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Econ-335 · economics · Economics

Sectoral shock propagation

LLM status Open unverified

Statement

Original subject group: Business cycles and macro dynamics

Audit classification: Background; proposed extension. The cited model already supplies conditional network-amplification results; this broader threshold specification is editorial. Verification concerns the cited version, not first priority or proof that this exact editorial specification remains unsolved on October 8, 2026.

Definitions and precise research target

Editorial mathematical specification.
Consider a finite economy with sectors $i=1,\ldots,n$. Sector $i$ combines labor and intermediate inputs using a constant-returns production function, has productivity $A_i$, and purchases an expenditure share $\omega_{ij}$ from sector $j$. Households have specified preferences and hold debt subject to a collateral constraint whose collateral is a declared function of sectoral profits. Fix the input-share matrix $\Omega=(\omega_{ij})$, initial balance sheets, substitution elasticities, and a rule for monetary and fiscal policy. Assume an equilibrium exists and specify an equilibrium selection if constraints generate multiplicity. For a productivity loss $s>0$ in sector $j$, let $Y_j(s)$ be equilibrium real aggregate output and define \[ R_j(s)=1-\frac{Y_j(s)}{Y_j(0)},\qquad s_j(q)=\inf\{s>0:R_j(s)\ge q\}, \] where $Y_j(0)>0$, and $q\in(0,1)$ is a predetermined recession threshold and the infimum is $+\infty$ if the set is empty. Characterize and quantitatively identify how network structure changes $s_j(q)$ when financial conditions and shock sizes are held fixed. Separate first-order production effects from nonlinear collateral amplification; establish which conclusions survive plausible substitutions and alternative collateral definitions. Validate amplification predictions against independently identified sectoral shocks. Existing sufficient-statistic results for particular models are admissible benchmarks, not evidence that this entire class is unsolved.

Specify the intervention as $A_j(s)=A_j(0)e^{-s}$, $s\ge0$, with all other productivities fixed, and evaluate the same selected equilibrium under each intervention. A network comparison is a pair $(\Omega,\Omega')$ with common primitives and a specified rule replacing missing inputs, not a derivative holding endogenous CES expenditure shares artificially fixed. In a Cobb--Douglas benchmark $y_i=A_i\ell_i^{\alpha_i}\prod_k m_{ik}^{\omega_{ik}}$, impose $\alpha_i>0$, $\omega_{ik}\ge0$, and $\alpha_i+\sum_k\omega_{ik}=1$; specify labor endowment, final demand, and the collateral inequality in real units. The threshold infimum need not be attained or monotone in the network. For multiple equilibria report the thresholds for every admissible selection or a worst-case threshold.

Variants and assumption changes

The following are editorial variants, not additional published open conjectures.
  1. Unconstrained benchmark. Remove borrowing constraints but retain the same technologies and final demand; compare $s_j(q)$ with its constrained counterpart and with the local elasticity $-\partial\log Y_j/\partial s|_{0}$. Local elasticities cannot determine a finite recession threshold without curvature restrictions.
  2. Endogenous substitution. Replace fixed Cobb--Douglas coefficients by CES nests with elasticities in a declared compact set. Determine bounds on the recession threshold when relative prices change shares and collateral values simultaneously.

References and source audit

The cited model already supplies conditional network-amplification results; this broader threshold specification is editorial.
  1. Jorge Miranda-Pinto; Eugenio Rojas; Felipe Saffie; Alvaro Silva. 2026. Connected for Better or Worse? The Role of Production Networks in Financial Crises. Cover and Abstract; Section1, printed pp.1--3; December2025 version of WP26-1. 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-335-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-335

Sectoral shock propagation: a research attempt

Status: unresolved. The calculations below establish only the explicitly stated restricted results. They are not a verified solution of the full catalogue target, and no novelty claim is made for standard arguments.

Research target and benchmark selection

For a log-productivity loss $A_j(s)=A_j(0)e^{-s}$, the target recession threshold is $s_j(q)=\inf\{s>0:1-Y_j(s)/Y_j(0)\geq q\}$, with $q\in(0,1)$. The full question combines input substitution, collateral, equilibrium selection and identification. A useful first step is an exactly soluble general-equilibrium benchmark, followed by a precise account of what its elasticity fails to identify.

The Boston Fed working paper cited in the statement was inspected at its abstract and introduction. It studies a particular collateral/network mechanism and expressly distinguishes production-side GDP effects from consumption amplification. The target here is aggregate real output; consumption losses must not be substituted for it. The calculations below do not purport to extend that paper's theorem to all CES or financing models.

Exact Cobb--Douglas production threshold

Consider competitive constant-returns firms \[ y_i=A_i\ell_i^{\alpha_i}\prod_k m_{ik}^{\omega_{ik}},\qquad \alpha_i>0,\quad\omega_{ik}\geq0,\quad\alpha_i+\sum_k\omega_{ik}=1. \] There is fixed labor supply $L>0$, no borrowing constraint, and a representative household with Cobb--Douglas final consumption weights $f_i\geq0$, $\sum_i f_i=1$. Fix the usual normalization of the final consumption index and its expenditure function $P$. Firms have zero profits, the household owns labor, and all goods and labor markets clear. Positive labor shares imply $\|\Omega\|_\infty<1$, so $G=(I-\Omega)^{-1}=\sum_{r\geq0}\Omega^r$ exists and has nonnegative entries.

