§ 7. The 44 Formulæ Compared

We have gone through the reasoning by which the conformity of one pair

(P1 =; T1 = Sp0Q1)

out of the 44 pairs of index numbers of prices and trade, given in the table, are tested and graded with respect to the eight tests. The table contains for the remaining 43 formulæ the results of similar reasoning. This reasoning is here omitted to save space. The mathematical reader who chooses can verify the results as tabulated. He can also prove the relationship by which it follows that the figure in any column for the odd tests corresponds to that in the neighboring antithetical column for the even tests. In consequence of this relationship the sum of any column for the odd tests equals the sum in the antithetical column for the even tests. In fact, the table is full of correspondences and relationships of many kinds.

The footings give us a means of comparing the merits of the various index numbers. These footings are intended to express, so far as may be, the fitness of the formulæ to serve as index numbers for price levels. Consequently the score for test 6 should be omitted from the footing, as test 6 has no value in regard to prices. (If it be desired to compare the scores of the correlative index numbers for quantities or trade, the score of test 6 will be included, but that of test 5 should then be omitted.)

Thus a perfect score would be seven. The highest score in the table is 5½, the lowest, 2.

It would, of course, be absurd to compare the merits of index numbers merely by their "score" in the table. This score is more or less arbitrary, and it treats all seven tests as equally important. Yet it affords at least some insight into the comparative characteristics of the 44 formulæ. It is noteworthy that, in general, the simplest formulæ have high scores and the most complicated have low scores. Thus formulæ 1 (Dutot), 7 (simple geometric), 9 (median), 11 and 12 (Scrope) have scores of 5 and 5½. The only others as high as 5 are "mixtures" of formulæ 11 and 12. The simple arithmetical (3) and simple harmonic (5) have a score of 4, which is fairly high. The more complicated forms which have fairly high scores are in several cases "mixtures," averages, or antitheses of simple formulæ 11 and 12.

The above comparisons treat all the other seven tests as of equal importance. But they are not of equal importance. Since opinions might differ as to the exact relative importance of the various tests, we shall not attempt to "weight" them. Nor will this be necessary in order to decide the question of most importance to us, viz., which of the 44 index numbers meet the tests most completely. Tests 3 and 4 are probably of little practical importance as compared with the remaining tests. Test 2, on the other hand, may be accorded chief importance, for reasons given in section 5 of this Appendix and in Chapter II. In order to select the best index numbers of prices, therefore, let us first rule out of the competition all the 18 formulæ which have "0" for test 2. We have left the following formulæ classified into two groups.

(Omitting test 6), score for formulæ which do not completely fail on test 2,

If, next, we rule out of the competition from among those which completely meet test 2, all except those which have scores of 4½ or above, we have left only formulæ numbers 10 and 11. Among the formulæ which only partially meet test 2 we may eliminate all which fail to exceed 4½ in total score; for, although those which reach 4½ tie formula 10, yet when all tests are counted as of equal importance, they are inferior in not completely meeting the most important test—test 2. Putting the matter in another way, we may say that if test 2 should be weighted more heavily than the other tests, the scores of those formulæ half meeting that test which now tie formulæ wholly meeting that test would fail to do so and would therefore drop out of competition with formulæ 10 and 11 in the first column.

Eliminating therefore from the second column all formulæ with scores of 4½ or less, we have as the only rivals of formulæ 10 and 11, formulæ 12, 17, and 21, having scores of 5½, 5, and 5 respectively. Our best formula, therefore, should be found among numbers 10, 11, 12, 17, 21. We shall therefore examine with particular care these five surviving competitors.

These all conform to tests 3, 4, and 8. Comparing them in other respects, we find:—

Tests 17 and 21 have scores identical in every instance, and may therefore be said to be tied.

Comparing tests 11 with 17 (or 21), we see that 11 excels in respect to the important test 2, and 17 in test 7. As test 2 is regarded as of more importance than test 7, we may safely give the preference to formula 11 over 17 (or 21). We therefore now strike out 17 and 21 from the competition.

We have left formulæ 10, 11, 12; comparing 10 and 11, we note that 10 excels in test 7, while 11 excels in tests 1 and 5. If we may be allowed here to exercise a comparative judgment, we shall say that the superiority in the one test 7 is more than offset by superiority in the two tests, 1 and 5. We therefore eliminate formula 10.

We now have left only the two formulæ, 11 and 12. There is not much to choose between them. While 12 has the higher score when all tests are counted as of equal importance, 11 excels in the most important test 2, and we are therefore inclined to give it the preference.

According to our judgment, therefore, test 11 emerges as the winner in the score contest. It has also the advantage of being among the very simplest formulæ and of having as its correlative formula for T the simplest of all formulæ for T, viz. T1 = Sp0Q1.

In nonmathematical language, the pair of formulæ 11 mean that the level of prices in any year is found by dividing the total value of the quantities sold in that year by what that value would have been at base prices, and that the trade index in any year is simply the value of the quantities sold in that year reckoned at base prices.

