Appendix: the race behind q_∞ — the GIC-policy inequality in slow motion

Status: expository appendix of the simplified capstone, referenced from the GIC-policy assumption block of §1. It proves nothing new: every claim below is a slow-motion restatement of results established in the main page ((2), (3), (25), Paragraph) or, for the supercritical counterexample, in the archived full treatment. Its arithmetic is certified by verify_race_intuition_checks.py (committed output verify_race_intuition_checks_out.txt). The question it answers is the one a reader should ask at §1: the standing assumption ℛÞ_Γ < 1 is not required for the gap to decay as a power law — any positive root of (25) delivers that — so what, exactly, does the inequality guarantee? The answer, developed below and stated sharply in the closing section: it guarantees that the power law you see is the buffer’s own shadow — that no matter how high wealth gets, the return to the buffer-stock region (§8’s return region around the target m̂) remains not merely present in behavior but its leading correction.

A.1 Two clocks, three rates¶

Why does a wealthy household’s consumption differ from the frictionless perfect-foresight rule at all? For one reason only: someday it will be back in the return region, where the constraint and precaution bite. The gap g(w̄) is the present shadow of that future episode. How dark the shadow is at wealth w̄ therefore factors into two separate questions, each governed by its own clock:

1. How far away in time is the return? Impatience runs the wealth ratio down geometrically — ln w̄ falls by −þ_g = ln(1/Þ_Γ) per period ((2)'s descent speed) — so the travel time is τ(w̄) ≈ ln w̄ / (−þ_g): wealth buys time only logarithmically. Call −þ_g the melt rate (an informal alias, used in this appendix only; the formal object is always −þ_g). Doubling wealth buys a fixed ln 2/(−þ_g) additional periods, not twice as many.

2. How much does an event that far away matter today? What the future episode does to a rich household now is deny it the ability to borrow against income accruing beyond the date the constraint starts to bind — it truncates the tail of human wealth. The present value of the income tail beyond date τ shrinks by the factor Γ/R = 1/ℛ per period of remoteness: income grows at Γ but the market discounts at R. So ln ℛ — the interest-minus-growth gap, (2)'s discounting speed — is the rate at which the future stops mattering. Call it the forget rate. (FHWC, ℛ > 1, says precisely that the future is forgotten at all — human wealth converges.)

Compose the clocks and the power law appears, with no further input ((3)): the relevance of the return, seen from wealth w̄, is

ℛ^{−τ(w̄)} = w̄^{−ln ℛ/(−þ_g)} = w̄^{−q_∞}   (ψ ≡ 1),

exponential forgetting read on a logarithmic wealth-to-time clock. The exponent is the ratio of the two rates — which is why it is a pure shape object, indifferent to whether the rates are quoted per quarter or per year (the units cancel).

The third rate enters with permanent shocks and gets its own section below: luck, ln E[ψ²] = σ_ψ² under lognormality — descent by ψ-streaks rather than by impatience.

A.2 Fast in time is slow in wealth: an impatient billionaire¶

The ratio reading produces an inversion that fights ordinary intuition: more impatience makes the exponent smaller — 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 ten times the buffer scale 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: temporally the return is eons away even at modest wealth, so its behavior looks frictionless almost immediately — a gap that dies fast in wealth. Households that converge fast in time are the ones whose gap decays slowly in wealth, and conversely. The half-lives at the estimated calibrations (quarterly rates; half-life = ln 2/rate, in years) make the race concrete:

Standing at w̄ = 10 (ten times the buffer scale): the HS-mean household is τ ≈ 46 years from the return region and still assigns it relevance weight ℛ^{−τ} = 10^{−0.43} ≈ 0.37; the archived GIC-CAP household is ≈ 1{,}600 years away and assigns it 10^{−14} — economically invisible (until luck is priced in; §A.3). Hence the impatient billionaire: an impatient billionaire is still, at the margin, a buffer-stock consumer — only a couple of decades from the return region, with behavior visibly shaped by it — while a sufficiently patient household of far more modest wealth already behaves as if frictionless. Impatience is what keeps the buffer-stock region relevant at high wealth; patience is what buries it.

The GIC-CAP row is the cautionary tale, told fully in the archived full treatment: HAFiscal’s estimation cap pins raw growth impatience to hold, barely (Þ_Γ = 0.999638 < 1) — a solution-method safeguard doing exactly the job it was designed for. But pinning Þ_Γ just under 1 kills the melt rate (half-life ≈ 479 years), and since the exponent is the ratio forgetting/descent, a dead melt rate is precisely what hands the race to forgetting. The GIC and the exponent condition pull on the same dial in opposite directions near the GIC boundary: the closer the calibration is allowed to drift to the raw-GIC knife-edge, the more supercritical its exponent. The cap guards the wrong dial for this purpose — which is exactly why the simplified capstone replaces proximity-to-GIC with the GIC-policy inequality as the maintained condition.

