Comparative Advantage and Optimal Trade Policy

Transcription

1 Comparatve Advantage and Optmal Trade Polcy Arnaud Costnot MIT Dave Donaldson MIT Iván Wernng MIT October 213 Jonathan Vogel Columba Abstract The theory of comparatve advantage s at the core of neoclasscal trade theory. Yet we know lttle about ts mplcatons for how natons should conduct ther trade polcy. For example, should mport sectors wth weaker comparatve advantage be protected more? Conversely, should export sectors wth stronger comparatve advantage be subsdzed less? In ths paper we take a frst stab at explorng these ssues n the context of a canoncal Rcardan model. Our man results mply that optmal mport tarffs should be unform, whereas optmal export subsdes should be weakly decreasng wth respect to comparatve advantage, reflectng the fact that countres have more room to manpulate prces n ther comparatve-advantage sectors. We thank Gene Grossman as well as semnar partcpants at Brown and Prnceton Unversty for useful comments. Rodrgo Rodrgues Adao provded superb research assstance. Costnot and Donaldson thank the Natonal Scence Foundaton (under Grant SES ) for research support. Vogel thanks the Prnceton Internatonal Economcs Secton for ther support.

2 1 Introducton Two of the most central questons n nternatonal economcs are Why do natons trade? and How should a naton conduct ts trade polcy? The theory of comparatve advantage s one of the most nfluental answers to the former queston. Yet, t has had lttle mpact on answers to the latter queston. Our goal n ths paper s to explore the relatonshp between comparatve advantage and optmal trade polcy. Our man result can be stated as follows. Optmal trade taxes should be unform across mported goods and weakly monotone wth respect to comparatve advantage across exported goods. Examples of optmal trade taxes nclude () a zero mport tarff accompaned by export taxes that are weakly ncreasng wth comparatve advantage or () a unform, postve mport tarff accompaned by export subsdes that are weakly decreasng wth comparatve advantage. Whle the latter pattern accords well wth the observaton that countres tend to protect ther least compettve sectors n practce, larger subsdes do not stem from a greater desre to expand producton n less compettve sectors. Rather they reflect tghter constrants on the ablty to explot monopoly power by contractng exports. Put smply, countres have more room to manpulate world prces n ther comparatve-advantage sectors. Our startng pont s a canoncal Rcardan model of trade. We focus on ths model because ths s the oldest and smplest theory of comparatve advantage as well the new workhorse model for quanttatve work n the feld; see Eaton and Kortum (212). We consder a world economy wth two countres, Home and Foregn, one factor of producton, labor, a contnuum of goods, and Constant Elastcty of Substtuton (CES) utlty, as n Dornbusch et al. (1977), Wlson (198), Eaton and Kortum (22), and Alvarez and Lucas (27). Labor productvty can vary arbtrarly across sectors n both countres. Home sets trade taxes n order to maxmze domestc welfare, whereas Foregn s passve. In the nterest of clarty we assume no other trade costs n our baselne model. In order to characterze the structure of optmal trade taxes, we use the prmal approach and consder frst a fcttous plannng problem n whch the domestc government drectly controls consumpton and output decsons. Usng Lagrange multpler methods, we then show how to transform ths nfnte dmensonal problem wth constrants nto a seres of smple unconstraned, low-dmensonal problems. Ths allows us to derve sharp predctons about the structure of the optmal allocaton. Fnally, we demonstrate how that allocaton can be mplemented through trade taxes and relate optmal trade taxes to comparatve advantage. Our approach s flexble enough to be used n more general envronments that fea- 1

3 ture non-ces preferences and trade costs. In both extensons, we are able to show that our man nsghts survve. In partcular, the predcton that optmal trade taxes are unform across mported goods and weakly monotone wth respect to comparatve advantage holds wthout further qualfcaton n a Rcardan model wth unform ceberg trade costs. Our approach can also be used for quanttatve work. We apply our theoretcal results to study the desgn of optmal trade polcy n a world economy comprsng two countres: the Unted States and the Rest of the World. We consder two separate exercses. In the frst one, all goods are assumed to be agrcultural goods, whereas n the second one, all goods are assumed to be manufactured goods. We fnd U.S. gans from trade under optmal trade taxes that are 2% larger than those obtaned under lassez-fare for the agrcultural exercse and 33% larger for the manufacturng exercse. Interestngly, a sgnfcant fracton of these gans arses from the use of trade taxes that are monotone n comparatve advantage. Under an optmal unform tarff, gans from trade for both the agrculture and manufacturng exercses would only be 9% larger than those obtaned under lassez-fare. Whle these two-country examples are admttedly stylzed, they suggest that the economc forces emphaszed n ths paper may be quanttatvely mportant as well. We hope that future quanttatve work, n the sprt of Ossa (211), wll further explore ths ssue n an envronment featurng a large number of countres and a rch geography of trade costs. Our paper makes two dstnct contrbutons to the exstng lterature. The frst one s to study the relatonshp between comparatve advantage and optmal trade taxes. In hs survey of the lterature, Dxt (1985) sets up the general problem of optmal taxes n an open economy as a fcttous plannng problem and derves the assocated frst-order condtons. As Bond (199) demonstrates, such condtons mpose very weak restrctons on the structure of optmal trade taxes. Hence, optmal tarff arguments are typcally cast usng smple general equlbrum models featurng only two goods or partal equlbrum models. In such envronments, characterzng optmal trade taxes reduces to solvng the problem of a sngle-good monopolst/monopsonst and leads to the predcton that the optmal tarff should be equal to the nverse of the (own-prce) elastcty of the foregn export supply curve. 1 In a canoncal Rcardan model, countres buy and sell many goods whose prces depend on the entre vector of net mports through ther effects on wages. Thus the (own- 1 Ths dea has a long hstory n the nternatonal trade lterature, gong back to Torrens (1844) and Mll (1844). Ths rch hstory s echoed by recent theoretcal and emprcal work emphaszng the role of termsof-trade manpulaton n the analyss of optmal tarffs and ts mplcaton for the WTO; see Bagwell and Stager (199), Bagwell and Stager (211) and Broda et al. (28). 2

