 21J5 2614 1621 14 069214115 55-1 2977 7117 461 150111 0 1411
l711I 2602 2 94717011 770 790 Kill lX0 15 0691 1141 175 729 21 lfgt65 16116 JHHH
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4554 505J67H A6K6 4691 4699 4706 22 06K6 I Jl l 1717 2014 l5HK 1XI94656 4664 4671I K 4641 41151K074756 4761 17144744 4750 2J 06KS 1319 250020694716 412 47 111 9 470 4719
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J906 4911 06M4 14212471 27714913 4916 170 4909 27 L14 10524196 489R 4901 49044191 06114925 4927 4929 4931 492 4914 4916 140K2467 27612M U13 1701 204849111 4920 4922
4941 4941 4945 4946 494K 4949 4951 4951 IJII JJ9629 068) 2462 2756204S49111 494025
UJO 1697 2042 2457 17504QSS 4964 lJK S4961 JO 06KJ4960 4961 4961
496 J951J4956 49576
7 10 06KI 10749 1 6K4497 241J i7044970 4971 49724967 496K 1021J966 60 0679 129611 4981
4979 4979 49SO 16714977 2 190 26604976 4977Igrave 12122000 4975
l Jfi) 1090
49864914 4914 4985 498S 4986 120 0677 11M9 21671651 19KO 2J58
49111 49H9 49M9 4990 _ 1 1810674 164S 1960 1 126 1576 49H I 49K2 4982 498)
49X7 49K7 49117 49111
N This tahlc gives the area in the riahtmiddothand tail or the distribution tic bull t 0) But since the normal di5tribumiddot Nolt 111e smaller probability shown at the head of each column is the area in one tail the 1in is symmetrical about r- Othe area in the ldtmiddothlnd tail is the same u the area in the corresponding right-hand larser probability is the area in both tails bullbullil rltgtr eumplc PI- 196 S z S 01-04750 TMrdore -196 S t S 196)-2(047S0)-09S Sourr From E S Pearwn and H 0 Hartley eds bull Blomtuila TabftJot Slulislidans vol I
ld ed table 12 Cbullmbridgc University Prns New York 1966 Reproduced by perminion of the middot
editors and trustees of Biomeria
bull bull bull bull 10 II 30 110 100 500
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195
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229
1 59
245
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142
194
1 94
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117
174
104
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109
284
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1 77
194 175
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149
157
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174 159
145 1 4
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179 147
682 AIPFNHIX ll
TARLE Ol l lpper percentage poinls or the F distribution (continut-d)
I I
STATISTICAl TA8LfS 681
df for dl lor llllmtntor N
100 N
I 16 111 I 12 I y I 2ltI 10 1 10 111 1 1 1 1 111 I Y 1 41 1 4 1 1-10 I 19 1 19 UHI middotW 1171411 147 145
I 176 IJl 170 IUI 167 165 I M 161 1 bull1 101 1 97 191 1 90 11111 1116Hi 4 l)
2 15 212
Ill 1117 201 I I I M4 I IIII I I II
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I 1 I J I 110 I Y 12Y I 211 I M I 7 1 7 I 6 1 15 1 11 U9 UK IJK 1middot17 U6
17M I 71 1 101 167 1 162 161 I I S1 n6 I 51 IU1115 I ll 1 4 1 1401 19 1 47 146 1 44
I 98
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2M9 2 66 UK 2 40 111 1 1 1 2-1 2 1 Ul167 middot 1 16 126 117 109 ill2 H I 472 4 12 01
114 I 12 111 1 10 129 I 211 UK 16 I 26 126 I 25 I 2 I 42 1 4 1 19 us I H 17 l l6 2 1 111 146 us 1 16 I 1611 165 1 61 I 5Y UK us IS-1 I S1 I S l I t HIIll 11111 186 2 II 1 2 1 11 2 1111 11111 Y I 151 107 199 190 1-JIS 1112 1111) 1 16 1 1S I 71 I 7 1 211 222 2111 215Ull OS 219U7 198 1 14 2 t7 2111 2511 lSO 242 1 16 2 225 22J 219 216 2 IJ UH l09 )02 196HI 129SSl 41gt4 41401
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H14 196 1 9 1 1 81 1 112 119 1 71 I 71 1 11 1 69 167 165 0)219 2 15 1 1 1 JN 295 1 1 1 2242)6 229 V5 1 1 144 IS 210 2 26 1 19 1 1 1 2 1 1 109 U6 U l
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270 2-17 210 2 2 1 IJ I I 207 2 Ul ] 0 1 147 J0 1 1 7 )07 298 2910 1 1 56 5 19 451 40
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166 161 155 150 146 14 117 115 1 12 1211 125 I Hl 217 109 101 191 IKOl 219 201 I 176 1 101 156 UJ I 1 41 I IX
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172 162 IS7 IS2 146 141 119 I J2 129 U6 1 22 119 m198 19J I 88 I 114 180OS H9 1 04 265 242 226 2 14 2lt16
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Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
bull bull bull bull 10 II 30 110 100 500
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4 20
116
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df for dl lor llllmtntor N
100 N
I 16 111 I 12 I y I 2ltI 10 1 10 111 1 1 1 1 111 I Y 1 41 1 4 1 1-10 I 19 1 19 UHI middotW 1171411 147 145
I 176 IJl 170 IUI 167 165 I M 161 1 bull1 101 1 97 191 1 90 11111 1116Hi 4 l)
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110 I 126 125 124 Ut 12 Ul 121 119 1 19 25IJ6 us 1 14 IJJ Ul U l140 11914425
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166 161 155 150 146 14 117 115 1 12 1211 125 I Hl 217 109 101 191 IKOl 219 201 I 176 1 101 156 UJ I 1 41 I IX
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152 146 1 41 1 111 1 1-1 I I 1211 I 24 122 uu 1 1 7 Ia Ill 2 I I 197 IKII IKO 1 75 170 166 161 160 157
172 162 IS7 IS2 146 141 119 I J2 129 U6 1 22 119 m198 19J I 88 I 114 180OS H9 1 04 265 242 226 2 14 2lt16
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260 2SO 241 2 14 227
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Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
Part I Answer Question 1 and either Question 2 or Question 3
l You are constructing a simple model of cotton supply for a short-term project you are working on and have formulated the following conceptual model
where yS is bales of cotton ps is the price paid to growers w1 is the labor wage rate w2 is the price of water w3 is pesticide price and DO is degree-days a measure of the growing conditions during the season
(a) Another variable you might include in this specification is annual rainfall (R) whose coefficient if included in the model would be as While you are sure that all the other variables in the model are part of the correct specification you are not sure about annual rainfall As an applied econometrician explain the tradeoffs made to the properties of your estimates if you include or exclude this variable from the specification
If you exclude a relevant variable you risk biasing regression coefficients if the omiued variable is correlated with included variables if you include an irrelevant variable your estimates remain unbiased but may be inefficient One can also always test for relevance of the included regressor(s) via -tests
(b) One must always be concerned with measurement error in empirical models Explain the effects on the properties of your estimates of (a) measurement error in the dependent variable vs (b) measurement error in one or more explanatory variables
Measurement error in the dependent variable will not bias coefficient estimates (provided right-hand variables are measured without error) Errors in measuring independent variables will bias coefficient estimates even if the measurement error is not correlated with the regression disturbance In this simplest case the coefficients will be biased toward zero or attenuated
(c) You estimate the model with R included and the results are
yS = -6354 + 8727ps - 05791w1 - 7907w2 - 09045w3 + 769000 + 0045R (-079) (236) (-1 77) (-234) ( - 190) ( 1 57) (045)
R2 = 0819 n = 56
where the Students-t statistics are given in parentheses and the following sums of squared errors result
middot Total SSY6609 Model SSRS412 Errors SSE 1 197
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
Jvar(fiJ)
(-09045))tJ114
In answering the following parts justify your answers with quantitative test results Qualitative answers may receive partial credit
(i) Is the variable R statistically significant at the a = 05 level
No-- the Student s-t statistic for a test of difference from zero is 045 while the critical value for a = 05 is approximately 201 (interpolating) for 49 degrees of freedom
(ii) Is the overall model significant That is is the hypothesis that a1 = a2 = OJ = a4 = as = a6 = 0 rejected
The F-statistic for model significance in terms of sums of squares is [SSRJ(k-1 )ISSEI(nshyk)]=[(54126)1( 1 19749)]=369 The critical F649 is approximately 230 Thus the null hypothesis of no significance is rejected the model is significant
