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Econ-251 · economics · Economics

Sudden-stop amplification

LLM status Open unverified

Statement

Original subject group: International finance and exchange rates

Audit classification: Background; proposed extension. The primary paper already derives model-specific network amplification and reports calibrated crisis effects. The broader external-financing intervention is editorial; its relaxed-constraint denominator may vanish.

Definitions and precise research target

Fix a small open economy with firms linked by an input-output matrix $A$, external debt denominated in stated currencies, banks and collateral constraints. A sudden-stop intervention reduces external financing supply by $u>0$ at date $t$, independently of specified domestic productivity news. Let $Y_h(u;A,b)$ be output at horizon $h$ under this intervention, where $b$ collects initial maturity, currency mismatch and collateral exposures. Define amplification relative to a counterfactual economy with relaxed financing constraints and the same technologies by \[\Gamma_h(A,b)=\frac{E[Y_h(0;A,b)-Y_h(u;A,b)]}{E[Y_h^0(0;A,b)-Y_h^0(u;A,b)]},\] when the denominator is positive and the relaxed-constraint economy retains a stated transmission channel; otherwise report the signs and the loss difference defined below. Superscript $0$ denotes that relaxed-constraint economy. The task is to identify amplification and characterize network features increasing crisis losses. Model endogenous exchange rates, collateral prices, substitution and lender risk-bearing; holding these fixed changes the target. Establish the financing shock's exclusion and avoid conflating capital withdrawal with investors responding to output declines. Report nonlinearities as constraints become binding, sensitivity to input substitutability and possible firm exit. Where only total responses are identified, distinguish them from model-dependent counterfactual amplification. An input-output correlation with crisis severity alone does not isolate a causal network mechanism.

Additional formal definitions and scope (editorial). Define $A_{ij}$ as the input share from sector $i$ in sector $j$, with nonnegative entries and feasibility conditions from the chosen production functions; Cobb--Douglas share restrictions and CES substitutability cannot be mixed implicitly. The sudden-stop intervention changes a named financing-supply limit or price equation. Relaxing collateral constraints can remove that shock's entire effect, so the denominator of $\Gamma_h$ may be zero. Use the ratio only when the frictionless benchmark retains a nonzero financing-price or expenditure channel, and specify that channel. Otherwise report the difference in losses $\mathcal A_h=E[Y_h(0)-Y_h(u)]-E[Y_h^0(0)-Y_h^0(u)]$. This is a model-specific amplification contrast, not an automatically identified causal network effect. Define the same exogenous shock, technology and initial resource state for both comparisons.

For the identification part, declare an admissible class $\Theta$ of structural models, including preferences, technologies, shock laws, measurement/sampling rules and any equilibrium selector. If $O$ denotes the complete observable history and $T(\theta)$ the target defined above, the identified set is $\mathcal I_T=\{T(\theta):\theta\in\Theta,\ \mathcal L_\theta(O)=\mathcal L_{\rm obs}(O)\}$; point identification means this set is a singleton. A $do(a)$ comparison replaces stated policy/technology equations while fixing the declared exogenous law and re-solving the remaining equations. These definitions do not assert that an identification theorem holds without further restrictions.

Variants and assumption changes

  1. Two-sector collateral model (established source subcase). Characterize the source documented network-dependent amplification within its stated tradable/nontradable model; its theoretical and calibrated answers are already available.
  2. External financing-supply intervention (editorial). Extend to many sectors and currency mismatch; use loss differences when removing constraints makes the frictionless shock response zero.

