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From unidimensional to multidimensional inequality: a review

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Abstract

We review results concerning the representation of partial orders of univariate distributions via stochastic orders and investigate their applications to some classes of stochastic dominance conditions applied in inequality and welfare measurement. The results obtained in an unidimensional framework are extended to multidimensional analysis. We discuss difficulties arising from aggregation of multidimensional distributions into synthetic indicators that value both inequality in the distribution of each attribute and the association between the attributes. We explore the potential for multidimensional evaluations that are based on the partial orders induced by different criteria of majorization and organize related and equivalent inequality and welfare representations.

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Notes

  1. For alternative reviews of multivariate inequality see Savaglio [82], Marshall et al. [64] and Aaberge and Brandolini [3].

  2. Both in Dahl [23] and Andreoli and Zoli [5] the criterion of matrix majorization is formalized through transformations obtained by post multiplication of a matrix by a row stochatic matrix. However, for consistency of notation in this work all matrices are transposed with respect to those considered in the mentioned papers and therefore the notion of matrix majorization should be based on transformations generated by pre-multiplication via column stochastic matrices.

  3. In order to obtain this result it is sufficient to set the integer n sufficiently large to guarantee that the product \(nz_{i}\) gives an integer number \(s_{i}\) for any unit i in the populations of both \(X^{\prime }\) and \(Y^{\prime }.\) Then, each unit i should be split in \(s_{i}\) units of equal weight 1/n.

  4. For general definitions of lift zonoid and Lorenz zonoid see Koshevoy and Mosler [57] and Mosler [70].

  5. These concepts are known in the literature under different denominations. They are known either as linear-combinations majorization and positive linear-combinations majorization or as directional majorization and positive directional majorization, see Bhandari [11], Joe and Verducci [51], and Tsui [94], for an overview see Marshall et al. [64] Ch.15. Koshevoy and Mosler [57] suggest the possibility of interpreting the directional majorization in terms of prices (both positive and negative), while Mosler [70] uses the term price majorization for the case where prices are non-negative. In our exposition we retain the explanation in terms of prices making explicit in the terminology when non-negative prices are considered, in this case we use the term positive price majorization.

  6. As an extreme case if \(p_{1}=-1\) and \(p_{2}=+1,\) then the potential budget distibution for Z is (0, 0);  and for Y is \((1,-1)\) where clearly the former is less unequal than the latter also in term of the PD principle of transfers. This result goes in the opposite direction of what required by Aversion to CIT.

References

  1. Aaberge, R.: Characterizations of Lorenz curves and income distributions. Soc. Choice Welf. 17, 639–653 (2000)

    Article  MATH  Google Scholar 

  2. Aaberge, R.: Ranking intersecting Lorenz curves. Soc. Choice Welf. 33(2), 235–259 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aaberge, R., Brandolini, A.: Multidimensional poverty and inequality. In: Atkinson, A.B., Bourguignon, F. (eds.) Handbook of Income Distribution, vol. 2, Ch.3, pp. 141–216. Elsevier, Amsterdam (2015)

  4. Andreoli, F.: Robust inference for inverse stochastic dominance. J. Bus. Econ. Stat. 36(1), 146–159 (2018)

    Article  MathSciNet  Google Scholar 

  5. Andreoli, F., Zoli, C.: Robust dissimilarity comparisons with categorical outcomes. Working Papers 502, ECINEQ, Society for the Study of Economic Inequality (2019)

  6. Arnold, B.C.: Majorization and the Lorenz curve. Lecture Notes in Statistics, vol. 43. Springer, New York (1987)

    Google Scholar 

  7. Athey, S.: Characterizing properties of stochastic objective functions. MIT Working Paper No. 96-1R, 1998 revised (2000)

  8. Athey, S.: Monotone comparative statics under uncertainty. Q. J. Econ. 117, 187–223 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Atkinson, A.B.: On the measurement of inequality. J. Econ. Theory 2, 244–263 (1970)

    Article  MathSciNet  Google Scholar 

  10. Atkinson, A.B., Bourguignon, F.: The comparison of multidimensioned distribution of economic status. Rev. Econ. Stud. 49, 183–201 (1982)

    Article  Google Scholar 

  11. Bhandari, S.K.: Multivariate majorization and directional majorization: positive results. Sankya Ser. A 50, 199–204 (1988)

