Supply-chain resilience versus efficiency
LLM status Open unverified
L23 — Organization of Production Economics catalogue openaigpt 6 astra ultra 25% cataloguednot recordedOpenStatement
Original subject group: Trade, globalization and geoeconomics
Audit classification: Published research agenda. The official supply-chain risk priority is verified. The supplier/inventory robust optimization and private-social comparison are editorial, with frontier and dependence assumptions now defined.
Definitions and precise research target
Consider firms choosing supplier shares $w$, inventories $b$ and substitution capabilities $s$ before uncertain input disruptions. Feasible configurations belong to $A$, defined by production technology, working-capital and contracting constraints. A disruption state $\omega$ specifies supplier availability and input costs; delivered output and consumption follow from equilibrium production and price responses. Let $C(a)$ be normal resource costs and $L(a,\omega)$ disruption losses for configuration $a=(w,b,s)$. For a publicly specified set $\mathcal P$ of shock distributions, seek \[J(a)=C(a)+\sup_{P\in\mathcal P}E_P[L(a,\omega)],\qquad J_*=\inf_{a\in A}J(a).\] The task is to identify the efficiency-resilience frontier and whether private firm choices attain the social optimum when shortages spill through production networks. Estimate switching costs, substitutability and correlated supplier risks rather than assuming suppliers fail independently. Compare diversification, inventory and domestic sourcing using the same loss convention. Include endogenous supplier prices and entry when diversification is adopted widely. Report how optimal decisions vary with upstream position and shock severity, and give bounds where rare-event probabilities are poorly measured. Domestic sourcing is not defined as resilient by assumption; concentrated domestic supply can face common shocks.Additional formal definitions and scope (editorial). Supplier shares satisfy $w_i\geq0$, $\sum_iw_i=1$; buffers $b_i\geq0$ have explicit storage, depreciation and financing costs; substitution capability $s$ has an investment cost and an identified production role. Let $\omega$ specify jointly dependent supplier disruptions and delivery fractions. Output loss is measured against the same normal-output or consumption baseline; include shortage externalities only once. The social cost objective includes normal costs and expected disruption loss, whereas individual firms minimize their own cost under market prices. Define the Pareto frontier in $(C(a),\sup_{P\in\mathcal P}E_PL(a,\omega))$ by eliminating configurations weakly worse in both coordinates and strictly worse in one. Compact feasibility and lower semicontinuous finite losses yield a standard existence result, leaving calibration, externality identification and comparative design as research targets. For a nonempty feasible set and a finite optimal value, the design target is an $\varepsilon$-optimal feasible rule for each specified $\varepsilon>0$: its cost is at most the infimum plus $\varepsilon$, or its welfare is at least the supremum minus $\varepsilon$. Report infeasibility or an unbounded value separately. An exact optimizer is requested only under additional attainment assumptions; it need not exist for the general rule class.
Variants and assumption changes
- Known independent disruptions (editorial). Restrict to a given production technology and independent supplier failures; compare inventory and diversification costs.
- Correlated network disruptions (editorial). Permit shared geopolitical/natural risks and equilibrium supplier entry/prices; characterize private versus social cost frontiers over an uncertainty set.
References and source audit
- Official January 9, 2026 web version; locator: Interconnected World / Reconfigured trade; 2026 supply-chain priority. Broad agenda; exact model editorial. URL: https://www.bankofengland.co.uk/research/bank-of-england-agenda-for-research.
- Bank of England. 2026. Bank of England Agenda for Research, 2025–2028. Interconnected World / Reconfigured trade; 2026 supply-chain priority. https://www.bankofengland.co.uk/research/bank-of-england-agenda-for-research.
Integrated refinement: ECON-080
Supply-chain resilience versus specialization gainsThis refinement adopts the additional source conventions (audited:conventions), subject to its local restrictions. Tail-risk versus ambiguity-based resilience. For feasible supplier shares and a specified correlated disruption law, consider the additional criterion \[ J(x)=\mathbb E[C_r(x,\omega)+L_r(x,\omega)] +\lambda\operatorname{CVaR}_\alpha(L_r),\qquad \operatorname{CVaR}_\alpha(L)=\inf_{a\in\mathbb R} \left\{a+\frac{\mathbb E[(L-a)_+]}{1-\alpha}\right\}. \] This definition handles atoms at the quantile. Require integrable loss, $\alpha\in(0,1)$ and $\lambda\geq0$; fix common shock scenarios, costs and output valuation. The tail-risk penalty is a preference criterion, not another expected physical loss. Compare it with the original worst-case ambiguity, expected-loss and reliability constraints. Supplier capacity, inventories, rerouting, substitution and timing constrain feasible diversification. Distinguish private sourcing losses from downstream and distributional social losses.
