Tari Wars in the Ricardian Model with a Continuum of Goods

Transcription

1 Tari Wars in the Ricardian Model with a Continuum of Goods Marcus Opp July 3, 27 Abstract This paper derives the optimum tari rate policy within the Ricardian framework of Dornbusch-Fischer-Samuelson (977) and analyzes the comparative statics of the associated Nash equilibrium. First, it is established that the optimum tari schedule involves a uniform tari rate (across goods) which is inversely related to the import demand elasticity of the other country. The impact of absolute productivity advantage and the size of the labor force on this tari rate are shown to be equivalent such that a single measure of relative e ective size summarizes their joint e ect. Using this measure, I show that a su ciently large economy will prefer the ine cient Nash equilibrium outcome over Free Trade due to its quasi-monopolistic power on world markets. The static Nash equilibrium analysis is shown to be directly relevant within the framework of a dynamic game which can be used to characterize self-enforcing trade agreements. JEL classi cation: C72, F, F3. Keywords: Optimum Tari Rates, Ricardian Trade Models, WTO, Gains from Trade PhD Program, Graduate School of Business at the University of Chicago, 587 South Woodlawn Avenue, Chicago, Illinois The author can be contacted via at mopp@chicagogsb.edu or phone/fax (773) I am grateful for the outstanding support I received from Bob Lucas throughout the course of writing this paper. Moreover, I would like to thank Robert Staiger, the co-editor, and two anonymous referees for a very thoughtful and detailed analysis of my paper. In addition, I am thankful to Fernando Alvarez, Christian Broda, Thomas Chaney, Doug Diamond, Pedro Gete, Samuel Kortum, Fernando Mejía, Christian Opp, Tiago Pinheiro and Ioanid Rosu for helpful suggestions. I acknowledge great feedback from the participants of the Economics Student Working Group and the Capital Theory Working Group at the University of Chicago. All remaining errors are mine.

2 Introduction "Full-blown" general equilibrium models featuring multiple goods and a production sector like the Eaton-Kortum model (22) have become standard to study the determinants of trade ows such as technology and geography. In contrast, strategic trade barriers in the form of tari rates are mostly studied within a two-good exchange economy setup or even assumed to be exogenously given. This paper aims to bridge that gap within a particularly simple general equilibrium framework namely the Ricardian Model à la Dornbusch-Fischer-Samuelson (henceforth labeled DFS) which allows me to study the role of technology in the form of absolute and comparative advantage and transportation cost for optimum tari policies. I follow the traditional economic approach to this subject by not explicitly considering political factors and viewing optimum tari rates as optimal strategic decisions within a single period non-cooperative game. While government actions may realistically involve political considerations (see Grossman and Helpman, 995), such a model extension does not o er a separate rationale for trade agreements which are designed to escape the terms-of-trade driven prisoner s dilemma in a Nash equilibrium (see Bagwell and Staiger, 999). 2 Within the just described framework the optimum tari rate schedule is uniform across goods. 3 The key factor for this result is that the Cobb-Douglas demand function implies a unitary income elasticity for each good and zero cross price elasticities, and is as such a special case for the analysis of optimum tari rates with multiple goods. By reducing the tari choice to a one-dimensional problem, I am able to derive a simple optimality condition which can be used to calculate optimum response functions for arbitrary speci cations of technology and taste within the DFS setup. The expression is shown to be inversely related to an appropriately de ned import demand elasticity of the other country. The main determinant of tari rates is given by productivity adjusted relative size ( GDP ratio). Larger economies (either driven by the size of the labor force, e.g. China, or productivity, e.g. Germany) will apply higher tari rates in the Nash equilibrium. The intuition behind my results is as follows: Small economies are heavily dependent on trade and therefore possess a relatively inelastic import demand function. This can be exploited by bigger and more productive countries through the lever of tari rates to achieve terms-of-trade e ects (intensive margin) while hardly increasing ine cient home production (extensive margin). The e ect of comparative advantage is also quite intuitive. Smaller comparative advantage i.e. a more similar production technology will imply a larger demand reduction for a xed increase in tari rates and therefore reduces the optimum tari rate. Natural trade barriers in the form of transportation cost limit the scope of tari rate policies to achieve terms-of-trade e ects and therefore reduce optimum tari rates, too. The welfare analysis implies that the market power of su ciently large economies is so strong that they can be better o in a Nash equilibrium of tari s than in a free trade regime. In such a situation, the small economy bears (more than) the full deadweight loss of the globally As the analysis of Alvarez and Lucas (25) shows a general equilibrium analysis can become quite complicated. 2 The fact that politicians seem to be relatively agnostic about terms-of-trade e ects does not invalidate the main forces of this paper as terms-of-trade e ects can be equivalently framed in terms of market access, which is a more frequently voiced concern of politicians. This insight is pointed out by Bagwell and Staiger (2). 3 It is interesting to note that the US senators Schumer and Graham proposed a at 27.5% import tari on all Chinese manufacturing goods. 2

