Source: Ballpark summary notebook Benhabib_et_al_2019_summary.ipynb, section “The Model” (Recursive Formulation). Paper: wealth distribution and mobility in the U.S. with stochastic earnings, heterogeneous returns, and differential saving / bequests.
Utility (explicit for YAML / EGM): CRRA flow utility $u(c) = \frac{c^{1-\sigma}}{1-\sigma}$ with risk aversion parameter $\sigma > 0$, $\sigma \neq 1$ (notebook form). Terminal warm-glow: $e(a_T) = A\frac{a_T^{1-\mu}}{1-\mu}$ with $\mu > 0$, $\mu \neq 1$.
Preferences (repeated for visibility): $u(c) = \frac{c^{1-\sigma}}{1-\sigma}$, $e(a_T) = A\frac{a_T^{1-\mu}}{1-\mu}$.
For $t = 1,\ldots,T-1$:
\[\begin{aligned} V_t(a) &= \max_{c,\, a'} \; u(c) + \beta V_{t+1}(a') \\ \text{s.t.} \quad a' &= (1+r)a - c + w_t \\ 0 &\le c \le (1+r)a + w_t \quad \text{(no borrowing; notebook states } 0 \le c \le a \text{ with } a \text{ interpreted as cash-on-hand / resources).} \end{aligned}\]Terminal period:
\[V_T(a) = \max_{c,\, a'} \; u(c) + e(a') \quad \text{subject to the same budget and } 0 \le c \le a.\](Notebook writes $V_T(a) = u(c) + e(a’)$ inside the max.)
Following the modular stage decomposition used in SolvingMicroDSOPs (Sections 12–13) and the DDSL cons_stage canonical template, the notebook’s single-period budget constraint factors into three perches connected by two transitions. Since return $r$ and wage $w_t$ are fixed within a life (no within-period shocks), there is no exogenous shock group and the arrival mover is a deterministic pass-through.
The notebook’s variable $a$ plays two distinct roles at different perches: it is carried-in assets at arrival and end-of-period savings at continuation. The intermediate quantity cash on hand $m \coloneqq (1+r)\,a + w_t$ is the true decision-perch state — the object the consumption policy maps from.
Perch decomposition:
| Perch | Objects (states / controls) | Key transition or Bellman step |
|---|---|---|
| Arrival ($\prec$) | $a_{\prec} \in \mathbb{R}_+$ (assets carried from prior period) | $\mathrm{g}{\prec\circ}$: $\; m = (1+r)\,a{\prec} + w_t$ — deterministic resource construction |
| Decision ($\circ$) | State $m \in \mathbb{R}_+$; control $c \in [0,\, m]$ | $\mathbb{B}$: $\; V(m) = \max_{c}!\big{u(c) + \beta\, V_{[\succ]}(m - c)\big}$ — Bellman optimisation; EGM inverts $u’$ to recover $c$ from $V’_{[\succ]}$ |
| Continuation ($\succ$) | $a_{\succ} \in \mathbb{R}_+$ (end-of-period savings) | $\mathrm{g}{\circ\succ}$: $\; a{\succ} = m - c$ — savings identity; wires to next period via $a_{\prec,t+1} = a_{\succ,t}$ |
Stage operator: $\mathbb{T}t = \mathbb{I}_t \circ \mathbb{B}_t$, where the arrival mover $\mathbb{I}_t$ is an identity ($V{[\prec]} = V$, no expectation) because there are no within-period shocks.
EGM channel (cf. solve_consumption_lifecycle in notebook_solvers.py): With CRRA utility, the marginal-value envelope $V’(m) = c^{-\sigma}$ and the inverse Euler equation $c_{[\succ]} = (\beta\, V’{[\succ]})^{-1/\sigma}$ provide the endogenous-grid step. The reverse transition $m{[\succ]} = a + c_{[\succ]}$ recovers the decision-perch grid from the poststate grid — matching the backward-induction loop in the notebook’s solve_consumption_lifecycle.
Composition across ages: The period template repeats the same stage for $t = 1, \ldots, T-1$ with age-varying wage $w_t$; inter-period connectors wire $a_{\succ,t} \mapsto a_{\prec,t+1}$ and $V_{[\succ],t} \mapsto V_{[\prec],t+1}$ as identities.
Given draws ${r^n, w^n}$ for dynasty $n$, policy induces a mapping $a^n = g(a^{n-1}; r^n, w^n)$. Tail results depend on $\mu$ vs $\sigma$ for existence and thickness of stationary wealth distribution.
The notebook uses $a$ both for the value function state and in the constraint “$0 \le c \le a$”; in standard notation this is often market resources or cash on hand at the start of the period. The transition should be read as: resources after labor and capital income, then choose $c$ and carry $a’$ to next period. As the stage decomposition above clarifies, the arrival perch carries raw assets $a_{\prec}$; the decision perch operates on cash-on-hand $m = (1+r)\,a_{\prec} + w_t$; and the continuation perch stores end-of-period savings $a_{\succ} = m - c$.
This file is the SMD-polished version produced in the instructor Matsya demo (same session as stage decomposition and YAML steps).