(iii) The correlation between the price of water (w2) and the price of pesticides (wJ) is zero Do these two variables have the same effect on cotton supply ie are their coefficients equal
The t-test for equality of coefficients is t=(fh - amp4) + var(fx4) - 2cov(ampJamp4middot Because OJ and a4 are uncorrelated cov(ampJfi4) = 0 Because the t-statistic reported for
-ampJ is -234 and t = ampJise(fxJ) se(fiJ) = amp3t = 79071-234 = 338 and var(fiJ) = 3382 = 1 14 Likewisese(fi4) = amp4t = - 9045- 1 90 = 476 and var(fx4) = 4762 = 227 So the t-statistic on the test of equality of coefficients is
t = (-7907 - + 227 - 2 o = 1138476 = 239
This value is clearly less than the critical t = 201 (interpolating) for N-k=49 degrees of freedom and a = 05 We fail to reject the null hypothesis of equality of the effects of the two variables
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
University of California Davis June 23 2000 Department of Agricultural and Resource Economics
MS Comprehensive Exam 2
Directions You have three hours to complete this exam Please write legibly Answers that cannot be read will be regarded as incorrect Use complete sentences and well thought out responses when writing out your answers All questions are weighted equally but the various parts within the question may not be There are two sections to this exam
Section 1 This section consists of questions 1) 2) 3) and 4) Answer question 1) and then select two of the remaining three to answer Do not answer all four questions Questions 3) and 4) begin
middot
on page 4
1 Below is a price dependent demand equation for the price of Merlot grapes The numbers in parentheses below the coefficients are the t-ratios The data used is annual time series
PG = -1899 - 0071 C + 195 Y + 687 PG_1 (1 0) (16) (22) (18)
R2 = 98 DW = 177 N = 14
The variables are defined as follows
PG - Average farm price of Merlot Grapes in Napa county
C - Expected Merlcit grape crush in Napa county Obtained by regressing actual yield on a time trend obtaining the predicted values of the yield and then multiplying these predicted values of yield times bearing acreage
Y - Disposable Personal Income
PG_1 - PG lagged one period
(a) Test the significance of Y at the 5 level (See tables at end) Explain
(b) Is the overall model significant at the 5 level
(c) We are considering testing whether the expected crush (C) has the same effect as the actual crush (C) How would you test this hypothesis Be specific
(d) Suppose you wish to test the hypothesis that C has a greater effect than C How would this test differ from the above equality test
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
(e) The above regression has a number of problems How many can you find What are the specific causes these problems What would you do to correct them
2 There are three arch problems in econometrics (a) autocorrelation (b) heteroscedasticity and (c) simultaneous equation bias For autocorrelation and one of either heteroscedasticity or simultaneous equation bias explain i) the cause or causes of the problem ii) the consequences of the problem iii) the tests to detect the problem and iv) the remedies for the problem Show in analytic detail how each problem effects the properties of OLS estimators and how the remedies mitigate these problems
See page 4 for problems 3 and 4
Section 2 This section consists of questions 5) 6) and 7) Answer question 5 and one of the remaining two questions
5 Consider a man who operates a vineyard part time The farmer has a fixed and exogenous total hours of work per year H These H hours are allocated between farming time h and off farm work as an economic consultant (let e represent hours worked as a consultant) for which the wage is w per hour
a Name and explain at least key three assumptions you would use to simplify the problem in order to answer the following questions
b Derive conditions for an optimal allocation of time to e when e gt 0
c Show conditions for e gt 0
d Illustrate your answers to b and c graphically
e Show graphically and mathematically the effects of an increase in the price of grapes on the amount of work off the farm
f Explain within this simple model what causes aggregate quantity of grapes to rise when the price of grapes rises Show this graphically
6 Consider two firms that contribute to point source water pollution Initially each contributes 15 units of pollution Total cost of pollution abatement for firm 1 is C1 = 2A and for firm 2 it is C2 = 1 5A Remember A measures the units of pollution reduced There is no fixed cost of abatement The regulator decides to reduce the amount of total pollution by 14 units from 30 to 16 We will not consider if this level of abatement is optimal or not because we will not consider the benefits of abatement
a Suppose the government does riot know the abatement cost schedules but decides to impose a pollution tax anyway The tax is set at $36 per unit of pollution Explain if the $36 tax will meet the pollution abatement target
b If the tax of $36 per unit of pollution is not the appropriate tax to achieve the target pollution what would be Explain
2
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
----------------
c
f
c Instead of a pollution tax the regulator decides on a pollution permit scheme Each firm is assigned 7 units of pollution that must be reduced No trading is allowed What is the total cost of abatement Is this system more or less efficient than the pollution tax system Explain
d Now the permits assigned in part c are tradable Is there an incentive to trade What trade will occur Explain
e What is the price of permits Explain
f What is total abatement cost with tradable permits Explain What advantage does the tradable permit system have over a tax scheme
g Now under the tradable permit system the amount of pollution reduction is increased from 14 to 20 and each firm is initially assigned 105 units What trades will occur now What is the cost of meeting the new standard
71 Adverse selection and moral hazard are major considerations in economic situations with costly information Discuss each of the following and explain how either adverse selection or moral hazard both or neither enter the problem
a Loan applicants are routinely asked to provide a credit history
b Home loans are usually made for less than the appraised value of the home
Insurance plans usually require a co-payment of some amount of cost of service
d Crop insurance typically pays only for a share of the loss above a certain threshold of loss
e Cars from private sellers are often cheaper than similar cars at used car lots
Cars sold to private buyers often return a higher price to a private seller than cars sold to a used car lot
g A cooperative may offer dairy farmers the same price for milk picked up at their farm independent of the location of their farm relative to the processing plant
72 Externalities often arise because of informational problems Do the following situations involve externalities Why or why not
h Noise from airplanes affects homeowners in a new subdivision recently built near the airport
i Deaths and injuries occur to occupants in cars driven by drunken drivers
j Farm operators are sometimes harmed when they use pesticides in an unsafe manner
3
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
Section 1
3 Given the primal production problem Max cx
Subject to A x $ b x 0 where c is a nxl vector of net revenues per unit x
A = m x n n gt m and rank A = m
Defining the optimal basic solution as x6 = B1 b
(a) Write out the Dual formulation of this problem using the symbol A for the vector of dual variables Give a brief economic explanation of the dual constraints that you have defmed
(b) Show that optimality of the primal basic solution implies that the dual problem constraints are feasible
( c ) If a single element bi of the right hand side vector in the problem in part (a) is parameterized over a range of values Show
(i ) Why the resulting derived demand function is a step function
(ii) That the dual value on the constraint is equal to the marginal contribution of the resource to the objective function
4
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
Section 1
4 Given a two crop PMP calibration problem with one binding constraint and an associated level of marginal crop production
Max v1x1 + v2x2 - ac1x1- ac2x2 Subject to x1 + x2 b
