Necessary and Sufficient Conditions for Existence and Uniqueness of Recursive Utilities

TL;DR

This paper establishes necessary and sufficient conditions for the existence and uniqueness of solutions in discrete-time recursive utility models, linking eigenvalue analysis and monotone operator theory, and confirms the convergence of iterative algorithms.

Contribution

It combines recent theoretical approaches to provide comprehensive criteria for solution existence, uniqueness, and algorithm convergence in recursive utility models.

Findings

Derived necessary and sufficient conditions for solutions

Proved convergence of natural iterative algorithms

Allowed nonstationary consumption processes

Abstract

We study existence, uniqueness and computability of solutions for a class of discrete time recursive utilities models. By combining two streams of the recent literature on recursive preferences---one that analyzes principal eigenvalues of valuation operators and another that exploits the theory of monotone concave operators---we obtain conditions that are both necessary and sufficient for existence and uniqueness of solutions. We also show that the natural iterative algorithm is convergent if and only if a solution exists. Consumption processes are allowed to be nonstationary.