    MathSciNet  MATH  Google Scholar 

  12. Basili, M., Casaca, P., Chateauneuf, Franzini, M.: Multidimensional Pigou-Dalton transfers and social evaluations functions. Theor. Decis. 83(4), 573–590 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ben Porath, E., Gilboa, I.: Linear measures, the Gini index, and the income-equality trade-off. J. Econ. Theory 18, 59–80 (1994)

    MathSciNet  MATH  Google Scholar 

  14. Ben-Porath, E., Gilboa, I., Schmeidler, D.: On the measurement of inequality under uncertainty. J. Econ. Theory 75, 194–204 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bosmans, K., Lauwers, L., Ooghe, E.: A consistent multidimensional Pigou–Dalton transfer principle. J. Econ. Theory 144(3), 1358–1371 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bourguignon, F., Chakravarty, S.R.: Multidimensional poverty orderings. DELTA Working Papers 2002-22, DELTA (Paris: Ecole Normale Supérieure) (2002)

  17. Chakravarty, S.R.: Ethical Social Index Numbers. Springer, Berlin (1990)

    Book  Google Scholar 

  18. Chakravarty, S., Muliere, P.: Welfare indicators: a review and new perspectives. 1. Measurement of inequality. Metron Int. J. Stat. 61, 457–497 (2003)

    MathSciNet  MATH  Google Scholar 

  19. Chakravarty, S., Muliere, P.: Welfare indicators: a review and new perspectives. 2. Measurement of poverty. Metron Int. J. Stat. 62, 247–281 (2004)

    MathSciNet  MATH  Google Scholar 

  20. Chateauneuf, A., Gajdos, T., Wilthien, P.H.: The principle of strong diminishing transfer. J. Econ. Theory 103, 311–333 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cowell, F.A.: Measurement of Inequality. In: Atkinson, A.B., Bourguignon, F. (eds.) Handbook of Income Distribution, vol. 1, chap. 2, pp. 87–166. North Holland, Amsterdam (2000)

  22. Cowell, F.A., Flachaire, E.: Statistical methods for distributional analysis. In: Atkinson, A.B., Bourguignon, F. (eds.) Handbook of Income Distribution, vol. 2, ch. 3, pp. 359–465. Elsevier, Amsterdam (2015)

  23. Dahl, G.: Matrix majorization. Linear Algebra Appl. 288, 53–73 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Dardanoni, V., Lambert, P.J.: Welfare rankings of income distributions: a role for the variance and some insights for tax reform. Soc. Choice Welf. 5, 1–17 (1988)

    Article  MathSciNet  Google Scholar 

  25. Dardanoni, V.: On multidimensional inequality measurement. In: Dagum, C., Lemmi, A. (eds.) Research on Economic Inequality: Income Distribution, Social Welfare, Inequality and Poverty, vol. 6, pp. 201–205. JAI Press Inc., Bingley (1996)

  26. Dasgupta, P., Sen, A.K., Starrett, D.: Notes on the measurement of inequality. J. Econ. Theory 6, 180–187 (1973)

    Article  MathSciNet  Google Scholar 

  27. Decancq, K.: Elementary multivariate rearrangements and stochastic dominance on a Fréchet class. J. Econ. Theory 147(4), 1450–1459 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Decancq, K., Decoster, A., Schokkaert, E.: The evolution in world inequality in well-being. World Dev. 37, 11–25 (2009)

    Article  Google Scholar 

  29. Denuit, M., Mesfioui, M.: Generalized increasing convex and directionally convex orders. J. Appl. Probab. 47, 264–276 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Donaldson, D., Weymark, J.A.: A single-parameter generalization of the Gini indices of inequality. J. Econ. Theory 22, 67–86 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  31. Dutta, I., Pattanaik, P.K., Xu, Y.: On measuring deprivation and the standard of living in a multidimensional framework on the basis of aggregate data. Economica 70, 197–221 (2003)

    Article  Google Scholar 

  32. Ebert, U.: Measurement of inequality: an attempt at unification and generalization. Soc. Choice Welf. 5, 147–69 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  33. Epstein, L.G., Tanny, S.M.: Increasing generalized correlation: a definition and some economic consequences. Can. J. Econ. 13, 16–34 (1980)

