Preferential Trade Agreements Harm Third Countries

Transcription

1 Preferential Trade Agreements Harm Third Countries Pascal Mossay y and Takatoshi Tabuchi z April 8, 24 Abstract We study market liberalisation under imperfect competition in the presence of price e ects. For this purpose, we build a three-country model of international trade under monopolistic competition. The neighbouring e ect translates how the size e ect propagates across countries. When some country increases in size, its relative wage increases, as well as that in a small and near country, while that in a large and distant country falls. We also show that a preferential trade agreement increases the relative wage, the welfare, and the terms-of-trade in the partner countries, where the integration e ect dominates, while lowering those in the third country. JEL Classi cation: F2, F5, R3 Key-words: monopolistic competition, market size e ect, preferential trade agreement We wish to thank Kazuya Kamiya, Masahiro Okuno-Fujiwara and Pierre Picard as well as participants at the Public Economic Theory Conference at Academia Sinica, the Public Economics Seminar at Keio University, the Economics Seminar at Swansea University, and the Seminar at HEC-Liège for their useful comments. This research has been supported by the COST Action IS4. y Department of Economics, University of Reading, Reading, UK and CORE. p.mossay@reading.ac.uk z Faculty of Economics, University of Tokyo. ttabuchi@e.u-tokyo.ac.jp

2 In imperfect competition models of international trade, the existence of a costlessly traded homogeneous good sector has often been assumed, especially when dealing with multi-country models. This leads to Factor Price Equalisation (FPE) across countries, which signi cantly simpli es the analysis. In that context, markets integrate via the relocation of rms and workers across countries, see Krugman (98), Baldwin et al. (23), Behrens et al. (27), Venables (987), or Ossa (2). By focusing on the consequences of production shifting and of the relocation of industry, that line of research abstracts completely from any price e ect present during the liberalisation process. In particular, it assumes away terms-of-trade considerations and their impact on welfare. Moreover, in the real world, FPE does not hold, even between developed countries. 2 In this paper, we address the consequences of market liberalisation in a framework dealing with size, neighbouring, price, and integration e ects. For this purpose, following Venables (987) and Ossa (2), among others, we build on Krugman s (98) new trade theory to construct a three-country model of international trade under monopolistic competition. In contrast to the existing literature, we relax the assumption of FPE by removing the costlessly traded good sector, so that prices and wages are endogenous, and price e ects are included into the analysis. As our framework deals with an arbitrary trade cost structure between countries, our results go beyond the analysis of speci c examples such as the symmetric or the hub-andspoke con gurations studied by Puga and Venables (997). Moreover, unlike in Ossa (2), no trade restriction is placed between countries. Hence, our model deals with general trade patterns allowing for asymmetries in country sizes and trade costs. FPE is a direct consequence of costless trade in the constant returns sector. Davis (998) shows how costly trade in both the constant and the increasing returns sectors substantially alters the equilibrium outcome. 2 The counterfactual prediction of FPE could be avoided by allowing for di erent productivities across countries in the homogeneous good sector. However, these productivities would not be endogenous. 2

3 The aim of this paper is twofold. First, we look at the role of country size on wages and welfare. Second, we analyse the impact of Preferential Trade Agreements (PTAs) on the partner countries and the left-out country. In this study, we consider the simplest setting allowing for third country e ects and terms-of-trade movements, namely a three-country Krugman model (98), for which a unique equilibrium is shown to exist. The rst set of results relate to size and neighbouring e ects. When some country increases in size, its relative wage increases, as well as that in a small and near country, while that in a large and distant country falls. This result extends the size e ect emphasized by Krugman (98) in the case of two countries by introducing a neighbouring e ect, which translates how the size e ect propagates across countries. The increase in market potential is larger in a neighbouring country than in a distant one. In terms of welfare, all countries gain from the increase in some country size because world production and consumption end up increasing. The second set of results relate to the consequences of PTAs. When some countries engage in a PTA, the integration e ect induces relative prices including relative wages to increase in the integrating area. By raising the export price in the partner countries, the e ect of a PTA is to improve the terms-of-trade in the integrating area, while lowering that in the excluded country. While a PTA is bene cial to the partner countries in utility terms, it is always detrimental to the third country because the latter one does not bene t from the integration e ect and is exposed to a negative price e ect: it has to import goods produced at higher costs in the integrating area. Third, a natural question is whether our PTAs results could hold in N-country environments. We provide several examples for which this is the case. A key property of Krugman s (98) framework is that shocks a ecting labour endowments and trade costs transmit either through terms-of-trade or production relocation e ects. In the 3

4 absence of production relocation e ects, Krugman s model behaves pretty much like a Ricardian or an Armington model, though they di er in their microfoundations. This similarity is now well understood, see Arkolakis et al. (22). That class of trade models also includes Eaton and Kortum (22) and Melitz (23) models. Many recent papers have built on that Armington structure. For instance, Chaney (28) has used Krugman s framework with asymmetric countries and trade costs. However, his framework is di erent from ours for the following reason. So as to deal with rm heterogeneity and to focus on extensive versus intensive margins of trade, he assumed the presence of an homogenous good which is freely traded. By using that good as the numeraire, wages are pinned down by production parameters, instead of being determined by general equilibrium conditions as is the case in our paper. Hsieh and Ossa (2) consider a multi-country Melitz model with multiple industries. Their objective is to estimate empirically the e ect of productivity shocks, not of size or trade cost shocks, on welfare. Moreover, in order to account for general equilibrium adjustments, they rely on a numerical approximation procedure. In the context of an Armington structure, Arkolakis et al. (22) have shown that the change in welfare associated with any foreign shock can be calculated from the change in the share of domestic expenditure. This important result provides a method for estimating welfare gains irrespective of the source of foreign shocks. However, their approach provides no indication on how these shares of domestic expenditures could respond to these various shocks in a general equilibrium environment. In contrast to their analysis, our work identi es the e ect all types of shocks including domestic ones on countries welfare in the context of Krugman s framework. By doing so, we also provide the corresponding implicit changes in the shares of domestic expenditure used by Arkolakis et al. (22). The role of trade policy intervention in imperfectly competitive markets is analysed in Bag- 4

