Problem 41 - Introductory Algebra for College Students

Question

b. Use your answer from part (a) to complete this statement: If workers are paid million available workers and \(-\) million workers \(-1\) will be hired. c. In 2007 , the federal minimum wage was set at \(\$ 5.15\) per hour. Substitute 5.15 for \(p\) in the demand model, \(p=-0.325 x+5.8,\) and determine the millions of workers employers will hire at this price. d. At a minimum wage of \(\$ 5.15\) per hour, use the supply model, \(p=0.375 x+3,\) to determine the millions of available workers. Round to one decimal place. e. At a minimum wage of \(\$ 5.15\) per hour, use your answers from parts (c) and (d) to determine how many more people are looking for work than employers are willing to hire.

Step-by-Step Solution

Answer

At an hourly wage of \$5.15, employers will hire approximately 2 million workers while there are about 5.7 million available workers in the market. Therefore, there are roughly 3.7 million more people looking for work than employers are willing to hire.

Step 1: Substitution in Demand Model

Substitute the minimum wage, which is \$5.15, into the demand model, \(p=-0.325x+5.8\). So, it becomes: \(5.15=-0.325x+5.8\).

Step 2: Solve for x in the Demand Model

To isolate 'x', first subtract 5.8 from both sides, then divide by -0.325. The equation is rearranged to: \(x=(5.8-5.15)/0.325\). After calculation, x is approximately 2. This means that employers will hire 2 million workers at a wage of \$5.15 per hour.

Step 3: Substitution into Supply Model

Use the same method to substitute \$5.15 into the supply model, \(p=0.375x+3\). This gives: \(5.15=0.375x+3\).

Step 4: Solve for x in the Supply Model

To isolate 'x', first subtract 3 from both sides, then divide by 0.375. So, the equation becomes: \(x=(5.15-3)/0.375\). After calculation, x is approximately 5.7. It means there are 5.7 million workers available at a wage of \$5.15 per hour.

Step 5: Determine the Difference

To find out how many more people are looking for work than employers are willing to hire, subtract the number of workers that employers will hire from the number of available workers. So, the calculation would be: \(5.7 - 2 = 3.7\).

Key Concepts

Demand Model Calculation

The demand model calculation is a tool used to determine how many workers employers are willing to hire at a given wage rate. This concept plays a crucial role in understanding labor markets, especially in relation to minimum wage impacts. In the exercise, employers decide on the number of workers to hire based on the demand model formula. For the scenario where the minimum wage is \(5.15 per hour, the demand model is expressed as:

• \( p = -0.325x + 5.8 \)

To find how many workers employers hire at this wage, we substitute \)5.15 for \(p\), giving us the modified equation:

• \( 5.15 = -0.325x + 5.8 \)

Rearranging the equation to find \(x\), which represents the millions of workers, involves the following steps:

• First, subtract 5.8 from both sides: \(5.15 - 5.8 = -0.325x\)
• Then, divide both sides by -0.325: \(x = (5.15 - 5.8)/-0.325\)
• Through this calculation, \(x\) equals about 2. This means 2 million workers are hired at this wage rate.

The demand model helps us understand employer choices in response to wage changes, indicating potential employment shifts with varying minimum wages.

Supply Model Calculation

Supply model calculation provides insight into the number of available workers at a certain wage rate. It is vital in examining labor supply dynamics in the context of minimum wages. In this exercise, the supply model helps us figure out how many workers would be willing to work at the \(5.15 wage rate. The supply model equation is:

• \( p = 0.375x + 3 \)

By substituting \)5.15 for \(p\), you obtain the amended equation:

• \( 5.15 = 0.375x + 3 \)

To solve for \(x\):

• Subtract 3 from each side: \(5.15 - 3 = 0.375x\)
• Divide the result by 0.375: \(x = (5.15 - 3)/0.375\)
• Calculating this gives \(x\) approximately 5.7, meaning 5.7 million workers are ready to work at this pay rate.

This equation reflects the worker supply available under given wage conditions, illustrating the potential labor supply surplus when wages reach specific levels.

Worker Availability and Employment

The analysis of worker availability and employment is essential to identify potential mismatches in the labor market. It directly correlates with understanding how many people are looking for jobs and how many positions are offered by employers at the minimum wage level.Based on our calculations, there are discrepancies between the number of workers willing to work and the number hired. At a wage of $5.15 per hour:

• Employers hire 2 million workers as determined by the demand model.
• There are 5.7 million workers available as calculated via the supply model.

To find the imbalance, you subtract the number of workers employed from those available:

• \(5.7 - 2 = 3.7\)

Thus, 3.7 million more people are seeking jobs than there are positions available at this wage. Such disparities provide critical insights into labor market dynamics, showing how a set minimum wage can create excess labor supply, affecting unemployment rates and economic stability.