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.