Question

Chapter 7, Labor Market Regulation (3 points):

• Consider an economy with the following Cobb-Douglas production function:

Y =k^1/3L^2/3
The economy has 1,000 units of capital and a labor force of 1,000 workers.

(a) Derive the equation describing labor demand in this economy as a function of the real wage and the capital stock (Hint: Review Chapter 3.)

(b) If the real wage can adjust to equilibrate labor supply and labor demand, what is the real wage? In this equilibrium, what are employment, output, and the total amount earned by workers?

(c) Now suppose that Congress, concerned about the welfare of the working class, passes a law requiring firms to pay workers a real wage of one unit of output. How does this real wage compare to the equilibrium wage?

(d) Congress cannot dictate how many workers firms hire at the mandated wage. Given this fact, what are the effects of this law? Specifically, what happens to employment, output, and the total amount earned by workers?

(e) Will Congress succeed in its goal of helping the working class? Explain.

(f) Do you think that this analysis provides a good way of thinking about a mini- mum wage law? Why or why not?

Answer 1

Production function: Y = K^1/3L^2/3

K = L = 1000

Let W be the nominal wage rate of a unit of labor and w be the real wage rate.

Let P be the price of the product.

A)

The company will hire labor until the value of Marginal Product of Labor equals the wage rate paid.

VMP_L = W

P*MP_L = W

MP_L = W/P

MP_L = w

For Marginal Product of labor differentiating production function with respect of L,

MP_L = w

2/3K^1/3L^-1/3 = w

Solving for L we get the labor demand function

L =

B)

In equilibrium, Labor demand equals the labor supply. Thus Put Value of L and K of 1000, we get

1000 = 8000/27w3

w* = 2/3 ( real wage)

Output Y = K^1/3L^2/3

Y* = (1000)^1/3(1000)^2/3

Y* = 1000

Labor earn 2/3rd of the output thus 2/3*1000 = 666.67

C)
Now the real wage is mandated to one unit of output. Since the equilibrium real wage is 2/3 unit of output. There is will an excess supply of labor demand and hence underemployment.

D)

Now w = 1

Put w = 1 in Labor demand function we get,

L =

L = 8*1000/27

L = 296.3

Put the new of L and K in the production function we have,

Y =  (1000)^1/3(296.3)^2/3

Y = 10*44.45

Y = 444.5

Labor earn 2/3rd of the output thus 2/3*444.5 = 296.33

E)

At this wage rate, there will be an excess supply of labor. Will the excess labor supply, unemployed people will bid down the wage until it reaches the equilibrium level. Hence the goal of helping working class will not succeed.

F)

Yes, this does provide an insight into minimum wage rate. A mandated wage rate above the equilibrium creates unemployment and thus is not sustainable in the long run.