we-solve (beta)

Econ-271 · economics · Economics

Global-value-chain shock propagation

LLM status Open unverified

Statement

Atlas ID: T4; original number: 84; field: International trade. Classification: Editorial research formulation (theoretical/empirical); not a certified unsolved theorem.

Definitions and assumptions

Let firms $i=1,\ldots,n$ trade intermediate goods at dates $t=0,\ldots,H$. Production satisfies $y_{it}\le F_i(k_{it},\ell_{it},x_{1it},\ldots,x_{nit};z_{it})$, where $x_{jit}\ge0$ is input from supplier $j$, $z_{it}$ is productivity and $F_i$ is a specified monotone technology. Inventories obey feasible stock-flow accounting; establishing a supplier relationship costs resources and takes a stated time. Firms choose sourcing, production, storage and investment to maximize finite expected discounted profits. Household demand, goods clearing, network formation and a specified equilibrium selection jointly determine quantities and prices.

Formal research question

For an exogenous disruption $z^1$ and a common baseline $z^0$, define aggregate value-added loss by \[ L_H(\theta,z^1)=\sum_{t=0}^{H}\delta^t [\mathrm{VA}_t(e_\theta(z^0))-\mathrm{VA}_t(e_\theta(z^1))],\quad 0<\delta\le1, \] using a common valuation convention and subtracting intermediate purchases to avoid double counting. Identify the technologies, inventory policies and supplier-switching costs needed to estimate the distribution of $L_H$ and its upper tail. Characterize conditions under which linear input-output propagation approximates the nonlinear equilibrium, and bound its approximation error as shock size, substitutability and available inventories vary.

Interpretation and scope

Fixed input-output coefficients are a benchmark rather than a description of all production. Bottlenecks may imply nondifferentiable responses, so a local derivative need not represent large-disruption losses. Domestic losses and world losses differ when activity moves internationally. A causal disruption design must exclude simultaneously changing final demand or explicitly model it. Network observation should cover exits, missing transactions and supplier substitution; otherwise apparent recovery may be sample attrition. For tail prediction, specify a shock distribution and report sensitivity to poorly observed dependence across suppliers. Welfare requires utility and incidence beyond the value-added loss statistic. Inventory dynamics must include depreciation and delivery lags, and intermediate purchases should be recorded at consistently defined transfer prices before comparing aggregate paths.

Source audit and status

[S1] and [S3], including linked source [43], concern production-sharing and value-added accounting; [S2] concerns organization and contracts. These sources do not establish a complete nonlinear shock-propagation solution with inventories and rewiring. This entry is an editorial extension, and the citations identify relevant foundations rather than certify a currently open theorem.

References

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  1. Robert C. Johnson and Guillermo Noguera (2012). Accounting for Intermediates: Production Sharing and Trade in Value Added. Journal of International Economics 86(2), 224–236. https://www.sciencedirect.com/science/article/pii/S002219961100122X. Locator and role: Publisher or author abstract / bibliographic record; Value-added rather than gross-trade accounting. Provides background or a benchmark result; does not establish that the editorial question remains unresolved in 2026.

  2. Pol Antràs and Davin Chor (2013). Organizing the Global Value Chain. Econometrica 81(6), 2127–2204. https://doi.org/10.3982/ECTA10813. Locator and role: Publisher or author abstract / bibliographic record; Contracting and organization along sequential production chains. Provides background or a benchmark result; does not establish that the editorial question remains unresolved in 2026.

  3. Zhi Wang, Shang-Jin Wei, and Kunfu Zhu (2013). Quantifying International Production Sharing at the Bilateral and Sector Levels. NBER Working Paper 19677. https://doi.org/10.3386/w19677. Locator and role: Publisher or author abstract / bibliographic record; Bilateral and sectoral production-sharing decompositions; resolves link [43]. Provides background or a benchmark result; does not establish that the editorial question remains unresolved in 2026.

