2020a Final 2005 Solutions

7/27/2019 2020a Final 2005 Solutions

1/19

Question 1: Agricultural Household Model

1a.

maxl,c,L

cl

s.t. : wLH

p( L

c) + wLS

LH + LU = Ll = LE LS LU

1b. The two resource constraints combine to become LH LS = L LU + l + LU LE = L + l LE .Making this substitution into the rst constraint yields:w (L + l LE ) p L c, or

wl + pc p L wL+ wLENote that the right-hand side of the problem only contains consumption variables and the left hand sideonly contains production variables. So, to solve the consumers problem you rst maximize pro t and thensolve the consumers problem in the usual way. This is an application of the separation result from theagricultural household model.

Pro t maximization:

maxL

p L wLDL =

p2 L = w

L = p2w2

Substituting L into the pro t equation yields p

p

2w

w

p

2w

2 = p

2

4w .

Next, we solve the consumers problem:max cl

s.t. : wl + pc p2

4w+ wLE

L = cl + p2

4w+ wLE wl pc

c = wl = p

wl + pc =p2

4w+ wLE

From the rst two foc we have wc = pl , so the budget constraint implies that:

2wl =p2

4w+ wLE

l =p2

8w2+

LE2

2 pc =p2

4w+ wLE

c =p

8w+

w2 p

LE .

1


2/19

1c. The consumer must hire outside labor if LU = LE l < L , or

LE p2

8w2+

LE2 < p2w

2

LE2 p

28w2

< p24w2

LE 0

2d. Note that the rm must choose q before learning the value of A. Thus the cost of producing q is2q

A 0 w1w2 or 2qA 0 + w1w2 depending on the value of the shock. Thus the rm maximizes:

p (q ) q 12 2q A0 w1w2 +12

2q A0 +

w1w2 p (q ) q 2q w1w2 12 (A0 ) +

12 (A0 + )

p (q ) q

2q w1w2

2 (A0 + ) + 2 ( A0 )

2 (A0 ) 2 (A0 + ) p (q ) q 2q w1w2 A0(A20 2) p (q ) q

2q w1w2A0 (2 /A 0)

So, the eff ect of the uncertainty is to make it as if the rm has cost function 2q w 1 w 2A 0 (2 /A 0 ) , which is greater

than what its cost function would be if A = A0 for sure, 2qA 0 w1w2 . Taking the derivative:

p0 (q ) q + p (q ) 2 w1w2

A0 (2 /A 0), with equality if q = 0 .

2e. Note that the marginal revenue is unchanged, but when A = A0 for sure the right-hand side is2 w 1 w 2

A 0 0 would be an equivalent Bernoulli utility.

b) For distributions with an innite number of outcomes the von Neumann-Morgenstern utility is still the mean of F . That is R xdF (x ).

c) If two distributions both have innite means a risk neutral decision makerhas no well-dened ranking between them. They are both better than any xedninte amount of wealth, no matter how large. We overcame this issue in theproof of the existence and nature of the von Neumann-Morgenstern representa-tion when we assumed that there was a best and a worst non-stochastic outcome.When this is the case we automatically have ruled out problems due to unbound-edly large possible outcomes and innite means. Of course it means that wedo not have a representation theorem for these problems without modifying ouraxioms somewhat - and we did not do that in this class.

d) When a risk neutral consumer has initial non-stochastic wealth w andhas the opportunity to acquire the random variable with distribution functionF (2k ) = P kj =1 12 j the consumer gladly would give up all of w for this acquisition.

