Answer to Question #157271 in Economics for Sahlah Ahmed

Question

A competitive frim has the following production function, q = −6K2 −3L2 −4KL +60K +

34L. Where p is the market price, w is the wage, and r is the rental rate of capital, then:

a. Write down the firm’s profit function. (2)

b. Derive the hotelling’s lemma (3)

c. Solve for the optimal K and L by assuming P=10, w=2 and r=3 (3)

d. Use the Hessian to check the second order conditions

Answer 1

$$\pi=pq-wL-rK$$

$$\pi=p(-6K^2-3L^2-4KL+60K+34L)-wL-rK$$

If p=10, w=2, r=3, then we will have profit

$$\pi=-60K^2-30L^2-40KL+597K+338L$$

$$\frac {\delta \pi}{\delta K}=-120K-40L+597$$

$$\frac {\delta \pi}{\delta L}=-60L-40K+338$$

$$-120K-40L+597=0$$

$$-40K-60L+338=0$$

$$L=2.98$$

$$K=3.98$$