Recursive Utility for Thompson Aggregators : Least Fixed Point, Uniqueness, and Approximation Theories

We reconsider the theory of Thompson aggregators proposed by Marinacci and Montrucchio (2010). We demonstrate the Koopmans equation has a unique utility function solution given a Thompson aggregator. Uniqueness holds only on the interior of the commodity space's positive cone. We verify Du's (1989, 1990) condition holds. We show iteration of the Koopmans operator with initial seed the zero function convergences uniformly to the operator's Least Fixed Point on each order interval in the commodity space's norm interior. This differs from Marinacci and Montrucchio's (2010, 2019) proofs as well as proofs given by Martins-da-Rocha and Vailakis (2010). Those papers rely on a form of the contraction mapping theorem where the space of possibly utility functions is endowed with the topology induced by the Thompson metric. Application of Du's theory works on the possible utility function space with its norm topology. Our approach combines order and metric structures to demonstrate uniqueness differently than in the existing literature