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Trade Wars with Trade Deficits

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Trade deficits raise a country's optimal tariff, and the paper's calibrated model shows the United States would gain from a trade war with China relative to existing tariff rates.

desk verdict The quantitative exercise is substantial and worth refereeing, but the paper's central theoretical derivation in equation (9) is algebraically wrong as printed and undercuts the abstract's main claim until fixed. read the letter →

arxiv 2411.15092 v2 pith:PFXV4LNS submitted 2024-11-22 econ.GN q-fin.EC

Pau Pujolas , Jack Rossbach This is my paper · ORCID
classification econ.GNq-fin.EC
keywords tradewarsoptimaltariffsdeficitstermsofappliedgeneralequilibriumU.S.–ChinawarNashinput-outputlinkages
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that trade deficits change the welfare consequences of tariffs: a country that imports more than it exports can push the price of its imports down and keep the tariff revenue, making its foreign partner's demand look less elastic. In a two-country model with CES preferences, larger deficits translate into higher welfare-maximizing tariffs. The authors then calibrate an 18-region, 22-sector model with input-output linkages and service sectors, and find that the United States—the deficit side of the world's largest bilateral imbalance—gains from a trade war with China relative to the pre-trade-war tariff baseline, while China loses. The U.S. gain is small (+0.008 percent of consumption), and both countries would be better off under free trade; eliminating the bilateral U.S.–China deficit turns the U.S. gain into a loss. The paper's point is that the imbalance, not tariffs alone, is what makes the war pay.

What carries the argument

The load-bearing object is the optimal-tariff condition in a two-country CES (constant elasticity of substitution) endowment economy, where the deficit country's welfare-maximizing tariff is $\tau_1 = \frac{1}{\lambda_2}\left[(\sigma_2(1+\tau_2)-1)\frac{c^2_1}{c^2_1+d} - \frac{d}{p c^1_2}\right]$. Here $d>0$ is the trade deficit, $\lambda_2$ is the partner's domestic absorption, $\sigma_2$ its import elasticity, $\tau_2$ its tariff, and $p$ the relative price of the imported good; the deficit appears in both the elasticity-dampening ratio and the extra terms-of-trade term. The paper embeds the same logic in a multi-region, multi-sector quantitative trade model with input-output linkages and services, solves best-response tariffs with a genetic algorithm, and iterates to a Nash equilibrium. Exogenously removing deficits in the calibrated model isolates the imbalance channel and flips the U.S. welfare sign.

What would settle it

Estimate how much the U.S.–China bilateral trade deficit falls when tariffs rise, using the 2018–2019 episode; if an endogenous-deficit version of the model turns the U.S. welfare change negative, or if the deficit shrinks enough to erase the +0.008 percent gain, the central claim fails.

Watch

Extended reading notes

Core claim

The central claim is that trade deficits raise optimal tariffs and can convert a trade war into a net gain for the deficit country. The mechanism appears in the paper's formula for the optimal tariff, where the deficit term $d$ enters twice: it dampens the partner's trade elasticity and retaliation terms, and it adds a terms-of-trade motive tied to the size of the deficit relative to imports. In the quantitative application, Nash-equilibrium tariffs average 10.41 percent for the United States and 17.04 percent for China from the pre-trade-war baseline; U.S. welfare rises by 0.008 percent and China's falls by 0.138 percent. When bilateral and aggregate deficits are eliminated, the U.S. change becomes $-0.019$ percent, so the sign of the U.S. result hinges on the imbalance. The paper also concludes that the tariffs actually imposed in 2018 made both countries worse off, and that free trade dominates the trade war for both countries.

Load-bearing premise

The model keeps each country's aggregate trade deficit fixed when tariffs change; if a tariff war instead shrank the deficit, the terms-of-trade gain that produces the U.S. advantage would weaken.