Supplied source audit for ECON-080
Local [S1], [S2], etc. in this audit and the references immediately below belong to ECON-080, independently of the original entry's reference numbering. [S1] and [S2] support supply-chain reorientation and resilience. This CVaR/network optimization is an editorial extension, not an exact open question stated by the trade review.References for integrated refinement ECON-080
[S1] David Atkin, Amit Khandelwal, Laura Boudreau, Rafael Dix-Carneiro, Isabela Manelici, Pamela Medina, Brian McCaig, Ameet Morjaria, Luigi Pascali, Heitor Pellegrina, Bob Rijkers, and Meredith Startz (2025). International Trade. VoxDevLit 4(2), February 2025.
Verified locator / source role: Primary publisher page checked for review identity, authors, version and background. The specific published agenda is corroborated in [S2]; the exact formal target is editorial.
https://voxdev.org/voxdevlit/international-trade
[S2] Oliver Hanney / VoxDev (living compilation). Evidence gaps: Key unanswered questions in development economics. Accessed 8 October 2026. The page currently covers 20 topics; the "16-topics URL is a historical slug.
Verified locator / source role: Topic heading: International Trade. Supports the broad research agenda; this is not a verbatim quotation or an attribution of the displayed mathematical target.
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.
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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.
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Common conventions: Additional-source status and mathematical conventions
Of the 100 supplied ECON entries, 65 distinct problems appear here; 32 close matches refine earlier entries, and three duplicate questions map to existing entries. Appendix app:audited-crosswalk lists every destination. The attachment's P/R/S/U distinctions and source audits are retained. Its precise mathematical formalizations are editorial unless the individual audit says otherwise. Candidate subsidy targets ECON-004--006 retain their U flags; the resolved approval-core question ECON-007 updates OP-0202 above. The following supplied conventions govern both this part and the attributed ECON refinements elsewhere in the catalogue, subject to local restrictions.
Parameterized questions. A problem family lists inputs that must be fixed before an instance is evaluated: population, horizon, policies, information, data law, model class and welfare weights. These are required choices, not quantities the citations are assumed to specify. Undefined applications should not be treated as identified estimands.
Indices and integrability. Sets of agents, firms and regions are finite unless a population measure is explicitly used. $i,j$ index units, $r$ regions, $t$ dates and $h$ horizons. $\mathbb E$ and $\Pr$ refer to the declared population and law; $\mathbf1\{A\}$ is an indicator. Every displayed expectation and discounted sum needs existence in the stated domain. Infinite horizons use a discount in $(0,1)$ and a convergence condition; finite horizons may use discount one.
Optimization and existence. A displayed $\max$, $\min$, or $\arg\max$ is conditional on attainment. For finite feasible menus this is immediate; general spaces need continuity and compactness/coercivity or another existence argument. Without attainment, the target is the supremum/infimum and arbitrarily near-optimal feasible choices. $\inf\varnothing=+\infty$. Empty feasibility and an unidentified target are different issues.
Potential outcomes. $Y_i(z)$ is the outcome under a specified intervention, not the observed conditional mean among people choosing $z$. In binary comparisons $\Delta Y_i=Y_i(1)-Y_i(0)$. Specify assignment, treatment versions, compliance, information and observation horizon. Interference is allowed under a full policy vector; compressed exposure notation needs a justified mapping. No interference, consistency and overlap are substantive assumptions, not automatic consequences of notation.
Populations and observation. Fix baseline eligibility and population weights. Conditioning on post-policy employment, survival, relocation or program uptake can change the population being compared. Death is not ordinary loss to follow-up; outcomes truncated by death need a composite convention, joint survival target, or explicitly defined survivor stratum. Fertility-changing interventions need a population functional rather than an unexplained fixed descendant cohort. Measurement and missingness models must be supplied.
