Question

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?

Answer 1

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.