## wpiea2025191-source-pdf

## Source details

**Canonical URL:** [wpiea2025191-source-pdf](https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025191-source-pdf.pdf)

## Other formats

- [Markdown version](/-/media/files/publications/wp/2025/english/wpiea2025191-source-pdf.pdf.md)
- [Structured JSON version](/-/media/files/publications/wp/2025/english/wpiea2025191-source-pdf.pdf.json)

---

### Main results and quantitative examples
- U.S. "loses1.2 percent of real income due to its trade deficit, wiping out half the gains from trade."
- China: "the gains from trade for China are 1.9 percent, and China gains anadditional.7 percent from being a surplus economy."
- PPP-adjusted real GDP per capita growth (2014–2024, IMF WEO database): "1.95 percent annually" for the U.S.; "7.13 percent annually" for China.
- Productivity advantage: "U.S. TFP exceeding E.U. TFP by about 20 percent (IMF, 2024)."

### Mechanisms: terms-of-trade effect and productivity effect
- Terms-of-trade effect:
  - Trade imbalances alter relative wages via market-clearing conditions, changing the terms of trade.
  - A trade deficit raises a country's relative wage and improves its terms of trade; a trade surplus lowers relative wages and worsens terms of trade.
  - Via m_nn in equation (16), imbalances raise real income in deficit economies and lower it in surplus economies.
- Productivity effect:
  - If scale economies concentrate in the traded sector (φ_G > 0, φ_S = 0), shifting labour to tradables raises aggregate productivity and real income via L_Gn in equation (16).
  - This productivity effect can redistribute gains from trade from deficit to surplus economies relative to counterfactually balanced trade.
- Net impact:
  - The sign and magnitude of net income changes from rebalancing depend on the initial trade balance, the strength of scale economies in tradables (φ_G), and the intertemporal elasticity of substitution.

### Welfare, policy counterfactuals, and financial autarky
- Static model limitations:
  - Static trade model with exogenous imbalances mechanically raises consumption for surplus economies and lowers it for deficit economies when shifting to balanced trade, complicating welfare assessment.
- Dynamic reinterpretation:
  - The generalized static gravity model is interpreted as the steady state of a dynamic OLG model where persistent trade imbalances can arise from differences in residents' lifetime income profiles and give a non-degenerate steady-state foreign asset distribution.
  - This framing allows modeling a shift to balanced trade as an endogenous outcome of policy-imposed de-facto financial autarky.
- Welfare implications:
  - With modest traded-sector scale economies and standard intertemporal substitution elasticities, economies with small trade-deficit-to-GDP ratios (including the U.S.) can gain from moving to balanced trade by withdrawing from international asset markets: welfare losses from financial autarky are offset by correction of traded-sector underproduction achieved by closing trade deficits.
  - Financial autarky reduces global welfare and is not the first-best policy response to distortions arising from relative scale intensity of traded production.
  - Market equilibrium is not Pareto efficient when φ_G > 0 and φ_S = 0 because firms do not internalise their effect on traded-sector productivity; underproduction in high-returns-to-scale sectors can make market frictions welfare-improving for some economies.

### Relation to prior literature and contributions
- Nests Eaton-Kortum and Dekle et al. (2007, 2008) when φ_G = 0 (constant returns) and reproduces their small-income-effects finding.
- Positive traded-sector scale economies introduce an additional productivity channel that can reverse qualitative effects of imbalances on real incomes and make quantitative effects up to an order of magnitude larger.
- Builds on Epifani and Gancia (2017) by embedding the mechanism that a trade surplus increases product variety in traded sectors into a generalized gravity framework.
- Extends static analyses by adding a dynamic block of "exact hat" equations that endogenise imbalances and permit formal welfare and spillover analysis for trade-balancing policies.

### Modeling framework and key equations (overview)
- Two-sector Armington setup with external economies of scale (EES):
  - Utility: C_n = (C_Gn^(1−σ))^(1−σ) (C_Sn^σ)^σ where σ∈[0,1).
  - Technologies:
    - q_Sn = Z_Sn L_Sn
    - q_Gn = Z_Gn (L_Gn^ν)^ν (J_Gn^(1−ν))^(1−ν)
  - External economies of scale:
    - Z_Sn = A_Sn L_Sn^φ_S
    - Z_Gn = A_Gn L_Gn^φ_G
  - Trade in sector G: origin-differentiated varieties, iceberg trade costs τ_n′n ≥ 1, CES aggregation with elasticity 1+θ.
- Equilibrium highlights:
  - Market-clearing relative-wage condition: equation (11).
  - Allocation of labour to tradables: equation (12).
  - Bilateral trade shares (gravity form): equation (13), with traded-sector labour productivity containing endogenous scale term L_Gn^(−φ_G/ν).
  - Real GDP expression (equation (16)) shows real income dependence on domestic production, m_nn and L_Gn (terms-of-trade and productivity channels).
- Exact-hat algebra and first-order approximation for shift to balanced trade:
  - Changes denoted ˆU = ˜U / U.
  - Key hat-equations: (17), (18), (19), (20), normalization (21).
  - First-order approximation (equation (22)) decomposes ln ˆC_n into direct consumption change, terms-of-trade income effect, and productivity income effect; includes t_n ≡ T_n / (W_n L_n) and term σ(φ_G/ν − φ_S) t_n.

