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

Multiple Choice

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. Let materials input be fixed now at $M = 2$ . What is the cost minimizing level of capital that the firm must use to produce a target level of output, $Q = 1600$ ?

A) $K = 5$
B) $K = 10$
C) $K = 20$
D) $K = 40$