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Bellman problem excerpt — Benhabib et al. (2019) lifecycle wealth model

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.

Economic setup (compressed)

• Finite life $T$; each period the agent chooses consumption $c_t$ and savings (next assets) $a’$ from market resources.
• No borrowing: $0 \le c \le a$ (resources in the notebook are denoted $a$ as beginning-of-period wealth).
• Idiosyncratic return $r$ and earnings profile ${w_t}_{t=1}^T$ fixed within a life, stochastic across generations; may correlate with parent’s draws.
• Terminal period: warm-glow bequest utility $e(a_T)$ on assets left.

Recursive formulation (as in the ballpark notebook)

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.)

Cross-generation dynamics

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.

Note for formalization

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.