Eur. Phys. J. B (2015) 88: 19
https://doi.org/10.1140/epjb/e2014-50661-7

Regular Article

The Immediate Exchange model: an analytical investigation

Department of Mathematics, ORT Braude College, Karmiel 2161002, Israel

^a e-mail: katriel@braude.ac.il

Received: 27 September 2014
Received in final form: 16 November 2014
Published online: 14 January 2015

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.