5 well and Staiger (29): the rationale for a trade agreement is to overcome the ine ciency related to the terms-of-trade externality. Here the terms-of-trade gain provides a strong incentive for countries to engage in bilateral trade agreements. This result is in line with that obtained in standard neoclassical trade models, e.g. Maggi and Rodriguez-Clare (998), and other new trade models in the presence of a freely traded homogeneous good, see the trade policy implications derived by Puga and Venables (997), as well as the model by Ossa (2) where the third country trades with one partner country only. However, in general, when trade is not restricted so that each country trades with any other one, a PTA may hurt some partner country in terms of welfare under FPE. This has been shown to happen when the hub e ect is large enough, see Behrens et al. (27). Here, in contrast to the models relying on FPE, the integration e ect always dominates the price e ect irrespective of country sizes and of the spatial distribution of resources across countries, so that welfare always increases in the integrating area. Under FPE, falling trade costs between countries concluding the PTA is accompanied by the relocation of rms to the PTA partners, while it does not lead to terms-of-trade movements. The relocation of rms from the third country to the PTA partners implies a worse access of the third country to the manufacturing varieties. In contrast, in this paper, another rationale for the lower utility in the third country is provided. The falling trade costs between PTA partners raise prices and wages in the integrating area relative to the price level in the third country. As a result, consumers in the left-out country su er a terms-of-trade loss because they have to import varieties produced at higher costs in the partner countries, which lowers their welfare. The literature on Free Trade Agreements (FTAs) has studied a wide range of determinants a ecting the incentives of countries to enter an FTA. By relying on a three-continent model with two countries on each continent, Baier and Bergstrand (24) developed a computational general approach to determine empirically how partner countries characteristics (e.g. the distance 5

6 between them, their market size, or their factor endowments) and also other factors (e.g. their remoteness from other countries) a ect the welfare of member countries. In that respect, the scope of our paper is more limited than theirs. In order to keep their sophisticated model tractable, asymmetries in transport costs are neglected. Because Krugman s structure is more stylized, we are able to provide a formal positive analysis for both member and non-member countries allowing for general asymmetries including those in trade costs. Egger and Larch (28) analysed empirically how a pre-existing PTA a ects the incentive to form a new PTA. More generally, Baldwin and Jaimovich (22) analyse spreading regionalism (i.e., the contagion nature of FTAs) based on a multi-country framework with monopolistic competition under the assumption of FPE. Our contribution with respect this -mainly empirical- literature on FTAs is theoretical. Our analysis provides clear directions for price and welfare changes allowing for third-country e ects (due to country or trade cost asymmetries) and terms-of-trade movements. According to us, this is a step forward in better understanding general third country e ects (among which FTA interdependence is an important particular case) in international trade. Our model derives implications not only for PTA members but also for left-out countries. The remainder of the paper is organised as follows. The three-country model of international trade under imperfect competition is introduced in Section. Section 2 presents some preliminary results of the model. In Section 3, we analyse the role of country size and the impact of a PTA on wages and welfare. Section 4 extends the PTA results to several N-country economies. Section 5 concludes.. The Model The economy consists of three countries and a manufacturing sector producing a di erentiated good. The mass of immobile workers in country i is denoted by L i. Without loss of generality, we assume that the total mass of workers is P 3 i= L i =. 6

7 The utility of an individual in country i is given by Dixit-Stiglitz preferences 3X Z U i j= q ji (v) dv A v2v j ; () where q ji (v) is the amount of variety v produced in country j and consumed in country i, V j is the set of varieties produced in country j, and > is the elasticity of substitution between any two varieties. The budget constraint of a worker in country i earning a wage w i is given by X Z j v2v j p ji (v)q ji (v)dv = w i ; (2) where p ji (v) is the delivery price of variety v produced in country j and consumed in country i. In order to simplify the notation, we drop the variety label v from now on. The maximisation of utility () subject to the budget constraint (2) yields the following worker s demand in country i for a variety produced in country j: with the price index P i in country i given by q ji = p ji w i (3) Pi P i = X k n k p ki! ; (4) where n k is the mass of rms located in country k. Assuming iceberg trade costs, ij > units of a variety have to be shipped from country i for one unit of that variety to reach country j(6= i). We also assume that these trade costs are symmetric ij = ji and ii =. The production technology requires a xed and a constant marginal labour requirements, labeled f and c respectively. 3 In order to satisfy the demand q ij L j in country j, each rm in 3 Because immobile labour is the only production factor, the equilibrium number of rms in each country turns out to be constant. As a result, there is no production relocation e ect à la Krugman (98). 7