Editorial extensions

If this is right

  • Relative to pre-trade-war tariff rates, the United States gains 0.008 percent of consumption from Nash-equilibrium tariffs with China, while China loses 0.138 percent.
  • Eliminating the bilateral U.S.–China deficit—with or without aggregate deficits—turns the U.S. welfare change negative (−0.019 percent), so the bilateral imbalance is the decisive factor.
  • Starting from free trade, both countries lose from the trade war, so free trade dominates the Nash war for both.
  • The United States would also gain from trade wars with Canada and India and lose against the European Union and Mexico; China would lose against almost all partners, with the United States the most damaging.
  • The 2018 tariff increases reduced welfare in both countries and were negatively correlated with U.S. Nash-tariff changes, implying the actual tariffs were not set to maximize welfare.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the trade deficit responds to tariffs—through exchange rates, savings, or relocation—the fixed-deficit assumption could shrink the U.S. gain; a dynamic model with persistent imbalances would test this directly.
  • The same calibration logic could be run on every bilateral pair to rank which deficit countries have the strongest incentive to start a trade war, turning the paper's two-country result into a cross-country prediction.
  • If deficits make trade wars attractive to deficit countries, then rising global imbalances should make multilateral tariff agreements harder to sustain, because large-deficit members have a credible outside option to escalate.
  • A sharper test of the political-economy reading would compare the sector pattern of actual tariff increases across deficit countries with the sector pattern of Nash-equilibrium tariffs implied by this model.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper studies how trade imbalances affect optimal tariff policy. In a two-country CES model, it claims that a trade deficit raises a country's optimal (welfare-maximizing) tariff, with the key algebraic result stated in equation (9). The paper then builds a multi-region, multi-sector Armington GE model with input-output linkages and service sectors, calibrates it to WIOD and TRAINS data, and computes optimal unilateral and Nash tariffs between the United States and China using a genetic algorithm. The main quantitative finding is that, relative to pre-trade-war tariff rates, the United States gains from a trade war with China, and that this gain is driven by the bilateral US-China imbalance: removing bilateral and aggregate deficits turns the US gain into a loss. The paper also evaluates the welfare effects of the 2018 tariff increases and of hypothetical trade wars with other partners, and reports robustness in a Caliendo-Parro Ricardian framework.

Significance. If the results hold, the paper would make a useful contribution: it provides one of the first computations of optimal unilateral and Nash tariffs in a quantitative trade model with many sectors and input-output linkages, and it draws attention to an underexplored interaction between trade imbalances and tariff policy. The policy-relevant conclusion that the US-China imbalance makes the US better positioned in a tariff war is clearly laid out and subjected to substantial robustness analysis, including alternative sources of imbalances and a Ricardian specification. The numerical exercises in Section 6 give a systematic picture of how optimal tariffs vary with bilateral and aggregate deficits, and the historical evaluation of the 2018 trade war is a nice consistency check. However, the paper's central theoretical result, as printed in equation (9), is algebraically inconsistent with equation (8) and with the textbook special case the text itself cites, so the theoretical mechanism claimed in the abstract is not currently established by the derivation.

major comments (3)
  1. [Section 2, Eq. (9)] The central theoretical result is not established by the derivation as printed. Substituting d=0 and τ2=0 into equation (9) gives τ1=(σ2−1)/λ2, whereas equation (8) with d=0 yields d log c1_2/d log p = λ2(σ2−1), so equation (3) implies τ1=1/[λ2(σ2−1)], the textbook case that the text itself identifies. Moreover, for d>0 the term c2_1/(c2_1+d) is less than 1 and the second term −d/(p c1_2) is negative and grows in magnitude with d, so the printed formula implies lower, not higher, optimal tariffs for a larger deficit. The abstract's claim that 'greater trade deficits imply higher optimal tariffs' is therefore not supported by the current derivation. The formula needs to be corrected (a derivation from equation (8) yields τ1=(c2_1+d)/[λ2((σ2−1)c2_1−d)], which does increase with d), and the ensuing paragraph explaining the sign of the deficit effect must be rewritten accordingly.
  2. [Section 5.3 and Section 7] The verification of the computed Nash tariffs is incomplete. The paper states that after convergence it verifies through exhaustive search that no single-sector deviations improve welfare, and Figure 7 shows only one-sector-at-a-time deviations. In the 22-dimensional tariff space, a profitable deviation could involve simultaneous changes in multiple sectors, and the reported check does not rule that out. Since Tables 2–5 are described as Nash equilibrium outcomes, this verification gap is load-bearing for the quantitative welfare comparisons. The authors should either provide a verification that covers multi-dimensional deviations (for example, random multi-sector perturbations or an analytic first-order condition check) or explicitly label the results as candidate Nash equilibria.
  3. [Section 4, Eq. (22), and Section 7] The aggregate trade deficits Di are treated as exogenous lump-sum transfers and are held fixed in all counterfactual exercises, including the Nash tariff experiments. The headline result that the United States gains from a trade war relative to the 2014 tariff baseline depends on the persistence of the US-China imbalance under tariff changes that are much larger than those observed in 2018–2019. The evidence from Furceri et al. (2018) cited in the Introduction concerns the actual 2018 tariff changes and does not directly justify holding deficits fixed at the Nash counterfactual. The paper should either endogenize aggregate deficits (for example through a simple intertemporal closure) or clearly state this limitation and provide sensitivity analysis showing how the Table 4 results depend on the fixity of Di.
minor comments (6)
  1. [Section 2, p. 7] There is a typo, 'elasticty' for 'elasticity', and the sign convention in equation (3) should be stated explicitly, since d log c1_2/d log p may be negative outside the relevant region.
  2. [Section 7, Table 4 text] The phrase 'the the' appears twice in the paragraph after Table 4; please proofread the manuscript.
  3. [Section 5.3, Algorithm 1] The convergence criterion in Algorithm 1 uses max(∥Δτ_i^*∥, ∥ΔW_i^*∥), but the termination rule in Appendix C is based on a geometric mean of welfare improvements; the notation should be aligned.
  4. [Figure 7] The caption should specify which country's sectoral tariff is being varied and should clarify that the baseline is the free-trade Nash equilibrium; the current legend is ambiguous.
  5. [Appendix C, Algorithm C.1] The pseudocode indentation makes it difficult to determine which steps are inside the while loop; please restructure the algorithm listing for readability.
  6. [Section 3, Figures 1 and 2] The weighting by 'destination GDP' is not formally defined; specify the weighting variable and sample used for the Comtrade and BACI series.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: optimal tariffs are solved numerically, not fitted, and the deficit-elimination results are comparative statics rather than construction.

