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. (2 points) 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. (2 points) 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. (2 points) 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. (2 points) 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 take the partial derivative of the production function with respect to labor (L) while holding other factors of production constant.

First, let's rewrite the production function in a more simplified form:
Y = A(K^(1/3))(L^(1/3))(H^(1/3))

Differentiating Y with respect to L, while holding K and H constant:
∂Y/∂L = 1/3 * A(K^(1/3))(H^(1/3)) * (L^(-2/3))

Simplifying:
MPL = ∂Y/∂L = (1/3) * A * (K^(1/3)) * (H^(1/3)) * (L^(-2/3))

An increase in the amount of human capital (H) does not directly affect the marginal product of labor (MPL). The MPL is dependent on the level of labor (L) and the other factors of production (K and H), but the increase in the amount of human capital does not specifically impact MPL.

B. To derive the expression for the marginal product of human capital (MPH), we take the partial derivative of the production function with respect to human capital (H) while holding other factors of production constant.

Differentiating Y with respect to H, while holding K and L constant:
∂Y/∂H = 1/3 * A(K^(1/3))(L^(1/3)) * (H^(-2/3))

Simplifying:
MPH = ∂Y/∂H = (1/3) * A * (K^(1/3)) * (L^(1/3)) * (H^(-2/3))

An increase in the amount of human capital (H) will decrease the marginal product of human capital (MPH). As more human capital is added, the marginal benefit decreases, resulting in diminishing returns to human capital.

C. The skilled wage rate will be the sum of the marginal product of labor (MPL) and the marginal product of human capital (MPH), while the unskilled wage rate will be equal to MPL.

Ratio of skilled wage rate to unskilled wage rate:
(Skilled Wage Rate)/(Unskilled Wage Rate) = (MPL + MPH) / MPL

D. The answer to Part C sheds light on the debate regarding government funding of college scholarships. The model suggests that the skilled wage rate, which includes the additional contribution of human capital (MPH), is higher than the unskilled wage rate, which only considers the marginal product of labor (MPL). This implies that having a college degree, representing an increase in human capital, leads to higher wages.

Therefore, government funding of college scholarships can help create a more egalitarian society by allowing individuals who would not have access to higher education otherwise to acquire the necessary human capital and increase their earning potential. This supports the argument for government-funded scholarships as a means to promote greater equality.

A. To derive the expression for the marginal product of labor (MPL), we need to differentiate the production function with respect to labor (L) while holding other factors of production constant. The Cobb-Douglas production function is:

Y = A(K)^(1/3)*(L)^(1/3)*(H)^(1/3)

Taking the partial derivative of Y with respect to L, we get:

∂Y/∂L = (1/3)*A(K)^(1/3)*(H)^(1/3)*(L)^(-2/3)

Therefore, the marginal product of labor (MPL) is:

MPL = ∂Y/∂L = (1/3)*A(K)^(1/3)*(H)^(1/3)*(L)^(-2/3)

Regarding the effect of an increase in the amount of human capital (H) on MPL, we see that it is multiplied by (H)^(1/3) in the production function. Therefore, as H increases, MPL increases. This suggests that an increase in human capital would lead to a higher productivity of labor, resulting in a higher marginal product of labor.

B. To derive the expression for the marginal product of human capital (MPH), we need to differentiate the production function with respect to human capital (H) while holding other factors of production constant. Applying the same steps as in Part A, we find:

MPH = ∂Y/∂H = (1/3)*A(K)^(1/3)*(L)^(1/3)*(H)^(-2/3)

Regarding the effect of an increase in the amount of human capital (H) on MPH, we see that it is multiplied by (H)^(-2/3) in the production function. Therefore, as H increases, MPH decreases. This suggests that the marginal product of human capital diminishes as the amount of human capital increases.

C. According to the question, 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 as follows:

Skilled wage rate / Unskilled wage rate = (MPL + MPH) / MPL

D. The ratio of the skilled wage rate to the unskilled wage rate calculated in Part C sheds light on the debate about government funding of college scholarships and its impact on equality.

If the skilled wage rate is significantly higher than the unskilled wage rate, it means that the additional income earned by individuals with college degrees (skilled workers) is larger than the income earned by individuals without college degrees (unskilled workers). This suggests that scholarships may contribute to increasing equality by providing access to college education for individuals who may not have had the financial means to pursue it otherwise.

However, if the skilled wage rate is not significantly higher than the unskilled wage rate, it implies that the additional income earned by skilled workers is not substantially greater than that of unskilled workers. In this case, scholarships may not have a pronounced effect on equality since the increase in human capital does not significantly impact productivity and income.

Based on this model, if the ratio of the skilled wage rate to the unskilled wage rate is high, it suggests that scholarships can help promote equality by providing access to education and increasing the productivity of individuals, leading to higher incomes. On the other hand, if the ratio is low, it suggests that scholarships may not have a significant impact on equality as they do not result in a substantial increase in productivity and income for individuals.