tari⁄wars in the ricardian model with a continuum of...
TRANSCRIPT
Tari¤Wars in the Ricardian Model with a Continuum of Goods
Marcus M. Opp�
November 1, 2009
Abstract
This paper describes strategic tari¤ choices within the Ricardian framework of Dorn-busch, Fischer, and Samuelson (1977) using CES preferences. The optimum tari¤ scheduleis uniform across goods and inversely related to the import demand elasticity of the othercountry. In the Nash equilibrium of tari¤s, larger economies apply higher tari¤ rates. Pro-ductivity adjusted relative size (� GDP ratio) is a su¢ cient statistic for absolute produc-tivity advantage and the size of the labor force. Both countries apply higher tari¤ ratesif specialization gains from comparative advantage are high and transportation cost arelow. A su¢ ciently large economy prefers the ine¢ cient Nash equilibrium in tari¤s over freetrade due to its quasi-monopolistic power on world markets. The required threshold sizeis increasing in comparative advantage and decreasing in transportation cost. I discuss theimplications of the static Nash equilibrium analysis for the sustainability and structure oftrade agreements.
JEL classi�cation: C72, F11, F13.
Keywords: Optimum Tari¤ Rates, Ricardian Trade Models, WTO, Gains from Trade
�University of California, Berkeley (Haas School of Business), 545 Student Services Bldg. #1900, Berkeley,CA 94720. The author can be contacted via email at [email protected] or phone (510) 643 0658. I amespecially grateful to Robert E. Lucas, Jr. for encouragement and guidance to further develop a term paper forhis class "Economic Trade and Growth" into this publication. 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 appreciate helpful comments by Fernando Alvarez, Christian Broda, Thomas Chaney, Pedro Gete, SamuelKortum, Dmitry Livdan, Luis-Fernando Mejía, Christian Opp, Tiago Pinheiro, Christis Tombazos and JohanWalden. I acknowledge great feedback from the participants of the Economics Student Working Group and theCapital Theory Working Group at the University of Chicago. All remaining errors are mine.
1
1 Introduction
Protectionist policies such as tari¤s still represent major impediments to free trade, as exem-pli�ed by the recent failure of the Doha Development Round. The temptation to impose tari¤srepresents a classical problem studied by economists such as Bickerdike (1907): A country canimprove its terms-of-trade via unilateral tari¤s at the cost of ine¢ cient resource allocation anda reduction in trade volume. The theory of optimum tari¤s, which trade o¤ these bene�tsand costs, has central (empirical) implications for two purposes: On the one hand, it providestestable predictions of actual tari¤ rates: As such, Broda, Limao, and Weinstein (2008) provideevidence that non-WTO countries apply tari¤ rates consistent with the theory. On the otherhand, it helps us understand the incentive problems that countries face when they enter legallynon-enforceable tari¤ agreements. According to Bagwell and Staiger (1999) the sole rationalefor trade agreements is to escape terms-of-trade driven prisoner�s dilemma situations.
For both of these purposes, understanding the determinants and implications of strategictari¤ choices within a general equilibrium production framework is central, but nonethelesslargely unstudied due to the level of complexity such an analysis entails.1 This paper aimsto bridge that gap within a particularly simple general equilibrium framework �namely theRicardian Model à la Dornbusch, Fischer, and Samuelson (1977) (henceforth labeled DFS).Their setup allows me to study the role of technology in the form of absolute and comparativeadvantage as well as transportation cost for optimum tari¤policies. Moreover, I can characterizethe intuitive repercussions of these tari¤ policies on the allocation of productive resources(e¢ ciency) and the distribution of welfare.
At the heart of the paper is a generalized DFS framework with CES preferences in whichtari¤ rate policies are endogenously determined by benevolent governments.2 Within this frame-work, the optimum tari¤ rate schedule is uniform across goods. This result holds for di¤erentexpenditure share parameters (across goods and countries), di¤erent elasticities of substitution(across countries) as well as arbitrary speci�cations of technology. Moreover, it is robust tothe inclusion of transportation cost. The result may be surprising to the reader of Itoh andKiyono (1987) who �nd that non-uniform export subsidies � interpretable as negative tari¤rates �are welfare-improving in the DFS model. The apparent contradiction can be resolved astheir carefully designed export-subsidy policy is not proved to be optimal, but solely welfare-enhancing relative to free trade. By reducing the potentially complicated tari¤ schedule choiceto a one-dimensional problem, I am able to derive an easily interpretable optimality conditionfor the tari¤ rate: The expression is inversely related to an appropriately de�ned import de-mand elasticity of the other country. Tari¤s can thus be interpreted as optimum markups onexport goods. A higher foreign elasticity of substitution among goods increases the foreignimport demand elasticity and hence reduces optimum tari¤ rates.
I prove existence of a unique "trembling-hand-perfect" Nash equilibrium of tari¤s in whichlarger economies apply higher tari¤s. Productivity adjusted relative size (� GDP ratio) is asu¢ cient size statistic for optimum tari¤ rates as higher average absolute productivity impactsthe optimum tari¤ rate exactly as a larger relative size of the labor force. The intuition behind
1 The analysis of Alvarez and Lucas (2007) shows how di¢ cult it is to obtain intuitive results within a generalequilibrium framework with strategic tari¤ choices. General equilibrium production models with multiplegoods and exogenous trade barriers have become standard to study the determinants of trade �ows (see Eatonand Kortum (2002))
2 Thus, the original DFS setup with Cobb-Douglas preferences is a special case.
2
my results is as follows: Small economies are heavily dependent on trade and therefore pos-sess a relatively inelastic import demand function. This can be exploited by larger economiesthrough the lever of tari¤ rates to achieve terms-of-trade e¤ects (intensive margin) while hardlyincreasing their already large domestic production (extensive margin). As a result of strategictari¤s, the terms-of-trade will (approximately) only re�ect di¤erences in productivity, but notdi¤erences in the size of the labor force. In contrast to free trade, a small country will not beable to capture the specialization gains that arise from the focus on the production of goodswith the highest comparative advantage.
Both countries apply higher tari¤ rates if specialization gains (comparative advantage) arehigh. This is because any given increase in tari¤s causes smaller deviations from e¢ cient pro-duction. Consider the limiting case, when both countries are very similar and thus comparativeadvantage is low. Then, even a small tari¤ can completely exhaust the gains from trade. Trans-portation cost have the opposite e¤ect of comparative advantage. Higher transportation coste¤ectively reduce the potential gains from trade. This makes tari¤s more costly and reducesequilibrium tari¤ rates.
The welfare analysis implies that a su¢ ciently large economy is better o¤ in a Nash equi-librium 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 ine¢ cient tari¤ equilibrium. The structureof the DFS framework enables me to show that the threshold size level is an increasing functionof comparative advantage and a decreasing function of transportation cost. Hence, if e¤ectivegains from trade are high (high comparative advantage, small transportation cost) and there-fore both countries apply higher tari¤ rates in equilibrium a country has to be larger to preferthe Nash equilibrium outcome over free trade.
The Nash equilibrium analysis can be used as a stepping stone to study self-enforcing tradeagreements in the spirit of Bagwell and Staiger (1990), Bagwell and Staiger (2003) and Bondand Park (2002). The static Nash equilibrium outcome determines the (o¤-equilibrium path)punishment payo¤s for deviating from a trade agreement. I extend the work of Mayer (1981)to a general equilibrium production setting and �nd that e¢ cient tari¤ combinations can im-plement any desired welfare transfer from one country to the other. Free trade agreements canonly be sustained without transfers if both governments are su¢ ciently patient and size asym-metries are not too large. In contrast, if one government is su¢ ciently impatient, the short-runtemptation to renege on agreements outweighs the long-run cost, so that the static Nash equi-librium occurs on the equilibrium path. To the extent that high discount rates re�ect politicaleconomy considerations as in Acemoglu, Golosov, and Tsyvinski (2008) or Opp (2008), politi-cal factors determine whether the Nash equilibrium outcome characterizes the non-observableoutside option or the actual equilibrium outcome. Thus, terms-of-trade considerations can berelevant in the sense of Bagwell and Staiger (1999) or Broda, Limao, and Weinstein (2008).
My paper is related to various lines of research. I follow the traditional economic approachto this subject by not explicitly considering political factors and viewing optimum tari¤ ratesas optimal strategic decisions within a single period non-cooperative game. While governmentactions may realistically involve political considerations (see Grossman and Helpman (1995)),such a model does not o¤er a separate rationale for trade agreements which are designed toescape the terms-of-trade driven prisoner�s dilemma in a Nash equilibrium (see Bagwell andStaiger (1999)). However, as pointed out above political factors may matter indirectly throughthe discount rate for the feasibility of trade agreements.
