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Journal of the Italian Statistical Society

, Volume 1, Issue 3, pp 359–376 | Cite as

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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References

  1. Benedetti C. (1962), Teorie e tecniche dei numeri indici.Metron, vol. 22 n. 1–2.Google Scholar
  2. Benedetti C. (1972), Vecchi e tradizionali indici dei prezzi ricondotti a moderni indici funzionali a costante utilità.Metron, 2, 67–86.Google Scholar
  3. Biggeri L. (1984), Teoria e pratica dei numeri indici: vecchi orientamenti e recenti sviluppi. In:Atti della XXXII Riunione Scientifica della Società Italiana di Statistica, Sorrento, vol. II, 327–352.Google Scholar
  4. Biggeri L. andGiommi A. (1987), On the accuracy and precision of the consumer Price, Indices, Methods, and applications to evaluate the influence of the sampling of households.ISI, Tokyo Session, (I.P. 12.1, pp. 137).Google Scholar
  5. Chisini O. (1929), Sul concetto di media.Periodico di matematica, n. 2.Google Scholar
  6. De Finetti B. (1931), Sul concetto di media.Giornale dell' Istituto italiano degli attuari.Google Scholar
  7. Diewert W. E. (1976), Exact and superlative index number. In:Journal of Econometrics, vol. 4, 115–145.Google Scholar
  8. Drechsler L. (1973), Weighting of Index numbers in multilateral international Comparisons.The Review of Income and Wealth, ser. 19, n. 1, 17–34.CrossRefGoogle Scholar
  9. Edgeworth F. Y. (1925), The plurality of index numbers.The Economic Journal, vol. XXXV, n. 139, September, 379–388.CrossRefGoogle Scholar
  10. Eichhorn W. andVoeller J. (1976),Theory of the price index: Fisher's test approach and generalisations. Lecture notes in economics and mathematical systems, Berlin: Springer-Verlag.Google Scholar
  11. Fisher I. (1911),Purchasing power of money, New York: Macmillan.Google Scholar
  12. Fisher I. (1922, 2a ed. 1926),The making of index numbers, Boston: Houghton Mifflin.Google Scholar
  13. Geary R. C. (1958), A note on the comparison of exchange rates and purchasing power between countries.Journal of the Royal Statistical Society, A, 121, 97–99.Google Scholar
  14. Gerardi D. (1982), Selected problems of the intercountry comparisons on the basis of experience of the EEC.The Review of Income and Wealth, vol. 28, n. 4, 381–406.CrossRefGoogle Scholar
  15. Gini C. (1931), On the circular test of the index number.Metron, vol. 4, n. 2, 3–24.Google Scholar
  16. Khamis S. H. (1972), A new system of index numbers for national and international purposes.Journal of the Royal Statistical Society, A, vol. 135, part 1, 96–121.Google Scholar
  17. Koves P. (1983),Index theory and economic reality, Budapest: Akademiai Kiado.Google Scholar
  18. Martini M. (1977), Il confronto spaziale degli aggregati economici: I) Una proposta di metodi generalizzati… In:Rivista di Statistica Applicata, vol. 10, n. 2, 75–94.Google Scholar
  19. Martini M. (1977), Il confronto spaziale degli aggregati economici: II) Una valutazione comparative dell'impiego di diversi… In:Rivista di Statistica Applicata, vol. 10, n. 3, 179–192.Google Scholar
  20. Martini M. (1990). Il confronto dei valori nello spazio. In:Statistica Economica (a cura di G. Marbach), Torino: Utet.Google Scholar
  21. Martini M. (1992),I numeri indice in un approccio assiomatico. Milano: Giuffrè.Google Scholar
  22. Parenti G. (1948),Lezioni di statistica economica, Genova: Libreria Mario Bozzi.Google Scholar
  23. Samuelson P. A. andSwamy S. (1974), Invariant economic index numbers and canonical duality: survey and synthesis.The American Economic Review, vol. 64, n. 4, 566–593.Google Scholar
  24. Sato K. (1974), Ideal index numbers that almost satisfy factor reversal test.The Review of Economics and Statistics, 56, 549–552.CrossRefMathSciNetGoogle Scholar
  25. Sato K. (1976), The idela log-change index number.The Review of Economics and Statistics, vol. 58, n. 2, 223–228.CrossRefMathSciNetGoogle Scholar
  26. Theil H. (1960), Best linear index numbers of prices and quantities.Econometrica, vol. 28, n. 2, 464–480.zbMATHCrossRefMathSciNetGoogle Scholar
  27. Theil H. (1973), A new index number formula.The Review of Economics and Statistics, vol. 55, n. 4, 498–502.CrossRefMathSciNetGoogle Scholar
  28. Tornquist L. (1937), The consuption price index of the Bank of Finland,Bank of Finland Monthly Bulletin, 10, 1–8, 1936 e 73–95.Google Scholar
  29. Vartia Y. O. (1976), Ideal log-change index numbers.Scandinavian Journal of Statistics 3, 121–126.MathSciNetGoogle Scholar
  30. Vogt A. (1979),Das statistische Indexproblem im zwei-Situationen-Fall, Zürich: Juris Druck-Verlag.zbMATHGoogle Scholar
  31. Walsh C. M. (1901),The Measurement of General Exchange-Value, New York: Macmillan.Google Scholar

Copyright information

© Società Italiana di Statistica 1992

Authors and Affiliations

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

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