We consider a model of economic growth with altruistic agents who care about their consumption and the disposable income of their offspring. The agents' consumption and the offspring's disposable income are subject to positional concerns. We show that, if the measure of consumption-related positional concerns is sufficiently low and/or the measure of offspring-related positional concerns is sufficiently high, then there is a unique steady-state equilibrium, which is characterized by perfect income and wealth equality, and all intertemporal equilibira converge to it. Otherwise, in steady-state equilibria, the population splits into two classes, the rich and the poor; under this scenario, in any intertemporal equilibrium, all capital is eventually owned by the households that were the wealthiest from the outset and all other households become poor.
THE RICH AND THE POOR IN A SIMPLE MODEL OF GROWTH AND DISTRIBUTION