3
The dominant part of the existing literature on tari¤ games uses a two-good exchangeeconomy setup to analyze the strategic choice of tari¤ rates. This approach is largely inspiredby Johnson (1953). He �nds that a Nash equilibrium of tari¤s does not necessarily result ina prisoner�s dilemma situation in which both countries are worse o¤ than under a free traderegime. Most of the subsequent papers focus on generalized preferences (Gorman (1958)), theimpact of relative size (Kennan and Riezman (1988)) or existence of equilibria (Otani (1980)).By incorporating a production sector into the analysis, Syropoulos (2002) is able to signi�cantlyimprove 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 levelthat will cause the bigger country to prefer a tari¤-equilibrium over free trade. As opposed tothe Ricardian framework employed in my paper the Heckscher-Ohlin theory explains gains fromtrade by di¤erences in factor endowments rather than technology across countries. My papercan be seen as a response to the concluding remarks of Syropoulos who states that "it wouldbe interesting to examine how technology a¤ects outcomes in tari¤ wars" and "it would beworthwhile to investigate whether the �ndings on the relationship between relative country sizeand tari¤ war outcomes remain intact in multi-commodity settings". In addition to technologydi¤erences and the multi-commodity framework, I also consider the role of transportation costfor tari¤ policies. If one added these generalizations to the Syropoulos setup and assumedinternational capital mobility, the models should produce very similar results.3
General treatments on the optimum tari¤ structure with multiple goods go back to Graa¤(1949) and more recently Bond (1990) and Feenstra (1986). Their analysis yields conditions(e.g. on the substitution matrix) which can generate more complicated tari¤ structures suchas subsidies for some goods. Due to the constant elasticity of substitution the optimum tari¤structure turns out to be particularly simple in the DFS setup and consistent with this literature.
McLaren (1997) develops an innovative paper in the spirit of Grossman and Hart (1986)that yields the counterintuitive result that small countries may prefer an anticipated trade warrelative to an anticipated trade negotiation.4 The key driver for this result is that an irreversibleinvestment in the export sector by the small country in period 1 will reduce the threat pointin negotiation talks in period 2 (after the investment is sunk). Despite the di¤erent underlyingeconomic intuition, it is con�rmed that small size brings about a strategic disadvantage.
My paper is organized as follows. In section 2 I characterize the equilibrium conditionsof the generalized DFS Model with CES preferences. In section 3 I prove the optimality ofa uniform tari¤ schedule and derive the optimum tari¤ rate formula. Section 4 presents theNash equilibrium analysis using Cobb-Douglas preferences and an intuitive speci�cation oftechnology. The implications of my static analysis for trade agreements within a dynamiccontext are discussed in section 5. Section 6 concludes. For ease of exposition, the proofs ofmost propositions are moved to the Appendix, while the intuition is delivered in the main text.The notation has been kept in line with the original DFS paper.
3 (Neo-) Heckscher-Ohlin models with di¤erences in technology and one mobile factor are reduced to a standardRicardian model (see Chipman (1971) for an elegant proof).
4 This situation can occur if the production technologies are similar in terms of absolute and comparativeadvantage.
4
2 DFS Framework
2.1 Setup
The purpose of this section is to compactly describe the general framework of the RicardianModel with a continuum of goods à la DFS. There are two countries (home and foreign) withrespective labor endowments L and L�. Note, that asterisks are used throughout the paper torefer to the foreign country. Tildes refer to the log transformation. Without loss of generality,I normalize the size of the labor force (L�) and wage rate (w�) of the foreign country to unitysuch that the relative size of the home country is simply given by its absolute size L and thewage rate of the home country is equal to the relative wage rate !.
Production Technology There exists a continuum of goods (z) which are indexed over theinterval [0; 1] and produced competitively with a linear technology in labor. The labor unitrequirement functions a (z), a� (z) specify how many labor units are required to produce oneunit of good z in the home country and foreign country, respectively. Both a (z) and a� (z) areassumed to be twice di¤erentiable and bounded over the domain [0; 1]. The ratio of a (z) anda� (z) is de�ned as A (z):
A (z) � a� (z)
a (z)(1)
Goods are ordered in terms of decreasing comparative advantage of the home country, whichimplies that the function A (z) is decreasing in z over the domain [0; 1]. The log transformationof A (z) �denoted as ~A (z) �yields the productivity (dis)advantage of the home country inpercentage terms:
~A (z) � log [A (z)] (2)
Preferences The representative agent of each economy possesses a demand function gener-ated by CES preferences. Moreover, I extend the original DFS analysis �which relied on aCobb-Douglas speci�cation �by allowing the utility function to di¤er across countries:
U (c) =�R 10 � (z)
1� c (z)
��1� dz
� ���1
U� (c�) =
�R 10 �
� (z)1�� c� (z)
���1�� dz
� �����1
(3)
Thus, the share parameters � (z) ; �� (z) and the elasticities of substitution �; �� need not becommon. I make the technical assumption that � (z) and �� (z) are Lebesgue measurable.
Trade barriers I allow for two types of trade barriers: Import tari¤s (t (z) ; t� (z)) andexogenous symmetric iceberg transportation cost � (see Samuelson (1952)).5 Thus, only afraction exp (��) of exported goods eventually arrives in the other country.6 Import tari¤s are5 The author has veri�ed that the analysis of this paper extends one-to-one to export tari¤s. Thus, Lerner�s"Symmetry-Result" holds in this setup with a continuum of goods.
6 � is assumed to be su¢ ciently small, such that trade in at least some products would take place in the absenceof import tari¤s.
5
applied to all goods that arrive in the import country and redistributed to the consumers. Ide�ne gross import tari¤s as:
T (z) � 1 + t (z) (4)
T � (z) � 1 + t� (z) (5)
2.2 Equilibrium Conditions
2.2.1 Production
Perfect competition and the linear production technology imply that �nal consumption pricesof domestically produced goods pD (z) are equal to production cost whereas prices of importgoods pI (z) also re�ect transportation cost and tari¤s.
pD (z) = !a (z)pI (z) = a� (z) exp (�)T (z)
(6)
Let I denote the set of import goods and D denote the set of domestically produced goods, i.e.:
I = fzj pD (z) � pI (z)g (7)
D = fzj pD (z) < pI (z)g (8)
For ease of exposition, I assume that the currently exogenous tari¤ rate policies are given byLebesgue-measurable functions that divide the set of goods into two connected sets, i.e. Iand D.7 Hence, the set I and D are completely determined by a cuto¤ good �z such that allgoods z < �z are produced domestically and all goods z � �z are imported. Analogously, theforeign country imports goods in the interval I� = [0; �z�] and produces goods domestically inthe interval D� = (�z�; 1].8 Therefore, e¢ cient production specialization implies the followingtwo equilibrium conditions:
A (�z) � !exp(�)T (�z)
A (�z�) � !a (�z�) exp (�)T � (�z�)(9)
2.2.2 Consumption and Balance of Trade
It is well known that the optimal consumption schedule �c (z) with CES preferences is given by:
�c (z) = y� (z)
p (z)� P 1��(10)
where y represents the per capita income of the home country and P an appropriately de�nedprice index:
P ��Z 1
0� (z) p (z)1�� dz
� 11��
(11)
7 As in Itoh and Kiyono (1987), this assumption is made for ease of exposition.8 Prices of domestically produced goods in the foreign country are given by p�D (z) = a� (z) whereas importgood prices are given by: p�I (z) = !a (z) exp (�)T � (z) :
6
Note that unless preferences are of Cobb-Douglas type (� = 1), the expenditure share b (z) doesnot only depend on the exogenous share parameter � (z) but also on the endogenous price ofgood z and the price index P :9
b (z) � � (z)
�p (z)
P
�1��(12)
Given e¢ cient production specialization the share of income spent on domestically producedgoods # is:
# (�z; p (z) ; P ) �Z �z
0b (z) dz (13)
Balance of trade requires that the value of imports by the home country at world market(pre-tari¤) prices must be equal to the value of imports by the foreign country:
Ly1� #�T
= y�1� #��T �
(14)
where �T = 1�#R 1�z
b(z)T (z)
dzand �T � = 1�#�R �z�
0b�(z)T�(z)dz
can be interpreted as average tari¤ rates. Using the
de�nitions of after-tax per capita income y = !�T
1+#�t and y� =
�T �
1+#��t� , the balance of tradeequilibrium condition can be rewritten as in the original DFS paper:
! =1 + #�t
1� #1� #�1 + #��t�
1
L(15)
3 Optimum Tari¤Rates
So far, the analysis has treated tari¤ rates similar to transportation cost, i.e. as exogenouslygiven trade barriers. However, in contrast to transportation cost, import tari¤s are choicevariables for the respective government and a¤ect national income.10 Due to the CES demandstructure, the indirect utility function of the home country�s representative agent V (fp (z)g ; y)can be written as:
V (fp (z)g ; y) = y
P(16)
It is assumed that each government tries to maximize the real income of the representative agentyP given the tari¤ rate decision of the other country and subject to the equilibrium conditionson !; �z and �z� given by balance of trade (equation 15) and the cut-o¤ good de�nitions (seeequation 9).
Proposition 1 In the generalized DFS framework with CES preferences, the optimum tari¤rate is uniform across all import goods.
Proof: It is useful to apply the log transformation of the government�s objective function.Using basic algebra the problem can be stated as:
maxfT (z)g
~V (fT (z)g) = ~! � �
1� � log (FD + FI)� log (FD + FIx) (17)
9 P is well speci�ed since � (z) ; a (z) ; a� (z) and T (z) are measurable functions.10 If tari¤ rebates were simply lost �as in the case of iceberg transportation cost �the optimum tari¤ rate wouldbe zero.