8 country i has to produce ij q ij L j units. Thus, the pro t of a rm in country i is given by i X j p ij q ij L j A w + c X j ij q ij L j A : (5) By plugging the worker s demand (3) into expression (5), pro t maximisation with respect to prices yields p ij = c ij w i: (6) By assuming free entry and exit of manufacturing rms, pro t (5) is zero. Given that p ij = p ii ij, we have (p ii cw i ) X j ij q ij L j = w i f: (7) Because labour inputs are given by the second bracketed terms in expression (5), the labour market clearing condition is n + c X j ij q ij L j A = L i : (8) Using relations (6), (7), and (8), the equilibrium number of rms is proportional to the number of workers as follows: n i = L i f (9) By substituting relations (6) and (9) into the pro t expression (5), we have X j ij L j w j Pk kj L k w k = w i, for i = ; 2; 3: () Wages w i are determined by these three equilibrium conditions. By Walras law, one of these conditions is redundant, so that labour in some country can serve as numéraire. The equilibrium utility in country i is given by U i = w i P i = w i c f Pk ki L k w k ; () where wages are evaluated at equilibrium (). 8

9 2. Preliminary Results First of all, as stated in previous section already, we assume that shipping a manufacturing variety from one country to another is costly, so that we exclude the case of costless trade ij =, i 6= j. ASSUMPTION For any distinct i; j; k 2 f; 2; 3g, ij >. Assumption implies costly international trade and excludes perfect integration between countries. This assumption is in no way restrictive given that otherwise, the number of countries would reduce to two or less. We now show that our model is general enough to encompass both the direct- and the indirect-shipping of goods. While ij represents the direct-trade cost between countries i and j, the product ik jk corresponds to the trade cost between these countries when the good is shipped via country k. The former trade cost is more costly than the latter one if, for instance, a very high tari is imposed between countries i and j. In this case, direct trade is more costly than trade via a third country. However, should this arises, it could only be so for one pair of countries, not more. To see this, denote the largest trade cost by ij so that kl ij, 8k; l, where i 6= j and k 6= l. In particular, this implies that ik ij < ij jk and jk ij < ij ik, for k di erent from i and j, which means that at least two triangle inequalities are always satis ed. In other words, direct trade cannot be more costly than indirect trade for two pairs of countries. This is because the presence of two very costly direct-shipping routes makes indirectshipping simply not possible in our three-country model, as it would involve using two non-costly shipping routes. 9

10 Two cases may arise: (i) ik < ij jk ; ij < ik jk ; and jk < ij ik (ii) ik ij jk ; ij < ik jk ; and jk < ij ik : In case (i), direct trade is less costly than trade via a third country for any pair of countries, so that it involves direct-shipping only, and the triangle inequality always holds. An example of this situation is when trade costs correspond to distance-related transport costs. In case (ii), direct trade is more costly than trade via a third country for one pair of countries (i; k) and the triangle inequality is violated for that pair of countries. In this latter case, we will assume that rms transport goods from country i to country k via country j rather than directly so that the e ective trade cost ik is given by ij jk. For example, if tari s between countries i and k are very high, then rms will avoid direct trade by shipping goods via the third country j in order to reduce trade costs. Note that case (ii) is more likely to occur in international trade than in interregional trade because within a country trade costs increase in the geographical distance. To make our model as general as possible and so as to encompass the possibility of indirectshipping, we de ne the e ective trade cost ik = minf ik ; ij jk g. This de nition can be rewritten in terms of the freeness of trade ij ij 2 (; ] between countries i and j in the following way. For any distinct i; j; k 2 f; 2; 3g, the e ective freeness of trade is given by ik = maxf ik ; ij jk g. From Assumption, without loss of generality, we can set 23 = maxf 23 ; 2 3 g; 2 = 2 > 3 23 ; and 3 = 3 > 2 23 : (2) From now on, in order to simplify the writing, we drop the notation and de ne the freeness of

11 trade matrix 3 by 3 = : C A We now propose a useful way of restating the wage and utility equations () and (). LEMMA Wages and utilities are determined by v i = X 3 j= ijl j v j ", i = ; 2; 3; (3) where v i stands either for w i ( ) 2 or Ui, with " = ( ) = and = (f) 2 [c=( ( ) )] 2. Proof. This result has been proved by Tabata et al. (23) in the case of an in nitedimensional economy. For the sake of clarity, we now show the validity of the proposed decomposition (3). By using the freeness of trade notation, wage equations () can be rewritten as w i = X ij L j w j j Pk jkl k w, i = ; 2; 3: k By making the following substitution w j = C X k jkl k w k ; where C is some positive constant, we get w i = X j ij L j w j w j =C = C X j ijl j w j : By de ning v i = wi, we have v i = C X j= ijl j v " j ; (4) where " = ( )= 2 (; ).

12 The uniqueness of the decomposition (4) follows directly from Tabata et al. (23) by adapting their argument to the case of a nite-dimensional economy. Moreover, using U i = w i =P i yields w i = C 2 U 2 i ; where = (f) 2 [c=( )] 2. This means that the equilibrium utility in country i depends on the local wage only. As a consequence, the equilibrium equations (4) also describe utilities by making the substitution v i = U 2 i and considering C =. In the rest of the paper, we normalize C =. To analyse the equilibrium conditions (3), it is convenient to de ne the map F as follows F i (v) v i Xj= ijz j =, i = ; 2; 3; (5) where z i denotes L i v " i where x i denotes z i =v i :. The Jacobian matrix D v F of these equations is given by + " x " 2 x 2 " 3 x 3 D v F = " B 2 x + " x 2 " 23 x 3 ; (6) A " 3 x " 23 x 2 + " x 3 The freeness of trade matrix and the Jacobian of the equilibrium map have a positive determinant. LEMMA 2 det 3 > and det(d v F ) >. Proof. First, notice that the determinant of the freeness of trade matrix can be written as det 3 = = ( )( 2 ) + ( )( 3 ) +( )( 23 ) + ( 2 )( 3 )( 23 ): 2

