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Econ-36-A2 · revision r1 · on Econ-36

GPT 6 Astra Pro attempt on Econ-36

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Contents
  1. C63-35: Certified global nonlinear heterogeneous-agent computation
  2. 1) FORMAL RESTATEMENT
  3. 2) QUICK LITERATURE/CONTEXT CHECK
  4. 3) ATTACK PLAN
  5. 4) WORK
  6. 5) VERIFICATION
  7. 6) FINAL
  8. References

C63-35: Certified global nonlinear heterogeneous-agent computation

Outcome: UNRESOLVED; conditional global certificates and sharp obstructions proved.

Tools: web browsing available; Python computation available. Literature checked on 9 October 2026.

1) FORMAL RESTATEMENT

For a declared nonempty class $\mathcal H$ of heterogeneous-agent economies, let $D_h$ be the full relevant state domain, including the full distribution state, and let $\mathcal R_h$ be the declared equilibrium residual operator on a specified normed function space. Equalities, inequalities, and complementarity must all be encoded: for instance an inequality $a\le0$ has violation $a^+=\max(a,0)$; nonnegative complementarity $a\ge0,b\ge0,ab=0$ can be encoded by $(-a)^+,(-b)^+$ and $\min(a^+,b^+)$. A zero residual must be proved equivalent to the desired equilibrium conditions.

The requested programme is to construct an algorithm such that for every $\varepsilon>0$, \[ \sup_{h\in\mathcal H}\|\mathcal R_h(\widehat f_h)\|_{D_h}\le\varepsilon, \] with explicit operation counts, error propagation into estimation, and, under appropriate stability hypotheses, \[ {\operatorname{dist}}(\widehat f_h,\mathcal E_h) \le C\|\mathcal R_h(\widehat f_h)\|_{D_h}, \qquad \mathcal E_h=\{f:\mathcal R_h(f)=0\}. \] The coefficient $C$ must be uniform if the guarantee is uniform over $h$.

Specification issue. The attachment is an editorial research family, not a fully specified operator, class $\mathcal H$, norm, or cost model. The minimal correction is to fix these objects and any claimed stability class. The conditional theorems below do so explicitly; they do not assert that arbitrary heterogeneous-agent equilibrium operators are contractions. Residual scaling is part of the data, not a harmless convention when a universal $C$ is requested.

2) QUICK LITERATURE/CONTEXT CHECK

Auclert--Bardóczy--Rognlie--Straub [ABRS] provide an established sequence-space Jacobian solution and estimation method. That is not by itself a certificate of a supremum residual bound over an arbitrary global nonlinear state domain. This report supplies conditional numerical-analysis statements, not a claim that current heterogeneous-agent computation lacks powerful existing methods.

3) ATTACK PLAN

Proof routes. Derive a uniform contraction-based residual bound, prove a small-gain condition for coupled household and aggregate blocks, and convert finite computations into supremum certificates using an explicit modulus of continuity.

Disproof routes. Test a scalar residual with a degenerate derivative, rescale an otherwise regular residual, and hide a narrow residual spike between finitely many sampled states. These constructions show why stability and global approximation assumptions cannot be omitted.

Landscape and tools. This is verified numerical analysis and equilibrium stability. Banach contraction supplies existence and an error bound; geometric-series estimates control inexact iteration; weighted product norms test coupled feedback; covering nets turn local evaluations into global estimates; residual counterexamples expose poor conditioning; triangle inequalities propagate moment error; and an identification modulus translates moment accuracy into parameter accuracy.

