Consider the following simplified economy described by a Cobb-Douglas production function with three factors of production: “K” is physical 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). “A” measures the level of technology and efficiency in the economy, and “Y” is real GDP. The aggregate production function is:
Y= A(K)^(1/3)(L)^(1/3)(H)^(1/3)
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 (MPL)?
B. Derive an expression for the marginal product of human capital (MPH). How does an increase in the amount of human capital affect the marginal product of human capital?
C. An unskilled worker earns the marginal product of labor, whereas a skilled worker earns the marginal product of labor plus the marginal product of human capital. Using your answers in Parts A and B above, find the ratio of the skilled wage rate to the unskilled wage rate.
D. Some people advocate government funding of college scholarships as a way of creating a more egalitarian (more equal) society. Others argue that scholarships help only those who are able to go to college. Does your answer to Part C shed light on this debate? Based on this model, which side is correct? Explain.
A. To derive the expression for the marginal product of labor (MPL), we need to calculate the partial derivative of the aggregate production function with respect to labor (L).
Taking the partial derivative of Y with respect to L:
∂Y/∂L = (∂/∂L)[A(K)^(1/3)(L)^(1/3)(H)^(1/3)]
Using the chain rule, the derivative becomes:
∂Y/∂L = (1/3)A(K)^(1/3)(H)^(1/3)[(L)^(-2/3)]
Simplifying the expression, we get:
∂Y/∂L = (1/3)Y/L
So the marginal product of labor (MPL) is (1/3)Y/L.
Now, let's consider the effect of an increase in the amount of human capital (H) on MPL. From the production function, we can see that an increase in H will directly increase Y. This means that Y/L, or output per worker, will also increase. Consequently, an increase in H will increase the marginal product of labor (MPL).
B. To derive the expression for the marginal product of human capital (MPH), we need to calculate the partial derivative of the aggregate production function with respect to human capital (H).
Taking the partial derivative of Y with respect to H:
∂Y/∂H = (∂/∂H)[A(K)^(1/3)(L)^(1/3)(H)^(1/3)]
Using the chain rule, the derivative becomes:
∂Y/∂H = (1/3)A(K)^(1/3)(L)^(1/3)[(H)^(-2/3)]
Simplifying the expression, we get:
∂Y/∂H = (1/3)Y/H
So the marginal product of human capital (MPH) is (1/3)Y/H.
Now, to analyze the effect of an increase in the amount of human capital (H) on MPH, we can observe that an increase in H will directly increase Y. However, since MPH is expressed as (1/3)Y/H, an increase in H will lead to a decrease in MPH, holding all other factors constant. This is because as H increases, the additional output generated by each additional unit of human capital becomes smaller, leading to diminishing marginal returns.
C. In this scenario, an unskilled worker earns the marginal product of labor (MPL), whereas a skilled worker earns the marginal product of labor plus the marginal product of human capital (MPL + MPH).
Using the expressions derived in Parts A and B, we can calculate the ratio of the skilled wage rate to the unskilled wage rate:
Skilled wage rate / Unskilled wage rate = (MPL + MPH) / MPL
Substituting the derived expressions:
Skilled wage rate / Unskilled wage rate = [(1/3)Y/L + (1/3)Y/H] / [(1/3)Y/L]
Simplifying the expression, we get:
Skilled wage rate / Unskilled wage rate = 1 + (H/L)
The ratio of the skilled wage rate to the unskilled wage rate is equal to 1 plus the ratio of human capital (H) to labor (L).
D. The answer to Part C sheds light on the debate regarding government funding of college scholarships. By looking at the derived ratio of the skilled wage rate to the unskilled wage rate, which is equal to 1 plus the ratio of human capital (H) to labor (L), we can infer that increasing the amount of human capital leads to higher wages for skilled workers compared to unskilled workers.
Based on this model, government funding of college scholarships can be seen as a policy that increases the amount of human capital in the economy. This would lead to a higher skilled wage rate relative to the unskilled wage rate. Therefore, those advocating for scholarships as a means of creating a more egalitarian society may be correct, as it would potentially reduce income inequality by increasing opportunities for individuals to obtain college degrees and improve their human capital.