Question

 

The Production Function of a perfectly competitive firm is Q = 80L +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 1400 and 1600 in the next decade, what would be your long-run strategy to optimize the resources of your company?

Answer 1

The Production Function of a perfectly competitive firm is Q = 80L + 12L2 -0.5L3,

a. When diminishing returns are existing, MPL reaches its maximum and then it starts falling

MPL = dQ/dL = 80 + 24L - 1.5L^2

Find L* so that dMPL/dL = 0

24 - 3L = 0

L = 24/3 = 8. Hence at L = 8, MPL is maximum and then it will start falling.

b.Between 0 to 8, MPL is rising so this is stage I. When MPL becomes zero, L is

80 + 24L - 1.5L^2 = 0

1.5L^2 - 24L - 80 = 0

The valid root is 18.83. Hence stage II beings at L = 8 and ends at L = 18.83. Stage III starts at L = 18.83 and goes till more labor is hired.

c. MPL x Price = Wage rate should be the hiring rule

(80 + 24L - 1.5L^2)*2 = 30

80 + 24L - 1.5L^2 = 15

1.5L^2 - 24L - 65 = 0

Valid rule is 18.36. Hence we hire L = 18.36

d. Q is maximum when MPL is 0. This is given by L = 18.83.

Maximum Q = 80*18.83 + 12*(18.83^2) -0.5*(18.83^3) = 2423

e. Q is 1500 when L is 10. Similarly for L = 11, Q is 1667. Accordingly, we can hire 10 labor units to optimize the resources in the long run.