Problems for the fourth seminar
ECON4230/35 Microeconomic Theory — Fall semester 2010
Solutions to the problems will be presented in week 43.
Please direct any questions to Geir B. Asheim (Room ES1152, Tel: 228 55498, email: g.b.asheim@econ.uio.no) [for Problems 1 and 2 to Kjell Arne Brekke (Room
ES1032, Tel: 228 41169, e-mail: k.a.brekke@econ.uio.no)]
Problem 1 (General equilibrium analysis) Varian Problems 17.4, 17.8.
Problem 2 (General equilibrium analysis)
(a) State the first and second welfare theorems.
Consider an exchange economy with two consumers and two goods.
(b) Determine the Marshallian demand functions of consumer A when this
consumer has the utility function
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2
uA (x1 , x2 ) = (x1 ) 5 (x2 ) 5 .
(c) Determine the Marshallian demand functions of consumer B when this
consumer has the utility function
uB (x1 , x2 ) = max{x1 , x2 } .
(d) Assume that the consumers’ initial endowments of good 1 sum to 12
and that the consumers’ initial endowments of good 2 also sum to 12.
Consider the feasible allocation where consumer A consumes 12 units of
good 1 and 2 units of good 2, and consumer B consumes 0 units of good
1 and 10 units of good 2. Is this allocation Pareto-efficient? Can this
allocation be implemented as a competitive equilibrium?
Problem 3 (General equilibrium analysis) Suppose we have two consumers A and B with identical utility functions
0.2
uA (x1 , x2 ) = uB (x1 , x2 ) = x0.6
1 x2 .
(a) Calculate the Marshallian demand function for each consumer.
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(b) Assume that the consumers have the following initial endowment:
ω A = (12, 0) and ω B = (0, 12) .
Calculate the market clearing prices and the equilibrium allocation.
(c) Find the set of Pareto efficient allocations given that the total amount
of good 1 is 12, and the total amount of good 2 is 12.
(d) Suppose that the social welfare function W is given by: W (uA , uB ) =
uA + uB . Find the welfare maximizing allocation given that the total
amount of good 1 is 12, and the total amount of good 2 is 12.
(e) Show that the welfare maximizing allocation is a competitive equilibrium
if the initial endowment is redistributed so that ω A = (8, 0) and ω B =
(4, 12).
Problem 4 (General equilibrium analysis) Suppose that the economy has
available a large fixed endowment of a divisible good and a single unit of an
indivisible good. Let there be two consumers.
Which of the following cases is/are Pareto efficient?
(i) Consumer 1 receives the indivisible good. The compensation he would
require to give up the good is larger than consumer 2’s willingness to pay
for the good.
(ii) Consumer 1 receives the indivisible good. The compensation he would
require to give up the good is equal to consumer 2’s willingness to pay
for the good.
(iii) Consumer 1 receives the indivisible good. The compensation he would
require to give up the good is less than consumer 2’s willingness to pay
for the good.
Problem 5 (General equilibrium analysis) Denote by x the economic
allocation of a society. Let W (x) be the welfare function of the society under consideration. Consider two feasible allocations x∗ and x0 . Consider the
following statement:
x∗ is Pareto efficient, x0 is not Pareto efficient, and W (x0 ) > W (x∗ ).
Can this statement be true? State briefly why it can be or cannot be true.
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