References and source audit

  1. NBER indexed record verifies four authors, 2025 WP 34604 and DOI. Boston Fed WP 26-1 (2026), December 2025 text, inspected with its distinct DOI 10.29412/res.wp.2026.01. Locator: Boston Fed version Abstract and Sections 2--5. Support: Model-specific network amplification results. NBER direct fetch failed; equivalent primary Boston Fed PDF accessible. URL: https://www.nber.org/papers/w34604.
  2. NBER indexed record verifies four authors, 2025 WP 34604 and DOI. Boston Fed WP 26-1 (2026), December 2025 text, inspected with its distinct DOI 10.29412/res.wp.2026.01. Locator: Boston Fed version Abstract and Sections 2--5. Support: Model-specific network amplification results. NBER direct fetch failed; equivalent primary Boston Fed PDF accessible. URL: https://www.bostonfed.org/-/media/Documents/Workingpapers/PDF/2026/WP2601.pdf.
  1. Jorge Miranda-Pinto; Eugenio Rojas; Felipe Saffie; Álvaro Silva. 2025. Connected for Better or Worse? The Role of Production Networks in Financial Crises. Boston Fed version Abstract and Sections 2--5. https://www.nber.org/papers/w34604. DOI: https://doi.org/10.3386/w34604.
  2. Jorge Miranda-Pinto; Eugenio Rojas; Felipe Saffie; Álvaro Silva. 2026. Connected for Better or Worse? The Role of Production Networks in Financial Crises. Boston Fed version Abstract and Sections 2--5. https://www.bostonfed.org/-/media/Documents/Workingpapers/PDF/2026/WP2601.pdf. DOI: https://doi.org/10.29412/res.wp.2026.01.

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-251-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-251

Sudden-stop amplification: rank-one feedback and nonidentification

Target and scope

The target compares a financing contraction with a relaxed-constraint counterfactual, including production and financial spillovers. The source [mprs] already establishes model-specific network amplification. We give an exact local-equilibrium benchmark, an edge intervention formula, and a nonidentification construction. No equality between an observed input share and an equilibrium response derivative is presumed.

A local system with an explicit relaxed comparator

Let $x\in\mathbb R_+^n$ be sectoral output losses at one fixed horizon. The relaxed economy obeys $x^0=z+Ax^0$, where $A\ge0$, $\rho(A)<1$ and $z\ge0$. The direct loss $z$ retains a financing-price or expenditure channel even when collateral constraints are relaxed. In the constrained economy an aggregate collateral loss $v^{\mathsf T}x$ induces additional contractions $u(v^{\mathsf T}x)$, with $u,v\ge0$: \[ x=z+Ax+uv^{\mathsf T}x. \] All coefficients are derivatives of a declared local equilibrium mapping, in consistent loss units; they need not equal raw production input shares. For nonnegative aggregation weights $w$, write $R=(I-A)^{-1}$ and assume $w^{\mathsf T}Rz>0$. Put $\theta=v^{\mathsf T}Ru$.
Theorem (Feedback threshold and amplification): If $\theta<1$, then $\rho(A+uv^{\mathsf T})<1$ and \[ x=Rz+{Ru\,v^{\mathsf T}Rz\over1-\theta},\qquad \Gamma={w^{\mathsf T}x\over w^{\mathsf T}x^0} =1+{(w^{\mathsf T}Ru)(v^{\mathsf T}Rz) \over(1-\theta)w^{\mathsf T}Rz}\ge1. \tag{1} \] If $\theta\ge1$ and $v^{\mathsf T}Rz>0$, no finite nonnegative loss vector solves the constrained equation.
Proof: The Neumann series gives $R\ge0$. Multiplying the equilibrium equation by $R$ yields $x=Rz+Ru(v^{\mathsf T}x)$; multiplying by $v^{\mathsf T}$ gives $(1-\theta)v^{\mathsf T}x=v^{\mathsf T}Rz$. This proves the formula when $\theta<1$, and the impossibility assertion when $\theta\ge1$. To prove stability, the same calculation gives $(I-A-uv^{\mathsf T})^{-1}=R+Ru v^{\mathsf T}R/(1-\theta)\ge0$. Let $y$ be this inverse applied to the all-ones vector. Then $y\ge R\mathbf1 \ge\mathbf1$ and $(A+uv^{\mathsf T})y=y-\mathbf1$. In the weighted sup norm with weights $y_i$, the nonnegative matrix's operator norm is at most $\max_i(1-1/y_i)<1$. Hence its spectral radius is below one. Formula (1) and its sign follow by aggregation. ∎

This threshold is a local linear-system condition, not a global crisis threshold after bankruptcies or production substitution alter coefficients. If the relaxed shock response is zero, the ratio is undefined. The same formula without division still supplies the loss difference $w^{\mathsf T}Ru\,v^{\mathsf T}Rz/(1-\theta)$.