Cost minimization gives unit-cost equations \[ \log p=-\log A+\alpha\log w+\Omega\log p+c, \] where $c_i$ is the constant determined by technology exponents. Set the nominal wage to one. Since nominal final expenditure is $L$, real final output is $Y=L/P$; $\log P=\sum_i f_i\log p_i+c_f$.

Proposition: Writing $d_j=f^{\mathsf T}Ge_j$, the shock response and threshold are exactly \[ \frac{Y_j(s)}{Y_j(0)}=e^{-d_js},\qquad s_j(q)=\frac{-\log(1-q)}{d_j}\quad(d_j>0). \] If $d_j=0$, the threshold is $+\infty$.
Proof: Subtract the baseline cost equations to obtain $\log p(s)-\log p(0)=sGe_j$. Therefore $\log P(s)-\log P(0)=d_js$, giving the output ratio. The continuous increasing loss $1-e^{-d_js}$ reaches $q$ at the displayed point. For $d_j=0$ it is identically zero. ∎
This is an exact finite-shock result only because all relevant prices enter logarithmically with fixed exponents. It does not rely on treating a CES expenditure share as fixed after intervention. The coefficient $d_j$ is also a sales-to-final-expenditure weight; it can exceed one because gross sectoral sales include intermediates.

Network comparison and a calculation

For a replacement network $\Omega'$, final weights and labor endowment held fixed, both thresholds are computable by their own inverse matrices. Technology constants and any altered labor shares must be recomputed in each economy; the ratio within each economy still has the displayed formula. It is not an economically feasible intervention to increase every input share while silently leaving the labor shares unchanged.

For two sectors let \[ \Omega=\begin{pmatrix}0&a\\b&0\end{pmatrix},\quad G=\frac1{1-ab}\begin{pmatrix}1&a\\b&1\end{pmatrix},\quad d_1=\frac{f_1+bf_2}{1-ab}. \] When $f_1=f_2=1/2$ and $a=b=1/2$, $d_1=1$. With no intermediate inputs, $d_1=1/2$. A ten-percent recession therefore requires a log productivity shock $-\log0.9$ in the first network and $-2\log0.9$ in the second. These are internally consistent technology comparisons, not estimated crisis effects.

The matrix differential $dG=G(d\Omega)G$ provides local comparative statics. Increasing one coefficient $\omega_{ab}$ gives $\partial d_j/\partial\omega_{ab}=(f^{\mathsf T}G)_aG_{bj}\geq0$, interpreted together with its required labor-share adjustment. A share-preserving reallocation between two suppliers can have either sign, since it is the difference of two such expressions. Network density alone does not order all thresholds.

A finite-threshold obstruction and conditional bounds

For any selected constrained equilibrium, put $z(s)=-\log(Y_j(s)/Y_j(0))$ and $Q=-\log(1-q)$. If $z$ is absolutely continuous, $z(0)=0$, and its derivative obeys $0<\underline d\leq z'(s)\leq\overline d$ up to the relevant crossing, then \[ Q/\overline d\leq s_j(q)\leq Q/\underline d. \] Integrating the derivative bounds proves this claim, provided they hold at least through $Q/\underline d$ and the selected output remains positive. This gives an actionable role to curvature and substitution restrictions. A local elasticity $z'(0)=d$ alone gives no such finite interval: the paths $z_1(s)=ds$ and $z_2(s)=ds+Ks^2$, $K>0$, agree to first order but their crossings can differ arbitrarily as $K$ grows.

To isolate a possible collateral mechanism, assume a reduced-form log-loss relation $z=ds+a(z-h)_+$, where $d>0$, $h\geq0$ is a collateral-loss buffer and $0\leq a<1$. The right side is a contraction in $z$, so the unique solution is $z=ds$ for $ds\leq h$ and $z=(ds-ah)/(1-a)$ otherwise. Consequently \[ s_j(q)=\frac{Q-a(Q-h)_+}{d}. \] The pre-binding elasticity equals $d$ regardless of $a$ whenever $h>0$, although large-shock thresholds decrease as feedback strengthens. This explicitly demonstrates how slack-state observations can miss amplification. The scalar relation is an illustrative closure, not derived here from the canonical collateral budgets; claiming it to be a general-equilibrium result would leave the main step unproved.

Identification and unresolved work

In the frictionless benchmark, known $\Omega$ and final weights identify the threshold, with no separately estimated curvature parameter. In the constrained problem, observationally identical slack-state histories may be compatible with different $a,h$ and thus different thresholds. Exogenous shocks large enough to explore the binding region, collateral valuations and balance-sheet data are required to distinguish such models. A CES extension needs its own equilibrium existence and derivative control, and may have multiple selected branches. The exact benchmark and conditional bounds are the achieved results; identification and robust finite-shock characterization for the original class remain unresolved.

Completion Estimate

44%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt exactly characterizes finite recession thresholds in an unconstrained Cobb-Douglas network and derives conditional derivative bounds and an illustrative collateral-feedback threshold. Full collateral-budget equilibrium derivation, endogenous CES substitution, selection uncertainty, and quantitative identification from independent sectoral shocks remain unresolved.

References

Jorge Miranda-Pinto, Eugenio Rojas, Felipe Saffie and Alvaro Silva, Connected for Better or Worse? The Role of Production Networks in Financial Crises, Boston Fed WP 26-1, December 2025 manuscript version, abstract and introduction inspected: https://www.bostonfed.org/-/media/Documents/Workingpapers/PDF/2026/WP2601.pdf.

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