Common conventions: Source mathematical conventions

Unless an entry states otherwise, agent and firm sets are finite. State and action spaces are measurable, strategies and outcome maps are measurable, probability laws are specified, and expectations exist for the targets under discussion. Infinite-horizon discount factors lie in $(0,1)$ and discounted objectives are finite. Norms, tolerances and complexity input sizes must be specified for computational comparisons. Constants or primitives that appear locally are inputs, not empirical facts asserted by this catalogue.

Identification. A statistical model $m\in\mathcal M$ implies an observable law $P_m$ and target $\theta(m)$. At law $P$, its identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Point identification holds when this set is a singleton, or equivalently when $P_{m_1}=P_{m_2}$ implies $\theta(m_1)=\theta(m_2)$ for all compatible models. Sharp bounds mean bounds attained, or approached when the boundary is not attained, by compatible models. Writing this set is a definition, not a solution: the substantive task is to establish informative restrictions and derive or estimate the set. An unrestricted-confounding model may give only trivial bounds.

Inference. Identification, estimability and valid uncertainty quantification are separate questions. Fix $\alpha\in(0,1)$. A confidence procedure $C_n$ over a declared law class $\mathcal P$ has asymptotic uniform coverage at level $1-\alpha$ when \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P}\Pr_P\{\theta(P)\in C_n\}\geq1-\alpha. \] For set-identified targets the coverage event must be defined separately, for example coverage of the full identified set. If an entry uses identification-indexed models instead of uniquely indexed laws, the same coverage convention is applied over those models. Pointwise limits do not establish uniform validity, and asymptotic validity does not guarantee finite-sample precision. Sample sizes, dependence structures and weak-identification sequences are part of each application.

Counterfactuals. $Y_i(a)$ is a potential outcome under a specified intervention, including its timing, implementation and versions. It is not $Y_i$ conditional on voluntarily choosing $a$. A full intervention vector permits interference; shorthand scalar treatment notation does not silently impose no interference. Assignment, selection, attrition, anticipation and measurement must be specified. A claim about an instrument's local effect is not automatically a population policy effect.

Economic equilibrium. A counterfactual model includes best responses, constraints, feasibility, market clearing, state transitions and expectations consistent with the stated information. When several equilibria exist, either a declared selector is an input or the target is set-valued. Comparisons across policy regimes must use the stated selection convention. Reduced-form relationships are not presumed invariant after an equilibrium-changing intervention.

Optimization and welfare. Optimization is over the stated feasible set. If attainment is not established, $\sup$ or $\inf$ and $\varepsilon$-optimal choices replace an asserted maximizer or minimizer. For example, with value $V=\sup_{a\in\mathcal A}W(a)<\infty$, an $\varepsilon$-optimal set is $\{a\in\mathcal A:W(a)\geq V-\varepsilon\}$ for $\varepsilon>0$. Benefits, costs, risk and health or environmental outcomes can be aggregated only using declared units and conversion weights. Interpersonal utility weights, the treatment of fiscal transfers and foreign profits, discounting, and the behavioral welfare criterion are explicit inputs. A comparative-static derivative additionally requires existence and appropriate differentiability of the selected counterfactual.

Sources. Local labels [S1], [S2], and so forth restart in every question. Locators identify the relevant abstract, model, section or result at the level actually checked; a bibliographic or abstract-level verification is not represented as inspection of an entire proof. Working papers are distinguished from later publications and changing versions. Source summaries are paraphrased. All URL checks and editorial formulations refer to the audit date stated above.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-271-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-271

A controlled nonlinear input--output approximation and its inventory boundary

Catalogue problem: F12-267. Status: Unresolved. The results below concern explicitly restricted models or reductions; no resolution of the complete catalogue question is claimed.

