Chapter 3, Problem 7PA

(a)

To determine

Derive the expression of marginal product of labor.

Explanation of Solution

Given information:

Production function is $\text{Y}={\text{K}}^{\frac{1}{3}}{\text{L}}^{\frac{1}{3}}{\text{H}}^{\frac{1}{3}}$

Calculation:

The production function is given below:

$\text{Y}={\text{K}}^{\frac{1}{3}}{\text{L}}^{\frac{1}{3}}{\text{H}}^{\frac{1}{3}}$ (1)

In Equation (1), K is the capital input, L is the labor input, and H is the human capital.

To derive the marginal products labor, differentiate the total output for labor. The derivation is shown below:

$\begin{array}{c}\text{MPL}=\frac{\partial \left(\text{Total production}\right)}{\partial \left(\text{Total labor}\right)}\\ =\frac{\partial \left({\text{K}}^{\frac{1}{3}}{\text{L}}^{\frac{1}{3}}{\text{H}}^{\frac{1}{3}}\right)}{\partial \text{L}}\\ =\left(\frac{1}{3}\right){\text{K}}^{\frac{1}{3}}{\text{H}}^{\frac{1}{3}}{\text{L}}^{\frac{-2}{3}}\end{array}$

The marginal product of labor is$\text{MPL}=\left(\frac{1}{3}\right){\text{K}}^{\frac{1}{3}}{\text{H}}^{\frac{1}{3}}{\text{L}}^{\frac{-2}{3}}$.

The increase in amount of human capital increases the MPL because it increases the productivity of labor.

Economics Concept Introduction

Marginal product of labor (MPL): Marginal product of labor is the additional output attained by employing an extra unit of labor.

(b)

To determine

Derive the expression of marginal product of human capital.

Explanation of Solution

Given information:

Production function is $\text{Y}={\text{K}}^{\frac{1}{3}}{\text{L}}^{\frac{1}{3}}{\text{H}}^{\frac{1}{3}}$

Calculation:

To derive the marginal products human capital, differentiate the total output for human capital. The derivation is shown below:

$\begin{array}{c}\text{MPH}=\frac{\partial \left(\text{Total production}\right)}{\partial \left(\text{Total human capital}\right)}\\ =\frac{\partial \left({\text{K}}^{\frac{1}{3}}{\text{L}}^{\frac{1}{3}}{\text{H}}^{\frac{1}{3}}\right)}{\partial \text{H}}\\ =\left(\frac{1}{3}\right){\text{K}}^{\frac{1}{3}}{\text{L}}^{\frac{1}{3}}{\text{H}}^{\frac{-2}{3}}\end{array}$

The marginal product of human capital is$\text{MPH}=\left(\frac{1}{3}\right){\text{K}}^{\frac{1}{3}}{\text{L}}^{\frac{1}{3}}{\text{H}}^{\frac{-2}{3}}$.

The increase in the amount of human capital decreases the marginal product of human capital, because of the diminishing return.

(c)

To determine

Income share (wage) paid for labor and human capital.

Explanation of Solution

Given information:

Production function is $\text{Y}={\text{K}}^{\frac{1}{3}}{\text{L}}^{\frac{1}{3}}{\text{H}}^{\frac{1}{3}}$

Calculation:

The income share paid for labor is the real wage. The fraction of income spend on labor resource divided by the total income is the labor share of income. How the labor share is calculated is shown below:

$\begin{array}{c}\text{Labor share}=\left[\frac{\left(\frac{1}{3}{\text{K}}^{\frac{1}{3}}{\text{H}}^{\frac{1}{3}}{\text{L}}^{\frac{-2}{3}}\right)\text{L}}{{\text{K}}^{\frac{1}{3}}{\text{H}}^{\frac{1}{3}}{\text{L}}^{\frac{-1}{3}}}\right]\\ =\frac{1}{3}\end{array}$

$\frac{1}{3}$share of the total output is the income share paid for labor.

Using the same logic, the share of income paid for human capital is shown below:

$\begin{array}{c}\text{Human capital share}=\left[\frac{\left(\frac{1}{3}{\text{K}}^{\frac{1}{3}}{\text{L}}^{\frac{1}{3}}{\text{H}}^{\frac{-2}{3}}\right)\text{H}}{{\text{K}}^{\frac{1}{3}}{\text{H}}^{\frac{1}{3}}{\text{L}}^{\frac{-1}{3}}}\right]\\ =\frac{1}{3}\end{array}$

$\frac{1}{3}$share of the total output is the income share paid for human capital.

(d)

To determine

Ratio of skilled wage to unskilled wage.

Explanation of Solution

The unskilled worker earns the marginal product of labor, and skilled worker earns the marginal product of labor plus the marginal product of human capital. The ratio of skilled wage to unskilled wage is shown below:

$\begin{array}{c}\frac{{\text{W}}_{\text{Skilled}}}{{\text{W}}_{\text{Unskilled}}}=\left(\frac{\text{MPL+MPH}}{\text{MPL}}\right)\\ =\frac{\left(\frac{1}{3}{\text{K}}^{\frac{1}{3}}{\text{L}}^{\frac{-2}{3}}{\text{H}}^{\frac{1}{3}}\right)+\left(\frac{1}{3}{\text{K}}^{\frac{1}{3}}{\text{L}}^{\frac{1}{3}}{\text{H}}^{\frac{-2}{3}}\right)}{\frac{1}{3}{\text{K}}^{\frac{1}{3}}{\text{L}}^{\frac{-2}{3}}{\text{H}}^{\frac{1}{3}}}\\ =1+\frac{\text{L}}{\text{H}}\end{array}$

The ratio of skilled wage to unskilled wage is $1+\frac{\text{L}}{\text{H}}$. This ratio is always greater than one. The reason is the wage rate of skilled worker is more than that of unskilled worker’s. If the human capital increases, it reduces this ratio, because the diminishing returns to human capital lower its return and increase the marginal product of unskilled worker.

(e)

To determine

Impact of proving scholarship.

Explanation of Solution

Generally, colleges provide scholarships for students with the aim of increasing the human capital. But providing more scholarships may lower the returns to education and it may increase the gap between wages paid for more educated workers to less educated workers.