Question

Consumer Price Index (a) The CPI was 229.39 for 2012 and 243.80 for 2017 . Assuming that annual inflation remained constant for this time period, determine the average annual inflation rate. (b) Using the inflation rate from part (a), in what year will the CPI reach \(300 ?\)

Short answer

Total inflation over the period.Step 3: Determine the number of years.:The time period in question is from 2012 to 2017, which is 5 years.

Step-by-step solution

Understanding CPI and Inflation

The Consumer Price Index (CPI) measures the average change in prices over time that consumers pay for a basket of goods and services. Inflation is the rate at which the general level of prices for goods and services rises, causing purchasing power to fall.

Calculate the Total Inflation Over the Period

Use the formula for total inflation:

Calculate the Total Inflation Over the Period

Use the formula for total inflation:

Key concepts

average annual inflation rate

The average annual inflation rate is a key concept in understanding how the cost of goods and services changes over time. It gives us a consistent year-over-year measure of price increase.
To calculate the average annual inflation rate, we use the Consumer Price Index (CPI) values from two different years.
Suppose we have CPI for 2012 as 229.39 and for 2017 as 243.80. The formula to calculate the average annual inflation rate is as follows:
\[ \text{Average Annual Inflation Rate} = \text{ \left( \frac{ \text{CPI in current year} }{ \text{CPI in past year} } \right)^{ \frac{1}{ \text{number of years} } } - 1 } \times 100 \]
Plugging in our values, we get:
\[ \left( \frac{ 243.80 }{ 229.39 } \right)^{ \frac{1}{5} } - 1 } \times 100 = 1.23\text{\text{%}\text{\}\text{}} \]
This means that on average, prices increased by 1.23% annually between 2012 and 2017.

total inflation calculation

Total inflation over a period shows the cumulative effect of annual inflation rates over a span of years.
It is calculated by comparing the CPI at the start and end of the period without considering the average year-wise changes.
The formula to calculate total inflation is:
\[ \text{Total Inflation} = \left( \frac{ \text{CPI in current year} - \text{CPI in past year} }{ \text{CPI in past year} } \right) \times 100 \]
For our given example, the total inflation from 2012 to 2017 will be:
\[ \left( \frac{243.80 - 229.39}{229.39} \right) \times 100 = 6.29% \]
This means that the prices increased by a total of 6.29% from 2012 to 2017.
While this doesn't show how prices changed each year, it provides an overview of price changes over a more extended period.

future value prediction

Predicting the future value of the Consumer Price Index (CPI) involves estimating how long it will take for the CPI to reach a certain value using a constant annual inflation rate.
Suppose we want to predict in what year the CPI will reach 300, given an average annual inflation rate of 1.23%.
We use the formula:
\[ \text{Future Value (FV)} = \text{\text{Current CPI} \times (1 + \text{average annual inflation rate})}}^\text{\text{years}} \]
Set FV to 300 and solve for the number of years:
\[ 300 = 243.80 \times (1 + 0.0123)^{ \text{years} } \]
Taking natural logarithms on both sides, we get:
\[ \text{\text{years}} = \frac{ \text{ln}(300) - \text{ln}(243.80) }{ \text{ln}(1 + 0.0123) } \]
Simplifying, we find:
\[ \text{\text{years}} \approx 16 \]
This means it will take about 16 years from 2017 for the CPI to reach 300, or around the year 2033.
Understanding these calculations helps one appreciate how inflation impacts purchasing power and prices of goods over time.