Interdependence of Trade Policiesin General Equilibrium∗
Mostafa Beshkar
Indiana University
Ahmad Lashkaripour
Indiana University
First version: June 2016This version: June 7, 2019
Abstract
Many of the major results from theories of international trade are obtained
within a General Equilibrium (GE) framework, but our understanding of trade
policy is still largely limited to partial equilibrium analyses. We characterize
optimal policy and policy interdependencies in a multi-industry GE model
that features factor market, input-output, and cross-demand linkages, and
show how GE considerations change the analysis of trade policies both quanti-
tatively and qualitatively. We find that: (i) The variation in optimal trade taxes
are substantially dampened when GE factor-market effects are taken into ac-
count; (ii) Input-output linkages introduce a new channel of international ex-
ternality by affording governments the ability to levy a tax on value-added
generated and consumed outside its jurisdiction; (iii) Negotiated tariff cuts in
a subset of industries lead to unilateral cuts in other industries; and (iv) A
free trade agreement may lead to the adoption of wasteful trade barriers by a
welfare-maximizing government. Fitting our model to trade data for 15 major
economic regions, we show that these effects are quantitatively significant.∗The first draft of this paper entitled “Trade Policy with Inter-sectoral Linkages” was presented at the SITE Summer
Workshop (June 2016). For their helpful comments and discussions, we are grateful to Pol Antras, Costas Arkolakis, KyleBagwell, Eric Bond, Lorenzo Caliendo, Angela Campbell, Arnaud Costinot, Svetlana Demidova, Farid Farrokhi, FilomenaGarcia, Grey Gordon, Michael Kaganovich, Sajal Lahiri, Nuno Limao, Volodymyr Lugovskyy, Kaveh Majlesi, GiovanniMaggi, Monika Mrvarcelo Olarreaga, Frederic Robert-Nicoud, Andres Rodriguez-Clare, Ali Shourideh, Anson Soderbery,Tommaso Tempesti, Ben Zissimos and participants at various seminars and conferences. We also thank Mostafa TanhayiAhari for his feedback and research assistance.
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1 Introduction
The analysis of trade policies is often complicated by general-equilibrium linkagesacross industries. Consider, for example, the recent tariffs imposed by the UnitedStates on steel imports. In addition to its local effects on steel producers and con-sumers, such a policy has two general ramifications on the rest of the economy.First, by reallocating resources across industries and modifying demand and tradepatterns, steel tariffs may affect the cost of inputs (labor, capital, intermediate in-puts, etc.) as well as the intensity of import competition in the rest of the economy.A second complication arises due to the interdependence of trade policies acrossindustries: In response to the general equilibrium effects of steel tariffs, the gov-ernment may be compelled to adjust its trade policy across all industries, therebycreating further welfare consequences.
The consequences of policy interdependence across sectors have largely es-caped notice in the optimal policy literature. To avoid the complications resultingfrom general-equilibrium interactions, most of the trade policy literature has fo-cused on partial equilibrium models. Several authors, including Ossa (2014) andCaliendo and Parro (2014), have advanced the analysis of trade policy in generalequilibrium by developing and adopting a computational approach. Beyond thesestudies, the analytics of optimal policy within general equilibrium multi-industrymodels remain largely unknown. An exception is Costinot, Donaldson, Vogel, andWerning (2015), who study optimal policy under Dornbusch, Fischer, and Samuel-son’s (1977) version of the Ricardian model.
Our objective in this paper is two-fold. First, we analytically characterize theoptimal trade policy in the presence of various cross-industry linkages, includ-ing general-equilibrium factor price linkages, input-output linkages, and cross-demand effects. Second, we study various trade policy interdependencies thatarise due to these general equilibrium linkages. Achieving this second objectiveinvolves characterizing the optimal policy under various external constraints onthe government’s policy space.1
In characterizing the optimal trade policy, we consider a competitive general-equilibrium model that features general (non-parametric) consumer preferences
1Constraints on the policy space may be imposed by incomplete trade agreements or politicaland institutional considerations, for example.
2
and production technologies. In this general setup, we characterize the optimalindustry-level export and import taxes as a function of two sufficient statistics: (i)own- and cross-price elasticities of demand, and (ii) trade tax pass-throughs netof wage effects. We use our analytical characterization to study the structure ofoptimal trade policy in three special cases.
We first consider a general Ricardian economy without input-output linkages.We show that, in this particular setup, the optimal import taxes are uniform acrossproducts, but the optimal export taxes are differential and vary with the own- andcross-price elasticities of foreign demand for the exported products. Our unifor-mity result extends the result in Costinot et al. (2015) to environments that feature ageneral demand system that admits any arbitrary pattern of cross-substitutabilitybetween products. We also extend our analysis to a multiple-country case andshow that the optimal import tariffs discriminate among exporting countries butremain uniform across products imported from the same country.
Second, we consider a Ricardian economy that features Input-Output Linkages.We first show that, here, the entire matrix of tax passthroughs is fully determinedby the global Input-Output (IO) matrix. Correspondingly, the optimal trade taxschedule can be fully characterized in terms of reduced-form demand elasticitiesand input-output shares.2 The resulting optimal tax formula indicates that optimalimport taxes are uniform only across final goods or intermediate goods that are notre-exported. However, optimal import taxes are differential across imported goodsthat are destined for re-exporting.
To understand this result, note that any tax levied on re-exported intermedi-ate goods is effectively a tax on a transaction among foreign entities, because suchcomponents are produced and eventually consumed abroad. Therefore, input-output linkages provide the government with additional taxing power beyond itsjurisdiction, which we call the extraterritorial taxing power. The departure from uni-form tariffs on re-exported intermediate goods reflects the government’s desire toexercise its extraterritorial taxing power on such trade flows.3
2To be more precise the optimal trade tax schedule also depends on observable expenditureand revenue shares.
3As Yi (2003) points out in a model with vertically-fragmented production, the effect of ex-ogenous trade costs are amplified when countries specialize in different stages of production, andintermediate goods cross national borders multiple times. Our analysis complements Yi (2003) byconsidering the effect of IO linkages on endogenous trade costs.
3
It is worth highlighting the difference between the externality generated byextraterritorial taxing power and the standard terms-of-trade externality empha-sized in the prior literature. The standard terms-of-trade argument concerns theprice of an exchange between a consumer and a producer, when only one of themis located in a foreign country. The extraterritorial taxing argument, however, con-cerns the government’s ability to tax a value-added that is produced and consumedoutside its jurisdiction. Accordingly, the presence of IO linkages could amplify theunilateral gains and the externalities associated with trade taxes.
Third, we consider the case where the economy features industry-specific fac-tors of production. In this case, the marginal product of labor is diminishing in theindustry-level output. Hence, due to general equilibrium demand linkages, a taxon one industry could alter the productivity of labor in all other industries. As aresult, import tariffs are differential across products and their variation dependson the entire schedule of home and foreign’s industry-level supply elasticities aswell as the cross-demand elasticities between industries.
Our second set of results characterize the interdependence of trade policies acrossindustries in the Ricardian model. In summary, we find that: (i) Import policy isan imperfect substitute for export policy; (ii) Under mild conditions, import tariffsacross industries are complementary; (iii) Non-Revenue Trade Barriers (NRTBs),also known as wasteful trade barriers, may be optimal in the absence of revenue-raising trade policy instruments such as tariffs.
Our result about the interdependence of import and export policies is akinto—but distinct from—the Lerner’s (1936) Symmetry Theorem. We find that, ingeneral, import policy is only an imperfect substitute for export policy. In the Ri-cardian model, this imperfect substitutability takes a sharper form: the equilibriumobtained under optimal import tariffs can be exactly replicated with a set of exportpolicies, but no set of import tariffs could replicate the equilibrium under the op-timal export taxes. Under reasonable scenarios, an important implication of thisresult is that the elimination of export subsidies would lead to an increase in tradevolume.4 This insight is in contrast to one obtained under a partial equilibriumanalysis in which the elimination of export subsidies will necessarily reduce trade
4Within our model, this result is valid under a scenario in which the government could not useexport taxes due to political or institutional constraints such as the constitutional ban on exporttaxes in the United States.
4
volumes.5
To obtain a general intuition about the cross-industry tariff complementarity, it isinstructive to consider the following scenario. Suppose that, initially, export policyinstruments are unavailable but the government could freely choose import tariffs.Then, suppose the home government enters an incomplete trade agreement thatrestricts import tariffs in a subset of industries. Our tariff complementarity resultindicates that, under this partial restriction, it is optimal for the home governmentto voluntarily lower its import tariffs on unrestricted industries.6
Our result concerning the optimality of NRTBs sheds fresh light on measuressuch as import bans and inefficient customs regulations (i.e., red tapes at the bor-der) that discourage imports but do not generate revenues. These measure arequite prevalent in practice. For example, in the wake of negotiated tariff cuts,many countries have opted for non-tariff barriers that do not generate any rev-enues for the governments (Goldberg and Pavcnik 2016). From the perspectiveof the standard terms-of-trade analysis, the adoption of NRTBs is hard to ex-plain because such measures reduce trade without compensating the resultingconsumption losses with a better terms of trade. Under a multi-industry general-equilibrium framework, however, NRTBs could improve a country’s welfare be-cause restricting imports in one industry improves a country’s terms of trade in allother industries by depressing foreign factor rewards. Therefore, if the consump-tion loss due to import restriction in an industry is sufficiently small, imposing anNRTB in that industry could be welfare-improving. We show that this conditionis satisfied in relatively homogenous sectors where imported varieties could beeasily substituted with domestic counterparts.
Finally, we provide a quantitative assessment of our findings by fitting ourmodel to trade and production data from 15 regions (spanning 40 countries) across16 industries. In this process we also demonstrate how our theory simplifies the
5This result also provides a novel perspective on the GATT/WTO’s ban on export subsidies.As reviewed by Lee (2016), the terms-of-trade literature has found it “quite difficult to justify theprohibition of export subsidies given the trade-volume-expanding nature of export subsidies.” Ourgeneral equilibrium analysis provides a potential explanation for this puzzle, because we show thatthe elimination of export subsidies will spur unilateral tariff cuts to a degree that leads to an overallincrease in trade volumes.
6This finding is in line with Martin and Ng’s (2004) observation that after entering the WTO,many developing countries started cutting their tariffs beyond their obligations under the agree-ment. Baldwin (2010) also highlights these unilateral tariff liberalizations, but provides an alterna-tive explanation based on the fragmentation of the production processes.
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conduct of quantitative analysis of trade policy.Our quantitative exercise indicates that unilateral gains from optimal trade pol-
icy are (one average) around 1% in terms of real GDP. However, the gains fromunilateral policy would be considerably lower (around 0.35% of real GDP) if thegovernment was restricted from using export policy. This finding gives furtherperspective on our proposition regarding the imperfect substitutability of importand export taxes. The gains from NRTBs, when revenue-raising taxes are absent,are smaller but not negligible. For Taiwan and Mexico, for instance, we estimatethat the gains from optimal NRTBs are around 0.11% and 0.12% of their GDPs,respectively.
The unilateral gains from optimal policy are considerably larger in the presenceof input-output linkages. Specifically, accounting for global input-output linkages,the average unilateral gains from optimal trade policy increases 60% (from 1%to 1.6% of GDP). The higher gains from trade policy are partly driven by the ex-traterritorial taxing power effect identified by our theory. That is, with IO linkages,countries can generate revenues by taxing transactions between consumers andproducers in the rest of the world.
To illustrate the role of policy interdependencies, we conduct a counterfactualanalysis corresponding to a hypothetical gradual trade agreement that is reminis-cent of the constraints introduced over time by the GATT and the WTO. Startingfrom the home government’s unconstrained optimal policy equilibrium, we in-troduce a sequence of partial restrictions on the government’s policy space andquantify its optimal response with respect to unrestricted industries. We conductour counterfactual analysis twice, once where the United States is treated as thehome country and another where the European Union is treated as home.
For our no-agreement baseline, we adopt an import tariff equal to the averageSmoot-Hawley tariffs of 59%, and calculate the optimal export policy. The firstsequence of liberalization that we consider is a ban on export policies, while importtariffs are left at the discretion of the home government. This scenario is in linewith the GATT and WTO’s relatively more stringent conditions on export subsidiesthan import tariffs. We calculate that the restriction on export policy will inducethe US and EU governments to decrease their import tariffs uniformly from 59% toaround 30%. As a result of restrictions on export policy, the gains from unilateraltrade policy for the US and EU are reduced by more than fifty percent.
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The second sequence of liberalization retains the ban on all export taxes, butalso restricts import tariffs in half of the traded industries. For both the EU andthe US, we compute that such a restriction would induce the government to lowerits import tariffs by a third in unrestricted industries. This strong tariff comple-mentarity could have important implications for the optimal sequencing of tradeliberalization.
The paper is organized as follows. After discussing the related literature in thesubsequent Section, we begin by laying down our general framework in Section3. In Section 4, we derive the unconstrained optimal tax/subsidy schedule undera general Ricardian model. In Section 5, we extend our analysis of optimal policyto environments with input-output linkages and specific factors of production. InSection 6, we analyze the interdependence of trade policies and the optimality ofNRTBs by introducing constraints to the optimal tax/subsidy problem. Section7 presents our quantitative analyses. Finally, in Section 8, we provide concludingremarks including a discussion on the implications of policy interdependencies fortrade negotiations.
2 Related Literature
In this Section, we review the literature on general equilibrium analysis of tradepolicy and policy interdependence, and discuss the relevance of our contributionsto these previous studies.
The Literature on Optimal Trade Policy
Our results regarding the optimal schedule of trade taxes cover the previ-ous general-equilibrium characterizations of the optimal trade policy includingCostinot et al. 2015, Opp 2010, and Itoh and Kiyono 1987. While Opp 2010 focuseson import tariffs and Itoh and Kiyono 1987 focus on export subsidies, Costinotet al. 2015 consider the simultaneous choice of import and export policies andshow the optimality of uniform import tariffs for the case where trade elasticitiesare the same across sectors and preferences are additively separable. We show thatthese results continue to hold in an environment with heterogenous trade elastici-ties across sectors and a general (not necessary separable) preference structure.
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Using the primal approach and assuming additively-separable utilities, CDVWshow that the messy problem of optimal trade policy in a multi-sector general-equilibrium Ricardian model (Dornbusch, Fischer, and Samuelson 1977, hence-forth, DFS) reduces to an elegant cell-problem in which the optimal trade policy foreach product may be found in isolation from the policy of other sectors.7 CDVWshow that the cell-approach is specially useful in dealing with two difficulty in theproblem of optimal trade taxes under DFS’s Ricardian model, namely, the infinite-dimensionality of the optimal tax problem (since there is a continuum of goods)and non-smoothness (since the world production frontier has kinks.)
The simplicity and elegance afforded by the cell-problem come at a cost sinceby imposing constraints on preferences and technologies, this approach limits thetype of general equilibrium linkages that can be analyzed. We suggest an alterna-tive method to deal with these difficulties, which involves laying out a more gen-eral model with countable products that are imperfectly substitutable. The assump-tion of imperfect substitutability of products obviates the need to worry about theeffect of policy on the extensive margin of trade. In the limit, when the number ofproducts and the degree of substitutability of products tend to infinity, the modeldelivers DFS as a special case.
The idea that optimal trade policy for a product should depend on the elasticityof its supply and demand was proposed by Bickerdike (1906) and was later popu-larized by others including Kahn (1947), who calculated the exact formula for op-timal import tariff to be equal to the inverse of the foreign export supply elasticity.This approach came under criticism due to its disregard for general-equilibriumeffects (Graaff 1949; Horwell and Pearce 1970; Bond 1990). Nevertheless, thosecriticisms were mostly suggestive and did not provide a practical framework toevaluate general-equilibrium effects of trade policy. The subsequent literature,perhaps for practical reasons, adopted Bickerdike’s “elasticity approach” to studythe variation in sectoral trade policies (e.g., Grossman and Helpman 1995; Brodaet al. 2008; Bagwell and Staiger 2011; and Beshkar et al. 2015).8 In this paper, byproviding an analytical characterization of optimal trade policy (both constrained
7If a problem can be formulated as a cell-problem, the Lagrange multiplier provides a sufficientstatistic for the effect of the rest of the economy on each cell.
8The existing general-equilibrium analyses of trade policy are either conducted for a small openeconomy (as in the tariff reform literature cited below), or a two-sector economy with only oneimport good and one export good (e.g., Bagwell and Staiger 1999, Limão and Panagariya 2007).
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and unconstrained), we offer a practical way to analyze trade policy in generalequilibrium.
We are unaware of any previous work that views NRTBs as a beggar-thy-neighbor policy. The existing studies of non-tariff barriers as policy variables–e.g.,Berry, Levinsohn, and Pakes (1999), Harrigan and Barrows (2009), and Maggi, Mrá-zová, and Neary (2017)—view them implicitly as an instrument to transfer wealthto interest groups without generating any welfare gains at the national level. Sim-ilar to our study of NRTBs, Maggi, Mrázová, and Neary (2017) analyze the useof wasteful trade barriers when the governments’ policy space is constrained bya trade agreement. They show that if tariff commitments could not be fully con-tingent on political realizations, the extent of tariff liberalization is limited by theneed to prevent such wasteful behavior. Our framework offers a complementaryperspective on NRTBs as instruments that could be potentially used to improve acountry’s terms of trade in expense of foreign countries.
Within a partial equilibrium framework and assuming free trade in intermedi-ate inputs, Blanchard et al. (2016) study the effect of IO linkages on the optimalfinal-goods tariffs. In addition to accounting for general equilibrium effects, we al-low for the imposition of trade taxes on intermediate inputs as well as final goods.We find that the optimal trade restriction on intermediate inputs could be evenhigher than the optimal import tariffs on imported final goods. Therefore, ouranalysis offers a caveat about the assumption of free trade in intermediate inputs.
A growing literature, including Demidova and Rodríguez-Clare (2009), Felber-mayr, Jung, and Larch (2013), Haaland and Venables (2016), Costinot, Rodríguez-Clare, and Werning (2016), and Caliendo, Feenstra, Romalis, and Taylor (2015) an-alyzes trade policy under the monopolistically competitive framework of Melitz(2003). All of these papers focus on models with a single tradable sector and, thus,their results are not readily comparable to our findings regarding the optimal pol-icy across multiple sectors. A partial exception is Costinot, Rodríguez-Clare, andWerning (2016) who study firm-specific policies and find that within the same sec-tor, optimal firm-specific tariffs are increasing in the productivity of the foreignfirms.
Our theory contributes to a growing literature that attempts to quantify thetrade policy equilibrium of optimizing governments (Perroni and Whalley, 2000;Ossa, 2011, 2012, 2014). This literature, which is aptly discussed by Ossa (2016)
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and Costinot and Rodríguez-Clare (2014), uses numerical optimization to find thetariff choice of optimizing governments. Numerical optimization is often plaguedwith the curse of dimensionality when many sectors are involved. Applying ourtheory to trade data, we show that our analytical formulas facilitate the compu-tational task in such cases. Moreover, we take a first step towards highlightingthe empirical significance of cross-price elasticity effects in the design of optimalpolicy.
Our characterization of optimal taxes, which can be interpreted as optimalmarkups in a monopoly problem, contributes to the analysis of multi-productmonopolies as studied by Armstrong and Vickers (2018) and Amir et al. (2016),among others.9 In comparison to the monopoly problem, the problem of optimaltrade policy introduces additional nuances that are caused by general-equilibriumeffects on wages, productivities, and income, which do not emerge in standardmulti-product monopoly problems. Moreover, with the introduction of input-output linkages, which affords governments an extraterritorial taxing power, thetrade policy problem can no longer be cast purely as a multi-product monopoly-monopsony problem.
The Literature on Policy Interdependence
The existing literature is mostly silent about trade policy interdependencies due toits focus on “optimal” policy–rather than the tradeoffs that policymakers face out-side the optimum– and partial equilibrium, which precludes interrelations acrosssectors. Partial exceptions include the literature on incomplete trade agreementsand the literatures on tariff complementarity in Free Trade Areas and the PiecemealTariff Reforms, which we now discuss.
In a model of incomplete trade agreements, Horn, Maggi, and Staiger (2010)show that governments will have an incentive to use domestic subsidies in re-sponse to negotiated tariff cuts. The increase in domestic subsidies after enteringa trade agreement tends to partially offset the benefits from negotiated trade liber-alization.
9Note that the problem of optimal trade policy in the absence of IO linkages resembles a multi-product monopoly problem (on the export side) and a multi-product monopsony problem (on theimport side). In particular, our finding that the sectoral variation in optimal export (import) policiesare determined by demand-side (supply-side) parameters, is reminiscent of the solution to themonopolist’s (monopsonist’s) problem.
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There is a literature on tariff complementarity in Free Trade Areas (FTA). Whilewe find that tariffs across sectors within a country are complementary, Richard-son (1993), Bond, Riezman, and Syropoulos (2004) and Ornelas (2005) find thatfor members of a Free Trade Area (FTA), internal and external tariffs are comple-mentary. In particular, they find that as a response to tariff cuts within an FTA,the member countries will voluntarily reduce their tariffs on imports from non-members. Similarly, in a North-South model, Zissimos (2009) considers tariff com-plementarities across countries within a region that compete for imports from therest of the world.
