Question

Given the Production Function of a perfectly competitive firm: Q = 30L + 12L2 – 0.5L3 , where Q = Output and L = labor input

a. At what value of L will Diminishing Returns take effect?

b. Calculate the range of values for labor over which stages I, II, and III occur?

c. Suppose that the wage rate is $30 and the price of output is $2 per unit. How many workers should the firm hire?

d. At what value of L will Q be at its maximum? What is the maximum amount of Q?

e. If demand forecasts predict an output level of between 900 and 1100 in the next decade, what would be your long-run strategy to optimize the resources of your company?

Answer 1

A) Q = 30L + 12L2 - 0.5L3

MPL = 30 + 24L - 1.5L2

The law takes effect when DMPL/dL = 0

dMPL = 24 - 3L = 0

3L = 24

L = 24/3 = 8

Law of diminishing returns takes effect when 4 laborers are used in production.

B) stages I occurs, when MPL increases

MPL = Q' = 30 + 24L - 1.5L2

MPL increases until MPL' = 0

so 24 - 3L = 0

L = 8

so stages I occurs, when L increases from 0 to 8.

stages II occurs, when MPL decreases, but MPL > 0

30 + 24L - 1.5L2 = 0

L = (24 + 27.49)/3 = 17

so, stages II occurs when from 8 to 17 workers are employed.

stages III occurs, when more than 17 workers are employed.

C)

MRPL = MPL x P

( 30 + 24L - 1.5L2) x 30 = 2 x 30

900 + 720L -45L2 = 60

15 + 12L - 0.75L2 = 0

L = ( 12 + 7.54) / 1.5 = 13 workers