Question

A Manufacturer has a weekly production function of Q=200KL - K² -20L² It is currently in the short run, with capital fixed at K = 10. It sells its product in a perfectly competitive market for $25. It can hire all the labor it needs for $3, 000 per week. a. find the marginal product of labor b. find the marginal revenue product of labor c. Find the optimum amount of labor d. The situation is now in the Long run; find the marginal product of capital e. If the rental rate of capital is $250, 000 per week (r=$250, 000), is the combination of K=10 and the amount of L found in C above optimal? If not in which direction should the firm go (more K or more L)?

Answer 1

(a)

Marginal product of labor (MPL) = Q / L = 200K - 40L = (200 x 10) - 40L = 2,000 - 40L

(b)

Marginal revenue product of labor (MRPL) = MPL x Output price = 25 x (2,000 - 40L) = 50,000 - 1,000L

(c)

Labor demand is optimal when MRPL = Wage rate

50,000 - 1,000L = 3,000

1,000L = 47,000

L = 47

(d)

Marginal product of capital (MPK) = Q / K = 200L - 2K

(e)

Input combination is optimal when (MPL / w) = (MPK / r).

When L = 47 and K = 10,

MPL = 2,000 - (40 x 47) = 2,000 - 1,880 = 20

MPK = (200 x 47) - (2 x 10) = 9,400 - 20 = 9,380

MPL / w = 20 / 3,000 = 1/1,500 = 0.0007

MPK / r = 9,380 / 250,000 = 0.0375

Since (MPL / w) < (MPK / r), input combination is not optimal. Firm should hire more labor and less capital.