full rationale

The paper's derivation chain is self-contained. The illustrative two-country model derives optimal tariffs from CES preferences, the trade balance condition, and the planner's first-order condition; the optimal tariff is solved from these equations, not fitted to the data. The quantitative model calibrates taste shares, elasticities (from external estimates by Fontagne et al.), IO shares, tariffs, and aggregate deficits from WIOD/TRAINS data, but the optimal unilateral and Nash tariffs are then computed by numerical optimization, and welfare changes are evaluated from the model's counterfactual equilibria rather than imposed to match any target. The experiments that eliminate bilateral or aggregate deficits are controlled comparative statics: they change the baseline and recompute optimal tariffs, so the finding that the U.S. gain 'hinges on' the bilateral imbalance is a model prediction, not a property forced by construction. The self-citations (Kehoe, Pujolas, and Rossbach 2017; Kehoe et al. 2024) are methodological references; the model equations are stated fully in the paper, and no uniqueness theorem or central premise is imported solely from prior work by the same authors. A skeptic's algebraic concern about equation (9) is a correctness or consistency issue, not circularity, because the quantitative results are solved numerically and do not reduce to equation (9) by construction. Therefore, no circular step is exhibited.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The central result depends on calibrated trade deficits, elasticity parameters, and the exogeneity of deficits. The paper provides robustness to some assumptions (e.g., iceberg costs, eliminating deficits) but does not model endogenous deficits or test the sensitivity to the full range of elasticity uncertainty.

