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On a Merton Problem with Irreversible Healthcare Investment

This paper proposes a tractable dynamic framework extending Merton's portfolio problem to include irreversible healthcare investment, formulating the joint optimization of consumption, portfolio choice, and investment timing as a stochastic control-stopping problem that is solved via a dual two-dimensional optimal stopping approach to derive regularity properties and characterize the optimal investment boundary.

Original authors: Giorgio Ferrari, Shihao Zhu

Published 2026-07-15

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Giorgio Ferrari, Shihao Zhu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine your life as a high-stakes video game where you have two main resources: Gold (your wealth) and Health Points (your vitality). In this game, your Health Points naturally drain away as you level up (get older), and the faster they drain, the sooner your character might "game over" (pass away).

Most players just spend their Gold on fun stuff (consumption) or try to gamble it on risky stocks to get more Gold (portfolio choice). But this paper asks a tricky question: When is the perfect moment to spend a chunk of your Gold on a "Preventive Health Upgrade"?

Think of this upgrade like buying a permanent subscription to a super-gym or a lifetime supply of vitamins. It costs a steady stream of Gold every second you keep it, but it slows down the drain on your Health Points and lowers your chance of an early game-over. The catch? You can't undo it. Once you buy the subscription, you're stuck paying for it forever. You can't say, "Oops, I bought the gym pass, but now I'm broke, so I'll cancel it." That's what the authors call an irreversible investment.

The Big Discovery: The "Rich-Enough" Health Line

The authors built a complex mathematical model (a mix of finance and biology) to figure out the best strategy. They didn't just guess; they used advanced math to prove that there is a specific trigger line for every level of health.

Here is the main finding, translated from "Math-Speak" to "Human-Speak":

You should only buy the health upgrade when your Gold pile gets big enough to cross a specific line that depends on how healthy you currently are.

If you are very healthy: You need a massive amount of Gold to trigger the purchase. Why? Because if you're already healthy, the "marginal benefit" of getting even slightly healthier is small. It's like buying a turbocharger for a car that's already running perfectly; it's expensive and the extra speed isn't worth the cost unless you have a lot of extra cash to spare.
If you are sick: You can trigger the purchase with much less Gold. Why? Because if your health is tanking, the upgrade saves you from a much bigger drop in vitality and keeps you alive longer. The "bang for your buck" is huge, so you're willing to spend your savings sooner.

The paper explicitly rules out the idea that you should just start investing in health immediately, or that you should wait until you are dying. It also argues against the idea that the decision is the same for everyone regardless of their current health status. The timing is everything, and it changes based on your current "Health Points."

The "Wait and See" Strategy

Before you cross that line, the paper suggests you play a bit differently than you would after you buy the upgrade.

Before the upgrade: As you get closer to the line (your Gold pile grows), you actually start spending less on fun stuff. You hold back your Gold, saving it up like a squirrel preparing for winter, because you know a big health bill is coming soon.
After the upgrade: Once you cross the line and the subscription starts, your spending habits change again. You've paid the entry fee, your health is stabilizing, and you can afford to spend a bit more freely again.

What the Numbers Say (The Simulation)

The authors didn't just theorize; they ran simulations using specific numbers to see how this plays out in real-world scenarios.

They used a risk-free interest rate of 0.048 (about 4.8%) and an expected risky return of 0.108 (about 10.8%) for the stock market.
They modeled health depreciation at a rate of 0.0055 per year.
They tested two types of players: a "Sick Agent" starting with a health level of 5 and a "Healthy Agent" starting at 150.

In their simulations, they found that for the "Sick Agent," the line to trigger the investment was much lower (around $65–$ 90 depending on the exact health level) compared to the "Healthy Agent," whose line was much higher (around $250–$ 500).

They also simulated what happens if the cost of the upgrade changes. If the monthly cost of the health subscription goes up (from 2 to 6 in their model units), the line moves up. This means you need even more Gold to justify the purchase, because the cost is higher and the benefit (due to diminishing returns) doesn't grow as fast.

