# Cobb-Douglas production function

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

Y = K1/3 L2/3.

The economy has 1,000 units of capital and a labor force of 1,000 workers.

a. What is the equation describing the demand for labor in this economy? (Hint: Review the appendix to 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 is 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 1 unit of output. How does this 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 minimum-wage law? Why or why not?

#### Solution Preview

Cobb Douglas

<br>------------

<br>Consider an economy with the following Cobb-Douglas production function:

<br>

<br>Y = K1/3 L2/3.

<br>

<br>The economy has 1,000 units of capital and a labor force of 1,000 workers.

<br>

<br>----------------------------------------------------------------------

<br>a. What is the equation describing the demand for labor in this economy? (Hint: Review the appendix to Chapter 3.)

<br>

<br> Y = K^(1/3)* L^(2/3) ( "^" is the exponent operator; K^(1/3) means "k raised to 1/3".

<br>The labor demand is given by taking the partial derivative of Y with respect to L (= dY/dL)

<br>This also gives the labor wage in terms of units of output (W/P)

<br>W/P = dY/dL = (2/3)* K^(1/3)* L^-(1/3) = (2/3)* K^(1/3)/L^(1/3) ---- (a)

<br>----------------------------------------------------------------------

<br>b. If the real wage can adjust to equilibrate labor supply and labor demand, what is the real wage? ...

#### Solution Summary

Apply the Cobb-Douglas production function.