free parameters (2)
  • Service sector Armington elasticity = 5.5
    Set to 5.5 for all service industries because Fontagné et al. (2022) did not estimate elasticities for services. This hand-chosen value affects the quantitative optimal tariffs and welfare results.
  • Sectoral Armington elasticities for 22 tradable sectors = Varies by sector (from Fontagné et al., 2022)
    Taken from external estimates based on product-level tariff variation. These values are not tuned by the authors but are load-bearing inputs to the optimal tariff calculation.
assumptions (7)
  • domain assumption Aggregate trade deficits Di are exogenous transfers in the household budget constraint and are fixed under counterfactual tariffs.
    Equation (14) adds Di to income, and the model solves counterfactuals holding Di constant. The main U.S.-China result depends on this fixity, and the paper does not model how tariffs might change the deficit.
  • domain assumption There are no unobserved iceberg trade costs in the main calibration (t_s_ij = 1).
    Section 5.2 sets t_s_ij = 1, which attributes all bilateral trade flow asymmetries to preference shocks (gamma_s_ij). Appendix B.2 relaxes this, but the baseline results rely on it.
  • domain assumption Governments set tariffs to maximize domestic welfare, ignoring political economy motives.
    The objective in equation (32) is consumer welfare only. The authors later argue that actual tariffs were politically motivated, so the normative benchmark is distinct from observed policy.
  • domain assumption The model is static; trade imbalances are treated as exogenous transfers rather than arising from intertemporal savings and investment decisions.
    The paper discusses dynamic models in the introduction but adopts a static framework where deficits are parameters. This is a deliberate simplification that may affect the policy implications.
  • domain assumption In the two-country illustrative model, the pattern of trade (country 1 imports good 2) is assumed unchanged by tariffs, so the exogenous deficit d is consistent with equilibrium.
    The derivation of equation (9) assumes country 1 continues to import good 2 and country 2 imports good 1 even when tariffs are applied. The paper later allows autarky in numerical exercises, but not in the main theoretical result.
  • domain assumption Sectoral elasticities from Fontagné et al. (2022) are valid for the applied general equilibrium model.
    The optimal tariff rates depend crucially on these elasticity estimates. If the true elasticities differ, the quantitative results would change.
  • ad hoc to paper The genetic algorithm converges to the global optimum of the welfare function, though only single-sector deviations are verified.
    The paper relies on the genetic algorithm to find best-response tariffs and iterates to a Nash equilibrium. It verifies local single-sector deviations but not joint multi-sector deviations, so the reported equilibrium might not be a true global Nash equilibrium.

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Cite this review

Pith. "Pith review of Trade Wars with Trade Deficits." pith.science (2026). https://pith.science/paper/PFXV4LNS

@misc{pith2026241115092,
  author       = {Pith},
  title        = {Pith review of: Trade Wars with Trade Deficits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PFXV4LNS}},
  note         = {Machine review of arXiv:2411.15092}
}
abstract

Trade imbalances significantly alter the welfare implications of tariffs. Using an illustrative model, we show that trade deficits enhance a country's ability to alter its terms of trade, and thereby benefit from tariffs. Greater trade deficits imply higher optimal, or welfare maximizing, tariffs. We compute optimal unilateral and Nash equilibrium tariffs between the United States and China $\unicode{x2014}$ the countries with the largest bilateral trade imbalance $\unicode{x2014}$ using a multi-region, multi-sector applied general equilibrium model with service sectors and input-output linkages, a computationally complex task. Free trade benefits both countries compared to a trade war. Relative to existing tariff rates, however, the United States gains from a trade war with China $\unicode{x2014}$ a result that hinges on their bilateral trade imbalance.

Figures

Figures reproduced from arXiv: 2411.15092 by the authors.

Figure 2
Figure 2. Mean Aggregate Imbalance as Percentage of Gross Domestic Product 2 3 4 5 6 7 Aggregate Imbalance (% of GDP) 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 Year Comtrade BACI We further report changes in aggregate trade imbalances, which we compute as Agg.Imbalanceit = [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Optimal Tariffs with Varying Trade Imbalances (Two Countries) 20 25 30 35 40 Optimal Tariff Rate (%) −10 −5 0 5 10 Country 1 Deficit (% of Expenditures) Unilateral: Country 1 Nash: Country 1 Nash: Country 2 23 [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Optimal Tariffs and Welfare Changes with Varying Trade Imbalances (Three Countries, Bilateral Deficits Only) −10 −5 0 5 10 Change in Welfare (%) 0 20 40 60 Optimal Tariff Rate (%) −60 −40 −.20 0 20 40 60 Country 1 Deficit with Country 2 (% of Expenditures) Tariff: Unilateral Tariff: Country 1 Tariff: Country 2 Welfare: Unilateral Welfare: Country 1 Welfare: Country 2 [PITH_FULL_IMAGE:figures/full_fig_p026_4.png] view at source ↗
Figures from the paper (3 more)
Figure 5
Figure 5. Figure 5: Optimal Tariffs with Varying Trade Imbalances (Aggregate and Bilateral Deficits with Multiple Countries) −10 −7.5 −5−2.5 0 2.5 Welfare Change (%) 5 10 15 20 25 Optimal Tariff Rate (%) −40 −20 0 20 40 Country 1 Bilateral Deficit with Country 2 (% of Expenditures) Tariff…
Figure 6
Figure 6. Figure 6: Welfare Impact of a 10 Percent Tariff with Varying Trade Imbalances −2 0 2 4 Change in Welfare (%) −40 −20 0 20 40 Country 1 Bilateral Deficit with Country 2 (% of Expenditures) Unilateral: Two Countries Retaliation: Two Countries Unilateral: Three Countries Retaliatio…
Figure 8
Figure 8. Figure 8: Welfare Implications of a Unilateral Tariff Rate Reduction by China −0.025 0.000 0.025 0.050 0.075 0.100 Welfare Change from Nash Baseline (%) 0 20 40 60 80 100 China Unilateral Tariff Rate Reduction (%) USA: Welfare vs Nash China: Welfare vs Nash Given that the United…