3 ine cient tari equilibrium. Another interesting feature of the Nash equilibrium is that the terms-of-trade will (approximatively) only re ect di erences in productivity, but not di erences in the size of the labor force. In contrast to the original DFS paper (with exogenous tari rates), a small country will not be able to capture the specialization gains that arise from the focus on the production of goods with the highest comparative advantage. The Nash equilibrium analysis can be used as a stepping stone to study the nature of selfenforcing trade agreements in the spirit of Bagwell and Staiger (99 and 995) and Bond and Park (22). The static Nash equilibrium outcome determines the (o -equilibrium path) punishment payo s for deviating from a trade agreement. Intuitively, free trade will never be sustainable (even as the discount factor goes to ) if the economies are su ciently asymmetric, i.e. if one economy exceeds the threshold size to win a tari war. Since the required threshold size is shown to be decreasing in transportation cost my analysis predicts that it is easier to sustain regional rather than global free-trade zones. 4 My paper is related to various lines of research. General treatments on the optimum tari structure with multiple goods go back to Graa (949/5) and more recently Bond (99) and Feenstra (986). Their analysis yields conditions (e.g. on the substitution matrix) which can generate more complicated tari structures such as subsidies for some goods. Due to the above described properties of Cobb-Douglas preferences the optimum tari structure turns out to be particularly simple in the DFS setup and consistent with this literature. However, my analysis seems to be in contradiction with the result obtained by Itoh and Kiyono (989) who nd export subsidies to be welfare-enhancing in the DFS model (i.e. with Cobb-Douglas preferences!). This apparent contradiction on the role of export subsidies interpretable as negative tari rates can be easily resolved as their paper does not suggest optimality but solely a welfare improvement of a (carefully designed) export-subsidy policy relative to free trade. The dominant part of the existing literature on tari games uses an exchange economy setup to analyze the strategic choice of tari rates. This approach is largely inspired by Johnson s seminal paper "Optimum Tari Rates" (958). He nds that a Nash equilibrium of tari s does not necessarily result in a Prisoner s Dilemma situation in which both countries are worse o than under a free trade regime. Most of the subsequent papers focus on generalized preferences (Gorman, 958), the impact of relative size (Kennan and Riezman, 988) or existence of equilibria (Otani, 98). By incorporating a production sector into the analysis, Syropoulos (22) is able to signi cantly improve upon the previous literature. Using a generalized Heckscher-Ohlin setup (2 countries, 2 commodities and homothetic preferences), he proves the existence of a threshold size level that will cause the bigger country to prefer a tari -equilibrium over free trade. 5 McLaren (997) develops an innovative paper in the spirit of Grossman/Hart (986) that yields the counterintuitive result that small countries may prefer an anticipated trade war relative to an anticipated trade negotiation. 6 The key driver for this result is that an irreversible 4 I am grateful to an anonymous referee for hinting at this interpretation. 5 As opposed to the the Ricardian framework employed in my paper the Heckscher-Ohlin theory explains gains from trade by di erences in factor endowments rather than technology across countries. My paper can be seen as a response to the concluding remarks of Syropoulos who states that "it would be interesting to examine how technology a ects outcomes in tari wars" and "it would be worthwhile to investigate whether the ndings on the relationship between relative country size and tari war outcomes remain intact in multi-commodity settings". 6 This situation can occur if the production is relatively similar in terms of absolute and comparative advantage. 3

4 investment in the export sector by the small country in period will reduce the threat point in negotiation talks in period 2 (after the investment is sunk). Despite the di erent underlying economic setup, it is con rmed that small size brings about a strategic disadvantage. My paper is organized as follows. In section 2 I will present the framework of the Ricardian Model and the standard general equilibrium results with exogenous taxes. In section 3 I will derive the optimum tari rate formula and discuss the comparative statics of the optimum response function. Section 4 will present the detailed features of the Nash equilibrium in tari s. The implications of my static analysis for trade agreements within a dynamic context are discussed in section 5. Section 6 concludes. For ease of exposition, the proofs of all key propositions are moved to the appendix, while the intuition is delivered in the main text. 2 DFS Framework 2. Setup The purpose of this section is to compactly describe the general framework of the Ricardian Model à la DFS. The setup consists of two competitive economies (home country and foreign country) with respective labor endowments L and L. Note, that asterisks are used throughout the paper to refer to the foreign country. Without loss of generality, I normalize the size of the labor force (L ) and wage rate (w ) of the foreign country to unity such that the relative size of the home country is simply given by its absolute size L and the wage rate of the home country is equal to the relative wage rate!. Technology There exists a continuum of goods (z) which are indexed over the interval [; ] and produced competitively with a linear technology in labor. The labor unit requirement function a (z) speci es how many labor units are required to produce one unit of good z in the home country. Analogously, a (z) represents the labor unit requirement function of the foreign country. The ratio of a (z) and a (z) is de ned as A (z): A (z) a (z) a (z) () Goods are ordered in terms of decreasing comparative advantage of the home country, which implies that the function A (z) is decreasing in z over the domain [; ]. The log transformation of A (z) denoted as ~ A (z) yields the productivity (dis)advantage of the home country in percentage terms: ~A (z) log [A (z)] (2) This function is assumed to be twice di erentiable and bounded over the domain [; ]. In order to analyze the comparative statics of technology it is useful to introduce the family of functions ~ A (zj ; ): ~A (zj ; ) = + ~ A N (z) (3) 4