### Numerical and interpretive takeaways
- Productivity effect can notably redistribute gains from trade toward surplus economies under plausible calibrations of traded-sector scale economies.
- Static counterfactuals that ignore dynamic asset-market channels may misstate welfare impacts; embedding the static model as a steady state of a dynamic OLG allows evaluating policy-imposed de-facto financial autarky and its welfare trade-offs.
- Financial autarky can improve some deficit economies' welfare by correcting traded-sector underproduction, but it reduces global welfare and is not first-best for addressing scale-related distortions.

### Data and calibration (Section 3.1)
- Sample and data:
  - Calibrated to 66 individual economies and the rest of the world from OECD ICIO (2017-19 averages).
  - Sectors D, E, F, L, O, P, Q, R, S and T defined as non-traded.
- Calibration choices:
  - Parameters σ and ν calibrated from OECD ICIO (2017-2019).
  - Trade elasticity θ based on Simonovska and Waugh (2014).
  - Normalisation: φ_S = 0; φ ≡ φ_G.
  - Comparative cases: φ = 0 and φ = .50 (plausible upper bound; Appendix B.1 discusses evidence and uncertainty).
- Empirical considerations:
  - Evidence on sectoral scale elasticities incomplete but plausibly consistent with values as high as φ = .50.
  - Goods account for about half of defined traded-sector value added; evidence on services scale economies weaker.
  - Experiments use different values of φ to reflect uncertainty; φ = .50 is an upper-bound scenario.

### Main quantitative results (Section 5.2)
- Major-economy calibration and setup:
  - Key parameters: θ= 4; φ=.50; γ=.50.
  - Welfare impact computed as sum of real GDP impact and loss from removing consumption smoothing under laissez-faire, using hat algebra from Sections 2.2 and 4.2.
- Major findings:
  - Major surplus economies are unambiguously worse off under balanced trade despite increased aggregate consumption because a negative income impact—driven by the dominant productivity effect—combines with loss of consumption smoothing.
  - Germany faces a permanent welfare loss of 10.4 percent.
  - Major deficit economies are marginally better off under balanced trade because income gains from exploiting scale economies in traded production offset loss from de-facto financial autarky.
  - U.S.: balanced-trade steady state implies a permanent 0.3 percent welfare gain.
- Distributional and global outcomes:
  - Distribution of welfare impacts follows consumption smoothing losses but is shifted due to income transfer from surplus to deficit economies (productivity effect).
  - Economies in tails of trade-balance distribution are worse off under balanced trade; this includes all surplus economies and many with large trade deficits relative to GDP. Several low-income economies (e.g., Senegal and Laos) are especially adversely affected.
  - Condition to benefit from balanced trade: small trade deficit in steady state relative to own GDP.
  - Global rebalancing via de-facto financial autarky leads to a net welfare loss of about 2 percent for the world as a whole; coordination to preserve financial openness is desirable globally.
- Parameter sensitivity and threshold behavior:
  - With φ=γ=.5, U.S. is better off under financial autarky than laissez-faire.
  - Holding γ constant, U.S. prefers laissez-faire if φ < .4.
  - With γ > 1, U.S. would prefer financial autarky even for φ = .25.
  - Financial autarky is never preferred for φ < .2, for any γ, because terms-of-trade effect dominates productivity effect in this region.

### Alternative policies and first-best interventions
- Changing trade balances via asset-market interventions is not first-best to correct scale-economy distortions.
- First-best policy: target distortions directly via sector-specific interventions (lump-sum taxation assumed feasible).
  - Sector-specific wage subsidies to internalise productivity externality:
    - W_Sn = 1/(1 +φ_S) W_n. (41)
    - W_Gn = ν/(ν+φ_G) W_n. (42)
  - These subsidies implement the first-best equilibrium in the model of Section 2.1.
- Feasibility constraints:
  - First-best may be infeasible if governments lack non-distortionary taxation or sufficient information; subsidies might violate international agreements; all economies would need to implement subsidies to avoid international distortions.

