Abstract
In recent work, Boltzmann and Fokker-Planck equations were derived for the "Yard-Sale Model" of asset exchange. For the version of the model without redistribution, it was conjectured, based on numerical evidence, that the time-asymptotic state of the model was oligarchy -- complete concentration of wealth by a single individual. In this work, we prove that conjecture by demonstrating that the Gini coefficient, a measure of inequality commonly used by economists, is an $H$ function of both the Boltzmann and Fokker-Planck equations for the model.
Highlights
Over 100 years ago, the Italian economist Vilfredo Pareto [3] made one of the first empirical studies of the distribution of wealth by undertaking a careful study of land ownership in Italy, Switzerland and Germany
In analogy with [6], he derived a Boltzmann equation for the Yard-Sale Model (YSM), but went on to show that in the limit of small Δw and frequent transactions, this reduced to a certain nonlinear, nonlocal Fokker– Planck equation
We have proven that the Gini coefficient G is a Lyapunov function of the Boltzmann equation for the Yard-Sale Model of asset exchange, as well as of the Fokker–Planck equation obtained in the limit of small transaction sizes
Summary
Over 100 years ago, the Italian economist Vilfredo Pareto [3] made one of the first empirical studies of the distribution of wealth by undertaking a careful study of land ownership in Italy, Switzerland and Germany. Agents with land holdings worth more than w as a function of w His studies led him to believe that this function, which we shall denote by A(w) has a universal form. To put Pareto’s observations in modern terms, we may note that 1 − A(w) is the cumulative distribution function (CDF) of economic agents, ordered by wealth. Recent work [1,2] has suggested that the shape of Pareto’s curve is due to the fact that wealth distributions satisfy a certain Boltzmann equation. Before describing this Boltzmann equation, we first examine another common quantification of inequality.
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