Note on the choice of Formula for Elementary Aggregates: the Geometric v
Arithmetic mean of price relatives. March 2012.
This note challenges the assumptions used to justify using a different formula
in the CPI and from that used in the RPI to measure price change for
corresponding components of consumers’ expenditure.
1.
A central argument for choosing the geometric mean of price changes for the CPI is that it
better represents consumer tendencies to substitute lower priced items to counter the effects of
inflation. This argument assumes that
a) The CPI should approximate a “cost-of-living” (Coli) or constant utility index rather than a “fixed
basket” Laspeyres index, and
b) The sampled prices used to construct elementary aggregates represent actual substitutions that
consumers make.
This note challenges these assumptions.
2.
A price index is designed to represent the impact of changing prices on persons or
households. A Laspeyres1 index is a measure of the changing cost of a “fixed” basket of goods and
services bought by households (in the UK) over a twelve month period. The index is an answer to
the question “What would be the current cost of buying what consumers chose to buy in a given
reference period?” It is a question that expresses a simple and easily understood notion of “impact”
and for which it is possible to produce an answer on a monthly basis. Most, if not all, National CPIs
are intended to be Laspeyres-type indices.
3.
A “Cost of living index”, (constant utility index) many theorists have argued would be a
superior measure of consumer price inflation and this should be the objective notwithstanding the
difficulties of compiling such an index. Such an index is an answer to the question “What would be
the current cost of maintaining the utility of what consumers chose to buy in a given reference
period?” Many national statistics offices, and The Retail Prices Index Advisory Committee, have
specifically rejected this and the idea was not accepted as a requirement for the European
harmonized indices of consumer prices. Nevertheless, strong voices and much theoretical work
have kept the debate open. The US is the main advocate of the Coli.
4.
There are severe practical difficulties in producing a monthly Coli but ways have been
suggested to adapt current operations to approximate such an index; e.g. by the use of the GM
instead of the AM of relatives at the basic (elementary aggregate) level where only price data are
available (no weights). The concept of “utility” was introduced by Bentham in the 1800 century but it
is open to various interpretations, it is not easily understood and has yet to be operationally defined.
1
Note; there are other “fixed basket” indices, the Paasche and Fisher indices, which involve “current” weights and are not
readily compiled other than in retrospect. They do not have the simple appeal of the Laspeyres index; maintaining a chosen
standard in the face of price inflation.
Its use in Coli theory involves the problematic assumptions of “rational consumers” and “efficient
markets”. There is also a potential contradiction between the notions of “substitution” and “quality
difference” of items covered by elementary aggregates. Coli theory assumes that, in a market
economy, price differences between similar products are due to quality differences. In an efficient
market, price differences reflect consumer judgements of quality differences. (This is used to justify
a hedonic approach to quality adjustment.) Were this the case then any substitution of lower priced
items would imply a quality reduction contrary to the widespread acceptance that price CPIs should
not reflect reduced quality; i.e. contrary to the requirement for a “fixed basket” (or constant utility).
5.
The sampling of prices for CPIs is not generally designed to represent product substitution.
Price indices are constructed from the prices for selected samples of items within narrow categories
of goods and services. Elementary aggregates represent consumption within geographic and/or
outlet types for selected products within a product category. There is generally some attempt made
to properly represent the various products within a category (e.g. woman’s dresses sold in
department stores) but not to restrict the selection to possible “substitutes”. Indeed any attempt to
represent wide price dispersion (as in women’s dresses) is likely to involve the selection of “nonsubstitutes”. It has been shown that wide price dispersion among reference prices tends to increase
the difference between the GM and the AM of price relatives. It has not been shown that consumers
substitute between products within elementary aggregates used in CPIs.
6.
Nevertheless, there is a problem with using the arithmetic mean of price relatives in
elementary aggregates that was recognized by the Retail Prices Index Advisory Committee back in
19772. In order to ensure that price change is measured with an up-to-date basket the RPI weights
are updated annually and price changes are taken by reference to the latest January prices. This
chaining of the index can lead to a progressive divergence between the ratio of arithmetic mean
prices and the arithmetic mean of price relatives. In order to avoid this, the ratio of mean prices was
recommended for elementary aggregates except where there was a very wide variation in prices
(non-homogeneous items). The geometric mean was not considered at that time and there was no
suggestion that product substitution was relevant.
“In the autumn of 1977 the RPI Advisory Committee agreed ....that the ratio of averages formula
should be used in the case of homogeneous items to combine price quotations within the new
strata. The Committee recognized that the arguments in favour of the ratio of averages formulӕ also
applied in the case of non-homogeneous items but felt that other considerations were important in
these cases and further investigations were necessary.” The effect of the changes over the period
Jan 1975 to January 1977 was calculated to reduce the increase in the all items RPI from 43.7 per
cent to 43.3 per cent. I regret to say that the “further investigations” were not carried out by 1997
whilst I was responsible for the index. I am not aware
7.
The foregoing is not an argument against the use of the GM but only against the justification
so far given for using that formula. A price index is a construct. It is not a measure of something real,
it is hypothetical. There are no fixed baskets, purchases cannot be repeated; goods, services, and
consumers, all change from day to day and month to month. The aim is to produce a measure that
will be widely accepted as best representing consumer experience. The choice of formula for use in
2
See “Recent Developments in the Retail Prices Index” by Carruthers, Sellwood and Ward.
The Statistician, Vol.29, No1.. 1980 Institute of Statisticians
elementary aggregates cannot be resolved by appeal to a theory that is not accepted by all or most
concerned. It is a matter of judgement whether the GM or other formula can overcome the
weaknesses identified in existing formulӕ. It is inappropriate to have two indices using different
formulӕ to measure ostensibly the same thing.
8. The CPI follows Eurostat’s regulations for Harmonized indices across EU member states. The aim
of the Harmonization process was to ensure that member states measured consumer price inflation
by the same methods so that they may be directly compared. Member states agreed as a criterion
for comparability that the annual rate of change for CPIs should not differ between member states
by more than 0.1% on account of methodology alone. That is to say, for example, that were French
CPI procedures to be followed in the UK the annual change in the resulting index would not differ
from the UK CPI by more than 0.1%. Following this common sense and well accepted notion of
comparability it would seem appropriate to agree a common formula for the CPI and the RPI for
measuring price changes in components of coverage they have in common; unless good reasons can
be found for doing otherwise.