Profits, Scale Economies, and

the Gains from Trade and Industrial Policy∗

Ahmad LashkaripourIndiana University

Volodymyr LugovskyyIndiana University

First version: August, 2016This version: April, 2020

Abstract

Trade taxes are often used as (a) a first-best instrument to manipulate the terms-of-trade, or (b) a second-best instrument to correct pre-existing market distortions.We analyze the (in)effectiveness of trade taxes at achieving these two policy goals.To this end, we derive sufficient statistics formulas for first-best and second-besttrade/domestic taxes in an important class of multi-industry quantitative trademodels featuring market distortions. Guided by these formulas, we estimate thekey parameters that govern the gains from policy in these frameworks. Our esti-mates indicate that (i) the gains from terms-of-trade manipulation are relativelysmall; (ii) trade taxes are remarkably ineffective at correcting domestic distortions;(iii) unilateral domestic policies are also ineffective as they worsen the terms-of-trade; but (iv) deep trade agreements that prohibit trade taxation and encouragea multilateral adoption of corrective policies, deliver welfare gains that are largerthan any non-cooperative policy alternative.

1 Introduction

Over the past few years, trade and industrial policy have reemerged at the cen-ter stage of heated political debates. The Trump administration recently initiated theNational Trade Council with the mission to restructure the United States’ trade and in-dustrial policy. This initiative itself was marketed as an effort to counter the Chinesetrade and industrial policy initiative known as “Made in China 2025.”1

∗We are grateful to Adina Ardelean, Dominick Bartelme, Kerem Cosar, Arnaud Costinot, Farid Far-rokhi, Harald Fadinger, Fabio Ghironi, David Hummels, Kala Krishna, Konstantin Kucheryavyy, NunoLimao, Gary Lyn, Ralph Ossa, James Rauch, Andrés Rodríguez-Clare, Kadee Russ, Peter Schott, Alexan-dre Skiba, Anson Soderbery, Robert Staiger, Jonathan Vogel and conference participants at the MidwestTrade Meetings, Chicago Fed WIE, UECE Lisbon Meetings, 2017 NBER ITI Summer Institute, 2019 WCTW,Indiana University, Purdue University, SIU, and University of Mannheim for helpful comments and sug-gestions. We thank Nicolas de Roux and Santiago Tabares for providing us with the data on the ColombianHS10 product code changes over time. We are grateful to Fabio Gomez for research assistance. Lugovskyythanks Indiana University SSRC for financial support. All remaining errors are our own.

1This level of interest in trade and industrial policy is not unprecedented. In 1791 Alexander Hamilton

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The revival of real-world interest in trade and industrial policy has re-ignited aca-demic interest in these topics, resurfacing some old but unresolved questions: (i) Howlarge are the gains from terms-of-trade (ToT) manipulation in the presence of marketdistortions and general equilibrium linkages? (ii) Are trade taxes an effective second-best instrument for correcting pre-existing market distortions? (iii) If not, should gov-ernments correct these distortions with unilateral domestic policy measures, or (iv)should they coordinate their corrective policies via deep trade agreements?

Despite all the recent advances in quantitative trade theory, addressing the abovequestions remains challenging for two reasons:

First, we lack a sharp analytical understanding of optimal first-best and second-bestpolicies in general equilibrium trade models with pre-existing market distortions. Theprevailing views on optimal trade/industrial policy are mostly based on two-good orpartial equilibrium trade models.2

Second, we lack compatible estimates for the parameters that govern the ToT andcorrective gains from policy. We possess multiple estimates for the trade elasticity thatgoverns the gains from ToT manipulation. Likewise, we possess multiple estimatesfor parameters governing pre-existing market distortions and the corrective gains frompolicy. However, we lack a unified technique that can estimate both of these parametersin a mutually consistent way.

In this paper, we aim to fill this gap along both dimensions. We first present ananalytical characterization of optimal trade/industrial policy in an important class ofmulti-industry quantitative trade models where market distortions arise from scaleeconomies or profits. Our characterization delivers simple, sufficient statistics formulasfor first-best and second-best trade and domestic taxes.

We then employ micro-level trade data to estimate the structural parameters thatgovern the gains from trade/industrial policy in our framework. Using these estimatesand our sufficient statistics formulas, we quantify the gains from trade/industrial pol-icy under various scenarios. Our analysis delivers several stark predictions: (i) thegains from ToT manipulation are relatively small; (ii) trade taxes are remarkably in-effective at correcting domestic distortions; (iii) most countries lose from a unilat-eral adoption of corrective domestic policies; but (iv) the gains from a “deep” tradeagreement that corrects market distortions multilaterally dominate those of any non-

approached Congress with a report (namely, “the Report on the Subject of Manufactures”) that encouragedthe implementation of protective tariffs and targeted subsidies. At that time, these policies were intendedto help the US economy catch up with Britain’s economy.

2Recently, several studies have taken notable strides towards characterizing the optimal trade andindustrial policies in modern quantitative trade models with distortions. Two prominent examples areCampolmi, Fadinger, and Forlati (2014) and Costinot, Rodríguez-Clare, and Werning (2016) who respec-tively characterize optimal policy in the two-sector Krugman and Melitz models featuring (i) a differen-tiated and a costlessly traded homogeneous sector, (ii) fixed wages, and (iii) zero cross-substitutabilitybetween sectors. As noted by Costinot et al. (2016), given the state of the literature and beyond thesespecial cases, “there is little that can be said, in general, about the optimal structure of macro-level taxes” whendealing with “arbitrarily many sectors.”

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cooperative policy alternative.Section 2 presents our theoretical framework. Our baseline model is a generalized

multi-industry Krugman (1980) model that features (i) a non-parametric utility aggre-gator across industries, and (ii) a nested CES utility aggregator within industries thatallows for the degrees of firm-level and country-level market power to diverge. Witha basic reinterpretation of parameters, our framework also nests (a) a multi-industryMelitz (2003) model with a Pareto productivity distribution, and (b) a multi-industryEaton and Kortum (2002) model with industry-level Marshallian externalities.

Section 3 derives sufficient statistics formulas for first-best and second-best trade anddomestic taxes in the aforementioned frameworks. To summarize these formulas, notethat a non-cooperative government can use taxes to correct two types of inefficiency:

i. A ToT inefficiency that concerns unexploited market power with respect to the restof the world, and

ii. An allocative inefficiency that concerns sub-optimal production in high-returns-to-scale (or high-profit) industries.3

The first-best optimal trade taxes are targeted solely at correcting the ToT inefficiency.They include export taxes that equal the national-level optimal mark-up on industry-level exports and import taxes that equal the national-level optimal mark-down onindustry-level imports. Likewise, the first-best domestic taxes only correct the alloca-tive inefficiency. They are Pigouvian subsidies that equal the inverse of the industry-level scale elasticity (or markup) and restore allocative efficiency in the local economy.

The second-best trade tax formulas apply to cases where governments cannot usedomestic subsidies, due to say institutional barriers. Second-best trade taxes featurea standard ToT-correcting component and an additional misallocation-correcting compo-nent. The latter component restricts imports and subsidizes exports in high-returns-to-scale industries in an attempt to mimic the restricted Pigouvian subsidies.

Second-best trade taxes, therefore, face a lesser-known trade-off: Correcting alloca-tive inefficiency with trade taxes often worsens the ToT inefficiency. This trade-off lim-its the effectiveness of trade taxes as a second-best corrective policy measure; beyondwhat is already implied by the well-known targeting principle. Similar arguments ap-ply to second-best import taxes, which are relevant when both export taxes and domesticsubsidies are restricted.4

The flip side of the above observation is that a unilateral adoption of corrective do-mestic policies can also undermine welfare. That is because adopting corrective Pigou-

3The link between the degree of scale economies and firm-level market power has deep roots in theliterature. Under free entry and monopolistic competition the following relationship holds (Hanoch (1975)and Helpman (1984)): elasticity of firm-level output w.r.t. input cost = firm-level markup.

4This scenario has been a focal point in the recent trade policy literature. We show that second-best op-timal import taxes feature an additional uniform component that accounts for ToT manipulation throughgeneral equilibrium wage effects. This uniform term is redundant when export taxes are available becauseof the Lerner symmetry, but plays a key role otherwise.

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vian subsidies without raising trade taxes can worsen a country’s ToT. The solutionto this problem is for countries to coordinate their corrective domestic policies via adeep trade agreement. In this process, they abolish trade taxes and forgo the gainsfrom trade taxation; but benefit from corrective subsidies in the rest of the world. Ifthe extent of allocative inefficiency is sufficiently large, such an agreement will deliverwelfare gains that dominate those of any non-cooperative policy. This is true even ifnon-cooperation does not trigger retaliation by trading partners.

Section 5 estimates the structural parameters that govern the gains from policy inour framework. Our sufficient statistics formulas indicate that the gains from policydepend on (i) industry-level trade elasticities that govern the extent of the ToT ineffi-ciency, and (ii) industry-level scale elasticities (or markups) that govern the extent ofallocative inefficiency. We develop a new methodology that simultaneously estimatesthese parameters using transaction-level trade data.

Our approach involves fitting a structural firm-level import demand function to theuniverse of Colombian import transactions from 2007 to 2013. Our data covers 226,288exporting firms from 251 different countries. The main advantage of our approachis its unique ability to separately identify the degree of firm-level market power (thatdetermines the scale elasticity) from the degree of national-level market power (thatdetermines the trade elasticity).

The firm-level nature of our empirical strategy exposes us to a non-trivial identifi-cation challenge. Standard estimations of import demand are often conducted at thecountry level and use tariffs as an exogenous instrument to identify the underlyingparameters. This identification strategy is not fully-applicable to our firm-level estima-tion. To attain identification, we instead compile a comprehensive database on monthlyexchange rates. We then interact aggregate movements in monthly bilateral exchangerates with (lagged) monthly firm-level export sales to construct a shift-share instrumentthat measures exposure to exchange rate shocks at the firm-product-year level.

Section 6 combines our micro-level estimates, our sufficient statistics formulas, andmacro-level data from the 2012 World Input-Output Database to quantify the gainsfrom policy for 32 major economies. Our main findings can be summarized as follows:

i. The pure gains from ToT manipulation are relatively small. Suppose a coun-try applies its first-best non-cooperative trade and domestic taxes and the rest ofthe world does not retaliate. In that case, the average country can raise its realGDP by 1.7% under restricted entry and by 3% under free entry. However, onlyone-third of these gains are driven by ToT considerations. The rest is driven byrestoring allocative efficiency in the domestic economy.

ii. Trade taxes are remarkably ineffective instrument at correcting domestic marketdistortions. Under restricted entry, second-best trade taxes can raise the real GDPby only 0.5% for the average country. That amounts to only 29% of the gains

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attainable under the fist-best policy schedule. Second-best import taxes are evenless effective, raising the real GDP by a mere 0.4%. As noted earlier, using tradetaxes to improve allocative inefficiency worsens the ToT. This tension renderstrade taxes ineffective at simultaneously targeting and correcting both the ToTand allocative inefficiencies.

iii. A unilateral adoption of corrective domestic policies can be even less effective. Itsreduces the average country’s real GDP by 0.36% under restricted entry and by1.18% under free entry. As noted earlier, these losses result from ToT-worseningeffects, which are relevant when corrective domestic policies are applied eitherunilaterally or are not accompanied by appropriate trade taxes.5

iv. A multilateral adoption of corrective policies via a deep trade agreement, deliverswelfare gains that are larger than the first-best non-cooperative policy for mostcountries. This is true even if non-cooperation does not trigger retaliation bytrading partners. To provide numbers, the gains from deep cooperation are onaverage 0.2% percentage points higher than the gains from the first-best non-cooperative policy. In other words, governments are better off forgoing the ToTgains from policy in return for importing more-efficiently-produced goods fromthe rest of the world.

Related Literature

Our paper contributes to the existing literature in a number of areas. Our first-besttax formulas generalize those in Campolmi et al. (2014), Costinot et al. (2016), Cam-polmi, Fadinger, and Forlati (2018), and Bartelme, Costinot, Donaldson, and Rodriguez-Clare (2018) by allowing for a large economy, arbitrarily many industries, and arbitrarycross-demand effects.6 We also present a novel characterization of optimal (first-best)policy in the presence of political economy pressures and input-output linkages.

Our second-best tax formulas, to the best of our knowledge, are the first to analyt-ically characterize second-best export and import taxes in general equilibrium quan-titative trade models with market distortions. Previous studies on second-best tradetaxes either neglect pre-existing market distortions, are partial equilibrium in nature,or focus on the local effects of piecemeal trade tax reforms.7

5This result highlights a benefit of international coordination that is typically overlooked in critiquesof global governance (e.g., Rodrik (2019)).

6To be clear, we generalize the macro-level optimal tax formulas in Costinot et al. (2016). The afore-mentioned study also derives optimal firm-level tax formulas, which are not accommodated by our paper.Campolmi et al. (2018) allow for an arbitrary firm-productivity distribution but abstract from generalequilibrium wage, income, and cross-industry demand effects Relatedly, Demidova and Rodriguez-Clare(2009) and Felbermayr, Jung, and Larch (2013) characterize optimal import taxes in a single industry, con-stant markup framework. In their setups, the market outcome is efficient (Dhingra and Morrow (2019)).Moreover, since the economy is modeled as one industry, import taxes and export taxes are equivalent bythe Lerner symmetry. So, the first-best allocation can be attained with only import taxes.

7See e.g., Bond (1990); Bagwell and Staiger (2012); Bagwell and Lee (2018); Caliendo, Feenstra, Roma-

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We build heavily on Kucheryavyy, Lyn, and Rodríguez-Clare (2016) when estab-lishing isomorphism between our baseline model and other workhorse models in theliterature. Kucheryavyy et al. (2016) identify the trade and scale elasticities as key de-terminants of the gains from trade. We estimate these two elasticities and demonstratethat the same elasticities govern the gains from optimal trade and domestic policies.

Our estimation of the scale and trade elasticities exhibits key differences from theprior literature. Our approach separately identifies the degree of firm-level power fromcountry-level market power. In comparison, the prior literature typically estimates thetrade elasticity with macro-level trade data.8 The scale elasticity is then normalized toeither zero or to the inverse of the trade elasticity, which can create an arbitrary linkbetween firm-level and country-level market power (Benassy (1996)).9

A notable exception is Bartelme et al. (2018), who concurrent with us have devel-oped a strategy to estimate the product of scale and trade elasticities. Their approachhas the advantage of detecting industry-level Marshallian externalities but relies on theassumption that there are no diseconomies of scale due to industry-specific factors ofproduction. Our approach cannot detect Marshallian externalities but has the advan-tage of separately identifying the trade elasticity from the scale elasticity and is robustto the presence of industry-specific factors of production.

Several studies have used the exact hat-algebra methodology to study the con-sequences of counterfactual tariff reductions (Costinot and Rodríguez-Clare (2014);Caliendo and Parro (2015); Ossa (2014, 2016); Spearot (2016)). We contribute to thesestudies by combining the exact hat-algebra technique with sufficient statistics tax for-mulas. Doing so simplifies the analysis of optimal policy, allowing us to bypass some ofthe most important computational challenges facing this literature.

Our quantitative analysis of deep trade agreements has little precedent in the liter-ature. Our closest counterpart is Ossa (2014), who quantifies the gains from an agree-ment whereby countries cooperate in setting second-best corrective import tariffs. Wecontribute to Ossa (2014) by analyzing cooperation in first-best corrective policies. Do-ing so enables us to highlight a previously-neglected tension between the correctiveand ToT gains from taxation.

Finally, our paper is related to a vibrant literature that studies the effects of tradeopenness on allocative efficiency (e.g., Bond (2013); De Blas and Russ (2015); Edmond,Midrigan, and Xu (2015); Baqaee and Farhi (2019)). These studies highlight that tariffliberalization can exacerbate pre-existing market distortions. Related to this argument,we highlight a previously-unknown tension between restricting trade with tariffs andpreserving allocative inefficiency in the local economy.

lis, and Taylor (2015); Demidova (2017); Beshkar and Lashkaripour (2019); Costinot and Rodríguez-Clare(2014); Costinot, Donaldson, Vogel, and Werning (2015).

8See Broda and Weinstein (2006), Simonovska and Waugh (2014), Caliendo and Parro (2015), Soder-bery (2015), and Feenstra, Luck, Obstfeld, and Russ (2018) for different estimation approaches.

9See Kucheryavyy et al. (2016) for a review on how the scale and trade elasticities are treated acrossdifferent workhorse trade models.

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2 Theoretical Framework

We consider a world economy consisting of a Home country (h) and an aggregate ofthe rest of the world, which we call Foreign ( f ). C = h, f denotes the set of countries.There are k = 1, ..., K industries in each country, which can differ in fundamentals suchas the degree of scale economies or trade elasticity. K denotes the set of industries.Each country i ∈ C is populated by Li individuals who are each endowed with oneunit of labor, which is the sole factor of production. All individuals are perfectly mobileacross the production of different goods and are paid a country-wide wage, wi, but areimmobile across countries.

The two country assumption, as shown in Section 4, can be relaxed with no essentialchange to our optimal tax formulas. We nonetheless maintain this assumption in ourbaseline setup for expositional purposes. Likewise, the assumption that labor is the solefactor of production can be relaxed to accommodate intermediate inputs and accountfor global production networks. As shown in appendix 4, our baseline results extendto this richer setup with little qualification.

2.1 Preferences

Cross-Industry Demand. The representative consumer in country i ∈ C maximizes anon-parametric utility function subject to their budget constraint. The solution to thisproblem delivers the following indirect utility as a function of income, Yi, and industry-level “consumer” prices indexes in market i, Pi ≡ Pi,k:

Vi(Yi,Pi

)= max

Qi

Ui(Qi)

s.t. ∑k∈K

∑j∈C

Pi,kQi,k = Yi. (1)

In the above problem, Qi ≡ Qi,k denotes the vector of composite industry-level con-sumption, and Pi ≡ Pi,k denotes the corresponding consumer price indexes of thesecomposite bundles. To avoid any confusion later on, we should emphasize that thetilde notation distinguishes between “consumer” and “producer” prices. The formerincludes taxes, whereas the latter does not. Problem 1 yields an industry-level Marshal-lian demand function, which we denote by Qi = Di

(Yi, Pi

). This function tracks how

given prices and total income, consumers allocate their expenditure across industries.

Within-Industry Demand. Each industry-level bundle aggregates over various country-level composite varieties: Qi,k = Ui,k

(Qhi,k, Q f i,k

). Each country-level composite va-

riety aggregates over multiple firm-level varieties: Qji,k = Uji,k(qji,k), where qji,k =qji,k (ω)

ω∈Ωj,k

is a vector with each element qji,k (ω) denoting the quantity consumed

of firm ω’s output. Ωj,k denotes the set of all firms serving industry k from country j.10

10As we elaborate later, in the presence of firm-selection effects, only a sub-set of firms in set Ωj,k serve

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As a key restriction, we assume that the within-industry utility aggregator, Ui,k (.), hasa nested-CES structure. This assumption enables us to abstract from variable markups,thereby directing attention to the scale-driven or profit-shifting effects of policy.

A1. The within-industry utility aggregator is nested-CES:

Qi,k =

(∑j∈C

Qρkji,k

)1/ρk

,

where ρk ∈ (0, 1). Qji,k denotes the CES-composite bundle of firm-level varieties from countryj:

Qji,k =

(∫ω∈Ωj,k

ϕji,k(ω)1−$k qji,k (ω)$k dω

)1/$k

.

where $k ∈ (0, 1) and ϕji,k(ω) is a constant demand-shifter that accounts for variety ω’sperceived quality in market i.

Defining εk ≡ ρk/(1− ρk) and ξk ≡ $k/(1− $k), the demand for national-level vari-ety ji, k (origin country j–destination country i–industry k) is given by

Qji,k =(

Pji,k/Pi,k)−εk−1 Qi,k, (2)

where Pji,k =(

∑ω∈Ωji,kϕji,k(ω) pji,k (ω)−ξk

)−1/ξkand Pi,k =

(∑j∈C P−εk

ji,k

)−1/εkdenote

the origin-specific and industry-level price indexes. Qi,k, as noted earlier, denotes theindustry-level demand that is given by Qi,k = Di,k

(Yi, Pi

). The demand facing individ-

ual firms from country j is also given by

qji,k (ω) = ϕji,k(ω)

(pji,k (ω)

Pji,k

)−1−ξk(

Pji,k

Pi,k

)−1−εk

Di,k(Yi, Pi

). (3)

Importantly, the above demand structure allows for the firm-level and national-level de-grees of market power to diverge, with ξk reflecting the degree of firm-level marketpower and εk reflecting the degree of national-level market power.

National-Level Demand Elasticities. Following Equation 2, the demand for aggre-gate variety ji, k is a function of total income in market i, Yi, and the vector of allorigin-specific consumer price indexes in that market, Pi,k ≡ Pi,g. Namely, Qji,k =

Qji,k(Yi, Pi,k

). To keep track of changes in national-level demand, we define the price

elasticities facing national-level variety ji, k as follows.

Definition D1. (i) [own price elasticity] ε ji,k ≡ ∂ ln Qji,k(.)/∂ ln Pji,k;(ii) [cross-price elasticity] ε

i,gji,k ≡ ∂ ln Qji,k(.)/∂ ln Pι,g for ι, g 6= ji, k.

each market. In that case, Ωji,k ⊂ Ωj,k will denote the set of firms serving market i from country j.

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Absent cross-substitutability between industries, the national-level demand elas-ticities are fully determined by parameter εk and national-level expenditure shares.Specifically, ε

i,gji,k = 0 if g 6= k, while

εi,kji,k =

−1− εk(1− λji,k) = j

εkλi,k 6= j,

where λji,k ≡ Pji,kQji,k/ ∑ Pi,kQi,k denotes the share of expenditure on aggregate va-riety ji, k. Present cross-substitutability between industries, the demand elasticity fea-tures an additional term that accounts for cross-industry demand effects.

2.2 Production and Firms

Economy j ∈ C is populated with a mass, Mj,k, of single-product firms in industryk that compete under monopolistic competition. Labor is the only factor of produc-tion. Firm entry into industry k is either free or restricted. Under restricted entry,Mj,k = Mj,k is invariant to policy. Under free entry, a pool of ex-ante identical firms canpay an entry cost wj f e

k to serve industry k from country j. After paying the entry cost,each firm ω ∈ Ωj,k draws a productivity z(ω) ≥ 1 from distribution Gj,k (z). Giventhe realization of z(ω), the marginal cost of producing and delivering goods to marketi is given by τji,kwj/z (ω), where τji,k denotes a flat iceberg transport cost. We sum-marize these production-side assumptions under the following umbrella assumptionregarding national-level producer price indexes.

A2. The “producer” price index of composite good ji, k is given by

Pji,k = (1 + 1/ξk) aji,kwj M−1/ξkj,k ,

where aji,k = τji,k[∫ ∞

1 zξk dGj,k(z)]− 1

ξk is invariant to policy.

The invariance of aji,k follows from the implicit assumption that there are no fixedcosts associated with serving individual markets. This assumption rules out selectioneffects but is not as restrictive as it may appear. As elaborated in Subsection 3.4, thepresent model is isomorphic to a model that admits firm-selection effects as long as theproductivity distribution, Gj,k(z), is Pareto.

Following Kucheryavyy et al. (2016), we refer to 1/ξk as the industry-level scaleelasticity. Given A2, 1/ξk is the elasticity at which the producer price index decreaseswith industry scale, as measured by Mi,k. In our baseline Krugman model, the industry-level scale elasticity is exactly equal to the industry-level markup.11 However, as shownin Section 3.4, our analytical characterization of optimal policy extends to alternative

11Some version of the equivalence between the scale elasticity and markup holds in any nonparametricmodel that assumes free entry and monopolistic competition. More specifically, under the aforementionedassumptions: elasticity of firm-level output w.r.t. input cost = firm-level markup (see Hanoch (1975) and Help-man (1984)).

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models where the scale elasticity and markup levels diverge. In these alternative mod-els, agglomeration externalities operate through physical productivity rather than thelove-for-variety.

We can summarize the supply-side of the economy in terms of a vector of producerprices and aggregate profit levels. Specifically, given the constancy of markups, pro-ducer prices can be reformulated as

Pji,k =

ρji,kwj if entry is restricted

$ji,kwjQ−1/(1+ξk)j,k if entry is free

, (4)

where $ji,k ≡ aji,k (1 + ξk) f ek and ρji,k ≡ aji,k (1 + 1/ξk) M−1/ξk

j,k are invariant to policy;and Qj,k ≡ ∑i aji,kQji,k denotes the effective output of country j in industry k. GivenA1 and A2, net profits (gross of entry costs) are a constant fraction of total revenue andgiven by

Πj =

∑k ∑i

(1

1+ξkPji,kQji,k

)if entry is restricted

0 if entry is free. (5)

2.3 The Instruments of Policy

The government in country i has access to a full set of revenue-raising trade anddomestic tax instruments. Namely,

i. industry-level import taxes, tji,k that are applied by country i to all varietiesimported from country j 6= i in industry k (with tii,k = 0);

ii. industry-level export taxes, xji,k that are applied by country j to all export va-rieties sold to market i 6= j in industry k (with xjj,k = 0);

iii. industry-level production taxes, sj,k, that are applied by country j on industryk’s output irrespective of the location of final sale; and

iv. industry-level consumption taxes, which are redundant given the availability ofthe other tax instruments and are normalized to zero hereafter.

