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:)). 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.
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机译：我们开发了非保守经济环境中财富演变的模型，并扩展了Degond等人开发的理论。 （2014 J. Stat。Phys。> 154 ，751–780（doi :)）。该模型考虑了在博弈论框架中相互作用的理性主体系统。这种演变推动了财富和经济配置变量中代理人的动态。选择成本函数代表每个代理商的规避风险策略。也就是说，代理人与市场互动的可能性越大，市场越容易预测，因此其个人风险就越小。这产生了一个有效的单颗粒剂密度的动力学方程，其中纳什平衡用作局部热力学平衡。我们考虑尺度分离的机制，其中大规模动力学由具有这种局部平衡的流体动力闭合给出。开发了一类广义碰撞不变式，以克服在财富分配演变的大规模动力学的流体动力学闭合推导中非保守性质的困难。结果是一个大规模的气体动力学类型方程式，用于计算代理的密度和平均财富。我们恢复了逆伽玛分布，该逆向伽玛分布先前在文献中已被视为对成本函数的特定选择的局部均衡。
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