Recursive Utiity and Thompson Aggregators

Abstract

We reconsider the theory of Thompson aggregators proposed by Marinacci and Montrucchio. First, we prove a variant of their Recovery Theorem estabilishing the existence of extremal solutions to the Koopmans equation. Our approach applies the constructive Tarski-Kantorovich Fixed Point Theorem rather than the nonconstructive Tarski Theorem employed in their paper. We verify the Koopmans operator has the order continuity property that underlies invoking Tarski-Kantorovich. Then, under more restrictive conditions, we demonstrate there is a unique solution to the Koopmans equation. Our proof is based on υ₀- concave operator techniques as first developed by Kransosels'kii. This differs from Marinacci and Montrucchio's proof as well as proofs given by Martins-da-Rocha and Vailakis.