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In this paper we aim to address two questions faced by a long-term investorwith a power-type utility at high levels of wealth: one is whether the turnpikeproperty still holds for a general utility that is not necessarilydifferentiable or strictly concave, the other is whether the error and theconvergence rate of the turnpike property can be estimated. We give positiveanswers to both questions. To achieve these results, we first show that thereis a classical solution to the HJB equation and give a representation of thesolution in terms of the dual function of the solution to the dual HJBequation. We demonstrate the usefulness of that representation with somenontrivial examples that would be difficult to solve with the trial and errormethod. We then combine the dual method and the partial differential equationmethod to give a direct proof to the turnpike property and to estimate theerror and the convergence rate of the optimal policy when the utility functionis continuously differentiable and strictly concave. We finally relax theconditions of the utility function and provide some sufficient conditions thatguarantee the turnpike property and the convergence rate in terms of bothprimal and dual utility functions.

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