13 Then, by using the triangle inequalities (2), we get det 3 >. Second, by using the expression (6) of the Jacobian matrix D v F, we have det(d v F ) = + "(x + x 2 + x 3 ) + " 2 [( 2 2)x x 2 + ( 2 3)x x 3 + ( 2 23)x 2 x 3 ] + " 3 x x 2 x 3 det 3 ; which implies that det(d v F ) > given that det 3 >. We now address the existence and uniqueness of equilibrium. PROPOSITION The model admits a unique equilibrium. Proof. First, rewrite F as I H so that the map H is de ned by Let =min ij and observe that H maps D = R 3 + into itself H i (v) X N j= ijl j v j ", i = ; 2; 3: h " 2 ; " " i h 2 " 2 ; " " i h 2 " 2 ; " " 2 i H i (v) X L jv " j j X L j " 2 " 2 = j H i (v) X L jv " j j X L j " " 2 = " " 2 ; j " 2 where the total labour constraint ( P j L j = ) has been used. This shows that an equilibrium exists by Brouwer xed-point theorem. Note that this rst part of the proof is similar to the method used by Tabata et al. (23) for an economy with an in nite number of locations. Second, the uniqueness of equilibrium follows from an homotopy argument. For this, consider the parametric dependence of v on " and de ne the homotopy K(v; ") = F (v) with ". We are interested in the structure of the equilibrium set v(") : K(v; ") =, 8". By the rst part of the proof, a solution v(") is known to exist, 8". We now study the behaviour of v(") by considering its graph in the plane (v; "). When " =, the solution v is unique given the 3

14 expression of the equilibrium map v i () = L i + P j6=i ijl j. The local behavior of v(") around " = is given by the implicit function theorem D " vj "= = (D v K) D " K "= : By Lemma 2, det D v Kj "= = det D v F j "= 6=, which means that the homotopy is regular when " = (i.e. locally, v can be seen as a function of "). Note that the homotopy remains regular as " increases away from zero as det D v Kj " = det D v F j 6=, 8". This ensures that the curve v(") de nes a smooth path starting at " =, thereby preventing the path to bifurcate or to exhibit an horizontal tangency in the plane (v; "). Regarding the global behaviour of v("), the path emanating from " = is necessarily unique. This is because any other path would intersect " =, which would imply multiplicity of equilibria when " =. While equations (5) will be used to study utilities, we still need to derive equations helpful to analyse relative wages across countries. By Walras law, variables v and v 2 can be expressed relative to v 3 by normalizing v 3 = : 8 >< G (v ; v 2 ) v ( 3 z + 23 z 2 + L 3 ) (z + 2 z L 3 ) >: G 2 (v ; v 2 ) v 2 ( 3 z + 23 z 2 + L 3 ) ( 2 z + z L 3 ); (7) where z i still denotes L i v " i. We now show that admissible relative v s belong to the triangle de ned by the sides l i (v ; v 2 ; ) = for i =, 2, 3 in the plane (v ; v 2 ) = (w ; w 2 ) in Figure. l i(v ; v 2 ; ) is obtained by by setting L i = and eliminating z j and z k from (3) for distinct i, j, k. The three corners of correspond to the cases L i =, 8i. This is because side (l i = ) corresponds to L i = meaning that the corner (l j = ) \ (l k = ) corresponds to L i = for distinct i, j, k. For instance, when L =, (v ; v 2 ) = 3 ; 2= 3, which corresponds to the corner opposite to side l =. Similarly, when L 2 =, (v ; v 2 ) = 2 = 23 ; 23, which corresponds to the corner opposite to side l 2 =. In the rest of the paper, we focus on the case L i >, 8i. Otherwise 4

15 the number of countries would reduce to two or less. When L i >, 8i, it must be that l i >, 8i, meaning that admissible (v ; v 2 ) s belong to the interior of triangle. 3. Comparative Statics First, we examine the general equilibrium impacts of an exogenous increase in country size on utilities and relative wages. We then study the consequences of a PTA by considering an exogenous increase in the freeness of trade between the countries concluding the PTA. 3.. Size e ect PROPOSITION 2 For any distinct i; j; k 2 f; 2; 3g, (i) d (w j =w i ) =dl j >, and (ii) d (w k =w i ) =dl j R if w k =w i Q jk = ij =. The proof is contained in Appendix A. Proposition 2(i) implies that the larger a country, the higher its relative wage. This result corresponds to the size e ect emphasized by Krugman (98, p. 954) in the case of two countries. When the size of the local market increases, local rms face lower average transportation costs. 5

16 In equilibrium, that competitive advantage is o set by higher relative local wages. Proposition 2(ii) shows that the size e ect may a ect neighbouring countries in a multi-country setting. This neighbouring e ect says that a country size increase will tend to increase (resp. lower) relative wages in other countries provided that they are low enough (resp. high enough). To interpret = this result, observe rst that w k =w i < jk = ij as long as country k is su ciently small and the freeness of trade between countries j and k su ciently high. This means that an exogenous increase in the population of some country tends to increase (resp. lower) the relative wage in a small and near country (resp. a large and distant country). The neighbouring e ect translates how the size e ect propagates across countries: when some country increases in size, the increase in market potential is larger in a neighbouring country than in a remote one. Also, this impact is larger on a small country than on a large one as smaller countries are more sensitive to foreign shocks. PROPOSITION 3 du i =dl j > ; 8i and j. The proof is contained in Appendix B. An increase in the local labour force is bene cial to all countries because world consumption ends up increasing. While the larger number of manufacturing workers leads to more local varieties and to higher output, the increase in demand raises the relative wage of local workers, which in turn makes local goods more expensive. Though the impact on relative wages in other countries may be positive or negative (depending on the sign of the neighbouring e ect), Proposition 3 shows that overall, all countries gain in terms of welfare when some country increases in size. We now look at the corresponding impact on trade ows. Let ij denote the share of country j s total expenditure on goods imported from country i. 6