Identification. A structural model $m\in\mathcal M$ implies observable law $P_m$ and target $\theta(m)$. Identification means \[ P_{m_1}=P_{m_2}\ \Longrightarrow\ \theta(m_1)=\theta(m_2) \quad\text{for all }m_1,m_2\in\mathcal M . \] Without identification, the target is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Writing $\theta(P)$ before establishing identification can hide incompatible latent models. Confidence sets additionally need a sampling law, confidence level, asymptotic sequence and uniformity class. Point identification, coverage and informative precision are separate properties.
Mediation. Total effects of randomized assignment are distinct from cross-world natural indirect effects. Belief/effort-channel effects require a meaningful mediator intervention and additional measurement, exclusion or mediation assumptions. Randomization of the initial policy alone does not supply those assumptions. Report bounds or interventional alternatives when the stronger target is unsupported.
Equilibrium. A counterfactual $W(z)$, $p(z)$, or $\mu_t^z$ requires individual optimization, feasibility, government/resource budgets, market clearing and state-transition laws. State equilibrium existence and selection, or report ranges over compatible equilibria. Initial-state integration includes subsequent endogenous transitions; current-state integration must use the policy-dependent distribution. Reduced-form invariance after policy is not assumed.
Welfare accounts. Scores, utility, health, dollars and ecological quantities require explicit conversion before addition. Fees, taxes, subsidies, wages and extortion payments are transfers between parties; distributional weights can make them welfare relevant, but their face value is not automatically a resource loss. State ownership, geographic boundaries, foreign-income weights and fiscal finance. Do not add profits to household welfare already including dividends, or count land capitalization and the underlying benefit twice. A benefit-cost expression must distinguish gross benefits from net welfare and subtract every real cost exactly once.
Derivatives and risk. Log derivatives require positive arguments; local responses require admissible perturbations and specified equilibrium branches. Differentiation under expectations needs regularity/domination, and boundaries may allow only one-sided derivatives. Risk uses a specified law; ambiguity uses a declared nonempty law set on a common history space. Adaptive policies must be nonanticipative. A robust criterion is an editorial choice unless attributed explicitly.
Scope overlaps. Allocation, collective choice, contracts, household bargaining and coordinated policy overlap economics with optimization and game theory. They are retained because they appeared in the original catalogue; no claim of a strictly game-theory-free selection is made.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
GPT 6 Astra Ultra attempt on Econ-123
Dependence-robust sourcing and exact inventory thresholds
Catalogue problem:L23-120. Status: Unresolved. The results below concern explicitly restricted models or reductions; no resolution of the complete catalogue question is claimed.Abstract
With two substitutable suppliers, binary failures, and known marginal failure probabilities, worst-case expected shortage is attained at maximal positive dependence. The result holds jointly for expected loss and a CVaR penalty. Equal-risk suppliers yield exact diversification and inventory thresholds, including a regime where robust diversification offers no protection. The model also gives a direct private--social comparison when shortages have downstream costs.A restricted sourcing economy
The catalogue asks for an efficiency--resilience frontier under supplier dependence, inventories, substitution costs, and network spillovers. The official agenda emphasizes supply-chain risks and their macroeconomic consequences [BoE]; the integrated trade review supplies broader context [Trade]. Our calculation treats one firm, two perfectly substitutable inputs, exogenous contract prices, and no supplier entry.Normal input demand is one. Shares are $w$ and $1-w$, with $w\in[0,1]$, and buffer stock is $b\in[0,1]$. Supplier $i$ fails completely when $Z_i=1$, with marginal probability $p_i$. Contracts cost $c_1w+c_2(1-w)$ in all states. Carrying the buffer has opportunity, storage, and depreciation cost $hb$, $h>0$. One unit of missing input causes real output loss valued at $v>0$, so \[ L(w,b,Z)=v\{wZ_1+(1-w)Z_2-b\}_+. \tag{1} \] The common output baseline is the same for all configurations. Dependence is unknown. Its only free parameter is \[ q={\mathbb{P}}(Z_1=Z_2=1)\in[\underline q,\overline q] \subseteq[\max(0,p_1+p_2-1),\min(p_1,p_2)]. \tag{2} \] The uncertainty set is nonempty and fixed publicly before choosing $w,b$.
