  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.
