Supplement: why the standard asset-pricing toolbox does not replace EGM-plus-theoretical-tails - Power-law decay of the buffer-stock consumption gap (simplified capstone under GIC-policy)

Status: supplemental document of the presentation of record, commissioned by the owner 2026-07-18 (“write a new supplemental document that explains why none of the standard asset-pricing tools will work (or work as well as the existing EGM solution methodology augmented with the theoretical tails)”). It distills an adversarially verified numerical exploration whose full record — scripts, committed outputs, and verifier re-runs — is archived in the full corpus as BufferStockTheory-Latest:theory/powerlaw-decay/asset_pricing_solvers_note.md and the BufferStockTheory-Latest:theory/powerlaw-decay/spike_pricing_* artifacts; this supplement is deliberately qualitative. Scope companion: the main document’s §0 states where the asset-pricing resemblance is valid at all (large $\wbar$ — the tail; not the ergodic region).

1. The two structural obstacles, and the one fatal region¶

Equation (1) resembles the pricing recursions the discrete-time asset-pricing numerical literature solves. Three things break the resemblance as a solution strategy:

The transition is policy-endogenous. Tomorrow’s state contains the unknown function itself ( $\wbarNxt = \ThornG\wbarNow + (\tranShkNxt - 1) + \Rcal\,g\Now(\wbarNow)$ ). The pricing literature’s state dynamics and cash flows are exogenous; every tool below inherits some cost from this difference.
The flow is only asymptotically known. The Kimball precautionary premium’s closed form is the $\wbar \to \infty$ limit; at moderate wealth the exact flow is implicitly defined by the unknown consumption function. A pricing solver needs its payoff stream as data; here the payoff is part of the unknown.
The bottom is not smooth territory. The ergodic set sits at small $\wbar$ , 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). Any method whose accuracy rests on global smoothness loses precisely where the solution lives. This is the fatal region: the tools below fail here even when they are serviceable in the tail.

2. Tool by tool¶

Discretized-state pricing (Mehra–Prescott; Tauchen; Tauchen–Hussey). Assemble the discounted transition matrix, solve the linear system. Standalone it is not even defined here: there is no transition matrix until a policy is frozen (obstacle 1), and no payoff vector until the flow is evaluated from a policy (obstacle 2). Frozen-policy inner solves do work — but that architecture is policy iteration around the same EGM objects, an accelerator of the incumbent, not an alternative to it. And its natural fixed-grid collocation form fails outright on zero-income-atom specifications: fixed nodes cannot adapt to the bottom geometry the atom creates, while EGM’s endogenous gridpoints adapt by construction. The bottom belongs to EGM.

Projection / Chebyshev collocation (Judd; Pohl–Schmedders–Wilms). The one tradition that can swallow obstacle 1 whole (the exact equation’s residual is just a nonlinear function of basis coefficients). But global polynomial bases purchase their convergence rate from smoothness, and §1.3 is the anti-smoothness fact: as $\pZero \downarrow 0$ a kink forms, and near-kink curvature grows without bound. A spectral basis on a kink loses its rate; splitting the domain at the kink and using theory-given forms in each piece is not an alternative — it is the incumbent architecture (grid in the middle, theoretical forms at the ends) wearing a different basis. The accuracy-audit discipline of this literature is worth keeping; the solver is not.

Parameterized expectations (den Haan–Marcet). Simulation-based: it learns the expectation where the simulated process spends time — the ergodic set. That is backwards for this corpus twice over: the theorem’s object is the far tail, which the ergodic process visits too rarely to sample (§8.1 quantifies how rarely), and the ergodic region itself is the non-smooth bottom where a regression-smooth conditional expectation is the wrong functional class. Usable at most as a body-region cross-check.

Acceleration of the fixed point (Anderson mixing; SQUAREM). These repair slow contraction — a spectral gap near one. The measured slowness of backward iteration here is not a spectral gap: it is transport — each sweep propagates information one $\ThornG$ -step down the wealth line, so the far field cannot be correct before the horizon $\ln(\wbar_{\text{top}}/\wbar_{\text{body}})/\ln(1/\ThornG)$ is paid. Acceleration cannot beat a causality bound, and the measured gains were nil to negative. (The one true accelerant is structural: impose the theorem’s tail at the top — the corpus’s own ‘tails’ rule — which slaves the far field to the interior analytically instead of waiting for transport.)

Log-linear and closed-form pricing (Campbell–Shiller; Burnside). Iterating the leading-order recursion forward is a growing-perpetuity sum whose term ratio is the multiplier $\lambda(1)$ ( $> 1$ under GIC-policy, so the perpetuity diverges — the divergent branch is the operative one); the closed forms of this tradition are, in this corpus’s clothing, the tail law itself — (24), the (archived) Gordon amplitude of the supercritical branch, the ‘tails’ boundary rule. They are the theory at the ends, already installed; they are not a solver for the middle.

Operator methods (Hansen–Scheinkman; Perron–Frobenius long-run factorization). The right theoretical frame, and the deepest connection: the root equation $\E[\psi^{1+q}] = \Rcal\,\ThornG^{q}$ is a principal-eigenvalue condition, with the power modes as eigenfunctions and $\qup$ the marginal one; the ‘tails’ top rule is the corresponding eigenfunction-transparent boundary. But the frame delivers exponents — never amplitudes, and nothing in the body. It names what the corpus proves; it does not compute what the corpus computes.

3. Why EGM-plus-theoretical-tails is the architecture that remains¶

The bottom: EGM’s endogenous gridpoints place the constraint boundary exactly and adapt to whatever geometry the income process creates there — the one region (§1.3) where every smooth-basis and fixed-grid method degrades, and the region the ergodic set occupies. No pricing-literature tool matches this; the failed fixed-node experiment in the exploration record is the demonstration.
The top: above the grid, the theorem’s $\qup$ tail is not an extrapolation heuristic but the proved eigen-direction — and imposing it is exactly what removes the near-unit boundary mode and the transport wait. The pricing literature’s own operator theory says this boundary rule is optimal; it is already the corpus’s.
Below the grid: the constraint-end theorem ( $\qdn = \rho$ , the MPC approach to $\bar{\kappa}$ ) supplies the principled bottom form — the mirror of the top tail. Since no affordable grid can chase the $\pZero \downarrow 0$ curvature down to the kink, the representation below the grid must come from theory; it does.
The middle: the ergodic region is solved, not approximated — which is the entire point of the in-grid moral: an in-grid representation plus the theorem’s tails is the whole solution, and no ad-hoc external criterion does any work.

What the exploration adds to this architecture is operational, not architectural: certify and stop on the Euler residual rather than a successive-change metric (the observed plateau is churn plus transport, not non-convergence), report rather than exclude the near-bottom certification band, and — if large-scale use ever wants the outer loop shortened — a frozen-policy linear inner solve is available as an accelerator of the same architecture. The asset-pricing analogy itself retires to the role the main document now assigns it: proving the tail.