A Twist in the Plot: The Wobbly Line

Here is a fun twist the paper uncovered. Usually, we expect things to be smooth and predictable. But the authors found that under certain specific (and somewhat extreme) parameter settings, the "trigger line" in the mathematical model can actually wiggle up and down over time before settling.

They showed a simulation where the mathematical boundary goes up and then down, reaching a peak at roughly 34 years. However, they are careful to note that this is a mathematical curiosity found in specific simulations, not necessarily how real life works. In their main, more realistic simulations, the line stays smooth and goes up as you get healthier. They did not prove that the line is always wobbly; they just showed that it can be, which is a surprise to standard math models.

What They Don't Know (Yet)

The paper is very clear about what it doesn't do.

It assumes you commit to a fixed rate of investment forever once you start. It does not solve the problem of if you can change how much you spend every month (like buying a gym pass for a year and then quitting). The authors suggest this is a future challenge for other researchers.
It does not tell you exactly how much to spend on stocks or food in every single second of your life. Instead, it gives you the rule (the boundary) to decide when to make the big health move.
It does not claim to have solved the mystery of human life or death. It simply provides a better map for making financial decisions when health is a factor.

The Bottom Line

This paper suggests that the best time to invest in your health isn't when you're broke, and it isn't when you're already perfect. It's a delicate balance: Wait until your wallet is heavy enough to handle the cost, but do it sooner if your health is fragile.

The authors used a mix of rigorous math and computer simulations to show that this "wait-for-the-right-moment" strategy is mathematically optimal. They didn't just guess; they derived a complex equation that acts like a GPS for your financial and health decisions, telling you exactly when to hit the "Buy Health" button based on your current Gold and Health Points.

Technical Summary: On a Merton Problem with Irreversible Healthcare Investment

1. Problem Formulation
The paper addresses a continuous-time stochastic control-stopping problem that integrates Merton's classic portfolio and consumption framework with the economics of health capital. The agent seeks to maximize the expected intertemporal utility derived from consumption and health status over a random time horizon, defined as the agent's time of death ( $\eta$ ).

The model features two state variables:

Wealth ( $X_t$ ): Evolves according to a standard diffusion process driven by a risk-free asset and a risky stock, subject to consumption ( $c_t$ ) and a fixed healthcare investment cost ( $I$ ) once initiated.
Health Capital ( $H_t$ ): Modeled as a deterministic process that depreciates at a constant rate $\delta$ . Crucially, the agent can choose an optimal stopping time $\tau$ to initiate an irreversible, continuous healthcare investment at a fixed rate $I$ . Upon investment, the drift of the health process increases by a constant $K$ (where $K = I^\beta$ with $0 < \beta < 1$ ), representing the production of health.

The force of mortality ( $M_t$ ) is endogenous and depends on the current health stock via the specification $M_t = m_0 + m_1 H_t^{-\kappa}$ . Consequently, the investment decision $\tau$ directly influences the distribution of the random horizon $\eta$ by improving health and reducing mortality risk. The objective is to determine the optimal consumption rate, portfolio allocation, and the optimal timing $\tau$ for initiating the healthcare investment.

2. Methodology
The authors employ a combination of martingale duality methods and free-boundary analysis to solve the coupled stochastic control-stopping problem. The mathematical approach proceeds in three distinct steps:

