Chapter 12.9, Problem 55E

Production functions Economists model the output of manufacturing systems using production functions that have many of the same properties as utility functions. The family of Cobb-Douglas production functions has the form P = f(K, L) = CK^a L^1–a, where K represents capital, L represents labor, and C and a are positive real numbers with 0 < a < 1. If the cost of capital is p dollars per unit, the cost of labor is q dollars per unit, and the total available budget is B, then the constraint takes the form pK + qL = B. Find the values of K and L that maximize the following production functions subject to the given constraint, assuming K ≥ 0 and L ≥ 0.

55.    Given the production function P = f(K, L) = K^aL^1–a and the budget constraint pK + qL = B, where a, p, q and B are given, show that P is maximized when K = aB/p and L = (1 – a)B/q.