Answer to Question #62978 in Microeconomics for christopher
A firm has the following short-run production function:
Q = 50L+6L 2 −0.5L 3
where Q = Quantity of output per week
L = Labor (number of workers)
a. When does the law of diminishing returns take effect?
b. Calculate the range of values for labor over which Stages I, II, and III occur.
c. Assume each worker is paid $10 per hour and works a 40-hour week. How many workers should the firm hire if the price of the output is $10? Suppose the price of the output falls to $7.50. What do you think would be the short-run impact on the firm’s production? The long-run impact?
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.
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