On a Merton Problem with Irreversible Healthcare Investment

TL;DR

This paper develops a dynamic model combining consumption, portfolio choice, and irreversible healthcare investment, analyzing how health investments impact mortality and financial decisions over time.

Contribution

It introduces a dual stochastic control-stopping framework for joint healthcare investment and financial decision-making, with novel analysis of the free boundary and numerical solutions.

Findings

Optimal healthcare investment occurs when wealth exceeds a certain age-dependent threshold.

The free boundary surface is Lipschitz continuous and characterized by a nonlinear integral equation.

Numerical illustrations demonstrate the impact of health and wealth on investment timing.

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.