Question

Let L represent the amount of labor that a firm employs and let Q represent the quantity of the firm's output. The firm's production function is Q = f(L) = 100 In(2L +1). a. Let 2 = 2L +1. Then Q = 100 In(2). Use the chain rule for derivatives to to find f'(L). b. Find f"(L). c. What do your answers to parts (a) and (b) tell you about the appearance of the marginal-product-of-labor (MPL) curve?

Answer 1

Answer

(a)

Q = 100ln(Z)

Using chain rule :

dQ/dL = (100/z)(dz/dL) (Note : d(log(x))/dx = 1/x)

Here z = 2L + 1 => dz/dL = 2

=> dQ/dL = (dQ/dz)(dz/dL) = (100/z)(dz/dL) = [100/(2L + 1)]*2

=> dQ/dL = 200/(2L + 1)

Hence, f '(L) = dQ/dL = 200/(2L + 1)

Hence f '(L) = 200/(2L + 1)

(b)

f ''(L) = d(f '(L))/dL = 200(-1/(2L + 1)²)*2

=> f ''(L) = -400/(2L + 1)²

(c)

Marginal Product of Labor(MPL) = dQ/dL = f '(L) = 200/(2L + 1) > 0. This means that For any value of L Marginal product of Labor is positive.

d(MPL)/dL = f ''(L) = -400/(2L + 1)² < 0. Hence If : increases MPL decreases.

Thus, MPL is a downward sloping function.

Hence, MPL curve is always above horizontal axis and is a downward sloping curve(where MPL is on vertical axis and L is on horizontal axis).