Abstract
This paper seeks to extend the unidimensional notion of Lorenz dominance to the multidimensional context. It formulates a definition of a multidimensional Lorenz dominance relation (MLDR) on the set of alternative distributions of well-being in an economy by incorporating a generalization of the well-known transfer principle of unidimensional theory suggested in recent literature. It also proposes two conditions which an MLDR may reasonably be required to satisfy. The paper notes that the existing literature does not seem to contain an example of an MLDR satisfying these two conditions and suggests one that does. The suggested MLDR does not seem to have appeared in the literature before.
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References
Arnold BC (1983) Pareto distribution. International Cooperative Publishing House, Burtonsville
Arnold BC (2005) The Lorenz curve: evergreen after 100 years. Paper presented at the international conference in memory of two eminent social scientists: C. Gini and M.O. Lorenz, Siena
Banerjee AK (2010) A multidimensional Gini index. Math Soc Sci 60:87–93
Dardanoni V (1996) On multidimensional inequality measurement. In: Dagum C, Lemmi A (eds) Income distribution, social welfare, inequality and poverty. Vol. 6 of Research on economic inequality, pp 201–207. JAI Press, Stamford
Debreu H, Herstein IN (1953) Non-negative square matrices. Econometrica 21:597–607
Decancq K, Lugo MA (2010) Weights in multidimensional indices of well-being: an overview. Working Paper, Centre for Economic Studies, Catholic University of Leuven, Leuven
Fleurbaey M, Trannoy A (2003) The impossibility of a Paretian egalitarian. Soc Choice Welf 21:319–329
Gajdos T, Weymark JA (2005) Multidimensional generalized Gini indices. Econ Theory 26:471–496
Gastwirth JL (1971) A general definition of the Lorenz curve. Econometrica 39:1037–1039
Hardy GH, Littlewood JE, Poliya G (1952) Inequalities, 2nd edn. Cambridge University Press, London
Horn RA, Johnson CR (1985) Matrix analysis. Cambridge University Press, Cambridge
Kolm SC (1977) Multidimensional egalitarianisms. Quart J Econ 91:1–13
Koshevoy G (1995) Multivariate Lorenz majorization. Soc Choice Welf 12:93–102
Koshevoy G, Mosler K (2005) Multivariate Lorenz dominance based on zonoids. Paper presented at the international conference in memory of two eminent social scientists: C. Gini and M.O. Lorenz, Siena,
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 35:319–329
List C (1999) Multidimensional inequality measurement: a proposal. Working paper in economics No. 1999-W27, Nuffield College, Oxford
Marshall AW, Olkin I (1979) Inequalities: theory of majorization and its applications. Academic Press, New York
Mosler K (2002) Multivariate dispersion, central regions and depth: the lift Zonoid approach. Springer-Verlag, Berlin
Moyes P (1999) Comparisons of heterogeneous distributions and dominance criteria. Economie et Pre’vision 138–139, 125–146 (in French)
Muller C, Trannoy A (2011) A dominance approach to the appraisal of the distribution of well-being across countries. J Public Econ 95:239–246
Ram R (1982) Composite indices of physical quality of life, basic needs fulfilment and income. J Dev Econ 11:227–247
Savaglio E (2006) Multidimensional inequality with variable population size. Econ Theory 28:85–94
Savaglio E (2007) n multidimensional inequality with variable distribution mean. Department of Economics, University of Siena, Siena, p 522
Savaglio E (2010) Multidimensional inequality with variable household weights. Department of Economics, University of Siena, Siena
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 24:355–381
Trannoy A (2006) Multidimensional egalitariansm and the dominance approach: a lost paradise. In: Farina F, Savaglio E (eds) Inequality and economic integration. Routledge, London, pp 284–302
Tsui K-Y (1995) Multidimensional generalizations of the relative and absolute inequality indices: the Atkinson–Kolm–Sen approach. J Econ Theory 67:251–265
Tsui K-Y (1999) Multidimensional inequality and multidimensional generalized entropy measures. Soc Choice Welf 16:145–177
Weymark JA (2006) The normative approach to measurement of multidimensional Inequality. In: Farina F, Savaglio E (eds) Inequality and economic integration. Routledge, London, pp 303–328
Acknowledgments
For insightful comments on an earlier version of the paper I am extremely grateful to two anonymous referees. Thanks are also due to the participants in the North American Summer Meeting of the Econometric Society, June 9–12, 2011, Washington University at St. Louis, MO where that version was presented. Needless to say, none of the above-mentioned is responsible for any error that the paper may contain.
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Banerjee, A.K. A multidimensional Lorenz dominance relation. Soc Choice Welf 42, 171–191 (2014). https://doi.org/10.1007/s00355-013-0722-6
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DOI: https://doi.org/10.1007/s00355-013-0722-6