The Immediate Exchange model: an analytical investigation

TL;DR

This paper provides an analytical proof that the wealth distribution in the Immediate Exchange model converges to a Gamma distribution with shape parameter 2, confirming previous simulation results and extending understanding to a mixed exchange model.

Contribution

It analytically justifies the convergence to Gamma distribution in the Immediate Exchange model and clarifies the distribution in the mixed model, which was previously only empirically observed.

Findings

We proved Gamma distributions with shape 2 are fixed points of the model.

The wealth distribution converges to these fixed points from any initial distribution.

The mixed model's equilibrium distribution is not a Gamma distribution, contrary to previous empirical fits.

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 {\it{not}} a Gamma distribution.