A factory produces product A according to the production function Q_A = 100X_A, where X_A denotes the amount of input and Q_A is the quantity produced. It also produces product B according to the production function Q_B = 200X_B - X_B², where X_B denotes the amount of input and Q_B 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 = 30 and X_B = 70.

Question

Quizplus · Accepted Answer

To find the profit-maximizing allocation of input X, we need to calculate the marginal products (MPs) of each product and compare them to their respective prices. The MP is the additional output produced by using an additional unit of input. For product A: MPA = dQA/dXA = 100 PriceA = $10 For product B: MPB = dQB/dXB = 200 - 2XB PriceB = $5 Setting the MPs equal to their respective prices, we get: 100 = 10X1 -> X1 = 10 200 - 2XB = 5X2 -> XB = 75 Therefore, X_A = X1 + X2 = 10 + 75 = 85 And X_B = X - X_A = 100 - 85 = 15 However, we need to make sure that the MPs are decreasing for both products at these input levels. If they're not, we need to adjust our allocation. For product A: dMPA/dXA = 0 Therefore, MPA is constant and we don't need to adjust our allocation. For product B: dMPB/dXB = -2 < 0 Therefore, MPB is decreasing with more input of B. We don't need to adjust our allocation. Therefore, the profit-maximizing allocation is X_A = 85 and X_B = 15.

A Factory Produces Product a According to the Production Function