# Does limited punishment limit the scope for cross retaliation?

I. INTRODUCTIONThe dispute settlement mechanism of the World Trade Organization (WTO) is often criticized for being indifferent to asymmetries among its members. In practice, developing countries typically find it difficult to retaliate against advanced economies; yet, the WTO legislation does not seem to make such retaliation easier. (1) An important example of this restriction is Article 22.3 in the Understanding on Rules and Procedures Governing the Settlement of Disputes (DSU), which confines all retaliations into the sector where the original breach has occurred, except in limited circumstances. Accordingly, when an advanced country discriminates against a manufactured or commodity export of a developing country, the latter is expected to retaliate by imposing tariffs on a manufactured or commodity import even if they have little market power in trade. Potentially more credible cross-retaliation threats are restricted by the DSU. To quote Anatole France, "In its majestic equality, the law forbids rich and poor alike to sleep under bridges, beg in the streets and steal loaves of bread." But, why is such cross retaliation discouraged for all nations by the DSU legislation?

This paper studies this issue by explicitly linking the DSU Article 22.3 with a limited punishment rule characterized by the General Agreement on Tariffs and Trade (GATT) Article XXVIII. In general, both rules are designed to limit the countermeasures upon a violation; however, the former rule specifies the limits of composition in retaliation, whereas the latter one designates the limits of retaliation magnitude. We show that, albeit seemingly unrelated, the limited cross-retaliation rule complements the limited punishment rule in permitting greater trade liberalization. Specifically, we show how the limited cross-retaliation rule also helps limit the incentives to violate the trade agreement when the limited punishment rule prevails.

The DSU Article 22.3 emphasizes that the suspension of concessions or other obligations should be implemented with respect to the same sector in which the initial violation or other nullification or impairment has occurred. The concept of" sector(s)" is defined with respect to goods as all goods, and with respect to services as a principal sector identified in the "Services Sectoral Classification List," which comprises 12 principal services categories. (2) Governments can seek to suspend concessions across sectors, or agreements, if within-sector punishment cannot be implemented in a practicable and effective manner. (3) However, the basic rationale for this Limited Cross Retaliation rule (LCR) is to ensure that retaliation across sectors and agreements remains an exception. There have been only three cases (out of nine requests) where the complainant government was authorized for cross-agreement retaliation in the entire GATT-WTO system history: the US-Internet Gambling Case (Antigua and Barbuda were authorized to retaliate in the Agreement on Trade-Related Aspects of Intellectual Property Rights [TRIPS]), the EC-Banana III Case (Ecuador was authorized to retaliate by $191 million annually in GATS and TRIPS) and the US-Cotton Case (Brazil was authorized to retaliate by $147 million in GATS and TRIPS).

The GATT Article XXVIII, on the other hand, determines an upper bound for countermeasures that is equivalent to the level of nullification or impairment resulting from the breach of agreement obligations. (4) In principle, this Withdrawal of Equivalent Concessions (WEC) rule dismisses the punitive character of countermeasures and, instead, defines them as procedures for inducing compliance with WTO obligations. Suspension of concessions or other obligations are, therefore, typically advised to be temporary and to be applied until the measures inconsistent with the WTO obligations are removed by the violating member or until a mutually agreeable solution is obtained. In practice, this implies that both the violator and complainant governments apply measures and countermeasures until a mutually satisfactory agreement is reached. (5)

In order to analyze the impact of these rules, we employ a two-country two-sector tariff-setting framework, where each country is an exporter and an importer in each sector. Goods in a given sector are substitutes in consumption, and goods across the sectors are independent. Our representation of preferences captures these demand substitutabilities and, in turn, generates welfare functions whereby tariffs in each sector are strategic substitutes. Tariffs across the sectors are independent. In the absence of a trade agreement, governments apply unilaterally optimal tariffs in each sector, which are globally inefficient. We then characterize alternative cooperation paths by formally introducing the LCR and WEC punishment rules. We first analyze the structure of cooperation under the benchmark Nash-reversion strategies. We then introduce both the LCR and the WEC rules. In this case, any deviation from the cooperative path in a given sector is punished by an equivalent deviation within the same sector (unlinked agreements). Formally, we characterize the punishment stage with simultaneous applications of the deviation tariff, which, we believe, is a good approximation to the "inducing compliance" interpretation of countermeasures. (6) Finally, we remove the LCR rule to allow cross retaliation, where an initial violation in a given sector might be punished by an equivalent deviation in the other sector (linkage case).

Our first main result shows that, given symmetric issues and identical cooperative tariffs, the magnitude of initial violations is greater under the linkage case. Hence, removing the LCR rule and linking the agreements reduces the self-enforcing level of cooperation. The idea here is that when tariffs are strategic substitutes an equivalent punishment within the same sector hurts the deviating country more than does cross retaliation across independent sectors. In order to reduce retaliation, governments, therefore, reduce the violation magnitude when both LCR and WEC rules prevail. The maximum level of cooperation in our model (the lowest self-enforcing-tariff) decreases in the magnitude of the deviation tariffs. Limiting cross retaliation, therefore, enhances cooperation between governments.

Our second main result actually shows that whenever it is possible governments will always choose cross retaliation over within-sector retaliation. After a violation occurs, the punishing government prefers avoiding within-sector punishment. The relative gain from increasing their own tariff in a sector where the trading partner has already raised its tariff is lower when tariffs are strategic substitutes; therefore, the punisher suspends its concessions in the other sector. The initial violator does not oppose this choice. This result is interesting in the sense that it points to a time-inconsistency problem: once a deviation occurs, both countries prefer the punishment path with less enforcement power. Therefore, our third main result shows that, for sufficiently patient governments, limiting the punishment by LCR and WEC rules together at the outset generates the preferred subgame perfect outcome.

We next consider how changes in the gains from trade affect the degree of cooperation (as measured by the lowest self-enforcing tariff). We first show that symmetric export-biased technological improvements in both countries reduce cooperation under the benchmark Nash-reversion strategies; however, it leaves tariffs unchanged under limited retaliation regimes (WEC, both with and without LCR). The intuition here is that under Nash-reversion strategies the punishment phase is not tied to the original deviation. As the purpose of an optimal tariff is to grab as much of the gains from trade as possible, the gain from an optimal deviation goes up by more than its long-run cost with Nash-reversion strategies so that the lowest self-enforcing tariff must increase. In a world with technological progress, this result provides a new justification for limited retaliation that is different from the information-based rationales given in Bagwell and Staiger (2005) or Beshkar (2010a and 2010b). We then analyze asymmetric export-biased technological improvements. Even though the resulting increase in the optimal deviation is mitigated under limited retaliation (as compared to Nash-reversion), its reduction is larger when the retaliation is in the same sector (as a result of the strategic substitutability of same sector tariffs). Hence, we again see the benefit of limiting cross retaliation when punishments are limited.

Johnson (1953-1954) provides the first formalization of the terms of trade rational for trade agreements and the strategic substitutability of tariffs. Recognizing that there are no international soldiers to enforce trade agreements, authors such as Dixit (1987) and Bagwell and Staiger (1990, 1999, 2002) began to look at trade agreements as self-enforcing outcomes in a repeated game framework. Bagwell and Staiger (2005) and Beshkar (2010a and 2010b) provide a rationale for limited punishments after limited deviations. The paper closest to ours is Zissimos (2007) who provides an excellent analysis of trade agreements under WEC strategies. He describes the equilibrium behavior of governments when tariffs are strategic substitutes and the limited punishment rule is applied. The focus of his paper is gradual trade liberalization and there is only one sector. We build on his analysis to analyze linkage across sectors.

This paper also relates to a small but distinguished literature on linkage in repeated games. Bernheim and Whinston (1990) show that firms act more cooperatively when there is multimarket interaction and firms and/or markets are asymmetric. This multimarket collusion effect arises from reciprocal exchange of unilateral concessions. Linkage, however, does not affect enforcement when markets and firms are symmetric. Spagnolo (1999) changes the last result by introducing interaction between the payoffs from independent markets via concavity in the firms' objective functions. The concavity generates scale economies and provides further collusion in both markets by reducing the incentives to act selfishly in both issues. Limao (2005) builds on Spagnolo to introduce explicit structural independence between two international issues. When tariffs and externality taxes are strategic complements, a simultaneous deviation in both policies grants the deviator less benefit than the sum of the gains in each policy independently. Therefore, the linkage incentive constraint is slack when evaluated at the no-linkage solution. A common element in these papers (see also Ederington 2001, 2003 and Conconi and Perroni 2002) is that linking the issues cannot reduce the enforcement. On the other hand, Ederington (2002) uses information asymmetry to show that linking might be detrimental when countries incorrectly observe cheating, and it might be beneficial when they fail to detect cheating. A major methodological departure of our paper is the lack of structural or informational interdependence between the issues. Our results solely depend on the strategic substitutability of tariffs between governments and the choice of punishment strategies.

The present paper also fits into a body of research that investigates the economic implications of the current legal and institutional framework in international economic relations. Bown (2004) analyzes the WTO dispute settlement process from an economic perspective, and Bown and Hoekman (2005, 2008) focus on the legal aspects of it.

In the next section, we describe the economy of each country. In the third section, we consider the tariff choices in the absence of a trade agreement. In the fourth section, we introduces trade agreements under differing enforcement strategies. In the fifth section, we show how the architects of the GATTAVTO were prescient in combining LCR with WEC. We consider some comparative statics in the sixth section and our conclusions are in the seventh section.

II. ECONOMIC ENVIRONMENT

We are interested in a two-country tariff-setting framework. To introduce cross retaliation requires that each country have at least two export goods. In addition, we follow Johnson's (1953-1954) seminal analysis of tariff games which suggests a strategic dependence between the tariffs chosen by each country. In a traditional two-good general equilibrium framework, the tariffs are strategic substitutes because the income effect from an increase in a foreign tariff generates a negatively sloped home tariff best response function. Our need to include at least four goods makes a traditional framework difficult to implement; however, home and foreign tariffs are generally independent in a partial-equilibrium framework. Our model reintroduces strategic dependence of the tariffs in a tractable structure with a partial-equilibrium intuition.