5 where = R ~ A (z) dz governs the average absolute productivity advantage of the home country and governs comparative advantage. By construction, we have: R ~ A N (z) dz = such that positive values of AN ~ (z) imply a comparative advantage in home country production for the respective good z: Changes in induce parallel shifts of A ~ (z) (holding xed) while changes of correspond to changes in the slope (holding average productivity at ). Higher values of magnify the comparative advantage e ects resulting in greater gains from trade. A particularly simple functional form is used to create the graphs of the paper (see Figure ): ~A (zj ; ) = + (:5 z) (4) This loglinear speci cation can be interpreted as a rst order approximation of the true function. 7 Preferences Like in the original DFS paper the representative agent of each economy possesses a Cobb-Douglas utility function over the range of goods: U (c) = U (c ) = R R b (z) log (c (z))dz b (z) log (c (z))dz (5) where the expenditure shares b (z) and b (z) satisfy: Z b (z) dz = Z b (z) dz = (6) The gures in the present text are calculated under the assumption that both countries value all goods equally, i.e. b (z) = b (z) = : Trade barriers I allow for two types of trade barriers: Import tari s (t; t ) and transportation cost (). The latter are in the form of symmetric iceberg transportation cost (Samuelson 954), which implies that only a fraction of exported goods eventually arrives in the other country. This fraction is given by exp ( ). Import tari s are applied to all goods that arrive in the import country and redistributed to the consumers. Since it turns out to be optimal to charge a uniform tari rate across all goods, I will use this result from the very beginning to simplify the analysis. Export tari s do not have to be explicitly considered due to Lerner s "Symmetry-Result". 2.2 General Equilibrium The purpose of this section is to highlight the main implications of the DFS model with exogenous tari s. The main contribution of the original paper is to show that the incorporation of trade barriers (tari s, transportation cost) into a Ricardian Model with a continuum of goods 7 An earlier version of this paper focused solely on this speci cation (see 5

6 gives rise to an endogenously determined non-traded sector in equilibrium. The equilibrium conditions on the three endogenous variables the equilibrium wage rate! and the boundary goods for domestic production in the home and foreign country (z; z ) can be summarized in the following system of equations (see derivation in Appendix A): f (x; q) = log (!) + log + t + log +t + log (L) h i = f 2 (x; q) = log (!) log ( + t) + A ~ N (z) = {z } h ~A(z) i f 3 (x; q) = log (!) + log ( + t ) + + A ~ N (z ) = {z } ~A(z ) (7) where: (z) = zr b (z) dz and (z ) = b (z) dz z R The rst equation is obtained by imposing balance of trade. Note that and represent the share of domestic income that each country spends on domestically produced goods. For notational convenience, the arguments z and z are omitted from the functions and throughout the text. The second and third equation pin down the boundary goods for domestic production at which the cost of domestic production equals the cost of importing the good adjusted for the tari fare and transportation cost. The home country produces goods on the interval [; z] and imports goods on the interval [z; ]: Likewise the foreign country imports goods on the interval [; z ] and produces goods domestically on the interval [z ; ]. One can represent the above described system more concisely by stacking the endogenous variables in the vector x = [log (!) ; z; z ] and the exogenous variables in the vector q = [t; t ; log (L) ; ]: f (x; q) = (8) The comparative statics can be obtained by applying the implicit function theorem. The total derivative of the i-th element of vector x with respect to the j-th element of vector is given by i; j element of the matrix: D q (q) = [D x f] D q f (9) where (q) is the solution for x and D x f; D q f represent the respective Jacobians. The algebra con rms the well understood positive e ect of tari s on the terms-of-trade. The economic intuition is as follows. By imposing a tari on import goods, the home country increases the nal consumption prices of foreign goods which in turn reduces demand for those goods in the home country. At the old equilibrium prices (before imposing the tari ) the balance of trade condition will no longer hold as the foreign country will demand too many import goods. In order to eliminate the excess demand for import goods in the foreign country, the terms-of-trade have to deteriorate (improve) from the perspective of the foreign (home) country. This causes the equilibrium wage rate of the home country to rise (intensive margin). Since the equilibrium wage rate rises relatively less than the price of import goods (i.e. the tari rate increase) the home country will increase home production as a response to its own tari increase (extensive margin). 6

