A factory produces product A according to the production function QA = 100XA,where XA denotes the amount of input and QA is the quantity produced.It also produces product B according to the production function QB = 200XB + XB2,where XB denotes the amount of input and QB is the quantity produced.The total available amount of input X is 100 units.The firm's profit-maximizing allocation of input X is _____. A)X_A = 60 and X_B = 40 B)X_A = 50 and X_B = 50 C)X_A = 0 and X_B = 100 D)X_A = 100 and X_B = 0 E)X_A = 45 and X_B = 55

Question

Quizplus · Accepted Answer

To maximize profits, the firm should allocate input to each product until the marginal product per unit of input is the same for both products. For product A: Marginal product of XA = 100, so the firm should allocate 50 units of XA to product A to reach a total production of QA = 100(50) = 5000. For product B: Marginal product of XB = 200 + 2XB, so the firm should allocate XB = 25 units of X to product B to reach a total production of QB = 200(25) + 25^2 = 2125. Thus, the allocation of input X is X_A = 50 and X_B = 50.

A Factory Produces Product a According to the Production Function