Where the base observed values are x1 x2 v are fixed revenues per acre and ac are the average costs of production per acre reported by the farmer
(a) Explain whether the base PMP solution can calibrate to the observed cropping levels without using additional information on the marginal activity
(b) State briefly how the problem can be respecified to calibrate using the basic PMP formulae
(c) Briefly compare and explain any differences in policy parameterization between the two models in parts ( a) and (b)
5
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
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9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
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220 12 208 204 1 9520 245349 310 287435 260271 25121
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230 280
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25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
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23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
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5208
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4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
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28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
955 879 859 857 855 853
577 569 563
579 495 477 474 456 443 436 453 439 387 384 377
559 474 397 379 373 364 357 344 34 1 334 323 384 335 315 363 314
348 333 277 254 398 336 290
475 349 243 234 13 303 234
334 239
454 259 233 449 363 235 445 359 249 245
355 254
39 199 237 234
24 179
339 177 274 199 175
173 334
29 333 235 194
234 179 174 253 1 75 147
1 55 135 237
158
Upper 5 Points of the F-Distrlbutlon
X83 15I 4 6 7 9 10 12 24 402 5 30 60 12020
2368 2389 25 1 1 1995 2157 2246 2340 24192302 2439 2459 25011 6 1 4 2405 2491 25222480 2533 2543I 1916 1925 1935 19372 1900 1930 1933 1943 19471851 1938 1941 1945 1945 1946 19481940 1949 1950
3 862889928 894 885912 901 881 874 870 8661013 864bull 659 609 604694 639 616 586 580600 596 591626 575 572771 566
541 488 482519 50556
661 468 462 450 446453 440 476514 421599 428 415 374400 394 381 370410 406 367
7 435 412 387 330351 338 327368 407532 4468 358 350 328369 344 322 308 304 301312 297339 293
9 386512 426 337 329 294 279286 283290348 323 307 301 275318 271
274 266 262270410 37110 496 322 314 307 285291302 298 258 u12
359 295 272484 320 309 301 285 279 265 261 257 253 249 245 240 326 2853 1 1 291300389 262 254 247251275 269 238280 230 318467 381 341 292 283 277 253 246 230271 225267 260 242 238 221
14 460 374 276 253270 265 260 2223 1 1 296 285 246 231 227235 218 213
21 1 207306 264368 329 279 271290 240 22015 254 248 229 225 216 259324 301 274285 26616 215249 242 224 219 211254 228 206 201
320 261 255296 27028117 238 231 223 210 206215219 201 1961 922 1 1215 206316 258 234 227 219293 277 251 246266 24118 441 202 197
193274 26319 352 313 290438 248 198242 216238 231 223 211 207 203 188
220 12 208 204 1 9520 245349 310 287435 260271 25121
235 228 190 184 307432 284 249 2423-17 268 257
282 246 240 218 192210 201 196205 1 89
232 225 187 18122 21
305 223 207 1942Hi 203 1983A-1 266 255 1 84 1781 81 I76
230 280
I)Q-1 8426
342 303 253 244 237 191264 227 220 213 196 186232 205 201 340 301 278 251 236 2 1 1 1 89 1 84 225 203262 242 230 218 1 94 1 98 173
25 -12-126
299 276 249 240 234 216 187 182224 209 201 196 192260 228 1 71 337 298 232 207222 215227 185423 259 247 239 195 190 180 169
1 671651 64
17127 421 335 296 231 211 179273 246 237 206 197 188 184257 225 193220 1771 87 182196229 212295 236 2 19 204 19122427 1 256 245
218 42028
181 175418 293 270 255 243 228 222 210 203 170190 185
292 174184 1 7930 4 17 332 269 242 227 209 201 193 1 89 162233253 21 216 1681 64192408 2lt15323 284 184
166
16940 225 200218 212 208 151261 159 1531 50
315 276 217 210 192204 1 99 16560 -100 1 20 392
184 139237 225 170 307 268 245 191 175217 209 202 183229 196 143161 125 300 260 221 210 194 1 75 201 183188 167X 384 139157 152 146 132 122 100
Vote m = degrees of freedom for the numerator n = degrees of freedom for the denominator
ource Handbook oThbles or Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
7453
5959
4437 695
23 3745
Percentage Points of the t-Distrlbutlon
1T = 04 025 01 005 df 2T = 08 05 02 01
1 0325 1000 3078 6314
2 289 0816 1886 2920
0025 005
12706
4303
001 002
31821
6965
0005 001
63657
9925
00025 0001 00005 0005 0002 0001 12732 31831 63662
14089 22327 315983 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 4773 5893 68696 265 718 1440 1943 2447 3143 3707 4317 7 263 711 1415 1895 2365 2998 3499 4029
5208
8 262 706 1397 1860 2306 2896 3355 3833 4501
4785 5408
9 261 703 1383 1833 2262 2821 3250 3690 5041
4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 3497 12 259
4025
13 259 694 1350 1771 26502160 3012 3372
1356 1782 2179 2681 3055 3428 3930 4318
14 258 692 1345 1761 2145 2624 2977 3326 3787
3852 4221 4140
15 0258 0691 16 258 690 17 257 689
18 257 688
19 257 688
20 0257 0687 21 257 686
1341
1337
1333
L330
1328
1325 1323
1753
1746
1740
1734
1729
1725
1721
2131
2120
2110
2101
2093
2086
2080
2602
2583 2567
2552
2539
2528
2518
2947
2921
2898
2878
2861
2845 2831
3286
3252
3222
3197 3174
3153
3135
3733
3686 3646
3610 3579
3552
3527
4073 4015
3965
3922 3883
3850
381922 256 686 1321 1717 2074 2508 2819 3119 3505 3792
256 685 1319 1714 2069 2500 2807 3104 3485 3767 24 256 685 1318 1711 2064 2492 2797 3091 3467
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 3725
27 256 684 1314 1703 2052 2473 2771 3057
26 256 684 1315 1706 2056 2479 2779 3067 3435 3707 3421 3690
28 256 683 29 256 683
30 0256 0683 40 255 681
60 254 679
120 254 677 253 674
1313
1311
1310
1303
1296
1289
1282
1701
1699
1697
1684
1671
1658
1845
2048
2045
2042
2021
2000
1980
1960
2467
2462
2457
2423
2390
2358
2326
2763
2756
2750
2704
2660
2617
2576
3047
3038
3030
2971
2915
2860
2807
3408
3396
3385 3307
3232
3160
3090
3674
3659
3646 3551
3460
3373
3291
Note 1 T = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and Pft lt - 2060 or t gt 2060) = 005 Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966 Reprinted with the pennission of the Biometrika Trustees
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
7453
4773
5959
3355
3497
14
15 3733
23
3435
Percentage Points of the t-Distribution
1T =OA 0-25 df_ 2T= 08 05
1 0325 1000
2 289 0816
0-1 02
3078 1886
o_o5 01
6314
2920
0-025 005
12706 4303
o_ol 002
31821
6965
0-005 0-01
63657
9925
0_0025 0001 0-0005 0005 0002 0001 12732 31831 63662
14089 22327 31598 3 277 765 1638 2353 3182 4541 5841 10214 12924 4 271 741 1533 2132 2776 3747 4604 5598 7173 8610
5 0267 0727 1476 2015 2571 3365 4032 5893 68696 265 718 1440 1943 2447 3143 3707 7 263 711 1415 1895 2365 2998 3499
4317 5208 4029 4785 5408
8 262 706 1397 1860 2306 2896 3833 4501 50419 261 703 1383 1833 2262 2821 3250 3690 4297 4781
10 0260 0700 1372 1812 2228 2764 3169 3581 4144 458711 260 697 1363 1796 2201 2718 3106 4025 443712 259 695 1356 1782 2 179 2681 3055 3428 3930 4318 13 259 694 1350 1771 2160 2650 3012 3372 3852 4221
258 692 1345 1761 2145 2624 2977 3326 3787 4140
0258 0691 1341 1753 2131 2602 2947 3286 4073 16 258 690 1337 1746 2120 2583 2921 3252 3686 4015 17 257 689 18 257 688 19 257 688
20 0257 0687 21 257 686
1333
1330
1328
1325
1323
1740
1734
1729
1725
1721
2110
2101
2093
2086
2080
2567
2552
2539
2528
2518
2898
2878 2861
2845
2831
3222 3646 3965 3197 3610 3922 3174 3579 3883
3153 3552 3850 3135 3527 3819
22 256 686 1321 1717 2074 2508 2819 3119 3505 3792 256 685 1319 1714 2069 2500 2807 3104 3485 3767
24 256 685 1318 1711 2064 2492 2797 3091 3467 3745
25 0256 0684 1316 1708 2060 2485 2787 3078 3450 372526 256 684 1315 1706 2056 2479 2779 3067 370727 256 684 1314 1703 2052 2473 2771 3057 3421 3690 28 29
256
256 683
683 1313
1311 1701
1699 2048 2045
2467
2462 2763 2756
3047
3038 3408
3396 3674
3659
30 40 60
120
0256
255
254
254
0683
681
679
677
1310
1303
1296
1289
1697
1684
1671
1658
2042
2021
2000
1980
2457
2423
2390
2358
2750
2704
2660
2617
3030
2971
2915 2860
3385 3307 3232
3160
3646
3551 3460
3373 253 674 1282 1645 1960 2326 2576 2807 3090 3291
Note IT = area under one tail 2T = area under both tails
For 25 degrees of freedom (df) P(t gt 2060) = 0025 and P(t lt - 2060 or t gt 2060) = 005
Source Biometrika Tables for Statisticians Vol I Edited by E S Pearson and H 0 Hartley 3rd edition 1966
