A Fokker-Planck model for wealth inequality dynamics

Abstract

Studying the mechanisms that govern the dynamics of the wealth distribution is essential for understanding the recent trend of growing wealth inequality. A particularly important explanation is Piketty's argument, giving credit to the seminal events of the first half of the 20th century for the relatively egalitarian second half of this century. Piketty suggested that these dramatic events were merely a perturbation imposed on the economy affecting the wealth structure, while in general, wealth inequality tends to increase regularly. We present a simple stochastic model for wealth and income based on coupled geometric Brownian motions and derive a Fokker-Planck equation from which the joint wealth-income distribution and its moments can be extracted. We then analyze the dynamics of these moments and hence of the inequality. Our analysis largely supports Piketty's argument regarding the irregularity of the 20th century, that wealth inequality inevitably tends to increase. We find, however, that even if wealth inequality will eventually go up, under plausible conditions, it can go down for periods of up to several decades.

Dynamics of wealth inequality

“…For sparse networks the limiting distribution is given by a generalised-inverse-gamma distribution [36]. The system displays a highly non-trivial pattern of relaxation to the limiting distribution [28,37,38]. For finite size the system does not reach a stationary state.…”

“…The regime of the model that corresponds to aggressive economy, J ij < 0, favouring the rich-get-richer (and poorget-poorer) interactions has not been studied yet. In this case the system is unstable [28,38] unless some regulatory, stabilising mechanism is introduced.…”

“…Research on wealth distribution is a very active branch of science both in economics [2,3,[6][7][8][25][26][27][28][29][30] and econophysics [9,14,16,20,[31][32][33][34][35][36][37][38][39]. An important contribution of econophysics to this field is the development of multi-agent models which are capable to describe statistical, collective effects and large-scale emergent phenomena.…”

Power accretion in social systems

“…This result is reminiscent of Piketty's claim that inequality will grow when the rate of return of capital r is larger than the rate of growth g of the whole economy, since r −1 and g −1 can be thought as the time scales associated to capital and economic growth. [2,32,47].…”

Rich or poor: Who should pay higher tax rates?

“…We provide in ref. [6] an incomplete and somewhat arbitrary list of works related to this issue; the interested reader may also wish to consult the works cited in these references. All of them follow the dynamic evolution of the wealth distribution among agents, not the distribution of wealth shares here introduced.…”