7
subject to:
g1 = ~L+ ~! + log (FIx)� log (FD + FIx) + log (F �D + F �Ix)� log (F �Ix) = 0
g2 = ~! � � � log (T (�z))� ~A (�z) � 0
g3 = ~! + � + log (T � (�z�))� ~A (�z�) � 0
where the functions FD and F �D are proportional to the share of domestically produced goods#; #� and FI ; F �I ; FIx are related to the share of imported goods inclusive of tari¤s and ex-tari¤s(indicated with x):
FD (!; �z) =
Z �z
0� (z) pD (z)
1�� dz (18)
FI (fT (z)g ; �z) =Z 1
�z� (z) pI (z)
1�� dz (19)
FIx (fT (z)g ; �z) =Z 1
�z� (z)
pI (z)1��
T (z)dz (20)
De�nitions are analogous for the foreign country. The Lagrangian ~� can be written as:
~� = ~! � �
1� � log (FD + FI)� log (FD + FIx)�3Xi=1
�igi (21)
where �i represents the Lagrange multiplier on equilibrium constraint gi. Note, that onlythe terms FI and FIx depend directly on the home country tari¤ schedule T (z). Thus, theargument of these functionals FI ; FIx is itself a function. It is useful to de�ne the functionsfj (z) = � (z) pj (z)
1�� such that Fj =Rj fj (z) dz. Now, the Euler-Lagrange equations for this
problem can be written as:
� �
1� �1
FD + FI
@fI (z)
@T (z)� 1
FD + FIx
@fIx (z)
@T (z)� �1
�1
FIx� 1
FD + FIx
�@fIx (z)
@T (z)= 0 (22)
where the partial derivatives of fI and fIx with respect to the tari¤ rate T (z) are given by:
@fI (z)
@T (z)=1� �T (z)
fI (z) (23)
@fIx (z)
@T (z)= � �
T (z)2fI (z) (24)
As@fI (z)
@T (z)@fIx(z)
@T (z)
= �T (z) 1��� we can rewrite the Euler-Lagrange equations as:
T (z) = (FD + FI)
�1� �1
FD + FIx+
�1FIx
�(25)
The right hand side is independent of z. So the left hand side must be independent of z aswell, i.e. T (z) = T 8z � �z. The solution that satis�es the Euler-Lagrange equation is a globalmaximum since the objective function is quasi-concave and the constraint set is convex.11 Thiscompletes the proof.11 Uniqueness is subject to a technical quali�cation. Any tari¤ rate T (�z) < T that satis�es equilibrium constraint2 would be optimal as well since good �z has measure 0. Generally, deviations from the optimum tari¤ ratepolicy for a �nite number of goods will not a¤ect welfare.Tari¤ rates for non-import goods are also not uniquely pinned down. For each good ~z < �z, there exists a lowerbound TL (~z) � T such that for any ~T � TL (~z) good ~z is not imported. For non-import goods tari¤ rates candi¤er from the optimum tari¤ rate even on a set of goods with positive mass.
8
Proposition 1 is important because it justi�es the focus on uniform tari¤ rates in this frame-work. It has to be stressed that uniformity has been proved under general labor unit requirementfunctions, di¤erent preferences across countries and with possibly di¤erent expenditure shareparameters � (z) across goods. Due to this result, tari¤ rate policies of each country can besimply summarized by only two variables t and t�, respectively.
Proposition 2 The optimum tari¤ rate in the DFS model tR can be expressed as follows:
tR =1 + #�t�
T �1
� b�(�z�)~A0(�z�)
11�#� + #
� (�� � 1)(26)
t�R =1 + #t
T
1
� b(�z)~A0(�z)
11�# + # (� � 1)
(27)
Proof: See Appendix A.2.
The optimum tari¤ rate trades o¤ the terms-of-trade improvements with the ine¢ cientexpansion of domestic production and the costly reduction in trade. The economic intuitionis as follows: By imposing a tari¤ on import goods, the home country increases the �nalconsumption prices of foreign goods which in turn reduces demand for those goods in the homecountry. At the old equilibrium prices (before imposing the tari¤) the balance of trade conditionwill no longer hold as the foreign country will demand too many import goods. In order toeliminate the excess demand for import goods in the foreign country, the terms-of-trade have toimprove from the perspective of the home country (intensive margin). Since the elasticity of the
equilibrium wage rate with respect to tari¤s is smaller than one�d log(!)d log(T ) < 1
�the home country
will ine¢ ciently expand home production to products with a lower comparative advantage(extensive margin).
In order to characterize the optimum tari¤ rate in terms of demand elasticities ("; "�), it isuseful to interpret the per-capita import demand m as a composite good for the continuum ofimport goods. The real per capita import demand of the home country is essentially given bythe left hand side of the balance of trade condition (see equation 14):
m = y1� #T
= !1� #1 + #t
(28)
The "price" of an import good in local currency is proportional to the inverse of ! such thatwe can de�ne the import demand elasticity as:
" =
���� d logmd log!�1
���� = d logm
d~!(29)
It can be shown that the optimum tari¤ rate of the foreign country (see Proof of Proposition2 in Appendix A.2) can be written as t�R =
1"�1 and analogously for the home country tR =
1"��1 .
This pricing formula is identical to the optimal markup that a monopolistic �rm charges over
marginal cost MC, i.e. p =�1 + 1
"�1
�MC. To see this analogy, it is useful to apply Lerner�s
symmetry result with regards to import and export taxes (see Lerner (1936)): The optimumimport tari¤ is equivalent to the optimal export tari¤. While goods are priced competitively
9
within each economy each country behaves like a monopolist for the goods it exports. In theabsence of retaliation a government can improve country welfare through tari¤s because itcan e¤ectively enforce markup pricing on world markets despite perfect competition within itscountry. Intuitively, the optimal tari¤ tR is smaller the greater the degree of substitutabilitybetween goods (��) (see equation 26). In the limit, as preferences of the trade partner becomelinear, the optimum tari¤ rate approaches 0.
4 Equilibrium Analysis
4.1 Speci�cation
The derived optimum tari¤ rate formulae (see equations 26 and 27) allow us to calculate op-timum tari¤ rate policies for arbitrary technology and taste speci�cations. For ease of expo-sition, I use simple speci�cations of technology and preferences to be able to make sharperpredictions about comparative statics in the DFS setup.12 Preferences are of Cobb-Douglastype (� = �� = 1) and all goods are valued equally (b (z) = � (z) = 1).13 These preferences im-ply that the optimum tari¤ rate only depends on the relative labor unit requirement functionA (z), but not separately on a (z) and a� (z).14 The share of income spent on domesticallyproduced goods #; #� is directly determined by the respective cut-o¤ goods:
# = �z (30)
#� = 1� �z� (31)
The relative labor unit requirement function is assumed to be exponentially a¢ ne, which shouldbe interpreted as a linear projection of the true relative productivities:
~A (z) � �+ =2� z; � 2 R and 2 R+ (32)
Thus, two parameters � and are su¢ cient determinants of the technology. It can be readilyveri�ed that the function is strictly positive and bounded over the speci�ed domain [0; 1]. Thee¤ect of di¤erent parameter choices for � and is illustrated in Figure 1. The parameter �can be interpreted as the absolute productivity advantage of the home country averaged over
all goods, i.e.1R0
(�+ =2� z) dz = �.15 In contrast, the parameter measures the degree of
comparative advantage (dispersion parameter). Larger values of generate a greater di¤usionof relative productivities across goods which implies greater gains from trade.16
12 A previous draft of the paper considered a more general speci�cation which I have dropped for simplicity. Theresults hold more generally for bounded technology speci�cations.
13 Empirical evidence (see Weder (2003)) suggests that countries have a taste bias in favor of the goods theyexport, i.e. goods with a higher comparative advantage. In contrast to Opp, Sonnenschein, and Tombazos(2009) I do not account for this empirical fact.
14 This is because the expenditure share b (z) is given by the exogenous share parameter � (z) for Cobb-Douglaspreferences.
15 Due to the simple a¢ ne structure this coincides with the productivity advantage at the median good (z = 0:5),i.e. ~A (0:5) = �:
16 The parameter is closely related to the variance parameter of individual productivities in the Eaton-KortumModel (see also Alvarez and Lucas (2007)).
10
The speci�cation generates easily interpretable expressions for the cuto¤ goods and the sizeof the non-traded sector N = �z � �z�:
�z = 12 +
�+�+log(T )�~!
�z� = 12 +
����log(T �)�~!
N = 2�+log T+log T �
(33)
Intuitively, the size of the non-traded sector N solely depends on the ratio of exogenousand endogenous trade barriers (2� + log T + log T �) to the potential gains from trade . Thus,barriers are prohibitive if 2� + log T + log T � > . In the following analysis, I assume that theexogenous transportation cost satisfy 2� < .
