Microeconomics, Block I Final Exam

(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)