7 Domestic production is increasing in trade barriers (t, t ; ) and increasing in productivity and size L. It seems quite intuitive that larger and more productive economies produce a greater range of goods. Interestingly, log(l) and have exactly the same impact on the boundary goods z and z which results from the linear technology (see derivation in Appendix C): This means that knowledge of the productivity adjusted relative size L exp () - i.e. e ective relative size - summarizes the separate e ects of size and productivity on the boundary goods. For the discussion of optimum tari rates this measure of e ective size, precisely its logarithm (c), will become of crucial importance. E ective relative size is also closely to the amount of goods that can be produced under autarky, i.e. the size of the economy (production capacity). 8 c log (L exp ) = log L + () For illustration purposes it is also convenient to de ne a normalized measure of c denoted as : R! [; ]: exp (c) (c) () + exp (c) This measure can be interpreted as the economic weight of a country, as the "weights" (c) and (c ) = ( c) add up to one. The in nitesimally small country gets a weight of zero, the in nitely large country a weight of one. 9 3 Optimum Response Function 3. Derivation So far, the analysis has treated tari rates similar to transaction cost, i.e. as exogenously given trade barriers. However, in contrast to transportation cost, import tari s are choice variables for the respective government and a ect national income. Both governments are assumed to choose the optimal tari rate for their respective country. As such, they try to maximize the utility of the representative agent given the tari rate decision of the other country. In general, tari rates could vary across goods. However, proposition provides a justi cation for my focus on uniform tari rates. Proposition If the demand function is of Cobb-Douglas type, it is optimal to impose a uniform tari rate across all import goods. Proof. See Appendix D.. The idea of the proof is relatively simple. First, I show that the marginal tari rate t M (at the boundary good) and an appropriately de ned average tari rate t are su cient statistics for an arbitrary taxation schedule t (z) to determine the vector of endogenous variables x (see Appendix A). Secondly, I split up the utility maximization problem over the schedule t (z) 8 For any speci c good the relative production capacity (number of available labor units / number of required labor units) of both countries is given by C (z) = L = L = LA(z): Taking logarithms and averaging over all a(z) a (z) goods implies that we can de ne c log (L) + : 9 The careful reader will notice that the function (x) is identical to the CDF of the logistic distribution. If tari rebates were simply lost as in the case of iceberg transportation cost the optimum tari rate would be zero. 7

8 into two parts by rst solving the subproblem of the welfare optimizing tari rate policy t (z) conditional on the choice of t M and t and then maximizing over the choice of t M and t. The solution to the subproblem results in uniform tari rates, which delivers the claimed result. Now, two variables (t; t ) summarize the tari rate policies of both countries, such that we can write the derived equilibrium utility of the representative agent in the home country V (t; t ;q) and the foreign country V (t; t ; q) as a function of the tari rates and the exogenous parameters. Formally, the optimum tari rate is determined by setting the partial derivative of the indirect utility function with respect to the tari rate equal = (2) The solution to this problem results in the following expression: Proposition 2 The optimum tari rate in the DFS model t opt can be expressed as follows: t opt = ( t opt = ( ) + t A ~ (z ) + t b (z ) ) + t + t ~A (z) b (z) = " = " (3) (4) In order to interpret this result in terms of import demand elasticities ("; " ), I introduce an appropriately de ned measure for this setup. The import demand M of the home country on the world market (valued in terms of its own currency) is given by: M = Ly + t = L! + t The "price" of an import good in local currency is proportional to the inverse of! such that we can de ne the import demand elasticity as: (5) h " = d log log d log! = log! = + t opt (6) This pricing formula is identical to the optimal markup that a monopolistic rm charges over marginal cost MC, i.e. p = + MC. To see this analogy, it is useful to apply " (D) the Lerner symmetry result, such that one can interpret the tari rate formula as the optimal export tari, i.e. the markup on cost. Thus, by applying optimum tari rates a country acts as monopolist for the goods it exports, which leads to a costly reduction in the volume of trade. From a computational point, these formulae are progress, as one can now calculate optimum response functions for arbitrary technology and taste speci cations (in conjunction with the other three equilibrium conditions) in the DFS model. Before turning to the general properties of the optimum response function in the next chapter, I want to note that the optimum tari only depends on the relative labor unit requirement function A (z), but not separately on the labor unit requirement functions a (z) and a (z). 8

