The Fractal Nature of Inequality in Societies with Equal Opportunities

In this paper we investigate wealth inequality/polarization properties related to the geometry of the support of the limit distribution of wealth in economies characterized by equal opportunities and uninsurable individual risk. We work out two simple successive generation models, one with stochastic human capital accumulation and one with R&D, and prove that intense technological progress makes the support of the wealth distribution converge to a fractal Cantor-like set. Such limit distribution implies the disappearance of the middle class, with a “gap” between two wealth clusters that widens as the growth rate becomes higher. Hence, we claim that in a highly meritocratic world in which the payoff of the successful individuals is high enough, and in which social mobility is strong, societies tend to become unequal and polarized. We also show that a redistribution scheme financed by proportional taxation does not help cure society’s inequality/polarization – on the contrary, it might increase it – whereas random taxation may well succeed in filling the gap by giving rise to an artificial middle class, but it hardly makes such class sizeable enough. Finally, we investigate how disconnection, a typical feature of Cantor-like sets, is related to inequality in the long run. JEL Classification Numbers: C61, O41