We consider an environment with two countries, two sectors, and two goods in each sector. Each country has an export good and an import good in each sector (a and b). The home country exports the * goods in each sector ([x.sub.a] and [x.sub.b]) and imports the y goods ([y.sub.a] and [y.sub.b]). There is also a numeraire good z? Consumer preferences are represented by a quasi-linear utility function, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the consumption of good [j.sub.i]; j [member of] [x,y] and i [member of] {a,b}. Subutility functions [u.sub.a](.) and [u.sub.b](.) are increasing and concave in each argument. Demand for the goods is related in each sector, but the sectors are independent. The subutility functions may be written as

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where A, B, and b are all positive constants. We start by assuming symmetry between the countries and sectors, but we relax this assumption in later sections. In this case, the goods are substitutes in consumption and the demand function for good y can be written as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with a similar expression for good x.

The numeraire good is produced under a constant return to scale technology using a single unit of labor per output. The labor supply in each country is sufficiently large, therefore, the numeraire is produced, and the wage is equal to one, in both countries. The other goods are produced under increasing marginal costs using labor only. Costs of production are given by the strictly convex functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the production of good [j.sub.i]. Producers maximize profits given technologies and equilibrium prices, which equalizes the producer price of a good to its marginal cost. The home marginal costs are lower for the x goods and higher for the y in each sector, so that home has a comparative advantage in the x goods. The cost functions may be written as (where a superscript star denotes Foreign country values):

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We start by assuming symmetry between the countries (given identical preferences, this assumption implies symmetric cost functions so that D = F) but we relax this assumption in later sections.

Governments choose tariffs, [[tau].sub.a] and [[tau].sub.b], on imported goods in each sector to maximize domestic welfare. (7) There are no export taxes or subsidies. Tariffs generate a wedge between domestic and international prices so that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Equilibrium prices in each sector are, therefore, only a function of market clearing conditions in their own sector: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Hence, prices, and quantities, can be written as a function of own sector tariff choices. The welfare of the home country can then be written as:

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that because the sectors are not related, the indirect utility function is separable in the policy variables.

III. UNILATERAL POLICY IN THE ABSENCE OF TRADE AGREEMENTS

In the absence of a trade agreement, governments maximize domestic welfare unilaterally given the policy variable chosen by the other government.

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is Home's import demand function, a prime indicates a derivative, and numbers in subscripts denote the ordered derivatives when the function has more than one parameter. Each term in the square brackets is equal to zero by the Envelope Theorem and the first order conditions from the producer and consumer maximization problems. The unilaterally optimal tariff is, therefore, given by:

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note that when goods are related in a given sector, Home's import demand is a function of the foreign tariff. Therefore, the unilaterally optimal tariff is also a function of the foreign tariff. The following proposition provides some useful characteristics of how welfare is affected by tariff policies. The proofs of all propositions are contained in the appendix.

PROPOSITION 1. Suppose the consumer utilities are given in quasilinear form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with the subutility functions given by Equation (1) and the cost functions given by Equation (2). The sector-specific social welfare function has the following characteristics when countries are symmetric.

(i.) [[??].sub.i] ([[tau].sub.i],[[tau].sup.*.sub.i]) is strictly concave and initially increasing in own tariff; strictly convex and decreasing in the Foreign tariff: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(ii.) Home and Foreign tariffs are strategic substitutes: [[??].sub.i12] ([[tau].sub.i], [[tau].sup.*.sub.i]) < 0.

(iii.) Free trade is the global optimum: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(iv.) [[??].sub.u11]([[tau].sub.i],[[tau].sub.i]) + [delta] x [[??].sub.22] ([[tau].sub.i],[[tau].sub.i]) < 0.

When tariffs are strategic substitutes, each government has less incentive to increase its tariff unilaterally when its exports are subject to greater tariffs in the destination country. The idea here is that both foreign and home tariffs reduce the relative price of the export good in the home country. A lower export price diminishes the effect of a tariff hike in the home country when the goods are substitutes in consumption. We also see that unilaterally optimal policies are not globally efficient. In the absence of cooperation between the governments, the applied tariffs are too high and the trade volume is too low as compared to the free trade levels. However, as cooperation is mutually beneficial and the interaction between governments is repeated, there is scope for a cooperative relationship. The next section will investigate alternative cooperation schemes.

IV. STRUCTURE OF COOPERATION IN TRADE AGREEMENTS

A trade agreement in sector i specifies a maximum tariff rate ([[tau].sup.c.sub.i]) to be applied by both governments in that sector. In the absence of an external enforcement mechanism, this cooperative tariff needs to be incentive compatible (i.e., a one shot gain by betraying at any point in time needs to be (weakly) lower than the cost of future punishments). Therefore, the actual punishment strategies determine the structure of cooperation. We focus on two types of punishment strategies in this paper. First, we investigate an unlinked trade agreement when a limited punishment strategy WEC and limited LCR is applied. Next, we consider a linked agreement and we remove the LCR restriction so that only WEC is applied, therefore, in the linked agreement we allow for cross-sector and cross-agreement retaliation. Our focus is to compare same-sector versus cross-sector retaliation under these WEC-limited punishment strategies.

Under limited punishment strategies, governments are restrained by the WEC rule in the spirit of GATT Article XXVIII: if any government applies a tariff greater than the agreed cooperative rate, [[tau].sup.d.sub.i] > [[tau].sup.c.sub.i], then the other government is allowed to retaliate only by the same amount, [[tau].sup.*d.sub.i] = [[tau].sup.d.sub.i] in the future periods as long as the initial deviation is no larger than the static Nash tariff, [[tau].sup.d.sub.i] [less than or equal to] [[tau].sup.n.sub.i]. Deviations greater than the static Nash tariff are considered egregious and both countries apply the static Nash tariffs forevermore. (8)

The LCR restriction maintains that the retaliation must be applied in the same sector (or agreement) as the original deviating tariff. The combination of WEC and LCR is termed the unlinked agreement. The linked agreement does not limit cross retaliation and only imposes WEC. Although the linked agreement does not require cross retaliation, we conduct our analysis under the assumption that it would be used when permitted. We confirm this hypothesis in Proposition 8 below.

The limited punishment is enforced by the threat of a more severe punishment. In particular, if a country deviates from the limited punishment, then this is considered egregious (it is, in effect, abandoning the trade agreement) and it triggers an infinite reversion to the static Nash tariffs. We show, in Proposition 7, that these self-enforcing punishment regimes can be sequentially rational and part of a subgame perfect equilibrium.

We consider the following timing of events:

1. In period 0, governments agree on a type of agreement [theta] [member of] {[[theta].sup.L], [[theta].sup.U]} (linked or unlinked), and then specify the cooperative tariff rates for each sector: [[tau].sup.c.sub.i], i [member of] {a,b}.

2. In the beginning of each period, t, governments observe the action history, then simultaneously announce the tariff rate to be applied: [[tau].sup.l.sub.i], [[tau].sup.*l.sub.i], I [member of]{a,b}.

3. Production and consumption take place upon observing the announced tariffs and prices adjust to clear markets.

Formally, the trade agreement uses history-dependent strategies. A history through period t provides the complete information of all previous tariff choices by both countries and also the type of the agreement, [H.sup.t] = {[theta], [[tau].sup.T.sub.i], [[tau].sup.*T.sub.i]} where [theta] [member of] {[[theta].sup.L], [[theta].sup.U]}, [[tau].sup.T.sub.i] = {[[tau].sup.1.sub.i], [[tau].sup.2.sub.i], ..., [[tau].sup.t-1.sub.i]}, and i[member of]{a,b}. The trade agreement then specifies a transformation rule that conditions the actions to be chosen in the current period t upon the observed history, T ([H.sup.t]) [right arrow] ([[tau].sup.t.sub.i]) [member of] [R.sup.2.sub.+].

In the cooperative phase both countries levy ([[tau].sup.c.sub.i], [[tau].sup.*c.sub.i]} and the value to the home country is [OMEGA] = [[??].sub.i] ([[tau].sub.i], [[tau].sup.*.sub.i]). We now analyze how [OMEGA] interacts with [PSI] ([p[tau].sub.i], [[tau].sup.*.sub.i], [delta]), which is the continuation value of a deviation and the subsequent punishment strategies. We perform this analysis for each of the considered punishment strategies and we describe the differing levels of cooperation ([[tau].sup.c.sub.i], [[tau].sup.*c.sub.i]} that are obtainable by each punishment regime.

In the repeated game implied by the trade agreement, we consider the discounted average payoffs [[??].sub.i], ([[tau].sub.i], [[tau].sup.*.sub.i]) = (1 - [delta]) [[??].sub.i] ([[tau].sub.i], [[tau].sup.*.sub.i]), where [delta] is the common factor by which governments discount future payoffs. Hence, starting in any period s we have [[infinity].summation over (t=s)] [[delta].sup.t-s] [[??.sub.i] ([[tau].sub.i], [[tau].sup.*.sub.i]) = [[??].sub.i] ([[tau].sub.i], [[tau].sup.*.sub.i]).

As a benchmark case, we start by analyzing the well-known Nash-reversion punishment strategies.

A. Cooperation under Nash Reversion Strategies

Given that the sectors are identical, there is no cost or benefit to linking in this regime. This claim is a version of the well-known Bernheim and Whinston (1990) irrelevance result. Hence, we make no distinction between the linked or unlinked case in the Nash-reversion regime. Under Nash reversion, governments apply cooperative tariffs as long as there is no deviation in the current history of the agreement in a given sector; however, both governments apply the static Nash equilibrium tariff forever upon observing a deviation at any point in time. The cooperative tariff rate in sector i, therefore, needs to satisfy the following incentive constraint

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where superscripts d, p, n, and c, denote deviation, punishment, Nash, and cooperative values, respectively. The left-hand side of this inequality is the normalized sum of the discounted payoff stream when the home government deviates in the current period and both governments apply Nash tariffs in the remaining periods.

PROPOSITION 2. The optimal deviation tariff under the Nash Reversion rule is a strictly decreasing function of the cooperative tariff: [[tau].sup.dN.sub.i] [equivalent to] [[tau].sup.d.sub.i] ([[tau].sup.*c.sub.i]) and {d[[tau].sup.dN.sub.i]/d[[tau].sup.*c.sub.i]) < 0.

