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Inequality indices and the starshaped principle of transfers

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Abstract

The evaluation of income distributions is usually based on the Pigou-Dalton (PD) principle which says that a transfer from any people to people who have less decreases economic inequality, i.e., increases the social evaluation index. We introduce two weaker principles of transfers which refer to a parameter θ. With the new principles, only those PD transfers increase the social evaluation index which take from the class of incomes above θ and give to the class below θ. The relative positions of individuals remain unchanged, and either no individual may cross the line θ (principle of transfers about θ) or some may do who have been situated next to it (starshaped principle of transfers at θ). θ may be a given constant, a function of mean income, or a quantile of the income distribution. The classes of indices which are consistent with these transfers are completely characterized, and examples are given.

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References

  • Amiel, Y., and F. Cowell (1992). “Measurement of income inequality: Experimental test by questionnai”.Journal of Public Economics 47, 3–26.

    Article  Google Scholar 

  • Arnold, B.C. (1987).Majorization and the Lorenz Order: A Brief Introduction. Springer Verlag, Berlin.

    MATH  Google Scholar 

  • Atkinson, A. B. (1970). “On the measurement of inequality”.Journal of Economic Theory 2, 244–263.

    Article  MathSciNet  Google Scholar 

  • Barlow, R.E., A.W. Marshall, and F. Proschan (1969). “Some inequalities for starshaped and convex functions”.Pacific Journal of Mathematics 29, 19–42.

    MathSciNet  MATH  Google Scholar 

  • Bonferroni, C. (1930).Elementi di Statistica Generale. Libreria Seeber, Firenze.

    Google Scholar 

  • Bonferroni, C. (1940). “Un indice quadratico di concentrazione”.Atti II Congresso UMI, pp 700–706.

  • Bresciani-Turroni, C. (1937). “On Pareto’s law”.Journal of the Royal Statistical Society A 3, 421–432.

    Google Scholar 

  • Castagnoli, E. (1990). “On order preserving functions”. In M. Fedrizzi and J. Kacprzyk, eds.,Proceedings of the 8th Italian Symposium on System Analysis and Decision Support in Economics and Technology, pp 151–165, Omnitech Press, Warszawa.

    Google Scholar 

  • Castagnoli, E., and P. Muliere (1990). “A note on inequality measures and the Pigou-Dalton principle of transfers”. In C. Dagum and M. Zenga, eds.,Income and Wealth Distribution, Inequality and Poverty, pp 171–182. Springer Verlag, Berlin.

    Google Scholar 

  • Chakravarty, S.R. (1990).Ethical Social Index Numbers. Springer Verlag, Berlin.

    Google Scholar 

  • Cowell, F.A. (1977).Measuring Inequality. Philips Allan Publ Ltd., Oxford.

    Google Scholar 

  • Dalton, H. (1920). “The measurement of the inequality of incomes”.The Economic Journal 30, 348–361.

    Article  Google Scholar 

  • Davies, J., and M. Hoy (1991). “Making inequality comparisons when Lorenz curves intersect”.Mimeo.

  • Dasgupta, P., A. Sen, and D. Starrett (1973). “Notes on the measurement of inequality”.Journal of Economic Theory 6,180–187.

    Article  MathSciNet  Google Scholar 

  • Ebert, U. (1984). “A family of absolute measures of inequality”.Methods of Operations Research 48, 129–144.

    MathSciNet  Google Scholar 

  • — (1984). “A family of aggregative compromise inequality measures”.International Economic Review 29, 363–376.

    Article  MathSciNet  Google Scholar 

  • Eichhorn, W. and W. Gehrig (1981). “Generalized convexity and the measurement of inequality”. In S. Schaible and W.T. Ziemba, eds.,Generalized Concavity in Optimization and Economics, pp 637–642. Academic Press, New York.

    Google Scholar 

  • Eichhorn, W. and W. Gehrig (1982). “Measurement of inequality in economics”. In B. Korte, ed.,Modern Applied Mathematics, Optimization and Operations Research, pp 658–693. North-Holland, Amsterdam.

    Google Scholar 

  • Eltetö, O. and E. Frigyes (1968). “New income inequality measures as efficient for casual analysis and planning”.Econometrica 36, 383–396.

    Article  Google Scholar 

  • Fields, G.S. and J.C.H. Fei (1978). “On inequality comparisons”.Econometrica 46, 303–316.

