### Problem Set 2: Due on May 14

Advanced Microeconomics I (Spring, 1st, 2013)

1. Question 1 (6 points)

(a) Suppose the utility function is continuous and strictly increasing. Then, show that the associated indirect utility function v(p, ω) is quasi-convex in (p, ω). (b) Show that the (minimum) expenditure function e(p, u) is concave in p.

(c) A real-valued function f (·) is called superaddittive if
f_{(x}^{1}_{+ x}^{2}_{) ≥ f (x}^{1}_{) + f (x}^{2}_{).}

Show that every cost function is superadditive in prices. Use this property to prove that the cost function is nondecreasing in input prices.

2. Question 2 (4 points)

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

(a) p^{0} = (1, 3), x^{0} = (4, 2); p^{1} = (3, 5), x^{1} = (3, 1).
(b) p^{0} = (1, 6), x^{0} = (10, 5); p^{1} = (3, 5), x^{1} = (15, 4).

(c) p^{0} = (1, 2), x^{0} = (3, 1); p^{1} = (2, 2), x^{1} = (1, 2).
(d) p^{0} = (2, 6), x^{0} = (20, 10); p^{1} = (3, 5), x^{1} = (25, 8).
3. Question 3 (3 points)

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

4. Question 4 (6 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 = (w^{1}, 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 w^{3} > ^{w}_{2}^{4}. Then, derive the cost function c(w, y).

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5. Question 5 (6 points)

Consumers have the following Cobb-Douglas utility functions:
U_{(x}1, x2_{) = α ln x}1 + (1 − α) ln x^{2}^{,}

where x^{t} is consumption in period t. Consumers vary in their preference parameter
α. We will refer to a consumer with parameter α as a type α consumer. There is a
single commodity. Types are distributed continuously over the interval [0, 1]. Type
α has density f (α) = 1 over this interval so that the total mass of types is 1. Each
type has the same income in each of two periods.

(a) If the interest rate is r, which types will be savers and which will be borrowers? (b) Calculate the amount of net saving for type α consumer, s(α).

(c) Solve for the total saving S by all types who save and the total borrowing B by all types who borrow. Depict S(r) and B(r) in a neat figure.

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