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National Sovereignty in an Interdependent
World
Kyle Bagwelly^ Robert W. Staigerz
November 2017
Abstract What are the sovereign rights of nations in an interdependent world, and to what extent do these rights stand in the way of achieving internationally e¢ cient outcomes? These two questions rest at the heart of contemporary debate over the role and design of international institutions as well as growing tension between globalization and the preser- vation of national sovereignty. In this chapter, we propose answers to these two questions. We do so by Örst developing formal deÖnitions of national sovereignty that build on fea- tures of sovereignty emphasized in the international political economy literature. We then utilize these deÖnitions to describe the degree and nature of national sovereignty pos- sessed by countries in a benchmark (Nash) world in which there exist no international agreements of any kind. And with national sovereignty characterized in this benchmark world, we then evaluate the extent to which national sovereignty is compromised by inter- national agreements with speciÖc design features. In this way, we delineate the degree of tension between national sovereignty and international objectives and describe how that tension can be minimized ñand sometimes in principle even eliminated ñthrough careful institutional design.
This is a revised and updated version of our working paper Bagwell and Staiger (2004). For many helpful comments on the previous version, we thank Robert Keohane, Stephen Krasner, Alberto Martin, Jon Peve- house, Jeremy Rabkin, Donald Regan, Guido Tabellini and seminar participants at Berkeley, Notre Dame, the Stockholm School and the NOITS Workshop in Copenhagen, and we thank the National Science Foundation (award number SES-0214021) for support. We are also grateful to Lili Yan Ing and Miaojie Yu for providing very helpful comments on this version. y
zDepartment of Economics, Stanford University; and NBER. Department of Economics, Dartmouth College; and NBER.
ìOf all the rights possessed by a nation, that of sovereignty is doubtless the most important.îEmmerich de Vattel in The Law of Nations, as quoted in Jeremy Rabkin, Why Sovereignty Matters, p. 27.
1. Introduction
What are the sovereign rights of nations in an interdependent world, and to what extent do these rights stand in the way of achieving internationally e¢ cient outcomes? These questions rest at the heart of contemporary debate over the role and design of international institutions as well as growing tension between globalization and the preservation of national sovereignty. But answers are elusive. This is attributable in part to the fact that national sovereignty is a complex notion, reáecting a number of di§erent features. And it is attributable as well to the fact that nations interact in increasingly complex and interdependent ways, making it di¢ cult to draw clear distinctions between international and domestic a§airs. In this chapter, we propose answers to these questions. We do so by Örst developing for- mal deÖnitions of national sovereignty that build on features of sovereignty emphasized in the international political economy literature. We then utilize these deÖnitions to describe the de- gree and nature of national sovereignty possessed by countries in a benchmark (Nash) world in which there exist no international agreements of any kind. And with national sovereignty characterized in this benchmark world, we evaluate the extent to which national sovereignty is violated by international agreements with speciÖc design features. In this way, we delineate the degree of tension between national sovereignty and international objectives and describe how that tension can be minimized ñand sometimes in principle even eliminated ñthrough careful institutional design.