4 prce) elastcty of the foregn export supply curve no longer provdes a suffcent statstc for optmal trade taxes. Nevertheless our analyss shows that for any wage level, optmal trade taxes must satsfy smple and ntutve propertes. What matters for one of our man results s not the entre schedule of own-prce and cross-prce elastctes faced by a country actng as a monopolst, whch determnes the optmal level of wages n a non-trval manner, but the cross-sectonal varaton n own-prce elastctes across sectors holdng wages fxed, whch s tghtly connected to a country s comparatve advantage. The paper most closely related to ours s Itoh and Kyono (1987). They show that n a Rcardan model wth Cobb-Douglas preferences, export subsdes that are concentrated on margnal goods are always welfare-enhancng. Though the logc behnd ther result s dstnct from ours a pont we come back to n Secton 4.3 t resonates well wth our fndng that, at the optmum, export subsdes should be weakly decreasng wth comparatve advantage, so that margnal goods should ndeed be subsdzed more. Our analyss goes beyond the results of Itoh and Kyono (1987) by consderng a Rcardan envronment wth general CES utlty and, more mportantly, by solvng for optmal trade taxes rather than provdng examples of welfare-enhancng polces. 2 Beyond generalty, our results also shed lght on the smple economcs behnd optmal trade taxes n a canoncal Rcardan model: taxes should be monotone n comparatve advantage because countres have more room to manpulate prces n ther comparatve-advantage sectors. More broadly, these results have mplcatons for the recent controversy regardng the consequences, or lack thereof, of mcro-level heterogenety for the welfare gans from trade; see Helpman (213) and Meltz and Reddng (213). In recent work, Arkolaks et al. (212) have shown that, dependng on how the queston s framed, answers to mcrolevel questons may be of no consequence for predctng how nternatonal trade affects welfare wthn a broad class of models. These results rely on calbratng certan macro responses, holdng them fxed across models. Helpman (213) and Meltz and Reddng (213) offer a dfferent perspectve, where these behavoral responses are not held fxed. Regardless of ths debatable choce, our paper emphaszes polcy margns that brng out the mportance of mcro structure. Our qualtatve results that trade taxes should be monotone n comparatve advantage and our quanttatve results that such trade taxes lead to substantally larger welfare gans than unform trade taxes llustrate that the desgn of and the gans assocated wth optmal trade polcy may crucally depend on the extent of mcro-level heterogenety. Here, mcro-level data matter, both qualtatvely and quanttatvely, for answerng the polcy-relevant queston: How should a naton conduct 2 Opp (29) also studes optmal trade taxes n a two-country Rcardan model wth CES utlty, but hs analyss focuses on optmal tarffs that are unform across goods. 3

5 ts trade polcy? 3 The second contrbuton of our paper s techncal. The problem of fndng optmal trade taxes n a canoncal Rcardan model s nfnte-dmensonal (snce there s a contnuum of goods), non-concave (snce ndrect utlty functons are quas-convex n prces), and non-smooth (snce the world producton possblty fronter has knks). To make progress on ths queston, we follow a three-step approach. Frst, we use the prmal approach to go from taxes to quanttes. Second, we dentfy concave subproblems for whch general Lagrangan necessty and suffcency theorems problems apply. Thrd, we use the addtve separablty of preferences to break down the maxmzaton of a potentally nfnte-dmensonal Lagrangan nto multple low-dmensonal maxmzaton problems that can be solved by smple calculus. Beyond the two extensons presented n ths paper, the same approach could be used to study optmal trade taxes n economes wth alternatve market structures, as n Bernard et al. (23) and Meltz (23), or multple factors of producton, as n Dornbusch et al. (198). Our approach s related to recent work by Amador et al. (26) and Amador and Bagwell (213) who have used general Lagrange multpler methods to study optmal delegaton problems, ncludng the desgn of optmal trade agreements, and to Costnot et al. (213) who have used these methods together wth the tme-separable structure of preferences typcally used n macro applcatons to study optmal captal controls. We brefly come back to the specfc dfferences between these varous approaches n Secton 3. For now, we note that lke n Costnot et al. (213), our approach heavly reles on the observaton, frst made by Everett (1963), that Lagrange multpler methods are partcularly well suted for studyng cell-problems,.e., addtvely separable maxmzaton problems wth constrants. Gven the mportance of addtvely separable utlty n the feld of nternatonal trade, we beleve that these methods could prove useful beyond the queston of how comparatve advantage shapes optmal trade taxes. We hope that our paper wll help make such methods part of the standard toolbox of trade economsts. The rest of our paper s organzed as follows. Secton 2 descrbes our baselne Rcardan model. Secton 3 sets up and solves the plannng problem of a welfare-maxmzng country manpulatng ts terms-of-trade. Secton 4 shows how to decentralze the soluton of the plannng problem through trade taxes and derve our man theoretcal results. Secton 5 establshes the robustness of our man nsghts to departures from CES utlty and the ntroducton of trade costs. Secton 6 apples our theoretcal results to the desgn of 3 Though we have restrcted ourselves to a Rcardan model for whch the relevant mcro-level data are heterogeneous productvty levels across goods, not frms, the exact same consderatons would make frm-level data crtcal nputs for the desgn of optmal polcy n mperfectly compettve models. 4

6 optmal trade taxes n the agrcultural and manufacturng sectors. Secton 7 offers some concludng remarks. All formal proofs can be found n the Appendx. 2 Basc Envronment 2.1 A Rcardan Economy Consder a world economy wth two countres, Home and Foregn, one factor of producton, labor, and a contnuum of goods ndexed by. 4 Preferences at home are represented by the Constant Elastcty of Substtuton (CES) utlty, U(c) u (c )d, / where c (c ) denotes domestc consumpton; u (c ) β (c 1 1/σ 1) (1 1/σ) denotes utlty per good; σ 1 denotes the elastcty of substtuton between goods; and (β ) are exogenous preference parameters such that β d = 1. Preferences abroad have a smlar form wth astersks denotng foregn varables. Producton s subject to constant returns to scale n all sectors. a and a denote the constant unt labor requrements at home and abroad, respectvely. Labor s perfectly moble across sectors and mmoble across countres. L and L denote labor endowments at home and abroad, respectvely. 2.2 Compettve Equlbrum We are nterested n stuatons n whch the domestc government mposes ad-valorem trade taxes cum subsdes, t (t ), whereas the foregn government does not mpose any tax. Each element t corresponds to an mport tarff f good s mported or an export subsdy f t s exported. Conversely, each element t corresponds to an mport subsdy or an export tax. Tax revenues are rebated to domestc consumers through a lump-sum transfer, T. Here, we characterze a compettve equlbrum for arbtrary taxes. Next, we wll descrbe the domestc government s problem that determnes optmal taxes. At home, domestc consumers choose consumpton to maxmze utlty subject to ther budget constrants; domestc frms choose output to maxmze profts; the domestc gov- 4 All subsequent results generalze trvally to economes wth a countable number of goods. Whenever the ntegral sgn appears, one should smply thnk of a Lebesgue ntegral. If the set of goods s fnte or countable, s equvalent to. 5

