Some economists believe that the U.S. economy as a whole can be modeled with the following production function, called the $\textit{Cobb$-$Douglas production function}$: $$Y=AK^{1/3}L^{2/3},$$ 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 $W/P$. (Note: The wage is labor compensation measured in dollars, whereas the real wage is labor compensation measured in units of output.)

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.