Problem Set 2: Due on May 10
Advanced Microeconomics I (Spring, 1st, 2012)
1. Question 1 (2 points)
Suppose the production function f satisfies (i) f (0) = 0, (ii) increasing, (iii) con- tinuous, (iv) quasi-concave, and (v) constant returns to scale. Then, show that f must be a concave function of x.
2. Question 2 (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(x1+ x2) ≥ f (x1) + f (x2).
Show that every cost function is superadditive in prices. Use this property to prove that the cost function is nondecreasing in input prices.
3. Question 3 (4 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). 4. Question 4 (4 points)
A consumer’s utility function is given as
u(x, y) = min(αx, βy)
where α, β > 0. Let p, q > 0 be the prices for good x and y respectively. Then, answer the following questions.
(a) Derive the Marshallian demand function. (b) Find the indirect utility function.
(c) Find the (minimum) expenditure function. (d) Derive the Hicksian demand function.
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5. Question 5 (4 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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