Reprinted with the pennission of the Biomet ka Trustees
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
I 7
955 879 639 577 575
579 519 495 488 474 453 443 440 453 439 394 377 370 379 373 357 341 327 323 350 344 339 335 275
333 277 274 295 279 475 349 3U 275 13
374 334 454 259 254 233 449 274 259 249 445 359
349 237
1 94 253 237 L73
249 274
175 335 334 333 255 235
179 174
392
139
Source Handbook o Thbles fo r Mathematics edited by Robert C West and Samuel M Selby 1970 Reprinted with the permission of the CRC Press Inc
middot - degrees of freedom for the denominator
Upper 5 Points of the F-Distribution
1 2 3 4 5 6 8 9 10 12 15 20 24 30 40 60 120 I 1614 1995 2157 2246 2302 2340 2368 2389 2405 2419 2439 2459 2480 2491 2501 2511 2522 2533 25432 1851 1900 1916 1925 1930 19333 1935 1937 1938 1940 194 1 1943 1945 1945 1946 1947 1948 1949 1950
694 1013
4 7 7 1 928 912 862 859 857 ssmiddots901 894 889 885 881
596 874 870 866 864 853659 626 616 609 604 600 591 586 580 572 569 566 563
514 5 661 67
599 428 421 541 505
406 482 477
364 468 462 456
374 450 446 436
412 397 476 415
351 410 400
334 387 381 367
293
559 474 435 387 328
368 364 312
344 297
338 3308 532 446 407 384 369 358 9 512 426 386 363 348 337 329 323 318 314
322 315 308 304 301 307 301 294 290 286 283 279 271
10 496ll 4 1 0 3 7 1 348
320 322
254314 307 265 261
302 298 291 285 270 266 262 258484 398 359 336 285
309 301 269I2 290 285 272 257 253 249 245 240
467 341 303 389 381
326 318 267
300 291 280 262 254 251 247 243 238 234 230 14
292 283 277 271 260 253 246 242 238 234 230 225 221460 311 296 285 276 270 265 260 253 246 239 235 231 227 222 218 21315 368 329 306 290 279 271 264 254 249
228 248 240 229 225 220 216 2 1 1 20716
270 266 261 255
363 324 301 285 245
242 235 224 219 215 2 1 1 206 201 441 355
17 18
320 296 281 238 231 223 219 215 210 206 201 196316 293 277 266 258 251 246 241 234 227 219 215 2ll 206 202 197 19219 438 352 313 290 274 263 254 248 242 238 231 223 216 211 207 203 1 98 193 188 20 435 21 432 347
3 1 0 287 271 260 251 245 239 235 228 220 212 208 204 199 195 190 184307 284 268 257 249 242 234
232 225 218 210 205 201 196 192 187 181 430 344 305 282 266 255 2462223 2AO 230 223 215 207 203 198
191 189 184 17842 342426 303 2 0 264uo 251 2Ui
244 242
232 227 220 213 205 201 196 186 18124 176301 27R 262 230 225 218 211 203 198 194 189
17 1
184 17925 424 339 299 276 260
247 240 234 228 224 216 209 201 196 192 187 182 177
273 2627 423 337 298 259 239 232 227 222 215 207 199 195 190
173 185 180 169
167 4 2 1 296 257 246 237 23128
29 225 220 213 206 197 193 1 88 184 179420 295 271 256 245 236 229 221 219 212 204 196 191 187 182 177 171 165293 270 243418
181 I75 170 164228 222 218 210 203 194 190 185 30 bullU7 332 292 269 253 242 233 227 221 216 209 201 193 189 184
169 174 168 162 1 64 158 1 5 1
40 408 323 284 261 245 234 225 218 212 208 200 192 184 179 165
60 400 us 276 253 237 225 2 1 7 2 1 0 204 199 192 184 175 170 159 153 147 139120 X 384
307 268 245 229 217 209 202 196 191 183 175 166 161 155 150 143 135 125300 260 237 221 210 201 194 188 1 83 175 167 157 152 146 132 122 100
Note m = degrees of freedom for the numerator bull
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
------------------------------ -
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam December I I 200 I
You have three hours for this exam Choose four of the following six questions to answer Read the questions through first and watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences
Question 1
(a) Given the linear program farm model specified below
Max p x - cx Subject to A x s b
Where x is an n x I vector of production activities p and c are conformable vectors of prices and variable cost coefficients respectively b is an m x I vector of fixed inputs and A is an n x m matrix of input requirement coefficients
(i) Define the DUAL to the above problem
(ii) Briefly comment on the connection between the dual problem and the Dual Complementry Slackness condition
( i i i) If you had to calculate the derived demand for a single input b describe the type of the function you would obtain from the problem
(a) Now use the information that The product prices p are changed according to an aggregate quantity-dependent demand p = + Ox _
Assume the farmers act as perfect competitors
(i) Show how you would reformulate your problem an give an analytic economic explanation of the new objective function (If stuck use graphs for partial credit)
(i i) Derive the dual to the problem in part b (i) Showing your steps or describe the principle steps in the derivation of the dual of a nonlinear problem
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
middot
Question 2
A peasant works on a small plot of land producing com C according to a diminishing returns production function
middot
where -
He sells the corn at a price Pc and then buys generic food in quantities F for which he pays PF per unit He maximizes utility which is Cobb-Douglas in leisure and food with
- middot coefficients and y respectively subject to a time constraint [T= +L) He is contemplating an alternative income earning activity namely working in a nearby village for a wage rate w
A Write down a simple economic model that would describe the choice between village or forest work for the peasant Show how the wage rate affects labor time in the village occupation How does it affect food consumption Utility
B Under what circumstances would we expect the individual to switch from farming to village labor work Would the peasant end up working more or fewer hours if he made the switch to the village job
2
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
Question 3 In the Nicholsen micro text a general production function specification
q = A(t)f(KL)
(where q represents output K capital and L labor) was totally differentiated to yield a productivity growth expression (in proportional terms) of the form
dln qdt = din Adt + (dqdK)qdKdt + (dqdL)qdLdt or
dln qdt = din Adt + eqKbulldln Kdt + eqKbulldln Ldt
where eqK for example is the output elasticity din qidln K=dqldKbull(Kfq)
This in turn can given some assumptions be written as
din qdt = din Adt + SKmiddotdln Kdt+ SLmiddotdln Ldt
where SK for example is the revenue share PKbullKlpqbullq
a) In graphical form what is being represented here If you could estimate the components of this expression what kind(s) of question(s) could be asked
b) What parts of this expression would be computable directly from the data and what would you need to estimate
a) What allows you to do the substitution required to get the shares (such as SK) from the output elasticities (such as EqK) What assumptions are required Are these assumptions also necessary if you estimate these values parametrically
d) What form would the output elasticities take if the functional form for f(KL) were assumed to be a simple Cobb-Douglas f(KL)=K0 Can you estimate the parameter values or output elasticities directly from the data How might one estimate them parametrically if a more complex assumption about the form of the production function were made
e) I f you wrote the production function as q=f(KLt) what difference would it make
f) Say that the values for the components of this expression are din qdt = 275 percent per year SK= 035 din Kdt = I 75 percent per year SL= 065 din Ldt = I percent per year
as in the Nicholsen text What can you say about each individual number (interpret them) and what can now be said to answer the overall question implied by this experiment
3
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
1 3
1 4
2 4
2 5
3 0
3 2
3 8
4 0
4 1
4 4
52 5 3 54 5 5 5 6
Q 3
Q3
4 7
Question 4 The Cache Creek Sand amp Gravel Co has signed six contracts to deliver gravel to six construction sites numbered S I S6 They operate three gravel quarries located along -
Cache Creek and denoted Q l - Q6 The objective is to minimize the total cost of transport
Use the Gams print-out below to answer the following questions
7 SETS J QUARRIES I Ql Q 2 Q3 I 8
9 K S I TES I 5 1 1 0
I
1 1 PARAMETERS C ( J ) QUARRY S U P PLY CAPACITY
I Q1 550
Q2 4 00
8 0 0 I1 5
1 6
1 8 PARAMETERS D ( K ) S I T E MINIMUM DEMAND 1 9 5 1 3 0 5 52 2 60 53 1 8 0 54 2 0 0 55 4 1 0 56 3 2 0 I 2 0
2 1 TABLE T ( K J ) TRUCKING COST FROM QUARRY TO S ITE 2 3 Q 1 Q2