Our specification of policy is quite flexible as it accommodates import or export sub-sidies (t < 0 or x < 0) as well as production subsidies (s < 0). A formal proof forthe redundancy of consumption taxes is provided in Appendix A. There is a simpleintuition behind this result: Country i ∈ h, f has access to 4 different tax instrumentsin each industry. These 4 tax instruments can directly manipulate 3 consumer priceindexes: Pji,k, Pij,k, and Pii.k. So, by construction 1 of the 4 tax instruments, in each in-dustry, is redundant. Here, we treat the consumption tax as a redundant instrument.12

The remaining tax instruments create a wedge between consumer price indexes,Pji,k and producer price indexes, Pji,k, as follows:

12With more than two countries (N > 2), Country i has access to 2(N− 1)+ 2 instruments per industry.These instruments can manipulate 2(N − 1) + 1 price variables, which implies the same redundancy.

10

Pji,k =(1 + tji,k

) (1 + xji,k

) (1 + sj,k

)Pji.k, ∀j, i ∈ C, k ∈ K. (6)

These tax instruments also generates/exhausts revenue for the tax-imposing country.The combination of all taxes imposed by country i ∈ C produce a tax revenue equal to

Ri = ∑k∈K

∑j∈C

(tji,kPji,kQji,k +

[(1 + xij,k)(1 + si,k)− 1

]Pij,kQij,k

). (7)

Tax revenues are rebated to the consumers in a lump-sum fashion. After we accountfor tax revenues, total income in country i equals wage income, wiLi, plus total profits,Πi, plus total tax revenues: Yi = wiLi + Πi +Ri, where Ri can be positive or negativedepending on whether country i’s policy consists of net taxes or subsidies.

2.4 Equilibrium

We assume throughout the paper that the necessary and sufficient conditions forthe uniqueness of equilibrium are satisfied.

A3. For any combination of tax instruments, equilibrium is unique.

Following Kucheryavyy et al. (2016), Assumption A3 is equivalent to assumingthat εk/ξk ≤ 1. Given the above assumption, any combination of taxes (x, t, and s) isconsistent with only one vector of wages, w. Accordingly, the set of feasible tax-wagecombinations can be defined as follows.

Definition. [Feasible Tax-Wage Combinations] The tax-wage combination A ≡ (x, t, s; w)

is feasible if (i) Producer prices for any good ji, k are characterized by 4; (ii) Consumer pricesfor any good ji, k are given by 6; (iii) Industry-level consumption choices are a solution to 1and demand for national-level varieties, Qji,k, is characterized by 2; (iv) Factor markets clear,wiLi + Πi = ∑n ∑k Pin,kQin,k, where profits are given by 5; and (v) Total income equals factorincome plus tax revenue, Yi=wiLi+Πi+Ri, whereRi is given by 7.

Following the above definition, we let F denote the set of all feasible tax-wage com-binations. For any feasible outcome, (x, t, s; w) ∈ F, we can express all equilibriumvariables including welfare as a function of taxes and wages:

Wi (x, t, s; w) ≡ Vi(Yi(x, t, s; w), Pi(x, t, s; w)

).

Noting this choice of notation, we present an intermediate result on the equivalence ofpolicy equilibria, which simplifies our subsequent analysis.

Lemma 1. [Equivalence of Policy Equilibria] For any a and a ∈ R+ (i) if A ≡ (1 +

ti, t−i,1 + xi, x−i, 1 + si, s−i; wi, w−i) ∈ F, then A′ ≡ (a(1 + ti), t−i, (1 + xi)/a, x−i, a(1 +si), s−i; awi/a, w−i) ∈ F. Moreover, (ii) the real welfare is preserved under allocations A andA′: Wi(A) = Wi(A′).

11

The above lemma is proven in Appendix B. It is a basic extension of the Lernersymmetry to an environment where domestic taxes are also applicable. It implies thatthere are multiple optimal tax combinations for each country i, which simplifies ourforthcoming task of characterizing the optimal policy. To elaborate, the contribution ofgeneral equilibrium wage and income effects to the optimal tax schedule is often sum-marized in uniform tax shifters. The redundancy established by Lemma 1, simplifiesthe handling of these uniform tax shifters to a great degree.

Table 1: Summary of Key Variables

Variable Description

Pji,k Consumer price index (exporter j–importer i–industry k)

Pji,k Producer price index (exporter j–importer i–industry k)

Yi Total income in country i

Πi Total profits in country i

Ri Total tax revenue in country i

wiLi Factor income in country i (wage×population size)

xk ≡ xh f ,k Home’s export tax in industry k (corresponding vector: x ≡ xk)

tk ≡ t f h,k Home’s import tax in industry k (corresponding vector: t ≡ tk)

sk ≡ sh,k Home’s production tax in industry k (corresponding vector: s ≡ sk)

λji,k Within-industry expenditure share (good ji, k): Pji,kQji,k/ ∑ Pi,kQi,k

rji,k Within-industry revenue share (good ji, k): Pji,kQji,k/ ∑ι Pjι,kQjι,k

λji,k Overall expenditure share (good ji, k): Pji,kQji,k/Yi

rji,k Overall revenue share (good ji, k): Pji,kQji,k/wjLj

ei,k Industry-level expenditure share (country i–industry k)

ξk Degree of firm-level market power in industry k ~ sub-national CES parameter

εk Trade elasticity in industry k ~ cross-national CES parameter

ε ji,k Own price elasticity of national-level demand: ∂ ln Qji,k/∂ ln Pji,k

εi,gji,k Cross price elasticity of national-level demand: ∂ ln Qji,k/∂ ln Pi,g

3 Sufficient Statistics Formulas for Optimal Policy

In this section, we derive sufficient statistics formulas for optimal trade and domes-tic taxes. These formulas are later employed to quantify the gains from policy. Beforederiving our formulas, let us briefly discuss the motives for taxation in our setup. Fromthe perspective of a non-cooperative government, taxes can correct two distinct types

12

of inefficiency:

a) A unilateral terms-of-trade (ToT) inefficiency that concerns unexploited export/importmarket power with respect to the rest of the world, and

b) An allocative inefficiency that results from differential profit margins or scale elas-ticities across industries.

Correcting the ToT inefficiency will always lead to a Pareto sub-optimal outcome. Thatis because such a correction has a beggar-thy-neighbor effect that transfers surplus fromthe rest of the world to the tax-imposing economy. Improving allocative inefficiency,however, is Pareto improving. There is sub-optimal global output in high-profit orhigh-returns-to-scale (high-1/ξ) industries. When this distortions is corrected, con-sumers all around the world reap the benefits.

Optimal Cooperative Policy. As a useful benchmark, we first characterize the opti-mal cooperative policy in our framework. Such a policy maximizes global welfare, asdefined by a population-weighted sum of welfare across all countries:

max(x,t,s;w)∈F

∑i∈C

αiWi (x, t, s; w) .

It is straightforward to show that the solution to above problem involves zero tradetaxes (t∗ji,k = x∗ji,k = 0 ∀ji, k) and corrective Pigouvian subsidies in all countries:

1 + s∗i,k =1

1 + 1/ξk∀i, k (8)

The above characterization applies to both the free and restricted entry cases, and isanalogous to the cooperative policy structure in Bagwell and Staiger (2001). The gainsfrom policy can be strictly different under the free and restricted entry cases, though.The gains are typically larger under the former where optimal policy corrects distor-tions due to sub-optimal entry decisions—we come back to this point in Section 6.

The fact that international cooperation restores allocative efficiency at the globallevel is an artifact of assuming a non-politically-weighted objective function. We can,thus, think of the policy schedule highlighted above as the outcome of deep coopera-tion or a deep trade agreement. As we discuss in Section 4, domestic taxes applied bypolitically-motivated governments can worsen allocative efficiency without necessarilyimproving one’s ToT.

3.1 First-Best Trade and Domestic Taxes

Now we turn to characterize Home’s optimal non-cooperative policy. We considera case where Home applies taxes tk ≡ t f h,k, xk ≡ xh f ,k, and sk ≡ sk, but the rest of theworld is passive, th f ,k = x f h,k = s f ,k = 0. So, the vectors t ≡ tk, x ≡ xk, and

13

s ≡ sk, hereafter, correspond to only the Home government’s policy instruments.With this choice of notation, Home’s optimal (first-best) non-cooperative taxes solvethe following problem:

max(t,x,s;w)∈F

Wh (t, x, s; w) (P1).

Below, we analytically solve Problem (P1) under both the restricted and free entrycases. These two cases bear some resemblance but also exhibit key differences.

3.1.1 The Restricted Entry Case

Under restricted entry, the direct passthrough of a tax onto own consumer price iscomplete and the cross-passthrough is zero: ∂ ln Pji,k(.)/∂ ln(1+ tji,k) = 1, but ∂ ln Pji,k(.)/∂ ln(1+tι,g) = 0 if ji, k 6= ι, g. So, aside from general equilibrium wage and profit-shifting ef-fects, the burden of tk falls entirely on Home consumers, while the burden of xk falls onForeign consumers.

Considering this, export taxes (xk) can be used to extract additional mark-ups fromForeign consumers on an industry-by-industry basis. Import taxes (tk), however, canonly impose a flat mark-down on foreign prices (Pf h,g) through general equilibriumwage effects. Production taxes can serve as a first-best instrument to eliminate alloca-tive inefficiency from markup heterogeneity. The following theorem summarizes theseresults, providing sufficient statistics formulas for optimal tax rates.

Theorem 1.A. [Optimal Policy Under Restricted Entry] With restricted entry, the optimalpolicy is unique up to two uniform tax shifters s and t ∈ R+, and is given by

[(1 + s∗k ) (1 + 1/ξk) / (1 + s)]k = 1

[(1 + t∗k ) / (1 + t)]k = 1

[1/ (1 + x∗k ) (1 + t)] k = E−1h f(

I + Eh f)

1

where Eh f ≡[ε

h f ,gh f ,k

]k,g

is a K× K matrix composed of reduced-form demand elasticities.13

The above theorem is proven in Appendix E. The uniform tax shifters, s, and t, canbe assigned any arbitrary value, which results from the multiplicity of optimal policyequilibria as highlighted by Lemma 1. In summary, the above theorem states that theoptimal domestic policy is a Pigouvian subsidy that solely restores allocative efficiencyin the economy. The optimal import tax can be normalized to zero by choice of t = 0and is essentially redundant. This redundancy is due to the Lerner symmetry, wherebythe uniform mark-down chargeable with import taxes can be perfectly mimicked with auniform increase in export taxes. The optimal export tax is equal to the optimal markup

13More generally, Eh f ≡ [λh f ,g/λh f ,kεh f ,kh f ,g]k,g. But when preferences are homothetic, the symmetry of the

Slutsky matrix entails that λh f ,gεh f ,kh f ,g = λh f ,kε

h f ,gh f ,k , which delivers the above specification.

14

of a country that resembles a multi-product monopolist internalizing all cross-demandeffects.

To gain a deeper intuition, consider a special case where there is zero cross-substitutabilitybetween industries. In that case, the formulas specified by Theorem 1.A reduce to14

1 + s∗k =1 + s

1 + 1/ξk, ∀k

1 + t∗k = 1 + t, ∀k

1 + x∗k = (1 + t)−1(

1 +1

εkλ f f ,k

), ∀k

with the export tax now corresponding to the optimal markup of a single product mo-nopolist facing a CES demand. A well-known special case of the above formula is thesingle-industry formula in Gros (1987), where, x∗ = 1/ελ f f .15 Based on the above,we can express the optimal tax schedule as a function of three sufficient statistics: (i)observable expenditure shares, λ f f ,k, (ii) industry-level trade elasticities, εk, and (iii)industry-level scale elasticities, 1/ξk. This feature greatly simplifies our quantitativeanalysis of optimal policy in Section 6.

Interestingly, the optimal trade taxes specified above are independent of ξk, whichmeans they do not depend of the degree of allocative inefficiency in the economy. Theoptimal trade tax formulas are actually identical to that arising from a perfectly com-petitive model. This outcome corresponds to the result in Kopczuk (2003), wherebywe can first eliminate the underlying market inefficiency with the appropriate policyinstruments and then solve Problem (P1) as if markets where efficient.

3.1.2 The Free Entry Case

Under free entry, taxes can affect Foreign prices through scale or firm-delocation ef-fects. Specifically, a tax on good ji, k, can affect the scale of production for that good aswell as all other goods through cross-demand and firm-entry effects. As a result, thepassthrough of taxes onto own price is incomplete (i.e., ∂ ln Pji,k (t, x, s; w) /∂ ln(1 +

tji,k) 6= 1) while the passthrough onto other international prices is possibly non-zero(i.e., ∂ ln Pji,k (t, x, s; w) /∂ ln(1 + tι,g) 6= 0).

These considerations present the Home economy with an ability to manipulate itsimport market power beyond what is possible under restricted entry. Through firm-delocation, import tax-cum-subsidies can now charge a direct mark-down on foreignproducer prices, Pf h,k. The optimal tax schedule, characterized by the following theo-rem, reflects this additional degree of import market power.

14Recall that with zero cross-substitutability, εh f ,k = −εkλ f f ,k − 1 and εh f ,kh f ,g = 0 if k 6= g.

15In the single-industry model of Gros (1987), the economy is efficient and import and export taxes areperfectly substitutable. The first-best can, thus, be attained with either only import taxes or only exporttaxes. Gros (1987) considers only import taxes and shows that t∗ = 1/ελ f f . Alternatively, if one considersonly export taxes, the optimal policy formula becomes x∗ = 1/ελ f f .

15

Theorem 1.B. [Optimal Policy Under Free Entry] Under free entry, Home’s optimal first-best policy is unique up to two uniform tax shifters s and t ∈ R+, and is given by

[(1 + s∗k )(1 + 1/ξk)/(1 + s)

]k = 1[

(1 + t∗k )/(

1− r f h,k1+ξk

)(1 + t)

]k= 1[

1/(1 + x∗k )(

1− r f h,k1+ξk

)(1 + t)

]k= E−1

h f(

I + Eh f)

1

,

where Eh f ≡[

1+r f h,k/(1+ξk)

1+r f h,g/(1+ξg)ε

h f ,gh f ,k

]k,g

is a K × K matrix composed of reduced-form demand

elasticities, scale elasticities, and revenue shares.16

As with Theorem 1.A, the above theorem characterizes optimal taxes in terms ofa set of sufficient statistics: (i) observable expenditure and revenue shares, λji,k, andrji,k, (ii) industry-level trade elasticities, εk, and (iii) industry-level scale elasticities, ξk.Also, similar to the restricted entry case, optimal domestic taxes are solely correctivePigouvian subsidies, while optimal trade taxes only target the ToT inefficiency.

Theorems 1.A and 1.B, however, exhibit a key difference: import taxes are neitherredundant nor uniform under free entry. That is because, under free entry, a negativeimport tax on goods f h, k can induce a relocation of firms to industry k in Foreign. Thisdevelopment increases M f ,k and lowers the producer index of Home’s imports, Pf h,k,the extent of which depends on 1/ξk. Such a reduction is akin to Home exploitingits monopsony power to charge an industry-specific mark-down on Pf h,k. The non-uniform component of the optimal import tax in Theorem 1.B corresponds to the opti-mal rate of this mark-down.17

We can once more gain extra intuition by focusing on the special case where there iszero cross-substitutability between industries. In this special case, Theorem 1.B yieldsthe following formulas:

1 + s∗k =1 + s

1 + 1/ξk, ∀k

1 + x∗k =

(1 +

1εkλ f f ,k

)(1−

r f h,k

1 + ξk

)−1

(1 + t)−1, ∀k

1 + t∗k = (1 + t)(

1−r f h,k

1 + ξk

), ∀k.

The above formula is itself a strict generalization of the formula derived by Bartelmeet al. (2018) for a small open economy (for which r f h,k ≈ 0 and λ f f ,k ≈ 1). The character-ization of t∗k also bears resemblance to the classic optimal tariff formula, as it essentiallyindicates that the optimal tariff is related to the inverse of Foreign’s backward-falling

16More generally, Eh f ≡ [1+r f h,k/(1+ξk)

1+r f h,g/(1+ξg)

λh f ,g

λh f ,kε

h f ,kh f ,g]k,g. But when preferences are homothetic, the symme-

try of the Slutsky matrix entails that λh f ,gεh f ,kh f ,g = λh f ,kε

h f ,gh f ,k , which delivers the above specification.

17It is important to note that Home’s optimal policy encourages entry into high-ξ industries beyondwhat is socially optimal for the Foreign economy.

16

supply curve. In the case where Home is not a major market for Foreign industriesand lacks industry-level import market power (i.e., r f h,k ≈ 0 for all k), the above for-mulas converge to those derived under restricted entry. So, even though the free andrestricted entry cases exhibit key differences, these differences diminish with Home’ssize relative to the rest of the world.18

3.2 Second-Best Trade Taxes

Suppose the Home government cannot raise domestic taxes due to institutional orpolitical constraints. In that case, it is optimal to use trade taxes as a second-best in-strument to tackle allocative inefficiency. In this section, we derive sufficient statisticsformulas for second-best optimal trade taxes in such instances. We have already high-lighted the role of cross-demand effects as well as the subtle distinctions that separatethe restricted and free entry cases. To streamline the presentation, though, we hereafterpresent results corresponding to the case of restricted entry and zero cross-industry de-mand effects. Formulas corresponding to free entry or arbitrary cross-demand effectsare relegated to the appendix.

When domestic taxes are unavailable, the Home government’s optimal policy prob-lem includes an added constraint that s = 0:

max(t,x,s;w|s=0)∈F

Wh (t, x, s; w) (P2).

Using the same set of techniques highlight earlier, we analytically solve Problem (P2)and derive sufficient statistics formulas for second-best optimal trade taxes. The fol-lowing proposition presents these formulas, with a formal proof provided in AppendixG.19

Theorem 2. Suppose the use of domestic taxes is restricted, the second-best optimal trade taxesare unique up to a uniform tax shifter t ∈ R+, and are given by:

1 + t∗k =(1 + ξk) [1 + εkλhh,k]

1 + ξk [1 + εkλhh,k](1 + t), ∀k

1 + x∗k =1 + 1/εkλ f f ,k

1 + 1/ξk(1 + t)−1. ∀k (9)

The above theorem indicates that, when the use of domestic taxes is restricted, (i)the optimal export tax equals the wedge between the firm-level and country-level mar-ket powers in industry k, and (ii) the optimal import tax is non-uniform and positivelyrelated to the industry-level scale elasticity, 1/ξk. Intuitively, the government’s objec-tive when solving (P2) is to mimic domestic Pigouvian subsidies with trade taxes. Todo so, it is optimal to raise import taxes and lower export taxes in high-returns-to-scale

18Another difference between the free and restricted entry cases is that the first-best tax schedule canbe replicated with a combination of trade taxes and entry subsidies.

19See Appendix E for a more general version of the formulas under arbitrary cross-demand effects.

17

industries relative to the first-best benchmark. While these measures promote produc-tion in high-1/ξ industries, they cannot perfectly reproduce the first-best outcome dueto a lack of sufficient policy instruments.20

Second-Best Import Taxes. Now suppose that in addition to domestic taxes, the useof export taxes is also restricted. The Home government’s optimal policy problem inthis case features two additional constraints, x = s = 0:

max(t,x,s;w|s,x=0)∈F

Wh (t, x, s; w) (P3).

Some variation of the above problem, which concerns second-best optimal import taxes,has been the focus of an expansive literature in trade policy. Using the same techniques,we can analytically solve Problem (P3) to produce sufficient statistics formulas for theoptimal second-best import taxes. The following proposition presents these formulas,with a formal proof provided in Appendix H.

Theorem 3. Suppose the use of both domestic and export taxes is restricted, the second-bestoptimal import taxes are uniquely given by:

1 + t∗k =(1 + ξk) [1 + εkλhh,k]

1 + ξk [1 + εkλhh,k]

1 + ∑gξg

1+ξg

λh f ,g

λh fεgλ f f ,g

∑gλh f ,g

λh fεgλ f f ,g

, ∀k

where λh f ,g ≡ λh f ,ge f ,g and λh f = ∑g λh f ,g.

The above optimal tariff formula is composed of and industry-specific componentand a uniform component in parenthesis. The industry-specific component bears thesame intuition as that provided under Theorem 2. The uniform component accounts forthe ability of import taxes to manipulate the ToT through general equilibrium wage ef-fects. These effects where previously redundant due to the availability of export taxes.The intuition is that, previously, the uniform tariff component could be perfectly mim-icked with an across-the-board shift in export taxes (see Lemma 1). Since export taxesare now restricted, the uniform tariff component becomes necessary for optimal ToTmanipulation.

The ToT versus Allocative Efficiency Trade-Off. Based on Theorems 2 and 3, second-best trade taxes pursue two distinct objectives: (i) to shrink exports in high-ε indus-tries, and (ii) to promote output in high-returns-to-scale (high-1/ξ) industries. Theformer corrects the ToT inefficiency, while the latter corrects allocative inefficiency.There is, however, a previously-unnoticed tension between these two objectives: If

20As noted earlier, the government needs at least three tax instruments (per industry) to achieve thefirst-best outcome. In homogeneous good models, though, trade taxes can fully mimic domestic taxes,because two instruments suffice to achieve the first-best.

18

Covk(εk, 1/ξk) < 0, reaching objective (ii) undermines objective (i). In other words,improving allocative efficiency with trade taxation worsens the ToT. Correcting alloca-tive inefficiency requires subsidizing the high-ε industries, while correcting the ToTinefficiency requires the opposite. This tension diminishes the effectiveness of tradetaxes as a second-best corrective policy measure, beyond what is implied by the target-ing principle (Bhagwati and Ramaswami (1963)).21

Proposition 1. If the industry-level trade and scale elasticities are negatively correlated (i.e.,Covk(εk, 1/ξk) < 0), then correcting allocative inefficiency with “trade taxes” worsens the ToTinefficiency. This tension diminishes the effectiveness of trade taxes as a second-best correctivepolicy measure, beyond what is already implied by the targeting principle.

One the flip side, the above proposition highlights the pitfalls of unilateralism whenadopting corrective domestic policies—an issue we cover next.

3.3 Unilateralism versus Multilateralism in Corrective Policies

Proposition 1 suggested that trade taxes are a poor second-best measure for restor-ing allocative efficiency when Covk(εk, 1/ξk) < 0. Here, we argue that a unilateraladoption of corrective domestic policies is also ineffective. However, a multilateraladoption of corrective policies can be so effective, it dominates any non-cooperativepolicy alternative. Even so, if non-cooperation does not trigger retaliation by tradingpartners.

The first claim follows from the fact that corrective domestic taxes subsidize bothexports and domestically-sold output in high-1/ξk industries. The ToT motives for tax-ation meanwhile require that countries tax exports in high-ε industries. So, if Covk(εk,1/ξk)<0,there exists an innate tension between the ToT and corrective motives for domestic tax-ation. This tension is, by construction, resolved if other countries also adopt correctivesubsidies that expand exports in high-1/ξk industries.

To establish the benefits of multilateralism, we can compare the gains from thefirst-best non-cooperative policy to the gains from deep cooperation. To this end, letWh(t∗h, x∗h, s∗h; s f = 0) denote Home’s welfare under the first-best non-cooperative pol-icy as characterized by Theorems 1.A or 1.B. Let Wh(0, 0, s∗h; s∗f ) denote Home’s welfareunder a deep trade agreement that prohibits trade taxes but implements Pigouvian sub-sidies in both Home and Foreign. We can compare welfare under these two scenariosusing the following decomposition:

Wh(0, 0, s∗h; s∗f )

Wh(t∗h, x∗h, s∗h; s f = 0)=

Wh(0, 0, s∗h; s∗f )

Wh(t∗h, x∗h, s∗h; s∗f )︸ ︷︷ ︸forgone ToT gains

×Wh(0, 0, s∗h; s∗f )

Wh(0, 0, s∗h; s f = 0)︸ ︷︷ ︸spilled-over corrective gains

.

21On the flip side, when trade taxes are restricted, governments will use domestic taxes as a second-bestinstrument to improve their ToT. Moving from the first-best to the second-best production tax-cum-subsidyschedule, in that case, improves the ToT inefficiency at the expense of worsening allocative efficiency.

19

The first component, labeled “forgone ToT gains” concerns the loss in welfare whentrade taxes are abolished as part of the deep agreement. The second component, la-beled “spilled-over corrective gains,” accounts for the gains that accrue to the Homeeconomy when the Foreign economy restores efficiency. The above decompositionis theoretically grounded in the fact that t∗ and x∗ only target the ToT inefficiency,whereas s∗ only targets allocative inefficiency.

Noting the above decomposition, consider two extreme cases: First, suppose Vark(1/ξk) ≈0, in which case Theorems 1.A and 1.B indicate that s∗ ≈ 0 by choice of shifter s. So,

there are zero “spilled-over corrective gains” in this extreme case, i.e.,Wh(0,0,s∗h ;s∗f )

Wh(0,0,s∗h ;s f =0) ≈ 1.Second, consider the case where εk → ∞, in which case t∗ ≈ x∗ ≈ 0 by choice of shifter

t. As a result, there are zero “foregone ToT gains” in this case, i.e.,Wh(0,0,s∗h ;s∗f )

Wh(t∗h ,x∗h ,s∗h ;s∗f )≈ 1.

