An extended Gini approach to inequality measurement

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.

[1]  P. Muliere,et al.  Target Shortfall Orderings and Indices , 2003 .

[2]  The Relative Deprivation Curve and Its Applications , 1984 .

[3]  John C. H. Fei,et al.  On Inequality Comparisons , 1978, Employment and Development.

[4]  M. Yaari,et al.  A controversial proposal concerning inequality measurement , 1988 .

[5]  U. Ebert,et al.  An axiomatic characterization of Yitzhaki’s index of individual deprivation , 2000 .

[6]  Claudio Zoli.  Inverse stochastic dominance, inequality measurement and Gini indices , 2002 .

[7]  S. Chakravarty,et al.  Measures of Deprivation and their Meaning in Terms of Social Satisfaction , 1999 .

[8]  Shlomo Yitzhaki,et al.  Relative Deprivation and the Gini Coefficient , 1979 .

[9]  Joseph L. Gastwirth,et al.  A General Definition of the Lorenz Curve , 1971 .

[10]  M. Landsberg,et al.  C. Berge, Espaces Topologiques — Fonetions Multivoques (Collection Universitaire de Mathématiques) XII + 272 S. m. 47 Abb. Paris 1959. Dunod Éditeur. Preis geb. 3,400 F , 1960 .

[11]  D. Slottje,et al.  WHY DO PEOPLE VIOLATE THE TRANSFER PRINCIPLE? EVIDENCE FROM EDUCATIONAL SAMPLE SURVEYS , 2004 .

[12]  S. Chakravarty,et al.  Individual welfare, social deprivation and income taxation , 2003 .

[13]  I. Olkin,et al.  MONOTONICITY OF RATIOS OF MEANS AND OTHER APPLICATIONS OF MAJORIZATION , 1965 .

[14]  Wulf Gaertner,et al.  Income inequality, risk, and the transfer principle: A questionnaire-experimental investigation , 2003, Math. Soc. Sci..

[15]  W. Bossert,et al.  Intermediate inequality: concepts, indices, and welfare implications , 1990 .

[16]  Frank A. Cowell,et al.  Thinking about Inequality , 1999 .

[17]  S. Chakravarty,et al.  INCOME INEQUALITY AND RELATIVE DEPRIVATION , 1995 .

[18]  W. Runciman.  Relative Deprivation and Social Justice , 1966 .

[19]  C. Seidl,et al.  Perceptional inequality and preferential judgements: An empirical examination of distributional axioms , 1994 .

[20]  J. C. Fei.  Equity Oriented Fiscal Programs , 1981 .

[21]  A. Sen,et al.  Notes on the measurement of inequality , 1973 .

[22]  I. Meilijson,et al.  A tale of two tails: An alternative characterization of comparative risk , 1990 .

[23]  N. Kakwani.  On a Class of Poverty Measures , 1980 .

[24]  I. Jewitt.  Choosing between risky prospects: the characterization of comparative statics results, and location independent risk , 1989 .

[25]  M. Yaari.  The Dual Theory of Choice under Risk , 1987 .

[26]  P. Bickel,et al.  Descriptive Statistics for Nonparametric Models. III. Dispersion , 1976 .

[27]  J. Quiggin.  Generalized expected utility theory : the rank-dependent model , 1994 .

[28]  C. Gini,et al.  On the measurement of concentration and variability of characters , 2005 .

[29]  A. Atkinson.  On the measurement of inequality , 1970 .

[30]  C. Berge,et al.  Espaces topologiques : fonctions multivoques , 1966 .

[31]  S. Chakravarty.  RELATIVE DEPRIVATION AND SATISFACTION ORDERINGS , 1997 .

[32]  I. Olkin,et al.  Inequalities: Theory of Majorization and Its Applications , 1980 .

[33]  Patrick Moyes.  Inequality Reducing and Inequality Preserving Transformations of Incomes: Symmetric and Individualistic Transformations , 1994 .

[34]  Patrick Moyes,et al.  Deprivation, welfare and inequality , 2007, Soc. Choice Welf..

[35]  P. Moyes,et al.  A note on minimally progressive taxation and absolute income inequality , 1988 .

[36]  K. Mosler,et al.  Inequality indices and the starshaped principle of transfers , 1996 .

[37]  Shlomo Yitzhaki,et al.  On an Extension of the Gini Inequality Index , 1983 .

[38]  Frank A. Cowell,et al.  Measurement of income inequality: Experimental test by questionnaire , 1992 .

[39]  John A. Weymark,et al.  GENERALIZED GIN 1 INEQUALITY INDICES , 2001 .

[40]  U. Jakobsson.  On the measurement of the degree of progression , 1976 .

[41]  Udo Ebert,et al.  Complaints and inequality , 2002, Soc. Choice Welf..

[42]  C. Fershtman,et al.  Social Status and Economic Performance: A Survey , 1998 .

[43]  S. Kolm.  The Optimal Production of Social Justice , 1969 .

[44]  H. Dalton.  The Measurement of the Inequality of Incomes , 1920 .

[45]  A. Sen.  On Economic Inequality , 1974 .

[46]  John A. Weymark,et al.  A single-parameter generalization of the Gini indices of inequality , 1980 .

[47]  Walter Bossert,et al.  Income Inequality Measurement: The Normative Approach , 1999 .

[48]  D. Thon.  Redistributive properties of progressive taxation , 1987 .

[49]  U. Ebert.  Measurement of Inequality: An Attempt at Unification and Generalization , 1988 .