Turnpike Property and Convergence Rate for an Investment Model with General Utility Functions

TL;DR

This paper investigates the turnpike property and convergence rates for long-term investment models with general utility functions, extending classical results to non-differentiable and non-strictly concave utilities.

Contribution

It establishes the turnpike property and estimates convergence rates for a broad class of utility functions, including non-differentiable and non-strictly concave cases, using dual and PDE methods.

Findings

Positive answers to the hold of the turnpike property for general utilities.

Representation of solutions to the HJB equation via dual functions.

Estimation of error and convergence rate of the optimal policy.

Abstract

In this paper we aim to address two questions faced by a long-term investor with a power-type utility at high levels of wealth: one is whether the turnpike property still holds for a general utility that is not necessarily differentiable or strictly concave, the other is whether the error and the convergence rate of the turnpike property can be estimated. We give positive answers to both questions. To achieve these results, we first show that there is a classical solution to the HJB equation and give a representation of the solution in terms of the dual function of the solution to the dual HJB equation. We demonstrate the usefulness of that representation with some nontrivial examples that would be difficult to solve with the trial and error method. We then combine the dual method and the partial differential equation method to give a direct proof to the turnpike property and to estimate the error and the convergence rate of the optimal policy when the utility function is continuously differentiable and strictly concave. We finally relax the conditions of the utility function and provide some sufficient conditions that guarantee the turnpike property and the convergence rate in terms of both primal and dual utility functions.