A firm's production function is given by Q = KL. The marginal products of labor and capital are, respectively, MPL = K and MPK = 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 find the combination that produces 1,800 units of output at the lowest cost.

The marginal product of labor (MPL) is the additional output produced by hiring one more unit of labor, holding all other inputs constant. MPL = K means that the marginal product of labor is equal to the amount of capital used.

Likewise, the marginal product of capital (MPK) is the additional output produced by using one more unit of capital, holding all other inputs constant. MPK = L means that the marginal product of capital is equal to the amount of labor used.

To produce 1,800 units of output, we can solve for K and L in the production function:

Q = KL 
1,800 = K L

We know that MPL = K and MPK = L, so we can rewrite the production function as:

Q = MPL * L 
1,800 = K L 
L = 1,800/MPL

And

Q = MPK * K 
1,800 = K L 
K = 1,800/MPK

Now we can plug in the values for MPL and MPK:

L = 1,800/K 
L = 1,800/20 
L = 90

K = 1,800/L 
K = 1,800/60 
K = 30

Therefore, the most efficient combination of labor and capital is (L, K) = (90, 30). To find the most efficient combination of labor and capital, we need to equate the marginal product of labor to the wage rate and the marginal product of capital to the rental rate. 
MPL = w = $10 
K = MPL = 60 
MPK = r = $20 
L = MPK = 30 
Thus, the most efficient combination of labor and capital is (L,K) = (30, 60).

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 $( L , K )$