    Article  Google Scholar 

  34. Faure, M., Gravel, N.: Equality among unequals. AMSE Working Papers No 1702, Aix-Marseille School of Economics, France (2017)

  35. Fields, G.S., Fei, C.H.: On inequality comparisons. Econometrica 46, 303–316 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  36. Fishburn, P.C.: Continua of stochastic dominance relations for bounded probability distributions. J. Math. Econ. 3, 295–311 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  37. Fishburn, P.C., Willig, R.D.: Transfer principles in income redistribution. J. Public Econ. 25, 323–328 (1984)

    Article  Google Scholar 

  38. Foster, J.: Inequality measurement. In: Young, H.P. (ed.) Fair allocation; Proceeding of Symposia in Applied Mathematics, vol. 33, pp. 31–68. The American Mathematical Society, Providence (1985)

  39. Foster, J., Greer, J., Thorbecke, D.: A class of decomposable poverty measures. Econometrica 52, 761–766 (1984)

    Article  MATH  Google Scholar 

  40. Foster, J., Shorrocks, A.F.: Poverty orderings. Econometrica 56, 173–177 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  41. Foster, J.E., Lopez-Calva, L., Szekely, M.: Measuring the distribution of human development: Methodology and an application to Mexico. J. Human Dev. 6, 5–29 (2005)

    Article  Google Scholar 

  42. Gajdos, T., Maurin, E.: Unequal uncertainties and uncertain inequalities: An axiomatic approach. J. Econ. Theory 116, 93–118 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  43. Gajdos, T., Weymark, J.A.: Multidimensional generalized Gini indices. Econ. Theor. 26(3), 471–496 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  44. Gastwirth, J.L.: A general definition of the Lorenz curve. Econometrica 39, 1037–1039 (1971)

    Article  MATH  Google Scholar 

  45. Gilboa, I., Schmeidler, D.: Maximin expected utility with non-unique prior. J. Math. Econ. 18, 141–153 (1989)

    Article  MATH  Google Scholar 

  46. Gini, C.: Di una misura di dissomiglianza tra due gruppi di quantità e delle sue applicazioni allo studio delle relazioni statistiche, Atti del R, Istitituto Veneto di Scienze Lettere e Arti LXXIII (1914)

  47. Gravel, N., Moyes, P.: Ethically robust comparisons of bidimensional distributions with an ordinal attribute. J. Econ. Theory 147(4), 1384–1426 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  48. Hadar, J., Russell, W.: Rules for ordering uncertain prospects. Am. Econ. Rev. 49, 25–34 (1969)

    Google Scholar 

  49. Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge University Press, London (1934)

    MATH  Google Scholar 

  50. Hicks, D.A.: The inequality-adjusted human development Index: a constructive proposal. World Dev. 25, 1283–1298 (1997)

    Article  Google Scholar 

  51. Joe, H., Verducci, J.: Multivariate majorization by positive combinations. In: Shaked, M., Tong, Y. (eds.) Stochastic Inequalities—IMS Lectures Notes, vol. 22, pp. 159–181 (1992)

  52. Kolm, S.C.: The optimal production of social justice. In: Margolis, J., Gutton, H. (eds.) Public Economics, pp. 145–200. Mcmillan, London (1969)

    Chapter  Google Scholar 

  53. Kolm, S.C.: Unequal inequalities I, II. J. Econ. Theory 12, 416–442 (1976a)

  54. Kolm, S.C.: Unequal inequalities I, II. J. Econ. Theory 13, 82–111 (1976b)

  55. Kolm, S.-C.: Multidimensional egalitarianism. Q. J. Econ. 91(1), 1–13 (1977)

    Article  MATH  Google Scholar 

  56. Koshevoy, G.: Multivariate Lorenz majorization. Soc. Choice Welf. 12, 93–102 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  57. Koshevoy, G., Mosler, K.: The Lorenz zonoid of a multivariate distribution. J. Am. Stat. Assoc. 91, 873–882 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  58. Koshevoy, G., Mosler, K.: Multivariate Lorenz dominance based on zonoids. Adv. Stat. Anal. 91, 57–76 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  59. Lambert, P.J.: The distribution and redistribution of income: A mathematical analysis, 2nd edn. Manchester University Press, Manchester (2001)