4) WORK

Theorem (Uniform contraction certificate): For each $h\in\mathcal H$, let $S_h$ be a nonempty closed subset of a Banach space $X_h$, and let $F_h:S_h\to S_h$ satisfy \[ \|F_h(f)-F_h(g)\|\le q\|f-g\|\qquad(f,g\in S_h) \] with the same $0\le q<1$ for every $h$. For the explicitly normalized residual $R_h(f)=f-F_h(f)$, there is a unique fixed point $f_h^*\in S_h$, and \[ \|f-f_h^*\|\le\frac{\|R_h(f)\|}{1-q}\qquad(f\in S_h). \]
Proof: Fix $h$ and $f_0\in S_h$ and iterate $f_{j+1}=F_h(f_j)$. Induction gives $\|f_{j+1}-f_j\|\le q^j\|f_1-f_0\|$. For $k>j$, the triangle inequality bounds $\|f_k-f_j\|$ by the tail of the resulting geometric series, which tends to zero. Closedness and completeness give a limit $f_h^*\in S_h$, and the Lipschitz inequality gives $F_h(f_h^*)=f_h^*$. Two fixed points have distance at most $q$ times that distance, and hence coincide. Finally, \[ \|f-f_h^*\|\le\|f-F_h(f)\|+\|F_h(f)-F_h(f_h^*)\| \le\|R_h(f)\|+q\|f-f_h^*\|. \] Move the final term to the left. The same $q$ gives the uniform constant. If an economic residual $\mathcal R_h$ differs from $R_h$, a separately proved uniform bound $\|R_h(f)\|\le L\|\mathcal R_h(f)\|$ is required before transferring this certificate. ∎
Proposition (Inexact global evaluation and iteration): Suppose $\widehat F_h(f)\in S_h$ and a verified oracle bound $\|\widehat F_h(f)-F_h(f)\|\le\delta$ holds globally for all required inputs. Then \[ \|f-f_h^*\|\le\frac{\|f-\widehat F_h(f)\|+\delta}{1-q}. \] For $f_{j+1}=\widehat F_h(f_j)$, \[ \|f_j-f_h^*\|\le q^j\|f_0-f_h^*\| +\frac{\delta(1-q^j)}{1-q}. \]
Proof: The first estimate follows from $\|f-F_h(f)\|\le\|f-\widehat F_h(f)\|+\delta$ and the theorem. The iteration satisfies $e_{j+1}\le q e_j+\delta$. Substitution of this inequality repeatedly yields the second formula. If a verified initial bound $e_0\le E_0$ is available, choosing $\delta\le(1-q)\varepsilon/2$ and $q^jE_0\le\varepsilon/2$ certifies error at most $\varepsilon$. For $0<q<1$ one can take $j\ge\max\{0,\lceil\log(2E_0/\varepsilon)/\log(1/q)\rceil\}$; for $q=0$ one oracle evaluation suffices when $\delta\le\varepsilon$. If one globally certified oracle evaluation costs at most $C_h(\delta)$ operations, the iteration cost is at most $jC_h(\delta)$ plus explicitly charged certificate and initialization costs. No cost bound is asserted before such an oracle is supplied. ∎
Proposition (A coupled-block small-gain criterion): Suppose $F=(H,G)$ on an invariant closed product domain and \begin{align*} \|H(x,y)-H(x',y')\|&\le a\|x-x'\|+b\|y-y'\|,\\ \|G(x,y)-G(x',y')\|&\le c\|x-x'\|+d\|y-y'\|, \end{align*} where $a,b,c,d\ge0$. For any $\lambda>0$, under the norm $\|(x,y)\|_\lambda=\max\{\|x\|,\|y\|/\lambda\}$, the map has Lipschitz constant at most \[ q_\lambda=\max\{a+b\lambda,\ d+c/\lambda\}. \] In particular, if $b,c>0$, $a,d<1$, and $bc<(1-a)(1-d)$, a $\lambda$ with $q_\lambda<1$ exists.
Proof: Writing $r=\|(x-x',y-y')\|_\lambda$ gives $\|x-x'\|\le r$ and $\|y-y'\|\le\lambda r$. Substitute these in the first block bound and in the second bound divided by $\lambda$. For strict contraction choose $c/(1-d)<\lambda<(1-a)/b$; this interval is nonempty exactly under the displayed product inequality. If $b=0$, take $\lambda$ large enough to make $d+c/\lambda<1$, provided $a,d<1$; if $c=0$, take $\lambda$ small enough for $a+b\lambda<1$. If both vanish, $q=\max(a,d)$. ∎

Thus contraction of a household block alone does not prove contraction of the entire general-equilibrium calculation: the feedback constants $b,c$ must also be controlled.

Theorem (From a finite net to a supremum certificate): Let $(D,d_D)$ be a compact metric state domain, $r:D\to{\mathbb{R}}^k$ an $L$-Lipschitz residual in a declared norm, and $\{s_1,\ldots,s_N\}$ an $\eta$-net. If certified computed values satisfy $\|\widehat r(s_j)-r(s_j)\|\le\delta$ and $\max_j\|\widehat r(s_j)\|\le\rho$, then \[ \sup_{s\in D}\|r(s)\|\le\rho+\delta+L\eta. \]
Proof: For each $s\in D$, choose $s_j$ with $d_D(s,s_j)\le\eta$. The triangle inequality gives $\|r(s)\|\le\|r(s)-r(s_j)\|+\|r(s_j)-\widehat r(s_j)\|+\|\widehat r(s_j)\|\le L\eta+\delta+\rho$. Taking the supremum proves the result. The net construction and evaluation cost is part of any algorithmic claim, not suppressed in this bound. ∎

For a distribution state, compactness in a declared weak or Wasserstein metric does not bound the covering number cheaply. Truncation, quadrature, interpolation, distribution discretization, and floating-point errors can be added to $\delta$ only after individual bounds in compatible norms have been proved. Testing a simulated path is not the same as supplying a net of the full state domain.

