The Generalized “Ideal” Index-Number Formula

THIS paper fills a gap left by earlier work on the design of a satisfactory "atomistic"l index-number system for cases in which two or more multiplicative factors contribute to a product having a unique index expression. In more explicit terms, the problem considered here is the development of a general formula meeting certain well-known requirements and at the same time satisfying the relationship An *Bn =C * -Vn where the n factors on the left are the appropriate indexes of the ai, bi, ci, * (i= 1, * , n), respectively, for the time period t1 with respect to the base period to, and V,, =Zalbic, ... /Eaoboco ... is the unique index of the v aibci * * * . The general formula derived here for A, Bn Cn * * * has some very desirable properties: it includes the Fisher "ideal" index in the special case of two factors (i.e., n=2), satisfies the time-reversal and factor-reversal tests, is determinate for all practical (i.e., non-negative) values of the variables, is reducible to an internal mean of relatives, and, in particular, meets the proportionality test. The theoretical merit of this general formula may best be appraised, perhaps, in the light of the earlier contributions. In 1931, J. K. Wisniewski derived a general expression which he considered to be an extension of Fisher's "ideal" index to n factors.2 In 1937, C. Gini criticized WiAniewski's formula, devised another for the case n=3, and stated that his own expression could be generalized.3 In the same year, J. K. Montgomery, starting alternatively from certain algebraic considerations and from differential expressions analogous to those introduced by F. Divisia, developed a novel general formula.4 Gini objected to WiAniewski's formula on the ground that it not a true average: it does not necessarily lie between the largest and smallest relatives of