ANSWERS TO ODD NUMBERED EXERCISES IN...

© John Riley 11 April 2011

Answers to Chapter 5 page 1

ANSWERS TO ODD NUMBERED EXERCISES IN CHAPTER 5

Section 5.1: The Robinson Crusoe Economy

Exercise 5.1-1: Equilibrium

(a) 1 2( ) ( 147, )x y y yω= + = + . Then 1 1147y x− = − . Substituting for y in the

production function, 1/ 22 2 12(147 )x y x= = − . Robinsons utility is therefore

1/ 21 2 1 12 (147 )U x x x x= = − Taking the logarithm and differentiating,

1 1

1 1ln147 2

UL x x∂

= − +∂ −

. Setting this equal to zero, *1 98x = , hence * *

1 1 1 49y x ω= − = −

and so *2 14x = .

(b) Let the price vector be p. The production set is 211 2 1 24{( , ) | 0}y y y y= + ≤Y .

Robinson then seeks the solution of the following problem.

21

1 24{ | 0}y

Max p y y yΠ = ⋅ + ≤

The constraint is binding at the maximum therefore we can substitute for 1y and write the

profit as follows.

212 1 2 2 24( )y p y p yπ = − +

FOC for profit maximization:

11 2 22

2

0p y pyπ∂= − + =

∂.

Hence coconut supply is 22

1

2 pyp

= . Substituting back into the production function, labor

demand is 22

1

2( )DpLp

= . For labor demand to be optimal we therefore require that

1 2/ 2 / 7p p = .



Exercise 5.1-3: Existence with a non-convexity

(a) The boundary of the production set, 21

1 28 0y yγ+ + = can be rewritten as follows.

2 18( )y y γ= − − .

The graph of this function is depicted below.

Adding the endowment (2 ,0)ω γ= shifts the boundary horizontally by 2γ .

(b) Crusoe’s consumption vector is x y ω= + where 211 28y yγ= − − and so 21

1 28x yγ= − .

Then Crusoe’s utility is

211 2 2 28( ) ( ) 2 ln lnU x U y y y y yω γ γ= + = + + = − + .

FOC

124

2

1 0yy

− + = .

(c) Hence *2 2y = and so * 1

1 2y γ= − − .

O

2 2,x y

1 1,x y γ− γ

Y

ωY +



( ) arg { | }y

y p Max p y y= ⋅ ∈Y

(d) First ignore the option of producing nothing. Then Robinson solves the following

maximization problem

2

211 2 2 28{ ( ) }

yMax p y p yγ− − + .

Solving, 2 2 14 /y p p= . The resulting profit is

2 22 1 1 1 2 1( ) 2 / (2( / ) )p p p p p p pγ γΠ = − = − .

This is positive if and only if 22 12( / ) 0p p γ− ≥ . Therefore the supply curve if the firm is

2

2 12 2

2 1 2 1

0 , 2( / )( )

4 / , 2( / )p p

y pp p p p

γγ

⎧ ≤= ⎨

≥⎩

Note that this is a supply correspondence as it is multi-valued at the critical point where 2

2 12( / )p p γ= .

(e) If there is a Walrasian equilibrium must be efficient. Since the efficient output of

coconuts is 2, it follows that

2 14 / 2p p = .

Thus if a WE exits, the WE price ration 12 1 2/p p = .

(f) The implied equilibrium profit is

2 11 2 1 1 2( ) (2( / ) ) ( )p p p p pγ γΠ = − = −

(g) Thus a WE exists if and only if 12γ ≤ .

5.2 General Equilibrium

Exercise 5.2-1: Cobb Douglas economy

(a) The FOC for cost minimization profit maximizing by a firm in industry j are



1 1 2 2

1j jf fj jr z r z

α α−= .

Rewriting this equation, 2 1

1 2

1fj j

fj j

z rz r

αα−

= , 1,...,f F= .

