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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