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Econ-424 · economics · Economics

Reshoring and friend-shoring welfare

LLM status Open unverified

Statement

Atlas ID: T8; original number: 88; field: International trade. Classification: Editorial research formulation (theoretical/empirical); not a certified unsolved theorem.

Definitions and assumptions

A firm selects supplier weights $x\in\Delta^J=\{x\ge0:\sum_jx_j=1\}$, capacity and inventories before stochastic disruptions $Z$ are realized. Model $\theta$ specifies production substitution, investment costs, delivery failures and the distribution $\mathbb P$ of $Z$, including cross-location dependence. Consumers choose feasible consumption, and firms and households determine a selected market equilibrium under sourcing policy $b$. Let $W_\theta(b;\mathbb P)$ be finite expected social utility net of real resilience costs under explicit welfare weights. Tariffs or subsidies must have a stated financing and revenue-recycling rule.

Formal research question

Let $\mathcal B$ be a compact, prespecified class of implementable reshoring or diversification rules and $\mathcal P$ a nonempty set of plausible disruption laws. Determine the robust policy problem \[ \sup_{b\in\mathcal B}\inf_{\mathbb P\in\mathcal P} [W_\theta(b;\mathbb P)-W_\theta(b^0;\mathbb P)], \] or its identified range when technological parameters are not point identified. Establish conditions for existence and characterize when geographic diversification, domestic redundancy or unconstrained sourcing attains the highest welfare. Estimate willingness to pay for reliability separately from market power, subsidies, insurance and firms’ subjective disaster beliefs.

Interpretation and scope

Diversification across strongly correlated suppliers may provide little insurance; reshoring can also concentrate common domestic hazards. A security objective must enter as an explicit payoff or constraint rather than an unexplained welfare gain. Rare disruption probabilities are weakly identified by short panels, so stress-test results depend on the stated law or ambiguity set. Private revealed willingness to pay need not equal social value when shortages spill over to other firms or taxpayers provide support. Comparing policies requires endogenous prices, entry and foreign retaliation. Existence needs continuity, compactness and controlled expectations; these are conditions to prove for a proposed model, not automatic properties of resilience.

Source audit and status

The probable source behind "Grossman–Helpman (2021)" is [S1], whose full authors also include Hugo Lhuillier; its published 2023 title adds "International." The fragment remains only provisionally disambiguated. [S2] documents sourcing changes and [S3] verifies an EU diversification objective. The atlas’s "production-network literature" is not a bibliographic source. These foundations do not certify current open status.

References

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  1. Gene M. Grossman, Elhanan Helpman, and Hugo Lhuillier (2021). Supply Chain Resilience: Should Policy Promote Diversification or Reshoring?. NBER Working Paper 29330; published with International added to the title in Journal of Political Economy 131(12), 3462–3496 (2023). https://www.nber.org/papers/w29330. Locator and role: Publisher or author abstract / bibliographic record; Supply-chain resilience and welfare of diversification versus reshoring. Probable disambiguation: original citation omitted Hugo Lhuillier and a title. The author-year fragment alone does not prove that this was the intended source.

  2. Laura Alfaro and Davin Chor (2023). Global Supply Chains: The Looming “Great Reallocation”. NBER Working Paper 31661. https://doi.org/10.3386/w31661. Locator and role: Publisher or author abstract / bibliographic record; Changing sourcing patterns and limits of direct bilateral trade data. Provides background or a benchmark result; does not establish that the editorial question remains unresolved in 2026.

  3. European Commission (2026). EU Trade Relations with China. Directorate-General for Trade and Economic Security, institutional webpage. https://policy.trade.ec.europa.eu/eu-trade-relationships-country-and-region/countries-and-regions/china_en. Locator and role: Publisher or author abstract / bibliographic record; Stated policy of reducing critical dependencies and diversifying where necessary. Verifies the institutional objective only; supplies no welfare estimate or probability distribution for geopolitical disruption. Accessed 8 October 2026.