17 COROLLARY (i) d ij =dl i >, 8i and j, and (ii) d kj =dl i < ; 8j and k 6= i. The proof is contained in Appendix B. So, when some country increases in size, all the import shares from that country increase while those from the other countries fall. Arkolakis et al. (22) determine how welfare changes can be inferred from changes in the share of domestic expenditures. Our results complement their result by determining how trade shares and welfare respond simultaneously to country size shocks in Krugman s model The Impact of a PTA In this section, we consider the scenario where countries j and k engage in a PTA. Market integration is studied by investigating the impact of an increase in the freeness of trade between PTA partners on relative wages and welfare. By neglecting the source of potential tari revenues for partner countries, our approach follows Venables (987), Behrens et al. (27), and Ossa (2). PROPOSITION 4 For any distinct i; j; k 2 f; 2; 3g, d (w j =w i ) =d jk >. The proof is contained in Appendix C. Proposition 4 states that a PTA between two countries via a reduction in their mutual trade cost increases their wages relative to that of the third country. The integration e ect due to a better market access between PTA partners induces the price index in the integrating area to fall and local consumption to rise. However, because supply is xed, the price e ect leads prices and relative wages in the integrating area to rise so as to restore equilibrium. Because the export price is proportional to the wage in the export country (see expression (6)), this price 7

18 e ect improves the terms-of-trade of the integrating area, while lowering that of the excluded country. The implication on welfare is derived in the following Proposition. PROPOSITION 5 For any distinct i; j; k 2 f; 2; 3g, (i) du j =d jk > and du k =d jk >. (ii) du i =d jk <. The proof is contained in Appendix D. Proposition 5(i) is intuitive: the terms-of-trade gain provides a strong incentive for countries to engage in bilateral trade agreements. This result is similar to that obtained in other new trade models in the presence of a freely traded homogeneous good, see the trade policy implications derived in the symmetric and hub-and-spoke con gurations by Puga and Venables (997), as well as the model by Ossa (2) where the third country trades with one partner country only. However, in general, when trade is not restricted so that each country trades with any other one, a PTA may hurt some partner country in terms of welfare under FPE. This has been shown to happen when the hub e ect is large enough, see the example with du j =d jk < provided by Behrens et al. (27, p. 637). In that case, although the two countries engaging in a PTA have a better market access and attract rms overall, rms in the smaller country move to the larger one (Baldwin and Robert-Nicoud, 2), which reduces the welfare in the smaller country concluding the PTA. Here, in contrast to the models relying on FPE, the integration e ect always dominates the price e ect irrespective of country sizes and of the spatial distribution of resources across countries, so that welfare always increases in the integrating area. Proposition 5(ii) is another important nding of this paper. It states that while a PTA is bene cial to PTA partners, it is always detrimental to the third country. This result is in line with those obtained under FPE, see Puga and Venables (997), Behrens et al. (27, Proposition 3), and Ossa (2, Proposition 3). However, here, another rationale for the lower utility in the 8

19 third country is provided. Under FPE, the falling trade costs between countries concluding the PTA are accompanied by the relocation of rms to PTA partners, while it does not lead to terms-of-trade movements. In other words, the relocation of rms from the third country to PTA partners implies a worse access of the third country to the manufacturing varieties. In contrast, in this paper, the falling trade costs jk between PTA partners raise prices and relative wages in the integrating countries j and k, with respect to the price level in the third country i (see Proposition 4). As a result, consumers in the third country do not bene t from the integration e ect and are exposed to a negative price e ect. They su er a terms-of-trade loss because they have to import the varieties produced at higher costs in the partner countries, which lowers their welfare. 4. Some Extensions to N Countries In this section, we extend our PTA results to several N-country economies. We rst consider the case of PTA partners and 2 trading symmetrically with N 2 symmetric third countries. Country sizes are given by L, L 2, L 3 = ::: = L N, and the freeness of trade matrix by = ; C A where 2 denotes the freeness of trade between countries and 2, 3 (resp. 23 ) that between country (resp. country 2) and any third country, and that between any two third countries. To keep the example tractable, we assume > maxf 2 ; 3 ; 23 g. As third countries are of 9

20 identical size, the equilibrium equations (3) can be extended to v v 2 v 3 = C B z + 2 z 2 + (N 2) 3 z 3 2 z + z 2 + (N 2) 23 z 3 3 z + 23 z 2 + [ + (N 3)]z 3 ; (8) C A where v 3 denotes v in any third country and z i still denotes L i v " i, i = ; 2; 3. PROPOSITION 6 Assume that PTA partners and 2 trade symmetrically with N 2 symmetric third countries. Then, (i) du i =d 2 >, i = ; 2 (ii) du 3 =d 2 < for any third country. The proof is contained in Appendix E. Our second extension explores a di erent special case. Here trade relationships between countries are assumed to be symmetric, while country sizes L i may be asymmetric. The freeness of trade between any two countries is given by. The equilibrium equations (3) can then be written as where z i still denotes L i v " i. v i = z i + NX z j, i = ; :::; N; (9) j6=i Our last extension considers the case of low or high product di erentiation between varieties (! + or! ). No restriction is placed on country sizes nor on trade costs. The equilibrium equations (3) are now given by v i = z i + NX ij z j, i = ; :::; N: The PTA results corresponding to these last extensions are now summarized. j6=i 2