We begin by describing a benchmark two-country model of international interdependence. In this benchmark model, an international ìexternalityî variable deÖnes the interdependence between the two countries, and this variable is modeled in a way that is general enough to allow the nature of this interdependence to take a variety of possible forms, ranging from international trade to the depletion of a common-pool resource to global climate change. Within this benchmark model, we develop a working deÖnition of sovereignty. Our starting point for deÖning sovereignty is the Westphalian norm of ìnon-intervention in the internal a§airs of other statesî(Krasner, 1999, p. 20). To make this norm operational, we
the matters that concern the internal a§airs of each country are that countryís choices among all its policy combinations that are consistent with a given contribution to and level of the externality variable, since its payo§ in this choice problem is independent of the actions of external actors. We put our deÖnition of sovereignty to work by evaluating according to this deÖnition the consequences of various forms of international agreements for national sovereignty. SpeciÖcally, we consider Örst whether it is possible to eliminate the ine¢ ciencies that arise in the Nash equilibrium with international agreements that are limited only to the external a§airs of each country, and thereby to navigate all the way to the international e¢ ciency frontier without violating national sovereignty by means of such agreements. Our Örst main result is that this is indeed possible within the benchmark model. That is, we show that it is always possible within the benchmark model to pick any point on the international e¢ ciency frontier that could be achieved by international negotiations over all policy instruments, and to achieve that point with international negotiations that are limited only to the external a§airs of each country, i.e., to the level of the externality variable and each countryís contribution to it. We next consider the way in which a countryís sovereignty is violated within our benchmark model when the country negotiates international commitments that concern its internal a§airs. Such commitments directly violate a countryís sovereignty, but we show that direct violations of sovereignty can also imply further indirect violations of sovereignty as well, under which government decisions that are not the subject of international negotiation are nevertheless distorted away from the decisions that would normally have been made under the domestic institutional arrangements of the country. We argue that this ìcontamination e§ectîgenerally prevents countries from containing violations of sovereignty caused by international agreements to narrow subsets of policy instruments. In fact, we establish as our second main result that within our benchmark model any international agreement that involves direct commitments over matters that are the internal a§airs of a country must in general violate that countryís sovereignty over at least as many policy instruments as it preserves. Our Örst two results highlight an important distinction between international agreements that mitigate international externalities and international agreements that erode national sov- ereignty, and indicate that it can be possible to have the former without the latter and so allow national sovereignty and international e¢ ciency to coexist in harmony. But questions remain as to (i) whether this harmony is likely to extend to environments beyond those captured by
our benchmark model, and (ii) the degree to which prominent channels of interdependence between countries can be identiÖed that Önd representation in the environment described by our benchmark model. These questions are taken up in the second half of the chapter. We consider a variety of settings that go beyond our benchmark model, and Önd that the harmony between national sovereignty and international e¢ ciency does not always survive in these extended settings. For example, we identify a conáict between national sovereignty and international e¢ ciency that arises whenever an externality variable is completely determined by the policy choices of a single country. This is because in this case the externality variable becomes this countryís internal a§airs according to our deÖnition, and subjecting the externality variable to the constraints of an international agreement (which would be necessary to achieve international e¢ ciency in this case) would therefore violate the countryís sovereignty. Another particularly salient extension of the benchmark model that we consider is to a world of ìsmallîcountries. When all countries are small in relation to the externality variable, we show that each countyís contribution to the externality variable again becomes its internal a§airs. This in turn implies that, when all countries are small, any international agreement that constrains countries from pursuing their Nash policy choices must violate their sovereignty. Accordingly, as we demonstrate, whether or not the harmony between national sovereignty and international e¢ ciency described above survives in a world of small countries hinges on whether governments agree or disagree in the Nash equilibrium over the direction that they would like the externality variable to move. If all governments agree, then the Nash equilibrium in the small-country case is ine¢ cient and an international agreement will be required to reach the e¢ ciency frontier, implying necessarily that national sovereignty and international e¢ ciency will stand in conáict in this case. However, if there is disagreement, then the Nash equilibrium in the small-country case is e¢ cient, and in this case the harmony between national sovereignty and international e¢ ciency identiÖed above survives in a world of small countries. Of course, which of these two cases is applicable will depend on the nature of the externality variable under consideration, but as we later demonstrate, the latter case has special signiÖcance in the context of international trade agreements.
Besides being of interest in their own right, these extensions of our benchmark model high- light an important feature of our approach: rather than tailoring our deÖnition of sovereignty on a case-by-case basis so that national sovereignty is necessarily in harmony with interna- tional e¢ ciency in all circumstances, we propose a formal deÖnition of sovereignty and then
rule coupled with a market access agreement can facilitate the attainment of internationally e¢ cient outcomes that do not compromise national sovereignty. In light of our Öndings, we discuss the basic harmony between the underlying GATT/WTO principles and the maintenance of national sovereignty, and we suggest that this harmony may be at risk as a result of changes that are occurring within the WTO.
