§ 4.Review of 44 Formulæ, heading the Table Columns

We shall now briefly review the formulæ of the table of selected index numbers. Each even-numbered formula is best regarded as derivable from the odd-numbered formula at its left as its antithesis. The odd formulæ are those constructed directly for the p's without reference to any average for the Q's; the even are constructed indirectly by reference to some average first assumed for the Q's. The latter are what Walsh had in mind under the name of "double weighting."

Formula (1) is simply the ratio of the sums of prices. It may also be considered as the ratio of the averages of prices in the two years considered, as is evident by writing it,—, where n is the number of commodities employed.

This formula was used by Dutot in 1738,and has been used recently by Bradstreet,who applied it practically.

Although it is a ratio of average prices, it may also be thrown into the form of a weighted arithmetical average of the price ratios,, etc., as the following transformation shows:—

The formula in this last form is evidently the weighted arithmetical average of the ratios in parenthesis, the weights being the prices p0, p'0, p''0,...of the year 0. A change in the units of quantity for the various goods would change these prices; thus a change from ounces to pounds would multiply the number expressing price by sixteen. Such a change in any price, such as p0, would entirely change the relative importance of the "weights," p0, p'0, etc. Consequently this system of weighting is, as Walsh says, quite accidental or haphazard.

The same formula is also a harmonic average, as the following transformations show:—

The last expression is evidently the reciprocal of a weighted arithmetical average of the price ratios in parenthesis. But these price ratios are the reciprocals of, etc. In other words, the formula is the reciprocal of a weighted arithmetical average of the reciprocals of the ratios p1/p0, etc. It is therefore the weighted harmonic average of these ratios p1/p0, etc., the weights being p1, p1', etc., or the prices of the year 1.

In short, formula (1) is both an arithmetical and a harmonic average of, etc., the weights being, in the first case, the terms of the denominator, and in the second, those of the numerator.

We have seen that formula (1) in the table, although primarily a ratio of averages of prices, may be also considered as an average of ratios of prices with arbitrary weighting.

Conversely we may, if we choose, regard every average of ratios as a ratio of averages by assuming arbitrary units for measuring commodities. It is evident that, if the unit of measure is increased in any ratio, the number expressing the price is decreased in the inverse ratio. If, therefore, we change the unit of measure of a commodity the price of which is at first expressed by p1 by dividing by the ratio p0, this price becomes p1/p0. Thus p1/p0 may be considered to be a price as well as a price ratio. Hence an average of

etc., may be regarded as an average of prices. The new units, instead of being pounds, yards, etc., are dollars-worth-in-the-base-year. With these units, the price in the base year is unity, for dividing the price, p0, in the original units by the factor p0, we obtain unity.

Hereafter, however, we shall treat all index numbers as averages of price ratios.

It is interesting to note that the antithesis of Dutot's or Bradstreet's formula (No. 2), found by dividing the fraction

by the correlative formula for Q1, viz., turns out to be that advocated by Drobisch,and earlier by Sir Rawson-Rawson.

Formula (3)is evidently the familiar simple arithmetical average,—

Formula (4), the antithesis of formula (3), gives, as the average price ratio, the ratio of total values Sp1Q1/Sp0Q0 corrected for change in the Q's by division by the arithmetical average ratio of the Q's.

Hereafter the even-numbered formulæ, being antitheses of the preceding odd formulæ, will be passed over unless there is, in any case, special reason for mention.

Formulæ (5), (7), and (9)are respectively the simple harmonic, simple geometric, and simple median averages. We note that the antithesis of (7), viz. (8), is one proposed by Nicholson and Walsh.

Formula (11)resembles Bradstreet's, except that the introduction of the Q's as multipliers prevents the weighting from being arbitrary; for the weights p0Q1, etc., unlike the weights p0, etc., are uninfluenced by a change in the units of measurement for commodities. Whether an article be measured in pounds or ounces will not affect the value of a given amount of it. The following transformations show that the formula is a weighted arithmetic mean:—

The last expression is evidently a weighted arithmetic average of the price ratios in the parentheses, the weights being p0Q1, p'0Q'1, etc., i.e. the values of the quantities in the year 1 reckoned at the prices of year 0.

But the same formula is also a harmonic average, as may be seen by transforming the denominator instead of the numerator as was done for formula (1). It is a weighted harmonic average, the weights being p1Q1, p'1Q'1, etc., or the values in the year 1.

In short, formula (11) or

is, like formula (1), both a weighted arithmetical and a weighted harmonic average of, etc., but the weights are different in the two cases.

(11) has the interesting property that its antithesis (12) is of the same form except that the subscripts for Q are now 0 in place of 1. Similar reasoning shows that this formula (12) is also both an arithmetical and an harmonic average, weighted according to the terms in its denominator and numerator respectively.

These two formulæ, (11) and (12), seem to be the favorites among writers on Index Numbers. Since the shortcomings of one are, in some cases, not shortcomings of the other, there have been many attempts to combine them into some composite. No. (13),for instance, is their simple arithmetical average. The antithesis of (13), viz. (14), turns out to be the simple harmonic average of (11) and (12). Number (15) is the simple geometric average of (11) and (12). This formula (15) has the distinction of being identical with its own antithesis (16). Numbers (17), (19), (21), and (23) are other attempts at combining (11) and (12), not by averaging them, as was the case with (13) and (15), but by averaging their coefficients, viz., Q1 and Q0, Q'1, and Q'0, etc. Two antitheses of these, namely (18) and (22), turn out to be formulæ proposed by Walsh, and a third (24) to be one proposed by Julius Lehr.

We have seen that the formulæ (11) and (12) considered as arithmetical averages have for weights

thus completing the four permutations of the subscripts, 01, 00, 11, 10. Number (29) represents a weighted arithmetical average in which the weights are derived from other considerations than the product of the prices and quantities of the base year (1). An instance is the method employed in some of the tables in the "Aldrich Report,"the weights being the percentage of consumption of various kinds in workingmen's budgets without reference to the base year or any other particular year.

Numbers (31) and (33) are weighted harmonic means in which the weights instead of being

thus completing for harmonic averages the same permutations of subscripts as before for arithmetical averages. We see then that the odd formulæ (11) to (33) inclusive are merely arithmetical averages or harmonic averages of p1/p0 etc., or else averages or mixtures of such averages.

Numbers (35), (37), (39), (41) are various forms of weighted geometric averages of those price ratios, the weights being

Number (43) is the ratio of the weighted geometric average of the prices in years 1 and 0, the weights being p1Q1, etc., for year 1 and p0Q0, etc., for year 0.

It will be seen that all of the 44 formulæ selected for the table are based on a few simple principles of averaging. Most are arithmetic, harmonic, or geometric averages or their combinations. Needless to say, numerous other and more complicated forms might be constructed.