# Price indices and the geometric mean

Author: Andrew McCulloch

First, I want to tidy up a point arising from the last piece I wrote about RPI and CPI. They are both measures of how prices change but they are calculated using different ‘baskets’ of prices. But there is another difference between them as well. The RPI is uses the Arithmetic Mean of its prices to work out its index; the CPI uses the Geometric Mean.

As I mentioned previously, the varying use of the CPI and RPI by the government seems to give the impression that the CPI is used to maximise incomes while the RPI is used minimise payments, and we can understand that there might be an incentive for government to behave in that way.

There is an economic rationale, however, for using the geometric mean when calculating a price index. Say I go to the shops and I buy a bag of Golden Wonder potatoes and a bag of Marris Piper potatoes both of which cost £1. If the price of Golden Wonder potatoes then rises to £2, the potato price index calculated using the arithmetic mean is 1.5 while the potato price index calculated using the geometric mean is 1.41. The use of the arithmetic mean assumes that consumers do not substitute between potato varieties: I now need to have £3 to be able to buy the same basket of goods that I could with £2 before the price rise. That may be a correct description of the way people behave, we still want our two bags of potatoes, but it assumes that people do not substitute between potato varieties in response to price changes. If people do substitute between potato varieties and they go to the shops with £3, then perhaps they can achieve greater satisfaction from their potato purchases by buying say ¾ of a bag of Golden Wonders and 1½ bags of Marris Pipers, still at a cost of £3. That is an extra ¼ bag. For the same money!

The price index calculated using the geometric mean of prices arises under a specific assumption about how people substitute between potato varieties. Specifically, it assumes that a price change leads to an equal and offsetting proportionate change in quantity such that the share of the budget is unchanged. If this is an accurate description of behaviour, then people only need 41% (rather than 50%) more income in order to be as satisfied with their potato purchases before and after the price change. That is if we give you £2.83 rather than £3, you spend 50 percent of your budget on Marris Pipers (1.415 of a bag) and the rest on Golden Wonders (0.707 of a bag) and you are just as happy as you were before the price change. So the use of the geometric mean to calculate the CPI can be defended on economic grounds - but it does depend on assumptions about the choices people make about potatoes (and other goods too, of course).

Another area where the geometric mean is useful is in dealing with numbers that are ratios of other numbers. For example, suppose we have the figures in the Table below for the percentage increase in the price of a company share as a fraction of the price in the previous year and we want the mean percentage increase over the 4 years. The arithmetic mean of the growth rates is 108.25 and the geometric mean is 108.20. If the initial share price was £50 the geometric mean, unlike the arithmetic mean, gives the right value for the final figure of the share price, although the intermediate values are not totally correct. So when economists talk about the average annual return on an investment, they are often referring to a geometric mean rather than an arithmetic mean.

### OhDearMe!

Both RPI and CPI are supposed to be measures of price, not of changing consumer habit. The golden rule of statistics is: "Always compare like with like". In calculating CPI we're now saying that price rises are less because people buy something different. That's perverse!

So, if a BMW costs £20,000 in January and a Ford costs £15,000 in December, we can say argue that the price of cars has fallen, can we?

And this logic has devastating consequences for the poor who already buy the cheapest products and don't have the opportunity to switch to a cheaper brand. Their only option is to go without. Great!

### Tachopro 2008

A abeyant applicant would be the redesigned Indica. The modifications fabricated to the car are appreciably impressive. No tweaking was fabricated to the air clarify but administration modifications and all-important upgrades fabricated the car "automotively palatable". Quite obviously, the Indica's new-found fit, performance, and accomplishment are advised to bang aficionados bendable spot.

### Frankman

Excellent.  It is a pity general commentators only ever say that the difference between CPI and RPI is because of "housing costs" when the major difference is mathematical.  The use of CPI is basically a convenient political fudge. If you ask the man in the street what inflation was he would talk about "average" price increases - and he would mean the arithmetical mean.

Substitution is catered for in RPI by changes to the basket of goods.  So using substitution to justify the use of CPI is another part of the same fudge.

### Mike

The substitution occurs, surely, because the initial equilibrium is that people spend half their budget for potatoes on Maris Piper and half on Golden Wonder. If the price of Golden Wonder goes up, you would expect the consumption of them to come down, and the consumption of Maris Piper to go up. Exactly how that level of substitution would occur, I can see is uncertain, but it doesn't seem unreasonable that customers would still want to spend half their budget on each.

I find it odd, though, that the article assumes the purchase of 2.12 bags of potatoes? Surely customers would probably still buy two bags of potatoes? The demand for potatoes overall is probably more inelastic than the demand for each variety; it's less of a wrench to buy a different variety, or even have more mashed rather than jacket potatoes, than to eat fewer poatoes overall. If so, and splitting the budget 50:50 between them, you would expect to see purchases of 1.33 bags of Maris Piper and 0.67 bags of Golden wonder, giving a total price of £2.67?

### Heather Hargrove

Quote: If the consumer regarded the goods as exactly equivalent and was willing to substitute in proportion, as this explanation of the use of the Jevons is suggesting, then why would the consumer ever buy the more expensive product?  Surely the logic would be that they would always substitute the cheaper good and therefore you should use the cheapest price!

