Inflation Rate

In one sentence: The inflation rate says how much more prices rose this year than last, as a share of last year’s prices, measured with a price index.

$\pi = \frac{\text{CPI}_{t} - \text{CPI}_{t-1}}{\text{CPI}_{t-1}}$

Read it out loud: “Pi equals CPI at time t, minus CPI at time t minus one, all divided by CPI at time t minus one. CPI is the consumer price index, a single number that tracks the average price of the things households buy. The inflation rate is the change in that number, as a share of the old number.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Index numbers: An index number shows how a quantity changes over time, and it is usually not a price in dollars. Statisticians choose a base period, give it a starting value, and express every other period relative to it. The numbers used on this page, such as 200.0 and 206.0, are made up. Only the ratio between two index values matters, so an index has no unit, and a rise from 200.0 to 206.0 is the same kind of change as a rise from 2.00 to 2.06.
Consumer price index (CPI): The consumer price index, or CPI, is an index number that tracks the average price of a fixed basket of goods and services bought by households, such as food, rent, transport and clothing. Each period, statisticians price the same basket again and compare the total cost with a base period. The CPI measures the overall price level, not the price of any one item, so one item can move much more than the index does.
Subscripts like $t$ and $t-1$: A subscript is a small letter or number written below a symbol to say which version of it you mean. Here $t$ stands for the current period, such as this year, and $t-1$ means one period earlier, such as last year. So $\text{CPI}_{t}$ is the index this year and $\text{CPI}_{t-1}$ is the index last year. The minus one is ordinary arithmetic: it is one step back in time.
Percent change: A percent change compares a new value with an old one. Subtract the old value from the new one, divide by the old value to get a share, and multiply by 100 to read it as a percent. For example, going from 200 to 206 is a change of 6. Then 6 divided by 200 is 0.03, which is a change of 3%.

What each symbol means

Symbol	Meaning
$\pi$	The inflation rate, the change in the price level over one period, written as a decimal and not as a percent. In the example it is 0.03, which is 3% per year. It has no money unit, because it is a share of last period’s index, and it always means per period.
$\text{CPI}_{t}$	The consumer price index in the current period $t$ , such as this year. It is a pure number with no unit, read against a base period. In the example this year’s value is 206.0, which is a made-up figure chosen for illustration.
$\text{CPI}_{t-1}$	The consumer price index one period earlier, which is last year in the example. It is the old price level that the change is measured against. The example uses 200.0 for this value, again a made-up figure.
$t$	The current period, such as this year or this month. The formula looks at the step from the earlier period $t-1$ to the current period $t$ , so the rate always describes one period of change.

The formula piece by piece

$\text{CPI}_{t}$: Start with this period’s price index. It is the overall price level now, a pure number from the consumer price index, with no unit attached.
$\text{CPI}_{t-1}$: Last period’s price index, which is the starting level. The change is measured against it, so it appears twice in the formula, once in the top line and once at the bottom.
$\text{CPI}_{t} - \text{CPI}_{t-1}$: Subtract to find the change. The result is how many index points the price level moved, positive for a rise and negative for a fall.
$\frac{\text{CPI}_{t} - \text{CPI}_{t-1}}{\text{CPI}_{t-1}}$: Divide the change by the starting level. This turns the change into a share of last period’s prices, a decimal that you can read as a percent by multiplying by 100.
$\pi = \frac{\text{CPI}_{t} - \text{CPI}_{t-1}}{\text{CPI}_{t-1}}$: The whole formula in one sentence: the inflation rate equals the change in the price index divided by last period’s price index.
$1 + \pi = \frac{\text{CPI}_{t}}{\text{CPI}_{t-1}}$: The same idea written as a growth factor: this period’s index divided by last period’s index equals one plus the inflation rate.

What the formula is doing, and why

The inflation rate answers one question: how much did the general level of prices rise over one period? To answer it, we need a single number that stands for the overall price level. The consumer price index, CPI, is that number. It prices a fixed basket of everyday goods and services, and each period’s value is compared with a base. The formula then compares this period’s index with last period’s index.

Why subtract? The difference $\text{CPI}_{t} - \text{CPI}_{t-1}$ is how many index points the price level moved. A rise of 6.0 points tells us prices went up, and a fall would tell us prices went down. Subtraction turns two levels into a change, and the sign of the answer tells us the direction as well as the size.

Why divide by the old value? A 6.0 point rise means something different when the index starts at 200.0 than when it starts at 20.0. Dividing by last period’s index, $\text{CPI}_{t-1}$ , turns the change into a share of what prices were before, so the answer is the same kind of number whatever the base. This is why the rate is always measured against the starting point, not the end point.