Based on these two extreme cases, we can deduce that

a) the foregone ToT gains are determined primarily by Avgk(εk);

b) the spilled-over corrective gains are determined primarily by Vark(1/ξk).

Consequently, if Avgk(εk) is sufficiently high and Vark(1/ξk) is sufficiently low, it isbeneficial for countries to forgo the ToT gains from taxation and join a deep tradeagreement that prohibits trade taxation but restores efficiency at a global level. Thesearguments are summarized in the following proposition.

Proposition 2. (a) When Covk(εk, 1/ξk) < 0, a unilateral adoption of corrective industrialpolicies (that is not paired with trade taxes) can worsen social welfare. However, when Avgk(εk)

is sufficiently high and Vark(1/ξk) is sufficiently low, (ii) a multilateral adoption of correctivepolicies via a deep trade agreement, delivers welfare gains that dominate those of any non-cooperative policy alternative. This latter assertion holds even if non-cooperation does not leadto retaliation by trading partners.

In light of Propositions 1 and 2, the optimal implementation of industrial policy de-pends critically on the values of εk and 1/ξk. The next section develops a new method-ology to estimate these key policy parameters. But before moving on to the estimation,let us briefly discuss how our theory extends to other canonical trade models and howthe trade-offs we identify extend to richer environments.

3.4 Application to other Canonical Trade Models

Before moving forward, let us briefly outline the breadth of theory. The optimaltax formulas specified by Theorems 1-3 apply to two other canonical trade models.Though, parameters ξk and εk in these formulas adopt different interpretations underthese alternative frameworks with different micro-foundations.

(a) The Eaton-Kortum model with Marshallian externalities. As in Kucheryavyyet al. (2016), consider a multi-industry Eaton and Kortum (2002) model where pro-duction in each industry is subject to agglomeration economies. Let ψk denote the

20

constant agglomeration elasticity in industry k, and let θk denotes the industry-levelEaton-Kortum Fréchet shape parameter. Theorem 1.B characterizes the optimal policyin this model under the following reinterpretation of parameters: 1/ξEK

k = ψk andεEK

k = θk. The ToT-allocative efficiency trade-off identified by Proposition 1 also ex-tends to this model if Covk(ψk, θk) < 0. The fact that our theory readily extends tothe Eaton-Kortum model is an artifact of the isomorphism established in Kucheryavyyet al. (2016). We extend this isomorphism result in Appendix C, demonstrating thatthe nested-CES import demand function implied by A1 may analogously arise fromwithin-industry specialization á la Eaton-Kortum.

(ii) The Melitz-Pareto model. Consider a multi-industry Melitz (2003) model, whichfeatures the same nested-CES demand function specified by A1. Suppose the firm-level productivity distribution is Pareto in each industry with a shape parameter, γk.Appendix D establishes that the Melitz-Pareto model is isomorphism to our baselineKrugman model as far macro-level representation is concerned. Considering this, The-orem 1.A characterizes the optimal policy in the Melitz-Pareto model with restricted en-try under the following reinterpretation of parameters: ξMelitz

k = (γk + 1)ξk/(γk − ξk)

and εMelitzk = γk/

[1 + γk

(1εk− 1

ξk

)]. Theorem 1.B characterizes the optimal policy in

the Melitz-Pareto model with free entry under the following reinterpretation of param-eters: ξMelitz

k = γk and εMelitzk = γk/

[1 + γk

(1εk− 1

ξk

)].

4 The Generality of the ToT-Allocative Efficiency Trade-off

Our theory identified a lesser-known trade-off between the misallocation- and ToT-improving motives for trade taxation. This section shows that the same trade-off extendsto richer environments with many countries, global input-output networks, and polit-ical economy pressures.

(ii) An Environment with Arbitrarily Many Countries. With arbitrarily many coun-tries, we can follow the same steps to derive sufficient statistics formulas for optimaltaxes.22 Doing so, as shown in Appendix J, yields the following optimal tax formulasunder restricted entry:

1 + s∗i,k =1 + s

1 + 1/ξk, ∀k

1 + t∗ji,k = 1 + t, ∀j, k

1 + x∗ij,k =(

1 +1

εk(1− λij,k)

)(1 + t)−1, ∀j, k.

The above formulas differ from the baseline formula in that optimal export taxes now

22These formulas rely on the observation that tji,k has (ii) a direct first-order effect on Wi , as well as(ii) a indirect first-order effect on Wi through its effect on wj/wi; while the effect of tji,k on Wi throughwn/wj is second-order in nature if n 6= i.

21

depend on the destination-specific import demand elasticity.23 But the trade-off high-lighted by Proposition 1 remains relevant nonetheless. The above formulas indicatethat correcting allocative inefficiency entails subsidizing high-1/ξ industries, while cor-recting the ToT efficiency entails shrinking exports in low-ε industries. So, if Covk(εk, 1/ξk) <

0, correcting allocative efficiency with trade taxes worsens the ToT inefficiency.

(ii) An Environment with Input-Output Networks. Now, consider an extension of ourbaseline model where production employs both labor and intermediate inputs that aredenoted by superscript I . Supposing entry is restricted, the producer price of good ji, kdepends on (i) the local wage rate and (ii) the price of all intermediate inputs availableto suppliers in country j: Pji,k = (1 + 1/ξk)τji,kCi,k(wj, PIj ) where PIj ≡ PIij,k. Withoutloss of generality, assume that each good ji, k can be used as either an intermediate in-put or a final consumption good and that taxes are applied on a good irrespective of theintended final use, i.e., PIij,k = Pij,k.24 By appealing to supply-side envelope conditions,we can derive the optimal policy under restricted entry as follows (see Appendix I):

1 + s∗k =1 + s

1 + 1/ξk

1 + t∗k = 1 + t

1 + x∗k =

(1 +

1− αh f ,k

λ f f ,kεk + αh f ,k

)(1 + t)−1 .

In the above formula αh f ,k ≡ ∑ r f h,gλIh f ,kg, where λIh f ,kg denotes the fraction of goodh f , k’s exports that are used as inputs in Foreign’s industry g. The above formula in-dicates that correcting the ToT distortion entails taxing more nationally-differentiated(low-ε) but downstream (low-α) industries. So, provided that downstream industriesare not systemically less differentiated, Proposition 1 maintains its validity. That is ifCovk(εk, 1/ξk) < 0, correcting allocative inefficiency with trade taxes worsens the ToTinefficiency.

(iii) An Environment with Political Economy Pressures. The biggest challenge toProposition 1 is perhaps the presence of political motives in taxation. To account forthese motives, we follow Ossa’s (2014) adaptation of Grossman and Helpman (1994).We consider the Cobb-Douglas-CES case of our model where social welfare in countryi is given by Wi ≡ Vi(.) = wi Li+Ri

Pi+ ∑k

Πi,kPi

. Instead of maximizing Wi, we assumethat the government in country i maximizes a politically-weighted welfare function,WP

i = wi Li+RiPi

+ ∑k ϑi,kΠi,kPi

; where ϑi,k is the political economy weight assigned to in-

23Following the discussion in Section 3, the same formula holds under free entry if country i is a smallopen economy: rji,k ≈ 0 if j 6= i.

24This assumption is just a matter of labeling the goods. We can always break up every industry k intoa final good version k′ and an intermediate good version k′′. Since we do not impose any restrictions onthe number of industries, all the results carry over to this economy, which allows for differential taxes onthe final versus intermediate-input version of industry k goods.

22

dustry k’s profits (with ∑k ϑi,k/K =1). It follows trivially from Theorem 1.A that theoptimal (first-best) policy in this setup is given by

1 + s∗k =1 + s

1 + 1/ξPi,k

, ∀k

1 + t∗k = 1 + t, ∀k

1 + x∗k = (1 + t)−1(

1 +1

εkλ f f ,k

), ∀k

where 1/ξPi,k =

1ϑi,k(1+ξk)−1 is the political economy-adjusted markup of industry k. Consid-

ering the above formulas, if Covk(ϑi,k, 1/ξk)<0, correcting the politically-weighted al-locative efficiency may entail taxing high-1/ξ industries. In that case, even if Covk(εk,1/ξk)<0,the corrective and ToT motives for trade taxation align. However, if Covk(ϑi,k, 1/ξk)≥0Proposition 1 remains valid despite political economy pressures.25

5 Estimating the Key Policy Parameters

It should be clear by now that computing the gains from policy requires credibleestimates for (i) the industry-level trade elasticities, εk, which reflect the degree ofnational-level market power in industry k; and (ii) the industry-level scale elasticities,1/ξk, which reflect the degree of firm-level market power in industry k. The trade lit-erature has paid considerable attention to estimating εk, but less attention has beendevoted to disentangling ξk from εk. Instead, ξk is often pinned down based on the fol-lowing two normalizations: (i) 1/ξk = 1/εk in monopolistically competitive models,and (ii) 1/ξk = 0 in perfectly competitive models.26 The latter normalization elimi-nates market imperfections. The former allows researchers to pin down ξk based onexisting estimates for the trade elasticity. But as noted by Benassy (1996), it also createsan arbitrary link between the market power of individual firms and the national-levelmarket power in each industry.27

To credibly analyze the ToT gains and corrective gains from policy, we need to (a)separately identify and estimate both εk and ξk, and (b) make sure that these estimatesare mutually-consistent. To this goal, we propose a methodology that simultaneously

25The above formula has another basic implication: When governments are politically-motivated, theirpolicy objectives may no longer align insofar as domestic policies are concerned. In particular, some gov-ernments may assign a greater weight to low-profit industries such as agriculture. These political econ-omy considerations can prompt governments to subsidize and promote the wrong domestic industriesfrom the perspective of the rest of the world. This concern is at the root of some existing opposition toindustrial policy.

26See Ossa (2016) and Costinot and Rodríguez-Clare (2014) for a detailed review of the literature adopt-ing these two normalizations.

27Outside of the trade policy literature, several studies, including De Loecker and Warzynski (2012),Edmond et al. (2015), and De Loecker, Goldberg, Khandelwal, and Pavcnik (2016), have used firm-levelproduction data to estimate the degree of firm-level market power within industries. These production-side approaches implicitly pin down ξk without determining the trade elasticity. The challenge, as notedabove, is to jointly estimate both εk and ξk with the same data.

23

estimates εk and ξk, thereby satisfying requirements (a) and (b).28 Our approach in-volves fitting the structural firm-level import demand function implied by A1 to theuniverse of Colombian import transactions from 2007–2013. We outline this approachbelow, starting with a description of the data used in our estimation.

Data Description. Our estimation uses data on import transactions from the Colom-bian Customs Office for the 2007–2013 period.29 The data include detailed informa-tion about each transaction, such as the Harmonized System 10-digit product category(HS10), importing and exporting firms,30 f.o.b. (free on board) and c.i.f. (customs,insurance, and freight) values of shipments in US dollars, quantity, unit of measure-ment (of quantity), freight in US dollars, insurance in US dollars, value-added tax inUS dollars, country of origin, and weight. A unique feature of this data set is that itreports the identities of all foreign firms exporting to Colombia, allowing us to defineimport varieties as firm-product combinations—in comparison, most papers focusingon international exports to a given location typically treat varieties as more aggregatecountry-product combinations. Table 5 (in the appendix) reports a summary of basictrade statistics in our data.31

When working with the above data set, we face the challenge that, for some prod-ucts, Colombia has been changing the HS10 classification between 2007 and 2013. For-tunately, the Colombian Statistical Agency, DANE, has kept track of these changes,32

and we utilized this information to concord the Colombian HS10 codes over time. Inthe process, we followed the guidelines outlined by Pierce and Schott (2012) for theconcordance of the U.S. HS10 codes over time.33 Overall, changes in HS10 codes be-tween 2007 and 2013 affect a very small portion (less than 0.1%) of our dataset.

5.1 Estimating Equation

Since we are focusing on one importer, we hereafter drop the importer’s subscripti and add a year subscript t to account for the time dimension of our data. With this

28In the presence of selection effects, our estimated parameters are necessary but not sufficient to pindown the trade and scale elasticities. In addition to these parameters, we also need information on theshape of the Pareto productivity distribution—see Appendix D.

29The data is obtained from Datamyne, a company that specializes in documenting import and exporttransactions in Americas. For more detail, please see www.datamyne.com.

30The identification of the Colombian importing firms is standardized by the national tax ID number.For the foreign exporting firms, the data provide the name of the firm, phone number, and address. Thenames of the firms are not standardized, and thus there are instances in which the name of a firm and itsaddress are recorded differently (e.g., using abbreviations, capital and lower-case letters, dashes, etc.). Wedeal with this problem by standardizing the spelling and the length of the names along with utilizing thedata on firms’ phone numbers. The detailed description of cleaning the exporters’ names is provided inthe Technical Appendix.

31Our estimation also employs data on monthly average exchange rates, which are taken from the Bankof Canada: http://www.bankofcanada.ca/rates/exchange/monthly-average-lookup/.

32We thank Nicolas de Roux and Santiago Tabares for providing us with this information.33To preserve the industry identifier of the product codes, and in contrast to Pierce and Schott (2012),

we try to minimize the number of the synthetic codes. The concordance data and do files are provided inthe data appendix.

24

slight change of notation, the demand facing firm ω located in country j and supplyingproduct k in year t is given by (see A1):

qj,kt (ω) = ϕj,kt(ω)

(pj,kt(ω)

Pj,kt

)−1−ξk(

Pj,kt

Pkt

)−1−εk

Qkt, (10)

Note that subscript k has been used thus far to reference industries. But in our empiricalanalysis, k will designate the most disaggregated industry/product category in ourdataset, which is an HS10 product category. The quadruplet “ωjkt” therefore denotesa unique imported variety corresponding to firm ω–country of origin j–HS10 product k–year t. Rearranging Equation 10, we can produce the following equation

qj,kt (ω) =ϕjkt(ω) pj,kt(ω)−εk−1

(pj,kt(ω)

Pj,kt

)εk−ξk

Pεk+1kt Qkt

=ϕjkt(ω) pj,kt(ω)−εk−1 λj,kt (ω)1− εk

ξk Pεk+1kt Qkt, (11)

where λj,kt (ω) denotes the share of expenditure on firm ω “conditional” on buying fromcountry j,

λj,kt (ω) ≡ ϕj,kt(ω)

(pj,kt(ω)

Pj,kt

)−ξk

=pj,kt(ω)qj,kt(ω)

∑ω′∈Ωj,ktpj,kt(ω′)qj,kt(ω′)

.

Taking logs from Equation 11 and letting X(ω) ≡ p(ω)q(ω) denote sales, yields thefollowing firm-level import demand function:

ln Xj,kt(ω) = −εk ln pj,kt(ω) +

(1− εk

ξk

)ln λj,kt (ω) + δkt + ϕj,kt(ω), (12)

where δkt ≡ ln P−εk−1kt Qkt can be treated as a product-year fixed effect. We assume

that ϕjkt(ω) = ϕj,k(ω) + ϕωjkt can be decomposed into a time-invariant firm×product-specific quality component, ϕj,k(ω), and a time-varying component ϕωjkt, that encom-passes (i) idiosyncratic variations in consumer taste, (ii) measurement errors, and (iii)omitted variables that account for dynamic demand optimization. To eliminate ϕj,k(ω)

from the estimating equation, we employ a first-difference estimator which also dropsobservations pertaining to one-time exporters. We deem the first-difference approachappropriate given the possibility that ϕωjkt’s are sequentially correlated. Stated in termsof first-differences, our estimating equation takes the following form

∆ ln Xj,kt(ω) = −εk∆ ln pj,kt (ω) +

(1− εk

ξk

)∆ ln λj,kt (ω) + ∆δkt + ∆ϕωjkt, (13)

25

where ∆ϕωjkt, crudely speaking, represents a variety-specific demand shock; and ∆δkt isa product-year fixed effect.34 Of the remaining variables, ∆ ln pj,kt(ω) and ∆ ln Xj,kt(ω)

are directly observable for each import variety, while changes in the within-nationalmarket share, ∆ ln λj,kt (ω), can be calculated using the universe of firm-level sales toColombia.35

As noted earlier, k indexes an HS10 product category in Equation 13. To conduct ourforthcoming quantitative analysis, however, we need to estimate ε and ξ for 14 broadly-defined industries based on the World Input-Output Database (WIOD) classification.Considering this, we pool all HS10 products belonging to the same WIOD industry κ

together, and estimate Equation 13 on this pooled sample assuming that εk and ξk areuniform across products within the same industry (i.e., ξk = ξκ and εk = εκ for allk ∈ Kκ). In principle, we can also estimate the import demand function separately foreach HS10 product category to attain HS10-level elasticities. However, such elasticitieswill be of little use for our quantitative policy analysis, as multi-country data on trade,production, and expenditure shares are scarce at such levels of disaggregation.

Before moving forward, a discussion regarding the role of εk/ξk in Equation 13 isin order. The spread εk/ξk reflects the wedge between the national-level and firm-leveldegrees of market power. Broadly speaking, variety ωjkt is (i) either imported from athick market like China in which case it competes with many other Chinese varieties,hence a low λj,kt (ω), or (ii) it is imported from a thin market like Taiwan where itcompetes with a few other Taiwanese varieties, hence a high λj,kt (ω). Controlling forprices, if varieties originating from thick markets generate lower sales, our import de-mand function identifies this as a case where 1 > εk/ξk > 0. That is, 1 > εk/ξk > 0corresponds to a situation where firm-level market power is less than the national-levelmarket power due to sub-national competition.36

5.2 Identification Strategy

The identification challenge we face is that ∆ ln pj,kt (ω) and ∆ ln λj,kt (ω) are en-dogenous variables that can covary with the demand shock, ∆ϕωjkt.37 Standard importdemand estimation techniques resolve a similar challenge by instrumenting for prices

34In the previous version of the paper, we included all observations that reported a non-missing∆ ln pj,kt. Some of our industry-level estimates, however, display sensitivity to outlier observations. Con-sidering this, we now trim the sample to exclude observations that report a price change, ∆ ln pj,kt, abovethe 99th percentile or below the 1st percentile of the industry.

35Some readers may notice parallels between our estimating equation and the nested demand functionanalyzed in Berry (1994); but given the structure of our data, we adopt a distinct identification strategy.

36Note that the case εk/ξk > 1 undermines assumption A3 regarding the uniqueness of the equilib-rium. As we will see shortly, our estimation always yields a value of εk/ξk < 1.

37An alternative issue is tat unit prices data may be contaminated with measurement errors, as they areaveraged across transactions and consumers. This type of measurement error, however, is fairly innocu-ous given the log-linear structure of our demand function (see Berry (1994)). This point notwithstanding,the instrumental variable approach presented in this section will handle measurement errors, providedthat lagged sales are not correlated with concurrent measurement errors.

26

with plausibly exogenous tariff rates.38 This strategy, however, does not apply to ourfirm-level estimation, because tariffs discriminate by country-of-origin but not acrossfirms from the same country.

To achieve identification, we need to construct a firm×origin×product×year-specificcost shifter that is uncorrelated with ∆ϕωjkt. To this end, we capitalize on the monthlyfrequency of import transactions in our data. We compile an external database on ag-gregate monthly exchange rates and interact the monthly variation in aggregate ex-change rates with the lagged monthly composition of firm-level exports to constructthe following shift-share instrument:

zj,kt(ω) =12

∑m=1

Xj,kt−1(ω; m)

Xj,kt−1(ω)∆Ejt(m).

In the above expression, ∆Ejt(m) ≡ Ejt(m)− Ejt(m− 1) denotes the change in countryj’s exchange rate with Colombia in month m of year t; Xj,kt−1(ω; m) denotes the monthm sales of firm ω (from origin country j in product category k) in the prior year, t− 1;and Xj,kt−1(ω) = ∑12

m=1 Xj,kt−1(ω; m) denotes firm ω’s total annual export sales in t− 1.Crudely speaking, zj,kt(ω) measures the firm×origin×product×year-specific exposureto exchange rate shocks. The idea being that aggregate exchange rate movements havedifferential effects on different firms depending on the monthly composition of theirprior export activity to Colombia.39 Encouragingly, this idea is backed by the fact that zand ∆ ln p exhibit a strong and statistically significant correlation in our data. AppendixL illustrates this point using the example of two U.S.-based exporting firms.

Our instrument utilizes lagged monthly sales, Xjkt−1(ω, m) = X( pjkt−1(ω, m); ...),which depend on lagged prices and other market-level indexes. Therefore, the exclu-sion restriction, Corr [z, ∆ϕ] = 0, hinges on two identifying assumptions:

(a1) Prior price-setting decisions are orthogonal to concurrent demand shocks:Corr

[pjkt−1(ω), ∆ϕωjkt

]= 0.

(a2) National-level exchange rate movements are orthogonal to variety-level de-mand shocks: Corr

[∆Ejt(m), ∆ϕωjkt

]= 0.

These assumptions can be challenged if there are cross-inventory linkages or if indi-vidual export varieties account for a significant fraction of a country’s total exports toColombia. We will discuss and address these issues further in Section 5.4.

Instruments for ∆ ln λj,kt (ω). Following Khandelwal (2010), we construct two stan-dard instruments for the annual variation in the within-national market shares: (i)

38A prominent example is Caliendo and Parro (2015) who use tariff data identify the trade elasticity.39One can draw parallels between our instrument and the widely-used Bartik instrument. The latter

builds on the idea that different regions are affected differentially by national labor market shocks de-pending on the lagged industry composition of their production.

27

annual changes in the total number of country j firms serving the Colombian mar-ket in product category k, and (ii) changes in the total number of HS10 product cate-gories actively served by firm ω in year t. These count measures will be correlated with∆ ln λj,kt (ω) but uncorrelated with ∆ϕωjkt if variety-level entry and exit occur prior to,or independent of, the demand shock realization of competing varieties. As noted byKhandelwal (2010), this assumption is rather standard in the literature that estimatesdiscrete choice demands curves—see also Berry, Levinsohn, and Pakes (1995).40

5.3 Estimation Results

The industry-level estimation results are reported in Table 2. We also report es-timation results corresponding to a pooled sample of all industries in Table 6 of theappendix. This table also compares the 2SLS and OLS estimates to ensure that ourIV strategy is operating in the expected direction. Our estimates point to an averagetrade elasticity of ε = 3.6 and an average scale elasticity of 1/ξ ≈ 0.19. Our pooledestimation yields a heteroskedasticity-robust Kleibergen-Paap Wald rk F-statistic of 95.Hence, we can reject the null of weak instruments given the Stock-Yogo critical values.A similar, albeit weaker, argument applies to the industry-level estimation.

The industry-level elasticities reported in Table 2 display considerable variationacross industries. The estimated scale elasticity or markup margin is highest in the’Electrical & Optical Equipment’ (1/ξ = 0.45) and ’Petroleum’ (1/ξ = 1.7) industries;both of which are associated with high R&D or fixed costs. The estimated scale elas-ticity is lowest in ’Agricultural & Mining’ (1/ξ = 0.14) and ’Machinery’ (1/ξ = 0.10)industries. Furthermore, with the exception of ’Agriculture & Mining,’ we cannot re-ject the prevalence of scale economies. The finding that returns-to-scale are negligiblein the agricultural sector aligns with a large body of evidence on the inverse farm-sizeproductivity (IFSP) relationship—see Sen (1962) and subsequent references to IFSP.

Our industry-level trade elasticity estimates, εk, also display some novel proper-ties. To our knowledge, our estimation is the first to identify the industry-level tradeelasticities using (i) firm-level data, and (ii) while controlling for intra-national cross-demand effects. Qualitatively speaking, our estimated trade elasticities are similar inmagnitude to those estimated by Simonovska and Waugh (2014) but slightly lowerthan those estimated by Caliendo and Parro (2015). Aside from the firm-level nature ofour estimation, these differences may be driven by the fact that instead of controllingfor f.o.b. prices with exporter fixed effects, we directly use data on f.o.b. price levels.

Importantly, our estimates indicate that εk/ξk < 1 in nearly all industries. Thisfinding rejects the arbitrary link often assumed between the firm-level and national-

40Although trade taxes count as a weak instrument in our firm-level estimation, we nonetheless includethem as an additional instrument to comply with the literature. These taxes include applied ad-valoremtariffs and and the Columbian value-added tax (VAT). We exclude the VAT in the ’Transportation’ and’Petroleum’ industries since the VAT in these industries discriminates by the method of delivery and levelof luxury, both of which may be correlated with ∆ϕωjkt.

28

level degrees of market power in the literature.41 Our estimates also indicate thatCovk(εk, 1/ξk) < 0, which empirically confirms the innate tension between the ToTand corrective motives for trade taxation, as highlighted Proposition1.

5.4 Challenges to identification

As noted earlier, our man identifying assumptions, (a1), and, (a2), can be challengedunder certain circumstances. Below, we discuss these challenges and present additionalevidence to address them.

Within-Cluster Correlation in Error Terms. Adao et al. (2019) show that identifica-tion based on shift-share instruments exhibits an over-rejection problem if regressionerrors are cross-correlated. In the context of our estimation, this problem will ariseif demand shocks are correlated across firm-origin-product-year varieties with a similarmonthly export composition. To handle this issue, we adopt a conservative two-wayclustering of standard errors by product-year and origin-product. Clustering standarderrors this way is akin to the correction proposed by Adao et al. (2019).