Question and scope

The catalogue asks for shock propagation through production networks with substitution, inventories, delivery lags and endogenous links, with losses measured in value added. We solve a frictionless, fixed-link CES benchmark and bound the error of its linear input--output approximation. We then show explicitly why inventory slack cannot be omitted from a uniform extension. Johnson and Noguera [jn] motivate value-added accounting; Antràs and Chor [ac] study an organizational margin that is held fixed here. Neither source is asserted to prove the results below.

A closed fixed-network economy

There are $N$ competitive constant-returns sectors and one representative household with labor endowment $L>0$. Labor is freely mobile and is the only primary factor; set its wage to one. Sector $i$ has primary cost share $a_i\in(0,1)$ and CES intermediate weights $b_{ij}\geq0$, $\sum_jb_{ij}=1$. An absent link has $b_{ij}=0$. Let $r_i=1-\sigma_i<1$, with elasticity $\sigma_i>0$. The calibrated unit-cost function is \begin{equation}\label{wsref-0-65-71-3a-63-6f-73-74} c_i(p,w;\varepsilon_i)=e^{\varepsilon_i}w^{a_i} \left(\sum_jb_{ij}p_j^{r_i}\right)^{(1-a_i)/r_i}. \end{equation} At $r_i=0$, use $e^{\varepsilon_i}w^{a_i}\prod_jp_j^{(1-a_i)b_{ij}}$. This is the dual cost of a Cobb--Douglas primary/intermediate composite with a CES intermediate technology, normalized to unit cost one at $p=w=1$, $\varepsilon=0$. A positive $\varepsilon_i$ is an adverse productivity shock. All active links adjust without setup costs or delivery lags. There are no inventories.

The household maximizes $\prod_i(C_i/\theta_i)^{\theta_i}$, with $\theta_i>0$, $\sum_i\theta_i=1$, and expenditure $L$. Firms earn zero profits. All sectors may be interpreted as nodes of a global economy with one integrated labor market; this restriction deliberately omits international wage differences and national welfare incidence.

Write $x_i=\log p_i$ and \[ F_i(x)=\frac{1-a_i}{r_i}\log\left(\sum_jb_{ij}e^{r_ix_j}\right), \quad \rho=\max_i(1-a_i)<1, \quad M=\max_i(1-a_i)|r_i|. \] Again $F_i(x)=(1-a_i)\sum_jb_{ij}x_j$ when $r_i=0$.

Lemma (Equilibrium closure): For every finite shock vector $\varepsilon\in{\mathbb{R}}^N$, the equation \[ x=\varepsilon+F(x) \] has a unique solution. Define the nonnegative matrix \[ B_{ij}(x)=\frac{\partial F_i(x)}{\partial x_j} =(1-a_i)\frac{b_{ij}e^{r_ix_j}}{\sum_kb_{ik}e^{r_ix_k}}. \] Its row sums are $1-a_i$. There is a competitive allocation with sector revenues \[ v=(I-B(x)^T)^{-1}\theta L, \] outputs $Q_i=v_i/p_i$, intermediate quantities $X_{ij}=B_{ij}(x)v_i/p_j$ used by sector $i$ from sector $j$, labor inputs $a_iv_i$, and final quantities $C_i=\theta_iL/p_i$. It clears goods and labor markets.
Proof: Each derivative row is nonnegative with sum at most $\rho$, so the mean value formula implies $\|F(x)-F(y)\|_\infty\leq\rho\|x-y\|_\infty$. Iteration from any finite vector is Cauchy: successive differences are bounded by a geometric sequence. Its limit solves the equation by continuity; contraction also proves uniqueness.