7 ernment balances ts budget; and the labor market clears: c argmax c { u ( c )d p (1 + t ) c d wl + T }, (1) q argmax q {p (1 + t ) q wa q }, (2) T = p t (c q ) d, (3) L = a q d, (4) where p (p ) s the schedule of world prces; w s the domestc wage; and q (q ) s domestc output. Smlarly, utlty maxmzaton by foregn consumers, proft maxmzaton by foregn frms, and labor market clearng abroad mply c argmax c { u ( c )d p c d w L }, (5) q argmax q {p q w a q }, (6) L = a q d, (7) where w s the foregn wage and q ( q ) s foregn output. Fnally, good market clearng requres c + c = q + q. (8) In the rest of ths paper we defne a compettve equlbrum wth taxes as follows. Defnton 1. A compettve equlbrum wth taxes corresponds to a schedule of trade taxes t (t ), a lump-sum transfer, T, a par of wages, w and w, a schedule of world prces, p (p ), a par of consumpton schedules, c (c ) and c ( c ), and a par of output schedules, q (q ) and q ( q ), such that condtons (1)-(8) hold. By Walras Law, compettve prces are only determned up to a normalzaton. For expostonal purposes, we set prces throughout our analyss so that the margnal utlty of ncome n Foregn, that s the Lagrange multpler assocated wth the budget constrant n (5), s equal to one. Hence the foregn wage, w, also represents the real ncome of the foregn consumer. 2.3 The Domestc Government s Problem We assume that Home s a strategc country that sets ad-valorem trade taxes t (t ) and a lump-sum transfer T n order to maxmze domestc welfare, whereas Foregn s passve. Formally, the domestc government s problem s to choose the compettve equlbrum wth taxes, (t, T, w, w, p, c, c, q, q ), that maxmzes the utlty of ts representatve con- 6

8 sumer, U (c). Ths leads to the followng defnton. Defnton 2. The domestc government s problem s max t,t,w,w,p,c,c,q,q U(c) subject to condtons (1)-(8). The goal of the next two sectons s to characterze how unlaterally optmal trade taxes,.e., taxes that preval at a soluton to the domestc government s problem, vary wth Home s comparatve advantage, as measured by the relatve unt labor requrements a /a. To do so we follow the publc fnance lterature and use the prmal approach as n, for nstance, Dxt (1985). 5 Namely, we wll frst approach the optmal polcy problem of the domestc government n terms of a relaxed plannng problem n whch domestc consumpton and domestc output can be chosen drectly (Secton 3). We wll then establsh that the optmal allocaton can be mplemented through trade taxes and characterze the structure of these taxes (Secton 4). 6 3 Optmal Allocaton 3.1 Home s Plannng Problem Throughout ths secton we focus on a fcttous envronment n whch there are no taxes and no compettve markets at home. Rather the domestc government drectly controls domestc consumpton, c, and domestc output, q, subject to the resource constrant, a q d L. (9) In other words, we gnore the equlbrum condtons assocated wth utlty and proft maxmzaton by domestc consumers and frms; we gnore the government s budget constrant; and we relax the labor market clearng condton nto nequalty (9). We refer to ths relaxed maxmzaton problem as Home s plannng problem. Defnton 3. Home s plannng problem s max w,p,c,c,q,q U(c) subject to condtons (5)-(9). 5 An early applcaton of the prmal approach n nternatonal trade can be found n Baldwn (1948). 6 As wll become clear, our man results do not hnge on ths partcular choce of nstruments. We choose to focus on trade taxes cum subsdes for expostonal convenence because they are the smplest tax nstruments requred to mplement the optmal allocaton. It s well-known that one could allow for consumpton taxes, producton taxes, or mport tarffs that are not accompaned by export subsdes. One would then fnd that constranng consumpton taxes to be equal to producton taxes or mport tarffs to be equal to export subsdes,.e. restrctng attenton to trade taxes cum subsdes, has no effect on the allocaton that a welfare-maxmzng government would choose to mplement. 7

9 In order to prepare our dscusson of optmal trade taxes, we wll focus on the foregn wage, w, net mports m c q, and domestc output, q, as the three key control varables of the domestc government. To do so, we frst establsh that the condtons for an equlbrum n the rest of the world namely, foregn utlty maxmzaton, foregn proft maxmzaton, and good and labor market clearng can be expressed more compactly as a functon of net mports and the foregn wage alone. Lemma 1. (w, p, m, c, q ) satsfes condtons (5)-(8) f and only f p = p (m, w ) mn { u ( m ), w a }, (1) c = c (m, w ) max { m, d (w a )}, (11) q = q (m, w ) max {, m + d (w a )}, (12) for all, wth d ( ) u 1 ( ), u ( m ) f m, and a q (m, w ) d = L, (13) p (m, w )m d =. (14) Accordng to Lemma 1, when Home s net mports are hgh, m + d (w a ) >, foregn frms produce good, the world prce s determned by ther margnal costs, w a, and foregn consumers demand d (w a ).7 Conversely, when Home s net mports are low, m + d (w a ) <, foregn frms do not produce good, foregn consumpton s equal to Home s net exports, m, and the world prce s determned by the margnal utlty of the foregn consumer, p (m, w ) = u ( m ). Equatons (13) and (14), n turn, derve from the foregn labor market clearng condton and the foregn consumer s budget constrant. Let p (m, w ) (p (m, w )), c (m, w ) (c (m, w )), and q (m, w ) (q (m, w )) denote the schedule of equlbrum world prces, foregn consumpton, and foregn output as a functon of Home s net mports and the foregn wage. Usng Lemma 1, we can characterze the set of solutons to Home s plannng problem as follows. Lemma 2. Suppose that (w, p, c, c, q, q ) solves Home s plannng problem. Then (w, m = c q, q ) solves max w,m,q u (q + m )d (P) 7 Recall that good prces are normalzed so that the margnal utlty of ncome n Foregn s equal to one. 8