This proposition is illustrated in Figure 1. It shows that when the magnitude of the punishment is independent from the magnitude of deviation, then governments maximize the stage game payoff regardless of how they discount future welfare. The deviation tariff is decreasing in the cooperative tariff at a given point in time only because Home's best response tariff in a static set up is decreasing in the Foreign tariff due to strategic substitutability.

B. Cooperation under Unlinked Limited Punishment Strategies

Incentive compatibility, therefore, needs to address two issues. First, each government decides on the optimal level of deviation given the cooperative tariff rate applied by the partner. Second, they decide whether it is optimal to deviate from the agreement using the optimal deviation tariff. For [[tau].sup.c.sub.i] > [[tau].sup.n.sub.i], the incentive constraint is the same as in Equation (6). More importantly, for [[tau].sup.d.sub.i] [less than or equal to] [[tau].sup.n.sub.i], the incentive constraint under the unlinked limited punishment rule reflects the effect of the deviation level on future punishments:

(7)[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As in the Nash Reversion case, the left-hand side of this inequality is the normalized sum of the discounted payoff stream when the home government deviates in the current period. As opposed to the former case, however, the initial deviation determines the payoff stream during the punishment phase under WEC rule. Notice that in the unlinked case, both the deviation and the punishment take place in the same sector and because of symmetry between the sectors it is immaterial which sector is chosen. Given the cooperative tariff rate, the [[tau].sup.dU.sub.i] solves:

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As seen in Figure 1, and described in Proposition 3 below, the best response tariff [[??].sup.dU.sub.i] includes [[tau].sup.dU.sub.i]; however, it is more complicated than [[tau].sup.dU.sub.i]. The following proposition provides the analytical results regarding the behavior of [[tau].sup.dU.sub.i] and the best response tariff, [[??].sup.dU.sub.i], in the Unlinked-WEC regime.

PROPOSITION 3. (i.) [[tau].sup.dU.sub.i] is a strictly decreasing function of the cooperative tariff rate and the discount factor: [[tau].sup.*c.sub.i] ([[tau].sup.*c.sub.i], [delta]) < 0 and [[tau].sup.dU.sub.i2] ([[tau].sup.dU.sub.i], [delta]) < 0.

(ii.) For any level of the cooperative tariff, xf ft) < xf ft).

(iii.) There exists a unique [[tau].sup.mcU.sub.i] such that [[tau].sup.dU.sub.i] ([[tau].sup.mcU.sub.i]) = [[tau].sup.mcU.sub.i]. If [[tau].sup.c.sub.i], [[tau].sup.mcU.sub.i] then [[tau].sup.c.sub.i] <[[tau].sup.mcU.sub.i] < [[tau].sup.dU.sub.i] ([[tau].sup.dU.sub.i]). If [[tau].sup.c.sub.i] > [[tau].sup.cmU.sub.i] then

(iv). If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This proposition shows first of all that when punishment is tailored to the initial deviation, the optimal deviation tariff is no longer a static best response to the current cooperative tariff rate. The optimality condition in this case states that a deviating government increases the deviation tariff until the marginal gain in the current payoff becomes equal to the losses in the future punishment phase. In this way, the optimal deviation takes account of its effect on the future punishment and, therefore, it is lower than it would be in the Nash-reversion regime.

The proposition is illustrated in Figure 1. We see there that both [[tau].sup.dU.sub.i] ([[tau].sup.c.sub.i]) and [[tau].sup.dN.sub.i] ([[tau].sup.c.sub.i]) are declining in x'. Furthermore, for any x'. we have that [[tau].sup.dU.sub.i] ([[tau].sup.c.sub.i]) [less than or equal to] [[tau].sup.dN.sub.i] ([[tau].sup.c.sub.i]). When each function crosses the 45-degree line, then the function provides a best response to itself. In the Nashreversion regime, [[tau].sup.dN.sub.i] crosses the 45-degree line at the static Nash tariff, [[tau].sup.dn.sub.i], because [[tau].sup.n.sub.i] is a static best response to itself. If [[tau].sup.dU.sub.i] ([[tau].sup.c.sub.i]) > [[tau].sup.n.sub.i], then the deviation is considered egregious and the punishment is given by the Nash regime. In this case, the best response [[tau].sup.dU.sub.i] ([[tau].sup.c.sub.i] jumps up to [[tau].sup.n.sub.i]. The bold part of the graph is the best response function. For very high [[tau].sup.c.sub.i], the best response would be a tariff reduction if it would be matched in the future; however. WEC only applies to tariff increases, so that the tariff reduction would not be matched. In this case, the best response is to match the tariff increase, but not to supersede it, and this part of the best response function is the bold part of the 45-degree line from where [[tau].sup.dU.sub.i] crosses it until it reaches the static Nash tariff. Finally, it is interesting to note how x[[tau].sup.dU.sub.i] and, therefore, the best response tariff [[??].sup.dU.sub.i] change with the discount factor [delta]. First notice that when countries care more about the future, and [delta] increases, the second term in [[PSI].sup.U.sub.i] has a higher weighting so that the [[tau].sup.dU.sub.i] shifts down. It eventually shifts down enough so that [[tau].sup.dU.sub.i] is always less than [[tau].sup.n.sub.i] and there is no discontinuity in the best response tariff. In the limit as [delta] approaches one, we can see from Equation (A2) in the appendix that the best response is free trade. Similarly, as [delta] approaches zero, the [[tau].sup.dU.sub.i] shifts up so that the best response is the static Nash tariff.

The following proposition is very useful because it shows that the most cooperative tariff in the Unlinked-WEC case can be characterized by where the optimal deviation tariff crosses the 45-degree line. This result is crucial to our later analysis. In particular, we will also show that the most cooperative tariff in the Linked-WEC case can be characterized in the same manner and, in this way, we will be able to compare the level of cooperation under the two regimes. In addition the proposition shows that for any level of the cooperative tariff, the optimal deviation tariff in the Unlinked-WEC case is less than in the Nash reversion case. (9)

PROPOSITION 4. There exists a unique most cooperative tariff under Unlinked-WEC strategies, [[tau].sup.mcU.sub.i] [equivalent to] [[tau].sup.dU.sub.i] ([[tau].sup.mcU.sub.i]), which is decreasing in the discount factor [delta].

C. Linking the Agreements under Limited Punishment Rule

In this section, we analyze how WEC as stated in the GATT Article XXVIII interacts with the WTO DSU Article 22.3, which allows for but also limits cross retaliation. In particular, we investigate the consequences of linking the agreements under the limited punishment rule in terms of its welfare and enforcement implications. Linking enables the governments to undertake cross retaliation (i.e., betrayal in one agreement generates a punishment phase in the other one). Our idea is rather general in that cross retaliation may entail cross-sector retaliation as in DSU Article 22.3 paragraph (b), or it may be cross-agreement retaliation as in DSU Article 22.3 paragraph (c). The key is that goods in the same sector exhibit strategic substitutability and goods across sectors (or agreements) are strategically independent. We continue to assume that the WEC rule is applicable only when the initial deviation is not egregious. We characterize two types of egregious imposed by the GATTAVTO, as a starting point and we use it to compare the role of cross retaliation under WEC. deviations, both of which call for different treatment in punishment stage. First, deviation in both policies or deviation from the punishment path generates Nash reversion in both policies. (10) Second, a deviating tariff greater than the static Nash tariff brings a cross retaliation using Nash Tariffs, (i.e., if the Home government applies [[tau].sup.d.sub.i] > [[tau].sup.c.sub.i]. then forever after the Home government will apply [[tau].sup.N.sub.i] and [[tau].sup.c.sub.-i], whereas the foreign government will apply [[tau].sup.*c.sub.i] and [[tau].sup.*N.sub.-i], i [member of] {a, i>}. (11) For a nonegregious deviation, we can, therefore, write the Linked-WEC incentive constraint as follows:

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Being symmetric, deviation in either sector is possible and both deviations are equal. We decided the most natural was to write the constraint for Home deviating in sector a. The deviation tariff is derived from:

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

A deviation in sector a generates a stream of gains in that sector; however, it also generates a future stream of losses in sector b due to cross retaliation by the partner. The first-order condition for this optimization problem is given by:

(11) [[??].sup.d.sub.a1] ([[tau].sup.d.sub.a] ([[tau].sup.*c.sub.a]), [[tau].sup.*c.sub.a]) + [delta] * [[??].sup.p.sub.b2] ([[tau].sup.c.sub.b], [[tau].sup.*d.sub.b] ([[tau].sup.c.sub.b])) = 0

where the second term shows the discounted change in sector b payoffs due to a marginal increase in the deviation tariff in sector a. The following proposition elaborates the characteristics of cooperation under a Linked-WEC agreement and is directly comparable to Propositions 3 and 4.

PROPOSITION 5. The Linked-WEC agreement has the following characteristics:

(i.) The deviation tariff, [[tau].sup.dL.sub.i], is strictly decreasing in the cooperative tariff rate, (d[[tau].sup.dL.sub.i] ([[tau].sup.*c.sub.i]) /d[[tau].sup.*c.sub.i]) < 0 and in the discount factor (d[[tau].sup.dL.sub.i] ([[tau].sup.*c.sub.i]) /d[delta]) < 0.

(ii.) There exists a most cooperative tariff [[tau].sup.mcL.sub.i] [equivalent to] [[tau].sup.dL.sub.i] ([[tau].sup.mcL.sub.i]), which decreases in the discount factor [delta].

(Hi.) If [[tau].sup.c.sub.i] [less than or equal to] [[tau].sup.dL.sub.i] ([[tau].sup.c.sub.i]) [less than or equal to] [[tau].sup.n.sub.i], then the best response [[??].sup.dL.sub.i] ([[tau].sup.c.sub.i]) = [[tau].sup.dL.sub.i] ([[tau].sup.c.sub.i]). If [[tau].sup.c.sub.i] > [[tau].sup.mcL.sub.i], then [[??].sup.dL.sub.i] ([[tau].sup.c.sub.i]) = [[tau].sup.c.sub.i]. If [[tau].sup.dL.sub.i] ([[tau].sup.c.sub.i]) > [[tau].sup.n.sub.i], then [[??].sup.dL.sub.i] ([[tau].sup.c.sub.i]) = [[tau].sup.dN.sub.i] ([[tau].sup.c.sub.i]).

