Answer:
a) MPL = 72L - 1.2L²
b) L = 45
c) L = 60
d) 0 < L < 60
Step-by-step explanation:
a. To find the marginal product of labor (MPL), we differentiate the production function with respect to labor (L):
MPL = dQ/dL = d/dL (36L² - 0.4L³)
MPL = 72L - 1.2L²
To find the average product of labor (APL), we divide the total product (Q) by the quantity of labor (L):
APL = Q/L = (36L² - 0.4L³)/L
APL = 36L - 0.4L²
b. To find the number of labor that maximizes average product (AP), we set the derivative of APL with respect to L equal to zero:
dAPL/dL = 36 - 0.8L = 0
0.8L = 36
L = 45
Therefore, the number of labor that maximizes average product is L = 45.
To find the maximum value of average product, substitute L = 45 into the APL function:
APL = 36(45) - 0.4(45)²
APL = 1620 - 810
APL = 810
The maximum value of average product is 810.
c. Stage III occurs when the marginal product of labor becomes negative. Therefore, we need to find the number of labor (L) where MPL = 0:
72L - 1.2L² = 0
L(72 - 1.2L) = 0
L = 0 or L = 60
Stage III starts to occur at L = 60.
To find the total production at that level of employment, substitute L = 60 into the production function:
Q = 36(60)² - 0.4(60)³
Q = 129,600 - 864,000
Q = -734,400
The total production at that level of employment is -734,400.
d. Rational production occurs when MPL > 0. To determine the range of labor where rational production happens, we need to find the critical points of MPL:
MPL = 72L - 1.2L²
Setting MPL = 0, we get:
72L - 1.2L² = 0
L(72 - 1.2L) = 0
L = 0 or L = 60
Therefore, rational production occurs when 0 < L < 60.
The range of labor where rational production happens is 0 < L < 60.
Hope this helps!