A firm's production process uses labor, L, and capital, K, to produce an output, Q according to the function Q = 25KL. Let labor be fixed at level $L = \bar { L }$ . What is the cost minimizing level of capital that the firm must use to produce a target level of output, $Q = \bar { Q }$ ?

Question

Quizplus · Accepted Answer

When labor is fixed at level $\bar{L}$, we can rewrite the production function as follows:
$$Q=25K\bar{L}$$
The cost minimizing level of capital will occur where the marginal product of capital (MPL) divided by the price of capital (w) is equal to the marginal product of labor (MPL) divided by the price of labor (r), which is a result of the firm's cost minimization problem. Since labor is fixed here, the MPL of labor is 0, and we only need to consider the MPL of capital:
$$MPL=\frac{\partial Q}{\partial K}=25\bar{L}$$
The price of capital can be assumed to be 1 (since there is no information given about the price of capital), so we just need to set MPL/w equal to MPL/r and solve for K:
$$\frac{MPL}{w}=\frac{MPL}{r}$$
$$25\bar{L}= \frac{\bar{Q}}{K}$$
$$K=\frac{\bar{Q}}{25\bar{L}}$$
Therefore, the cost minimizing level of capital is $K=\frac{\bar{Q}}{25\bar{L}}$, which is answer choice C.

A Firm's Production Process Uses Labor, L, and Capital, K L=LˉL = \bar { L }L=Lˉ

A Firm's Production Process Uses Labor, L, and Capital, K $L = \bar { L }$