Proposition 5 is illustrated in Figure 2. We see there that [[tau].sup.dL.sub.i] is negatively sloped as is [[tau].sup.dU.sub.i]. Furthermore, we see that the best response, [[??].sup.dL.sub.i], has the same shape and the same three sections as does [[??].sup.c.sub.i]. Finally, we see that where [[tau].sup.dL.sub.i] crosses the 45-degree line determines the most cooperative tariff in the Linked-WEC regime, [[tau].sup.mcL.sub.i]. The most important part of Figure 2 is that [[tau].sup.dL.sub.i] ([[tau].sup.dL.sub.i]) > [[tau].sup.dU.sub.i] ([[tau].sup.c.sub.i]) for all [[tau].sup.c.sub.i], so that [[tau].sup.mcL.sub.i] > [[tau].sup.mc.sub.U]. This last claim is the most important result of our paper and is the subject of the following proposition.

PROPOSITION 6. (Main Result 1) For every level of the cooperative tariff, the optimal deviation tariff under the Linked-WEC agreement is greater than the one under the Unlinked-WEC agreement, [[tau].sup.dL.sub.i] ([[tau].sup.c.sub.i]) > [[tau].sup.dU.sub.i] ([[tau].sup.c.sub.i]). Linkage, therefore, reduces cooperation in a given sector: [[tau].sup.mcL.sub.i] > [[tau].sup.mcU.sub.i].

The idea behind this comes from the strategic substitutability of tariffs in the same sector. Foreseeing a reciprocating punishment in the same sector generates a reduction in the same-sector optimal deviation. Across sectors, or agreements, there is strategic independence between the goods and the optimal deviation tariff is not mitigated by the strategic substitutability effect; therefore, for an equal deviation, the punishment hurts the deviating country by a larger amount in the Unlinked-WEC agreement and the optimal deviation is lower. Hence (because the most cooperative tariff can be described as an invertible, and monotonie increasing function of the optimal deviating tariff), we have that the most cooperative tariff is larger in the Linked-WEC than in the Unlinked-WEC regime.

V. THE ORDER OF PREFERRED RETALIATION IN THE DSU ARTICLE 22.3

In this section, we analyze whether, given a deviation, each country prefers same-sector or cross-sector retaliation. We then consider whether their history dependent preferred action generates the best outcome for the entire trade agreement. As a first step in this analysis, we need to show that both punishment paths are subgame perfect. We next analyze which path is preferred after a deviation and which is preferred for the entire agreement.

A. Subgame Perfection of the Linked-WEC and Unlinked-WEC regimes

We now show that both the Linked-WEC and Unlinked-WEC trade agreement strategies and payoffs are subgame perfect. In particular, we show that after any deviation, each country would adhere to the punishment strategies given by the chosen regime. We then provide conditions on the patience necessary to support cooperation in each regime. We also provide an alternative proof for our main result from Proposition 6: For any cooperative tariff, the required patience is larger in the Linked-WEC regime.

PROPOSITION 7. (i.) For any cooperative tariff [[tau].sup.c.sub.i], there exists a [[delta].sup.L] ([[tau].sup.c.sub.i]) [member of] (0,1), such that for all [delta] [greater than or equal to] [[delta].sup.L] ([[tau].sup.c.sub.i]), the Linked-WEC trade agreement strategies and payoffs constitute a subgame perfect equilibrium.

(ii.) For any cooperative tariff [[tau].sup.c.sub.i], there exists a [[delta].sup.U] ([[tau].sup.c.sub.i]) [member of] (0,1), such that for all [delta] [greater than or equal to] [[delta].sup.U] ([[tau].sup.c.sub.i]), the Unlinked-WEC trade agreement strategies and payoffs constitute a subgame peifect equilibrium. (iii.) For any ([[tau].sup.c.sub.i]), [[delta].sup.L] ([[tau].sup.c.sub.i]) > [[delta].sup.U] ([[tau].sup.c.sub.i]).

Proposition 7 shows that the Linked-WEC and Unlinked-WEC strategies are subgame perfect if countries are sufficiently patient. Still, it does not show that the most cooperative tariffs derived in Propositions 4 and 5 can be supported by subgame perfect WEC regimes for any value of the discount factor. Even though the most cooperative Linked-WEC and Unlinked-WEC tariffs are declining in the discount factor, it is possible that for a very low discount factor both WEC regimes can reduce the tariff slightly from the static Nash equilibrium tariff but the proposed punishment supporting this reduction is not subgame perfect. Still, it is important to note that Unlinked-WEC strategies are subgame perfect for a wider range of discount factors than are the Linked-WEC strategies. (12)

B. DSU Article 22.3

Although for sufficiently patient governments both punishment paths are subgame perfect, the governments may prefer one path over the other after a deviation. In the next proposition we show that this is indeed the case. For the same reason that the optimal deviation is higher in the Linked-WEC regime, both countries prefer the Linked-WEC regime after a deviation. In particular, the strategic substitutability of within-sector tariffs generates lower payoffs when the punishment phase occurs only in one sector. The necessary step in the proof of the following proposition is the following property of submodular functions.

Definition. (Topkis 1998, 43) A real valued function fix) : [R.sup.n] [right arrow] R is supermodular in x [member of] X, if:

f(x')+f(x") [less than or equal to] f(min (x'x")) + f (max (x'x"))

for all x', x" [member of] X. It is strictly supermodular if the inequality is strict. It is (strictly) submodular if -f(x) is (strictly) supermodular.

For continuously differentiable functions, supermodularity and submodularity reduce to strategic complementarity and strategic substitutability. In the case of our welfare function, shows that ([[partial derivative].sup.2]v(.)/[partial derivative] [tau][partial derivative][tau]*) < 0, so that [[??].sub.i] ([[tau].sub.i],[[tau].sup.*.sub.i]) is submodular in {[[tau].sub.i], [[tau].sup.*.sub.i]}.

PROPOSITION 8. (Main result 2) After any deviation from the cooperative path by either country, both the deviating country and the retaliating country prefer the continuation path given by punishments in the Linked-WEC regime.

Proposition 8 is our second main result in that it shows that countries prefer the punishment path with less enforcement power. Although the Unlinked-WEC regime generates a higher level of cooperation (a lower most cooperative tariff), both countries would prefer the less punitive Linked-WEC regime after a deviation.13 In particular, the punishing country would choose to punish by cross retaliation if possible and the deviating country would welcome this punishment choice. Given that countries recognize their time inconsistency in the punishment regime choice it would make sense for them to limit their retaliation regime from the outset. Remember that in the initial stage of the trade agreement countries decide on the Linked or Unlinked regime. The following proposition, which is a corollary of Propositions 6 and 8 states that in the symmetric case considered so far, the countries would choose to limit their punishment options and choose the Unlinked-WEC regime from the outset.

PROPOSITION 9. (Main result 3) If countries place enough value on future payoffs, then in any subgame perfect equilibrium of the entire trade agreement the regime choice is always the Unlinked-WEC regime: [theta] = [[theta].sup.U].

Propositions 8 and 9 justify the order of cross retaliation given in the WTO DSU Article 22.3 that was described in the introduction. Countries are encouraged to choose same-sector retaliation if feasible, and only if not feasible can they consider cross-sector (and very rarely cross-agreement) retaliation. In the symmetric case considered here same-sector retaliation is feasible and this limitation is welfare enhancing.

VI. ASYMMETRIES AND COMPARATIVE STATICS

In this section, we consider technological improvements that change the magnitude of comparative advantage and we analyze how these changes affect the most cooperative outcomes in the different trade agreement regimes. First, we introduce a symmetric change in the degrees of comparative advantage and then we consider asymmetric changes.

When these changes create asymmetries, we need to be certain that the previous results of our model obtain in the absence of symmetry. First note that the functional forms are all twice continuously differentiable and the results are all based on first and second derivative conditions. Hence, the results hold for small asymmetries.

We start by considering identical symmetric increases in comparative advantage in both countries and in both sectors: d[D.sub.i] = d[F.sub.i] > 0. Next we consider asymmetric changes, whereby d[D.sub.i] > 0 and d[F.sub.i] = 0, or d[D.sub.i] = 0 and d[F.sub.i] > 0 for i [member of] {a,b}. Notice that these changes can be interpreted as export-biased technological improvements, where the relative cost of production in the exporting country decreases. As we show in the following proposition, for a symmetric change there is no difference between the enforcement capability of the Linked-WEC and Unlinked-WEC regime: however, they both dominate the Nash-reversion regime. In particular, the most cooperative tariff does not change in either WEC regime and it increases in the Nash regime.

PROPOSITION 10. A symmetric increase in comparative advantage in both countries and in both sectors, d[D.sub.i] = d[F.sub.i] > 0, decreases cooperation under Nash-reversion strategies. Cooperation under Unlinked-WEC and Linked-WEC strategies remain unchanged.

This result provides a surprising justification of the WEC rule. In the Nash-reversion regime, the future punishment is not directly tied to the original deviation. Being as the original deviation is an optimal tariff, and the point of an optimal tariff is to capture as much of the gains from trade as possible, then it is clear that the optimal deviation in the Nash regime will fluctuate with the degree of gains from trade, or comparative advantage. Hence, changes in the gains from trade require flexibility in the trade agreement to avoid generating serious trade wars (Bagwell and Staiger 1990). In the WEC regimes, on the other hand, a larger Home deviating tariff allows the Foreign trading partner to also capture more of the gains from trade generated by Home's symmetric export-biased technological improvement. In the particular case considered here, these effects are offsetting so the net effect is zero. Still, the intuition suggests that the result should hold in a more general model.

We now consider asymmetric changes and we analyze their affect on the most cooperative tariff in the Linked- and Unlinked-WEC regimes. We again use an increase in the cost disadvantage of the importing country to represent an export-biased technological improvement in the exporting country (or an increase in the gains from trade). We could also consider the opposite change, or an import-biased technological improvement. Our main goal in this paper is to compare Linked- and Unlinked-WEC and in the particular case of a change in only one country we can show that Linked-WEC regime generates wider fluctuations than does the Unlinked-WEC regime.

