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 = 25LK with marginal products MP_L = 25K and MP_K = 25L. 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 demand for labor can be derived from the marginal product of labor (MPL) and the wage rate (w), such that MPL/w = MPK/r.

Substituting the given marginal products, we get:

25K/w = 25L/r

Solving for L gives:

L = (25 r/w) * K

Substituting Q = 25LK, we get:

Q = 25 * [(25r/w) * K] * K

Simplifying, we get:

Q = (625r/w) * K^2

Solving for K, we get:

K = sqrt((wQ)/(625r))

Substituting this back into the expression for labor demand:

L = (25r/w) * sqrt((wQ)/(625r))

Simplifying, we get:

L = sqrt((rQ)/(25w))

Therefore, the correct choice is C.

Suppose That a Firm's Production Function of Output Q Is L=wQ25rL = \sqrt { \frac { w Q } { 25 r } }L=25rwQ​​

Suppose That a Firm's Production Function of Output Q Is $L = \sqrt { \frac { w Q } { 25 r } }$