4.2 Optimum Response Function
In this speci�c setup the optimum tari¤ rate expression becomes:
tR = �z�1 + (1� �z�) t�
T �(34)
t�R = (1� �z) 1 + �zt1 + t
(35)
Note that �z and �z� are functions of the tari¤ rates as well as the parameters for size andtechnology. It can be shown (see Appendix A.1) that productivity adjusted size c = � + ~Lis a su¢ cient statistic for the parameters L and � in determining the cuto¤ goods �z and �z�.Therefore, tari¤ rates are also just a function of e¤ective relative size c.17
In order to make sharper predictions about the properties of the best response functions, Ineed to make a mild technical assumption.
Assumption 1 The foreign country�s tari¤ rate satis�es: t� (2�z� � 1) < 1
A violation of assumption 1 necessarily requires an unrealistic combination of high foreigntari¤ rates (more than 100%) and a large foreign import sector (more than 50% of the goodsare imported).18
Proposition 3 The optimum response function tR has the following properties:- Decreasing in the other country�s tari¤ rate dtR
dt� < 0
- Decreasing in the transportation cost dtRd� < 0
- Increasing in the dispersion parameter dtRd > 0
- Increasing in a country�s relative production capacity dtRdc > 0
17 E¤ective relative size is closely related to the amount of goods that can be produced under autarky, i.e. thesize of the economy (production capacity). For any speci�c good the relative production capacity (number ofavailable labor units / number of required labor units) of both countries is given by C (z) = L
a(z)= L�
a�(z) = LA(z):
Taking logarithms and averaging over all goods implies that we can de�ne c � log (L) + �:18 Recall, that t; t� > 0 and 0 � �z� � �z � 1 and that 1� �z and �z� can be interpreted as the size of the respectiveimport sectors. Numerical results show that < 7:4 is a su¢ cient condition for the validity of assumption1. A value of = 7:4 would imply that the ratio of the relative labor unit requirement function at the twoendpoints A(0)=A(1) is equal to exp(7:4) � 1635.
11
Proof: See Appendix B.3.
Thus, tari¤s are strategic substitutes (see Figure 2) with an elasticity d log TRd log T � smaller than 1
(see Appendix B.4). The intuition behind this result is that higher foreign tari¤ rates increasetrade barriers that exogenously increase the size of the non-traded sector N = 2�+log T+log T �
:Hence, the room for imposing additional own trade restrictions without heavily reducing oreven shutting down trade is limited.
From the perspective of the home country, transportation cost � are very similar to theforeign country�s tari¤ rate t�: Both e¤ectively represent exogenous barriers to trade. Con-sequently, increases in transportation cost must reduce the optimum tari¤ rate of the homecountry. A �atter relative labor unit requirement function (smaller ) implies that any giventari¤ rate will result in a greater (ine¢ cient) expansion of the domestic sector, i.e. the cut-o¤good is more responsive to tari¤s. Smaller specialization gains thus render domestic and foreignproduction to be more substitutable. Therefore, the optimum tari¤ rate must be smaller.
The positive marginal impact of the size (c) re�ects that larger economies have a smallerimport 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 smallerthe greater the economic weight of one country. Consider the case when the home economy isvery large such that almost all goods are produced domestically even without tari¤s. In thiscase, the cost of tari¤s � ine¢ cient expansion of home production � is only of second-orderconcern. In contrast, the in�nitesimally small country will not impose a tari¤ as the importdemand elasticity of the large country approaches in�nity.19
4.3 Nash Equilibrium
4.3.1 Existence
In a Nash equilibrium both economies�tari¤ rate choice represents an optimum response to thetari¤ rate of the other country. The optimum response functions for two parameter constella-tions 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 oc-cur, if both countries choose a prohibitively high tari¤ rate tproh > � 2�, i.e. a tari¤ rate thatexhausts the gains from trade adjusted for transportation cost. There exists a continuum ofNo-trade equilibria all of which are uninteresting, as applying a prohibitive tari¤ rate is weaklydominated. Formally, these equilibria could be ruled out using the "trembling hand perfection"re�nement. In the following analysis I will only consider interior Nash equilibria.
Proposition 4 There exists a unique interior Nash equilibrium in tari¤s that Pareto dominatesany No-Trade Nash equilibrium.
Proof: See Appendix B.4.
The existence of a unique interior Nash equilibrium is established by applying the Contrac-tion Mapping Theorem. The contraction property follows directly from the fact that the slope19 This follows from the boundedness of the technology speci�cation: A violation of this property such as in theEaton-Kortum speci�cation of technology can imply strictly positive tari¤ rates even for the in�nitesimallysmall country.
12
of the optimum response function is less than one (see Figure 2).
4.3.2 Tari¤ Rates
The comparative statics of the optimum response function with respect to the exogenous para-meters c; � and are preserved in the Nash equilibrium of tari¤s:
Proposition 5 The Nash equilibrium tari¤ rate tN has the following properties:- Decreasing in transportation cost dtN
d� < 0
- Increasing in the dispersion parameter dtNd > 0
- Increasing in a country�s relative production capacity dtNdc > 0
Proof: See Appendix B.5.
Intuitively, the comparative statics of the Nash equilibrium tari¤ rate can be decomposedinto the direct e¤ect of the optimum response function and the feedback e¤ect through thestrategic tari¤ choice of the other country. In case of the size parameter c, the feedback e¤ectampli�es the direct e¤ect: As the home economy becomes relatively larger the foreign economywill apply lower tari¤ rates which in turn increases the tari¤ rate of the home country due tostrategic substitutability (see Proposition 3). In the case of the dispersion parameter andtransportation cost �, the feedback e¤ect counters the direct e¤ect, but is outweighed by thedirect e¤ect.
Figure 3 plots the comparative statics where : R ! [0; 1] denotes a normalized measureof size c:
(c) � exp (c)
1 + exp (c)(36)
This measure can be interpreted as the economic "weight" of a country, as the weights ofeach country (c) and (c�) = (�c) sum up to one. The in�nitesimally small country has aweight of zero, the in�nitely large country a weight of one.20 The model theoretic foundation forthis measure is based upon the �rst order Taylor series approximation of the Nash equilibriumtari¤ rate about the point = 0 (see Appendix B.6).21
tN � (c)� 2� ��
(37)
The (almost) linear graphs in Figure 3 reveal that the �rst-order approximation captures therelevant characteristics of the comparative statics.
4.3.3 Terms-of-Trade
Since terms-of-trade e¤ects are central to our analysis, it is helpful to revisit a key result fromthe DFS model under free trade: Countries with a relatively small size of the labor force L facesigni�cantly better terms-of-trade, a measure of relative welfare under free trade. The intuitionfor this result is that small countries can specialize their production on goods with the highest
20 The careful reader will notice that the function (x) is identical to the CDF of the logistic distribution.21 Formally, I take the limit as approaches 0 �xing the ratio of �
at some value r < 1
2.
13
comparative advantage whereas the large country needs to supply low-comparative-advantage-goods as well.
In a Nash equilibrium of tari¤s, the small country can no longer focus on the productionof goods with the highest comparative advantage as endogenous tari¤ rates create a sizeablenon-traded sector (see upper panel of Figure 4). In addition, the small country�s terms-of-tradedeteriorate as it applies disproportionately lower tari¤ rates (see Proposition 5). In brief, thespecialization bene�ts that can be reaped under free trade are almost perfectly o¤set by thestrategic choice of tari¤ rates in a Nash equilibrium (see lower panel of Figure 4). The exactequilibrium outcome can be well approximated using the �rst-order approximation of tari¤s(see Appendix B.6):
~! � � (38)
N � 1
2+�
(39)
The terms-of-trade only re�ect di¤erent in absolute productivity �, whereas the size of thenon-traded sector only depends on the ratio of transportation cost � to comparative advantage .22
4.3.4 Welfare
Since the Nash equilibrium of tari¤s generates globally ine¢ cient production patterns the rep-resentative agent of the world economy must be worse o¤ than under free trade. In order tomeasure welfare losses for each country I determine the required rate of consumption growth �that equalizes the utility obtained in the Nash equilibrium (N) and free trade (F ).23 Therefore� is de�ned as:
~VF�e�cF
�� t = 0; t� = 0� � ~VN (cN j t = tN ; t� =t�N ) (40)
Due to the simple form of the utility function the required growth rate of consumption is givenby the di¤erence of the utility levels:
�V = ~VN � ~VF (41)
The exponentially a¢ ne speci�cation generates intuitive expressions for the derived utilities(see Appendix B.2):
~VN =
2(1� #N )2 + log
�TN
1 + #N tN
�(42)
~VF =
2(1� #F )2 (43)
The free trade welfare ~VF interacts comparative advantage (gains from trade) with the size ofthe import sector (1� #F ). This provides another illustration how small economies bene�t fromspecialization. In the absence of transportation cost, the in�nitesimally small economy�s derivedutility approaches
2 . The derived Nash equilibrium welfare ~VN consists of two components.The �rst component is similar to the free trade welfare. However, as the import sector is
22 The accuracy of these approximations is greater as approaches 0. In the limit, the statements hold exactly.23 This welfare measure has been proposed by Lucas (1987) and Alvarez and Lucas (2007).
