Inflation and Percent Change

Inflation is a percent change in a price index. A pay rise is a percent change in a wage. A grocery item’s jump is a percent change in that sticker. None of them is automatically the others. “Real” change is the attempt to look at one of them after taking an index into account, so that a bigger number of euros is not confused with more loaves of bread.

Nominal change first

Do not skip the plain formula. A rent that moves from €900 to €990 rose by 90/900 = 10%. That 10% is nominal. It is what your bank transfer does. It is true whether or not a statistics office published anything that month. The percentage change calculator is this step. Finish it before you adjust anything.

A price index is also a percent change

Suppose a consumer price index goes from 110 to 114.6. Inflation over the period is (114.6 − 110) / 110 = 4.18%. Agencies often publish that percent for you, which saves the division and does not change its meaning. The index level is a made-up base (often 100 in some year). Only the percent change, or the ratio of two levels, is comparable.

To express an old euro amount in the purchasing power of a later index:

later equivalent = old amount × (later index / earlier index)

€900 in the earlier period, indexes 110 then 114.6: 900 × (114.6/110) ≈ €937.64. That is the rent that would have merely kept up with this index. The actual new rent is €990, which is 990 / 937.64 − 1 ≈ 5.6% above the inflation-adjusted old rent. You can also divide factors: the rent factor is 1.10, the price factor is 114.6/110 = 1.0418, and 1.10 / 1.0418 − 1 ≈ 5.6%. Same real increase. The rent got more expensive even in index terms, by about five and a half percent, on top of the inflation that was already in the air.

Subtracting rates

nominal − inflation ≈ real is the shortcut. 10% − 4.18% = 5.82%, close to 5.6%, and a bit high. The shortcut pretends the inflation applies to the old price only, not to the path between. The error is about the product of the two small rates (roughly 0.10 × 0.0418 = 0.4 percentage points), which is why textbooks say “fine when both rates are small.”

The third row is where subtraction becomes a different story. A 50% nominal jump with 20% inflation is a 25% real increase (1.50 / 1.20 = 1.25), not 30%. Hyperinflation makes the shortcut unusable. Household rates usually sit in the first rows, where you will not ruin a decision by subtracting — and you will look more careful if you divide.

The last row is a real decline: prices rose faster than the thing you are tracking. Wages do this in bad years. Calling an 8% raise “a raise” is nominally true and really incomplete if the index rose 10%.

Baskets are not your life

A national index averages thousands of prices with weights from a typical consumption survey. If your rent is 40% of spending and the index weights shelter at 15%, your personal inflation can be far from the headline even when the statisticians are doing their job. Compute the percent change of the two or three bills that dominate — rent, childcare, energy — with the ordinary formula, and let the index handle the rest of the basket as a benchmark rather than as a verdict.

Indexes also differ on purpose: consumer prices, producer prices, a single city’s housing index, a wage index. Dividing a wage by a producer-price index answers a question you may not have asked. Match the index to the story. For “can I buy the same household basket,” a consumer price index is the usual choice. For “did this house beat local property prices,” use a housing series. Do not divide a salary by a stock index and call it inflation.

Chained periods

Year 1 inflation 8%, year 2 inflation 3%. The two-year price factor is 1.08 × 1.03 = 1.1124, or 11.24%, not 11%. Adding the published annual rates is the shortcut again, and over many years it drifts. When you have index levels, ignore the annual headlines and divide the latest level by the earliest. When you only have annual rates, multiply the factors. This is the same “do not add successive percents” rule as in stacked discounts, wearing a macroeconomic coat.

When the adjustment is worth doing

Do it when you compare money across several years: a salary then and now, a project budget, a parent’s anecdote about what bread cost. Do it when a nominal raise is small enough that inflation might have eaten it. Skip it when both numbers are from the same week. A 10% off sale is not an inflation calculation.

A wage and a basket, side by side

In year 0 you earn €2,000 a month and a defined basket of groceries costs €400. In year 3 you earn €2,240 and the same basket costs €460.

• Wage factor: 2240/2000 = 1.12, so +12% nominal.
• Basket factor: 460/400 = 1.15, so +15% on that basket.
• Real wage against this basket: 1.12 / 1.15 − 1 ≈ −2.6%.

You were paid more euros and you can buy slightly less of that basket. A national CPI of +10% over the same years would tell a kinder story (1.12 / 1.10 − 1 ≈ +1.8%), because the official basket is not your groceries. Both can be published without one of them being a lie. Say which basket you divided by. If you only subtract (12 − 15 = −3) you are close to −2.6 and you should still show the factors when the rates leave the single digits.

Old menu prices people repeat at dinner (“it was 80 cents”) are the same exercise with worse memory. Write the two prices, run percent change, then decide whether an index for the intervening years exists and is relevant. Nostalgia is not a denominator.

Keep the nominal percent in the sentence even after you compute the real one. “Rent rose 10% on the lease, about 5.6% after this index” is two facts. Dropping the 10% hides what the contract did. The index explains the contract. It does not replace it.