Oligarchy as a phase transition: The effect of wealth-attained advantage in a Fokker–Planck description of asset exchange

The “Yard-Sale Model” of asset exchange is known to result in complete inequality—all of the wealth in the hands of a single agent. It is also known that, when this model is modified by introducing a simple model of redistribution based on the Ornstein–Uhlenbeck process, it admits a steady state exhibiting some features similar to the celebrated Pareto Law of wealth distribution. In the present work, we analyze the form of this steady-state distribution in much greater detail, using a combination of analytic and numerical techniques. We find that, while Pareto’s Law is approximately valid for low redistribution, it gives way to something more similar to Gibrat’s Law when redistribution is higher. Additionally, we prove in this work that, while this Pareto or Gibrat behavior may persist over many orders of magnitude, it ultimately gives way to gaussian decay at extremely large wealth. Also in this work, we introduce a bias in favor of the wealthier agent–what we call Wealth-Attained Advantage (WAA)–and show that this leads to the phenomenon of “wealth condensation” when the bias exceeds a certain critical value. In the wealth-condensed state, a finite fraction of the total wealth of the population “condenses” to the wealthiest agent. We examine this phenomenon in some detail, and derive the corresponding modification to the Fokker–Planck equation. We observe a second-order phase transition to a state of coexistence between an oligarch and a distribution of non-oligarchs. Finally, by studying the asymptotic behavior of the distribution in some detail, we show that the onset of wealth condensation has an abrupt reciprocal effect on the character of the non-oligarchical part of the distribution. Specifically, we show that the above-mentioned gaussian decay at extremely large wealth is valid both above and below criticality, but degenerates to exponential decay precisely at criticality.