Suppose that a firm's production function of output $Q$ is a function of only two inputs, labor (L) and capital $( K )$ and can be written $Q = 25 L K$ with marginal products $M P _ { L } = 25 K$ and $M P _ { K } =$ $25 L$. Letting the wage rate for labor be $w$ and the rental rate of capital be $r$, the equation for the firm's demand for labor would be:
A) $L = \sqrt { \frac { w Q } { 25 r } }$
B) $L = \sqrt { \frac { 25 r } { w Q } }$
C) $L = \sqrt { \frac { r Q } { 25 w } }$
D) $L = \sqrt { \frac { 25 w } { r Q } }$

Question

Quizplus · Accepted Answer

The firm's demand for labor is determined by equating the marginal product of labor to the wage rate, so:
$$M P _ { L }= \frac{\text{change in } Q}{\text{change in } L}=25K=w$$
Solving for L:
$$L=\frac{w}{25K}$$
Since we know that $Q=25LK$, we can substitute for $K$ to get:
$$L=\sqrt{\frac{rQ}{25w}}$$ The demand for labor equation can be derived by setting the marginal product of labor equal to the wage rate, which gives us: 
$$MPL = 25K = w$$
$$K = \frac{w}{25}$$
Substituting this into the production function, we get:
$$Q = 25LK = 25L(\frac{w}{25}) = Lw$$
Rearranging this equation to solve for L gives us:
$$L = \frac{Q}{w}$$
Substituting in the demand for capital (which can be derived similarly using the marginal product of capital and the rental rate equation), we get:
$$L = \sqrt{\frac{rQ}{25w}}$$
Therefore, the correct answer is C.

Suppose That a Firm's Production Function of Output QQQ Is a Function of Only Two Inputs, Labor (L) and a Function

Suppose That a Firm's Production Function of Output $Q$ Is a Function of Only Two Inputs, Labor (L) and a Function