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

We propose a method for quantifying inequality within a system of coupled Fokker–Planck-type equations, which model the evolution of probability densities for two populations interacting pairwise by economic motivations. The macroscopic dynamics of their mean values follows a Lotka–Volterra system of ordinary differential equations. Therefore, unlike classical models of wealth distribution, which converge toward a steady equilibrium profile, the oscillatory behavior of the mean values only leads to the formation, within the Fokker–Planck system, of time-dependent quasi-equilibria. This makes measuring the evolution in time of inequality in the system challenging. An insightful perspective on the problem is commonly gained through the Gini index. However, the coefficient of variation offers an alternative, mathematically convenient inequality measure that retains the essential characteristics of the Gini index. Numerical experiments confirm that, despite the system’s oscillatory nature, inequality initially tends to decrease. • We measure inequality in a system of coupled Fokker–Planck-type equations describing the evolution of the distribution densities of a multi-agent society interacting with the bank system via deposits and loans. • The evolution of the mean values is shown to obey a Lotka–Volterra system. • A comparison between the oscillations of the mean values and those of the coefficients of variation shows that the evolution of the system leads to a reduction of the inequality of the multi-agent society.