  4. Unresolved original citation: "production-network literature." No author, title or year was supplied, so no unique source can be assigned.

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-424-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-424

Robust diversification with uncertain cross-country disruption dependence

Problem ID: F13-420. Status: Unresolved. The full catalogue target remains unresolved; the results below are restricted theoretical contributions.

A restricted welfare problem with an exact solution

The catalogue's target compares each sourcing policy to the same baseline under the same disruption law, then takes the worst law. We solve that problem in a two-supplier subclass. This supplies explicit conditions under which diversification helps or fails; it does not estimate the general equilibrium value of reshoring. The source by Grossman, Helpman and Lhuillier [source] motivates the distinction between private and social resilience incentives. Our quadratic shortage model is a separately specified restriction.

A planner purchases a unit of input capacity, share $x\in[0,1]$ from supplier 1 and $1-x$ from supplier 2, at real unit costs $c_1,c_2\ge0$. Failures $Z_1,Z_2\in\{0,1\}$ destroy the corresponding shares; no post-disruption substitution is possible. Shortage is \[ S(x,Z)=xZ_1+(1-x)Z_2. \] Expected welfare, apart from a constant, is \[ W(x;P)=-c_1x-c_2(1-x)-L{\mathbb{E}}_P S(x,Z)^2, \qquad L>0. \tag{1} \] The welfare coefficient $L$ is a stated social valuation of shortage. It is not inferred from firms' choices unless their objectives coincide with (1). Policy can implement any $x$ directly; taxes, market power and endogenous investment are excluded here.

The marginal failure probabilities $p_1,p_2\in[0,1]$ are known, while the joint failure probability $q={\mathbb{P}}(Z_1=Z_2=1)$ lies in a declared nonempty interval \[ q\in[\underline q,\overline q]\subseteq [\max(0,p_1+p_2-1),\min(p_1,p_2)]. \tag{2} \] Every such $q$ defines a feasible binary joint distribution with probabilities $q,p_1-q,p_2-q,1-p_1-p_2+q$. This checks that the ambiguity set consists of actual probability laws.

A closed-form robust policy

Take baseline $x^0=0$, meaning all capacity comes from supplier 2. Write $\Delta c=c_1-c_2$ and $A=p_1+p_2-2\overline q\ge0$.
Theorem (Worst dependence and optimal diversification): The robust gain is \[ G(x)=\inf_{q\in[\underline q,\overline q]} \{W(x;q)-W(0;q)\} =\{2L(p_2-\overline q)-\Delta c\}x-LAx^2. \tag{3} \] If $A>0$, its unique maximizer on $[0,1]$ is \[ x^*=\left[\frac{2L(p_2-\overline q)-\Delta c}{2LA} \right]_{[0,1]}, \tag{4} \] where brackets denote clipping to the interval. If $A=0$, the objective is linear and its endpoint or full-interval maximizers follow from its slope.
Proof: Since $Z_j^2=Z_j$, \[ {\mathbb{E}} S^2=p_1x^2+p_2(1-x)^2+2qx(1-x). \] The baseline expected shortage loss is $Lp_2$, independent of $q$. For $x\in[0,1]$, higher $q$ weakly lowers the policy gain because the coefficient is $-2Lx(1-x)\le0$. Thus the worst law uses $q=\overline q$, with equality across all laws at the endpoints. Expand the resulting quadratic to obtain (3). When $A>0$ it is strictly concave, so its unconstrained stationary point, clipped to the feasible interval, is the unique maximizer. When $A=0$ there is no quadratic term. ∎

Existence here is proved by the explicit continuous quadratic on a compact interval, not inferred from compactness alone. The baseline is feasible, so the optimized robust improvement is always at least zero. A minimum robustness requirement or a positive lower bound on domestic shares would alter that statement by potentially excluding the baseline.