The chapter proceeds as follows. Section 2 describes the two-country benchmark model. Section 3 develops our formal deÖnition of sovereignty, characterizes the nature and degree of sovereignty in the Nash equilibrium, and relates this characterization to notions of sovereignty in the international political economy literature. Section 4 considers how national sovereignty is a§ected under international agreements that adopt alternative designs within the benchmark model, while section 5 considers the issue of sovereignty within a number of extensions of the benchmark model. Section 6 establishes that the benchmark model and all its results can be given a trade interpretation, and extends the modeling environment to a multilateral setting to consider the implications of a non-discrimination rule for our sovereignty results. Section 7 concludes. An Appendix contains proofs not included in the body of the chapter.
2. A Benchmark Model
In this section we describe a benchmark model of international interdependence that is general enough to allow interdependence to take a wide variety of forms. Our benchmark model has two countries (territories), referred to respectively as the home and foreign country, in which private agents (home and foreign citizens) reside. Each country has a government, and each government is endowed with a set of (tax and/or regulatory) policy instruments, represented by the 1 I vector i for the home government and the 1 I^ vector i^ for the foreign government, that are applied by each government to activities within its territory. The objectives of the home and foreign governments are represented by the respective functions G(i; x~(i; i)) and G(i; x~(i; i)), with the equilibrium level of the ìexternalityîvariable x~(i; i) entering into each government objective function and embodying the nature of the policy spillovers between the two countries. The ability to represent government objectives in this way reáects an essential assumption of our benchmark model, namely, that there exists a well-deÖned channel (e.g., the level of a price or the quantity of a pollutant) through which the e§ect of each governmentís policy choices on the other governmentís objectives (the externality) travels. Aside from global concavity assumptions on the G and G^ functions to ensure that second-
order conditions are globally satisÖed, the only additional structure we impose in the benchmark model is that ~x(i; i) is a well-behaved function deÖned implicitly according to
f (g(i;x); g(i; x); x) = 0 ; (2.1) with fggik 6 = 0 for some k and fg g im 6 = 0 for some m;
where here and throughout we use subscripts on a function to denote the partial derivative of that function with respect to the subscripted argument. In e§ect, g represents the home countryís ìcontributionîto the determination of the equilibrium level of the externality variable ~x, a contribution that is assumed to be impacted by at least one policy instrument of the home government (i.e., gik 6 = 0 for some k); and this contribution is deÖned for a given level of the externality once the home-country policy instruments are determined. An analogous interpretation holds for g. The function f then aggregates the contributions of the home and foreign country to determine the equilibrium level of the externality x~ according to f () = 0, under the assumption that f is impacted by changes in either countryís contribution (i.e., fg 6 = 0 6 = fg ).
As we conÖrm in section 6 below, this structure is consistent with a setting in which the interdependence across countries is purely pecuniary and takes the form of international trade, with (2.1) then amounting to a market-clearing condition. But this structure is general enough as well to include many other forms of interdependence.
For example, x might represent the density of the Ösh population in a common Öshery, with g representing the home catch when the home áeet operates in the policy environment i and faces a Ösh population density x, and with g^ representing the foreign catch when the foreign áeet operates in the policy environment i^ and faces a Ösh population density x. In this setting, it would be natural that gx > 0 and g x > 0. The equilibrium density of the Ösh population, given the policy environment faced by home and foreign áeets, ~x(i; i), is then determined according to (2.1). Alternatively, x might represent the temperature of the globe, with g representing the home countryís carbon output when the home industry operates in the policy environment i and faces a global temperature x, and with g^ representing the carbon output of the foreign country when the foreign industry operates in the policy environment i^ and faces a global temperature x. Here it could be that gx R 0 and g x R 0 depending on circumstances. The equilibrium temperature of the globe, given the policy environment faced by home and foreign
The joint solutions to (2.2) and (2.3) deÖne the Nash equilibrium of the benchmark model, which throughout we assume exists and is unique.
