Measuring inflation — CPI and RPI — Economics

Before this

Real and nominal GDP
Percentage increase and decrease · coming soon
Mean, median and mode · coming soon

This leads to

Three official answers to one question

In November 2022, the United Kingdom's Office for National Statistics (ONS) published the inflation figures for October. It gave three numbers for the rise in prices over the previous 12 months. The Consumer Prices Index (CPI) said 11.1%. The Retail Prices Index (RPI) said 14.2%. A third index, CPIH, said 9.6%.

All three were built from the same prices, collected in the same month. Yet they differ by more than four percentage points. So which one is "the" inflation rate?

Inflation is not a number you can read from a price tag. It is built, and every step of the building involves a choice. This lecture follows those steps, and shows why the three numbers differ and why the difference matters for real money.

What inflation is, and what it is not

Inflation is a sustained rise in the general price level. The price level is the average price of the goods and services that households buy.

"General" means most prices, not one. If a poor harvest makes coffee dearer, that is a change in a relative price, not inflation. "Sustained" means it continues over months and years.

The inflation rate is the percentage change in the price level over 12 months. An inflation rate of 3% means that the same things cost about 3% more than they did a year ago.

So to measure inflation, you need one number for "the price level", every month. Nobody can record every price, so statisticians use a sample.

Step one: a fixed basket

Statisticians choose a basket of goods and services: a list of items that together represent what households buy. In the UK in 2026, the basket has 760 representative items. The ONS reviews it every year. For 2026 it added 27 items, such as alcohol-free beer, and removed 19 (ONS, March 2026).

Within a year, the basket is fixed: the same items, in the same quantities and of the same quality. If the contents changed, the cost of the basket would change even if no price had moved. A fixed basket means that a change in its cost measures only a change in prices.

Collectors record prices in shops around the country and online. In 2022, that meant around 180 000 price quotes a month (ONS). From February 2026, the ONS also uses supermarket scanner data for groceries.

Step two: index numbers

The cost of a basket in dollars is hard to compare across years. So the cost is turned into an index number. Choose a base period and give it the value 100. Every other period is measured against it:

index = (cost of the basket now \div cost of the basket in the base period) \times 100

An index of 108 means that the basket costs 8% more than in the base period. The UK's CPI uses 2015 as its base, so 2015 = 100. In October 2022 the CPI was 126.2 (ONS). On average, prices were about 26.2% higher than in 2015.

Worked Example 1 — From basket cost to inflation rate (invented)

A basket costs $250 in the base year, $270 one year later, and $283.50 the year after. Find the index numbers and the inflation rate in each year.

Base year: index 100, by definition.

Year 1: $\frac{270}{250} \times 100 = 108$ . Inflation in year 1 is 8%.

Year 2: $\frac{283.50}{250} \times 100 = 113.4$ .

Now be careful. The index rose by 5.4 points, from 108 to 113.4. But the inflation rate compares this year with last year, not with the base:

\frac{113.4}{108} = 1.05, so inflation in year 2 is 5.0%.

Check with the dollar costs: $\frac{283.50}{270} = 1.05$ . ✓ A rise of 5.4 index points is a rise of 5%, because each point is now a smaller part of a bigger index.

Step three: weights

Suppose bread and concert tickets both rise by 10%. Do they matter equally? No. Households spend far more on bread than on concerts, so a fair average must count each price change by how much households spend on it.

So each item gets a weight: its share of total household spending. In the UK's CPIH in 2026, food and non-alcoholic drinks together have a weight of 8.7% (ONS). The weights are updated every year, from surveys of what households actually buy.

Inflation from a weighted basket

If item $i$ has weight $w_{i}$ (its share of spending, with all the weights adding to 1) and its price changes by $p_{i} %$ , then

$inflation rate = w_{1} p_{1} + w_{2} p_{2} + \dots + w_{n} p_{n}$

Each term $w_{i} p_{i}$ is that item's contribution, in percentage points.

Why does this work? Suppose a household spends 30% of its budget on food, and food prices rise by 8%. The food part of the budget now costs 8% more. That adds $0.30 \times 8% = 2.4%$ to the cost of the whole budget. Each item adds its own share in the same way.

Worked Example 2 — A three-item basket (invented)

Households spend 30% on food, 40% on rent and 30% on transport. In one year, food prices rise by 8%, rents by 5% and transport by 12%. Find the inflation rate.

Food: $0.30 \times 8 = 2.4$ percentage points.

Rent: $0.40 \times 5 = 2.0$ percentage points.

Transport: $0.30 \times 12 = 3.6$ percentage points.

