Voting on federations

Research in Economics 59 (2005) 1–21 www.elsevier.com/locate/yreec

Voting on federations Sergio Currarini Dipartimento di Scienze Economiche, Universita di Venezia, S. Giobbe 873, 30121 Venezia, Italy Received 8 October 2004; accepted 15 December 2004

Abstract In this paper, we study the problem faced by a set of sovereign countries deciding whether to form a federation. Incentives to merge stem from the externality of domestically produced international public goods. Each country’s decision is taken by a domestic voting committee (parliament). If the federation forms, federal decisions are taken by a federal voting committee, made of the union of the national parliaments of federal members. We rely on results by [Kaneko, M., 1977a. The ratio equilibrium and a voting game in a public goods economy. Journal of Economic Theory 16, 123–136; Kaneko, M., 1997b. The ratio equilibria and the core of the voting game G(N,W) in a public goods economy. Econometrica 45, 1589–1594) and Hirokawa, M., 1992. The equivalence of the cost share equilibria and the core of a voting game in a public goods economy, Social Choice and Welfare 9, 63– 72. International Journal of Game Theory 11, 385–393] to identify voting outcomes at both domestic and federal levels with ratio equilibrium allocations for the respective economies. We show that although the efficient full federation (encompassing all countries) might not form in general, it always does when federal voting bears on the variations of public goods production with respect to the status quo pre-federal equilibrium, and taxes are defined on individual incomes at the pre-federal equilibrium. q 2005 University of Venice. Published by Elsevier Ltd. All rights reserved. Keywords: Federations; Voting; Political economy; Coalitions JEL classification: C71; D6; H41; H7

1. Introduction In this paper, we study the following problem. A set of independent countries, each producing a pure international public good out of a private good, consider whether to form

E-mail address: [email protected] 1090-9443/$ - see front matter q 2005 University of Venice. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.rie.2004.12.001

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a federation. Incentives to merge stem from the international externalities produced by domestically produced public goods, such as, for example, the regulation of polluting emissions or the design of tax systems. Within each country, a committee (parliament) takes decisions by voting. In the absence of a federation, these committees independently set domestic policies, consisting of production levels of the public goods and a tax system covering production costs. A prefederal equilibrium state of the economy is therefore one at which all domestic policies are mutually consistent in the Nash sense. More precisely, each country’s production and distribution choices select a domestic voting equilibrium given the choices of the other countries. Taking this pre-federal equilibrium as given, each domestic committee decides (by voting) whether to join the federation, possibly with different rules from those applied to purely domestic decisions. If a federation is formed, resource allocation within the federation is determined by means of a centralized voting process, run by a voting committee consisting of the union of all the domestic committees of federal members. A natural example is the European Parliament, including representations of all domestic political coalitions of members of the European Union. Our aim is to analyze the incentives of individual countries to form a federation. Given the inefficiency of the status quo, aggregate incentives are present whenever the federal voting process attains an efficient allocation of resources within the federation. In our model, where federal voting equilibria (discussed further on in this introduction) always selects a Pareto Optimum of the economy, such aggregate incentives remain until the full federation (encompassing all countries) is formed. Even in this case, however, individual incentives of voters might be incompatible with the formation of the full federation and, therefore, with the achievement of a social optimum. This conflict between private and aggregate incentives stems from the impossibility of distributing the efficiency gains in any arbitrary way. Indeed, the set of feasible distributions is determined by the features of the voting equilibrium that determines federal resource allocation. We will show by means of two simple examples how this potential conflict can be imputed to the unequal distribution of preferences and/or incomes. The main point of the paper is that this conflict is resolved if federal decisions take as a reference point (in terms of income levels) the status quo pre-federal equilibrium, and compute productions costs in terms of variations of production levels with respect to it. An example of cost sharing practice inspired by the rule proposed here are cooperative reductions of transboundary polluting emissions, whose magnitude is often related to the status quo emission levels. We show in our main proposition that if such rules are expected to be applied at the federal decision level, all countries agree to join the full federation, provided this is their only alternative to the status quo. If the economy satisfies constant returns to scale and the international public good accumulates in a linear way as a result of national productions, we also show that the full federation forms even when countries are allowed to form smaller federations as well as to remain at the status quo pre-federal equilibrium. Before presenting the model, we wish to briefly discuss the way in which the various voting processes and the federation formation stage are modeled in the paper. We consider

