An H Theorem for Boltzmann’s Equation for the Yard-Sale Model of Asset Exchange: The Gini Coefficient as an H Functional

Abstract

In recent work (Boghosian, Phys Rev E 89:042804–042825, 2014; Boghosian, Int J Mod Phys 25:1441008–1441015, 2014), 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.