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 for a monthly rate of 0.8% is approximately 9.93%. The annual rate is not equal to 12 times the monthly rate (9.6%) due to the compounding effect.

Step 1: Understand the given information

We are given the monthly inflation rate as \(0.8\%\). We need to find its corresponding annual rate, assuming the monthly rate remains constant.

Step 2: Convert the percentage rate to decimal

To convert the percentage rate to decimal, divide it by \(100\). So, the monthly rate in decimal form is \(0.008\).

Step 3: Calculate the annual rate using the compound interest formula

The formula for calculating the annual rate with compounding is given by \((1 + r_m)^{12} - 1\), where \(r_m\) is the monthly rate in decimal form. To find the corresponding annual rate, insert the monthly rate into the formula: \((1 + 0.008)^{12} - 1\)

Step 4: Calculate the result

Calculate the expression \((1 + 0.008)^{12} - 1\): \((1.008)^{12} - 1 \approx 0.0993\)

Step 5: Convert the annual rate back to percentage

To convert the annual rate back to percentage, multiply it by \(100\): \(0.0993 \times 100 \approx 9.93\%\)

Step 6: Compare the annual rate with 12 times the monthly rate

The annual rate is \(9.93\%\), and if we multiply the monthly rate by \(12\), we get \(0.8\% \times 12 = 9.6\%\). The annual inflation rate is not equal to 12 times the monthly rate, because the effect of compounding is taken into account when calculating the annual rate.

Step 7: Interpret the result

If the monthly inflation rate remains constant at \(0.8\%\), its corresponding annual inflation rate is approximately \(9.93\%\). The annual rate is not exactly 12 times the monthly rate (\(9.6\%\)) due to the compounding effect.

Key Concepts

Inflation Rate

The inflation rate is an important measure that lets us know how much prices are rising over time. When we talk about inflation, we're often referring to how much more you'll need to pay for goods and services compared to before. Understanding inflation is crucial because it affects buying power and the cost of living.

Inflation is generally reported as a percentage. For instance, if the inflation rate is 2%, this means prices have, on average, increased by 2% over a certain time period. It's a common way for governments and economists to keep track of how the economy is doing.

Inflation can be measured in different intervals:

• **Monthly inflation rate:** This tells us how much prices rise from one month to the next.
• **Annual inflation rate:** This shows the price change over a full year.

To calculate the annual inflation from the monthly rate, the effect of compounding is applied. This means we account for not just a simple multiplication, but for the way inflation builds upon each previous month.

Consumer Price Index

The Consumer Price Index (CPI) is a tool used to measure the average change in prices over time that consumers pay for goods and services. Essentially, it's a metric that helps us understand inflation from the perspective of everyday shopping.

The CPI includes a basket of goods and services that many people commonly use, such as food, clothing, rent, and transportation. Here’s why CPI is essential:

• It reflects changes in the cost of living.
• It's used to adjust salaries and pensions to maintain purchasing power.
• It helps policymakers make economic decisions.

Changes in CPI over a specified period indicate how much prices have shifted, serving as a proxy for inflation. If the CPI goes up significantly, it means your money buys less than it did before.

Percentage to Decimal Conversion

Converting percentages to decimals is an essential mathematical skill, especially when dealing with financial calculations, like those related to inflation and interest rates.

Here's how it works:

• Take the percentage value, for example, 0.8%.
• To convert it to a decimal, divide the percentage by 100. This shifts the decimal point two places to the left.
• For instance, 0.8% becomes 0.008 as a decimal.

This conversion is crucial in calculations involving growth or change over time, such as computing compound interest or determining annual inflation from a monthly rate. By working with decimals rather than percentages, the mathematical formulas become more manageable and less prone to error.