Journal of the Italian Statistical Society

, Volume 1, Issue 3, pp 359–376

# A general function of axiomatic index numbers

• Marco Martini
Article

## Summary

This paper describes an axiomatic approach to index number theory. A general bilaeral formula which generates a set of indices of prices and quantities is proposed. This formula is written as a geometric mean, weighted with logarithmic means of relative values, and includes all of those indices and related cofactors which satisfy the following axiomatic properties: strong identity, commensurability, linear homogeneity and associativity (or monotonicity). Moreover, two subsets of indices are identified: the first subset includes those which can be expressed either as geometric means or as expenditure ratios; the corresponding bounds are given by the Laspeyres' and Paasche's indices. The second subset also satisfies the desired properties of base and factor reversibility; in this case the bounds are given by Sato Vartia's and Fisher's indices. In addition it is shown that the intersection between the two subsets identifies a new bilateral ideal index which satisfies the axiomatic and reversibility properties (base and factor) and may also be written as an expenditure ratio. All other formulas, which do not belong to the family of the proposed indices, violate some of the axiomatic properties. For multilateral comparisons, a mixed system of direct and indirect indices which satisfies the transitivity condition is proposed.

## Keywords

index numbers axiomatic approach log change index base and factor reversibility

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© Società Italiana di Statistica 1992

## Authors and Affiliations

• Marco Martini
• 1
1. 1.Università degli Studi di MilanoMilanoItalia

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