Problem
Consider a firm that produces output using a Cobb-Douglas combination of capital and labor: Y = KαL1-α, 0 < α>< 1. suppose that the firm's price is fixed in the short run; thus it takes both the price of its product, p, and the quantity, y, as given. input markets are competitive; thus the firm takes the wage, w, and the rental price of capital,>K, as given.
(a) What is the firm's choice of L given P, Y, W, and K?
(b) Given this choice of L, what are profits as a function of P, Y, W, and K?
(c) Find the first-order condition for the profit-maximizing choice of K. Is the second-order condition satisfied?
(d) Solve the first-order condition in part (c) for K as a function of P, Y, W, and rK. How, if at all, do changes in each of these variables affect K?