Revisiting the Merton Problem: from HARA to CARA Utility

Abstract

This paper revisits the classical Merton problem on the finite horizon with the constant absolute risk aversion utility function. We apply two different methods to derive the closed-form solution of the corresponding Hamilton–Jacobi–Bellman (HJB) equation. An approximating method consists of two steps: solve the HJB equation with the hyperbolic absolute risk aversion utility function first and then take the limits of the risk aversion parameter to negative infinite. A direct method is also provided to derive another closed-form solution. Finally, we prove that the solutions obtained from different methods are equivalent. In addition, a sufficient condition is proposed to guarantee the optimal consumption is nonnegative and such a condition also leads to the verification theorem. A great advantage of our derived solution is that optimal policies can now be quantitatively scrutinized and discussed from both mathematical and economic viewpoints.