Worked dependence checks

Suppose $p_1=p_2=p$ and all feasible dependence structures are allowed. Then $\overline q=p$, corresponding to perfectly common failures. Now $A=0$ and the shortage term in (3) disappears entirely: $G(x)=-\Delta c x$. Diversification has no guaranteed reliability benefit. This is not a claim that actual suppliers fail together; it is the implication of allowing that law in the ambiguity set.

If instead independence is known, $q=p^2$. With equal costs and $p\in(0,1)$, (4) gives $x^*=1/2$. Its improvement is \[ G(1/2)=\frac L2p(1-p)>0. \tag{5} \] For $p=0.1$ and $L=100$, the gain is $4.5$ in the model's money units. Under unrestricted dependence the guaranteed gain for the same portfolio is zero. Marginal disruption rates alone therefore do not determine the value of geographical diversification.

For a more reliable but costlier domestic supplier, take $p_1<p_2$ and unrestricted feasible dependence, so $\overline q=p_1$. Then \[ x^*=\left[1-\frac{\Delta c}{2L(p_2-p_1)}\right]_{[0,1]}. \] Positive domestic sourcing is optimal exactly when $\Delta c<2L(p_2-p_1)$; full domestic sourcing requires $\Delta c\le0$ in this quadratic model. The latter conclusion reflects divisible shares and the chosen smooth shortage penalty, not a universal result about national security. Fixed redundancy costs, indivisible plants or catastrophic threshold losses would create different corner conditions.

The baseline must stay inside the same worst-case comparison

In general, \[ \inf_P[W(x;P)-W(x^0;P)]\ne \inf_P W(x;P)-\inf_P W(x^0;P). \tag{6} \] For a two-law numerical check, let the policy have welfare $(0,10)$ across the laws and the baseline $(5,0)$. The policy improvements are $(-5,10)$, whose infimum is $-5$, while both separate welfare infima are zero. This explicit contrast shows why equality cannot be presumed. Optimizing the two worst cases separately implicitly allows different adverse laws for the policy and baseline, whereas the catalogue compares them under one common law.

In (3) the comparison is handled correctly because the selected baseline's shortage law depends only on $p_2$. For an interior baseline $x^0$, the coefficient of $q$ is $-2L\{x(1-x)-x^0(1-x^0)\}$, whose sign may change with the candidate policy. The worst law can then be the lower rather than upper endpoint of (2). Optimizing policies against a fixed upper-correlation scenario would not automatically solve the stated robust-gain problem.

Social versus private values and identification limits

If a firm internalizes shortage penalty $L_P$ while society values it at $L_S$, applying (4) with the respective coefficients produces different sourcing choices. Observing the firm's $x$ does not identify $L_S$; it may identify at most a combination of private loss, costs and subjective probabilities. A subsidy can implement a different private optimum by changing $\Delta c$, but its financing and distortionary costs must be included before calling the resulting change a social gain. Transfers alone are absent from (1) because only real sourcing costs enter.

Estimating a narrow interval for $q$ requires joint failure observations or defended stress assumptions. A short panel with no simultaneous failures does not prove $q=0$. For $T$ independent periods, the probability of observing no joint failure is $(1-q)^T$, which remains substantial for rare $q$. This elementary calculation explains why dependence sensitivity can dominate a point calibration.

The remaining full problem includes substitutable inputs, inventory, capacity investment, equilibrium prices, foreign responses and legally implementable policy instruments. Equations (3)--(5) settle only the declared divisible two-supplier model. They provide concrete existence, optimality and dependence conditions without claiming empirical welfare rankings for reshoring or friend-shoring.

Completion Estimate

45%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt exactly solves robust two-supplier sourcing under uncertain failure dependence and establishes when diversification or domestic sourcing improves declared welfare. General equilibrium prices, inventory and investment choices, substitution, foreign responses, and legally implementable financing instruments remain absent.

References

  • [source] G. M. Grossman, E. Helpman and H. Lhuillier, Supply Chain Resilience: Should Policy Promote Diversification or Reshoring?, NBER Working Paper 29330 (2021). Primary indexed paper checked for the resilience-policy setting. https://www.nber.org/papers/w29330.

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