10 subject to a q d L, (15) a q (m, w ) d L, (16) p (m, w )m d. (17) Conversely, suppose that ( w, m, q ) solves (P). Then there exsts a soluton to Home s plannng problem, ( w, p, c, c, q, q ), such that p = p ( m, w ), c = m + q, c = c ( m, w ), and q = q ( m, w ). The frst nequalty (15) corresponds to the resource constrant at home and does not mert further comment. The fnal two nequaltes, (16) and (17), are the counterparts of equatons (13) and (14) n Lemma 1. One can thnk of Inequalty (17) as Home s trade balance condton. It characterzes the set of feasble net mports. If Home were a small open economy, then t would take p (m, w ) as exogenously gven and the soluton to (P) would concde wth the free trade equlbrum. Here, n contrast, Home nternalzes the fact that net mport decsons affect world prces, both drectly through ther effects on the margnal utlty of the foregn consumer and ndrectly through ther effects on the foregn wage, as reflected n nequalty (16). Two techncal aspects of Home s plannng problem are worth mentonng at ths pont. Frst, n spte of the fact that the foregn consumer s budget constrant and the foregn labor market clearng condton must bnd n a compettve equlbrum, as shown n Lemma 1, the soluton to Home s plannng problem can be obtaned as the soluton to a new relaxed problem (P) that only features nequalty constrants. Ths wll allow us to nvoke Lagrangan necessty theorems n Secton 3.2. Second, Home s plannng problem can be decomposed nto an nner and an outer problem. Defne W as the set of values for w such that there exst mport and output levels m, q that satsfy (15)-(17). The nner problem takes w W as gven and maxmzes over mport and output levels, V(w ) max m,q u (q + m )d (P w ) subject to (15)-(17). The outer problem then maxmzes the value functon from the nner problem over the foregn wage, max w W V(w ). It s the partcular structure of the nner problem (P w ) that wll allow us to make progress n characterzng the optmal allocaton. In the next two subsectons, we wll 9

11 take the foregn wage w as gven and characterze the man qualtatve propertes of the solutons to (P w ). Snce such propertes wll hold for all feasble values of the foregn wage, they wll hold for the optmal one, w arg max w W V(w ), and so by Lemma 2, they wll apply to any soluton to Home s plannng problem. 8 Of course, for the purposes of obtanng quanttatve results we also need to solve for the optmal foregn wage, w, whch we wll do n Secton 6. Two observatons wll facltate our analyss of the nner problem (P w ). Frst, as we wll formally demonstrate, (P w ) s concave, whch mples that ts solutons can be computed usng Lagrange multpler methods. Second, both the objectve functon and the constrants n (P w ) are addtvely separable n (m, q ). In the words of Everett (1963), (P w ) s a cell-problem. Usng Lagrange multpler methods, we wll therefore be able to transform an nfnte dmensonal problem wth constrants nto a seres of smple unconstraned, low-dmensonal problems. 3.2 Lagrangan Formulaton The Lagrangan assocated wth (P w ) s gven by L (m, q, λ, λ, µ; w ) u (q + m ) d λ a q d λ a q (m, w ) d µ p (m, w )m d, where λ, λ, and µ are the Lagrange multplers assocated wth constrants (15)-(17). As alluded to above, a crucal property of L s that t s addtvely separable n (m, q ). Ths mples that n order to maxmze L wth respect to (m, q), one smply needs to maxmze the good-specfc Lagrangan, L (m, q, λ, λ, µ; w ) u (q + m ) λa q λ a q (m, w ) µp (m, w )m, wth respect to (m, q ) for almost all. In short, cell problems can be solved cell-by-cell, or n the present context, good-by-good. Buldng on the prevous observaton, the concavty of (P w ), and Lagrangan necessty and suffcency theorems Theorem 1, p. 217 and Theorem 1, p. 22 n Luenberger (1969), respectvely we obtan the followng characterzaton of the set of solutons to (P w ). 8 Ths s a key techncal dfference between our approach and the approaches used n Amador et al. (26), Amador and Bagwell (213), and Costnot et al. (213). The basc strategy here does not consst n showng that the maxmzaton problem of nterest can be studed usng general Lagrange multpler methods. Rather, the core of our approach les n fndng a subproblem to whch these methods can be appled. Secton 5 llustrates the usefulness of ths approach by showng how our results can easly be extended to envronments wth weakly separable preferences. 1

12 Lemma 3. For any w W, ( m, q ) solves (P w ) f and only f ( m, q ) solves max L (m, q, λ, λ, µ; w ) (P ) m,q for almost all, wth the Lagrange multplers (λ, λ, µ) such that constrants (15)-(17) hold wth complementary slackness. Let us take stock. We started ths secton wth Home s plannng problem, whch s an nfnte dmensonal problem n consumpton and output n both countres as well as world prces and the foregn wage. By expressng world prces, foregn consumpton and foregn output as a functon of net mports and the foregn wage (Lemma 1), we then transformed t nto a new plannng problem (P) that only nvolves the schedule of domestc net mports, m, domestc output, q, and the foregn wage, w, but remans nfntely dmensonal (Lemma 2). Fnally, n ths subsecton we have taken advantage of the concavty and the addtve separablty of the nner problem (P w ) n (m, q ) to go from one hgh-dmensonal problem wth constrants to many two-dmensonal, unconstraned maxmzaton problems (P ) usng Lagrange multpler methods (Lemma 3). The goal of the next subsecton s to solve these two-dmensonal problems n (m, q ) takng the foregn wage, w, and the Lagrange multplers, (λ, λ, µ), as gven. Ths s all we wll need to characterze qualtatvely how comparatve advantage affects the soluton of Home s plannng problem and, as dscussed n Secton 4, the structure of optmal trade taxes. Once agan, a full computaton of optmal trade taxes wll depend on the equlbrum values of (λ, λ, µ), found by usng the constrants (15)-(17) and the value of w that maxmzes V(w ), calculatons that we defer untl Secton Optmal Output and Net Imports Our objectve here s to fnd the soluton ( m, q ) of max L (m, q, λ, λ, µ; w ) u (q + m ) λa q λ a m,q q (m, w ) µp (m, w )m. Whle the economc ntuton underlyng our results wll become transparent once we rentroduce trade taxes n Secton 4, we focus for now on the smple maths through whch ( m, q ) comes about. We proceed n two steps. Frst, we solve for the output level q (m ) that maxmzes L (m, q, λ, λ, µ; w ), takng m as gven. Second, we solve for the net mport level m ( that maxmzes L m, q (m ), λ, λ, µ; w ). The optmal output level s then smply gven by q = q ( ) m. 11