21 PROPOSITION 7 When trade relationships are symmetric or when product di erentiation among varieties is either very low or very high, (i) du i =d ij > (ii) du k =d ij < for any distinct countries i, j, and k. The proof is contained in Appendix F. 5. Conclusion In this paper, we have built a three-country model of international trade under monopolistic competition. In contrast to the existing literature which relies on FPE across countries, our approach accounts for prices e ects and terms-of-trade movements. We have determined the role of country size on relative wages and welfare. When some country increases in size, its relative wage increases, as well as that in a small and near country, while that in a large and distant country falls. The size e ect, emphasized by Krugman in the case of two countries, propagates across countries, giving rise to a neighbouring e ect: the increase in market potential is larger in a neighbouring country than in a distant one. We have also determined the impact of a PTA on the participating countries and the left-out country. A PTA increases the relative wage, the welfare, and the terms-of-trade in the integrating area, while lowering those in the third country. These PTA results have been extended to a number of multi-country economies involving country size or trade cost asymmetries. Further research aiming at a better understanding of third-country e ects is still needed. In particular, quantifying the changes derived in the present work would be especially important as it could only enlarge the set of theoretically-informed PTAs propositions to be tested empirically. Also, extending our approach to multi-industry environments would be particularly relevant. 2

22 Appendix A: Proof of Proposition 2 It is su cient to show the signs of d(w 2 =w 3 )=dl and d(w 2 =w 3 )=dl 2 as the other results follow from a symmetric argument. Consider the relative wages equations (7). The corresponding Jacobian matrix D v G is given by L 3 + ( ") 3 z + 23 z 2 + "x "( 2 23 v )x 2 "( 2 3 v 2 )x L z + ( ") 23 z 2 + "x 2 where x i still denotes z i =v i. We observe that det (D v G) >. C A ; (2) det (D v G) = [L 3 + ( ") 3 z + 23 z 2 + "x ][L z + ( ") 23 z 2 + "x 2 ] " 2 ( 2 3 v 2 )( 2 23 v )x x 2 > 23 z 2 3 z + "x "x 2 " 2 ( 2 2x x v v 2 x x 2 ) = ( " 2 ) 3 23 z z 2 + " 2 ( 2 2)x x 2 > : (2) The inverse of the Jacobian matrix, det(d v G) D v G, is given by L z + ( ") 23 z 2 + "x 2 "( 2 23 v )x 2 "( 2 3 v 2 )x L 3 + ( ") 3 z + 23 z 2 + "x C A : (22) (i) By applying the implicit function theorem to the relative wage equilibrium conditions (7), we get dv = (D v G) D L2 G = (D v G) v " B 2 dl 2 23 v 23 v 2 C A d(v 2 =v 3 ) dl 2 = v " 2 [D vg 2 ( 2 23 v ) + D v G 22 ( 23v 2 )] = v " 2 det(d v G) g (v ; v 2 ); where g (v ; v 2 ) ( 23 v 2 ) ( + "x " 3 z ) "x ( 2 3 v 2 )( 2 23 v ): 22

23 We show that g (v ; v 2 ) > in the interior of triangle. This is to equivalent to show that the curve de ned by the expression g (v ; v 2 ) = does not intersect triangle, except at corner ( 2 = 23 ; 23 ). For this purpose, we evaluate g along the three sides of triangle. First, solving l (v ; v 2 ) = for v 2 and plugging the solution into g lead to g (v ; v 2 )j l = = ( 23v 2 ) ( ) : Because v 2 [ 2 = 23 ; 3 ] on the side (l = ) of triangle, we get g (v ; v 2 )j l =, where the equality holds only if v = 2 = 23. Second, solving l 2 (v ; v 2 ) = for v 2 and plugging the solution into g lead to g (v ; v 2 )j l2 = = ( 2 23 v ) "x det : Because v 2 [ 3 ; 2 = 23 ] on the side (l 2 = ) of triangle, we get g (v ; v 2 )j l2 =, where the equality holds only if v = 2 = 23. Third, solving l 3 (v ; v 2 ) = for v 2 and plugging the solution into g lead to g (v ; v 2 )j l3 = = ( 3v ) [( ) 23 + " 3 x det 3 ] ( ) > 3 3 for all v 2 [ 3 ; 3 ] on the side (l 3 = ) of triangle. Hence, g is positive along the three sides of triangle except at the vertex (= 2 ; 23 = 2 ). By a continuity argument, g is positive inside. (ii) Similarly, we get dv = (D v G) D L G = (D v G) v " B 3 v 2 3 v 2 C A : Then, d(v 2 =v 3 ) dl = v " [D vg 2 ( 3v ) + D v G 22 ( 2 3 v 2 )] = v " ( 2 3 v 2 ) (L 3 + det(d v G) 3 z + 23 z 2 ) : 23

24 So d(v 2 =v 3 )=dl? if v = 3. Appendix B: Proofs of Proposition 3 and Corollary Regarding the proof of Proposition 3, it su ces to show that du =dl > and du 3 =dl > as the other results follow from a symmetric argument. By applying the implicit function theorem to the utility equilibrium equations (5), we get D L v = (D v F ) D L F = (D v F ) ( z L ) 2 3. First, we derive the inverse of the Jacobian matrix D v F. Direct computations allow to write det(d v F )D v F as + " (x 2 + x 3 ) + " 2 ( 2 23)x 2 x 3 "x 2 [ 2 + "( )x 3 ] "x 3 [ 3 + "( )x 2 ] "x [ B 2 + "( )x 3 ] + " (x 3 + x ) + " 2 ( 2 3)x 3 x "x 3 [ 23 + "( )x "x [ 3 + "( )x 2 ] " x 2 [ 23 + "( )x ] + " (x + x 2 ) + " 2 ( 2 2)x x 2 ] (23) : C A Second, we get dv dl = z L (D v F + 2D v F 2 + 3D v F 3 ) = z L det(d v F ) [ + "( 2 2)x 2 + "( 2 3)x 3 + " 2 x 2 x 3 det 3 ] > ; where Lemma 2 has been used. We also have dv 3 dl = z L (D v F 3 + 2D v F D v F 33 ) = z L det(d v F ) [ 3 + "( )x 2 ] > by making use of the triangle inequality 3 > So as to prove Corollary, the trade share ij is rewritten by using Lemma as follows ij = n i pij P j = L i ij w i Pk L k kj w k wi = L i ij wj : 24