### Micro-foundations: learning-by-doing, Krugman and Melitz sketches
- Learning-by-doing externality in traded production:
  - Z̄_Gnt+1 / Z̄_Gnt = L̄_Gnt λ_Gnt (Ā_Gn Z̄_Gnt)^β (equation (43)); λ > 0, β > 0.
  - In steady state this corresponds to (6) with φ = λ/β.
- Krugman model (equations (44)–(51)):
  - Composite labour, monopolistic competition, free entry; equilibrium prices and price index incorporate origin-scale L_Gn′^(−φ/ν) and φ ≡ ν/(χ−1).
  - Isomorphisms to aggregated hat-equations (19) and (20) obtained.
- Melitz model (equations (52)–(64)):
  - Heterogeneous producers, Pareto productivity, fixed and marketing costs, congestion parameter ψ ∈ [0,1].
  - Two scale effects: origin production scale lowers supply costs; destination absorption scale raises number of active origin varieties.
  - Special case ψ = 1 removes destination-scale mechanism and recovers isomorphisms to (19) and (20).

### Data and calibration — static global rebalancing and sensitivity (additional details)
- Data choices and sample construction:
  - OECD ICIO (2017-2019 three-year averages); folded 10 economies into rest-of-world → 66 individual economies.
  - Non-traded ISIC sectors D, E, F, L, O, P, Q, R, S, T account for less than 2.5 percent of world trade in 2017-19; their share of world final spending is .49 → used to parameterise σ.
  - Input-output linkages yield ν = .44 baseline.
  - Baseline trade elasticity θ = 4; alternative θ = 2 explored.
  - Evidence suggests goods scale elasticity ≈ .25 (baseline); traded services and aggregation drive φ calibration.
- Additional calibration for steady-state rebalancing:
  - Intertemporal elasticity of substitution γ = .50 baseline.
  - Interest calibration: R = 1.02 ≈ 1.81 (annual interest rate 2 percent with model period = 30 years; R represents interest-growth differential in de-trended model).
  - Calibration implies "average" agents get 57 percent of income in first half of lives.
- Sensitivity experiments — key findings:
  - Lowering θ from 4 to 2: strengthens terms-of-trade effect slightly; productivity effect unchanged; productivity effect remains dominant. Example (θ = 2; φ = .50): half of surplus economies see real GDP decline > 1.1 percent from shifting to balanced trade; half of deficit economies experience increase > .6 percent.
  - Reducing intermediate input share 1 − ν by one half (i.e., adjusting ν) weakens productivity effect significantly; minimal impact on terms-of-trade effect.
  - Productivity effect depends critically on relative scale elasticities across traded and non-traded sectors and on input-output linkage strength ν.
  - Baseline goods-scale elasticity fixed at .25 in exercises; variation in traded-services and non-traded elasticities drives aggregate outcomes.

*Source: wpiea2025191-source-pdf (https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025191-source-pdf.pdf)*

### 2.3 percent using standard sufficient statistics.

### wpiea2025191-source-pdf - 2.3 percent using standard sufficient statistics.

### Main results and quantitative examples
- The paper finds that the U.S. "loses1.2 percent of real income due to its trade deficit, wiping out half the gains from trade."  
- By contrast, "the gains from trade for China are 1.9 percent, and China gains anadditional.7 percent from being a surplus economy."  
- A contextual comparison: between 2014 and 2024 the PPP-adjusted real GDP per capita grew by "1.95 percent annually" for the U.S. and "7.13 percent annually" for China, according to the IMF World Economic Outlook database. The U.S. retains a productivity advantage with "U.S. TFP exceeding E.U. TFP by about 20 percent (IMF, 2024)."

### Mechanisms: terms-of-trade effect and productivity effect
- Terms-of-trade effect:
  - Trade imbalances alter relative wages via market-clearing conditions, which change the terms of trade.
  - A trade deficit raises a country's relative wage and improves its terms of trade; a trade surplus lowers relative wages and worsens terms of trade.
  - Through m_nn in equation (16), imbalances raise real income in deficit economies and lower it in surplus economies.
- Productivity effect:
  - When scale economies are concentrated in the traded sector (φ_G > 0, φ_S = 0), shifting labour to tradables raises aggregate productivity and real income via L_Gn in equation (16).
  - This productivity effect can redistribute gains from trade from deficit to surplus economies relative to counterfactually balanced trade.
- Net impact:
  - The sign and magnitude of net income changes from rebalancing depend on the initial trade balance, the strength of scale economies in tradables (φ_G), and the intertemporal elasticity of substitution.

