Wealth condensation in a multiplicative random asset exchange model

摘要

Random Asset Exchange (RAE) models, despite a number of simplifying assumptions, serve the purpose of establishing direct relationships between microscopic exchange mechanisms and observed economical data. In this work a conservative multiplicative RAE model is discussed in which, at each timestep, two agents “bet” for a fraction f of the poorest agent's wealth. When the poorest agent wins the bet with probability p, we show that, in a well defined region of the (p,f) phase space, there is wealth condensation. This means that all wealth ends up owned by only one agent, in the long run. We derive the condensation conditions analytically by two different procedures, and find results in accordance with previous numerical estimates. In the non-condensed phase, the equilibrium wealth distribution is a power law for small wealths. The associated exponent is derived analytically and it is found that it tends to -1 on the condensation interface. I turns out that wealth condensation happens also for values of p much larger than 0.5, that is under microscopic exchange rules that, apparently, favor the poor. We argue that the observed “rich get richer” effect is enhanced by the multiplicative character of the dynamics.