25 The above proof of Proposition 3 ensures that dw j =dl i > since dv j =dl i >. We then have d kj dw k = L k dl kj i wk + w j dl i w k wj + dw j dl i! < ; 8j and k 6= i: Moreover, d ij =dl i = P k6=i d kj=dl i > as trade shares add up to. Appendix C: Proof of Proposition 4 It su ces to show that d (w 2 =w 3 ) =d 2 > as the other results follow from a symmetric argument. By applying the implicit function theorem to the relative wage equilibrium conditions (7), we have dv = (D v G) D 2 G = (D v G) B 2 z 2 z C A : Notice that det(d v G) D v G has already been derived in expression (22) in Appendix A. Using x i = z i =v i and (3), we get d(v 2 =v 3 ) d 2 = D v G 2 z 2 + D v G 22 z = x det(d v G) g 2(v ; v 2 ); where g 2 (v ; v 2 ) " v 2 + det 3 + " v +" v 2 2 2" v2 " (det 3 ) : We show that g 2 (v ; v 2 ) > in the interior of triangle. This is to equivalent to show that the curve 2 de ned by the expression g 2 (v ; v 2 ) = does not intersect triangle. For this purpose, we evaluate g 2 along the three sides of triangle. First, solving l (v ; v 2 ) = for v 2 and plugging the solution into g 2 lead to g 2 (v ; v 2 )j l = = g 3 (v ) ( ) 2 ; 25

26 where g 3 (v ) " " v 2 h + ( ) 2 " i v : Because g 3 (v ) >, g 3 ( 3) > and g 3 ( 3 ) >, we have g 3 (v ) > for all v 2 [ 3 ; 2 = 23 ] on the side (l = ) of triangle. Second, solving l 2 (v ; v 2 ) = for v 2 and plugging the solution into g 2 lead to g 2 (v ; v 2 )j l2 = = g 4(v ) " 3v 2 2 " " v 2 : 3 Because g 4 (v ) <, g 4 ( 3 ) > and g 4 ( 3 ) >, we have g 4(v ) > for all v 2 [ 3 ; 3 ] on the side (l 2 = ) of triangle. where Third, solving l 3 (v ; v 2 ) = for v 2 and plugging the solution into g 2 lead to g 2 (v ; v 2 )j l3 = = g 5 (v ) ( ) 2 ; g 5 (v ) " " v 2 h + ( ) 2 " i v : We get g 5 ( 2 = 23 ) > and g 5 (3 ) >, Because sgng 5 (v ) =sgn( 3 23 ), two cases may arise. (a) If 3 23, then g 5 is concave, and thus g 5 (v ) > for all v 2 [ 2 = 23 ; 3 ] on the side (l 3 = ) of triangle. (b) If 3 > 23, then g 5 is convex. Solving g 5 (v ) = with respect to v and plugging the solution into g 5 leads to g 6 (") = 4" A 23 A ( ) 2 " ": 26

27 Therefore, if A >, then g 6 (") > holds on side (l 3 = ). Next, we get g 5( 2 = 23 ) = 23 ( ) A 2 A 2 23 ( ) ": Therefore, if A 2 >, then the curve 2 does not intersect (l 3 = ) for all v 2 [ 2 = 23 ; 3 ]. We readily have A R, " R " A 2 R, " Q " 2 < " < " 2 < implying that A > or A 2 > holds for all v 2 [ 2 = 23 ; 3 ]. Hence, the curve 2 does not intersect (l 3 = ) inside triangle. Appendix D: Proof of Proposition 5 It su ces to show that du =d 2 > and du 3 =d 2 < as the other results follow from a symmetric argument. By applying the implicit function theorem to the equilibrium equations (5), we get D 2 v = (D v F ) D 2 F = (D v F ) z 2 z. B. We make use of expression (23) of (D v F ) derived in the proof of Proposition 3 in Appendix First, note that (D v F ) i;j = "x i[ ij + "( ij ik jk )x k ]= det(d v F ) < ; for any distinct i, j, k, given the triangle inequality ij > ik jk. This implies that dv 3 =d 2 = (D v F 3 z 2 + D v F 32 z ) <. Second, dv =d 2 = (D v F z 2 + D v F 2 z ). Since D v F = f + "x 3 + "x 2 [ + "( 2 23)x 3 ]g= det(d v F ) >, dv =d 2 = (D v F z 2 +D v F 2 z ) corresponds to the sum of a positive and a negative term. However, by replacing the utility equilibrium conditions (5), dv =d 2 27