51 6 9 5
52 7 9 1 0
2 6 5 3 7 8 4
2 7 54 8 8 7 2 8 55 3 1 0 8
2 9 5 6 1 1 4 1 0
3 1 VARIABLES X ( J K ) QUANTITIES QUARRY TO SITE
TCOST TOTAL COST 3 4
3 7 EQUATIONS
QUARRY ( J ) QUANTITY OUT OF QUARRY 3 9 DEMAND ( K ) QUANTITY LOWER BOUND
COSTS SUM OF COSTS
4 2 QUARRY ( J ) SUM ( ( K ) X ( J K) ) =L= C ( J ) 4 3 DEMAND ( K ) SUM ( ( J ) X ( J K ) ) =G= D ( K)
COSTS SUM ( ( K J ) T ( K J ) X ( J K ) ) =E= TCOST 4 5
4 6 MODEL CACHE ALL
4 8 SOLVE CACHE U S I NG L P M I N I M I ZING TCOST
4 9
bull bull bull bull MODEL STATUS 1 O PTIMAL
OBJECTIVE VALUE 8 2 5 5 0000
EQU QUARRY QUANT ITY OUT OF QUARRY LOWER LEVEL U P PER MARGINAL
Ql - I NF 5 5 0 0 0 0 5 5 0 0 0 0 - 3 0 0 0
Q2 - I NF 4 0 0 0 0 0 4 0 0 0 0 0 - 1 0 00
Q3 - I N F 7 2 5 0 0 0 8 0 0 0 00
4
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
Q 3 S 4
middotbull
EQU DEMAND QUANT ITY LOWER BOUND
LOWER LEVEL UPPER MARGINAL
S 1 3 0 5 000 3 0 5 0 0 0 + I NF 5 0 0 0
S 2 2 60 000 2 6 0 0 0 0 + I N F 1 0 0 0 0
S 3 1 8 0 00 0 1 8 0 0 0 0 + I N F 4 00 0
S 4 2 0 0 0 0 0 2 0 0 0 0 0 + I N F 7 0 0 0
S 5 4 1 0 00 0 4 1 0 00 0 + I N F 6 00 0
S 6 3 2 0 00 0 3 2 0 0 0 0 + I N F 5 00 0
- - - - VAR X QUAN T I T I ES QUARRY TO S ITE
LOWER LEVEL UPPER MARGINAL
Q l S 1 +INF 4 0 0 0
Q l S 2 1 4 0 00 0 + I N F
Q1 S 3 + I N F 6 0 0 0
Q 1 S 4 + I N F 4 0 00
Q l S 5 4 1 0 00 0 + I N F
Q l S 6 + I N F 9 000
Q2 S 1 + I N F 5 000
Q 2 S 2 8 0 00 0 + I N F
Q2 S 3 + I N F 5 0 00
Q2 S 4 + I N F 2 0 00
Q2 S 5 + I N F 5 0 00
Q2 S 6 3 2 0 0 0 0 + I N F
Q 3 S 1 3 0 5 0 0 0 + I N F
Q3 S 2 4 0 0 0 0 + I N F
Q3 S 3 1 8 0 0 0 0 + I N F
2 0 0 000 +INF
Q3 S 5 + I N F 2 000
Q 3 S 6 + I N F 5 000
I I f you were forced to deliver ALL the contracted gravel for site 4 from quarry I how much extra would it cost the company
2 If Yolo county let you expand the capacity of one of the quarries
(i) Which quarry would you expand and why
(ii) How much could you profitably spend per ton capacity on expansion
3 I f the trucking costs from quarry I to sites 3 and 4 decreased by $4 ton how many of the optimal activities would change and why
4 Use the numerical results to show that Site 2 i s in spatial equilibrium at the optimal solution
5
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
Question 5
Assume that you are a farmer growing grapes for wine production
Under the first few years of your production the grapes are of only average quality and they are sold into a competitive market with many buyers As your vines mature the quality of the grapes improves
Show how the opportunity to sell your grapes to a buyer with market control would affect your profit opportunities Assume in particular that you have an opportunity to sell to a wine producer who is a monopolist Compare and contrast your options and discuss how market power in the output market affect the demand for your grapes
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
-49
37
Question 6
Say that you have estimated a cost function of the form (as was done by Berndt and Wood 1 975)
In TC = In CXo + In Y + aK In PK + a L in PL+ aE In PE + aM In PM + YKK (In PKi + YKL In PK In PL + YKE In PK In PE + YKM In PK In PM + YLL (In
_
Pd + YLE In PL In PE2 2+ YLM In PL In PM + _ YEE (In PE) + YEM In PE In PM + _
_
YMM (In PM)
where TC is total cost Y is output PK is the price of capital PL the price of labor PE the price of energy and PM the price of materials
and have generated the parameter estimates (with t statistics in parentheses)
aK 0564 (36571) YKM -0 1 53 (- 1 30 1 )
aL 2539 ( 1 1 2359) YLL 0739 ( 10 1 59)
aE 0442 (38078) YLE -0043 (- 1 438)
aM 6455 1 73348) YLM -0697 (-5942)
YKK 0254 (3 455) YEE 02 1 4 (2393)
YKL 0001 ( 022) YEM -0068 (-527)
YKE -0 1 02 (-2444) YMM 09 1 8 (3243)
and the price elasticities of demand (for 1 97 1 )
EKK -44 EEK - 1 7 EKL 40 EEL 20 EKE - 1 6 EEE EKM 30 EEM 46
ELK 05 EMK 02 ELL -45 EML 1 8 Eu 03 EME 03 ELM EMM -24
where ELL for example is Clln LClln PL and L is the demand for labor (not the share)
7
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
middot
a) What kinds of questions can be asked using such a model What graphs might be used to represent these questions
b) Estimating equations for such a model are typically written in the form of input shares based on Shephards lemma Write out the estimating equation for L and indicate how the parameters can be interpreted How (statistically) significant do the relationships appear to be
c) Based oil the elasticity estimates reported above can you say anything about whether required curvature conditions for the cost function to be valid are satisfied
d) Can you say anything about substitutability between the inputs Complementarity
e) From the cost function and its parameter estimates can you say anything about returns to scale If not why not and what extensions would you make to consider returns to scale What types of returns to scale questions might you want to ask
f) Can you ask questions about technical change with this function and estimates If so what types of questions If not what extensions to the model would be necessary to ask such questions
In Bhuyan and Lopez ( 1 997) they added an extra estimating equation to a similar cost function model which they called a share of output that looked something like
Sy = (av + Yvv In Y + IYvt In Pt + PTy T )( 1 -lt1gtT])
where Sv=PvYTC (total revenues divided by total costs) P is the price of input i T is a time counter (representing technology) ltIgt is a conjectural elasticity and T] is the output demand elasticity
I n the Berndt-Wood notation this would instead become
Sv = I ( l -lt1gt11)
g) Why does this equation look so different from the one in the Bhuyan and Lopez paper
h) Why would you want to estimate such an equation What type of question(s) is it designed to answer And what kind of function(s) would it come from
8
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
1(
middot
a)
e)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 22 200 I
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) count equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
Suppose that you are contemplating buying a trucking company Before you buy you need to determine what your cost function will be Your output is total truck miles per day M Truck miles per day depend (via an identity) on average speed S per truck (measured in miles per hour) times total hours H driven Your costs consist of drivers wages ($20 per hour) plus gasoline costs ($1 per gallon) Gasoline consumption per truck mile is given by
middot
G=2+005)S
Write down the total cost function expressing total costs as an explicit function of both wage and gasoline components of cost
b) Now assume that you wish to direct your drivers to maintain an average speed to minimize total costs Find the optimal speed
c) Now derive the cost function under the assumption that speed is chosen optimally
d) Show how a speed limit of 55 mph would affect the trucking companys cost function
f)
Show how the cost function is affected by an increase in the gasoline price
Are there any incentive problems associated with hiring drivers at a fixed wage As an owner of the firm what kinds of actions could you take to take to ensure that your drivers operate in your best interests
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
r
bull
Question 2
Given a standard PMP calibration problem with one binding resource constraint and observed levels of crop production f- The LP model with calibration constraints added is defmed as
Max V cx middot
Subject to Ax 5 b
I X 5 X + pound
x O
Where v IS a n x I vector of gross revenues c is a n x I vector of average costs x is a n x I vector of activity levels A is an m x n matrix of leontieff production coefficients b is a m x I vector of input resources available is a n x I vector of observed activity levels and E is a corresponding vector of small perturbation variables
a) Analytically explain whether a PMP calibrated model solution will optimize at the observed cropping levels without using additional information on the marginal aty
b) If you knew the elasticity of supply of the marginal crop show how you could use this information in the calibration of the model
c) Compare and explain the properties of the two models calibrated by the two methods in part (a) and (b) Will the dual values for the binding constraint be the same for the models in part (a) and part (b) Explain analytically