### Welfare, policy counterfactuals, and financial autarky
- Static model limitations:
  - The static trade model with exogenous imbalances mechanically raises consumption for surplus economies and lowers it for deficit economies when shifting to balanced trade, complicating welfare assessment.
- Dynamic reinterpretation:
  - The generalized static gravity model is recast as the steady state of a dynamic model (OLG) where persistent trade imbalances can arise from differences in residents' lifetime income profiles and give a non-degenerate steady-state foreign asset distribution.
  - This allows modeling a shift to balanced trade as an endogenous outcome of policy-imposed de-facto financial autarky.
- Welfare implications:
  - With modest traded-sector scale economies and standard intertemporal substitution elasticities, economies with small trade-deficit-to-GDP ratios (including the U.S.) can gain from moving to balanced trade by withdrawing from international asset markets: welfare losses from financial autarky are offset by correction of traded-sector underproduction achieved by closing trade deficits.
  - However, financial autarky reduces global welfare and is not the first-best policy response to distortions arising from relative scale intensity of traded production.
  - The market equilibrium is not Pareto efficient when φ_G > 0 and φ_S = 0 because firms do not internalise their effect on traded-sector productivity; this leads to underproduction in high-returns-to-scale sectors. Introducing market frictions can be welfare-improving for some economies when traded-sector scale economies are strong.

### Relation to prior literature and contributions
- The framework nests Eaton-Kortum and Dekle et al. (2007, 2008) results (constant returns φ_G = 0) and reproduces their small-income-effects finding in that special case.
- When traded-sector scale economies are positive, the model introduces an additional productivity channel that can reverse the qualitative effect of imbalances on real incomes and make quantitative effects up to an order of magnitude larger.
- Builds on Epifani and Gancia (2017) by embedding the mechanism that a trade surplus increases product variety in traded sectors into a generalised gravity framework applicable to many heterogeneous economies.
- Extends static analyses by adding a dynamic block of "exact hat" equations that endogenise imbalances and permit formal welfare and spillover analysis for trade-balancing policies.

### Modeling framework and key equations (overview)
- Two-sector Armington setup with external economies of scale (EES):
  - Utility: C_n = (C_Gn^(1−σ))^(1−σ) (C_Sn^σ)^σ where σ∈[0,1).
  - Technologies:
    - q_Sn = Z_Sn L_Sn
    - q_Gn = Z_Gn (L_Gn^ν)^ν (J_Gn^(1−ν))^(1−ν)
  - External economies of scale specification:
    - Z_Sn = A_Sn L_Sn^φ_S
    - Z_Gn = A_Gn L_Gn^φ_G
  - Trade in sector G: origin-differentiated varieties, iceberg trade costs τ_n′n ≥ 1, CES aggregation with elasticity 1+θ.
- Equilibrium characterization highlights:
  - Market-clearing relative-wage condition: equation (11).
  - Allocation of labour to tradables: equation (12).
  - Bilateral trade shares (gravity form): equation (13), where traded-sector labour productivity contains an endogenous scale term L_Gn^(−φ_G/ν).
  - Real GDP expression (equation (16)) which shows real income dependence on domestic production and on m_nn and L_Gn, encapsulating terms-of-trade and productivity channels.
- Exact hat algebra for counterfactuals:
  - Changes relative to initial equilibrium denoted ˆU = ˜U / U.
  - Key hat-equations: (17), (18), (19), (20), and normalization (21).
  - First-order approximation for the special case of a shift to balanced trade (equation (22)):
    - ln ˆC_n ' t_n + ln ˜Y_n ' t_n + (1−σ) (1−m_nn) / ν ln ˆS_n − σ(φ_G/ν − φ_S) t_n,
    - where t_n ≡ T_n / (W_n L_n). The terms correspond to direct consumption change, indirect terms-of-trade income effect, and indirect productivity income effect respectively.
- Key comparative statics from equation (22):
  - If the non-traded sector is large (large σ) and home bias is strong (m_nn close to 1), terms-of-trade effects on real income and consumption are small.
  - Trade elasticity θ does not play a first-order role for income responses to a shift to balanced trade.
  - Productivity effect magnitude is governed by φ_G relative to φ_S.

### Numerical and interpretive takeaways
- The productivity effect can cause notable redistribution of gains from trade toward surplus economies under plausible calibrations of traded-sector scale economies.
- Static counterfactuals that ignore dynamic asset-market channels may misstate welfare impacts; embedding the static model as a steady state of a dynamic OLG allows evaluating policy-imposed de-facto financial autarky and its welfare trade-offs.
- Financial autarky can improve some deficit economies' welfare by correcting traded-sector underproduction, but it reduces global welfare and is not first-best for addressing scale-related distortions.

