Consider an economy with the following CobbDouglass producti

Consider an economy with the following Cobb-Douglass production function:

Y=5K^(1/3)L^(2/3)

a. Derive the equation describing labor demand in this economy as a function of the real wage and capital(The answer is: MPL=real wage)

b. The economy has 27,000 units of capital and a labor force of 1000 workers. Assuming that factor prices adjust to equilibrate supply and demand, calculate the real wage, total output, and the total amount earned by workers

c. Now suppose that congress, concerned about the welfare of the working class, passes a law setting minimum wage that is 10 percent above the wage you derived in part b. Assuming that congress cannot dictate how many workers are hired at the mandated wage, what are the effects of this law? Specifically, calculate what happens to the real wage, employment, output , and the total amount earned by workers.

Solution

Y = 5K^(1/3)L^(2/3)

(a)

Labor demand curve is derived as

MPL = w

dY / dL = w

5 x (2/3) x (K / L)^1/3 = w

10 x (K / L)^1/3 = 3w [Labor demand function]

(b) K = 27,000 & L = 1,000

Y = 5 x (27,000)^1/3(1,000)2/3 = 5 x 30 x 100 = 15,000 (Output)

From part (a),

10 x (K / L)^1/3 = 3w

10 x (27,000 / 1,000)^1/3 = 3w

10 x 3 = 3w

w = 10

Total amount earned by workers = w x L = 10 x 1,000 = 10,000

(c)

At this higher minimum wage, demand for labor will fall and unemployment will rise. Employment falls, output falls.

Revised wage = $10 x 1.1 = $11

3 x 10 x 1.1 = 10 x (K / L)^1/3

3.3 = (K / L)^1/3

(K / L) = 35.937

L = K / 35.937 = 27,000 / 35.937 = 751.31 (employment decreases)

Y = 5 x (27,000)^1/3 x (751.31)^2/3 = 5 x 30 x 82.64 = 12,396.69 (Y falls)

Total amount earned by workers = w x L = 11 x 751.31 = 8,264.41 (Decreases)