Revisiting the Merton Problem: from HARA to CARA Utility

Guiyuan Ma () and Song-Ping Zhu ()
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Guiyuan Ma: Xi’an Jiaotong University
Song-Ping Zhu: University of Wollongong

Computational Economics, 2022, vol. 59, issue 2, No 8, 686 pages

Abstract: 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.

Keywords: Optimal investment and consumption problem; Exponential utility; Hamilton-Jacobi-Bellman (HJB) equation; Closed-form solutions; Horizon effect (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10614-021-10102-z

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