Using the form of the expected utility from part b), R xdF (x ) = P1

j =11

2 j 2j

=1 .

e) When the Bernoulli utility is ln x we can get an innite expected utilityby putting the payos at x j = e2

j

,and using the same geometrically decliningprobabilities. Then expected utility is P

1

j =11

2 j ln(e2 j ) = P

1

j =11

2 j 2j = 1

since, as above, every term in the sum is 1.

f) There is a problem with using utilities that can take an innite expectedvalue for some probability distributions, since these distributions can thereforenot be ranked or we have to say they are all equally good, since the expectedutility is + 1 . If we take any distribution with innite mean and increase allpayos, say by $1, we have made a rst-order stochastically dominating shift in

the distribution but the expected utility does not rank it as a strict improvement.Therefore, either we have to use our theory only on some subset of distributions,or we have to use Bernoulli utilities that are bounded above.

g) The same problem as in part f) arises when utility is bounded below. Inthe case of Bernoulli utility 1(1 ) (x + w)

1 with > 1 we can arrange for the

1

Question 3.

4


5/19

expected utility to be 1 using a parallel construction to that above. Takethe probabilities to decline like 2 j . Then we want the Bernoulli utilities to be

2j . That is,1

(1 )(x j + w)1 = 2j

To ndx j explicitly, write this expression as

(x j + w)1 = ( 1)2j

Then take the logarithm of both sides (base 2)

(1 )log2(x j + w) = log 2( 1) + j

which can be rewritten as

x j + w = 2 ( 1)1

(1 ) 2j

( 1)

Therefore expected utility theory with utilities of the constant relative riskaversion form is incompatible with a theory of risk bearing that asks for all distributions to be ranked in a manner that respects a strict preference for rst-order stochastic dominace .

2 5


6/19

Page 1 of 14

4. [20 points] Consider an economy consisting of:

I consumers, with utility over some consumption good x and a numeraire good m as follows:

( ) ( ) ( ) ( ), where 0, 0 at all 0i i i i i i i iu x m m f x f x f x x = + > < J firms (where J is a large number and all firms are price-takers) that produce x

using units of m with twice-differentiable strictly convex cost function c j(q j)(where q j is the quantity of x produced by firm j).

Price vector ( pm, px) = (1, p)

4a) List the conditions that must be satisfied in order for an allocation( )* * * *1 1, , , , , I J x y yK K and price p* to constitute a competitive equilibrium in thistwo-good economy. (5 points)

(i) Profit maximization : for each firm j, q j*

solves

( )*0

Max j

j j jq p q c q

(4.1)

which gives necessary and sufficient FOC for each firm j:

( )* * *, with equality if 0 j j p c q q > (4.2)

(ii) Utility maximization : For each consumer i, xi* solves:

( )

( )( )* * * *Max

subject to 1

i ii i x X

i i i ij j j j j

m f x

p x m p q c q

+

+ + (4.3)

where X i is the set of all consumption bundles, i is the individualsendowment of the numeraire, and ij is consumer is ownership share in firm j.(Since it was not specified that individuals were endowed with levels of thenumeraire and that individuals own firms, it was also an acceptable answer to

put wi on the right hand side of the budget constraint, where wi =exogenously given wealth of consumer i.)

The necessary and sufficient FOC for each consumer i is:

( )* * *, with equality if 0i i f x p x > (4.4)

(iii) Market clearing : In the market for good x:


7/19


8/19

Page 3 of 14

5. [35 points] The small country of Crimsonia produces two goods food and books that it sells on the world market. Production of each of these goods requires three inputs:land, labor and machinery. Specifically, production of one unit of food requires 2 units of land, 3 units of labor and 1 unit of machinery. Production of one book requires 2 units of each input (the books are very fancy). The food and book industries have constant returns

to scale technology, and Crimsonia is endowed with 48 units of land, 63 units of labor,and 40 units of machinery.

5a) Will all three of the inputs be completely used up in the process of making foodand clothingi.e. will all of the resource constraints bind in equilibrium? Whyor why not? (5 points)

No. There are three resource constraints in this economy:

2 2 48 F B+ (5.1)3 2 63 F B+ (5.2)

2 40 F B+ (5.3)where F is amount of food produced and B is number of books produced. These are threelinearly independent conditions in two unknowns. Therefore, all three cannot hold withequality.

