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Inverse stochastic dominance, inequality measurement and Gini indices

  • Inequality Measurement, Decomposition and Redistribution
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Abstract

We investigate the relationship between the third degree inverse stochastic dominance criterion introduced in Muliere and Scarsini (1989) and inequality dominance when Lorenz curves intersect. We propose a new definition of transfer sensitivity aimed at strengthening the Pigou-Dalton Principle of Transfers. Our definition is dual to that suggested by Shorrocks and Foster (1987). It involves a regressive transfer and a progressive transfer both from the same donor, leaving the Gini index unchanged. We prove that finite sequences of these transfers and/or progressive transfers characterize the third degree inverse stochastic dominance criterion. This criterion allows us to make unanimous inequality judgements even when Lorenz curves intersect. The Gini coefficient becomes relevant in these cases in order to conclusively rank the distributions.

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References

  • Aaberge, R. (2000): “Ranking Intersecting Lorenz Curves.” Discussion Paper 271, Statistics Norway Research Department.

  • Atkinson, A. B. (1970): “On The Measurement Of Inequality”.Journal of Economic Theory 2: 244–263.

    Article  Google Scholar 

  • Atkinson, A. B. (1973):More on the measurement of inequality. Mimeo.

  • Ben Porath, E. and Gilboa, I. (1994): “Linear Measures, The Gini Index, and The Income-Equality Trade-Off.”Journal of Economic Theory 18: 59–80.

    Google Scholar 

  • Blackorby, C., and Donaldson, D. (1978): “Measures Of Relative Inequality and Their Meaning In Terms Of Social Welfare.”Journal of Economic Theory 64: 443–467.

    Google Scholar 

  • Blackorby, C. and Donaldson, D. (1984): “Ethically Significant Ordinal Index Of Relative Inequality.” InAdvances in Econometrics, edited by R. L. Basman, and G. F. Rhodes, volume 3, pages 131–147. London: JAI Press.

    Google Scholar 

  • Bossert, W., and Pfingsten, A. (1990): “Intemediate Inequality: Concepts, Indices and Welfare Implications.”Mathematical Social Sciences 19: 117–134.

    Article  Google Scholar 

  • Chakravarty, S. R. (1988): “On Quasi Orderings Of Income Profiles.” University of Paderborn,Methods of Operations Research 60, XIII Symposium on Operations Research, 455–473.

  • Chateauneuf, A., Gajdos, T., and Wilthien, P. H. (1999): “The Principle Of Strong Diminishing Transfer.” Working Paper, forthcoming in Journal of Economic Theorey.

  • Chateauneuf, A., and Wilthien, P. H. (2000): “Third Inverse Stochastic Dominance, Lorenz Curves and Favourable Double Transfers.” Working Paper CERMSEM.

  • Dalton, H. (1920): “The Measurement Of The Inequality Of Incomes.”Economic Journal 20: 348–361.

    Article  Google Scholar 

  • Dardanoni, V., and Lambert, P. J. (1988): “Welfare Rankings Of Income Distributions: A Role For The Variance and Some Insights For Tax Reform.”Social Choice and Welfare 5: 1–17.

    Article  Google Scholar 

  • Dasgupta, P., Sen, A. K., and Starrett, D. (1973): “Notes On The Measurement Of Inequality.”Journal of Economic Theory 6: 180–187.

    Article  Google Scholar 

  • Davies, J., and Hoy, M. (1994): “The Normative Significance Of Using Third-Degree Stochastic Dominance In Comparing Income Distributions.”Journal of Economic Theory 64: 520–530.

    Article  Google Scholar 

  • Davies, J., and Hoy, M. (1995): “Making Inequality Comparisons When Lorenz Curves Intersect.”American Economic Review 85: 980–986.

    Google Scholar 

  • Donaldson, D., and Weymark, J.A. (1980): “A Single-Parameter Generalization Of The Gini Indices Of Inequality.”Journal of Economic Theory 22: 67–86.

    Article  Google Scholar 

  • Donaldson, D., and Weymark, J. A. (1983): “Ethically Flexible Gini Indices For Income Distributions In The Continuum.”Journal of Economic Theory 29: 353–358.

    Article  Google Scholar 

  • Dutta, B., and Esteban, J. (1992): “Social Welfare And Equality.”Social Choice and Welfare 9: 267–276.

    Article  Google Scholar 

  • Ebert, U. (1987): “Size And Distribution Of Income As Determinants Of Social Welfare.”Journal of Economic Theory 41: 23–33.

    Article  Google Scholar 

  • Ebert, U. (1988): “Measurement Of Inequality: An Attempt At Unification And Generalization.”Social Choice and Welfare 5: 147–169.

    Article  Google Scholar 

  • Eichhorn, W. (1988): “On A Class Of Inequality Measures.”Social Choice and Welfare 5: 171–177.

    Article  Google Scholar 

  • Fields, G. S., and Fei, C. H. (1978): “On Inequality Comparisons.”Econometrica 46: 303–316.

    Article  Google Scholar 

  • Fishburn, P. C., and Willig, R. D. (1984): “Transfer Principles In Income Redistribution.”Journal of Public Economics 25: 323–328.

