A New Inequality Measure that is Sensitive to Extreme Values and Asymmetries

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Facultad de Ciencias Económicas y Empresariales. Instituto Complutense de Análisis Económico (ICAE)
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There is a vast literature on the selection of an appropriate index of income inequality and on what desirable properties such a measure (or index) should contain. The Gini index is, of course, the most popular. There is a concurrent literature on the use of hypothetical statistical distributions to approximate and describe an observed distribution of incomes. Pareto and others observed early on that incomes tend to be heavily right-tailed in their distribution. These asymmetries led to approximating the observed income distributions with extreme value hypothetical statistical distributions, such as the Pareto distribution. But these income distribution functions (IDFs) continue to be described with a single index (such as the Gini) that poorly detects the extreme values present in the underlying empirical IDF. This paper introduces a new inequality measure to supplement, but not to replace, the Gini that measures more accurately the inherent asymmetries and extreme values that are present in observed income distributions. The new measure is based on a third-order term of a Legendre polynomial from the logarithm of a share function (or Lorenz curve). We advocate using the two measures together to provide a better description of inequality inherent in empirical income distributions with extreme values.
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Arfken, George, 1985, Mathematical methods for physicists, third edition, Academic Press, Inc. San Diego. Basmann, R. and D. Slottje, (1987), “A new index of income inequality,” Economics Letters 24: 385-389. Basmann, R., K. Hayes, and D. Slottje, (1991), “The Lorenz curve and the mobility function,” Economics Letters, 35: 105-111. Boushey, H., J. Delong, and M. Steinbaum, (2017), After Piketty, Harvard University, Cambridge, MA. Choo, Hakchung, and Hang Ryu, 1994, Gini coefficient, Lorenz curves, and Lorenz dominance effect: An application to Korean income distribution data, Journal of Economic Development 19, No.2, 47-65. Coles, S. (2001), An introduction to Statistical Modeling of Extreme Values, Springer-Verlag. Cowell, F. (2011), Measuring Inequality, 3rd Edition, Oxford: Oxford U. Press. Cowell, F. and E. Flachaire (2002), “Sensitivity of Inequality Measures to Extreme Values,” LSE STICERD Paper No. DARP 60. Cowell, F. and E. Flachaire (2007), “Income Distributions and Inequality Measurement: the Problem of Extreme Values,” Journal of Econometrics, 141: 1044-1072. Maasoumi, E. (1986), "The Measurement and Decomposition of Multidimensional Inequality," Econometrica, 54: 991-998. Maasoumi, E. (1989), "Continuously Distributed Attributes and Measures of Multivariate Inequality," Journal of Econometrics, 42: 131-144. McDonald, J.B. (1984), “Some Generalized Functions for the Size Distributions of Income,” Econometrica, 52: 647–663. McDonald, J., J. Sorenson and P. Turley (2013), “Skewness and Kurtosis Properties of Income Distribution Models,” Review of Income and Wealth, 59: 360–374. Milne, W. (1949), Numerical Calculus, Princeton University Press, Princeton. Pareto, V. (1876), Cours d'Économie Politique Professé a l'Université de Lausanne. Piketty, T. (1995), “Social Mobility and Redistributive Politics”, Quarterly Journal of Economics, 110: 551-584. Piketty, T. (2014), Capital in the Twenty-First Century, , Harvard University Press, Cambridge. Ryu, H. (1993), "Maximum entropy estimation of density and regression functions", Journal of Econometrics, 56: 397-440. Ryu, H. (2013), “A bottom poor sensitive Gini coefficient and maximum entropy estimation of income distributions, Economics Letters, 118: 370-374. Ryu, H. and D. Slottje, (1996), "Two Flexible Functional Form Approaches for Approximating the Lorenz Curve", Journal of Econometrics, 72: 251-274. Ryu, H. and D. Slottje, (1998), Measuring Trends in U.S. Income Inequality, Theory and Applications, Springer, New York. Ryu, H. and D. Slottje (2017), “Maximum Entropy Estimation of Income Distributions from Basmann’s WGM Class,” Journal of Econometrics, 199 (2): 221-231. Slottje, D. (1987), “Relative Price Changes and Inequality in the Size Distribution of Various Components of Income,” Journal of Business and Economic Statistics, 5: 19-26. Yitzhaki, S. (2013), More than a dozen ways of spelling Gini, ch-2 in The Gini Methodology, Springer, 11-13. Zellner, A. and R. Highfield, (1988), “Calculation of maximum entropy distributions and approximation of marginal posterior distributions,” Journal of Econometrics, 37: 195-209.