PROPOSITION 11. A small export-biased (or import-biased) technological improvement in only one country generates a larger change in the most cooperative tariff when countries abide by the Linked-WEC regime as opposed to Unlinked-WEC regime.

Proposition 11 shows that when the same-sector goods are strategic substitutes, then linking agreements generates wider fluctuations in the most cooperative tariffs. This result occurs because technology changes generate changes in comparative advantage and the gains from trade. These changes in the gains from trade alter the benefits from an optimal deviation tariff and, therefore, change the level of obtainable cooperation. The key is that the optimal deviation tariff is mitigated in the WEC regimes, because the level of deviation affects the permissible retaliation. This effect is captured by [delta] in the above derivatives. In the Linked-WEC regime, retaliation takes place in the other sector so the deviation does not impinge on the benefit of the deviation. In the Unlinked-WEC regime, this retaliation takes place in the same sector and because of strategic substitutability the future benefit of the deviating tariff is declining in the level of the retaliation. This effect is captured by b in the above derivatives. If b = 0, then the goods are independent and the two regimes are identical. As b increases the substitutability increases and the difference between the regimes grows.

VII. CONCLUSION

In this paper, we consider two prominent institutional rules in the international trading system that are designed to limit the countermeasures upon a violation. One rule limits the composition of retaliation and the other limits the magnitude of retaliation. Although seemingly unrelated, the limited cross-retaliation rule complements the limited punishment rule in constraining the scope and the magnitude of punishment in international trade disputes. Specifically, we elaborate a mechanism through which the limited cross-retaliation rule also helps limit the incentives to violate the trade agreement when the limited punishment rule prevails.

We start by showing that if the import and export goods are substitutes in consumption, then the underlying preferences generate a welfare function whereby tariffs are strategic substitutes. Given strategical substitutability, the limited retaliation rule reduces the deviation magnitude by a larger amount when cross retaliation is not allowed. On the other hand, once a limited deviation has occurred countries prefer cross-retaliation over same-sector retaliation. This preference can create a problem in that countries will expect cross retaliation ex post and increase the level of deviation ex ante. By subordinating more distant cross retaliations, the WTO DSU Article 22.3 was prescient in foreseeing this time-inconsistency problem and, in fact, cross-agreement retaliation has only been permitted three times in the history of the WTO (out of nine requests).

APPENDIX

PROPOSITION 1.

Proof. Suppose the subutility functions are of the form given in Equation (I):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Solving for the consumer and producer maximization problems and plugging in the indirect utility function provides us the following value:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Now we can show the changes in social welfare as a response to a marginal change in policy variables.

(i.) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(ii.) [[partial derivative].sup.2] [[??].sub.i] ([[partial derivative].sub.i], [[partial derivative].sup.*.sub.i])/ [partial derivative][[tau].sup.*.sub.i] [partial derivative][[tau].sub.i]

(iii.) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(iv.)[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

PROPOSITION 2.

Proof. The optimal deviation solves the first-order condition from the maximization of [[PSI].sup.N.sub.i] given in Equation (6):

(A1) ([partial derivative][[PSI].sup.N.sub.i]/[partial derivative][[tau].sup.*.sub.i]) = [[??].sup.di.sub.1], ([[tau].sup.d.sub.i], [[tau].sup.*c.sub.i]) = 0

Totally differentiating this condition with respect to [[tau].sup.*rc.sub.i] and [[tau].sup.d.sub.i] and rearranging yields:

d[[tau].sup.dN.sub.i]/d[[tau].sup.*c.sub.i] = (- [[??].sup.,d.sub.i12] ([[tau].sup.d.sub.i][[tau].sup.*c.sub.i])/[[??].sup.di.sub.11] ([[tau].sup.d.sub.i], [[tau].sup.*c.sub.i]) < 0

where, the denominator is negative by concavity of the indirect utility function (Proposition l.i) and the numerator is positive since the tariffs are strategic substitutes (Proposition 1.ii).

PROPOSITION 3.

Proof, (i.) The first-order condition for the maximization problem (8) is given by:

(A2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which provides the solution for the optimal deviation tariff as [[tau].sup.dU.sub.i] [equivalent to] [[tau].sup.dU.sub.i] ([[tau].sup.*c.sub.i], [delta]). Totally differentiating this first-order condition with respect to the cooperative tariff and the deviating tariff and using the implicit function theorem yields:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the denominator is the second-order condition and negative, and the numerator is positive by strategic substitutability.

Similarly, totally differentiating this first-order condition with respect to the discount factor and the deviating tariff and rearranging yields:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the term in brackets in the numerator is negative by the property that free trade is globally efficient.

(ii.) To show this, we compare the first-order conditions for the optimal deviation tariffs under Nash reversion and Unlinked-WEC strategies, (Al) and (A2), respectively. The term in brackets in the latter one is smaller than zero, therefore [[??].sup.d.sub.i1] ([[tau].sup.dN.sub.i]([[tau].sup.*c.sub.i]), [[tau].sup.*c.sub.i]) = 0 < [[??].sup.d.sub.i1] ([[tau].sup.dU.sub.i] ([[tau].sup.*c.sub.i]), [[tau].sup.*c.sub.i]) for identical cooperative tariff rates. This implies that [[tau].sup.dU.sub.c] ([[tau].sup.*c.sub.i]) < [[tau].sup.dU.sub.i] ([[tau].sup.*c.sub.i]) at all identical [[tau].sup.*c.sub.i] by concavity of the indirect utility function in its own tariff (Proposition 1. ii).

(iii.) From part (i.) [[tau].sup.dsub.i] is monotonie decreasing in [[tau].sup.*c.sub.i]. Hence, when [[tau].sup.c.sub.i] - [[tau].sup.mcU.sub.i] we have that [[tau].sup.dU.sub.i] ([[tau].sup.c.sub.i]) = [[tau].sup.c.sub.i]. From part (i.) for [[tau].sup.c.sub.i] < [[tau].sup.mcU.sub.i], we have [[tau].sup.c.sub.i] < [[tau].sup.mc.sub.i] < [[tau].sup.dU.sub.i] ([[tau].sup.c.sub.i]) and for [[tau].sup.c.sub.i] > [[tau].sup.mcU.sub.i], we have [[tau].sup.c.sub.i] > [[tau].sup.mc.sub.i] > [[tau].sup.dU.sub.i] ([[tau].sup.c.sub.i]).

(iv.) Tariff reductions are not subject to retaliation (reciprocation), therefore, there is no profitable one shot deviation for [[tau].sup.c.sub.i] > [[tau].sup.mcU.sub.i]. Hence, [[??].sup.dU.sub.i] ([[tau].sup.c.sub.i]) = [[tau].sup.c.sub.i]. in that region. If [[tau].sup.dU.sub.i] ([[tau].sup.c.sub.i]) > [[tau].sup.c.sub.i] ([[tau].sup.c.sub.i]), then the deviation is egregious and the punishment is given by the Nash-reversion regime, therefore, the best response is given as in Proposition 2: [[??].sup.dU.sub.i] ([[tau].sup.c.sub.i]) = [[tau].sup.dN.sub.i] ([[tau].sup.c.sub.i]). If [[tau].sup.c.sub.i]. [less than or equal to] [[tau].sup.dU.sub.i] ([[tau].sup.c.sub.i]) [less than or equal to] [[tau].sup.dN.sub.i] ([[tau].sup.c.sub.i]). then the best response is given by Equation (8) so that [[??].sup.dU.sub.i] ([[tau].sup.c.sub.i]) = [[tau].sup.dU] ([[tau].sup.c.sub.i]).

PROPOSITION 4.

Proof, (i.) In order to prove the existence of a unique most cooperative tariff, we will show that [[PSI].sup.U.sub.i] = [[OMEGA].sup.U.sub.i] at all [[tau].sup.c.sub.i] [greater than or equal to] [[tau].sup.mcU.sub.i] (from Proposition 3.iii) and show that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for [[tau].sup.c.sub.i] < [[tau].sup.mcU.sub.i], which shows that [[PSI].sup.U.sub.i] > [[OMEGA].sup.U.sub.i] for all [[tau].sup.c.sub.i] < [[tau].sup.mcU.sub.i]. Hence, [[tau].sup.mcU.sub.i] is the lowest self-enforcing tariff in the Unlinked-WEC regime. Using the first-order condition and the Envelope Theorem, this condition is reduced to showing:

(A3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now, as [[??].sup.d.sub.i2] ([[tau].sup.dU.sub.i] ([[tau].sup.8*c.sub.i]), [[tau].sup.*c.sub.i]) < [[??].sup.d.sub.i2] ([[tau].sup.c.sub.i], [[tau].sup.*c.sub.i]) by strategic substitutability of tariffs, we will replace the left-hand side of this inequality and rearrange to get:

0 < [[??].sup.c.sub.i1] ([[tau].sup.*c.sub.i], [[tau].sup.*c.sub.i]) + [delta] x [[??].sup.x.sub.i2] ([[tau].sup.c.sub.i]), [[tau].sup.*c.sub.i]).

For [[tau].sup.c.sub.i] < [[tau].sup.mcU.sub.i], the above expression can be expressed as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The term inside the brackets is negative because of Proposition I.iv which shows that [[??].sup.c.sub.i11] + [delta] x [[??].sup.c.sub.i22] < 0 and Proposition I.ii which shows that [[??].sup.c.sub.i12] < 0. Therefore, the above condition is satisfied, completing the proof.

PROPOSITION 5.

Proof, (i.) The first part follows from totally differentiating the first-order condition (11) with respect to the cooperative, and the deviating, tariff and using the implicit function theorem to obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the cross partiais are negative by strategic substitutability. The second derivative with respect to own tariff is negative by concavity, moreover, from Proposition l.iv, it dominates the positive sign of the second derivative with respect to the foreign tariff. Therefore, the numerator is positive and denominator is negative. Similarly, totally differentiating the first-order condition (11) with respect to the discount factor and the deviating tariff yields:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(ii.) We will use two lemmas in order to prove part (ii.).

