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Robust inequality comparisons


This paper is concerned with the problem of ranking Lorenz curves in situations where the Lorenz curves intersect and no unambiguous ranking can be attained without introducing weaker ranking criteria than first-degree Lorenz dominance. To deal with such situations, Aaberge (Soc Choice Welf 33:235–259, 2009) introduced two alternative sequences of nested dominance criteria for Lorenz curves, which proved to characterize two separate systems of nested subfamilies of inequality measures. This paper uses the obtained characterization results to arrange the members of two different generalized Gini families of inequality measures into subfamilies according to their relationship to Lorenz dominance of various degrees. Since the various criteria of higher degree Lorenz dominance provide convenient computational methods, these results can be used to identify the largest subfamily of the generalized Gini families, and thus the least restrictive social preferences, required to reach unambiguous ranking of a set of Lorenz curves. We further show that the weight-functions of the members of the generalized Gini families offer intuitive interpretations of higher degree Lorenz dominance, which generally has been viewed as difficult to interpret because they involve assumptions about third and higher derivatives. To demonstrate the usefulness of these methods for empirical applications, we examine the time trend in income and earnings inequality of Norwegian males during the period 1967–2005.


  1. 1.

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

    Article  Google Scholar 

  2. 2.

    Aaberge, R.: Axiomatic characterization of the Gini coefficient and Lorenz curve orderings. J. Econ. Theory 101, 115–132 (2001)

    Article  Google Scholar 

  3. 3.

    Aaberge, R.: Asymptotic distribution theory of empirical rank-dependent measures of inequality. In: Nair, V. (ed.) Advances in Statistical Modeling and Inference—Essays in Honor of Kjell A. Doksum, World Scientific (2006)

  4. 4.

    Aaberge, R.: Gini’s nuclear family. J. Econ. Inequal. 5, 305–322 (2007)

    Article  Google Scholar 

  5. 5.

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

    Article  Google Scholar 

  6. 6.

    Aaberge, R., Atkinson, A.B.: Top incomes in Norway. In: Atkinson, T., Piketty, T. (eds.) Top Incomes: A Global Perspective, pp. 448–482. Oxford University Press, Oxford (2010)

    Google Scholar 

  7. 7.

    Aaberge, R., Bhuller M., Langørgen A., Mogstad M.: The distributional impact of public services when needs differ. J. Public Econ. 94, 549–562 (2010)

    Article  Google Scholar 

  8. 8.

    Almås, I., Havnes T., Mogstad M.: Baby booming inequality? Demographic change and inequality in Norway, 1967–2000. CESifo Working Paper 3200 (2010)

  9. 9.

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

    Article  Google Scholar 

  10. 10.

    Atkinson, A.B.: Multidimensional deprivation: contrasting social welfare and counting approaches. J. Econ. Inequal. 1, 51–65 (2003)

    Article  Google Scholar 

  11. 11.

    Atkinson, A.B.: More on the measurement of inequality. J. Econ. Inequal. 6, 277–283 (2008)

    Article  Google Scholar 

  12. 12.

    Bossert, W.: An approximation of the single-series Ginis. J. Econ. Theory 50, 82–92 (1990)

    Article  Google Scholar 

  13. 13.

    Csörgö, M., Gastwith, J.L., Zitikis, R.: Asymptotic confidence bands for the Lorenz and Bonferroni curves based on the empirical Lorenz curve. J. Stat. Plan. Inference 74, 65–91 (1998)

    Article  Google Scholar 

  14. 14.

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

    Article  Google Scholar 

  15. 15.

    Davies, J.B., Hoy, M.: Making inequality comparisons when Lorenz curves intersect. Am. Econ. Rev. 85, 980–986 (1995)

    Google Scholar 

  16. 16.

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

    Article  Google Scholar 

  17. 17.

    Fjærli, E., Aaberge, R.: Tax reforms, dividend policy and trends in income inequality empirical evidence based on Norwegian data. Statistics Norway, Discussion Paper, no. 284 (2000)

  18. 18.

    Gottschalk, P., Smeeding, T.: Cross-national comparisons of earnings and income inequality. J. Econ. Lit. 35, 633–687 (1997)

    Google Scholar 

  19. 19.

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

    Google Scholar 

  20. 20.

    Kolm, S.C.: The optimal production of social justice. In: Margolis, J., Guitton, H. (eds.) Public Economics. Macmillan, New York (1969)

    Google Scholar 

  21. 21.

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

    Article  Google Scholar 

  22. 22.

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

    Article  Google Scholar 

  23. 23.

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

    Article  Google Scholar 

  24. 24.

    Schechtman, E., Shelef, A., Yitzhaki, S., Zitikis, R.: Testing hypotheses about absolute concentration curves and marginal conditional stochastic dominance. Econom. Theory 24, 1044–1062 (2008)

    Article  Google Scholar 

  25. 25.

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

    Google Scholar 

  26. 26.

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

    Article  Google Scholar 

  27. 27.

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

    Article  Google Scholar 

  28. 28.

    Yaari, M.E.: A controversial proposal concerning inequality measurement. J. Econ. Theory 44, 381–397 (1988)

    Article  Google Scholar 

  29. 29.

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

    Article  Google Scholar 

  30. 30.

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

    Article  Google Scholar 

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Correspondence to Magne Mogstad.

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Aaberge, R., Mogstad, M. Robust inequality comparisons. J Econ Inequal 9, 353–371 (2011).

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  • The Lorenz curve
  • Lorenz dominance
  • Rank-dependent measures of inequality
  • The Gini coefficient
  • Generalized Gini families of inequality measures

JEL Classification

  • D31
  • D63