d) If the PMP models in part (a) (b) are used to generate points on a derived demand function for the constraining input by reducing the right hand side constraint value and a supply function by changing the price of the marginal crop Describe analytically
(i) The difference in the resulting derived demand function between the LP model and the part (a) PMP model
(ii) The difference in the supply function for the marginal crop between the Part (a) PMP model and the part (b) PMP model
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
a)
c)
Question 3
In 1888 the US Senate adopted a resolution appointing tive senators to investigate whether there exists or has existed any combination of any kind by reason of which the prices of beef and beef cattle have been so controlled or affected as to diminish the price paid the producer without lessening the cost of meat to the conswner
Briefly in words What are the fundamental economic problems that would cause this kind of concern How do our theoretical economic models need to be adapted to consider this issue and why might it be considered a problem Do these types of concerns necessarily imply that something is wrong with the beef market Who might be helped or harmed by the implied market power
b) Theoretically What is the market failure that is implied here Show using graph(s) or equation(s) (or a combination if you wish) what characteristics a market that stimulated this kind of concern might have In what sense is your specification different from a standard perfectly competitive model Distinguish what might be happening in output as compared to input markets
Model and empirical implementation If you were assigned to carry out such an investigation what would you be looking for What part of the graphs or equations you wrote down for (b) would you want to quantify What kinds of measures (broadly speaking - not detailed equations) would you be trying to generate What kinds of equations would you want to specify and estimate
d) In Bhuyan and Lopez (1 997) a fundamental estimating equation in their model was
where Sy=PQC (total revenues divided by total costs) Q is output Wi is the price of input i T is a time counter ctgt is a conjectural elasticity and 11 is the output demand elasticity If you estimated this equation what parts of it would help you to answer the questions you posed above What would you be trying to measure by estimating this equation and what kinds of function(s) does it come from
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
bull
Question 4
Consider a production process for producing a single output Q such as plywood that is sold in a competitive market at some price P Asswne that the product is produced with a single input labor (L) with a production function Q=f(L) There is also a fixed cost F associated with production Now consider the following two ways to orgailize the firm
Conventional Managers of the firm maximize profits from production which consists of gross revenues less labor costs and fixed costs The firm hires labor at some competitive wage w
Cooperative The workers run the firm The laborers decide how much labor to employ and each member of the coop gets an equal share of total profits which consist of gross revenues less fixed costs
Develop two behavioral models that allow you to compare the two different institutional structures for managing a firm Use your models to deduce some predictions that could be empirically tested For example how would the two firms differ regarding total labor employment What are the respective elasticities of labor utilization with respect to output prices What about the supply response effect of fixed costs And can anything be said about wages or the elasticity of labor supply with respect for wages
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
I
Question 5
Given that you have product demand functions for j regions xd- = amiddot + 0 5 smiddot pd J J bull 1 J
and supply functions for i regions ( note j and i are often the same region) XSi = bi + 05 gi PSi
If the transaction costs of trading from i to j are known and defined as Cij and xdi XSi are the demand and supply quantities in regions i and j respectively and pdi and pdi are the respective demand and supply prices
a) Analytically formulate the simplest model that will maximize the benefits of trade between regions i and j
b) What are the analytical first order conditions for this model
c) Show how can you use the frrst order conditions to obtain the quantities traded between regions and show that the optimal trades are consistent with trade theory
d) If a nonlinear programming problem is generally specified as minimizing Minimize f(x) Subject to X E n where x = nx I vector and il is the feasible solution set
(i) State the first order conditions for a local minimum for the above problem Define the terms used
(ii) If there exists another feasible solution at point x(a) where x(a) is at a distance a (a gt 0) in a direction d from the minimum point Xo Formally show that the frrst order conditions ensure that the value of f(x) is a local minimum
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
middot bull
Question 6
Susan Capalbo (a UCD graduate) estimated a number of production-oriented models to determine how well they represented agricultural production processes and how they compared in terms of satisfying required conditions to make the model consistent with economic theory One of the models (like the paper by Berndt and Wood you saw in your econometrics class) resulted in estimating share equations derived using Shephards lemma For a translog function these equations are
S = a + 1 yj In W + po In Q + a T
where S is the cost share (ith input cost divided by total cost) W is the price of input i Q is output and T is a time counter The estimation results for two of the resulting equations those for capital (K) and labor (L) were
SK = 0292 - 0045 In WL- 0050 In Ww + 0160 In WK- 0065 1n WM - 027 1 In Q + 0005 T (0005) (0014) (0005) (00 1 1 ) (0006) (0084) (0002)
SL = 0220 + 0133 In WL - 0070 In Ww - 0045 ln WK - 0017 In WM + 0497 1n Q - 0014 T (0007) (0022) (0006) (00 1 4) (00 1 0) (0 1 25) (0002)
where standard errors are in the parentheses and LD=l31d M=materials
a) What relationships are we trying to quantify here in terms of your standard graphs from intermediate economic theory
b) What optimization behavior by firms is implied How would you write out the optimization problem
c) Why might a translog functional form be assumed for analysis compared to say a linear or Cobb-Douglas form And why do shares end up on the left hand side of the estimating equations with this form
d) What questions might you use this type of model to answer
e) In terms of interpretation of the results (i) Why is the coefficient on In WL in the SK equation equal to that on In WK in
the SL equation (ii) Do the positive signs on In WK in the SK equation and In WL in the SL
equation mean anything (iii) What do the different signs on the coefficients for In Q in the two equations
imply In the paper Susan also indicated that from the total cost function the elasticity (illn TCilln Q)1 was insignificantly different from zero Is this related to these coefficient estimates What does this elasticity represent and what might its insignificance imply
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
a)
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam June 192002
You have three hours for this exam Choose four of the following six questions to answer Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one If you dont get the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with full sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Question 1
I Production econometrics emphasizes smooth functional forms for the inputs and outputs of a farm Farming systems models such as linear programming and integer programming emphasize discrete choices and step functions for inputs and outputs
If the truth is closer to the view of farming systems models what general conclusions from the production econometrics approach are most suspect and which are most likely to be valid
b) If the truth is closer to the view of production econometrics what general conclusions from farming systems models are most suspect and which are most likely to be valid
