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Third-degree stochastic dominance and inequality measurement

Abstract

We investigate the third-degree stochastic dominance order, which is receiving increasing attention in the field of inequality measurement. Observing that this partial order fails to satisfy the von Neumann–Morgenstern independence property in the space of random variables, we introduce the concepts of strong and local third-degree stochastic dominance, which do not suffer from this deficiency. We motivate these two new binary relations and characterize them in the spirit of the Lorenz characterization of the second-degree stochastic order, comparing our findings with the closest results in inequality literature.

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References

  1. 1.

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

    Article  Google Scholar 

  2. 2.

    Atkinson, A.B.: More on the Measurement of Inequality. Mimeo (1973)

  3. 3.

    Baucells, M., Shapley, L.S.: Multiperson Utility. Mimeo, UCLA (1998)

  4. 4.

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

    Article  Google Scholar 

  5. 5.

    Chew, S.H.: A generalization of the quasilinear mean with applications to the measurement of income inequality and decision theory resolving the Allais paradox. Econometrica 51, 1065–1092 (1983)

    Article  Google Scholar 

  6. 6.

    Chiu, W.H.: Intersecting Lorenz curves, the degree of downside inequality aversion, and tax reforms. Soc. Choice Welf. 28, 375–399 (2007)

    Article  Google Scholar 

  7. 7.

    Davies, J.B., Hoy, M.: The normative significance of using third-degree stochatic dominance in comparing income distributions. J. Econ. Theory 64, 520–530 (1994)

    Article  Google Scholar 

  8. 8.

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

    Google Scholar 

  9. 9.

    Davies, J.B., Hoy, M.: Flat rate taxes and inequality measurement. J. Public Econ. 84, 33–46 (2002)

    Article  Google Scholar 

  10. 10.

    Dekel, E.: An axiomatic characterization of preferences under uncertainty: weakening the independence axiom. J. Econ. Theory 40, 304–318 (1986)

    Article  Google Scholar 

  11. 11.

    Dubra, J., Maccheroni, F., Ok, E.A.: Expected utility theory without the completeness axiom. J. Econ. Theory 115, 118–133 (2004)

    Article  Google Scholar 

  12. 12.

    Feldstein, M.S.: On the theory of tax reform. J. Public Econ. 6, 77–104 (1976)

    Article  Google Scholar 

  13. 13.

    Fishburn, P., Vickson, R.G.: Theoretical foundations of stochastic dominance. In: Withmore, G., Findlay, M. (eds.) Stochastic Dominance, pp. 39–113. Lexington Books, London (1978)

    Google Scholar 

  14. 14.

    Gajdos, T.: Single crossing Lorenz curves and inequality comparisons. Math. Soc. Sci. 47, 21–36 (2004)

    Article  Google Scholar 

  15. 15.

    Guesnerie, R.: On the direction of tax reform. J. Public Econ. 7, 179–202 (1977)

    Article  Google Scholar 

  16. 16.

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

    Google Scholar 

  17. 17.

    Iochum, B.: Cônes Autopolaires et Algèbres de Jordan. Lecture Notes in Mathematics N° 1049. Springer Verlag, New York (1984)

    Google Scholar 

  18. 18.

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

    Article  Google Scholar 

  19. 19.

    Le Breton, M.: Essais sur les Fondements de l’Analyse Economique de l’Inégalité, Thèse de Doctorat d’Etat, Université de Rennes (1986)

  20. 20.

    Le Breton, M.: Stochastic dominance: a bibliographical rectification and a restatement of Whitmore’s theorem. Math. Soc. Sci. 13, 73–79 (1987)

    Article  Google Scholar 

  21. 21.

    Marshall, A.W., Olkin, I.: Inequalities: Theory of Majorization and Its Applications. Academic Press, London (1979)

    Google Scholar 

  22. 22.

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

    Article  Google Scholar 

  23. 23.

    Shorrocks, A., Foster, J.E.: Transfer sensitive inequality measures. Rev. Econ. Stud. 54, 485–497 (1987)

    Article  Google Scholar 

  24. 24.

    Trannoy, A., Lugand, C.: L’Evolution de l’Inégalité des Salaires dues aux Différences de Qualifications: Une Etude d’Entreprises Françaises de 1976 à 1987. Economie et Prévision 102103, 205–220 (1992)

    Google Scholar 

  25. 25.

    Weymark, J.A.: Undominated directions of tax reform. J. Public Econ. 16, 343–369 (1981a)

    Article  Google Scholar 

  26. 26.

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

    Article  Google Scholar 

  27. 27.

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

    Article  Google Scholar 

  28. 28.

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

    Article  Google Scholar 

  29. 29.

    Zoli, C.: Inverse stochastic dominance, inequality measurement and Gini indices. In: Moyes, P., Seidl, C., Shorrocks, A.F. (eds.) Inequalities: Theory, Measurement and Applications, pp. 119–161, Journal of Economics, Supplement 9 (2002)

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Correspondence to Eugenio Peluso.

Additional information

A preliminary version of this paper was presented at the second Canazei Winter School on Inequality and Collective Welfare Theory (IT2). We would like to thank all participants for their comments and suggestions. We are especially grateful to Rolf Aaberge, Peter Lambert, Maria G. Monti, Ernesto Savaglio, John Weymark, Claudio Zoli and two anonymous referees who provided very detailed and insightful comments.

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Le Breton, M., Peluso, E. Third-degree stochastic dominance and inequality measurement. J Econ Inequal 7, 249–268 (2009). https://doi.org/10.1007/s10888-008-9077-0

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Keywords

  • Inequality measurement
  • Stochastic dominance
  • Lorenz order

JEL Classification

  • D31
  • D63