LEMMA 1. [partial derivative][[OMEGA].sup.L.sub.i]/[partial derivative][[tau].sup.c.sub.i] < 0 and [partial derivative][[PSI].sup.L.sub.i]/[partial derivative][[tau].sup.c.sub.i]. < 0 for all [[tau].sup.c.sub.i] < [[tau].sup.mcL.sub.i].

To show this, remember that [partial derivative][[OMEGA].sup.L.sub.i]/[partial derivative][[tau].sup.c.sub.i] = [[??].sup.c.sub.i1] ([[tau].sup.c.sub.i], [[tau].sup.*c.sub.i]) + [[??].sup.c.sub.i2] ([[tau].sup.c.sub.i], [[tau].sup.*c.sub.i]) < 0 for all [[tau].sup.c.sub.i] > 0 by Proposition 1.iii. Using the first-order condition for [[tau].sup.dL.sub.i] from Equation (11) and the Envelope Theorem, we get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The sign of the first-bracketed term is given by the global efficiency of free trade (Proposition 1.iii). To see the sign of the second-bracketed term remember that [[??].sup.c.sub.i2] ([[??].sup.p.sub.b1], [[tau].sup.*d.sub.i]) < [[??].sup.c.sub.b1] ([[tau].sup.c.sub.b], [[tau].sup.*c.sub.b]) and [[??].sup.d.sub.a2] ([[tau].sup.c.sub.a], [[tau].sup.*c.sub.a]) < [[??].sup.c.sub.a2] ([[tau].sup.c.sub.a], [[tau].sup.*c.sub.a]) by strategic substitutability. We sum these two inequalities up to get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all t' > 0. Now, since the above inequality is true, then [delta] x [??][b.sup.p.sub.1] ([[tau].sup.c.sub.b], [[tau].sup.*d.sub.b]) + [[??].sup.d.sub.a2] ([[tau].sup.d.sub.a], [[tau].sup.*c.sub.a]) < 0 must also be true because [[??].sup.p.sub.b1] ([[tau].sup.c.sub.b], [[tau].sup.*d.sub.b]) > 0 and [delta] [member of] [0,1].

LEMMA 2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all [[tau].sup.c.sub.i] [member of] [0, [[tau].sup.mcL.sub.i]] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all [[tau].sup.c.sub.i] [greater than or equal to] [[tau].sup.mcL.sub.i], so that [[PSI].sup.L.sub.i] > [[OMEGA].sup.L.sub.i] for all [[tau].sup.c.sub.i] [member of] [0, [[tau].sup.mcL.sub.i]] and equal for all [[tau].sup.c.sub.i] [greater than or equal to] [[tau].sup.mcL.sub.i].

Using the derivatives from the previous part, and rearranging, the required condition for this claim can be written as

(A4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Remember that [[??].sup.p.sub.b1] ([[tau].sup.c.sub.b], [[tau].sup.*d.sub.b]) < [[??].sup.c.sub.b1] ([[tau].sup.c.sub.b], [[tau].sup.*c.sub.b]) and [[??].sup.d.sub.a2] ([[tau].sup.d.sub.a], [[tau].sup.*c.sub.a]) < [[??].sup.c.sub.a2] ([[tau].sup.c.sub.a], [[tau].sup.*c.sub.a]) by strategic substitutability. Therefore, we replace the left-hand side of the above inequality. It is sufficient to prove the following condition:

(A3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Rearranging, we get:

(A6) 0 < [[??].sup.c.sub.b1] ([[tau].sup.c.sub.b], [[tau].sup.*c.sub.b]) + [delta] x [[??].sup.c.sub.a2] ([[tau].sup.c.sub.a], [[tau].sup.*c.sub.a])

Using the first-order condition, remember that the following holds at the intersection of the 45-degree line and the optimal deviation tariff line, [[??].sup.d.sub.a1] ([[tau].sup.d.sub.a], [[tau].sup.*c.sub.a]) + [delta] x [[??].sup.p.sub.b2] ([[tau].sup.c.sub.b], [[tau].sup.*d.sub.b]) = 0. However, since [[tau].sup.d]([[tau].sup.mcL]) = [[tau].sup.mcL] at the intersection and sectors are symmetric, we can rewrite the first-order condition as [[??].sup.c.sub.b1] ([[tau].sup.mcL.sub.b], [[tau].sup.*mcL.sub.b]) + [delta] x [[??].sup.c.sub.a2] ([[tau].sup.mcL.sub.a], [[tau].sup.*mcL.sub.a]) = 0

For [[tau].sup.c]. < [[tau].sup.mcL.sub.i] and symmetric issues, this can be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is satisfied because the term in brackets is negative because of Proposition I.iv which shows that [[??].sup.c.sub.i11] + [delta] x [[??].sup.c.sub.i22] < 0 and Proposition 1.ii which shows that [[??].sup.c.sub.i12] 2 < 0.

The proof to part (iii.) is identical to the similar section in Proposition 3.

PROPOSITION 6.

Proof. We compare the first-order conditions for separated and linked agreements to elaborate the results. Reorganize the conditions to get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We prove the first part, [[tau].sup.dL.sub.i]([[tau].sup.c.sub.i]) > [[tau].sup.dU.sub.i] ([[tau].sup.c.sub.i]), by contradiction. Suppose not, so that [[tau].sup.dL.sub.i] ([[tau].sup.c.sub.i]) [less than or equal to] [[tau].sup.dU.sub.i] ([[tau].sup.c.sub.i]). Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by strategic substitutability of tariffs and symmetry of issues. This shows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] using the first-order conditions. However, by concavity of payoffs, this implies that [[tau].sup.dU.sub.a]([[tau].sup.*c.sub.i]) < [[tau].sup.dL.sub.a] ([[tau].sup.*c.sub.i]), a contradiction. Therefore, the assumption is not correct.

The second part follows from the definition of most cooperative tariff and the result in the first part. Remember, [[tau].sup.mc.sub.i] [equivalent to] [[tau].sup.d.sub.i] ([[tau].sup.mc.sub.i]) in both linked and nonlinked agreements. The result in the first part. [[tau].sup.dU.sub.a] ([[tau].sup.*c.sub.a]) < [[tau].sup.dL.sub.a] ([[tau].sup.*c.sub.a]), implies that [[tau].sup.mcU.sub.a] < [[tau].sup.mcL.sub.a] because [[tau].sup.d.sub.i] is monotonously decreasing in [[tau].sup.*c.sub.i] in both cases.

PROPOSITION 7.

Proof, (i.) There are three cases to consider. First, if [[tau].sup.c.sub.i] < [[tau].sup.mcL.sub.i] then [[tau].sup.c.sub.i]. < [[tau].sup.mcL.sub.i] < [[tau].sup.dL.sub.i] ([[tau].sup.c.sub.i]) by Proposition 5. Now, if [[tau].sup.dL.sub.i] ([[tau].sup.c.sub.i]) [less than or equal to] [[tau].sup.n.sub.i]. then, given the symmetry of sectors, and that [[tau].sup.dL.sub.i] ([[tau].sup.c.sub.i]) is an optimal deviation we have that [[??].sup.dL.sub.-i] ([[tau].sup.c.sub.-i]) = [[tau].sup.dL.sub.i] ([[tau].sup.c.sub.i]) is a best response in the other sector as well. Still, if [[tau].sup.*d] ([x.sup.dL.sub.i] ([x.sup.*c.sub.i])) > [[tau].sup.*c.sub.i] then a country may consider deviating from the punishment path; however, this would generate [[tau].sup.n] in both sectors (and lower per period payoffs) forevermore. Hence, if countries care sufficiently about future payoffs, then they would not make this deviation. Second, when [[tau].sup.n.sub.i] < [[tau].sup.dL.sub.i] ([[tau].sup.c.sub.i]). the Linked-WEC strategies call for the punisher to choose [[tau].sup.c.sub.-i] and the deviator to choose [[tau].sup.n.sub.i]. A country may consider deviating from the punishment path to [[tau].sup.dL.sub.i] ([[tau].sup.c.sub.i]) > [[tau].sup.n.sub.i] or to [[tau].sup.d] ([[tau].sup.n.sub.i]) > [[tau].sup.c.sub.i], but again this would generate ([[tau].sup.n.sub.i], [[tau].sup.n.sub.-i]) in both sectors which is not preferred if 5 is sufficiently high. Finally, if [[tau].sup.c.sub.i]. > [[tau].sup.mcL.sub.i]. then [[??].sup.dL.sub.i] ([[tau].sup.c.sub.i]) = [[tau].sup.c.sub.i] > [[tau].sup.mcL.sub.i] > [[tau].sup.dL.sub.i] ([[tau].sup.c.sub.i]), because there is no beneficial deviation when [[tau].sup.dL.sub.i] < [[tau].sup.c.sub.i]. Therefore. [[??].sup.*dL.sub.-i] ([[tau].sup.c.sub.i-]) = [[??].sup.dL.sub.i] ([[tau].sup.c.sub.i]) = [[tau].sup.c.sub.i]. = [[tau].sup.c.sub.-i] is a best response in the continuation game.

To help see that the Linked-WEC retaliation, strategies are subgame perfect, consider a contradiction. Suppose then that there exists a [[tau].sup.pd.sub.i] [not equal to] [[tau].sup.p.sub.i] and/or a [[tau].sup.cd.sub.i] [not equal to] [[tau].sup.c.sub.i], where [[tau].sup.p.sub.i] is the strategy specified in the punishment path ([[tau].sup.dL.sub.i] or [[tau].sup.n.sub.i]) such that the following holds:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

However, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] because [[tau].sup.p.sub.i] [less than or equal to] [[tau].sup.n.sub.i]). Therefore, the above inequality is not satisfied for sufficiently patient governments, a contradiction. We can denote the necessary patience such that countries would adhere to the Linked-WEC retaliation strategies as [[delta].sup.SPL] ([[tau].sup.c.sub.i]).

