The Mathematics of Inflation: Why Prices Compound

When inflation falls from 9% back to 3%, many people expect prices to go back down. They don't. A lower inflation rate only means prices are rising more slowly, on top of everything that has already been added.

Inflation isn't a series of separate yearly bumps. It's compound growth applied to prices, and like all compound growth it gets big over long stretches of time.

"Low" Inflation Still Doubles Prices

A 2% or 3% inflation rate sounds harmless. Run it through an exponent and see what happens:

At 3% a year, which is only slightly above most central banks' 2% target, your cash loses more than half its purchasing power in 30 years. That's roughly the length of a career.

The Core Formula

If a price is P₀ today and inflation runs at rate i per year, then after t years:

P(t) = P₀ × (1 + i)^t

It's exactly the compound interest formula with the rate relabelled. Purchasing power works the other way, as exponential decay:

Real value = Nominal value / (1 + i)^t

That's why economists say inflation is a hidden tax on cash. It's a guaranteed negative compound return on money that sits still.

Why Rates Don't Add, They Multiply

Suppose prices rise 9% one year, then 3% the next. Total inflation isn't 12%:

1.09 × 1.03 = 1.1227 → 12.27%

The 3% in year two applies to prices that were already 9% higher. Over many years that small gap between adding and multiplying becomes large. To find the average yearly rate across several years, you need the geometric mean, not the ordinary average:

Average rate = (1.09 × 1.03)^(1/2) − 1 ≈ 5.96%

The arithmetic mean says 6.00%. The difference looks tiny, but the geometric mean is the correct number to compound forward.

The Rule of 70 (and 72)

For quick mental math, divide 70 (or 72) by the inflation rate to estimate how many years it takes prices to double:

Doubling time ≈ 70 / i

• At 2%: 70 / 2 = 35 years
• At 3.5%: 70 / 3.5 = 20 years
• At 7%: 70 / 7 = 10 years

It works because the exact doubling time is ln(2) / ln(1 + i), and for small i, ln(1 + i) ≈ i. Since ln(2) ≈ 0.693, the rule uses 69.3, rounded to 70 or to the more divisible 72. You can check the logarithms with the logarithm calculator.

Measuring Inflation: Index Numbers

Inflation is measured with a price index such as the U.S. Consumer Price Index (CPI), published monthly by the Bureau of Labor Statistics. The index tracks the cost of a fixed basket of goods relative to a base period.

The inflation rate between two dates is simply:

Inflation = (CPI_later / CPI_earlier) − 1

Two real data points: the CPI-U annual average was 38.8 in 1970 and 258.8 in 2020. Divide them and prices rose by a factor of about 6.67 over 50 years. Solve for the yearly rate:

6.67^(1/50) − 1 ≈ 3.87% per year

A dollar in 1970 bought what about $6.67 bought in 2020. And in June 2022, U.S. year-over-year CPI inflation hit 9.1%, its highest reading in more than four decades.

An Insider Reference: When Compounding Runs Wild

Economist Steve Hanke of Johns Hopkins University has catalogued episodes of hyperinflation, defined as monthly inflation above 50%. His estimate for Zimbabwe's peak in November 2008 was a monthly rate of about 79.6 billion percent, meaning prices doubled roughly every 24.7 hours.

Hyperinflation is the same exponential formula with a huge i. At 50% a month, prices grow by a factor of 1.5¹² ≈ 130 in a single year.

Milton Friedman famously argued that "inflation is always and everywhere a monetary phenomenon." Whatever you think of that claim, the arithmetic of how price increases stack on top of each other isn't up for debate.

Two Concepts Worth Knowing

Real vs. Nominal Returns

If your savings earn 5% while inflation is 3%, your real return isn't exactly 2%. The Fisher equation gives the precise answer:

(1 + real) = (1 + nominal) / (1 + inflation)
real = 1.05 / 1.03 − 1 ≈ 1.94%

Continuous Compounding

Economists often use log differences to measure inflation, because ln(P₁) − ln(P₀) adds cleanly across periods. Twelve monthly log changes sum to the annual log change, which regular percentages don't do.

Quick Answer: Why Does Inflation Compound?

Inflation compounds because each year's price rise is applied to prices that already include every earlier rise. After t years at rate i, prices equal P₀ × (1 + i)^t. At 3% a year, prices roughly double every 23 years.

Try Them Yourself

• Compound Interest Calculator: set the "interest" rate to an inflation rate
• Logarithm Calculator: compute exact doubling times
• Financial Formulas: present value and real-return formulas
• Statistics Formulas: arithmetic vs. geometric mean
• Retirement Calculator: see how inflation shapes a long-term plan
• Scientific Calculator: run (1 + i)^t for your own numbers

Pick a price you remember from 20 years ago, find today's price, and work backwards to the annual rate. It's a fun way to feel what compounding does.