There is a tendency for the purchasing power of money to decrease over time; this is called inflation. Sometimes the rate of inflation will change rapidly, but often it stays roughly constant for several years. Inflation follows the same model as continuous compounding.

For example, let’s suppose inflation rate is 5% from 2012 to 2017. If something cost $100 in January 2012, then 5 years later, in January 2017, we expect it to cost $100×(1.05)⁵ = $127.63. Of course, individual items do not increase at a uniform rate, but this is a useful approximate guide to the cost of living.

As inflation is not constant, governments often calculate tables to show the purchasing power of today’s dollar in earlier years. In the United States, the Consumer Price Index (CPI) is calculated each month by finding the cost of a standard set of items (food, housing, vehicles, and so on). There are in fact several CPIs constructed. We shall always refer to the CPI-U, an index that reflects the cost of living in urban areas (about 80% of America). There is also a CPI-W, for wageearners,  and there are other indices. Tables of the CPI-U are available online at

                                      http://stats.bls.gov/cpi/#tables.

The total CPI is divided by the average for 1982–1984 (the base period),  and multiplied by 100. For example, the CPI for February 2006 was 198.7, so a collection of goods that cost $198.70 in February 2006 would have cost an average of about $100 in the base period. The average for a year is also published; the average for 1988 was 118.3; the figures for May and June were 118.0 and 118.5 respectively.

This can be used to compare two different years. The CPI for June 2002 was 179.9. The ratio 179.9/118.5 = 1.518... provides a comparison between June 2002 and June 1988 prices: if something cost $1,000 in 1988, our best guess is that it would cost about $1,518 in 2002. These figures are approximate, because the prices of different items do not increase at the same rate. However, it is reasonable to say “the cost of living was about 50% higher in 2002 than in 1988.” A speaker in 2002 might say, “A dollar today is worth about two-thirds of what it was worth in 1988.”

Suppose the cost of a major item at time A is $XA. Suppose the CPI at time A is CA, and at time B it is CB. Then your estimate of the cost at time B is

                                                     (C_B /C_A)  XA.

Sample Problem 1.1 A house cost $150,000 in June 1988. What would you expect a similar house to have cost in February 2006?

Solution. If the house cost $150,000 in mid-1988, it is equivalent to a house that cost 198.7 118.5 × $150,000 = $251,518.99 in February 2006. Your realistic answer might be “about $250,000.”

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