A firm faces the production function Q = 10L2K2. The wage rate is 2 and the rental rate of capital is 1. What are the optimal amounts of L and K if the firm’s objective is to produce Q = 25, 000?
Production function is Q = 10L^2K^2. If the wage rate PL = 2 and the rental rate of capital Pk = 1, then to find the optimal amounts of L and K, we need to find marginal products of labor and capital MPL and MPk, if the firm’s objective is to produce Q = 25,000, using optimal amounts. MPL = Q'(L) = 20K^2*L MPk = Q'(K) = 20K*L^2 In optimal situation MPL/PL = MPk/Pk, so: 20K^2*L/2 = 20K*L^2/1 10K = 20L K = 2L If K = 2L, then 10L^2*(2L)^2 = 25,000, 40L^4 = 25,000, L^4 = 625, L = 5 workers, so K = 2*5 = 10 units.