Question

3/4 1/4 The production function for a firm is p(x, y) = 108x "y", where x and y are the number of units of labor and capital utilized. Suppose that labor costs $243 per unit and capital costs $625 per unit and that the firm decides to produce 40,500 units of goods. Determine the following.

(A) Determine the amounts of labor and capital that should be utilized in order to minimize the cost. That is, find the values of x and y that minimize 243x + 625y, subject to the constraint 40,500 - 108x3/41/4 = 0 x= y= (Type exact answers in simplified form.)
(B) Find the value of at the optimal level of production A = (Type an exact answer in simplified form.)
(C) Find the expressions for marginal productivity of labor and marginal productivity of capital. Then find the ratio of marginal productivity of labor to marginal productivity of capital at the optimum level of production and the ratio of unit price of labor to unit price of capital and determine the relationship between the ratios. marginal productivity of labor = marginal productivity of capital = Determine the marginal productivity of labor and marginal productivity of capital at the optimum level of production and then compare the ratios. marginal productivity of labor marginal productivity of capital unit price of labor unit price of capital (Simplify your answers.)

Answer 1

The value of the marginal product of labor is 108y, and the profit-maximizing level of employment for a firm in a perfectly competitive labor market is approximately 0.000457.

Step-by-step explanation:

a. To determine the value of the marginal product at each level of labor, we need to calculate the partial derivative of the production function with respect to labor (x). Differentiating the production function p(x, y) = 108xy with respect to x, we get:

∂p/∂x = 108y

Thus, the value of the marginal product of labor is 108y.

b. In a perfectly competitive labor market, the profit-maximizing level of employment occurs when the market wage equals the marginal revenue product of labor (MRP). The MRP can be calculated using the derivative of the production function. In this case, since the production function is p(x, y) = 108xy, the MRP of labor is given by:

MRP = ∂p/∂x * Px = 108y * 243 = 26244y

Since the market wage is $12, we can equate the MRP to the wage to find the profit-maximizing level of employment:

26244y = 12

y = 12/26244 ≈ 0.000457

Therefore, the firm's profit-maximizing level of employment is approximately 0.000457.