Proposition (Three explicit obstructions): Without additional hypotheses, neither a linear residual error bound nor a finite-sample supremum certificate is valid.
Proof: First let $R(x)=x^2$ on $[-1,1]$ with root set $\{0\}$. A bound $|x|\le C|R(x)|$ for all small nonzero $x$ would imply $1\le C|x|$, contradicted by $0<|x|<1/C$. Thus even a smooth scalar residual with a unique root need not have a linear error bound.

Second, for $R_\eta(x)=\eta x$, $\eta>0$, the exact ratio of error to residual is $1/\eta$. A class containing arbitrarily small $\eta$ has no uniform constant. For $R(x)=x(x-1)$, a zero residual at $x=1$ also cannot certify distance to the selected root $0$; an estimate must concern the root set unless a selection has been justified.

Third, consider any deterministic finite-query procedure for continuous functions on $[0,1]$. Run it on the zero function and record its finitely many query points. Choose an open interval avoiding these points and a continuous triangular bump supported in that interval, of height $M>0$. The bump and zero give identical answers at every query; by induction the adaptive query transcript is identical. Nevertheless their suprema differ by $M$, which can be arbitrarily large. Therefore finite evaluations without a known modulus or another global enclosure do not prove a uniform supremum bound. This is a deterministic information obstruction, not a claimed randomized lower bound under an unstated oracle model. ∎

Proposition (A transparent estimation-error propagation bound): Suppose a numerical equilibrium error $\|\widehat f_\theta-f_\theta\|\le\delta_f$ holds uniformly in $\theta$, and a population moment functional obeys $\|m(f)-m(g)\|\le L_m\|f-g\|$. Its moment error is at most $L_m\delta_f$. More generally, let $m(\theta_0)=b$, suppose $\|m(\theta)-m(\theta_0)\|\ge\kappa\|\theta-\theta_0\|$ on the declared parameter domain, and assume \[ \sup_\theta\|\widehat m(\theta)-m(\theta)\|\le\eta_{\rm num}, \quad \|\widehat b-b\|\le\eta_{\rm data}, \quad \|\widehat m(\widehat\theta)-\widehat b\|\le\rho. \] Then $\|\widehat\theta-\theta_0\|\le(\eta_{\rm num}+\eta_{\rm data}+\rho)/\kappa$.
Proof: The moment bound is the stated Lipschitz inequality. For the parameter claim, write $m(\widehat\theta)-m(\theta_0)$ as the sum of its numerical error, its numerical fitting residual, and the data-target error. The triangle inequality bounds its norm by the numerator. The identification inequality gives the asserted result. This is a deterministic or eventwise guarantee according to the meaning of the three input bounds; it does not manufacture a confidence statement. Likelihood error would likewise require its own continuity or density-lower-bound assumptions. ∎

5) VERIFICATION

For $F(f)=f/2+1/4$, the root is $f^*=1/2$. At $f=3$, the residual is $1.25$ and the error is $2.5$, attaining the contraction error bound exactly. A block test with $(a,b,c,d)=(0.2,0.1,0.15,0.3)$ and $\lambda=1$ gives $q=0.45$. For $R(x)=x^2$, the error/residual ratios at $x=10^{-1},10^{-2},10^{-3}$ are $10,100,1000$. These calculations are included in the script.

The global certificates require self-mapping and completeness, not merely a Jacobian norm near a computed point. Approximation iterates must remain in the certified domain. No uniqueness of equilibrium is assumed for the generic residual family; the contraction theorem proves uniqueness only in its explicit subclass. Binding constraints are included in the residual definition but do not automatically preserve a smooth inverse or contraction. Every constant claimed uniform must be bounded over the specified class, including covering and oracle costs.

6) FINAL

{LABEL: UNRESOLVED.}

Strongest checked partial result. We prove uniform global residual-to-solution and inexact-iteration certificates for a specified contraction class, a coupled-block criterion, finite-net supremum certification, and a conditional moment-to-parameter bound. Scalar and finite-query constructions rigorously refute unconditional versions of the desired guarantees.

Exact first gap for the intended economic application. For a concrete rich nonlinear heterogeneous-agent model class, construct a globally valid equilibrium operator and verify a uniform stability estimate on its full invariant domain, together with a finite-cost globally certified evaluation oracle. Neither condition is supplied by the editorial problem statement or proved for that entire economic class here.

Next three moves. (1) Bound the four household/aggregate feedback constants on an economically meaningful invariant domain and test the small-gain inequality. (2) Construct a distribution-state enclosure with explicit covering and quadrature error costs. (3) Where contraction fails, prove a metric-regularity error bound across occasionally binding constraints and track its constant into the actual estimation criterion.

Likely minimal obstruction. A near-singular market-clearing response, a binding-constraint regime switch, or two equilibria in the same certification domain can invalidate a uniform selected-solution guarantee even when household optimization is easy. The scalar residual examples show the exact mathematical failure that must be ruled out.

References