On a Merton Problem with Irreversible Healthcare Investment

We propose a tractable dynamic framework for the joint determination of optimal consumption, portfolio choice, and irreversible healthcare investment. Our model is based on Merton's portfolio and consumption problem, where, in addition, the agent can optimally choose the time at which she undertakes healthcare investment at a fixed continuous rate. Health depreciates with age and directly affects the agent's force of mortality, so that investment in 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 a two-dimensional optimal stopping problem with interconnected dynamics. Regularity of the optimal stopping value function is derived, and the related free boundary is characterized through a nonlinear integral equation, which we compute numerically. In the original coordinates, the agent thus invests in healthcare whenever her wealth exceeds a health-dependent transformed version of the optimal stopping boundary. We also provide numerical illustrations of the optimal strategies and discuss some financial implications.