The Consumer Price Index ($CPI$) is a statistical estimate of the change of prices of goods and services bought for consumption. It is generally calculated by collecting the prices of a sample of representative items over a specific period of time. It is then give by a function $C(t)$ of time. The inflation rate is the percentage of rate of change of the CPI: $$I(t)=\frac{1}{C}\frac{dC}{dt}$$
(a) Can you explain in words what the inflation rate is measuring? What would a positive inflation rate mean?
(b) Suppose that the function $$C(t)=-\frac{1}{5}t^3+3t^2+100$$ gives the CPI of an economy for $0 \leq t \leq 9$, where $t$ is measured in years and $t=0$ corresponds to the year 2004. Find the inflation rate in 2009.
Hi there, so I have no idea what I'm doing for part a... Any help would be great with regards to part a. I just don't get what it means? For part b, I just want to know that I'm doing the right thing. So what I did is I found $I(t) = \frac{\frac{3}{5}t^2+6t}{-\frac{1}{5}t^3+3t^2+100}$. Plugging in the $t=(2009-2004)=5$, I got a $\frac{1}{10}$ inflation rate. Is this the right answer? Should I write this answer as $ 10percent $? Thanks to anyone in advance :)
a) Inflation rate is measuring the change of the CPI in a period of time and comparing it against the CPI at the beginning of the period $\Delta t$:
$\frac {\frac {\Delta C}{\Delta t}}{C}=\frac{C(n+1)-C(n)}{C(n)}$
A positive rate means prices are increasing.
b) Calculate the CPI over time until year 2009 and calculate 2009 inflation:
CPI (2004) = 100
CPI (2005) = -1/5+3+100 = 102.8
CPI (2006) = -1.6+12+100 = 110.4
CPI (2007) = -5.4+27+100 = 121.6
CPI (2008) = -12.8+48+100 = 135.2
CPI (2009) = -25+75+100 = 150
Inflation (2009) = $\frac{CPI(2009)-CPI(2008)}{CPI(2008)}$=$\frac{14.8}{135.2}$=0.109467 or 10.95%