The Power-Law Approach of the Consumption Function to its Perfect-Foresight Asymptote

0. The theorem in plain economics: precautionary saving as a discounted stream of premia, built backward (a reader’s guide)¶

The problem. The consumer solves the standard buffer-stock problem of BST — CRRA utility, permanent and transitory income shocks, geometric discounting — whose complete statement is the paper’s problem section. This document works with the normalized problem under the assumptions listed in §1 ((A1)–(A6), as itemized in statement.md §1; each element deep-linked there to its anchor in the published BST draft).

The question. A very wealthy consumer facing labor-income risk behaves almost like a perfect-foresight consumer: income risk is trivial relative to wealth, so the precautionary motive nearly vanishes. How fast does it vanish as wealth grows? That rate matters computationally — every solution algorithm must assume something above its grid top — and theoretically: the buffer-stock canon proves the ratio version of the approach (BST’s limiting MPC bounds) but neither the level nor the rate (§3.3).

The answer in one sentence. Precautionary saving — the shortfall of consumption below the perfect-foresight rule — dies off as a POWER of wealth ((25), proved in §§5–6: Theorem 1, Paragraph; constructed transparently in §2) — the same functional family as the Pareto laws economists know from wealth, firm-size, and city-size distributions (Gabaix 1999, 2009, 2016) — with an exponent $\min(1, \qup)$ built from patience, returns, and growth ((2)), and (when $\qup > 1$) a level $B$ (Theorem 1, (39)) that is nothing more exotic than a growing-perpetuity (Gordon) value.

The route. §1 states the model and the perfect-foresight benchmark; §2 builds the theorem by backward induction from a perfect-foresight resumption date $T$ — the presentation’s spine; §3 states the claim formally and places it in the literature; §§4–7 are the four rigor engines that certify the spine’s steps; §8 defines and measures the journey the spine describes; §9 cashes the theorem out computationally; §10 tallies the engines; §11 documents figures and reproduction.

The buffer-stock reading. The household’s buffer stock is its end-of-period assets $a$. These households are impatient: absent the precautionary motive they would run their assets down to zero (under BST’s maintained zero-income event $\pZero > 0$, which makes the natural borrowing limit zero; with $\theta_{\min} > 0$ they would borrow, toward the natural limit), so the assets they do hold are a buffer against the income risk. The buffer stock has a target $\hat{a} = \hat{m} - \cFunc(\hat{m})$: at $\hat{a}$, market resources are expected to reproduce themselves — $\Rcal\,\hat{a} + 1 = \hat{m}$ at $\psi \equiv 1$, the asset-side form of BST’s buffer-stock target (§3’s The target has the ψ-general form). In this document’s region the buffer stock stands far above $\hat{a}$ and the household runs it down — normalized PF wealth descends at rate $\ThornG$ per period — while the behavior that would hold the buffer stock at its target (maintenance) is target-zone behavior, priced where the journey ends. What the theorem tracks along the descent is precautionary saving: $\psavFunc(m)$, the eXtra saving relative to the perfect-foresight rule, which along the path satisfies — to leading order; the exact version, with its $O(1/\wbar)$ remainders sized, is §5’s one-step identity —

$\bufAdjFunc$ is the Kimball precautionary premium (KPP), and its dating deserves care. It is generated by next period’s shock $\tranShkNxt$ ($\sigma^2 = \operatorname{Var}\theta$): prudence — Kimball (1990)'s $-\uFunc'''/\uFunc'' = (\rho+1)/c$ — is evaluated where that shock will strike, tomorrow’s PF position (wealth $\ThornG\wbarNow$, consumption $\kap\ThornG\wbarNow$), and multiplies the consumption risk the MPC lets through, $\tfrac{1}{2}\kap^2\sigma^2$; the product is $\bufAdjFunc(\wbarNow)$, a function of today’s state only because tomorrow’s PF position is today’s state drifted once. It is then priced into today’s saving from one period away — whence its own $\Rcal^{-1}$, the same discount tomorrow’s expected saving carries. Unrolling the recursion, $x$ is the discounted stock of premia still to be provided along the journey, rung by rung: the KPP is always positive here (the forcing floor of §5, which is why $x \geq 0$), and while the premium itself need not rise monotonically as wealth falls, the sum of the premia still to be provided grows with every backward step of the journey.