9 3.2 Comparative Statics This section aims to lay out the general properties of the optimum response function g (t ) of the home country. The foreign country s optimum response function g (t) follows by symmetry. Thus, the fourth equilibrium condition (in addition to the conditions outlined in equation 7) is given by: f 4 (x; q) = t + ( (z )) + (z )t A ~ (z ) +t b (z ) = (7) Again the comparative statics are obtained via the implicit function theorem. Now, the vector of endogenous variables x includes the tari rate t. In order to be able to sign the derivatives of the optimum response function I need to make a mild technical assumption: < Since the DFS framework neither speci es the sign of the second derivatives of the relative labor unit requirement function nor the rst derivatives of the expenditure share function b (z ), the sign cannot be determined in general (see discussion in Appendix B). However, numerical exercises with various functional forms reveal that this assumption is purely technical and should not at all be considered as restrictive. Hence, for the rest of the analysis I will assume that the assumption is satis ed. This implies: Proposition 3 The optimum response function has the following properties: - Decreasing in the other country s tari rate - Decreasing in the transportation cost - Increasing in the dispersion parameter - Increasing in a country s relative @c < Assumption is necessary and su cient, Assumption is su cient. Proof. See Appendix D.3. Thus, tari s are strategic substitutes (see Figure 2). The intuition behind this result is that higher foreign tari rates increase trade barriers that exogenously reduce trade volume (like transportation cost) such that the room for imposing own trade restrictions without heavily reducing or even shutting down trade is limited. Proposition 4 The strategic substitutability is less than perfect, i.e. d log ( + t opt ) d log ( + t ) < Proof. See Appendix D.4. For t relatively small the term + (z )t is approximately equal to one such that the term ( ) dominates the comparative statics. This assumption will always be satis ed if one uses the technology speci cation +t underlying the Eaton-Kortum model (22). 9

10 Proposition 4 implies, that an increase in the tari rate of the foreign country will lead to a decrease in the home country tari rate that does not fully compensate for the trade barrier imposed by the foreign country, i.e. the non-traded sector (goods in the range between z and z) will expand. It can be shown, that strategic substitutability is increasing in size, i.e. larger economies respond stronger to an increase in the tari rate of the other country. An increase in natural trade barriers (transportation cost) has qualitatively the same negative e ect on the optimum tari rate as an increase in the other country s tari rate. A atter relative labor unit requirement function (lower ) implies smaller specialization advantages which causes domestic and foreign production to be more similar and hence more substitutable. Hence, a xed increase in tari rates results in a larger demand reduction for atter functions ~A (z) such that the optimum tari rate will be smaller. The positive marginal impact of the parameter c implies that countries with a higher effective size can exploit the import dependency of the other country (and the associated lower import demand elasticity). The relationship is monotone, as the harmful side-e ects of tari s (ine cient expansion of home production and price increase of import products) weigh smaller and the terms-of-trade e ect becomes stronger the greater the economic weight of one country. 2 Moreover, it is possible to determine the optimum tari rate policy of a "small" economy: 3 Corollary As the home economy becomes in nitesimally small, the tari rate approaches zero: lim c! g (t ) = Proof. In the limit the foreign country s consumption share of domestically produced goods approaches ; i.e. lim c! = and lim c! z =. Therefore, we have: t opt = lim! ( ) + t A ~ () A + t b = ~ () () b lim () (! ) = 4 Nash Equilibrium 4. Existence A Nash equilibrium in tari rates is obtained, once both countries have no incentive to deviate from their chosen tari rate. Formally, a Nash equilibrium is characterized by an additional equilibrium condition that ensures the optimality of the foreign country s tari rate t, the fth element of the vector of endogenous variables x. f 5 (x; q) = t + ( (z )) + (z )t +t ~ A (z ) b (z ) = (8) 2 Consider the extreme case when the home country is in nitely large/productive such that all goods are produced domestically even without tari s. 3 The boundedness assumption plays a key role. A violation of boundedness such as in the Eaton-Kortum speci cation of technology can imply strictly positive tari rates even for the in nitesimally small country.

