Suppose a firm’s production function is: f(E,K) = 40K0.25E0.75 The corresponding

Question

Suppose a firm’s production function is: f(E,K) = 40K0.25E0.75 The corresponding marginal product functions are: MPK = 10K-0.75E0.75 and MPE = 30K0.25E-0.25 Suppose initially that the hourly wage is $9 and the price of each unit of capital is $15.

a. If the firm is maximizing profits, what is the firm’s optimal capital-labor ratio?

b. If the firm has hired 25 machines and 50 workers, is it maximizing profits? Why or why not?

c. If the price of labor increases to $15 and the firm increases its capital stock to 12 units from 10 units, how many workers will the firm choose to hire? Does the scale or substitution effect dominate in this case?

Explanation / Answer

Solution :

The cost-minimizing combination of capital and labor is the one where

MRTS = MPK / MPE

MRTS =   30K0.25E-0.25 / 10K-0.75 E0.75

30K0.25E-0.25 / 10K-0.75 E0.75 = 15/9

30K1 E-1 / 10 = 15/9

30K / 10E = 15/9

K/E = 150 / 270

The optimal capital to labor ratio is K/E ratio = 0.55

b) K/L = 25 X $ 15 / 50 X $ 9

K/L = 375/ 450

K/L = 0.833

The capital to labor ratio is greater than the optimal capital - labor ratio , therefore the firm is not maximizing profit .

c) The optimal capital labor ratio is 0.55

0.55 = 12 X $ 15 / 15 X L

8.25 L = 180

L = 21.81 ( Rounded to 22)

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