Worst-case dependence
Because $L\ge0$, a minimizing CVaR threshold can be restricted to $a\ge0$: for $a<0$ its objective is ${\mathbb{E}} L/(1-\alpha)-\alpha a/(1-\alpha)$, decreasing as $a$ rises to zero. For $a\ge0$, \[ (L-a)_+=v\{wZ_1+(1-w)Z_2-b-a/v\}_+. \] If $b+a/v\le1$, the lemma applied to this larger buffer shows its expectation is nondecreasing in $q$. If $b+a/v>1$, it is zero. Thus the threshold objective is nondecreasing in $q$ for every admissible $a$, and taking an infimum preserves this order. Both terms of the combined criterion are therefore maximized by $\overline q$. ∎
This assertion uses substitutability and a convex shortage loss. It is not a universal claim that positive dependence is worst for every network production technology.
Exact equal-risk frontier and inventory rules
Take $p_1=p_2=p\in(0,1)$ and $c_1=c_2=c$. Write $q_* =\overline q$. For each fixed buffer, expected loss is a convex symmetric function of $w$: each positive-part term in (3) is convex and its coefficient $p-q_*$ is nonnegative. Thus $w=1/2$ is always a minimizer, possibly among many. At this choice, \[ \frac1v{\mathbb{E}}_{q_*}L= \begin{cases} p-(2p-q_*)b,&0\le b\le1/2,\\ q_*(1-b),&1/2\le b\le1. \end{cases} \tag{4} \]If dependence is unrestricted beyond equal marginals, $q_*=p$ is feasible. Then both suppliers fail together under the worst law, and \[ \sup_q{\mathbb{E}} L=vp(1-b) \] for every share $w$. Diversification has no robust shortage benefit. With different normal costs, optimal sourcing selects a cheapest supplier; an all-domestic supplier has no special protection in this model. The optimal buffer is zero or one according as $h$ is above or below $vp$, with indifference at equality. Thus an assumption of independent failures, which would set $q=p^2$, can change the optimal buffer and can create an apparent diversification benefit absent under the stated ambiguity set.
Private costs, social spillovers, and computation
Suppose the firm internalizes only $v_P$ per unit of shortage while downstream users lose an additional $v_E\ge0$, and the two losses do not overlap. Then the social loss coefficient is $v_S=v_P+v_E$. Applying the buffer theorem with these two values gives exact private and social rules under identical shocks and carrying costs. For instance, when $q_*=p$ and \[ v_Pp<h<v_Sp, \] the firm holds no buffer while the social optimum fully buffers. This is a model-specific shortage externality, not a claim that all inventories are underprovided. Endogenous supplier prices and inventory financing would alter these thresholds.For unequal marginals and unequal costs, (3) still gives a finite, directly solvable design problem. The square $[0,1]^2$ in $(w,b)$ is divided by $b=w$ and $b=1-w$ into finitely many polygons. On each polygon, (3) plus the affine normal costs is affine, so a minimizer occurs at one of its vertices (unless the objective is flat on a face). Enumerating those vertices and evaluating the objective is an exact constant-size algorithm. A convex quadratic adjustment cost in shares can be included by minimizing the resulting quadratic separately on each polygon; it should not be omitted if supplier switching is costly in the application.
Remaining empirical and equilibrium questions
The results are conditional on measured marginal risks and a declared dependence interval. The width of that interval is economically decisive: rare joint disruptions may be nearly absent from a finite historical sample without being implausible. No estimate of the interval or causal effect of reshoring is provided here. Input complementarity, multi-tier propagation, dynamic replenishment, supplier capacity and entry, and aggregate price adjustment remain outside the proof. Those components, and the empirical distinction between real output loss and a tail-risk preference penalty, are required to answer the complete catalogue agenda.References
- [BoE] Bank of England (2026). Bank of England Agenda for Research, 2025--2028. "The Interconnected World," "Reconfigured trade," and its 2026 supply-chain priority. https://www.bankofengland.co.uk/research/bank-of-england-agenda-for-research.
- [Trade] D. Atkin et al. (2025). International Trade. VoxDevLit 4(2), February 2025. This review motivates the trade setting; the two-supplier optimization above is a separately declared model. https://voxdev.org/voxdevlit/international-trade.