Q = 50L + 6L^2 − 0.5L^3
a. The law of diminishing returns take effect when the marginal product of production start to decrease as the amount of a single factor of production increases, so MPL decreases, while L increases.
b. Stage I occur, when MPL increases, so:
MPL = Q' = 50 + 12L - 1.5L^2
MPL increases until MPL' = 0, so 12 - 3L = 0, L = 4 units.
So, stage I occurs, when L increases from 0 to 4.
Stage II occur, when MPL decreases, but MPL >0, so:
50 + 12L - 1.5L^2 = 0,
1.5L^2 - 12L - 50 = 0,
L = (12 + 21.1)/3 = 11 workers.
So, stage II occurs, when from 4 to 11 workers are employed.
Stage III occur, when more than 11 workers are employed.
c. If each worker is paid $10 per hour and works a 40-hour week and the price of the output is $10, then the firm should hire such number of workers, for which MRPL = w, MRPL = MPL*P, so:
(50 + 12L - 1.5L^2)*10 = (10*40),
50 + 12L - 1.5L^2 = 40,
10 + 12L - 1.5L^2 = 0,
1.5L^2 - 12L - 10 = 0,
L = (12 + 14.3)/3 = 8.8 or 9 workers.
If the price of the output falls to $7.50, then in the short-run the firm’s production will decrease. In the long-run the firm will earn zero profits.