We next characterize the international e¢ ciency frontier and evaluate the e¢ ciency prop- erties of the Nash equilibrium. We deÖne the international e¢ ciency frontier with respect to the objectives of each government. Accordingly, the international e¢ ciency frontier solves the following program:
Program 2 : max i;i;x G(i; x) s.t. (i) G(i; x) G, and (ii) f (g(i;x); g(i; x); x) = 0:
Using Program 2, and letting k = 1 denote a domestic and foreign policy instrument for which gik 6 = 0 and g i k 6 = 0, it is direct to derive the Örst-order conditions that characterize the international e¢ ciency frontier:
Gik = gik g^ Gi^1 i 1
for k = 2; :::; I; (2.4)
G i k =
g i k G i 1 gi 1 for^ k^ = 2; :::; I
; and (2.5)
Gi 1 G i 1 + Gi 1 G x x~i 1 + G i 1 Gx x~i 1 = 0; (2.6)
along with the complementary slackness conditions ensuring that the Kuhn-Tucker multiplier on constraint (i) of Program 2 is non-negative:
G Gx x
x + fx] G xfggi 1 ^0 if^ G
x 6 = 0 and (2.7)
Gi 1 f (^) g g i k G ik fggi 1 ^0 for each^ k^ for which^ G
i k 6 = 0:
The e¢ ciency properties of the Nash equilibrium may now be assessed. Using (2.2) and (2.3), whose joint solutions deÖne the Nash equilibrium in our benchmark model, we may state:
Proposition 1. The Nash equilibrium of the benchmark model is ine¢ cient if and only if Gx 6 = 0 and G x 6 = 0 at the Nash policy choices.
Proof: See Appendix.^2 According to Proposition 1, the Nash equilibrium is ine¢ cient ñand hence a potential role for international agreements arises in our benchmark model ñif and only if Gx 6 = 0 and G x 6 = 0 at the Nash policy choices. We will refer to the case where Gx 6 = 0 and G x 6 = 0 at the Nash policy choices as the case where the home and foreign countries are mutually interdependent. As our central purpose is to consider the implications of voluntary (i.e., mutually beneÖcial) international agreements for national sovereignty, and as the case of mutually interdependent countries is the only case in which voluntary international agreements can arise, this case will be a primary focus of the analysis to follow.
3. What is Sovereignty?
With the essential elements of our benchmark model described, we now turn to develop a deÖnition of sovereignty within the context of this model. DeÖning sovereignty is not a simple task. On the one hand, to be operational, our deÖnition of sovereignty must be amenable to formal analysis. On the other hand, to be relevant, our deÖnition of sovereignty must capture elements that feature prominently in the common usage of this term. This latter requirement is particularly di¢ cult, because the international political economy literature within which sovereignty has been most discussed is not always clear about the precise meaning of the term and, when clear, does not always adopt a uniform meaning. In fact, Krasner (1999) identiÖes four distinct ways in which the term ìsovereigntyîhas been commonly used in this literature. Krasner refers to these as domestic sovereignty, international legal sovereignty, interdependence sovereignty, and Westphalian sovereignty. Domestic sovereignty refers to the organization and e§ectiveness of political authority within the state. International legal sovereignty refers to the mutual recognition of states. Interdependence sovereignty refers to the scope of activities over which states can e§ectively exercise control. And Westphalian sovereignty reáects as its central premise the rule of nonintervention in the internal a§airs of other states.
Our starting point for deÖning sovereignty is the Westphalian norm of ìnon-intervention in the internal a§airs of other statesî (Krasner, 1999, p. 20). To formalize this deÖnition, we (^2) The required conditions for ine¢ ciency of the Nash equilibrium in the benchmark model would be weakened if the ability to make explicit international transfers were introduced. This can be done without changing the nature of any of our results, but we prefer to keep explicit international transfer instruments out of our model for simplicity. We also consider several extensions of the benchmark model in a later section where the Nash ine¢ ciencies take more complicated forms.
