ADVANCED MICROECONOMICS I HOMEWORK 3

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ADVANCED MICROECONOMICS I

HOMEWORK 3

Exercise 1. Please show that c(w,y) = yc(w, 1) holds if a production function is homogeneous of degree one.

Exercise 2. If f is continuous, strictly increasing, and strictly quasiconcave, please show that the profit function is

1. homogeneous of degree one in (p, w) .

2. increasing in p, and decreasing in w.

3. convex in (p, w) .

4. Hotelling’s lemma:?(?)p(π)  = y(p, w); 一 ?(?)w(π)i   = xi(p, w).

Exercise 3 (Output Price Uncertainty). Assume that a firm is risk neutral with respect to profits and that if there is any uncertainty in prices, production decisions are made after the resolution of such uncertainty. Suppose that the firm faces a choice between two alternatives. In the first, the output price is a random variable, but the firm knows its probability distribution. In the second, the output price is nonrandom and equal to the expected price in the first alternative. Show that a firm that maximizes expected profits will prefer the first alternative (the uncertain prices) over the second.

Exercise 4.  Suppose the inverse demand function is P(Q) = 1 一 Q, Q = s1 qi, firms have different unit costs and c1 < c2 <．．．< cn, what is the Cournot equilibrium?

Exercise 5.  Suppose two symmetric firms with constant unit cost c produce differentiated goods. The inverse demand function is

pi(qi , qj) = a 一 qi 一 sqj , i, j = 1, 2

Please solve for the Cournot equilibrium and Bertrand equilibrium.

Exercise 6. The inverse demand function of an industry is P(Q) = 1 一 Y, Y = y1 +y2 , the firms’s cost functions are c1 (y1 ) = y1(2) and c2 (y2 ) = 2y2(2), respectively. Suppose firms collude, how much should each firm produce?