11 The optimum response functions for two parameter constellations are depicted in Figure 2. The intersection point constitutes a Nash equilibrium in tari rates which is characterized by strictly positive trade ows. No-Trade Nash equilibria can occur, if both countries choose a prohibitively high tari rate t proh, i.e. a tari rate that exhausts the specialization advantage adjusted for transport cost. Formally a prohibitive tari rate (given that the other country imposes no tari barriers) satis es the following relation: 4 log ( + t proh: ) > ~ A () ~ A () 2 (9) There exists a continuum of No-trade equilibria all of which are uninteresting, as applying a prohibitive tari rate is weakly dominated. In the following analysis I will only consider interior Nash equilibria. Proposition 5 There exists a unique interior Nash equilibrium in tari s that Pareto dominates any No-Trade Nash equilibrium Proof. See Appendix E.. The existence of a unique equilibrium is established by applying the Contraction Mapping Theorem. The contraction property follows directly from the fact that the slope of the optimum response function is less than one (see Proposition 4). 4.2 Tari Rates The comparative statics in the Nash equilibrium are driven by the direct e ects of the exogenous parameters on the optimum response function as well as the feedback e ect through the strategic tari choice of the other country. In case of the size parameter c, the feedback e ect ampli es the original e ect. This is because as the home economy becomes relatively larger (and thus the foreign economy smaller) the foreign economy will apply lower tari rates which in turn increases the tari rate of the home country due to strategic substitutability (see proposition 4). This results in the following claim: Proposition 6 The Nash equilibrium tari rate is increasing in relative e ective size. In the case of transportation cost parameter, the feedback e ect counters the direct e ect. Holding the other country s tari rate xed, an increase in transportation cost lowers the optimum tari rate of both countries. On the other hand, due to strategic substitutability a lower tari rate of the foreign country increases the home country tari rate. Without further restrictions the net e ect could go either way. However, in all relevant scenarios the direct e ect is expected to dominate because precisely when the sensitivity to the foreign country s tari rate decision is large when the home economy is large it is also the case that the (small) foreign country will hardly adjust its tari rate to changes in transportation cost (see Figure 4 The restriction can be obtained by subtracting the equilibrium condition on the boundary good for the foreign country from the boundary good condition of the home country, i.e. f 3 (x; q) f 2 (x; q) ; and using the no-trade condition on the boundary goods (z = and z = ): For the log-linear technology speci cation, a log tari rate greater than 2 is prohibitive.

12 3). 5 Moreover note, that at most one country may increase its tari rate as a response to an increase in transportation cost. Therefore, for economies of equal size (c = ) the net e ect will be unambiguous and both countries will lower their tari rate (as a response to an increase in transportation cost) due to symmetry. The Nash equilibrium comparative statics of the dispersion parameter features the same tension between the direct and the feedback e ect. While the direct e ect implies higher tari s for both countries if the gains from trade are greater the feedback e ect goes the opposite way. As for the case of transportation cost, the direct e ect is expected to dominate (see Figure 3). 4.3 Terms-of-Trade and Welfare Since the improvement of terms-of-trade is central to our analysis, it is helpful to revisit a key result from the original DFS model (with exogenous tari s). Small countries (i.e. low relative size L) face signi cantly better terms-of-trade and bene t the most from free trade. The intuition for this result is that small countries can specialize their production on goods with the highest comparative advantage whereas the large country has to supply the remainder of the goods (for a formal derivation see Appendix C). However, the strategic disadvantage of being small mitigates this positive e ect in the Nash equilibrium. As shown in the previous section, smaller (larger) countries apply lower (higher) equilibrium tari rates. Since the relative wage rate is an increasing function of the own tari rate and a decreasing function of the other country s tari rate, relative size will in uence the wage rate positively through this channel. Whether this strategic e ect possesses signi cant impact depends on the speci cation of technology. The log-linear approximation (see Figure 4) reveals that the large specialization bene ts for the small country (driven by comparative advantage) are almost completely wipedout in the Nash equilibrium such that the terms of trade will only represent the absolute technology advantage. The just described terms-of-trade e ect will also play an important role for the welfare analysis. Since the Nash equilibrium is not Pareto optimal both countries could increase their welfare by removing all tari s and split up the surplus. In order to measure welfare losses for each country I determine the required log rate of consumption growth that equalizes the utility obtained in the Nash equilibrium (N) and free trade (F ). 6 Therefore is de ned as: V F e c F t = ; t = V N (c N j t = t N ; t =t N) (2) Due to the simple form of the utility function the required growth rate of consumption is given by the di erence of the utility levels: V = V N V F (2) The following lemma describes intuitive properties of the welfare measure V (see Figure 5). Lemma V is a function of c; and and has the following limit properties: ) lim c! V (c; ; ) < 2) lim c! V (c; ; ) (c;;) 3) lim < 5 Numerical results for various functional forms (see subsequent section) con rm this claim. 6 This welfare measure has been proposed by Lucas (987) and Alvarez, Lucas (25). 2

