Multidimensional Inequality-Sensitive Development Ranking

  • Asis Kumar BanerjeeEmail author
Part of the Themes in Economics book series (THIE)


This chapter starts by stating a number of properties that any multidimensional development index may be intuitively expected to satisfy. It then investigates whether (or under what conditions) we can rank any pair of economies X and Y unambiguously in terms of their levels of development, i.e. whether we can check the veracity of the statement that X is at least as developed as Y as per all development indices having the desired properties. It turns out that, as in the unidimensional case, here too there would be some grey areas of non-comparability. However, the notion of fuzzy binary elations can again be used to formulate fuzzy versions of the proposed conditions on the development index and to obtain fuzzy rankings of economies. These, in turn, can be utilised to obtain crisp (i.e. non-fuzzy) development ranking rules that would reduce these grey areas. An appendix shows that the development ranking method proposed in the chapter can be used to formulate a unifying approach to the problem of obtaining multidimensional versions of various specific unidimensional inequality indices. In the existing literature the multidimensional versions of different unidimensional indices have been characterised by different sets of conditions with no apparent linkage between them. This appendix uses a procedure for obtaining vector representations of distribution matrices to suggest a simple procedure for the task under consideration.


  1. Albert A, Zhang L (2010) A novel definition of multivariate coefficient of variation. Biometrical J 52(5):667–675CrossRefGoogle Scholar
  2. Atkinson AB (1970) On the measurement of inequality. J Econ Theory 2(3):244–263Google Scholar
  3. Atkinson AB, Bourguignon F (1982) The comparison of multidimensional distributions of economic status. Rev Econ Stud 49(2):183–201CrossRefGoogle Scholar
  4. Banerjee AK (2010) A multidimensional Gini index. Math Soc Sci 60(2):87–93CrossRefGoogle Scholar
  5. Banerjee AK (2018a) Normative properties of multidimensional inequality indices with data-driven dimensional weights: the case of a Gini index. Int J Econ Theory 14(3):279–288CrossRefGoogle Scholar
  6. Banerjee AK (2018b) Multidimensional indices with data-driven dimensional weights: a multidimensional coefficient of variation. Arthaniti 17(2):140–156Google Scholar
  7. Banerjee AK (2019a) Measuring multidimensional inequality: a Gini index. In: Gagari Chakrabarti G, Sen C (eds) The globalization conundrumdark clouds behind the silver. Lining Springer, Singapore, pp 65–78Google Scholar
  8. Banerjee AK (2019b) Economic properties of statistical indices: the case of a multidimensional Gini index. J Quant Econ 17(1):41–56CrossRefGoogle Scholar
  9. Boland PJ, Proschan F (1988) Multivariate arrangement increasing functions with applications in probability and statistics. J Multivar Anal 25(2):286–292CrossRefGoogle Scholar
  10. Bourguignon F (1999) Comment on Maasoumi (1999). In Silber J (ed) Handbook of income inequality measurement. Kluwer Academic Publishers, Boston, MA, pp 477–484Google Scholar
  11. Dardanoni V (1996) On multidimensional inequality measurement. In: Dagim C, Lemmi A (eds) Research on economic inequality, vol 6. JAI Press. Stamford, CT, pp 201–207Google Scholar
  12. Decancq K, Lugo MA (2012) Inequality of well-being: a multidimensional approach. Economica 79(316):721–746Google Scholar
  13. Ebert U (2000) Sequential generalized Lorenz dominance and transfer principles. Bull Econ Res 52(2):113–122CrossRefGoogle Scholar
  14. Ebert U, Moyes P (2003) Equivalence scales reconsidered. Econometrica 71(1):319–343CrossRefGoogle Scholar
  15. Epstein LG, Tanny SM (1980) Increasing generalized correlation: a definition and some economic examples. Can J Econ 13(1):16–34CrossRefGoogle Scholar
  16. Fleurbaey M, Trannoy A (2003) The impossibility of a Paretian egalitarian. Soc Choice Welf 21(2):243–263CrossRefGoogle Scholar
  17. Gajdos T, Weymark JA (2005) Multidimensional generalized Gini indices. Econ Theory 26(3):471–496CrossRefGoogle Scholar
  18. Gravel N, Moyes P (2006) Ethically robust comparisons of distributions of two individual attributes. IDEP Discussion Paper 06-05, CNRS-EHESS-Universites Aix-Marseille II et IIIGoogle Scholar
  19. Kolm S-C (1977) Multidimensional egalitarianisms. Q J Econ 91(1):1–13CrossRefGoogle Scholar
  20. Koshevoy GA, Mosler K (1997) Multivariate Gini indices. J Multivar Anal 60(2):252–276CrossRefGoogle Scholar
  21. Lasso de la Vega C, Urrutia A, de Sarachu A (2010) Characterizing multidimensional inequality measures which fulfil the Pigou-Dalton bundle principle. Soc Ch Welf 35(2):319–329CrossRefGoogle Scholar
  22. List, C (1999) Multidimensional inequality measurement: A proposal. Nuffield College, Oxford, Working Paper in Economics No. 1999-W27Google Scholar
  23. Reyment RA (1960) Studies on Nigerian upper cretaceous and lower tertiary ostracoda, Part 1. Stockholm Contributions in Geology, University of Stockholm, StokholmGoogle Scholar
  24. Sen A (1997) On economic inequality. Oxford University Press, OxfordGoogle Scholar
  25. Shorrocks AF (1983) Ranking Income Distributions. Economica 50(197):3–17Google Scholar
  26. Tsui Kai-Y (1995) Multidimensional generalizations of the relative and absolute inequality indices: The Atkinson-Kolm-Sen approach. J Econ Theory 67(1):251–265CrossRefGoogle Scholar
  27. Tsui K-Y (1999) Multidimensional inequality and multidimensional generalized entropy measures. Soc Ch Welf 16(1):145–157CrossRefGoogle Scholar
  28. Van Valen L (1974) Multivariate structural statistics in natural history. J Theor Biol 45(1):235–247CrossRefGoogle Scholar
  29. Van Valen L (2005) The statistics of variation. In: Hallgrimsson B, Hall BK (eds) Variation: a central concept in biology. Elsevier, Boston, pp 29–47CrossRefGoogle Scholar
  30. Voinov VG, Nikulin MS (1996) Unbiased estimators and their applications, vol 2. Kluwer, DordrechtGoogle Scholar
  31. Weymark JA (2006) The normative approach to measurement of multidimensional Inequality. In: Farina F, Savaglio E (eds) Inequality and economic integration. Routledge, London, pp 303–328Google Scholar

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© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.University of CalcuttaKolkataIndia

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