The Lorenz Curve and the Concentration Curve
The Lorenz and the concentration curves play important roles in the areas of GMD and the related measures such as Gini covariance, Gini correlation, Gini regression, and more. In this chapter we introduce the curves, discuss their properties, and show their connections to the Gini world. In addition, in order to be able to analyze the parallel concepts that are common in the variance world we investigate the equivalents of the Lorenz and the concentration curves that are relevant to the variance and the covariance, respectively. Those parallel curves share some properties among themselves. Therefore one can deduct from the concentration curve about some properties of the covariance and not only about the Gini covariance. In addition we present the relationships between the concepts of second-degree stochastic dominance and welfare dominance on one hand and the concentration and Lorenz curves on the other hand. These relationships enable the Gini methodology to serve as an analytical tool for statistical analyses and to be compatible with economic theory, a property that holds for the variance as well, but only for specific distributions.
KeywordsRegression Coefficient Utility Function Horizontal Axis Concentration Curve Gini Coefficient
- Blitz, R. C., & Brittain, J. A. (1964). An extension of the Lorenz diagram to the correlation of two variables. Metron, XXIII(1–4), 137–143.Google Scholar
- Hart, P. E. (1975). Moment distributions in economics: An exposition. Journal of the Royal Statistical Society, A, 138(Part 3), 423–434.Google Scholar
- Lambert, P. J. (2). The distribution and redistribution of income. Manchester: Manchester University Press. Third edition.Google Scholar
- Levy, H. (2006). Stochastic dominance: Investment decision making under uncertainty (2nd ed.). New York: Springer Verlag.Google Scholar
- Lorenz, M. O. (1905). Methods for measuring concentration of wealth. Journal of American Statistical Association, 9(June), 209–219. New Series No. 70.Google Scholar
- Marshall, A. W., & Olkin, I. (1979). Inequalities: Theory of majorization and its application. New York: Academic Press.Google Scholar
- Mayshar, J., & Yitzhaki, S. (1995). Dalton-improving indirect tax reforms. American Economic Review, 85(4 (September)), 793–808.Google Scholar
- Suits, D. (1977). Measurement of tax progressivity. American Economic Review, 67, 747–752.Google Scholar
- Yitzhaki, S. (1982a). Stochastic dominance, mean variance, and the Gini’s mean difference. American Economic Review, 72, 178–185.Google Scholar
- Yitzhaki, S. (1998). More than a dozen alternative ways of spelling Gini. Research on Economic Inequality, 8, 13–30.Google Scholar
- Yitzhaki, S., & Golan, Y. (2010). Monotonic and non-monotonic relationship in the labor market. Economic Quarterly, 56(2), 143–168 (Hebrew).Google Scholar
- Yitzhaki, S. (1999). Necessary and Sufficient Conditions for Dominance using Generalized Lorenz Curves, in Slottje, D. J. (ed.) Advances in Econometrics, Income Distribution and Scientific Methodology: Essays in Honor of Camilo Dagum, Heidelberg:Physica-VerlagGoogle Scholar
- Yitzhaki, S., & Mayshar, J. (2002). Characterizing efficient portfolios. Available at: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=297899. Unpublished.
- Yitzhaki, S., & Olkin, I. (1988). Concentration curves. WP no. 179, Dept. of Economics, The Hebrew University and technical report no.230, 1987, Dept. of Statistics, Stanford University. http://statistics.stanford.edu/~ckirby/techreports/NSF/OLK%20NSF%20230.pdf.
- Yitzhaki, S., & Olkin, I. (1991). Concentration indices and concentration curves. In Karl Mosler and Marco Scarsini (Eds.), Stochastic orders and decisions under risk, 19 (pp. 380–392). Institute Of Mathematical Statistics: Lecture-Notes Monograph Series.Google Scholar
- Yitzhaki, S., & Semrod, J. (1991). Welfare dominance: An application to commodity taxation. American Economic Review, 81(3 (June)), 480–496.Google Scholar