Therefore 2 2

1 1

fj j

fj j

z zz z

= and so f fz zθ= for some 0fθ > where 1

1F

ff

θ=

=∑ . Hence

1 11 2 1 2

1 1 1( ) ( ) ( ) ( )j j j j

F F Ff f f f

j j j j j jf f f

q q z z z zα α α αθ θ θ− −

= = =

= = =∑ ∑ ∑

11 2( ) ( )j j

j jz zα α−=

(b) 1 1 2 2 1 1 11 1 21 2 2 12 2 22ln ln ( ln (1 ) ln ) ( ln (1 ) lnU a q a q a z z a z zα α α α= + = + − + + −

Utility is maximized subject to the constraint 11 12 1z z ω+ = , 21 22 2z z ω+ = .

Note that the optimization problem separates into two simpler optimization problems.

The first is

11 12

1 1 11 2 2 12 11 12 1,{ ln ln | }

z zMax a z a z z zα α ω+ + ≤ .

This is easily solved.

* * 1 1 1 2 2 111 12

1 1 2 2 1 1 2 2

( , ) ( , )a az za a a a

α ω α ωα α α α

=+ +

.

Similarly,

* * 1 1 1 2 2 121 22

1 1 2 2 1 1 2 2

(1 ) (1 )( , ) ( , )(1 ) (1 ) (1 ) (1 )

a az za a a a

α ω α ωα α α α

− −=

− + − − + −

(c) From (a), 1 2

1 1* *11 21

1r r

z zα α=

− hence 1 1

* *11 21

1( , )rz zα α−

= is a WE.

The cost function for industry j is 11 2( , ) ( ) ( )1

j jj j

j j

r rC q r qα α

α α−=

−.



11 2( ) ( )1

j jj j j

j j

r rMC AC p α α

α α−= = =

−

Exercise 5.2-3: Robinson meets Friday

(a) Since both consumers have the same preferences consider this as a representative

agent problem. Since 1 1 1x zω= − and 1/42 1x z= , the utility of the representative agent is

1/41 1 1 1 1 1ln( ) 4 ln( ) ln( ) lnU z a z z a zω ω= − + = − + .

It is readily confirmed that * 11 1

aza

ω=

+ and hence * 1/41 1( , ( ) )

1 1ax

a aω ω

=+ +

.

(b) The representative consumer has the following budget constraint.

1 1 2 2 1 1 ( )p x p x p pω+ ≤ +Π .

He therefore solves

1 2 1 1 2 2 1 1{ln 4 ln | ( )}x

Max x a x p x p x p pω+ + ≤ +Π

FOC

1 1 2 2

1 4ap x p x

λ= = .

Rearranging

1 2

1 2

1 4( ) ( )

p pa

x x

=

The WE must be efficient. Hence *x x= and so * *1 2

1 4( , )apx x

= is a WE price vector.

(c) Friday has no profit. His endowment of time is half the total endowment. His optimal

consumption bundle is therefore

11 2 1 1 2 2 1 12arg {ln 4 ln | ( )}F

xx Max x a x p x p x p ω= + + ≤ .

FOC

1

1 1 2 2 1 1 2 2 1 12

1 4 1 4 1 4F F F F

a a ap x p x p x p x pω

+ += = =

+.



Therefore

11 2(1 4 )Fx

aω

=+

.

Hence Robinson’s demand for leisure is *1 1 1 1

1 1( )1 2(1 4 )

R Fx x xa a

ω= − = −+ +

. Note that

this approaches 112ω as 0a → . Moreover it is readily checked that 1

Rx is an increasing

function of a for a greater than zero and sufficiently small. But Robinson’s endowment

of time is only 112ω . Thus the representative agent approach must be modified.

The only other possibility is that Robinson consumes all this tile in the form of leisure.

Then Friday is the only consumer supplying labor. We have already solved for his labor

supply. The WE price ratio must be chosen so that the firm’s demand for labor

1

1/4 11 2 1 1 1 1 1 12arg { }D F F

zz Max p z p z z xω= − = = − .

5.3 Existence

Exercise 5.3-1: Continuity of the supply function

We first show that is unique. Since Y is compact and p y⋅ is continuous, there is a

solution to the maximization problem { | }y

Max p y y⋅ ∈Y . To prove that there is a unique

solution, we suppose instead that there are at least two solutions 0y and 1y . Then for

every convex combination yλ is also a solution, since 0 0 0(1 )p y p y p y p yλ λ λ⋅ = − ⋅ + ⋅ = ⋅ . Since Y is strictly convex, every convex

combination intyλ ∈ Y . Then for sufficiently small 0δ >> , y yλ δ= + ∈Y . Then 0p y p y p p yλ δ⋅ = ⋅ + ⋅ > ⋅ . But then 0y is not a solution to the optimization problem

after all. Thus the profit maximizing response is a supply function ( )y p .By the Theorem

of the Maximum (I), this function is continuous.