The theory of piecemeal tariff reform (Hatta 1977; Fukushima 1979; Andersonand Neary 1992, 2007; Ju and Krishna 2000) is another strand of the literature thattouches on the issue of policy interdependence. This literature is primarily con-cerned with welfare-enhancing tariff reforms that are revenue-neutral (or revenue-enhancing) in a small open economy. A general finding of the piecemeal reformliterature is that compressing the variation of existing tariffs in developing coun-tries—by reducing the highest tariff rates and increasing the lowest ones—couldincrease welfare without decreasing revenues. Although we focus on an entirelydifferent problem in this paper, our finding about the optimality of uniform tariffsresonates with this literature’s recommendation for tariff reforms.
As in this paper, Bagwell and Lee (2015) provide a perspective on the WTO’sban on export subsidies. Within a heterogenous-firm model, Bagwell and Lee(2015) show that if import tariffs (as well as transportation costs) are very low,then an export subsidy may benefit a country at the expense of its trading partners.Their finding suggests that a ban on export subsidies is useful only after substan-tial liberalizations have been reached through previous negotiations. By contrast,our analysis suggests that a ban on export subsidies is useful even without anyrestrictions on import tariffs.
Another related literature studies issue linkages in international relations. Thisliterature considers various conditions under which there might be an interdepen-dence between trade policies and non-trade policies—such as environmental poli-cies (Ederington, 2001, 2002; Limão, 2005), production subsidies (Horn, Maggi, andStaiger, 2010), and intellectual property protection. These papers draw conclusionsabout whether these non-trade issues should be linked to trade agreements (seeMaggi 2016 for a review).
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3 The Economic Environment
The global economy consists of i = 1, ..., N countries (with C denoting the setof countries) and k = 1, ..., K industries (with K denoting the set of industries).With the exception of Section 4.2, our analysis focuses on a two-country case, i.e.,C = {h, f }, where h is referred to as the Home country and f is referred to as theForeign country representing an aggregate of the rest of the world.
Each country i is populated with Li workers that are perfectly mobile acrossindustries but immobile across countries. In the non-Ricardian extension of themodel, each industry k in country i is also endowed with Si,k units of an industry-specific factor of production, which is combined with labor in production.
In a typical industry k, country j ∈ C produces a differentiated variety foreach destination market, i ∈ C, which we denote by ji, k (supplier j–destinationi–industry k). Since no restrictions are imposed on the size or the number of indus-tries, our framework can be alternatively viewed as one concerning product-leveltaxes.
3.1 Preferences
The consumers in country i choose the vector of consumption quantities, qi ≡{qji,k}, to maximize a general utility function, Ui (qi), subject to the budget con-straint. The optimal choice of the consumers yields an indirect utility function,
Vi
(Yi, pi,k
)≡ max
qiUi (qi)
s.t. ∑k∈K
∑j∈C
(pji,kqji,k
)= Yi, (1)
and a Marshallian demand function qi = Di (pi, Yi), which summarizes thedemand-side of the economy as a function of total income Yi and the vector ofconsumer prices pi ≡ { pji,k} in country i. We define the price and income elastici-ties associated with demand function Di (.) as follows:
D1. [Marshallian Demand Elasticities](i) [own price elasticity] ε ji,k ≡ ∂ ln qji,k/∂ ln pji,k;(ii) [cross-price elasticity] ε
ι,gji,k ≡ ∂ ln qji,k/∂ ln pι,g for ι, g , ji, k;
(iii) [income elasticity] ηji,k ≡ ∂ ln qji,k/∂ ln Yi.
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Throughout the paper, we restrict our attention to well-behaved demand func-tions that are continuous and locally non-satiated. We also assume that demandfor each traded variety exhibits an elastic region where | ε ji,k |> 1. As in monopolyproblems, this condition will be necessary for obtaining a bounded solution foroptimal trade taxes.
3.2 Technology
We assume that firms are competitive and technologies exhibit constant returnsto scale. The general case of our framework allows for production to employ (i)labor, (ii) intermediate inputs, and (iii) industry-specific factors of production.Accordingly, a cost-minimizing producer supplying good ji, k faces the followingnon-parametric marginal cost function, which pins down their competitive “pro-ducer” price, pji,k, as a function of input prices and output level:
pji,k = Cji,k
(wj, pIj ; qj
). (2)
To elaborate, the marginal cost is a function of the labor wage in economy j, wj;the vector of intermediate input prices employed by producers in country j, pIj ≡{ pI`j,g}`,g; and the producer’s output schedule, qj ≡ {qjι,g}ι,g. This last argumentaccounts for (i) the presence of industry-specific factors of production, which leadsto a marginal cost that is increasing in output, as well as (ii) a finite elasticity oftransformation between output produced for different markets—we discuss themicro-foundation underlying Equation 2 in more detail in Section 5.2.
We begin our analysis in Section 4 with the basic Ricardian case of the abovestructure. In that case, Cji,k (.) = aji,kwj, with aji,k being a constant (policy-invariant) unit labor cost. Then, in Section 5, we consider the most general caseof our model that admits both input-output linkages and industry-specific factorsof production.
3.3 Policy Instruments
The government in country i has access to a full set of industry-level exporttax/subsidy instruments (denoted by xij,k) and import tax/subsidy instruments(denoted by tji,k). Moreover, the government could impose Non-Revenue Trade
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Barriers (NRTBs, denoted by τji,k) which are frictions in the form of iceberg trans-port costs that impede imports without raising revenues. NRTBs account for poli-cies such as red-tape barriers at the border, frivolous regulations, and other poli-cies that act as coveted protectionism. Together, these policy instruments createa wedge between the consumer price, pji,k, and the producer price, pji,k, of eachgood ji, k as follows:
pji,k =(1 + tji,k
) (1 + xji,k
) (1 + τji,k
)pji,k,
where tji,k denotes the import tariff applied by country i on good ji, k; xji,k denotesthe export tax applied by country j on good ji, k; and τji,k denotes the NRTB appliedby country i on good ji, k. The combination of these tax instruments raises thefollowing tax revenue for the government in country i:
Ri = ∑k∈K
∑j∈C
[tji,k
(1 + xji,k
)pji,kqin,k + xij,k pij,kqij,k
]. (3)
Throughout this paper, we assume that domestic policies are unavailable, i.e.,tii,k = xii,k = τii,k = 0 for all i and k. We also focus on cases where “only” theHome country, indexed h, sets trade taxes. All other countries (denoted as Foreign)are assumed to follow a passive Laissez-Faire policy. In the two-country case, forinstance, this assumption entails that x f h,k = th f ,k = τh f ,k = 0 for all k.
3.4 Equilibrium
Now, we define equilibrium in the two-country case where C = {h, f }, noting thatan analogous definition applies to the multi-country extension. Provided that theequilibrium is unique, a combination of policies imposed by the Home country,namely, xh f ≡ {xh f ,k}, t f h ≡ {t f h,k}, and τ f h ≡ {τf h,k}, is consistent with onlyone equilibrium wage vector, w ≡ {wi}. Since Foreign does not impose taxes byassumption, we can uniquely characterize policy outcomes in terms of xh f , t f h,τ f h, and w (with w itself implicitly depending on the trade taxes). Consideringthis, we formally define the set of feasible wage-policy combinations as follows.
D2. [Feasible Wage-Policy Combinations]Suppose Foreign does not impose taxes (i.e., x f h = th f = τh f = 0). A vector of Home
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taxes and wages A ≡(xh f , t f h, τ f h; w
)constitutes a feasible combination if (i) the pro-
ducer price for any good ji, k is characterized by
pji,k = Cji,k
(wj, pIj ; qj
);
(ii) the consumer price for any good ji, k is given by
pji,k =(1 + xji,k
) (1 + tji,k
) (1 + τji,k
)pji,k;
(iii) consumption choices are a solution to 1 in country i ∈ {h, f }:
qi = Di (Yi, pi) ;
(iv) factor markets clear in country i ∈ {h, f }
wiLi + Πi = ∑n
∑k
pin,kqin,k;
where Πi denotes the surplus paid to the industry-specific factors; and (v) Total incomeequals factor income plus tax revenue in country i ∈ {h, f }
Yi = wiLi + Πi +Ri,
whereRi is given by Equation 3.
We hereafter denote the set of all feasible wage-policy combinations by A. Sinceonly home imposes taxes, we can simplify the notation by letting tk ≡ t f h,k andxk ≡ xh f ,k and τk ≡ τf h,k denote Home’s trade taxes and NRTBs, with t, x, τ
denoting the corresponding vectors. Relatedly, for any feasible wage-policy com-bination, (x, t, τ; w) ∈ A, we use
Wh (x, t, τ; w) ≡ Vh (Yh(x, t, τ; w), ph(x, t, τ; w))
to denote Home’s welfare under that policy. In the two-country case of our model,we recurringly appeal to the Lerner symmetry theorem. So, to fix minds, wepresent a formal statement of the Lerner symmetry in the following.
Lemma 1. [The Lerner Symmetry] A = (x, t, τ; w) and A′ = (x′, t′, τ; 1) represent
15
identical equilibria iff 1 + x′ = (1 + x)w fwh
1 + t′ = whw f
(1 + t).
An immediate corollary of the Lerner symmetry is that when both export andimport taxes are available, we can normalize wages in both economies and stillidentify one of the multiple optimal wage-policy combinations. This result, how-ever, follows only if a full set of export and import tax instruments are availableto the Home government. As we will see in Section 6, once the policy space isrestricted, we can no longer normalize both wh and w f . Therefore, under partialrestrictions on the policy space, general equilibrium changes in wh/w f becomeconsequential to the structure of optimal policy.
4 Optimal Trade Policy in a Ricardian Model
We begin our analysis by characterizing the optimal trade policy of the Homecountry under Ricardian technologies in a two-country setup where C = {h, f }.We then transition to the multiple-country case of the Ricardian model in Subsec-tion 4.2. In the Ricardian model, the set of feasible wage-policy combinations, A, isgiven by D2, with the specific restriction that labor is the only factor of productionand the unit labor cost is constant. That is, Πi = 0 for all i, and
pji,k = aji,kwj, ∀j, i ∈ C; ∀k ∈ K
where aji,k is invariant to policy. Correspondingly, Home’s unconstrained optimaltrade policy is a solution to the following problem:
max(x,t,τ;w)∈A
Wh (x, t, τ; w) (4)
It follows trivially that, when revenue-raising taxes are available, the optimalNRTB is zero, τ∗ = 0. To solve for the optimal revenue-raising trade taxes (x∗
and t∗), we treat Foreign labor as the numeraire (i.e., w f = 1) and invoke demand-side envelop conditions. Doing so, we show that x∗ and t∗, should simultaneously
16
solve the 2×K system of First Order Conditions (FOC), corresponding to K importtax instruments,
∑g
[tgr f h,g
d ln q f h,g (x, t, τ; w)
d ln (1 + tk)
]= τ
d ln whd ln (1 + tk)
∀k ∈ K, (5)
and K export tax instruments10
∑g
[(1
1 + xg− 1)
λh f ,gd ln qh f ,g (x, t, τ; w)
d ln (1 + xk)
]= λh f ,k + τ
d ln whd ln (1 + xk)
∀k ∈ K.
(6)In the above expressions, λh f ,k ≡ ph f ,kqh f ,k/Yf denotes the share of country f ’sexpenditure on variety h f , k; r f h,k ≡ p f h,kq f h,k/w f L f denotes the share of country
f ’s output generated from sales of product f h, k; and τ ≡(
∂Wh∂ ln wh
/ ∂Vh∂Yh
)L−1
f is anaggregate term that accounts for effect of wh on aggregate welfare.
The left-hand side of both equations accounts for the marginal revenue lossholding prices constant. This effect is regulated by two distinct linkage betweenindustries. First, cross-price elasticity effects, whereby a tax on variety f h, k (or h f , k)can modify the demand for all other varieties and the tax revenues collected inall other industries. Second, income effects, whereby the tax revenue collectedfrom good f h, k (or h f , k) can alter the entire demand schedule, qh = Dh (Yh, ph),through its effect on aggregate income, Yh.
The right-hand side of FOCs 5 and 6 accounts for the terms-of-trade gains frompolicy. In the case of import taxes, the only source of terms-of-trade gains arewage effects. That is, Home can raise its wage (relative to Foreign) by exercisingits collective export/import market power. Export taxes, meanwhile, allow Hometo exercise its export market power in a more local fashion. That is, using exporttaxes, Home can directly extract monopoly markups from foreign consumers—aneffect that is picked up by the additional term, λh f ,k, on the right-hand side ofEquation 6.
Given that import taxes can only improve the terms-of-trade through their ef-
10The number of export and import tax instruments need not to be equal, as our model allows forsome tax instruments to be redundant. Suppose Home exports in only K′ industries, with λh f = 0in the remaining K−K′ industries. In that case for a non-exported industry k, if ∂λh f ,k/∂1+ xk = 0,then 1 + xkis a redundant instrument. Similar arguments apply to import taxes.
17
fect on the aggregate wage rate, it follows immediately that the optimal importtax should be uniform across industries. Moreover, since Equation 5 describes theoptimal import tax for any given vector of applied export taxes, the optimality ofuniform import taxes holds irrespective of what export taxes are applied.11
Lemma 2. Under Ricardian technologies, for an arbitrary vector of applied export taxes,the optimal import taxes are uniform across industries.
To provide further intuition, note that in a Ricardian economy, the marginal costnet of wage is invariant to policy. So, import taxes cannot affect neither consumerprices nor producer prices in the Foreign market beyond wage effects. Put differ-ently, Home’s import policy does not generate a local-price externality in Foreign,beyond wage effects. As a result, the optimal import policy is a uniform importtax that allows Home to manipulate its collective market power while introducingminimal allocative inefficiency in the local economy—we elaborate more on thispoint in Section 6.12
To further simplify the FOCs 5 and 6, we can appeal to the Lerner sym-metry. That is, following Lemma 1, we can set d ln wh/d ln (1 + tk) =
d ln wh/d ln (1 + xk) = 0 in FOCs 5 and 6 to identify one of the multiple opti-mal policy combinations. Once this particular solution is identified, the remainingsolutions can be determined with an across the board shift in all import and ex-port taxes. Taking these steps delivers the following theorem, which characterizesHome’s optimal trade policy as a function of reduced-form demand elasticities andexpenditure shares.
Theorem 1. The unconstrained optimal policy under Ricardian technologies consists ofzero NRTBs, a uniform tariff, t∗, and a variable industry-level export tax,
(1 + x∗k ) (1 + t∗) =εh f ,k
1 + εh f ,k + ξh f ,k. (7)
where[ξh f ,k
]k = 1
[Ξ−1 − IK
]accounts for cross-price elasticity effects between indus-
11A formal proof is provided in Appendix A.12The above intuition is similar to that highlighted by Costinot et al. (2015). However, as we
note later in Section 5, in more general environments, the intuition behind the uniformity of importtaxes is more nuanced. For example, optimal import tariffs can be non-uniform even when theForeign economy has a Ricardian production structure but the Home economy does not.
18
tries, with Ξ ≡[λh f ,gε
h f ,kh f ,g/λh f ,kεh f ,g
]k,g
.13
As noted earlier, due to the Lerner symmetry, the optimal policy is indeter-minate and unique only up to a uniform tariff, t∗. The optimal policy may, forinstance, consist of a high uniform import tax and industry-level export subsidies,or a low uniform import tax and industry-level export taxes. Also, note that withzero cross-substitutability between industries (i.e., Ξ = IK ⇐⇒ ξh f ,k = 0), the op-timal export tax formula reduces to the familiar single-product optimal monopolymarkup, εh f ,k/
(1 + εh f ,k
).
An attractive feature of Theorem 1 is that it characterizes the optimal policyin terms of estimable and observable statistics. In the words of Piketty and Saez(2013), such sufficient statistic formulas have two broad merits. First, “this ap-proach allows us to understand the key economic mechanisms behind the formu-las.” Second “the ’sufficient statistics’ formulas are also often robust to changingthe primitives of the model.” In the present context, the formula characterizedby Theorem 1 can be empirically evaluated with readily-available trade statistics.Moreover, as shown later in the Section 7, the above theorem greatly simplifies thequantitative analysis of trade policy in Ricardian gravity models.
4.1 Special Cases of the Ricardian Model
Two canonical models in international trade, namely, the multi-industry competi-tive gravity model and Dornbusch, Fischer, and Samuelson (1977), are special casesof the general Ricardian framework discussed above. In this subsection, we useTheorem 1 to derive the optimal trade tax formula for each of these special cases.
(i) The Multi-Industry Gravity Model (Costinot et al. 2011). Suppose that
Ui = ∏k Qei,ki,k , where Qi,k =
(∑j=h, f χji,kqρk
ji,k
)1/ρk. It immediately follows that
εh f ,k = −1− εkλji,k/ei,k, where εk ≡ ρk/ (1− ρk). The Cobb-Douglass assumptioneliminates cross-price elasticity effects, so that ξh f ,k = 0. Plugging these valuesinto the optimal tax formula (Equation 7) yields
(1 + x∗k ) (1 + t∗) = 1 + ei,k/εkλ f f ,k.
131 denotes K× 1 vector of ones, while Ξ is a K× K matrix.
19
That is, the optimal trade tax consists of a uniform tariff, t∗, and a industry-specificexport tax that varies primarily with the industry-level trade elasticity, εk. If theeconomy is modeled as a single industry, the above formula reduces to Gros’ (1987)optimal tariff formula, t∗ = 1/ελ f f .
(ii) Dornbusch et al. (1977) The Dornbusch et al. (1977) (DFS) model ana-lyzed in Costinot et al. (2015) is a limiting case of the gravity model, but with
a CES upper-tier utility aggregator. That is, Ui =(
∑k Q(σ−1)/σi,k
)σ/(σ−1), where
Qi,k = limρk→1
(∑j=h, f qρk
ji,k
)1/ρk. As shown in Appendix A.1, our optimal trade tax
formula (7) implies the following limit-pricing solution for the DFS model:
(1 + x∗k ) (1 + t∗) =
σ
σ−1 if σσ−1 ≤
a f f ,kw fah f ,kwh
,a f f ,kw fah f ,kwh
if σσ−1 >
a f f ,kw fah f ,kwh
.
That is, the optimal export tax is equal to the optimal monopoly markup in the caseof strong comparative advantage industries, and a limit-pricing markup in the caseof weak comparative advantage industries. Correspondingly, the optimal exporttax is weakly increasing in the degree of comparative advantage, a f f ,kw f /ah f ,kwh,which is the pattern emphasized in Costinot et al. (2015). On a related note, since1/λ f f ,k is increasing in the degree of comparative advantage, a f f ,kw f /ah f ,kwh, thepositive association between optimal export taxes and comparative advantage isalso implicit in the gravity model. However, in that case, the importance of com-parative advantage for optimal policy diminishes the lower the trade elasticity, εk.
4.2 Multiple Countries
Now, we turn to the case where the world economy consists of N > 2 countries.That is, the Home country trades with multiple partners, and can impose discrimi-natory taxes on goods imported from or exported to different countries. The prob-lem facing the Home country is similar to 4; but now the Home economy sets(N − 1)× K import tax instruments, t = {tjh,k}j,k, and (N − 1)× K export tax in-struments, x = {xhi,k}i,k.
Since we are dealing with more than 2 countries, we can no longer appeal tothe Lerner symmetry to normalize the vector of wages. Instead, Home’s import
20
taxes can improve the terms-of-trade through their effect on the wage rate in N− 1different economies. As a result, Home’s import policy has a local-price externalityacross goods produced in different foreign countries.14 However, Home is still unableto change the relative prices of goods produced from the same country, i.e., it canimpose no local price externality across products from the same supplier. That beingthe case, the optimal tariff remains uniform across goods imported from the samecountry, but can vary across different countries. The following proposition, whichis formally proven in Appendix D, outlines this claim.
Proposition 1. In a multi-country Ricardian model, the optimal tariff on products im-ported from any given country i is uniform: t∗jh,k = tjh for all k ∈ K. However, differentialtariffs may be optimal on products imported from different countries.
The degree to which import taxes discriminate between exporters depends onthe size and the openness of the Home country. A small economy’s trade tax, forinstance, has a negligible effect on the relative wage of other countries. As a result,for such a country, the optimal import tax will be uniform and the optimal exporttax will be characterized by Theorem 1. For a large economy, however, the degreeof import tax discrimination across trading partners can be significant.
Proposition 1 can facilitate the quantitative analysis of tariffs in a multi-countrysetup. As we will elaborate later, solving computationally for optimal import taxesin a multi-country model involves a non-linear optimization over (N − 1)× K tar-iff rates. If the number of countries and industries is large, the computation willbe hindered by the curse of dimensionality. Proposition 1, however, will allowresearchers to shrink the state space by a factor of K (i.e., solve for N − 1 tariffsinstead of (N − 1)× K ).
5 Optimal Policy with Cross-Product Price Linkages
We now turn to characterizing the optimal trade policy in our general model,which allows for input-output linkages and industry-specific factors of produc-tion. Following the discussion in Section 3, competitive producer prices in this
14See Bagwell and Staiger’s (1999) discussion of local-price externality for the case of a two-goodmultiple-country model.
21
general setup are given by Equation 2. That is,
pji,k = Cji,k
(wj, pIj ; qj
),
where pIj = { pI`j,k} is a vector describing the price of all intermediate inputs avail-able to producers in economy j, and qj = {qji,k} is a vector describing economy j’soutput across all industries. Also, note that, in this general setup, each economyi exhibits a surplus, Πi, that is paid to the specific factors of production in thateconomy.
