The Life Objective Function

$$\max_{\{x_t\}_{t=0}^{T}} \; \mathbb{E}\!\left[\,\sum_{t=0}^{T} \pi(t) \cdot J(S_t,\, x_t)\,\right]$$

At every age $t$, choose decisions $x_t$ that maximize the probability-weighted sum of joy over your expected life. The earlier you act, the more of the sum you still have ahead of you.

Component 1 — The Joy Function

$$J(S_t, x_t) \;=\; \lambda_h h_t \;+\; \lambda_w w_t \;+\; \lambda_r r_t \;+\; \lambda_p p_t$$

Subject to $\,\lambda_h + \lambda_w + \lambda_r + \lambda_p = 1\;$ and $\;\lambda_i > 0$. Only you know your $\lambda$s -- and they change as you age.

$h_t$ health

tends to drift downward; enjoy its high value now

$w_t$ wealth

tends to accumulate; but can't offset everything

$r_t$ relationships

compounding returns; neglect decays it fast

$p_t$ purpose

the multiplier everything else runs through

Component 2 — The Mortality Discount

$$\pi(t) \;=\; \prod_{s=0}^{t-1}\!\bigl(1 - \mu(s)\bigr),$$ $$\mu(t) \;\approx\; \alpha\, e^{\,\beta t}$$

$\pi(t)$ is your survival probability at age $t$. It starts near 1 and decays. $\mu(t)$ is the Gompertz mortality rate -- it compounds exponentially with age. This is the decay term. It is not negotiable, but it is foreseeable.

Component 3 — State Transition (The SDA Part)

$$S_{t+1} \;=\; S^M\!\bigl(S_t,\; x_t,\; W_{t+1}\bigr)$$

$S_t = (h_t,\, w_t,\, r_t,\, p_t)$ is your state vector today. $x_t$ is the decision you make. $W_{t+1}$ is exogenous information -- the world's move, not yours. Your job is to optimize over what you control.

What This Actually Tells You

Timing is not neutral. Because $\pi(t)$ decays, a unit of joy at 35 is worth more to the sum than a unit of joy at 75. This is not pessimism. It is the math of a finite horizon -- and it argues for front-loading the things money can't buy, like relationships and purpose, not deferring them.

Wealth cannot substitute for health past a threshold. $w_t$ tends to rise while $h_t$ tends to fall. But $\lambda_h$ and $\lambda_w$ are your weights -- if you underinvest in health today, no future $w_t$ accumulation fully compensates inside $J$.

$x_t$ is the only lever you hold. You cannot change $S_t$ right now -- it is already realized. You cannot control $W_{t+1}$. You can only choose $x_t$, which shapes the transition to $S_{t+1}$. The Stoic point exactly: control what you can, accept the rest as $W$.

Your $\lambda$s are the real problem to solve. Most people never explicitly define what they are actually maximizing. Without knowing your weights, the objective function is just decoration. The Six Steps start with the metaproblem -- and this is the metaproblem of your life.

Continue to Decay Curves