© John Riley Summer 2012

CHAPTER 5 ADDITIONAL EXERCISES

1. Pareto Efficiency with transferable utility1

Consider a two consumer exchange economy with aggregate endowment ω . Consumer A has utility function 1 2 3( ) ( , ), { , }h h h h hU x ax u x x h A B= + ∈ where ( )hu ⋅ is homogeneous of

degree 1 and for all x , 2 3 2 3( , ) ( , )A BMRS x x MRS x x> .

(a) Let 2 3( , )S S ω ω= be the utility possibility set for ( , )A Bu u given the endowments 2 3( , )ω ω . Show that a function that is homogeneous of degree 1 is concave. Hence show that S is a convex set. Explain also why ( )h hMRS x increase along th boundary of this set as Au increases.

(b) Let 0 0 0( , )A Bu u u= and 1 1 1( , )A Bu u u= be two strictly positive boundary points of S . Appeal to

Proposition5.6-2 to show that every convex combination 0 11 )u u uλ λ λ= ( − + lies in the interior of S.

(c) Show that if 1 1( , )A Bω ω is sufficiently large, then a necessary and sufficient condition for

,A Bx x to be Pareto efficient is that

2 3 2 3( , ) ( ( , ), ( , ) arg { | }A B

A B A A A B B B A B

x xu u u x x u x x Max u u u S≡ ∈ + ∈

(d) Explain why this maximization problem has a unique solution.

(e) Draw a neat figure with Au on the horizontal and Bu on the vertical axis. Depict the set S , the point ( , )A Bu u and the set of PE allocations.

(f) Let ( , )hv p y be the indirect utility function for hu under the normalization 3 1p = . Explain

why it can be written as follows 2 2( , ) ( )h hv p y p yφ= .

(g) Henceforth consider the special case 11 2( )

h hh b bu x x x −= where A Bb b> . If 1 1( , )A Bω ω is sufficiently large, solve for the WE price of commodity 2 and hence commodity 1.

(h) Can you also characterize the WE allocation?

1 This is a somewhat longer and more challenging question.

© John Riley Summer 2012

2. Walrasian Equilibrium with three commodities2

(a) A consumer has utility function 1321( , , ) ( ) ( )

1xxU x a ax β ββ

β β−= +

− where 0a > and

(0,1).β ∈ The price of commodity 3 is normalized at 1. If the consumer must spend y on commodities 2x and 3x and chooses optimally, show that her resulting utility is

1 1 2( , ) /u x y ax y p β= + .

(b) Consider a two consumer exchange economy with aggregate endowment 2(1, ,1)ω and price

vector 1 2( , ,1)p p . Consumer A has utility function ( , , )AU x a b and consumer B has utility

function ( , ,1 )BU x a b− where 1/ 2b > . Show that if the equilibrium price of commodity 2 is not equal to 1 then the equilibrium consumption vectors cannot both be strictly positive.

HINT: Consider the indirect utility function 1( , ), { , }h h hu x y h A B∈ .

(c) If the equilibrium price of commodity 2 is 1 show that equilibrium spending on commodity 2 is 1 for each consumer. Show also that the equilibrium price of commodity 1 is 1p a=

(d) Given the price vector ( ,1,1)p a= explain why the endowment 1 2 3( , )A A Aω ω ω+ must lie in the shaded region for there to be an equilibrium in which both consumption vectors are strictly positive.

2 This is a somewhat longer and more challenging question.

BO

AO 1x

y

a

a

1

© John Riley Summer 2012

Henceforth consider the model with 2 1ω = .

(e) Show that unless 2 3 2 3 1A A B Bω ω ω ω+ = + = there cannot be a WE with 2 1p = if a is sufficiently small.

(f) Show that for any endowment for consumer A above the shaded region the WE price 2 1p >

and Consumer B’s WE consumption of commodity 1 is zero.

(g) Characterize the equilibrium if consumer A’s endowment lies below the shaded region?

(h) If 0a = and 0.5b > then preferences are homothetic and consumer A has a stronger preference for commodity 2. Suppose that 2 3 3 0.5A A Bω ω ω> = = . As A’s endowment of

commodity 2 rises the WE price of commodity 2p rises. Thus the direct effect of the higher endowment is offset by the deteriorating terms of trade. Assuming that the endowments lie in the shaded region, contrast this with the outcome when 0a >