13 Proof. See Appendix E.3. First of all, since this measure only captures changes between free trade and the Nash equilibrium, it is driven by the strategic choices of tari rates. As tari rates only depend on size L and productivity through their e ect on the parameter c, so does the welfare measure V. The rst limit property states that the in nitesimally small economy will surely be worse o in the Nash equilibrium than under free trade where it faces the best terms-of-trade. In contrast, the in nitely large economy will be equally well o in the Nash equilibrium as under free trade because welfare converges to the autarky level in both situations. The last property states that the in nitely large country would bene t from an extension of the economy of its trading partner (from which monopoly rents can be extracted). As in Syropoulos (22), these limit properties (together with continuity) are su cient to establish existence of a unique threshold size c T at which a country is indi erent between the Nash equilibrium outcome and and Free Trade: Proposition 7 A country prefers the Nash-Equilibrium outcome over Free Trade if its e ective relative size c exceeds the threshold level c T (; ). Since the structure of the proof is essentially identical to Syropoulos treatment, a rigorous proof of existence and uniqueness is omitted. A graphical illustration is provided in Figure 5. The idea is as follows. Properties 2) and 3) ensure that it is possible to be better o in the Nash equilibrium than under free trade (i.e. V > for c very large). By property ), the small country will be worse o in the Nash equilibrium such that V is negative for small c: The existence of the threshold size level follows by continuity. If the expenditure functions b (z) and b (z) coincide, then the threshold size level of both countries will be identical by symmetry. The threshold size level is greater than, because a Nash equilibrium induces Pareto ine cient production. Hence, when the countries are of equal size they will be both worse o than in the free trade scenario. Such a Prisoner s Dilemma situation will always occur provided that the size asymmetries are not too great, i.e. whenever jcj < c T. In addition to the veri cation of Syropoulos core existence result (albeit in a di erent framework), I am able to analyze the impact of transportation cost and comparative advantage on the threshold size level c T : Two easily interpretable assumptions are required: Assumption (c T < Assumption (c T > Assumption 2 implies that an increase in comparative advantage will imply greater welfare gains in the Free-Trade situation than in the Nash equilibrium (for a country at the threshold size). Assumption 3 implies that transportation cost are relatively more harmful under free trade than in the Nash equilibrium (for a country at the threshold size). For the rest of this paper, I will assume that these assumptions are satis ed: 3

14 Proposition 8 The threshold level c T is an increasing function of and a decreasing function of. Proof. The threshold size level is characterized by the following implicit function: V (c T ; ; ) = By the implicit function theorem, we @c i (ct ; ; {z } > 2 (c T ; ; {z } (c T ; ; {z } > {z } see Assumptions The partial derivative with respect to size must be positive at the threshold size level which follows directly from the discussion of Proposition 7 and Figure 5. For the log-linear speci cation of technology, the required threshold c T size turns out to be a particularly simple linear function of and (see Figure 6). 5 Implications for Self-Enforcing Trade Agreements The goal of this section is to discuss the general implications of the static Nash equilibrium analysis for cooperative trade agreements within a dynamic context. Rather than solving for the optimal dynamic contract, I want to highlight the key intuitions that can be obtained from the static analysis and point to the relevant extensions in a dynamic setup. 7 It is well understood that any trade agreement has to be self-enforcing due to the lack of international courts with real enforcement power. Thus, a sustainable (subgame perfect) contract requires that the short-run bene t from a unilateral tari choice (i.e. setting the static best response tari s given by equations 3 and 4) must be overwhelmed by the long-run cost resulting from the retaliation of the other country. 8 Due to the lack of commitment such a punishment by the other country has to be itself credible, i.e. subgame perfect. The static Nash equilibrium outcome can be used as such an o -equilibrium path threat point to induce cooperation on the equilibrium path. 9 Due to asychronous nature of the cost and bene t of reneging on a contract, impatience plays a central role for the Pareto set of equilibrium contracts. The folk theorem implies that as the discount factor approaches one, virtually any tari pair on the contract curve the set of internationally e cient tari combinations that yield higher utility than the Nash outcome (see Mayer, 98) becomes self-enforcing. 2 However, the static analysis reveals that this contract curve may not include reciprocal free trade in which case free trade is not sustainable (even in the limit). Such a situation will always occur in the presence of su cient size asymmetries, 7 A rigorous derivation of the optimal dynamic contract using the ingenious machinery of Abreu et al. (99) is interesting enough in its own right and beyond the scope of this paper. 8 A summary of this literature is found in chapter 6 of Bagwell and Staiger (22). 9 Friedman s (97) formal analysis of oligopolistic structures uses this idea rst. Bagwell and Staiger (22) show that there is also a limited role for on-equilibrium path retaliation within the GATT framework. This idea is not explored further in this paper. 2 E cient tari combinations satisfy: ( + t) ( + t ) =. 4

