INDEX NUMBERS
[30]Index numbers are used to indicate changes in the value of money.
The objects for which this measurement is undertaken are thus well
stated by Sir R. Giffen (Second Report of the committee appointed for
the purpose of investigating the best method of ascertaining and
measuring variations in the value of the monetary standard. Report of
the British Association, 1888): (1) The fixation of rents or other
deferred payments extending over long periods of time, for which it has
been desired to obtain a currency of a more stable sort than money is
supposed to be. (2) To enable comparisons to be made between the value
of money incomes in different places, which is often an object of great
practical interest; not only individuals contemplating residential
changes, but also governments and other large spending bodies, spending
money in widely distant places, having to consider this question. (3) To
enable historians and other students making comparisons between past and
present to give an approximate meaning to the money expressions which
they deal with, and say roughly what a given fine, or payment, or amount
of national revenue or expenditure in a past age would mean in modern
language. To which some would add: (4) To afford a measure of the extent
to which trade and industry have been injuriously affected by a
variation in prices; and of the correction which it would be desirable
to apply to the currency.
An index number is constructed by combining several items, each of which
is a ratio between the price of a certain article at a particular date
under consideration (_e. g._, last year or month) and the price of the
same article at a period taken as base or standard (_e. g._, 1867-77, in
the index number constructed by Mr. Sauerbeck, _Journal of the
Statistical Society_, 1886 and 1893). These ratios are generally
expressed as percentages. _E. g._, the percentage for _flour_ in 1885,
as given by Mr. Sauerbeck, is 63; meaning that the price of flour in
1885 is to the average price of the same article in 1867-77 as 63:100.
The term index number is sometimes applied (_e. g._, by Mr. Sauerbeck,
_op. cit._) to each of these items, as well as to their combination.
The percentages are usually compounded by taking an AVERAGE of them. But
a result of equal generality may be obtained by taking their sum. One of
the best-known index numbers, that of the _Economist_, is thus
constructed. Twenty-two articles having been selected, the price of each
article at the current date compared with its price at the standard
period (1845-50) is expressed as a percentage; and the sum of these
percentages is put as the index number. Thus the _Economist_ index
number for the year 1873 is 2947; such a sum is easily reduced to the
form of an average by simple division (_e. g._, 2947 / 22 = 134).
Accordingly in what follows it will be sufficient to consider the latter
form only.
The construction of an index number presents the following problems:
(_a_) What are the commodities of which the prices are to be taken?
(_b_) How are the prices to be ascertained? (_c_) How are the ratios
between the prices of each article at the current and the standard dates
to be combined?
The answers to these questions vary according to the purpose in hand....
As appropriate to the first purpose, a standard of deferred payments,
two methods present themselves, viz., to arrange that the debtor should
pay, the creditor receive, either (1) the same quantity of goods and
services, the same amount of utility, so to speak; or (2) the product of
the same quantity of labour--or more exactly effort and sacrifice.
Of these methods the former has been more generally accepted. It is
adopted for instance by the British Association Committee already
referred to, as _par excellence_ the measure of the change in the value
of the monetary standard. The former method is indeed more intelligible.
However, in favour of the latter there are some weighty considerations
and authorities. It seems to be the nearest possible approach to
Ricardo's conception of a commodity invariable in value, "which at all
times requires the same sacrifice of toil and labour to produce it."
(_Principles_, iii. ch. xx., "On Value and Riches," cp. Mill, bk. iii.
ch. xv., "On a Measure of Value.") "A standard," says Mr. Leonard
Courtney, "should be something which as far as possible involves the
same labour and the same sacrifice in obtaining it" (_Nineteenth
Century_, March, 1893). Prof. Marshall, in his evidence before the royal
commission on gold and silver, says, speaking of appreciation of gold:
"When it is used as denoting a rise in the real value of gold, I then
regard it as measured by the diminution in the power which gold has of
purchasing labour of all kinds--that is, not only manual labour, but the
labour of business men and all others engaged in industry of any kind"
(Question 9625).
If the first method is adopted, the answers to the questions above set
are as follows: (_a_) The commodities of which the prices are to be
taken should be articles of consumption rather than materials and
implements. Payments for personal services should be included, but not
wages in general. (_b_) Retail prices should be used. (_c_) The proper
combination of the ratios is an average of the kind technically called
_weighted_.... The general principle according to which the weights are
to be assigned is that they should represent the importance of each
commodity to the consumer. But this idea may be embodied in different
plans.
1. One plan is to assign as the weight of each percentage, or ratio
between prices, the value of the corresponding commodity at the initial
or standard period. According to this plan the index number is the ratio
between these two values: the quantities initially consumed at the
prices of the current date, and the same quantities at the standard
prices. This method is exemplified by Sir R. Giffen's estimate of the
change in the value of money between 1873 (and 1883) and _earlier_
years, in his report on prices of exports and imports, 1885, table v.
2. Another plan is to assign, as the relative importance of each
percentage, its value at the particular epoch, the current year. This
plan is adopted by Mr. Palgrave in his memorandum on _Currency and
Standard of Value_ ... in the third report of the royal commission on
depression of trade and industry, table xxvii.
