Microeconomics, Block I Final Exam

Microeconomics, Block I
Final Exam
Piero Gottardi October 28, 2009
[Instructions. You have two hours. You are allowed to bring to the exam a pocket
calculator. (Please turn off your mobile phone.) Exchanging information or communicating
with other people, as well as any other form of cheating, implies the immediate disqualification
of your exam. The weight attributed to each question for the final grade is indicated
in brackets. Full marks correspond to 60 points. Good luck!]
A) Consumer and Producer Theory
1. (14) Consider a consumer with preferences described by U(x1, x2) = x1 + (x2) 1/2 and
income m.
(a) (8) Find the consumer’s demand function x(p, m) as well as his compensated
demand function x(p, u) [Do not worry about the boundary of the consumption
set: do as if negative consumption for good 1 is allowed]
(b) (6) Consider ¯p = (4, 1), ¯m = 8 and ū = 3. Verify that x1(¯p, ¯m) = x1(¯p, ū). Compute
then the derivatives of these two functions with respect to p1, at this point:
∂x1(¯p, ¯m)/∂p1 and ∂x1(¯p, ū)/∂p1. Verify that ∂x1(¯p, ¯m)/∂p1−∂x1(¯p, ū)/∂p1 has a
negative sign: explain why it is so in this case and what is the economic meaning
of this difference.
2. (7) Show that if a consumer has homothetic preferences his demand is linear in income:
x(p, αm) = αx(p, m) for all m, p ≫ 0, m > 0.
B) General Equilibrium under Certainty
1. (25) Consider a pure exchange economy with two (types of) agents, A and B, and
two commodities, 1 and 2. Agent A has preferences U A (x1, x2) = 3 log x1 + log x2 and
an initial endowment ω A = (0, 8), while B’s preferences are U B (x1, x2) = min {x1, x2}
and her initial endowment is ω B = (8, 0).
(a) (3) Construct an Edgeworth box diagram showing the feasible allocations of this
economy. Represent in the diagram the initial endowment allocation and the
preferences (shape of indifference curves and direction of preference) for each
consumer’s type.
1

(b) (3) Consider the prices p1 = 1 = p2. Are these equilibrium prices? Explain why
they are, or aren’t.
(c) (8) Find the competitive equilibria (prices and allocation) of this economy (if
they exist) and show them clearly on the diagram. Are they Pareto optimal?
(d) (6) Consider the allocation where A gets (x A 1 , x A 2 ) = (4, 4) (and B gets the rest).
Is this allocation Pareto optimal? If so, find the amount t ∈ R of the transfer of
commodity 1 from B to A that yields this allocation as a competitive equilibrium.
Compute the associated equilibrium prices.
(e) (5) Suppose the equilibrium found in c. constitutes the autarky equilibrium of
country I, populated by the agents of type A and B described above. There is
then country II, populated by identical agents of another type, say C. Explain
why, at the free trade equilibrium between countries I and II, either type A or
type B gains with respect to the autarky equilibrium; it cannot be that they both
loose nor that they both gain.
2. (11) Consider an economy, populated by H consumers, each of them with utility
function U h (xh 1, xh 2) = xh 1 + xh 1/2 h
2 (as in Exercise A.1) and income m .
(a) (6) Show that a representative consumer exist globally. That is, show that
there exists a utility function U(x) and income level m such that the demand
function x(p, m), obtained by maximizing U(x) subject to the budget constraint
p · x ≤ m, equals the aggregate demand function of the economy, H h=1 xh (p, mh )
for all p ≫ 0 and for all (mh ) H h=1 . [Again do not worry about the boundary of the
consumption set: do as if negative consumption for good 1 is allowed]
(b) (5) Suppose each consumer is endowed with an amount (ω h 1, ω h 2) of the two
goods, thus his income is the market value of his endowment m h = p1ω h 1 + p2ω h 2.
Explain why the economy described has a unique competitive equilibrium and
how then the equilibrium prices can be found.
C) General Equilibrium Under Uncertainty
1. (9) Consider a pure exchange economy under uncertainty with two types of consumer
(H = 2), a single commodity (L = 1) and S = 2 states of nature. The two consumers
have the same preferences E log(x) = π(1) log x(1)+π(2) log x(2), with identical beliefs
π(1) = 1/2. Type 1 consumer has then endowments equal to 10 units in state 1 and 4
units in state 2 while type 2 has endowments equal to 2 in state 1 and to 4 in state 2.
(a) (5) Find the Pareto effi cient allocations of this economy
(b) [IGNORE THIS LAST QUESTION] (4) Discuss the relationship between the
value of the prices (p(1), p(2)) which support any Pareto effi cient allocation as
a competitive equilibrium with complete contingent markets and the probability
is always greater, equal, or smaller
beliefs (π(1), π(2)). Can you tell whether p(1)
p(2)
than π(1)
? Explain.
π(2)