Question

Some economists believe that the U.S. economy as a whole can be modeled with the following production function, called the Cobb-Douglas production function: Y = AK13L23 where Y is the amount of output, K is the amount of capital, L is the amount of labor, and A is a parameter that measures the state of technology. For this production function, the marginal product of labor is: MPL = 23A(KL)13 Suppose that the price of output (P) is $2, A is 3, K is 1,000,000, and L is 1,000. The labor market is competitive, so labor is paid the value of its marginal product. The amount of output produced (Y) is , and the dollar value of output (PY) is $ . The wage (W) is $ , and the real wage (WP) is . (Note: The wage is labor compensation measured in dollars, whereas the real wage is labor compensation measured in units of output.)

What is the labor share (WLPY)? (Note: The labor share is the fraction of the value of output that is paid to labor.)

Answer 1

The labor share (WLPY) is approximately 33.33%, meaning that about 33.33% of the value of output is paid to labor as compensation.

Step-by-step explanation:

The labor share (WLPY) is the fraction of the value of output that is paid to labor. To calculate the labor share, we need to find the value of labor compensation (W) and divide it by the dollar value of output (PY).

Price of output (P) = $2

A = 3

K = 1,000,000

L = 1,000

`Y = AK^((1/3))L^((2/3))`

First, let's calculate the output (Y):

`Y = AK^((1/3))L^{(2/3)`

`Y = 3 * (1,000,000)^((1/3)) * (1,000)^((2/3))`

Y = 3 * 100 * 10 ≈ 3000

Next, let's calculate the value of output (PY):

PY = P * Y

PY = $2 * 3000 = $6000

The marginal product of labor (MPL) is given by:

`MPL = 2/3 * 3 * (1,000,000)^((1/3)) * (1,000)^{(-1/3)`

MPL = 2 * 100 * 0.1 = 20

Since labor is paid the value of its marginal product, the wage (W) is $20.

Now, let's calculate the labor share (WLPY):

Labor Share (WLPY) = (W / PY) * 100

Labor Share (WLPY) = ($20 / $6000) * 100

Labor Share (WLPY) ≈ 0.3333 * 100 ≈ 33.33%

Therefore, the labor share (WLPY) is approximately 33.33%, meaning that about 33.33% of the value of output is paid to labor as compensation.

Answer 2

The labor share in a Cobb-Douglas production function is calculated by finding the total wages paid to labor as a proportion of the total dollar value of output produced. This involves using the production function to determine the output, calculating the wage based on the marginal product of labor and the price of output, and then determining the labor share from these calculations.

Step-by-step explanation:

Calculating the Labor Share in a Cobb-Douglas Production Function

Under the Cobb-Douglas production function, the amount of output (Y) and the value of the output (PY) can be calculated using the given variables. The wage (W) equals the marginal product of labor (MPL) times the price of output (P). Therefore, the labor share (WLPY) is the ratio of total wages paid to the value of total output produced.

Given that P = $2, A = 3, K = 1,000,000, and L = 1,000, we first find the value of Y by plugging these values into the Cobb-Douglas production function Y = AK^(1/3)L^(2/3). Next, the dollar value of output is calculated by multiplying Y by P. To find W, we use the given marginal product of labor formula and multiply it by P. Then, to calculate the real wage (W/P), we divide W by P. Lastly, the labor share is obtained by dividing total wages (WL) by the dollar value of output (PY), providing the proportion of the output value that is paid to labor.