of policy instruments within the territory of that government. For example, domestic authority may be concentrated in the hands of one individual, whose preferences are then the government objective function for that country. If that individual is subjected to lobbying by interest groups, then these interest groups also comprise a part of the domestic authority structure, and the government objective function for that country will reáect as well the ináuence wielded by these interest groups. Alternatively, domestic authority may be dispersed in the hands of the electorate and take the form of a direct democracy, in which case under appropriate assumptions the preferences of the median voter are then the government objective function for that country. Or domestic authority over di§erent policies may be allocated across di§erent domestic institutions: as long as coordination across domestic institutions (e.g. bargaining among them) is possible, our representation of government objectives allows a valid description of the domestic policy environment in this setting as well. The point is, each governmentís objective function as we have deÖned it reáects both the underlying preferences of the citizens of that country and the domestic authority structures under which those preferences are translated into choices over policy instruments in that country. According to the meaning of Westphalian sovereignty in the international political economy literature, then, within the context of our model the internal a§airs of the home (foreign) country are embodied in its choice of G (G). As for what constitutes intervention by external actors, Krasner (1999, Chapters 6 and 7) observes that coercion (as in international armed conáict) has frequently resulted in constitu- tional changes that explicitly alter the domestic institutions of a country and thereby violate its Westphalian sovereignty. Such explicit changes in domestic institutions could be interpreted within our benchmark model as alterations in the G and/or G^ functions, reáecting for example the forced removal of a dictator and the introduction of democratic institutions. However, our focus here is not on armed conáict, but rather on voluntary international agreements. Hence, violations of Westphalian sovereignty that would result in changes in the G and/or G^ functions are not our central concern.
But Krasner (1999, p. 22) observes that invitation (as in international contracts and conven- tions) can also violate Westphalian sovereignty, not necessarily by explicitly altering domestic institutions, but by ì...subjecting internal authority structures to external constraints.îRabkin (1998, p. 34) puts the point slightly di§erently: (Westphalian) sovereignty is violated by inter- national commitments that ìdistort or derange the normal workings of our own system...î. In e§ect, international commitments need not alter the domestic institutions of a country in order
to violate its Westphalian sovereignty: international commitments that distort the operation of domestic institutions will also violate Westphalian sovereignty.
Implicit in the above discussion is the notion that speciÖc commitments arising out of vol- untary international agreements would not ordinarily be viewed as violations of Westphalian sovereignty. For Westphalian sovereignty to be violated by invitation, a ìdeeperî interven- tion into the internal a§airs of the state is required. Even here though, both Krasner (1999) and Rabkin (1998) suggest that there are limits to the appropriate matters for international agreement, and that a nationís Westphalian sovereignty would be violated by negotiated com- mitments over policies that cross these limits and stray into ìsu¢ ciently domesticî a§airs. For instance, Krasner (1999, pp. 146-148) observes that the IMF routinely violates the norm of Westphalian sovereignty, in part because ì...A country entering into negotiations with the IMF could basically consider any aspect of its domestic economic policy open to discussion.î Similarly, in the Preface to his book, Rabkin (1998, p. x) states that e§orts to delineate the appropriate limits of international commitments are ì...particularly urgent now because, in the absence of any clear understandings on the matter, we seem to be letting international agree- ments and international authorities determine more and more of our policies.îNeither Krasner nor Rabkin o§er a precise method for deÖning the limits of proper subjects of international negotiation, though Rabkin (pp. 69-70) proposes several criteria. Summarizing, three key features of Westphalian sovereignty that seem especially relevant in the context of voluntary international agreements can be identiÖed from our discussion of the international political economy literature to this point: Örst, commitments that result from voluntary international agreements do not necessarily violate Westphalian sovereignty; second, international commitments over policies that concern ìsu¢ ciently domesticî a§airs (i.e., internal a§airs) do violate Westphalian sovereignty; and third, international commitments that distort the normal operation of domestic institutions also violate Westphalian sovereignty. We wish to construct a deÖnition of sovereignty that reáects these three features.
