Suppose a firm is a monopolist in its output market and a perfect competitor in its input market.The demand for its output is $Q _ { D } = 1800 - P$ .The firm's production is given by $Q = 10 \sqrt { L }$ and the market wage is $50.The profit-maximizing level of labor input is
A)100
B)1,000
C)3,600
D)5,000

Question

Quizplus · Accepted Answer

To find the profit-maximizing level of labor input, we need to calculate the marginal revenue product (MRP) of labor and equate it with the wage rate.
The MRP of labor is given by the marginal product of labor (MPL) multiplied by the marginal revenue (MR) of the output. We can find MPL as the derivative of the production function with respect to L:
MPL = 5 / sqrt(L)
To find MR, we can take the derivative of the demand function with respect to Q and multiply it by Q:
MR = (dQ/dP) * P / Q = (d(1800-Q)/dQ) * P / Q = (1800-2Q)/Q
Now we can find MRP:
MRP = MPL * MR = 5 / sqrt(L) * (1800-2(10 sqrt(L)))/(10 sqrt(L)) = 450/sqrt(L) - 90
Setting MRP equal to the wage rate of $50, we get:
450/sqrt(L) - 90 = 50
Solving for L, we get L = 1600. This is the labor input that maximizes profits, which corresponds to a production level of Q = 400. Note that this is less than the output level that would maximize revenue (900), but as a monopolist, the firm wants to restrict output to charge a higher price and increase profit.

Suppose a Firm Is a Monopolist in Its Output Market QD=1800−PQ _ { D } = 1800 - PQD​=1800−P

Suppose a Firm Is a Monopolist in Its Output Market $Q _ { D } = 1800 - P$