Exercise 27

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 \(P = bL^\alpha K^{1-\alpha}\) follows from certain economic assumptions, where b and \(\alpha\) are positive constants and \(\alpha < 1\). 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 \(L = \frac{\alpha p}{m}\) and \(K = \frac{(1-\alpha)p}{n}\)