This general case features a rich set of cross-product price linkages that whereabsent in baseline Ricardian model. To be specific, in the Ricardian model, a taxon a given product affected the price of other products only through its impact oncountry-level wages. Here, however, a change in the price of a product may alsohave a more direct effect on the price of other products either through input-output(IO) linkages or through its effect on the demand schedule, qj.
To handle cross-price linkages, we characterize the optimal policy in termsof (i) reduced-form demand elasticities and (ii) trade tax passthroughs, both ofwhich are estimable statistics. To fix minds, define the passthrough of taxes on toconsumer prices (net of wage effects) as follows:
σι,gji,k ≡
∂ ln pji,k (t, x; w)
∂ ln(1 + tι,g
) =∂ ln pji,k (t, x; w)
∂ ln(1 + xι,g
) .
To elaborate, σι,gji,k captures the passthrough of a trade tax on good ι, g to the “con-
sumer” price of good ji, k, netting out the effect of that tax on country-level wages.Analogously, we use σ
f h,kf h,g = σ
f h,kf h,g − 1 {g = k} to denote the passthrough of taxes
onto “producer” prices. Recall that in the Ricardian model, σι,gji,k = 1 if ji, k = ι, g,
while σι,gji,k = 0 if ji, k , ι, g. But, here, the own-passthrough of trade taxes maybe
incomplete and cross-passthrough of trade taxes may be non-zero.As in the Ricardian case, the Lerner Symmetry implies that if (1 + t∗, 1 + x∗)
is a vector of optimal trade taxes, then vector ((1 + t∗) (1 + t), (1 + x∗) /(1 + t)),where t ∈ R+, also constitutes an optimal policy combination. Considering thisand for notational convenience, we express the optimal trade policy formula in
22
terms of Tk and Xk, which are defined as1 + Tk ≡ 1 + t∗k / (1 + t) ,
1 +Xk ≡ 1/[(1 + t)
(1 + x∗k
)].
Using the above definition and invoking demand- and supply-side envelop con-ditions, we can write the FOC corresponding to the import tax on industry k asfollows,
∑g
(−Tgr f h,g
∂ ln q f h,g (t, x; w, Y)∂ ln (1 + tk)
+Xgλh f ,g∂ ln qh f ,g (t, x; w, Y)
∂ ln (1 + tk)
)=
∂TOTh∂ ln (1 + tk)
∀k,
(8)where the right-hand side denotes the terms-of-trade gains from the import tax,which we formally define as,15
∂TOTh∂ ln (1 + tk)
≡∑g
(λh f ,g
∂ ln ph f ,g
∂ ln (1 + tk)− r f h,g
∂ ln p f h,g
∂ ln (1 + tk)
)(9)
= ∑g
(σ
f h,kh f ,gλh f ,g − σ
f h,kf h,gr f h,g
).
Importantly, the terms-of-trade gains (net of wage effects) are fully characterizedby the sub-matrix of import tax passthroughs, σ f h ≡
[σ
f h,kι,g
]ιg,k
. The left-hand side
of Equation 9, meanwhile, represents the trade volume loss from the import tax.Given that
∂ ln qji,g (t, x; w, Y)∂ ln (1 + tk)
= ∑s∈K
∑`∈C
ε`i,sji,gσ
f h,k`i,s ,
the trade volume loss can also be fully characterized in terms of pass-throughs,σ f h ≡
[σ
f h,kι,g
]ιg,k
, as well as import and export demand elasticities, ε f h ≡[ε
ι,gf h,k
]k,ιg
and εh f ≡[ε
ι,gh f ,k
]k,ιg
.
15Recall that σf h,kf h,g = ∂ ln p f h,g/∂ ln(1 + tk) denotes the pass-through of taxes onto “producer”
prices.
23
Similarly, the FOC corresponding to the export tax on industry k is given by
∑g
(Tgr f h,g
∂ ln q f h,g (t, x; w, Y)∂ ln (1 + xk)
−Xgλh f ,g∂ ln qh f ,g (t, x; w, Y)
∂ ln (1 + xk)
)=
∂TOTh∂ ln (1 + xk)
∀k,
(10)where, as before, the right-hand side denotes the terms-of-trade gains from theexport tax:
∂TOTh∂ ln (1 + xk)
≡∑g
(λh f ,g
∂ ln ph f ,g
∂ ln (1 + xk)− r f h,g
∂ ln p f h,g
∂ ln (1 + xk)
)(11)
= ∑g
(σ
h f ,kh f ,gλh f ,g − σ
h f ,kf h,gr f h,g
).
Also, as with the case of import taxes, the left-hand side of Equation 10 representsthe trade volume loss from export tax, xk, which again can be fully characterizedin terms of the passthroughs and demand elasticities. Combining Equations 9 and10, the following theorem provides an analytical characterization of optimal policyas a function of trade shares, λ, revenue shares, r, demand elasticities, ε, and taxpassthroughs, σ. The former two statistics are directly observable, while the lattertwo can be locally estimated.
Theorem 2. [Optimal Trade Taxes under General Price Linkages]The optimal trade tax schedule (T ,X ) is implicitly given by[
−r f h ◦ ε f hσ f h λh f ◦ εh f σ f h
r f h ◦ ε f hσh f −λh f ◦ εh f σh f
] [TX
]=
[∇ln 1+tTOTh
−∇ln 1+xTOTh
],
where εji =[ε
ι,gji,k
]k,ιg
is a K× 4K sub-matrix of demand elasticities and σ ji =[σ
ji,kι,g
]ιg,k
is a 4K× K sub-matrix of tax pass-throughs.16
Based on the above theorem, the optimal trade tax on any subset of industries isuniform if the tax has a zero passthrough (net of wage effects) onto Foreign prices,{p f h,k} and { ph f ,k}. This result applies to both export and import taxes, and issimply reflective of the fact that in the zero passthrough case, the only purpose for
16Note that σ ji where ji ∈ C× C are different block elements of the pass-through matrix: σ =[σ ji]
ji. Also, ◦ denotes the Hadamard or entry-wise product.
24
trade taxes is to increase Home’s labor wage relative to Foreign. This particularobjective is best achieved through a uniform import/export tax.17
Relatedly, the Ricardian model studied in Section 4 is a special case of theabove theorem, where the own passthrough trade taxes is complete, but thecross passthrough of trade taxes is zero. As a result, ∇ln 1+tTOTh = 0 and∇ln 1+xTOTh = λh f , which leads to a uniform import tax, and a differential ex-port tax that is proportional to the inverse of the foreign country’s import demandelasticity for each product.
In the next two subsections, we derive the matrix of pass-throughs,
σ = [ σhh σ f h σh f σ f f ],
and use Theorem 2 to characterize the optimal policy in two special cases. First,a Ricardian model with general input-output linkages. Second, a generalizedspecific-factors model where the marginal cost of production (net of labor wage) isincreasing in output.
Before moving forward, however, it is worth noting that when export taxes areunavailable, Theorem 2 implies the following formula for optimal import tariffs,
1 + t∗ = (1 + t)[1−
(r f h ◦ ε f hσ f h
)−1∇ln 1+tTOTh
],
where the industry-specific component (the term in bracket) accounts for Home’sindustry-level monopsony power, while the uniform term accounts for Home’saggregate monopsony power due to general equilibrium wage effects. Unlike thebenchmark case, however, the exact value of t is critical here, and is determined bythe following equation:18
1 + t =
(εh f −∑
g
(tg − t
)r f h,g ε f h,g
)/(1 + εh f
),
17Theorem 2 also indicates that optimal trade taxes do not explicitly depend on neither the ag-gregate labor demand elasticities nor the income elasticities of demand. This outcome is a directbyproduct of the Lerner symmetry theorem. In particular, accounting for general equilibrium in-come and wage effects leads to a uniform shift in all export or import taxes. But given the Lernersymmetry, when both export and import taxes are available, a uniform shift in either tax instrumentis redundant.
18See Appendix E for a formal derivation.
25
where εh f = ∑k ∑g
(rh f ,krh f
εh f ,gh f ,k
)denotes the elasticity of Foreign’s demand for Home
labor—see Section 6 for a formal definition. Considering this, the above expressioncorresponds to the optimal markup on Home’s wage rate, with a correction fornon-uniformity. Moreover, based on the above, when export taxes are restricted,general equilibrium wage effects dampen the cross-industry heterogeneity in op-timal import tariffs. So, even though tariffs are non-uniform, they are less hetero-geneous than traditional theories would suggest.
5.1 Input-Output Linkages
Below, we derive the passthrough matrix and the optimal trade tax schedule in aRicardian model with input-output linkages. In this case, the production of eachgood employs labor and intermediate inputs from (possibly) all industries. Statedformally, the production of good ji, k (export j–importer i–industry k) is charac-terized by qji,k = qji,k(Lji,k, qIji,k), where Lji,k is the amount of labor employed in
the production of good ji, k; and qIji,k ≡ {qj,gji,k},gis the vector of intermediate input
quantities, with qj,gji,k denoting the quantity the intermediate input ι, g in the pro-
duction of good ji, k.19 Such a production setup implies that the competitive priceset by cost-minimizing firms (net of taxes) should exhibit the following formula-tion, which is a special case of Equation 2:
pji,k = Cji,k
(wj, pIj
)∀j, i ∈ C; k ∈ K (12)
That is, the producer of good ji, k is a function of the local wage rate, wj, andthe (tax-inclusive) price of all intermediate inputs, pIj ≡ { pIj,g}. For notationalconvenience, we assume that the price of a product in a given market is the samewhether it is used as an intermediate input (indexed I) or a consumption good(indexed C), namely, pIi = pCi = pi for i = h, f .20 Correspondingly, the consumerprice of goods ji, k can be determined exclusively as a function of the taxes and the
19In the above notation, qι,gji,k = 0 if ι , j, by construction. For instance, variety f h, g which is sold
by foreign firms to the home country cannot be directly employed as an input by foreign firms, butvariety f f , g that is sold in the foreign market can be.
20By the choice of parameters in the preferences, this structure still allows for different prices ofintermediate and consumption goods, which may occur due to, for example, differential taxationof intermediate and consumption goods.
26
wages as follows:
pji,k (t, x; w) =(1 + tji,k
) (1 + xji,k
)Cji,k
(wj, pj
)∀j, i ∈ C; k ∈ K (13)
where, by assumption, x f h,k = th f ,k = 0, while tk ≡ t f h,k and xk ≡ xh f ,k for all k.
To characterize the entire 4K × 4K matrix of pass-throughs, σ ≡[σ
ι,gji,k
]jik,ιg
, we
apply the implicit function theorem to Equation 13. We also appeal to Shepard’slemma that ∂ ln pji,k/∂ ln pIι,g = pι,gqι,g
ji,k /pji,kqji,k, where the right-hand side αι,gji,k ≡
pι,gqι,gji,k /pji,kqji,k denotes the jik× ιg’th element of the global (4K× 4K) IO matrix,
A ≡[α
ji,kı,g
]ıg,jik
. Following these two steps, we can produce the following lemma,
which states that (beyond wage effects) the pass-through of taxes on to consumerprices is fully determined by the IO matrix.
Proposition 2. In a Ricardian Model with input-output linkages, the matrix of trade taxpassthroughs is exclusively characterized by the global input-output matrix:
σ = (I − A)−1 .
Note that σ = [ σhh σ f h σh f σ f f ] is closely related to the Domar weightscharacterized by Baqaee and Farhi (2017).21 That σ does not depend on thedemand-side of the economy is an artifact of the Ricardian supply structure. Basedon this assumption, trade taxes affect the unit labor cost only through the their ef-fect on input prices. Once we relax the Ricardian supply structure, which is donein the following section, σ depends on the entire schedule of demand elasticities.
Optimal Tariff on Intermediate vs. Final Products
We now use the passthrough matrix σ and Theorem 2 to determine how opti-mal trade taxes differ across intermediate and final goods. To fix minds, note thatour general IO structure allows us to have two versions of the same good: a finalgood version and an intermediate input version. Thus, it can accommodate thecase where differential taxes are applied to the same product, depending on theintended final use.
21Since there are no misallocations in our Ricardian economy, the revenue-based and cost-basedinput-output matrixes and the corresponding Domar weights are identical in our setup.
27
Our main claim is that the optimal import tax on any given good dependscrucially on whether it will be re-exported as part of another product. To seethis, suppose Home imposes an import tax tk on good, f h, k, but does not ex-port goods that employ f h, k as an intermediate input. In that case, Proposition2 implies that σ
f h,kh f ,g = σ
f h,kf h,g = 0. That is, tk cannot influence consumer and pro-
ducer prices in Foreign beyond general-equilibrium wage effects. Correspond-ingly, ∂TOTh/∂ ln (1 + tk) = 0 and the trade volume losses due to tk are only directeffects, i.e., ∂ ln q f h,k/∂ ln
(1 + tg
)= ε
h f ,kf h,g and ∂ ln qh f ,k/∂ ln
(1 + tg
)= 0. Consid-
ering this, Theorem 2 immediately implies that Tk = 0 for product f h, k that is notre-exported through input-output linkages.22
The following proposition formally outlines this claim.
Proposition 3. In a Ricardian model with input-output linkages, the optimal import tariffis uniform across all imported final goods and intermediate goods that are not re-exported.
Now, consider imported intermediate inputs that are re-exported as part of an-other (more downstream) product. In this case, an import tax on an intermediateinput f h, k could influence the consumer and producer prices in the rest of theworld beyond general-equilibrium wage-effects. In particular, the consumer priceof any exported good h f , g (namely, ph f ,g) that uses f h, k as an input will be af-fected by tk. Similarly, the producer price of a foreign-produced good f h, s, whichuses h f , g as an intermediate input, will be affected indirectly by tk. As a result,∂ ln TOTh/∂ ln (1 + tk) , 0, which leads to non-uniformity in import taxes.
Following up on the above argument, re-exporting gives the Home govern-ment the power to effectively tax transactions outside its territory. This extrater-ritorial taxing power creates a policy externality that is distinct from the terms-of-trade and local-price effects identified in the prior literature ((Bagwell and Staiger,1999)). To elaborate, suppose Home imposes an import tax on tires (good f h, k)and exports them back as part of a fully-assembled vehicle to Foreign (good h f , g).The price of this exported vehicle can be decomposed into a domestic value-added component, pVA
h f ,g (i.e., the price without the tires), and a foreign-produced
22Another special case where taxes are identical to the Ricardian case, is one where (i) the input-output structure is symmetric α
j,ki,g = αk for all , j, ι, k and g. In that case, σ
ı,gji,k = σk if ι = j and
σı,gji,k = σ′k if ι , j , with σk − σ′k = αk), and (ii) preferences are quasi-linear and additively separable
(i.e., εi,gji,k = 0 if g , k and ε f h,k = −εhh,k
f h,k). For this special case, it is straightforward to show thatTk = 1 and Xk = 1/εh f ,k .
28
and foreign-consumed component, pFf f ,k (i.e., the price of tires). Stated formally,
ph f ,k = pVAh f ,g + pF
f f ,k. Hence, supposing that the tires or the vehicle are not used asintermediate inputs in any other good, the terms-of-trade effect of such an importtax (tk) can be stated as
∂TOTh∂ ln (1 + tk)
≡ λh f ,g∂ ln ph f ,g
∂ ln (1 + tk)= λVA
h f ,g
∂ ln pVAh f ,g
∂ ln (1 + tk)︸ ︷︷ ︸standard TOT effect
+ λFh f ,g
∂ ln pFf f ,k
∂ ln (1 + tk)︸ ︷︷ ︸extraterritorial taxing power effect
where λVAh f ,g ≡ λh f ,g
(pVA
h f ,g/ ph f ,g
)and λF
h f ,g ≡ λh f ,g
(pF
f f ,k/ ph f ,g
). The second
term in the above equation corresponds to rents accruing to the Home govern-ment through taxing a transaction between tire producers and consumers who arelocated outside of its territory. If tires were instead assembled on the car in For-eign, the Home government would not have such extraterritorial taxing power.Such extraterritorial taxing power can perhaps explain why the gains from policyare significantly larger in the presence of IO linkages–a claim we formally docu-ment in Section 7.
5.2 Generalized Specific-Factors Model
Now we consider a generalized specific factors model in which producer pricesare given by the following special case of Equation 2:
pji,k = Cji,k
(wj; qj,k
)i, j ∈ C, k ∈ K. (14)
That is, the competitive price of goods ji, k is a function of country j’s output levelin industry k, namely, qj,k = {qji,k}i. Before characterizing the passthrough matrixin this case, let us briefly discuss the micro-foundation that rationalizes the aboveequation.
Generally speaking, the upward-sloping supply curve underlying Equation 14may arise due to (i) a finite elasticity of transformation between different out-put varieties from the same country, and/or (ii) industry-specific factors of pro-duction. To elaborate, suppose that country j’s composite output in industryk is characterized by Qj,k = Qj,k
(qjh,k, qj f ,k
), which allows for a finite elastic-
ity of transformation between output produced for different markets. A well-
29
known special case of this specification is the constant elasticity of transforma-tion (CET) production possibility frontier popularized by Powell and Gruen (1968):
Qj,k =(
∑i=h, f χji,kqρkji,k
)1/ρk, where ρk > 1. The composite output Qj,k is itself pro-
duced using labor, Lj,k, that is perfectly mobile across industries, and an industryk-specific factor, Sj,k, that is immobile, i.e., Qj,k = Fj,k(Lj,k,Sj,k), where Fj,k (.) is anon-parametric production function.
Given the above production structure, cost-minimizing firms set a competitiveprice that is a function of their output schedule and the local wage rate, as in Equa-tion 14. A classic special case covered by Equation 14, is the the standard Ricardo-Viner model, in which the output produced for domestic and foreign markets areperfectly substitutable (i.e., ρk → 1 in the CET case).
Noting that Cji,k (.) describes the supply curve of variety ji, k, the supply-sideof this economy can be fully summarized in terms of the following reduced-form(inverse) supply elasticities.
D3. [Supply Elasticities][own-supply elasticity] γji,k ≡ ∂ ln Cji,k
(wj; qj,k
)/∂ ln qji,k
[cross-supply elasticity] γjι,kji,k ≡ ∂ ln Cji,k
(wj; qj,k
)/∂ ln qjι,k.
To attain further perspective on the above elasticities, note two special cases.First, the standard Ricardo-Viner model where γ f h,k = γ
f h,kf f ,k simply equals the in-
verse of Foreign’s export supply (or excess supply) elasticity. Second, the CETmodel without industry-specific factors of production where the supply elasticityassumes a more structural interpretation, γji,k =
(1− rji,k
)/ (ρk − 1) for all ji, k.23
Considering D3, we now proceed to characterizing the passthrough matrix inthe generalized specific factors model. To this end, we follow similar steps to thosetaken in the case of IO linkages. We apply the implicit function theorem to thefollowing equation, which characterizes consumer prices:
pji,k (t, x; w) =(1 + tji,k
) (1 + xji,k
)Cji,k
(wj; qj,k
)∀j, i ∈ C; k ∈ K.
Doing so, yields a 4K× 4K matrix of pass-throughs,
σ = (I − Σ)−1 (15)
23rji,k = rji,k/ ∑ι rjι,k denotes the within-industry output share.
30
where Σ =[γ
`,gι,g εn`,s
`,g
]ιg,n`s
is fully determined by reduced-form demand and sup-
ply elasticities. Noting that σ = [ σhh σ f h σh f σ f f ], we can plug σ f h andσh f calculated using the above equation into Theorem 2 to calculate the optimaltrade tax schedule exclusively as a function of reduced-form supply and demandelasticities as well as observable expenditure and revenue shares.
In the standard Ricardo-Viner case, where there is zero cross-substitutabilitybetween industries and γji,k ≡ γ
ji,kj f ,k = γ
ji,kjh,k for all j, i, and k, Equation 15 implies
that σjι,kji,k = 1/
(1− ε jι,kγjι,k
)if j, k = , g and σ
ι,gji,k = 0 if j, k , , g. Plugging
these expressions into Theorem 2, implies that Tk = γ f h,k and Xk = 1/εh f ,k, whichin turn yields the following familiar-looking optimal tax formula for the Ricardo-Viner case:
1 + t∗k = (1 + t)(1 + γ f h,k
),
(1 + x∗k ) (1 + t) =εh f ,k/(1 + εh f ,k
). (16)
What is perhaps notable about the above formula, is that it can be obtained withoutabstracting from either general equilibrium wage effects or general equilibrium in-come effects. The only necessary assumption is that cross-demand elasticities bezero between industries and all trade tax instruments be available to the govern-ment.
Importantly, Equation 2 and Theorem 2 indicate that cross-substitutability be-tween Ricardian and non-Ricardian industries can lead to the non-uniformity ofoptimal import tariffs even across Ricardian industries. To elaborate, supposethe Foreign economy employs Ricardian production technologies, but Home’s in-dustries employ specific factors of production and exhibit upward-sloping supplycurves. Then, a tax on import good f h, k can affect the demand for good hh, g.This effect can, in turn, alter the consumer price of Home’s exports to Foreign inindustry g, ph f ,g—i.e., σ
f h,kh f ,g > 0. Considering this, and based on Theorem 2, the
non-uniform component of tk will be non-zero (i.e., Tk , 0), even though the For-eign supply curve is flat in all industries.