To accomplish this, we maintain as our essential focus the Westphalian norm of non- intervention in the internal a§airs of other states. And below we adopt a deÖnition of non- intervention that is well-reáected in the discussion above. But in proposing a formal deÖnition of internal a§airs, we augment the Westphalian emphasis on authority over the determination of institutions and policies, and add to this an emphasis on authority and control over the determination of outcomes and therefore payo§s as well, all evaluated from the perspective
institutions/policies with a focus as well on authority over outcomes/payo§s, our approach may facilitate a more direct link to issues of accountability in an interdependent world than do any of the existing notions of sovereignty taken separately, and may thus be of some interest in its own right. For example, a government might maintain authority over its institutions and policies (and therefore maintain Westphalian sovereignty) and yet claim that it cannot be held accountable for its choices, as a consequence of a ìrace to the bottomî that external constraints have forced upon it. But in matters where the government maintains control over outcomes/payo§s as well (and hence in matters that are also the countryís internal a§airs ac- cording to our deÖnition), this possibility of avoiding accountability by appealing to external constraints does not arise. As a consequence, our approach to deÖning national sovereignty can be used to forge a tighter link between international agreements that can be said to avoid an erosion of sovereignty and those that can be said to avoid an erosion of accountability.
3.1. Sovereignty DeÖned
We now proceed to develop our formal deÖnitions in detail. To this end, we return to the benchmark model presented in section 2, and consider the Nash policy choices made by each government. It is with respect to Nash choices, unconstrained by any international commit- ments, that a nationís internal a§airs can be deÖned. As noted above, we adopt a deÖnition of internal a§airs that equates the internal a§airs of a country with (i) the choice of domestic authority structures under which the preferences of its citizens are translated into choices over policy instruments in that country, as embodied in the home and foreign government objective functions G and G^ respectively, and (ii) the matters in which its government has control or ìsole authorityîover outcomes/payo§s (in the Nash equilibrium).
To develop this deÖnition, we consider alternative representations of each governmentís best- response policy choice problem (that is, alternative representations of Program 1 and Program 1 ^ in section 2) that partition this problem into a sequence of sub-problems, each of which can be considered a choice problem of its own, and where the solution to any such representation
Jacksonís approach is to propose an updated concept of Westphalian sovereignty that he terms ìsovereignty- modern,îand which is meant to be more consistent with international e¢ ciency and the need for international policy coordination in the modern world. Our approach provides a formal deÖnition of sovereignty which achieves some of what Jackson has in mind, in that according to our deÖnition the domain of sovereignty will evolve as the nature of international interdependence evolves; but we do not tailor our deÖnition of sovereignty on a case-by-case basis to necessarily be in harmony with international e¢ ciency, and instead evaluate formally the circumstances when a tradeo§ between maintaining national sovereignty and achieving international e¢ ciency can be avoided versus when this tradeo§ will necessarily arise.
yields the original best-response policy choices of the government characterized by (2.2) and (2.3). Our approach is to use these partitions to identify the maximal set of choice problems over which a government can be said to exercise sole authority in the Nash equilibrium, and thereby (together with its choice of government objective function) identify the countryís internal a§airs according to our deÖnition.
We begin by deÖning a partition of a governmentís best-response policy choice problem:
DeÖnition 1. A partition P of a governmentís best-response policy choice problem is any sequence of choice problems whose solution yields the best-response policy choices of the gov- ernment.
According to its deÖnition, each element of P is a choice problem for the government, and when the government has solved each of these choice problems it arrives at the original best-response policies.^5 We also need a deÖnition of authority:
DeÖnition 2. A government has sole authority in a choice problem if and only if its payo§ in that choice problem is independent of the actions of ìexternal actors.î
By the actions of external actors, we mean the setting of all policy instruments by the govern- ment of the other country and all decisions by private agents in the other country. Accordingly, a government has authority in a choice problem provided that its policy choices, together with the decisions of private agents operating within its borders, fully determine its payo§s in that choice problem.
