Measuring Development pp 133-154 | Cite as

# Multidimensional Lorenz Dominance

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## Abstract

In this chapter, we begin to take cognisance of the *multidimensional* nature of development. However, we approach the matter in steps. Since we desire an inequality-sensitive measure of development, we first consider the problem of how to measure *multidimensional inequality*. This is the subject matter of this chapter. As in the unidimensional case, here again, inequality can be measured in various ways. Again, however, there is no guarantee that if inequality as measured by a particular multidimensional inequality index is seen to decrease, the same will be true of inequality measured by a different index. This suggests that a more appropriate procedure would be to look for a way to extend the concept of Lorenz dominance from the unidimensional context to the multidimensional one. The chapter starts by stating a number of conditions or properties that one would intuitively expect any notion of multidimensional Lorenz dominance to satisfy and using these conditions to formulate a definition of a *multidimensional Lorenz dominance* *relation*. It then examines a number of “candidate” relations that have been proposed in the literature and shows that all of these fail to satisfy the definition formulated here. The question, therefore, arises as to whether there exists a Lorenz dominance relation as defined here. The chapter gives an affirmative answer to the question.

## References

- Aaberge R, Brandolini A (2015) Multidimensional poverty and inequality. In: Atkinson AB, Bourguignon F (eds) Handbook of income distribution, Vol. 2A, North-Holland, Amsterdam, pp 141–216Google Scholar
- Albert A, Zhang L (2010) A novel definition of the multivariate coefficient of variation. Biometrical J 52(5):667–675Google Scholar
- Arnold BC (1983) Pareto distribution. International Cooperative Publishing House, Burtonsville, MDGoogle Scholar
- Arnold BC (2008) The Lorenz curve: evergreen after 100 years. In: Betti G, Lemmii A (eds) Advances in Income inequality and concentration measures. Routledge, London, pp 12–24Google Scholar
- Atkinson AB, Bourguignon F (1982) The comparison of multidimensional distributions of economic status. Rev Econ Stud 49(2):183–201CrossRefGoogle Scholar
- Banerjee AK (2010) A multidimensional Gini index. Math Soc Sci 60(2):87–93CrossRefGoogle Scholar
- Banerjee AK (2014) A multidimensional Lorenz dominance relation. Soc Choice Welf 42(1):171–191CrossRefGoogle Scholar
- 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
- Banerjee AK (2018b) Multidimensional indices with data-driven dimensional weights: a multidimensional coefficient of variation. Arthaniti 17(2):140–156Google Scholar
- Banerjee AK (2019a) Measuring multidimensional inequality: a gini index. In: Gagari Chakrabarti G, Sen C (eds) The globalization conundrum
*—*dark clouds behind the silver lining. Springer, Singapore, pp 65–78Google Scholar - Banerjee AK (2019b) Economic properties of statistical indices: the case of a multidimensional Gini index. J Quant Econ 17(1):41–56CrossRefGoogle Scholar
- Bhandari SK (1988) Multivariate majorization and directional majorization: positive results. Sankhya A 50(2):199–204Google Scholar
- Boland PJ, Proschan F (1988) Multivariate arrangement increasing functions with applications in probability and statistics. J Multivar Anal 25(2):286–292CrossRefGoogle Scholar
- Dardanoni V (1996) On multidimensional inequality measurement. In Dagum C, Lemmi A (eds) Research on economic inequality 6:202–207Google Scholar
- Debreu G, Herstein IN (1953) Non-negative square matrices. Econometrica 21(4):597–607CrossRefGoogle Scholar
- Decancq K, Lugo MA (2012) Inequality of well-being: A multidimensional approach. Economica, 79(316):721–746Google Scholar
- Epstein LG, Tanny SM (1980) Increasing generalized correlation: a definition and some economic examples. Can J Econ 13(1):16–34CrossRefGoogle Scholar
- Fleurbaey M, Trannoy A (2003) The impossibility of a Paretian egalitarian. Soc Choice Welf 21(2):243–263CrossRefGoogle Scholar
- Gajdos T, Weymark JA (2005) Multidimensional generalized Gini indices. Econ Theory 26(3):471–496CrossRefGoogle Scholar
- Gastwirth JL (1971) A general definition of the Lorenz curve. Econometrica 39(6):1037–1039CrossRefGoogle Scholar
- Hardy GH, Littlewood JE, Poliya G (1952) Inequalities, 2nd edn. Cambridge University Press, LondonGoogle Scholar
- Horn RA, Johnson CR (2013) Matrix analysis, 2nd edn. Cambridge University Press, LondonGoogle Scholar
- Joe H, Verducci J (1993) Multivariate majorization by positive combinations. In: Shaked M, Tong YL (eds) Stochastic inequalities. Lecture Notes Monograph Series, Institute of Mathematical Statistics, Hayward, CA, pp 159–181Google Scholar
- Kolm S-C (1977) Multidimensional egalitarianisms. Quart J Econ 91(1):1–13CrossRefGoogle Scholar
- Koshevoy G (1995) Multivariate Lorenz majorization. Soc Choice Welf 12(1):93–102CrossRefGoogle Scholar
- Koshevoy G, Mosler K (2007) Multivariate Lorenz dominance based on zonoids. Asta-Adv Stat Anal 91(1):57–76CrossRefGoogle Scholar
- Lasso de la Vega C, Urrutia A, de Sarachu A (2010) Characterizing multidimensional inequality measures which fulfil the Pigou-Dalton bundle principle. Soc Choice Welf 35(2):319–329CrossRefGoogle Scholar
- List, C (1999) Multidimensional inequality measurement: a proposal. Nuffield College, Oxford, Working Paper in Economics No. 1999-W27Google Scholar
- Marshall AW, Olkin I (1979) Inequalities: theory of majorization and its applications. Academic Press, New YorkGoogle Scholar
- Mosler K (2002) Multivariate dispersion, central regions and depth: the lift zonoid approach. Springer, BerlinCrossRefGoogle Scholar
- Sarabia JM, Jorda V (2013) Modelling bivariate Lorenz curves with applications to multidimensional inequality in well-being. Paper presented at the Fifth Meeting of ECINEQ, Bari, Italy, 22–24 JulyGoogle Scholar
- Savglio E (2011) On multidimensional inequality with variable distribution mean. J Math Econ 47(4–5):453–461CrossRefGoogle Scholar
- Taguchi T (1972) On the two-dimensional concentration surface and extensions of concentration coefficient and Pareto distribution to the two-dimensional case (On an application of differential geometric methods to statistical analysis, I). Ann Inst Stat Math 24(2):355–381CrossRefGoogle Scholar
- Tsui K-Y (1999) Multidimensional inequality and multidimensional generalized entropy measures. Soc Choice Welf 16(1):145–157CrossRefGoogle Scholar
- 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