31
6 Interdependence of Trade Policies
Trade policy interdependence concerns the effect of policy choices in one area onthe tradeoffs that policymakers face in other areas. The existence of these interde-pendencies is not a controversial idea among economists. In fact, one of the best-known results in international economics—the Lerner’s (1936) Symmetry Theo-rem—establishes a strong link between import and export policies. Nor is the im-portance of these interdependencies too hard to notice: Many disputes in the WTOare about alleged use of policy instruments that are not restricted by the WTO buthave the effect of replicating trade taxes/subsidies. Nevertheless, the current liter-ature provides very little insight about the nature of these interdependencies andtheir potential political and economic implications.
In the context of our model, political considerations or international tradeagreements may limit the government’s freedom in choosing their trade policyfrom set A. In many instances, governments may be prohibited from conductingexport policy or from setting import taxes in select industries.24 In the presenceof general equilibrium linkages, these partial restrictions can influence the govern-ment’s choice of optimal policy with respect to unrestricted instruments.
In what follows, we outline three novel trade policy interdependencies. Tostreamline the presentation, we hereafter restrict attention to the Ricardian caseof the model; noting that most of the trade-offs we identify are not an artifact ofthe Ricardian supply structure and prevail in the more general case of our frame-work. We present our results using a sequence of hypothetical partial liberalizationepisodes. In each episode, a set of previously-available policy instruments are re-stricted and the government sets the unrestricted instruments optimally.
6.1 Optimal Policy when Export Taxes are Restricted
Suppose Home enters a trade agreement that prohibits all export taxes, but leavesimport taxes at the discretion of the government. Home’s optimal policy problem
24Two notable features of the GATT/WTO agreement resemble the partial policy restorationsemphasized here. First, the GATT/WTO has only gradually introduced more constraints on thegovernments’ policy space over time. Earlier GATT/WTO negotiations were focused on tariff cutsin a few industries, leaving import tariffs in many other industries at the discretion of governments.Another notable feature of the GATT/WTO agreement is its rather strict stance towards exportpolicy in comparison to import policy.
32
under this partial restriction can be stated as follows:
max(0,t,τ;w)∈A
Wh (0, t, τ; w)
Since export taxes are restricted, the optimal tax combination is now unique andwages can no longer be normalized in both countries. Instead, due to the Ricar-dian supply structure, Home’s import taxes can only improve its terms-of-tradethrough their effect on the relative wage, wh/w f . As we show below, given thesecircumstances, Home’s optimal import tax is uniform and determined by the elas-ticity of Foreign demand for Home’s labor.
To make this point formally, let Lji = Lji (w; t, x) denote country j’s demand forcountry i’s labor. The elasticity of country j’s demand for country i’s labor demandcan, thus, be defined as follows.
D4. [Elasticity of Labor Demand] ε ji ≡ ∂ lnLji (w; t, x) /∂ ln wj =
∑k ∑g
(rji,krji
εji,gji,k
).
In the above definition, the last line follows from the Ricardian supply struc-ture, which indicates that Lji=∑k qji,k/aji,k. One straightforward interpretation ofε ji is that it reflects country j’s collective export market power. As noted by thefollowing proposition, Home’s optimal import tax is determined solely by εh f .
Proposition 4. When export taxes are restricted, the optimal policy consists of a uniformimport tariff that reflects the home country’s collective export power
1 + t∗ =εh f
1 + εh f.
The intuition behind the uniformity of optimal import taxes is similar to whatwe provided earlier. But the intuition behind the formula characterizing the op-timal tax level is the following. A uniform import tax (which is isomorphic to auniform export tax) is akin to a markup charged by Home on its labor wage rate.The optimal markup level is, thus, determined by the elasticity of Foreign demandfor Home’s labor. In the widely-used multi-industry gravity model outlined inSection 4.1, the optimal import tax formula reduces to
1 + t∗ = 1 +1
∑k
(rh f ,krh f
εkλ f f ,k
) ,
33
where εk denotes the trade elasticity in industry k. A well-know special case of theabove formula is the single-industry optimal tariff formula, t∗ = 1/ελ f f , popular-ized by Gros (1987).25
Proposition 4 points to a rather surprising corollary. Given the Lerner symme-try, Proposition 4 implies that the effect of the optimal import tax can be exactlyreplicated with a uniform export tax. The opposite, however, is not true as theoptimal export tax is typically non-uniform according to Theorem 1.
Corollary. In a Ricardian economy, import taxes are only an imperfect substitute for ex-port taxes. But export taxes can perfectly reproduce any welfare outcome that is attainablewith import tariffs.
The intuition behind the above corollary is simple. In a Ricardian economy,import taxes can only improve Home’s terms-of-trade through their effect on theeconomy-wide wage rate. Export taxes can do that, but they can also improvethe terms-of-trade through their direct effect on consumer prices in Foreign. As aresult, export taxes are more potent of a policy instrument in a Ricardian economy.These results, have an immediate implication for the design of trade agreements.That is, an incomplete agreement that restricts only import tariffs is ineffective,because the restricted import tariffs can be perfectly substituted with unrestrictedexport taxes. However, restricting export taxes can effectively lower the degree ofprotection, even without imposing restrictions on import taxes.
6.2 Optimal Policy when a Subset of Industries are Restricted
Now, we consider a second sequence of liberalization whereby import taxes arerestricted in a subset of industries. To be specific, in this second sequence, Homeis still obliged to set zero export taxes in all industries, i.e., xk = 0 for all k. Inaddition, it is also restricted to setting zero import taxes in a subset of industries.To streamline the presentation, we let KR denote the set of import-tax-restrictedindustries and let KL denote the set of unrestricted industries, with KR ∪KL =
25Based on the above formula, optimal tariffs increase with the level of trade openness, 1/λ f f ,k.This prediction, however, hinges on the convexity of the CES demand, whereby | εh f ,k | is strictlydecreasing in the level of trade in industry k. Conversely, if the underlying demand was sub-convex, then more trade openness would entail a lower optimal tariff.
34
K, by construction. We also temporarily focus on the special case where tradedindustries exhibit a zero income elasticity.26
As demonstrated in Appendix B, Home’s optimal import tax in unrestrictedindustries can be characterized as follows:
1 + t∗k = (1 + t)
1 + ∑g,k
(1 + tg
1 + t− 1) r f h,gε
f h,kf h,g
r f h,kε f h,k
, k ∈ KL, (17)
where t is a uniform term that accounts for general equilibrium wage effects:
1 + t =εh f + ∑g
[(tg − t
) r f h,gr f h
ε f h,g
]1 + εh f
. (18)
Based on the above formula, the post-partial-liberalization import taxes are gener-ally non-uniform. Furthermore, as the set of restricted industries shrinks to zero,the optimal import tax specified by the above proposition converges to the uni-form import tax formula specified by Proposition 4—that is, if KR = ∅, then1 + t∗k = 1 + t = εh f /
(1 + εh f
)for all k.
Based on Equation 17, partial liberalization affects the optimal import tax in un-restricted industries through two distinct channels. The first driver of interdepen-dence between restricted and unrestricted tariffs are cross-price elasticity effects.To elaborate, consider the case where industries are gross substitutes: ε
f h,kf h,g > 0
for all k and g. In that case, the second parenthesis in Equation 17 is equal to“one” without partial restrictions but smaller than “one” otherwise. Hence, partialrestrictions lower the optimal import tax in unrestricted industries through cross-elasticity effects. The intuition being that restricting import taxes in a subset ofindustries decreases the volume of trade in unrestricted industries. The reductionin trade volume, in turn, reduces the marginal revenue from taxation and entails alower optimal tax.
General equilibrium wage effects, which operate through t, are the seconddriver of tariff interdependence. Noting that ε f h,g < 0, it follows immediately
26This assumption is consistent with a quasi-linear utility aggregator across industries, withtrade costs being prohibitively high in the linear industry.
35
thatεh f + ∑g
[(tg − t
) r f h,gr f h
ε f h,g
]1 + εh f
<εh f
1 + εh f,
where the right-hand side corresponds to the optimal import tax level withoutpartial restrictions (Proposition 4). The above expression simply indicates that theuniform component of the optimal import tax declines in face of partial liberal-ization. The intuition behind this second channel can be stated as follows. Whentariffs are restricted in some industries, the wage-driven component of the optimalimport tax, t, introduces a relative price distortion between restricted and unre-stricted industries. To partially countervail this distortion, t assumes a lower valueunder the partial restriction.
The tariff interdependence arising from a reduction in t is subject to one basicqualification. The elasticity of labor demand, εh f , can itself vary with a changein the underlying trade taxes. So, to ensure that t reduces in face of the partialrestrictions, we need ∂ | εh f | /∂wh to be sufficiently small. That is, the declinein wh/w f due to partial liberalization, should not lead to a too large of a declinein | εh f |. This will the case if the demand for labor is sufficiently concave. Thefollowing Proposition summarizes these arguments.
Proposition 5. If (i) ∂ | εh f | /∂wh is sufficiently small, and (ii) industries are grosssubstitutes, then tariffs are complementary across industries. That is, restricting tariffs ina subset of industries lowers the optimal tariff in unrestricted industries.
The above proposition is significant because negotiating tariff cuts can be costly,and more so for certain industries. Based on Proposition 5, it may be optimal fortrade agreements to focus on tariff reductions in a subset of low-negotiation-costindustries. Once tariffs are lowered in these industries, governments will volun-tarily lower their tariffs in the non-negotiated industries.
It should be noted that the assumption placed on ∂ ln | εh f | /∂wh by Proposi-tion 5 is weaker than it may appear. In our multi-industry framework, two factorsaffect the convexity of demand for labor. On one hand, a drop in wh alters thecomposition of demand in favor of high-elasticity industries. This effect alwayscontributes to a lower ∂ ln | εh f | /∂wh. On the other hand, a drop in wh can alsoalter the demand elasticity level, εh f ,k, per industry, with the direction of this latterchange depending on the underlying demand function. Considering this, Proposi-
36
tion 5 simply holds if composition effects are sufficiently large. Later in Section 7,we show that in a standard multi-industry gravity model fitted to data, the condi-tions outlined by Proposition 5 are satisfied, and that industry-level import taxesexhibit strong complementarity.
Importantly, the above interdependence results are derived under the assump-tion that export tax instruments are restricted. A considerably weaker version oftariff complementarity arises when export taxes are not restricted. Specifically,suppose Home is obliged to set zero import taxes in a subset KR of industries.Then following Theorem 1, it is optimal for Home to lower the the tariff to zero inthe unrestricted industries, and apply a uniform upward shift to all export taxes.By applying these changes, the import tax restriction will have essentially no effecton Home’s rate of protection. By contrast, the tariff complementarity outlined byProposition 5 induces Home to voluntarily lower its effective rate of protection.In other words, Foreign experiences a welfare gain from the complementarity-induced reduction in import taxes.
6.3 Optimal NRTBs when Revenue-Raising Taxes are Restricted
Finally, we consider a third sequence of liberalization where all revenue-raisingtrade taxes are restricted. In this case, the Home government can only erect non-revenue trade barriers (NRTBs) that restrict imports without generating any rev-enues. It is well known that under standard partial-equilibrium or one-industrygeneral-equilibrium trade policy models, there are no gains from erecting NRTBs.However, in our multi-industry general equilibrium framework, we find thatNRTBs could improve the Home country’s welfare at the expense of the Foreigncountry.
We model NRTBs as wasteful iceberg transport costs that do not generate rev-enues for the government or utility for consumers.27 The government’s problemis to choose product-specific NRTBs, {τk} to maximize its welfare.28 We find thatoptimal NRTBs are (i) strictly positive in industries where demand for imports
27Import quotas and voluntary export restraints could also be considered non-revenue tradebarriers, but both are restricted under trade agreements. We focus on wasteful iceberg transportcosts to better represent hidden trade barriers such as border red-tapes and frivolous regulations.
28Stated formally, the optimal NRTBs, τ, are chosen to maximize Wh (0, 0, τ; w) subject to(0, 0, τ; w) ∈ A.
37
from Foreign is sufficiently elastic, and (ii) zero in other industries (see AppendixC). Moreover, if ε f h,k is non-decreasing in q f h,k, the optimal NRTB is prohibitivelylarge in high-ε industries.29 That is,
τ∗k =
∞ if ε f h,k < 1 + εh f + ε f h
0 if ε f h,k > 1 + εh f + ε f h
,
where, as earlier, ε ji denotes the elasticity of labor demand, which is weightedaverage of industry-level demand elasticities. The above formula indicates thatin a single-industry model, the optimal NRTB is always zero since ε f h,k = ε f h >
1 + ε f h + εh f . Similarly, the optimal NRTB will be zero in all industries if wageswhere assumed to be invariant to policy as in partial equilibrium models. Butonce we accommodate general equilibrium wage effects and allow for multipleindustries, there is an incentive for setting NRTBs, which is summarizes by thefollowing proposition.
Proposition 6. Absent revenue-raising taxes, it is optimal to impose a prohibitively highNRTB on imports with sufficiently high demand elasticities. The optimal NRTB on allother imports and exports will be zero.30
There is a simple logic behind the above result. Erecting NRTBs on a subset ofproducts reduces the Home consumers’ welfare with respect to those products, butimproves Home’s terms of trade with respect to all other imports. In high-ε indus-tries, the gains from importing Foreign varieties are relatively small, because a highε indicates strong substitutability between imported and domestic varieties. Onthe other hand, restricting imports in high-ε industries can reduce foreign wagesand, as a result, the price of imports in all other product categories. These generalequilibrium wage effects can be large enough to offset the modest welfare loss dueto a price increase in the NRTB-restricted industries.
In the widely-used multi-industry gravity model (outlined in Section 4.1), weshow that the optimal NRTB is positive for industries that are sufficiently large(high-eh,k), sufficiently productive (high-λhh,k), and feature a sufficiently large trade
29The condition that ∂ε f h,k/∂q f h,k ≥ 0 is widely-known as Marshall’s Second Law of Demand,and is satisfied in an important class of trade models.
30Proof is provided in Appendix C.
38
elasticity, εk.31 Using trade elasticities estimated by Caliendo and Parro (2014),our quantitative analysis in Section 7 indicates that these conditions are typicallysatisfied in the the ’Mining’, ’Petroleum’, and ’Wood’ industries.
6.4 Interdependence with Multiple Countries
The multi-country version of our framework features a margin of policy interde-pendence that is absent in the two-country model. This margin is closely relatedto the Most-Favored Nation (MFN) clause of the WTO, which requires that mem-ber countries apply their import tariffs non-discriminatorily against all other WTOmembers. Even in the absence of the MFN clause, governments may find it ad-ministratively convenient to impose tariffs that are independent of national origin.
To present this particular type of interdependence, recall that unconstrained op-timal import tariffs are uniform across products of the same exporter (Proposition1). However, as shown in Appendix D, when countries are bound to MFN tar-iffs, it becomes optimal to vary the tariff rate across products. The intuition is thatMFN tariffs do not allow Home to discriminate among various exporters based onits bilateral export market power. As a result, it is optimal for Home to vary itsimport tariff rate across industries based on its average bilateral market power inthat industry. More specifically, optimal MFN tariffs should be higher in industrieswhere imports are predominantly from exporters towards which Home possessesrelatively more market power.
7 Quantitative Analysis
In this section, we fit our theoretical model to industry-level trade and productiondata. In doing so, we pursue two basic objectives. First, we want to quantifythe gains from optimal policy and the degree of policy interdependence acrossindustries. Second, we wish to highlight how our analytical formulas streamlinethe computational analysis of trade policy.
31In theory, one can easily construct an example where industry k is subject to a positive NRTB.To this end, consider the K-industry gravity model in Section 4.1. If ei,k’s are uniform across in-dustries and Home and Foreign are symmetric, then rji/,k = rji/K for all k, and for the NRTB inindustry k to be positive it suffices that εk > 2 ∑g εg/K.
39
Data. Our main data source is the 2012 edition of the World Input-OutputDatabase (WIOD, Timmer et al. 2012). The WIOD database covers 35 industriesand 40 countries, which account for more than 85% of world GDP, plus an aggre-gate of the rest of the world. The countries in the sample include all 27 membersof the European Union and 13 other major economies, namely, Australia, Brazil,Canada, China, India, Indonesia, Japan, Mexico, Russia, South Korea, Taiwan,Turkey, and the United States. The 35 industries in WIOD database include 15tradable industries and 20 service-related industries—see Tables 2 and 3 for a thor-ough description of countries and industries used in the analysis.
For each country in the sample, the WIOD reports total national output, wiLi aswell as total national expenditure, Yi, and allows for the construction of industry-level output shares, rji,k, and expenditure shares, λji,k. Moreover, the data identifiesthe global input-output matrix, A. We complement the WIOD with industry-leveltrade elasticity estimates from Caliendo and Parro (2014).
To be consistent with our analytical framework, we restructure the WIODdatabase in two dimensions. First, we merge all the service industries into a singleaggregated non-traded sector. Second, we purge the data from trade imbalances.In this process, we closely follow the methodology in Costinot and Rodríguez-Clare (2014), who apply Dekle et al.’s (2007) hat-algebra methodology to purge the2008 edition of the WIOD.32
Competitive Gravity Model without IO Linkages. We first consider the two-country competitive gravity model outlined in Section 4.1. As in Costinot andRodríguez-Clare (2014), we assume that the status-quo is free trade. That is, theWIOD data describes a global economy where countries set near-zero trade taxes.The standard Ricardian gravity model features a Cobb-Douglas utility aggrega-tor across industries and a CES demand structure within industries. In that case,V(Yi, Pi
)= Yi/Pi, where the aggregate price index in economy i ∈ C is given by
Pi = ∏k∈K
(∑j∈C
χji,k p−εkji,k
)−ei,k/εk
, (19)
32A similar approach is also applied by Ossa (2014) to eliminate trade imbalances from the GTAPdatabase.
40
with χji,k being a structural parameter that accounts for productivity levels andtrade barriers. Accordingly, bilateral expenditure shares given by
λji,k =χji,k p−εk
ji,k
∑`∈C χ`i,k p−εk`i,k
ei,k ∀j, i ∈ C, (20)
where ei,k = ∑j λji,k denotes country i’s total expenditure share on industry k. Thedemand elasticities are also equal to ε ji,k = −1− εkλii,k/ei,k, εii,k
ji,k = εkλii,k/ei,k, and
εi,gji,k = 0 if g , k. Given the above demand structure, we can use the hat-algebra
methodology in combination with Theorem 1 to solve for Home’s optimal tradetaxes, t∗f h and x∗h f . Importantly, this approach eliminates the need to (i) estimatethe structural parameters χji,k, and (ii) perform a global optimization problem.Instead, t∗f h and x∗h f can be determined as a function of observables by merelysolving a system of non-linear equations. The following proposition states thisclaim formally, using the hat-algebra notation that z ≡ z′/z denotes the ratio ofcounterfactual-to-factual value for any variable z.
Proposition 7. Suppose the observed data is generated by the Ricardian model outlinedin Section 4 and functional forms 19 and 20. Then, the optimal trade tax can be fullycharacterized by solving the following system of equations
(1 + xh f ,k
)/ (1 + t) = 1 + 1/εkλ f f ,kλ f f ,k; t f h,k = t
x f h,k = 0, th f ,k = 0
λji,k =[wjLj
(1 + tji,k
) (1 + xji,k
)]−εkPεk
i,k
P−εki,k = ∑j
([wjLj
(1 + tji,k
) (1 + xji,k
)]−εkλji,k
)wiLiwiLi = ∑k ∑j
[λij,kλij,kYjYj/
(1 + xij,k
) (1 + tij,k
)]YiYi = wiLiwiLi + ∑k ∑j
(tji,k
1+tji,kλji,kλji,kYiYi +
xij,k1+xij,k
λij,kλij,kwjLjwjLj
),
in terms of {xh f ,k}, {λji,k}, {Pi,k}, {wiLi}, and {Yi}, as a function of (i) observed expen-diture shares, {λji,k}; (ii) observed national output/income levels, Yi = wiLi ; and (iii)industry-level trade elasticities, {εk}.33
33To handle the multiplicity of optimal trade taxes, we choose a value of t = 0.59 based on theaverage Smoot–Hawley tariff rates.
41
To put the above proposition in perspective, it is worth comparing it to the stan-dard approach highlighted in Costinot and Rodríguez-Clare (2014). By appealingto the hat-algebra methodology, the standard approach eliminates the need to es-timate the structural parameters χji,k. However, it involves solving a constrainedglobal optimization over 2K + 4 choice variables. In comparison, Proposition 7 re-duces the computational task to simply solving a system of 4K + 4 equations andunknowns.
Competitive Gravity Model with IO Linkages Using the structure of the WIOD,we also analyze a Ricardian gravity model with IO linkages. To this end, we as-sume a Cobb-Douglas production function, so that the elements of the global IOmatrix become invariant to trade taxes. In that case, using Theorem 2 and Lemma2, we can produce an analog of Proposition 7 in the presence of IO linkages. Doingso, allows us to solve for the optimal trade taxes without conducting a constrainedglobal optimization. In the interest of brevity, the details of this proposition arerelegated to Appendix G.
7.1 The Gains from Optimal Policy
After solving for optimal trade taxes, we can compute the gains from policy asWh = Yh/
(∏k Peh.k
h,k
). We do so for 14 major economies—in each case we calculate
the optimal policy for that country with respect to an aggregate of the rest of theworld. To put these gains in perspective, we also compare them to the gains fromoptimal policy when the Home government is restricted to (i) only import taxes,and (ii) only NRTBs. The computed gains from policy are reported in Table 1, bothin the absence and in the presence of IO linkages.