3. According to another plan, the index number is the ratio between the
following two values: the quantities consumed at the current date at the
current prices, and the same quantities at standard prices. This plan is
adopted by Mr. Sauerbeck (_Journ. Stat. Soc._, 1886, p. 595).
4. Or, instead of taking either the initial quantities or those of the
current date, a mean between the two may be taken. This is the plan
adopted by the British Association Committee. They estimate "the average
national expenditure on each class of article at present and for the
last few years"; and put for the relative importance of each commodity a
round number corresponding to that estimate. Thus the estimated
expenditure per annum on _wheat_ is L60,000,000, and on _meat_
L100,000,000: that is respectively 6.5 per cent., and 11 per cent. of
the sum of the corresponding estimates for all the commodities utilized
by the committee. As convenient approximations, the weights five and ten
are recommended by the committee.
If the index number based on labour ... rather than on consumption, is
adopted as the standard for deferred payments, it would be proper by
analogy to take as the measure of appreciation or depreciation the
change in the pecuniary remuneration of a certain set of services,
namely all, or the principal, which are rendered in the course of
production throughout the community during a year, either at the initial
or the current epoch; or some expression intermediate between the two
specified. But it may be doubted whether the statistics requisite for
this method are available.
With regard to the second and third of the purposes above enumerated,
the determination of the comparative value of money at distant places
and remote times--one or other of the two methods indicated would seem
to be theoretically proper.
For the fourth purpose, the regulation of currency, the proper
construction of the index number would seem to be as follows: (_a_) The
"articles" of which the prices are taken into account should be both
commodities and services; (_b_) both wholesale and retail prices should
be used; (_c_) the relative importance of each article should be
proportioned to the demand upon the currency which it makes. But here as
in other parts of the subject theory halts a little, and statistics lag
far behind theory.
Considering the theoretical doubts and statistical difficulties which
attend the determination of _weights_ proper to each purpose, there is
much to be said in favour of assigning equal relative importance to all
the items; especially if care is taken to include many articles such as
_corn_, _cotton_, etc., which for any of the purposes which may be
contemplated must be of first-rate importance. Such is the character of
some of the principal index numbers which have been constructed--those
of the _Economist_, of Jevons, of Soetbeer, and of Mr. Sauerbeck.
In the construction of such an index number the use of the arithmetic
mean is not imperative. Jevons employs the geometric mean. His reasons
for preferring it are not very clear (the "Variation of Prices,"
_Currency and Finance_, p. 120).... The geometric mean has also the
advantage of being less liable than the ordinary average to be unduly
affected by extremely high prices (_Report of the British Association_,
1887, p. 283). The great objection to the geometric mean is its
cumbrousness.
There is another kind of mean which has some of the advantages of the
geometric, and is free from its essential disadvantage; namely, the
median ... which is formed by arranging the items in the order of
magnitude, and taking as the mean that figure which has as many of the
items above as below it. For instance the median of the forty-five
percentages on which Mr. Sauerbeck's index number is based was, for
1892, 66; while the arithmetic mean was 68. It is difficult to see why
the latter result is preferable to the former; if what is required is an
_index_ of the change in general prices, not specially referred to any
particular purpose, such as of securing a constant benefit to a legatee.
The perplexity of a choice between such a variety of methods is much
reduced by the two following considerations. _First_, beggars cannot be
choosers. The paucity of statistical data (see the report drawn up by
Sir R. Giffen in the _Report of the British Association_ for 1888, p.
183) restricts the operation. Thus for the purpose of index numbers
based on consumption ... retail prices are theoretically appropriate;
but "practically it is found that only the prices of leading
commodities, capable of being dealt with in large wholesale markets, can
be made use of" (Giffen, _loc. cit._). _Second_, the difference between
the results of different methods is likely to be less than at first
sight appears. For instance, the probable difference between the index
number constructed by the British Association committee, and six others
which have been proposed by high authorities--supposing the different
methods to be applied to the same data, viz., the prices of twenty-one
articles specified by the Committee may thus be expressed. The
discrepancy which is as likely as not to occur between the committee's
and other results is from 2 to 2.5 per cent. The discrepancy which is
very unlikely to occur is from 8 to 11 per cent. (_Report of the British
Association_ for 1888, p. 217). In fact, the index number for the year
1885, as determined from the same data by seven different methods,
proved to be 70, 70.6, 73, 69, 72, 72, 69.5 (_ibid._, p. 211).
The practical outcome of these two considerations is thus well expressed
by Giffen (_loc. cit._ p. 184), "The articles as to which records of
prices are obtainable being themselves only a portion of the whole,
nearly as good a final result may apparently be arrived at by a
selection without bias, according to no better principle than
accessibility of record, as by a careful attention to weighting....
Practically the committee would recommend the use of a weighted index
number of some kind, as, on the whole, commanding more confidence.... A
weighted index number, in one aspect, is almost an unnecessary
precaution to secure accuracy, though, on the whole, the committee
recommend it."
FOOTNOTES:
[30] _Dictionary of Political Economy_, edited by R. H. T. Palgrave.
Vol. II, pp. 384-7. Macmillan and Company, Limited. London. 1912.