Problems 5-9 - Mihaylo Faculty Websites - Mihaylo …mihaylofaculty.fullerton.edu/sites/esolberg/Econ_315... · Web viewWhat do you notice about the trade-off the consumer is willing

Chapter 5 Individual Choice and Demand

5.1 Rocky Rhodes spends all of his weekly income on ice cream sundaes and Evian water. Currently, market prices are $3 for a sundae and $2 for a bottle of Evian. If Rocky spends all of his income on sundaes, he can buy 40 per week.

(a) What is Rocky’s weekly income (M)? M = $___________.

(b) Write the equation for Rocky’s budget line. Graph the budget line on the graph below.

S

E

10 20 30 40 50 60

10

20

30

40

50

60

(c) What is the slope of this budget line? How would you interpret this number? __________ __________________________________________________________________________________________________________________________________________________

(d) Consider the following bundles of sundaes and bottles of Evian. Are they on the budget line, in the attainable set, or outside the attainable set?

Bundle A: 45 bottles of Evian and 10 sundaes. _____________________.

Bundle B: 15 bottles of Evian and 25 sundaes. _____________________.

Bundle C: 35 bottles of Evian and 20 sundaes. _____________________.

42 Chapter 5 Exercises

5.2 Jane has a monthly entertainment budget of $120 that she spends on compact discs (C) and Nitwitto video games (N). Her budget line appears in the graph below.

C

N

2

4

6

2 4 6 8 10 12

(a) What are the prices of CDs and Nitwitto games? pC = $________. pN = $_________.

(b) Can Jane afford to buy 3 Nitwitto games and 5 CDs per month?

(c) Suppose Jane’s monthly entertainment budget increases to $150. Draw in her new budget line on the same graph. Can she now afford to buy 3 Nitwitto games and 5 CDs per month?

(d) If Jane’s income stayed at its original level of $120, show how her budget line would change if the price of a CD fell to $10. How would it change if instead the price of Nitwitto games rose to $40? In each case, what is the equation of the budget line?

Chapter 5 Exercises 43

5.3 On the following figure, the prices of two goods X and Y are pX = 2 and pY = 4 and the

consumer is choosing bundle A:

X

Y

20 40 60 80

20

40

60

80

A

(a) The consumer’s money income must be M = ______________________________.

(b) The maximum level of X that can be consumed is M/pX = _______________. The maximum level of Y that can be consumed is M/pY = _______________.

(c) Given current prices and income, graph the consumer’s budget line.

(d) Now suppose that pX changes by pX = 2. If the consumer is simultaneously given a change in income of M = 80, the maximum level of X that can be consumed will become (M + M)/(pX + pX) = _______________. The maximum level of Y that can be consumed becomes (M + M)/pY = _______________. Graph this new budget line on the same diagram.

(e) According to the uniqueness assumption, the consumer will choose some new bundle other than A. The non satiation assumption requires that the new bundle be on the new budget line. This new bundle must be to the (right, left) of A. Label this new bundle C and indicate its placement on the new budget line.

5.4 Given the following information:A = {X = 20, Y = 20} is chosen when pX = $3, pY = $3B = {X = 30, Y = 10} is chosen when pX = $3, pY = $6

(a) Is A revealed preferred to B consistent with the revealed preference approach to consumer choice? Why or why not? _______________________________________________.

(b) Is B revealed preferred to A consistent with the revealed preference approach to consumer choice? Why or why not? _______________________________________________.


5.5 Dapper Jim regularly buys Armani suits, among other things. The graph below shows Jim’s Price Expansion Path (PEP) for Armani suits. Initially, Jim is at A.

Armani suits

all other goods

$20,000

10 16 20

PEP

A

B

C


(a) What is the price of an Armani suit initially? pA = $___________.

(b) Notice that when the price of a suit falls to $1,250, Jim moves from A to B, where he spends less on all other goods and buys more suits. Since his income has not changed, the amount of money Jim is spending on Armani suits must be (rising, falling).

(c) Between A and B, where the PEP is falling, Jim’s demand for Armani suits must be (elastic, inelastic).

(d) When the price of a suit falls to $1,000, Jim moves from B to C, where he spends more on all other goods, though still buying more Armani suits. Since his income has not changed, the amount of money Jim is spending on Armani suits must now be (rising, falling).

(e) Between B and C, where the PEP is rising, Jim’s demand for Armani suits must be (elastic, inelastic).

(f) Are these results consistent with what you have learned about elasticity and linear demand curves? Explain. ________________________________________________________.

5.6 Jack N. DeBoxe spends all of his weekly lunch budget of $25 on hamburgers and egg rolls. Hamburgers cost $2. Egg rolls cost $3 per order. Jack has been buying, on average, 5 burgers and 5 orders of egg rolls each week. Suppose that the price of egg rolls rises to $3.50 and, at the same time, Jack’s mom increases his weekly lunch allowance by $2.50. This exactly compensates him for the higher price of egg rolls, allowing him to buy the same food bundle as before. Do you think his consumption pattern will change? Why or why not? __________________________ ____________________________________________________________________________.

5.7 In Chapter 3, we derived the market demand of Huey, Dewey, and Louie for Animal Crackers. This market demand appears in the table below.

Huey’s Dewey’s Louie’s MarketPrice Demand Demand Demand Demand$2.50 0 0 0 0$2.00 10 0 0 10$1.50 12 8 0 20$1.00 14 12 8 34$0.50 16 16 10 42$0.00 18 20 12 50

(a) What is Huey’s reservation price for Animal Crackers? pr,H = $___________. What is Dewey’s reservation price? pr,D = $__________. What is Louie’s reservation price? pr,L = $__________.


(b) As the price of Animal Crackers falls from $2 to $1.50, the change in market Qd is a combination of a change on Huey’s (extensive, intensive) margin and Dewey’s (extensive, intensive) margin.

5.8 Andrew O’Gratin, a poor Irish boy, spends a large portion of his income on potatoes, which cost $2 per pound. Two years ago when his yearly income rose from $1,000 to $2,000, Andrew’s consumption of potatoes increased from 200 to 300 pounds. Last year, however, Andrew’s income rose from $2,000 to $2,500, and he consumed only 250 pounds of potatoes.

(a) Draw Andrew’s budget lines, and carefully construct his Income Expansion Path (IEP).

potatoes

all other goods

500 1000 1250

1000

2000

2500

250 750

(b) Are potatoes a normal good or an inferior good for Andrew? Explain. _____________.

5.9 Consider the utility function U = XY for a person who consumes only goods X and Y:

(a) What is this person’s utility if he consumes:Bundle A, consisting of 20 units of Y and 5 units of X? UA = _________.Bundle B, consisting of 10 units of Y and 10 units of X? UB = _________.Bundle C, consisting of 5 units of Y and 20 units of X? UC = _________.


(b) Using these points, draw one smooth indifference curve, labeling the bundles A, B, and C.

X

Y

5

10

15

20

25

30

5 10 15 20 25 30

(c) If this person can consume a combination bundle D, made up of one-half A and one-half C, he will get _____ units of X and ____ units of Y. His utility will be equal to U = ______.

(d) Draw an indifference curve through this new combination of goods X and Y.

(e) Is this result consistent with the notion that indifference curves are convex? Explain. ___ _______________________________________________________________________.

5.10 Draw indifference curves, and indicate which direction represents higher levels of utility, for each of the following situations.

(a) “I only buy my shoes as a pair — one left, and one right.”

(b) “I can buy Morton Salt either in Supermarket A or in Supermarket B.”

(c) Superman hates Kryptonite, but loves all other goods.

(d) Bugs doesn’t care if he has any peas or not, but always prefers to have more carrots.

(e) Phil Anthropist likes to give some of his income to the poor, but only to people poorer than himself. If someone has more income than Phil, he couldn’t care less about giving charity to that person.

right

shoe

s

left shoes

salt

from

B

salt from A

all o

ther

goo

ds

Kryptonite

carro

ts

peas

othe

rs' i

ncom

e

Phil's income

(a) (b) (c) (d) (e)


5.11 Since utility is an ordinal measurement designed to rank bundles, the numerical assignment of utility can be completely arbitrary as long as it is consistent with the consumer’s ranking. Consider the following bundles of Goods X and Y that a consumer can buy:

Bundle X Y U = XY V = (XY)1/2 W = XY – 50 A 11.0 11.0B 12.5 8.0C 9.0 16.0

(a) If utility is U = XY, calculate the utility of each bundle. Which bundle gives the highest level of utility? Rank the three bundles in terms of utility. __________________.