Step 1: Dual Transformation. The original problem is transformed into a dual problem using standard martingale methods. This reduces the consumption-portfolio component to a pure optimal stopping problem. The dual value function, denoted $J(z, h)$ , depends on a dual variable $z$ (related to the Lagrange multiplier of the budget constraint) and the health capital $h$ . The dual problem is formulated as an infinite-horizon, two-dimensional optimal stopping problem with interconnected dynamics between the dual variable and the health process.
Step 2: Analysis of the Dual Stopping Problem. The authors analyze the regularity of the dual value function $\hat{J}(z, h) = J(z, h) - \tilde{W}(z, h)$ $\hat{J} (z, h) = J (z, h) - \tilde{W} (z, h)$ , where $\tilde{W}$ $\tilde{W}$ represents the payoff from immediate investment.
- They establish the continuity of $\hat{J}$ and prove that the stopping region is closed.
- A key technical challenge addressed is the lack of general monotonicity of the free boundary with respect to health. Unlike standard optimal stopping problems, the interaction between the mortality channel and the investment-benefit channel prevents the application of standard monotonicity arguments.
- To characterize the boundary, the authors introduce a time-inhomogeneous reformulation. By fixing the initial health level and viewing the problem as a function of time and the dual variable, they establish $C^{1,2}$ -regularity in the continuation region and a smooth-fit property at the boundary.
- Using a weak version of Dynkin's formula, they derive a nonlinear integral equation that characterizes the free boundary.
Step 3: Return to Primal Variables. Using the duality relations and the regularity results, the optimal strategies in the original wealth-health coordinates are recovered. The optimal investment time is shown to be the first time the agent's wealth hits a health-dependent boundary.

3. Key Results

Optimal Investment Rule: The agent optimally initiates irreversible healthcare investment when her wealth first reaches a threshold $\hat{b}(h)$ , which is a function of her current health status. This threshold is the inverse image of the dual stopping boundary under the duality mapping.
Boundary Characteristics:
- Under the baseline calibration, the investment boundary $\hat{b}(h)$ is increasing in health. Healthier agents require a higher level of wealth to justify the investment. This is because the marginal utility gain and mortality risk reduction from a fixed improvement in health are larger for agents in poor health.
- The boundary is generally not monotone with respect to time in the dual formulation, though it remains increasing with respect to health in the primal formulation under standard parameterizations.
Impact on Consumption and Portfolio:
- Pre-investment: As wealth approaches the investment boundary, the consumption-wealth ratio decreases (precautionary saving to fund the future investment cost), and the portfolio share increases.
- Post-investment: Once the investment is made, the consumption-wealth ratio increases, and the portfolio share exhibits a downward jump before gradually increasing again.
- Health Dependence: Unlike the standard Merton benchmark where ratios are often independent of health, the availability of the investment option makes consumption and portfolio policies significantly more sensitive to health status. Healthier agents consume a larger fraction of wealth and hold different portfolio allocations compared to the no-investment benchmark.

4. Numerical Illustrations and Sensitivity
The paper provides numerical solutions to the derived nonlinear integral equation to illustrate the optimal strategies.

Parameter Sensitivity: The study examines how the investment boundary responds to changes in the investment intensity ( $I$ $I$ ), the Cobb-Douglas parameter ( $\alpha$ $α$ ), and the subjective discount rate ( $\rho$ $ρ$ ).
- Higher investment intensity or a higher weight on consumption ( $\alpha$ ) shifts the boundary upward, delaying investment.
- The effect of the discount rate ( $\rho$ ) differs by health status: for sick agents, higher impatience lowers the investment threshold (due to the immediate utility gain of health), whereas for healthy agents, it raises the threshold.
Non-monotonicity: The authors demonstrate that under specific, albeit less realistic, parameter configurations, the transformed dual boundary can be non-monotone, highlighting the complexity of the interaction between mortality and investment dynamics.

5. Significance and Contribution
The paper claims a twofold contribution:

Economically: It introduces a timing margin for irreversible healthcare investment into a continuous-time consumption-portfolio framework. Unlike previous models (e.g., Grossman, [22]) that focus on continuous adjustment of health spending, this model specifically addresses the when of initiating sustained preventive care. It elucidates the trade-off between the immediate wealth cost of investment and the long-term reduction in mortality risk.
Mathematically: The paper analyzes a two-dimensional optimal stopping problem with coupled state dynamics where the free boundary is not necessarily monotone. It establishes the necessary regularity properties to represent the stopping region and derives a nonlinear integral equation for the boundary, solving it numerically to study the sensitivity of the optimal strategy.

The authors position their work as a tractable extension of the Merton problem that captures the empirical relevance of threshold-based health spending decisions, while acknowledging that future work could extend the framework to singular control problems allowing for flexible investment sizes.