Effect of tax dynamics on linearly growing processes under stochastic resetting: a possible economic model

Abstract

We study a system of $N$ agents, whose wealth grows linearly, under the effect of stochastic resetting and interacting via a tax-like dynamics -- all agents donate a part of their wealth, which is, in turn, redistributed equally among all others. This mimics a socio-economic scenario where people have fixed incomes, suffer individual economic setbacks, and pay taxes to the state. The system always reaches a stationary state, which shows a trivial exponential wealth distribution in the absence of tax dynamics. The introduction of the tax dynamics leads to several interesting features in the stationary wealth distribution. In particular, we analytically find that an increase in taxation for a homogeneous system (where all agents are alike) results in a transition from a society where agents are most likely poor to another where rich agents are more common. We also study inhomogeneous systems, where the growth rates of the agents are chosen from a distribution, and the taxation is proportional to the individual growth rates. We find an optimal taxation, which produces a complete economic equality (average wealth is independent of the individual growth rates), beyond which there is a reverse disparity, where agents with low growth rates are more likely to be rich. We consider three income distributions observed in the real world and show that they exhibit the same qualitative features. Our analytical results are in the $N\to\infty$ limit and backed by numerical simulations.