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Rh of units of the commodities in the standard budget, and P₁, P₂, P₃ .. . the prices per unit at the date taken as starting point, and we write Q₁×P₁ = E₁, Q₂×P₂ = E₂. . .. where E₁, E₂, E₃. . . are the expenditures on the commodities, then E = E₁ + E₂. . . = Q₁×P₁ + Q₂×P₂ . . . is the whole expenditure at the first date on the standard budget. Let p₁, p₂, p₃. . . be the prices per unit at a subsequent date; then Q₁×p₁ = e₁, Q₂×p₂ = e₂. . . are the presumed expenditures, and e = e₁ + e₂ +. . . = Q₁×p₁ + Q₂×p₂ +. . . is the whole expenditure. The ratio of the cost of the standard budget at the second date to that at the first, is $e⁄E$ = $Q₁×p₁ + Q₂×p₂ +. ..⁄Q₁×P₁ + Q₂×P₂ +. ..$ = $E₁×r₁ + E₂×r₂ +. ..⁄E₁ + E₂ +. ..$ where r₁ = p₁/P₁ (the price ratio for the first-named commodity at the two dates), r₂ = p₂/P₂. . . . The last expression shows that by this method the ratio of the costs of living is a weighted average in which the price ratios are weighted by the expenditures at the first date; hence we only need to know these expenditures and ratios, and not the actual quantities nor prices. In the official measurement in the United Kingdom only the quantities E and r are in fact used; this method is very convenient in dealing with rent (for which there is no natural unit of quantity) and with clothing (for which a general price ratio is obtained without any definition of unit). The general theory of weighted averages shows that a considerable roughness in the estimation of the smaller expenditures is smoothed out in the process of averaging, but that it is important to obtain precision in the case of large items, such as clothing, treated in a single entry, and rent. It is important, however, that the r's should be accurately known when they differ much from one another, and the quality of the commodities that are priced should be the same at both dates.

The index number for the second date is $e⁄E$×100, and the percentage increase is ($e⁄E$−1)×100.

Case (c). It must be granted that when the cost of living is compared at two places or at two dates we ought not to assume that precisely the same quantities of the same commodities are purchasable in both cases, and in order to make a strict numerical comparison we need a test of equality of standard if not a means of comparing two standards. The problem so stated has not yet been completely solved. A measurement could be made on a strictly nutritive basis and the cost of purchasing in the most economic way the amount of calories (including the necessary protein) considered proper to health and efficiency could be ascertained in both countries or at both periods; but this would only give a theoretic solution, since it ignores the influence of custom and taste in diet, and, in fact, in developed countries relatively few people have been compelled to purchase their nutriment in the cheapest possible way. The actual practical question in England in 1921 was what was the cost of maintaining the pre-war standard of living in nutritive power and satisfaction or pleasure derived from food and clothing, allowance being made for changes in prices and available qualities. This statement introduces the vague word satisfaction, which it is not practicable to define exactly, though some mathematical methods based on economic principles have been suggested for ascertaining its equality in two cases.

It has been suggested (Bowley, “Measurement of Cost of Living,” Journal of the Royal Statistical Society, May 1919, p. 354, and “Cost of Living and Wage Determination,” Economic Journal, March 1920, p. 117) that an approximation could be reached by devising “a diet, based on available supplies, as nutritious, digestible and not less attractive than the pre-war diet, and estimate at what price it could now be obtained,” or “to frame a new budget of goods obtainable, and, in fact, purchased, by housekeepers with the same skill of adjusting purchases to desires as in the case of the earlier budgets. Instead of measuring satisfaction by formula, we may recognize that it is subjective and a matter of opinion, and obtain from representative working-class women a budget which in their opinion would now give the same variety and pleasure as a selected budget of 1914, care being taken that the energy value is the same. The result would give a new conventional budget, the ratio of whose cost to (that of) the pre-war budget would give a rough measure of the change of. . . the cost of living.” It should be added that this solution would only be definite if the “satisfaction” was obtained as cheaply as possible, it being assumed that before the war given sums of money were laid out to the best advantage. This method would only be satisfactory if fairly close agreement was obtained as to the equality of the new with the old standard.