Finally, note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] from Proposition 1. so that abiding by the cooperative path specified by the agreement and receiving [[OMEGA].sup.L.sub.i] = [[??] ([[tau].sup.c.sub.b] [[tau].sup.*c.sub.b],) + [[??].sub.b]([[tau].sup.c.sub.b], [[tau].sup.*c.sub.b])] must be greater than deviating and receiving [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] it 5 is sufficiently close to one. We denote this necessary discount factor as [[delta].sup.L] ([[tau].sup.c.sub.i]). We can also write [[delta].sup.L>N] ([[tau].sup.c.sub.i]) as the necessary discount factor so that a limited deviation in one sector followed by the Linked-WEC retaliation is preferred to a maximal deviation in both sectors followed by Nash reversion. In addition, we can write [[delta].sup.N] ([[tau].sup.c.sub.i]) as the necessary discount factor to support cooperation by the threat of Nash reversion.

(ii.) First note that if [[tau].sup.dU.sub.i] [greater than or equal to] [[tau].sup.n.sub.i] the Unlinked-WEC strategies specify [[tau].sup.n.sub.i] forevermore. By definition [[tau].sup.n.sub.i] is a best response to [[tau].sup.n.sub.i]. If [[tau].sup.dU.sub.i] < [[tau].sup.n.sub.i]. then we need to show that [[tau].sup.*dU.sub.i] is a best response to [[tau].sup.dU.sub.i]. and once in the punishment stage, neither government has an incentive to deviate from it by applying a greater tariff. For [[tau].sup.c.sub.i] < [[tau].sup.mcU.sub.i], the deviation tariff is greater than the most cooperative tariff [[tau].sup.dU.sub.i] < [[tau].sup.mcU.sub.i] < [[tau].sup.dU.sub.i] ([[tau].sup.c.sub.i]) by Proposition 3. However, for [[tau].sup.dU.sub.i] ([[tau].sup.c.sub.i]) > [[tau].sup.mcU.sub.i], [[tau].sup.*d.sub.i] ([[tau].sup.dU.sub.i] ([[tau].sup.c.sub.i]) = [[tau].sup.dU.sub.i] ([[tau].sup.c.sub.i]) is a best response by Proposition 3. Finally, if [[tau].sup.c.sub.i] > [[tau].sup.mcL.sub.i] then [[??].sup.dL.sub.i] ([[tau].sup.c.sub.i]) = [[tau].sup.c.sub.i] > [[tau].sup.mc.sub.i] > [[tau].sup.dL.sub.i] ([[tau].sup.c.sub.i]) and [[tau].sup.*d.sub.i] ([[??].sup.dL.sub.i] ([[tau].sup.c.sub.i]) = [[tau].sup.c.sub.i]. is the best response.

In order to see that neither government has an incentive deviate from the punishment path, we check the incentive constraint. Suppose not. so that there exists a [[tau].sup.cpd.sub.i] [not equal to] [[tau].sup.p.sub.i] which satisfies the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

however. [[??].sup.p.sub.i]([[tau].sup.p.sub.i], [[tau].sup.*p.sub.i]) > [[??].sub.i]([[tau].sup.n.sub.i], [[tau].sup.*n.sub.i]) for [[tau].sup.p.sub.i] < [[tau].sup.n.sub.i] (and equal for [[tau].sup.p.sub.i] = [[tau].sup.n.sub.i], but then there is no profitable deviation in the punishment phase), therefore, this condition is not satisfied for sufficiently patient governments, a contradiction.

Finally, note that from Proposition 1 we have [[??].sub.i]([[tau].sup.d.sub.i], ([[tau].sup.*c.sub.i]), [[tau].sup.*d.sub.i] ([[tau].sup.c.sub.i])) [[??].sub.i] ([[tau].sup.c.sub.i], [[tau].sup.*c.sub.i]). Hence, examining Equation (7) shows that for 5 close to one we must have [[PSI].sup.U.sub.i] [less than or equal to] [[OMEGA].sup.U.sub.i]. We denote this necessary discount factor as [[delta].sup.U] ([[tau].sup.c.sub.i]).

(iii.) From Equation (7), we can write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Similarly, from Equation (10) we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where we use the symmetry between the sectors. Note that by Proposition 1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, the denominator of [[delta].sup.U] ([[tau].sup.c.sub.i]) is larger than that of [[delta].sup.L] ([[tau].sup.c.sub.i]). In addition, note that by Proposition 6, [[tau].sup.dL.sub.i] ([[tau].sup.*c.sub.i]) [greater than or equal to] [[tau].sup.dU.sub.i] ([[tau].sup.*c.sub.i]) so that [[??].sup.dL.sub.i] ([[tau].sup.dL.sub.i] ([[tau].sup.*c.sub.i]), [[tau].sup.*c.sub.i]) [greater than or equal to] [[??].sup.dU.sub.i] ([[tau].sup.dU.sub.i]([[tau].sup.*c.sub.i]). [[tau].sup.*c.sub.i]). Hence, the numerator of [[delta].sup.L]([[tau].sup.c.sub.i]) is larger than that of (x1).

PROPOSITION 8.

Proof. The continuation path in the punishment stage of the Linked-WEC regime has discounted average payoffs of

(1 - [delta])[[[??].sup.p.sub.a]([[tau].sup.d.sub.i][[tau].sup.*c.sub.i],) + [[??].sup.p.sub.a]([[tau].sup.c.sub.b], [[tau].sup.*c.sub.b])].

The continuation path in the punishment stage of the Unlinked-WEC regime has discounted average payoffs of

(1 - [delta])[[[??].sup.p.sub.a]([[tau].sup.d.sub.i][[tau].sup.*d.sub.a],) + [[??].sup.c.sub.b]([[tau].sup.c.sub.b], [[tau].sup.*c.sub.b])].

From Proposition 1, we know that ([[partial derivative].sup.2] [[??].sub.i] ([[tau].sub.i], [[tau].sup.*.sub.i])) /[partial derivative][[tau].sup.*.sub.i][partial derivative][[tau].sub.i] < 0, which implies that the increase in welfare from levying a larger tariff is less when the trading partner's tariff is also larger, or that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] This last expression can be rewritten as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] symmetry between sectors a and b is equivalent to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Hence, we have shown that for any pair of tariffs {[[tau].sup.d], [[tau].sup.c]}, the punishment stage in the Linked-WEC regime generates higher discounted average payoffs.

PROPOSITION 9.

Proof. From Proposition 1, we know that lower cooperative tariffs generate higher welfare. From Proposition 6, we know that the Unlinked-WEC regime generates lower cooperative tariffs. From Proposition 8, we know that, following any deviation, countries will choose linked punishments. Hence, welfare is improved by setting [theta] = [[theta].sup.U] in the initial period.

PROPOSITION 10.

Proof We start by proving the following lemma which establishes the comparative static properties of cost changes on the optimal deviation tariff in each regime.

LEMMA 3. (i.) In the Nash-reversion regime the optima] Home deviation tariff increases for d[F.sub.i] > 0 and remains unchanged for d[D.sub.i] > 0; whereas the Foreign deviation tariff remains unchanged for d[F.sub.i] > 0 and increases for d[D.sub.i] > 0.

(ii.) In the Unlinked-WEC regime, the optimal Home deviation tariff increases for d[F.sub.i] > 0 and decreases for d[D.sub.i] > 0 by the same amount. Similarly, the foreign deviation tariff decreases for d[F.sub.i] > 0 and increases for d[D.sub.i] >0 by the same amount.

(iii.) In the Linked-WEC regime, the optimal Home deviation tariff increases for d[F.sub.i] >0 and decreases for d[D.sub.i] >0 by the same amount. Similarly, the foreign deviation tariff decreases for d[F.sub.i] > 0 and increases for d[D.sub.i] >0 by the same amount.

Proof. We use the specific form of quasi-linear consumer utilities defined by Equation (1) and convex cost technologies defined by Equation (2). We also impose the value c = 1/2 for simplicity.

(i.) In order to prove the first part, we use the first-order condition for the optimal deviation in the Nash-reversion regime given in Equation (Al). Totally differentiating this equation with respect to [[tau].sup.dNR.sub.i] and F or D and using the Implicit Function Theorem yields:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(ii.) The second part follows from a similar application of the Implicit Function Theorem on the first-order condition for the optimal deviation in the Unlinked-WEC regime given in Equation (A2):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(iii.) The third part follows from a similar application of the Implicit Function Theorem on the first-order condition for the optimal deviation in the Linked-WEC regime given in Equation (11):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In order to see the changes in the most cooperative tariff under the Linked-WEC and Unlinked-WEC strategies, remember that [[tau].sup.dU.sub.i]([[tau].sup.mcU.sub.i]) = [[tau].sup.mcU.sub.i] and [[tau].sup.dL.sub.i] ([[tau].sup.mcL.sub.i]) = [[tau].sup.mcL.sub.i]. Propositions 3.iii and 4 show that the most cooperative tariff is defined at the intersection of optimal deviation tariff and 45-degree line. Therefore, a greater [[tau].sup.d.sub.i] for a given [[tau].sup.c.sub.i]. implies a greater [[tau].sup.mc.sub.i]. Things are different, however, in the Nash-reversion regime because the intersection of [[tau].sup.dN.sub.i] with the 45-degree line designates the static Nash tariff and not the cooperative tariff. We use the incentive constraint to show the positive correlation between the deviation tariff and the most cooperative tariff in this case. Remember that the most cooperative tariff is defined at the point where the incentive constraint just binds under the Nash-reversion strategy:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

now, differentiate this with respect to the most cooperative, and the deviation, tariff to get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where both the numerator and denominator are negative. In order to see this, we can rewrite the denominator as follows: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Remember that 0 < [[??].sup.c.sub.i1]([[tau].sup.cm.sub.i], [[tau].sup.*mc.sub.i]) + [delyta] x [[??].sup.c.sub.i2] ([[tau].sup.cm.sub.i], [[tau].sup.*.sub.mc]) as shown in the proof of Proposition 5, and [[??].sup.d.sub.i2] ([[tau].sup.d.sub.i], [[tau].sup.*mc.sub.i]) < [[??].sup.c.sub.i2], ([[tau].sup.mc.sub.i], [[tau].sup.*mc.sub.i]) by strategic substitutability.