14
smaller in the Nash equilibrium (#N > #F ), the �rst term must be strictly smaller than under
free trade. The second term log�
TN1+#N tN
�is strictly positive and re�ects gains from imposing
import tari¤s.
The following lemma describes intuitive properties of the welfare measure �V (see Figure5).
Lemma 1 �V is a function of c; � and and has the following limit properties:1) limc!0�V (c; ; �) < 02) limc!1�V (c; ; �) = 0
3) limc!1@�V (c; ;�)
@c < 0
Proof: See Appendix B.7.
Any welfare change between free trade and the Nash equilibrium of tari¤s must be causedby the endogenous choice 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 propertystates that the in�nitesimally small economy will surely be worse o¤ in the Nash equilibriumthan under free trade where it faces the best terms-of-trade. In contrast, the in�nitely largeeconomy will be equally well o¤ in the Nash equilibrium as under free trade because welfareconverges to the autarky level in both situations. The last property states that the in�nitelylarge country would bene�t from an expansion of the economy of its trading partner (fromwhich monopoly rents can be extracted).
In conjunction with continuity, these limit properties are su¢ cient to establish existence ofa unique threshold size level cT at which a country is indi¤erent between the Nash equilibriumoutcome and and free trade (see Syropoulos (2002)).
Proposition 6 A country prefers the Nash-Equilibrium outcome over free trade if its e¤ectiverelative size c exceeds the threshold level cT ( ; �) > 0.
Since the structure of the proof is essentially identical to Syropoulos�treatment, a rigorousproof 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 theNash equilibrium than under free trade (i.e. �V > 0 for c very large). By property 1), thesmall 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.
By symmetry, the threshold size level of each economy is identical. The threshold size level isgreater than 0, because a Nash equilibrium induces Pareto ine¢ cient production. Hence, whenthe countries are of equal size (c = 0) 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 arenot too great, i.e. whenever jcj < cT .
Due to the simple general equilibrium structure it is possible to analyze how the thresholdsize level is in�uenced by transportation cost and comparative advantage:
Proposition 7 The threshold size level cT is an increasing function of and a decreasingfunction of �.
15
Proof: See Appendix B.8.
Recall that if gains from trade increase, either through higher specialization bene�ts orlower transportation cost �, Nash equilibrium tari¤s of both countries increase (see Proposition5). On the margin, an increase in (decrease in �) will make a country with threshold sizecT prefer free trade over the Nash equilibrium outcome. The graph of the threshold size as afunction of and � is plotted in Figure 6. In the limit, as the gains from trade vanish ( ! 0),the home country prefers the Nash equilibrium outcome over free trade if its economy is 50%larger than the one of the country or equivalently if its economic weight is greater than 3
5 .24
lim !0
cT = log
�3
2
�(44)
5 Implications for Self-Enforcing Trade Agreements
The goal of this section is to discuss implications of the static Nash equilibrium analysis forcooperative trade agreements within a dynamic context. Rather than solving for the set ofoptimal dynamic contracts, I want to highlight conjectures that can be obtained from the staticanalysis and point to the relevant extensions in a dynamic setup.25
It is well understood, that any trade agreement has to be self-enforcing due to the lack ofinternational courts with real enforcement power. Thus, a sustainable (subgame perfect) agree-ment requires that the short-run bene�t from imposing an optimum tari¤ rate (see equation26) must be overwhelmed by the long-run cost resulting from a tari¤ war.26 Due to the lackof commitment such a punishment by the other country has to be itself credible, i.e. subgameperfect. The static Nash equilibrium outcome studied in this paper can be used as such an o¤-equilibrium path threat point to sustain e¢ cient equilibrium allocations.27 In the �rst part ofthis section I characterize e¢ cient allocations and show that any desired redistributive transfercan be implemented through e¢ cient tari¤ combinations in the spirit of Mayer (1981). Thisinsight simpli�es the subsequent discussion of the dynamic implications of my static analysis.
5.1 E¢ cient Allocations and Transfers
In the absence of transport cost, the Pareto-e¢ cient free trade production allocation is fullycharacterized by the boundary good �zF which satis�es:
�zF =1
2+1
�c� log �zF
1� �zF
�(45)
24 In the limit ( ! 0; � = 0) the cut-o¤ goods evaluated at the threshold size level cT = log�32
�are given by
zN =45, z�N =
310, zF = z�F = (cT ) =
35. See Appendix B.8 for the results when � > 0:
25 A complete characterization of the Pareto e¢ cient dynamic contracts using the recursive machinery of Abreu,Pearce, and Stacchetti (1990) is beyond the scope of this paper.
26 A summary of this literature is found in chapter 6 of Bagwell and Staiger (2002).27 In his analysis of oligopolies Friedman (1971) suggested this punishment equilibrium �rst. More sophisticatedpenal codes (see Abreu (1988)) may exist. Bagwell and Staiger (2002) show that there is also a limited role foron-equilibrium path retaliation within the GATT framework. This idea is not explored further in this paper.
16
Cooperation gains from e¢ cient free-trade production could be split up between countriesthrough good transfers.28 Proposition 8 implies that these transfers can also be implementedby e¢ cient tari¤ combinations:
Proposition 8 E¢ cient tari¤ combinations TE = 1T �Eimply:
a) E¢ cient free trade production: �zE = �zFb) Terms-of-trade satisfy: ~!E = log (TE) + ~!Fc) Relative welfare satis�es: ~VE � ~V �E = log (TE) +
�12 � �zF
�Proof: E¢ cient tari¤s TE = 1
T �Egenerate identical cut-o¤ goods for both countries:
NE = �zE � �z�E =log TE + log T
�E
=log TE + log
1TE
= 0 (46)
Substituting the terms-of-trade expression (equation 15) into the de�nition of the cuto¤ good�zE (equation 33) implies:
�zE =1
2+1
�c� log �zE
1� �zE
�= �zF (47)
Therefore, the terms-of-trade satisfy:
~!E = log
0@ �zF1� �zF
1 + �zF (TE � 1)1 + (1� �zF )
�1TE� 1�1A� ~L = ~!F + log (TE) (48)
The indirect utility of both countries can be written as:
~VE =
2(1� �zF )2 + log
�TE
1 + �zF tE
�(49)
~V �E =
2�z2F + log
�T �E
1 + (1� �zF ) t�E
�(50)
Simple algebra generates the desired result.
The proposition implies that positive tari¤s of the home economy (TE > 1) are isomorphicto real transfers of export goods from the foreign country to the home country under free tradeproduction. Therefore, the dynamic decision problem of each government can be framed solelyas a sequence of tari¤ rate decisions fTsg1s=1. Tari¤ induced transfers are uniform across exportgoods due to a proportional decrease of the relative wage rate.
5.2 Tari¤Agreements
The folk-theorem (see Fudenberg and Maskin (1986)) implies that as the discount factor ofall players approaches unity any individually rational allocation becomes incentive compatible.Applied to this setup, a trade agreement with e¢ cient production (see previous section) canalways be achieved provided that both governments are su¢ ciently patient. Thus, the Nash
28 Note, that within this general equilibrium framework side payments must involve physical transfers from onecountry to the other.
17
equilibrium outcome only matters through the bounds it imposes on the distribution of rentsfrom e¢ cient production. However, reciprocal free trade without transfers (TE = T �E = 1) canonly be sustained if both countries are below the threshold size. This is important to the extentthat there are political or other constraints (outside of the model) that rule out transfers fromthe small to the large economy. In this case Proposition 7 implies that free-trade agreementsare more likely to occur if specialization bene�ts are higher or countries are closer (� lower).29
Gradualism in tari¤ agreements can be analyzed as optimal dynamic contracts (see alsoBond and Park (2002)). Contract dynamics can either be driven by exogenous dynamics inmodel parameters or can result endogenously even in a stationary environment. An exampleof the former case is expected relative growth (change in c) of one economy (such as China orIndia). Since a growing economy�s outside option of a tari¤ war is becoming relatively moreattractive over time tari¤ agreements are expected to become more favorable to ensure dynamicsustainability. Di¤erences in discount factors such as in Acemoglu, Golosov, and Tsyvinski(2008) or Opp (2008) can give rise to non-trivial dynamics even in stationary settings. Inthe political economy literature self-interested governments with short time horizons are oftenmodeled as e¤ectively impatient. On the one hand, it is desirable to grant the impatientgovernment initially more favorable contract terms ("teaser rates"). On the other hand, thedynamic provision of incentives requires that the agreement cannot become too unfavorableover time to ensure incentive compatibility. The trade-o¤ between impatience and incentivecompatibility is the cornerstone of the optimal dynamic contract.
Free trade agreements are not sustainable if one economy is su¢ ciently impatient. In thiscase the short-run temptation to renege on the agreement is greater than the (discounted) long-run bene�t from free trade. Therefore, the static Nash equilibrium outcome should characterizethe actual behavior of governments, not just the o¤-equilibrium path threat point. Empiricalstudies such as Broda, Limao, and Weinstein (2008) and Magee and Magee (2008) comparetheir results relative to optimum static tari¤s. This benchmark equilibrium selection can betheoretically justi�ed if governments are su¢ ciently impatient.