13 Snce L (m, q, λ, λ, µ; w ) s strctly concave and dfferentable n q, the optmal output level, q (m ), s gven by the necessary and suffcent frst-order condton, u ) (q (m ) + m λa, wth equalty f q (m ) >. The prevous condton can be rearranged n a more compact form as q (m ) = max {d (λa ) m, }. (18) Note that the domestc resource constrant (15) must be bndng at any soluton of (P w ). Otherwse the domestc government could strctly ncrease utlty by ncreasng output. Thus λ must be strctly postve by Lemma 3, whch mples that q (m ) s well-defned. Let us now turn to our second Lagrangan problem, fndng the value of m that maxmzes L ( m, q (m ), λ, λ, µ; w ). The same arguments as n the proof of Lemma 3 mply that the prevous Lagrangan s concave n m wth two knks. The frst one occurs at m = M I d (w a ) <, when Foregn starts producng good ; see equaton (12). The second one occurs at m = M II d (λa ) >, when Home stops producng good ; see ( equaton (18). Accordngly, we cannot search for maxma of L m, q (m ), λ, λ, µ; w ) by lookng for statonary ponts. But ths techncalty s of lttle consequence for our approach, the end goal of whch s the maxmzaton of the Lagrangan wth respect to m, not the locaton of ts statonary ponts. ( To study how L m, q (m ), λ, λ, µ; w ) vares wth m, we consder separately the three regons parttoned by the two knks: m < M I, MI m M II, and m > M II. Frst, suppose that m < M I. In ths regon, equatons (1), (12), and (18) mply ( L m, q (m ), λ, λ, µ; w ) = u (d (λa )) λa d (λa ) + λa m µm u ( m ). CES utlty further mples u ( ) ( ) c = β c 1/σ. Thus, L s strctly ncreasng f m ( ), m I and strctly decreasng f m ( m I, ) ( ) MI, wth m I σ σ λa σ 1 µβ. Furthermore, by defnton of M I d (w a ) = (w a /β ) σ, the nterval ( m I, ) MI s non-empty f a a < A I σ 1 µw σ λ. When the prevous nequalty s satsfed, the concavty of L mples that Home exports m I unts of good, as llustrated n Fgure 1a, whereas Foregn does not produce anythng. Second, suppose that m [ M I, MII ]. In ths regon, equatons (1), (12), and (18) 12

14 L L m I MI M II m M I M II m (a) a /a < A I. (b) a /a [ A I, A II). L L M I M II m M I M II m III m (c) a /a = A II. (d) a /a > A II. Fgure 1: Optmal net mports. mply L ( m, q (m ), λ, λ, µ; w ) = u (d (λa )) λa d (λa ) + (λa (λ + µw ) a ) m λ a d (w a ), whch s strctly decreasng n m f and only f a a < A II λ +µw λ. When a a [ A I, A II), the concavty of L mples that Home wll export M I unts of good, as llustrated n Fgure 1b. For these goods, Foregn s at a tppng pont: t would start producng f Home s exports were to go down by any amount. In the knfe-edge case, a a = A II, the Lagrangan s flat between M I and M II so that any mport level between M I and M II s optmal, as llustrated n Fgure 1c. In ths stuaton, ether Home or Foregn may produce and export good. Fnally, suppose that M II m. In ths regon, equatons (1), (12), and (18) mply L (m, q (m ), λ, λ, µ; w ) = u (m ) (λ + µw ) a m λ a d (w a ), 13

15 whch s strctly ncreasng f m ( M II, m II ) and strctly decreasng f m ( m II, ), wth m II ( d (λ + µw ) a ). Furthermore, by defnton of M II d (λa ), ( M II, m II ) s non-empty f a a > A II λ +µw λ. When ths nequalty s satsfed, the concavty of L mples that Home wll mport m II unts of good, as llustrated n Fgure 1d. We summarze the above observatons n the followng proposton. Proposton 1. If ( m, ) q solves (P ), then optmal net mports are such that: (a) m = m I, f a /a < A I ; (b) m = M I, f a /a [ A I, A II) ; (c) m [ M I, ] MII f a /a = A II ; and (d) = m II, f a /a (λ, λ, µ) defned above. m > A II, where m I, MI, mii, M II, A I, and A II are the functons of w and Proposton 1 hghlghts the mportance of comparatve advantage,.e., the crosssectoral varaton n the relatve unt labor requrement a /a, for the structure of optmal mports. In partcular, Proposton 1 mples that Home s a net exporter of good only f a /a < A II. Usng Lemmas 2 and 3 to go from (P ) to Home s plannng problem, ths leads to the followng corollary. Corollary 1. At any soluton to Home s plannng problem, Home produces and exports goods n whch t has a comparatve advantage, a /a < A II, whereas Foregn produces and exports goods n whch t has a comparatve advantage, a /a > A II. Accordng to Corollary 1, there wll be no pattern of comparatve advantage reversals at an optmum. Lke n a free trade equlbrum, there exsts a cut-off such that Home exports a good only f ts relatve unt labor requrement s below the cut-off. Of course, the value of that cut-off as well as the export levels wll, n general, be dfferent from those n a free trade equlbrum. 4 Optmal Trade Taxes We now demonstrate how to mplement the soluton of Home s plannng problem usng trade taxes n a compettve equlbrum. 4.1 Wedges Trade taxes cause domestc and world prces to dffer from one another. To prepare our analyss of optmal trade taxes, we therefore start by descrbng the wedges, τ, between the margnal utlty of the domestc consumer, u (c ) = β ( ) c 1 σ, and the world prce, 14

16 p, that must preval at any soluton to Home s plannng problem: τ u p ( c ) 1. (19) By Lemma 1, we know that f ( w, p, c, c, q, q ) solves Home s plannng problem and hence satsfes condtons (5) (8) then p = p ( m, w ). By Lemma 2, we also know that f ( w, p, c, c, q, q ) solves Home s plannng problem, then ( w, m = c q, q ) solves (P). In turn, ths mples that ( m = c q, q ) solves (P w ) for w = w, and by Lemma 3, that ( m, q ) solves (P ) for almost all. Accordngly, the good-specfc wedge can be expressed as τ = u ( q ( m ) + m ) ( p m, w ) 1, for almost all, wth p (m, w ) and q (m ) gven by equatons (1) and (18) and m satsfyng condtons (a)-(d) n Proposton 1. Ths further mples τ = σ 1 σ µ 1, f a λ a w a 1, λ w + µ 1, a f A I < a a < A I σ 1 σ µ w λ ; A II µ w +λ λ ; f a a > A II. Snce A I < A II, we see that good-specfc wedges are (weakly) ncreasng wth a /a. For goods that are exported, a /a < A II, the magntude of the wedge depends on the strength of Home s comparatve advantage. It attans ts mnmum value, σ 1 σ µ 1, for goods such that a /a < A I and ncreases lnearly wth a /a for goods such that a /a (A I, A II ). For goods that are mported, a /a > A II, wedges are constant and equal to ther maxmum value, λ w + µ Comparatve Advantage and Trade Taxes Let us now demonstrate that any soluton ( w, p, c, c, q, q ) to Home s plannng problem can be mplemented by constructng a schedule of trade taxes, t = τ, and a lump-sum transfer, T = p τ m d. Snce the domestc government s budget constrant s satsfed by constructon and the resource constrant (9) must bnd at any soluton to Home s plannng problem, equatons (3) and (4) trvally hold. Thus we only need to check that we can fnd a domestc wage, w, such that the condtons for utlty and proft maxmzaton by domestc consumers and frms at dstorted local prces p ( ) 1 + t,.e., condtons (1) and (2), are satsfed as well. 15 (2)

