Let P(K, L) be a production function, where K and L represent the capital and labor required for a certain manufacturing procedure. Suppose we wish to maximize P(K, L) subject to a cost constraint, C(K, L) = A, for constant A. Use the method of Lagrange multipliers to show that optimal production is attained when

that is, when the ratio of marginal production from capital to the marginal cost of capital equals the ratio of marginal production of labor to the marginal cost of labor.

aP K ac ak L ac L