We next deÖne internal a§airs and external a§airs, conditional on the partition P under consideration:
DeÖnition 3. For any partition P of its governmentís best-response policy choice problem, a countryís P-internal a§airs consist of (i) the choice of domestic authority structures under which the preferences of its citizens are translated into choices over policy instruments in that country, as embodied in its government objective function, and (ii) the collection of choice problems in P over which the government has sole authority. The countryís P-external a§airs are the remaining choice problems in P. (^5) Notice that, as the home governmentís best-response policy choices are deÖned by the vector of domestic policies i that solve 2.2, the choice problems in any partition P of the home governmentís best-response policy choice problem cannot include (trivial or otherwise) choices over foreign policies in i, and similarly the choice problems in any partition P of the foreign governmentís best-response policy choice problem cannot include (trivial or otherwise) choices over home policies in i.
the partition P^ is equivalent to the collection of choice problems ~s contained in the partition P^ ~ provided that there exists a collection of choice problems s~^0 such that ff Pn~ ~sg [ ~s^0 g is also a partition of the governmentís best-response policy choice problem and a choice problem is in ^s if and only if it is also in s~^0. We may now deÖne a minimal partition:
DeÖnition 4. A minimal partition P^ of a governmentís best-response policy choice problem is a partition for which the governmentís P^-external a§airs are a subset of its P-external a§airs for all P.
And with this we arrive at our preferred deÖnition of internal and external a§airs:
DeÖnition 5. If there exists a minimal partition P^ of its governmentís best-response policy choice problem, then a countryís internal a§airs consist of (i) the choice of domestic authority structures under which the preferences of its citizens are translated into choices over policy instruments in that country, as embodied in its government objective function, and (ii) its P^ ^-internal a§airs; and the countryís external a§airs are its P^-external a§airs.
Notice that it is possible that a set of partitions may qualify as minimal partitions, but each partition in this set must (i) be payo§ equivalent, because all partitions must lead to the same best-response policy choices, and (ii) deliver the same characterization of internal and external a§airs according to deÖnitions 4 and 5. Hence, this possible non-uniqueness is immaterial for our purposes.^7
To make use of deÖnition 5, we must establish the existence of minimal partitions of the home and foreign government best-response choice problems in our benchmark model. To this
greater than the cardinality of ~s. This possibility would simply reáect that ^s had not been written in a form that had allowed direct comparison with ~s on a choice-problem by choice-problem basis (and that by such rewriting it became possible to show that in fact 7 s^ was a subset of ~s according to our deÖnition) As a general matter, we do not have an existence proof for a minimal partition, which is why we state the deÖnition of internal and external a§airs conditional on the existence of a minimal partition. But as we next show, we are able to establish existence in the formal setting of our benchmark model.
end, consider the alternative 2-step representation of Program 1:
Program 1^0 : Step 1 : For a given (g; x) : max i G(i; x) s.t. [g(i; x) g] = 0: Step 2 : For a given i^ : max g;x L(^{(g; x); g; x) s.t. f (g; g(i; x); x) = 0;
where ^{(g; x) is the solution from Step 1 and L is the Step-1 Lagrangean.^8 The Step-1 choice problem in Program 1^0 is solved conditional on a given level of the ìexternalityîvariable x and the home-countryís ìcontributionîg to the externality variable, and has the home government making its preferred choices over domestic policy instruments i so as to deliver this contribution. The Step-2 choice problem has the home government then making its preferred choices over g and x subject to the constraint placed on its choices which is implied by a vector of foreign policy instruments i. Our Örst result is that Program 1^0 is indeed an equivalent way of characterizing the home- governmentís best-response policies in Program 1. We record this in:
Lemma 1. Program 1^0 is a partition of Program 1.
Proof: See Appendix. We prove Lemma 1 by establishing that the Örst-order conditions associated with Program 10 are given by (2.2), the Örst-order conditions associated with Program 1, hence the solution of Program 1^0 yields the best-response policy choices of the home government. While we have developed this partition from the perspective of the home governmentís problem Program 1, an exactly analogous partition (which we denote henceforth by Program 1^0 ) can be developed for the foreign governmentís problem Program 1. For future reference, we denote by P 0 the partition of the home-governmentís best-response policy choice problem deÖned by Program 10 , and by P 0 the partition of the foreign-governmentís best-response policy choice problem deÖned by Program 1^0.
We next use Program 1^0 (and the analogous Program 1^0 of the foreign government) to assess the internal a§airs of each country according to the partitions P 0 and P 0 . It is immediate (^8) It should be understood that the set of g and x for which Step-1 is solved are those for which there exists a vector of instruments i such that [g(i; x) g] = 0 is satisÖed.