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voting procedures taking place in small voting committees, such as parliaments, in which communication among voters is costless and unrestricted. In particular, we wish to allow voters to trade votes for money transfers, and to form coalitions in order to propose or object to proposals on the floor. These situations do not fulfill the conditions for median voter’s results, since the possibility of side payments breaks down the unidimensionality of the issue space. Moreover, in federal decision making, we wish to allow voting to bear on production bundles of public goods, as well as on the relative cost distribution parameters. This choice differentiates this paper from other papers on the stability of federations based on an analysis of the political process (see, for instance, Le Breton and Weber, 2003). Motivated by these remarks, we will represent voting processes by means of cooperative games, in which the players’ set is the voting committee and in which only certain types of coalitions (named winning) may object or propose alternatives. Voting equilibria are here identified with the core of the voting game, i.e. on the set of alternatives which are not objected by any winning coalition (Peleg, 1984). If coalitions can form freely and at no cost, voting equilibria stand as the only stable parliamentary decisions. Our characterization of voting equilibria relies on a result by Kaneko (1977a,b), more recently generalized by Hirokawa (1992), proving that the core of a voting game bearing on production of public goods and relative cost sharing coincides with the set of ratio equilibria of the associated public goods economy.1 We therefore identify voting equilibria (both at domestic and at federal level) with the ratio equilibria of appropriately defined economies, and study the independent choice of countries at the federation formation stage. To model this stage, we do not use an explicit game theoretic representation of countries’ decision making. We do so for the sake of simplicity; as argued in Section 4.3, our main results can be restated is terms of an application of the closed membership (or gamma) coalition formation game of Hart and Kurz (1983). This type of coalition formation model has been employed by Burbidge et al. (1997) to analyze the formation of a federation in a tax competition problem. They find that the formation of the full federation occurs only when there are two countries, while when the number of countries is greater, examples can be built in which the unique equilibrium coalition structure is one with a smaller federation forming. In their model, the specific payoff imputation within the federation is not induced by a political process but is built to satisfy the axioms of the Nash bargaining solution of an appropriately defined coalitional bargaining situation. Such partial federation are not ruled in the present context, where, however, the full federation is always stable. The paper is organized as follows. Section 2 presents the economy. Section 3 characterizes the status quo by means of a concept of pre-federal equilibrium. Section 4 studies the federation formation stage, and contains the main results. Section 5 concludes the paper.

1

This result is obtained under specific assumptions on voting rules that we impose in the present paper.

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2. The economy We consider the economy E in which n agents are organized into m countries. We denote by I the set of all agents in the economy and by K the set of countries. We let Bk denote the set of agents in country k, with gm kZ1 Bk Z I and BjhBkZ: for all jsk, and with nk denoting the cardinality of Bk. There are m public goods and one private good. Each agent i2I is endowed with ui units of the private good. Each public good is produced by one (and only one) country out of the individually owned private good endowments. The cost of producing the amount qk of the k-th public good is given by Ck(qk). Each agent i has a utility function ui(q, xi), where q denotes the vector (q1,.,qm) and xi denotes the amount of private good consumed by agent i. We make the following assumptions. For all iZ1,.,n, and for all kZ1,.,m: Assumption 1. ui(q, xi) is monotonically increasing in all arguments, strictly increasing in xi, continuously differentiable and concave; Assumption 2. uiO0; Assumption 3. Ck(qk) monotonically increasing, convex and continuously differentiable, with Ck(0)Z0 and limqk/0 Ck0 ð0Þ Z 0: mCn such that Definition 1. An allocation for the economy E is a pair ðq; xÞ 2RC n X iZ1

ðui K xi ÞR

m X

Ck ðqk Þ:

kZ1

Definition 2. A Pareto Optimum of the economy E is an allocation (q, x) such that there is no other allocation (q 0 , x 0 ) such that ui(q 0 ,x 0 i)>ui(q, xi) for all iZ1,2,.,n. Special Pareto Optima are the ratio equilibria of the economy E, at which agents pay for each public good according to their relative willingness to pay. Ratio equilibria are a natural Lindahlian concept for environments in which the public good is produced by a central agency, such as a government or a federation, and are defined as follows. For i2I and k2K, let pki 2½0; 1 be a cost-sharing ratio expressing the share of the total production cost for the k-th public good imputedP to agent i. If the vector of public goods q is produced, m k the total cost paid by agent i is kZ1 pi Ck ðqk Þ. Let Dn denote the unitary (nK1)m dimensional simplex and, for p2(Dn) , let ( ) m X m n k pi Ck ðqk Þ C xi % ui Bi ðpÞ Z ðq; xi Þ 2RC !RC : kZ1

