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[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 £60,000,000, and on meat £100,000,000: 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."