Why the result is a decimal. Dividing a small number by a bigger one gives a fraction, such as 6.0 divided by 200.0, which equals 0.03. Multiplying by 100 reads it as a percent, 3%. Compare periods of the same length, because a year-to-year index gives a yearly rate and a month-to-month index gives a monthly one.

What the rate does not measure. The rate describes the average price level, so one item can rise much faster or slower than the rate. A rent rise of 10% in a basket where everything else stays flat moves the index only a little, because rent is one part of the basket.

An everyday picture

Think of a school canteen that sells the same set of snacks every week: a sandwich, a drink and a muffin. Each week a clerk adds up what the whole set would cost. If the set cost 200 coins last week and 206 coins this week, the set has become 6 coins more expensive, and 6 out of 200 is 3%. The clerk’s total is like the CPI, and the 3% is like the inflation rate for that week. The clerk does not care about the price of one muffin alone, only about the cost of the whole set compared with last week.

Worked example: How much did the grocery bill’s price level rise?

A family tracks its regular grocery shopping as a basket, and a statistics agency tracks the average price of baskets like it with the CPI. The index values in this example are made up: last year the index was 200.0, and this year it is 206.0. The family’s regular grocery trip cost 50 dollars last year. We want the inflation rate for the year, and then the new cost of the same trip.

Write the formula with the two index values. This year’s index, $\text{CPI}_{t}$ , is 206.0, and last year’s index, $\text{CPI}_{t-1}$ , is 200.0. Putting them into the formula gives $\pi = \frac{206.0 - 200.0}{200.0}$ , which is a change divided by the old value.
Find the change in index points. Subtracting last year’s index from this year’s index shows how far the index moved: $206.0 - 200.0 = 6.0$ points. The answer is positive, so prices went up over the year.
Divide the change by last year’s index. The 6.0 points are compared with the starting level of 200.0, so $\frac{6.0}{200.0} = 0.03$ . As a decimal, 0.03 means 3 parts of every 100 parts of last year’s price level, which is a rise of 3 in every 100.
Turn the decimal into a percent. Multiplying by 100 moves the decimal point two places to the right, so $0.03 \times 100 = 3\%$ . The inflation rate for the year is therefore 3%, which means general prices rose by 3% over the year.
Apply the rate to the grocery trip. The growth factor is one plus the rate, so 1.03 means 3% more. A trip that cost 50 dollars last year now costs $50 \times 1.03 = 51.5$ dollars, so the family pays 1.5 dollars more for the same goods.

Answer: The inflation rate is 3% per year in this example. The index rose 6.0 points, from 200.0 to 206.0, and the grocery trip that cost 50 dollars last year now costs 51.5 dollars. The index values are made up, while real CPI figures are published by statistics agencies.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Dividing by this year’s index instead of last year’s. The rate is measured against the starting level, so the bottom of the fraction must be $\text{CPI}_{t-1}$ . Dividing by 206.0 would give about 2.91%, which is not the rate this formula describes.
Reading the rate as the change in index points. The 6.0 points is a change in the index, not a percent. The rate is those points divided by last year’s index value.
Thinking inflation means prices are high. Inflation is the rate at which prices rise. Prices can be high while inflation is zero, when the index stays flat from one year to the next.
Forgetting to multiply by 100 when you want a percent, or to add 1 when you want the growth factor $1 + \pi$ for working out a new price.

Where you will meet it

Setting yearly raises in wage and pension agreements, which are often linked to a price index so that pay keeps its buying power.
Restating old prices, salaries and incomes in today’s money, so that figures from different years can be compared fairly.
Central banks watching how fast prices are rising when they decide on interest rates and other policy tools.
Household budgets, where the change in a grocery bill over one year gives a rough personal inflation rate for that family.

Questions people ask

Is inflation the same as prices being high?

No. Inflation is the speed at which the price level rises, so it is a rate per period. Prices can be very high while inflation is zero, if they have stopped rising. A flat price level has zero inflation, even when everything is expensive.

Why does the index start at a base value like 100?

The base is a reference point that makes the index easy to read. A value of 110 against a base of 100 means prices are 10% above the base period. The base is a convention that statistics agencies choose, and the inflation rate is the same whichever base is used, because the ratio of two index values does not change when the whole index is scaled.

Why do official inflation numbers differ from my own grocery bill?

The CPI averages a whole basket of goods and services, weighted by how much households spend on each item. Your own shopping may contain different items or buy them in different amounts, so your personal change can be higher or lower than the average.

Fisher Equation: The Fisher equation says the real growth of your money is what is left of your interest after prices have risen.
Present Value: Present value says how much money you must put aside today, at a steady interest rate, so that it grows to a known amount by a later date.
Compound Interest: Compound interest says a savings balance grows faster over time, because each round of interest is added to the balance and then earns interest of its own.