On average, the gains from unilateral export policy dominate those of importpolicy by a factor of 3. The gains from unilateral NRTBs are relatively modestcompared to both export and import taxes. For countries like Mexico and Taiwan,however, the gains from NRTBs are non-trivial and stand around 0.11-0.12%.
A notable observation, here, is that the welfare gains from trade policy are con-siderably larger in the presence of IO linkages. This outcome is perhaps related tothe extraterritorial taxing power, identified in Section 5.1, whereby the Home econ-omy generates revenue by effectively taxing transactions between Foreign suppli-
42
Table 1: The gains from trade policy: % change in real GDP
without IO Linkages with IO Linkages
Country Export tax Import tax NRTB Export tax Import tax NRTB
AUS 0.30% 0.17% 0.03% 0.59% 0.30% 0.00%E.U. 0.91% 0.47% 0.04% 1.54% 0.89% 0.05%BRA 0.56% 0.25% 0.02% 1.00% 0.45% 0.03%CAN 1.39% 0.37% 0.04% 2.71% 0.71% 0.03%CHN 0.61% 0.35% 0.02% 1.71% 1.05% 0.05%IDN 0.48% 0.22% 0.02% 1.17% 0.50% 0.00%IND 0.43% 0.27% 0.01% 1.20% 0.53% 0.01%JPN 1.00% 0.41% 0.03% 1.58% 0.71% 0.05%KOR 2.56% 0.72% 0.08% 4.01% 1.88% 0.12%MEX 2.19% 0.51% 0.11% 1.82% 0.78% 0.05%RUS 0.40% 0.16% 0.02% 0.67% 0.29% 0.03%TUR 1.26% 0.34% 0.01% 1.58% 0.22% 0.01%TWN 1.57% 0.55% 0.12% 1.97% 1.16% 0.07%USA 0.56% 0.23% 0.01% 1.14% 0.39% 0.01%Average 0.99% 0.35% 0.04% 1.59% 0.69% 0.04%
ers and consumers.
7.2 Quantifying the Degree of Policy Interdependencies
Next, we use our quantitative model to determine how sizable trade policy inter-dependencies are in practice. To this end, we start from the unconstrained optimalpolicy equilibrium calculated in the previous step, and sequentially introduce a setof new restrictions on Home’s policy space. Namely,
i. In the first sequence, we suppose export taxes are restricted in all industriesbut import taxes remain unrestricted.
ii. In the second sequence, we suppose export taxes remain restricted and im-port taxes are also restricted in a subset of industries.
In each sequence, we compute how the partial restriction affects the Home gov-ernment’s optimal choice with respect to unrestricted policies. We conduct ouranalysis twice: once where the United States (US) is treated as the Home country
43
and the rest of the world is treated as Foreign; and another where the EuropeanUnion (EU) is treated as the Home country.
The results displayed in Figure 1 indicate that, without any restrictions, theoptimal policy involves a uniform tariff and an export tax that varies primarilywith the industry-level trade elasticity.34 After export tax-cum-subsidies are fullyrestricted, it is optimal for both the US and EU to voluntary lower their importtaxes in all industries. More importantly, this partial restriction on export policylowers the US and EU’s effective rate of protection.
More interesting is perhaps the second sequence where import taxes are re-stricted in a subset of (high trade elasticity) industries. In this case, partial restric-tion lowers the optimal import tax in restricted industries from around 30% to 10%in both cases. From the perspective of Proposition 5, these tariff complementarityeffects are driven solely by a reduction in the wage-driven term, t. While cross-substitutability between industries can magnify the degree of tariff complemen-tarity, they are absent here due to the Cobb-Douglas assumption. So, our currentanalysis presents a lower bound on the degree of tariff complementarity. Finally,as illustrated in 2, introducing input-output linkages preserves and even magnifiesthe interdependence patterns highlighted above.
7.3 Optimal Policy with Multiple Countries
Finally, we analyze the case where the Home country can impose differential tradetaxes on multiple Foreign countries. While we do not have an explicit formula forthe optimal tax in the multi-country case, we have proven that optimal tariffs areuniform on exports from a given country. This result streamlines the computa-tional analysis quite significantly; but we still have to conduct a constrained globaloptimization to determine the optimal policy.
Our multi-country analysis features 15 countries: Australia, Brazil, China, In-donesia, India, Japan, Korea, Mexico, Russia, Taiwan, Turkey, the European Union,the United States, and an aggregate of the rest of the world. We conduct two sep-arate analyses. First, we treat the United States as the Home country and computeits optimal trade taxes with respect to the remaining 14 countries. Second, we treat
34When all instruments are available, the value of t is redundant. This, leads to an indeterminacyof optimal policy, which we handle by setting t = 0.
44
Figure 1: The interdependence of polices without input-output linkages
45
Figure 2: The interdependence of polices with input-output linkages
46
the European Union as the Home country.Figure 3 displays the multi-country optimal tax levels and compares them those
produced using the two-country benchmark—each dot in the graph correspondsto an optimal tax rate for one particular import/export partner.
Evidently, optimal trade taxes discriminate between various partners, but onlyto a modest degree. There is a simple intuition behind this result. Given the factualtrade levels, even the United Stated and the European Union are relatively smalleconomies compared to the rest of the world.35 As a result, their optimal policywith respect to other countries is determined primarily by the industry-level tradeelasticities that are the same across all exporters and importers. Another takeawayfrom this comparison is that perhaps the two-country model of optimal policy pro-vides a reasonable approximation for optimal policy in the multi-country setup.
8 Concluding Remarks
We provide a general equilibrium analysis of optimal trade policy and trade policyinterdependence. Our framework accommodates input-output linkages, specificfactors of production, cross-demand effects, and multiple countries. To conclude,we highlight some of the important implications of our results.
First, we identify (i) reduced-form demand elasticities and (ii) trade tax pass-throughs (net of wage effects), as the two sufficient statistics that characterize theoptimal trade tax schedule. Importantly, both of these statistics can be estimatedusing standard customs data and input-output matrixes. Relatedly, we find thatoptimal trade taxes are uniform across industries if their pass-through onto For-eign prices is zero, net of wage effects. Otherwise, optimal trade taxes vary withindustry-level demand and supply characteristics.
Second, we highlight the importance of policy interdependence (i.e., comple-mentarity or substitutability of various policy instruments) in analyzing tradeagreements. These interdependencies are especially important for the optimal se-quencing of trade liberalization. Our theory suggests that an optimal sequence
35Note that that in our setup, due to presence of product differentiation, even a small openeconomy possess export market power. So, optimal tariffs are significant even though they do notsignificantly discriminate between different suppliers. See Alvarez and Lucas (2007) for a similarargument.
47
Figure 3: Optimal Policy: 15-Country vs. 2-Country Model
48
should first restrict export taxes/subsidies, then import taxes in a subset of indus-tries, and finally all import taxes and non-revenue import barriers.
Third, we offer novel insights about the implications of IO linkages for optimaltrade policy. Most importantly, we show that input-output linkages present thegovernments with a new international externality that we called “extraterritorialtaxing power.” That is, by taxing re-exported intermediate goods, governmentscan tax transactions among foreign entities, which involve components producedand eventually consumed abroad. In line with this observation, we showed quan-titatively that accounting for IO linkages increases the unilateral gains from protec-tion substantially. The flip-side of this finding is that multilateral gains from tradeagreements could be much larger in the presence of IO linkages, which justifies aparticular attention to intermediate-input trade in multilateral trade negotiations.
Finally, our general equilibrium analysis of policy across multiple industriescould guide future empirical studies of trade policy. In standard empirical workson trade policy, researchers usually use comparative static results from a partial-equilibrium model to explain cross-industry variation in policies. Interpretingcomparative static results as cross-industry variation is less than satisfactory be-cause it ignores cross-industry linkages, which are quantitatively important in thedata. Our theory, by comparison, directly characterizes the cross-industry varia-tion in optimal trade policies, which could be used as a guide for future empiricalwork in this area.
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54
Appendix
A Optimal Trade Taxes in a Ricardian Economy
The optimal trade tax problem of the home country can be formulated as
max(t,x,τ;w)∈A
Wh (t, x, τ; w) ,
where the set of feasible wage-policy combinations, A, is given by a special case of D2,where pji,k = aji,kwj and Πi = 0 for all i and j∈ C. Since in the presence of revenue-raising taxes the optimal NRTB is zero, we drop τ when referencing a feasible wage-policy combination in our proof. That is, we express all equilibrium outcomes in terms ofa revenue-raising wage-policy combination, (t, x; w), with Foreign labor assigned as thenumeraire (i.e., w f = 1).
Step 1: Deriving the F.O.C. for Import Taxes. We can express the F.O.C. with respect tothe tariff in sector k as follows
dWh (t, x; w)
d (1 + tk)=
∂Vh (Yh, ph)
∂Yh
[∂Yh
∂ (1 + tk)+
∂Yh∂wh
dwhd (1 + tk)
]+
∑g
∑j= f ,h
(∂Vh (Yh, ph)
∂ pjh,g
[∂ pjh,g
∂ (1 + tk)+
∂ pjh,g
∂wh
dwhd (1 + tk)
])= 0,
Noting (i) the zero cross-passthrough of taxes onto consumer prices, i.e.,∂ pji,g/∂ (1 + tk) = 0, if ji, g , f h, k, and (ii) the complete passthrough of taxesonto own prices, i.e., ∂ p f h,k/∂ (1 + tk) = 1; we can simplify the above condition as
dWh (t, x; w)
d (1 + tk)=
∂Vh∂Yh
∂Yh∂ (1 + tk)
+∂Vh/∂ p f h,k
∂Vh/∂Yh
∂ p f h,k
∂ (1 + tk)+
∂Vh∂wh∂Vh∂Yh
dwhd (1 + tk)
= 0
55
where ∂Vh∂wh
= ∂Yh∂wh
∂Vh∂Yh
+∑g ∑j=h, f∂Vh
∂ pjh,g
∂ pjh,g∂wh
. In the above equation, ∂Yh∂(1+tk)
can be expressedas:
∂Yh (t, x; w)
∂ (1 + tk)=
∂
∂ (1 + tk)
{wiLi + ∑
g
[tg p f h,gq f h,g + xg ph f ,gqh f ,g
]}(21)
=p f h,kq f h,k + ∑g
(tg p f h,g
∂q f h,g
∂ p f h,k
)∂ p f h,k
∂ (1 + tk)+ ∑
g
(tg p f h,g
∂q f h,g
∂Yh
)∂Yh
∂ (1 + tk).
Note that due to the Lerner symmetry we can set dwh/d (1 + tk) to zero to identify oneof the multiple optimal policy combinations. However, to thoroughly demonstrate thispoint and to also be in sync with subsequent proofs, we formally derive and substitutethis term. To this end, we apply the implicit function theorem to the balance trade condi-tion, Dh (t, x; w) ≡ ∑g
[(1 + xg
)ph f ,gqh f ,g − p f h,gq f h,g
]= 0, which yields the following:
dwhd (1 + tk)
=−∂Dh (t, x; w) /∂ ln (1 + tk)
∂Dh (t, x; w) /∂wh=−∑g
[p f h,g
∂q f h,g∂ p f h,k
∂ p f h,k∂(1+tk)
+ p f h,g∂q f h,g∂Yh
∂Yh∂(1+tk)
]∂Dh (t, x; w) /∂wh
.
Plugging the expressions for ∂Yh∂(1+tk)
and dwhd(1+tk)
back into the the F.O.C. implies the fol-lowing optimality condition:
dWh (t, x; w)
d (1 + tk)=
∂Vh∂Yh
{tk p f h,k
∂q f h,k
∂ p f h,k
∂ p f h,k
∂ (1 + tk)+ p f h,kq f h,k + ∑
g,k
(tg p f h,g
∂q f h,g
∂ p f h,k
)∂ p f h,k
∂ (1 + tk)
+∑g
(tg p f h,g
∂q f h,g
∂Yh
)∂Yh
∂ (1 + tk)+
∂Vh/∂ p f h,k
∂Vh/∂Yh
∂ p f h,k
∂ (1 + tk)
−∂Vh∂w / ∂Vh
∂Yh∂Dh∂w
[∑g
(p f h,g
∂q f h,g
∂ p f h,k
)∂ p f h,k
∂ (1 + tk)+ ∑
g
(p f h,g
∂q f h,g
∂Yh
)∂Yh
∂ (1 + tk)
] = 0
Applying Roy’s identity,(∂Vh/∂ p f h,k
)/ (∂Vh/∂Yh) = −q f h,k; defining
τ ≡(
∂Vh∂w / ∂Vh
∂Yh
)/ ∂Dh
∂w >0; and noting that ∂ ln p f h,k/∂ ln (1 + tk) = 1, we can furthersimplify the F.O.C. as
∑g
[(τ − tg
)p f h,gq f h,g
(∂ ln q f h,g
∂ ln Yh
∂ ln Yh∂ ln (1 + tk)
+∂ ln q f h,g
∂ ln p f h,k
)]= 0,
Recalling our Definition D1 that (i) ε f h,k ≡ ∂ ln q f h,k/∂ ln p f h,k, (ii) εf h,gf h,k ≡
∂ ln q f h,k/∂ ln p f h,g, and (iii) η f h,k ≡ ∂ ln q f h,k/∂ ln Yh, we can further simplify the F.O.C.
56
as a function of reduced-form elasticities,
∑g
[(1− 1 + τ
1 + tg
)(ε
f h,kf h,g + η f h,g
∂ ln Yh∂ ln (1 + tk)
)λ f h,g
]= 0, for all k.
The trivial solution to the above system of K first-order conditions is tk = τ for all k. Wecan also characterize conditions that ensure that the trivial solution, tk = τk, is the uniquesolution. To this end, define the K × K matrix B ≡
[λ f h,g
(ε
f h,kf h,g + η f h,g
∂ ln Yh∂ ln(1+tk)
)]k,g
, and
the K × 1 vector ω as ω ≡[1− 1+τ
1+tk
]k. The system of F.O.C.s can, thus, be expressed
as Bω = 0. For ω = 0 to be the unique (and trivial) solution to Bω = 0, it suffices thatdet B , 0.
The second-order condition for optimality is also satisfied provided that ε f h,k < 0,
εf h,gf h,k > 0, and η f h,g > 0. Specifically, under the aforementioned sign of demand elastici-
ties, one can easily verify that (i) if tk < τ | tg = τ, ∀g , k then,
∂Wh (t, x; w)
∂ (1 + tk)= ∑
g
((tg − τ
)p f h,gq f h,g
[ε
f h,kf h,g +
∂ ln Yh∂ ln (1 + tk)
η f h,g
])> 0
and (ii) if tk > τ | tg = τ, ∀g , k then
∂Wh (t, x; w)
∂ (1 + tk)= ∑
g
((tg − τ
)p f h,gq f h,g
[ε
f h,kf h,g +
∂ ln Yh∂ ln (1 + tk)
η f h,g
])< 0.
Hence, the solution tk = τ for all k, is also a welfare-maximizing solution to the F.O.C.Note that the optimal import tax is tk = τ for all k, irrespective of the applied export
taxes. When export taxes are available and set optimally, the value of τ is irrelevant tothe multiplicity of optimal tax schedules. However, when export taxes are unavailable orset sub-optimally, the exact value τ is relevant. We can derive τ, in these circumstances,along the following lines
τ =
∂Vh∂wh
/ ∂Vh∂Yh
∂Dh∂wh
=
∂Yh∂wh
+ ∑k
(∂Vh/∂ phh,k∂Vh/∂Yh
∂ phh,k∂wh
)∂(∑k p f hqh f ,k)
∂wh− ∂(∑k ph f qh f ,k)
∂wh
=
∂Yh∂wh−∑k
(qhh,k
∂ phh,k,∂wh
)∂(∑k p f hqh f ,k)
∂wh− ∂(∑k ph f qh f ,k)
∂wh
=
whLh + ∑k
(tk
∂(p f h,kq f h,k)∂ ln wh
+ xk∂(ph f ,kqh f ,k)
∂ ln wh
)−∑k phh,kqhh,k
∂(∑k p f hqh f ,k)∂wh
− ∂(∑k ph f qh f ,k)∂wh
,
where the second line follows for Roy’s identity. Noting the pji,kqji,k = wjLji,k (w; t, x)
57
and ∑k pji,kqji,k = wjLji (w; t, x) (by the definition of labor demand), the above expressioncan be reformulated as
τ =whLh f + τ
∂ ln w fL f h∂ ln wh
+ ∑k
(xkwhLh f ,k
∂ ln whLh f ,k∂ ln wh
)∂w fL f h
∂wh− ∂ ∑k(1+xk)whLh f ,k
∂wh
=1 + ∑g
[xg
rh f ,krh f
(1 + εh f ,k
)]−εh f − 1
,
where the last line follows from Definition D4 that
εh f ≡ ∂ lnLh f (w; t, x) /∂ ln wh
= ∑k
rh f ,k
rh fεh f ,k = ∑
k∑g
rh f ,k
rh fε
h f ,gh f ,k,
denotes the elasticity of Foreign’s demand for Home’s labor, with r f h,k ≡ p f h,kq f h,k/whLh
being the share of Home’s (non-tax) revenue generated from sales to Foreign in industryk. Rearranging the above equation, expresses the optimal import tax for any given vectorof export taxes/subsidies:
1 + t∗ =εh f −∑k
[xk
rh f ,krh f
(1 + εh f ,k
)]1 + εh f
. (22)
When export taxes are restricted, the above formula reduces to that specified by Proposi-tion 4: 1 + t∗ = εh f /(1 + εh f ).
Step 2: Deriving the F.O.C. for Export Taxes. Noting our notation for consumer pricesthat ph f ,k = (1 + xk) ph f ,k, the F.O.C. with respect to the export tax in sector k can beexpressed as follows
dWh (t, x; w)
d (1 + xk)=
∂Vh∂Yh
[∂Yh
∂ (1 + xk)+
∂Yh∂wh
dwhd (1 + xk)
]+∑
g∑
j= f ,h
[∂Vh
∂ pjh,g
∂ pjh,g
∂ (1 + xk)+
∂Vh∂ pjh,g
∂ pjh,g
∂wh
dwhd (1 + xk)
]= 0
in the above expression, (i)∂ pjh,g∂1+xk
= 0 for all g because the effect of export taxes on Home
prices are only through their effects on wages, and (ii)∂ p f h,g∂wh
= 0 for all g, by normal-ization of foreign wage to 1. Plugging these values in to the above equation, yields the
58
following simplified F.O.C.,
dWh (t, x; w)
d ln (1 + xk)=
∂Vh∂Yh
{∂Yh
∂ ln (1 + xk)+
(∂Vh∂wh
/∂Vh∂Yh
)dwh
d ln (1 + xk)
}= 0 (23)
where, as before, ∂Vh∂wh
= ∂Yh∂wh
∂Vh∂Yh
+ ∑g ∑j=h, f∂Vh
∂ pjh,g
∂ pjh,g∂wh
. The term ∂Yh/∂ ln 1 + xk in Equa-tion 23 can be calculated as,
∂Yh (t, x; w)
∂ ln (1 + xk)=
∂
∂ (1 + xk)
{wiLi + ∑
g
(tg p f h,gq f h,g + xg ph f ,gqh f ,g
)}
= ph f ,kqh f ,k + ∑g
(xg ph f ,gqh f ,g
∂ ln qh f ,g
∂ ln ph f ,k
)∂ ln ph f ,k
∂ ln (1 + xk)+
∑g
(tg p f h,gq f h,g
∂ ln q f h,g
∂ ln Yh
)∂ ln Yh
∂ ln (1 + xk). (24)
Also, as with the case of import taxes, an expression for dwh/d ln 1+ xk can be derived byapplying the implicit function theorem to the balance trade condition:
dwhd ln (1 + xk)
=∂Dh (t, x; w) /∂ ln (1 + xk)
∂Dh (t, x; w) /∂wh=
∂(
∑g p f h,gq f h,g − ph f ,gqh f ,g
)/∂ ln (1 + xk)
∂Dh (t, x; w) /∂wh
=∑g
(p f h,gq f h,g
∂ ln q f h,g∂ ln Yh
)∂ ln Yh
∂ ln(1+xk)− ph f ,kqh f ,k −∑g
(ph f ,gqh f ,g
∂ ln qh f ,g∂ ln ph f ,k
)∂ ln ph f ,k
∂ ln(1+xk)
∂Dh (t, x; w) /∂wh.
