Consider the following short-run production function: q = 5L2 – (1/3) L3.
a. At what level of L do diminishing marginal returns begin? Show your derivation.
b. At what level of L do diminishing returns begin? Show your derivation.
c. At what level of L does the marginal product of labor equal the average product of labor? Show your derivation.
The short-run production function is given by q = 5L^2 – (1/3)*L^3 where L is the labor and q is the output.
The derivative of q with respect to L gives the marginal return, or the change in output with change in labor employed.
q' = 10*L - L^2
When the derivative of q' is negative a stage of diminishing marginal returns is reached.
q'' = 10 - 2L
10 - 2L < 0
10 < 2L
L > 5
At values of L greater than 5, the marginal returns start to diminish.
The returns start to diminish when q'<0 or
10*L - L^2 < 0
L^2 > 10*L
L > 10
The average product of labor is q/L
APL = (5L^2 – (1/3)*L^3)/L
APL = 5L - L^2/3
The average product of labor is equal to the marginal product of labor when
5*L - L^2/3 = 10*L - L^2
(2/3)*L^2 = 5*L
L = (5*3)/2 = 7.5
At L = 7.5, the average product of labor is equal to marginal product of labor.