Derivation of wealth distributions from biased exchange of money
Derivation of wealth distributions from biased exchange of money
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DOI：
10.3934/krm.2023007
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发表时间：
2021-05
影响因子：
1
通讯作者：
Fei Cao;Sébastien Motsch
中科院分区：
文献类型：
--
作者：
Fei Cao;Sébastien Motsch
在手稿中，我们有兴趣使用动力学理论来更好地理解财富分配的时间演变及其大规模行为，如不平等的演变（例如基尼指数）。我们研究了三种类型的动态表示无偏，贫偏和富偏动态。在粒子级，一个代理人根据其财富被随机挑选，其中一个美元在人口中重新分配。为了证明所谓的混沌传播，我们使用耦合技术[48]和基于鞅的方法[36]确定每个动力学的极限，因为个体的数量接近无穷大。借助极限方程，我们识别并证明了无偏动态和弱偏动态都收敛于特定平衡点。然而，在富偏置动力学中，我们观察到一个更复杂的行为，其中出现了色散波。虽然色散波在时间上消失了，但它也积累了所有的财富，导致基尼系数接近1（其最大值）。我们用数值方法描述了色散波的行为，但需要进一步的分析研究来直接从动力学中推导出这种色散波。
In the manuscript, we are interested in using kinetic theory to better understand the time evolution of wealth distribution and their large scale behavior such as the evolution of inequality (e.g. Gini index). We investigate three type of dynamics denoted unbiased, poor-biased and rich-biased dynamics. At the particle level, one agent is picked randomly based on its wealth and one of its dollar is redistributed among the population. Proving the so-called propagation of chaos, we identify the limit of each dynamics as the number of individual approaches infinity using both coupling techniques [48] and martingale-based approach [36]. Equipped with the limit equation, we identify and prove the convergence to specific equilibrium for both the unbiased and poor-biased dynamics. In the rich-biased dynamics however, we observe a more complex behavior where a dispersive wave emerges. Although the dispersive wave is vanishing in time, its also accumulates all the wealth leading to a Gini approaching 1 (its maximum value). We characterize numerically the behavior of dispersive wave but further analytic investigation is needed to derive such dispersive wave directly from the dynamics.