Question

Consider a short-run production function Q(L), where L is labor input. Think of the case when L is large enough so that the marginal product of labor is decreasing. If the average product of labor equals the marginal product of labor, which of the following statement is true?

A) the average product of labor is at a maximum.

B) the marginal product of labor is at a maximum.

C) Both A and B above.

D) Neither A nor B above.

Write down the mathematical expression of dAP/dL and use this expression to support your answer.

Answer 1

Average Product(AP) = Q/L where Q = Q(L)

Optimize : AP = Q/L

First order condition :

d(AP)/dL = 0 => d(Q/L)/dL = 0 => [L - Q(dL/dQ)]/L² = 0 {Formula : Quotient rule : d(A/B)/dx = [B(dA/dx) - A(dB/dx)]/B²}

=> L - Q(dL/dQ) = 0 => dQ/dL = Q/L . As dQ/dL = Marginal product of labor(MP).

Thus, Q/L = dQ/dL => AP = MP

Initially firm enjoys increasing returns to production and thus MP increases initially and thus AP increases but is below MP. When firm continues to hire L, then point will come when MP starts decreasing(like here). Thus MP will starts decreasing and thus MP can cut AP(i.e. AP = MP) only when AP is at its maximum. We have seen from above optimization that when AP is optimized then AP = MP and thus this will be a maximum.

So when AP = MC we have AP at its maximum.

(Note : When MP increases MP is always greater than AP, so MP cannot be at its maximum when it cuts AP. Thus option (B) is also incorrect)

Hence, the correct answer is (A) the average product of labor is at a maximum.