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On the Kolkata index as a measure of income inequality

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  • Suchismita Banerjee
  • Bikas K. Chakrabarti
  • Manipushpak Mitra
  • Suresh Mutuswami

Abstract

We study the mathematical and economic structure of the Kolkata (k) index of income inequality. We show that the k-index always exists and is a unique fixed point of the complementary Lorenz function, where the Lorenz function itself gives the fraction of cumulative income possessed by the cumulative fraction of population (when arranged from poorer to richer). We show that the k-index generalizes Pareto's 80/20 rule. Although the k and Pietra indices both split the society into two groups, we show that k-index is a more intensive measure for the poor-rich split. We compare the normalized k-index with the Gini coefficient and the Pietra index and discuss when they coincide. We establish that for any income distribution the value of Gini coefficient is no less than that of the Pietra index and the value of the Pietra index is no less than that of the normalized k-index. While the Gini coefficient and the Pietra index are affected by transfers exclusively among the rich or among the poor, the k-index is only affected by transfers across the two groups.

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  • Suchismita Banerjee & Bikas K. Chakrabarti & Manipushpak Mitra & Suresh Mutuswami, 2019. "On the Kolkata index as a measure of income inequality," Papers 1905.03615, arXiv.org, revised Oct 2019.
    • Handle: RePEc:arx:papers:1905.03615
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    1. Eliazar, Iddo, 2015. "The sociogeometry of inequality: Part II," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 426(C), pages 116-137.
    2. Eliazar, Iddo, 2015. "The sociogeometry of inequality: Part I," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 426(C), pages 93-115.
    3. Chatterjee, Arnab & Ghosh, Asim & Chakrabarti, Bikas K., 2017. "Socio-economic inequality: Relationship between Gini and Kolkata indices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 466(C), pages 583-595.
    4. Eliazar, Iddo I. & Sokolov, Igor M., 2010. "Measuring statistical heterogeneity: The Pietra index," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(1), pages 117-125.
    5. Rolf Aaberge, 2000. "Characterizations of Lorenz curves and income distributions," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 17(4), pages 639-653.
    6. Gastwirth, Joseph L, 1971. "A General Definition of the Lorenz Curve," Econometrica, Econometric Society, vol. 39(6), pages 1037-1039, November.
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    1. Ghosh, Asim & Chakrabarti, Bikas K., 2021. "Limiting value of the Kolkata index for social inequality and a possible social constant," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 573(C).
    2. Grachev, Gennady A., 2022. "Size distribution of states, counties, and cities in the USA: New inequality form information," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 592(C).
    3. Ruz, Soumendra Nath, 2023. "Amazing aspects of inequality indices (Gini and Kolkata Index) of COVID-19 confirmed cases in India," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 632(P2).
    4. Christopher W. Kulp & Michael Kurtz & Charles Hunt & Matthew Velardi, 2023. "The distribution of wealth: an agent-based approach to examine the effect of estate taxation, skill inheritance, and the Carnegie Effect," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 18(2), pages 397-415, April.

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