Oligarchy as a Phase Transition: The effect of wealth-attained advantage in a Fokker-Planck description of asset exchange (1511.00770v2)

Abstract: In earlier work, we derived a nonlinear, nonlocal Fokker-Planck equation for the Yard-Sale Model of asset exchange. In the absence of redistribution, we showed that the Gini coefficient is a Lyapunov functional for this model, tending to one in the time-asymptotic limit, corresponding to maximal inequality. When a one-parameter model of redistribution is introduced, we showed that the model admits a steady state similar to Pareto's Law. In this work, we analyze the form of this distribution in greater detail, both analytically and numerically. We find that, while Pareto's Law is approximately valid for low redistribution, it gives way to something like Gibrat's Law at higher redistribution. We also prove that, while this Pareto or Gibrat behavior persists over many orders of magnitude, it ultimately gives way to gaussian decay at extremely large wealth. Following the work of Moukarzel et al., we introduce a bias in favor of the wealthier agent. We derive the corresponding modification to the Fokker-Planck equation, and we show this leads to wealth condensation when the bias exceeds a critical value. Earlier work took the bias to be a discontinuous function of the wealth differential between the two transacting agents, and reported a first-order phase transition to absolute oligarchy. By contrast, in this work we take the bias to be a continuous function of the wealth differential, and consequently we observe a second-order phase transition with a region of coexistence between the oligarch and a distribution of non-oligarchs. We additionally show that the onset of wealth condensation has a 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.

• The paper models wealth distribution dynamics using a Fokker-Planck equation, showing how wealth-attained advantage can drive a phase transition to oligarchy.
• The model reveals that wealth-attained advantage above a critical threshold triggers a phase transition, leading to partial wealth condensation and oligarchy.
• Implications suggest effective redistribution policies are essential to counteract wealth-attained advantage and prevent extreme inequality by promoting diverse wealth distributions.

Overview of "Oligarchy as a Phase Transition: The Effect of Wealth-Attained Advantage in a Fokker-Planck Description of Asset Exchange"

This paper explores the dynamics of wealth distribution using a mathematical model known as the Yard-Sale Model (YSM) of asset exchange, enhanced by introducing a parameter for Wealth-Attained Advantage (WAA). The paper employs a nonlinear, integrodifferential Fokker-Planck equation to describe how wealth asymmetry emerges and evolves over time. The authors aim to understand under what conditions wealth becomes concentrated within a small fraction of agents, known as "wealth condensation," and to investigate the transition from egalitarian distributions to oligarchies.

Key Results and Claims

1. Wealth Dynamics Without Redistribution:
   • Without any mechanism for wealth redistribution, the system modeled by YSM results in an increase in inequality, characterized by the Gini coefficient approaching one, signifying that all wealth is eventually held by a single agent.
2. Introduction of Redistribution:
   • By introducing a model of redistribution akin to the Ornstein-Uhlenbeck process, the system achieves a steady-state wealth distribution. For low redistribution levels, this distribution resembles the Pareto law, transitioning to a form similar to Gibrat's law with increased redistribution. Ultimately, extremely large wealth exhibits gaussian decay in the distribution.
3. Wealth-Attained Advantage (WAA):
   • The introduction of WAA as a continuous, smooth function of wealth disparity between agents leads to a significant behavioral change. When WAA exceeds a critical threshold, the system undergoes a second-order phase transition, resulting in partial wealth condensation where a substantial portion of the total wealth is held by the wealthiest agents, rather than absolute condensation into a single agent's hands.
4. Criticality and Wealth Condensation:
   • At critical points, peculiar phenomena emerge: the decay of wealth distribution transitions from gaussian to exponential as redistribution increases and meets a threshold, beyond which wealth condensation occurs. This region shows coexistence between an oligarchic fraction and a diverse wealth distribution among non-oligarchs.

Implications

• Theoretical Insights:

This research advances theoretical understanding of wealth distribution dynamics, providing a mathematically rigorous explanation of how and why wealth concentration occurs under varying economic policies, particularly in terms of redistribution and inherent bias towards wealth accumulation.

• Economic Implications:

For policymakers, these findings offer crucial insights into preventing extreme wealth inequality by adjusting redistribution measures. The model quantitatively highlights the importance of redistribution in maintaining diverse wealth distributions and suggests the sensitivity of economies to both redistributive policies and inherent structural biases favoring wealth accumulation.

Future Research Directions

The authors propose extending the model by incorporating additional economic factors such as production, consumption, and more nuanced redistribution models to enhance its applicability to real-world economies. Understanding transient behaviors in the face of dynamic modern economies could also benefit from this foundational work, prompting further investigation into temporal evolutions of wealth distributions in a stochastic framework.

In summary, this paper contributes significantly to quantitative economic modeling by illustrating the conditions under which economies may transition to oligarchic states, thereby offering an analytic framework for examining policies that can mitigate inequality and promote wealth distribution diversity.