Cross-Inventory Effects. Lags in inventory clearances can challenge our identifyingassumptions on two fronts. First, firms’ optimal pricing decisions may be forward-looking, which violates assumption (a1). To address this concern, we reconstruct ourshift-share instrument using 4 lags instead of 1. If inventories clear in at most 4 years,we can deduce that pricing decisions do not internalize expected demand shocks be-yond the 4-year mark. Hence, Corr

[pjkt−4(ω), ∆ϕωjkt

]= 0, and the new instrument

will satisfy the exclusion restriction. The trade-off is that using such an instrumentamounts to losing more observations, as the instrument can be constructed for firmsthat continuously export in the 4 different years. The top panel of Figure 3 (in Ap-pendix M) compares estimation results under this alternative instrument to the baselineresults. The ordering and magnitude of the estimated elasticities are rather preservedacross industries. More importantly, the new estimation retains the negative correla-tion between εk and 1/ξk, which is the key assumption underlying Proposition 1.

Second, with cross-inventory effects, ∆ϕωjkt may encompass omitted variables thatgovern firms’ dynamic inventory management decisions. One of these omitted vari-ables is presumably the exchange rate. If so, Corr

[∆Ejt(m), ∆ϕωjkt

]6= 0, and our

identifying assumption (a2) will be violated. To address this concern, we reestimateEquation 13 while directly controlling for the annual change in the exchange rate, ∆Ejt.

41In other words, our estimation rejects the independence of irrelevant alternatives (IIA) because productvarieties or technologies are less differentiated intra-nationally than inter-nationally. While the IIA as-sumption has garnered considerable attention in the industrial organization literature, the trade literaturehas only recently tested this assumption against data. Redding and Weinstein (2016) estimate an interna-tional demand system that relaxes the IIA assumption by accommodating heterogeneous taste across con-sumers. Adao, Costinot, and Donaldson (2017) estimate a trade model that (unlike standard CES models)permits varieties from certain countries to be closer substitutes. Our results contribute to this emergingliterature by highlighting another aspect of the trade data that is at odds with the IIA assumption.

29

Table 2: Industry-level estimation results

Estimated Parameter

Sector ISIC4 codes εk 1− εkξk

1/ξk Obs. WeakIdent. Test

Agriculture & Mining 100-1499 6.212 0.125 0.141 11,962 2.51(2.112) (0.142) (0.167)

Food 100-1499 3.333 0.117 0.265 20.042 6.00(0.815) (0.050) (0.131)

Textiles, Leather & Footwear 100-1499 3.413 0.293 0.207 126,483 63.63(0.276) (0.020) (0.022)

Wood 100-1499 3.329 0.101 0.270 5,962 1.76(1.331) (0.181) (0.497)

Paper 100-1499 2.046 0.187 0.397 37,815 2.65(0.960) (0.216) (0.215)

Petroleum 100-1499 0.397 0.302 1.758 4,035 2.03(0.342) (0.081) (1.584)

Chemicals 100-1499 4.320 0.085 0.212 134,413 42.11(0.376) (0.027) (0.069)

Rubber & Plastic 100-1499 3.599 0.418 0.162 107,713 7.22(0.802) (0.041) (0.039)

Minerals 100-1499 4.561 0.153 0.186 28,197 3.19(1.347) (0.096) (0.129)

Basic & Fabricated Metals 100-1499 2.959 0.441 0.189 155,032 16.35(0.468) (0.024) (0.032)

Machinery 100-1499 8.682 0.130 0.100 266,628 8.54(1.765) (0.080) (0.065)

Electrical & Optical Equipment 100-1499 1.392 0.369 0.453 260,207 17.98(0.300) (0.015) (0.099)

Transport Equipment 100-1499 2.173 0.711 0.133 86,853 5.09(0.589) (0.028) (0.036)

N.E.C. & Recycling 100-1499 6.704 0.049 0.142 70,974 8.51(1.133) (0.100) (0.289)

Notes. Estimation results of Equation (13). Standard errors in parentheses. The estimation is conducted with HS10product-year fixed effects. All standard errors are simultaneously clustered by product-year and by origin-product,which is akin to the correction proposed by Adao, Kolesár, and Morales (2019). The weak identification teststatistics is the F statistics from the Kleibergen-Paap Wald test for weak identification of all instrumented variables.The test for over-identification is not reported due to the pitfalls of the standard over-identification Sargan-HansenJ test in the multi-dimensional large datasets pointed by Angrist, Imbens, and Rubin (1996).

Even if changes in inventory-related demand depend on the changes in exchange rate,we can still assert that E

[zj,kt(ω) ∆ϕωjkt | ∆Ejt

]= 0; i.e., the exclusion restriction is sat-

isfied with the added control for ∆Ejt. The middle panel of Figure 3 (in Appendix M)compares estimation results under this alternative specification to the baseline results.Once again, the ordering and magnitude of the estimated elasticities are preservedacross industries. Also, the new estimation retains the negative correlation between εk

30

and 1/ξk.

Export Varieties with Significant Market Share. Our identification can come underchallenge if individual varieties account for a significant fraction of a country’s salesto Colombia. In such a case, variety-specific demand shocks can influence the bilat-eral exchange between the Colombian Peso and the origin country’s currency, therebyviolating identifying assumption (a2). This concern, however, does not apply to oursample of exporters. The variety with the highest 99th percentile within-national mar-ket share accounts for only 0.1% of the origin country’s total exports to Colombia. Thevariety with the highest 90th percentile within-national market share accounts for only0.0008% of the origin country’s total exports to Colombia.

One may still be concerned about large multi-product firms that export multipleproduct varieties to Colombia in a given year. Consider, for instance, a multi-productfirm ω that exports goods k and g to Colombia in year t. If demand shocks are corre-lated across the varieties supplied by this firm (i.e., E

[∆ϕωjkt ∆ϕωjgt

]6= 0), Assumption

(a2) may be violated despite each variety’s market share being infinitesimally small. Toaddress this issue, we reestimate Equation 13, on a restricted sample that drops ex-cessively large firms with a total within-national market share that exceeds 0.1%. Thebottom panel of Figure 3 (in Appendix M) compares estimation results under this moreconservation sample selection to the baseline results. Once again, the ordering andmagnitude of the estimated elasticities are rather preserved across industries. Also, thenew estimation retains the negative correlation between εk and 1/ξk.

5.5 Plausibility of Estimates

Before moving on to the quantitative analysis, let us briefly discuss the plausibilityof our estimates. We do so by exploring their macro-level implications and by bench-marking them against alternative estimates in the literature.

Plausibility from the Lens of Macro-Level Predictions. Our estimated elasticitiesdeliver sharp predictions about the cross-national income-size elasticity. As pointedout by Ramondo, Rodríguez-Clare, and Saborío-Rodríguez (2016), the factual relation-ship between real per capita income and population size (i.e., the income-size elastic-ity) is negative and statistically insignificant. Quantitative trade models featuring thenormalization ε/ξ = 1, however, predict a strong and positive income-size elasticitythat remains significant even after the introduction of domestic trade frictions. Ra-mondo et al. (2016) call this observation the income-size elasticity puzzle. Consideringthis puzzle, in Appendix N we compute the income-size elasticity implied by our es-timated value of ε/ξ ≈ 0.67. Encouragingly, we find that our estimated value for ε/ξ

completely resolves the aforementioned puzzle. In other words, our micro-estimatedelasticities are consistent with the macro-level cross-national relationship between pop-ulation size and real per capita income.

31

Comparison to Industry-Level Estimates in the Literature. Reassuringly, our esti-mates align closely with well-known industry-level case studies. Take, for example,our elasticity estimates for the ’Petroleum’ industry, which appear somewhat low. First,our estimate for εk aligns with the consensus in the Energy Economics literature that,at the national level, demand for petroleum products is price-inelastic.42 Second, ourestimated 1/ξk for the ’Petroleum’ industry closely resembles existing estimates in theIndustrial Organization literature. Considine (2001), for instance, estimates 1/ξ ≈ 2using detailed data on the U.S. petroleum industry. Moreover, our finding that the’Petroleum’ industry is the most scale-intensive industry is consistent with the findingin Antweiler and Trefler (2002), which is based on more aggregated data. Likewise,consider the case of ’Transportation’ or auto industry, where our estimated 1/ξk = 0.13implies an optimal markup of 13%. This estimate aligns with existing estimates fromvarious industry-level studies. Recently, Cosar, Grieco, Li, and Tintelnot (2018) haveestimated markups for the auto industry that range between roughly 6% to 13%. Pre-viously, Berry et al. (1995) have estimated markups of around 20% in the U.S. autoindustry using data from 1971-1990.

Comparison to Bartelme et al. (2018). Concurrent with us, Bartelme et al., (2018,BCDR) have developed an alternative methodology to estimate the scale elasticity. Inparticular, they estimate the elasticity of export sales with respect to industry-level em-ployment, namely αBCDR

k . From the lens of gravity trade models, this elasticity assumesthe following interpretation:

αBCDRk = (ψk + 1/ξk) εk − βk,

where ψk denotes the industry-level agglomeration elasticity and βk denotes the shareof industry-specific factors in total production.

BCDR’s estimation implicitly assumes that there are no are industry-specific fac-tors of production, which amounts to βk = 0. They then take estimates for the tradeelasticity, εk, from other sources in the literature and calculate the scale elasticity as1/ξk + ψk = αBCDR

k /εk. The advantage of their approach is that it detects Marshallian(or agglomeration) externalities. The advantages of our approach are two-fold: (a) weseparately identify εk from 1/ξk;43 and (b) our estimates are robust to the presence ofindustry-specific factors of production, i.e., our identification does not rely on βk = 0.44

42See Pesaran, Smith, and Akiyama (1998), which is the main reference in this literature, as well asFattouh (2007) for a survey of that literature

43As noted in Appendix D, in the presence of selection effect, our approach can identify the scaleelasticity only up-to an externally chosen trade elasticity. Second, our approach can identify the scaleelasticity even when there are diseconomies of scale at the industry-level.

44Our estimation of 1/ξk and εk relies solely on assumption A1. Our estimates are, therefore, com-patible with a non-parametric production function that admits multiple specific or non-specific factors ofproduction. Assumption A2 and A3 only matter for how our estimated parameters map into optimal taxformulas.

32

These differences can also explain why we estimate a larger scale elasticity thanBCDR in some industries. Consider, for instance, the ’Petroleum’ industry where weestimate a considerably larger scale elasticity than BCDR. This difference can be drivenby the ’Petroleum’ industry employing a large amount of industry-specific factors ofproduction, like natural resources or offshore platforms. This situation corresponds toa high βk that can attenuate the estimates in BCDR relative to ours.

6 Quantifying the Gains from Policy

As a final step, we quantify the gains from trade and industrial policy across 32major economies. To this end, we combine our micro-level estimates for εk and ξk,our sufficient statistic tax formulas from Section 3, and macro-level data on productionand expenditure from the 2012 World Input-Output Database (WIOD, Timmer, Erum-ban, Gouma, Los, Temurshoev, de Vries, Arto, Genty, Neuwahl, Francois, et al. (2012)).The original WIOD database covers 35 industries and 40 countries that account formore than 85% of world GDP, plus an aggregate of the rest of the world. The coun-tries in the sample include all 27 members of the European Union plus 13 other majoreconomies, namely, Australia, Brazil, Canada, China, India, Indonesia, Japan, Mexico,Russia, South Korea, Taiwan, Turkey, and the United States. The 35 industries in theWIOD database include the 15 industries listed in Table 2, plus 20 service-related in-dustries that are listed in Table 7 of the appendix. Our analysis aggregates the originalWIOD sample into 32 countries plus an aggregate of the rest of the world (as listed inTable 3) and 16 industries, which include 15 traded industries and an aggregate of allservice-related industries.45 To make the data consistent with our model, we closelyfollow the methodology outlined in Costinot and Rodríguez-Clare (2014) to purge thedata from trade imbalances.

Our goal is to calculate the optimal policy and the corresponding gains for each ofthe 32 economies in our sample. So, in each iteration of our analysis, we treat one ofthe 32 economies as “Home,” and aggregate the rest of the world into one country andtreat it as “Foreign.” We then compute the gains from optimal policy for the designatedHome country. To simplify this task, we appeal to our sufficient statistics tax formulasderived in Section 3.

6.1 Mapping the Sufficient Statistics Formulas to Data

The sufficient statistics formulas, presented under Theorems 1-3, simplify the taskof computing the gains from optimal policy. The alternative approach is to use globaloptimization techniques. But as implied by the analysis in Costinot and Rodríguez-Clare (2014), this approach can become increasingly burdensome when dealing withmany free-moving tax instruments.46 To tractably map our optimal tax formulas to

45Our shrinking of the sample along these dimensions is akin to Costinot and Rodríguez-Clare (2014).See their appendix for a discussion of why shrinking the WIOD sample in this regard is appropriate.

46This is perhaps why the optimal tariff analysis in Costinot and Rodríguez-Clare (2014) restricts at-

33

data, we assume that the cross-industry utility aggregator is Cobb-Douglas:

Ui (Qi) = ∏k

Qei,ki,k . (14)

With the above assumption, the reduced-form demand elasticities are given by 2. Wealso assume that the status-quo is a tax-free equilibrium.

We present our quantitative approach using the conventional exact hat-algebra no-tation (x ≡ x′/x). Our approach combines the optimal tax formulas specified by The-orems 1-3 with a set of equilibrium conditions, all expressed in hat-algebra notation.This procedure yields a system of equations, the solution of which determines theoptimal tax rates, (t∗, x∗, s∗), and their corresponding welfare effects. The followingpropositions lays out this approach for the restricted entry case. It specifies a systemthat solves the first-best non-cooperative tax schedule characterized by Theorem 1.A.An analogous proposition is presented for the free case entry in Appendix O.

Proposition 3. Suppose the observed data is generated by a model that exhibits the cross-industry utility aggregator 14. Then, the optimal (first-best) taxes and their effects on wages,profits, and income can be fully characterized by solving the following system of equations:

xh f ,k =1

εkλ f f ,kλ f f ,k; t f h,k = 0; 1 + sh,k =

11+1/ξk

∀k ∈ K

x f h,k = 0, th f ,k = 0; sk = 0 ∀k ∈ K

λji,k =[(1 + tji,k)(1 + xji,k)(1 + sj,k)wj

]−εk ˆPεki,k ∀j, i ∈ C, k ∈ K

ˆP−εki,k = ∑j

([(1 + tji,k)(1 + xji,k)(1 + sj,k)wj

]−εk λji,k

)∀j, i ∈ C, k ∈ K

wiwiLi = ∑k ∑j

(ξkλij,kλij,kej,kYjYj

(1+ξk)(1+xij,k)(1+si,k)(1+tij,k)

)∀i ∈ C

ΠiΠi = ∑k ∑j

(λij,kλij,kej,kYjYj

(1+ξk)(1+xij,k)(1+si,k)(1+tij,k)

)∀i ∈ C

RiRi = ∑k ∑j

(tji,k

1+tji,kλji,kλji,kei,kYiYi +

(1+xij,k)(1+si,k)−1(1+xij,k)(1+si,k)(1+tij,k)

λij,kλij,kej,kYjYj

)∀i ∈ C

YiYi = wiwiLi + ΠiΠi + RiRi ∀i ∈ C

.

The above system involves 3K independent unknowns, xh f ,k, wi, and Yi,and 3K independent equations. Moreover, it can be solved with knowledge of only (a)observable expenditure shares λji,k, and ei,k, wage income, wiLi, aggregate profits, Πi,and total income Yi = wiLi + Πi, all of which are reported by the WIOD. Solving thesystem also requires knowledge of (b) industry-level trade and scale elasticities, εk and1/ξk, which were estimated in the previous section.

The next proposition outlines a procedure to solve for the second-best optimal tradetaxes characterized by Theorem 2. Recall that these are the optimal non-cooperativeexport and import taxes when the government is prohibited from using domestic taxes.

tention to a uniform tariff applied to all industries (see Page 227 and the discussion following Figure 4.1)

34

Proposition 4. Suppose the observed data is generated by a model that exhibits the cross-industry utility aggregator 14. The second-best optimal trade taxes and their effects on wages,profits, and income can be fully characterized by solving the following system of equations:

xh f ,k =

(1 + 1

εkλ f f ,kλ f f ,k

)(1 + 1/ξk)

−1; t f h,k =(1+ξk)[1+εkλhh,kλhh,k]1+ξk(1+εkλhh,kλhh,k)

; sh,k = 0 ∀k ∈ K

x f h,k = 0, th f ,k = 0; sk = 0 ∀k ∈ K

λji,k =[(1 + tji,k)(1 + xji,k)(1 + sj,k)wj

]−εk ˆPεki,k ∀j, i ∈ C, k ∈ K

ˆP−εki,k = ∑j

([(1 + tji,k)(1 + xji,k)(1 + sj,k)wj

]−εk λji,k

)∀j, i ∈ C, k ∈ K

wiwiLi = ∑k ∑j

(ξkλij,kλij,kej,kYjYj

(1+ξk)(1+xij,k)(1+si,k)(1+tij,k)

)∀i ∈ C

ΠiΠi = ∑k ∑j

(λij,kλij,kej,kYjYj

(1+ξk)(1+xij,k)(1+si,k)(1+tij,k)

)∀i ∈ C

RiRi = ∑k ∑j

(tji,k

1+tji,kλji,kλji,kei,kYiYi +

(1+xij,k)(1+si,k)−1(1+xij,k)(1+si,k)(1+tij,k)

λij,kλij,kej,kYjYj

)∀i ∈ C

YiYi = wiwiLi + ΠiΠi + RiRi ∀i ∈ C

,

The system specified by Proposition 2 involves 4K independent unknowns, xh f ,k,t f h,k, wi, and Yi, and 4K independent equations. Moreover, like the previoussystem, it can be solved with knowledge of only observables and industry-level pa-rameters, εk, and ξk, which were estimated in the previous section.

Our last proposition outlines a procedure to solve for the second-best optimal im-port taxes characterized by Theorem 3. Recall that these are optimal non-cooperativeimport taxes when the government is prohibited from using both domestic and exporttaxes.

Proposition 5. Suppose the observed data is generated by a model that exhibits the cross-industry utility aggregator 14. The second-best optimal trade taxes and their effects on wages,profits, and income can be fully characterized by solving the following system of equations:

t f h,k =(1+ξk)[1+εkλhh,kλhh,k]1+ξk(1+εkλhh,kλhh,k)

(1 + τ) ; xh f ,k = 0; ; sh,k = 0 ∀k ∈ K

x f h,k = 0, th f ,k = 0; sk = 0 ∀k ∈ K

1 + τ = ∑g

[(1 + εg

λh f ,gλh f ,ge f ,g

λh f λh fλ f f ,gλ f f ,g)(1 + 1/ξg)−1

]/ ∑g

(εg

λh f ,gλh f ,ge f ,g

λh f λh fλ f f ,gλ f f ,g

)λji,k =

[(1 + tji,k)(1 + xji,k)(1 + sj,k)wj

]−εk ˆPεki,k ∀j, i ∈ C, k ∈ K

ˆP−εki,k = ∑j

([(1 + tji,k)(1 + xji,k)(1 + sj,k)wj

]−εk λji,k

)∀j, i ∈ C, k ∈ K

wiwiLi = ∑k ∑j

(ξkλij,kλij,kej,kYjYj

(1+ξk)(1+xij,k)(1+si,k)(1+tij,k)

)∀i ∈ C

ΠiΠi = ∑k ∑j

(λij,kλij,kej,kYjYj

(1+ξk)(1+xij,k)(1+si,k)(1+tij,k)

)∀i ∈ C

RiRi = ∑k ∑j

(tji,k

1+tji,kλji,kλji,kei,kYiYi +

(1+xij,k)(1+si,k)−1(1+xij,k)(1+si,k)(1+tij,k)

λij,kλij,kej,kYjYj

)∀i ∈ C

YiYi = wiwiLi + ΠiΠi + RiRi ∀i ∈ C

.

35

The above system also involves 3K independent unknowns, t f h,k, wi, and Yi,and 3K independent equations, and can be solved with knowledge of only observablesand industry-level parameters, εk, and ξk. Once we solve the above three systems, thegains from optimal policy can be calculated as Wi = Yi/ ∏k

(ˆPei,ki,k

).

To take stock, Propositions 1-3 take an important step in simplifying the task ofcomputing the gains from policy. The alternative to these propositions is to determinethe optimal policy schedule using constrained global optimization techniques.47 Asnoted earlier, these techniques can become difficult-to-implement when dealing withmany free-moving tax variables. An accurate implementation of these techniques alsorequires specialized commercial solvers like SNOPT or KNITRO. Propositions 1-3, how-ever, allow us to bypass the full-blown global optimization procedure, which leads toa notable increase in both computational speed and accuracy.

6.2 The Gains from First- and Second-Best Non-Cooperative Policies

Table 3 reports the gains from non-cooperative policies both under free and re-stricted entry. In all cases, the welfare gains are computed assuming the rest of theworld does not retaliate. The first column of each panel reports the gains from the first-best non-cooperative tax schedule (Theorems 1.A and 1.B). The second column reportsthe gains from second-best optimal import and export taxes (Theorem 2). The thirdcolumn reports the gains from second-best optimal import taxes (Theorem 3). We candraw three main conclusions from the results in Table 3:

a) The gains from ToT manipulation are relatively small. This is partly evident fromthe fact that when governments are restricted to only trade taxes, the resultinggains are significantly smaller than the first-best case. We can also directly verifythat pure ToT gains account for less than 1/3 of the total gains from first-best policy,which average around 1.7% under restricted entry and 3% under free entry.

b) Trade taxes are a poor second-best substitute for domestic Pigouvian taxes. Thisconclusion can be drawn from the fact that trade taxes alone can barely repli-cate a third of the welfare gains attainable under the first-best policy. Under re-stricted entry, for instance, the fist-best policy increases welfare by 1.67% on aver-age, whereas second-best trade taxes increase welfare by only 0.54%. Second-bestimport taxes (that are not paired with export taxes) are even less effective. Simi-lar results apply to the free entry case. This outcome is driven by our estimatedelasticities implying Covk(εk, 1/ξk) < 0. So, per Proposition 1, there is an innatetension between improving the ToT and restoring allocate efficiency with tradetaxes. This tension renders second-best trade taxes as largely ineffective.

c) The gains from policy are larger under free entry than restricted entry, despite

47Such a problem can be formulated as an MPEC (Mathematical Programming with Equilibrium Con-straint) problem–see Ossa (2014).

36

similar optimal tax rates. This nuanced observation suggests that firm-delocationgains from policy dominate the profit-shifting gains. One can perhaps extrapolatefrom this result that the gains from trade agreements under free entry will be evenlarger than those estimated by the restricted entry models in Ossa (2014, 2016).

Table 3: The gains from optimal non-cooperative policies

Restricted Entry Free Entry

CountryAll Taxes

(First-Best)Only

Exp/Imp TaxesOnly

Import TaxesAll Taxes

(First-Best)Only

Exp/Imp TaxesOnly

Import Taxes

AUS 0.91% 0.29% 0.25% 1.97% 0.85% 0.14%AUT 1.36% 0.79% 0.62% 2.07% 1.25% 0.72%BEL 1.34% 0.68% 0.49% 1.77% 0.79% 0.49%BRA 1.69% 0.31% 0.27% 3.47% 1.15% 0.37%CAN 1.53% 0.62% 0.49% 2.61% 1.25% 0.57%CHN 1.74% 0.57% 0.50% 2.82% 0.96% 0.56%CZE 1.84% 1.03% 0.77% 3.04% 1.86% 0.99%DEU 1.83% 0.86% 0.65% 3.28% 1.60% 0.80%DNK 1.03% 0.54% 0.42% 1.84% 0.96% 0.45%ESP 1.46% 0.55% 0.45% 2.55% 1.19% 0.50%FIN 1.65% 0.60% 0.51% 2.69% 1.18% 0.57%FRA 1.85% 0.57% 0.44% 3.98% 1.77% 0.64%GBR 0.97% 0.53% 0.42% 1.35% 0.77% 0.46%GRC 1.75% 0.60% 0.57% 3.66% 1.75% 0.89%HUN 2.41% 1.15% 0.86% 4.74% 2.88% 1.46%IDN 1.12% 0.42% 0.37% 1.37% 0.50% 0.41%IND 2.08% 0.45% 0.39% 4.08% 1.41% 0.41%IRL 1.75% 0.99% 0.85% 2.63% 1.52% 0.94%ITA 1.43% 0.52% 0.43% 2.47% 1.12% 0.51%JPN 1.60% 0.43% 0.37% 3.07% 0.94% 0.43%KOR 2.79% 1.08% 0.92% 4.61% 2.13% 0.98%MEX 2.04% 0.82% 0.63% 4.40% 2.21% 1.07%NLD 1.14% 0.61% 0.44% 1.37% 0.75% 0.46%POL 1.94% 0.83% 0.66% 3.38% 1.74% 0.85%PRT 1.60% 0.62% 0.54% 2.91% 1.39% 0.66%ROM 1.75% 0.74% 0.64% 3.18% 1.60% 0.88%RUS 2.69% 0.51% 0.45% 5.43% 1.92% 0.60%SVK 1.73% 0.93% 0.69% 2.65% 1.55% 0.86%SWE 1.13% 0.72% 0.57% 1.35% 0.90% 0.66%TUR 1.43% 0.61% 0.52% 2.24% 1.08% 0.61%TWN 2.37% 0.99% 0.82% 4.79% 2.70% 1.43%USA 1.55% 0.39% 0.33% 3.02% 1.13% 0.40%Average 1.67% 0.54% 0.45% 3.02% 1.21% 0.54%

37

6.3 The Pitfalls of Unilateralism in Corrective Policies

As noted in Section 2, a unilateral adoption of corrective domestic subsidies canworsen welfare, if (i) these subsidies are not paired with appropriate trade tax mea-sures, or (ii) or the same corrective subsides are not implemented in the rest of theworld. The intuition is that a unilateral application of Pigouvian subsidies will im-prove allocative inefficiency but will worsen the ToT. Table 4 displays the extent of thistrade-off for the average country. The unilateral case corresponds to a scenario wherea country unilaterally imposes Pigouvian subsidies without erecting any trade taxes.The multilateral case corresponds to a coordinated adoption of Pigouvian subsidies viadeep trade agreements.

