Problem Set 3: Due on May 28
Advanced Microeconomics I (Spring, 1st, 2013)
1. Question 1 (5 points)
A real-valued function f (x) is called homothetic if f (x) = g(h(x)) where g : R → R is a strictly increasing function and h is a real-valued function which is homo- geneous of degree 1. Suppose that preferences can be represented by a homothetic utility function. Then, prove the following statements.
(a) The marginal rate of substitution between any two goods depends only on the ratio of the demands consumed. That is M RSij is identical whenever xi
xj takes the same value.
(b) The cross price derivatives of Marshallian demands are identical, i.e.,
∂xi(p, I)
∂pj =
∂xj(p, I)
∂pi . 2. Question 2 (4 points)
Consider the following vNM utility function, u(x) = α + βx12.
(a) What restrictions must be placed on parameters α and β for this function to express risk aversion?
(b) Given the restrictions derived in (a), show that u(x) displays decreasing abso- lute risk aversion.
3. Question 3 (5 points)
Consider the following three lotteries, L1, L2 and L3:
L1 :
50 dollars with probability 12 150 dollars with probability 12 L2 : 100 dollars with probability 23 200 dollars with probability 13
L3 :
50 dollars with probability 13 150 dollars with probability 59 300 dollars with probability 19 Answer the following questions:
(a) Suppose that a decision maker prefers for sure return of 90 dollars rather than L1. Then, can we conclude that she is (i) risk averse or (ii) not risk loving? Explain.
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(b) Show that any risk averse decision maker whose preference satisfies indepen- dence axiom must prefer L2 to L3.
4. Question 4 (5 points)
Consider the following (social welfare) maximization problem where ui is a strictly increasing and continuous function for all i ∈ I and (λ1,· · · , λI) ∈ RI+\ {0}.
max
x I
X
i=1
λiui(xi)
s.t.
I
X
h=1
xkh ≤ ek for k = 1, ..., n.
(a) Show that any solution of the above maximization problem (you may denote x∗) must be Pareto efficient.
(b) Find an example of Pareto efficient allocation that cannot be the solution of the maximization problem whichever (λ1,· · · , λI) ∈ RI+\ {0} will be chosen. 5. Question 5 (6 points)
Consider an exchange economy with two goods, x and y. Suppose that individuals have the following symmetric utility functions and symmetric initial endowments:
u(x, y) = x2+ y2 (ωx, ωy) = (1, 1)
(a) Assume there are only two individuals in this economy. Then, draw the Edgworth-box and show the contract curve. Find a competitive equilibrium if it exists. If there is no equilibrium, explain the reason.
(b) Now suppose there are n(> 2) individuals. Then, can we find a competitive equilibrium? (How) Does your answer depend on n?
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