Problem 37 - Calculus Early Transcendentals

Question

The U.S. government reports the rate of inflation (as measured by the Consumer Price Index) both monthly and annually. Suppose that, for a particular month, the monthly rate of inflation is reported as \(0.8 \%\). Assuming that this rate remains constant, what is the corresponding annual rate of inflation? Is the annual rate 12 times the monthly rate? Explain.

Step-by-Step Solution

Answer

Answer: The corresponding annual inflation rate is approximately 9.97%.

Step 1: Understand the problem

The problem states that the monthly inflation rate is 0.8% and the rate remains constant throughout the year. We need to find the corresponding annual inflation rate.

Step 2: Convert to decimal

Convert the monthly inflation rate from percentage to decimal format. 0.8% = 0.008

Step 3: Calculate the annual inflation rate

Use the compound interest formula to calculate the annual inflation rate: \(A = P(1 + r)^n\) Where: A = the future value of the investment/loan, including interest P = the principal investment amount (the initial deposit or loan amount) r = the annual interest rate (decimal) n = the number of times that interest is compounded per time period For our problem, P = 1, r = 1 + the monthly inflation rate, and n = 12 (number of months in a year). We will solve for A: \(A = 1(1 + 0.008)^{12}\)

Step 4: Compute the result

Calculate the value of annual rate: \(A = (1 + 0.008)^{12}\) \(A ≈ 1.0997\) Now, convert the result to a percentage and subtract 100% to get the annual inflation rate: Annual inflation rate ≈ (1.0997 - 1) × 100% ≈ 9.97%

Step 5: Compare the rates

If we naively multiply the monthly rate by 12, we get: 0.008 × 12 = 0.096 or 9.6% Comparing these two values, we see that the annual rate calculated using the compound interest formula is 9.97%, which is not equal to 9.6%. So, the annual rate is not 12 times the monthly rate.

Step 6: Conclusion

The corresponding annual inflation rate, assuming a constant monthly rate of 0.8%, is approximately 9.97%. The annual rate is not simply 12 times the monthly rate.

Key Concepts

Compound Interest Formula

The compound interest formula is a powerful tool for calculating how quantities grow over time when they accrue interest on both the initial principal and the accumulated interest from previous periods. This concept is crucial when computing the annual inflation rate when only a monthly rate is provided. To use this formula, you should first convert the percentage rate into a decimal. For instance, a monthly rate of 0.8% becomes 0.008 in decimal form.

The general structure of the compound interest formula is:

• \[ A = P(1 + r)^n \]
• Where \( A \) is the amount after the interest is applied, \( P \) is the principal amount, \( r \) is the interest rate per period in decimal, and \( n \) is the number of compounding periods.

In the context of inflation, \( P \) could be considered as 1, since we're often calculating the percentage of increase, and the rate \( r \) represents the monthly rate converted to decimal. For an annual calculation, \( n \) would be set to 12, corresponding to twelve months in a year. Understanding this formula helps us appreciate how different rates and compounding periods affect growth.

Consumer Price Index

The Consumer Price Index (CPI) is an essential economic indicator that measures changes in the price level of a basket of consumer goods and services. Essentially, it reflects the purchasing trends and inflation experienced by households. By tracking the CPI, analysts can determine the inflation rate—a critical measure for gauging economic health.

CPI is used primarily because it provides a comprehensive indication of price changes over time. It consists of:

• A fixed basket of goods and services.
• Regular price checks across various retailers and service providers.
• Statistical weights that represent the importance or share of each item in total consumer spending.

When the CPI rises, it indicates an increase in inflation, meaning goods and services are more expensive now compared to the past. Conversely, a decrease in CPI suggests deflation. Annual and monthly inflation rates derived from the CPI help inform government policy, business decisions, and economic planning. Understanding CPI is crucial for anyone studying how inflation impacts economies.

Monthly Inflation Rate

The monthly inflation rate is an expression of how the price level for a typical basket of goods and services changes from one month to the next. It's an essential metric because it provides frequent updates on inflation trends, allowing businesses and policymakers to make timely decisions.

When reported, the monthly inflation rate is typically given as a percentage. For example, a monthly inflation rate of 0.8% means that, on average, prices increased by 0.8% over the previous month. This measure is different from the annual inflation rate, which examines how prices have changed over a full year.

To convert a monthly rate to an annual rate using the compound interest concept, we assume the monthly rate remains constant throughout the year. However, it's important to remember that multiplying the monthly rate by 12 does not yield the correct annual inflation rate. Instead, we use the compound interest formula to accurately reflect the compound nature of inflation, leading to a slightly higher effective annual inflation rate. Understanding monthly inflation helps individuals and businesses alike to budget, plan, and anticipate changes in purchasing power.