The results in Table 4 suggest that the ToT losses from applying unilateral Pigouviansubsidies outweigh the corrective gains for the average country. This result highlightsthe importance of international coordination in domestic policies. The failure of inter-national coordination can deter most countries from undertaking corrective subsidiesthat are ultimately beneficial—a point that has been overlooked by existing critiques ofglobal governance (e.g., Rodrik (2019)).

Table 4: The gains from corrective subsidies: unilateral vs. multilateral adoption

Restricted Entry Free Entry

Unilateral Multilateral Unilateral Multilateral

% ∆Wavg -0.36% 1.67% -1.18% 3.02%

6.4 The Gains from “Deep” Trade Agreements

Finally, we compare the gains from deep agreements to those attainable under non-cooperative policy measures. Deep agreements, as in Bagwell and Staiger (2001), corre-spond to a scenario where all countries agree to (i) adopt corrective domestic policies,and (ii) abolish trade taxes. By joining such an agreement, countries forgo the ToTgains from non-cooperative taxation, but in return benefit from corrective subsidies inthe rest of the word. We wish to quantify how this trade-off is borne out in practice.48

The results are displayed in Figure (1).49 The x-axis in the aforementioned figurecorresponds to the gains from deep cooperation. The y-axis corresponds to the gainsfrom the optimal (first-best) non-cooperative policy in the absence of retaliation. Formost economies, the gains from deep cooperation dominate those of the optimal non-cooperative policy. What is perhaps surprising is that this outcome emerges even whennon-cooperation does not trigger retaliation by trading partners. It simply reflects thefact that the ToT gains from taxation are relatively small, whereas the extent of alloca-

48We compute the gains from deep cooperation by using the closed-form formula specified by Equation8 and applying the exact hat-algebra methodology.

49The graph excludes Ireland, which experiences negative gains from cooperation.

38

Figure 1: Deep cooperation vs. first-best non-cooperative policy

AUSAUT

BEL

BRA

CANCHNCZE

DEU

DNK

ESPFIN

FRA

GBR

GRC

HUN

IDN

IND

IRLITA

JPN

KORMEX

NLD

POL

PRTROM

RUS

SVK

SWE

TUR

TWN

USA

0

2

4

6

% G

ain

s fr

om

No

n−

Co

op

erat

ive

Po

licy

0 2 4 6

% Gains from Deep Cooperatiion

Free Entry

AUS

AUTBEL

BRA

CAN

CHNCZEDEU

DNK

ESP

FIN

FRA

GBR

GRC

HUN

IDN

IND

ITA

JPN

KOR

MEX

NLD

POL

PRT

ROM

RUS

SVK

SWE

TUR

TWN

USA

1

2

3

% G

ain

s fr

om

No

n−

Co

op

erat

ive

Po

licy

0 1 2 3

% Gains from Deep Cooperation

Restricted Entry

tive inefficiency in the global economy is sizable. So, for most countries, it is beneficialto forgo the ToT gains in exchange for importing more efficient varieties. Also inter-estingly, the gains from deep cooperation favor countries that have a comparative dis-advantage in high-returns-to-scale (or high-profit) industries, e.g., Greece, Russia, andMexico. The intuition is that these countries depend relatively more on imported vari-eties in high-returns-to-scale industries, and, under deep cooperation, these industriesare subsidized across the globe.

Plausibility of the Estimated Gains. Our finding that the gains from restoring al-locative efficiency are large sits well with the findings in Baqaee and Farhi (2017) thateliminating sectoral markup-heterogeneity in the U.S. economy can raise real GDP by2.3%.50 Bartelme et al. (2018), however, estimate smaller gains from similar policies.To understand these differences, we can appeal to the Hsieh and Klenow (2009) exactformula for distance from the efficient frontier. Suppose εk = ε is uniform across in-dustries, and that industry-wide productivity levels and 1/ξk’s have a joint log-normaldistribution. The true distance from the frontier can be approximated to a first-orderas L≈ 1

2 εVar(ln(ψk + 1/ξk)), where ψk denotes the elasticity of Marshallian externali-ties. Our analysis like Baqaee and Farhi (2017) sets ψk = 0, and measures the degreeof allocative inefficiency as LLL≈ 1

2 εVar(ln(1/ξk)). This approach can lead to an over-statement of L if ψk is negatively correlated with firm-level market power, 1/ξk.51 Incomparison and as noted in Section 5.4, the degree of allocative efficiency in BDCR’s

50This number corresponds to the average of the numbers reported in the last column of Table 2 inBaqaee and Farhi (2017).

51Another issue is that we are assuming away selection effects in our quantitative analysis. In thepresence of selection effects, we can still use our estimates for εk and ξk to identify the scale elasticityup-to an externally chosen trade elasticity. Doing so, however, may lead to a lower or higher L.

39

analysis is measured as LBCDR≈ 12 εVar(ln(1/ξk + ψk − βk

ε )), where βk is the share ofindustry-specific factors in production. This approach can understate Lwhen there aresignificant diseconomies of scale due to a high βk.

7 Concluding Remarks

For centuries scale economies have served as the backbone of many arguments con-cerning trade and industrial policy. Despite the immense conceptual interest, we knowsurprisingly little about the empirical relevance of scale economies for the optimal de-sign of and the gains from policy. Against the backdrop of this disconnect, we took apreliminary step toward identifying the force of industry-level scale economies usingmicro-level trade data. Our estimates indicated that, due to significant cross-industryheterogeneity in scale economies, corrective domestic taxes deliver greater gains thanforeign trade taxes for most economies. Furthermore, the gains from a deep multilat-eral agreement that eliminates inefficiency at the global level dominate the gains fromany unilateral policy.

While we highlighted several macro-level implications, our micro-estimates havean even broader reach. Two implications, which were left out in the interest of space,merit particularly close attention. First, our scale elasticity estimates can help disentan-gle the relative contribution of scale economies and Ricardian comparative advantageto industry-level specialization. This is an old question, which, from an empirical per-spective, we know surprisingly little about.

Second, our estimates can shed fresh light on the puzzlingly large income gap be-tween rich and poor countries. Economists have always hypothesized that a fractionof this income gap is driven by rich countries specializing in high returns-to-scale in-dustries. An empirical assessment of these hypotheses, however, has been impeded bya lack of comprehensive estimates for industry-level scale elasticities. Our micro-levelestimates pave the way for an empirical exploration in this direction.

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44

Appendix: For Online Publication

A The Redundancy of Consumption Taxes

Consider any arbitrary combination of taxes that includes (i) a domestic productiontax, sk, (ii) a domestic consumption tax, ck, (iii) an import tax, tk, and (iv) an export tax,xk, in industry k. This complete set of taxes produces the following wedges betweenproducer and consumer prices in both markets:

Phh,k = (1 + ch,k) (1 + sh,k) Phh,k

Pf h,k = (1 + th,k) (1 + ch,k) Phh,k

Ph f ,k = (1 + xh,k) (1 + sh,k) Phh,k

Pf f ,k = Pf f ,k

Our claim here is that the same wedges can be replicated without appealing to theconsumption tax, ck. This claim can be established by considering the following set ofproduction, export, and import taxes: (i) 1+ s′h,k = (1 + ch,k) (1 + sh,k); (ii)

(1 + t′h,k

)=

(1 + th,k) (1 + ch,k); and (iii)(

1 + x′h,k

)= (1 + xh,k) / (1 + ch,k). It is straightforward

to see that these “three” tax instruments, by construction, reproduce the same wedgebetween producer and consumer prices as those expressed above:

Phh,k =(

1 + s′h,k

)Phh,k

Pf h,k =(

1 + t′h,k

)Phh,k

Ph f ,k =(

1 + x′h,k

) (1 + s′h,k

)Phh,k

Pf f ,k = Pf f ,k

.

That being the case, consumption taxes are redundant in the presence of the other threeinstruments. The same can be said about any other tax instrument. More specifically,the effect of an import tax can be perfectly replicated with a consumption, production,and export tax combination. However, due to product differentiation, if one of the fourtax instruments is restricted, the argument falls apart. For instance, if production taxesare restricted, export and import taxes cannot fully replicate the effect of a domesticconsumption tax.

45

B Proof of Lemma 1

Consider the policy-wage combinations A = (s, t, x; w) and A′ = (s′, t′, x′; w),where (a and a ∈ R+)

1 + x′i = (1 + xi) /a 1 + t′−i = 1 + t−i

1 + t′ = a (1 + t) 1 + x′−i = 1 + x−i

1 + s′ = a (1 + s) 1 + s′−i = 1 + s−i

w′i = (a/a)wi w′−i = w−i

.

Our goal is to prove that (i) if A ∈ F then A′ ∈ F, and (ii)Wi(A) = Wi(A′) for alli. To prove, these claims it suffices to establish two intermediate claims, C1 and C2.Claim C1 follows trivially from the Marshallian demand function being homogeneousof degree zero. i.e., Dji,k(Yi, Pi) = Dji,k(aYi, aPi):

[C1]

Ph (A′) = aPh (A)

P f (A′) = Ph (A)

Yh (A′) = aYh (A)

Yf (A′) = Yf (A)

=⇒ Qji,k(A′) = Qji,k(A) ∀ji, k

Claim C2 follows (after some algebra) from Assumption A2 and the balanced tradecondition, ∑k

[Ph f ,kQh f ,k/(1 + th f ,k)

]= ∑k

[Pf h,kQ f h,k/(1 + t f h,k)

]:

[C2] Qji,k(A′) = Qji,k(A) ∀ji, k =⇒

Ph (A′) = aPh (A)

P f (A′) = Ph (A)

Yh (A′) = aYh (A)

Yf (A′) = Yf (A)

Together, Claims C1 and C2 indicate that combinations A and A′ correspond to thesame allocation, Qji,k, and yield the same welfare levels due to the indirect utilitybeing homogeneous of degree zero. i.e., Vi(Yi, Pi) = V(aYi, aPi):

Wi(A′) = Vi(Yi(

A′)

, Pi(

A′)) = V(Yi (A) , Pi (A)) = Wi(A).

C Nested-Eaton and Kortum (2002) Framework

Here we show that the nested CES import demand function specified by A1, canalso arise from within-product specialization à la Eaton and Kortum (2002). To thisend, suppose that each industry k is comprised of a continuum of homogenous goodsindexed by ν. The sub-utility of the representative consumer in country i with respect

46

to industry k is a log-linear aggregator across the continuum of goods in that industry:

Qi,k =∫ 1

0ln qi,k(ν)dν

As in our main model, country j hosts Mj,k firms indexed by ω, with Ωj,k denotes theset of all firms serving industry k from country j.52 Each firm ω supplies good ν tomarket i at the following quality-adjusted price:

pji,k(ν; ω) = pji,k (ω) /ϕ(ν; ω),

where pji,k (ω) is a nominal price (driven by production costs) that applies to all goodssupplied by firm ω in industry k, while the quality component, ϕ(ν; ω), is good×firm-specific. Suppose for any given good ν, firm-specific qualities are drawn independentlyfrom the following nested Fréchet joint distribution:

Fk(ϕ(ν)) = exp

− N

∑i=1

(∑

ω∈Ωi,k

ϕ(ν; ω)−ϑk

) θkϑk

,

The above distribution generalizes the basic Fréchet distribution in Eaton and Kortum(2002). In particular, it relaxes the restriction that productivities are perfectly correlatedacross firms within the same country. Instead, it allows for sub-national productivitydifferentiation and also for the degrees of cross- and sub-national productivity differ-entiations (ϑk and θk, respectively) to diverge. A special case of the distribution whereϑk −→ ∞ corresponds to the standard Eaton and Kortum (2002) specification.

The above distribution also has deep theoretical roots. The Fisher–Tippett–Gnedenkotheorem states that if ideas are drawn from a (normalized) distribution, in the limit thedistribution of the best draw takes the form of a general extreme value (GEV) distribu-tion, which includes the above Fréchet distribution as a special case. A special applica-tion of this result can be found in Kortum (1997) who develops an idea-based growthmodel where the limit distribution of productivities is Fréchet, with ϕω,k reflecting thestock of technological knowledge accumulated by firms ω in category k.

Given the vector of effective prices, the representative consumer in county i (who isendowed with income Yi) maximizes their real consumption of each good, qi,k(ν) =

ei,kYi/ pi,k(ν), by choosing pi,k(ν) = minω pji,k (ω). That being the case, the con-sumer’s discrete choice problem for each good ν can be expressed as:

minω

pji,k (ω) /z(ν; ω) ∼ maxω

ln z(ν; ω)− ln pji,k (ω) .

To determine the share of goods for which firm ω is the most competitive supplier,

52The implicit assumption here is that entry is restricted, so that Mj,k is exogenous.

47

we can invoke the theorem of “General Extreme Value.” Specifically, define G(pi) asfollows

Gk(pi) =N

∑j=1

∑ω∈Ωj,k

exp(−ϑk ln pji,k (ω))

θkϑk

=N

∑j=1

∑ω∈Ωj,k

pji,k (ω)−ϑk

θkϑk

.

Note that Gk(.) is a continuous and differentiable function of vector pi ≡ pji,k (ω)and has the following properties:

a) Gk(.) ≥ 0;

b) Gk(.) is a homogeneous function of rank θk: Gk(ρpi) = ρθk Gk(pi) for any ρ ≥ 0;

c) limpji,k(ω)→∞ Gk(pi) = ∞, ∀ω;

d) the m’th partial derivative of Gk(.) with respect to a generic combination of mvariables pji,k (ω), is non-negative if m is odd and non-positive if m is even.

Manski and McFadden (1981) show that if Gk(.) satisfies the above conditions, andϕ(ν; ω)’s are drawn from

Fk(ϕ(ν)) = exp(−Gk(e− lnϕ)

)= exp

− N

∑j=1

∑ω∈Ωj,k

ϕ(ν; ω)−ϑk

θkϑk

,

which is the exact same distribution specified above, then the probability of choosingvariety ω is equal to

πji,k(ω) =

(pji,k(ω)

θk

)∂Gk(pi)∂pji,k(ω)

Gk(pi)=

pji,k (ω) pji,k (ω)ϑk−1(

∑ω′∈Ωj,kpji,k (ω

′)−ϑk) θk

ϑk−1

∑Nn=1

(∑ω′∈Ωj,k

pji,k (ω′)−ϑk) θk

ϑk

Rearranging the above equation yields the following expression:

πji,k(ω) =

(pji,k (ω)

Pji,k

)−ϑk(

Pji,k

Pi,k

)−θk

,

where Pji,k ≡[∑ω′∈Ωj,k

pji,k (ω′)−ϑk

]−1/ϑkand Pi,k ≡

[∑ P−θk

ji,k

]− 1θk . Given the share of

goods sourced from firm ω, total sales of firm ω to market i, in industry k can be calcu-

48

lated as:

pji,k(ω)qji,k(ω) = pji,k(ω)πji,k(ω)ei,kYi

pji,k(ω)

=

(pji,k (ω)

Pji,k

)−ϑk(

Pji,k

Pi,k

)−θk

ei,kYi,

which is identical to the nested-CES function specified by A1, with ξk = ϑk and εk = θk.

D Firm-Selection under Pareto

In this appendix, we outline the isomorphism between our baseline model and onethat admits selection effects. In doing so, we borrow heavily from Kucheryavyy et al.(2016). The two key assumptions here are,

a) the firm-level productivity distribution, Gi,k(z), is Pareto with an industry-levelshape parameter γk, and

b) the fixed “marketing” cost is paid in terms of labor in the destination market.

Following Kucheryavyy et al. (2016), we also assume that cross-industry utility aggre-gator is Cobb-Douglas with ei,k denoting the constant share of country i’s expenditureon industry k. Following Kucheryavyy et al. (2016), we can define the effective supplyof production labor in country i as

Li =

[1−∑

kei,k

(γk − ξk

γk(ξk + 1)

)]Li,

with the labor market clearing condition given by

∑ wiLi,k = wi Li.

First, we can appeal to the well-known result that the profit margin in each industry isgiven by the following expression:

µk ≡∑n Pin,kQin,k

wiLi,k=

γk − ξk

(γk + 1)ξk.

Second, following the steps in Appendix B.2 of Kucheryavyy et al. (2016), we can ex-press the national-level demand as

Qji,k =

(Pji,k

Pi,k

)−θk

Qi,k

49

where θk ≡ γk/[1 + γk

(1εk− 1

ξk

)]. Abstracting from taxes, the national-level price

indexes in the above expression are given by,

Pji,k = ρji,kwj M− 1

γkj,k = ρji,kwjQ

− 11+γk

j,k ,

where ρji,k is composed of structural parameters that are invariant to policy.53 Corre-

spondingly, Pi,k =(

∑ P−θkji,k

)−1/θkis the industry-level consumer price index that shows

up in indirect utility Vi(.). Let ψk ≡ ∂ ln Pij,k/∂ ln Mi,k denote the scale elasticity and µk

denote the constant firm-level markup as defined above. In baseline Krugman model,ψk = µk = 1/ξk. Combining the above expressions, the Melitz-Pareto model is isomor-phic to the baseline model but with the following interpretation of the scale elasticity,markup margin, and reduced-form demand elasticities:

ψk = 1/γk, ∀k

µk = (γk − ξk)/(γk + 1)ξk, ∀k

εi,kji,k = −1− θk

(1− λji,k/ei,k

), ∀k; = j

εi,kji,k = θkλi,g/ei,k, ∀k; 6= j

To pin down all the above elasticities, we also need an estimate for θk, which is essen-tially the trade elasticity under firm selection. Doing so, requires estimating ∂ ln Qji,k =

∂ ln(1 + tji,k

), assuming that tariffs are imposed on cost rather than revenue. To con-

clude, the above arguments establish the claim in the main text that Theorem 1.A char-acterizes the optimal policy in the Melitz-Pareto model under restricted entry underthe following reinterpretation of parameters: 1/ξMelitz

k = (γk − ξk)/(γk + 1)ξk andεMelitz

k = θk = γk/[1 + γk

(1εk− 1

ξk

)]. Theorem 1.B characterizes the optimal policy

in the Melitz-Pareto model under free entry under the following reinterpretation ofparameters: 1/ξMelitz

k = 1/γk and εMelitzk = θk =γk/

[1 + γk

(1εk− 1

ξk

)].

E Proof of Theorem 1.A

The solve for the optimal tax schedule, we take the dual approach. We can simplifythe problem by formulating the optimal tax problem as on where government directlychooses optimal consumer prices P∗hh,k, P∗f h,k, P∗h f ,k. This choice implicitly pins down

53Unlike Pi,k, the national-level indexes, Pji,k, are not the same as the the CES price indexes defined inthe main text, but this is not problematic from the point of the isomorphism result we are seeking.

50

the optimal tax schedule

1 + s∗k ≡ P∗hh,k/Phh,k ∀k

1 + t∗k ≡ P∗f h,k/Pf h,k ∀k

1 + x∗k ≡P∗h f ,k/Phh,k

P∗hh,k/Phh,k∀k

To cast the problem in this way, note that any feasible combination (t, x, s; w) ∈ F

corresponds to a feasible price-wage combination (P; w), where P ≡ Pji,k. Accord-ingly, the optimal tax problem is characterized by the following system of first orderconditions (F.O.C.s):

∇PhhWh(P; w) = 0

∇P f hWh(P; w

)= 0

∇Ph fWh(P; w

)= 0

,

Our objective is to solve this system subject to (P; w) ∼ (t, x, s; w) ∈ F. Focusing on thefeasible set of tax-wage combination means that we can invoke envelope conditions toevaluate the F.O.C.s. We, hereafter, treat labor in Foreign as the numeraire, i.e., w f = 1.Also, to simplify on notation, we let µk = 1/ξk denote the constant industry-levelmarkup margin throughout the proof.

F.O.C. with respect to Phh,k

The F.O.C. with respect to Phh,k features terms accounting for direct effects and gen-eral equilibrium wage effects:

dWh(P; wh)

dPhh,k=

∂Vh(.)∂Yh

∂Yh

∂Phh,k+

∂Vh(.)∂Phh,k

+ ∑g

∂Wh

∂Pf f ,g

∂Pf f ,g

∂Phh,k

+

(∂Vh(.)

∂Yh

∂Yh

∂wh+ ∑

i=h, f∑g

[∂Vh(.)∂Pih,g

∂Pih,g

∂wh

])dwh

dPhh,k= 0.

Note that by choice of numeraire, Pf f ,g is invariant to policy, i.e., ∂Pf f ,g/∂Phh,k = 0.

We can also appeal to Roy’s identity, ∂Vh/∂Pih,k∂Vh/∂Yh

= −Qih,k, to further simplify the aboveequation as follows:

dWh(P; wh

)d ln Phh,k

=∂Vh

∂Yh

∂Yh

∂ ln Phh,k− Phh,kQhh,k

+

(∂Yh

∂ ln wh− ∑

i=h, f∑g

[Pih,gQih,g

∂ ln Pih,g

∂ ln wh

])d ln wh

d ln Phh,k

= 0. (15)

51

To determine ∂Yh/∂ ln Phh,k, note that total income is the sum of factor income, profits,and tax revenue,

Yh(P; wh

)= whLh + ∑

g∑

i

(µg

1 + µgPhi,gQhi,g

)+Rh(P; wh), (16)

where

Rh(P; wh) = ∑g

[(Phh,g − Phh,g)Qhh,g + (Pf h,g − Pf h,g)Q f h,g + (Ph f ,g − Ph f ,g)Qh f ,g

].

(17)Taking a basic partial derivative from Equation 16, yields the following

∂Yh(P; wh

)∂ ln Phh,k

=∑g

(µg

1 + µgPhh,gQhh,g

(∂ ln Qhh,g

∂ ln Phh,k+

∂ ln Qhh,g

∂ ln Yh

∂ ln Yh

ln Phh,k

))+Phh,kQhh,k + ∑

g

((Phh,g − Phh,g)Qhh,g

[∂ ln Qhh,g

∂ ln Phh,k+

∂ ln Qhh,g

∂ ln Yh

∂ ln Yh

ln Phh,k

])+∑

g

((Pf h,g − Pf h,g)Q f h,g

[∂ ln Q f h,g

∂ ln Phh,k+

∂ ln Q f h,g

∂ ln Yh

∂ ln Yh

ln Phh,k

])

=∑g

([µg

1 + µg+ (

Phh,g

Phh,g− 1)

]Phh,gQhh,gεhh,k

hh,g

)+ Phh,kQ f h,k + ∑

g

[(Pf h,g − Pf h,g)Q f h,gεhh,k

f h,g

]+∑

g

((Pf h,g − Pf h,g)Q f h,gQ f h,gη f h,g +

[µg

1 + µgPhh,g + (Phh,g − Phh,g)

]Qhh,gηhh,g

)∂ ln Yh

∂ ln Phh,k.

where η f h,g ≡ ∂ ln Q f h,g/∂ ln Yh denotes the income elasticity of demand. Similarly,dwh/dPhh,k can be calculated by applying the implicit function theorem to the balancedtrade condition,

Dh(P; wh

)≡∑

k

(Pf h,kQ f h,k − Ph f ,kQh f ,k

)= 0,

which yields the following:

d ln wh

dlnPhh,k=

[∂

∂ ln Phh,k∑g

(Pf h,gQ f h,g − Ph f ,gQh f ,g

)]/∂Dh(P; wh

)∂ ln wh

=

[∑g

(Pf h,gQ f h,g

∂ ln Q f h,g

∂ ln Phh,k

)+ ∑

g

(Pf h,gQ f h,gη f h,g

) ∂ ln Yh

∂ ln Phh,k

]/∂Dh(.)∂ ln wh

=

[∑g

(Pf h,gQ f h,gεhh,k

f h,g

)+ ∑

g

(Pf h,gQ f h,gη f h,g

) ∂Yh

∂ ln Phh,k

]/∂Dh(.)∂ ln wh

.