The scope of the asset-pricing resemblance — narrower than it first appears, and the rest of this document should be read accordingly. The resemblance is an asymptotic statement: it holds for LARGE $\wbar$, and its value is that it proves the power-law tail — the Gordon-perpetuity mechanics of §2 and the engines of §§4–6 are the payoff, and for that purpose the analogy is kept. It is NOT valid where the solution lives: the ergodic set (§8.1) sits at SMALL $\wbar$, and there the approximation fails on every count. The displayed formula for $\bufAdjFunc$ is only the $\wbar \to \infty$ limit — at smaller $\wbar$, especially near $\wbar \approx h$ (bank balances $\bNrm = m - 1$ near zero), the terms the derivation drops are no longer small and no closed form replaces them; even where the formula applies, the premium’s shape (a flow proportional to $1/\wbar$) is nothing like a standard payoff stream; and in the limit $\pZero \downarrow 0$ the consumption function’s curvature $\cFunc''$ becomes UNBOUNDED near the point where the liquidity constraint would bind (BST proves this), so no smooth pricing approximation can survive at the bottom. That is what necessitates numerical solution of the model in the ergodic region — the EGM methodology with the theorem’s tails (§9); the supplement why the standard asset-pricing toolbox does not replace it records the tool-by-tool case. The pointer: §2 gives the full and careful exposition, deriving the recursion and the premium’s formula by backward recursion from a final period $T$ at which risk ends, one newly risky period per step. Last, the premium’s positivity is a high-wealth theorem; the buffer-stock adjustment it finances is signed in general — a household whose buffer stock exceeds what it wants provides less, one short of it provides more. With that scope on the table: for the tail, (1) strongly resembles an asset-pricing recursion, and the Gordon-perpetuity mechanics below are the payoff of taking the resemblance seriously — there, and only there.

(Objects, for this section: $\psavFunc(m) := \bar{\cFunc}(m) - \cFunc(m)$ — precautionary saving, the eXtra saving induced by precaution, the shortfall of consumption below the perfect-foresight rule. The PF-wealth-coordinate form is the gap $g(\wbar)$ ((7)); $\wbar(m) := \bNrm + h$, with $\bNrm$ BST’s bank balances; §1 displays the definitions.)

None of this is posited: §2.3 derives the recursion, discount and adjustment included, from two Euler equations and a second-order expansion ((20)), and Lemma 5.1 ((29), §4) is its exact form (the translation box there spells out the precise discount and per-period flow). The adjustment is $\approx \kap(\rho+1)\sigma^2/(2\ThornG\wbar)$ with $\wbar$ PF total wealth — the Kimball precautionary premium (KPP) — the precautionary counterpart (Kimball 1990) of the Arrow–Pratt risk premium: absolute risk over PF wealth, hence tiny for the rich; (30), §3.4; the discounting per step is $1/\Rcal$, with the state drifting down to $\ThornG \wbar$ (the wealth recursion (8)). Everything in the theorem is bookkeeping on this recursion, and §2.4 does the bookkeeping in a single geometric sum ((21)): the exponent is about which adjustments dominate the present value (near ones or far ones), the amplitude $B$ is a perpetuity value, and the trichotomy is Gordon-growth’s convergent / $r = g$ / divergent trichotomy.

What is $\qup$? (and why $\min(1, \qup)$)¶

First the raw material — BST’s patience factors, written with the Old English thorn letter: $\Thorn := (R\beta)^{1/\rho}$ is the absolute patience factor (the growth factor of perfect-foresight consumption); dividing by what consumption must keep up with gives return patience $\ThornR := \Thorn/R$ (RIC: $\ThornR < 1$) and growth patience $\ThornG := \Thorn/\Gamma$ (GIC: $\ThornG < 1$ — the impatience that makes wealth descend). §1 anchors each to its BST definition. Now two per-period log-rates, both familiar:

• $\ln \Rcal = \ln R - \ln \Gamma$ — the interest-minus-growth gap: how fast discounting compounds along the journey;

• $\GPRte := \ln \ThornG$ — the log growth-patience rate, NEGATIVE under GIC, so $-\GPRte = \ln(1/\ThornG) > 0$ is the per-period descent rate: how fast an arbitrarily rich household runs its wealth down — its $\wbar$ falls by the factor $\ThornG$ (about $-\GPRte$ percent) per period — toward §8’s return region around the target.

Their ratio is the star of the theorem (at $\psi\equiv1$; in general the root of (26)):

The intuition. The household’s state is $m$; perfect-foresight wealth is a function of it, $\wbar(m) := \bNrm + h = m - 1 + h$ — the same state relabeled into PF-wealth units (§1’s convention). The bar is BST’s upper-bound mark, not a mean: $\wbar$ is not a constant. A change of units, not of position: no consumer ever “begins at $m = \wbar$”. A wealthy household runs its buffer stock down, so its $m$ falls period by period; measured as $\wbar$, the fall is asymptotically exactly geometric — $\wbarNxt \approx \ThornG\,\wbarNow$, $\ln\wbar$ dropping $-\GPRte$ per period — which is exactly why the engines use these units. The journey ends not at a point but in §8’s return region around BST’s buffer-stock target $\hat{m}$: bounded shocks confine the recurrent range of wealth to a compact interval whose upper edge, in PF-wealth units, §8 computes as $\zeta = C_0/(1-\ThornG)$ (8.1 How rarely is the tail visited? The dual (Kesten) root quantifies how rarely even that edge is visited from above). The time remaining from the current position is therefore $\trvTime(\wbar) = \ln(\wbar/\zeta)/(-\GPRte)$ periods — LOGARITHMIC in the current state (made exact by §5.2’s shells and Paragraph's shell-to-wealth conversion; later sections write $\trvTime(\wbar) \approx \ln\wbar/(-\GPRte)$, absorbing the edge’s additive constant $\ln\zeta$ into the amplitudes they track). Anything discounted exponentially in TIME therefore becomes a power law in WEALTH:

A power law in wealth is exponential discounting read on a logarithmic clock. (The same log-clock conversion generates the Pareto tails of the random-growth literature — there applied to cross-sections of cities and firms, here to a single household’s policy function.) So $\qup$ answers: by the time the big precautionary action arrives — near the target — how discounted is it, per unit of log-wealth?

