On a Merton Problem with Irreversible Healthcare Investment

Abstract: We propose a tractable dynamic framework for the joint determination of optimal consumption, portfolio choice, and healthcare irreversible investment. Our model is based on a Merton's portfolio and consumption problem, where, in addition, the agent can choose the time at which undertaking a costly lump sum health investment decision. Health depreciates with age and directly affects the agent's mortality force, so that investment into healthcare reduces the agent's mortality risk. The resulting optimization problem is formulated as a stochastic control-stopping problem with a random time-horizon and state-variables given by the agent's wealth and health capital. We transform this problem into its dual version, which is now a two-dimensional optimal stopping problem with interconnected dynamics and finite time-horizon. Regularity of the optimal stopping value function is derived and the related free boundary surface is proved to be Lipschitz continuous, and it is characterized as the unique solution to a nonlinear integral equation, which we compute numerically. In the original coordinates, the agent thus invests into healthcare whenever her wealth exceeds an age- and health-dependent transformed version of the optimal stopping boundary. We also provide the numerical illustrations of the optimal strategies and some financial implications are discussed.