§ 9. Summary
20th Century Irving Fisher EnglishThe conclusions of this Appendix may be briefly stated as follows:—
- Any sum of products of two factors each, such as SpQ, may be converted into any one of three forms: (1) PT, in which P is an average of the ratios of the p's to some base p0's, and T is the quotient
(2) A Q, in which Q is an average of the ratios of the Q's to some base Q0's, and A is the quotient;
(3) PQSp0Q0.
Of the foregoing three formulæ only the last is symmetrical in the sense that the p's and Q's are treated alike.
P and T (or P and Q) are said to be correlative, and any particular formula for either implies a particular correlative formula for the other.
Two correlative formulæ for P and Q are, in general, quite unlike each other. If like formulæ be constructed for P and Q, the correlate of Q constitutes a new formula for P, said to be antithetical to the original formula for P, and vice versa.
There are an indefinite number of formulæ for P, of which 44 are given in the table; and there are at least 8 important tests to which each formula may conform in one of three degrees (1)wholly, or for the ratio of P1 to P2, each being relative to a third year as a base; (2) partially, or for the ratio P1 to unity; and (3) not at all.
The eight tests are of proportionality as to prices or quantities (1 and 2); determinateness as to prices or quantities (3 and 4); withdrawal or entry as to prices and quantities (5 and 6); changing base as to prices and quantities (7); and changing units as to prices and quantities (8).
The tests arrange themselves in pairs, one of each pair relating to the p's in the same manner as the other is related to the Q's; but each test has significance both with respect to the p's and the Q's.
The formulæ arrange themselves in antithetical pairs.
Of any four neighboring compartments in the table, relating to two correlative rows (tests) and two antithetical columns (formulæ), the diagonals will have the same "scores."
No known form of index number P conforms perfectly to all the eight tests when a common base year is employed, but several conform well, the best being formula No. 11,
But if we are content with year-to-year comparisons, renouncing comparisons in terms of a third year, there are two formulæ which conform perfectly, viz. formulæ 17 and 21.
Practically, however, formula 11 is superior to 17 or 21, and formula 9 (median)—when properly "weighted"—is superior to 11.
For practical purposes, therefore, unless the expense and labor of computation can be disregarded, the median (with its two neighboring quartiles) is recommended, with a simple system of weights (whole numbers) based on expenditures, and changing from time to time for the sake of making better year-to-year comparisons.