Another method has been used in the case of comparison of the cost of living in two places. In 1905 the Labour Department of the Board of Trade (United Kingdom) initiated inquiries about the cost of living in the United Kingdom, United States, France, Belgium

and Germany, and obtained budgets of expenditure in each country; the results are published in the official papers Cd.3864, Cd.5609, Cd.4512, Cd.5065 and Cd.4032. A comparison was made between the cost of living in the United Kingdom and in each other country on a double basis, as follows:—it was found that an English housewife purchasing in 1909 in the United States a week's supply of food as customary in England would have spent 38% more in the first-named country, the ratio of the costs of living being on this basis 100:138; on the other hand, an American housewife purchasing in England a week's supply of food as customary in America would have found her expenses reduced in the ratio 125:100 (Cd.5609, pp. lxvi., lxvii.). If these ratios had been reciprocate, either would measure the difference in the cost of living (so far as food is concerned); as it is, their divergence illustrates the want of definiteness in the problem. Now it is quite possible to obtain in any country a current budget to be compared with a pre-war budget and the method just described can be applied. Thus, in the Journal of the Royal Statistical Society, May 1919, p. 344, details are given of the standard pre-war British budget and of the average of budgets collected by an official committee on the cost of living in the last year of the war, in which the standard of living had been modified and had fallen somewhat. A housewife purchasing in 1918 the same qualities and quantities of food as in 1914 would have increased her expenditure in the ratio 100:212, while if she had purchased in 1914 the same qualities and quantities as in 1918 the ratio of the earlier to the later expenditure would have been 100:202. Both these are possible measurements (the first being identical with case a above), and where the difference between them is so moderate an intermediate number, such as the arithmetic or geometric mean (which are nearly coincident), 100:207 makes a plausible measurement of the change.

Another method, allied to that just described, gives perhaps the most practical solution, though its adequacy can hardly be proved from theoretic conditions. Obtain typical budgets of expenditure at two dates; compile a new or mean standard of quantities which item by item are the averages of the entries in the budgets; thus, if in one the consumption of 33 lb. of bread is stated, in the other 35 lb., enter 34 lb. in the mean standard; now find the cost of the mean standard at each date and take the ratio of these costs as the measurement of the change in the cost of living. In the example just used this ratio was found to be 100:204. (On the methods formerly used for this problem, see Palgrave's Dictionary of Political Economy, vol. iii., article “Wages, Nominal and Real,” p. 640.)

If all prices rose in the same ratio the methods now described would necessarily yield the same result; the need for choice arises from inequalities of increase, including the case where the goods are no longer in the market as one where the price is indefinitely great. Now if at one date purchases are made so as to maximize the satisfaction in the outlay of the week's housekeeping allowance, as we may reasonably assume, and prices rise irregularly, it is evident that somewhat less will be bought of the commodities which have risen most and more of those which have risen least if a maximum is still obtained, and that consequently the increase in the expenditure necessary to obtain the same satisfaction as before is less than the increase if exactly the same quantities had been purchased. For example, if oranges are doubled in price and bananas increased only by one-half, more bananas and fewer oranges will be purchased.

If with the notation used above we also write q₁, q₂, q₃. . . for the quantities purchased at the second date, the measurement obtained by using these quantities is $q₁p₁ + q₂p₂ +. ..⁄q₁P₁ + q₂P₂ +. ..$ = $1⁄100$I₂ (say) instead of $Q₁p₁ + Q₂p₂ +. ..⁄Q₁P₁ + Q₂P₂ +. ..$ (as above) = $1⁄100$I₁ (say). If the small letters refer to a second place (instead of date), then as between England and America I₁ = 138 in the illustration I₂ = 125. For two dates the method illustrated from expenditure on food in England gives I₁ = 212 and I₂ = 202, and the suggested index number is I₃ = $1⁄2$(I₁ + I₂) = 207. The other method recommended is to take I₄ = $1⁄2(Q₁+q₁)p₁ + 1⁄2(Q₂+q₂)p₂ +. ..⁄1⁄2(Q₁+q₁)P₁ + 1⁄2(Q₂+q₂)P₂ +. ..$ × 100. It is easily shown that I₄ is always intermediate between I₁ and I₂, and by a more troublesome analysis that I₄ is less than I₁ when prices in general are rising and quantities consumed of individual goods have increased or diminished according as their prices have risen more or less than the average as measured by I₁; in fact

where 100r = I₁, and the factors in each term of the numerator are both positive or both negative under the conditions named. Hence, I₄ satisfies many of the fundamental conditions of the measurement required. Bowley (Stat. Journal, loc. cit., p. 351) suggests as