Replacing the expressions for dwh/d ln (1 + xk) and ∂Yh/∂ ln (1 + xk) into Equation 23;defining τ ≡
(∂Vh∂w / ∂Vh
∂Yh
)/ ∂Dh
∂w , as before; and Noting that ∂ ln ph f ,k/∂ ln (1 + xk) = 1, theF.O.C. reduces to
dWh (t, x; w)
d ln (1 + xk)=
∂Vh∂Yh
{ph f ,kqh f ,k + ∑
g
(xg ph f ,gqh f ,g
∂ ln qh f ,g
∂ ln ph f ,k
)+ ∑
g
(tg p f h,gq f h,g
∂ ln q f h,g
∂ ln Yh
)∂ ln Yh
∂ ln (1 + xk)−
τ
(∑g
(p f h,gq f h,g
∂ ln q f h,g
∂ ln Yh
)∂ ln Yh
∂ ln (1 + xk)− ph f ,kqh f ,k −∑
g
(ph f ,gqh f ,g
∂ ln qh f ,g
∂ ln ph f ,k
))}=
∂Vh∂Yh
{(1 + xk) (1 + τ) + ∑
g
([(1 + xg
)(1 + τ)− 1
] ph f ,gqh f ,g
ph f ,kqh f ,k
∂ ln qh f ,g
∂ ln ph f ,k
)−
∑g
([tg − τ
] p f h,gq f h,g
ph f ,kqh f ,k
∂ ln q f h,g
∂ ln Yh
)∂ ln Yh
∂ ln (1 + xk)
}ph f ,kqh f ,k = 0
59
Step 3: Solving for the Optimal Trade Tax Combination. Noting that the optimality ofimport tariffs entails that t∗g = τ ∀g, the F.O.C. for optimal export policy reduces to:
(1 + xk) (1 + τ) + ∑g
([(1 + xg
)(1 + τ)− 1
] ph f ,gqh f ,g
ph f ,kqh f ,k
∂ ln qh f ,g
∂ ln ph f ,g
)= 0
Recalling Definition D1 that (i) εh f ,k ≡ ∂ ln qh f ,k/∂ ln ph f ,k, (ii) εh f ,kh f ,g ≡ ∂ ln qh f ,g/∂ ln ph f ,k,
and noting that (iii)ph f ,gqh f ,gph f ,kqh f ,k
= 1+xk1+xg
λh f ,gλh f ,k
, the above condition can be written in terms oftrade shares and reduced-form elasticities as follows:
(1 + xk) (1 + τ)
{(1 + εh f ,k
)+ ∑
g,k
([1− 1(
1 + xg)(1 + τ)
]λh f ,g
λh f ,kε
h f ,kh f ,g
)}= εh f ,k. (25)
Defining
ξh f ,k ≡ ∑g,k
[1− 1(
1 + xg)(1 + τ)
]λh f ,g
λh f ,kε
h f ,kh f ,g,
Equation 25 yields the following formula for the optimal import and export taxes
(1 + x∗k ) (1 + τ) =εh f ,k
1 + εh f ,k + ξh f ,k. (26)
The term ξh f ,k accounts for cross-price elasticity effects. To see this, note that if cross-price
elasticities are zero (i.e., εh f ,kh f ,g = 0 for all g , k), then ξh f ,k = 0. In order to calculate ξh f ,k
based on cross-price elasticities, we can rewrite the equation expressing ξh f ,k as follows:
ξh f ,k = −∑g,k
[ξh f ,g + 1
] λh f ,g
λh f ,k
εh f ,kh f ,g
εh f ,k.
The vector[ξh f ,k
]k, therefore, solves ∑g
(ξh f ,k + 1
)λh f ,gε
h f ,kh f ,g/λh f ,kεh f ,g = 1; namely,
[ξh f ,k
]k,1 =
[Ξ−1 − IK
]1K,
where Ξ ≡[λh f ,gε
h f ,kh f ,g/λh f ,kεh f ,k
]k,g
is a K × K matrix and 1 ≡ [1]k is a K × 1 vector.
Since ∑g λh f ,gεh f ,kh f ,g = −λh f ,k − ∑g λhh,gε
h f ,khh,g < 0, then Ξ is strictly diagonally dominant.
Therefore, Ξ−1 exists and is monotone, which ensures that ξh f ,k + 1 > 0 (Berman andPlemmons (1994)). That is to say, with zero import tariffs an export subsidy is neveroptimal.
60
A.1 The DFS Case
We now show how to derive the optimal tax scheme for the DFS model as a special caseof our optimal tax formula. To this end, first note that cross-price elasticities are zero inthe DFS model due to the assumption that the expenditure share of each is infinitesimal.Therefore, the optimal trade taxes in the DFS model are given by
(1 + x∗k ) (1 + t) =εh f ,k
1 + εh f ,k.
Assuming CES preferences with elasticity of substitution σ, the trade elasticity at an in-terior solution will be given by εh f ,k = −σ, and the optimal tariff, at an interior solution,will be given by σ
σ−1 . An interior solution is obtained if and only if
1 ≥a f f ,kw f
ah f ,kwh≥ σ
σ− 1,
that is, iff the mark-up that is induced by the tariff is not larger than the ratio of foreign toHome cost of producing goods k. At a corner solution, i.e., for
a f f ,kw fah f ,kwh
< σσ−1 , the optimal
markup takes a limit-pricing form, i.e., x∗k =a f f ,kw fah f ,kwh
. We can also establish this claim moreformally, by deriving the optimal monopoly markup (in the limit) as industries becomehomogeneous. Namely, by showing that
1 + x∗k = limεk→∞
εh f ,k
1 + εh f ,k= lim
εk→∞1 +
1εkλ f f ,k + (σ− 1) λ f f ,k
. (27)
To elaborate, our claim is that based on the above equation, if 1 ≥ ah f ,kwha f f ,kw f
≥ σ−1σ , then
1 + x∗k =a f f ,kw f
ah f ,kwh.
Noting that limκ→0 κ ln aκ = 0, to establish the above claim, it suffices to show that 1 +
x∗k =a f f ,kw fah f ,kwh
[1− 1
εkln akεk
]is a solution to Equation 27, when ak ≡
a f f ,kw fah f ,kwh
−1
1−(σ−1)a f f ,kw fah f ,kwh
. To
establish this claim, notice that conditional trade shares in industry/good k are given by
λ f f ,k =
(a f f ,kw f
)−εk(a f f ,kw f
)−εk +([1 + xk] ah f ,kwh
)−εk=
((1+xk)ah f ,kw
a f f ,kw f
)εk
1 +((1+xk)ah f ,kw
a f f ,kw f
)εk.
61
Plugging the above formula and our guess for the export tax, 1 + x∗k =a f f ,kw fah f ,kwh
[1− 1
εkln akεk
], into Equation 27 yields the following
limεk→∞
1 +1
(εk − σ + 1) λ f f ,k + σ− 1= 1 + lim
εk→∞
1
εk
((1+xk)ah f ,kw
a f f ,kw f
)εk+ σ− 1
= 1 +1
limεk→∞ εk
[1− 1
εkln akεk
]εk= 1 +
11ak+ (σ− 1)
=a f f ,kw f
ah f ,kwh= lim
εk→∞1 + x∗k ,
where the second line uses the fact that limκ→∞ κ(
1− ln aκκ
)κ= 1a . That is, 1 + x∗k =
limεk→∞a f f ,kw fah f ,kwh
[1− 1
εkln akεk
], is the solution implied by Equation 27. Correspondingly,
ifah f ,kwha f f ,kw f
< σ−1σ , then 1 + x∗k = σ
σ−1 is the implied solution of Equation 27, given thatλ f f ,k(x∗k ) = 0 and λh f ,k(x∗k ) = 1.
B Constrained Optimal Policy & Interdependence
In this Section we characterize the optimal policy when trade taxes are restricted in a sub-set of industries. Here, we analyze both export and import taxes. Our results concerningthe optimal import tax is used to derive Proposition 5.
Optimal Import Tax when a Subset of Industries are Restricted. Recall from AppendixA that the F.O.C. for the tariff in industry k is given by
dWh (t, x; w)
d ln (1 + tk)= p f h,kq f h,k
∂ ln p f h,k
∂ ln (1 + tk)+[
1− τ ∑g
( p f h,gq f h,g
Yh
∂ ln q f h,g
∂ ln Yh
)]Yh
∂ ln Yh∂ ln (1 + tk)
− τ ∑g
(p f h,gq f h,g
∂ ln q f h,g
∂ ln p f h,k
)∂ ln p f h,k
∂ ln (1 + tk)= 0.
where, as before, τ≡(
∂Vh∂wh
/ ∂Vh∂Yh
)/ ∂Dh
∂wh. Applying the implicit function theorem to Yh =
whLh + ∑g(tg p f h,gq f h,g + xg ph f ,gqh f ,g
), we will have
Yh∂ ln Yh
∂ ln (1 + tk)=
p f h,gq f h,g + ∑g
(tg p f h,gq f h,g
∂ ln q f h,g∂ ln p f h,k
)∂ ln p f h,k
∂ ln(1+tk)
1−∑g
(tg
p f h,gq f h,gYh
∂ ln q f h,g∂ ln Yh
) .
62
Considering the above equation and to account for income effects, we can define
Υ ≡1−∑g
(tg
p f h,gq f h,gYh
∂ ln q f h,g∂ ln Yh
)1−∑g
(τ
p f h,gq f h,gYh
∂ ln q f h,g∂ ln Yh
) =1−∑g
(tg
1+tgλ f h,gη f h,g
)1−∑g
(τ
1+tgλ f h,gη f h,g
) ,
which allows us to further simplify the F.O.C. as
(1− Υ) (1 + tk) p f h,kq f h,k + ∑g
[(tg − Υτ
)p f h,gq f h,gε
f h,kf h,g
]= 0.
Rearranging the above expression implies the following formula for optimal tariff in in-dustry k as a function of applied tariffs in other industries:
1 + t∗k = (1 + Υτ)
(1 +
1− Υε f h,k
)−11 + ∑
g,k
( 1 + tg
1 + Υτ− 1) λ f h,gε
f h,kf h,g
λ f h,kε f h,k
.
Note that when traded industries exhibit a zero income elasticity, which is the case in ouranalysis in Section 6, then Υ = 1 and the above equation reduces to 17.
Optimal Export Tax when a Subset of Industries are Restricted.. Correspondingly, asshown in Appendix A, the F.O.C. for the export tax in industry k implies
dWh (t, x; w)
d ln (1 + xk)=
[1− τ ∑
g
( p f h,gq f h,g
Yh
∂ ln q f h,g
∂ ln Yh
)]Yh
∂ ln Yh∂ ln (1 + xk)
− τ ∑g
(ph f ,gqh f ,g
∂ ln qh f ,g
∂ ln ph f ,k
)∂ ln ph f ,k
∂ ln (1 + xk)= 0.
Defining Υ and τ as before, the above expression can be stated as[ph f ,kqh f ,k + ∑
g
(xg ph f ,gqh f ,g
∂ ln qh f ,g
∂ ln ph f ,k
)∂ ln ph f ,k
∂ ln (1 + xk)
]1Υ−
τ
(− ph f ,kqh f ,k −∑
g
(ph f ,gqh f ,g
∂ ln qh f ,g
∂ ln ph f ,k
)∂ ln ph f ,k
∂ ln (1 + xk)
)= 0.
Rearranging the above equation yields
(1 + xk) (1 + Υτ)(1 + εh f ,k
)1 + ∑g,k
[1− 1(1 + xg
)(1 + Υτ)
]λh f ,g
λh f ,k
εh f ,kh f ,g
1 + εh f ,k
= εh f ,k,
63
which, in turn, implies the following formula for optimal export tax in industry k as afunction of applied taxes in other industries:
(1 + xk) (1 + Υτ) =εh f ,k
1 + εh f ,k
1 + ∑g,k
[1− 1(1 + xg
)(1 + Υτ)
]λh f ,g
λh f ,k
εh f ,kh f ,g
1 + εh f ,k
−1
.
Solving for 1 + τ. To finalize the characterization of the restricted optimal taxes, wealso need to characterize τ ≡
(∂Vh∂wh
/ ∂Vh∂Yh
)/ ∂Dh
∂wh. To this end, we can follow the same steps
presented in Appendix A. That is, defining X ≡ pq, τ can be expressed as
1 + τ = 1+whLh
∂ ln(whLh)∂ ln wh
+ ∑k
(tk p f h,kq f h,k
∂ ln(p f h,kq f h,k)∂ ln wh
+ xk ph f ,kqh f ,k∂ ln(ph f ,kqh f ,k)
∂ ln wh− ∂Vh/∂ phh,k
∂Vh/∂Yhphh,k
∂ phh,k∂ ln wh
)∑k(
p f h,kq f h,k) ∂ ln ∑k(p f h,kq f h,k)
∂ ln wh−∑k
(ph f ,kqh f ,k
) ∂ ln ∑k( ph f ,kqh f ,k)∂ ln wh
=1 +∑k(
ph f ,kqh f ,k)+ ∑k
(tk p f h,kq f h,k
∂ ln(p f h,kq f h,k)∂ ln wh
+[ph f ,k − ph f ,k
]qh f ,k
∂ ln(ph f ,kqh f ,k)∂ ln wh
)∑k(
p f h,kq f h,k) ∂ ln ∑k(p f h,kq f h,k)
∂ ln wh−∑k
(ph f ,kqh f ,k
) ∂ ln ∑k( ph f ,kqh f ,k)∂ ln wh
=∑k
[(1 + tk)
r f h,kr f h
∂ ln(p f h,kq f h,k)∂ ln wh
]− ∑g(ph f ,gqh f ,g)
∑g( ph f ,gqh f ,g)∑k
[rh f ,krh f
(∂ ln(ph f ,kqh f ,k)
∂ ln wh− 1)]
∂ ln ∑k(p f h,kq f h,k)∂ ln wh
− ∂ ln ∑k( ph f ,kqh f ,k)∂ ln wh
=∑k
[(1 + tk)
r f h,kr f h
∂ lnL f h,k(w)
∂ ln wh
]− (1 + x)−1 ∑k
[rh f ,krh f
∂ lnLh f ,k(w)
∂ ln wh
]∂ lnL f h(w)
∂ ln wh− ∂ ln(1+x)−1Lh f (w)
∂ ln wh
=(1 + t) ε f h + (1 + x)−1 εh f
1 + εh f + ε f h.
where (i) 1 + x ≡ ∑g( ph f ,kqh f ,g)∑g(ph f ,gqh f ,g)
= ∑k
[r f h,kr f h
(1 + xk)], (ii) 1 + t ≡ ∑k
[(1 + tk)
r f h,kr f h
ε f h,kε f h
], and
(vi) ε ji,k ≡ ∑g εji,gji,k and ε ji ≡ ∑g
rji,grji
ε ji,g denote the elasticity of labor demand per D4.36
Also, note that in deriving the above expression we use the fact that∂ ln(ph f ,kqh f ,k)
∂ ln wh= 1 +
∂ lnLh f ,k(w)
∂ ln wh, as well as the fact that (absent of income effects)
∂ lnL f h,k(w)
∂ ln wh= − ∂ lnL f h,k(w)
∂ ln w f=
−ε f h,kFinally, note that when export taxes are zero (which is case in our analysis in Section6), x = 0 and the formula for τ reduces to that presented in the text—albeit in the maintext we used t instead of τ to label the uniform term.
36The above equation implicitly assumes that ∂ (1 + x) /∂wh ≈ 0.
64
C Optimal NRTBs
The optimal NRTB problem of the home country can be formulated as
max(0,0,τ;w)∈A
Wh (0, 0, τ; w) .
To tighten the notation, we henceforth use Wh (τ; w) ≡ Wh (0, 0, τ; w) to denote welfarearising from the imposition of NRTBs when revenue-raising taxes are restricted. TheF.O.C. with respect to the NRTB in industry k can be stated as
dWh (τ; w)
d (1 + τk)=
∂Vh∂Yh
[∂Yh
∂ (1 + τk)+
∂Yh∂wh
dwhd (1 + τk)
]+∑
g∑
j= f ,h
[∂Vh
∂ pjh,g
∂ pjh,g
∂ (1 + τk)+
∂Vh∂ pjh,g
∂ pjh,g
∂wh
dwhd (1 + τk)
]= 0.
Noting that (i) ∂Yh/∂ (1 + τk) = 0, (ii) ∂ phh,g/∂ (1 + τk) = 0 for all g, (iii)∂ p f h,g/∂ (1 + τk) = 0 if g , k or, and (iv) applying the implicit function theorem to derivedwh/d (1 + τk), the F.O.C. can be written as:
dWh (τ; w)
d ln (1 + τk)=
∂Vh∂Yh
∂Vh/∂ p f h,k
∂Vh/∂Yh
∂ p f h,k
∂ ln (1 + τk)−
∂Vh∂w / ∂Vh
∂Yh∂Dh(τ;w)
∂w
(p f h,q
∂q f h,k
∂ p f h,k
∂ p f h,k
∂ ln (1 + τk)
) = 0,
where, as before, ∂Vh∂wh
= ∂Yh∂wh
∂Vh∂Yh
+ ∑g ∑j=h, f
(∂Vh
∂ pjh,g
∂ pjh,g∂wh
). The above condition can be sim-
plified as follows:
dWh (τ; w)
d ln (1 + τk)= −∂Vh
∂Yhp f h,kq f h,k
(1 + τε f h,k
)= 0.
65
Given its definition, τ can be calculated as
τ =
∂Vh∂wh
/ ∂Vh∂Yh
∂Dh(τ;w)∂wh
=
∂Yh∂wh
+ ∑k
(∂Vh/∂phh,k∂Vh/∂Yh
∂phh,k∂wh
)∂(∑k p f h,kq f h,k)
∂wh− ∂(∑k ph f ,kqh f ,k)
∂wh
=
∂Yh∂wh−∑k
(qhh,k
∂phh,k,∂wh
)∂(∑k p f h,kq f h,k)
∂wh− ∂(∑k ph f ,kqh f ,k)
∂wh
=Yh
∂ ln Yh∂ ln wh
−∑k
(phh,kqhh,k
∂ ln phh,k∂ ln wh
)∑k(
p f h,kq f h,k) ∂(∑k p f h,kq f h,k)
∂wh−∑k
(ph f ,kqh f ,k
) ∂(∑k ph f ,kqh f ,k)∂wh
=whLh −∑k phh,kqhh,k(
ph f ,kqh f ,k) ( ∂(∑k p f h,kq f h,k)
∂wh− ∂(∑k ph f ,kqh f ,k)
∂wh
) =
(ph f ,kqh f ,k
)(
ph f ,kqh f ,k) ( ∂(∑k p f h,kq f h,k)
∂wh− ∂(∑k ph f ,kqh f ,k)
∂wh
)=
1∂L f h(w;τ)
∂wh− ∂Lh f (w;τ)
∂wh
= − 11 + ε f h + εh f
,
with the last line following from our earlier derivation that (per D4) ∂L f h (w; τ) /∂wh =
−ε f h and ∂ lnLh f (w; τ) /∂ ln wh = 1 + εh f . Plugging τ from above expression back intothe F.O.C., implies the following:
∂Wh∂ ln(1+τk)
> 0 if ε f h,k < 1 + εh f + ε f h∂Wh
∂ ln(1+τk)< 0 if ε f h,k > 1 + εh f + ε f h
Note the imposition of τk reduces q f h,k. So if demand is super-convex, i.e., ∂ε f h,k/∂q f h,k >
0, then the above conditions imply following formula for optimal NRTBs:
τk =
∞ if ε f h,k < 1 + εh f + ε f h
0 if ε f h,k > 1 + εh f + ε f h
.
The condition that ∂ε f h,k/∂q f h,k > 0 is widely-known as Marshall’s Second Law of De-mand, and is satisfied in an important class of trade models.
66
D Optimal Trade Taxes with Multiple Foreign Countries
Country i’s optimal trade tax solves the following problem
maxti,xi
Vi(Yi, pi)
s.t.
wjLj = ∑k∈K ∑`∈C
(pj`,kqj`,k
)∀j ∈ C
Yj = wjLj + ∑k∈K ∑`∈C
(t`j,k p`j,kq`j,k + xj`,k pj`,kqj`,k
)∀j ∈ C
pj`,k =(1 + xj`,k
) (1 + tj`,k
)pj`,k ∀j, ` ∈ C
pj`,k = aj`,kwj ∀j, ` ∈ C
t`j,k = xj`,k = 0 ∀j , i
Below, we show that the solution to the above problem features a uniform import tax oneach supplier. Then, we solve a more restrictive case of the problem to show that MFNtariffs are non-uniform.
Uniformity of Import Taxes [Proposition 1]. Note that we want to solve for countryi’s optimal import tax schedule. As in the baseline model, the welfare in country i,can be expressed as Wi = ∂Vi (Yi, pi), where Yi = wiLi + ∑k
(tji,k pji,kqji,k + xij,k pij,kqij,k
).
Correspondingly, Wi is uniquely determined by the vector of import and export taxes,ti = {tji,k} and xi = {xij,k}, plus the vector of country-level wages, w = {wj}:
Wi (ti, xi; w) = Vi (Yi (ti, xi; w) , pi (ti, xi; w))
Defining Di (ti, xi; w) = wiLi − ∑k ∑` (pi`,kqi`,k), the equilibrium vector of aggregatewages, w, solves the following system of equations:
D1 (t1, x1; w) = 0...
DN (tN, xN; w) = 0
(28)
67
Keeping the above observation in mind, we can write the F.O.C. with respect to tji,k as
dWi (ti, xi; w)
d(1 + tji,k
) =∂Vi (Yi, pi)
∂Yi
[∂Yi
∂(1 + tji,k
) + ∂Yi
∂wi
dwi
d(1 + tji,k
)]
+ ∑g∈K
∑`∈C
(∂Vi (Yi, pi)
∂ p`i,g
[∂ p`i,g
∂(1 + tji,k
) + ∂ p`i,g
∂w`
dw`
d(1 + tji,k
)])
=∂Vi (Yi, pi)
∂Yi
{∂Yi
∂(1 + tji,k
) + ∂Vi (Yi, pi) /∂ pji,k
∂Vi (Yi, pi) /∂Yi
∂ pji,g
∂(1 + tji,k
) + ∑`
(∂Wi (.)