• $\qup > 1$ (discounting beats the descent): faraway adjustments are negligible; the shortfall is dominated by the adjustment provided here and now, which is $\propto 1/\wbar$ (Kimball). Realized exponent: 1 (Theorem 1).

• $\qup < 1$ (descent beats discounting): the shortfall is dominated by the large adjustments awaiting near the target, discounted back. Realized exponent: $\qup$ (Paragraph).

• Hence $q = \min(1, \qup)$: the shortfall is a sum of both components, and the slower-fading one wins asymptotically (Paragraph). And it can never fade faster than $1/\wbar$ — a prudent consumer is always at least providing today’s adjustment (the $1/\wbar$ floor on the forcing: stage-A Lemma 6.1 / stage-B Lemma B-6.1, restated with Lemma 5.1 in §4) — so the decay is a power law, not an exponential (Paragraph).

• Permanent shocks make the descent NOISY (a random-walk component in log PF wealth). Runs of small permanent-income realizations ($\psi < 1$) shrink the denominator of the normalized ratio and so inflate $\wbar$: excursions to high NORMALIZED perfect-foresight wealth are produced by bad income luck, not good — the household is income-poor and ratio-rich. Such runs weaken the effective descent toward the target, weakening the effective discounting and LOWERING $\qup$ below the deterministic ratio (§7, Theorem 2; the general-$\psi$ root solves (26)): for the estimated high-school calibration (HS-mean, §3.1), $\ln \Rcal/(-\GPRte) = 0.43$ but $\qup = 0.38$; at the college patience ceiling the deterministic ratio is $\approx 14$, yet $\qup = 1.47$.

Half-lives, and an impatient billionaire. (2)'s ratio carries an inversion worth making explicit: more impatience means a smaller exponent — the gap fades more SLOWLY up the wealth ladder. There is no paradox. An impatient household converges quickly in time, and that is exactly why its gap dies slowly in wealth: even from $\wbar = 10$ the return region is only decades away, so the destination still shapes behavior far up the ladder. A nearly-patient household is the mirror image — the return temporally remote even at modest wealth, behavior that looks frictionless almost immediately. The per-quarter rates at the recurring calibrations (§3.1), converted to half-lives ($\ln 2/\text{rate}$, in years):

Standing at $\wbar = 10$: the HS-mean household is $\approx 46$ years from the return region and still assigns it relevance weight $\Rcal^{-\trvTime} = 10^{-0.43} \approx 0.37$; the ceiling household is $\approx 1{,}600$ years away and assigns it 10^-14 — economically invisible until the $\psi$-tilt of (26) is priced in ($\qup = 1.47$: weight 10^-1.5 — twelve orders of magnitude of Jensen). Hence: an impatient billionaire is still, at the margin, a buffer-stock consumer — a couple of decades from the return region, behavior visibly shaped by it — while a patient household of far more modest wealth already behaves as if frictionless. Households that converge fast in time are the ones whose gap decays slowly in wealth, and conversely. (The simplified capstone’s race appendix replays this race in slow motion — including a two-line lognormal reading of (26) that reproduces the exact roots of all four calibrations to a few parts in ten thousand.)

A characteristic root, and a saddle path. Economists solve linear rational-expectations models by finding the characteristic roots and discarding explosive solutions (Blanchard–Kahn). Equation (26) is exactly the characteristic equation of the linearized Euler dynamics in log wealth: the homogeneous solutions of the shortfall recursion are the powers $\wbar^{-q}$ with $q$ solving (26), and the exponent-detection theorem of §6 (Paragraph) is the corresponding stability selection — compensate by any wrong exponent and the transformed problem explodes or dies; only $\min(1, \qup)$ is the saddle path.

(For readers who do not live in linear rational-expectations models: the ‘homogeneous solutions’ are the family of alternative decay shapes the recursion tolerates on its own before economics is imposed; almost all of them either explode along the induction or violate a boundary requirement, and ‘the saddle path’ is the one shape that survives both tests. §2.4 performs this selection explicitly, with no linear-systems vocabulary required.)

