Chapter 14.8, Problem 27E

The total production P of a certain product depends on the amount L of labor used and the amount K of capital investment, in Sections 14.1 and 14.3 we discussed how the Cobb-Douglas model P = bL^αK^1-α follows from certain economic assumptions, where b and α are positive constants and α < 1. If the cost of a unit of labor is m and the cost of a unit of capital is n, and the company can spend only p dollars as its total budget, then maximizing the production P is subject to the constraint mL + nK = p. Show that the maximum production occurs when

$L=\frac{\alpha P}{m}\text{and}K=\frac{(1-\alpha )p}{n}$