(b) Suppose that utility were instead measured as V = (XY)1/2. Which bundle now gives the highest level of utility? Rank the three bundles in terms of utility. _________________.

(c) Suppose that utility were instead measured as W = XY – 50. Which bundle now gives the highest level of utility? Rank the three bundles in terms of utility. _________________.

(d) Is there any difference in your rankings when the different utility functions are used? Why or why not? _____________________________________________________________.

5.12 Below is shown a budget line for Cindy Pauper, who spends her monthly income of $1,000 on food and other goods. A unit of food costs $4:

Food

Other goods

$1,000

(a) If Cindy buys only food, how many units can she afford? F = M/pF = _______. Show this point on your graph.

(b) If Cindy spends $400 on food, keeping $600 for other goods, she can buy how many units of food? F = _______. Show this point on your graph.


(c) With the government’s new food stamp program, Cindy can buy $800 worth of food stamps for only $400. This program forbids the use of food stamps for purchases other than food, and it also outlaws the reselling of food stamps. If Cindy spends $400 on food stamps, how many units of food can she now buy? F = ________. Show this point on your graph.

(d) If Cindy buys only food, how much food can she now buy each month? F = __________.

(e) Graph Cindy’s new budget line with the food stamp program.

(f) How would the budget line be different with no restrictions on the re-sale of food stamps?

(g) Using indifference curves, show that if Cindy has a low preference for food, she is forced to a lower level of utility by the government’s restriction on the re-sale of food stamps.

5.13 Consider the following utility functions:(i) U = XY (where MUX = Y and MUY = X)(ii) V = 8X + 4Y (where MUX = 8 and MUY = 4)

(a) Do indifference curves associated with these utility functions exhibit a diminishing marginal rate of substitution (MRS)? Explain. _________________________________________.

(b) For each utility function, graph the indifference curve representing a utility level of 100. What do you notice about the trade-off the consumer is willing to make between Goods X and Y to remain at the same level of utility?

X

Y

X

YU = XY U = 8X + 4Y

5.14 Rich is buying 3 pounds of steak per week and 3 pounds of hamburger. The marginal

utility of steak is MUS = 3, and the marginal utility of hamburger is MUH = 1. The price of steak

is $4 per pound and the price of hamburger is $2 per pound. Rich’s total weekly income is $18.


(a) Graph Rich’s budget line (putting hamburger on the X-axis and steak on the Y-axis) and show his current consumption bundle. Label his current bundle A.

X

Y

(b) At his current level of consumption, what is Mr. Rich’s MRS? __________. What is the ratio of hamburger-to-steak prices? _________________

(c) Is Mr. Rich getting the highest utility possible for his $18? Why, or why not? ________ ____________________________________________________.

(d) If Mr. Rich spends $4 more on steak, he can buy _________ more pound(s). The marginal utility gained will be equal to ________. To do this, he must give up _________ pounds of hamburger. The marginal utility lost will be equal to ___________.

(e) By buying (more, less) steak, Mr. Rich can increase his utility.

(f) Draw in two indifference curves, one passing through Mr. Rich’s current consumption bundle and another that indicates where Mr. Rich’s utility maximizing point might be.

5.15 Sam Square has the following utility function, U(X, Y) = X2 + 2XY + Y2. Sam’s marginal utilities are: MUX = 2X + 2Y, MUY = 2X + 2Y. Lynn Kneaire has the following utility function, W(X, Y) = X + Y. Lynn’s marginal utilities are: MUX = 1, MUY = 1.

(a) Sam’s marginal rate of substitution is MRSXY = _____________________.

(b) Lynn’s marginal rate of substitution is MRSXY = _____________________.

(c) Do Sam and Lynn have the similar tastes? How do you know? ___________________.

(d) Show that U(X, Y) is a monotonic transformation of W(X,Y).

5.16 Wally Wheeler has the following utility function: U(X, Y) = 4X + 4Y – X2 – Y2. The marginal utilities for X and Y are: MUX = 4 – 2X, MUY = 4 – 2Y.

(a) Complete the following table:


X Y U(X, Y) X Y U(X, Y)2 0 1 22 4 2 30 2 2 14 2 3 2

(b) Plot the smooth indifference curves U = 4 and U = 7 on the following figure:

X

Y

0 1 2 3 4 5

1

2

3

4

5

(c) Set MUX = 0 and MUY = 0 simultaneously and find the critical values X* = ___________ and Y* = __________. At this point utility is U* = ______________________________.

(d) On your graph, identify the economic region where MUX ≥ 0 and MUY > 0.

(e) The critical point (X*, Y*) is called a ___________ __________. At this point, Wally is said to be ___________________ with respect to both X and Y simultaneously.

(f) Suppose that X is hamburgers and Y is ounces of water. Water is provided at zero price, but hamburgers have price pX = $4. If Wally has $4 to spend, how many hamburgers will he buy? _______________. How much water will he drink? ______________.

(g) If the price of hamburgers falls to pX = $2, Wally will buy _____________ hamburgers. How much water will Wally drink? _______________.

(h) If the price of hamburgers falls to pX = 50¢, Wally will buy _____________ hamburgers. How much water will Wally drink? _______________.

(i) If both X and Y are free, how many hamburgers will Wally eat? ______________. How much water will he drink? ______________.


(j) What must be true, according to economic theory, for Wally’s utility function to be plausible? _______________________________________________________________.

5.17 Given the following indifference map, this consumer is currently choosing A along an initial budget line. She is offered a choice between an increase in income (M > 0) that makes C affordable or a decrease in the price of X (pX < 0) that makes B affordable. Which

alternative does she prefer? Why?

X

Y

A B

C

5.18 Given the utility function U = X2Y so that MUX = 2XY and MUY = X2:

(a) MRSXY = MUX/MUY = ____________________.

(b) Given pX = 2, pY = 4, and M = 180, write the equation for the budget line:

pXX + pYY = M = ___________________________________.

(c) Utility maximization requires tangency of an isoquant with the budget line, and so two equations must be satisfied: (i) MRSXY = pX/pY, (ii) pXX + pYY = M. Solve these

simultaneously for X and Y. Call these quantities Xa and Ya. On the following diagram, draw the current budget line through the end points M/pX and M/pY, and indicate the bundle

A = {Xa, Ya}. On the X axis, label the quantity Xa.


X

Y

15 30

5

10

15

20

30

35

40

45

50

25

45 60 9075

(d) Now suppose that the price of X changes by pX = 2. The tangency equations become:

(i) MRSXY = (pX + pX)/pY

(ii) (pX + pX)X + pYY = M

Given the new set of prices, once again substitute into the tangency equations and solve for X and Y. Call these quantities Xb and Yb. On the same diagram as before, draw the new budget line through the new end points, M/(pX + pX) and M/pY, and indicate the new

bundle as B = {Xb, Yb}. On the X axis, label the quantity Xb.

(e) One way to compensate the consumer for the loss in real income caused by the price change is to shift the budget line after the price change back to the original bundle. This requires a lump-sum change in M in the amount M = XapX = __________________.

(f) After compensating for the price change, the tangency equations become:

(i) MRSXY = (pX + pX)/pY

(ii) (pX + pX)X + pYY = M + M


Substitute the MRSXY and the current prices and income into these two equations and solve

them simultaneously for X and Y. Call these quantities Xc and Yc. On the same diagram, draw the third budget line through the end points (M + M)/(pX + pX) and (M + M)/pY,

and indicate the bundle C = {Xc, Yc}. On the X axis, label the quantity Xc.

(g) The total effect of the price change is Xb – Xa = _________________. The substitution effect is Xc – Xa = _________________. The price-induced income effect is Xb – Xc = __________________. Verify that the total effect is equal to the sum of the substitution and income effects of the price change. _______________________________________.

