On recursive utilities with non-affine aggregator and conditional certainty equivalent

Łukasz Balbus

doi:10.1007/s00199-019-01221-8

Abstract

In this paper, we consider the problem of the existence and the uniqueness of a recursive utility function defined on intertemporal lotteries. The purpose of this paper is to provide the results of the existence and the uniqueness of a recursive utility function. The utility function is obtained as the limit of iterations on a nonlinear operator and is independent on initial starting points, with iterations converging at an exponential rate. We also find the maximum utility and an optimal strategy by means of iterations of the Bellman operator.

Highlights

Dynamic models in economics often assume the utility functions defined over sequences of random consumptions are represented by a time-additive expected overall utility which discounts future temporal utilities at a constant rate
We introduce the notion of a generalized mean-valued operator referred as Conditional Certainty Equivalent (CCE)
The paper contains the proof of the existence and the uniqueness of a recursive utility in the model with a nonlinear aggregator and CCE

Summary

Introduction

Dynamic models in economics often assume the utility functions defined over sequences of random consumptions are represented by a time-additive expected overall utility which discounts future temporal utilities at a constant rate. While others prefer to know the realization of uncertainty at a later date This situation cannot be captured within the standard utility framework [see Kreps and Porteus (1978), Chew and Epstein (1989) as well as Klibanoff and Ozdenoren (2007)]. The set of fixed points of this operator, we apply a key theorem in Guo et al (2004) on the cone of nonnegative functions This theorem gives sufficient conditions for the existence and the uniqueness of fixed points, as well as provides the results on the global convergence of iterations.. This theorem gives sufficient conditions for the existence and the uniqueness of fixed points, as well as provides the results on the global convergence of iterations.2 This approach is new in the literature. The proofs of all Lemmas and Propositions can be found in “Appendix”

Fixed point theorems on solid normal cones

Basic notations and definitions in a space of functions and measures

Description of the model

Existence and global attracting property of a recursive utility function

Bellman equation and the existence of an optimal program

Final conclusions