A.3 Luck credit: the exact frontier, and a two-line quadratic¶

With permanent shocks the descent is noisy: as §2’s intuition section records, runs of small permanent-income realizations (ψ < 1) inflate the normalized ratio — the household is income-poor and ratio-rich — so travel times to the return region become random. The gap at w̄ is then an average over descent scenarios of the discounted relevance, E[ℛ^{−τ}], and the average is convexity-dominated: a scenario in which the return arrives early contributes weight near 1, a late one contributes nearly 0, so the rare early-arrival branches rule the expectation — E[ℛ^{−τ}] far exceeds ℛ^{−E[τ]} (Jensen). Noise therefore weakens effective discounting and lowers the root below the deterministic ratio. That is the entire content of the tilt in the eigen-equation (25), E[ψ^{1+q}] = ℛ Þ_Γ^q. Under lognormal mean-one ψ (ln E[ψ^{1+q}] = ½σ_ψ² q(1+q)), taking logs of (25) gives the whole story in one line — with melt = −þ_g, forget = ln ℛ, and luck = ln E[ψ²] = σ_ψ²:

melt·q + ½·luck·q(q+1) = forget.

The linear term is descent by impatience; the quadratic term is descent by luck — the Jensen credit. This two-line approximation reproduces the frozen exact-atom roots of all four calibrations to a few parts in ten thousand (certified in the battery; luck = ln 1.002492 = 0.2489 %/qtr, the ladder figure’s frontier value):

Read at q = 1, the quadratic gives the exact resonance frontier of §1’s note in rate form: q_∞ < 1 if and only if

melt + luck > forget      (⟺  ℛÞ_Γ < E[ψ²]),

descent-by-impatience plus descent-by-luck must outrun forgetting. GIC-policy is the no-credit-for-luck version: it demands melt > forget on its own (ℛÞ_Γ < 1), so that q_∞ < 1 holds whatever the permanent-shock distribution — which is why it is sufficient but not strictly necessary, exactly as §1 states. The band between the solid and dashed lines of Figure 1 is the luck-credit region: calibrations there fail GIC-policy yet still have q_∞ < 1, by grace of their own σ_ψ².

Luck credit is also what rescues GIC-CAP from absurdity — and quantifies why it still fails. Its deterministic ratio is ≈ 14; luck hauls the root down to 1.4735. At w̄ = 10 that raises the gap from 10^{−14} to 10^{−1.47} ≈ 0.03 — twelve orders of magnitude of Jensen — yet the frontier inequality still fails, because at the q = 1 gauge the whole descent budget is melt + luck = 0.036 + 0.249 ≈ 0.29 %/qtr against forget = 0.51 %/qtr. Luck can carry most of the distance; it cannot carry all of it.

A.4 What, exactly, does GIC-policy guarantee?¶

Not the power law: (25) has a positive root, and the gap decays as a power of wealth, under the maintained conditions alone — supercritically, resonantly, or subcritically — and under any finite exponent the buffer-stock region “still matters” at every wealth level, in the weak sense that a power law never reaches zero. The content of the standing assumption is sharper, and it has three faces, in decreasing order of importance:

1. Dominance — the buffer’s shadow remains the leading correction. By Paragraph the shortfall from frictionless behavior is the sum of two components: the destination component — the discounted memory of the return region, ∝ w̄^{−q_∞} — and the here-and-now component, the Kimball adjustment to current-period risk, ∝ 1/w̄. The slower-fading component wins asymptotically. q_∞ < 1 is precisely the statement that the destination component wins: however high wealth gets, the leading departure from frictionless behavior is still the anticipated return to the buffer-stock region — not local prudence. That is the sharp version of “the buffer-stock region still matters”: it matters most — the impatient billionaire of §A.2, at the margin, is still a buffer-stock consumer. When the inequality fails (GIC-CAP, in the archive), the ordering flips: the local term dominates, the realized exponent saturates at 1, and the visible tail of the consumption function no longer reads out the buffer’s shadow at all.

2. Identification — what you fit is what the theory names. Under GIC-policy the realized decay exponent is q_∞ itself (no min(1, q_∞) clip), and the mode separation 1 − q_∞ > 0 is the purification rate of §3.1: the measured tail slope converges to the theorem’s root, at a knowable pace.

3. Nondegeneracy — the simple rule stays simple. The resonance knife-edge q_∞ = 1 (decay w̄^{−1} ln w̄, not a pure power) and the supercritical amplitude theory are both excluded, so the power-law policy rule needs no case analysis — the point of §1’s gloss that GIC-policy “simplifies the construction of a simple power-law policy rule”.

On necessity, the precise statement: GIC-policy is sufficient, never necessary — §A.3’s luck credit can hold q_∞ < 1 with ℛÞ_Γ somewhat above 1. But among conditions that use no information about the permanent-shock distribution it is exact: with ψ ≡ 1 (or declining to lean on σ_ψ², whose estimated value is small and model-dependent), ℛÞ_Γ < 1 is necessary and sufficient for q_∞ < 1. It is the distribution-free frontier, which is what a standing assumption ought to be.

A.5 Verification¶

verify_race_intuition_checks.py (stdlib-only; committed output verify_race_intuition_checks_out.txt) certifies every number above from the frozen factors: (C1) melt/forget rates and half-lives of the §A.2 table; (C2) the deterministic ratios, including HS-mean 0.43 and GIC-CAP ≈ 14; (C3) the w̄ = 10 vignette weights; (C4) the quadratic roots of §A.3 against the frozen exact-atom roots (tolerance 5×10⁻⁴); (C5) the q = 1 gauge arithmetic — the trio passes melt > forget (COL-TOP by 1.2×10⁻⁴ in factor form), GIC-CAP fails even melt + luck > forget; (C6) the twelve orders-of-magnitude Jensen figure at w̄ = 10.