Chapter 5.7, Problem 69AYU

(a)

To determine

To calculate: The average annual inflation rate, if the CPI for $2012$ is $229.39$ and for $2017$

is $243.80$

Answer to Problem 69AYU

Solution:

The average annual inflation rate is $1.23\%$

Explanation of Solution

Given information:

The CPI for $2012$ is $229.39$ and for $2017$ is $243.80$

Formula used:

If the rate of inflation averages $r\%$ 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 $n$ year period.

Calculation:

The CPI uses the base period $2012$ for comparison (the CPI for this years is $229.39$).

The CPI for $2017$ was $243.80$

This means that $229.39$ in $2012$ had the same purchasing power as $243.80$ in $2017$

Using the formula for CPI, we get

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

$\Rightarrow 243.80=229.39{\left(1+\frac{r}{100}\right)}^5$

$\Rightarrow {\left(\frac{243.80}{229.39}\right)}^{\frac{1}{5}}=1+\frac{r}{100}$

$\Rightarrow \frac{r}{100}=1.01225-1$

$\Rightarrow r=1.23\%$

Thus, the average annual inflation rate is $1.23\%$

(b)

To determine

To calculate: The year when CPI will reach $300$, if the CPI for $2012$ is $229.39$ and the average annual inflation rate is $1.23\%$.

Answer to Problem 69AYU

Solution:

The CPI will reach $300$ in year $2034$ if the CPI for $2012$ is $229.39$ and the average annual inflation rate is $1.23\%$

Explanation of Solution

Given information:

The CPI for $2012$ is $229.39$ and the average annual inflation rate is $1.23\%$

Formula used:

The formula for CPI is

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

Calculation:

Considering the base year $2012$ with $CP{I}_0=229.39$ and the rate of inflation $1.23\%$ from part (a).

Let’s calculate $n$ for CPI $300$

Then, the CPI index will be

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

$\Rightarrow 300=229.39{\left(1+\frac{1.23}{100}\right)}^n$

$\Rightarrow \frac{300}{229.39}={\left(\frac{101.23}{100}\right)}^n$

$\Rightarrow \ln \left(\frac{300}{229.39}\right)=n\ln \left(\frac{101.23}{100}\right)$

$\Rightarrow n=\frac{\ln \left(\frac{300}{229.39}\right)}{\ln \left(\frac{101.23}{100}\right)}$

$\Rightarrow n=21.95\approx 22$

The CPI will reach $300$ after $22$ years from $2012$

Thus, the CPI will reach $300$ in year $2012+22=2034$.