Exercise 8

Exercise Session 8, November 17th , 2006 Mathematics for Economics and Finance Prof: Norman Schürho¤ TAs: Zhihua Chen (Cissy), Natalia Guseva 1. Consider a consumer who seeks to maximize her utility subject to budget constraint. There are two goods x1 and x2 with prices p1 and p2 : Income of consumer is I and utility function is u(x1 ; x2 ) = x1 + ln x2 : Note that consumer can not consume negative amount of goods. 2. Suppose a …rm has its sales policies determined by a manager whose objective function is to maximize revenue, without letting pro…t drop below some …xed level. There is an advertising cost a > 0 that a¤ects positively the revenue. Let R(y; a) = ay 2 be the …rm’s revenue when production p level is y, C(y) = y is the cost of manufacturing y units of output: Solve the maximization problem of the …rm. 3. Consider a pro…t-maximizing …rm with production function f (l; k) where l; k are the amount of labor and capital employed by the …rm (in units of output). The …rm’s pro…t equals (l; k) = pf (l; k)

wl

rk;

where p is the price of output, w is the real wage, and r is the real rental price of capital. The …rm takes p, w, and r as given. Assume that the Hessian matrix of f is negative de…nite. a) Show that if the wage increases by a small amount, the …rm employs less labor. b) Show that if the wage increases by a small amount and the …rm is constrained to maintain the same amount of capital k 0 , the …rm will reduce labor l by less than in a). 4. Exercise 3, Session 7.

1