3. The formula effect

Consumer price indices in the UK typically make use of three different formulae to combine individual price quotes. The Retail Prices Index (RPI) uses the Carli and Dutot formulae, whereas the Consumer Prices Index including owner occupiers’ housing costs (CPIH) and Consumer Prices Index (CPI) use the Jevons formula in place of the Carli.

The mathematical properties of the Carli mean that price inflation calculated using it cannot be lower than inflation using the Jevons on the same basket of goods and services. There is evidence that the choice of formula has a large impact on the inflation rate. In the recent past, this “formula effect” has on average added 0.7 percentage points to the annual RPI measure of inflation compared with the CPIH.

There is no straightforward test for identifying the best formula to use. It is possible to produce hypothetical scenarios in which any of the formulae can be shown to produce reasonable or unreasonable results. A judgement therefore needs to be taken in the round on the basis of a wide range of evidence and analysis.

In January 2013, the Consumer Prices Advisory Committee concluded that “the statistical properties of the Carli meant it was unsuitable” for use in combining individual price quotes.

In 2015, the independent Review of UK Consumer Price Statistics by Paul Johnson explored the Carli and alternatives including the Jevons. Chapter 10 of that report provides a detailed exploration of the evidence around the Carli, including a number of different approaches for examining the appropriateness of different formulae. Taking into account all the evidence, the Review concludes that:

“Carli should not be used in any index aiming to achieve a good estimate of changes in consumer prices” and further that it “is not suitable for use”.

Both the Consumer Prices Advisory Committee and the Independent Review looked at international practice, noting that other countries had moved away from the use of Carli. For example, Canada had stopped using the Carli in 1978 and of 30 countries surveyed by Office for National Statistics (ONS) in 2012 (PDF, 152KB), none used the Carli. The United Nations Practical Guide to Producing Consumer Price Indices says:

“A key result is that the Carli formula for the arithmetic average of price relatives has an upward bias relative to the trend in average item prices. In particular the Carli suffers from lack of transitivity i.e. when prices return to an earlier level the chained index doesn’t. Consequently, it is a formula to be avoided and some judge that it should be prohibited.”

The issue of transitivity or “price bounce” is described in Annex A.

The Carli formula is no longer international best practice. This is the main reason why the RPI had its National Statistics status removed in 2013 following an assessment by the UK Statistics Authority (PDF, 99KB).

There have been some subsequent arguments in favour of Carli. The ONS’s Technical Advisory Panel on Consumer Prices reviewed some of the issues in January 2016, when a paper by Dr Mark Courtney was considered, but again concluded that, in the context of price collection in practice, the Carli formula is less suitable than the Jevons and Dutot formulae.

In light of this evidence the ONS view is that the Carli is not a suitable method and using it is likely to result in an upward bias to measures of inflation.

The impact of the formula effect was brought into focus in 2010, when ONS introduced a number of changes to the collection of clothing prices, mainly to increase the sample size and better reflect consumer spending patterns. These changes nearly doubled the formula effect and have acted to illustrate the weaknesses of the Carli. Some clothing items showed implausible levels of inflation over this period.

Figure 2 shows the change in the typical price of items of clothing between January 2010 and January 2018, compared with the increase in the price index using the Carli and Jevons methods. Each marker is for a particular clothing item within the wider clothing category. The expected result would be for the typical prices and the indices to grow at the same rate (the “expected” line in the figure). We would expect some variation, however, due to chain-linking and the formula used. In addition, changes to the sample would also show up as deviations from the expected line.

For the Jevons index, the different items are fairly close to the expected line. On the other hand, when using Carli the price index rise tends to be much higher than the rise in the typical price. We see around 20 clothing items where the Carli index suggests prices have doubled (increased by over 100%), despite the typical price of an item increasing by no more than 50% or so. The same price data are providing very different results due to the formula used.

In our view, it is clear that the changing clothing price collection has exposed serious shortcomings in the Carli formula.