Recursive Utility and Thompson Aggregators I: Constructive Existence Theory for the Koopmans Equation

Abstract

We reconsider the theory of Thompson aggregators proposed by Marinacci and Montrucchio [34]. We prove a variant of their Recovery Theorem establishing the existence of extremal solutions to the Koopmans equation. We apply the constructive Tarski-Kantorovich Fixed Point Theorem rather than the nonconstructive Tarski Theorem employed in [34]. We also obtain additional properties of the extremal solutions. The Koopmans operator possesses two distinct order continuity properties. Each is sufficient for the application of the Tarski-Kantorovich Theorem. One version builds on the order properties of the underlying vector spaces for utility functions and commodities. The second form is topological. The Koopmans operator is continuous in Scott's [40] induced topology. The least fixed point is constructed with either continuity hypothesis by the partial sum method. This solution is a concave function whenever the Thompson aggregator is concave and also norm continuous on the interior of its effective domain.