CHANGE IN A FIXED AGGREGATE (b) The method generally used in the more objective problem of measuring the change of cost of a fixed aggregate of goods is as follows : the quantities of defined commodities which are produced or consumed in a year, or exported or imported, or are purchased in a week by a family, are estimated and the prices per unit are ascertained in a base year or period and also in the year to which the measurement is to refer. This budget of quantities is valued at the base year prices and again at the prices of the year in question ; the latter total expressed as a percentage of the first gives the required index number. The computation is often effected by writing down the relative expenditure (quantity X price) on each commodity in the base year, applying to each the percentage that its price in the second year forms of that in the first, and adding the products. This sum (divided by the total of the base year's relative expenditures) gives the same index number as be fore. Cost of living index numbers are computed by this method. In this form the index number appears as a "weighted average," where the weights are the relative expenditures and the things weighted are the price-relatives. From the theory of weighted averages it is known that considerable roughness in the weights has little effect on the result. It may also be regarded as an aver age of price-relatives which form a sample of a larger group than that included, and therefore—as under (a)—its precision depends on the square root of the number of independent relatives and, inversely, on their dispersion about their average.
The weakness of this method is that, when we make a compari son between two years or two places, the relative expenditures generally differ and each scheme of expenditure appears to have an equal claim to be included. This difficulty may be met theo retically by computing the index twice, first with the expenditures in year or place A and then with those in B and averaging the result. For comparison between places this method is applied ; but it is seldom that the necessary "weights" can be obtained for more than one year, and till there is a further census of production or a new collection of working class budgets the double computa tion cannot be made. Further, if the weights are available for three years or places, A, B and C, the index for B in reference to A multiplied by that for C in reference to B does not by any weighted average formula give that for C in reference to A (=loo), as it should. This so-called "condition transitive" is not satisfied by any formula that is symmetrical with regard to weights. The algebra of the method is as follows:— Write ,Q,, . . . and . . . for quantities and prices in A, • • , UPI, bP2 • . . in B. The "forward" number is I.= Ioo/aQ,•bPi=E.Q,•aPi and the "backward" number is The Geometric is Prof. Irving Fisher's "Ideal index number" Making of Index Numbers, 1922, p. 22o). The arithmetic 2 and the form I ood 2 (aQ,-}-bQl) •bP1=Z i bQl) •aPi, may be conveniently used. The three forms of average may be expected to give nearly identical results. in which the quantities at the two dates are averaged, is perhaps the simplest in idea. Each average lies between I. and Ib, and is greater than I. if an increase of prices of a commodity above the general crease from year A to a later year B is correlated with a relative decrease of the quantity purchased, as may be expected if there is a possibility of substitution of one commodity for another out any general change of standard (Bowley, Statistical Journal, pp. 343 seq.).
In their measurement of the change of import or export index prices from year A to the consecutive year B, the Board of Trade uses the formula and for comparison with the next year C the formula I ooX bQl F+ PQl °P' which does not equal too obtained by direct re-valuation of the C quantities by the A prices and, if there is a very rapid change of and prices, may differ considerably from it. This is an of the "step-by-step" or "chain" method of index numbers. Note that from A to B, 100 is an index number of quantity aQraP1 of trade (prices constant), and 1 oo bQ' • is the index number • of value of trade. The product of these measurements of price and quantity is the index of value (X I oo) .
Since no form of index number satisfies all the conditions which can properly be laid down (see e.g., Gini, in Metron, 1924, pp. 81 and 134), we must select in each case the form most suited for the particular purpose in hand (regard being had to the data available) and it follows that we can only expect precision when different relevant forms give approximately the same result, for which the conditions are that the dispersion of prices from their average should be small and the number of constituent elements should be considerable. See COST OF LIVING: PRICES, STATISTICS OF. (A. L. B.)