Plugging the expressions for ∂Yh/∂ ln Phh,k and dwh/dlnPhh,k back into the F.O.C., defin-

ing τ ≡(

∂Vh∂Yh

∂Yh∂wh

+ ∑i ∑g∂Vh

∂Pih,g

∂Pih,g∂ ln wh

)/ (∂ ln Dh/∂wh), we can write the F.O.C. (Equation

52

15) as

∑g

([1− 1

1 + µg

Phh,g

Phh,g

]Phh,gQhh,gεhh,k

hh,g

)+ ∑

g

[(1− (1 + τ)

Pf h,g

Pf h,g

)Pf h,gQ f h,gεhh,k

f h,g

]

+∑g

([1− (1 + τ)

Pf h,g

Pf h,g]Pf h,gQ f h,gη f h,g +

[1− 1

1 + µg

Phh,g

Phh,g

]Phh,gQhh,gηhh,g

)∂ ln Yh

ln Phh,k= 0

Dividing the above equation by Yh and noting that λji,k ≡ Pji,kQji,k/Yi, 1+ sk ≡ Phh,k/Phh,k,and 1 + tk ≡ Pf h,k/Pf h,k, yields the following simplified expression:

∑g

([1−

1 + sg

1 + µg

]λhh,gεhh,k

hh,g

)+ ∑

g

[(1− 1 + τ

1 + tg

)λ f h,gεhh,k

f h,g

]+∑

g

((1− 1 + τ

1 + tg

)λ f h,gη f h,g +

[1 + sg

1 + µg− 1]

λhh,gηhh,g

)∂ ln Yh

ln Phh,k= 0

We can further simplify the above expression, we can rely on the observations that (i)there are multiple optimal tax combinations per Lemma 1, (ii) the term in parenthesisin the second line of the above equation, namely,

Ψ (t, x, s) ≡∑g

((1− 1 + τ

1 + tg

)λ f h,gη f h,g +

[1 + sg

1 + µg− 1]

λhh,gηhh,g

),

is not industry-specific. Hence, by choice of s in Lemma 1, there exists an optimal taxcombination for which Ψ (t∗, x∗, s∗) = 0. We can first restrict attention to this particularsolution, and then trivially identify the other solutions with an across-the-board shiftin all tax instruments. Doing so, the F.O.C. with respect to Phh,k reduces to

∑g

([1−

1 + sg

1 + µg

]λhh,gεhh,k

hh,g

)+ ∑

g

[(1− 1 + τ

1 + tg

)λ f h,gεhh,k

f h,g

]= 0 (18)

F.O.C. with respect to Pf h,k

The F.O.C. with respect to Pf h,k also features terms accounting for direct effects andgeneral equilibrium wage effects:

dWh(Ph; wh)

dPf h,k=

∂Vh(.)∂Yh

∂Yh

∂Pf h,k+

∂Vh(.)∂Pf h,k

+ ∑g

∂Wh

∂Pf f ,g

∂Pf f ,g

∂Pf h,k

+

(∂Vh(.)

∂Yh

∂Yh

∂wh+ ∑

i=h, f∑g

[∂Vh(.)∂Pih,g

∂Pih,g

∂wh

])dwh

dPf h,k= 0.

53

Appealing to Roy’s identity, ∂Vh/∂Pih,k∂Vh/∂Yh

= −Qih,k, and given that ∂Pf f ,g/∂Pf h,k = 0, wecan simplify the above equation as follows:

dWh(P; wh

)d ln Pf h,k

=∂Vh

∂Yh

∂Yh

∂ ln Pf h,k− Pf h,kQ f h,k

+

(∂Yh

∂ ln wh− ∑

i=h, f∑g

[Pih,gQih,g

∂ ln Pih,g

∂ ln wh

])d ln wh

d ln Pf h,k

= 0. (19)

Taking a basic partial derivative from Equation 16, yields

∂Yh(P; wh

)∂ ln Pf h,k

=∑g

(µg

1 + µgPhh,gQhh,g

(∂ ln Qhh,g

∂ ln Pf h,k+

∂ ln Qhh,g

∂ ln Yh

∂ ln Yh

ln Pf h,k

))

+Pf h,kQ f h,k + ∑g

((Phh,g − Phh,g)Qhh,g

[∂ ln Qhh,g

∂ ln Pf h,k+

∂ ln Qhh,g

∂ ln Yh

∂ ln Yh

ln Pf h,k

])

+∑g

((Pf h,g − Pf h,g)Q f h,g

[∂ ln Q f h,g

∂ ln Pf h,k+

∂ ln Q f h,g

∂ ln Yh

∂ ln Yh

ln Pf h,k

])

=∑g

([µg

1 + µg+ (

Phh,g

Phh,g− 1)

]Phh,gQhh,gε

f h,khh,g

)+ Pf h,kQ f h,k + ∑

g

[(Pf h,g − Pf h,g)Q f h,gε

f h,kf h,g

]+∑

g

((Pf h,g − Pf h,g)Q f h,gQ f h,gη f h,g +

[µg

1 + µgPhh,g + (Phh,g − Phh,g)

]Qhh,gηhh,g

)∂ ln Yh

∂ ln Pf h,k.

where η f h,g ≡ ∂ ln Q f h,g/∂ ln Yh denotes the income elasticity of demand. Similarly,dwh/dPhh,k can be calculated by applying the implicit function theorem to the balancedtrade condition,

Dh(P; wh

)≡∑

k

(Pf h,kQ f h,k − Ph f ,kQh f ,k

)= 0,

which yields the following:

d ln wh

dlnPf h,k=

[∂

∂ ln Pf h,k∑g

(Pf h,gQ f h,g − Ph f ,gQh f ,g

)]/∂Dh(P; wh

)∂ ln wh

=

[∑g

(Pf h,gQ f h,g

∂ ln Q f h,g

∂ ln Pf h,k

)+ ∑

g

(Pf h,gQ f h,gη f h,g

) ∂ ln Yh

∂ ln Pf h,k

]/∂Dh(.)∂ ln wh

=

[∑g

(Pf h,gQ f h,gε

f h,kf h,g

)+ ∑

g

(Pf h,gQ f h,gη f h,g

) ∂Yh

∂ ln Pf h,k

]/∂Dh(.)∂ ln wh

.

Plugging the expressions for ∂Yh/∂ ln Phh,k and dwh/dlnPhh,k back into the F.O.C., defin-

ing τ ≡(

∂Vh∂Yh

∂Yh∂wh

+ ∑i ∑g∂Vh

∂Pih,g

∂Pih,g∂ ln wh

)/ (∂ ln Dh/∂wh), we can write the F.O.C. (Equation

54

19) as

∑g

([1− 1

1 + µg

Phh,g

Phh,g

]Phh,gQhh,gε

f h,khh,g

)+ ∑

g

[(1− (1 + τ)

Pf h,g

Pf h,g

)Pf h,gQ f h,gε

f h,kf h,g

]

+∑g

([1− (1 + τ)

Pf h,g

Pf h,g]Pf h,gQ f h,gη f h,g +

[1− 1

1 + µg

Phh,g

Phh,g

]Phh,gQhh,gηhh,g

)∂ ln Yh

ln Phh,k= 0

Dividing the above equation by Yh and noting that λji,k ≡ Pji,kQji,k/Yi, 1+ sk ≡ Phh,k/Phh,k,and 1 + tk ≡ Pf h,k/Pf h,k, yields the following simplified expression:

∑g

([1−

1 + sg

1 + µg

]λhh,gε

f h,khh,g

)+ ∑

g

[(1− 1 + τ

1 + tg

)λ f h,gε

f h,kf h,g

]+ Ψ (t, x, s)

∂ ln Yh

ln Phh,k= 0

As noted earlier, there exists an optimal tax combination for which Ψ (t∗, x∗, s∗) = 0.Restricting attention to this particular solution, the F.O.C. with respect to Pf h,k reducesto

∑g

([1−

1 + sg

1 + µg

]λhh,gε

f h,khh,g

)+ ∑

g

[(1− 1 + τ

1 + tg

)λ f h,gε

f h,kf h,g

]= 0 (20)

F.O.C. with respect to Ph f ,k

Noting that ∂Pf f ,g/∂Pf h,k = 0, the F.O.C. with respect to Ph f ,k can be stated as

dWh(P; wh)

dPh f ,k=

∂Vh(.)∂Yh

∂Yh

∂Ph f ,k+

(∂Vh(.)

∂Yh

∂Yh

∂wh+ ∑

i=h, f∑g

[∂Vh(.)∂Pih,g

∂Pih,g

∂wh

])dwh

dPh f ,k= 0

As before, we can derive ∂Yh/∂ ln Ph f ,k by taking a derivative from Equation 16, whichyields

∂Yh(P; wh

)∂ ln Ph f ,k

=∑g

(µg

1 + µg

(Ph f ,gQh f ,g

∂ ln Qh f ,g

∂ ln Ph f ,k+ Phh,gQhh,g

∂ ln Qhh,g

∂ ln Yh

∂ ln Yh

ln Ph f ,k

))

+Ph f ,kQh f ,k + ∑g

((Ph f ,g − Ph f ,g)

∂Qh f ,g

∂ ln Ph f ,k

)

+∑g

((Phh,g − Phh,g)

∂Qhh,g

∂ ln Yh+ (Pf h,g − Pf h,g)

∂Q f h,g

∂ ln Yh

)∂ ln Yh

ln Ph f ,k

=∑g

([µg

1 + µg+ (

Ph f ,g

Ph f ,g− 1)

]Ph f ,gQh f ,gε

h f ,kh f ,g

)+ Ph f ,kQh f ,k + ∑

g

[(Ph f ,g − Ph f ,g)Qh f ,gε

h f ,kh f ,g

]+∑

g

((Pf h,g − Pf h,g)Q f h,gQ f h,gη f h,g +

[µg

1 + µgPhh,g + (Phh,g − Phh,g)

]Qhh,gηhh,g

)∂ ln Yh

∂ ln Ph f ,k.

55

Similarly, dwh/dPh f ,k can be calculated by applying the implicit function theorem tothe balanced trade condition, Dh

(P; wh

), which yields the following:

d ln wh

dlnPh f ,k=

[∂

∂ ln Ph f ,k∑g

(Pf h,gQ f h,g − Ph f ,gQh f ,g

)]/∂Dh(P; wh

)∂ ln wh

=

[∑g

(Pf h,gQ f h,gη f h,g

) ∂ ln Yh

∂ ln Ph f ,k− Ph f ,kQh f ,k −∑

g

(Ph f ,gQh f ,g

∂ ln Qh f ,g

∂ ln Ph f ,k

)]/∂Dh(.)∂ ln wh

=

[∑g

(Pf h,gQ f h,gη f h,g

) ∂Yh

∂ ln Ph f ,k− Ph f ,kQh f ,k −∑

g

(Ph f ,gQh f ,gε

h f ,kh f ,g

)]/ ∂Dh(.)∂ ln wh

.

Plugging the expressions for ∂Yh/∂ ln Ph f ,k and dwh/dlnPh f ,k back into the F.O.C.; adopt-

ing our earlier definition τ ≡(

∂Vh∂Yh

∂Yh∂ ln wh

+ ∑i ∑g∂Vh

∂Pih,g

∂Pih,g∂ ln wh

)/ (∂Dh/∂ ln wh); and fo-

cusing in the solution for which Ψ (t∗, x∗, s∗) = 0, we can write the F.O.C. as

λh f ,k + ∑g

[1− 1

(1 + xk)(1 + τ)(1 + sg)(1 + µg)

]λh f ,gε

h f ,kh f ,g = 0, (21)

where in deriving the above expression, we use the fact Ph f ,k/Ph f ,k = (1 + xk)(1 + sk),and λh f ,g ≡ Ph f ,gQh f ,g/Yh.

Simultaneously Solving the System of F.O.C.

Combining Equations 18, 20, and 21 yields the following system of F.O.C.:

∑g

([1−

1 + sg

1 + µg

]λhh,gεhh,k

hh,g

)+ ∑

g

[(1− 1 + τ

1 + tg

)λ f h,gεhh,k

f h,g

]= 0 [Phh,k]

∑g

([1−

1 + sg

1 + µg

]λhh,gε

f h,khh,g

)+ ∑

g

[(1− 1 + τ

1 + tg

)λ f h,gε

f h,kf h,g

]= 0 [Pf h,k]

λh f ,k + ∑g

[1− 1

(1 + xk)(1 + τ)(1 + sg)(1 + µg)

]λh f ,gε

h f ,kh f ,g = 0 [Ph f ,k]

Solving the above system implies that 1 + sg = 1 + µg and 1 + tg = 1 + τ. The term τ

is, however, redundant due to multiplicity of the optimal policy up to two uniform taxshifters (Lemma 1).54 So, noting that µk ≡ 1/ξk, for any s and t ∈ R+, the followingformula characterizes an optimal tax combination:

(1 + sk) (1 + s) = 1 + 1/ξk ∀k ∈ K

1 + tk = 1 + t ∀k ∈ K

[1/ (1 + xk) (1 + t)]k = E−1h f(

I + Eh f)

1

,

54Considering the notation in Lemma 1, by choice of s we were able to set Ψ (t∗, x∗, s∗) = 0, and bychoice of t we are able to set τ = 0. These choices identify one of the multiple solutions. The remainingsolutions then differ from this particular solution in only the uniform tax-shifters s and t.

56

where Eh f ≡[

λh f ,gλh f ,k

εh f ,kh f ,g

]g,k

is a K× K matrix composed of reduced-form demand elas-

ticities and expenditure shares. Under Homothetic preferences, λh f ,gλh f ,k

εh f ,kh f ,g = ε

h f ,gh f ,k, which

implies that Eh f ≡[ε

h f ,gh f ,k

]g,k

.

F Proof of Theorem 1.B

The optimal tax problem under free entry is characterized by the following systemof first order conditions (F.O.C.s):

∇PhhWh(P; w) = 0

∇P f hWh(P; w

)= 0

∇Ph fWh(P; w

)= 0

,

Our objective is to solve this system subject to (P; w) ∼ (t, x, s; w) ∈ F. As noted in theprevious appendix, focusing on the feasible set of tax-wage combination means thatwe can invoke envelope conditions to evaluate the F.O.C.s. We, hereafter, treat labor inForeign as the numeraire, i.e., w f = 1. Also, to simplify on notation, we let ψk ≡ 1/ξk

denote the industry-level scale elasticity throughout the proof.

F.O.C. with respect to Phh,k

The F.O.C. with respect to Phh,k features terms accounting for direct effects and gen-eral equilibrium wage effects:

dWh(P; wh)

dPhh,k=

∂Vh(.)∂Yh

∂Yh

∂Phh,k+

∂Vh(.)∂Phh,k

+ ∑g

∂Wh

∂Pf f ,g

∂Pf f ,g

∂Phh,k

+

(∂Vh(.)

∂Yh

∂Yh

∂wh+ ∑

i=h, f∑g

[∂Vh(.)∂Pih,g

∂Pih,g

∂wh

])dwh

dPhh,k= 0.

where P ≡ Pji,k. The above problem is greatly simplified by the observation that

∂Wh/∂Pf f ,g = 0. Appealing to Roy’s identity that ∂Vh/∂Pih,k∂Vh/∂Yh

= −Qih,k, we can simplifythe above equation as follows:

dWh(P; wh

)d ln Phh,k

=∂Vh

∂Yh

∂Yh

∂ ln Phh,k− Phh,kQhh,k

+

(∂Yh

∂ ln wh− ∑

i=h, f∑g

[Pih,gQih,g

∂ ln Pih,g

∂ ln wh

])d ln wh

d ln Phh,k

= 0. (22)

To determine ∂Yh/∂ ln Phh,k, note that total income is the sum of factor income, profits,and tax revenue,

Yh(P; wh

)= whLh +Rh(Ph; wh), (23)

57

whereRh(.) is given by 17. Taking a basic partial derivative from Equation 23, yields

∂Yh(P; wh

)∂ ln Phh,k

= −∑g

∑i

(Qhi,g

∂Phi,g

∂ ln Qhh,g

[∂ ln Qhh,g

∂ ln Phh,k+

∂ ln Qhh,g

∂ ln Yh

∂ ln Yh

ln Phh,k

])+Phh,kQhh,k + ∑

g

((Phh,g − Phh,g)Qhh,g

[∂ ln Qhh,g

∂ ln Phh,k+

∂ ln Qhh,g

∂ ln Yh

∂ ln Yh

ln Phh,k

])+∑

g

((Pf h,g − Pf h,g[1 +

∂ ln Pf h,g

∂ ln Q f h,g])Q f h,g

[∂ ln Q f h,g

∂ ln Phh,k+

∂ ln Q f h,g

∂ ln Yh

∂ ln Yh

ln Phh,k

])

Given Phi,k = ρhi,k (∑ι τhι,kQhι,k)− ψk

1+ψk , we can simplify the first line in the above equa-tion as

−∑g

∑i

(Qhi,g

∂Phi,g

∂ ln Qhh,g

∂ ln Qhh,g

∂ ln Phh,k

)= ∑

g

(ψg

1 + ψgPhh,gQhh,g

∂ ln Qhh,g

∂ ln Phh,k

)

Plugging the above equation back into the expression for ∂Yh/∂ ln Phh,k, yields

∂Yh(P; wh

)∂ ln Phh,k

=∑g

([ψg

1 + ψg+ (

Phh,g

Phh,g− 1)

]Phh,gQhh,gεhh,k

hh,g

)+

+Phh,kQhh,k + ∑g

[(Pf h,g − Pf h,g[1 +

∂ ln Pf h,g

∂ ln Q f h,g])Q f h,gεhh,k

f h,g

]+∑

g

((Pf h,g − Pf h,g)Q f h,gη f h,g +

[Phh,g −

ψg

1 + ψgPhh,g

]Qhh,gηhh,g

)∂ ln Yh

∂ ln Phh,k.

where η f h,g ≡ ∂ ln Q f h,g/∂ ln Yh denotes the income elasticity of demand. Similarly,dwh/dPhh,k can be calculated by applying the implicit function theorem to the balancedtrade condition,

Dh(P; wh

)≡∑

k

(Pf h,kQ f h,k − Ph f ,kQh f ,k

)= 0,

which yields the following:

d ln wh

dlnPhh,k=

[∂

∂ ln Phh,k∑g

(Pf h,gQ f h,g − Ph f ,gQh f ,g

)]/∂Dh(P; wh

)∂ ln wh

=

[∑g

(Pf h,gQ f h,g[1 +

∂ ln Pf h,g

∂ ln Q f h,g]∂ ln Q f h,g

∂ ln Phh,k

)+ ∑

g

(Pf h,gQ f h,gη f h,g

) ∂ ln Yh

∂ ln Phh,k

]/∂Dh(.)∂ ln wh

=

[∑g

(Pf h,gQ f h,g[1 +

∂ ln Pf h,g

∂ ln Q f h,g]εhh,k

f h,g

)+ ∑

g

(Pf h,gQ f h,gη f h,g

) ∂Yh

∂ ln Phh,k

]/∂Dh(.)∂ ln wh

.

Plugging the expressions for ∂Yh/∂ ln Phh,k and dwh/dlnPhh,k back into the F.O.C., defin-

ing τ ≡(

∂Vh∂Yh

∂Yh∂wh

+ ∑i ∑g∂Vh

∂Pih,g

∂Pih,g∂ ln wh

)/ (∂ ln Dh/∂wh), we can write the F.O.C. (Equation

58

22) as

∑g

([1− 1

1 + ψg

Phh,g

Phh,g

]Phh,gQhh,gεhh,k

hh,g

)+ ∑

g

[(1− (1 + τ)[1 +

∂ ln Pf h,g

∂ ln Q f h,g]Pf h,g

Pf h,g

)Pf h,kQ f h,kεhh,k

f h,g

]

+∑g

([1− (1 + τ)

Pf h,g

Pf h,g]Pf h,gQ f h,gη f h,g +

[1− 1

1 + ψg

Phh,g

Phh,g

]Phh,gQhh,gηhh,g

)∂ ln Yh

ln Phh,k= 0

Dividing the above equation by Yh and noting that λji,k ≡ Pji,kQji,k/Yi, 1+ sk ≡ Phh,k/Phh,k,1 + tk ≡ Pf h,k/Pf h,k, and

∂ ln Pf h,g

∂ ln Q f h,g=

∂ ln(∑ι τf ι,gQ f ι,g

)− ψg1+ψg

∂ ln Q f h,g= −

ψg

1 + ψgr f h,g

yields the following simplified expression:

∑g

([1−

1 + sg

1 + ψg

]λhh,gεhh,k

hh,g

)+ ∑

g

[(1− 1 + τ

1 + tg

[1 + ψgr f f ,g

1 + ψg

])λ f h,gεhh,k

f h,g

]+∑

g

((1− 1 + τ

1 + tg

)λ f h,gη f h,g +

[1 + sg

1 + ψg− 1]

λhh,gηhh,g

)∂ ln Yh

ln Phh,k= 0

We can further simplify the above expression, we can rely on the observations that (i)there are multiple optimal tax combinations per Lemma 1, (ii) the term in parenthesisin the second line of the above equation, namely,

Ψ (t, x, s) ≡∑g

((1− 1 + τ

1 + tg

)λ f h,gη f h,g +

[1 + sg

1 + ψg− 1]

λhh,gηhh,g

),

is not industry-specific. Hence, by choice of s in Lemma 1, there exists an optimal taxcombination for which Ψ (t∗, x∗, s∗) = 0. We can first restrict attention to this particularsolution, and then trivially identify the other solutions with an across-the-board shiftin all tax instruments. Doing so, the F.O.C. with respect to Phh,k reduces to

∑g

([1−

1 + sg

1 + ψg

]λhh,gεhh,k

hh,g

)+ ∑

g

[(1− 1 + τ

1 + tg

[1 + ψgr f f ,g

1 + ψg

])λ f h,gεhh,k

f h,g

]= 0 (24)

59

F.O.C. with respect to Pf h,k

The F.O.C. with respect to Pf h,k also features terms accounting for direct effects andgeneral equilibrium wage effects:

dWh(P; wh)

dPf h,k=

∂Vh(.)∂Yh

∂Yh

∂Pf h,k+

∂Vh(.)∂Pf h,k

+ ∑g

∂Wh

∂Pf f ,g

∂Pf f ,g

∂Pf h,k

+

(∂Vh(.)

∂Yh

∂Yh

∂wh+ ∑

i=h, f∑g

[∂Vh(.)∂Pih,g

∂Pih,g

∂wh

])dwh

dPf h,k= 0.

Noting that ∂Wh/∂Pf f ,g = 0 and appealing to Roy’s identity that ∂Vh/∂Pih,k∂Vh/∂Yh

= −Qih,k, wecan simplify the above equation as follows:

dWh(P; wh

)d ln Pf h,k

=∂Vh

∂Yh

∂Yh

∂ ln Pf h,k− Pf h,kQ f h,k

+

(∂Yh

∂ ln wh− ∑

i=h, f∑g

[Pih,gQih,g

∂ ln Pih,g

∂ ln wh

])d ln wh

d ln Pf h,k

= 0. (25)

Taking a basic partial derivative from Equation 23, yields

∂Yh(P; wh

)∂ ln Pf h,k

=∑g

([ψg

1 + ψg+ (

Phh,g

Phh,g− 1)

]Phh,gQhh,gε

f h,khh,g

)

Pf h,kQhh,k + ∑g

[(Pf h,g − Pf h,g[1 +

∂ ln Pf h,g

∂ ln Q f h,g])Q f h,gε

f h,kf h,g

]+ ∑

g

((Pf h,g − Pf h,g)Q f h,gη f h,g +

[Phh,g −

ψg

1 + ψgPhh,g

]Qhh,gηhh,g

)∂ ln Yh

∂ ln Phh,k.

where η f h,g ≡ ∂ ln Q f h,g/∂ ln Yh denotes the income elasticity of demand. Similarly,dwh/dPhh,k can be calculated by applying the implicit function theorem to the balancedtrade condition, which yields the following:

d ln wh

dlnPf h,k=

[∂

∂ ln Pf h,k∑g

(Pf h,gQ f h,g − Ph f ,gQh f ,g

)]/∂Dh(P; wh

)∂ ln wh

=

[∑g

(Pf h,gQ f h,g

[1 +

∂ ln Pf h,g

∂ ln Q f h,g

]ε

f h,kf h,g

)+ ∑

g

(Pf h,gQ f h,gη f h,g

) ∂Yh

∂ ln Pf h,k

]/∂Dh(.)∂ ln wh

.

Plugging the expressions for ∂Yh/∂ ln Phh,k and dwh/dlnPhh,k back into the F.O.C., defin-

ing τ ≡(

∂Vh∂Yh

∂Yh∂wh

+ ∑i ∑g∂Vh

∂Pih,g

∂Pih,g∂ ln wh

)/ (∂ ln Dh/∂wh), we can write the F.O.C. (Equation

60

25) as

∑g

([1− 1

1 + ψg

Phh,g

Phh,g

]Phh,gQhh,gε

f h,khh,g

)+ ∑

g

[(1− (1 + τ)

[1 +

∂ ln Pf h,g

∂ ln Q f h,g

]Pf h,g

Pf h,g

)Pf h,gQ f h,gε

f h,kf h,g

]

+∑g

([1− (1 + τ)

Pf h,g

Pf h,g]Pf h,gQ f h,gη f h,g +

[1− 1

1 + ψg

Phh,g

Phh,g

]Phh,gQhh,gηhh,g

)∂ ln Yh

ln Phh,k= 0

Dividing the above equation by Yh and noting that λji,k ≡ Pji,kQji,k/Yi, 1+ sk ≡ Phh,k/Phh,k,and 1 + tk ≡ Pf h,k/Pf h,k, yields the following simplified expression:

∑g

([1−

1 + sg

1 + ψg

]λhh,gε

f h,khh,g

)+ ∑

g

[(1− 1 + τ

1 + tg

[1 + ψgr f f ,g

1 + ψg

])λ f h,gε

f h,kf h,g

]+ Ψ (t, x, s)

∂ ln Yh

ln Phh,k= 0

As noted earlier, there exists an optimal tax combination for which Ψ (t∗, x∗, s∗) = 0.Restricting attention to this particular solution, the F.O.C. with respect to Pf h,k reducesto

∑g

([1−

1 + sg

1 + ψg

]λhh,gε

f h,khh,g

)+ ∑

g

[(1− 1 + τ

1 + tg

[1 + ψgr f f ,g

1 + ψg

])λ f h,gε

f h,kf h,g

]= 0 (26)

F.O.C. with respect to Ph f ,k

Noting that ∂Wh/∂Pf f ,g = 0, the F.O.C. with respect to Ph f ,k can be stated as

dWh(P; wh)

dPh f ,k=

∂Vh(.)∂Yh

∂Yh

∂Ph f ,k+

(∂Vh(.)

