Fokker-Planck Equations for Weatlh Distribution Dynamics

Jeudi 7 avril 2016 14:15-15:15 - Bruce Boghosian - Tufts University

Résumé : Today there are 62 people in the world who have as much combined wealth as half the human population. In 2010, that figure was 388, so it has dropped by a factor of more than six in six years. These figures make clear that wealth distributions are dynamic and that wealth is concentrating. Understanding how and why this is happening, and what if anything needs to be done about it is an interdisciplinary problem that will ultimately involve mathematicians, economists, political scientists, and specialists in ethics, justice, and public policy.
Asset-exchange models are mathematical models of wealth distribution that have received much attention in recent years. They model an economy by a collection of economic agents, who exchange wealth in pairwise transactions according to idealized rules. Their evolution may be described by stochastic differential equations, or by nonlinear and nonlocal Boltzmann equations or Fokker-Planck equations [Boghosian, 2014].
One of the simplest yet most intriguing asset-exchange models, called the Yard-Sale Model, predicts the unbounded concentration of wealth ; indeed, the Gini coefficient, a basic measure of wealth inequality, has been shown to be a Lyapunov functional of this model [Boghosian, Johnson, Marcq, 2015]. Though this is by itself unrealistic, when an Ornstein-Uhlenbeck model of redistribution is added, the Yard-Sale Model exhibits stable steady-state behavior that is similar in form to the famous Pareto distribution. When transactions in the model are biased in favor of the wealthier party, the model also exhibits a phase transition known as « wealth condensation » [Bouchaud, Mézard, 2000], in which a finite fraction of the population’s wealth falls into the hands of the wealthiest agent. It has been argued that this transition provides a first-principles explanation of the phenomenon of oligarchy.
In this presentation, the Yard-Sale Model will be described, followed by a model for redistribution and a model for wealth-attained advantage [Boghosian, Devitt-Lee, Johnson, Marcq, Wang, 2016]. A Fokker-Planck equation for the wealth probability density function will be derived, and both theoretical and numerical studies of the asymptotics of its solution will be discussed in some detail, for low, intermediate and high ranges of wealth. The phenomenon of wealth condensation will be described by a distributional solution to this Fokker-Planck equation, involving the appearance of a singular distribution that describes a vanishingly small number of agents, holding an amount of wealth that tends to infinity, in such a way that the total wealth held by these agents is finite.

Lieu : Bât 425, salle 113-115

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