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k-Generalized Statistics in Personal Income Distribution


  • F. Clementi
  • M. Gallegati
  • G. Kaniadakis


Starting from the generalized exponential function $\exp_{\kappa}(x)=(\sqrt{1+\kappa^{2}x^{2}}+\kappa x)^{1/\kappa}$, with $\exp_{0}(x)=\exp(x)$, proposed in Ref. [G. Kaniadakis, Physica A \textbf{296}, 405 (2001)], the survival function $P_{>}(x)=\exp_{\kappa}(-\beta x^{\alpha})$, where $x\in\mathbf{R}^{+}$, $\alpha,\beta>0$, and $\kappa\in[0,1)$, is considered in order to analyze the data on personal income distribution for Germany, Italy, and the United Kingdom. The above defined distribution is a continuous one-parameter deformation of the stretched exponential function $P_{>}^{0}(x)=\exp(-\beta x^{\alpha})$\textemdash to which reduces as $\kappa$ approaches zero\textemdash behaving in very different way in the $x\to0$ and $x\to\infty$ regions. Its bulk is very close to the stretched exponential one, whereas its tail decays following the power-law $P_{>}(x)\sim(2\beta\kappa)^{-1/\kappa}x^{-\alpha/\kappa}$. This makes the $\kappa$-generalized function particularly suitable to describe simultaneously the income distribution among both the richest part and the vast majority of the population, generally fitting different curves. An excellent agreement is found between our theoretical model and the observational data on personal income over their entire range.

Suggested Citation

  • F. Clementi & M. Gallegati & G. Kaniadakis, 2006. "k-Generalized Statistics in Personal Income Distribution," Papers physics/0607293,, revised Feb 2007.
  • Handle: RePEc:arx:papers:physics/0607293

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    References listed on IDEAS

    1. Andrea Brandolini, 1999. "The Distribution of Personal Income in Post-War Italy: Source Description, Data Quality, and the Time Pattern of Income Inequality," Giornale degli Economisti, GDE (Giornale degli Economisti e Annali di Economia), Bocconi University, vol. 58(2), pages 183-239, September.
    2. Geoff Willis & Juergen Mimkes, 2004. "Evidence for the Independence of Waged and Unwaged Income, Evidence for Boltzmann Distributions in Waged Income, and the Outlines of a Coherent Theory of Income Distribution," Microeconomics 0408001, EconWPA.
    3. Makoto Nirei & Wataru Souma, 2007. "A Two Factor Model Of Income Distribution Dynamics," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 53(3), pages 440-459, September.
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    1. Sarabia, José María & Jordá, Vanesa, 2014. "Explicit expressions of the Pietra index for the generalized function for the size distribution of income," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 416(C), pages 582-595.
    2. Bourguignon, Marcelo & Saulo, Helton & Fernandez, Rodrigo Nobre, 2016. "A new Pareto-type distribution with applications in reliability and income data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 457(C), pages 166-175.
    3. Tapiero, Oren J., 2013. "A maximum (non-extensive) entropy approach to equity options bid–ask spread," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(14), pages 3051-3060.
    4. Elvis Oltean, 2016. "Modelling income, wealth, and expenditure data by use of Econophysics," Papers 1603.08383,
    5. Costas Efthimiou & Adam Wearne, 2016. "Household Income Distribution in the USA," Papers 1602.06234,
    6. F. Clementi & M. Gallegati & G. Kaniadakis, 2012. "A generalized statistical model for the size distribution of wealth," Papers 1209.4787,, revised Dec 2012.
    7. Masato Okamoto, 2014. "A flexible descriptive model for the size distribution of incomes," Economics Bulletin, AccessEcon, vol. 34(3), pages 1600-1610.
    8. Masato Okamoto, 2012. "Evaluation of the goodness of fit of new statistical size distributions with consideration of accurate income inequality estimation," Economics Bulletin, AccessEcon, vol. 32(4), pages 2969-2982.
    9. F. Clementi & M. Gallegati, 2016. "New economic windows on income and wealth: The k-generalized family of distributions," Papers 1608.06076,
    10. Anwar Shaikh, Amr Ragab, 2007. "WP 2007-3 An International Comparison of the Incomes of the Vast Majority," SCEPA working paper series. SCEPA's main areas of research are macroeconomic policy, inequality and poverty, and globalization. 2007-3, Schwartz Center for Economic Policy Analysis (SCEPA), The New School.
    11. Chami Figueira, F. & Moura, N.J. & Ribeiro, M.B., 2011. "The Gompertz–Pareto income distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(4), pages 689-698.
    12. Victor M. Yakovenko & J. Barkley Rosser, 2009. "Colloquium: Statistical mechanics of money, wealth, and income," Papers 0905.1518,, revised Dec 2009.

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