Theoretical Analysis and Simulations of the Generalized Lotka-Volterra Model

TL;DR

This paper analyzes the generalized Lotka-Volterra model to understand wealth distribution dynamics, revealing a power-law distribution with an exponent linked to economic parameters, supported by theoretical and simulation results.

Contribution

It provides an analytical expression for the power-law exponent in wealth distribution models, connecting it to economic parameters and validating findings with simulations.

Findings

Wealth distribution follows a power-law with exponent related to social security and investment ratio.

The exponent is insensitive to economic saturation variations.

Empirical wealth distributions in various countries align with the model's predictions.

Abstract

The dynamics of generalized Lotka-Volterra systems is studied by theoretical techniques and computer simulations. These systems describe the time evolution of the wealth distribution of individuals in a society, as well as of the market values of firms in the stock market. The individual wealths or market values are given by a set of time dependent variables $w_i$, $i=1,...N$. The equations include a stochastic autocatalytic term (representing investments), a drift term (representing social security payments) and a time dependent saturation term (due to the finite size of the economy). The $w_i$'s turn out to exhibit a power-law distribution of the form $P(w) \sim w^{-1-\alpha}$. It is shown analytically that the exponent $\alpha$ can be expressed as a function of one parameter, which is the ratio between the constant drift component (social security) and the fluctuating component (investments). This result provides a link between the lower and upper cutoffs of this distribution, namely between the resources available to the poorest and those available to the richest in a given society. The value of $\alpha$ %as well as the position of the lower cutoff is found to be insensitive to variations in the saturation term, that represent the expansion or contraction of the economy. The results are of much relevance to empirical studies that show that the distribution of the individual wealth in different countries during different periods in the 20th century has followed a power-law distribution with $1 < \alpha < 2$.