The Fractal Nature of Inequality in Societies with Equal Opportunities ∗ Guido Cozzi † Fabio Privileggi ‡ November 18, 2007 Abstract 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 merito- cratic 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 redistribu- tion 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 in- equality in the long run. JEL Classification Numbers: C61, O41 Keywords: Wealth Inequality, Growth, Technological Change, Iterated Function System and its Attractor, Fractal, Cantor Set, Invariant Distribution, Polarization/Pulverization. 1 Introduction How do we predict a growing and unequal society’s wealth distribution to look like? In this paper we construct two variants of a simple competitive economy with successive gener- ations and uninsurable individual risk to show how easily the support of their limit distribution of individual relative wealth levels can look like a peculiar geometric object called Cantor set, provided that the exogenous growth rate is high enough. A Cantor set is a fractal on the real line, that is, a totally disconnected set with self-similar structure with an evident characteristic: it exhibits a “hole” in the middle. Our definition of (extreme) inequality is based on such hole, which may abviously be interpreted as the lack of a middle class, which, in turn, is often identified with the term ‘polarization’ by the mainstream literature on inequality. * Preliminary draft, not to be quoted. † Dept. of Economics, University of Glasgow and Universitry of Macerata; e-mail: G.Cozzi@lbss.gla.ac.uk. ‡ Dept. of Public Policy end Public Choice, University of Eastern Piedmont; e-mail: fabio.privileggi@sp.unipmn.it. 1