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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 functionf(x) is calledhomotheticif f(x) =g(h(x)) whereg :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 isM 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 byf(x1, x2, x3, x4) = min{x 1 3 1x

2 3

2, x3+ 2x4}. Letw= (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 > w 4

2 . Then, derive the cost function c(w, y).

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