The price of food and books is determined on the world marketi.e. exogenously, fromthe point of view of Crimsonia. Currently, the price of food is $12 per unit, and the priceof a book is $9.

5b) How much food and how many books will be produced? (5 points)

Since we know that at most two of the constraints will bind, we can try to solve this problem for the three different cases of the land, labor and machinery endowmentconstraints not binding. Note that when the resource constraint is not binding for a

particular input, the price of that input will be zero in equilibrium.

To solve for production levels, we can utilize the fact that, given CRS production, thefirms must earn zero profits. The zero profit condition can be written as cost of one unit =

price of that unit.

Let input prices for land, labor and machinery, respectively, be w H , w L, and w M .

Case 1: Land constraint doesnt bind ( w H =0):

3 12

2 2 9 L M

L M

w w

w w

+ =+ = (5.4)


9/19

Page 4 of 14

This gives us15 /4 3 /4 L M w w= = (5.5)

So, the prices for the two inputs with non-zero prices are greater than zero and thereforefeasible. Now we need to check to see if the resource use is feasible. Using the (binding)

resource constraints for labor and machinery, we can calculate food and book productionfor this case:

3 2 632 40

F B

F B

+ =+ =

(5.6)

This yields:

23/2 57/4 F B= = (5.7)

Finally, we can check to see if this case is feasible by checking to see if the resourceconstraint for land is satisfied:

?

(23 / 2) (2) (57 / 4)*(2) 48 but this does not hold, since 51.5 48

+ >

So, this case is not feasible.

Case 2: Labor constraint doesnt bind ( w L=0):

2 12

2 2 9 H M

H M

w w

w w

+ =+ = (5.8)

This gives us15/2 3 H M w w= = (5.9)

We cannot have negative input prices, so we know that this case is not feasible.

Case 3: Machinery constraint doesnt bind ( w M =0):

2 3 12

2 2 9 H L

H L

w w

w w

+ =+ = (5.10)

This gives us3 /2 3 H Lw w= = (5.11)

So, the prices for the two inputs with non-zero prices are greater than zero and thereforefeasible. Now we need to check to see if the resource use is feasible. Using the (binding)


10/19

Page 5 of 14

resource constraints for labor and machinery, we can calculate food and book productionfor this case:

2 2 483 2 63 F B

F B

+ =+ =

(5.12)

This yields:

15 9 F B= = (5.13)

Finally, we can check to see if this case is feasible by checking to see if the resourceconstraint for machinery is satisfied:

?

(15) (1) (9) *(2) 40this does hold, since 33 40

+

So, this case is feasible.

Therefore, (5.13) is the answer. 15 units of food and 9 books are produced.

5c) What are the equilibrium input prices? How much of each input will be usedin the production of food? In the production of books? (10 points)

Based on Case 3 above, the equilibrium input prices are w H =3/2, w L=3, w M =0.

The resource constraints for land and labor bind, so 48 units of land and 63 units of labor will be used. Also, as calculated above, 33 units of machinery will be used.

Due to changes in international demand, the price of a book rises to $10, while the priceof food remains constant.

5d) What general equilibrium comparative statics theorem would help you topredict the effects of this change in output price on the equilibrium values youcalculated above? What do you predict will happen, on the basis of thistheorem? Now do the calculations to determine whether or not your predictionis correct. (Note: partial credit will be awarded for correctly explaining how

one would go about calculating the effect.) (5 points)

The Stolper-Samuelson theorem would lead one to believe that we should see the price of land rise and the price of labor fall, since land is used relatively more intensively thanlabor in book production than in food production.

To determine whether or not this is the case, one should first check to see if, under thenew prices, machinery is still the input whose constraint does not bind. In the example


11/19


12/19

Page 7 of 14

Production Possibilities Set for Crimsonia

0

5

10

15

20

25

30

35

40

45

0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 3 0 3 1 3 2

Books

F o o

d

Land Labor Machinery

The solution to this problem can be seen graphically above. The dashed line is an iso-revenue line, K = P book *B + P food * F where K is a constant. The line has slope - P book /P food .

