Abstract
It is well-known that, when the Lorenz curves do not cross, the ranking of distributions provided by the Gini index is identical to the one implied by the Lorenz criterion. This does not preclude inequality as measured by the Gini index to increase while the Lorenz curves cross. A suitable modification of the Gini coefficient allows the Lorenz quasi-ordering to coincide with the ranking generated by the application of unanimity over the class of extended Gini indices. Recently the Lorenz quasi-ordering and the underlying principle of transfers have come under attack, while new criteria – the differentials, deprivation and satisfaction quasi-orderings – have been proposed for providing unambiguous rankings of distributions. We suggest to weaken the principle of transfers by imposing additional restrictions on the progressive transfers, which take into account the positions on the income scale of the donors and beneficiaries. We identify the subclasses of extended Gini indices that satisfy these weaker versions of the principle of transfers and we show that the application of unanimity among these classes generate rankings of distributions that coincide with those implied by the differentials, deprivation and satisfaction quasi-orderings.
Similar content being viewed by others
References
Aaberge, R.: The meaning and measurement of duration dependence in hazard rate models. Discussion Paper, No. 319, Statistics Norway (2002)
Amiel, Y., Cowell, F.A.: Measurement of income inequality: experimental test by questionnaire. J. Public Econ. 47, 3–26 (1992)
Amiel, Y., Cowell, F.A.: Thinking about Inequality. Cambridge University Press, Cambridge, UK (1999)
Amiel, Y., Cowell, F.A., Slottje, D.: Why do people violate the transfers principle? Evidence from educational sample surveys. Res. Econ. Inequal. 11, 1–16 (2004)
Atkinson, A.B.: On the measurement of inequality. J. Econ. Theory 2, 244–263 (1970)
Berge, C.: Espaces Topologiques. Fonctions Multivoques, 2nd edn. Dunod, Paris (1966)
Bickel, P.J., Lehmann, E.L.: Descriptive statistics for non-parametric model III. Dispersion. Ann. Stat. 4, 1139–1158 (1976)
Blackorby, C., Bossert, W., Donaldson, D.: Income inequality measurement: the normative approach. In: Silber, J. (ed.) Handbook of Income Inequality Measurement, pp. 133–157. Kluwer, Dordrecht, The Netherlands (1999)
Bossert, W., Pfingsten, A.: Intermediate inequality: concepts, indices and welfare implications. Math. Soc. Sci. 19, 117–134 (1990)
Chakravarty, S.R.: Relative deprivation and satisfaction orderings. Keio Econ. Stud. 34, 17–32 (1997)
Chakravarty, S.R., d’Ambrosio, C., Muliere, P.: Target shortfall orderings and indices. DIW Working Paper, No. 340, Berlin (2003)
Chakravarty, S.R., Chattopadhyay, N., Majumder, A.: Income inequality and relative deprivation. Keio Econ. Stud. 32, 1–15 (1995)
Chakravarty, S.R., Moyes, P.: Individual welfare, social deprivation and income taxation. Econ. Theory 21, 843–869 (2003)
Chakravarty, S.R. and Mukherjee, D.: Measures of deprivation and their meaning in term of social satisfaction. Theory and Decision. 47, 88–100 (1999)
Chateauneuf, A., Moyes, P.: Does the Lorenz curve really measure inequality? GRAPE Discussion Paper, Université Montesquieu Bordeaux IV, France (2005)
Chateauneuf, A., Wilthien, P.-H.: Third inverse stochastic dominance, Lorenz curves and favourable double-transfers, CERMSEM Discussion Paper, Université Paris I Panthéon-Sorbonne (1999)
Cowell, F.A., Ebert, U.: Complaints and inequality. Soc. Choice Welf. 23, 71–90 (2004)
Dalton, H.: The measurement of the inequality of incomes. Econ. J. 30, 348–361 (1920)
Dasgupta, P., Sen, A.K., Starrett, D.: Notes on the measurement of inequality. J. Econ. Theory 22, 67–86 (1973)
Donaldson, D., Weymark, J.A.: A single-parameter generalization of the Gini indices of inequality. J. Econ. Theory 6, 180–187 (1980)
Ebert, U.: Measurement of inequality: an attempt at unification and generalization. Soc. Choice Welf. 5, 59–81 (1988)
Ebert, U., Moyes, P.: An axiomatic characterization of the Yitzhaki’s index of individual deprivation. Econ. Lett. 68, 263–270 (2000)
Fei, J.C.H.: Equity oriented fiscal programs. Econometrica 49, 869–881 (1980)
Fields, G.S., Fei, J.C.H.: On inequality comparisons. Econometrica 46, 305–316 (1978)
