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Problem Set 3: Due on May 24

Advanced Microeconomics I (Spring, 1st, 2012)

1. Question 1 (4 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

x_j ^takes the same value.

(b) The cross price derivatives of Marshallian demands are identical, i.e.,

∂x_{i(p, I)}

∂p_j ⁼

∂x_{j(p, I)}

∂p_i ^. 2. Question 2 (6 points)

Assume that a consumer has preferences over the single good x and all other goods m represented by the utility function, u(x, m) = ln(x) + m. Let the price of x be p, the price of m be unity (= 1), and let income be ω.

(a) Derive the Marshallian demands for x and m.

(b) Use the Slutsky equation to decompose the effect of an own-price change on the demand for x into an income and substitution effect.

(c) Suppose that the price of x rises from p⁰ to p¹(> p⁰). Calculate the (change of) consumer surplus, compensating variation, and equivalent variation. 3. Question 3 (6 points)

Consider the following (social welfare) maximization problem where ui is a strictly increasing and continuous function for all i ∈ I and (λ1,· · · , λI) ∈ R^I+\ {0}.

max

x

∑I i=1

λ_iu_i(xi)

s.t.

∑I h=1

x^k_{h ≤ e}^k for k = 1, ..., n.

(a) Under the above assumptions on ui for all i ∈ I, prove the equivalence between strong Pareto efficiency and weak Pareto efficiency.

(b) Show that any solution of the above maximization problem (you may denote x^∗) must be Pareto efficient.

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(c) Find an example of Pareto efficient allocation that cannot be the solution of the maximization problem whichever (λ1,_{· · · , λI) ∈ R}^I₊\ {0} will be chosen. 4. Question 4 (4 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^{, ω}y) = (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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