Efficient use of the inputs requires that production be maximized given the resourceendowments. Graphically, this means that we will want to be on the highest possible iso-revenue line. The Production Possibilities Frontier is kinked and has 3 different slopes:for the region corresponding to 9 books or fewer, the slope is -2/3, between 9 books and16 books, the slope is -1 and between 16 and 20 books the slope is -2. For the original

prices, the highest iso-revenue line will intersect the PPF only at point A in the diagramabove because its slope is -3/4, which is between -2/3 and -1. If its slope were between -1and -2, then the highest iso-revenue line would intersect the PPF only at point B. After the price change, the slope of the iso-revenue line is -5/6, which is still between -2/3 and -1, so production levels do not change.

A

Shaded area is the ProductionPossibilities Set: the set where allresources constraints are satisfiedsimultaneously

The slope of the dashed

line is - P book / P food

B


13/19

Page 8 of 14

International prices return to their initial level, but a natural disaster has made some of Crimsonias land unusable. The endowment of land has decreased to 44 units.

5f) What general equilibrium comparative statics theorem would help you topredict the effects of this change in endowment of land on the equilibrium

values you calculated above? What do you predict will happen, on the basis of this theorem? Now do the calculations to determine whether or not yourprediction is correct. (Note: partial credit will be awarded for correctlyexplaining how one would go about calculating the effect.) (5 points)

The Rybczsinski Theorem suggests that production of books should decrease and production of food increase, since land is used more intensively in book production thanit is in food production. We can verify this by recalculating production using the newresource constraint for land and the old one for labor (which hasnt changed):

2 2 443 2 63 F B F B

+ =+ = (5.16)

This yields:

19 3 F B= = (5.17)

which corresponds to our prediction.

Note, again, that we need first to verify that the change in land endowment does not leadto a shift in input mix that would cause either land or labor to become the factor not

completely used up. In the example above, this will not be the case: machinery will still be the over-abundant factor. You could verify this rigorously by looking at all three casesagain, or by noting that in the diagram above, the decrease in land corresponds to aninward shift of the land constraint, but does not affect the basic shape and slopecharacteristics of the PPF.


14/19

Page 9 of 14

6. [30 points] In the country of Harvardiana, there are 10 citizens, all of whom havequasi-linear utility with respect to consumption of a pure public good x and a numerairegood m. The individual utility functions are:

( ) ( ), * ln for 1, ,10i i iu x m m i x i= + = K

where ii x= .

Initially, the government is the sole producer of the public good. It has costs c(q) = q2,where q is the quantity provided of the public good. You (the ruler of Harvardiana) are incharge of setting the production level of the public good. Being a benevolent ruler, youwant to achieve a Pareto Optimal allocation.

6a) What problem will you solve? What is the Pareto Optimal allocation? (5points)

Since consumers have quasi-linear utility, the Utility Possibilities frontier will be a line(further, it will have slope -1 since one unit of numeraire good corresponds to the sameamount of utility for all consumers). To be specific, given any initial allocation of utility,a social planner can always redistribute utility one for one by redistributing thenumeraire.

Consequently, to achieve a Pareto Optimum, the social planner will want to ensure thatthe allocation of the public good is generating as much utility as possible. (Whatever numeraire good is left over can be distributed in whatever way is deemed desirable.)This, in turn requires that the allocation maximizes the Marshallian aggregate surplus(recall that in the quasi-linear case, the Marshallian aggregate surplus is a valid utilitymeasure).

