Suppose a firm uses both labor L and capital K as inputs to production. Its production function is of the Cobb-Douglas form, i.e.

F(K,L)=K1/3L2/3

The firm charges a price P for every good it sells, pays a nominal rental rate R to every unit of capital it hires and pays a nominal wage W to every unit of labor it hires. Assume the supply of capital is inelastic LS and the supply of labor is inelastic LS=8

a. Express the firm's real revenues, real costs, and real profits in terms of the variables defined above.

b. Assume the firm operates in goods markets, capital markets, and labor markets that are perfectly competitive - that is, it takes R/P and W/P as given. Derive two conditions that will need to hold in order for the firm to maximize its profits ( Set the partial derivative of profits with respect to L equal to zero and the partial derivative of profits with respect to K equal to zero). The first of these conditions should include the real wage rate W/P as well as K and L . This is the labor demand equation. The second condition should involve the real rental rate of capital R/P as well as K and L . This is the capital demand equation.

c. Focus first on the capital market: plot the capital demand function when L=8 (You need at least three points to plot this curve). On the same graph plot the capital supply function. Compute the equilibrium rental price of capital and show it on the graph.

d. Focus now on the labor market: plot the labor demand function when K=27 (You need at least three points to plot this curve). On the same graph plot the labor supply function. Compute the equilibrium real wage and show it on the graph.

e. In equilibrium, how many goods will that firm's supply to the market for goods?

f. Find real labor income PWL , real capital income PRK and use them to compute the firm's real profits in equilibrium.