Consider a firm that produces output using a Cobb-Douglas combination of capital and labor (assuming constant returns to scale). Suppose that the price of the firm's product is fixed in the short run and that it also takes the quantity, Y, as given. Input markets are competitive; thus the firm takes the wage, W, and the rental price of capital, r, as given.
a) What is the firm's choice of L given Y and K?
b) Given this choice of L, what are profits as a function of P, Y, W, r, and K?
c) Find the first-order condition for the profit-maximizing choice of K.
d) Solve the first-order condition in part (c) for K as a function of P, Y, W, and r. How, if at all, do changes in each of these variables affect K? Explain intuitively.