Multiplicative Asset Exchange with Arbitrary Return Distributions

Abstract

The conservative wealth-exchange process derived from trade interactions is modeled as a multiplicative stochastic transference of value, where each interaction multiplies the wealth of the poorest of the two intervening agents by a random gain eta=(1+kappa), with kappa a random return. Analyzing the kinetic equation for the wealth distribution P(w,t), general properties are derived for arbitrary return distributions pi(kappa). If the geometrical average of the gain is larger than one, i.e. if >0, in the long time limit a nontrivial equilibrium wealth distribution P(w) is attained. Whenever <0, on the other hand, Wealth Condensation occurs, meaning that a single agent gets the whole wealth in the long run. This concentration phenomenon happens even if the average return of the poor agent is positive. In the stable phase, P(w) behaves as w^{(T-1)} for w -> 0, and we find T exactly. This exponent is nonzero in the stable phase but goes to zero on approach to the condensation interface. The exact wealth distribution can be obtained analytically for the particular case of Kelly betting, and it turns out to be exponential. We show, however, that our model is never reversible, no matter what pi(kappa) is. In the condensing phase, the wealth of an agent with relative rank x is found to be w(x,t) \sim e^{x t } for finite times t. The wealth distribution is consequently P(w) \sim 1/w for finite times, while all wealth ends up in the hands of a single agent for large times. Numerical simulations are carried out, and found to satisfactorily compare with the above mentioned analytic results.