Calculation issues

Calculation issues – the historical background

An ideal index is worked out using a simple method that is easy for all to understand and accurately portrays the price rises actually experienced by the total population.

It might be thought that the solution to this is simple. You just measure the price rise each month for each item and compute the index. Indeed the very first price indices created in the 18th Century did just this. A French economist called Dutot suggested that the best way to calculate a price index was to take an average of the prices and compare it to the average seen in the same places in the period before. This method called the “ratio of the averages” is also sometimes referred to as the Dutot index.

At a similar time (1764), an Italian economist Giovanni Carli published what is believed to be the first historical price index based on three key commodities: grain, wine and oil . He used a slightly different method. Instead of taking the average each month across the shops, he worked out the change in price for each location and then averaged those. This method has become known as the “average of the relatives” or the Carli index.

Either of these would satisfy being a method that would be easy to understand. However it might be argued that Carli’s average change for each shop better represents what would happen to the population, assuming a good sampling frame of stores is adopted.

However as statisticians started to use these formulae they became aware of differences between them. Dutot’s simple averages gave more weight to locations where items cost more and this could distort the data if prices of an item varied a lot. However other theorists pointed out that Carli’s method of averaging the changes could create strange results over a period of time too . If prices reverted to their original levels the sum of all the monthly changes might be greater than zero* – this phenomenon is known as the price reversibility problem or the price bouncing issue.

Many alternatives were proposed to get around these issues. The one that gained most attention was suggested by an English economist, William Jevons, in 1863. He suggested a similar approach to Dutot but, instead of taking a simple average of the store prices, he suggested taking a “geometric mean” of them (also known as the Jevons index). Geometric means are calculated by multiplying all the numbers together and then taking the nth root of them—hardly what most would describe as a simple to understand calculation.

It too had some issues; for instance, it was impossible to average prices which included free products. However it did fix the issue identified with the Carli method of price reversibility over time. But this came with a side effect. The resulting average also had the property of always being lower than Carli’s simpler average. Moreover if there was a wide range in the price increases seen between different stores the reducing effect could be very large. Compared with the Dutot method, the Jevons scores were also normally lower too, but the effect was more variable and it was possible for Jevons to be higher.

Statisticians favouring the Jevons approach justified the lower price rises the method created on the basis that they were taking account substitution effects. They argued that if the prices of an item went up people would not experience that price rise fully as some would switch to cheaper products. They said that was why the Jevons was lower and why it was more accurate.

Substitution is already covered elsewhere in price index calculations and does not need further adjustment. But, more importantly, if the price of a product goes up, then the prices have gone up and this should be reflected in a cost of living index. If people then decide to cut back their spending by buying an inferior quality product, that is their decision, but the item they used to buy is now indisputably more expensive. Furthermore, how the psychological dilemma of substitution could be estimated by taking a mathematical nth root was never explained.

In addition, in the recent debate about switching to Jevons in the UK in 2012, the Institute of Fiscal Studies pointed out the supposed advantage of Jevons over Carli on reversibility grounds was actually not that important as any price index in totality is not reversible (because of the way it is made up of a weighted average of many different items). Despite this, the Jevons method has become very popular as you will see.

Calculation issues – the world today

When it created RPI in the UK, the ONS took a pragmatic view and used a combination of simple averages (Dutot) where price variability was not that great and used the average of the relatives (Carli) when it was greater (e.g. for clothes and food). In contrast, CPI(H) primarily uses geometric means (Jevons). Note, ONS will not divulge the exact which method they use for each item or even the total numbers of items being averaged using each method**.

Other methods of calculation are also used in CPI(H) and RPI for a few items such as the Laspeyres Index and straight averages.

Some other countries, such as Germany and Japan, use just the simple Dutot method to calculate inflation. Unlike the UK, they tend to be very specific over the items selected by their price checkers and so face less price variability than the ONS (which can tell the checkers to just find things like “any ladies’ blouse”).

Indeed, until the 1990s, most countries used Dutot or a combination of the Dutot and Carli methods, but many have now switched to the Jevons method. It can only be speculated whether this was part of a desire to appear to be bringing inflation down following its peak in the early 1980s. The switch was given a further impetus by the EU in 1996 when it sanctioned the use of only the Dutot and Jevons methods when compiling its harmonised inflation measure (HICP).

Most countries have justified their switch to Jevons either on the grounds of it dealing with substitution better (e.g. France), the reversibility issues of Carli (e.g. Canada) or that it is now the internationally accepted method (which was ONS’s main argument in 2012) . Outside the UK, relatively little work has been published on the effects of switching to Jevons. However some estimate that it helped reduce US inflation rates by 0.3 per cent after their switch from Dutot and in other countries by up to 0.2 per cent.

Jevons has a larger impact than this in the UK. The so called “formula effect” of using Jevons in CPI(H) (as opposed to Carli and Dutot in RPI) has been around 1 per cent from 2011-2014***. To demonstrate the impact, if inflation in the UK was 3 per cent, it would have been reported as only being 2 per cent if you use the Jevons geometric mean as CPI does.

This may not seem much, but of bigger concern is the cumulative impact of this adjustment. For example, from January 2000 to April 2013 RPI inflation was recorded as increasing prices by a half. With CPI, prices went up by only just over a third. To make this clear, if your income had been indexed by the CPI instead of the RPI over that period, you would now be around 10 per cent worse off (i.e. 137/150).

Source: ONS (Jan 2000–Apr 2013)

* In certain theoretical circumstances, a situation might occur where you get the wrong answer if you use an average of the price relatives (i.e. the Carli mean of the shifts). Take the following example. Say prices are just being checked in two shops A and B. If in month the cost in shop A is £2 and in shop B £3, the average price would be £2.50. In the second month, say the price in shop A increases to £3 and declines in shop B to £2. The average is still £2.50 i.e. inflation should be zero. However if you average the two relative price changes (i.e. +50 per cent and -33 per cent respectively), you do not get zero.

** The author had a number of email conversations with the ONS in December 2014 where they confirmed that they do not officially publish this information. They provided an approximate expenditure weighting split but this differed markedly to some figures in another ONS document published in 2012 as part of the RPI consultation. It is difficult to understand why ONS cannot share this factual information. It unfortunately undermines perceptions of the independence and validity of their statistics.

*** Source: ONS. Note, the formula effect used to average around 0.6 per cent until the ONS made ”improvements” in the way they measured clothing prices in 2010. See: Morgan, D., & Gooding, P., 2011, “CPI and RPI: increased impact of the formula effect in 2010”, ONS.