Question

. rhis problem requires the use of calculus.) Consider a Cobb-Douglas production function with three nputs. K is capital (the number of machines). L is labor (the number of workers), and H is human capital (the number of college degrees among the workers).The production function is a. Derive an expression for the marginal product of labor. How does an increase in the amount of human capital affect the marginal product of labor? b. Derive an expression for the marginal product of human capital. How does an increase in the amount of human capital affect the mar- ginal product of human capital? c. What is the income share paid to labor? What is the income share paid to human capital? In the national income accounts of this economy, what share of total income do you think workers would appear to receive? (Hint: Consider where the return to human capital shows up) d. An unskilled worker earns the marginal product of abor, whereas a skilled worker earns the marginal product of labor plus the marginal product of human capital. Using your answers to parts (a) and (b). find the ratio of the skilled wage to the unskilled wage. How does an increase in the amount of human capital affect this ratio? Explain. e. Some people advocate government funding of college scholarships as a way of creating

Answer 1

A)

$MPL = 1/3 \frac{(K^{1/3}H^{1/3})}{L^{2/3}}$

Increase in human capital raises MPL because Human Capital and MPL are directly related.

B)

$MPH = 1/3 \frac{(K^{1/3}L^{1/3})}{H^{2/3}}$

Increase in human capital lowers MPH because Human capital and MPH are inversely related.

C)

Income share paid to labor = Income share paid to human capital = 1/3 of Total Income

( Income share decides by the superscript given in equation because it shows the contribution of each factor of production in the process)

All workers will receive 2/3 of total income because they put their labor and human capital in this process.

D)

$W_{skilled}/W_{unskilled} = (MPL + MPH) / MPL = 1 + (L/H)$. When H increases, this ratio falls because the diminishing returns to human capital lower its return, while at the same time increasing the marginal product of unskilled workers.