Chapter 5.7, Problem 69AYU

Problems 69-72 require the following discussion. The consumer price index (CPI) indicates the relative change in price over time for a fixed basket of goods and services. It is a cost of living index that helps measure the effect of inflation on the cost of goods and services. The CPI uses the base period 1982-1984 for comparison (the CPI for this period is 100). The CPI for March 2015 was $236.12$. This means that $\$100$ in the period 1982-1984 had the same purchasing power as $\$236.12$ in March 2015. In general, if the rate of inflation averages $r$ percent per annum over $n$ years, then the CPI index after $n$ years is $CPI=CP{I}_0{\left(1+\frac{r}{100}\right)}^n$ where $CP{I}_0$ is the CPI index at the beginning of the $n\text{-year}$ period. Source: U.S. Bureau of Labor Statistics

Consumer Price Index

(a) The CPI was $214.5$ for 2009 and $236.7$ for 2014. 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?

To determine

To find:

a. The CPI was $214.5$ for 2009 and $236.7$ for 2014. Assuming that annual inflation remained constant for this time period, determine the average annual inflation rate.

Answer to Problem 69AYU

Solution:

a. $r=1.99\%$

Explanation of Solution

Given:

The CPI was $214.5$ for 2009 and $236.7$ for 2014.

Calculation:

a. The CPI was $214.5$ for 2009 and $236.7$ for 2014. Assuming that annual inflation remained constant for this time period, determine the average annual inflation rate.

$CP{I}_0=214.5, CPI=236.7$ and $n=5$

$CPI = CP{I}_0{\left(1+\frac{r}{100}\right)}^n$

$236.7 = 214.5{\left(1+\frac{r}{100}\right)}^5$

$1.1035={\left(1+\frac{r}{100}\right)}^5$

$\left(1+\frac{r}{100}\right)={\left(1.1035\right)}^{\frac{1}{5}}$

$1+\frac{r}{100}=1.0199$

$\frac{r}{100}=0.0199$

$r=1.99\%$

To determine

To find:

b. Using the inflation rate from part (a), in what year will the CPI reach 300?

Answer to Problem 69AYU

Solution:

b. $\text{17}\text{.03}$ years

Explanation of Solution

Given:

The CPI was $214.5$ for 2009 and $236.7$ for 2014.

Calculation:

b. What year will the CPI reach 300?

$CP{I}_0=214.5, CPI=300\text{and}r=1.99$

$CPI = CP{I}_0{\left(1+\frac{r}{100}\right)}^n$

$300=214.5{\left(1+\frac{1.99}{100}\right)}^n$

$1.3986= {\left(1+\frac{1.99}{100}\right)}^n$

$\ln 1.3986=\ln {\left(1+\frac{1.99}{100}\right)}^n$

$\ln 1.3986=n\ln \left(1+\frac{1.99}{100}\right)$

$\ln 1.3986=n\ln 1.0199$

$n=17.03$