6 Conclusion
This paper has analyzed the strategic choice of tari¤ rates in a generalized Ricardian model à laDornbusch, Fischer, and Samuelson (1977) with CES preferences. These standard preferences inthe international trade literature generate a particularly simple tari¤ structure: The optimumtari¤ schedule is uniform across goods. While this result cannot explain the observed dispersionof tari¤ rates across goods it is useful from a theoretical perspective even for di¤erent underlyingmodels of trade.
The general equilibrium framework of DFS generates novel comparative statics predictionswith regards to the role of comparative advantage and transportation cost in strategic settings.Higher gains from trade (higher comparative advantage, lower transportation cost) increaseNash equilibrium tari¤ rates as any given tari¤ rate will lead to smaller deviations from e¢ cientproduction specialization. Within the model this implies that the cut-o¤ good for domesticproduction is less responsive to tari¤s in settings with high comparative advantage or lowtransportation cost. Su¢ ciently large economies (in terms of productivity adjusted size) are
29 This may explain the regional focus of most free trade agreements such as the NAFTA or within the EU.
18
better o¤ in a Nash equilibrium than under free trade. The required threshold size is higherwhen Nash equilibrium tari¤s of both countries are higher, i.e. when comparative advantage ishigh or transportation cost are low.
The results of this paper suggest two lines of future research. On the empirical side, it wouldbe interesting to examine the testable predictions with regards to the exogenous parameters inthe model in the spirit of Broda, Limao, and Weinstein (2008). The advantage of using a generalequilibrium framework is that demand / supply elasticities are endogenously determined andneed not be estimated.30
A rigorous understanding of the static Nash equilibrium outcome can be viewed as a steppingstone to the more complicated analysis of self-enforcing trade agreements within a dynamiccontext. For example, it would be interesting to study the e¤ect of growth and increasesin specialization gains on trade agreements.31 Likewise, it seems worthwhile to analyze theimpact of business cycle shocks by extending the partial equilibrium framework of Bagwell andStaiger (2003) to a general equilibrium setup. This line of research might produce interestingimplications about the relationship between (free) trade, growth and technology di¤usion.
References
Abreu, D. (1988): �On the Theory of In�nitely Repeated Games with Discounting,�Econo-metrica, 56(2), 383�96.
Abreu, D., D. Pearce, and E. Stacchetti (1990): �Toward a Theory of Discounted Re-peated Games with Imperfect Monitoring,�Econometrica, 58(5), 1041�63.
Acemoglu, D., M. Golosov, and A. Tsyvinski (2008): �Political Economy of Mecha-nisms,�Econometrica, 76(3), 619�641.
Alvarez, F., and R. E. Lucas, Jr. (2007): �General equilibrium analysis of the Eaton-Kortum model of international trade,�Journal of Monetary Economics, 54(6), 1726 �1768.
Bagwell, K., and R. Staiger (2003): �Protection and the Business Cycle,�Advances inEconomic Analysis & Policy, 3(1), 1139�1139.
Bagwell, K., and R. W. Staiger (1990): �A Theory of Managed Trade,�American Eco-nomic Review, 80(4), 779�95.
Bagwell, K., and R. W. Staiger (1999): �An Economic Theory of GATT,� AmericanEconomic Review, 89(1), 215�248.
Bagwell, K., and R. W. Staiger (2002): The economics of the world trading system / KyleBagwell and Robert W. Staiger. MIT Press, Cambridge, Mass.
Bickerdike, C. F. (1907): �Review of A.C. Pigou�s Protective and Preferential Import Du-ties,�Economic Journal, 17, 98�108.
30 Di¤erent estimation methods for elasticities can lead to drastic di¤erences in results. For example, Magee andMagee (2008) obtain very small elasticity estimates relative to Broda, Limao, and Weinstein (2008).
31 Predictable trends in specialization gains may result from production technologies that feature "learning bydoing" such as in Devereux (1997).
19
Bond, E. W. (1990): �The Optimal Tari¤ Structure in Higher Dimensions,� InternationalEconomic Review, 31(1), 103�16.
Bond, E. W., and J.-H. Park (2002): �Gradualism in Trade Agreements with AsymmetricCountries,�Review of Economic Studies, 69(2), 379�406.
Broda, C., N. Limao, and D. E. Weinstein (2008): �Optimal Tari¤s and Market Power:The Evidence,�American Economic Review, 98(5), 2032�65.
Chipman, J. S. (1971): �International Trade with Capital Mobility: A Substitution Theorem,�in Trade, Balance of Payments and Growth, ed. by J. Bhagwati, R. W. Jones, R. A. Mundell,and J. Vanek, pp. 201�37. North-Holland, Amsterdam.
Devereux, M. B. (1997): �Growth, Specialization, and Trade Liberalization,� InternationalEconomic Review, 38(3), 565�85.
Dornbusch, R., S. Fischer, and P. A. Samuelson (1977): �Comparative Advantage,Trade, and Payments in a Ricardian Model with a Continuum of Goods,�American EconomicReview, 67(5), 823�39.
Eaton, J., and S. Kortum (2002): �Technology, Geography, and Trade,� Econometrica,70(5), 1741�1779.
Feenstra, R. C. (1986): �Trade policy with several goods and market linkages,�Journal ofInternational Economics, 20(3-4), 249�267.
Friedman, J. W. (1971): �A Non-cooperative Equilibrium for Supergames,�Review of Eco-nomic Studies, 38(113), 1�12.
Fudenberg, D., and E. Maskin (1986): �The Folk Theorem in Repeated Games with Dis-counting or with Incomplete Information,�Econometrica, 54(3), 533�54.
Gorman, W. M. (1958): �Tari¤s, Retaliation, and the Elasticity of Demand for Imports,�The Review of Economic Studies, 25(3), 133�162.
Graaff, J. d. V. (1949): �On Optimum Tari¤ Structures,�The Review of Economic Studies,17(1), 47�59.
Grossman, G. M., and E. Helpman (1995): �Trade Wars and Trade Talks,�The Journal ofPolitical Economy, 103(4), 675�708.
Grossman, S. J., and O. D. Hart (1986): �The Costs and Bene�ts of Ownership: A Theoryof Vertical and Lateral Integration,�The Journal of Political Economy, 94(4), 691�719.
Itoh, M., and K. Kiyono (1987): �Welfare-enhancing Export Subsidies,� The Journal ofPolitical Economy, 95(1), 115�137.
Johnson, H. G. (1953): �Optimum Tari¤s and Retaliation,�The Review of Economic Studies,21(2), 142�153.
Kennan, J., and R. Riezman (1988): �Do Big Countries Win Tari¤ Wars?,� InternationalEconomic Review, 29(1), 81�85.
20
Lerner, A. P. (1936): �The Symmetry between Import and Export Taxes,�Economica, 3(11),306�313.
Lucas, Jr., R. E. (1987): Models of Business Cycles. Basil Blackwell, New York.
Magee, C. S. P., and S. P. Magee (2008): �The United States is a Small Country in WorldTrade,�Review of International Economics, 16(5), 990�1004.
Mayer, W. (1981): �Theoretical Considerations on Negotiated Tari¤ Adjustments,�OxfordEconomic Papers, 33(1), 135�153.
McLaren, J. (1997): �Size, Sunk Costs, and Judge Bowker�s Objection to Free Trade,�TheAmerican Economic Review, 87(3), 400�420.
Opp, M. M. (2008): �Expropriation Risk and Technology,�Unpublished dissertation, Univer-sity of Chicago, Booth School of Business.
Opp, M. M., H. F. Sonnenschein, and C. G. Tombazos (2009): �Rybczynski�s Theoremin the Heckscher-Ohlin World �Anything Goes,�Journal of International Economics, 79(1),137 �142.
Otani, Y. (1980): �Strategic Equilibrium of Tari¤s and General Equilibrium,�Econometrica,48(3), 643�662.
Samuelson, P. A. (1952): �The Transfer Problem and Transport Costs: The Terms of TradeWhen Impediments are Absent,�The Economic Journal, 62(246), 278�304.
Syropoulos, C. (2002): �Optimum Tari¤s and Retaliation Revisited: How Country SizeMatters,�The Review of Economic Studies, 69(3), 707�727.
Weder, R. (2003): �Comparative home-market advantage: An empirical analysis of Britishand American exports,�Review of World Economics (Weltwirtschaftliches Archiv), 139(2),220�247.
