A firm's production function is given by $Q = K L$. The marginal products of labor and capital are, respectively, $M P _ { L } = K$ and $M P _ { K } = L$. The wage rate of labor 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 yields a cost of exactly $\$ 1000$ ?
A) $( 10,45 )$
B) $( 25,50 )$
C) $( 50,25 )$
D) $( 20,40 )$

Question

Quizplus · Accepted Answer

Given the cost constraint of $1000 and the prices for labor and capital, the most efficient combination is found by setting the cost equal to $1000: $10L + 20K = 1000$. Solving this equation with the given options, only option C, $(50, 25)$, satisfies the condition when substituting $L = 50$ and $K = 25$ into the equation, resulting in $10(50) + 20(25) = 1000$.

A Firm's Production Function Is Given By Q=KLQ = K LQ=KL

A Firm's Production Function Is Given By $Q = K L$