17 Consder frst the problem of a domestc frm. At a soluton to Home s plannng problem, we have already argued n Secton 3.3 that for almost all, u ( ) q + m λ a, wth equalty f q >. By defnton of τ, we also know that u ( q + m t = τ, then p ) = u ( c ) = p ( 1 + τ ). Thus f ( ) 1 + t λ a, wth equalty f q >. (21) Ths mples that condton (2) s satsfed, wth the domestc wage n the compettve equlbrum gven by the Lagrange multpler on the labor resource constrant, w = λ. Let us turn to the domestc consumer s problem. By defnton of τ, f t = τ, then u ( ) c = p Thus for any par of goods, 1 and 2, we have ( ) 1 + t. ( ) u 1 c ( 1 ) = 1 + t p 1 1 u 2 c 1 + t 2 2 p. (22) 2 Hence, the domestc consumer s margnal rate of substtuton s equal to the domestc relatve prce. By Lemma 1, we know that trade must be balanced at a soluton to Home s plannng problem, p m d =. Together wth T = p τ m d = p t m d, ths mples ( ) 1 + t c d = ( ) 1 + t q d + T. (23) p p Snce condtons (4) and (21) mply ( ) p 1 + t q d = λ L, equaton (23) mples that the domestc consumer s budget constrant must hold for w = λ. Combnng ths observaton wth equaton (22), we can conclude that condton (1) must hold as well. At ths pont, we have establshed that any soluton ( w, p, c, c, q, q ) to Home s plannng problem can be mplemented by constructng a schedule of trade taxes, t = τ, and a lump-sum transfer, T = p τ m d. Snce Home s plannng problem s a relaxed verson of the domestc government s problem, ths mmedately mples that ( t, T, w, p, c, c, q, q ) s a soluton to the orgnal problem. Conversely, suppose that ( t, T, w, p, c, c, q, q ) s a soluton to the domestc s government problem, then ( w, p, c, c, q, q ) must solve Home s plannng problem and, by condton (1), 16

18 t t t = AII A I 1 t = A I A II 1 A I A II a a A I A II a a (a) Export taxes. (b) Export subsdes and mport tarffs. Fgure 2: Optmal trade taxes. the optmal trade taxes t must satsfy ( ) t = u c 1, νp wth ν > the Lagrange multpler on the domestc consumer s budget constrant. By = 1 ν equaton (19), ths mples that 1 + t (1 + τ ). Combnng ths observaton wth equaton (2), we obtan the followng characterzaton of optmal trade taxes. Proposton 2. At any soluton of the domestc government s problem, trade taxes, t, are such that: (a) t = (1 + t) (A I /A II ) 1, f a /a < A I ; (b) t = (1 + t) (( a /a ) /A II ) 1, f a /a [ A I, A II] ; and (c) t = t, f a /a > A II, wth t > 1 and A I < A II. Proposton 2 states that optmal trade taxes vary wth comparatve advantage as wedges do. Trade taxes are at ther lowest values, (1 + t) (A I /A II ) 1, for goods n whch Home s comparatve advantage s the strongest, a /a < A I ; they are lnearly ncreasng wth a /a for goods n whch Home s comparatve advantage s n some ntermedate range, a /a [ A I, A II] ; and they are at ther hghest value, t, for goods n whch Home s comparatve advantage s the weakest, a /a > A II. Snce only relatve prces and hence relatve taxes matter for domestc consumers and frms, the overall level of taxes s ndetermnate. Ths s an expresson of Lerner symmetry, whch s captured by the free parameter t > 1 n the prevous proposton. Fgure 2 llustrates two polar cases. In Fgure 2a, there are no mport tarffs, t =, and all exported goods are subject to an export tax that rses n absolute value wth comparatve advantage. In Fgure 2b, n contrast, all mported goods are subject to a tarff t = AII A I 17 1, whereas

19 exported goods receve a subsdy that falls wth comparatve advantage. For expostonal purposes, we focus n the rest of our dscusson on the soluton wth zero mport tarffs, t =, as n Fgure 2a. To gan ntuton about the economc forces that shape optmal trade taxes, consder frst the case n whch foregn preferences are Cobb-Douglas, σ = 1, as n Dornbusch et al. (1977). In ths case, A I = so that the frst regon, a /a < A I, s empty. In the second regon, a /a [ A I, A II], there s lmt prcng: Home exports the goods and sets export taxes t < t = such that foregn frms are exactly ndfferent between producng and not producng those goods,.e., such that the world prce satsfes p = λ a /ν(1 + t ) = w a. The less productve are foregn frms relatve to domestc frms, the more room Home has to manpulate prces, and the bgger the export tax s (n absolute value). Fnally, n the thrd regon, a /a > A II, relatve prces are pnned down by the relatve unt labor requrements n Foregn. Snce Home has no ablty to manpulate these relatve prces, a unform mport tarff (here normalzed to zero) s optmal. In the more general case, σ 1, as n Wlson (198), Eaton and Kortum (22), and Alvarez and Lucas (27), the frst regon, a /a < A I, s no longer necessarly empty. The ntuton, however, remans smple. In ths regon the domestc government has ncentves to charge a constant monopoly markup, proportonal to σ /(σ 1). Specfcally, the rato between the world prce and the domestc prce s equal to 1/(1 + t ) = σ σ 1 ν. µ In the regon a /a [ A I, A II], lmt prcng s stll optmal. But because A I s ncreasng n σ, the extent of lmt prcng, all else equal, decreases wth the elastcty of demand n the foregn market. 4.3 Dscusson Proposton 2 accords well wth the observaton that governments often protect a small number of less compettve ndustres. Yet n our model, such targeted subsdes do not stem from a greater desre to expand producton n these sectors. On the contrary, they reflect tghter constrants on the ablty to explot monopoly power by contractng exports. Accordng to Proposton 2, Home can only charge constant monopoly markups for exported goods n whch ts comparatve advantage s the strongest. For other exported goods, the threat of entry of foregn frms leads markups to declne together wth Home s comparatve advantage. An nterestng ssue s whether the structure of optmal trade taxes descrbed n Proposton 2 crucally reles on the assumpton that domestc frms are perfectly compettve. Snce Home s government behaves lke a domestc monopolst competng à la Bertrand 18