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
Question 2
Various types of technical change have impacted agricultural production processes and markets in the past few decades One of these is the development of genetically modified crops through biotechnology Say that this technical change takes the form of a seed that is more productive How might one represent the impact on farmers production and profitability from adoption of such seeds In particular
a) Using math or diagrams or a combination outline in general how an economist might think of technical change affecting firms (farms) production processes
b) How might you set up an equation (or equations) to represent the impact of technical change (or productivity) on firms
c) How might you estimate such an equation (or equations) That is could you estimate it nonparametrically Parametrically What is the difference
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
c) Explain in words how positive mathematical programming smooths the output and input curves of traditional programming models
d) I f you wanted to represent and estimate technical change impacts specifically from biotech adoption how might you do so What kinds of functions equations and estimating methods would you use What kinds of issues might you want to think about and incorporate in your model and estimating procedures to appropriateiy measure such impacts
e) How would you model the welfare impacts of this technical change What equations would you wish to model and measure and what types of measures might be used to represent the (consumer and producer) welfare effects
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
Question 3
A farmer with initial wealth of $ 1 00 must decide which crop to plant this year Her choice is influenced by the weather - which is Good with probability 07 and Bad with probability 03 She must choose one (and only one) of the following two options
Option A Plant corn and earn a profit of $800 with certainty Option B Pant cotton and earn a profit of $ 1 500 if the weather is Good but
lose $ 100 if the weather is Bad
Her preferences over risky terminal wealth W (initial wealth plus farm profit) are -
described by the following Von Neumann - Morganstern utility index
U(W) = W
a) What is the expected profit associated with each option
b) Which option will the farmer choose
c) Do this farmers preferences over risk exhibit increasing decreasing or constant absolute risk aversion
d) If the only difference across farmers is initial wealth (ie they have the same preferences and technology) would you expect all farmers to make the same
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
Extra How would you expect the distribution of wealth to evolve over time given your answers to ( 1 -d) and how would the existence of insurance
choice Explain your answer How would you expect the distribution of wealth to evolve over time
e) Assume (for part e only) that the farmer chooses Option B What is the optimal amount of insurance (rn) she will purchase
f) Does the availability of insurance affect the farmers crop choice (compared to a world without insurance) If so how If not why not
g)
markets (e-f) affect this distribution
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
a)
c)
Question 4
A small underdeveloped country with a single port and one poor road into the hinterland produces two crops peanuts which it processes into oil at the port and primarily exports and maize which it consumes as its main staple and which it must import to some extent
Describe the locational patterns of peanut and maize production and oil and rnai7e consumption as a function of the dstance from the port given that each location feels relative transport costs differently (A graph showing a farmers crop choices as a function of farm-gate prices would be useful)
b) One political party argues that the country depends too much on the outside world for its food and its market for peanut oil It proposes a self-sufficiency campaign that would close the country to trade At which locations might farmers support this political party (A graph of the market-level effects of the trade restriction and another related to the graph in part (a) would be useful for this)
Another political party says the countrys main problem is its poor road It has approached you as a consultant to determine whether a new road should be built and if built how long it should be One of the many models you might employ to conduct such a costbenefit study is a matllematical programming model of a farmers crop choices Describe the components of such a model including the characteristics of the road Who would be the optimizer implicit in the model How would the model be used to determine whether a new road would be beneficial
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
mathematical_Programllil)g_moiel
II WIW I V I
d) What are the two or three most important parameters or technical coefficients you would need for your model Explain why they are important in the context of a generic
Extra Explain as you would to a minister untrained in technical economics why a model is needed rather than j ust a survey of current road use
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
Question 5
A large food processor such as Fresh Choice might be expected not to behave as a perfect competitor but to have market power Market power in both output and input markets might in fact be exhibited because Fresh Choice may be a primary demander of fresh produce as well as a primary supplier of packaged salads The extent of the resulting market power might be a concern i f you as a policy maker believe this is causimiddotng consumers to pay inflated prices for their packaged salads or farmers to receive excess ively low prices for the ir produce relative to the prices that would prevail in competitive markets l f you wanted to detem1ine the amount of market power Fresh Cho ice has what would you do In particular
a) Using math or diagrams or both show how an economist m ight represent the economic behavior ( i nput demand output supply) of the Fresh Choice company if perfect competition in output and input markets could be assumed
b) How is the representation of firm decisions a ffected if it is recognized that i mperfect competition exists in the output market How would i t differ if Fresh Choice were a monopo l ist as compared to an o l igopo l ist
g) Di scuss the potential impact of credit rationing on the relationship between farm size and productivity
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
I
c) Similarly how would your answer to (a) differ with imperfect input markets What are the differences between monopsony and oligopsony
d) How might one measure and interpret the extent of imperfect competition in the output and input markets What kinds of equations would be relevant and what kinds of measures would represent the extent of imperfect competition
Extra How are imperfect competition in output and input markets different and how are they similar
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
Now assume that farmers must borrow to pay for fertilizer The net interest rate is 1 (Thus the total amount to be repaid equals (1 +i)qN)
d) Derive the input demand function for fertilizer under this new assumption Does this assumption affect the relationship between farm size and
productivity Explain
Credit markets have often been described as imperfect One manifestation of this imperfection is that poor farmers may be rationed by banks To capture this
assume that farmers with farm sizes smaller than L face a maximum loan size
equal to uL where a is a constant Farmers with L L can borrow as much as they want
e) Write down the constrained optimization problem and the associated
Lagrangean for small farmers (ie those with L lt L )
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
results
Question 6
Profit maximizing farmers produce a single output according to the following technology
Q = Nf1 L-P where Q is the quantity of output N is kilograms of fertilizer L is farm size in acres and p is a constant with 0 lt P lt I
