A firm's production function is given by $Q = K ^ { 2 } L$. The marginal products of labor and capital are, respectively, $M P _ { L } = K ^ { 2 }$ and $M P _ { K } = 2 K L$. Further, the wage rate is $w = \$ 10$ and the rental rate of capital is $r = \$ 20$. What is the most efficient combination of labor and capital $( L , K )$ that also results in a total cost level of exactly $\$ 1,200$ ?
A) $( 20,50 )$
B) $( 25,47.5 )$
C) $( 30,45 )$
D) $( 40,40 )$

Question

Quizplus · Accepted Answer

The most efficient combination of labor and capital is determined by minimizing the total cost subject to the given wage and rental rates. The total cost is given by $C = wL + rK$. Substituting the given wage and rental rates, we get $C = 10L + 20K$. Setting $C = 1200$ and solving for $L$ and $K$ gives us the combination that also satisfies the marginal product conditions for efficiency. The combination $(40,40)$ is the only one that meets these criteria within the given options, as it balances the cost of labor and capital under the given constraints.

A Firm's Production Function Is Given By Q=K2LQ = K ^ { 2 } LQ=K2L

A Firm's Production Function Is Given By $Q = K ^ { 2 } L$