Thompson aggregators, Scott continuous Koopmans operators, and Least Fixed Point theory

Abstract

We reconsider the theory of Thompson aggregators proposed by Marinacci and Montrucchio (Marinacci and Montrucchio, 2010). We prove the existence of a Least Fixed Point (LFP) solution to the Koopmans equation. It is a recursive utility function. Our proof turns on demonstrating the Koopmans operator is a Scott continuous function when its domain is an order bounded subset of a space of bounded functions defined on the commodity space. Kleene’s Fixed Point Theorem yields the construction of the LFP by an iterative procedure. We argue the LFP solution is the Koopmans equation’s principal solution. It is constructed by an iterative procedure requiring less information (according to an information ordering) than approximations for any other fixed point. Additional distinctions between the LFP and GFP (Greatest Fixed Point) are presented. A general selection criterion for multiple solutions for functional equations and recursive methods is proposed.