21
A General Proofs
A.1 Partial Derivatives of Equilibrium Conditions
The equilibrium conditions with uniform tari¤s can be written as:
g1 = ~L+ ~! + log (FIx)� log (FD + FIx) + log (F �D + F �Ix)� log (F �Ix) = 0
g2 = ~! � � � log (T )� ~A (�z) = 0
g3 = ~! + � + log (T �)� ~A (�z�) = 0
where the functions FD; FI and FIx are de�ned in the Proof of Proposition 1 (equations 18 to20). The partial derivative of equation i with respect to variable x is denoted by gix:
Term Expression Term Expressiong1�z �b (�z) T
(1+#t)(1�#) g2�z � ~A0 (�z)g1�z� �b� (�z�) T �
(1+#�t�)(1�#�) g2t � 1T
g1t � �#1+#t g3�z� � ~A0 (�z�)
g1~! 1� (1� �) #T1+#t � (1� �
�)�#�T �
1+#�t� g3t�1T �
The total derivatives of ~! and �z with respect to the tari¤ rate choice can be obtained using theimplicit function theorem:
d~!
dt=(g1�zg2t � g1tg2�z) g3�z�
DD(51)
d�z
dt=g1�z�g2t + (g1t � g2t) g3�z�
DD(52)
where: DD = �g1�z�g2�z � g1�zg3�z� + g2�zg3�z� > 0. Moreover, the partials with respect to e¤ectiverelative size are given by:
d�z
dc=g3�z�
DD(53)
d�z�
dc=
g2�zDD
(54)
A.2 Tari¤Rate Formula: Proof of Proposition 2
The derived utility function can be expressed as (see equation 17):
~V = ~! � �
1� � log (FD + FI)� log (FD + FIx) (55)
where the functions FD; FI and FIx are de�ned in the Proof of Proposition 1 (equations 18 to20). The partials of the objective function with respect to the relevant variables are given by:h
@ ~V@t
@ ~V@�z
@ ~V@~!
i=�� �T #t
1�#1+#t �b (�z) t
1+#t (1� #) 1+�#t1+#t
�(56)
where I have used ~! � � � log (T (�z))� ~A (�z) = 0. The �rst-order condition implies:
@ ~V
@t+@ ~V
@�z
d�z
dt+@ ~V
@~!
d~!
dt= 0 (57)
22
Using the comparative statics results for d�zdt and
d~!dt from Appendix A.1 simple algebraic ma-
nipulations yield:
tR =1 + #�t�
T �1
� b�(�z�)~A0(�z�)
11�#� + #
� (�� � 1)(58)
The optimum tari¤ rate for the foreign country follows by symmetry. It can also be derived viathe elasticity formula:
t�R =1
"� 1 (59)
The import demand elasticity of the home country " = d logmd log! is given by:
" =d log
�! FIx(�z)FD(!;�z)+FIx(�z)
�d~!
= 1� @ log (FD (!; �z) + FIx (�z))
@~!+@ log
�FIx(�z)
FD(!;�z)+FIx(�z)
�@�z
@�z
@~!
(60)
= 1 + (� � 1) FDFD + FIx
� g1�zg2�z
= 1 + (� � 1)# T
1 + #t� b (�z)~A0 (�z)
T
(1 + #t) (1� #) (61)
where I have used the fact that:FD
FD + FIx=
#T
1 + #t(62)
B Cobb-Douglas Speci�cation
B.1 Equilibrium Conditions
For the speci�cation used in section 4, we obtain the following equilibrium conditions:
g1 (x; q) = log (!) + log�1+#�t�
1�#��+ log
�1�#1+#t
�+ log (L) = 0
g2 (x; q) = log (!)� log (T )� ����+
2 � �z�
= 0g3 (x; q) = log (!)+ log (T �) + ��
��+
2 � �z�� = 0
g4 (x; q) = t� (1� #�) 1+#�t�T � = 0
g5 (x; q) = t� � (1� #) 1+#tT = 0
(63)
The partial derivatives are given by:
Term Expression Sign Term Expression Signg1�z � T
(1+#t)(1�#) < 0 g3t�1T � > 0
g1�z� � T �
(1+#�t�)(1�#�) < 0 g3 �12 + �z
� ?
g1t � #1+#t < 0 g4�z� � 1�t
�(1�2#�)T � < 0�
g1t�#�
1+#�t� > 0 g4t�1�#�1+t�
t1+#�t� > 0
g2�z > 0 g4 f3�z� f1�z�
= � t < 0
g2t � 1T < 0 g5�z 1�t(1�2#)T > 0�
g2 �12 + �z ? g5t
1�#1+t
t�
1+#t > 0
g3�z� > 0 g5 f2�z f1�z
= � t�
< 0
The signs of g4�z� and g5�z follow directly from Assumption 1.
23
B.2 Derived Utility
In the Cobb-Douglas case, we can rewrite the derived utility (equation 17) as :
~V = (~! � �) (1� #)�Z 1
�z
~A (z) dz + #T � log (1 + #t)�Z 1
0log (a (z)) dz (64)
Realize thatR 10 log (a (z)) dz is just a constant and therefore irrelevant. Moreover, use ! =
log (T ) + �+ ~A (�z) to obtain equivalent preferences ~v :
~v = ~A (�z) (1� #)�Z 1
�z
~A (z) dz + log
�T
1 + #t
�(65)
Using the exponentially a¢ ne production technology generates the derived utility expressionsin the Nash equilibrium and free trade.
~v =
2(1� �z)2 + log
�T
1 + #t
�(66)
~vF =
2(1� �zF )2 (67)
B.3 Proof of Proposition 3: Properties of Optimum Response Function
Let x denote the vector of endogenous variables x =�~! �z �z� t
�0and q the vector of
exogenous variables q =�~L � � t�
�0. The �rst four elements of the function g (see
Appendix B.1) de�ne the equilibrium conditions.32 Moreover, let DXg represent the Jacobianof the function g with respect to x and Dqg the Jacobian with respect to q.
DXg =
26641 g1�z g1�z� g1t1 g2�z 0 g2t1 0 g3�z� 00 0 g4�z� 1
3775 and Dqg =
26641 0 0 0 g1t�
0 �1 �1 g2 00 �1 1 g3 g3t�
0 0 0 g3�z� g1�z�
g4t�
3775 (68)
det (DXg) = �g1�z�g2�z| {z }<0
� g1�zg3�z�| {z }<0
+ g2�zg3�z�| {z }>0
� g1�zg2tg4�z�| {z }<0
+ g2�zg1tg4�z�| {z }>0
> 0 (69)
The derivative of the endogenous variables x with respect to q are given by the implicit functiontheorem:
Dqx = � (DXg)�1Dqg (70)
The fourth row vector yields the comparative statics of the best response tari¤ rate:
dtRdc =
�g4�z�g2�zdet(DXg)
> 0dtRd� =
(�2g1�z+g2�z)g4�z�det(DXg)
< 0
dtRdt� =
g4t�g2�zg3�z��+g4�z��g1�zg3t��g2�z 1�#�
(1+#�t�)(1+t�)
�det(DXg)
< 0
dtRd =
""��1+ (("�1)g2 �g3 ")g4�z�+1
det(DXg)> 0
(71)
32 The �fth equilibrium condition is only relevant for the Nash equilibrium analysis in which the foreign country�stari¤ rate is endogenous.
24
where "; "� > 1 represent the respective import demand elasticities and � the Marshall-Lernercondition:
" = 1� g1�zg2�z
> 1 (72)
"� = 1� g1�z�
g3�z�> 1 (73)
� = "+ "� � 1 > 0 (74)
B.4 Proof of Proposition 4: Existence of Nash Equilibrium
Lemma 2 �t;t� =���d log(TR)d log(T �)
��� < 1Proof: We can rewrite det (Dxg) (see Appendix B.3) as:
det (DXg) = g2�z
�g3�z�� �
g4�z�
T
�"� 1 + #T
1 + #t
��(75)
�t;t� =dTRdT �
T �
TR=
tg3�z�� � g4�z��1+#�t�
1�#� ("� 1) + 1�
1 + #�t�
1� #�| {z }>1
Tg3�z�� � g4�z�
0BB@1+#�t�1�#� ("� 1) +
1 + #�t�
1� #�# (1 + t)
1 + #t| {z }>1
1CCA< 1 (76)
The numerator is strictly smaller than the denominator, so that the ratio is less than one.
Let (S; �) de�ne the complete metric space with S = [0; �2�] and � = jx� yj and de�ne theoperator �� : S ! S with ��x = ~�� (~� (x)) where ~� represents the optimum response functionfor log tari¤ rates log(T ). �� is a contraction since the optimum response functions ~� (log (T �))and ~�� (log (T )) are continuous functions with slope uniformly less than one in absolute value(by Lemma 2 ). Hence, we can invoke the Contraction Mapping Theorem which guarantees theexistence of a unique �xed point in S. This �xed point constitutes the Nash Equilibrium tari¤rate of the foreign country. An analogous argument with �x = ~� (~�� (x)) yields the uniqueNash Equilibrium tari¤ rate log(TN ) of the home country. An interior Nash equilibrium Paretodominates the No-Trade Nash equilibrium (autarky), since this option (no trade) is availablein the action set of each country (choosing a prohibitive tari¤ rate of � 2�).
B.5 Proof of Proposition 5: Properties of Nash Equilibrium Tari¤s
The proof is analogous to the one of Proposition 3. The tari¤ rate of the foreign countryt� is now endogenous and given by the 5th equilibrium condition (see Appendix B.1). Thecomparative statics are again obtained by the implicit function theorem.