Inflation rate = 2.4 + 2.0 + 3.6 = 8.0%.

Check with a budget of $1000. Food: $300 → $324. Rent: $400 → $420. Transport: $300 → $336. Total: $1000 → $1080, which is 8% more. ✓ Notice that rent has the largest weight but the smallest contribution after food, because its price rose least.

This keeps a promise from the real and nominal GDP lecture. The GDP deflator weights prices by what the country produces, and leaves out imports. The CPI weights prices by what households buy, including imports. When imported fuel becomes dearer, the CPI feels it at once, and the deflator may hardly move.

Whose basket is it?

The CPI weights come from the total spending of all households added together. A household that spends twice as much has twice as much influence on the weights. (Statisticians call this "plutocratic" weighting.) So the official basket is the spending of the whole country. It is not the basket of any one real household.

A household that spends a large share on the fastest-rising items faces a higher inflation rate than the official one. Try it in the model below, with invented prices for a year in which energy prices jump.

Real-world model — your own inflation rate (invented prices)

Choose a household, or set the spending shares yourself. "Other" is whatever is left. Compare the household's inflation rate with the spending-weighted rate. Price changes this year: food +15%, energy +60%, transport +8%, rent +5%, other +4%.

Households:

Bars: each kind of spending's contribution, in percentage points. Dashed line: the spending-weighted rate.

In an invented year, prices rise as follows: food 15%, energy 60%, transport 8%, rent 5% and everything else 4%. Four households spend their money differently, so they face different inflation rates.

Household	Food, energy, transport, rent (%)	Rate
All, spending-weighted	11, 5, 14, 10	≈ 8.7%
Low-income renter	18, 10, 8, 25	≈ 12.2%
High-income commuter	9, 4, 20, 5	≈ 8.1%
Retired home-owner	15, 9, 10, 0	≈ 11.1%

Inflation rates for four invented households in the same invented year. The rest of each budget is "other" spending, which rose by 4%. Energy rose by 60%, so households that spend a large share on energy face much higher inflation.

What should you have seen? The low-income renter spends twice the official share on energy, so energy alone adds 6 percentage points to that household's rate. The commuter spends more on transport, which rose little, and faces less inflation than the official rate. The model leaves a lot out: its prices are invented, it has only five kinds of spending, and real households switch to cheaper goods when prices rise.

Real data show the same pattern. In October 2022, the ONS estimated CPI inflation of 11.9% for low-income households and 10.5% for high-income households, against 11.1% for all households. The main reason was the larger share of low-income budgets spent on energy and food (ONS, 2022).

Inside one item: the formula effect

Weights work for items, such as "rice". But one item has prices from many shops, and the ONS does not know how much each shop sells. So the prices inside one item are combined without weights, and the way they are combined matters a great deal.

This is the hard part of the lecture. Spend time here. Start with a prediction.

Predict, then reveal — has the price of rice risen?

A shop sells two kinds of rice, A and B, equally often. Last year A cost $2 and B cost $4. This year A costs $4 and B costs $2. By how much has the price of rice risen?

Last year the two kinds cost $2 and $4. This year they cost $4 and $2. The shop is selling the same two prices as before, so the price of rice has not risen. But an average of the two percentage changes, +100% and −50%, gives +25%. That is the flaw in the arithmetic mean of price ratios.

Why does averaging the percentage changes go wrong? Because percentages are not symmetric. A price that doubles has risen by 100%. A price that halves back has fallen by only 50%. Going up and coming back down by the same factor gives changes of different sizes, so their ordinary average is pushed upwards.

There are three common ways to combine the prices for one item. Each uses the price ratio, the new price divided by the old:

Average of the ratios (the Carli formula): $\frac{2 + 0.5}{2} = 1.25$ , so "up 25%".
Ratio of the average prices (the Dutot formula): average price $\frac{4 + 2}{2} = 3$ this year and $\frac{2 + 4}{2} = 3$ last year, so "no change".
Geometric mean of the ratios (the Jevons formula): $2 \times 0.5 = 1$ , so "no change".

The geometric mean multiplies the ratios and then takes the root. Multiplying treats "× 2" and "× ½" as exact opposites, so they cancel. That is it. That is the whole idea behind what statisticians call the formula effect.

The RPI uses the Carli and Dutot formulas. The CPI uses Jevons in place of Carli. According to the ONS (2018), the formula effect has on average added about 0.7 percentage points to annual RPI inflation compared with CPIH. The Carli formula is no longer international best practice. It was the main reason why the RPI lost its status as a National Statistic in 2013.

Stop and think

Could the arithmetic mean of a set of price ratios ever be smaller than their geometric mean?