An exact edge intervention

For any nonnegative stable response matrix $B$, let $Q=(I-B)^{-1}$. Increase edge $(i,j)$ by $\delta\ge0$, and suppose $\delta Q_{ji}<1$. The edge may represent a financing or production-response link; its meaning must be fixed before interpreting the intervention.
Proposition: The perturbed matrix is stable and its aggregate-loss increase is \[ w^{\mathsf T}\{(I-B-\delta e_ie_j^{\mathsf T})^{-1}-(I-B)^{-1}\}z ={\delta(w^{\mathsf T}Qe_i)(e_j^{\mathsf T}Qz)\over1-\delta Q_{ji}}. \] In particular the local derivative at zero is the product of upstream aggregate sensitivity and downstream shock exposure, not a function of the edge weight alone.
Proof: Apply the preceding theorem with $A=B$, $u=\delta e_i$, $v=e_j$. The denominator condition is exactly its stability condition. Substitute into the rank-one inverse formula, aggregate, and differentiate the scalar rational function at zero. ∎

The formula is suitable for a finite edge modification only while the other coefficients remain invariant as specified. A policy that changes exchange rates, lender balance sheets or feasible substitution may change both $B$ and $z$, requiring a new equilibrium calculation.

The relaxed response is not observed from total losses

Theorem (An unbounded observational-equivalence family): Observing the full sectoral response $x=(1,1)^{\mathsf T}$ to one normalized financing intervention does not identify amplification. For any $b\in[0,1)$, let \[ B_b=\frac b2\begin{pmatrix}1&1\\1&1\end{pmatrix},\qquad z_b=(1-b)(1,1)^{\mathsf T}. \] The constrained response solves $x=z_b+B_bx$ for every $b$. With relaxed response matrix zero, the same model's relaxed direct response is $x_b^0=z_b$. For $w=(1/2,1/2)^{\mathsf T}$, the compatible amplification is $\Gamma_b=1/(1-b)$, ranging over $[1,\infty)$.
Proof: The eigenvalues of $B_b$ are $b,0$, so all these models are stable. This belongs to the earlier rank-one feedback class with $A=0$, $u=(b/2)(1,1)^{\mathsf T}$ and $v=(1,1)^{\mathsf T}$. Substitution verifies the identical observed sectoral response. Removing the specified network-feedback equations leaves the nonzero direct channel $z_b$, whose aggregate response is $1-b$; the constrained aggregate response is one. The ratio follows and diverges as $b\uparrow1$. All models can also share an arbitrary common baseline output; the example compares local response slopes around it. The direct shock loading is an unobserved structural object allowed to vary across these models. ∎

Remaining gap

Multiple excluded financing instruments with known loadings, or independent measurement of network response coefficients, can rule out parts of the family. They are substantive identifying restrictions, not supplied by a single observed sudden stop. Nonlinear default, currency mismatch, endogenous exchange rates and a justified relaxed-equilibrium intervention remain outside the local theorem. The note therefore derives conditional amplification and an identification obstruction while retaining unresolved status for F32-247.

Completion Estimate

35%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt derives a stable rank-one amplification formula and edge-intervention sensitivity, then proves that observed sectoral losses can leave amplification unboundedly unidentified. Nonlinear default, currency mismatch, endogenous exchange rates and a structurally justified relaxed-financing counterfactual remain outside the local benchmark.

References

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