(h) On the same diagram as before, indicate the magnitude and direction of the substitution, income, and total effects of the price change.

5.19 Cookie Monster loves to eat cookies. In fact, the more cookies he eats, the more he wants. In other words, the more cookies he eats, the harder it is for him to substitute any other goods (or income) for cookies and remain indifferent. Cookie Monster’s indifference map and his budget line are shown on the graph below.

other goods

cookies

budget line

A

B

C

(a) Where is Cookie Monster’s utility maximization point, A, B, or C?

(b) Does the marginal rate of substitution decrease as more cookies are consumed? _______.

(c) What could happen to Cookie Monster’s demand for cookies if they increased in price? Illustrate your answer on the graph.

(d) How realistic does this indifference map seem? What type of consumer behavior does it imply? _______________________________________________________________.


5.20 On the following figures, draw an indifference map, budget lines, and the price expansion path to represent the following situations:

(a) Tonya Hardknee has decided to budget a fixed amount of money to buy ice skates and within a certain range of prices will spend neither more nor less than this amount on skates.

(b) A diabetic’s demand for insulin is completely price inelastic within a broad price range.

Hardknee's Diabetic's

ice skates insulin

othe

r goo

ds

othe

r goo

ds

5.21 In an effort to control U.S. consumption of gas and to reduce dependence on foreign oil, the president has decided to institute a mandatory gas rationing plan. Each week every consumer will be sent a certain number of ration tickets. These tickets will entitle the consumer to buy up to X* gallons of gas (at the normal price), where X* is a quantity based on the average weekly gas consumption in the U.S. Obviously, a consumer doesn’t have to buy X* gallons — unused coupons will simply get thrown away. Assume that there is no resale of coupons allowed. Using indifference curve analysis, illustrate the following cases:

Gus Guzzler

othe

r goo

ds

gas

Wally Walker

othe

r goo

ds

gasX* X*

(a) Label a bundle A that is chosen by Gus Guzzler, who is hurt by this plan. Draw in Gus’ current indifference curve.

(b) Label a bundle B that is chosen by Wally Walker, who is not hurt by this plan. Draw in Wally’s current indifference curve.


(c) Now suppose that a black market for gas coupons develops, and that the black market price of gas is higher than the normal price. Draw in the new budget line for each consumer. (The new budget line still passes through the original endowment point. Do you know why?)

(d) The black market clearly benefits Wally Walker. He can reach a higher level of utility by (buying, selling) some of his gas coupons. Show this on Wally’s graph.

(e) Surprisingly, the black market can even benefit Gus Guzzler. He can reach a higher of utility by (buying, selling) some gas coupons on the black market. Show this on Gus’ graph.

(f) The presence of a black market, in this case, will (increase, decrease) society’s welfare.

5.22 During the 1980 presidential campaign, one candidate proposed a 50¢ per gallon “motor fuels conservation tax,” coupled with a reduction in the taxes employees pay toward Social Security. For analytical convenience, suppose the government returns the proceeds of the gasoline tax to consumers in the form of lump-sum payments. Moreover, suppose that no one is intended to be worse (or better) off because of the excise tax-rebate plan — fuel conservation is the only goal. Many critics argued that the lump-sum payment would cancel the consumer’s incentive to economize on fuel. We can use indifference curve analysis to analyze the plan and to determine whether the critics are right. Assume that a typical consumer is initially at A where she is maximizing her utility (U*). Her income is M = $15,000 and current consumption is Ga gallons of gas.

other goods

gas

M

M/pG

U*

A

Ga

(a) If the maximum possible level of gas consumption is 15,000 gallons, what is the price of gas? pG = $_________.


(b) When the 50 cent per gallon tax is imposed, the budget line rotates in. Draw in the new budget line that reflects the higher price of gasoline. Indicate on your graph the maximum level of gas consumption as M/(pG + 50¢).

(c) Indicate the new utility maximizing bundle B and the new indifference curve Ub.

(d) To compensate for the negative income effect of the higher gas price, a lump-sum payment will then be given to the consumer. The transfer is intended to shift the budget line until it just touches the original indifference curve ensuring that total satisfaction remains constant. Draw in this shifted budget line, and label the point where it touches U* as point C.

(e) The movement from B to C represents the (income, substitution) effect. Therefore, the movement from A to C must represent the (income, substitution) effect.

(f) Were the critics of the gas tax-rebate plan correct? Do consumers lose the incentive to conserve fuel? Why or why not? __________________________________________.

5.23 Some lobster fishermen have been observed to consume more lobster when the price of lobster rises! Does this mean that the law of demand doesn’t apply to them? Consider the case where a lobster fisherman’s only source of income is derived from lobsters sold at market. The money earned can be spent on other goods. He is endowed with human capital that allows him to catch 100 lobsters per month. If L represents the number of lobsters consumed (eaten), the level of expenditure on other goods is C = pL100 – pLL where L ≤ 100.

(a) If the price of lobster is pL = $5, graph the fisherman’s budget line on the figure below.


L

C

1000

500

400

950

800

10 205 100

(b) At the price pL = $5, he consumes 20 lobsters. What, then, is his consumption expenditure on other goods? _______________. Label that bundle A on the graph.

(c) If the price of lobster increases to pL = $10, graph the fisherman’s new budget line. Suppose that he continues to consume 20 lobsters. What, then, is his consumption expenditure on other goods? ________________. Label that bundle B on the graph.

(d) Decompose the total effect of the price change into substitution and income effects on your graph. Identify the substitution effect as occurring between A and a bundle C, and identify the income effect as occurring between C and B.

(e) Does the fisherman view lobsters as a normal or inferior good? How can you tell?

(f) Does this fisherman consume more lobster when the price of lobster rises? __________.

(g) Can you imagine a situation where his consumption of lobster would be greater than 20 at the higher price? Explain.

5.24 A negative income transfer (NIT) is a formula income-transfer plan that has been proposed as a replacement for categorical welfare programs. The NIT is attractive because of its simplicity. It would be less costly to administer than categorical programs. Eligibility is determined solely by a person’s, or family’s, level of income (M). A person with income above a break-even level (B) would not be eligible. Persons with income below B will be given a transfer payment (paid a negative income tax) determined by multiplying the amount their


income is below the break-even level (B – M) by a tax rate t, payment = t(B – M); M < B, 0 < t < 1. The income guarantee is G = tB. Thus, all three of these parameters may not be set independently — one is always dependent on the other two.

The primary concern with using the NIT is its potential effects on labor supply. If G is set too high, there may be a wholesale withdrawal from the labor force. If t is set too high (B set too low), there will be no incentive for a welfare recipient to accept anything other than a high-paying job. Our consumption-leisure model can be used to evaluate the potential disincentives on work effort. The composite good is numeraire, and the budget line is

C = N + wLwhere L = T – l is labor time, T is total time, and l is leisure time, and N is non-work income. To keep the analysis simple, let’s assume that the NIT is the only source of non-work income. When M < B, non-work income is t(B – wL), and so the budget line for someone eligible for the NIT is

C = t(B – wL) + wL = G + (1 – t)wLWe will use the following figure to evaluate the incentive effects of the NIT. The break-even level of consumption corresponds to point B and the guaranteed level of consumption corresponds to G.

Implementation of the NIT can be thought of as composed of two steps: First, a positive guarantee is established with a 100 percent marginal tax rate, G > 0 and t = 1, (B = G > 0). This means that work income reduces the welfare payment dollar for dollar. Second, reducing the marginal tax rate to less than 100 percent so that B > G > 0 and 0 < t < 1. That is, the guarantee is held constant but the tax rate is reduced — a dollar of work income reduces the welfare payment by less than a dollar.


time constraintC

l = T – L

slope = –(1 – t)w

slope = –w

without NIT

with NIT and 0 < t < 1

G

B

A

T0

G

B

(a) What would the budget line look like when G > 0 and t = 1? Show this budget line on the graph.

(b) Suppose that a person initially at A is now offered a guarantee of G when t =1. What choice will be made if more is preferred to less? _______________________________.

(c) Assuming that leisure is a normal good, indicate a plausible choice under total implementation (B > G > 0 and 0 < t < 1) as point C on the graph. Use revealed preference to show that C must be to the right of A. Divide the total effect from A to C into substitution and income effects. Indicate the change due to the substitution effect as a movement from A to a bundle called D, and the change due to the income effect as a movement from D to C.

