Measuring inequality in society-oriented Lotka--Volterra-type kinetic equations

TL;DR

The paper addresses measuring inequality in a two-population system described by coupled Fokker--Planck equations whose mean quantities obey Lotka--Volterra dynamics. It introduces the time evolution of the coefficient of variation as a high-signal metric to track inequality in oscillatory, nonstationary settings, linking it to the Gini index through analytic relationships and quasi-equilibria. The deposit--loan kinetic model is used to illustrate how distributions evolve under interactions, risk, and redistribution, showing that inequality tends to decrease despite oscillations, especially under moderate risk where Gamma quasi-equilibria are informative. This work provides a practical framework for inequality assessment in socio-economic systems lacking a stationary state and demonstrates the usefulness of kinetic methods in interpreting macro-level inequality from micro-level interactions.

Abstract

We present a possible approach to measuring inequality in a system of coupled Fokker-Planck-type equations that describe the evolution of distribution densities for two populations interacting pairwise due to social and/or economic factors. The macroscopic dynamics of their mean values follow a Lotka-Volterra system of ordinary differential equations. Unlike classical models of wealth and opinion formation, which tend to converge toward a steady-state profile, the oscillatory behavior of these densities only leads to the formation of local equilibria within the Fokker-Planck system. This makes tracking the evolution of most inequality measures challenging. However, an insightful perspective on the problem is obtained by using the coefficient of variation, a simple inequality measure closely linked to the Gini index. Numerical experiments confirm that, despite the system's oscillatory nature, inequality initially tends to decrease.

Figures (4)

• Figure 1: Evolution of the system for $p=\frac{1}{2}$ in both populations. Panel on the left: evolution of coefficients of variation, $c_f(t)$ and $c_g(t)$. Panel on the right: evolution of coefficients of variation, $c_f(t)$ and $c_g(t)$, along with mean values, $m_f(t)$ and $m_g(t)$. Initial data $(m_f(0), m_g(0), c_f(0), c_g(0))=(4,\, 3,\, 2,\, 1)$.
• Figure 2: Evolution of the system for $p=\frac{1}{2}$ for the population and $p=1$ for the second population. Panel on the left: evolution of coefficients of variation, $c_f(t)$ and $c_g(t)$. Panel on the right: evolution of coefficients of variation, $c_f(t)$ and $c_v(t)$, along with mean values, $m_f(t)$ and $m_g(t)$. Initial data $(m_f(0), m_g(0), c_f(0), c_g(0))=(4,\, 3,\, 2,\, 1)$.
• Figure 3: Evolution of coefficients of variation, $c_f(t)$ and $c_g(t)$, with a reduction of values $\sigma_f$ and $\sigma_g$ of the microscopic interactions. Panel on the right: $p=\frac{1}{2}$ for both populations. Panel on the left: $p=\frac{1}{2}$ for the first population and $p=1$ for the second population. Initial data $(m_f(0), m_g(0), c_f(0), c_g(0))=(4,\, 3,\, 2,\, 1)$
• Figure 4: Evolution of the system for $p=\frac{1}{2}$ in both populations. Comparison of the coefficients of variation between the solutions of the system and the quasi-equilibrium solutions. Initial data $(m_f(0), m_g(0), c_f(0), c_g(0))=(4,\, 3,\, 2,\, 1)$.