*Source: wpiea2025191-source-pdf (https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025191-source-pdf.pdf)*

### 3.1  Data and calibration

### 3.1  Data and calibration

### Data and sample
- Calibrated to data for 66 individual economies and the rest of the world from the OECD Inter-Country Input-Output Database (ICIO; OECD, 2023).
- Data are averaged for the years 2017-19.
- Sectors D, E, F, L, O, P, Q, R, S and T are defined as non-traded (as noted in Table 1).

### Calibration approach and key parameter choices
- Most parameters are conventionally calibrated; parameters σ and ν are calibrated based on OECD ICIO data averaged for 2017-2019 (Table 1).
- Trade elasticity θ is based on Simonovska and Waugh (2014).
- Normalisation and notation for scale elasticities:
  - φ_S = 0 (normalised).
  - φ ≡ φ_G (measure of the strength of trade-sector scale economies relative to non-traded production).
- Comparative cases included:
  - φ = 0 (limit case for comparison with Dekle et al. (2007, 2008)).
  - φ = .50 designated as a plausible upper bound for traded-sector scale elasticity (experiments use different values of φ; Appendix B.1 discusses empirical evidence and uncertainty).
- The range of scale elasticities reported in Table 1 is based on model parameter relationships and empirical estimates (see Appendix B.1 for details).

### Empirical considerations and interpretation
- The strength of the productivity effect depends on the scale elasticity in traded relative to non-traded production.
- Available empirical evidence on sectoral scale elasticities is incomplete but plausibly consistent with values as high as φ = .50 (Appendix B.1).
- Although evidence supports sizeable scale economies in goods production, goods account for only about half of what the authors define as traded-sector value added.
- Evidence on scale economies in services is weaker and lacks systematic information on the distribution between traded and non-traded services.
- High values of φ are plausible if traded services are similarly scale intensive as goods; otherwise lower values of φ are appropriate.
- To reflect uncertainty from scarce empirical evidence on services scale elasticities, the paper experiments with different values of φ and designates φ = .50 as a plausible upper bound.

*Source: wpiea2025191-source-pdf - 3.1  Data and calibration*

### 5.2  Results

### wpiea2025191-source-pdf - 5.2  Results

### Major economies
- Calibration and setup:
  - Key parameters: θ= 4; φ=.50; γ=.50.
  - Welfare impact computed as the sum of the real GDP impact and the loss from removing consumption smoothing opportunities available under laissez-faire, using hat algebra from Sections 2.2 and 4.2.
- Main findings:
  - Major surplus economies are unambiguously worse off under balanced trade despite increased aggregate consumption, because a negative income impact—due to the dominant productivity effect—combines with a loss of consumption smoothing.
  - Germany faces a permanent welfare loss of 10.4 percent.
  - Major deficit economies are marginally better off under balanced trade because income gains from exploiting scale economies in traded production offset the loss from moving to de-facto financial autarky.
  - For the U.S., the balanced-trade steady state implies a permanent 0.3 percent welfare gain.
- Mechanism and interpretation:
  - The initial laissez-faire equilibrium is second-best: multi-sector models typically underproduce in high-returns-to-scale sectors.
  - For some deficits (e.g., U.S. and India under this calibration), the welfare gains from using international asset markets are small relative to gains from partially correcting traded-sector underproduction under balanced trade.

### All economies
- Distributional outcomes:
  - The distribution of welfare impacts follows the distribution of consumption smoothing losses but is shifted north west through the origin due to income transfer from surplus to deficit economies (the productivity effect).
  - Economies in the tails of the trade-balance distribution are made worse off by balanced trade; this includes all surplus economies and many economies with large trade deficits relative to GDP.
  - Several low-income economies, such as Senegal and Laos, are especially adversely affected.
- Conditions for benefiting from balanced trade:
  - An economy needs a small trade deficit in steady state relative to its own GDP to benefit.
  - The laissez-faire international bond interest rate is a weighted average of economies’ lifetime income profiles and is similar to autarky interest rates of the largest economies. Large economies benefit least from consumption smoothing and thus have small foreign asset positions and imbalances relative to GDP.
- Global aggregate effect:
  - Global rebalancing via de-facto financial autarky leads to a net welfare loss of about 2 percent for the world as a whole.
  - Coordination around policies that preserve financial openness is desirable from a global standpoint.

### Alternative parameter calibrations (scenarios)
- Key governing parameters: φ (concentration of scale economies in traded production) and γ (intertemporal elasticity of substitution).
- U.S. examples and threshold behavior:
  - With φ=γ=.5, the U.S. is better off under financial autarky than in the laissez-faire steady state.
  - Holding γ constant, the U.S. prefers laissez-faire if φ falls below .4.
  - With γ >1, the U.S. would prefer financial autarky even for φ=.25.
  - Financial autarky is never preferred for values of φ < .2, no matter γ, because the terms-of-trade effect from balancing trade dominates the productivity effect; a deficit economy suffers a real income loss from financial autarky in this region.

