Problem

The Consumer Price Index (CPI) indicates the rela…

Problem 69 Easy Difficulty

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 January 2006 was $198.3 .$ This means that IOO in the period $1982-1984$ had the same purchasing power as $\$ 198.30$ in January $2006 .$ In general, if the rate of inflation averages rper annum over $n$ years, then the CPI index after n years is
$$
\mathrm{CPI}=\mathrm{CPl}_{0}\left(1+\frac{r}{100}\right)^{n}
$$
where $C P I_{0}$ is the CPI index at the beginning of the $n$ -year period.
(a) The CPI was 163.0 for 1998 and 215.3 for $2008 .$ 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 $300 ?$

Answer

(a) 2.82% (b) In 2020 or after 12 yr

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Chapter 5

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Section 7

Financial Models

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Video Transcript

So this problem We're looking at another linear application. We were actually told in this case, right that we're talking about things in terms of tea, right where t represents our time, the number of years since 1980. And our output represents our C p i. R. Consumer price index. So one of things that we know is it was in 1983 and again, we're talking about years since 1980. So three years after 1980 the CPI I was 100 in 19. I'm sorry in 2008 So 2000 and eight. And so think about this, right? If we want to know how many years it was 28 years from 1980 the index would be to 15.3. Now we have two points. And considering that it's approximately linear, we could write an equation to represent the situation. So we're gonna look at the changes of the wise over the changes of the exes, which will give us our slope. So in this case, I'm going to take to 15.3 and subtract 100 which is 1 15.3. Then I'm gonna take 28 minus three, which is 25. So that's going to give us an approximate slope of 4.612 So why equals 4.612 x plus b when he used the point here. So 100 equals 4.612 times three plus b mhm. So that's 100 is equal to 13.836 plus B. So 100 minus that. That 13 number is going to give us a B value. So our equation will be 4.612 x plus 86.164 This is the answer to the first part of the question. Second, part of the question is based on our answer. What was the predicted value of the C P in the year 2000? Well, we know that the year 2000 is 20 years from 1980. So if I plug in 20 into this equation, why equals 4.612 times 20 plus 86.164 so 4.612 times 20 plus 86.164 So that tells us the c p I in 2000 would have been 178.4 now. What compared to see actual estimate? The actual estimate was 1 72.2. That's pretty close. It's not perfect, but it's pretty close. So that's the answer to Part B and finally in part C. It's has described the rate at which the annual rate in which it's changing what it's saying is the c P I will increase 4.612 points per year now. Is that exact? No. It's a good approximation of what's happening, so 4.612 points increased per year.

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