Problem Set 3: General Equilibrium
1. Consider a two-consumer (a and b) and two-good economy. Let ua
(xa
1; xa
2) = (xa
1)1=3
(xa
2)2=3
,
!a
= (3; 1), ub
(xb
1; xb
2) = (xb
1)1=3
(xb
2)2=3
and !b
= (1; 1). Take good one as the numeraire.
(a) Find the competitive equilibria. Graph then in the Edgeworth box.
(b) Find the set of interior Pareto e¢ cient allocations. Graph it in the Edgeworth
box.
2. Robinson Crusoe lives on an island and derives utility u(x; y) =
p
xy from goods x and
y. Their respective production functions from the two factors of production, bamboo
shoots b and coconuts c, are x = [minfb; cg]2
and y =
p
bc. Robinson is endowed with
one unit of each of b and c.
(a) Find the optimal allocation of b and c to their productive uses.
(b) Can this outcome arise from a competitive allocation (i.e., if each good is produced
by a price-taking pro…t-maximizing …rm, and Robinson is a price-taking supplier
of both factors and also the owner of both …rms)? Explain.
3. Consider an economy with two consumers, two …rms, and two commodities: time
and consumption good. The consumers have to sleep 12 hours a day and hence each
consumer is endowed with 12 hours of waking time and has preferences over consumption
(x1) and leisure (x2) given by the utility function u1
(x1
1; x1
2) = (x1
1)
3=4
(x1
2)
1=4
for
consumer 1 and u2
(x2
1; x2
2) = (x2
1)
1=4
(x2
2)
3=4
for consumer 2. Each …rm produces the
consumption good (y1) out of labor ( y2) using the production possibility set characterized
by y1 2
p
y2 0. Each consumer owns one of the two …rms.
(a) Solve for the competitive equilibrium in this economy when the wage rate is normalized
to 1 (including the price of consumption, consumption bundles and production
plans). Do both consumers work in equilibrium?
(b) Find the set of interior Pareto-e¢ cient allocations in this economy. For simplicity,
ignore the fact that in reality a person can consume no more than 24 hours of
leisure per day.
4. Consider an exchange economy with I + 1 consumers and L goods. Let (!i
1; ::; !i
L) denote
consumer i’s initial endowment vector and ui
denote his utility function. Suppose
(x; p) is a competitive equilibrium of this economy, where x = (x1
; ::; xI
; !I+1
). That
is, consumer I +1 consumes his initial endowment !I+1
at the competitive equilibrium.
Now consider the situation when consumer I +1 does not show up in the market. That
is, consider the I-person exchange economy where the agents i = 1; ::; I have the same
initial endowment !i
and the utility function ui
as before. Is the allocation (x1
; ::; xI
)
a competitive equilibrium allocation of this economy?
5. Consider an economy composed of 2I + 1 consumers. Of these, I each own one right
shoe and I + 1 each own a left shoe. Shoes are indivisible. Everyone has the same
utility function given by minfL; Rg, where L and R are the quantities of left and right
shoes consumed, respectively. Free disposal is not allowed.
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(a) Show that any allocation of shoes that is matched (i.e., every individual consumes
the same number of each kind) with the exception of one consumer who is consuming
the remaining left shoe is Pareto e¢ cient and conversely.
(b) Let pL and pR be the respective prices of the two kinds of shoes. Find the competitive
equilibria of this economy.
6. Consider the following statements each of which may be true or false. If a statement
is true, prove it. If it is false, give a counterexample. Your examples may be based
on “pictures”, but make sure they are clear and precise. You can assume that a
competitive equilibrium exists and it is unique. You can also assume that agents have
locally non-satiated and continuous preferences.
(a) Consider two separate exchange economies. The competitive mechanism is used
in both economies. Suppose these two economies merge. Then there is no agent
who strictly prefers the initial competitive allocation to the competitive allocation
under the merged economy. That is, no one su¤ers from “globalization”.
(b) Consider two separate exchange economies. The competitive mechanism is used in
both economies. Suppose these two economies merge. Then there is at least one
agent who is at least as well o¤ under the competitive allocation of the merged
economy as he/she is in the initial “autarkic”competitive allocation. That is, not
everyone loses from “globalization”.
(c) Consider an exchange economy in which the competitive mechanism is used to determine
the allocation. Suppose an agent destroys a part of his initial endowment
before the market-clearing prices are determined. Then he necessarily su¤ers from
this act. That is, he necessarily ends up with a bundle that is worse than what
he would receive in the case he didn’t destroy any of his endowment.
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