Exercise 5.3-3: Existence and Non-existance with a minimum consumption threshold



(a) This is a standard case of two consumers with different homothetic preferences.

From Section 5.2, the PE allocations are a depicted below and the WE price ratio 1 2/p p

rises along the PE allocations as the utility of consumer A increases. The highest price

ratio is where Alex has the entire endowment ( , )ω ω . Since his marginal rate of

substitution is /A Aα β along the 45 line, this is the maximum price ratio. Making an

identical argument for Bev, the minimum price ratio is /B Bα β .

Figure: Edgeworth box diagram

(b) The only part of the Edgeworth Box or which neither consumer has a consumption

below γ is the inner depicted below. Redefining units, 1 2ˆ ˆh hhU x xα β= . The budget

constraint h hp x p ω⋅ ≤ ⋅ becomes

1 1 2 2 1 1 2 2 ˆˆ ( ) ( ) ( ) ( )h h h h h hp x p x p x p p pγ γ ω γ ω γ ω⋅ = − + − ≤ − + − = ⋅ .

We can therefore appeal to the answer to part (a) for all the points in the inner square of

the Edgeworth Box.

AO

BO



Figure: Edgeworth box diagram

By the first welfare theorem, if there is a WE allocation , it must be PE. Thus there are no

WE for endowments outside the shaded region.

Exercise 5.3-5: Continuum of Equilibria with no consumption Threshold

(b) The Edgeworth-Box Diagram is depicted below.

For PE allocations outside the shaded region the price is determined by the slope of the

high wealth consumer. Along the 45 line the slope of Alex’s indifference curve is 2

while the slope of Bev’s is ½.

For the PE allocations inside the shaded region we can redefine units so that 2i iz x= − .

Preferences are homothetic in z thus we can appeal to our earlier results and conclude

that along the PE allocations the WE price ratio rises from ½ to 2.

AO

BO



Figure:Edgeworth box diagram

Arguing as in the previous exercise, there is a continuum of equilibria for one endowment

point.

SECTION 5.4

Exercise 5.4-1: A Public “Bad”

(a) The problem is to solve ( , )

{ ( , ) | ( , ) 0}y b

Max U y b G y b ≤ . The Lagrangian is

( , ) ( , )U y b G y bλ= −L

The FOC are therefore

( , ) ( , ) 0, 1,2U Gy b y b jλ∂ ∂ ∂= − = =

∂ ∂ ∂j j j

Ly y y

and

( , ) ( , ) 0.U Gy b y bb b b

λ∂ ∂ ∂= − =

∂ ∂ ∂L

(2, 2)

(3,3)



(b) Introduce prices 1p and 2p for the two commodities and the pollution tax t. The

frm’s optimization problem is ( , )

{ ) | ( , ) 0}y b

Max p y tb G y b⋅ − ≤ . Form the Lagrangian

( , )p y tb G y bμ= ⋅ − −L . The FOC are therefore

( , ) 0, 1, 2jGp y b jμ∂ ∂

= − = =∂ ∂j j

Ly y

and ( , ) 0Gt y bb b

μ∂ ∂= − − =

∂ ∂L .

The consumer equates his marginal utility per dollar..

1 2

1 2

U Uy yp p

∂ ∂∂ ∂

= .

(c) It is easy to confirm that these conditions are the FOC for the optimum if jj

U py

ν∂=

∂

and /μ ν λ=

(d) The analysis generalizes directly. The new optimization problem is

2

( , ) 1

{ ( , ) | , ( , ) 0, 1, 2}f f

y b f

Max U y b b b G y b f=

≤ ≤ =∑

(e) The government intervention creates a new commodity “pollution rights” and sets the

endowment of this commodity at the optimal level. To price taking firms, a “pollution

right” is in effect a third input. Markets clear when demand for this right equals the fixed

supply so the additional market replaces the need for a direct tax.

