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Econ-413 · economics · Economics

Long-run reshoring effects

LLM status Partially solved unverified

Statement

Original subject group: Trade, globalization and geoeconomics

Audit classification: Published research agenda. The 2026 primary draft explicitly asks about long-term capital/labor/productivity effects; the official agenda supports reconfigured trade. Capacity, persistence and rerouting definitions are editorial.

Definitions and precise research target

Fix industries $j$, countries and a tariff or sourcing-restriction policy path $a$. Let $K^d_{j,t+h}(a)$ be productive domestic capacity, defined by a prespecified engineering or production metric rather than nominal sales. Let $V^d_{j,t+h}(a)$ be real domestic value added and $C^{down}_{t+h}(a)$ downstream production costs. Define reshoring effects relative to an otherwise specified baseline path $0$: \[\Delta K_j(h)=E[K^d_{j,t+h}(do(a))-K^d_{j,t+h}(do(0))].\] The task is to identify when positive capacity and value-added changes persist after the initial policy shock, alongside employment, productivity and downstream-cost effects. Specify the long horizon $h$, policy permanence and capacity utilization. Distinguish relocation to the home country from import diversion through third countries, transshipment or increased imported inputs embedded in domestic assembly. Account for retaliation, foreign investment, exchange rates and endogenous entry. Identify or bound counterfactual capacity when policy selects industries already expected to expand or decline. Compare local adjustment estimates with national general-equilibrium outcomes. Evidence of reduced direct imports from a targeted partner does not establish durable domestic productive-capacity growth or a positive net welfare effect.

Additional formal definitions and scope (editorial). A policy path specifies its announcement, implementation, expected duration and any reversal date. Productive capacity is an engineering maximum at a stated utilization convention; realized output is capacity times endogenous utilization and productivity. Domestic value added subtracts imported and domestic intermediate purchases consistently and is deflated with a declared price index. Long-run persistence means a specified finite horizon or a limit $\liminf_{h\to\infty}\Delta K_j(h)>0$, with stationary or balanced-growth normalization required before taking the limit. Define relocation by establishments or production stages, and identify ultimate country of origin to separate transshipment. A tariff-induced investment increase in protected firms may coexist with downstream contractions; national welfare requires all sectors and fiscal financing in the same equilibrium.

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. Finite-horizon capacity response (editorial). Fix policy permanence and a measurable capacity definition, then compare domestic investment with rerouted imports.
  2. Persistent general-equilibrium reshoring (source-motivated extension). Include entry, downstream costs and foreign investment and use a long-horizon or balanced-growth limit; evaluate national welfare alongside capacity.

References and source audit

  1. Official January 9, 2026 web version; locator: Interconnected World / Reconfigured trade. Broad agenda; exact model editorial. URL: https://www.bankofengland.co.uk/research/bank-of-england-agenda-for-research.
  2. Brookings cover verifies Fajgelbaum and Khandelwal, March 2026 conference draft; NBER indexed metadata independently verifies WP 35064 and DOI. Locator: Section 7, printed p.49 (PDF p.50), explicit further-research list. Support: Dynamic and distributional tariff agenda. Full conference draft accessible. URL: https://www.brookings.edu/wp-content/uploads/2026/03/1_Fajgelbaum-Khandelwal_unembargoed.pdf.
  1. Bank of England. 2026. Bank of England Agenda for Research, 2025–2028. Interconnected World / Reconfigured trade. https://www.bankofengland.co.uk/research/bank-of-england-agenda-for-research.
  2. Pablo D. Fajgelbaum; Amit Khandelwal. 2026. Tariffs in 2025: Short-Run Impacts on the U.S. Economy. Section 7, printed p.49 (PDF p.50), explicit further-research list. https://www.brookings.edu/wp-content/uploads/2026/03/1_Fajgelbaum-Khandelwal_unembargoed.pdf. DOI: https://doi.org/10.3386/w35064.

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-413-A1 · Partial result · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-413

Long-run reshoring effects

Question and output of this attempt

The target is durable domestic productive capacity after a specified sourcing policy, with downstream costs and equilibrium responses kept distinct. The result here is a stock-flow persistence theorem, a quantitative extrapolation bound, and a measurement counterexample. These identify what additional evidence would be needed; they do not infer a long-run tariff effect from recent imports.

A capacity law that separates permanence from a temporary surge

For one sector define engineering capacity $K_t$ by a fixed physical standard, with gross investment $I_t\ge0$ and depreciation $0<\delta\le1$: \[ K_{t+1}=(1-\delta)K_t+I_t. \] Use the same initial state and depreciation in policy and baseline economies, and let $x_t=K_t^a-K_t^0$, $b_t=I_t^a-I_t^0$. The exact difference law is \[ x_h=(1-\delta)^h x_0+ \sum_{s=0}^{h-1}(1-\delta)^{h-1-s}b_s. \tag{1} \]
Theorem (Persistence needs a persistent investment mechanism): With $x_0=0$, if $b_s=0$ for all $s\ge H$, then $x_{H+k}=(1-\delta)^k x_H\to0$. If $b_s\to b_\infty$ and the sequence is bounded, then $x_h\to b_\infty/\delta$. More generally if $\liminf b_s\ge\underline b$ and $\limsup b_s\le\overline b$, then \[ \frac{\underline b}{\delta}\le\liminf x_h \le\limsup x_h\le\frac{\overline b}{\delta}. \tag{2} \]
Proof: Iterating the recurrence proves (1). Once the investment gap is zero, the first assertion follows by direct iteration. For the remaining assertions, fix $\varepsilon>0$ and choose $S$ after which investment gaps lie between $\underline b-\varepsilon$ and $\overline b+\varepsilon$. The finite pre-$S$ contribution decays geometrically. The tail weights have sum tending to $1/\delta$, so lower and upper bounds follow; let $\varepsilon\downarrow0$. Setting both bounds to $b_\infty$ proves convergence. ∎
Thus an irreversible sunk investment is not by itself permanent capacity: depreciation must be replaced. Persistent gains can arise from entry, productivity, agglomeration or a lasting policy, but those channels must sustain investment or change the law of motion. The special case $\delta=0$ permits a temporary investment pulse to survive forever and is excluded deliberately. A balanced-growth normalization has a different effective depreciation and must be derived before applying the theorem.