    Article  MathSciNet  MATH  Google Scholar 

  • Foster, J.E., and A.F. Shorrocks (1987). “Transfer sensitive inequality measures”.Review of Economic Studies 54, 485–497.

    Article  MathSciNet  MATH  Google Scholar 

  • Gehrig, W. (1983). “On a characterization of the Shannon measure”.Utilitas Mathematica 24, 67–85.

    MathSciNet  MATH  Google Scholar 

  • — (1984). “Two characterizations of Theil’s concentration measure”.Methods of Operations Research 48, 97–112.

    MathSciNet  MATH  Google Scholar 

  • Gini, C. (1912). “Variabilità e mutabilità: contributo allo studio delle distribuzioni e delle relazioni statistiche”.Studi Economico Giuridici dell’Università di Cagliari 3, 1–158.

    Google Scholar 

  • Hardy, G.H., J.E. Littlewood and G. Polya (1934).Inequalities. Cambridge Univ. Press, London.

    Google Scholar 

  • Harrison, E., and C. Seidl (1994). “Perceptional inequality and preferential judgements: An empirical examination of distributional axioms”.Public Choice 79, 61–81.

    Article  Google Scholar 

  • Harsanyi, J.C. (1953). “Cardinal utility in welfare economics and in the theory of risk-taking”.Journal of Political Economy 61, 434–435.

    Article  Google Scholar 

  • Kolm, S. (1969). “The optimal production of social justice”. In J. Margolis and H. Guitton, eds.,Public Economics, pp 145–200. Macmillan, London.

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Kohm, S.Ch. (1976b). “Unequal inequalities II”.Journal of Economic Theory 13, 82–111.

    Article  MathSciNet  Google Scholar 

  • Kuznets, S. (1959).Six Lectures on Economic Growth. Free Press of Glencoe, Ilinois.

    Google Scholar 

  • Landsberger, M. and I. Meilijson (1990). “Lotteries, insurance and star-shaped utility functions”.Journal of Economic Theory 52, 1–17.

    Article  MathSciNet  MATH  Google Scholar 

  • Marshall, A. and I. Olkin (1979).Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.

    MATH  Google Scholar 

  • Marshall, A.W., D.W. Walkup, and J.B. Wets (1967). “Order-preserving functions; applications to majorization and order statistics”.Pacific Journal of Mathematics 23, 569–584.

    MathSciNet  MATH  Google Scholar 

  • Menezes, C., C. Geiss, and J. Tressler (1980). “Increasing downside risk”.American Economic Review 70, 921–923.

    Google Scholar 

  • Muliere, P., and G. Parmigiani (1993). “On quasi-means”.Utilitas Mathematica 43, 79–87.

    MathSciNet  MATH  Google Scholar 

  • Piesch, W. (1975).Statistische Konzentrationsmaße. J.C.B. Mohr, Tübingen.

    Google Scholar 

  • Pietra, G. (1915). “Delle relazioni tra gli indici di variabilità, nota I”.Atti del R. Istituto Veneto di Scienze Lettere ed Arti 74, 775–804.

    Google Scholar 

  • Rothschild, M., and J.E. Stiglitz (1973). “Some further results on the measurement of inequality”.Journal of Economic Theory 6, 188–204.

    Article  MathSciNet  Google Scholar 

  • Schutz, R.R. (1951). “On the measurement of income inequality”.American Economic Review 41, 107–122.

    Google Scholar 

  • Seidl, C. (1988). “Poverty measurement: A survey”. In D. Bös, M. Rose and C. Seidl, eds.,Welfare and Efficiency in Public Economics, pp 71–147. Springer, Berlin.

    Google Scholar 

  • Shorrocks, A.F. (1980) “The class of additively decomposable inequality measures”.Econometrica 48, 613–625.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Seminar für Wirtschafts-und Sozialstatistik, Universität zu Köln, Albertus-Magnus-Platz, D-50923, Köln

    Karl Mosler

  2. Dip. di Economia Politica e Metodi Quantitativi, Università di Pavia, Via San Felice, 5, I-27100, Pavia

    Pietro Muliere

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  1. Karl Mosler
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  2. Pietro Muliere
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Research supported by the Vigoni program of the Deutscher Akademischer Austauschdienst and the Conferenza Permanente dei Rettori delle Università Italiane.

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Mosler, K., Muliere, P. Inequality indices and the starshaped principle of transfers. Statistical Papers 37, 343–364 (1996). https://doi.org/10.1007/BF02926113

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  • DOI: https://doi.org/10.1007/BF02926113

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