Problem 31 - Basic Statistics for Business and Economics with Student

Question

The Consumer Price Index is reported monthly by the U.S. Bureau of Labor Statistics. It reports the change in prices for a market basket of goods from one period to another. The index for 1994 was 148.2 , by 2006 it increased to 200.6 . What was the geometric mean annual increase for the period?

Step-by-Step Solution

Answer

The geometric mean annual increase is approximately 2.52%.

Step 1: Understand the Problem

We are given the Consumer Price Index (CPI) for two years, 1994 and 2006. The CPI for 1994 is 148.2 and for 2006 is 200.6. We need to calculate the geometric mean annual increase over this period.

Step 2: Determine the Formula

The formula for the geometric mean annual increase rate from the initial year to the final year is given by:\[\text{Geometric Mean Rate} = \left( \frac{\text{Final CPI}}{\text{Initial CPI}} \right)^{\frac{1}{n}} - 1\]where \(n\) is the number of years between the two periods.

Step 3: Find the Number of Years

The period from 1994 to 2006 spans 12 years. This will be the value of \(n\) in our formula.

Step 4: Plug Values into the Formula

Substitute the known values into the formula. The initial CPI is 148.2, the final CPI is 200.6, and \(n = 12\):\[\text{Geometric Mean Rate} = \left( \frac{200.6}{148.2} \right)^{\frac{1}{12}} - 1\]

Step 5: Calculate the Division Inside the Bracket

Calculate \( \frac{200.6}{148.2} \):\[\frac{200.6}{148.2} = 1.35396\]

Step 6: Compute the Exponentiation

Calculate the power of the result from Step 5 raised to \( \frac{1}{12} \):\[1.35396^{\frac{1}{12}} \approx 1.0252\]

Step 7: Subtract 1

Now, subtract 1 from the result obtained in Step 6 to find the geometric mean rate:\[1.0252 - 1 = 0.0252\]

Step 8: Convert to Percentage

Convert the decimal result to a percentage by multiplying by 100:\[0.0252 \times 100 \approx 2.52\%\]

Step 9: Conclusion

The geometric mean annual increase rate of the Consumer Price Index from 1994 to 2006 is approximately 2.52%.

Key Concepts

Consumer Price Index

The Consumer Price Index (CPI) is a critical economic indicator used to measure changes in the price level of a market basket of consumer goods and services over time. The U.S. Bureau of Labor Statistics publishes it monthly to help understand inflation patterns. CPI tracks the average price change relative to a base period, showing where aspects of the cost of living might be rising or falling.

• A higher CPI indicates increased prices, suggesting inflation.
• A lower CPI shows reduced prices, suggesting deflation.

This index is pivotal for adjusting income eligibility levels for government benefits and planning salary adjustments.

Annual Increase

Understanding the annual increase in CPI helps in analyzing economic trends over specific periods. It shows how much the price levels for consumer goods and services have risen each year on average. In the exercise provided, we are interested in finding out how much the CPI has increased annually from 1994 to 2006.
The process involves calculating the average rate at which prices have increased within the given years. This information enables analysts to predict inflation trends and make economic decisions. Analyzing CPI over multiple years helps in understanding whether price increases are stable or erratic.

Geometric Mean Formula

The geometric mean formula is essential for finding the average rate of growth over time, especially when dealing with percentages like those found in CPI calculations. Unlike the arithmetic mean, which is used for adding numbers over time, the geometric mean adds a crucial layer by calculating the average factor by which numbers are multiplied.
The formula used for calculating the geometric mean annual increase rate is:

\[\text{Geometric Mean Rate} = \left( \frac{\text{Final CPI}}{\text{Initial CPI}} \right)^{\frac{1}{n}} - 1\]

Here, \(n\) represents the number of years, and the formula essentially balances out the compound growth effects over the selected period.

CPI Calculation

Calculating CPI changes over time entails using the geometric mean formula to determine how prices have evolved annually. To find the annual increase, follow these steps:

• Identify the initial and final CPI values.
• Calculate the number of years between the two periods.
• Plug the values into the geometric mean formula.

In our exercise, the values are an initial CPI of 148.2 in 1994 and a final CPI of 200.6 in 2006.

After finding the ratio \(\frac{200.6}{148.2}\) and taking it to the power of \(\frac{1}{12}\), we obtain the geometric mean rate. The outcome helps gauge the average annual change, which was determined to be approximately 2.52%.