∂w`
dw`
d(1 + tji,k
))} = 0
Invoking Roy’s identity,∂Vi(Yi,pi)/∂ pji,k∂Vi(Yi,pi)/∂Yi
= qji,k, and noting that
∂Yi
∂ ln(1 + tji,k
) = pji,kqji,k + ∑`∈C
∑g∈K
[t`i,k p`i,kq`i,k
(ε
ji,k`i,g + η`i,g
∂ ln Yi
∂ ln(1 + tji,k
))]
Plugging the above equation back into the F.O.C. and defining ∆ji,k`i,g ≡ ε
ji,k`i,g +
η`i,g∂ ln Yi
∂ ln(1+tji,k), will yield the following optimality condition
dWi (ti, xi; w)
d ln(1 + tji,k
) =∂Vi (Yi, pi)
∂Yi
[pji,kqji,k − pji,kqji,k + ∑
`∈C
∑g∈K
(t`i,k p`i,kq`i,k∆ji,k
`i,g
)− ∑
`∈C
(νi`
d ln w`
d ln(1 + tji,k
))] ,
where νi` ≡ ∂Wi/∂ ln w`. Applying the implicit function theorem to the System of Equa-tions 28, we can solve for d ln w/d ln 1 + ti as follows:
d ln w1d ln(1+t1i,k)
· · · d ln w1∂ ln(1+tNi,k)
... . . . ...d ln wN
d ln(1+t1i,k)· · · d ln wN
d ln(1+tNi,k)
=
(∂ ln D∂ ln w
)−1
∂ ln D1
∂ ln(1+t1i,k)· · · ∂ ln D1
∂ ln(1+tNi,k)... . . . ...
∂ ln DN∂ ln(1+t1i,k)
· · · ∂ ln DN∂ ln(1+tNi,k)
,
Letting τ`i denotes element `i of matrix (∂ ln D/∂ ln w)−1, the above system implies thatfor every ` ∈ C
d ln w`
d ln(1 + tji,k
) = ∑`∈C
(τ`n ∑
g∈K
(pni,gqni,g∆ji,k
ni,g
)).
Plugging the above expression back into the F.O.C. implies the following
∑n∈C
∑g∈K
(tni,g pni,gqni,g∆ji,k
ni,g
)− ∑
`∈C
(ν` ∑
n∈C
(τ`n ∑
g∈K
pni,gqni,g∆ji,kni,g
))= 0.
68
The above expression can in turn be rearranged as
∑n∈C
∑g∈K
[(tni,g − ∑
`∈Cτ`nνi`
)pni,gqni,g∆ji,k
ni,g
]= ∑
n∈C
∑g∈K
[(tni,g − τni
)pni,gqni,g∆ji,k
ni,g
]= 0,
where τni ≡ ∑`∈C τ`nνi`, or, equivalently
T i∆i = 0,
where T i =[tni,g − τni
]ng∈C×K
and ∆i =[
pni,gqni,g∆ji,kni,g
]ng,jk∈C×K
are respectively 1× N ·K and N · K× N · K matrixes. If det ∆i , 0, then T i = 0 is the unique solution to the abovesystem, which implies that the optimal import tax is uniform across products originatingfrom the same exporting country:
t∗ji,k = τji, ∀j, k
MFN Tariffs. No suppose the country i is bound by the MFN clause, whereby it hasto impose the same tariff ti,k on industry k irrespective of origin country. In that case,following the same steps as above the F.O.C. can be stated as
∑n∈C
(∑
g∈K
ti,g ∑j
(pji,gqji,g∆ni,k
ji,g
))− ∑
n∈C
(τni ∑
g∈K
∑j
(pji,gqji,g∆ni,k
ji,g
))= 0.
The above equation that unless for all k and k′ ∈ K,
∑j,n∈C ∑g∈K
(τni pji,gqji,g∆ni,k
ji,g
)∑j,n∈C ∑g∈K
(τni pji,gqji,g∆ni,k
ji,g
) =∑j,n∈C ∑g∈K
(τni pji,gqji,g∆ni,k′
ji,g
)∑j,n∈C ∑g∈K
(τni pji,gqji,g∆ni,k′
ji,g
) ,
The optimal MFN tariff is non-uniform across industries. That is, unless industries aresymmetric the optimal tariffs is non-uniform.
69
E Proof of Theorem 2
The optimal trade tax problem of the home country can be formulated as
max(t,x,τ;w)∈A
Wh (t, x, τ; w) ,
where the set of feasible wage-policy combinations, A, is given by D2. Trivially, whenrevenue-raising taxes are available, the optimal NRTB is zero, i.e., τ = 0. So, hereafter wecan restrict our attention to characterizing the optimal revenue-raising taxes, t∗ and x∗.
To handle the complex nature of the above problem, we proceed in several steps. First,we characterize the optimal policy as a function of the general equilibrium trade tax pass-throughs and reduced-form demand elasticities. Note that the pass-through of taxes ontoconsumers were always equal to “one” in the Ricardian mode, os this first step was pre-viously irrelevant. To fix minds, we define the pass-through of taxes (net of wage effects)as follows:
σι,gji,k ≡
∂ ln pji,k (t, x; w)
∂ ln(1 + tι,g
) =∂ ln pji,k (t, x; w)
∂ ln(1 + xι,g
) .
To elaborate, σι,gji,k denotes the pass-through of a tax on good ι, g to the consumer price of
good ji, k, net of wage-driven effects.Second, to handle the effects of policy on “aggregate” variables (e.g., wage and in-
come), we invoke the Lerner symmetry. Specifically, we first identify a solution to theabove problem for which the “aggregate” term corresponding to general equilibrium in-come effects drops out of the FOCs. Once this particular solution is identified, we canconstruct the remaining solutions by an across-the-board multiplication of the export andimport tax vectors.
Finally, as with the Ricardian case, we invoke a set of supply-side and demand-sideenvelop conditions to account for general equilibrium behavioral responses. It is this lat-ter steps that allows us to characterize the optimal tax schedule without imposing strongparametric assumptions on the supply or demand-sides of the economy.
70
Step 1: Deriving the F.O.C. for Import Taxes. The F.O.C. with respect to sector k’s tariffcan be expressed as
dWh (t, x, w)
d (1 + tk)=
∂Vh∂Yh
[∂Yh
∂ (1 + tk)+
∂Yh∂wh
dwhd (1 + tk)
]+
∑g
∑j= f ,h
(∂Vh
∂ pjh,g
[∂ pjh,g
∂ (1 + tk)+
∂ pjh,g
∂wh
dwhd (1 + tk)
])= 0,
It should be noted upfront that the difference between the present setup and the pureRicardian case, is that the (conditional) tariff pass-through σ
f h,kjh,g = ∂ ln pjh,g/∂ ln (1 + tk)
can be non-zero (even if g , k) due to (i) the upward sloping supply curve in industryg, plus (ii) the cross-substitutability between industry k and industry g goods. PluggingYh = whLh + Πh +RX
h +RMh , into the F.O.C. yields the following:
dWh (t, x, w)
d (1 + tk)=
∂Vh∂Yh
{∂Πh
∂ (1 + tk)+
∂RXh
∂ (1 + tk)+
∂RMh
∂ (1 + tk)
+ ∑g
∑j= f ,h
(∂Vh/∂ pjh,g
∂Vh/∂Yh
∂ pjh,g
∂ (1 + tk)
)+
∂Vh/∂wh∂Vh/∂Yh
dwhd (1 + tk)
}= 0,
where ∂Vh/∂wh = ∂Yh/∂wh + ∑g(∂Vh/∂ pjh,g
) (∂ pjh,g/∂wh
). The above F.O.C. is charac-
terized by five different elements that can be characterized as follows. First, The effect oftariffs on producer surplus, ∂Πh/∂1 + tk, can be expressed as
∂Πh (t, x, w)
∂ ln (1 + tk)= ∑
g∑
i=h, f
(∂Πhi,g
∂phi,g
∂phi,g
∂ ln (1 + tk)
)+ ∑
s∑g
∑j=h, f
(∂Πhi,g
∂ pIjh,s
∂ pIjh,s
∂ ln (1 + tk)
)
= ∑g
∑i=h, f
(phi,gqhi,g
∂phi,g
∂ ln (1 + tk)
)−∑
s∑
j=h, f
(pIjh,sq
Ijh,s
∂ pjh,s
∂ ln (1 + tk)
),
where the last line follows from Hotelling’s lemma that ∂Πji,g/∂pji,g = qji,k and
∑g ∂Πhi,g/∂pjh,g = qIjh,g. Second, the effect of import taxes on import tax revenues,
71
∂RM/∂ (1 + tk), can be expressed as
∂RM (t, x, w)
∂ ln (1 + tk)= ∑
g
∂tg p f h,gq f h,g
∂ ln (1 + tk)
=∑g
(tg p f h,gq f h,gη f h,g
) ∂ ln Yh∂ ln (1 + tk)
+ p f h,kq f h,k
+∑g
(tg p f h,gq f h,g
[∂ ln p f h,g
∂ ln (1 + tk)+ ∑
s∑
j
(∂ ln q f h,g
∂ ln pjh,s
∂ ln pjh,s
∂ ln (1 + tk)
)]).
Third, the effect of import taxes on export tax revenues, ∂RX/∂ (1 + tk), can be expressedas
∂RX (t, x, w)
∂ ln (1 + tk)= ∑
g
∂xg ph f ,gqh f ,g
∂ ln (1 + tk)
=∑g
(xg ph f ,gqh f ,g
[∂ ln ph f ,g
∂ ln (1 + tk)+ ∑
s∑
j
(∂ ln qh f ,g
∂ ln pj f ,s
∂ ln pj f ,s
∂ ln (1 + tk)
)]).
Fourth, the effect of taxes on the consumer prices can be simplified using Roy’s identity,∂Vi/∂ pji,g∂Vi/∂Yi
= qji,g, as follows
∑g
∑j= f ,h
(∂Vh/∂ pjh,g
∂Vh/∂Yh
∂ pjh,g
∂ ln (1 + tk)
)= −∑
g∑
j= f ,h
(pCjh,gqjh,g
∂ pjh,g
∂ ln (1 + tk)
)
Finally, the effect of tariffs on wages can be determined by applying theimplicit function theorem to the balanced trade condition, Dh (t, x, w) =
∑g(
p f h,gq f h,g −(1 + xg
)ph f ,gqh f ,g
). Doing so, implies dwh
d(1+tk)= − ∂Dh(.)
∂(1+tk)/ ∂Dh(.)
∂wh.
Hence, defining τ ≡(
∂Vh∂wh
/ ∂Vh∂Yh
)/ ∂Dh
∂wh,
∂Vh/∂wh∂Vh/∂Yh
dwhd ln (1 + tk)
= −τ
{∑g
(p f h,gq f h,gη f h,g
) ∂ ln Yh∂ ln (1 + tk)
+∑g
(p f h,gq f h,g
[∂ ln p f h,g
∂ ln (1 + tk)+ ∑
s∑
j
(∂ ln q f h,g
∂ ln pjh,s
∂ ln pjh,s
∂ ln (1 + tk)
)])
−∑g
(ph f ,gqh f ,g
[∂ ln ph f ,g
∂ ln (1 + tk)+ ∑
s∑
j
(∂ ln qh f ,g
∂ ln pj f ,s
∂ ln pj f ,s
∂ ln (1 + tk)
)])}
To simplify the above expressions, we can employ our notation for the tax pass-through, σ
f h,kji,g ≡ ∂ ln pji,g/∂ ln 1 + tk, and the Marshallian demand elasticity, ε
ι,gji,k ≡
72
∂ ln qji,k/∂ ln pι,g. We can also use the fact that the pass-through of taxes onto “producer”prices (pι,g = pι,g/
(1 + tι,g
) (1 + xι,g
)) are, by construction, described by the following:
∂ ln pι,g
∂ ln (1 + tk)=
σf h,kι,g ι, g , f h, k
σf h,kι,g − 1 ι, g = f h, k
.
Doing as stated above and noting that ∂Vh/∂Yh > 0 and qji,k = qIji,k + qCji,k, the F.O.C. canbe expressed as follows:
∑g
∑i=h, f
(phi,gqhi,gσ
f h,khi,g
)−∑
g∑
j
(pjh,gqjh,gσ
f h,kjh,g
)+∑
g
[xg ph f ,gqh f ,g
(σ
f h,kh f ,g + ∑
s∑
jε
j f ,sh f ,gσ
f h,kj f ,s
)]
+p f h,kq f h,k + ∑g
[tg p f h,gq f h,g
(σ
f h,kf h,g + ∑
s∑
jε
jh,sf h,gσ
f h,kjh,s
)]
−τ
(−p f h,kq f h,k + ∑
g
[p f h,gq f h,g
(σ
f h,kf h,g + ∑
s∑
jε
jh,sf h,gσ
f h,kjh,s
)])
+τ ∑g
(ph f ,gqh f ,g
[σ
f h,kh f ,g + ∑
s∑
jε
j f ,sh f ,gσ
f h,kj f ,s
])
+∑g
([tg − τ
]p f h,gq f h,gη f h,g
) ∂ ln Yh∂ ln (1 + tk)
= 0
Given the Lerner symmetry and the multiplicity of the optimal trade tax, there alwaysexists a solution to the above problem where ∑g
([tg − τ
]p f h,gq f h,gη f h,g
)= 0. Hence-
forth, we restrict our attention to solving for this particular solution. Once we do that, theremaining solutions can be identified with a basic multiplicative transformation of theimport and export tax vectors. Considering this and rearranging the above expression,yields the following F.O.C.
∑g
(ph f ,gqh f ,gσ
f h,kh f ,g − p f h,gq f h,gσ
f h,kf h,g
)+ (1 + τ) p f h,kq f h,k
+∑g
((tg − τ
)p f h,gq f h,g
[σ
f h,kf h,g + ∑
s∑
jε
jh,sf h,gσ
f h,kjh,s
])
+∑g
([(1 + τ)
(1 + xg
)− 1] (
σf h,kh f ,g + ∑
s∑
jε
j f ,sh f ,gσ
f h,kj f ,s
)ph f ,gqh f ,g
)= 0.
73
Finally, dividing the above expressions by 1 + τ and Yf = w f L f (noting that p f h,kq f h,k =
r f h,kYf and ph f ,gqh f ,g = λh f ,gYf ) leads us the following optimality condition:
∑g
[(1 + tg
1 + τ− 1)(
∑s
∑j
εjh,sf h,gσ
f h,kjh,s
)r f h,g
+
(1− 1
(1 + τ) (1 + xg)
)(∑
s∑
jε
j f ,sh f ,gσ
f h,kj f ,s
)λh f ,g
]= r f h,k −∑
g
(λh f ,gσ
f h,kh f ,g − r f h,gσ
f h,kf h,g
)(29)
Step 2. Deriving the F.O.C. for Export Taxes. The F.O.C. with respect to sector k’s exporttax can be stated as
dWh (t, x, w)
d (1 + xk)=
∂Vh∂Yh
[∂Yh
∂ (1 + xk)+
∂Yh∂wh
dwhd (1 + xk)
]+∑
g∑
j= f ,h
[∂Vh
∂ pjh,g
∂ pjh,g
∂ (1 + xk)+
∂Vh∂ pjh,g
∂ pjh,g
∂wh
dwhd (1 + xk)
]
Adopting our earlier definition for ∂Vh/∂wh and noting that Yh = whLh +Πh +RXh +RM
h ,the above condition can be reformulated as
dWh (t, x, w)
d (1 + xk)=
∂Vh∂Yh
{∂Πh
∂ (1 + xk)+
∂RXh
∂ (1 + xk)+
∂RMh
∂ (1 + xk)
+ ∑g
∑j= f ,h
(∂Vh/∂ pjh,g
∂Vh/∂Yh
∂ pjh,g
∂ (1 + xk)
)+
∂Vh/∂wh∂Vh/∂Yh
dwhd (1 + xk)
}= 0,
As before, the above condition is composed of five elements that can be expressed as fol-lows. First, The effect of export taxes on producer surplus, ∂Πh/∂1+ xk, can be expressedas
∂Πh (t, x, w)
∂ ln (1 + xk)= ∑
g∑
i=h, f
(∂Πhi,g
∂phi,g
∂phi,g
∂ ln (1 + xk)
)+ ∑
s∑g
∑j=h, f
(∂Πhi,g
∂ pIjh,s
∂ pIjh,s
∂ ln (1 + xk)
)
= ∑g
∑i=h, f
(phi,gqhi,g
∂phi,g
∂ ln (1 + xk)
)−∑
s∑
j=h, f
(pIjh,sq
Ijh,s
∂ pjh,s
∂ ln (1 + xk)
),
where the last line follows from Hotelling’s lemma that ∂Πji,g/∂pji,g = qji,k and
∑g ∂Πhi,g/∂pjh,g = qIjh,g. Second, the effect of export taxes on import tax revenues,
74
∂RM/∂1 + xk, can be expressed as
∂RM (t, x, w)
∂ ln (1 + xk)= ∑
g
∂tg p f h,gq f h,g
∂ ln (1 + xk)
=∑g
(tg p f h,gq f h,gη f h,g
) ∂ ln Yh∂ ln (1 + xk)
+∑g
(tg p f h,gq f h,g
[∂ ln p f h,g
∂ ln (1 + xk)+ ∑
s∑
j
(∂ ln q f h,g
∂ ln pjh,s
∂ ln pjh,s
∂ ln (1 + xk)
)])
Third, the effect of export taxes on export tax revenues, ∂RX/∂ (1 + xk), can be expressedas
∂RX (t, x, w)
∂ ln (1 + xk)= ∑
g
∂xg ph f ,gqh f ,g
∂ ln (1 + xk)
= ph f ,kqh f ,k + ∑g
(xg ph f ,gqh f ,g
[∂ ln ph f ,g
∂ ln (1 + xk)+ ∑
s∑
j
(∂ ln qh f ,g
∂ ln pj f ,s
∂ ln pj f ,s
∂ ln (1 + xk)
)])
Fourth, the effect of on the consumer prices can be simplified using Roy’s identity,∂Vi/∂ pji,g∂Vi/∂Yi
= qji,g, as follows
∑g
∑j= f ,h
(∂Vh/∂ pjh,g
∂Vh/∂Yh
∂ pjh,g
∂ ln (1 + xk)
)= ∑
g∑
j= f ,h
(pjh,gqjh,g
∂ pjh,g
∂ ln (1 + xk)
)
Finally, the effect of export taxes on wages can be determined by applying the im-plicit function theorem to the balanced trade condition, Dh = ∑g p f h,gq f h,g − ph f ,gqh f ,g.Doing so, implies dwh
d(1+xk)= − ∂Dh
∂(1+xk)/ ∂Dh
∂wh. Hence, adopting our earlier definition,
τ ≡(
∂Vh∂wh
/ ∂Vh∂Yh
)/ ∂Dh
∂wh,
∂Vh/∂wh∂Vh/∂Yh
dwhd ln (1 + xk)
= −τ
{∑g
(p f h,gq f h,gη f h,g
) ∂ ln Yh∂ ln (1 + xk)
+∑g
(p f h,gq f h,g
[∂ ln p f h,g
∂ ln (1 + xk)+ ∑
s∑
j
∂ ln q f h,g
∂ ln pjh,s
∂ ln pjh,s
∂ ln (1 + xk)
])
−∑g
(ph f ,gqh f ,g
[1 +
∂ ln ph f ,g
∂ ln (1 + xk)+ ∑
s∑
j
∂ ln qh f ,g
∂ ln pj f ,s
∂ ln pj f ,s
∂ ln (1 + xk)
])}
75
As before, we can simplify the above expressions by using our notation that σh f ,kji,g ≡
∂ ln pji,g/∂ ln (1 + xk) and ει,gji,k ≡ ∂ ln qji,k/∂ ln pι,g; plus the fact that
∂ ln pι,g
∂ ln (1 + xk)=
σh f ,kι,g ι, g , h f , k
σh f ,kι,g − 1 ι, g = h f , k
.
Doing so and noting that ∂Vh/∂Yh > 0, the F.O.C. can be stated as follows:
−∑g
∑j=h, f
(pjh,gqjh,gσ
h f ,kjh,g
)+
(∑g
∑i=h, f
phi,gqhi,gσh f ,khi,g
)− ph f ,kqh f ,k + ph f ,kqh f ,k
−xk ph f ,kqh f ,k + ∑g
(xg ph f ,gqh f ,g
[σ
h f ,kh f ,g + ∑
s∑
j=h, fε
j f ,sh f ,gσ
h f ,kj f ,s
])
+∑g
[tg p f h,gq f h,g
(σ
h f ,kf h,g + ∑
s∑
j=h, fε
jh,sf h,gσ
h f ,kjh,s
)]
−τ ∑g
(p f h,gq f h,g
[σ
h f ,kf h,g + ∑
s∑
j=h, fε
jh,sf h,gσ
h f ,kjh,s
])
+τ ∑g
(ph f ,gqh f ,g
[σ
h f ,kh f ,g + ∑
s∑
j=h, fε
j f ,sh f ,gσ
h f ,kj f ,s
])
+∑g
([tg − τ
]p f h,gq f h,gη f h,g
) ∂ ln Yh∂ ln (1 + xk)
= 0.