### Alternative policies (policy recommendations)
- First-best vs second-best:
  - Changing trade balances through interventions in asset markets is not first-best for correcting production and price distortions that arise with scale economies.
  - The first-best policy targets the source of distortions directly via sector-specific interventions.
- Example (Armington model with EES) assuming lump-sum taxation is feasible:
  - Implement sector-specific wage subsidies so firms in sectors S and G face net wages characterised by:
    - W_Sn = 1/(1 +φ_S) W_n. (41)
    - W_Gn = ν/(ν+φ_G) W_n. (42)
  - These subsidies cause firms to internalise the productivity externality from their labour choices and implement the first-best equilibrium in the model of Section 2.1.
- Feasibility and constraints:
  - First-best policy may be infeasible if governments lack access to non-distortionary taxation or sufficient information to calibrate subsidies.
  - It would require relatively high subsidization of traded-sector production if scale economies are concentrated there, potentially violating international agreements.
  - All economies would need to implement the subsidies to avoid creating international distortions; partial implementation would correct domestic distortions but create international relative-price distortions.

### Conclusion (summary of substantive implications)
- Under plausible calibrations with external economies of scale concentrated in traded production, trade imbalances can redistribute gains from trade from deficit to surplus economies.
- When observed imbalances are steady-state outcomes of consumption smoothing via international asset markets, some major deficit economies may benefit from financial autarky because deficits exacerbate second-best underproduction of traded goods.
- Uncertainty remains about the relative strength of EES in traded versus non-traded services; more evidence on EES in services is critical for refining assessments of macroeconomic effects of persistent imbalances.
- If spillovers from EES in services are large, it strengthens the case for policy cooperation to support balanced growth of international trade.

*Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025191-source-pdf.pdf*

### Section 2.1 or the Eaton-Kortum model sketched here is to assume a learning-

### wpiea2025191-source-pdf - Section 2.1 or the Eaton-Kortum model sketched here is to assume a learning-

### Learning-by-doing externality in traded production
- Productive ideas in the traded sector are generated according to:
  - Z̄_Gnt+1 / Z̄_Gnt = L̄_Gnt λ_Gnt (Ā_Gn Z̄_Gnt)^β (equation (43) in source).
  - λ > 0 reflects the strength of the learning-by-doing externality.
  - β > 0 captures the rate at which ideas are getting harder to find.
  - This corresponds to a sectoral analogue of the semi-endogenous growth equation.
  - In steady state with Z̄_Gnt+1 = Z̄_Gnt = Z̄_Gn and L̄_Gnt+1 = L̄_Gnt = L̄_Gn this corresponds to (6) with φ = λ/β.

### Krugman model — assumptions and equilibrium characterization
- Technologies and market structure:
  - Economy-n output in sector G produced with:
    - q_Gn = A_Gn (L̄_Gn^ν)^ν (J_Gn^{1−ν})^{1−ν} (equation (44)).
  - Composite labour input assembled from an endogenous mass H_Gn of providers:
    - L̄_Gn = [∫_0^{H̄_Gn} l_Gn(h)^{(χ−1)/χ} dh]^{χ/(χ−1)} (equation (45)), with χ > 1.
  - Labour providers are monopolistic competitors; free entry after paying fixed cost Ψ_n in units of domestic labour.
- Equilibrium by symmetry (optimal pricing and free entry):
  - w_Gn = w_Gn(h) = χ/(χ−1) W_n.
  - l_Gn = l_Gn(h) = ν(χ−1)/χ p_Gn q_Gn W_n H_Gn.
  - p_Gn = (χ/(χ−1))^ν W_n^ν P_Gn^{1−ν} H_Gn^φ A_Gn, where φ ≡ ν/(χ−1).
  - Free entry zero-profit conditions:
    - W_n L_Gn = W_n H_Gn (l_Gn + Ψ_n) = (1 − 1/χ) ν p_Gn q_Gn + W_n H_Gn Ψ_n = ν p_Gn q_Gn (equation (48)).
    - H_Gn = (1/χ) L_Gn / Ψ_n (equation (49)).
  - Price index and trade share expressions:
    - P_Gn = Ξ_K [∑_{n′=1}^n (τ_{n′n} m_{n′n}^{1−ν} θ_n′n′ L_{Gn′}^{−φ/ν} W_{n′}/Ā_{Gn′}^{1/ν})^{−θ}]^{−1/θ} (equation (50)).
    - m_{n′n} given by the corresponding normalized expression (equation (51)).
    - Ā_Gn ≡ A_Gn / Ψ_n^φ; Ξ_K is a constant.
  - The isomorphisms to (19) and (20) are immediate.