Foster, J.E.: Inequality measurement. In: Young, H.P. (ed.) Fair Allocation. In: American Mathematical Society Proceedings of Applied Mathematics, vol. 33, pp. 38–61 (1985)
Gaertner, W., Namezie, C.: Income inequality, risk, and the transfer principle: A questionnaire–experimental investigation. Math. Soc. Sci. 45, 229–245 (2003)
Gastwirth, J.L.: A general definition of the Lorenz curve. Econometrica 39, 1037–1039 (1971)
Gini, C.: On the measurement of concentration and variability of characters. Metron 63, 3–38 (2005)
Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge, UK (1952)
Harrison, E., Seidl, C.: Perceptional inequality and preference judgements: an empirical examination of distributional axioms. Public Choice 79, 61–81 (1994)
Jakobsson, U.: On the measurement of the degree of progression. J. Public Econ. 5, 161–168 (1976)
Jewitt, I.: Choosing between risky prospects: the characterization of comparative statics results and location independent risk. Manag. Sci. 35, 60–70 (1989)
Kakwani, N.C.: On a class of poverty measures. Econometrica 48, 437–446 (1980)
Kakwani, N.C.: The relative deprivation curve and its applications. J. Bus. Econ. Stat. 2, 384–405 (1984)
Kolm, S-.C.: The optimal production of social justice. In: Margolis, J., Guitton, H. (eds.) Public Economics, pp. 145–200. Macmillan, London (1969)
Kolm, S-.C.: Unequal inequalities I. J. Econ. Theory 12, 416–442 (1976)
Landsberger, M., Meilijson, I.: A tale of two tails: an alternative characterization of comparative risk. J. Risk Uncertain. 3, 65–82 (1990)
Magdalou, B.: A Theoretical and Experimental Contribution to the Measurement of Inequality (in French). Doctoral Dissertation, Université Montpellier I (2006)
Magdalou, B., Moyes, P.: Deprivation, welfare and inequality. GRAPE Discussion Paper, Université Montesquieu Bordeaux IV (2006)
Marshall, A.W., Olkin, I.: Inequalities: Theory of Majorization and Its Applications. Academic, New York (1979)
Marshall, A.W., Olkin, I., Proschan, F.: Monotonicity of ratios of means and other applications of majorization. In: Shisha, O. (ed.) Inequalities, pp. 177–190. Academic, New York (1967)
Mosler, K. and Muliere, P.: Inequality indices and the starshaped principle of transfers. Stat. Pap. 37, 343–364 (1996)
Moyes, P.: A note on minimally progressive taxation and absolute income inequality. Soc. Choice Welf. 5, 227–234 (1988)
Moyes, P.: Inequality reducing and inequality preserving transformations of incomes: symmetric and individualistic transformations. J. Econ. Theory 63, 271–298 (1994)
Preston, I.: Ratios, differences and inequality indices. Institute for Fiscal Studies Working Paper No. W90–9 (1990)
Quiggin, J.: Generalized Expected Utility Theory. The Rank-dependent Model. Kluwer, Boston, MA (1993)
Runciman, W.G.: Relative Deprivation and Social Justice. Routledge and Kegan Paul, London (1966)
Sen, A.K.: On Economic Inequality. Clarendon, Oxford (1973)
Thon, D.: Redistributive properties of progressive taxation. Math. Soc. Sci. 14, 185–191 (1987)
Weiss, Y., Fershtman, C.: Social status and economic performance: a survey. Eur. Econ. Rev. 42, 801–820 (1998)
Weymark, J.A.: Generalized Gini inequality indices. Math. Soc. Sci. 1, 409–430 (1981)
Yaari, M.E.: The dual theory of choice under risk. Econometrica 55, 99–115 (1987)
Yaari, M.E.: A controversial proposal concerning inequality measurement. J. Econ. Theory 44, 381–397 (1988)
Yitzhaki, S.: Relative deprivation and the Gini coefficient. Q. J. Econ. 93, 321–324 (1979)
Yitzhaki, S.: On an extension of the Gini inequality index. Int. Econ. Rev. 24, 615–628 (1983)
Zoli, C.: Inverse stochastic dominance, inequality measurement and Gini indices. In: Moyes, P., Seidl, C., Shorrocks, A.F. (eds). Inequalities: Theory, Experiments and Applications. Journal of Economics, vol. 9 (Suppl.), pp. 119–161 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Moyes, P. An extended Gini approach to inequality measurement. J Econ Inequal 5, 279–303 (2007). https://doi.org/10.1007/s10888-006-9051-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10888-006-9051-7
Key words
- progressive transfers
- income differentials
- deprivation
- satisfaction
- Lorenz dominance
- extended Gini social evaluation functions