∂Yh

∂Yh

∂wh+ ∑

i=h, f∑g

[∂Vh(.)∂Pih,g

∂Pih,g

∂wh

])dwh

dPh f ,k= 0

As before, we can derive ∂Yh/∂ ln Ph f ,k by taking a derivative from Equation 23, yield-ing the following

∂Yh(P; wh

)∂ ln Ph f ,k

= −∑g

∑i

[Phi,gQh f ,g

∂ ln Phh,g

∂ ln Qh f ,g

(∂ ln Qhh,g

∂ ln Ph f ,k+

∂ ln Qhh,g

∂ ln Yh

∂ ln Yh

ln Ph f ,k

)]

+Ph f ,kQh f ,k + ∑g

((Ph f ,g − Ph f ,g)Qh f ,g

∂ ln Qh f ,g

∂ ln Ph f ,k

)−∑

g

(Pf h,gQ f h,g

∂ ln Pf h,g

∂ ln Q f f ,g

∂ ln Q f f ,g

∂ ln Ph f ,k

)

+∑g

((Phh,g − Phh,g)Qhh,g

∂ ln Qhh,g

∂ ln Yh+ (Pf h,g − Pf h,g)Q f h,g

∂ ln Q f h,g

∂ ln Yh

)∂ ln Yh

ln Ph f ,k.

To simplify the last sum in line 2, we can use the fact that Pf h,gQ f h,g∂ ln Pf h,g∂ ln Q f f ,g

= Pf f ,gQ f f ,g∂ ln Pf h,g∂ ln Q f h,g

as well as the following well-known result from consumer theory:

∑g

Pf f ,gQ f f ,g∂ ln Q f f ,g

∂ ln Ph f ,k= −Ph f ,kQh f ,k −∑

gPh f ,gQh f ,g

∂ ln Qh f ,g

∂ ln Ph f ,k.

61

Plugging these equations back into the expression for ∂Yh/∂ ln Ph f ,k yields the follow-ing:

∂Yh(P; wh

)∂ ln Ph f ,k

=∑g

([ψg

1 + ψg+ (

Ph f ,g

Ph f ,g− 1)

]Ph f ,gQh f ,gε

h f ,kh f ,g

)

+Ph f ,kQh f ,k(1 +∂ ln Pf h,k

∂ ln Q f h,k) + ∑

g

[(Ph f ,g(1 +

∂ ln Pf h,g

∂ ln Q f h,g)− Ph f ,g)Qh f ,gε

h f ,kh f ,g

]+∑

g

((Pf h,g − Pf h,g)Q f h,gη f h,g +

[ψg

1 + ψgPhh,g + (Phh,g − Phh,g)

]Qhh,gηhh,g

)∂ ln Yh

∂ ln Ph f ,k.

Similarly, dwh/dPh f ,k can be calculated by applying the implicit function theorem tothe balanced trade condition, Dh

(P; wh

), which yields the following:

d ln wh

dlnPh f ,k=

[∂

∂ ln Ph f ,k∑g

(Pf h,gQ f h,g − Ph f ,gQh f ,g

)]/∂Dh(P; wh

)∂ ln wh

=

[∑g

(Pf h,gQ f h,g

[∂ ln Pf h,g

∂ ln Q f f ,g

∂ ln Q f f ,g

∂ ln Ph f ,k+ η f h,g

∂ ln Yh

∂ ln Ph f ,k

])

− Ph f ,kQh f ,k −∑g

(Ph f ,gQh f ,g

∂ ln Qh f ,g

∂ ln Ph f ,k

)]/∂Dh(.)∂ ln wh

.

Simplifying the term Pf h,gQ f h,g∂ ln Pf h,g∂ ln Q f f ,g

∂ ln Q f f ,g

∂ ln Ph f ,kalong the same lines outlined earlier, we

can rewrite the expression for d ln wh/dlnPh f ,k as

d ln wh

dlnPh f ,k=

[∑g

(Pf h,gQ f h,gη f h,g

) ∂Yh

∂ ln Ph f ,k

− Ph f ,kQh f ,k(1 +∂ ln Pf h,k

∂ ln Q f h,k)−∑

g

(Ph f ,gQh f ,g(1 +

∂ ln Pf h,g

∂ ln Q f h,g)ε

h f ,kh f ,g

)]/∂Dh(.)∂ ln wh

.

Plugging the expressions for ∂Yh/∂ ln Ph f ,k and dwh/dlnPh f ,k back into the F.O.C.; adopt-

ing our earlier definition τ ≡(

∂Vh∂Yh

∂Yh∂ ln wh

+ ∑i ∑g∂Vh

∂Pih,g

∂Pih,g∂ ln wh

)/ (∂Dh/∂ ln wh); focusing

in the solution for which Ψ (t∗, x∗, s∗) = 0; and noting that 1 +∂ ln Pf h,g∂ ln Q f h,g

=1+ψgr f f ,g

1+ψg; we

can write the F.O.C. as(1 + ψkr f f ,k

1 + ψk

)λh f ,k +∑

g

[1 + ψgr f f ,g

1 + ψg− 1

(1 + xg)(1 + τ)(1 + sg)(1 + µg)

]λh f ,gε

h f ,kh f ,g = 0,

(27)where in deriving the above expression, we use the fact Ph f ,k/Ph f ,k = (1 + xk)(1 + sk),and λh f ,g ≡ Ph f ,gQh f ,g/Yh.

62

Simultaneously Solving the System of F.O.C.

Combining Equations 24, 26, and 27 yields the following system of F.O.C.:

[Phh,k] ∑g

([1−

1 + sg

1 + µg

]λhh,gεhh,k

hh,g

)+ ∑

g

[(1− 1 + τ

1 + tk

)λ f h,gεhh,k

f h,g

]= 0 [Phh,k]

[Pf h,k] ∑g

([1−

1 + sg

1 + µg

]λhh,gε

f h,khh,g

)+ ∑

g

[(1− 1 + τ

1 + tk

(1 + ψgr f f ,g

1 + ψg

))λ f h,gε

f h,kf h,g

]= 0

[Ph f ,k] λh f ,k + ∑g

[1− 1

(1 + xk)(1 + τ)(1 + sg)(1 + ψg)

(1 + ψg

1 + ψgr f f ,g

)]λh f ,gε

h f ,kh f ,g

(1 + ψg)(1 + ψkr f f ,k)

(1 + ψk)(1 + ψgr f f ,g)= 0

Solving the above system implies that 1 + sg = 1 + ψg and 1 + tg = 1 + τ. The termτ is, however, redundant due to multiplicity of the optimal policy up to two uniformtax shifters (Lemma 1). So, noting that ψk ≡ 1/ξk, for any s and t ∈ R+, the followingformula characterizes an optimal tax combination:

(1 + sk) (1 + s) = 1 + 1/ξk ∀k ∈ K

1 + tk = (1 + t)(

1+r f f ,k/ξk1+1/ξk

)= (1 + t)

(1− r f h,k

1+ξk

)∀k ∈ K[

1/(

1− r f h,k1+ξk

)(1 + xk) (1 + t)

]k= E−1

h f(

I + Eh f)

1

,

where Eh f ≡[

1−r f h,k/(1+ξk)

1−r f h,g/(1+ξg)

λh f ,gλh f ,k

εh f ,kh f ,g

]g,k

is a K×K matrix composed of reduced-form de-

mand elasticities and expenditure shares. Under Homothetic preferences, λh f ,gλh f ,k

εh f ,kh f ,g =

εh f ,gh f ,k, which implies that Eh f ≡

[1−r f h,k/(1+ξk)

1−r f h,g/(1+ξg)ε

h f ,gh f ,k

]g,k

.

G Proof of Theorem 2

As before, we simplify on notation by letting µk = 1/ξk denote the constant markupthroughout. Without domestic taxes the F.O.C. with respect to trade taxes xk and tk

(Equations 18 and 21) can be stated as follows:

∑g

[(1− 1

1 + µg

)Phh,gQhh,gε

f h,khh,g

]+ ∑

g

[(1− 1 + τ

1 + tg

)Pf h,gQ f h,gε

f h,kf h,g

]= 0

Ph f ,kQh f ,k + ∑g

([1− 1

(1 + µg)(1 + xg) (1 + τ)

]Ph f ,gQh f ,gε

h f ,kh f ,g

)= 0

Given that λji,g = Pji,gQji,g/Yi and ∑g ∑j λjh,gεf h,kjh,g = −λ f h,k, the first equation can be

simplified as

∑g

[1 + τ

1 + tg

λ f h,g

λ f h,kε

f h,kf h,g

]= −1−

(∑g

11 + µg

λhh,g

λ f h,kε

f h,khh,g

).

63

Writing the above equation in matrix form; combining it withe F.O.C. for the exporttax; and accounting for the multiplicity of the trade tax, implies the following optimaltax formula for any t ∈ R+:[

1 + t∗k1 + t

]k= −E−1

f h

(I + E f h

hhm)

[1

(1 + x∗k ) (1 + t) (1 + µk)

]k= E−1

h f(

I + Eh f)

1

where E f h ≡[

λ f h,g

λ f h,kε

f h,kf h,g

]g,k

, E f hhh ≡

[λhh,g

λh f ,kε

f h,khh,g

]g,k

, and m ≡[

11+µk

]. Finally, assuming

zero cross-substitutability between industries, i.e., εf h,kf h,g = 0 and ε

f h,khh,g if g 6= k, and

noting that µk ≡ 1/ξk, the above formula reduces to:

1 + t∗k =(1 + ξk) [1 + εkλhh,k]

1 + ξk (1 + εkλhh,k)(1 + t);

1 + x∗k =1 + 1/εkλhh,k

1 + 1/ξk(1 + t)−1.

H Proof of Theorem 3

As before, we simplify on notation by letting µk = 1/ξk denote the constant markupthroughout. Following Section E, when sk = xk = 0, the F.O.C. with respect to tk

(Equation 18) can be stated as

∑g

[1 + τ

1 + tg

λ f h,g

λ f h,kε

f h,kQf h,g

]= −1−

(∑g

11 + µg

λhh,g

λ f h,kε

f h,khh,g

).

Rearranging the above expression while noting that εf h,kf h,g = −1− εkλhh,k if g = k and

εf h,kf h,g = 0 if g 6= k implies the following optimal tax formula:

1 + t∗k =(1 + µk) [1 + εkλhh,k]

1 + µk + εkλhh,k(1 + τ).

However, since the use of export taxes is restricted , τ is no longer redundant andshould be formally characterized. To so so, we can appeal to the definition of τ asfollows:

τ = −∂Wh/∂wh

∂Dh/∂wh= −

∂Wh/∂w f

∂Dh/∂w f=

∂Vh∂Yh

∂Yh∂w f

+ ∑i ∑g∂Vh

∂Pih,g

∂Pih,g∂w f

∂∂wh

∑k[Ph f ,kQh f ,k − Pf h,kQ f h,k

] .

Given that ∂Vh(.)/∂Pih,g = −Qih,g, ∂ ln Pf i,g/∂ ln w f =1, and Yh = whLh + Πh, we canrewrite the above expression as

64

τ =∑g ∑i

(µg

1+µgPhi,gQhi,gε

f i,ghi,g

)+ ∑k

(tgPf h,gQ f h,gεhh,k

f h,g

)−∑k Pf h,kQ f h,k

∑k

[Pf h,kQ f h,k(1 + ε f h,k)− Ph f ,kQh f ,k(ε

f f ,kh f ,k + γh f ,k)

] .

To simplify the above expression we can appeal to the F.O.C., which states that

∑ (tk − τ) Pf h,kQ f h,kε f h,k = −∑g

[µg

1 + µgPhh,gQhh,gε

f h,khh,g

].

Plugging the above expression back into Equation and noting that εf i,ghi,g = εkλ f i,g,

γh f ,k = 1, and ∑k Pf h,kQ f h,k = ∑ Ph f ,kQh f ,k, yields the following expression for τ:

τ =∑g ∑i

(µg

1+µgPhi,gQhi,gεgλ f i,g

)−∑g

(µg

1+µgPhh,gQhh,gεgλ f h,g

)−∑k

(Ph f ,kQh f ,k

)−∑k Ph f ,kQh f ,kεkλ f f ,k

.

Plugging the above equation back in the optimal tax formula and noting that Ph f ,kQh f ,k

∑g Ph f ,gQh f ,g=

λh f ,k

λh fand µk ≡ 1/ξk yields the sufficient statistics tax formulas specified under Theorem

3:

1 + t∗k =(1 + ξk) [1 + εkλhh,k]

1 + ξk (1 + εkλhh,k)

1 + ∑g

[ξk

1+ξg

λh f ,g

λh fεgλ f f ,g

]∑g

λh f ,g

λh fεgλ f f ,g

.

I Optimal Tax Formula with IO Linkages

We can appeal to the theorem in Kopczuk (2003), whereby one can first eliminatethe underlying market inefficiency with the appropriate domestic tax and then solvefor optimal taxes as if markets where efficient. The optimal policy will then be the sumof corrective taxes and taxes corresponding to an efficient market. In the present con-text, we apply this result by solving for optimal taxes while abstracting from profits,i.e., Yh(P; wh) = whLh +Rh(Ph; wh). After solving for the optimal taxes under effi-cient markets, we amend them with corrective Pigouvian taxes that eliminate markupheterogeneity.

To streamline the notation, we let αj,gi,k denote the cost-based input-output shares: α

j,gi,k

denotes the (possibly variable) share of intermediate inputs from Country j× Industryg in the output of Country i× Industry k. By Shepherd’s Lemma:

∂ ln Pij,k(P; wh)

∂ ln PIji,g= α

j,gi,k , ∀, j, i ∈ C; g, k ∈ K.

Also, as in the previous proofs, we simplify the notation by letting µk ≡ 1/ξk denotethe constant industry-level markup margin.

65

F.O.C. with respect to Phh,k

The F.O.C. with respect to Phh,k features terms accounting for direct effects and gen-eral equilibrium wage effects:

dWh(P; wh)

dPhh,k=

∂Vh(.)∂Yh

∂Yh

∂Phh,k+

∂Vh(.)∂Phh,k

+ ∑g

∂Wh

∂Pf f ,g

∂Pf f ,g

∂Phh,k

+

(∂Vh(.)

∂Yh

∂Yh

∂wh+ ∑

i=h, f∑g

[∂Vh(.)∂Pih,g

∂Pih,g

∂wh

])dwh

dPhh,k= 0.

where P ≡ Pji,k. The above problem is greatly simplified by the observation that∂Pf f ,g/∂Phh,k = 0—i.e., holding Pf h,k, Ph f ,k, and wh fixed, Phh,k has no effect on Pf f ,k.

Appealing to Roy’s identity that ∂Vh/∂Pih,k∂Vh/∂Yh

= −QCih,k, we can simplify the above equationas follows:

dWh(P; wh

)d ln Phh,k

=∂Vh

∂Yh

∂Yh

∂ ln Phh,k− Phh,kQChh,k

+

(∂Yh

∂ ln wh− ∑

i=h, f∑g

[Pih,gQCih,g

∂ ln Pih,g

∂ ln wh

])d ln wh

d ln Phh,k

= 0. (28)

To determine ∂Yh/∂ ln Phh,k, note that total income is the sum of factor income, profits,and tax revenue,

Yh(P; wh

)= whLh +Rh(Ph; wh), (29)

whereRh(.) is given by 17. Taking a basic partial derivative from Equation 29, yields

∂Yh(P; wh

)∂ ln Phh,k

= −∑g

∑i

(Qhi,gPhi,g

∂ ln Phi,g

∂ ln PIhh,g

)+ Phh,kQhh,k

+∑g

((Phh,g − Phh,g)Qhh,g

[∂ ln Qhh,g

∂ ln Phh,k+

∂ ln Qhh,g

∂ ln Yh

∂ ln Yh

ln Phh,k

])+∑

g

((Pf h,g − Pf h,g)Q f h,g

[∂ ln Q f h,g

∂ ln Phh,k+

∂ ln Q f h,g

∂ ln Yh

∂ ln Yh

ln Phh,k

])

Noting that by Shepherd’s Lemma, ∂ ln Pin,g(P; wh)/∂ ln PIji,k = αj,ki,g, we can simplify the

first line in the above equation as

−∑g

∑i

(Qhi,gPhi,g

∂ ln Phi,g

∂ ln PIhh,g

)= −∑

g

(αh,k

h,gQhi,gPhi,g

)= −Phh,kQIhh,k

66

Plugging the above equation back into the expression for ∂Yh/∂ ln Phh,k, yields

∂Yh(P; wh

)∂ ln Phh,k

=PChh,kQhh,k + ∑g

[(Pf h,g − Pf h,g)Q f h,gεhh,k

f h,g + (Phh,g − Phh,g)Qhh,gεhh,khh,g

]+∑

g

((Pf h,g − Pf h,g)Q f h,gη f h,g +

[Phh,g − Phh,g

]Qhh,gηhh,g

) ∂ ln Yh

∂ ln Phh,k.

where η f h,g ≡ ∂ ln Q f h,g/∂ ln Yh denotes the income elasticity of demand. Similarly,dwh/dPhh,k can be calculated by applying the implicit function theorem to the balancedtrade condition,

Dh(P; wh

)≡∑

k

(Pf h,kQ f h,k − Ph f ,kQh f ,k

)= 0,

which yields the following:

d ln wh

dlnPhh,k=

[∂

∂ ln Phh,k∑g

(Pf h,gQ f h,g − Ph f ,gQh f ,g

)]/∂Dh(P; wh

)∂ ln wh

=

[∑g

(Pf h,gQ f h,gεhh,k

f h,g

)+ ∑

g

(Pf h,gQ f h,gη f h,g

) ∂Yh

∂ ln Phh,k

]/∂Dh(.)∂ ln wh

.

Plugging the expressions for ∂Yh/∂ ln Phh,k and dwh/dlnPhh,k back into the F.O.C., defin-

ing τ ≡(

∂Vh∂Yh

∂Yh∂wh

+ ∑i ∑g∂Vh

∂Pih,g

∂Pih,g∂ ln wh

)/ (∂ ln Dh/∂wh), we can write the F.O.C. (Equation

28) as

∑g

([1−

Phh,g

Phh,g

]Phh,gQhh,gεhh,k

hh,g

)+ ∑

g

[(1− (1 + τ)

Pf h,k

Pf h,k

)Pf h,kQ f h,kεhh,k

f h,g

]

+∑g

([1− (1 + τ)

Pf h,k

Pf h,k]Pf h,gQ f h,gη f h,g +

[1−

Phh,g

Phh,g

]Phh,gQhh,gηhh,g

)∂ ln Yh

ln Phh,k= 0

yields the following simplified expression:

∑g

([1−

(1 + sg

)]λhh,gεhh,k

hh,g

)+ ∑

g

[(1− 1 + τ

1 + tg

)λ f h,gεhh,k

f h,g

]+∑

g

((1− 1 + τ

1 + tg

)λ f h,gη f h,g +

[(1 + sg

)− 1]

λhh,gηhh,g

)∂ ln Yh

ln Phh,k= 0

We can further simplify the above expression, we can rely on the observations that (i)there are multiple optimal tax combinations per Lemma 1, (ii) the term in parenthesisin the second line of the above equation, namely,

Ψ (t, x, s) ≡∑g

((1− 1 + τ

1 + tg

)λ f h,gη f h,g +

[(1 + sg

)− 1]

λhh,gηhh,g

),

67

is not industry-specific. Hence, by choice of s in Lemma 1, there exists an optimal taxcombination for which Ψ (t∗, x∗, s∗) = 0. We can first restrict attention to this particularsolution, and then trivially identify the other solutions with an across-the-board shiftin all tax instruments. Doing so, the F.O.C. with respect to Phh,k reduces to

∑g

([1−

(1 + sg

)]λhh,gεhh,k

hh,g

)+ ∑

g

[(1− 1 + τ

1 + tg

)λ f h,gεhh,k

f h,g

](30)

F.O.C. with respect to Pf h,k

The F.O.C. with respect to Pf h,k also features terms accounting for direct effects andgeneral equilibrium wage effects:

dWh(P; wh)

dPf h,k=

∂Vh(.)∂Yh

∂Yh

∂Pf h,k+

∂Vh(.)∂Pf h,k

+ ∑g

∂Wh

∂Pf f ,g

∂Pf f ,g

∂Pf h,k

+

(∂Vh(.)

∂Yh

∂Yh

∂wh+ ∑

i=h, f∑g

[∂Vh(.)∂Pih,g

∂Pih,g

∂wh

])dwh

dPf h,k= 0.

Noting that ∂Pf f ,g/Pf h,g = 0 and appealing to Roy’s identity that ∂Vh/∂Pih,k∂Vh/∂Yh

= −QCih,k, wecan simplify the above equation as follows:

dWh(P; wh

)d ln Pf h,k

=∂Vh

∂Yh

∂Yh

∂ ln Pf h,k− Pf h,kQCf h,k

+

(∂Yh

∂ ln wh− ∑

i=h, f∑g

[Pih,gQCih,g

∂ ln Pih,g

∂ ln wh

])d ln wh

d ln Pf h,k

= 0. (31)

Taking a basic partial derivative from Equation 29, yields

∂Yh(P; wh

)∂ ln Pf h,k

=∑g

([Phh,g

Phh,g− 1

]Phh,gQhh,gε

f h,khh,g

)+Pf h,kQChh,k + ∑

g

[(Pf h,g − Pf h,g)Q f h,gε

f h,kf h,g

]+ ∑

g

((Pf h,g − Pf h,g)Q f h,gη f h,g +

[Phh,g − Phh,g

]Qhh,gηhh,g

) ∂ ln Yh

∂ ln Phh,k.

where η f h,g ≡ ∂ ln Q f h,g/∂ ln Yh denotes the income elasticity of demand. Similarly,dwh/dPhh,k can be calculated by applying the implicit function theorem to the balanced

68

trade condition:

d ln wh

dlnPf h,k=

[∂

∂ ln Pf h,k∑g

(Pf h,gQ f h,g − Ph f ,gQh f ,g

)]/∂Dh(P; wh

)∂ ln wh

=

[∑g

(Pf h,gQ f h,gε

f h,kf h,g

)+ ∑

g

(Pf h,gQ f h,gη f h,g

) ∂Yh

∂ ln Pf h,k

]/∂Dh(.)∂ ln wh

.

Plugging the expressions for ∂Yh/∂ ln Phh,k and dwh/dlnPhh,k back into the F.O.C., defin-

ing τ ≡(

∂Vh∂Yh

∂Yh∂wh

+ ∑i ∑g∂Vh

∂Pih,g

∂Pih,g∂ ln wh

)/ (∂ ln Dh/∂wh), we can write the F.O.C. (Equation

31) as

∑g

([1−

Phh,g

Phh,g

]Phh,gQhh,gε

f h,khh,g

)+ ∑

g

[(1− (1 + τ)

Pf h,k

Pf h,k

)Pf h,kQ f h,kε

f h,kf h,g

]

+∑g

([1− (1 + τ)

Pf h,k

Pf h,k]Pf h,gQ f h,gη f h,g +

[1−

Phh,g

Phh,g

]Phh,gQhh,gηhh,g

)∂ ln Yh

ln Phh,k= 0

Dividing the above equation by Yh and noting that λji,k ≡ Pji,kQji,k/Yi, 1+ sk ≡ Phh,k/Phh,k,and 1 + tk ≡ Pf h,k/Pf h,k, yields the following simplified expression:

∑g

([1− (1 + sg)

]λhh,gε

f h,khh,g

)+ ∑

g

[(1− 1 + τ

1 + tg

)λ f h,gε

f h,kf h,g

]+ Ψ (t, x, s)

∂ ln Yh

ln Phh,k= 0

As noted earlier, there exists an optimal tax combination for which Ψ (t∗, x∗, s∗) = 0.Restricting attention to this particular solution, the F.O.C. with respect to Pf h,k reducesto

∑g

([1− (1 + sg)

]λhh,gε

f h,khh,g

)+ ∑

g

[(1− 1 + τ

1 + tg

[1 + ψgr f f ,g

1 + ψg

])λ f h,gε

f h,kf h,g

]= 0 (32)

F.O.C. with respect to Ph f ,k

Noting that ∂Wh/∂Pf f ,g = 0, the F.O.C. with respect to Ph f ,k can be stated as

dWh(P; wh)

dPh f ,k=

∂Vh(.)∂Yh

∂Yh

∂Ph f ,k+

(∂Vh(.)