### Melitz model — assumptions and equilibrium characterization
- Technologies and market structure:
  - Output now origin- and destination-specific:
    - q_Gnn′ = A_Gn (L̄_Gnn′^ν)^ν (J_Gnn′^{1−ν})^{1−ν} (equation (52)).
  - Economy n assembles sector-G goods from all origins:
    - X_Gn = (∑_{n′=1}^N x_Gn′n^{(η−1)/η})^{η/(η−1)} (equation (53)), with η > 1.
    - q_Gnn′ = τ_{nn′} x_Gnn′; p_Gn q_Gn ≡ ∑_{n′=1}^N p_Gnn′ q_Gnn′ (equation (54)).
  - Composite labour input L̄_Gnn′ assembled from mass H_Gnn′ with elasticity χ > 1 (equation (55)).
  - Labour providers differ in productivity a drawn from Pareto with shape φ > χ−1 and location a_n > 0; entry pays fixed cost Ψ_n; additional fixed marketing cost Ψ_nn′ X_ψGn′ to serve destination n′, paid in units of good destined for n′. Parameter ψ ∈ [0,1] represents congestion in destination markets.
- Equilibrium characterization:
  - Optimal pricing:
    - w_Gnn′(a) = χ/(χ−1) W_n a.
    - l_Gnn′(a) = ν(χ−1)(φ−χ+1) a / [χ χ φ a_n a^{∗ −(φ−χ+1)}_{nn′}] p_Gnn′ q_Gnn′ W_n H_Gn (equation (56)).
  - Price expression:
    - p_Gnn′ = (χ/(χ−1))^ν (φ/(φ−χ+1))^{ν(1−χ)} τ_{nn′} (a^{∗ φ−χ+1}_{nn′} / a_n^φ H_Gn)^{ν/(χ−1)} W_n^ν P_Gn^{1−ν} A_Gn (equation (57)).
    - a^{∗}_{nn′} is productivity cut-off for providers from n serving destination n′; H_Gn is total active providers in n.
  - Cut-off determined by:
    - H_Gnn′ = (a^{∗}_{nn′} / a_n)^{−φ} H_Gn.
    - H_Gn = (φ−χ+1)ν/(χφ) q_Gnn′ / (Ψ_nn′ X_ψGn′) (equation (58)).
  - Ex-post profits:
    - ν/χ p_Gnn′ q_Gnn′ − H_nn′ p_Gnn′ Ψ_nn′ X_ψGn′ = (χ−1)ν/(χφ) p_Gnn′ q_Gnn′ (equation (59)).
  - Free entry zero expected profits:
    - W_n L_Gn = W_n [∑_{n′=1}^N ∫_0^{H̄_Gnn′} l_Gnn′(h) dh + H_Gn Ψ_n] = (1 − χ^{-1}/(χφ)) ν p_Gn q_Gn + W_n H_Gn Ψ_n = ν p_Gn q_Gn (equation (60)).
    - H_Gn = (χ−1)/(χφ) L_Gn / Ψ_n (equation (61)).
  - Combined expressions yield p_Gnn′ solution (equation (62)) and price index/trade share:
    - P_Gn = Ξ_M {∑_{n′=1}^N [τ̄_{n′n} m_{n′n}^{1−ν} θ_{n′n′} X_{Gn′}^{−(1−ψ)/φ (φ/(χ−1) − 1)} L_{Gn′}^{−φ/ν} W_{n′}/Ā_{Gn′}^{1/ν}]^{−θ}}^{−1/θ} (equation (63)).
    - m_{n′n} analogous normalization (equation (64)).
    - Definitions: τ̄_{nn′} ≡ τ_{nn′} Ψ_{nn′}^{(ν/φ)[φ/(χ−1)−1]}; Ā_Gn ≡ A_Gn a_n^ν / Ψ_n^φ; φ ≡ ν/φ; Ξ_M constant; θ ≡ φ(η−1)/{νη + φ[1 − νη/(χ−1)]}.
  - Special case ψ = 1 removes destination-scale mechanism and recovers isomorphisms to (19) and (20).
- Structural insight:
  - Two scale effects in Melitz model:
    - Origin production scale L_Gn′ reduces costs of supplying destinations.
    - Destination absorption scale X_Gn raises the subset of active origin varieties supplying that destination (shut down if ψ = 1).
  - If χ−1 > νη is sufficient to ensure conventional sign of trade elasticity (note in source).

