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Kinetic theory models for the distribution of wealth: power law from overlap of exponentials

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  • Marco Patriarca
  • Anirban Chakraborti
  • Kimmo Kaski
  • Guido Germano

Abstract

Various multi-agent models of wealth distributions defined by microscopic laws regulating the trades, with or without a saving criterion, are reviewed. We discuss and clarify the equilibrium properties of the model with constant global saving propensity, resulting in Gamma distributions, and their equivalence to the Maxwell-Boltzmann kinetic energy distribution for a system of molecules in an effective number of dimensions $D_\lambda$, related to the saving propensity $\lambda$ [M. Patriarca, A. Chakraborti, and K. Kaski, Phys. Rev. E 70 (2004) 016104]. We use these results to analyze the model in which the individual saving propensities of the agents are quenched random variables, and the tail of the equilibrium wealth distribution exhibits a Pareto law $f(x) \propto x^{-\alpha -1}$ with an exponent $\alpha=1$ [A. Chatterjee, B. K. Chakrabarti, and S. S. Manna, Physica Scripta T106 (2003) 367]. Here, we show that the observed Pareto power law can be explained as arising from the overlap of the Maxwell-Boltzmann distributions associated to the various agents, which reach an equilibrium state characterized by their individual Gamma distributions. We also consider the influence of different types of saving propensity distributions on the equilibrium state.

Suggested Citation

  • Marco Patriarca & Anirban Chakraborti & Kimmo Kaski & Guido Germano, 2005. "Kinetic theory models for the distribution of wealth: power law from overlap of exponentials," Papers physics/0504153, arXiv.org, revised May 2005.
  • Handle: RePEc:arx:papers:physics/0504153
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    Cited by:

    1. Adams Vallejos & Ignacio Ormazabal & Felix A. Borotto & Hernan F. Astudillo, 2018. "A new $\kappa$-deformed parametric model for the size distribution of wealth," Papers 1805.06929, arXiv.org.
    2. Patriarca, Marco & Chakraborti, Anirban & Germano, Guido, 2006. "Influence of saving propensity on the power-law tail of the wealth distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 369(2), pages 723-736.
    3. Angle, John, 2006. "The Inequality Process as a wealth maximizing process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 367(C), pages 388-414.
    4. Vallejos, Adams & Ormazábal, Ignacio & Borotto, Félix A. & Astudillo, Hernán F., 2019. "A new κ-deformed parametric model for the size distribution of wealth," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 514(C), pages 819-829.
    5. J. M. Pellon-Diaz & A. Aragones-Munoz & A. Sandoval-Villalbazo & A. Diaz-Reynoso, 2011. "Gaussian Noise Effects on the Evolution of Wealth in a Closed System of n-Economies," Papers 1102.1713, arXiv.org, revised Feb 2011.
    6. Anirban Chakraborti & Ioane Muni Toke & Marco Patriarca & Frédéric Abergel, 2011. "Econophysics review: II. Agent-based models," Post-Print hal-00621059, HAL.
    7. Chakrabarti, Anindya S. & Chakrabarti, Bikas K., 2010. "Statistical theories of income and wealth distribution," Economics - The Open-Access, Open-Assessment E-Journal (2007-2020), Kiel Institute for the World Economy (IfW Kiel), vol. 4, pages 1-31.
    8. Kiran Sharma & Anirban Chakraborti, 2016. "Physicists' approach to studying socio-economic inequalities: Can humans be modelled as atoms?," Papers 1606.06051, arXiv.org, revised Aug 2018.
    9. Maciej Jagielski & Ryszard Kutner, 2013. "Modelling the income distribution in the European Union: An application for the initial analysis of the recent worldwide financial crisis," Papers 1312.2362, arXiv.org.
    10. Rem Sadykhov & Geoffrey Goodell & Denis de Montigny & Martin Schoernig & Philip Treleaven, 2023. "Decentralized Token Economy Theory (DeTEcT)," Papers 2309.12330, arXiv.org, revised Jan 2024.
    11. Rem Sadykhov & Geoffrey Goodell & Philip Treleaven, 2024. "DeTEcT: Dynamic and Probabilistic Parameters Extension," Papers 2405.16688, arXiv.org, revised Dec 2024.
    12. Els Heinsalu & Marco Patriarca, 2014. "Kinetic models of immediate exchange," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 87(8), pages 1-10, August.
    13. Anindya S. Chakrabarti, 2011. "Firm dynamics in a closed, conserved economy: A model of size distribution of employment and related statistics," Papers 1112.2168, arXiv.org.
    14. Victor M. Yakovenko & J. Barkley Rosser, 2009. "Colloquium: Statistical mechanics of money, wealth, and income," Papers 0905.1518, arXiv.org, revised Dec 2009.

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