Chapter 5.7, Problem 71AYU

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 If the average annual inflation rate is $3.1\%$, how long will it take for the CPI index to double? (A doubling of the CPI index means purchasing power is cut in half.)

To determine

To find: Consumer Price Index If the average annual inflation rate is $3.1\%$, how long will it take for the CPI index to double? (A doubling of the CPI index means purchasing power is cut in half).

Answer to Problem 71AYU

Solution:

$\text{22}\text{.7}$ years

Explanation of Solution

Given:

$CP{I}_0=P, CPI=2P, r=3.1\%$ and $n=5$

Calculation:

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

$2P= P{\left(1+\frac{3.1}{100}\right)}^n$

$2={\left(1.031\right)}^n$

$\ln 2 =\ln {\left(1.031\right)}^n$

$\ln 2=n\ln 1.031$

$n=22.7$ years