Recall that our aim is to initially identify the solution where ∑g([
tg − τ]
p f h,gq f h,gη f h,g)=
0. Considering this, the last term drops out and the above expression reduces to
∑g
(ph f ,gqh f ,gσ
h f ,kh f ,g − p f h,gq f h,gσ
h f ,kf h,g
)+∑
g
[(tg − τ
) (∑
s∑
j=h, fε
jh,sf h,gσ
h f ,kjh,s
)p f h,gq f h,g
]
+∑g
([(1 + xg) (1 + τ)− 1
] (∑
s∑
j=h, fε
j f ,sh f ,gσ
h f ,kj f ,s
)ph f ,gqh f ,g
)= 0.
76
Rearranging the above equation and dividing by Yf = w f L f and 1 + τ yields the follow-ing optimality condition:
−∑g
[(1 + tg
1 + τ− 1)(
∑s
∑j
εjh,sf h,gσ
h f ,kjh,s
)r f h,g
+
[(1
(1 + τ) (1 + xg)− 1)(
∑s
∑j
εj f ,sh f ,gσ
h f ,kj f ,s
)]λh f ,g
]= ∑
g
(σ
h f ,kh f ,gλh f ,g − σ
h f ,kf h,gr f h,g
)(30)
Step 3: Simultaneously Solving the System of F.O.C. As a final step, we simultaneouslysolve the system of F.O.C.s for all tax instruments. To simplify the process, we solve thesystem of F.O.C.s in terms of 1 + Tk ≡ (1 + tk) / (1 + τ) and 1 +Xk ≡ 1/ (1 + xk) (1 + τ).Noting that, by solving solving for Tk and Xk , we also automatically pin down the opti-mal tax schedule: 1 + tk = (1 + τ) (1 + Tk)
1 + xk = 1/ (1 + τ) (1 +Xk).
Given the definition for Tk, and Xk, Equation 29, which describes the set of FOCs w.r.t.import taxes, can be simplified as follows
∑g
(−Tgr f h,g ∑
s∈K
∑`∈C
ε`h,sf h,gσ
f h,k`h,s +Xgλh f ,g ∑
s∈K
∑`∈C
ε` f ,sh f ,gσ
f h,k` f ,s
)= r f h,k +∑
g
(σ
f h,kh f ,gλh f ,g − σ
f h,kf h,gr f h,g
)(31)
Similarly, Equation 30, which describes the set of FOCs w.r.t. export taxes, adopts thefollowing simplified expression
∑g
(Tgr f h,g ∑
s∈K
∑`∈C
ε`h,sf h,gσ
h f ,k`h,s −Xgλh f ,g ∑
s∈K
∑`∈C
ε` f ,sh f ,gσ
h f ,k` f ,s
)= −∑
g
(σ
h f ,kh f ,gλh f ,g − σ
h f ,kf h,gr f h,g
)(32)
The gains intuition about the above equations, note that
∑s∈K
∑`∈C
ε`ι,sι,gσ
ji,k`ι,s =
∂ ln qι,g (t, x; w, y)∂ ln
(1 + tji,k
) =∂ ln qι,g (t, x; w, y)
∂ ln(1 + xji,k
) .
So, the therm on the left-hand side of both the above equations can be viewed as the tradevolume loss (at current prices) from taxes.The right-hand sides of both equations, mean-while, correspond to the terms-of-trade gains from policy net of wage effects. More specif-ically, the right-hand side expression in Equation 31, corresponds to a trade-weighted
77
(tk-induced) change in the relative price of Home’s exports:
∂TOTh∂ ln (1 + tk)
≡∑g
(λh f ,g
∂ ln ph f
∂ ln (1 + tk)− r f h,g
∂ ln p f h
∂ ln (1 + tk)
)= rh f ,k + ∑
g
(σ
f h,kh f ,gλh f ,g − σ
f h,kf h,gr f h,g
)Likewise, the right-hand side expression in Equation 32, corresponds to a trade-weighted(xk-induced) change in the relative price of Home’s exports:
∂TOTh∂ ln (1 + xk)
≡∑g
(λh f ,g
∂ ln ph f
∂ ln (1 + xk)− r f h,g
∂ ln p f h
∂ ln (1 + xk)
)= λh f ,k −∑
g
(σ
h f ,kh f ,gλh f ,g − σ
h f ,kf h,gr f h,g
)Given the definitions for ∂TOTh/∂ ln (1 + tk) and ∂TOTh/∂ ln (1 + xk), we can write thesystem of F.O.C.s described by Equations 29 and 30 in matrix-form as follows[
−r f h ◦ ε f hσ f h λh f ◦ εh f σ f h
r f h ◦ ε f hσh f −λh f ◦ εh f σh f
] [TX
]=
[∇ln 1+tTOTh
−∇ln 1+xTOTh
], (33)
where εji =[ε
ι,gji,k
]k,ιg
is a K × 4K matrix of demand elasticities; σ ji =[σ
ji,kι,g
]ιg,k
is a
4K × K matrix of tax pass-throughs; while T ≡ [Tk]k, X ≡ [Xk]k, ∇ln 1+tTOTh ≡[∂ ln TOTh/∂ ln (1 + tk)]k, ∇ln 1+xTOTh ≡ [∂ ln TOTh/∂ ln (1 + xk)]k , and λh f ≡
[λh f ,k
]k
are K× 1 vectors.
Characterizing τ when Export Taxes are Restricted
To characterize the uniform component of the optimal import tax, τ, we follow same stepsconducted in Appendix A. Specifically, along the same line of arguments presented there,τ can be expressed
τ ≡∂Vh
∂ ln wh/ ∂Vh
∂Yh∂Dh(.)∂ ln wh
=
∂Yh∂ ln wh
−∑k
(qhh,k
∂ phh,k∂ ln wh
)∂whL f h(t;w)
∂ ln wh− ∂w fLh f (t;w)
∂ ln wh
,
where Lji (t; w) denotes the demand for country j’s labor in market i (i.e., wjLji =
∑k pji,kqji,k), and the last equality is derived using Roy’s identity that ∂Vh/∂phh,k∂Vh/∂Yh
= −qhh,k.To simplify the above expression, we note that (i) Yh = whLh + Πh + ∑k tk p f h,kq f h,k, (ii)
78
∂ ln phh,k∂ ln wh
=∂ ln ph f ,k∂ ln wh
= 1, as well as (iii) by Hoteling’s lemma,
∂Πh∂ ln wh
= ∑k
∑i
(phi,k
∂Πhi,k
∂phi,k
∂ ln phi,k
∂ ln wh+
∂Πhi,k
∂ ln wh
)= ∑
k∑
i(phi,kqhi,k)− whLh
Plugging these expressions back into the initial formula for τ, yields the following:
τ =∑k ∑i (phi,kqhi,k) + ∑k
(tk
∂p f h,kq f h,k∂ ln wh
)−∑k (phh,kqhh,k)
w fL f h∂ ln w fL f h
∂ ln wh− whLh f
∂ ln whLh f∂ ln wh
=whLh f + ∑k
(tk p f h,kq f h,k
∂ ln q f h,k∂ ln wh
)w fL f h
∂ ln w fL f h∂ ln wh
− whLh f∂ ln whLh f
∂ ln wh
=1−∑k
[(tk − τ) r f h,k ε f h,k
]−εh f − 1
where the last line follows from that fact that (i) whLh f = ∑k ∑i (phi,kqhi,k) −∑k (phh,kqhh,k); (ii) w fL f h = whLh f by the balanced trade condition; and (iii)∂ lnL f h,k(.)/∂ ln wh = −ε f h,k, and ε f h ≡ L f h(.)/∂ ln w f = ∑k r f h,k ε f h,k per D4. Giventhe above equation, we can thus express 1 + τ as
1 + τ =εh f + ∑k
[(tk − τ) r f h,k ε f h,k
]1 + εh f
,
which is the expression presented as Equation 18 in Section 6.2.
F Characterizing the Pass-Throughs
Below, we characterize the general equilibrium pass-through of a trade tax 1 + tji,k (or1 + xji,k) on to consumer prices, pi, in both countries. To clarify the notation, 1 + tji,k (or1 + xji,k) corresponds to an import (or export) tax on variety ji, k. In our analysis we areinterested in home’s import tax 1+ tk ≡ 1+ t f h,k and home’s export tax 1+ xk = 1+ xh f ,k.
F.1 Ricardian Model with IO Linkages
To determine ∂ ln pι,g/∂ ln(1 + tji,k) = ∂ ln pι,g/∂ ln(1 + xji,k) for any two varieties ι, gand ji, k, we can applying the implicit function theorem to the price function, F (t, x, p) ≡
79
{Fι,g (t, x, p)
}ιg,37 which (in the presence of IO linkages) is defined as follows:
Fι,g (t, x, p) ≡ ln pι,g − ln(1 + tι,g
) (1 + xι,g
)pι,g
(pIj , wj
)= 0, ∀, ι ∈ C; g ∈ K
where pi ≡{
pji,k}
j∈C,k∈Kis the 2K× 1 vector of all “consumer” prices in country i, while
t =[tji,k]
jik, x =[xji,k
]jik, and p =
[pji,k
]jik are 4K× 1 vectors that contain all tax and price
combinations in both the Home and Foreign economies (i = h and f ). Considering theabove, the pass-through elasticity, σ
ji,kι,g ≡ ∂ ln pι,g/∂ ln 1+ tji,k (which is a partial elasticity
conditional on the wage rate, wj), can be expressed as follows
σ ≡[
∂ ln pι,g
∂ ln(1 + tji,k
)]ιg,jik
=
[∂ ln pι,g
∂ ln(1 + xji,k
)]ιg,jik
=(∇ln pF
)−1∇1+tF =(∇ln pF
)−1 ,
where σ, ∇ln pF, and ∇1+tF are 4K× 4K matrixes, and the last line follows from the factthat ∇1+tF = I. To characterize
(∇ln pF
)−1, we can invoke Shepherd’s Lemma. That is,
∂ ln pι,g (.)∂ ln pIji,k
=pIji,kqι,g
ji,k
pι,gqι,g, ∀, j, i ∈ C; g, k ∈ K
where αji,kι,g ≡ pIji,kqι,g
ji,k /pι,gqι,g denotes the share of intermediate input ji, k in the totalproduction cost of output variety ι, g (with qι,g
ji,k denoting the amount of ji, k-type inputs
used in the production of ι, g). Note that, by construction, αji,kι,g = 0 if ι , j. Considering
the above equation, it is also straightforward to verify that
σ =(∇ln pF
)−1= (I − A)−1 ,
where A ≡[α
ji,kı,g
]ıg,jik
is the full 4K × 4K input-output matrix of world production. So,
altogether, the pass-through of tax 1 + tji,k (or 1 + xji,k) on to consumer prices, is fullycharacterized by the jik’th column of σ, which is itself a function of the elements of theinput-output matrix:
σ ji,k = (I − A)−1 1ji,k,
where 1ji,k is a 4K× 1 vector where the jik’th element is “one” and the remaining elementsare “zero.” Correspondingly, σ ji =
[σ ji,k
]k is a 4K × K matrix comprised of all K σ ji,k
37Note that F (.) is a 4K× 1 vector.
80
vectors. To apply Theorem 2, we specifically need (i) σ f h that describes the pass-throughof home’s import taxes, as well as (ii) σh f that describes the pass-through of home’sexport taxes.
Note that we, hereafter, assume that I − A satisfies the Hawkins–Simon conditionsand is non-singular. Moreover, note that note that since all α
ι,gji,k < 1 for all ι, g and ji, k,
we can write I − A as:
(I − A)−1 = I + A2 + A3 + A4 + ...
Given the above expression it is straightforward to show that if good ji, k, is a “final”good (so that the jik’th column of A is zero), then σ ji,k is the identity vector. That is, allelements of σ ji,k are zero expect for the jik’th element which is equal to one.
F.2 Generalized Specific-Factors Model
S done above, to determine ∂ ln pι,g/∂ ln(1 + tji,k) = ∂ ln pι,g/∂ ln(1 + xji,k) for any twovarieties ι, g and ji, k, we can applying the implicit function theorem to the price function,F (t, x, p) ≡
{Fι,g (t, x, p)
}ιg,38 which is defined as follows:
Fι,g (t, x, p) ≡ ln pι,g− ln(1 + tι,g
) (1 + xι,g
)aι,g
(qh,g (ph) , q f ,g
(p f))
wj = 0, ∀, ι ∈ C; g ∈ K
where pi ≡{
pji,k}
j∈C,k∈Kis the 2K× 1 vector of all “consumer” prices in country i, while
t =[tji,k]
jik, x =[xji,k
]jik, and p =
[pji,k
]jik are 4K× 1 vectors that contain all tax and price
combinations in both the Home and Foreign economies (i = h and f ). Considering theabove, the pass-through elasticity, σ
ji,kι,g ≡ ∂ ln pι,g/∂ ln 1+ tji,k (which is a partial elasticity
conditional on the wage rate, wj), can be expressed as follows
σ ≡[
∂ ln pι,g
∂ ln(1 + tji,k
)]ιg,jik
=
[∂ ln pι,g
∂ ln(1 + xji,k
)]ιg,jik
=(∇ln pF
)−1∇1+tF =(∇ln pF
)−1 ,
where σ, ∇ln pF, and ∇1+tF are 4K× 4K matrixes, and the last line follows from the factthat ∇1+tF = I. It is also straightforward to verify that
σ =(∇ln pF
)−1= (I − Σ)−1 ,
38Note that F (.) is a 4K× 1 vector.
81
where Σ =[γ
`,gι,g εn`,s
`,g
]ιg,n`s
. So, altogether, the pass-through of tax 1 + tji,k (or 1 + xji,k)
on to consumer prices, is fully characterized by the jik’th column of σ as as function ofsupply and demand elasticities. In particular,
σ ji,k = (I − Σ)−1 1ji,k,
Recall that the two cases we are interested in are (i) vector σ f h,k describing the pass-through of home’s import tax, 1 + tk ≡ 1 + t f h,k, and (ii) vector σh f ,k describing thepass-through of home’s export tax, 1 + xk ≡ 1 + xh f ,k, for every industry k.
F.2.1 Special Cases: Ricardo-Viner model with Zero Cross-Demand Effects
The Ricardo-Viner model is a special case where γji,k ≡ γji,kj f ,k = γ
ji,kjh,k for all j, i, and k. But,
in line with traditional trade policy literature, let us also assume that cross-price elas-ticities are zero (i.e., preference are additively separable and feature a non-traded quasi-linear sector à la Broda et al. (2008)). In this special case, the pass-though is characterizedby the following expression:
σji,kι,g =
0 j, g , , k
ε ji,kγji,k1−ε ji,kγji,k
j, g = , k; i , ι
11−ε ji,kγji,k
ji, g = ι, k
Given the expression for σji,kι,g , the terms-of-trade effects can be immediately calculated as
follows: TOTt,k = r f h,k
(1− σ
f h,kf h,k
)=
ε f h,kγ f h,k1−γ f h,kε f h,k
r f h,k
λh f ,k − TOTx,k = λh f ,kσh f ,kh f ,k = 1
1−γh f ,kεh f ,kλh f ,k
.
Finally, since εji,kι,g = 0 if i, k , ι, g, plugging the above expressions into Equation 33, yields
the following formulas for Tk and XkTk =[r f h,k
(1− σ
f h,kf h,k
)]/(
r f h,kε f h,kσf h,kf h,k
)= γ f h,k
Xk =(
λh f ,kσh f ,kh f ,k
)/(
λh f ,kε f h,kσh f ,kh f ,k
)= 1/εh f ,k
.
82
The above equation, in turn, leads us to the following familiar-looking optimal tax for-mula: 1 + tk = (1 + τ)
(1 + γ f h,k
)1 + xk = 1/ (1 + τ)
(1 + 1/εh f ,k
) .
In other words, Home’s optimal import tax is equal to the inverse of Foreign’s exportsupply elasticity, γ f h,k. The above formula clearly outlines the symmetry between exportand import taxes. In fact, in the special case where εh f ,k → ∞ (which is analog to theRicardian assumption of γ f h,k = 0, but imposed on the demand side), the export tax isuniform while the import tax is not.
G Quantitative Methodology with IO Linkages
In this appendix, we present an analog to Proposition 7, but in the presence of IO linkages.To be able to conduct the quantitative analysis, we impose the additional restrictions thatthe global IO matrix, A = [α
ι,gji,k ]jik,ιg , is invariant to policy. Considering this, the first step
is to use the global IO matrix from the WIOD to compute the pass-through matrix. This,can be simply done as follows:
σ = (I − A)−1
Then, we can appeal to the following proposition to solve for Home’s optimal import andexport tax levels as a solution to system of non-linear equations.
Proposition 8. Suppose the observed data is generated by a Ricardian model with input-outputlinkages and functional forms 19 and 20. Then, the optimal trade tax can be fully characterized by
83
solving the following system of equations
1 + t f h,k
1 + xh f ,k
=
1 + Tk
1/ (1 +Xk)
;
th f ,k
x f h,k
= 0 ∀k ∈ K
T
X
=
(r f h ◦ r f h)◦ a f h
f h
(λh f ◦ λh f
)◦ a f h
h f(r f h ◦ r f h
)◦ ah f
f h
(λh f ◦ λh f
)◦ ah f
h f
−1 b f h
λh f − bh f
ai
ji =[εkλii,kλii,kσ
ι,gii,k −
(1 + εkλii,kλii,k
)σ
ι,gji,k
]g,k
∀j, , i, ι ∈ C
bji =[∑g σ
ji,kh f ,gλh f ,gλh f ,g − σ
ji,kf h,gr f h,gr f h,g
]k
∀j, i ∈ C
λji,k =[wjLj
(1 + tji,k
) (1 + xji,k
)]−εkPεk
i,k ∀j, i ∈ C; ∀k ∈ K
rji,k = λji,kYi/[(
1 + xji,k) (
1 + tji,k)]
∀j, i ∈ C; ∀k ∈ K
wiLiwiLi = ∑k ∑j
[(1−∑,ι,g α
ι,gji,k
)λji,kλji,kYiYi/
(1 + xji,k
) (1 + tji,k
)]∀i ∈ C
YiYi = wiLiwiLi + ∑k ∑j
(tji,k
1+tji,kλji,kλji,kYiYi +
xij,k1+xij,k
λij,kλij,kYjYj
)∀i ∈ C
P−εki,k = ∑j
[wjLj ∑,ι,g
(1 + σ
ι,gji,k
[(1 + tι,g
) (1 + xι,g
)− 1])]−εk
λji,k ∀i ∈ C; ∀k ∈ K
in terms of {t f h,k}, {xh f ,k}, {λji,k}, {rji,k}, {Pi,k}, {wiLi}, and {Yi}, as a function of (i) taxpass-through, {σι,g
ji,k } that are fully determined by the global IO matrix, A = [αι,gji,k ]jik,ιg, (ii)
observed expenditure shares, {λji,k}; (iii) observed output shares, {λji,k};(iv) observed nationaloutput and income levels, wiLi and Yi; as well as (v) industry-level trade elasticities, [εk]k.
Note that one we solve for the optimal policy, e can immediately calculate the gainsfrom policy as Yi/ ∏k Pei,k
i,k .
H Additional Tables
84
Table 2: List of countries in quantitative analysis
Country name WIOD code Basic aggregationAustralia AUS AustraliaBrazil BRA BrazilCanada CAN CanadaChina CHN ChinaIndonesia IDN IndonesiaIndia IND IndiaJapan JPN JapanKorea KOR KoreaMexico MEX MexicoRussia RUS RussiaTurkey TUR TurkeyTaiwan TWN TaiwanUnited States USA United StatesAustria AUT
European Union
Belgium BELBulgaria BGRCyprus CYPCzech Republic CZEGermany DEUDenmark DNKSpain ESPFinland FINFrance FRAUnited Kingdom GBRGreece GRCHungary HUNIreland IRLItaly ITANetherlands NLDPoland POLPortugal PRTRomania ROMSlovakia SVKSlovenia SVNSweden SWEEstonia ESTLatvia LVALithuania LTULuximburg LUXMalta MLTRest of the World RoW Rest of the World
85
Table 3: List of industries in quantitative analysis
WIOD Sector Sector’s Description Trade Ealsticity(Caliendo-Parro)
1 Agriculture, Hunting, Forestry and Fishing 8.112 Mining and Quarrying 15.723 Food, Beverages and Tobacco 2.55
4 Textiles and Textile Products 5.56Leather and Footwear5 Wood and Products of Wood and Cork 10.836 Pulp, Paper, Paper , Printing and Publishing 9.077 Coke, Refined Petroleum and Nuclear Fuel 51.088 Chemicals and Chemical Products 4.759 Rubber and Plastics 1.66
10 Other Non-Metallic Mineral 2.7611 Basic Metals and Fabricated Metal 7.9912 Machinery, Nec 1.5213 Electrical and Optical Equipment 10.6014 Transport Equipment 0.3715 Manufacturing, Nec; Recycling 5.00
16
Electricity, Gas and Water Supply
100
ConstructionSale, Maintenance and Repair of Motor Vehicles andMotorcycles; Retail Sale of FuelWholesale Trade and Commission Trade, Except of MotorVehicles and MotorcyclesRetail Trade, Except of Motor Vehicles and Motorcycles; Repairof Household GoodsHotels and RestaurantsInland TransportWater TransportAir TransportOther Supporting and Auxiliary Transport Activities; Activitiesof Travel AgenciesPost and TelecommunicationsFinancial IntermediationReal Estate ActivitiesRenting of M&Eq and Other Business ActivitiesEducationHealth and Social WorkPublic Admin and Defence; Compulsory Social SecurityOther Community, Social and Personal ServicesPrivate Households with Employed Persons
86
