An economist tracks the Consumer Price Index (CPI), a dimensionless indicator of average consumer prices. On January 1 of a particular year, the CPI is 180. Two years later, the CPI has increased to 198. The economist writes in a report:
“The instantaneous rate of change of the CPI is proportional to the current CPI value.”
Let P(t) denote the CPI t years after January 1 of the initial year.
(1) Translate the economist’s statement into an appropriate differential equation for P(t).
(2) Determine the proportionality constant k by using the CPI data provided for t=0 and t=2. Express k as a decimal rounded to four significant figures.
(3) Using your model, compute the annual percentage rate of inflation implied by the value of k. Clearly explain how this rate is obtained.
(4) Predict the CPI five years after the initial date. Then interpret what this value indicates about overall price changes during the five-year period.