title: Evolution of wealth in a non-conservative economy driven by local Nash equilibria.
abstract: We develop a model for the evolution of wealth in a non-conservative economic environment, extending a theory developed in Degond et al. (2014 J. Stat. Phys. 154, 751-780 (doi:10.1007/s10955-013-0888-4)). The model considers a system of rational agents interacting in a game-theoretical framework. This evolution drives the dynamics of the agents in both wealth and economic configuration variables. The cost function is chosen to represent a risk-averse strategy of each agent. That is, the agent is more likely to interact with the market, the more predictable the market, and therefore the smaller its individual risk. This yields a kinetic equation for an effective single particle agent density with a Nash equilibrium serving as the local thermodynamic equilibrium. We consider a regime of scale separation where the large-scale dynamics is given by a hydrodynamic closure with this local equilibrium. A class of generalized collision invariants is developed to overcome the difficulty of the non-conservative property in the hydrodynamic closure derivation of the large-scale dynamics for the evolution of wealth distribution. The result is a system of gas dynamics-type equations for the density and average wealth of the agents on large scales. We recover the inverse Gamma distribution, which has been previously considered in the literature, as a local equilibrium for particular choices of the cost function.
authors.affiliation: Department of Mathematics, Imperial College London, London SW7 2AZ, UK pdegond@imperial.ac.uk.
authors.forename: Pierre
authors.initials: P
authors.lastname: Degond
authors.affiliation: Departments of Physics and Mathematics, Duke University, Durham, NC 27708, USA.
authors.forename: Jian-Guo
authors.initials: JG
authors.lastname: Liu
authors.affiliation: School of Mathematics and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA.
authors.forename: Christian
authors.initials: C
authors.lastname: Ringhofer
keywords: Fokker–Planck equation; Pareto distribution; general collision invariants; mean field games; multi-agent market models; price strategies