Chapter 7, Problem 5PA

(a)

To determine

Derive the equation that describes labor demand as a function of real wage and capital stock.

Explanation of Solution

Given information:

Production function is $\text{Y}=5{\text{K}}^{\frac{1}{3}}{\text{L}}^{\frac{2}{3}}$ (1)

Calculation:

The quantity of labor that a profit-maximizing firm hires at given wage rate determines the demand for labor. The firm will hire the labor until the real wage rate is equal to the marginal product of labor (MPL). Symbolically, it is represented below:

$\text{MLP}=\frac{\text{W}}{\text{P}}$

To derive MPL, differentiate the total output for labor.

$\begin{array}{c}\text{MPL}=\frac{\partial \left(\text{Total production}\right)}{\partial \left(\text{Total labor}\right)}\\ =\frac{\partial \left({\text{5K}}^{\frac{1}{3}}{\text{L}}^{\frac{-2}{3}}\right)}{\partial \text{L}}\\ =\left(\frac{10}{3}\right){\text{K}}^{\frac{1}{3}}{\text{L}}^{\frac{-1}{3}}\end{array}$

Marginal product of labor is $\text{MPL}=\left(\frac{10}{3}\right){\text{K}}^{\frac{1}{3}}{\text{L}}^{\frac{-1}{3}}$.

Equate the real wage to marginal product of labor.

$\left(\frac{10}{3}\right){\text{K}}^{\frac{1}{3}}{\text{L}}^{\frac{-1}{3}}=\frac{\text{W}}{\text{P}}$

 Solving demand for labor, the equation can be written as follows:

$\text{L}=\frac{1,000}{27}\text{K}{\left(\frac{\text{W}}{\text{P}}\right)}^{-3}$ (2)

This equation  states that the labor demand is a function of real wage rate and capital stock.  An increase in real wage reduces the labor demand.

Economics Concept Introduction

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

(b)

To determine

Real wage rate, total output, and amount earned by worker.

Explanation of Solution

Given information:

Capital stock is 27,000 units.

Labor force is 1,000 workers.

Calculation:

To solve for real wage rate $\left(\frac{\text{W}}{\text{P}}\right)$, the equilibrium condition can be used. The equilibrium situation is  stated below:

$\text{L}=\frac{1,000}{27}\text{K}{\left(\frac{\text{W}}{\text{P}}\right)}^{-3}$

Substitute the respective values in Equation (2).

$\begin{array}{c}1,000=\frac{1,000}{27}\text{27,000}{\left(\frac{\text{W}}{\text{P}}\right)}^{-3}\\ \left(\frac{\text{W}}{\text{P}}\right)=10\end{array}$

In the equilibrium situation, the profit-maximizing firm will hire 1,000 laborers at real wage rate of 10 units of output.

To calculate the total output, substitute the respective values in Equation (1).

$\begin{array}{c}\text{Y}=5{\text{K}}^{\frac{1}{3}}{\text{L}}^{\frac{2}{3}}\\ =5{\left(27,000\right)}^{\frac{1}{3}}{\left(1,000\right)}^{\frac{2}{3}}\\ =15,000\end{array}$

Total output is 15,000 units.

(c)

To determine

Real wage rate, output, and demand for labor.

Explanation of Solution

Given information:

Minimum wage rate is increased to 10%.

Calculation:

The new real wage rate is 11$\left(10\times 1.1\right)$. Thus, to calculate the new labor demand, the respective values are substituted in the given equation:

$\begin{array}{c}\text{L}=\frac{1,000}{27}\text{27,000}\times {\left(\text{11}\right)}^{-3}\\ =37.037037\times 27,000\times 7.513148\\ =751.3\end{array}$

The profit-maximizing firm hires 751.3 workers at 11 wage rates.

To calculate the new total output, substitute the respective values in Equation (1).

$\begin{array}{c}\text{Y}=5{\text{K}}^{\frac{1}{3}}{\text{L}}^{\frac{2}{3}}\\ =5{\left(27,000\right)}^{\frac{1}{3}}{\left(751.3\right)}^{\frac{2}{3}}\\ =12,397\end{array}$

Total output is 12,397 units.

(d)

To determine

Explain whether the goal of congress will succeed or not.

Explanation of Solution

Among 1000 labor forces, 751.3 laborers earn one more unit of output, but at the same time, 24.7 laborers also experienced involuntary unemployment.  Thus, the law of congress makes an average impact.

(e)

To determine

Explain whether this analysis is a good way to think about the minimum wage law.

Explanation of Solution

As per this analysis, introduction of this law increased the wage rate but also lead to structural unemployment.