20 aganst foregn frms, one may conjecture that f each good were produced by only one domestc frm, then Home would no longer have to use trade taxes to manpulate prces: domestc frms would already manpulate prces under lassez-fare. Ths conjecture, however, s ncorrect for two reasons. The frst reason s that although the government behaves lke a monopolst, the domestc government s problem nvolves non-trval general equlbrum consderatons. Namely, t nternalzes the fact that by producng more goods at home, t lowers foregn labor demand, whch must cause a decrease n the foregn wage and an mprovement of Home s terms-of-trade. These consderatons are captured by the foregn resource constrant (16) n Home s plannng problem. As we dscuss n more detals n Secton 6.1, provded that the Lagrange multpler assocated wth that constrant, λ, s not zero, the optmal level of the markup charged by the domestc government wll be dfferent from what an ndvdual frm would have charged,.e., σ / (σ 1). The second reason s that to manpulate prces, Home s government needs to affect the behavors of both frms and consumers: net mports depend both on supply and demand. If each good were produced by only one domestc frm, Home s government would stll need to mpose good-varyng consumpton taxes that mmc the trade taxes descrbed above (plus output subsdes that reflect general equlbrum consderatons). Intutvely, f each good were produced by only one domestc frm, consumers would face monopoly markups n each country, whereas optmalty requres a wedge between consumer prces at home and abroad. As mentoned n the Introducton, our fndngs are related to the results of Itoh and Kyono (1987). They have shown that n the Rcardan model wth Cobb-Douglas preferences consdered by Dornbusch et al. (1977), export subsdes may be welfare enhancng. A key feature of the welfare-enhancng subsdes that they consder s that they are not unform across goods; nstead, they are concentrated on margnal goods. Ths s consstent wth our observaton that, at the optmum, export taxes should be weakly decreasng (n absolute value) wth Home s relatve unt labor requrements, a /a, so that margnal goods should ndeed be taxed less. The economc forces behnd ther results, however, are orthogonal to those emphaszed n Proposton 2. Ther results reflect the general-equlbrum consderatons alluded to above: by expandng the set of goods produced at home, the domestc government can lower the foregn wage and mprove ts terms-of-trade. In contrast, the heterogenety of taxes across goods n Proposton 2 derves entrely from the structure of the nner problem (P w ), whch takes the foregn wage as gven. Ths mples, n partcular, that Proposton 2 would stll hold f Home were a small country n the sense that t could not affect the foregn wage. 19

21 5 Robustness In ths secton we ncorporate general preferences and trade costs nto the Rcardan model presented n Secton 2. Our goal s twofold. Frst, we want to demonstrate that Lagrange multpler methods, and n partcular our strategy of dentfyng concave cellproblems, reman well suted to analyzng optmal trade polcy n these alternatve envronments. Second, we want to show that the central predctons derved n Secton 4 do not crucally hnge on the assumpton of CES utlty or the absence of trade costs. To save on space, we focus on sketchng alternatve envronments and summarzng ther man mplcatons. 5.1 Preferences Whle the assumpton of CES utlty s standard n the Rcardan lterature from Dornbusch et al. (1977) to Eaton and Kortum (22) t mples strong restrctons on the demand-sde of the economy: own-prce elastctes and elastctes of substtuton are both constant and pnned down by a sngle parameter, σ. In practce, prce elastctes may vary wth quanttes consumed and substtuton patterns may vary across goods. For nstance, one would expect two varetes of cars to be closer substtutes than, say, cars and bkes. Here we relax the assumptons of Secton 2 by assumng that: () Home s preferences are weakly separable over a dscrete number of sectors, s S {1,..., S}; and that: () subutlty wthn each sector, U s, s addtvely separable, though not necessarly CES. Specfcally, we assume that Home s preferences can be represented by the followng utlty functon, U = F(U 1 (c 1 ),..., U S (c S )), where F s a strctly ncreasng functon; c s (c ) I s s the consumpton of goods n sector s, wth I s the set of goods that belongs to that sector; and U s s such that U s (c s ) = I s u s(c )d. Foregn s preferences are smlar, and astersks denote foregn varables. Secton 2 corresponds to the specal)/ case n whch there s only one-sector, S = 1, and U s s CES, u s (c ) β s (c 1 1/σs 1 (1 1/σ s ). 9 9 The analyss n ths secton trvally extends to the case n whch only a subset of sectors have addtvely separable utlty. For ths subset of sectors, and ths subset only, our predctons would reman unchanged. 2

22 For expostonal purposes, let us start by consderng an ntermedate scenaro n whch utlty s not CES, but we mantan the assumpton that there s only one sector, S = 1. It should be clear that the CES assumpton s not crucal for the results derved n Sectons In contrast, CES plays a key role n determnng the optmal level of net ) σ, n Secton 3.3 and, n turn, n establshng that trade taxes mports, m I = ( σ λa σ 1 µβ are at ther lowest values, (1 + t) (A I /A II ) 1, for goods n whch Home s comparatve advantage s the strongest n Secton 4.2. Absent CES utlty, trade taxes would stll be unform for goods n whch Home s comparatve advantage s the weakest and lnear n a /a for goods n whch Home s comparatve advantage s n some ntermedate range. But for goods n whch Home s comparatve advantage s the strongest, the optmal trade tax would now vary wth the elastcty of demand, reflectng the ncentves to charge dfferent monopoly markups. Now let us turn to the general case wth multple sectors, S 1. Wth weakly separable preferences abroad, one can check that foregn consumpton n each sector, c s ( c ) I, must be such that s c s argmax c { U s ( c) I s p s c d E s }, Accordngly, by the same argument as n Lemma 1, we can wrte the world prce and foregn output for all s S and I s as p s (m, w, E s ) mn { u s ( m ) ν s (E s ), w a }, (24) and q s (m, w, E s ) max {m + d s (w a /νs (E s )), }, (25) where ν (E s ) s the Lagrange multpler assocated wth the constrant, I s p s c d E s. In ths stuaton, Home s plannng problem can stll be decomposed nto an outer problem and multple nner problems, one for each sector. At the outer level, the government now chooses the foregn wage, w, together wth the sectoral labor allocatons n Home and Foregn, L s and L s, the sectoral trade defcts, D s, subject to aggregate factor market clearng and trade balance. At the nner level, the government treats L s, L s, D s, and w as constrants and maxmzes subutlty sector-by-sector. More precsely, Home s plannng problem can be expressed as max {L s,l s,d s } s S,w W F(V1 (L 1, L 1, D 1, w ),..., V S (L S, L S, D S, w )) 21