denote the budget set of player i given p. mCn Definition 3. A ratio equilibrium of E is a triple ðq; x; pÞ 2RC !ðDn Þm such that (q, ~ x~i Þ for all i2I and all ðq; ~ x~i Þ 2Bi ðpÞ. xi)2Bi(p) for all i2I and ui ðq; xi ÞR ui ðq;

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We denote by RðEÞ the set of ratio equilibria of the economy E. The nonemptiness of the set RðEÞ follows from Assumptions 1–3 and Theorem 1 in Kaneko (1977a). Note that ratio equilibria of E do not account for the way in which countries are structures. In fact, it can be interpreted as one way in which a benevolent supernational authority could impute the costs of the efficient productions of public goods across countries and, within countries, across agents. If we denote by EK ðqKk Þ the sub-economy obtained by restricting the economy E to the kth country and for a vector of externally produced public goods qKk, the concept of ratio equilibrium can be directly applied to EK ðqKk Þ, denoting one way of decentralizing the efficient production of public good in country k; existence follows again from Kaneko (1977a) and Assumptions 1–3.

3. Equilibrium without federations 3.1. Domestic voting When no federation exists, each country sets its domestic policy by voting, independently of the other countries Within country k, the committee Bk (domestic parliament) votes over a production level qk of the domestically produced public good and over a cost sharing vector rk Z ðri Þi2Bk . In this process, voters can form coalitions in order to make proposals or to object to proposals currently on the floor. Not all coalitions are allowed to make or object to proposals: voting rules define a set of ‘winning coalitions’ that can actively participate in the parliamentary debate. In the case of simple majority rule, the winning coalitions are all the subset of voters which include a majority of voters in the committee. We point out that since we allow for international pure public goods, the way in which domestic voters rank alternatives may depend on the level of public goods produced by the other countries. This type of voting situations can be described (see Kaneko (1977a,b) and more recently Hirokawa (1992)) in terms of a voting game G(Bk,Wk, qKk), where Bk is the domestic committee, Wk 32Bk the set of winning coalitions and qKk the level of public goods produced outside country k.2 We will assume the following properties on the set Wk. 1. non-dictatorship: Bk\{i}2Wk for all i2Bk; 2. properness: S2Wk/Bk\S;Wk; 3. monotonicity: (T2Wk and T3S)/S2Wk. Note that Properties 1 and 3 imply that Bk2Wk. Properties 1 and 3 rule out the existence of voters that are needed in order to form a winning coalition. Property 2 rules out the existence of ties. Since national parliaments are relatively small committees, we will assume that private arrangements are possible among parliamentary members in order to transfer private good 2

In the quoted papers the term qKk is missing since the committee decided over all public goods produced in the economy.

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in exchange for votes. Under these conditions, a winning coalition S2Wk will be able to induce the state of the economy (q 0 k, x 0 k) by proposing an alternative (q 0 k, x 0 k) such that: X X xi0 % ui K ri0 Ck ðqk0 Þ; i2S

i2S

xh0 Z uh K rh0 Ck ðqk0 Þ c h 2Bk nS: If some other alternative (qk, rk) is on the floor, coalition S2Wk will use (q 0 k, x 0 k) as a counterproposal if: ui ðqk0 ; qKk ; ; xi0 ÞO ui ðqk ; qKk ; ; xi Þ c i 2S: A voting equilibrium is therefore defined as an alternative against which no winning coalition has a feasible and preferred counterproposal.3 The set of feasible counterproposals available to winning coalitions must be carefully defined. It is well known that nondictatorial voting rules do not admit voting equilibria if the set of such counterproposals is unrestricted (and therefore coincides with the set of feasible allocations for the economy). In the present context, this happens if we allow winning coalitions to make an unrestricted use of the cost-sharing parameter rk. To rule out this typical ‘cycling’ problem, Kaneko (1977b) has defined a special ‘minority protection’ rule, basically requiring that the counterproposal imputes to the objecting coalition a cost at least as large as the cost that would be imputed by the distributive parameter on the floor. More precisely, given that the proposal (qk, rk) is on the floor, coalition S2Wk can only make use of counterproposals (q 0 k, r 0 k) that satisfy the following condition: X X ri Ck ðqk Þ% ri0 Ck ðqk Þ: (1) i2S

