Answer to Question #157254 in Economics for siya

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

$$\pi=TR-TC=pq-(wL+rK)$$

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

The change in the profit function at the price of the i product is equal to the supply function of this product.

This statement is Hotteling's lemma. Often used for a group of products, rather than for a single product.

$$\pi=10(-6K^2-3L^2-4KL+60K-34L)-2L-3K$$

$$\pi=-60K^2-30L^2-40KL+597K-342L$$

$$\frac {\delta \pi}{\delta K}=-120K-40L+597=0$$

$$\frac {\delta \pi}{\delta L}=-60L-40K-342=0$$

$$L=3.06$$

$$K=8.84$$