25
B.6 First-Order Approximation
B.6.1 Limit Nash Equilibrium
I consider the limit as approaches 0 setting � = r where r < 12 is constant. Nash equilibrium
tari¤ rates are zero in the limit (see equations 26 and 27). In the limit, scaled tari¤ rates aregiven by:
lim !0
log (TN )
= �z� and lim
!0
log (T �N )
= 1� �z (77)
The non-traded sector satis�es: (see equation 33):
lim !0
�z � �z� = lim !0
log TN � log T �N + 2�
= 1� lim !0
(�z � �z�) + 2r (78)
Rearranging yields:
lim !0
�z � �z� = 1
2+ r (79)
As comparative advantage approaches 0, the terms of trade just re�ect absolute productivityadvantage:
lim !0
~! = � (80)
Substituting equation 80 into the balance of trade condition implies:
� = lim !0
log
��z�
1� �z
�+ log lim
!0
�1 + �z�t
1 + (1� �z�) �t�
�� ~L (81)
which can be simpli�ed to:
c = log lim !0
��z�
1� �z
�(82)
Setting �z� = �z��12 + r
�implies the closed-form expressions for �z and �z� using (c) = exp(c)
1+exp(c) :
�zN =1
2+ r + (c)
�1
2� r�
(83)
�z�N =
�1
2� r� (c) (84)
B.6.2 Limit Free Trade
As approaches 0, the free trade balance of trade condition can be simpli�ed analogously tothe Nash equilibrium limit (see equation 82). Since tari¤ rates are 0 equation 33 now impliesthat �z� = �z � 2r. This implies:
�zF = (c) + 2r (1� (c)) (85)
�z�F = (c) (1� 2r) (86)
26
B.6.3 First-Order Approximation
The �rst-order Taylor series expansion about = 0 for the Nash equilibrium tari¤ rate is nowgiven by:
tN ( ) = t (0) + lim !0
t0N (0) +O� 2�
(87)
=
�1
2� r� (c) +O
� 2�
(88)
t�N ( ) =
�1
2� r�(1� (c)) +O
� 2�
(89)
Using the �rst-order approximation the non-traded sector becomes:
N =log (1 + tN ) + log (1 + t
�N ) + 2�
� 1
2+�
(90)
The equilibrium log wage rate can be determined via the second equilibrium condition g2:
~! = log (1 + tN ) + � + �+
2� �z (91)
� �z�+
2+ � + �� �z = �+
2+ � � N � � (92)
B.7 Proof of Lemma 1: Welfare Comparison
If we subtract the derived utility under free trade (equation 43) from the Nash equilibriumutility (42) we obtain:
�V (c; ; �) =
2
h(1� �zN )2 � (1� �zF )2
i+ log
�TN
1 + �zN tN
�(93)
Since the tari¤ rate choice T and the threshold goods �zN , �zF are just a function of e¤ectiverelative size c, �V does not depend separately on � and L. The �rst two properties claimed inLemma 1 are discussed in the text. The third property limc!1
@�V (c; ;�)@c < 0 is proved here:
d
dc�V =
@�V
@�zN
d�zNdc
+@�V
@�zF
d�zFdc
+@�V
@tN
dtNdc
(94)
Since d�zNdc > 0, we obtain:
sign�limc!1
@�V (c; ; �)
@c
�= sign
limc!1
@�V
@�zN+ limc!1
@�V
@�zF
d�zFdcd�zNdc
+ limc!1
@�V
@tN
dtNdcd�zNdc
!(95)
There exists a constant B such that max�limc!1
���� d�zFdcd�zNdc
���� ; limc!1 ���� dtNdcd�zNdc
����� < B. Moreover, it is
easy to show that:limc!1
#N = limc!1
�zN = limc!1
#F = limc!1
�zF = 1 (96)
27
andTerm Expression limc!1@�V@�zN
�� (1� �zN ) + tN
1+�zN tN
�� tNTN
@�V@�zF
(1� �zF ) 0@�V@tN
1��zNT (1+�zt) 0
(97)
After simple algebraic manipulations we obtain:
sign�limc!1
@�V (c; ; �)
@c
�= sign
�� tNTN
�(98)
This completes the proof.
B.8 Proof of Proposition 7: Threshold Size Comparative Statics
B.8.1 Threshold Size Level
In order to obtain closed form expressions for the threshold size level and for the derivatives ofinterest I consider the limit as approaches 0 and setting � = r . It is useful to realize that:
TN1 + �zN tN
= 1 + (1� �zN )tN
1 + �zN tN(99)
which allows us to rewrite equation 93 as:
(1� �zN )2 � (1� �zF )2 +2
log
�1 + (1� �zN )
tN1 + �zN tN
�= 0 (100)
In the limit, second order terms can be ignored:
lim !0
h(1� �zN )2 � (1� �zF )2
i+ 2 lim
!0(1� �zN ) lim
!0
tN lim !0
1
1 + �zN tN= 0 (101)
Using lim !0tN = �z and lim !0
11+�zN tN
= 1 implies:
(1� �zN )2 � (1� �zF )2 + 2 (1� �zN ) �z�N = 0 (102)
The derived closed form expressions for �zN ; �zF and �z�N (see Appendix B.6) allow us to solve forcT :
lim !0
cT = log
�3
2
�(103)
The associated variables of interest are:
lim !0 cT = log (3=2)lim !0 (cT ) = 3
5lim !0 zN (cT ) = 4
5 +25r
lim !0 z�N (cT ) = 310 �
35r
lim !0 zF (cT ) = 35 +
45r
lim !0 z�F (cT ) = 35 �
65r
(104)
28
B.8.2 Comparative Statics
In order to obtain the derivatives, we have to use again the implicit function theorem:
dcTd
= ��d�V (cT ; ; �)
dc
��1 �d�V (cT ; ; �)d
�(105)
dcTd�
= ��d�V (cT ; ; �)
dc
��1 �d�V (cT ; ; �)d�
�(106)
From the discussion of Proposition 6 (see also Figure 5) it must be true that the derivative of�V with respect to size c (see equation 94) evaluated at the threshold size cT is positive. Theremaining derivatives of interest are:
d�V
d =@�
@ +@�
@�zN
d�zNd
+@�
@tN
dtNd
+@�
@�zF
d�zFd
(107)
d�V
dc=
@�
@�zN
d�zNdc
+@�
@tN
dtNdc
+@�
@�zF
d�zFdc
(108)
where the partials are given by:
@�@�zN
= �� (1� �zN ) + tN
1+�zN tN
�< 0
@�@�zF
= (1� �zF ) > 0@�@tN
= 1��zNTN (1+�ztN )
> 0
@�@ = 1
2
�(1� �zN )2 � (1� �zF )2
�< 0
(109)
The other terms follow directly from the comparative statics of the Nash equilibrium. Usingthe limiting expressions of equation 104 one obtains the following expressions:
lim !0
dcTd�
= ��19
75+2
5r
�< 0 (110)
lim !0
dcTd
=53
300+r
5
2> 0 (111)
The signs of the derivatives are not a¤ected if > 0 as the absolute value of the derivatives isincreasing in . This completes the proof.
29
C Figures
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
z
A(z
)
[µ, γ] = [-0.5, 1]
[µ, γ] = [0, 1]
[µ, γ] = [0, 2]
Figure 1: Relative Productivity Function for Various Parameter Values
30
0 0.2 0.4 0.6 0.80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
γ =0.4
γ =0.4
γ =0.8
γ =0.8
NE
NE
Tariff Rate of Foreign Country log(T∗
R)
Tariff
Rate
ofH
om
eC
ountr
ylo
g(T
R) tR(t∗), γ =0.4
t∗
R(t), γ =0.4
tR(t∗), γ =0.8
t∗
R(t), γ =0.8
Figure 2: Optimum Response Functions and Nash Equilibria (c = 0)
31
0 0.5 10
0.1
0.2
0.3
0.4
0.5
Economic Weight Ψ()
TariffRate
= 04 = 06 = 08
0 0.1 0.2 0.3 0.40
0.05
0.1
0.15
0.2
0.25
Transportation Cost
TariffRate
= 04 = 06 = 08
Figure 3: Comparative Statics Nash Equilibrium (left panel: � = 0; right panel: c = 0)
32
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Economic Weight of Home Country Ψ()
Boundary
Goods
0 0.2 0.4 0.6 0.8 1
-0.2
-0.1
0
0.1
0.2
− 2 +
+ 2−
Economic Weight of Home Country Ψ( L)
Term
sofTrade
Nash Equilibrium
Free Trade
Figure 4: The E¤ect of Size on Cut-o¤ Goods and Terms-of-Trade: Nash Equilibrium vs. FreeTrade (� = 0; = 0:4; � = 0)
33
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
Ψ(− ) Ψ( )
Economic Weight of Home Country Ψ()
WelfareChange∆
Prisoner’s DilemmaHome CountryForeign Country
Figure 5: Welfare and E¤ective Size: Nash Equilibrium vs. Free Trade ( = 0:6; � = 0)
34
0 0.05 0.1 0.15 0.20.4
0.45
0.5
0.55
log(32)
Transportation Cost
ThresholdSize
= 04 = 06 = 08
Figure 6: E¤ects of Transportation Cost and Comparative Advantage on the Threshold Size
35