Let p and q denote the prices o f output and fertilizer respectively Finally assume that land is a fixed factor of production (ie it cannot be rented or sold so that L is not a choice variable)
a) Write down the farmers optimization problem and derive the FONC (first order necessary condition) for the optimal level of fertilizer Interpret the FONC
b) Derive the input demand function for fertilizer
c) Defme productivity as output per acre Under the assumptions above is produciivity increasing decreasing or constant in farm size Prove your
l
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
University of California Davis Department of Agricultural and Resource Economics
MS Comprehensive Exam September 22 2002
You have three hours for this exam There are two parts to the exam Answer two questions from Part I and two questions from Part 11 Watch the time carefully each question (but not parts within the questions) counts equally so dont get bogged down with any one lf you dont gel the whole answer within the designated time frame just do what you can and move on The logic used to answer the question is an important part of your grade so be sure to clearly specify your reasoning with ful l sentences Also make sure your writing is legible answers that cannot be read will be assumed to be wrong
Part I middot Answer two of the following three questions
I Suppose the utility function for two goods X and Y is given by
U(XY) = In X+ In Y
where the budget constraint is X + PrY = I
a Derive the Marshall ian demand functions for Xand Y
b Obtain the own-price and income elasticities for good X
c Show that the demand functions derived in part (a) are homogeneous of degree zero in all prices and income
d How would the demand functions change i f the utility function were
U(X Y) = XY Explain
e Provide an economic interpretation of the Lagrangian multiplier Show your work
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
a
b
2 Arnold Wienerschnitzel likes money which we will denote by Af and power which middot
=we will denote by P His utility function is U(MP) M + P Arnold is endowed with a total of T hours He can spend his time in two ways - making movies or being a politician Let T11 be the number of hours he spends making movies and Tp be the number of hours he spends as a politician As a movie star Arnold earns a wage of $W per hour As a politician Arnold generates power according to the increasing concave function f(Tp)
Set up Arnolds optimization problem Make sure you clearly specify the choice variables objective function and constraint(s)
Let r and r denote an optimal solution to Arnolds time allocation problemmiddot
Write down the necessay conditions for T and r to indeed be optimal
c What conditions must hold such that i t is optimal for Arnold to dedicate some time to being both a movie star and a politician Please discuss the economic intuition behind your conditions
d What conditions must hold such that it is optimal for Arnold to spend all of his
time as a movie star and none as a politician (ie r = T and r = 0 ) Again
discuss the economic intuition behind your conditions (You might find it useful to draw a graph for parts c and d)
e Now as in part c assume it is optimal for Arnold to spend some time in each activity How would a decrease in the wage Arnold earns as a movie star affect Arnold s optimal time allocation Show your result formally (mathematically) and discuss your result intuitively
2
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
3 Suppose that the Mondavi Center i s the only supplier of classical music concerts in Davis Mr Mondavi (the owner) knows that faculty members have a demand for
classical music concerts given by QF = 48- P while students have a demand given
by Qs = 48- 2P The cost to Mr Mondavi of producing classical music concerts is
C(Q) = 1 2Q regardless of whether the concert is attended by faculty or students
a Suppose Mr Mondavi i s able to tell whether the buyer of a ticket i s faculty or a student What will be the profit maximizing price and quantity set by Mr Mondavi in this market (You might find it useful to draw graphs for both parts (a) and (b))
b Now suppose the state of Califomia declares it i llegal to charge different prices to students and faculty or musical events Does this change your answer to part a If so what is the new equilibrium price and quantity
c What if any is the change in consumer surplus and which consumers benefit Provide an intuitive discussion
d Does price discrimination always hurt consumers Provide an intuitive discussion
3
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
rejected
a
Part II Answer two of the following three questions
4 Given a log-linear demand function
where the dependent variable is per capita quantity demanded of beef p2 is the price
of beef p3p4 and I are prices of substitutes and complements and per capita
income respectively
Set up a test statistic and describe in detail how you would test for homogeneity of degree zero in all prices and income
b Suppose that income is omitted from the above demand function What effect would this have on the properties of the ordinary least squares estimators Be precise
c Suppose you tested for autocorrelation and the null hypothesis of no autocorrelation in the disturbance terms What would be your next step And what estimation procedure would you use Be precise
d What impact would multicollinearity have on the estimated standard errors of your coefficients
4
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
a
5 The Pan-Pacific Gas and Electric Company commissions you to do a study of income and electricity use as a basis for designing new energy conservation programs Specifically they want you to estimate and test for a significant effect of income (Y) on electricity demand in kilowatt hours KWH) You conduct a survey of 1 000 customers and decide to estimate a regression equation for KWH demand of the following form
KWH = ao + bY + u
(The subscript i refers to person i)
After taking ARE256 you suspectthere may be a heteroscedasticity problem with this regression Please
1 Define the problem 1 1 Tell whether or not the problem would result in estimated marginal
propensities to consume KWH that are not BLUE and explain your answer H I Propose a solution that would make your estimators have more desirable
properties-and tell us what these properties are
b Sadly consumer income i s measured with error-that is instead ofY you have
y = Y + v
where v is the measurement error which has zero mean and variance cr
1 If you use ordinary least squares to estimate the KWH regression will your estimated marginal propensities to consume KWH be BLUE Why or why not Prove it-and if there is a bias show us what it will be
1 1 Please propose a solution that would make your estimators have more desirable properties and tell us what these properties are
5
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6
d
0
6 Consider the following regression equation to estimate the aggregate demand for turkey in the United States
where M1 is the total US demand for turkey (in tons) Y is total income of US households (thousands of $) P is the price per pound of turkey (in dollars) all in month t D is a dummy variable equal to I if the month is November and zero
otherwise and u is a stochastic error with a mean of 0 and a constant variance of cr2
a There are at least two things likely to be wrong with this regression model Because of this we cannot expect ordinary least squares estimators to be BLUE What are these problems why do they create difficulties for our OLS estimators and how would you fix them Discuss two potential problems
1 Potential Problem I 11 Potential Problem 2
b Suppose you fix this regression equation to solve these problems estimate it with data covering 504 months and obtain an R2 of 025 Knowing that the critical value for a 95 F test with numerator degrees of freedom equal to 4 and denominator degrees of freedom equal to 500 F45oo is approximately 239 please perform a test of whether your estimated regression significantly explains the variation of turkey demand around its mean
c Please provide an economic interpretation of the coefficient 5 What would you expect its sign to be-and why
Please explain in detail how would you test which has a larger effect on income-an additional $ 1 000 in income or a $ 1 increase in the price of turkey
6