Abstract
We study the Immediate Exchange model, recently introduced by Heinsalu and Patriarca [Eur. Phys. J. B 87, 170 (2014)], who showed by simulations that the wealth distribution in this model converges to a Gamma distribution with shape parameter 2. Here we justify this conclusion analytically, in the infinite-population limit. An infinite-population version of the model is derived, describing the evolution of the wealth distribution in terms of iterations of a nonlinear operator on the space of probability densities. It is proved that the Gamma distributions with shape parameter 2 are fixed points of this operator, and that, starting with an arbitrary wealth distribution, the process converges to one of these fixed points. We also discuss the mixed model introduced in the same paper, in which exchanges are either bidirectional or unidirectional with fixed probability. We prove that, although, as found by Heinsalu and Patriarca, the equilibrium distribution can be closely fit by Gamma distributions, the equilibrium distribution for this model is not a Gamma distribution.
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A. Chatterjee, B.K. Chakrabarti, Eur. Phys. J. B 60, 135 (2007)
B.K. Chakrabarti, A. Chakraborti, S.R. Chakravarty, A. Chatterjee, Econophysics of Income and Wealth Distributions (Cambridge University Press, 2013)
A. Ghosh, A.S. Chakrabarti, A.K. Chandra, A. Chakraborti, Themes and Applications of Kinetic Exchange Models: Redux, in Econophysics of Agent-Based Models, edited by F. Abergel, H. Aoyama, B.K. Chakraborti, A. Chakraborti, A. Ghosh (Springer, 2014), p. 99
M. Patriarca, E. Hainsalu, A. Chakraborti, Eur. Phys. J. B 73, 145 (2010)
M. Patriarca, A. Chakraborti, Am. J. Phys. 81, 618 (2013)
V.M. Yakovenko, J.B. Rosser, Rev. Mod. Phys. 81, 1703 (2009)
J.C. Ferrero, Physica A 341, 575 (2004)
A.A. Drăgulescu, V.M. Yakovenko, Eur. Phys. J. B 20, 585 (2001)
E. Heinsalu, M. Patriarca, Eur. Phys. J. B 87, 170 (2014)
A.A. Drăgulescu, V.M. Yakovenko, Eur. Phys. J. B 17, 723 (2000)
A. Salem, T. Mount, Econometrica 42, 1115 (1974)
A. Chakraborti, B.K. Chakrabarti, Eur. Phys. J. B 17, 167 (2000)
M. Lallouache, A. Jedidi, A. Chakraborti, Sci. Culture 76, 478 (2010)
J. Angle, Soc. Forces 56, 293 (1986)
J. Angle, Physica A 386, 388 (2006)
I. Martínez-Martínez, R. López-Ruiz, Int. J. Mod. Phys. C 24, 1250088 (2013)
G. Katriel, Acta Appl. Math. (2014), DOI: 10.1007/s10440-014-9971-3
A. Chatterjee, B.K. Chakrabarti, S.S. Manna, Physica A 335, 155 (2004)
S. Sinha, The rich are different! Pareto law from asymmetric interactions in asset exchange models, in Econophysics of Wealth Distributions, edited by A. Chatterjee, S. Yarlagadda, B.K. Chakrabarti (Springer, 2005), p. 177
J.L. López, R. López-Ruiz, X. Calbet, J. Math. Anal. Appl. 386, 195 (2012)
R. López-Ruiz, X. Calbet, Exponential wealth distribution: a new approach from functional iteration theory, ESAIM: Proceedings 36, 183 (2012)
G. Katriel, Appl. Anal. 93, 1256 (2014)
D. Matthes, G.J. Toscani, Stat. Phys. 130, 1087 (2008)
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Katriel, G. The Immediate Exchange model: an analytical investigation. Eur. Phys. J. B 88, 19 (2015). https://doi.org/10.1140/epjb/e2014-50661-7
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DOI: https://doi.org/10.1140/epjb/e2014-50661-7