Applying formula 11 to the equation of exchange, we have—

We wish now to emphasize once more the virtues of this formula 11 in respect to test 2. The equation of exchange, stated above, is intended to show how prices are affected by changes in M, M', V, V' or the Q's. It is evident from the original form (1) of this equation that a proportional change in the M and M' (if the V's and Q's remain unchanged) will affect all the p1's in exactly the same ratio, or else raise some prices more and others enough less than this ratio to compensate in the sense that the equation of exchange will be preserved. In some sense, therefore, the general level of prices varies exactly with M and M'. Form (3) enables us to express this proportionality by formulating the price level as the fraction.

This varies directly with the M's.

In precisely the same way we are enabled to state that a uniform change in the two V's, or any change in the left side of the equation as a whole, will affect prices in precisely the same ratio (the Q1's being assumed constant). We may also say that a uniform change in the Q1's will affect T1 in exactly the same ratio, and P1 in exactly the inverse ratio (assuming the left side of the equation to be unchanged). In fact, if we use formula 11 to express the average price ratio, we are able to state in all cases (so long only as the Q1's change in unison or not at all) that prices rise or fall "on the average" directly as the left side of the equation, and inversely as the Q1's.

As noted, these are the basic theorems for which the equation of exchange stands. We would naturally like to remove the restriction as to the Q1's changing uniformly. We should consider an index number perfect (so far as needed in the equation of exchange) if we could assert of it the same theorem of proportion as above, without the restriction as to the Q1's changing uniformly, so that we might substitute an average change in the Q's in place of a uniform change. No such index is found in the table, and no such index seems possible. Practically this conclusion does not greatly matter, for we are interested in prices far more than in quantities, the latter being chiefly important as supplying weights for the price indexes. As we have already noted, Edgeworth has shown that considerable variation in weighting is of comparatively little practical importance.

The chief use of index numbers is to compare successive years, not years remotely distant from each other. We are not so much interested in comparing the prices of 1909 and 1910 each with those of 1873 as we are in comparing them with each other. In fact, the chief use of 1873 as a base year is to enable us to compare any other two years with each other. But only a few index numbers which afford a true comparison between any year and another year as the base will give a true comparison between any two years, each in terms of a third year as the base. These few index numbers are those which completely meet the base-shifting test 7.In the table the only formulæ which come up to this requirement are formulæ numbered 1, 2, 7, 8, 43, 44, to all of which there are serious objections on other grounds. Formulæ 1 and 2 are very arbitrary, having "haphazard weighting"; formulæ 43 and 44 have the lowest scores in the table; formula 7 has no system of weighting; and formula 8 becomes zero if a single quantity, as Q, should disappear from a year's sales.

The question therefore arises, why should we, as has usually been done, construct out index numbers with reference to a fixed base in terms of which we indirectly compare two given years? Why not make the comparison directly? The indirect comparison introduces an error in all cases except of those formulæ which conform to test 7. In these cases the indirect comparison cannot, of course, give any better result than the direct comparison, while in all other cases the direct comparison is better.

It seems, therefore, advisable to compare each year with the next, or, in other words, to make each year the base year for the next. Such a procedure has been recommended by Marshall, Edgeworth, and Flux.It largely meets the difficulty of non-uniform changes in the Q's, for any inequalities for successive years are relatively small.

Such successive index numbers, each on the basis of 100 per cent for the previous year, will, if multiplied together, give a chain of index numbers showing the fluctuations from year to year, like any ordinary series, but much more suitable for comparison of neighboring years.

Let us now reëxamine the comparative merits of index numbers on the supposition that they are to be used only for successive years, that is, for comparison between each year and the previous year as a base. In this case we do not need to distinguish between a "partial" and a "complete" fulfillment of the tests. We may therefore now substitute "1" for every "½" Omitting, as before, all formulæ which fail to meet test 2, we have the following results:—

We note that formulæ 11 and 12 have scores of 6 each, while their average 15 (and 16) and their mixture 18 and also 22 have the same score, but that formulæ 17 and 21, which are mixtures of 11 and 12, have perfect scores, 7. Each of these two formulæ uses as weights the average of the weights used in formulæ 11 and 12. Theoretically, therefore, we find two formulæ which fit all tests perfectly so far as year-to-year comparisons of prices are concerned.

Where, therefore, great accuracy is desired and there exist abundant funds to provide for the laborious computations necessary, we may recommend the use of formula 17 or 21. This presupposes that statistics are available for the Q's, which is not usually the case.

Thus far our conclusions therefore are (1) that theoretically formula 11 is the best when each year is expressed in terms of a common base; (2) that (also theoretically) formula 17 and 21 are slightly superior when each year is expressed in terms of the preceding year as base, and that these two meet all tests for year-to-year comparisons.