Show the answer

No. For positive numbers, the arithmetic mean is always at least as large as the geometric mean. The two are equal only when all the numbers are equal. So whenever prices for one item move by different amounts in different shops, the Carli average is higher. This is why the formula effect almost always pushes the RPI above the CPI. It is a fact of arithmetic.

Here is the whole process in one picture.

How a price index is built. The averaging inside each item uses no weights. The weights enter only when the items are combined. The CPI and the RPI differ at both steps.

Three indices: CPI, CPIH and RPI

Now the three numbers from the hook can be explained.

	CPI	CPIH	RPI
Owner-occupiers' housing	left out	included, by "rental equivalence"	included: mortgage interest and house depreciation
Formula inside an item	Jevons and Dutot	Jevons and Dutot	Carli and Dutot
Status	Bank of England target	ONS lead measure	legacy index, not a National Statistic

The CPI follows the international standard used across Europe, so it can be compared with other countries. Since December 2003, the Bank of England's target has been CPI inflation of 2%. Since 2011, most UK benefits have risen each year in line with the CPI (Department for Work and Pensions).

CPIH adds council tax and the housing costs of people who own their homes. These are measured by asking: "how much would I pay to rent a home like mine?" This is rental equivalence. CPIH has been the ONS's lead measure of inflation since March 2017. In October 2022, these housing costs rose by only about 3.6% (ONS). They are a large part of the basket, so CPIH (9.6%) came out below the CPI (11.1%).

The RPI was first calculated in 1947. It includes mortgage interest payments, so when interest rates rise, the RPI rises even if no shop price has changed. In 2022, the Bank of England was raising interest rates. Together with the formula effect, that put the RPI (14.2%) well above the CPI. But the RPI is not always the highest: in 2009, falling mortgage payments briefly made RPI inflation negative. The RPI is still used in some contracts, such as government bonds linked to inflation. Its methods are due to change to those of CPIH in February 2030 (UK Statistics Authority).

A real case: when the index decides the payment (UK, 2022–23)

The policy. In the UK, many payments are linked to an index by a fixed rule. Most benefits and the state pension rise each April by the CPI inflation rate of the previous September. Regulated rail fares in England, about 45% of all fares, were normally raised by the RPI inflation rate of the previous July.

The prediction. CPI inflation in September 2022 was 10.1%. RPI inflation in July 2022 was 12.3%. If the rules were followed, benefits would rise by 10.1% and rail fares by about 12.3%. Fares would rise faster, partly because they were linked to the higher index.

What happened. Benefits and the state pension rose by 10.1% in April 2023, as the rule said (Department for Work and Pensions, 2023). Rail fares did not follow their rule. The government announced in December 2022 that, "for this year only", the rise would be capped at 5.9%, the growth of average earnings in July 2022. That was 6.4 percentage points below the RPI figure (Department for Transport, 2022).

The verdict. The choice of index matters for real money. In September 2022, RPI inflation (12.6%) was 2.5 percentage points above CPI inflation (10.1%), and on payments worth billions, that gap is worth billions. But a rule linked to an index is still a political choice. When the formula gave an answer the government thought too costly for passengers, it overrode it. A good evaluation asks two questions: which index is the rule linked to, and will the rule be followed?

What a price index cannot see

Even a well-built index has limits. Four matter most.

Switching. When beef becomes expensive, many households buy more chicken. A fixed basket ignores this, so it can overstate the rise in the cost of living.
Quality. A phone at the same price as five years ago may do far more. Adjusting for quality is an estimate.
New goods. A new product enters the basket only at a yearly review, so its early price falls may be missed.
Averages hide differences. One national figure covers every region and every kind of household.

Three classic mistakes

Trap 1 — Reading index points as percentages

An index rising from 120 to 126 has risen by 6 points, but by 5%, because 6 is 5% of 120. Only when the index starts at 100 are points and percentages the same. Always divide by the starting value.

Trap 2 — "Inflation fell, so prices fell"

If inflation falls from 10% to 4%, prices are still rising, only more slowly. Prices fall only when the inflation rate is negative. A fall in the rate from one positive number to a smaller one is called disinflation.

Trap 3 — Averaging price changes without weights

If food rises 10% and holidays fall 10%, inflation is not 0% unless households spend exactly the same on each. Each price change must be multiplied by its weight, its share of spending, before adding.

Try it yourself

Work each one on paper first. Open a hint only when you are stuck.

A basket costs $400 in the base year, $418 a year later and $434.72 the year after (invented). Find the index numbers, and the inflation rate in each year. By what percentage have prices risen since the base year?

