Riv.Mat.Univ.Parma (8) 1 (2009)

B. Düring, D. Matthes and G. Toscani

A Boltzmann-type approach to the formation of wealth distribution curves

Pages 199-261
Received: 9 October 2008
Accepted: 2 December 2008
Mathematics Subject Classification (2000): 91B60 - 82C40 - 35B40

Keywords: Econophysics, Boltzmann equation, wealth and income distributions, Pareto distribution, mixtures.

Abstract: Kinetic market models have been proposed recently to account for the redistribution of wealth in simple market economies. These models allow to develop a qualitative theory, which is based on methods borrowed from the kinetic theory of rarefied gases. The aim of these notes is to present a unifying approach to the study of the evolution of wealth in the large-time regime. The considered models are divided into two classes: the first class is such that the society's mean wealth is conserved, while for models of the second class, the mean wealth grows or decreases exponentially in time. In both cases, it is possible to classify the most important feature of the steady (or self-similar, respectively) wealth distributions, namely the fatness of the Pareto tail. We shall also discuss the tails' dynamical stability in terms of the model parameters. Our results are derived by means of a qualitative analysis of the associated homogeneous Boltzmann equations. The key tools are suitable metrics for probability measures, and a concise description of the evolution of moments. A recent extension to economies, in which different groups of agents interact, is presented in detail. We conclude with numerical experiments that confirm the theoretical predictions.

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