Every term in $\max \mathbb{E}\!\left[\sum_t \pi(t) \cdot J(S_t)\right]$ has a shape over time.
Here is what those shapes look like — and what they mean for $x_t$ today.
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 mostly predetermined. Others are almost entirely determined by choices. Knowing the difference is half the work.
Peaks in your late 20s, declines slowly until your 50s — then the slope steepens. High $\lambda_h$ early in life pays the most. Exercise and sleep are decisions that shape this curve.
The sigmoid: slow accumulation that accelerates through your 40s, peaks around retirement, then draws down. The only state variable that mostly takes care of itself if you show up for work.
The gap between these two curves is almost entirely a function of $x_t$ choices. 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.
Wealth and health move in opposite directions. This is the central scheduling tension. The state where you have the most money ($w_t$ high) is exactly the state where $h_t$ and $r_t$ have been quietly eroding. The objective function does not care about your net worth in isolation. It cares about $\pi(t) \cdot J(S_t)$ — the full product.
Set your $\lambda$ weights below. The chart shows how your choices about relationships and purpose shift the total contribution to your life objective function — weighted by survival.