The spectral radius of $B$ is at most $\rho$, so the inverse defining $v$ is the convergent nonnegative series $\sum_{k\geq0}(B^T)^k$. It solves $v=\theta L+B^Tv$. Differentiating the homogeneous unit cost in (eq:cost) gives the stated conditional factor demands per unit of revenue: intermediate expenditure shares are $B_{ij}$ and labor share is $a_i$. These cost-minimizing inputs produce $Q_i$ at cost $p_iQ_i=v_i$, because the price equation is $p_i=c_i(p,1;\varepsilon_i)$. Goods clearing follows by dividing each coordinate of $v=\theta L+B^Tv$ by its price. Finally \[ \sum_i a_iv_i=\boldsymbol{1}^T(I-B^T)v =\boldsymbol{1}^T\theta L=L, \] so the household's labor endowment is exhausted and its expenditure equals income. Cobb--Douglas demand is exactly $C_i$. Thus prices, optimal choices and resource clearing are all specified. ∎

A second-order error bound

The baseline input-share matrix is $A_{ij}=(1-a_i)b_{ij}$. Define the linear log-price approximation \[ x^{\mathrm{lin}}=(I-A)^{-1}\varepsilon. \] This is a price version of the Leontief propagation operator; it should not be confused with summing gross output changes as aggregate losses.
Theorem (Uniform nonlinear approximation): For every $\varepsilon\in{\mathbb{R}}^N$, \[ \|x\|_\infty\leq\frac{\|\varepsilon\|_\infty}{1-\rho},\qquad \|x-x^{\mathrm{lin}}\|_\infty \leq\frac{M\|\varepsilon\|_\infty^2}{2(1-\rho)^3}. \] If all $r_i=0$, the log-price approximation is exact for shocks of any finite size.
Proof: Since $F(0)=0$, the equilibrium equation and contraction bound imply $\|x\|_\infty\leq\|\varepsilon\|_\infty+\rho\|x\|_\infty$. For a direction $h$, the second directional derivative is \[ D^2F_i(y)[h,h]=(1-a_i)r_i{\operatorname{Var}}_{\pi_i(y)}(h_j), \] where $\pi_{ij}(y)=b_{ij}e^{r_iy_j}/\sum_kb_{ik}e^{r_iy_k}$. A random variable in $[-\|h\|_\infty,\|h\|_\infty]$ has variance at most $\|h\|_\infty^2$: subtract its interval midpoint and use variance no greater than mean squared distance from that midpoint. Hence $|D^2F_i(y)[h,h]|\leq M\|h\|_\infty^2$. Taylor's formula with its integral remainder gives \[ F(x)=Ax+R(x),\qquad \|R(x)\|_\infty\leq M\|x\|_\infty^2/2. \] Subtracting the linear equation gives $x-x^{\mathrm{lin}}=(I-A)^{-1}R(x)$. Since $\|A\|_\infty\leq\rho$, the inverse has norm at most $(1-\rho)^{-1}$. Substitute the first bound. When every $r_i=0$, $F$ is linear and the remainder is identically zero. ∎

The bound grows as the primary-factor share falls and as substitution departs from Cobb--Douglas. It is an explicit sufficient error guarantee, not a claim that this dependence is minimax optimal.

