Every term in $\max_{\pi} \mathbb{E}\!\left[\sum_t \pi(t) \cdot J(S_t,\, X_t^\pi)\right]$ has a shape over time.
Here is what those shapes look like — and what they mean for the policy you choose.
The Gompertz mortality rate $\mu(t) = \alpha e^{\beta t}$ compounds exponentially with age. The survival probability $\pi(t)$ starts near certainty and stays deceptively high through your 40s and 50s. Then it falls fast. This is the discount factor on every unit of future joy.
The discount is generous for a long time — then it isn't. Notice how $\pi(t)$ barely moves from age 20 to 55. The false comfort of that plateau is one of the most dangerous features of the human life objective function. The steep part comes late, and it comes fast. Every year you delay front-loading what matters, $\pi(t)$ quietly shrinks the weight on that future payoff.
Each component of $J(S_t)$ has a different expected trajectory. Some are largely predetermined. Others are almost entirely shaped by the policy you run. Knowing the difference is half the work.
Both curves share the same shape — the gap is the cost of neglect. Poor sleep, sedentary habits, and deferred care compress the peak and steepen the decline. The neglected curve hits the floor a decade earlier.
The sigmoid shape holds either way — but neglect cuts the ceiling roughly in half and accelerates draw-down after retirement. Compounding works both directions: early investment decisions echo for decades.
The gap between these two curves is almost entirely a function of the policy you run. Neglect compounds the same way investment does — just in the wrong direction.
The dip in the 30s is real and common. The recovery is possible but not automatic. Purpose is the multiplier everything else runs through — low $p_t$ blunts even a high-health, high-wealth state.
All four state variables have an invested and a neglected trajectory. The gap in each is not luck — it is the difference between a policy that actively manages $h_t$, $w_t$, $r_t$, and $p_t$ and one that lets them drift. The objective function does not care which variable you are neglecting. It cares about $\pi(t) \cdot J(S_t,\, X_t^\pi)$ — the full weighted product, every year.
Set your $\lambda$ weights below. The chart shows how your policy — invested or neglected across all four state variables — shifts the total survival-weighted contribution to your life objective function. Your $\lambda$s determine which gaps hurt the most.