∂Yh

∂Yh

∂wh+ ∑

i=h, f∑g

[∂Vh(.)∂Pih,g

∂Pih,g

∂wh

])dwh

dPh f ,k= 0

69

As before, we can derive ∂Yh/∂ ln Ph f ,k by taking a derivative from Equation 29, yield-ing the following

∂Yh(P; wh

)∂ ln Ph f ,k

= −∑g

[(Ph f ,gQh f ,g

∂ ln Ph f ,g

∂ ln PIh f ,g+ Phh,gQhh,g

∂ ln Phh,g

∂ ln PIh f ,g

)]

+Ph f ,kQh f ,k + ∑g

((Ph f ,g − Ph f ,g)Qh f ,g

∂ ln Qh f ,g

∂ ln Ph f ,k

)−∑

g

(Pf h,gQ f h,g

∂ ln Pf h,g

∂ ln PIh f ,g

)

+∑g

((Phh,g − Phh,g)Qhh,g

∂ ln Qhh,g

∂ ln Yh+ (Pf h,g − Pf h,g)Q f h,g

∂ ln Q f h,g

∂ ln Yh

)∂ ln Yh

ln Ph f ,k.

To simplify the last sum in line 2, we can use the fact that Pf h,gQ f h,g∂ ln Pf h,g∂ ln Q f f ,g

= Pf f ,gQ f f ,g∂ ln Pf h,g∂ ln Q f h,g

as well as the following well-known result from consumer theory:

∑g

Pf f ,gQ f f ,g∂ ln Q f f ,g

∂ ln Ph f ,k= −Ph f ,kQh f ,k −∑

gPh f ,gQh f ,g

∂ ln Qh f ,g

∂ ln Ph f ,k.

Plugging these equations back into the expression for ∂Yh/∂ ln Ph f ,k yields the follow-ing:

∂Yh(P; wh

)∂ ln Ph f ,k

=Ph f ,kQh f ,k + ∑g

[(Ph f ,g − Ph f ,g)Qh f ,gε

h f ,kh f ,g

]+ ∑

g

(Pf h,gQ f h,gαh,k

f ,g

)+∑

g

((Pf h,g − Pf h,g)Q f h,gη f h,g +

[Phh,g − Phh,g

]Qhh,gηhh,g

) ∂ ln Yh

∂ ln Ph f ,k.

Similarly, dwh/dPh f ,k can be calculated by applying the implicit function theorem tothe balanced trade condition, Dh

(P; wh

), which yields the following:

d ln wh

dlnPh f ,k=

[∂

∂ ln Ph f ,k∑g

(Pf h,gQ f h,g − Ph f ,gQh f ,g

)]/∂Dh(P; wh

)∂ ln wh

=

[∑g

(Pf h,gQ f h,g

[αh,k

f ,g + η f h,g∂ ln Yh

∂ ln Ph f ,k

])

− Ph f ,kQh f ,k −∑g

(Ph f ,gQh f ,g

∂ ln Qh f ,g

∂ ln Ph f ,k

)]/∂Dh(.)∂ ln wh

.

Plugging the expressions for ∂Yh/∂ ln Ph f ,k and dwh/dlnPh f ,k back into the F.O.C.; adopt-

ing our earlier definition τ ≡(

∂Vh∂Yh

∂Yh∂ ln wh

+ ∑i ∑g∂Vh

∂Pih,g

∂Pih,g∂ ln wh

)/ (∂Dh/∂ ln wh); and fo-

70

cusing in the solution for which Ψ (t∗, x∗, s∗) = 0; we can write the F.O.C. as

λh f ,k + ∑g

[1− 1

(1 + xg)(1 + τ)(1 + sg)(1 + µg)

]λh f ,gε

h f ,kh f ,g −

∑k Pf h,gQ f h,ggαh,kf ,g

Ph f ,kQh f ,k=

λh f ,k + ∑g

[1− 1

(1 + xg)(1 + τ)(1 + sg)(1 + µg)

]λh f ,gε

h f ,kh f ,g −∑ r f h,gλIh f ,kg = 0

(33)

where where λIh f ,kg denotes the fraction of good h f , k’s exports that are used as inputsin Foreign’s industry g. Moreover, in deriving the above expression we use the factPh f ,k/Ph f ,k = (1 + xk)(1 + sk), and λh f ,g ≡ Ph f ,gQh f ,g/Yh.

Simultaneously Solving the System of F.O.C.

Combining Equations 30, 32, and 33 Solving the above system implies that 1+ sg =

1 and 1 + tg = 1 + τ. The term τ is, however, redundant due to multiplicity of theoptimal policy up to two uniform tax shifters (Lemma 1). So, introducing the Pigouviandomestic tax, the following formula characterizes an optimal tax combination for anys and t ∈ R+:

(1 + s∗k

)(1 + s) = 1 + 1/ξk ∀k ∈ K

1 + tk = 1 + t ∀k ∈ K

1 + x∗k =(

1 + 1−αh f ,kλ f f ,kεk+αh f ,k

)(1 + t)−1

,

where αh f ,k ≡ ∑ r f h,gλIh f ,kg. Considering the above theorem, the optimal policy is fullydetermined by industry-level expenditure and revenue shares, as well as 1/ξk, εk,and the global IO matrix, α. Here, however, the corrective gains from policy no longerdepend on only Vark(ln 1/ξk), but also depend on the centrality-adjusted variance of1/ξk’s–see Liu (2018) for a more formal discussion. Similarly, the terms-of-trade gainsfrom policy depend on a IO-weighted average of the trade elasticities. That is becauseIO linkages provide the Home country with extraterritorial taxing power—see Beshkarand Lashkaripour (2019). Finally, imposing more structure on the IO matrix, α, à laCaliendo and Parro (2015), we can use the above optimal tax formulas to produce ananalog of Proposition 3 under IO linkages.

J Optimal Tax Formula with Arbitrarily Many Countries

Suppose there are arbitrarily many countries. Then, the effect of country i’s importtax on own welfare can be expressed as

dWi(.)d ln(1 + tji,k)

=∂Wi(.)

∂ ln(1 + tji,k)+ ∑

6=j

∂Wi(.)∂ ln w

d ln w

d ln(1 + tji,k).

71

It is straightforward to show that ∂Wi(.)∂ ln w

d ln w

d ln(1+tji,k)∝ λiri,kλji,k. Based on actual trade

data, λiri,k/λiirii,k ≈ 0 for 6= i; so, changes in welfare can be approximated as

dWi(.)d ln(1 + tji,k)

≈ ∂Wi(.)∂ ln(1 + tji,k)

+∂Wi(.)∂ ln wi

d ln wi

d ln(1 + tji,k).

The same applies to other types of taxes. Wee can, thus, cast the Country i’s optimalpolicy problem as one that maximizes Wi(P, wi) by choosing Pji,k, Pij,ksubject to fea-sibility. Following the same steps as in Appendix E, the system of F.O.C. correspondingto this problem can be expressed as:

∑g

([1−

1 + si,g

1 + 1/ξg

]λhh,gε

ji,kii,g

)+ ∑

g∑j 6=i

[(1− 1 + τ

1 + tji,g

)λji,gε

ji,kji,g

]= 0 [Pji,k]

λij,k + ∑g

[1− 1

(1 + xij,k)(1 + τ)(1 + si,g)(1 + µg)

]λh f ,gε

h f ,kh f ,g = 0 [Pij,k].

Noting that in absence of cross-industry demand effects,

εi,kji,k =

−1− εk(1− λji,k) = j

εkλi,g 6= j,

the above system of F.O.C.s implies the following policy schedule for country i

1 + s∗i,k =1 + s

1 + 1/ξk, ∀k

1 + t∗ji,k = 1 + t, ∀j, k

1 + x∗ij,k =(

1 +1

εk(1− λij,k)

)(1 + t)−1, ∀j, k.

K Technical Appendix: Cleaning the data on the identity/nameof exporting firms

Utilizing the information on the identity of the foreign exporting firm is a criticalpart of our empirical exercise. Unfortunately, the names of the exporting firms in ourdataset are not standardized. As a result, there are instances when the same firm isrecorded differently due to using or not using the abbreviations, capital and lower-case letters, spaces, dots, other special characters, etc. To standardize the names of theexporting firms, we used the following procedure.55

1. We deleted all observations with the missing exporting names and/or zero tradevalues.

2. We capitalized firms names and their contact information (which is either email

55The corresponding Stata code is in the cleanFirmsNames.do.

72

Table 5: Summary Statistics of the Colombian Import Data.

Year

Statistic 2007 2008 2009 2010 2011 2012 2013

F.O.B. value (billion dollars) 30.77 37.26 31.39 38.41 52.00 55.79 56.92C.I.F. valueF.O.B. value 1.08 1.07 1.05 1.06 1.05 1.05 1.05C.I.F. + tax value

F.O.B. value 1.28 1.21 1.14 1.19 1.15 1.18 1.15

No. of exporting countries 210 219 213 216 213 221 224

No. of imported varieties 483,286 480,363 457,000 509,524 594,918 633,008 649,561

Notes: Tax value includes import tariff and value-added tax (VAT). The number of varieties correspondsto the number of country-firm-product combination imported by Colombia in a given year.

or phone number of the firm).3. We eliminated abbreviation “LLC,” spaces, parentheses, and other special char-

acters (. , ; / @ ‘ - & “) from the firms names.4. We eliminated all characters specified in 3. above and a few others (# : FAX) from

the contact information.5. We dropped observations without contact information (such as, "NOTIENE",

"NOREPORTA", "NOREGISTRA," etc.), with non-existent phone numbers (e.g., “0000000000”,“1234567890”, “1”), and with six phone numbers which are used for multiple firmswith different names (3218151311, 3218151297, 6676266, 44443866, 3058712865, 3055935515).

6. Next, we kept only up to first 12 characters in the firm’s name and up to first12 characters in the firm’s contact information (which is either email or phone num-ber). In our empirics, we treat all transaction with the same updated name and contactinformation as coming from the same firm.

7. We also analyzed all observations with the same contact information, but slightlydifferent name spelling. We only focused on the cases in which there are up to threedifferent variants of the firm name. For these cases, we calculated the Levenshteindistance in the names, which is the smallest number of edits required to match onename to another. We treat all export observations as coming from the same firm if thecontact phone number (or email) is the same and the Levenshtein distance is four orless.

L Illustrative Example for our Instrumental Variable

This section presents an example that highlights the workings of the shift-shareinstrument constructed in 5. The example corresponds to U.S. exports in productcategory HS8431490000 (PARTS AND ATTACHMENTS OTHER FOR DERRIKS ETC.)—aproduct category that features one of the most frequently imported varieties: machineparts from “CATERPILLAR.” The left panel of Figure 2 displays how both the exchangerate and the average import tax rate paid by U.S. based firms varied considerably on a

73

Figure 2: Monthly variations in national exchange rate, firm-specific exports, and import tax within a selectedproduct-country-year category.

.85

.9

.95

1

1.05

1.1

JAN 2009 APR 2009 JUL 2009 OCT 2009

Exchange Rate Trade Tax

0.60

0.80

1.00

1.20

1.40

1.60

JAN 2009 APR 2009 JUL 2009 OCT 2009

CATERPILLAR MACHINERY COR. OF AMERICA

Export Sales

Notes: The left panel plots monthly variations in exchange rate and (value-weighted) average importtax for US-based firms within product category HS8431490000-year 2009. The right panel plots monthlymovements in export sales for the two biggest US firms in product HS8431490000 in year 2009— namely,Caterpillar and Machinery Corp. of America.

monthly basis in 2009. The right panel plots the monthly variation in the export sales ofthe two largest U.S. based firms within category HS8431490000 (namely, “CATERPIL-LAR” and “MACHINERY CORP. OF AMERICA”). Given that the monthly composition ofexports from “CATERPILLAR” and “MACHINERY CORP. OF AMERICA” are markedlydifferent, the two firms are affected differently by aggregate movements in the monthlyexchange rate.

M Robustness Checks: Import Demand Estimation

This appendix reports three robustness checks that we described in Section 5. Thefirst check addresses the possibility that firms set prices in forward-looking manner.To restate the issue, when there are lags in inventory clearances, firms’ optimal pric-ing decisions may be forward-looking. If true, such price-setting behaviors can violateassumption (a1). To address this concern, we reconstruct our shift-share instrumentusing 4 lags instead of 1. If inventories clear in at most 4 years, we can deduce that pric-ing decisions do not internalize expected demand shocks beyond the 4 year mark. Asa result, Corr

[pjkt−4(ω, m), ∆ϕωjkt

]= 0, and this more-stringent instrument will sat-

isfy the exclusion restriction. The top panel in Figure 3 compares the estimated εk and1/ξkunder the new and baseline estimations. Evidently, the ordering and magnitudeof the estimated elasticities is rather preserved across industries. More importantly, thenew estimation retains the negative correlation between εk and 1/ξk, which is the keyassumption in Proposition 1.

The second check addresses the possibility that, in the presence of cross-inventory

74

effects, ∆ϕωjkt may encompass omitted variables that concern firms’ dynamic inventorymanagement decisions. These decisions internalize exchange rate movements, whichmay violate the identifying assumption (a2), i.e., Corr

[∆Ejt(m), ∆ϕωjkt

]6= 0. To ad-

dress this concern, we reestimate the firm-level import demand function while directlycontrolling for changes on the annual exchange rate. In that case, E

[zjk,t(ω) ∆ϕωjkt | ∆Ejt

],

and the exclusion restriction will be satisfied insofar as dynamic demand optimizationis a concern. The middle panel in Figure 3 compares the estimated εk and 1/ξkunder thenew and baseline estimations. Evidently, the ordering and magnitude of the estimatedelasticities is rather preserved across industries. More importantly, the new estimationretains the negative correlation between εk and 1/ξk, which is the key assumption inProposition 1.

The third check addresses large multi-product firms that export multiple productvarieties to Colombia in a given year. Suppose a multi-product firm ω exports manyproducts including products k and g to Colombia in year t. If demand shock are corre-lated across the varieties supplied by this firm (i.e., E

[∆ϕωjkt ∆ϕωjgt

]6= 0), Assumption

(a2) may be violated despite each variety’s market share being infinitesimally small. Toaddress this issue, we reestimate the firm-level import demand function on a restrictedsample that drops excessively large firms with a within-national market share that ex-ceeds 0.1%. The bottom panel in Figure 3 compares the estimated εk and 1/ξkunder thenew and baseline estimations. Evidently, the ordering and magnitude of the estimatedelasticities is rather preserved across industries. More importantly, the new estimationretains the negative correlation between εk and 1/ξk, which is the key assumption inProposition 1.

N Examining the Plausibility of Estimates

In this section, we look beyond policy implications and ask if our estimates can helpresolve the income-size elasticity puzzle. This puzzle, as noted by Ramondo et al. (2016),concerns the fact that a large class of quantitative trade models, including Krugman(1980), Eaton and Kortum (2001), and Melitz (2003), predict an income-size elasticity(i.e., the elasticity at which real per capita income increases with population size) that iscounterfactually high. One straightforward remedy for this counterfactual prediction isintroducing domestic trade frictions into the aforementioned models. This treatment,however, is only a partial remedy. As shown by Ramondo et al. (2016), even aftercontrolling for direct measures of internal trade frictions, the predicted income-sizeelasticity remains counterfactually strong.

To test macro-level predictions, we first produce economically-representative es-timates for εk and 1/ξk. We do so by pooling data across all manufacturing and non-manufacturing industries and estimating Equation 13 on theses two pooled samples.Theestimation results are reported in Table 6, and imply that ε ≈ 3.6 and ε/ξ ≈ 0.66 acrossmanufacturing industries. For the sake of comparison, the same table also reports esti-

75

Figure 3: Robustness checks to address challenges to the identification of εk and 1/ξk

76

Table 6: Pooled estimation results

Manufacturing Non-Manufacturing

Variable (log) 2SLS OLS 2SLS OLS

Price, −ε -3.588*** 0.202*** -4.658*** 0.104***(0.221) (0.002) (0.504) (0.004)

Within-national share, 1− ξ/ε 0.333*** 0.814*** 0.190*** 0.802***(0.012) (0.001) (0.028) (0.003)

Weak Identification Test 94.84 ... 24.16 ...Under-Identification P-value 0.00 ... 0.00 ...Within-R2 ... 0.77 ... 0.73N of Product-Year Groups 23,683 10,461Observations 1,126,976 205,580

Notes: *** denotes significant at the 1% level. The Estimating Equation is (13). Standard errors in brackets arerobust to clustering within product-year. The estimation is conducted with HS10 product-year fixed effects.The reported R2 in the OLS specifications correspond to within-group goodness of fit. Weak identification teststatistics is the F statistics from the Kleibergen-Paap Wald test for weak identification of all instrumented vari-ables. The p-value of the under-identification test of instrumented variables is based on the Kleibergen-PaapLM test. The test for over-identification is not reported due to the pitfalls of the standard over-identificationSargan-Hansen J test in the multi-dimensional large datasets pointed by Angrist et al. (1996).

mates produced using the standard OLS estimator.To understand the income-size elasticity puzzle, consider a single-industry version

of the model presented in Section 2. Such a model implies the following expressionrelating country i’s real income per worker or TFP (Wi = wi/Pi) to its structural effi-ciency, Ai, population size, Li, trade-to-GDP ratio, λii, and a measure of internal tradefrictions, τii:

Wi = γ Ai Lψi λ− 1

ε−1ii τ−1

ii . (34)

The standard Krugman model assumes extreme love-of-variety (or extreme scale economies),which implies ψ = 1/(ε − 1) and precludes internal trade frictions, which results inτii = 1. Given these two assumptions, we can compute the real income per worker pre-dicted by the standard Krugman model and contrast it to actual data for a cross-sectionof countries.

For this exercise, we use data on the trade-to-GDP ratio, real GDP per worker, andpopulation size for 116 countries from the PENN WORLD TABLES in the year 2011.Given our micro-estimated trade elasticity, ε− 1, and plugging τii = 1 as well as ψ =

1/(ε− 1) into Equation 34, we can compute the real income per worker predicted bythe Krugman model. Figure 4 (top panel) reports these predicted values and contraststhem to factual values. Clearly, there is a sizable discrepancy between the income-sizeelasticity predicted by the standard Krugman model (0.36, standard error 0.03) and thefactual elasticity (-0.04, standard error 0.06). To gain intuition, note that small countriesimport a higher share of their GDP (i.e., posses a lower λii), which partially mitigates

77

their size disadvantage. However, even after accounting for observable levels of tradeopenness, the scale economies underlying the Krugman model are so strong that theylead to a counterfactually high income-size elasticity.

One solution to the income-size elasticity puzzle is introducing internal trade fric-tions into the Krugman model (i.e., relaxing the τii = 1 assumption). Ramondo et al.(2016) perform this task using direct measures of domestic trade frictions. Their calibra-tion is suggestive of τii ∝ L0.17

i . Plugging this implicit relationship into Equation 34 andusing data on population size and trade openness, we compute the model-predictedreal income per worker and contrast it with actual data in Figure 4 (middle panel). Ex-pectedly, accounting for internal frictions shrinks the income-size elasticity. However,as pointed out by Ramondo et al. (2016), the income-size elasticity remains puzzlinglylarge.

We ask if our micro-estimated scale elasticity can help resolve the remaining income-size elasticity puzzle. To this end, in Equation 34, we set the scale elasticity to ψ =

α/(ε − 1) where α is set to 0.65 as implied by our micro-level estimation. Then, us-ing data on population size and trade-to-GDP ratios, we compute the real income percapita predicted by a model that features both domestic trade frictions and adjustedscale economies. Figure 4 plots these predicted values, indicating that this adjustmentindeed resolves the income-size elasticity puzzle. In particular, the income-size elas-ticity predicted by the Krugman model with adjusted scale economies is statisticallyinsignificant (0.02, standard error 0.03), aligning very closely with the factual elasticity.

O Mapping Tax Formulas to Data under Free Entry

In this appendix, we present an analog to Proposition 3, but under free entry. Tothis end, we appeal to Theorem 1.B which implies that under the CES-Cobb-Douglasutility aggregator:

1 + x∗h f ,k = 1 +1

εkλ f f ,k

1 + t∗f h,k = 1−r f h,k

1 + ξk

1 + s∗h,k =1

1 + 1/ξk.

Combining the above tax formula with equilibrium conditions and adopting the hat-algebra notation, x ≡ x′/x, yields the following proposition.

Proposition 6. Suppose the observed data is generated by a model that exhibits the cross-industry utility aggregator 14. Then, the optimal (first-best) taxes and their effects on wages,

78

Figure 4: The income-size elasticity

−4

−3

−2

−1

0

1R

eal

per

cap

ita

Inco

me

(log, U

S=

1)

−4 −2 0 2 4 6

Population (log)

Model Prediction Data

Standard Krugman Model

−4

−3

−2

−1

0

1

Rea

l per

cap

ita

Inco

me

(log, U

S)

−4 −2 0 2 4 6

Population (log)

Model Prediction Data

Krugman w/ doemstic trade costs

−4

−3

−2

−1

0

1

Rea

l per

cap

ita

Inco

me

(log, U

S=

1)

−4 −2 0 2 4 6

Population (log)

Model Prediction Data

Krugman w/ domestic trade costs & estimated scale elasticity

79

profits, and income can be fully characterized by solving the following system of equations:

xh f ,k =1

εkλ f f ,kλ f f ,k; t f h,k = −

r f h,kr f h,k1+ξk

; 1 + sh,k =1

1+1/ξk∀k ∈ K

x f h,k = 0, th f ,k = 0; sk = 0 ∀k ∈ K

λji,k = r1/ξkji,k

[(1 + tji,k)(1 + xji,k)(1 + sj,k)wj

]−εk ˆPεki,k ∀j, i ∈ C, k ∈ K

ˆP−εki,k = ∑j

(r1/ξk

ji,k

[(1 + tji,k)(1 + xji,k)(1 + sj,k)wj

]−εk λji,k

)∀j, i ∈ C, k ∈ K

wiwiLi = ∑k ∑j

(λij,kλij,kej,kYjYj

(1+µk)(1+xij,k)(1+si,k)(1+tij,k)

)∀i ∈ C

ΠiΠi = ∑k ∑j

(µkλij,kλij,kej,kYjYj

(1+µk)(1+xij,k)(1+si,k)(1+tij,k)

)∀i ∈ C

RiRi = ∑k ∑j

(tji,k

1+tji,kλji,kλji,kei,kYiYi +

(1+xij,k)(1+si,k)−1(1+xij,k)(1+si,k)(1+tij,k)

λij,kλij,kej,kYjYj

)∀i ∈ C

YiYi = wiwiLi + ΠiΠi + RiRi ∀i ∈ C

rji,krji,k =

(λji,kλji,kei,kYiYi

(1+xji,k)(1+sj,k)(1+tji,k)

)/(

∑ιλjι,kλjι,keι,kYιYι

(1+xjι,k)(1+sj,k)(1+tjι,k)

)∀j, i ∈ C, k ∈ K

.

The above system solves t f h,k, xh f ,k, sh,k, λji,k, r f f ,k, ˆPi,k, wi, and Yi, asa function of (i) industry-level scale elasticities, 1/ξk; (ii) industry-level trade elasticities,εk; (iii) observed expenditure shares, λji,k;(iv) observed revenue shares, rji,k; and (v)total expenditure and wage revenue, wiLi and Yi.

Note that once we solve for the optimal policy, we can immediately calculate thegains from policy as Yi/ ∏k

ˆPei,ki,k .

P WIOD Industry Details

In this appendix we present a figure illustrating our aggregation of WIOD indus-tries into 16 industries. To summarize, we aggregate the ’Agriculture’ and Mining’ intoone non-manufacturing industry. We also follow Costinot and Rodríguez-Clare (2014)and (a) aggregate the ’Textile’ and ’Leather’ industries into one industry, and (b) lumpall service-related industries together and treat as one quasi-non-tradable sector.

80

Table 7: List of industries in quantitative analysis

WIOD Sector Sector’s Description Trade Ealsticity Scale Ealsticity

1 Agriculture, Hunting, Forestry and Fishing 6.212 0.141

2 Mining and Quarrying 6.212 0.141

3 Food, Beverages and Tobacco 3.333 0.265

4Textiles and Textile Products

3.413 0.207Leather and Footwear

5 Wood and Products of Wood and Cork 3.329 0.270

6 Pulp, Paper, Paper , Printing and Publishing 2.046 0.397

7 Coke, Refined Petroleum and Nuclear Fuel 0.397 1.758

8 Chemicals and Chemical Products 4.320 0.212

9 Rubber and Plastics 3.599 0.162

10 Other Non-Metallic Mineral 4.561 0.186

11 Basic Metals and Fabricated Metal 2.959 0.189

12 Machinery, Nec 8.682 0.100

13 Electrical and Optical Equipment 1.392 0.453

14 Transport Equipment 2.173 0.133

15 Manufacturing, Nec; Recycling 6.704 0.142

16

Electricity, Gas and Water Supply

100 0

Construction

Sale, Maintenance and Repair of Motor Vehicles and Motorcycles;Retail Sale of Fuel

Wholesale Trade and Commission Trade, Except of Motor Vehiclesand Motorcycles

Retail Trade, Except of Motor Vehicles and Motorcycles; Repair ofHousehold Goods

Hotels and Restaurants

Inland Transport

Water Transport

Air Transport

Other Supporting and Auxiliary Transport Activities; Activities ofTravel Agencies

Post and Telecommunications

Financial Intermediation

Real Estate Activities

Renting of M&Eq and Other Business Activities

Education

Health and Social Work

Public Admin and Defence; Compulsory Social Security

Other Community, Social and Personal Services

Private Households with Employed Persons

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