    Google Scholar 

  60. LeBreton, M.: Inequality, poverty measurement and welfare dominance: an attempt at unification. In: Eichhorn, W. (ed.) Models and Measurement in of Welfare and Inequality, pp. 120–140. Springer, Berlin (1994)

  61. Maasoumi, E.: The measurement and decomposition of multidimensional inequality. Econometrica 54, 991–997 (1986)

    Article  MATH  Google Scholar 

  62. Maccheroni, F., Muliere, P., Zoli, C.: Inverse stochastic orders and generalized Gini functionals. Metron Int. J. Stat. 63(3), 529–559 (2005)

    MathSciNet  MATH  Google Scholar 

  63. Marshall, A.W., Olkin, I.: Inequalities: Theory of majorization and its applications. Academic Press, New York (1979)

    MATH  Google Scholar 

  64. Marshall, A.W., Olkin, I., Arnold, B.: Inequalities: Theory of majorization and its applications. Springer, New York (2011)

    Book  MATH  Google Scholar 

  65. Marinacci, M., Montrucchio, L.: Ultramodular functions. Math. Oper. Res. 30(2), 311–332 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  66. Mehran, F.: Linear measures of income inequality. Econometrica 44, 805–809 (1976)

    Article  MathSciNet  Google Scholar 

  67. Meyer, M., Strulovici, B.: Increasing interdependence of multivariate distributions. J. Econ. Theory 147(4), 1460–1489 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  68. Mosler, K.: Multidimensional welfarism. In: Eichhorn, W. (ed.) Models and Measurement of Welfare and Inequality, pp. 808–820. Springer-Verlag, Berlin (1994a)

    Chapter  Google Scholar 

  69. Mosler, K.: Majorization in economic disparity measures. Linear Algebra Appl. 199, 91–114 (1994b)

    Article  MathSciNet  MATH  Google Scholar 

  70. Mosler, K.: Multivariate dispersion, central regions and depth. Lecture Notes in Statistics. Springer, New York (2002)

    Book  MATH  Google Scholar 

  71. Mosler, K., Muliere, P.: Welfare means and equalizing transfers. Metron Int. J. Stat. 56, 11–52 (1998)

    MathSciNet  MATH  Google Scholar 

  72. Muliere, P., Scarsini, M.: A note on stochastic dominance and inequality measures. J. Econ. Theory 49, 314–323 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  73. Muller, C., Trannoy, A.: Multidimensional inequality comparisons: A compensation perspective. J. Econ. Theory 147(4), 1427–1449 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  74. Müller, A.: Stochastic orders generated by integrals: a unified study. Adv. Appl. Probab. 29, 414–428 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  75. Müller, A., Stoyan, D.: Comparisons Methods for Stochastic Models and Risks. Wiley, Chichester (2002)

    MATH  Google Scholar 

  76. Müller, A., Scarsini, M.: Stochastic comparisons of random vectors with a common copula. Math. Oper. Res. 26, 723–740 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  77. Müller, A., Scarsini, M.: Fear of loss, inframodularity, and transfers. J. Econ. Theory 147(4), 1490–1500 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  78. Rubinstein, A., Fishburn, P.C.: Algebraic aggregation theory. J. Econ. Theory 38(1), 63–77 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  79. Rothschild, M., Stiglitz, J.: Increasing risk: I. A definition. J. Econ. Theory 2, 225–243 (1970)

    Article  MathSciNet  Google Scholar 

  80. Rothschild, M., Stiglitz, J.E.: Some further results on the measurement of inequality. J. Econ. Theory 6, 188–204 (1973)

    Article  MathSciNet  Google Scholar 

  81. Saposnik, R.: Rank-dominance in income distributions. Public Choice 36, 147–51 (1981)

    Article  Google Scholar 

  82. Savaglio, E.: Multidimensional inequality: A survey. In: Farina, F., Savaglio, E. (eds.) Inequality and Economic Integration. Routledge, London (2002)

    Google Scholar 

  83. Scarsini, M.: Multivariate convex orderings, dependence, and stochastic equality. J. Appl. Probab. 35, 93–103 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  84. Sen, A.K. (1973): On economic inequality. Claredon Press, Oxford (1997) [expanded edition with the annexe “OnEconomic Inequality After a Quarter Century” by Foster, J. AndSen, A.K.]

