Gini’s nuclear family

Abstract

The purpose of this paper is to justify the use of the Gini coefficient and two close relatives for summarizing the basic information of inequality in distributions of income. To this end we employ a specific transformation of the Lorenz curve, the scaled conditional mean curve, rather than the Lorenz curve as the basic formal representation of inequality in distributions of income. The scaled conditional mean curve is shown to possess several attractive properties as an alternative interpretation of the information content of the Lorenz curve and furthermore proves to yield essential information on polarization in the population. The paper also provides asymptotic distribution results for the empirical scaled conditional mean curve and the related family of empirical measures of inequality. Keywords:The scaled conditional mean curve, measures of inequality, the Gini coefficient, the Bonferroni coefficient, measures of social welfare, principles of transfer sensitivity, estimation, asymptotic distributions. JEL classification: D3, D63.Acknowledgement: This paper was written when the author was visiting ICER in Torino. ICER and the Norwegian Research Council are gratefully acknowledged for providing financial support. I wish to thank Jacques Silber, Terje Skjerpen and two referees for helpful comments.

Accounting for family background when designing optimal income taxes: a microeconometric simulation analysis

“…(1 Aaberge (2007) proves that the family of inequality measures { } (in)equality. Moreover, these three measures of inequality also prove to supplement each other with regard to sensitivity to transfers at the lower, the central and the upper part of the income distribution.…”

On the generalization and decomposition of the Bonferroni index

“…The Bonferroni curve which, like the Lorenz curve, is bounded by the unit square is also strongly related to the shape of the underlying distribution curve F: when F is convex (strongly skewed to the left), the Bonferroni curve is concave, and when F is concave (strongly skewed to the right), the Bonferroni curve is convex. Aaberge (2007) proved also that the Bonferroni index satisfies the principle of diminishing transfers (see Kolm, 1976, andShorrocks andFoster, 1987) for all strictly log-concave distributional functions and the principle of positional transfer sensitivity (see Mehran, 1976) for all distributional functions. Finally the Bonferroni index may also be interpreted in terms of relative deprivation (see Chakravarty, 2007).…”

“…It is only in recent years that some authors attempted to rehabilitate this measure of inequality (e.g. Tarsitano, 1990;Aaberge, 2000;Giorgi and Crescenzi, 2001;Piesch, 2005;Aaberge, 2007;Chakravarty, 2007;Bárcena and Imedio, 2008;Imedio-Olmedo et al, 2011). Aaberge (2007) in particular drew our attention to several attractive properties of the Bonferroni index, or more precisely of what he called "scaled conditional mean curve" which is just another name for the Bonferroni (1930) curve, from which the definition of the Bonferroni index is derived.…”

“…Tarsitano, 1990;Aaberge, 2000;Giorgi and Crescenzi, 2001;Piesch, 2005;Aaberge, 2007;Chakravarty, 2007;Bárcena and Imedio, 2008;Imedio-Olmedo et al, 2011). Aaberge (2007) in particular drew our attention to several attractive properties of the Bonferroni index, or more precisely of what he called "scaled conditional mean curve" which is just another name for the Bonferroni (1930) curve, from which the definition of the Bonferroni index is derived. He thus stressed that since the scaled conditional mean for a given percentile p is the ratio of the mean income of the poorest 100p% of the population and the overall mean, it yields important information on poverty, assuming the poverty line has been determined.…”

“…He thus stressed that since the scaled conditional mean for a given percentile p is the ratio of the mean income of the poorest 100p% of the population and the overall mean, it yields important information on poverty, assuming the poverty line has been determined. Aaberge (2007) also emphasized the fact that in the case of a uniform distribution, the scale conditional mean curve becomes the diagonal and represents hence a useful reference line (in addition to the perfect equality line or to the case of maximal inequality). Aaberge showed in particular that if the Bonferroni curve intersects the diagonal once from below (single intersection), the corresponding distribution exhibits lower inequality than the uniform distribution below the intersection point and higher inequality than the uniform distribution above this intersection point.…”