Is a \$15 hourly wage today better than \$10 an hour twenty years ago? It depends on how much prices rose in between. The Inflation, Deflation and Disinflation and the price index let you compare dollars from different years.
Key idea
To express an amount in base-year dollars, divide by the index over 100: $$\text{real value} = \dfrac{\text{nominal value}}{\text{index}/100}$$ To move an amount from year A's dollars into year B's dollars: $$\text{amount}_B = \text{amount}_A \times \dfrac{\text{index}_B}{\text{index}_A}$$
Why it works
An index of 150 means prices are 1.5 times base-year prices. So a \$30 wage at that time buys what \$20 bought in the base year: $30 \div 1.5 = 20$. Dividing takes the price rise out. Multiplying instead would put a second dose of inflation in, making the old wage look even bigger, which is backwards. The ratio form works the same way: it scales by how much prices changed between the two years.
Worked examples
Worked example: Base-year dollars
A worker earns \$48,000. The price index is 120. What is the salary in base-year dollars?
- $\dfrac{48{,}000}{120/100} = \dfrac{48{,}000}{1.2} = 40{,}000$.
- The salary buys what \$40,000 bought in the base year.
Answer: \$40,000
Worked example: Then and now
A movie ticket cost \$4 in 1990, when the index was 130. The index is 260 today. What is the 1990 price in today's dollars?
- Prices doubled: $\dfrac{260}{130} = 2$.
- $4 \times \dfrac{260}{130} = 8$ dollars.
- If tickets now cost more than \$8, they rose faster than prices in general.
Answer: \$8
Worked example: Did the raise keep up?
A wage went from \$15 (index 100) to \$18 (index 125). Did the real wage rise?
- In base-year dollars: $\dfrac{18}{1.25} = 14.40$.
- $14.40 < 15$: the real wage fell, even though the paycheck number grew 20%.
Answer: The real wage fell
Common mistake
Comparing amounts from different decades in nominal dollars, or multiplying by the index when you should divide. Check the direction: if prices rose since year A, an old amount should become a bigger number in today's dollars, and a recent amount a smaller number in old dollars.