A very interetsing article which generally confirms my knowledge, such as it was, of the substitution argument. I have three comments/questions. Firstly the Jevons Formula seems to me to be very specifically a binary formulation between two goods. It would surely be much more difficult for a consumer, or even a cohort of consumers, to make the substitutions across a range of products so as to equalise the post inflation expenditures on each good in the range. It is impractical. Secondly, and this is probably the same point in a different form, the use of the geometric mean in CPI isn't actually measuring the amount of substitution going on. It is merely a proxy estimator of the substitution. If people do not in fact substitute in the way assumed, or only substitute partially, then surely the use of the geometric mean gives a wholly inaccurate measure of inflation as perceived by the consumer, which is in fact real inflation. Thirdly, and this is a key point, the assumption that substitution can maintain satisfaction is unproven, and in my view flawed. It may be theoretically possible but I have never seen an example which uses same quality goods. Your example here uses Golden Wonder and Maris Piper potatoes. Not only are Maris Piper better but the two potatoes are used for different proposes. Not onlt that but that is not the way supermarkets sell potatoes. They have levels of quality and the potatoes tend to be similar prices (prices set artificially by the shops) and there is little or no variation between potatoes in the smae price range. This is not a free untrammelled nmarket we are dealing with. The example on Wikipedia uses romaine and iceberg lettuce. Romaine are usually more expensive than iceberg but they are a superior lettuce. A shift to iceberg from romaine is definitely a loss of quality and a loss of satisfaction. Realistically retailers market quality goods at a higher price that lower quality goods almost without exception. It is about marketing not supply and demand. A substitution of lower priced goods for higher priced goods would therefore always in practical terms involve a loss of quality. It follows that if substitution were happening as the ONS etc implies quality would be constantly dropping and have been doing so for years. This is patently not the case.

### Rod

Quote:

This still does not explain why the use of geometric mean reflects the substitution of cheaper goods. You may as well say "allow 10% off for substitution". Can anyone  explain please in terms an accountant with a Maths background can understand ?

Why oh why do articles etc by at least some economists make a huge unexplained leap from what has been clearly explained to a mere assert action? A Maths background would illustrate you do not make an unsubstantiated leap. I would not accuse all economists of this but those that do give a bad impression.

### Rod

How on earth do you get from the formula to the substitution?

### Dieter Jens

I am afraid that while you describe the geometric mean correctly it should only be used where rates, percentages etc are involved cumulatively. That is to say where the graph of items is not a straight line but curves upward.

With the CPI calculation there is no logic in talking about substitution of goods, even if there was full satisfaction with the alternative choice of goods, which of course is not guaranteed.

The main problem though is that in calculating CPI we should be talking about just that, the prices and their changes, not consumer behaviour. This behaviour stops after deciding the basic basket of goods. Thereafter a price increase of potatoes is a price increase of potatoes irrespective of what the consumer may choose to do after he sees the increase.

No, I am afraid the substitution of goods is only an excuse. The CPI by using the geometric mean will always be lower than the RPI. Substitutions give a lower result but is not mathematically connected to the geometric mean. And of course is convenient politically for the government.

### Cemlyn

This still does not explain why the use of geometric mean reflects the substitution of cheaper goods. You may as well say "allow 10% off for substitution". Can anyone  explain please in terms an accountant with a Maths background can understand ?

### David Quinn

A very interetsing article which generally confirms my knowledge, such as it was, of the substitution argument. I have three comments/questions. Firstly the Jevons Formula seems to me to be very specifically a binary formulation between two goods. It would surely be much more difficult for a consumer, or even a cohort of consumers, to make the substitutions across a range of products so as to equalise the post inflation expenditures on each good in the range. It is impractical. Secondly, and this is probably the same point in a different form, the use of the geometric mean in CPI isn't actually measuring the amount of substitution going on. It is merely a proxy estimator of the substitution. If people do not in fact substitute in the way assumed, or only substitute partially, then surely the use of the geometric mean gives a wholly inaccurate measure of inflation as perceived by the consumer, which is in fact real inflation. Thirdly, and this is a key point, the assumption that substitution can maintain satisfaction is unproven, and in my view flawed. It may be theoretically possible but I have never seen an example which uses same quality goods. Your example here uses Golden Wonder and Maris Piper potatoes. Not only are Maris Piper better but the two potatoes are used for different proposes. Not onlt that but that is not the way supermarkets sell potatoes. They have levels of quality and the potatoes tend to be similar prices (prices set artificially by the shops) and there is little or no variation between potatoes in the smae price range. This is not a free untrammelled nmarket we are dealing with. The example on Wikipedia uses romaine and iceberg lettuce. Romaine are usually more expensive than iceberg but they are a superior lettuce. A shift to iceberg from romaine is definitely a loss of quality and a loss of satisfaction. Realistically retailers market quality goods at a higher price that lower quality goods almost without exception. It is about marketing not supply and demand. A substitution of lower priced goods for higher priced goods would therefore always in practical terms involve a loss of quality. It follows that if substitution were happening as the ONS etc implies quality would be constantly dropping and have been doing so for years. This is patently not the case.

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