Problem 50 - Applied Calculus

Question

The US Consumer Price Index (CPI) is a measure of the cost of living. The inflation rate is the annual relative rate of change of the CPI. Use the January data in Table \(1.24^{47}\) to estimate the inflation rate for each of years \(2007-2012\) $$\begin{array}{c|c|c|c|c|c|c} \hline \text { Year } & 2007 & 2008 & 2009 & 2010 & 2011 & 2012 \\ \hline \text { CPI } & 202.416 & 211.08 & 211.143 & 216.687 & 220.223 & 226.655 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

Inflation rates for 2008 to 2012: 4.27%, 0.03%, 2.62%, 1.63%, 2.92%.

Step 1: Define Inflation Rate Formula

The inflation rate for a year is computed as the percentage change in CPI from the previous year to the current year. The formula is \( \text{Inflation Rate} = \left( \frac{\text{CPI in Current Year} - \text{CPI in Previous Year}}{\text{CPI in Previous Year}} \right) \times 100 \% \).

Step 2: Calculate Inflation Rate for 2008

For 2008: \( \text{Inflation Rate} = \left( \frac{211.08 - 202.416}{202.416} \right) \times 100 \% = 4.27\% \).

Step 3: Calculate Inflation Rate for 2009

For 2009: \( \text{Inflation Rate} = \left( \frac{211.143 - 211.08}{211.08} \right) \times 100 \% = 0.03\% \).

Step 4: Calculate Inflation Rate for 2010

For 2010: \( \text{Inflation Rate} = \left( \frac{216.687 - 211.143}{211.143} \right) \times 100 \% = 2.62\% \).

Step 5: Calculate Inflation Rate for 2011

For 2011: \( \text{Inflation Rate} = \left( \frac{220.223 - 216.687}{216.687} \right) \times 100 \% = 1.63\% \).

Step 6: Calculate Inflation Rate for 2012

For 2012: \( \text{Inflation Rate} = \left( \frac{226.655 - 220.223}{220.223} \right) \times 100 \% = 2.92\% \).

Key Concepts

Consumer Price Index

The Consumer Price Index, often abbreviated as CPI, is an essential economic indicator. It measures the average change in prices paid by consumers for goods and services over time. Think of it as a giant shopping basket that includes various items—like food, clothing, gasoline, and housing costs—that people regularly purchase.

The CPI is represented as a number and is especially useful to understand inflation rates. For example, if one year the CPI is 200 and next year it’s 210, it means that overall, prices have increased, indicating inflation. It's important because it provides insights into the cost of living changes. When analyzing economic health, economists and policymakers use the CPI to make informed decisions.

Percentage Change

Percentage change is a mathematical concept used to compare the difference between two values. It describes the change in value as a percentage of the original value. To calculate it, you take the difference between the new and old values, divide that by the old value, and then multiply by 100.

In the context of the CPI, the percentage change helps to calculate the inflation rate. It shows how much prices have increased (or decreased) in percentage terms from one year to the next. This concept is particularly useful for understanding just how significant price movements are in terms that are easy to interpret, as percentages are universally understandable.

Cost of Living

The cost of living refers to the amount of money required to sustain a certain standard of living. It takes into account everyday expenses, such as food, housing, healthcare, and transportation. Changes in the cost of living are often linked to inflation rates, as inflation indicates that the same basket of goods and services costs more over time.

As the CPI is a measure of the cost of living, it reflects how inflation affects everyday expenditures. A rise in CPI indicates an increased cost of living, while a falling CPI suggests a decrease. Understanding the cost of living is important for wage earners because it impacts their purchasing power—how much they can buy with their income.

Annual Rate Change

The annual rate change measures the rate at which something changes over the course of a year. When dealing with inflation, the annual rate change specifically refers to the change in the inflation rate over a year, often calculated using the CPI.

This change indicates how fast or slow the cost of living is increasing or decreasing over a year. Using the annual rate change, individuals and businesses can make projections or set budget plans by understanding whether they can expect prices to rise or fall in the near future. Policymakers and economists also utilize this metric to make decisions about adjusting interest rates or other economic policies.