Suppose that a firm uses only capital, K, and labor, L, in its production process. At the firm's current long-run combination of capital and labor, it uses positive amounts of both inputs and measures the marginal products as $M P _ { X } = 15$ and $M P _ { L } = 10$ . The rental rate of capital is r = 6 and the current wage rate for labor is w = 3. The firm
A) is currently minimizing total cost in the long run.
B) could lower cost by increasing the usage of capital and decreasing the usage of labor.
C) could lower cost by increasing the usage of labor and decreasing the usage of capital.
D) cannot lower cost without also lowering the level of output.

Question

Quizplus · Accepted Answer

The firm can calculate the cost-minimizing combination of capital and labor using the formula for the marginal rate of technical substitution (MRTS) which is the ratio of the marginal products of labor and capital. MRTS = MPL/MPK. 
In this case,
MRTS = 10/15 = 2/3
The firm should hire labor until the wage rate divided by the marginal product of labor is equal to the MRTS (i.e., w/MPL = MRTS),
3/10 = 2/3
Multiplying both sides by MPL, we get
MPL = (2/3) x 10 = 6.67
This means the firm is currently using too little labor and should increase its usage. Similarly, the firm should use less capital since the ratio of the rental rate of capital to the marginal product of capital is higher than the MRTS. (i.e., r/MPK > MRTS),
6/15 > 2/3
Multiplying both sides by MPK, we get
MPK < (2/3) x 6 = 4
This means the firm is currently using too much capital and should decrease its usage. Therefore, option C is the best choice.

Suppose That a Firm Uses Only Capital, K, and Labor MPX=15M P _ { X } = 15MPX​=15

Suppose That a Firm Uses Only Capital, K, and Labor $M P _ { X } = 15$