The estimated calibrations. Three calibrations recur in the figures and numerical checks. They arise in the context of the HAFiscal paper (https://llorracc.github.io/HAFiscal-Latest/), which estimates discount-factor distributions by education group for U.S. households: HS-mean (the mean discount factor of the High-School group), COL-TOP (the top atom of the College group’s discount-factor distribution — the most patient estimated type), and GIC-CAP (College fundamentals with the discount factor at the estimation’s cap on $\beta$. The cap is a solution-method safeguard, not a point of theoretical interest in itself: for the solution algorithm to perform reliably, the search region for candidate parameters must stop a small gap short of the knife-edge value at which the Growth Impatience Condition fails; in practice HAFiscal sets the cap at 0.9995 times that GIC-failing value. GIC-CAP is thus as patient as the estimation permits — deliberately just below the knife-edge, never at it), plus an auxiliary resonance calibration (code id AUX-RES: College fundamentals with $\beta$ root-found so $\qup = 1$ exactly — the Gordon $r = g$ knife-edge, constructed for the theory’s checks rather than reached by the estimation). The parameters are quarterly. One BST ingredient enters silently: the mortality risk BST introduces late (its Mortality section — a constant survival probability $\mathcal{L}$, default $\mathcal{L} = 1$; script $\mathcal{L}$, distinct from §6’s fraktur functional $\Lfun$). Under the Blanchard–Yaari no-leakage scheme (estates return to survivors as an actuarially fair excess return) mortality enters the solution phase through two substitutions — the discount $\hat{\beta} = \beta\,\mathcal{L}$ (HARK’s DiscFacEff) and the annuitized return $\hat{R} = R/\mathcal{L}$ — whose product is invariant, $\hat{\beta}\,\hat{R} = \beta R$: the Euler equation, $\Thorn$, $\ThornG$, the GIC, and the descent rate are all unchanged, while the budget side shifts ($\Rcal \to R/(\mathcal{L}\Gamma)$, $h \to 1/(1 - \mathcal{L}\Gamma/R)$, $\kap \to 1 - \Thorn\mathcal{L}/R$, and $\qup$ rises). At the default $\mathcal{L} = 1$ everything reduces to the model as written. Where mortality has real consequences is the simulation phase — the ergodic distribution is mortality-financed, and the two mortality channels pull the effective wealth support in opposite directions (killing thins it; the annuity return fattens it — measured in verify_sim_stability_checks.py) — which is why §8.1 and §9 invoke it and why the simulation-phase analysis lives in the ergodic-coverage companion (ergodic_coverage.md). In the frozen verification code these appear under their original identifiers CAL-HS, CAL-CTOP, CAL-CCAP, which are pinned instruments and keep their names.

Translation table.

1. Model, conditions, and the perfect-foresight benchmark¶

The model in full (each element deep-linked to its anchor in the published BST draft; the statement is BST’s, so a reader can check any piece against the source).

The symbol correspondence. This corpus (and its HARK consumers) writes relative risk aversion as $\rho$ and permanent income growth as $\Gamma$; BST’s published draft writes $\gamma$ and $\mathcal{G}$ for the same objects. Everything else is shared: $\Thorn := (R\beta)^{1/\rho}$ (the absolute patience factor), $\Rcal := R/\Gamma$, $\kap$ and $\bar{\kappa}$ (the limiting MPC bounds), $\pZero$ (the zero-income probability), $\psi$ and $\theta$ (permanent and transitory shocks). One structural note: BST separates the transitory shock into the zero-income atom and the employed-state shock ($\xi$ vs $\theta$ there); this corpus writes the all-in transitory shock as the single symbol $\theta$ — so the corpus’s $\theta$ is BST’s $\xi$, and $\sigma^2 := \operatorname{Var}\theta$ includes the atom.

Preferences. CRRA period utility $\uFunc(c) = c^{1-\rho}/(1-\rho)$ with discount factor $\beta > 0$ (BST, Setup). BST maintains $\rho > 1$, and the existence routing below inherits that; the decay arguments of §§3–5 use only $\rho > 0$ once existence is in hand (assumption (A1) below).

The income process (Friedman–Muth; BST, Assumption assn-shocks, with the atom displayed at its transitory-shock definition). Permanent shocks $\psi$: $\E[\psi] = 1$, supported on $[\psi_{\min}, \psi_{\max}] \subset (0, \infty)$. Transitory shocks: income is zero with probability $\pZero > 0$ — the zero-income event, part of the assumption itself, not an optional garnish — and otherwise equals the employed-state shock (mean one, supported on a compact interval bounded away from zero, per BST) scaled by $1/(1-\pZero)$; the all-in shock $\theta$ then satisfies $\E[\theta] = 1$ and $0 \leq \theta \leq \theta_{\max} < \infty$. Both shocks are bounded and mean-one in levels — this general statement, BST’s own (Assumption assn-shocks’s bounded-support requirement), is what every theory section below argues from; the equiprobable discretizations of lognormal shocks used by the figures are implementations consistent with it (bounded support excludes unbounded specifications but is consistent with all standard discretizations of unbounded distributions onto finite grids). One clause on what the atom buys: the results here do not all need $\pZero > 0$ — with $\theta_{\min} > 0$ existence routes through MST (2020) instead (below) — but $\pZero > 0$ is BST’s maintained specification, it is what the estimated calibrations use, and it is load-bearing for the form of WRIC, for the natural borrowing constraint (the discussion following Assumption assn-shocks), and for the MPC upper bound $\bar{\kappa} = 1 - \pZero^{1/\rho}\ThornR$.