28 can be expressed in terms of z s only dv d 2 = z 2 v 2 v 3 det(d v F ) f( ") 2 3 z 2 + ( + ") 23 z ( + ") 23 z 2 3+ [( + ") 3 + ( ") 2 23 ]z z 2 + [( + ") 2 + ( " 2 ) 2 23]z 2 z 3 + [( " 2 ) 2 + ( + " 2 ) 3 23 ]z z 3 g: By direct inspection of this expression, each term in the sum is positive. Hence dv =d 2 >. Appendix E: Proof of Proposition 6 From the utility equilibrium conditions (8), the equilibrium map F can be de ned by F (v) v v 2 v 3 C A z + 2 z 2 + (N 2) 3 z 3 2 z + z 2 + (N 2) 23 z 3 = : (24) C A 3 z + 23 z 2 + [ + (N 3)]z 3 The corresponding Jacobian matrix D v F is given by + "x " 2 x 2 (N 2)" 3 x 3 " B 2 x + "x 2 (N 2)" 23 x " 3 x " 23 x 2 + "[ + (N 3)]x 3 ; (25) C A where x i = z i =v i : By applying the implicit function theorem to the equilibrium equations (24), we get D 2 v = (D v F ) D 2 F = (D v F ) z 2 z. First, we show that det(d v F ) = + "[x + x 2 + x 3 + (N 3)x 3 ] + " 2 g 7 (N) + " 3 x x 2 x 3 g 8 (N) > ; where g 7 (N) = x [x 2 ( 2 2) + x 3 ( + (N 3) (N 2) 2 3] + x 2 x 3 [ + (N 3) (N 2) 2 23] g 8 (N) = (N 3)( 2 2) (N 2)( ) + 2(N 2) : 28

29 Since g 7 (3) = x [x 2 ( 2 ) 2 + x 3 ( 2 3)] + x 2 x 3 ( 2 23) > and g 7 (N + ) g 7 (N) = x 3 [x ( 2 3) + x 2 ( 2 23)] >, we have g 7 (N) >. From the proof of Lemma 2, g 8 (3) = det 3 > so that g 8 (N + ) g 8 (N) = ( 2 2) = ( 3 ) ( ) + ( 23 ) ( ) + ( 2 ) ( ) > as > ( 3 ) ( 23 ) >. Second, as dv 3 =d 2 = [(D v F ) 3 z 2 + (D v F ) 32 z ], we get dv 3 d 2 = " det(d v F ) [ 3 + ( )"x 2 ]x z 2 "[ 23 + "( )]x 2 z < by making use of the triangle inequalities 3 > 2 23 and 23 > 2 3. Third, as dv =d 2 = [(D v F ) z 2 + (D v F ) 2 z ], we get dv d 2 = det(d v F ) [+"x 3(+(N 3))] (z 2 + "x 2 z 2 "x 2 z 2 )+(N 2)" 2 x 2 x 3 23 (z 3 z 2 23 ) : By using the utility equilibrium conditions (24), and substituting x i by z i =v i, we have v 2 v 3 det(d v F ) z 2 dv d 2 = ( ") 2 3 z 2 + ( + ") 23 z (N 2)( + ")[ + (N 3)] 23 z f( " 2 )[ + (N 3)] 2 + (N 2)( + " 2 ) 3 23 gz z 3 + [( + ") 3 + ( ") 2 23 ]z z 2 + f( + ") 2 [ + (N 3)] + (N 2)( " 2 ) 2 23gz 2 z 3 : By inspection of this expression, we get dv =d 2 >. By a symmetry argument, we also have dv 2 =d 2 >. Appendix F: Proof of Proposition 7 29

30 First, we consider the case of symmetric trade relationships. It su ces to show that du 3 =d 2 < and du =d 2 >. From the utility equilibrium conditions (9), the equilibrium map F can be de ned by NX F i (v) v i z i z j =, i = ; :::; N: (26) j6=i The corresponding Jacobian matrix D v F is given by + "x "x 2 "x N "x + "x 2... "x "xn "x + "x N ; (27) C A where x i = z i =v i : (i) By applying the implicit function theorem to the equilibrium equations (26), we get dv 3 =d 2 = z 2 D v F 3 + z D v F 32. Any o -diagonal of D vf can be shown to be negative. This because these elements can be written as D v F i;j6=i = "x j det(d v F ) Y [ + "( )x k ]; (28) k6=i;j where it can be shown that det(d v F ) >. Hence dv 3 =d 2 <. (ii) Similarly, dv =d 2 = z 2 D v F + z D v F2 holds. We have D v F = + "x 2 det(d v F ) > + "x 2 det(d v F ) Y [ + "( )x i ] + X i6=;2 j6=;2 "x j det(d v F ) Y [ + "( )x k ] k6=;j Y [ + "( )x i ]: (29) i6=;2 3

31 From expressions (28) and (29), we have dv d 2 > z 2 + "x 2 det(d v F ) Y "x 2 [ + "( )x i ] z det(d v F ) i6=;2 = [z 2 ( + "x 2 ) z "x 2 ] = 2 > ; 4 ( ") z + X j6=;2 det(d v F ) Y [ + "( )x k ] k6=;2 Y [ + "( )x i ] i6=;2 3 z j + ( + ") z 2 5 z 2 v 2 det(d v F ) Y [ + "( )x i ] i6=;2 where the last equality is due to x i = z i =v i and equation (26). Second, under su ciently high di erentiation of varieties ("! ), we have D v F i;j6=i = where x i = z i =v i. This implies that dv k =d ij <. "x j + O " 2 + " P k x k + O (" 2 ) < ; Hence We also have D v F i;i = + " P k6=i x k + O " 2 + " P k x k + O (" 2 ) : dv i = z j D v Fii + z i D v Fij d ij z j > : Finally, under su ciently low di erentiation of varieties ("! ), we get! and ij! for all i 6= j. De ning ij ij and letting! lead to D v F i;j6=i = Y N ijx i k6=i;j ( + x k ) + O 2 < ; NY ( + x i ) + O 2 i= 3

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