### Data and calibration — static global rebalancing
- Data:
  - OECD Inter-Country Input-Output Database (ICIO; OECD, 2023), averaged 2017-2019.
  - Three-year average used to smooth short-run fluctuations.
  - Bilateral expenditure shares {m_{n′n}}, world GDP shares {y_n}, and trade balances relative to world GDP {T_n} for 76 economies plus a rest-of-world residual.
  - Folded four highly oil-dependent economies (Brunei, Belarus, Kazakhstan, Saudi Arabia) and six small service-dependent economies (Ireland, Luxembourg, Malta, Singapore, Switzerland and Taiwan) into rest-of-world → sample of 66 individual economies.
- Calibration — key parameter choices and empirical inputs:
  - Non-traded sectors identified: ISIC sectors D, E, F, L, O, P, Q, R, S, T.
    - These sectors account for less than 2.5 percent of world trade in 2017-19.
    - Their share of world final spending in OECD Database is .49 → used to parameterise σ.
  - Input-output linkages in traded production:
    - Share of value added in total global output for traded sectors yields ν = .44 as baseline.
  - Trade elasticity:
    - Baseline aggregate trade elasticity θ = 4 (value commonly used based on Simonovska and Waugh (2014)).
    - Alternative lower value θ = 2 explored in Appendix C.1.
  - Relative traded-sector scale elasticity:
    - Evidence suggests goods scale elasticity ≈ .25 (used as baseline).
    - Table B1 aggregates literature estimates; sector group shares in world final absorption (2017-2019): primary = .03; manufacturing = .21; traded services = .26; non-traded private services = .23; non-traded public services = .26.
    - For aggregation used in later tables: goods (A01-C33) = .24; traded services = .26.
  - Calibration implications (Table B2 summary):
    - Constant returns in all services production with goods φ_goods = .25 gives small aggregate scale economies and small productivity effect.
    - If traded-services scale economies match goods (.25) and non-traded constant returns, aggregate scale economies are large.
    - To keep aggregate scale economies plausible, diseconomies in non-traded required if traded services have strong scale economies — amplifies productivity effect.
    - Upper-bound calibration in body of paper normalises non-traded scale elasticity to zero (implies unrealistically large aggregate scale economies) but is immaterial for experiments that keep total labour endowments constant.
- Initial world GDP shares:
  - Run hat algebra with status-quo trade balances T̃_n = T_n to generate initial world GDP shares used in counterfactuals.
  - These initial shares are highly, but not perfectly, correlated with OECD Database world GDP shares and vary with calibrations of σ and ν.
- Additional calibration for steady-state global rebalancing:
  - Intertemporal elasticity of substitution γ:
    - Adopt γ = .50 as baseline (justified as conservative for multi-decade model periods).
  - Lifetime income profiles and world interest rate:
    - R = 1.02 ≈ 1.81, corresponding to annual interest rate 2 percent with model period = 30 years (note in source that R represents interest-growth differential in de-trended model).
    - Given γ and R, ω_n can be backed out from trade balance data using equation (65).
    - Calibration implies in the “average” economy agents get 57 percent of their income in the first half of their lives.

### Additional and alternative counterfactuals — key results
- Varying the trade elasticity (Appendix C.1):
  - Lowering θ from 4 to 2:
    - Strengthens the terms-of-trade effect slightly; productivity effect unchanged.
    - Productivity effect remains dominant; overall impact of imbalances on real incomes is reduced but still sizeable.
    - Example (θ = 2; φ = .50): half of surplus economies see real GDP decline > 1.1 percent from shifting to balanced trade; half of deficit economies experience increase > .6 percent.
- Varying the strength of input-output linkages (Appendix C.2):
  - Reduce intermediate input share 1 − ν by one half (i.e., set ν = .72 baseline experiment reported) and re-run trade-balancing experiment:
    - Productivity effect is significantly weaker; minimal impact on terms-of-trade effect.
    - Interpretation: stronger input reliance amplifies productivity effects via feedback where increased output raises labour productivity, reduces input costs, further raises productivity; weakening input reliance weakens this amplification.
- Quantitative sensitivity:
  - The productivity effect depends critically on relative scale elasticities across traded and non-traded sectors and on the strength of input-output linkages (ν).
  - Goods-scale elasticity fixed at .25 in exercises; variation in traded-services and non-traded elasticities drives differences in aggregate outcomes.
  - Baseline parameters used in main text unless otherwise specified.

*Source: wpiea2025191-source-pdf - Section 2.1 or the Eaton-Kortum model sketched here is to assume a learning- (IMF Working Paper No. WP/2025/191).*

---


_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025191-source-pdf.pdf_