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### Problem Set 2: Posted on November 4

Advanced Microeconomics I (Fall, 1st, 2014)

1. Question 1 (7 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 (8 points)

The consumer buys the bundles xi at prices pi for i = 0, 1. Separately for parts (a) to (d), state whether these indicated choices satisfy the weak axiom of revealed preference:

(a) p0 = (1, 3), x0 = (4, 2); p1 = (3, 5), x1 = (3, 1). (b) p0 = (1, 6), x0 = (10, 5); p1 = (3, 5), x1 = (15, 4).

(c) p0 = (1, 2), x0 = (3, 1); p1 = (2, 2), x1 = (1, 2). (d) p0 = (2, 6), x0 = (20, 10); p1 = (3, 5), x1 = (25, 8). 3. Question 3 (6 points)

Prove that if a firm exhibits increasing returns to scale then average cost must strictly decrease with output.

4. Question 4 (9 points)

A firm has a production function given by f (x1, x2, x3, x4) = min{x

1 3

1x

2 3

2, x3+ 2x4}.

Let w = (w1, w2, w3, w4) ≫ 0 be factor prices and y be an (target) output.

(a) Does the production function exhibit increasing, constant or decreasing returns to scale? Explain.

(b) Calculate the conditional input demand function for factors 1 and 2. (c) Suppose w3 > w24. Then, derive the cost function c(w, y).

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