i2S

Note that condition (1) rules out counterproposals which are purely based on redistributive interests, as those that leave the level of public good unaffected and shift the cost of production to the minority. Under this restriction on feasible objections, and under the three properties of the set Wk listed above, the set of voting equilibria of the game G(Bk,Wk, qKk) coincides with the set of ratio equilibria of the economy Ek(qKk).4 3.2. Pre-federal equilibrium Given that countries independently and simultaneously choose their own domestic policies at the voting equilibria described in Section 3.1, a state of equilibrium is attained when each country is simultaneously at a voting equilibrium. 3

In more technical terms, the set of voting equilibria coincide with the core of the game G(Bk,Wk, qKk). See Theorems 1 and 2 in Hirokawa (1992). These results require some additional conditions that rule out boundary cases of zero public good production. These cases are here ruled out by our assumption on marginal costs at zero production of public goods. 4

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Definition 4. An pre-federal equilibrium is a vector ðq 1 ; .; q m ; r1 ; .; rm Þ such that the pair ðq j ; rj Þ is a voting equilibrium of the game Gj ðBj ; Wj ; qKj Þ, for all jZ1,2,.,m.  rÞ is an preBy the characterization results discussed in Section 3.1, the vector ðq; federal equilibrium if and only if the triple ðq j ; xj ; rj Þ, where xj is the private good allocation induced by the pair ðqj ; rj Þ, is a ratio equilibrium of the economy Ej ðqKj Þ, for all jZ1,.,m. We will show that if each public good is non inferior in the country producing it, then the economy E admits a pre-federal equilibrium. Assumption 4. The cross derivative (v2ui(q,xi))/(vqkvxi) is non-negative for all feasible allocations (q, x) of the economy E, for all i2Bk and for all kZ1,2,.,m. Proposition 1. Under Assumptions 1–3 there exists a pre-federal equilibrium. Due to their non-cooperative nature, pre-federal equilibria are not globally efficient for the economy E; they are however locally efficient, in the sense that they attain a Pareto Optimal allocation in each sub-economy. 4. Forming a federation We are now ready to address the main question of the paper: given that a status quo prefederal equilibrium has been attained, what are the incentives of countries to form a federation, given that both domestic and federal decisions are taken by voting? It is well understood that aggregate (or social) incentives are provided by the suboptimality of the pre-federal equilibrium; if federal decision making was designed in such a way to attain an efficient allocation of resources, all agents in the economy could gain from the federation. This may not, however, be sufficient to guarantee that the federation actually forms. In particular, it may result impossible to distribute the perspective efficiency gains associated with the formation of a federal system in the appropriate way. These limits to redistribution may be intrinsic in the properties of the decision making institution used at the federal level. For instance, if federal decisions are made by voting, voting equilibria are bound to impose tight constraints on federal resource allocation, in the same fashion as we have seen for the case of domestic decision making. 4.1. Federal decision making We assume that if a set of countries T4K forms a federation, federal decisions on a vector qT of produced public goods and on a distributive vector pT, sharing production costs across all voters in the federation, are taken by voting. In the federal committee, the members of national parliaments vote in a centralized process on a ‘one agent—one vote’ basis. The federal legislative committee is therefore made of the union of the national committees, i.e. by the set BThgj2TBj.5 5

The characterization of voting outcome which is the object of this section carries over to more complex federal voting rules, taking into account the jurisdictional structure (see Currarini, 2002).