For example, with $\delta=0.1$ and an investment gap of one for five periods, $x_5=(1-0.9^5)/0.1=4.0951$. Twenty periods after the investment gap stops, the remaining capacity gap is $4.0951(0.9)^{20}$, about $0.498$; its limit is zero. An investment gap permanently equal to one instead converges to ten. These paths agree during the first five observations yet have different long-run limits. No statistical precision on those first five observations distinguishes the two continuation mechanisms without further restrictions.

A finite-horizon identified interval from investment bounds

Suppose each future $b_s$ is bounded by $[\ell_s,u_s]$ independently of the others and $\delta$ is known. Positivity of the weights in (1) gives sharp bounds \[ \sum_{s=0}^{h-1}(1-\delta)^{h-1-s}\ell_s \le x_h\le \sum_{s=0}^{h-1}(1-\delta)^{h-1-s}u_s. \tag{3} \] Sharpness follows by choosing every investment gap at the corresponding endpoint, provided the baseline levels are sufficient to keep gross investment nonnegative. If financing or entry restrictions couple the $b_s$, (3) stays valid but can be loose. For a discounted capacity statistic $\sum_h\beta^h x_h$, absolute summability follows when investment gaps are uniformly bounded and $\beta<1$; finite sums may then be interchanged with the geometric tail using domination.

For multiple sectors, let $x_{t+1}=Ax_t+Bz_t$ describe a declared stable local approximation. A certified induced norm bound $\|A\|\le r<1$, input error $\|z_t-\widehat z_t\|\le\eta$, and state-model error at most $\epsilon$ per period yield \[ \|x_h-\widehat x_h\|\le r^h\|x_0-\widehat x_0\| +(\|B\|\eta+\epsilon)\frac{1-r^h}{1-r}. \tag{4} \] Subtract recurrences, take norms and iterate to prove it. A spectral-radius claim alone does not establish the same bound in the chosen physical norm; transient amplification can be large. Nor does a local linearization without a uniform remainder bound certify a global long-run path.

Output, domestic value added and imports do not identify capacity

Let observed production be $Q_t=u_tK_t$ with utilization $u_t\in[0,1]$. Suppose observed output rises from one to two. Model A has $K_0=K_1=2$ and $u_0=1/2,u_1=1$; model B has $K_0=1,K_1=2$ and $u_0=u_1=1$. Both fit the output change, while only B has a capacity increase. If a baseline capacity census also records two, use future capacity three with utilization $2/3$ in B and unchanged capacity two in A; both again fit the observed output. Physical plant data, operating hours and production engineering are therefore separate measurements from sales or shipments.

Likewise a fall of 100 units in imports from a targeted partner can be offset by 100 units shipped through a third country with unchanged ultimate origin, by 100 units of new domestic production, or by reduced domestic consumption. Origin and input-output records distinguish these cases; a bilateral customs total does not. Domestic value added also differs from gross assembly output. If gross output is $Q$ and intermediate inputs have real cost $M$, the accounting identity is $V=Q-M$ in consistently valued units; increased imported components can make $Q$ rise while $V$ barely moves. Deflation and domestic intermediate counting must avoid counting the same value added repeatedly across stages.

Causal assignment and welfare are further layers

Observed investment differences between protected and unprotected sectors combine treatment effects and policy selection. In a two-period design, $\operatorname{DiD}(I)$ equals the desired treated-sector effect plus its untreated investment-trend difference from controls. Bounds on that difference can enter the $[\ell_s,u_s]$ inputs of (3), but observations after policy selection do not justify zero trend bias. Anticipated tariff expiration or retaliation changes $b_s$ before implementation and belongs in the intervention path.

Even an identified capacity gain is not a welfare conclusion. Downstream input costs, financing, foreign responses and household ownership income enter the same resource-consistent equilibrium. A tariff's receipts are transfers within the declared welfare boundary, whereas real installation and operating costs are resources. A model can increase protected capacity while reducing national consumption possibilities. The unresolved extension is an identified general equilibrium investment/entry response with credible continuation restrictions, not the stock-flow algebra proved here.

Completion Estimate

30%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt proves stock-flow capacity persistence, finite-horizon investment bounds, and stable-model extrapolation controls while showing output and bilateral imports cannot identify reshoring. Endogenous investment, entry, retaliation, downstream effects, and credible continuation restrictions remain necessary to identify durable general-equilibrium policy effects.

References

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