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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    Allen, T., Arkolakis, C., and Takahashi, Y. (2019). ‘Universal Gravity’. Journal of Political Economy, volume 128, no. 2, 393–433. Amiti, M., Redding, S. J., and Weinstein, D. E. (2019). ‘The Impact of the 2018 Tariffs on Prices and Welfare’. Journal of Economic Perspectives, volume 33, no. 4, 187–210. Armington, P . S. (1969). ‘A Theory of Demand for Pro...

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    Table B.3 reports optimal unilateral and Nash equilibrium tariffs for the United States with and without the restriction that China is forced to enact uniform tariffs itself

    Furthermore, we reach the same conclusion that the United States gains from a trade war relative to the pre-trade war baseline, but experiences small welfare losses relative to a free-trade baseline. Table B.3 reports optimal unilateral and Nash equilibrium tariffs for the United States with and without the restriction that China is forced to enact unifor...

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    The MIT Press. Leontieff, W. (1936). ‘Note on Oure Theory of Capital Transfer’. In ‘Explorations in Economics: Notes and Essays Contributed in Honor of F. W. Taussig’, McGraw Hill. Li, L., Jamieson, K., DeSalvo, G., Rostamizadeh, A., and Talwalkar, A. (2018). ‘Hyper- band: A Novel Bandit-Based Approach to Hyperparameter Optimization’. Journal of Machine L...

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    Finally, the goods market clear, so that total output equals final demand plus inter- mediate input usage, ys i = cs i + N ∑ n=1 Z ms ni(ωs)dωs

    πs inXs i τs ij , (43) where equation (43) is the CP model counterpart of equation (21), and to write the aggregate trade imbalance as Di = S ∑ s=1 J ∑ j=1 πs inXs i τs ij − J ∑ j=1 πs jnXs i τs ji ! , (44) where equation (44) is the CP model counterpart of equation (22). Finally, the goods market clear, so that total output equals final demand plus inter...

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    Tariff revenues, trade deficits, income, and final consumption share parameters are calibrated with the same exact equations and data as in the baseline model. The only remaining objects to calibrate are the trade shares, πs in, which are calibrated to match expenditure shares in the data (expenditure flow divided by expenditures summed across sources, in...

    work page 2008

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    We also report the mean absolute and percentage difference between the numerical elasticities for the United States and China

    The numerical elasticities are computed at the baseline and reported for eight different scenarios (whether bilateral or deficits are present and whether the benchmark has free-trade or observed tariffs). We also report the mean absolute and percentage difference between the numerical elasticities for the United States and China. For the percentage differ...

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    We then initialize a population, P, of solution can- didates, τ{p,g} USA ≡ {τ1 USA ,CHN , ...,τ22 USA ,CHN }{p,g}, where p ∈ P represents candidate p of generation g. After including any user-specified seed candidates — for example from a previous execution of the genetic algorithm — we randomly generate candidate solu- tions until we have P/2 candidates ...

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    Even without the hybrid approach, choosing smaller stall limits and re-initializing the algorithm multiple times can yield improved performance

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    do 7: Fitness Scaling (Rank Fitness Scaling) Compute candidate fitness: F{p,g} 8: Rank candidates τ{p,g} USA by W{p,g} USA (rank 1 for highest welfare). 9: Compute F{p,g} ≡ 1/ √ r{p,g}, where r{p,g} is rank of p within g − 1 10: Include top Pelite = 16 ranked candidates as eli...

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    the baseline model

    trade model and uses the exact-hat notation from Dekle et al. (2008). A.1 Model Differences We closely follow the structure and model of the CP model, therefore our focus here is on how this framework differs, and is similar to, our baseline Armington model in Section 4 (hence...

    2008

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