The production function of a competitive firm is described by the equation y = 8x^1/2 1x^1/2 2.The factor prices are p1 = $1 and p2 =$4 and the firm can hire as much of either factor it wants at these prices.The firm's marginal cost is A)constant and equal to .50. B)constant and equal to 3. C)increasing. D)decreasing. E)None of the above.

Question

Quizplus · Accepted Answer

Marginal cost is equal to the change in total cost divided by the change in output. In this case, the total cost is given by the equation TC = p1x1 + p2x2, where p1 and p2 are the prices of factors 1 and 2, respectively, and x1 and x2 are the quantities of factors 1 and 2 used. The change in total cost is equal to dTC = p1dx1 + p2dx2, and the change in output is equal to dy = 8x1^(1/2)x2^(1/2)dx1 + 8x1^(1/2)x2^(1/2)dx2. Substituting these expressions into the equation for marginal cost, we get MC = dTC/dy = (p1dx1 + p2dx2)/(8x1^(1/2)x2^(1/2)dx1 + 8x1^(1/2)x2^(1/2)dx2) = (p1/8x1^(1/2)x2^(1/2)) + (p2/8x1^(1/2)x2^(1/2)). Since p1 = $1 and p2 = $4, we have MC = (1/8x1^(1/2)x2^(1/2)) + (4/8x1^(1/2)x2^(1/2)) = 5/8x1^(1/2)x2^(1/2). This expression is constant, so the marginal cost is constant and equal to 5/8x1^(1/2)x2^(1/2).

The Production Function of a Competitive Firm Is Described by the Equation