Budget transitions and normalization — the objects $a$, $k$, $b$, $m$, self-contained. BST’s within-period sequence, in levels: the consumer ends period $t$ holding assets $A_t$ (market resources minus consumption); those assets become next period’s kapital, $K_{t+1} = A_t$ — the same resources, renamed at the date they earn their return (BST’s $k_{t+1} = a_t$; without this definition the word “kapital” for date $t+1$ resources saved at $t$ would be mysterious); the return makes bank balances $B_{t+1} = R\,K_{t+1}$; adding the period’s labor income gives market resources $M_{t+1}$, from which consumption $C_{t+1}$ is chosen. Labor income is BST’s two-part process: permanent income grows by $P_{t+1} = \Gamma\,\permShkNxt\,P_t$ ($\Gamma \geq 1$ predictable growth, $\psi$ a mean-one permanent shock), and the period’s receipt is $P_{t+1}\,\tranShkNxt$ ($\theta$ the mean-one transitory shock of (A3), zero-income event included). Dividing every level by permanent income — lowercase letters are these normalized ratios throughout, and the $\Gamma\permShkNxt$ divisor below is exactly that renormalization crossing the period boundary — the problem is BST’s normalized Bellman equation: choose $\cNrmNow \in (0, \mNrmNow]$ (as in the statement page) with transitions

where $\aNrmNow$ is end-of-period assets, $\kNrmNxt$ next period’s kapital, $\bNrmNxt$ next period’s bank balances — the returns realized on the kapital, the market-wealth leg of $\wbar$ ((6)) — and $\tranShkNxt$ next period’s transitory income. The first-order condition is BST’s normalized Euler equation, $\cNrmNow^{-\rho} = \beta R\,\E_{\prdt}[(\Gamma\permShkNxt)^{-\rho} \cFunc(\mNrmNxt)^{-\rho}]$. At $\psi \equiv 1$ the growth-adjusted return is the constant $\Rcal = R/\Gamma$ and $\bNrmNxt = \Rcal\,\aNrmNow$.

Human wealth and the perfect-foresight benchmark. Normalized human wealth $h$ — BST’s hNrm, the PDV of labor income including the current period’s — has infinite-horizon limit $h = 1 + \Rcal^{-1} + \Rcal^{-2} + \cdots = 1/(1-\Rcal^{-1}) = R/(R-\Gamma)$ under FHWC (one line from BST’s finite-horizon definition). BST’s perfect-foresight consumption function is $\bar{\cFunc}(m) = (\bNrm + h)\,\kap$ with limiting MPC $\kap := 1 - \ThornR$, written in the decomposition its finite-horizon form displays — BST’s own $\bNrm := m - 1$ (bank balances) — so $\bar{\cFunc} = \kap\,\wbar$ in the $\wbar$ variable displayed just below ((6)), and precautionary saving is $\psavFunc(m) := \bar{\cFunc}(m) - \cFunc(m)$ — the object of the theorem. In the PF-wealth coordinate the same quantity is the gap $g(\wbar) := \kap\wbar - \cFunc(m(\wbar))$, with $g(\wbar(m)) = \psavFunc(m)$ — the engines’ object, because the PF drift is exactly multiplicative there (at zero gap (8) reads $\wbarNxt = \ThornG\,\wbarNow$); economics-facing prose keeps the household’s state $m$. The one bookkeeping trap of the correspondence: $h$’s leading term is this period’s income, so the PF rule carries the -1; dropping it shifts $x$ by $\kap$ (checked to machine precision).

The perfect-foresight benchmark, displayed (the three defining displays the rest of the document consumes; the conditions named here carry BST anchors in the table below). Under GIC, RIC, and FHWC the normalized consumption function $\cFunc(m)$ approaches its perfect-foresight asymptote from below:

where $h$ is BST’s human wealth — the PDV of labor income including the current period’s unit of it, which is why the rule (BST’s own, with $\kap$ from its limiting-MPC definition) carries the -1. The theorem is about the rate and form of that approach. Write

— the consumer’s total (human plus market) perfect-foresight wealth, viewed at the decision moment after this period’s returns have been realized on the kapital saved last period ($\kNrmNow = a_{t-1}$): $\bNrm$ is BST’s bank balances ($\bNrm = \Rcal\,\kNrmNow$ at $\psi \equiv 1$ — market resources net of the unit of current income that $h$ already counts; BST’s own decomposition). The bar follows the convention BST itself states for $\bar{\cFunc}$ — “the overbar signifies that $\bar{\cFunc}$ will be an upper bound as we modify the problem to incorporate constraints and uncertainty; analogously, $\kap$ is the MPC’s lower bound” — bar = upper bound, underline = lower bound, matching $\kap$; $w$ itself is unused in BST, so no clash. One caution: bars so often mark CONSTANTS (means, bounds) that $\wbar$ tempts a constant reading — it is not one: $\wbar(m) = m - 1 + h$ moves one-for-one with the state; after its definition, $\wbar$ denotes its value — the engines’ coordinate. In this variable the PF rule is simply $\bar{\cFunc} = \kap\,\wbar$, and

— the same quantity as precautionary saving: $g(\wbar(m)) = \psavFunc(m)$.