Finally looking at the results of Lemma 3, we see that symmetric changes in D and F are offsetting in both countries in either WEC regime; however, it generates a tariff increase in the Nash regime.

PROPOSITION 11.

Proof. Consider, for example, an export-biased technological improvement in the home country: d[D.sub.i] > 0. The other cases are similar. From Lemma (3.ii), this increase generates changes in the optimal deviation tariff of [partial derivative][[tau].sup.dU.sub.i] /[partial derivative][D.sub.i] = - ([partial derivative][[tau].sup.*dU.sub.i] /[partial derivative][D.sub.i]) = -1/ (6 - [delta] + 2b[delta]) < 0 in the unlinked case and [partial derivative][[tau].sup.dL.sub.i] /[partial derivative][D.sub.i] = - ([partial derivative][[tau].sup.*dL.sub.i] /[partial derivative][D.sub.i]) = -1 / (6 - [delta]) < 0 in the linked case. For b > 0, we have that [absolute value of -1/6-[delta]+2b[delta]] < [absolute value of -1/6-8] and the difference is increasing in b. Finally, from Propositions 3.iii and 4 we know that [[tau].sup.mcU.sub.i], ([[tau].sup.dU.sub.i]) and [[tau].sup.mcL.sub.i] ([[tau].sup.dL.sub.i]) increase at the same rate.

ABBREVIATIONS

GATT: General Agreement on Tariffs and Trade

LCR: Limited Cross Retaliation

TRIPs: Agreement on Trade-Related Aspects of Intellectual Property Rights

WEC: Withdrawal of Equivalent Concessions

WTO: World Trade Organization

doi:10.1111/ecin.12431

RICHARD CHISIK and HARUN ONDER *

* We would like to thank the editor, two anonymous referees, and seminar participants at Canadian Economic Association Annual Meetings, Mid-West International Economics and Economic Theory Meetings, Society for the Advancement of Economic Theory Meetings, Stanford Institute for Theoretical Economics Summer Workshop. Ryerson University, and the University of Toronto for useful comments and suggestions. We alone are responsible for any remaining errors. The findings, interpretations, and conclusions expressed in this publication are those of the author(s) and should not be attributed in any manner to The World Bank, its Board of Executive Directors, or the governments they represent.

Chisik: Associate Professor, Department of Economics, Ryerson University, Toronto, Ontario M5B 2K3, Canada. Phone (416) 979-5000, E-mail rchisik@ryerson.ca

Onder: Senior Economist, Macroeconomics and Fiscal Management, The World Bank. Washington. DC 20433. Phone 1 (202M73-5196, E-mail honder@worldbank.org

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Bown, C.P.. and M. Ruta. "The Economics of Permissible WTO Retaliation." WTO Staff Working Paper ERSD2008-04, 2008.

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Shadikhodjaev. S. Retaliation in the WTO Dispute Settlement System. The Netherlands: Kluwer Law International, 2009.

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SUPPORTING INFORMATION

Additional Supporting Information may be found in the online version of this article:

Appendix SI. A simple example with discrete actions

Figure S1. Interaction in two sectors

Figure S2. A generic prisoners' dilemma game

(1.) For instance, in the 2004 U.S.--Offset Act (Byrd Amendment) case, out of eight parties who were authorized for retaliation, with the exception of Mexico, only the high-income parties (European Community, Canada, and Japan) actually implemented the retaliatory actions. Other developing countries, including Brazil, Chile, and India refrained from doing so to avoid retribution.

(2.) The Understanding on Rules and Procedures Governing the Settlement of Disputes in WTO Analytical Index provides a detailed discussion on judicial interpretation of these definitions. In a hypothetical case, services that are close substitutes in consumption such as tourist guides services and museums and cultural services could fall into different principal sectors: the former into the "Tourism and Travel Related Services (9.A.)" and the latter into the "Recreational, Cultural. and Sporting Services (10.C.)." Therefore, a hypothetical retaliation in one of them in response to a breach in the other could be considered a cross retaliation, judicially speaking. We thank an anonymous referee for pointing the need to clarify the definition of sectors.

(3.) Cross retaliation in sectors other than the one where the dispute originated is allowed for in the DSU article 22, paragraph 3; however, it specifically subordinates more distant cross retaliations to those that are in the same sector or at least the same agreement. In particular, paragraph 3(b) allows cross retaliation in other sectors (of the same agreement) only if same sector retaliation, as described in paragraph 3(a), "is not practical or effective" (WTO 2007). Paragraph 3(c) allows for cross retaliation in other covered agreements (such as GATS or TRIPS) only if cross retaliation as allowed for in 3(b) "is not practical or effective."

(4.) For a detailed analysis of permissible retaliation in international trading system from a legal perspective see Shadikhodjaev (2009), for economic interpretations see Bown and Ruta (2008) and Bagwell (2008).

(5.) In the EC-Banana I case, the United States was authorized to retaliate by $191.4 million annually, effective April 1999, after the EC failed to comply with WTO arbitration. The EC measures that favored bananas from former colonial states as well as the U.S. countermeasures, in the form of a withdrawal of tariff concessions, lasted until July 2001, at which point, the parties reached a bilateral agreement. A similar interaction is also observable in pre-WTO trade wars. In November 1985, the United States increased the tariff on EC egg-pasta exports from 0.25% to 25% (as a response to the EC's failure to comply in its regulations against U.S. citrus exports). The EC counter retaliated by increasing the tariffs on U.S. lemon exports to 20%. This retaliation and counter retaliation lasted until August 1986 (Lawrence 2003), at which point an agreement was reached. Although these pre-WTO retaliations are clearly in the same food product sector and would adhere to the most stringent reading of article 22 paragraph 3(a). WTO era retaliation is generally permitted among all goods widely within an agreement, so that paragraph 3(a) is no more than a suggestion and cross retaliation is more likely to signify across sectors as in paragraph 3(b) or across agreements as in paragraph 3(c). Still, the results contained in this paper lend normative support for adhering to a stricter reading of the suggestion in paragraph 3(a).

(6.) See footnote 5.

(7.) Given the equilibrium prices of the goods and their income, consumers maximize their utilities by choosing optimal consumption bundles in each sector: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where the price of the numeraire good is normalized to one. Consumer income is given by the sum of wage earnings, profit share, and redistributed tariff revenues. Two stage budgeting provides the following demand structure: ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(8.) We could instead interpret the WEC rule as providing an upper bound on permissible retaliation. For deviations less than the Nash tariff countries would want to set the retaliatory tariff equal to the deviating tariff and the bound would bind, but for deviations greater than the Nash tariff countries would choose to retaliate with the Nash tariff so this upper bound would not be binding.

(9.) We cannot characterize the most cooperative tariff in the Nash-reversion case using this method, because the Nash reversion tariff is not a function of the cooperative (or deviating) tariff. As our interest is to compare the role of cross retaliation when WEC is imposed we do not look for an alternative method to compare the maximum level of cooperation in the Nash-reversion regime to those in the two WEC regimes. Although the stronger punishment of the Nash-reversion regime may be able to temporarily enforce a lower cooperative tariff than either of the linked or unlinked WEC regimes, it would be less than optimal when there are economic fluctuations and incomplete information, so that a punishment regime closer to WEC would be preferred (for formalizations of this idea, see Bagwell and Staiger (2005) and Beshkar (2010a and 2010b). We, therefore, take WEC.

(10.) This restriction allows us to make a meaningful consideration (for any value of the discount factor) of cross retaliation when retaliation is limited. Still, as we show below, (in footnote 13) if countries care enough about the future, then they would prefer a deviation in one sector followed by equivalent cross retaliation, rather than a deviation in both sectors followed by two sectors of equivalent retaliation. Furthermore, we require some government patience in order to establish subgame perfection of the limited punishment strategies (see Proposition 7), therefore, our main results would still obtain if we dispensed with this assumption on egregious deviations.

(11.) The arguments in footnotes 6 and 8, along with the fact that [[tau].sup.d.sub.i] ([[tau].sup.mc.sub.i]) < [[tau].sup.N.sub.i] in both the linked and unlinked regime, suggests that our results are not dependent on the classification of either type of egregious deviation and we could disregard these assumptions by adopting a more cumbersome model presentation.

(12.) In Appendix SI (Supporting Information), we analyze our model for three discrete tariffs (Low, Middle, and High) and we derive versions of our results in terms of the necessary discount factors (see Chisik and Onder 2016). In that discrete version, the necessary discount factors to support the WEC strategies as subgame perfect are less than those necessary to support the most cooperative tariff, so that whenever the most cooperative tariff is supported by WEC strategies, then these strategies are subgame perfect. This result is intuitive as the incentive to deviate from the relatively larger punishment tariffs should be less than the incentive to deviate from the lower cooperative tariffs. Hence, a lower discount factor is necessary to support adherence to the punishment regime than to the cooperative one. Furthermore, as in the continuous version contained in this paper, the Linked-WEC strategies require a larger discount factor. Finally, in the discrete version, when the discount factor falls below the level necessary to support full cooperation in either of the WEC regimes or the Nash-reversion regime, there is still a wide range of discount factors that support a limited deviation followed by a limited retaliation (i.e., WEC) as a subgame perfect outcome and this range of discount factors is larger if countries are restricted to Unlinked-WEC strategies.

(13.) In the proof of Proposition 8, we show that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and from Proposition 1, we know that [[??].sup.c.sub.b]([[tau].sup.c.sub.b] [[tau].sup.*c.sub.b]) > [[??].sup.p.sub.b] ([[tau].sup.d.sub.b] [[tau].sup.*d.sub.b]). As long as countries are sufficiently patient, we must then have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Sufficiently patient countries would, therefore, prefer a limited deviation in one sector followed by cross retaliation rather than limited deviation in both sectors followed by equivalent retaliation in both sectors even if such a deviation was not considered egregious.

Caption: FIGURE 1 Structure of Deviation under Nash Reversion and Unlinked-WEC Strategies

Caption: FIGURE 2 Deviation under Linked-WEC Strategy

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Author: | Chisik, Richard; Onder, Harun |
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Publication: | Economic Inquiry |

Date: | Jul 1, 2017 |

Words: | 14682 |

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