(d) Does the income effect reinforce the substitution effect under total implementation? Why or why not? ______________________________________________________________.

(e) Will work effort increase or decrease for this person? Why? _____________________ ______________________________________________________________________.

(f) Suppose that the NIT was used to replace an existing payment of G dollars. Will work effort increase or decrease for such a person? Why? _____________________________ _______________________________________________________________________.


5.25 Alan and Betty have identical tastes (in consumption) but have different endowments (of goods). Alan’s utility function is function UA = , and Betty’s utility function is UB = . However, Alan owns X = 5 and Y = 20 units of the two goods, and Betty owns X = 20 and Y = 5 units. The Edgeworth box is below, where the endowment point is labeled E.

Betty's

XA

YA

XB

YB

5

10

15

20

5

5

5

10

10

10

15

15

15

20

20

20

E

A

B

C

D

Alan's

(a) Alan’s and Betty’s world has X= _____ units of X and Y = _____ units of Y.

(b) Without trade, Alan’s utility is UA = _______, and Betty’s utility is UB = ________. Graph each of their current iso-utility curves on the diagram.

(c) If Alan and Betty trade so that they end up at C, Alan will have X = _____ and Y = _____ units of the two goods and Betty will have X = ______ and Y = ______ units. Alan’s utility will be UA = ________, and Betty’s utility will be UB = _______. Are they better off or worse off with this trade? ____________________.

(d) Starting at E, if they trade so that they end up at B, Alan will have X = ____ and Y = _____ units of the two goods and Betty will have X = _____ and Y = _____ units. Alan’s utility will be UA = ________, and Betty’s utility will be UB = ________. Are they better off or worse off with this trade? ____________________.

(e) Starting at E, if they trade so that they end up at A, Alan will have X = _____ and Y = ______ units of the two goods and Betty will have X = _____ and Y = _____ units. Alan’s


utility will be UA = ________, and Betty’s utility will be UB = ________. Are they better off or worse off with this trade? ____________________.

(f) Starting at E, if they trade so that they end up at D, Alan will have X = _____ and Y = _____ units of the two goods and Betty will have X = _____ and Y = _____ units. Alan’s utility will be UA = ________, and Betty’s utility will be UB = ________. Are they better off or worse off with this trade? ____________________.

(g) On the basis of this analysis, identify the contract curve on the graph.

(h) If the price ratio is pX/pY = 1, can you identify the equilibrium trading bundle? ________.

5.26 On the island of Morafruti, natives consume only papayas, mangos, and kumquats. Prices of goods and consumption patterns of a typical consumer in 1990 and 1991 are:

1990: Papayas Mangos Kumquats

Q 200 180 300p $2 $4 $1.50

1991: Papayas Mangos Kumquats

Q 220 150 350p $2.25 $4.75 $1.60

(a) Calculate the change in the general price level using the Laspeyres price index. _________.

(b) Calculate the change in the general price level using the Paasche price index. __________.

(c) If you were an employee of the Morafruti government, would you rather have your annual cost-of-living wage adjustment based on a Laspeyres index or a Paasche index? Why?

5.27 A young economist was preparing to travel to Kuwait for a special one-year consulting mission. He was worried, however, that his real income would fall while he was in the Middle East. In trying to establish what income he should receive, he made careful note of his consumption pattern for the past year in the United States, U.S. prices for the past year, and expected Kuwaiti prices (in dollars) for the upcoming year. Moreover, he insisted that his U.S. income be adjusted using the Paasche price index to determine his Kuwait salary. What advice, if any, would you give the Kuwaiti government concerning this economist?

Chapter 6 Time and Intertemporal Decisions

6.1 Seymour Cash just invested $100 in a savings account offering 6 percent interest. After 1 year, he will have $________; after 2 years, he will have $_______; and after 10 years, Seymour will have $________. If the rate of interest were 9 percent, after 1 year Seymour’s money would have grown to $_________; after 2 years, to $________; and after 10 years, to $_________. Clearly, the higher the rate of interest, the (faster, slower) your money will grow; that is, the (higher, lower) is its future value.

6.2 Louie, the hot dog vendor, finally decided to stick some of his money in the bank after all. In January 1990, Louie somewhat nervously deposited $1,000 at Pacific Insecurity Bank. By January 1992, Louie’s investment had grown to $1,150.25. At what annual rate of interest had his money been growing?

6.3 Donald has just inherited $1 million from his Uncle Scrooge. Unfortunately, he can’t collect it until his 21st birthday, which is 8 years away, and until then, the money will sit in a non-interest bearing account.

(a) What is the present value (PV) of the $1,000,000, if the going market interest rate is 6 percent? $____________.

(b) What is the PV of the $1 million if Donald can’t collect it until his 25th birthday — 12 years from now? $___________.

(c) What is the PV of the $1 million if he can collect it in 8 years, but the going market interest rate is 8 percent? $____________.

(d) The (lower, higher) the interest rate, or the (farther, closer) in the future money is to be received, the less is the PV of some future amount.

6.4 In certain Latin American countries, the annual market rate of interest is as high as 300 percent. If someone in one of these countries promises to pay you $1,000 in 5 years, what is the present value of this payment? $_______________________________________________.

6.5 Ima Spendthrift currently has an initial income endowment of $200 in the current time period (t1) and $250 next period (t2). The market rate of interest is 8 percent.

(a) If Ima borrows against her future income, so that she can maximize consumption in t1, she can consume $200 plus the PV of $250, which together equal $________. Why can’t she borrow and consume the whole $250 that she knows she’ll receive in t2?

(b) If Ima wants to wait until t2 to consume, she can consume next year’s $250 plus this year’s $200 plus the interest that $200 can earn. All together this equals $_______.

(c) Draw Ima’s budget line for current and future consumption on the following figure. Compute the slope. Verify that the slope is equal to –(1 + i).

C1

C2

(d) On the same graph, draw an indifference curve consistent with the notion that Ima is a saver (lender). Draw one consistent with the notion that she is a borrower.

(e) If the interest rate falls to 6 percent, Ima could forgo consumption in t1 and have how much to spend at the end of t2? $_________. If she borrows against her future income, to maximize consumption in t1, Ima can consume $_______ in the first period. Draw in the new budget line on the same graph.

(f) At the lower interest rate, the opportunity cost of consuming $1 more today is (lower, higher) than before. (Less, More) future consumption is sacrificed.

(g) At the lower interest rate, which Ima can clearly reach a higher indifference curve — Ima Borrower or Ima Lender?

6.6 Given a person who is endowed with current income M1 = 30 and future income of M2 = 40 (both in thousands of dollars), Bundle A is chosen as illustrated on the following figure. Use the information in that figure to answer the following questions:

(a) What is the current interest rate? ________________________.

(b) This person is currently choosing A where current consumption is C1a = 35. The future

level of consumption permitted is C2a = ____________________________________.

(c) At A, the level of (saving, borrowing) is Sa = M1 – C1a = ______________________.

(d) At A, how much must be paid out of future income? __________________________.

(e) Now suppose that the budget line rotates so that B is chosen where C1b = 25. What is the

interest rate now? _____________________________________________.

(f) At B, the level of future consumption is C2b = ________________________________.

(g) At B, the level of (saving, borrowing) is Sb = M1 – C1b = ______________________.


10 20 30 40

10

20

30

40

60

70

80

90

100

50

50 60 70 80C1

C2

endowment

A

B

25 35

for lower interest rate

for higher interest rate

(h) On the following figure, plot the level of saving or borrowing at each of the two interest rates. Assume that the S(i) curve is linear and graph it.

S

i

0 1 2 3 4 5–1–2–3–4–5

0.100.200.300.400.500.600.700.800.901.00

(i) This person has switched from being a (borrower, lender) to being a (borrower, lender) as the interest rate (rose, fell).


6.7 On the following figure, a person is endowed with bundle E. This person is currently choosing bundle A.

C1

C2

E

A

B

C

(a) If the interest rate falls and B is chosen, can you tell whether this person is better off or worse off than at A? Why or why not? ________________________________________.

