A firm's production function is given by Q = KL. The marginal products of labor and capital are, respectively, MP_L = K and MP_K = L. The wage rate of labor is w = $10 and the rental rate of capital is r = $20. The firm wants to produce 1,800 units of output. What is the most efficient combination of labor and capital $ (L, K) $ ? A) (10, 20) B) (20, 90) C) (60, 30) D) (90, 20)

Question

Quizplus · Accepted Answer

To find the most efficient combination of labor and capital, we need to use the formula for marginal rate of technical substitution (MRTS), which is given by: 
MRTS = MPL/MPK = w/r

Plugging in the given values, we get: 
MRTS = K/L x $10/$20 = K/2L

To produce 1,800 units of output, we can set up the following equation: 
1,800 = KL

Solving for one variable in terms of the other, we get: 
K = 1,800/L

Substituting this into the MRTS equation, we get: 
MRTS = (1,800/L)/2L = 900/L^2

To maximize efficiency, we want to choose the combination of labor and capital that equates the MRTS to the ratio of factor prices (w/r). In this case, that would be: 
900/L^2 = 10/20 
L = 60

Substituting back into the production function, we get: 
1,800 = K(60) 
K = 30

Therefore, the most efficient combination of labor and capital is (L, K) = (60, 30), which is choice C.

A Firm's Production Function Is Given by Q = KL (L,K) (L, K) (L,K)

A Firm's Production Function Is Given by Q = KL $(L, K)$