The conditions, each with BST’s name, label, and live anchor:

What the theorems maintain (Stage A / Stage B). Stage A ($\psi \equiv 1$): (A1) CRRA $\rho > 0$; (A2) constant $R, \Gamma, \beta$; (A3) the income process above with $\psi \equiv 1$ — i.i.d. $\theta$, $\E[\theta] = 1$, $\sigma^2 \in (0,\infty)$, support $[\theta_{\min}, \theta_{\max}]$ with $0 \leq \theta_{\min} < 1 < \theta_{\max} < \infty$; $\theta_{\min} = 0$ — BST’s maintained $\pZero > 0$ zero-income atom (Assumption assn-shocks) — is permitted throughout and is what the estimated calibrations use; (A4) FHWC; (A5) RIC; (A6) GIC(-Raw). Stage B adds i.i.d. $\psi \perp \theta$, $\E[\psi] = 1$, $\operatorname{supp}\psi \subseteq [\psi_{\min}, \psi_{\max}] \subset (0,\infty)$, and FVAC$^\psi$. Unbounded-above $\theta$ is covered by the L4′ extension ($\E[\theta^k] < \infty$ for some $k > \max(1,\qup)+1$) — the one deliberate departure from bounded support (8.1 How rarely is the tail visited? The dual (Kesten) root, item 3).

Existence, the atom, and the bounds. With $\pZero > 0$ the non-degenerate limiting solution exists by BST’s contraction theorems under WRIC + FVAC (the Boyd weighted-norm route — the atom is native to BST’s machinery); with $\theta_{\min} > 0$ one may instead invoke MST (2020) Theorem 2.2 (BST’s own remark maps the $\pZero = 0$ normalized problem into the MST framework as a special case). This routing is load-bearing and inherited by every proof in the program. The solution obeys BST’s MPC bounds $\kap m \leq \cFunc(m) \leq \bar{\kappa} m$ with $\bar{\kappa} = 1 - \pZero^{1/\rho}\ThornR$, and $\cFunc(m)/m \to \kap$ as $m \to \infty$ — the ratio limit whose level-and-rate refinement is this paper (§3.3).

The target. Under WRIC + FVAC + GIC-Mod there is a unique target $\hat{m} > 0$ with $\E_{\prdt}[\mNrmNxt/m] = 1$ at $\hat{m}$ (BST’s buffer-stock-target theorem), computable from the implicit equation $(\hat{m} - \cFunc(\hat{m}))\,\Rcal\,\E[\psi^{-1}] + 1 = \hat{m}$. An agent failing GIC-Mod has no target — the precise content behind the GIC-CAP name (§0); at the ceiling calibration even the mean log-drift is positive, so normalized wealth drifts up without bound (§8.1). Bounded support of the shocks (Assumption assn-shocks) is what makes the return region around $\hat{m}$ (§8) a compact interval; BST’s own bounded-support discussion makes the same point — it is what permits the contraction-rate arguments characterizing the stable target to be made globally rather than only asymptotically.

The imported ladder (all PROVEN and review-hardened in stage_A_proof.md §§1–5 / stage_B_proof.md §§B1–B3; statements only):

• L0–L2′: existence/uniqueness/Euler (routing above); $c$ strictly increasing, strictly concave (Carroll–Kimball 1996); the sandwich $0 \leq x \leq \bar{x} := \kap (h-1)$; $x$ non-increasing and convex; $c/m \to \kap$, $c' \downarrow \kap$.

• L3 (level convergence): $g(\wbar) \to 0$ — two lines from the gap equation, and not stated anywhere in the published canon (§3.3).

• The exact PF-wealth recursion, with the a.s. excursion bound $|\wbarNxt - \ThornG \wbarNow| \leq C_0$ (Cor 3.2):

  $$\wbarNxt = \ThornG\, \wbarNow + (\tranShkNxt - 1) + \Rcal\, g\Now(\wbarNow).$$

  (8)

  (Today’s saving is invested: the $\Rcal\,g\Now(\wbarNow)$ term relaxes tomorrow’s budget — remember it for the translation box in §4.)

• Lemma A5: (26) has a unique positive root; $\Lfun(q) := \ln \E[\psi^{1+q}] - \ln\Rcal - q\ln\ThornG$ is convex with $\Lfun(0) = -\ln\Rcal < 0$.

(Lemma 5.1 and Corollary 5.2 — the exact one-step law and its monotone comparison sandwich, the induction invariant that certifies the spine — complete the imported ladder; they are stated in §4, where they are consumed.)

2. The spine: equation (1) by backward induction from $T$¶

The theorem’s construction order is backward. Fix a date $T$ at which the consumer faces — or believes she faces — no further uncertainty, so that she resumes the perfect-foresight rule from $T$ on; every earlier period faces the transitory risk. Stepping backward from $T$, each step adds one newly risky period, and §0’s discounted stock of premia grows exactly as the pricing recursion (1) says it should. This section constructs that recursion — the discount, the drift, and the adjustment all derived, not posited — and unrolls it to the published constants. The setting is the Stage-A world ($\psi \equiv 1$: transitory shocks only, per §1); §2.6 records that the construction is $\psi$-general. Everything here is exact to leading order in $1/\wbar$, with the dropped terms itemized in §2.5 and certified by the rigor engines of §§4–7 (the map: (EQ-1) ↦ Lemma 5.1’s (29) and Cor 5.2’s sandwich; the unrolled sum ↦ §5’s shells; the truncation at the log-clock exit ↦ §8’s return region). The spine’s named battery is verify_eq1_backward_checks.py (§11), which implements the backward iteration exactly and checks every displayed constant.