(b) If the interest rate falls and C is chosen, can you tell whether this person is better off or worse off than at A? Why or why not? ________________________________________.

6.8 A, B, and C are three investments. A yields $120 at the end of the first year and yields $10 at the end of the second and third years. B yields $10 at the end of the first two years and yields $140 at the end of the third year. C yields $50 at the end of year 1, 2, and 3.

(a) Calculate the Net Present Value (NPV) of each investment alternative: if the market rate of interest (i) is 6 percent; if i = 9 percent; if i = 12 percent. Record your answers in the table:

Net Present Valuesi = 6% i = 9% i = 12%

Investment A

Investment B

Investment C

(b) When the rate of interest is 6 percent, alternative (A, B, C) has the highest NPV.

(c) When the rate of interest is 9 percent, alternative (A, B, C) has the highest NPV.

(d) When the rate of interest is 12 percent, alternative (A, B, C) has the highest NPV.


(e) This example shows that the Net Present Value criterion can be (sensitive, insensitive) to the rate of interest used.

6.9 There are many different market interest rates — passbook savings account rates, checking account rates, government securities rates, to name a few. Which interest rate is most appropriate? What factors are important to consider in deciding which rate to use?

6.10 The real rate of interest is often approximated by the formula = i – , where i = the market rate of interest and = the rate of inflation. The correct formula for calculating the real rate of interest, however, is given by: = (i – )/(1 + ).

(a) In the U.S. the rate of inflation is 3 percent and the market rate of interest is 7 percent. Calculate the real rate of interest and approximate using the approximation formula. Compare your two answers.

(b) In Brazil, the rate of inflation is 60 percent and the market rate of interest is 80 percent. Calculate the real rate of interest and approximate using the approximation formula. Compare your two answers.

(c) This approximation for calculating the real rate of interest is clearly more accurate when the rate of inflation is fairly (small, large).

6.11 Congratulations! You’ve just won $1 million in the lottery. The money will be given to you in 20 yearly payments of $50,000. What is the present value (PV) of your lottery win if the current market interest rate is 5 percent? $_________. If i = 10 percent? $_________. If i = 15 percent? $__________. As the interest rate rises, the present value of an annuity (rises, falls).

6.12 Which would you rather receive: $1,000/year for 20 years, or an annual payment of $800 forever? (Assume a market interest rate of 8 percent.) Would your answer be different if the interest rate is 10 percent? _____________________________________________________.

6.13 Lucky Larry just won a lottery that pays $2,000 for 50 years. With the market interest rate currently around 5 percent, Larry claims that the PV of this $100,000 is about $2,000/.05, or about $40,000. How good is this approximation? Use the correct formula to determine how close the approximation is. ____________________________________________________.

6.14 (a) If the market interest rate is 6%, what is the present value (PV) of a bond paying an annual

coupon of $50 for 10 years (including the 10th year), at which time it can be redeemed for $1,000 (its face value)? PV = $_________.

(b) At i = 0.08 it changes to PV = $_________. At i = 0.10, PV = $_________. Clearly, the present value of a bond varies (directly, inversely) with the market interest rate.


6.15 Mr. Investor Stallone currently has a portfolio consisting of, among other things, Asset A, which has a current market price of $100/share, and Asset B, which pays a guaranteed 10 percent rate of return. The price of Asset A next period is expected to be $105.

(a) In the current period, Mr. Stallone can sell 10 shares of Asset A for $1,000. If he invests this money in Asset B, he’ll have $_________ next period. He can then buy back his 10 shares of Asset A for $________, leaving him with a profit of $__________.

(b) If enough people try to do what Mr. Stallone is thinking of doing (sell Asset A), the demand for Asset A will (increase, decrease), and so its price will (rise, fall).

(c) The price of A will eventually fall to $_________, where A offers a rate of return of 10 percent, the same as B.

(d) Suppose instead that Asset A is selling for $90 in the current period. Now Mr. Stallone can buy 10 shares of A for $900, selling $900 worth of B, and forgoing $_______ in interest. In one year, these shares will be worth $_________, resulting in a profit of $_________.

(e) If enough people do this (buy A), the demand for A will (increase, decrease), and its price will (rise, fall), eventually reaching $__________.

(f) Whether A is overpriced or under priced, arbitrage activity drives its price to $__________, which is equal to its present value.

6.16 A firm is considering two investment projects. Project Alpha requires an initial capital outlay of $20,500, and will generate net revenues of $8,000 at the end of years 1, 2, and 3. For Project Beta, capital equipment priced at $27,000 must be purchased initially. This project will generate net revenues of $5,000 at the end of years 1, 2, and 3, and in addition, at the end of year 3, the capital equipment will have a scrap value of $20,000.

(a) What is the Net Present Value (NPV) of Alpha if the market rate of interest is: 7 percent; 9 percent?

(b) What is the Net Present Value (NPV) of Beta if the market rate of interest is: 7 percent; 9 percent?

(c) As interest rates rise, many projects will seem (less, more) desirable, and some projects will switch from having a (positive, negative) NPV to a (positive, negative) NPV.

(d) Calculate the internal rate of return for each project.

6.17 A company is evaluating two projects: X and Y. Each require an initial outlay of $950,000. Project X will generate net revenues of $100,000 at the end of years 1 and 2, and $1 million at the end of the 3rd year. Project Y will generate net revenues of $400,000, $380,000, and $350,000 at the end of years 1, 2, and 3, respectively.

(a) If the market rate of interest is 4 percent, which project has a higher NPV? ________.


(b) Calculate the internal rate of return (m) for each project. Which has the higher m? ______.

(c) Even though Project (X, Y) has the higher internal rate of return, at the current market interest rate, Project (X, Y) has the higher NPV. The generally accepted rule is to choose the investment with the greater (net present value, internal rate of return). This investment rule always leads to the correct decision. The company should, therefore, choose Project (X, Y).

6.18 People over the age of 40 rarely attend college. Is this consistent with the model of investment in human capital? Explain.

6.19 In the application in the textbook that deals with investments in education and training, a distinction is made between general training and specific training. Why would an employer be reluctant to hire a person for a job that entailed a considerable amount of specific on-the-job training if the person had a high probability of leaving the firm after a year or two? How can this help explain the empirically observed flatter age/earnings profile of female workers relative to male workers?

6.20 In the country of Isopoor, there are only two small business firms: I. O. U. Corporation and Invisible Inc. They are each considering 3 new projects — the capital needed and the internal rates of return (m) for each project are given in the table below. Both firms use borrowed funds exclusively for their investment projects.

Project IOU Corp. Project Invisible Inc.A $10,000; m = 12% D $20,000; m = 14%B $6,000; m = 10% E $12,000; m = 10%C $2,000; m = 8% F $5,000; m = 6%

(a) In most cases, firms will undertake all projects that yield an internal rate of return at least as high as the current market rate of interest. If i = 14 percent, only one project will be undertaken in Isopoor: Project ____, by (IOU, Invisible Inc.). The firm will borrow $___________ to initiate this project.

(b) If i drops to 12 percent, another project will be undertaken, Project ____, by (IOU, Invisible Inc.), bringing the total amount borrowed up to $____________.


(c) As i falls, (more, less) investment projects will be undertaken. Construct the Marginal Efficiency of Investment (MEI) curve for Isopoor. This curve represents the demand for _________________, as well as the supply of ___________________.

i(100%)

I(1,000s)20 40 60

68

10

12

14

6.21 By late 1991 and early 1992, interest rates had fallen to historical lows in the U.S., largely in response to a very sluggish economy. One of the policy questions was how to stimulate the economy. One option seriously considered was an investment tax credit similar to the one that worked so well during the Kennedy administration. If passed, what do you think would happen to the demand for investment? What would happen to market interest rates? Illustrate your answer on the following figure.

i

$

S(i)

I(i)

i1

I1 = S10


6.22 Adam and Eve live together in the Garden of Eden. Each has a demand for apples that can be represented by the following equation: p = 30 – Q. Derive an equation for the market demand when apples are: a private good; a public good. Graph each market demand.

p

Q

p

Q

private public

6.23 A typical individual’s demand curve for mosquito control on the tiny island-nation of Oamitchi can be represented by the following equation: p = 1 – (1/100)Q, where Q = the number of units of mosquito control provided and p = price per unit of control (in dollars).

(a) If there are 1,000 individuals on the island, derive and graph the market demand curve.


Q

p

100

200

300

400

600

700

800

900

1000

10 20 30 40

500

50 60 70 80 90 1000

(b) The efficient amount of mosquito control (or any public good) is that amount for which the total willingness to pay (which is measured by the market demand) is just equal to the marginal cost (MC) of providing that good — MC is the extra cost of producing an additional unit of the good. If government can provide mosquito control at a constant MC of $100 per unit, what is the efficient amount of mosquito control? Assume that each individual will pay 1/1,000 of the program costs and receive 1/1,000 of the benefits. Q =_________.

(c) What is the total cost of providing this amount of mosquito control? $_________. Indicate the total cost on your graph. What is the per-person cost? $____________.

(d) What is the total benefit of providing this amount of mosquito control? $_________. Indicate the total benefit on your graph. What is the per-person benefit? $___________.

(e) What is the consumer surplus associated with providing this amount of mosquito control? $__________. Indicate the consumer surplus on your graph. What is the per-person consumer surplus? $____________.


Now consider the existence of 10 free riders: people who say they don’t need mosquito control and therefore don’t want to pay for it. (They will in reality enjoy the benefits of the mosquito control program, however.)

(f) What is the new market demand (for 990 individuals)? P = _____________. What is the efficient amount of mosquito control now? Q = __________.

(g) What is the total cost of providing this amount of mosquito control? $________. What is the per-person cost (for 990 individuals)? $___________.

Free riders enjoy the benefits of the program, and so the original market demand (for all 1,000 individuals) must be used to calculate total benefits and consumer surplus.

(h) What are the total benefits now? $__________. What are the total benefits per person? $__________. Note that for the free riders this is also consumer surplus.

(i) What is total consumer surplus for the non free riders now? $_________. What is the per-person consumer surplus (for non free riders)? $___________.

(j) For the free riders, consumer surplus has increased by $_________. For the non-free riders, consumer surplus has decreased by $__________. There would seem to be a (strong, weak) incentive to be a free rider, and a (strong, weak) incentive to discourage free riders.

Chapter 7 Information, Risk, and Uncertainty

7.1 In 1991, the U.S. Congress proposed legislation that would put a ceiling on credit card interest rates. Suppose that credit card customers can be split into two groups: Group 1 customers always pay off their monthly balance and never have to pay finance charges; and Group 2 customers always carry a high monthly balance. Which of these two groups will more attracted to the offer of a low credit-card interest rate? _________________. Is there an adverse selection problem at work here? Explain. __________________________________________ ____________________________________________________________________________.

7.2 An interesting application of the adverse selection problem relates to the market for used cars. Suppose that cars in Citrusville are of two types: poor-quality cars (lemons), worth $4,000 to their current owners, and good cars, worth $8,000 to their owners. All together, there are 200 cars of each type. The demand for poor-quality used cars is QL = 220 – 0.004pL, and the demand for good-quality used cars is QG = 240 – 0.004pG.

(a) If buyers are able to distinguish between the two types of used cars, there will be two separate markets. What would be the price of each type of car? pL = $____; pG = $____.

(b) If buyers are unable to differentiate between good cars and lemons, however, this equilibrium cannot be sustained. Lemon owners will try to sell their cars in the good-car market, for how much? _____________. This will force the price in the good-car market to (rise, fall).

(c) Any price greater than $8,000 yields a total demand of no more than _________ cars. How many cars will be supplied at this price? ___________. Who will be disappointed in this situation? _____________________.

(d) Given the excess supply described in part (c), there will be downward pressure on prices. At any price below $8,000, however, how many good-quality cars will enter the market? _______________________________.

(e) This analysis leads one to conclude that the used car market could end up with only (lemons, good-quality cars) in it.

7.3 Recall Fisher’s separation theorem that states: Given a perfect lending market, individuals choose among assets by comparing present values using a market rate of interest, and the choices are not influenced by preferences about the timing of consumption. Consider the principal-agent problem as it applies to the separation of ownership and control in a large corporation. In this application, the stockholders (the principals) delegate production and investment decisions to the company’s managers (the agents). These agents, however, may not

know anything about the principals’ preferences, other than that they (the stockholders) want to maximize their profits. What relevance, if any, does Fisher’s theorem have to this situation?

7.4 Using Frank Knight’s classification between decisions made under risk or uncertainty, what type of decision-making is implied by the following situations?

(a) You are considering the purchase of Stock A. You believe that, over the next year, there’s a 60 percent chance the stock will gain $200/share and a 40 percent chance that it will lose $250/share. ___________.

(b) You have invested $10,000 in your friend’s new business. If the business is a success, your friend will return your $10,000 plus $5,000 more. If the business doesn’t make it, you’ll get back nothing. Your friend thinks that there is a good chance of success. _____________.

(c) If you place a bet on red at the roulette wheel in Las Vegas, the probability that you will win is 18 out of 38. (There are 36 red, 36 black, and two green numbers on a roulette wheel.) _______________.

(d) Your brother-in-law has asked you to invest in his Lost Gold Mine. He thinks that there’s a 75 percent chance you’ll make a fortune; on the other hand, there’s always that 25 percent chance that the mine deal will go belly-up, leaving you with little of your original investment. ________________.

7.5 Barry Cautious says that he is indifferent between receiving a sure $10,000 and a lottery that involves a 50 percent chance of winning $25,000 and a 50 percent chance of getting nothing. We can construct Barry’s utility function by assigning arbitrary values to the best and worst outcomes, say U(25,000) = 1 and U(0) = 0.


0.5

1.0

M (000s)

U(M)

0.0105 15 20 25

(a) Calculate the utility of receiving the certain $10,000. ____________________________.

(b) Using the three utility-money combinations you now have, diagram Barry’s utility function on the following figure.

(c) What is the expected monetary value (EMV) of the lottery offered to Barry? Indicate this EMV and Barry’s utility from part (a) on your graph. EMV = _____________________.

(d) Is Barry risk-averse? How can you tell? ____________________________________.

(e) Suppose now that Barry is offered another lottery, which involves a 1/3 chance of winning $10,000 and a 2/3 chance of winning $25,000. What is the EMV of this lottery? EMV = _______________________.

(f) Calculate the expected utility (EU) associated with this lottery. Indicate this utility, along with the EMV from part (e), on your graph. EU = _____________________________.

7.6 In the textbook application, “Reservation Price for Buying Insurance,” it was demonstrated that a risk-averse person would pay up to $500 to insure against an expected loss of $800. This was presented graphically in Figure 7-5. As an extension of that application, consider the following two cases: Joe Cool, a completely risk-neutral dude, whose utility function can be expressed simply as U = M; and Eva Kneivel, a risk-loving daredevil lass, whose utility function is U = M2 (with M = money income). As in the text, suppose that each person has invested $900 and faces a 50 percent chance of losing $800, so that the expected value (EV) of the gamble is $500.


(a) Find the expected utility (EU) of the gamble for Joe Cool. _______________________.

(b) Solve for Joe’s reservation price, the value R that yields the same utility as the EU when insurance is not bought. In other words, Joe Cool is willing to pay up to $___________ to avoid the risk of the $800 loss.

(c) The expected payoff by the insurance company is equal to $_______________. Could the insurance company cover its costs in this situation? Why or why not? _______________ _______________________________________________________________________.

(d) Find the expected utility (EU) of the gamble for Eva Kneivel. _____________________.

(e) Solve for Eva’s reservation price, the value R that yields the same utility as the EU when insurance is not bought. In other words, Eva Kneivel is willing to pay up to $_________ to avoid the risk of the $800 loss.

(f) Could the insurance company cover its costs in this situation? Do you think that most people are probably risk-averse or risk-lovers? ______________________.

(g) Eva’s utility function is graphed below. Show her EU from the 50 percent gamble of losing $800. What is the certainty equivalent of this gamble? __________________. Show it on the following graph.

M

U(M)

10,000

500,000

810,000

100 500 900

a

b

U

7.7 Ms. Bea Careful, who is risk-averse (with U = ), and Mr. I. M. Wilder, who is a risk-lover (with U = M2), have just won $5 each and are offered a chance at double-or-nothing. The double-or-nothing game will be a simple coin toss: Heads, you get $10, and tails, you lose your $5, and are left with nothing. The expected value (EV) of this game, of course, is exactly equal to their current winnings of $5.


(a) For Bea, the expected utility (EU) of this game is _______________________. She (will, won’t) play the game.

(b) For I. M., the expected utility (EU) of this game is ________________________. He (will, won’t) play the game.

(c) Suppose that the coin to be used is not a fair coin — it’s been tampered with so that the probability of “heads” P(H) is not equal to 1/2. How high would P(H) have to be before Bea Careful would play this game? ______________. How low would P(H) have to fall before I.M. Wilder wouldn’t play? _______________.

7.8 The adverse selection problem, which was examined for the used-car market in problem 7.2, has another interesting application in the market for insurance. In this case, the problem arises because the insurance company knows only about riskiness on average for the population as a whole. We can examine the insurance company’s problem using the graph below. Assume that all people have identical utility functions and initially have identical endowments of $50,000. These people also face a potential disaster that would cause a loss of $40,000.

M

U(M)

1810 30 42 50

a

b

c

U

d

e

f

gh

There are two distinct (and equal-sized) groups in this population, however. Type A people are nervous and accident-prone, making them high risks; their probability of suffering the loss is 0.8. Type B (low-risk) people are more relaxed; their probability of suffering the loss is 0.2.

(a) A typical, uninsured high-risk person (Type A) faces a gamble between $10,000 (with probability P = 0.8) and $50,000 (P = 0.2). What is the expected monetary value (EMV) of


this gamble? EMV = _________________. Indicate this person’s expected utility (EU) from the gamble on the graph.

(b) What would be an actuarially fair insurance premium for this high-risk person? (A fair premium equals the expected value of the loss.) _________. Would this person be willing to pay the premium? Why or why not? _______________________________________.

(c) A typical, uninsured low-risk person (Type B) faces a gamble between $10,000 (with probability, P = 0.2) and $50,000 (P = 0.8). What is the expected monetary value (EMV) of this gamble? EMV = ________________. Indicate this person’s expected utility (EU) from the gamble on the graph.

(d) What would be an actuarially fair insurance premium for this low-risk person? _________. Would this person be willing to pay the premium? Why or why not? _________________ ____________________________________________________.

(e) Unfortunately, the insurance company may find it impossible to distinguish between low- and high-risk people, and may be forced to offer fair insurance by charging everyone an average premium of $20,000. With this offer, all people (low and high-risk) would have an expected monetary value of $______________, and a certain utility level corresponding to point ____ on the graph.

(f) The (low-, high-) risk group clearly benefits here, moving from a utility level at point ____ to point g. The (low-, high-) risk group is forced to a lower level of utility, however, than what they receive even without insurance, which is at point ____.

(g) The insurance company, under these circumstances, will get only what type of customers? _______________________. Can the insurance company make money in this situation? Explain. _______________________________________________________________.

(h) Is there any way the insurance company might distinguish between the two risk groups? _______________________________________________________________________.

7.9 The problem of moral hazard arises whenever individuals, because they are insured, behave in ways that raise the probability of the unfavorable event. For example, people who have complete home theft insurance might be (more, less) likely to invest in a home security system. This type of consumer behavior makes it impossible for insurance companies to offer premiums that are fair in expected value terms. What can the insurance company do in such cases?

7.10 Professor Dropsy has just bought a dozen eggs at the supermarket. There is a 50-50 chance that he will drop his shopping basket, breaking all of the eggs, on any one trip home.

(a) If Prof. Dropsy takes all 12 eggs home in one trip, what is the expected number of eggs to make it home? __________________________________.

(b) If instead he makes two trips, taking 6 eggs home each time, what is the expected of eggs getting home safely? __________________________________.


(c) In each case, what is the probability of arriving home with no eggs? ________________.

(d) For each case, calculate the variance, using the formula 2 = ∑[xi – E(xi)]2f(xi). What is the standard deviation in each case? ________________________________________.

(e) For each option, calculate the coefficient of variation, v = /µ. _____________________.

(f) If Prof. Dropsy follows the mean-variance rule, should he diversify his holdings (of eggs), or put all of his eggs in one basket? __________________________________________.

(g) Do people diversify in real life? _____________________________________________.

7.11 Delbert Unwin Bell took a dislike to his name in his youth, and so he now goes by his initials, D. U. Bell. His friends call him “Dub.” Dub Bell has $10,000 to invest. He can put his money in T-bills that are yielding a 6 percent return, or he can buy a risky asset that has an expected return of 30 percent. However, the risky asset has a history of variation in its actual return. Its variance is 4 percent. Dub measures the risk of any asset by its standard deviation.

Dub has a third alternative. He can divide his money between the two assets. When asked how he will decide how much to put into each, he said: “Well, if you double the risk you’ll have to double the return to keep me happy.” Given this information, we can apply our model of consumer choice to Dub’s portfolio decision.(a) What are the values of and that describe the risk-less asset? C = _____, C = ______.

Denote this combination as bundle C, for certainty, on the following figure. What are the values of and that describe the risky asset? R = _____, R = ______. Denote this combination as bundle R, for risky, on the following figure.

(in %)

(in %)

6

12

18

24

30

1.00.5 1.5 2.0


(b) Draw Dub’s budget line between bundles C and R. What is the slope of this line? _______ ___________________. What does this slope represent? __________________________.

(c) Dub’s utility function can be written as U = / because when both and are doubled, his utility remains constant. Thus, the marginal rate of substitution is MRS = / also. (Recall that the MRS can be measured by the ratio of marginal utilities, the MRS has been defined as a positive value, and is a “bad” not a good.) Write an equation that will ensure the equality of the MRS and the slope of the budget line. _____________________________.

(d) Write an equation for Dub’s budget line, and substitute the appropriate known values into it. ________________________________________________________________________.

(e) Utility maximization requires that an indifference curve be tangent to the budget line. Tangency is given by the simultaneous solution to the equal slopes condition and the equation for the budget line. Solve those two equations for P, the desired risk for the optimal portfolio. (This is a bit tricky because is a “bad.”) P = __________.

(f) After you have found P, the risk of the optimal portfolio, what is the required expected return? P = __________. Designate the optimal portfolio as P* on the graph.

(g) The expected return to a portfolio P made up of R and C is P = kR + (1 – k)C. To achieve the portfolio P*, what value of k is required? k = __________.

(h) How much of his $10,000 should Dub invest in T-bills? $_____________. How much should he invest in the risky asset? $____________.

7.12 Suppose that you are very risk-averse, and would like to hold a two-asset portfolio, where the payoffs of the assets are inversely related. Generally, holding such a diversified portfolio can help you reduce risk. Consider the following joint distribution data on Assets A & B, and on Assets C & D and determine which pair of assets has the stronger negative association.

Percent gainin share

A B f(A, B) (A – A) (A – A)2 (B – B) (B – B)2 (A – A)(B – B)

10 1 0.28 2 0.36 6 0.44 7 0.1


Percent gainin share

C D f(C, D) (C – C) (C – C)2 (D – D) (D – D)2 (C – C)(D – D)

20 3 0.216 4 0.312 7 0.4

8 10 0.1

(a) Calculate the means for Asset A and Asset B: µA = _______; µB = ________.

(b) Calculate the variances for Assets A and B, using the table to help direct your effort: A

2 = ____________; B2 = _____________.

(c) Calculate the covariance between the percentage gains on A and B: AB = ________.

(d) Calculate the means for Asset C and Asset D: µC = ________; µD = _________.

(e) Calculate the variances for Assets C and D, using the table to help direct your effort: C

2 = ___________; D2 = ____________.

(f) Calculate the covariance between the dollar gains on Assets C and D: CD = _________.

(g) After calculating these covariances, you decide to pick the portfolio of Assets C and D, since it has the larger covariance. Is this a wise choice? Calculate the correlations for each asset pair: AB = ______________; CD = ______________.