15 i.e. if one economy exceeds the threshold size level c T. 2 As the threshold size c T is decreasing in transportation cost, it is easier to sustain free-trade agreements among countries which are geographically closer. Thus, my analysis is consistent with the regional focus of most free trade zones across the world such as NAFTA or the EU. An interesting implication of the repeated games analysis is that the e cient contracts can feature non-trivial dynamics even in a physically stationary environment. The e cient intertemporal provision of incentives without commitment requires the agent with the strongest incentive to deviate to obtain back-loaded payo s. 22 Applied to our setting, we can state that the welfare-maximizing contract for the small country (i.e. the contract on the Pareto frontier which is chosen when the small country has full bargaining power) necessarily involves frontloaded bene ts to the small and back-loaded bene ts to the large country. This is consistent with Bond and Park s analysis (22), who use the gradual nature of the trade liberalization between Poland and the EU as an empirical example. Despite these interesting predictions within a stationary environment, the assumption of stationarity in a clearly non-stationary world prevents the analysis of important dynamics that matter for trade agreements. 23 Especially of interest is how the anticipated strong growth of the BRIC countries, most notably China, in uences trade agreements. 24 The dynamics speci cation does not require separate processes for productivity and the demographics L, as both can be conveniently summarized by e ective relative size c. Intuitively, one would expect that anticipated strong growth diminishes the usefulness of providing front-loaded contracts to such economies, as they will have greater incentives to renege on a contract in the future. Moreover, it is interesting to study the e ect of comparative advantage through time series variations in the dispersion parameter. The static analysis suggests, that in times when specialization gains are high (and expected to stay high) free trade agreements should be easier to implement as the threshold size c T has been shown to be an increasing function of. The intuitive appeal is clear: larger specialization gains are more bene cial under cooperation than in the ine cient Nash equilibrium. Moreover, under the assumption that gains from trade follow a predictable trend, one can explain the gradual nature of trade agreements most notably the GATT rounds of negotiations Conclusion The analysis of this paper has suggested that the rigorous understanding of the static Nash equilibrium outcome can be viewed as a stepping stone to the more complicated analysis of self-enforcing trade agreements within a dynamic context. In order to study the important impact of growth dynamics on the sustainability and evolution of trade agreements it seems 2 This is a su cient condition, i.e. even when one country s relative size is slightly smaller than c T, free trade may not be sustainable as the small country does not apply a zero tari rate when its relative country size is c T : 22 The intuition goes back to the seminal paper of Harris and Holmstrom (982) within the labor market context. 23 In this spirit, Bagwell and Staiger (995) use a dynamic partial equilibrium model to explain the countercyclical nature of trade barriers. 24 The term BRIC countries refers to the emerging market economies Brazil, Russia, India and China. 25 This idea is somewhat a reduced form implication of the analysis by Devereux (997) who assumes that production technologies exhibit "learning by doing" features which generate predictable specialization gains. 5

16 worthwhile to prefer a more complicated dynamic game setup over a strictly repeated game framework. This line of research might produce interesting implications about the relationship between trade, growth and technology di usion. Within a multiperiod setup, it is also possible to meaningfully relax the assumption of period-by-period balanced trade and replace it by a present value constraint. Speci cally, one can address the interaction of the optimal dynamic government debt and trade policy in an environment without commitment. Comparative advantage on the production side might not only be helpful for enforcing trade agreements, but also for inducing repayment of sovereign debt. 26 This intution goes back to Bernheim and Whinston (99) who nd that cooperative behavior is easier to sustain among rms with multimarket contact. Instead of abandoning the static framework, one could understand the derived Nash equilibrium tari rates as empirical predictions for observed tari rates. 27 In order to meaningfully test whether applied tari rates are increasing in the size of the economy and comparative advantage and decreasing in transportation cost (distance) historical tari data of Non-WTO countries needs to be used. This would be very much in the spirit of Broda et al. (26). 28 Interesting datasets can also be obtained by going back further in time. For example, at the beginning of the 9 th century countries like Germany consisted essentially of a myriad of independent states ranging from the small free city of Frankfurt to the large state of Prussia all with a separate tari system This analysis is relevant in the light of the famous Bulow and Rogo (989) result who show that sovereign debt is not sustainable without e ective sanctions. Such sanctions can be achieved through trade policy. 27 On the theory side, one might want to extend the static analysis by allowing for more general classes of preferences such as in Wilson (98). Another idea is to abandon the 2 country framework as in Alvarez and Lucas (25) which would also allow the examination of multilateral enforcement of trade agreements in the spirit of Maggi (999). 28 The authors, however, do not nd convincing support for a (linear) relation between size and applied tari rates in their sample. Another prediction of my analysis uniformity of tari rates is probably more of technical rather than practical relevance. 29 A common system was not in place before the creation of the "Deutscher Zollverein" in

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