23 subject to s S L s = L, s S L s = L, s S D s =, where the sector-specfc value functon s now gven by V s (L s, L s, D s, w ) max m s,q s I s u s (m + q )d subject to I s a q d L s, I s a qs (m, w, w L s D s ) d L s, I s p s(m, w, w L s D s )m d D s. Gven equatons (24) and (25), the sector-specfc maxmzaton problem s of the same type as n the baselne case (program P). As n Secton 3.2, we can therefore reformulate each nfnte-dmensonal sector-level problem nto many two-dmensonal, unconstraned maxmzaton problems usng Lagrange multpler methods. Wthn any sector wth CES utlty, all of our prevous results hold exactly. Wthn any sector n whch utlty s not CES, our prevous results contnue to hold subject to the qualfcaton about monopoly markups dscussed above. 5.2 Trade Costs Trade taxes are not the only forces that may cause domestc and world prces to dverge. Here we extend our model to ncorporate exogenous ceberg trade costs, δ 1, such that f 1 unt of good s shpped from one country to another, only a fracton 1/δ arrves. In the canoncal two-country Rcardan model wth Cobb-Douglas preferences consdered by Dornbusch et al. (1977), these costs do not affect the qualtatve features of the equlbrum beyond gvng rse to a range of commodtes that are not traded. We now show that smlar conclusons arse from the ntroducton of trade costs n our analyss of optmal trade polcy. 22

24 We contnue to defne world prces, p, as those prevalng n Foregn and let { δ, f m φ (m ), 1/δ, f m <, (26) denote the gap between domestc and world prces n the absence of trade taxes. As n our benchmark model, the domestc government s problem can be reformulated and transformed nto many two-dmensonal, unconstraned maxmzaton problems usng Lagrange multpler methods. In the presence of trade costs, Home s objectve s to fnd the soluton ( m, ) q of the good-specfc Lagrangan, max m,q L (m, q, λ, λ, µ; w ) u (q + m ) λa q where p (m, w ) and q (m, w ) are now gven by λ a q (m, w ) µp (m, w )φ (m ) m, p (m, w ) mn { u ( m φ (m )), w a }, (27) q (m, w ) max { m φ (m ) + d (w a ), }. (28) Compared to the analyss of Secton 3, f Home exports m > unts abroad, then Foregn only consumes m /δ unts. Conversely, f Home mports m > unts from abroad, then Foregn must export m δ unts. Ths explans why φ (m ) appears n the two prevous expressons. The ntroducton of transportaton costs leads to a new knk n the good-specfc Lagrangan. In addton to the knks at m = M I/δ d (w a )/δ and m = M II d (λa ), there s now a knk at m =, reflectng the fact that some goods may no longer be traded at the soluton of Home s plannng problem. As before, snce we are not lookng for statonary ponts, ths techncalty does not complcate our problem. When maxmzng the good-specfc Lagrangan, we smply consder four regons n m space: m < M I/δ, MI /δ m <, m < M II, and m M II. As n Secton 3, f Home s comparatve advantage s suffcently strong, a /a 1 δ AI 1 δ σ 1 σ µw λ, then optmal net mports are m = δ 1 σ m I ( σ λa σ 1 µβ ) σ δ 1 σ. Smlarly, f Foregn s comparatve advantage s suffcently strong, a /a > δa II δ λ +µw λ, then optmal net mports are m = δ σ m II ( d (λ + µw ) δa ). Relatve to the benchmark model, there ( s now a range of goods for whch comparatve advantage s ntermedate, a /a 1δ A II, δa II), n whch no nternatonal trade takes place. For gven values of 23

25 the foregn wage, w, and the Lagrange multplers, λ, λ, µ, ths regon expands as trade costs become larger,.e., as δ ncreases. Buldng on the prevous observatons, we obtan the followng generalzaton of Proposton 1. Proposton 3. Optmal net ( mports are such that: [ (a) m ] = δ 1 σ m I, f a /a A I ; (b) m = 1 δ MI, f a /a A I, 1 δ AII) ; (c) m 1δ M I, f a /a = 1 δ AII ; (d) m =, ( f a /a 1δ A II, δa II) ; (e) m [, M II ] f a /a = δa II ; and ( f ) m = δ σ m II, f a /a > δa II. Usng Proposton 3, t s straghtforward to show, as n Secton 4.1, that wedges across traded goods are (weakly) ncreasng wth Home s comparatve advantage. Smlarly, as n Secton 4.2, one can show that any soluton to Home s plannng problem can be mplemented usng trade taxes and that the optmal taxes vary wth comparatve advantage as wedges do. In summary, our man theoretcal results are also robust to the ntroducton of exogenous ceberg trade costs. 6 Applcatons To conclude, we apply our theoretcal results to two sectors: agrculture and manufacturng. Our goal s to take a frst look at the quanttatve mportance of optmal trade taxes for welfare, both n an absolute sense and relatve to smpler unform mport tarffs. In both applcatons, we compute optmal trade taxes as follows. Frst, we use Proposton 1 to solve for optmal mports and output gven arbtrary values of the Lagrange multplers, (λ, λ, µ), and the foregn wage, w. Second, we use constrants (15)-(17) to solve for the Lagrange multplers. Fnally, we fnd the value of the foregn wage that maxmzes the value functon V(w ) assocated wth the nner problem. Gven the optmal foregn wage, w, and the assocated Lagrange multplers, ( λ, λ, µ ), we fnally compute optmal trade taxes usng Proposton Agrculture In many ways, agrculture provdes the perfect envronment n whch to explore the quanttatve mportance of our results. From a theoretcal perspectve, market structure s as close as possble to the neoclasscal deal. From a measurement perspectve, the scentfc knowledge of agronomsts provdes a unque wndow nto the structure of comparatve advantage, as dscussed n Costnot and Donaldson (211). Fnally, from a polcy 24