A firm's production process uses labor, $L$, and capital, $K$, and materials, $M$, to produce an output, $Q$ according to the function $Q = K L M$, where the marginal products of the three inputs are $M P _ { L }$ $= K M , M P _ { K } = L M$, and $M P _ { M } = K L$. The wage rate for labor is $w = 2$, the rental rate of capital is $r$ $= 1$, and the cost of materials is $m = 4$ per unit. What is the long run cost-minimizing level of capital that the firm must use to produce a target level of output, $Q = 1000$ ?
A) $K = 5$
B) $K = 10$
C) $K = 20$
D) $K = 40$

Question

Quizplus · Accepted Answer

In order to determine the cost-minimizing level of inputs for a firm, we need to use the formula for the cost of production, which is:

$C = wL + rK + mM$

Where $w$ is the wage rate, $r$ is the rental rate, and $m$ is the cost per unit of materials.

We can substitute in the marginal product functions given in the problem to get:

$C = 2L + K + 4LM$

We can then use the production function given in the problem to solve for one of the inputs in terms of the other two.

$Q = KLM \Rightarrow \frac{Q}{LM} = K$

So we can substitute this into the cost function:

$C = 2L + \frac{Q}{LM} + 4LM$

We can then take the derivative of the cost function with respect to $L$ and set it equal to zero to find the level of labor that minimizes costs:

$\frac{dC}{dL} = 2 - \frac{QM}{L^2}$

Setting this equal to zero, we get:

$2 = \frac{QM}{L^2}$

Solving for $L$, we get:

$L = \sqrt{\frac{QM}{2}}$

We can then use the production function and the value of $L$ to solve for the optimal level of capital:

$Q = KLM \Rightarrow K = \frac{Q}{LM} = \frac{Q}{L\sqrt{QM/2}} = \sqrt{\frac{2Q}{M}}$

Plugging in $Q = 1000$ and $M = 1$, we get:

$K = \sqrt{2000} = 20$

Therefore, the long run cost-minimizing level of capital that the firm must use to produce a target level of output, $Q = 1000$ is K = 20. To minimize costs in the long run, the firm will choose the input levels such that the ratio of the marginal product to the input price is equal for all inputs. This means, 
$$\frac{MP_L}{w} =\frac{MP_K}{r} = \frac{MP_M}{m}$$
Using the given marginal product functions, we get:
$$\frac{K M}{2} = \frac{L M}{1} = \frac{K L}{4}$$
Simplifying:
$$K = 2L$$ 
$$L = 4M$$
Plugging this into the production function:
$$Q = (2L)(L)(4M) = 8 L^2 M$$ 
And substituting the given target level of output, $Q = 1000$, we get: 
$$1000 = 8 L^2 M$$ 
$$125 = L^2 M$$
Given that the cost of materials is $4 per unit, we can solve for the optimal level of M:
$$M = \frac{125}{L^2}$$
Substituting this into the labor-capital ratio:
$$L = 4M = \frac{500}{K^2}$$
Substituting this back into the production function:
$$Q = KLM = K \left(\frac{500}{K^2}\right) \left(\frac{125}{K^4}\right) = \frac{62500}{K^2}$$
Solving for K when Q = 1000:
$$1000 =\frac{62500}{K^2}$$
$$K = 25$$
Finally, since capital is twice the level of labor, the optimal level of labor should be:
$$L = \frac{K}{2} = 12.5$$
Therefore, the answer is C) $K = 20$ because it is the closest given option to 25.

A Firm's Production Process Uses Labor LLL , and Capital KKK , and Materials

A Firm's Production Process Uses Labor $L$ , and Capital $K$ , and Materials