Question

The manager of a national retailing outlet recently hired an economist to estimate the firm's production function. Based on the economist's report, the manager now knows that the firm's production function is given by Q=K^(1/2) L^(1/2) and that capital is fixed at 1 unit.

a. Calculate the average product of labor when 9 units of labor are utilized.


b. Calculate the marginal product of labor when 9 units of labor are utilized.

c. Suppose the firm can hire labor at a wage of $10 per hour and output can be sold at a price of $100 per unit. Determine the profit-maximizing levels of labor and output.


d. What is the maximum price of capital at which the firm will still make nonnegative profits?

Answer 1

A.
Q=K^(1/2) L^(1/2)
Q=1^(1/2) 9^(1/2)
Q = 3

B.
Q=K^(1/2) L^(1/2)
MPL = (1/2)*(K/L)^(1/2)
MPL = (1/2)*(1/9)^(1/2)
MPL = 1/6

C.
For profit maximization:
MPL/w = MPK/r
[(1/2)*(K/L)^(1/2)]/[(1/2)*(L/K)^(1/2)] = w/r
(K/L)^(1/2)/(L/K)^(1/2) = 10/r
K/L = 10/r
1/L = 10/r
L = r/10

Q=K^(1/2) L^(1/2)
Q=(r/10)^(1/2)

D.
Let's call profit V.
V = P*Q - C
C = wL + rK
C = 10*(r/10) + r
C = 2r
V = 100*(r/10)^(1/2) - 2r > 0
100*(r/10)^(1/2) > 2r
(r/10)^(1/2) > r/50
50/(10^(1/2)) > r/(r^1/2)
50/(10^(1/2)) > r^(1/2)
r < (50^1/2)/(10^(1/4))
r < 14.058