The production function below shows the different levels of output obtained with
ID: 1104749 • Letter: T
Question
The production function below shows the different levels of output obtained with various levels of labor input per week.
Q = 360L + 9L2 – 0.2L3
a)What is the maximum level of production (Q) in units that this firm can attain per week?
b)At what level of labor utilization (L) is the average productivity of labor maximized? At what level of production (units of Q) is the average productivity of labor maximized?
c)Beyond what level of labor utilization (L) and units produced per week (Q) does the Law of Diminishing Marginal Returns (or productivity) begin to take effect?
Explanation / Answer
Q = 360L + 9L2 - 0.2L3
(a) Production is maximized when dQ/dL = 0
360 - 18L - 0.6L2 = 0
600 - 30L - L2 = 0
L2 + 30L - 600 = 0
Solving this quadratic equation using online solver,
L = 13.72 (or L = - 43.72, which is negative and inadmissible)
When L = 13.72,
Q = (360 x 13.72) + (9 x 13.72 x 13.72) - (0.2 x 13.72 x 13.72 x 13.72) = 4,939.2 + 1,694.15 - 516.53
= 6,116.82
(b)
APL = Q / L = 360 + 9L - 0.2L2
APL is maximized when dAPL / dL = 0
9 - 0.4L = 0
0.4L = 9
L = 22.5
When L = 22.5, Q = (360 x 22.5) + (9 x 22.5 x 22.5) - (0.2 x 22.5 x 22.5 x 22.5) = 8,100 + 4,556.25 - 2,278.125
= 10,378.125
(c) MPL = dQ / dL = 360 + 18L - 0.6L2
MPL is zero when dMPL / dL = 0
18 - 1.2L = 0
1.2L = 18
L = 15
Q = (360 x 15) + (9 x 15 x 15) - (0.2 x 15 x 15 x 15) = 5,400 + 2,025 - 675 = 6,750
Therefore, when L is higher than 15 units and Q is higher than 6,750 units, diminishing marginal returns set in.
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