    Article  Google Scholar 

  • Foster, J. (1985): “Inequality Measurement.” InFair Allocation, Proceedings of Symposia in Applied Mathematics, edited by H. P. Young, volume 33, pages 31–68. Providence, R.I.: American Mathematical Society.

    Google Scholar 

  • Gastwirth, J. L. (1971): “A General Definition Of The Lorenz Curve.”Econometrica 39: 1037–1039.

    Article  Google Scholar 

  • Kakwani, N. C. (1980): “On A Class Of Poverty Measures.”Econometrica 48: 437–446.

    Article  Google Scholar 

  • Kolm, S. C. (1969) “The Optimal Production Of Social Justice.” InPublic Economics, edited by J. Margolis, and H. Gutton, pages 145–200. London: Mcmillan.

    Google Scholar 

  • Kolm, S. C. (1976a): “Unequal Inequalities I.”Journal of Economic Theory 12: 416–442.

    Article  Google Scholar 

  • Kolm, S. C. (1976b): “Unequal Inequalities II.”Journal of Economic Theory 13: 82–111.

    Article  Google Scholar 

  • Lambert, P. J. (1993):The Distribution and Redistribution of Income: A Mathematical Analysis. Manchester: Manchester University Press.

    Google Scholar 

  • Mehran, F. (1976): “Linear Measures Of Income Inequality.”Econometrica 44: 805–809.

    Article  Google Scholar 

  • Menezes, C., Geiss, C., and Tressler, J. (1980): “Increasing Downside Risk.”American Economic Review 70: 921–932.

    Google Scholar 

  • Mosler, K., and Muliere, P. (1998): “Welfare Means and Equalizing Transfers.”Metron 55: 13–52.

    Google Scholar 

  • Moyes, P. (1987): “A New Concept Of Lorenz Domination.”Economic Letters 23: 203–207.

    Article  Google Scholar 

  • Muliere, P., and Scarsini, M. (1989): “A Note On Stochastic Dominance and Inequality Measures.”Journal of Economic Theory 49: 314–323.

    Article  Google Scholar 

  • Pigou, A. C. (1912):Wealth and Welfare. New York: MacMillan.

    Google Scholar 

  • Quiggin, J. (1993):Generalized Expected Utility Theory. The Rank Dependent Model. Dordrecht: Kluwer.

    Book  Google Scholar 

  • Rothschild, M., and Stiglitz, J. E. (1973): “Some Further Results On The Measurement Of Inequality.”Journal of Economic Theory 6: 188–204.

    Article  Google Scholar 

  • Safra, Z., and Segal, U. (1998): “Constant Risk Aversion.”Journal of Economic Theory 83: 19–42.

    Article  Google Scholar 

  • Sen, A. K. (1973):On Economic Inequality. Oxford: Clarendon Press. [1997 expanded edition with the annex “On Economic Inequality After A Quarter Century” by Foster, J., and Sen, A.K.].

    Book  Google Scholar 

  • Shorrocks, A. F. (1983): “Ranking Income Distributions.”Economica 50: 3–17.

    Article  Google Scholar 

  • Shorrocks, A. F., and Foster, J. E. (1987): “Transfer Sensitive Inequality Measures.”Review of Economic Studies 14: 485–497.

    Article  Google Scholar 

  • Wang, S. S., and Young, V. R. (1998): “Ordering Risks: Expected Utility Theory Versus Yaari's Dual Theory Of Risk.”Insurance: Mathematics and Economics 22: 145–161.

    Google Scholar 

  • Weymark, J. A. (1981): “Generalized Gini Inequality Indices.”Mathematical Social Sciences 1: 409–430.

    Article  Google Scholar 

  • Whitmore, G. A. (1970): “Third-Degree Stochastic Dominance.”American Economic Review 50: 457–459.

    Google Scholar 

  • Yaari, M. E. (1987): “The Dual Theory Of Choice Under Risk.”Econometrica 55: 99–115.

    Article  Google Scholar 

  • Yaari, M. E. (1988): “A Controversial Proposal Concerning Inequality Measurement.”Journal of Economic Theory 44: 381–397.

    Article  Google Scholar 

  • Yitzhaki, S. (1982): “Stochastic Dominance, Mean Variance, and Gini's Mean Difference.”American Economic Review 72: 178–185.

    Google Scholar 

  • Yitzhaki, S. (1983): “On An Extension Of The Gini Inequality Index.”International Economic Review 24: 617–628.

    Article  Google Scholar 

  • Zoli, C. (1997): “Intersecting Generalized Lorenz Curves and The Gini Index.” Quaderni del Dipartimento di Economia Pubblica e Territoriale Universitá di Pavia n. 2/1997.

  • Zoli, C. (1999): “Intersecting Generalized Lorenz Curves and The Gini Index.”Social Choice and Welfare 16: 183–196.

    Article  Google Scholar 

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  1. School of Economics, University of Nottingham, NG7 2RD, Nottingham, UK

    Claudio Zoli

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  1. Claudio Zoli
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Correspondence to Claudio Zoli.

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Zoli, C. Inverse stochastic dominance, inequality measurement and Gini indices. J. Econ. 77 (Suppl 1), 119–161 (2002). https://doi.org/10.1007/BF03052502

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