(h) Clearly, Assets A and B are (more, less) strongly correlated, even though their covariance is, in absolute value terms, (smaller, larger). The (covariance, correlation) is often a better indicator of the strength of an association between variables, since it is independent of the measurement scales used.

7.13 The capital asset pricing model (CAPM) relates j, the expected return of the jth asset, to m, the expected return to the market taken as a whole:

j = c + j(m – c)

where c is a risk-free return. The beta of the jth asset is defined as

j = mj

m2

where mj is the covariance of the return of the market with the return to the asset, and 2m is

the variance of the market return.

(a) Can the beta of an asset be negative? If so, what does that mean? ___________________ _______________________________________________________________________.


(b) Can the beta of an asset be zero? If so, what does that mean? Does that imply that the asset has no risk? _____________________________________________________________.

(c) Can the beta of an asset be positive? If so, what does that mean? ___________________ _______________________________________________________________________.

7.14 Considerable theoretical and empirical work has been published about the efficient market hypothesis (EMH), the idea that the stock market (or any securities market) is efficient in adjusting to all relevant information about individual investments and the economy as a whole. An important implication of this hypothesis is that even expert investors cannot achieve above-normal profits by following any particular investment strategy.

(a) Consider the market for gambling on National Football League (NFL) games. Is this market similar to the stock market? Do both exhibit characteristics of a perfectly competitive market? ______________________________________________________.

(b) Many supposed experts produce pamphlets and books — backed by considerable statistical evidence — that for a small price provide bettors with one or more profitable betting systems. In light of what the EMH implies, do you think that these betting systems will work? _________________________________________________________________.

7.15 In Chapter 6 (problem 6.16) we calculated a firm’s Net Present Value (NPV) for two 3-year projects: Project Alpha and Project Beta. The required outlays, and yearly revenues, associated with each project are shown in the table below, along with each project’s NPV at a discount rate of 7 percent.

Project Project Initial outlay $20,500 $27,000

Net revenues:Year 1 8,000 5,000Year 2 8,000 5,000Year 3 8,000 5,000

Scrap value:Year 3 0 20,000

Net present value $494.53 $2,447.54

Suppose that Project Beta is to be located in another country, where an unstable political climate will subject the investment to considerable risk. Management has calculated the following risk-adjusted factors for the project:


Risk-adjustment Risk-adjustedPeriod factor rate of return

1 2%2 4%

3 6%

(a) Calculate the risk-adjusted discount rates for Project Beta and fill in the table above.

(b) Calculate the risk-adjusted NPV for Project Beta. NPV = _______. Is it still preferred to Project Alpha? ___________.

(c) Suppose that after Project Beta is completed, the capital equipment can be sold for its scrap value but will have to be sold in the foreign country. Adjust the present value of the equipment’s scrap value and calculate the new NPV for Project Beta. NPV = ________. Is it still preferred to Project Alpha? _________. If Project Alpha could not be implemented, would you recommend Project Beta at all? _________________________.

7.16 Heddy Jergensen is a business major at a mid-western university. Her father, Ole Jergensen, is a wheat farmer in South Dakota. Wheat farming is a risky business because you have to decide how much to plant long before you know the price of wheat. Heddy has learned about futures contracts in her economics class, and she tells her father how to reduce the risk associated with not knowing the future price of wheat. The current price of wheat is $6.00 per bushel.

(a) If Ole enters into a contract to sell wheat in the future at $6.00, how much potential revenue will he lose if the future spot price rises to $6.25 per bushel? _______________________.

(b) If Ole contracts to sell wheat at $6.00 per bushel, what will be the change in potential revenue if the future spot price falls to $5.75 per bushel? ___________________________.

(c) Suppose that at the time Ole contracts to sell wheat at $6.00 in the future, he also contracts to buy wheat in the future at $6.05. Now what happens to his potential revenue if the future spot price is $6.25? _______________________________.

(d) Suppose instead that the future spot price is $5.75 per bushel. Now what is the change in potential revenue? ____________________________.

(e) Given the hedge just described, and set up by Heddy Jergensen, daughter of Ole the hedger, there will be a net (increase, decrease) in potential revenue whether the future spot price changes from $6.00 to $6.25 or from $6.00 to $5.75. What will determine whether Ole will undertake such a hedge? ___________________________________________________.


7.17 Speculators make money by buying when prices are low and selling when prices are high. What effect does their activity have on the overall level of price fluctuations in a market?

7.18 Consider the following payoff matrix, which depicts various rates of return (in percents) possible from four different investment plans (I1 through I4), given four possible future states of the world (S1 through S4). Since there is no universally valid decision rule for making decisions

under uncertainty, several rules have been developed.

States of the world

Inve

stm

ent p

lans

S2 S3 S4

I1

I2

I3

I4

S1

0 5 15 0

5

5 5 5 5

0

00 0 20

10 10

(a) Suppose that the maximin rule is followed: choose the action that yields the best of the worst payoffs. If I1 is chosen, what is the worst possible outcome? ________. What if I2 is chosen? _________. Which investment plan will be chosen, following this (pessimistic) maximin rule? _________.

(b) Another possible decision rule is the maximax rule: choose the action that yields the best of the best outcomes. If I1 is chosen, what is the best possible outcome? ________. What if I2 is chosen? ________. Which investment plan will be chosen, following this (optimistic) maximax rule? _________.

(c) Another possible decision rule is the minimax regret rule, which, unlike the minimax and maximax, takes intermediate payoffs into account. It does this by minimizing the opportunity cost of an incorrect action. If S2 occurs, for example, the greatest possible payoff is 10, when I2 is chosen. The regrets associated with I1, I3, and I4 are therefore 5, 5, and 10, respectively. Complete the following regret payoff table for all of the states of nature. Which investment plan will be chosen, using the minimax regret rule? _________.


States of the world

Inve

stm

ent p

lans

S2 S3 S4

I1

S1maximum

regret

I2

I3

I4

(d) A more sophisticated way to play this game involves assigning probabilities to each possible S. The Laplace rule says that equal probabilities should be assigned to the states of nature if there exists no better information. Here, each S is expected to occur 1/4 of the time. What is the expected value (EV) of selection I1? _______. The Laplace rule directs

us to choose the action with the highest EV. Which one will be chosen, following this rule? ________.

(e) Suppose that better information, in the form of subjective probabilities, can be applied to the four states of the world. Then, the Bayesian rule can be used. Like the Laplace rule, it says to choose the action with the highest EV. If the probabilities of Si are as follows: P(S1) = 1/8, P(S2) = 1/2, P(S3) = 1/4, and P(S4) = 1/8, which will be chosen now?

__________.

7.19 The Duke of Oil has the option of buying the rights to one of two North Sea oil platforms. Each of these is believed to be capable of producing large amounts of oil. The expected monetary payoffs (in millions of dollars) from buying either Platform A or Platform B are given in the table below. The Duke’s advisors believe that there is a 5/8 chance that Platform A has access to the larger reserves, and a 3/8 chance that Platform B does.

Plat

form A

B

36

16

25

49

B has larger reserves

A has larger reserves


(a) Given the table of money payoffs, construct the Duke’s payoff table of utilities if his utility function is U = , where M is the money payoff.

Plat

form A

B

B has larger reserves

A has larger reserves

(b) If the Duke buys Platform A, what will be his expected utility? EU = ________. If he buys Platform B, what will be his expected utility? EU = ________. Which alternative gives him the maximum EU? ________.

(c) Now suppose that perfect information is available, in the form of a detailed geologic report, that reveals which platform has access to the greater oil reserves. If it turns out to be Platform A, then of course the Duke will buy Platform A, and likewise for Platform B. What is the expected utility of a perfect prediction in this situation? ________________.

(d) You have now calculated both the maximum EU and the EU of a perfect prediction. The difference between the two is called the value (in utility terms) of the information, or the utility of perfect information. What is that value here? _______________.

(e) How much money will the Duke be willing to pay for this geologic report; in other words, what is the value of perfect information? ________________________________.


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Problems 5-9 - Mihaylo Faculty Websites - Mihaylo …mihaylofaculty.fullerton.edu/sites/esolberg/Econ_315... · Web viewWhat do you notice about the trade-off the consumer is willing