2.1 The terminal date, and the two identities that do all the work¶

The normalized transitory-only model of §1: CRRA $\uFunc$, $\uFunc'(c) = c^{-\rho}$; period budget/transition $\mNrmNxt = \Rcal\, \aNrmNow + \tranShkNxt$ with $\aNrmNow = \mNrmNow - \cFunc\Now(\mNrmNow)$, $\Rcal := R/\Gamma > 1$ (FHWC), $\E[\theta] = 1$, $\sigma^2 := \operatorname{Var}\theta$, bounded $\theta$. The normalized Euler operator is

Perfect-foresight (PF) objects ((5), (6)): with $\kap = 1 - \ThornR$ ($\ThornR = (R\beta)^{1/\rho}/R < 1$, RIC), $h = \Rcal/(\Rcal - 1)$, and perfect-foresight total wealth $\wbar = \bNrm + h$ ($\bNrm$ BST’s bank balances, §1),

(The $\wbar$-recursion is (8) at zero gap: $\wbarNxt = \Rcal(1-\kap)\wbar + [\Rcal(1-h) + h]$ and the bracket vanishes identically by $h = \Rcal/(\Rcal-1)$; $\Rcal(1-\kap) = \Rcal\ThornR = \ThornG$.)

Two elementary identities carry the whole derivation:

(I1) is the PF Euler equation itself: $\ThornG^{\rho}\,(\bar{c}_{\prdNxt}/\bar{c}_{\prdt})^{-\rho} = 1$ with PF consumption growth $\bar{c}_{\prdNxt}/\bar{c}_{\prdt} = \ThornG$. (I2) is RIC bookkeeping: $\kap = 1 - \ThornR$ and $\ThornG/\Rcal = \ThornR$.

The terminal assumption. At date $T$ the agent faces — or believes she faces — no further uncertainty, and therefore behaves according to the PF rule from $T$ on:

Before $T$ she faces the transitory risk. We iterate backward.

2.2 One step back ($T-1$): the doubly-differentiated Euler subtraction¶

(The section title unpacked: write the Euler equation twice — once for the agent who knows next period’s shock will be at its mean, once for the agent facing the risk — Taylor-expand both to second order in the shock, and subtract. Everything zeroth- and first-order cancels; the surviving second-order term is the premium, and the surviving first-order-in-the-gap terms are the recursion.)

Write the two Euler equations at $T-1$ — the agent who KNOWS $\theta_T = 1$ (she is the PF agent, $\bar{\cFunc}$), and the agent who faces the risk ($\cFunc_{T-1} = \bar{\cFunc} - \psavFunc_{T-1}$):

Let $\bar{m}_{\prdNxt} := \Rcal(m - \bar{\cFunc}(m)) + 1$ be tomorrow’s perfect-foresight market resources (its PF wealth is $\wbar_T = \ThornG\wbar$), and abbreviate $x := \psavFunc_{T-1}(m)$. The stochastic argument is $\bar{m}_{\prdNxt} + \Rcal x + (\theta_T - 1)$.

Expand (EE-sto) to second order in $(\theta_T - 1)$ around $\E[\theta] = 1$ — the double differentiation of the spine’s name. With $f(\theta) := \uFunc'(\cFunc_T(\bar{m}_{\prdNxt} + \Rcal x + (\theta - 1)))$:

so, taking expectations, the first-order term dies because $\E[\theta_T - 1] = 0$, and at $T$ the second term of $f''$ dies because $\cFunc_T'' = 0$ exactly:

Expand the remaining argument in $\Rcal x$ (small; verified ex post) and subtract (EE-det). On the left, $\uFunc'(\bar{c} - x) - \uFunc'(\bar{c}) = (-\uFunc''(\bar{c}))\,x + O(x^2)$. The zeroth-order terms cancel by (EE-det) — this is where “the first-order terms disappear”: what survives is first order in $x$ and second order in $(\theta_T - 1)$:

Both sides in CRRA, evaluated on the PF ray ($\bar{c} = \kap\wbar$, $\cFunc_T(\bar{m}_{\prdNxt}) = \kap\wbar_T = \kap\ThornG\wbar$): $-\uFunc''(c) = \rho c^{-\rho-1}$, $\uFunc'''(c) = \rho(\rho+1)c^{-\rho-2}$. Divide through by $\rho(\kap\wbar)^{-\rho-1}$ and use (I1) twice ($\ThornG^{\rho}\,\ThornG^{-\rho-1} = \ThornG^{-1}$):

By (I2), the bracket is exactly $\Rcal/\ThornG$ — the messy marginal-utility coefficients collapse — and:

This is the time-$(T-1)$ version of equation (1): a pure power law in $\wbar$ with exponent 1 and an explicit amplitude. Its reading: $\tfrac{1}{2}\kap^2(\rho+1)\sigma^2/c$ is the Kimball precautionary premium in marginal-utility units (prudence $-\uFunc'''/\uFunc'' = (\rho+1)/c$ times the risk $\kap^2\sigma^2/2$ the MPC passes through); dividing by the local marginal-utility gradient and discounting turns it into consumption units. Note that (19) equals $(1/\Rcal)\cdot[\kap(\rho+1)\sigma^2/(2\wbar_T)]$ — the adjustment priced at tomorrow’s PF wealth $\wbar_T = \ThornG\wbar$, then discounted one period at the interest-minus-growth factor $\Rcal$. The bracket’s feedback term $\kap\Rcal/\ThornG$ has its own economics: consuming $x$ less today banks $\Rcal x$ more for tomorrow, which lowers tomorrow’s marginal utility and so offsets part of today’s precautionary wedge; identity (I2) says this offset is exactly what converts the naive discount into the risk-free one.

2.3 The induction step: equation (1) exactly, with its discount and adjustment derived¶

Now step to $T-2$ (and generically to any $t < T$), assuming the shortfall $\psavNxt$ exists and is small in the bootstrap sense of §2.5 (it is: it was just constructed, of order $\wbar^{-1}$). Repeat the subtraction. Three — not two — pieces survive, all first order in the small quantities $\{\psavNow, \psavNxt\}$ or second order in $(\tranShkNxt - 1)$:

1. the feedback (as in §2.2): $\ThornG^{\rho}\,\uFunc''(\bar{\cFunc}(\bar{m}_{\prdNxt}))\,\kap\,\Rcal\,\psavNow$ — today’s shortfall banks assets;

2. the injection (as in §2.2): $\ThornG^{\rho}\,\tfrac{1}{2}\sigma^2\,\bigl[\,\uFunc'''(\bar{c}_{\prdNxt})\,\kap^2 + \uFunc''(\bar{c}_{\prdNxt})\,\cFunc\Nxt''\,\bigr]$; the $\cFunc''$ piece now reads $-\uFunc''\cdot\psavFunc\Nxt'' = O(\wbar^{-\rho-1}\cdot\psavFunc\Nxt'')$ — two orders in $\wbar$ below the $\uFunc'''$ term (§2.5), dropped;

3. the propagation — new: tomorrow’s consumer is herself short of HER PF rule, so $\uFunc'(\cFunc\Nxt) = \uFunc'(\bar{c}_{\prdNxt} - \psavNxt)$ contributes $\ThornG^{\rho}\,(-\uFunc''(\bar{\cFunc}(\bar{m}_{\prdNxt})))\,g\Nxt(\bar{m}_{\prdNxt})$, the continuation shortfall priced in marginal utility. ($\E[g\Nxt(\bar{m}_{\prdNxt} + (\tranShkNxt - 1))] = g\Nxt(\bar{m}_{\prdNxt})$ to leading order; the Jensen correction is second order, §2.5.)

The same division by $\rho(\kap\wbar)^{-\rho-1}$, the same (I1)–(I2) collapse, and — writing everything in PF-wealth coordinates, $\psavFunc\Now(m) = g\Now(\wbar)$ with the argument naming the coordinate, $\bar{m}_{\prdNxt}$'s PF wealth $= \ThornG\wbar$ — the recursion is

— shortfall today = (discounted at $1/\Rcal$) shortfall tomorrow at the deterministically drifted PF wealth $\ThornG\wbar$, plus the discounted adjustment. This is §0’s pricing recursion (1) — the expectation $\E_{\prdt}[\psavFunc(\mNrmNxt)]$ there evaluated at its certainty equivalent, the drifted PF wealth — with every term now derived: the discount is the interest-minus-growth factor $1/\Rcal$ (not $\beta$, not $R\beta\Gamma^{-\rho}$ — identities (I1)–(I2) do the conversion into consumption units along the PF ray); the state drifts by $\ThornG$ per step (the wealth recursion (8)); and the flow is the Kimball precautionary premium of §3.4, with $\wbar\,\bufAdjFunc(\wbar) = c_J := \kap(\rho+1)\sigma^2/(2\ThornG)$ identically in this leading-order form — exactly the forcing limit (30) of the rigorous lemma. The rigorous counterpart is Lemma 5.1’s exact identity (29) (§4), $(1 + \etaL(\wbar))\,\Rcal\,g(\wbar) = (1 + \etaR(\wbar))\,\E_{\prdt}[g(\wbarNxt)] + \bufAdjFunc(\wbar)$, whose monotone comparison form (31)/(32) is (EQ-1) with the honest $\pm C_0$ PF-wealth-drift slack; the remainders $\etaL$, $\etaR$ collect exactly the terms §2.5 drops.

2.4 Unrolling backward: the Gordon trichotomy with the published constants¶

Iterate (EQ-1) backward from $\psavFunc_T \equiv 0$, $k = T - t$ steps, using $\bufAdjFunc(\ThornG^{\,j}\wbar) = c_J\,\ThornG^{-j}/\wbar$: