Question

Workers are compensated by firms with "benefits" in addition to wages and salaries. The most prominent benefit offered by many firms is health insurance. Suppose that in 2000 , workers at one steel plant were paid $$\$ 20$$ per hour and in addition received health benefits at the rate of $$\$ 4$$ per hour. Also suppose that by 2010 workers at that plant were paid $$\$ 21$$ per hour but received $$\$ 9$$ in health insurance benefits. LO17.1 a. By what percentage did total compensation (wages plus benefits) change at this plant from 2000 to \(2010 ?\) What was the approximate average annual percentage change in total compensation? b. By what percentage did wages change at this plant from 2000 to \(2010 ?\) What was the approximate average annual percentage change in wages? c. If workers value a dollar of health benefits as much as they value a dollar of wages, by what total percentage will they feel that their incomes have risen over this time period? What if they only consider wages when calculating their incomes? d. Is it possible for workers to feel as though their wages are stagnating even if total compensation is rising?

Short answer

Total compensation increased by 25%, while wages increased by 5%. If workers value benefits equally, they'd feel a total income increase of 25%, but only 5% if considering wages alone. Wage stagnation feelings could arise despite rising total compensation.

Step-by-step solution

Calculate Total Compensation for 2000 and 2010

Total compensation is the sum of wages and benefits. For 2000, the total compensation is \(20 + 4 = 24\) dollars per hour. For 2010, it is \(21 + 9 = 30\) dollars per hour.

Calculate Percentage Change in Total Compensation

Use the formula for percentage change: \( \text{Percentage Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\% \). Apply this for total compensation from 2000 to 2010: \( \frac{30 - 24}{24} \times 100\% = 25\% \).

Calculate Average Annual Percentage Change in Total Compensation

The average annual percentage change can be approximated using the formula for compound annual growth rate (CAGR): \( \text{CAGR} = \left(\frac{\text{Ending Value}}{\text{Beginning Value}}\right)^{\frac{1}{\text{Number of Years}}} - 1 \). Hence, \( \left(\frac{30}{24}\right)^{\frac{1}{10}} - 1 \approx 0.0223 \) or \( 2.23\% \) per year.

Calculate Percentage Change in Wages Only

Using the same percentage change formula as in Step 2, for wages only: \( \frac{21 - 20}{20} \times 100\% = 5\% \).

Calculate Average Annual Percentage Change in Wages

Apply the CAGR formula for the wages: \( \left(\frac{21}{20}\right)^{\frac{1}{10}} - 1 \approx 0.00488 \) or \( 0.49\% \) per year.

Determine Workers' Perceived Income Change Including Benefits

If workers value benefits equally to wages, the total perceived income change is the total compensation change, which is \( 25\% \).

Determine Workers' Perceived Income Change Excluding Benefits

If workers consider only wages, their income would appear to have increased by \( 5\% \) over the period.

Discuss Wage Stagnation Despite Rising Compensation

Workers could feel wage stagnation because while their total compensation increased by \( 25\% \), their actual take-home (wages) only saw a \( 5\% \) increase. This week wage growth may be perceived as stagnation, especially given the greater increase in non-cash benefits.

Key concepts

Wage Stagnation

Wage stagnation is a concept referring to the slow or no growth in employee wages over a certain time period. In the given example, between 2000 and 2010, workers at a steel plant experienced only a 5% increase in their hourly wage, from $20 to $21. Despite an increase in total compensation including health insurance benefits, employees might feel as if their wages are stagnating. This perception occurs because their actual cash take-home pay has only marginally increased.

When we factor in things like inflation, the cost of living, and other economic changes, even a small percentage increase like 5% might not be enough to maintain the purchasing power of 2000 in 2010. This makes wage stagnation a significant issue, as workers need their wages to grow at a pace that sustains their lifestyle over the years.

• Slow growth in hourly wages
• Purchasing power considerations
• Impact on lifestyle

Health Insurance Benefits

Health insurance benefits are an essential part of employee compensation packages. These benefits can include coverage for medical expenses, dental care, and more, and play a crucial role in enhancing the overall compensation package a worker receives. In the scenario presented, the workers' health insurance benefits increased significantly from $4 to $9 per hour between 2000 and 2010.

While direct wages only increased by $1 per hour, the benefits package saw a notable rise. This highlights the shift some companies make by investing more in employee benefits rather than direct wages. For employees who value a dollar of benefits the same as a dollar of wages, this change can be seen as a significant increase in their overall income. However, if employees only focus on the wage component, they might not fully appreciate this increase.

• Increase in health benefits from $4 to $9 per hour
• Role in total compensation
• Impact on perceived income

Percentage Change Calculations

Percentage change calculations are fundamental for understanding how amounts have evolved over time. The percentage change formula: \[\text{Percentage Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\%\]is applied twice in the example provided — once for total compensation and once for wages.

For total compensation, which increased from \(24 to \)30 per hour, the percentage change is calculated as:\[\frac{30 - 24}{24} \times 100\% = 25\%\]This indicates a 25% rise in total compensation. Conversely, when calculating wages only (\(20 to \)21 per hour), the percentage change is:\[\frac{21 - 20}{20} \times 100\% = 5\%\]This shows a smaller increase of just 5%, helping illustrate the disparity between wage growth and overall compensation growth.

• Formula usage
• Application in total compensation
• Application in wage calculation

Compound Annual Growth Rate (CAGR)

The Compound Annual Growth Rate (CAGR) is used to measure the mean annual growth rate over a specific time period longer than one year. It's a useful formula because it accounts for compounding, offering a smoothed average growth rate.

The formula for CAGR is:\[\text{CAGR} = \left(\frac{\text{Ending Value}}{\text{Beginning Value}}\right)^{\frac{1}{\text{Number of Years}}} - 1\]For total compensation at the plant, the CAGR from 2000 to 2010 is:\[\left(\frac{30}{24}\right)^{\frac{1}{10}} - 1 \approx 2.23\% \text{ per year}\]This indicates that on average, the total compensation grew by about 2.23% annually. For wages, using the same formula from \(20 to \)21, the CAGR is:\[\left(\frac{21}{20}\right)^{\frac{1}{10}} - 1 \approx 0.49\% \text{ per year}\]Such analysis highlights the slow pace of wage growth compared to overall compensation growth, even when looked at annually.

• Smooths over annual fluctuations
• Provides average growth rates
• Comparison of total compensation vs wage growth