SECTION 5.5

Exercise 5.5-1: Robinson Cruse Economy

(a) Define 1 22 ln ln lnu U x x= = + . Since utility is strictly increasing 2 2x y= . Hence

11 12ln(108 ) ln12 lnu z z= − + + .

FOC

31

2 2

1 1

1108 108z z

= =−

.



Hence *1 36z = and so * (72,72)x = .

(b) The profit of the firm is 1 1

1/2 1/221 1 2 1 1 1 1

1

( ) { 12 } { 12 }z z

pp Max p z p z p Max z zp

Π = − + = − +

FOC

1/221

1

1 6 0p zp

−− + = .

Hence 221

1

( ) 36( )pz pp

= .

(c) The WE is a PE allocation. Let p be the equilibrium price vector. Then *

1 1( ) 36z p z= = and so (1,1)p = is WE price vector. Equilibrium profit is ( ) 36pΠ = .

The budget constraint of the representative consumer is

1108 ( )p x p p⋅ ≤ +Π

Setting p p= this can be rewritten as follows.

1 2 144x x+ ≤

(d) Alternatively, let r be the real interest rate. That is, by forgoing 1 unit of the

commodity in period 1, a consumer earns 1 r+ units in period 2. In the first period the

firm borrows 1z and in the second earns 1/2112z . The maximized present value of the firm

is therefore 1/2

11

12( ) { }1

zr Max zr

Π = − ++

Note that this is exactly the same as in the WE with spot and futures markets if

1

2

1 prp

+ = . Thus the equilibrium “real” interest rate is zero.

Exercise 5.5-3: Rational expectations equilibrium

(a) Let 1 2( , )t t tx x x= be period t consumption. Then

11 11 11x zω= − , 12 12 12x zω= − , 21 11 122x z ω= + and 22 12 224x z ω= +



Substituting into the utility function

111 12 11 1222 ln(120 ) ln(120 ) ln(2 120) ln(4 120)U z z z z= − + − + + + + .

FOC

11 11 11

2 1 0120 2 120

Uz z z∂

= − + ≤∂ − +

, with equality if 11 0z > .

21 21 21

1 4 0120 4 120

Uz z z∂

= − + =∂ − +

, with equality if 21 0z > .

It is readily checked that 11

0Uz∂

<∂

for all 11 0z ≥ and that the second condition is satisfied

if 12 45z = . Thus the optimal consumption vector is * *1 2( , )x x =(120,120,75,300)

(b) First consider an economy with spot and futures prices. The FOC for utility

maximization is

1 12 2

* * * *11 21 12 22

2 1 1 1 1 1( , , , ) ( , , , )60 120 75 300

U px x x x x

λ∂= = =

∂.

If 11 1p = , then 160

λ = . Hence 1 4 12 5 8(1, , , )p = .

(c) If the interest rate is zero, the spot price vector is 11 2(1, )p = and the future spot price

vector is 4 12 2 2 5 8(1 ) ( , )fsp r p p= + = = . Note that the spot prices are no guide to future spot

prices. Thus the representative consumer has no easy way of predicting the future spot

prices in this economy.

Exercise 5.5-5: Walrasian equilibrium in a two-period model

(a) 11 21 12 22( ) ln 2ln ln 2 lnU x x x x xδ δ= + + +

If there is no storage, the gradient vector must be proportional to the WE price vector at

the endowment. Therefore

12

11 21 21 22

1 2 2( ) ( , , , ) ( ,1, , 2 )U px

δ δω δ δ λω ω ω ω

∂= = =

∂.

Therefore 12( ,1, , 2 )p δ δ= is a WE price vector.



(b) The spot price vector is 11 2( ,1)p = and the futures price vector is 2 ( , 2 )p δ δ= . The

future spot price vector is 12 2 2(1 ) (1 )( , 2 ) (1 )2 ( ,1)fsp r p r rδ δ δ= + = + = + . Thus the spot

prices and future spot prices are identical if (1 )2 1r δ+ = , that is 1 1/ 2r δ+ = .