Hint 1
Index = cost ÷ base-year cost × 100.

Hint 2
For the second year's inflation, compare with the year before, not with the base year.

Full answer
Indices: 100, $\frac{418}{400} \times 100 = 104.5$ , and $\frac{434.72}{400} \times 100 = 108.68$ . Inflation in year 1 is 4.5%. In year 2: $\frac{434.72}{418} = 1.04$ , so 4.0%. Since the base year, prices have risen by 8.68%, not 4.5% + 4.0% = 8.5%, because the second rise is a percentage of a bigger number. Check: $1.045 \times 1.04 = 1.0868$ . ✓
Each month, Amara spends $300 on food, $500 on housing and $200 on transport (invented). Over a year, food prices rise by 10%, housing by 4%, and transport prices fall by 5%. What is Amara's personal inflation rate?

Hint 1
Turn the spending into weights first: each amount divided by the total.

Hint 2
The weights are 0.3, 0.5 and 0.2. A fall counts as a negative change.

Full answer
Total spending is $1000, so the weights are 0.3, 0.5 and 0.2. Contributions: $0.3 \times 10 = 3.0$ , $0.5 \times 4 = 2.0$ , $0.2 \times (- 5) = - 1.0$ . Inflation = 3.0 + 2.0 − 1.0 = 4.0%. Check in dollars: $330 + $520 + $190 = $1040, which is 4% more than $1000. ✓
A country's CPI (2015 = 100) rises from 120 to 126 in one year (invented). A newspaper says: "Prices rose by 6% this year." (a) Is the newspaper right? (b) A shopping trip cost $50 in 2015. Roughly what does it cost now?

Hint 1
Divide the change by the starting value of the index.

Hint 2
An index of 126 means 126% of the base-year cost.

Full answer
(a) No. The index rose by 6 points, but $\frac{6}{120} = 0.05$ , so prices rose by 5%. (b) An index of 126 means prices are 1.26 times the 2015 level, so the trip costs about $1.26 \times $50 = $63$ . It is "about", because this household's basket may not match the CPI basket. Check: $\frac{126}{120} = 1.05$ . ✓
A government pays $80 billion a year in benefits (invented amount). It must raise them by an inflation index. Use the UK's real rates for September 2022: CPI 10.1% and RPI 12.6%. (a) How much more would the RPI cost than the CPI? (b) Give one argument for each index. (c) Low-income households faced inflation above the CPI rate in late 2022. What does this suggest about using the CPI for benefits?

Hint 1
Find each rise as a percentage of $80 billion, then subtract.

Hint 2
Think about the formula effect, mortgage interest, and whose basket each index follows.

Full answer
(a) CPI: $0.101 \times 80 = $8.08$ billion. RPI: $0.126 \times 80 = $10.08$ billion. The RPI would cost $2.0 billion more, which is 2.5% of $80 billion. ✓ (b) For the CPI: it uses a formula that statisticians accept, it is the measure the central bank targets, and it costs taxpayers less. For the RPI: it was higher, so it would protect people more if their costs were rising faster than the CPI. However, part of the RPI's extra rise came from the formula effect and mortgage interest, which many benefit recipients do not pay. (c) The CPI follows the spending of all households, weighted by spending. Low-income households spend more of their budget on energy and food, so in 2022 their inflation was higher (11.9% against 11.1% in October). Linking benefits to the CPI then left them slightly worse off in real terms. A fairer link might be an index built from the spending of the people who receive the benefit.

The bigger idea

Every inflation figure is the result of choices: which items, whose spending, which formula, and what to do about housing. Each choice helps some groups and costs others. That does not make the figures dishonest: in the UK an independent statistics office produces them and publishes its methods. It means that a careful economist asks which index lies behind a number before using it. The causes of inflation, its effects, and the interest rates set by central banks, in the next lectures, all start from a number like the CPI.

Key concepts in this lesson

Inflation is a sustained rise in the general price level, and the inflation rate is the percentage change in a price index over 12 months.
A price index follows the cost of a fixed basket of goods and services, written as an index number with the base period equal to 100.
Each item is weighted by its share of household spending, so the inflation rate is the sum of each weight times that item's price change.
Inside one item, the RPI averages price ratios (Carli), while the CPI uses a geometric mean (Jevons); the Carli average is never lower, which is the formula effect.
The CPI leaves out owner-occupiers' housing, CPIH adds it by rental equivalence, and the RPI includes mortgage interest, so the three can differ by several percentage points.
A household's own inflation rate depends on its basket, so one national figure can understate inflation for households that spend more on the fastest-rising items.