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 = AK1/3L2/3,

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 = (2/3) A(K/L)1/3.

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.

a. Calculate the amount of output produced Y and the dollar value of output PY.

b. Calculate the wage W and the real wage W/P.

c. Calculate the labor share (the fraction of the value of output that is paid to labor), which is (WL)/(PY).

d. Calculate what happens to output Y, the wage W, the real wage W/P, and the labor share (WL)/(PY) in each of the following scenarios:

i. Inflation increases P from 2 to 3.

ii. Technological progress increases A from 3 to 9.

iii. Capital accumulation increases K from 1,000,000 to 8,000,000.

iv. A plague decreases L from 1,000 to 125.

e. Despite many changes in the U.S. economy over time, the labor share has been relatively stable. Is this observation consistent with the Cobb-Douglas production function? Explain.