(c) Storing 11z units of commodity 1 and 21z units of commodity 2 yields a profit of

12 11 11 22 21 21( ) ( )p p z p p z− + − .

The optimum is a WE. Thus if the optimum involves storage, profit must be maximized

at 0z >> . Then 12 11p p= and 22 21p p=

(d) If there is no storage the WE prices are 12( ,1, , 2 )p δ δ= . Storing 11z units of

commodity 1 and 21z units of commodity 2 yields a profit of

112 11 11 22 21 21 11 212( ) ( ) ( ) (2 1)p p z p p z z zδ δ− + − = − + − .

Thus it is profit-maximizing to do no storage if and only if 12δ ≤ .

SECTION 5.6: Equilibrium with Constant Returns to Scale Exercise 5.6-1: Market Adjustment

Let input 1 be labor so that commodity A is more labor intensive. Let r be the Walrasian

Equilibrium input price vector. If the supply of input 1 increases there is excess supply

resulting in a downward shift in 1r . This lowers the average and marginal cost of

production in both industries so both industries have an incentive to expand. The

additional labor input raises the marginal product of input 2 in both industries so there is

excess demand for input 2. The price of input 2 must therefore rise until it is profitable

for one of the industries to contract. Since industry 1 is more input 1 intensive, its unit

cost is lowered more thus it is industry 2 that contracts. The contraction of industry 2 and

expansion of industry 1 continues. This places upward pressure on the price of input 1

and downward pressure on the price of input 2. By Proposition 5.6.3, if both goods are

produced, then, in the long-run, both input prices must return to their original levels.

If the increase in supply of labor is large, industry B contracts completely so the

economy is specialized in the production of commodity A. The equilibrium cost line is



tangent to the isoquant at z . With constant returns to scale the slope is constant along a

ray thus, as labor increases, the wage-rental rate must fall.

Exercise 5.6-3: Corollary of the Rybczynski Theorem

From the derivation of the Rybczynski Theorem 1 1 1A A B Bq v q v z+ = . Hence

1 1 1A A B Bdq v dq v dz+ =

From the Rybczynski Theorem 0Bdq < . Therefore 1 1A Adq v dz> . It follows that

1 1 1 1

1 1 1

1A A A B B

A A A A A

z dq z q v q vq dz q v q v

+> = >

Exercise 5.6-5: Which Input Prices?

(a) Since ( ) (8,8) 1A A AF v F= = this is the unit isoquant. Note that 1A Ar v p⋅ = = so this is

the dollar isoquant. For commodity B, it is a standard result that for a symmetric Cobb-

Douglas production function, the cost shares are equal. Since the total cost of each input

are ½ when the input vector is (5, 20)Bv = this is indeed efficient. Since output is 1 and

Figure: Input price change with specialization

1z

2z

z



the price of commodity B is one, this is the dollar isoquant for commodity B. The dollar

isoquants for both commodities are depicted below along with the dollar cost line.

(b)

(b) The second equilibrium cost line is depicted below. Since the isoquants are

symmetric the equilibrium input price vector must be 1 140 10( , )r′ = .

(c) Given the unit input requirements of part (a), total demand for inputs is

1z

2z

8

20

5

Figure: Dollar isoquants and iso cost line

1z

2z

8 20

5

Figure: Second equilibrium iso-cost line



8 5 8 58 20 8 20

AA B

B

qq q

q⎡ ⎤⎡ ⎤ ⎛ ⎞ ⎛ ⎞

= +⎜ ⎟ ⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠ ⎝ ⎠⎣ ⎦

.

Thus the input demands are a positively weighted average of vectors with slopes of 1

and 4. With the unit input requirements of part (c) total demand for inputs is

8 20 8 208 5 8 5

AA B

B

qq q

q⎡ ⎤⎡ ⎤ ⎛ ⎞ ⎛ ⎞

= +⎜ ⎟ ⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎝ ⎠ ⎝ ⎠⎣ ⎦

Thus the input demands are a positively weighted average of vectors with slopes of 1 and

¼. Since the supply ratio is ½, only the second case is consistent with equilibrium.

(e) Now the supply ratio is 2 so only the first case is consistent with equilibrium. Thus 1 1

10 40( , )r = .

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