Formally, the problem to solve is:

( ) 20

Max lniiq m i q q+

which is equivalent to solving the following problem:

( ) 20

Max lniqi q q

(6.1)

The first order condition for this problem is:

2 , with equality if 0o ooii

q qq

> (6.2)

Solving this for oq gives us:


15/19


16/19

Page 11 of 14

Producers will maximize their individual profits, given an exogenously determined price(they are price-takers). So each will solve:

2*

0Max

j

j

jq

q p q

N

(6.6)

The first order condition is:

** *2 , with equality if 0 j j

q p q

N > (6.7)

6d) What is the market-clearing condition? (3 points)

The market clearing condition is that total amount supplied in equilibrium equals totalamount demanded:

* * * *i ji j

x x q q = (6.8)

6e) How much of the public good will be produced in equilibrium? How manyconsumers will pay for the public good? Which consumers? (5 points)

To simplify matters, we can deal on the production side simply with aggregate supply.Each firm solves condition (6.7), which means that each produces the same amount of the

public good:

* * 2 ** * *2 so, which give us

2 2 j

j

q Np N p p q q

N = = =

This means that the equilibrium price must satisfy:

**

2

2q p

N = (6.9)

From the consumer side, we know that each consumer will solve condition (6.5). Notethat the denominator of the left hand side of this condition is simply x*, and so is thesame for all consumers. Also note that the left hand side of (6.5) will be different for eachconsumer, since the numerator is i. In particular, each successive consumer will have ahigher value of the left hand side than the previous consumer. Specifically, consumer 1svalue will be 1/ x*, consumer 2s will be 2/ x*, consumer 3s will be 3/ x*, etc. This meansthat only one consumer can satisfy condition (6.5) with equality, and that will beconsumer 10 .


17/19

Page 12 of 14

We can see that this is true by asking what would happen if it were not true. For example,what if the condition held with equality for consumer 9? In this case, the left-hand side of the condition would be greater than p* for consumer 10, which violates the condition. Wecan also ask, what if the condition were not solved for any consumer? This would mean

that p* > the left-hand side for all consumers. This would mean that x* = 0. But as x* approaches 0, the left hand side of the condition approaches infinity, and p* is finite, sothis could not occur.

To solve for q* and p* we can first use the result that only consumer 10 will buy a positive quantity of the public good combined with condition (6.5) to get the following:

**

10 p

x= (6.10)

use the market clearing condition (6.8) and also equation (6.9) we have:

*

* 2

10 2qq N

= (6.11)

which allows us to solve for quantity and price:

* * 2 55 andq N p N

= = (6.12)

6f) How does the Pareto Optimal quantity compare to the private equilibrium quantity?Represent the two equilibria on a graph with quantity of public good on the x axisand price/marginal benefit on the y-axis. (4 points)

The Pareto Optimal quantity is greater than the private equilibrium quantity unless N>2.This is actually the result of a mistake in the question, which was meant to result in thePareto Optimal quantity being unambiguously greater than the private quantity. As aresult, grading was very lenient on these parts. In any event, the typical picture of thedifference between the private and Pareto Optimal allocations is:


18/19

Page 13 of 14

6g) Explain how you could use a per-unit subsidy to make the private decision-making process result in a Pareto Optimal outcome. (Precise calculation of thesubsidy is not required). (4 points)

A subsidy could be used to make individuals take into account in their own privateconsumption decision the benefits accruing to others as a result of that privateconsumption decision. Since the private decision results in the individual setting her ownmarginal utility equal to price, and the Pareto Optimal decision sets the sum of marginalutilities equal to price, a per-unit subsidy in the amount of the sum of all other consumers marginal benefits at the optimal quantity would bring the private and ParetoOptimal conditions into line with each other.

6h) What information and capabilities would you need in order to actually designand implement such a scheme? (3 points)

Among the kinds of information and capabilities one would need to design andimplement such a scheme are:

1. complete knowledge of individuals marginal benefit schedules2. an ability to reliably identify each individual using non-falsifiable information (so

that individuals could not masquerade as other individuals)

i u i (x)

u i (x)

Price /MarginalBenefit

po

p*

q* qo Quantity

c(q)


19/19

P 14 f 14

3. possibly a lot of moneythe subsidy could end up being very large4. a way to raise and distribute this money without distorting incentives in the

economy

Documents

2020a Final 2005 Solutions