On a Kinetic Model for a Simple Market Economy

Abstract

In this paper, we consider a simple kinetic model of economy involving both exchanges between agents and speculative trading. We show that the kinetic model admits non trivial quasi-stationary states with power law tails of Pareto type. In order to do this we consider a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution of wealth among individuals. For this equation the stationary state can be easily derived and shows a Pareto power law tail. Numerical results confirm the previous analysis.

On Steady Distributions of Kinetic Models of Conservative Economies

“…We derive a sufficient criterion under which a unique non-trivial stationary state exists, and provide criteria under which these steady states do or do not possess a Pareto tail. In particular, we prove the absence of Pareto tails in pointwise conservative models, like the one in [11], while models with speculative trades introduced in [14] develop fat tails if the market is "risky enough". The results are derived by a Fourier-based technique first developed for the Maxwell-Boltzmann equation [26,1,39], and from a recursive relation which allows to calculate arbitrary moments of the stationary state.…”

“…As specific examples, we study models with fixed saving propensity introduced by A. Chakraborty and B.K. Chakrabarthi [11], as well as models involving both exchange between agents and speculative trading as considered by S. Cordier, L. Pareschi and one of the authors [14]. We derive a sufficient criterion under which a unique non-trivial stationary state exists, and provide criteria under which these steady states do or do not possess a Pareto tail.…”

Analysis of a model for wealth redistribution

“…Although (13) constitutes only a lower bound on the speed of convergence, numerical experiments indicate [8] that the temporal rate in this estimate is sharp. The size of s 1 > 0 -which depends on the specific realization of the experiment -is seemingly decisive for the relaxation process.…”

“…In several models for open economies, where wealth increases due to trade interactions, the money distribution curve attains a self-similar profile with a Pareto tail, see e.g. [13,21,23]. Likewise, a combination of trades with risky investments, which preserve money in the statistical mean, but not in individual trades, are able to generate Pareto behavior [19].…”

A kinetic approach to the study of opinion formation

“…For example, kinetic-type equations have been introduced in order to describe a simple market economy with a constant growth mechanism [5,6,16], showing the formation of steady states with (overpopulated) Pareto tails.…”