Solved: 5355. Production functions Economists model the

Chapter 12, Problem 54

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5355. 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 1K, L2 = CKa L1 - a, where K represents capital, L represents labor, and C and a are positive real numbers with 0 6 a 6 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. P = f 1K, L2 = 10K1>3 L2>3 for 30K + 60L = 360