Value added, welfare and tail guarantees

At baseline every commodity price is one. Define fixed-baseline-price aggregate value added by \[ \mathrm{VA}(\varepsilon)=\sum_iQ_i-\sum_{i,j}X_{ij}. \] Intermediate outputs and uses cancel through goods clearing, so \begin{equation}\label{wsref-0-65-71-3a-76-61} \mathrm{VA}(\varepsilon)=\sum_iC_i=L\sum_i\theta_i e^{-x_i}. \end{equation} This differs from current-price nominal value added $\sum_i a_iv_i=L$, and from the household consumption index $L e^{-\theta^Tx}$. All three have now been distinguished.
Corollary (Loss in the catalogue's accounting convention): If $\varepsilon\geq0$ coordinatewise, let \[ \ell(\varepsilon)=L-\mathrm{VA}(\varepsilon),\qquad \ell^{\mathrm{lin}}(\varepsilon)=L\theta^T(I-A)^{-1}\varepsilon. \] Then \[ \left|\ell(\varepsilon)-\ell^{\mathrm{lin}}(\varepsilon)\right| \leq KL\|\varepsilon\|_\infty^2, \quad K=\frac12\left\{\frac{M}{(1-\rho)^3} +\frac1{(1-\rho)^2}\right\}. \] The same bound holds for the loss in the household consumption index relative to the same linear approximation.
Proof: Starting the fixed-point iteration at zero gives nonnegative iterates, since $F$ is coordinatewise increasing and $F(0)=0$. Hence $x\geq0$. For $z\geq0$, Taylor's formula gives $|1-e^{-z}-z|\leq z^2/2$. Apply this coordinatewise in (eq:va), average with the nonnegative weights $\theta$, and add the error from Theorem Theorem (Uniform nonlinear approximation). For the consumption index use the same scalar inequality at $z=\theta^Tx$, bounded by $\|x\|_\infty$. ∎
Proposition (Finite-horizon and distributional implication): Consider independent-in-technology dates $t=0,\ldots,H$ with the same primitives and arbitrary jointly distributed nonnegative shocks satisfying $\|\varepsilon_t\|_\infty\leq\eta_t$ almost surely. There is no intertemporal storage or saving; each date has the equilibrium above. For discount factor $\delta\in[0,1]$, put $Z=\sum_t\delta^t\ell(\varepsilon_t)$ and $Z_0=\sum_t\delta^t\ell^{\mathrm{lin}}(\varepsilon_t)$. Then \[ |Z-Z_0|\leq D:=KL\sum_{t=0}^H\delta^t\eta_t^2 \quad\hbox{almost surely}. \] Their distribution functions satisfy $F_{Z_0}(z-D)\leq F_Z(z)\leq F_{Z_0}(z+D)$. For every $u\in(0,1)$ their generalized $u$-quantiles differ by at most $D$.
Proof: Sum the pathwise bounds in Corollary Corollary (Loss in the catalogue's accounting convention). The events $\{Z_0\leq z-D\}\subseteq\{Z\leq z\}\subseteq\{Z_0\leq z+D\}$ imply the distribution inequalities. The pointwise inequalities $Z_0-D\leq Z\leq Z_0+D$ imply the same ordering of generalized quantiles, by their definition $\inf\{z:F(z)\geq u\}$, proving the final assertion. Temporal independence of shocks is unnecessary. ∎

An inventory obstruction to a uniform smooth extension

Proposition (Vanishing buffer slack): Consider one date, a final producer with capacity and demand one, a one-for-one intermediate requirement, initial usable inventory $I\in(0,1/2)$, and an exogenous delivery $1-s$, $s\in[0,1]$. Delivery and inventory costs are sunk; final output has positive value and unused terminal inventory has zero value. The profit-maximizing output loss is \[ g_I(s)=(s-I)_+. \] Its first-order approximation around $s=0$ is zero. No constant $C$ independent of $I$ can satisfy $|g_I(s)-g_I(0)-g_I'(0)s|\leq Cs^2$ for all such $I,s$.
Proof: Available input is $1-s+I$, so optimal output is $\min\{1,1-s+I\}$; the positive output value makes this capacity feasible maximum optimal. Loss is therefore the displayed positive part. It vanishes in a neighborhood of zero, giving derivative zero. At $s=2I$, loss is $I$, so the proposed bound would require $C\geq1/(4I)$ for every $I>0$, which is impossible. ∎

This example specifies its terminal condition rather than assigning an unpriced value to remaining stock. It shows why a useful error bound with inventories must control distance to depletion thresholds or use a piecewise approximation. The fixed-link theorem does not solve endogenous relationship formation, lags, capacity rationing, inventory investment or correlated firm failure. Identifying those primitives and their equilibrium selection remains the main empirical and theoretical gap in the original question.

Completion Estimate

45%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt closes a fixed-network CES equilibrium, bounds nonlinear input-output and value-added approximation error and transfers those bounds to loss tails. Identifying losses with inventory dynamics, delivery lags, capacity rationing and endogenous supplier rewiring still requires a substantially richer equilibrium model and controls near depletion thresholds.

References

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