Question

A manager hires labor and rents capital equipment in a very competitive market. Currently the wage rate is \(12 per hour and capital is rented at \)8 per hour. If the marginal product of labor is 60 units of output per hour and the marginal product of capital is 45 units of output per hour, is the firm using the cost minimizing combination of labor and capital? If not, should the firm increase or decrease the amount of capital used in its production process?

Short answer

The firm should decrease its capital by 5 units.

Step-by-step solution

Explaining the cost minimization combination

The cost minimization combination requires the ratio of marginal productivity of labor and capital to be equal to the wage and rent ratio.

$\begin{array}{rcl}\left({MRTS}_{KL}\right)& =& \frac{w}{r}\\ \frac{M{P}_L}{M{P}_K}& =& \frac{w}{r}\\ \frac{M{P}_L}{w}& =& \frac{M{P}_K}{r}\\ & & \end{array}$

Substituting the values of the variables:

$$\begin{gathered} \frac{M{P}_L}{w}=\frac{M{P}_K}{r} \\ \frac{60}{12}<\frac{45}{8} \\ \text{5<5.625} \end{gathered}$$

Thus, the firm isn't using the best combination of inputs because the ratio isn't equal.

Suitable Combination

The cost minimization will require the firm to decrease 5 units of capital to reach the equality.

$$\begin{gathered} \frac{M{P}_L}{w}=\frac{M{P}_K}{r} \\ \frac{60}{12}=\frac{40}{8} \\ \text{5 = 5} \end{gathered}$$

Thus, it can be seen that the condition gets satisfied at 40 units of capital. It means the firm should decrease its capital by 5 units.