  85. Sen, A.K.: Poverty: an ordinal approach to measurement. Econometrica 44, 219–231 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  86. Shaked, M., Shanthikumar, J.G.: Stochastic Orders and Their Applications. Academic Press, Boston (1994)

    MATH  Google Scholar 

  87. Shorrocks, A.F.: Ranking income distributions. Economica 50, 3–17 (1983)

    Article  Google Scholar 

  88. Shorrocks, A.F., Foster, J.E.: Transfer sensitive inequality measures. Rev. Econ. Stud. 14, 485–497 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  89. Silber, J.: Handbook of Income Inequality Measurement. In: Silber, J. (ed.). Kluwer Academic, Boston (1999)

  90. Tchen, A.: Inequalities for distributions with given marginals. Ann. Probab. 8(4), 814–827 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  91. Thistle, P.D.: Ranking distributions with generalized Lorenz curves. South. Econ. J. 56(1), 1–12 (1989)

    Article  Google Scholar 

  92. Thistle, P.D.: Negative moments, risk aversion and stochastic dominance. J. Financ. Quant. Anal. 28, 301–311 (1993)

    Article  Google Scholar 

  93. Tsui, K.-Y.: Multidimensional generalizations of the relative and absolute inequality indices: The Atkinson–Kolm–Sen approach. J. Econ. Theory 67, 251–265 (1995)

    Article  MATH  Google Scholar 

  94. Tsui, K.-Y.: Multidimensional inequality and multidimensional generalized entropy measures: an axiomatic approach. Soc. Choice Welf. 16, 145–158 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  95. UN Development Programme: Human development indices and indicators. UNDP, New York (2018)

  96. Wang, S.S., Young, V.R.: Ordering risks: Expected utility theory versus Yaari’s dual theory of risk. Insur. Math. Econ. 22, 145–161 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  97. Weymark, J.A.: Generalized Gini inequality indices. Math. Soc. Sci. 1, 409–430 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  98. Weymark, J.: The normative approach to the measurement of multidimensional inequality. In: Farina, F., Savaglio, E. (eds.) Inequality and Economic Integration. Routledge, London (2004)

    Google Scholar 

  99. Whitmore, G.A.: Third-degree stochastic dominance. Am. Econ. Rev. 60(3), 457–59 (1970)

    Google Scholar 

  100. Yaari, M.E.: The dual theory of choice under risk. Econometrica 55, 95–115 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  101. Yitzhaki, S.: On an extension of the Gini inequality index. Int. Econ. Rev. 24, 617–628 (1983)

    Article  MATH  Google Scholar 

  102. Zoli, C.: Intersecting generalized Lorenz curves and the Gini index. Soc. Choice Welf. 16, 183–196 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  103. Zoli, C.: Inverse stochastic dominance, inequality measurement and Gini indices. J. Econ. 77, 119–161 (2002). [Supplement # 9, P. Moyes, C. Seidl and A.F. Shorrocks (Eds.), Inequalities: Theory, Measurement and Applications]

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Economics, University of Verona, Via Cantarane, 24, 37129, Verona, Italy

    Francesco Andreoli & Claudio Zoli

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  1. Francesco Andreoli
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Andreoli F. research has also been undertaken at the Luxembourg Institute of Socio-Economic Research, LISER, his research acknowledges financial support from the French National Agency for Research under the project The Measurement of Ordinal and Multidimensional Inequalities (grant ANR-16-CE41-0005-01) and the Luxembourg Fonds National de la Recherche (IMCHILD grant INTER/NORFACE/16/11333934). This research is also part of the project MOBILIFE (grant RBVR-17KFHX) supported by the University of Verona. This paper is partially built on a revised and updated version of C. Zoli lecture notes on “multidimensional inequality” presented at Canazei winter school on Inequality and Social Welfare Theory. We are grateful to the editor and the anonymous referees for helpful suggestions on earlier versions of the paper. The usual disclaimer applies.

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Andreoli, F., Zoli, C. From unidimensional to multidimensional inequality: a review. METRON 78, 5–42 (2020). https://doi.org/10.1007/s40300-020-00168-4

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