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We can apply to the federal voting procedure the analysis developed for national voting procedures. In particular, we denote by G(BT,WT, qKT) the voting game in which BT is the committee and WT 32BT is the set of winning coalitions, and qKT is a given vector of public goods production by countries outside T. The same properties of non dictatorship, properness and monotonicity assumed for domestic voting are also assumed for the set WT. Also, the same minority protection rule, expressed by condition (1) for domestic voting games, is restated here as follows: given that the proposal (qT, pT) is on the floor, coalition S2WT can object only by means of counter-proposals (q 0 T, p 0 T) which satisfy the following condition: XX XX pki Ck ðqk Þ% pi0 kCk ðqk Þ: (2) i2S k2T

i2S k2T

The set of voting equilibria of the game G(BT, WT, qKT) coincides here with the set of ratio equilibria of the economy 3T (qKT). The allocation of resources associated with the formation of federation T is obtained by imposing a consistency requirement of the type imposed on the pre-federal equilibrium (see Section 3.2). Definition 5. A partial federal equilibrium with respect to T is a vector ðq^ T ; p^ T ; ðq^ j Þj2KnT ; ð^r j Þj2KnT Þ such that ðq^ T ; p^ T Þ is a voting equilibrium of the game GðBT ; WT ; q^KT Þ and ðq^j ; r^j Þ is a voting equilibrium for the game GðBj ; Wj ; q^Kj Þ for all j2K\T. Given the characterization of voting equilibria, every partial federal equilibrium induces a ratio equilibrium both within the federation and within each excluded country. 4.2. The federation formation stage Each country compares the prevailing pre-federal equilibrium allocation with the opportunity of forming a federation T, switching therefore to the federal regime of resource allocation given by the federal equilibrium with respect to T. There are two main problems in modelling the choice faced by countries. First, there may be multiple ratio equilibria within the federation for each possible choice of outside countries. Second, even when the ratio equilibrium is unique, there may exist multiple partial federal equilibria. Unlike in the case of pre-federal equilibria, multiplicity would constitute a problem here: since decisions must be taken before the federation has formed, the expected equilibrium must be specified and must be the same for all countries in the federation. The possibility of multiple equilibria will be explicitly take care of in the following definitions of stability. A first stability requirement we wish to impose is that countries join federations voluntarily, i.e. preferring the federal equilibrium to the status quo. The following definition expresses this requirement imposing that federations form by unanimity rule. Definition 6. Federation T is individually rational if for all j2T there exists no winning  xi ÞO ui ðq; ^ x^i Þ for all i2Sj, where ðq; ^ xÞ ^ is the allocation coalition Sj2Wj such that ui ðq;  xÞ  is induced by the induced by the partial federal equilibrium with respect to T and ðq; status quo pre-federal equilibrium.

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Note that the above definition is the least favorable (with respect to the possible multiplicity of equilibria) to the formation of a federation; for domestic winning coalitions to make an objection it is sufficient that in at least one federal equilibrium all coalitional members are worse off that at the pre-federation equilibrium. On the opposite extreme would be the optimistic definition, requiring that objecting coalitions are better off in all federal equilibria. The next examples show that there may be no individually rational federation under either definitions. In the first case, the failure to form the federation is due to heterogeneity in agents’ valuation of the public goods; in the second, it is due to heterogeneity in national incomes. P C xi for i2Bk, let Ck(qk)Zqk, let Example 1. Let KZ{1, ui ðq; xi ÞZ aki log qj P P 2}. Let ukRAk and let Ak Z i2Bk aki , for kZ1, 2, with AZ kZ1;2 Ak . The optimal aggregate level of public good is Q*ZA, while the equilibrium ratios are given by ðaki =AÞi2I . Assuming that A1OA2, country 1 is the only contributor to the public good in the prefederal equilibrium; by comparison of utilities in the ratio and pre-federal equilibria, the first is individually rational for country 2 if and only if log(A/A1)R1. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Example 2. Let KZ{1, 2}; let ui ðq; xi ÞZ ðq1 C q2 Þxi ; ci 2I; let also uj denote the aggregate endowment of country j, with uZu1Cu Pareto Optimal P2. The unique allocation satisfies Samuelson’s Condition MRSðq ; x ÞZ ðuK gðq1 C q2 Þ =2ðq1 C q2 ÞÞZ g, yielding q1 C q2 Z u=3g. The ratio equilibrium shares are ðui =uÞi2I . The pre-federal equilibrium is obtained by imposing the mutual Pareto Optimality of domestic choices, yielding q1 Z ð1=5gÞð3u1 K 2u2 Þ and q 2 Z ð1=5gÞð3u2 K 2u1 Þ, with  q1 C q2 Z ð1=5gÞu. By comparing each agent’s utility at the pre-federal equilibrium qZ and at the federal voting equilibrium (i.e. the ratio equilibrium of the whole economy) we obtain that the latter is preferred by all the agents in both countries if and only if 0.36u%u1%0.64u.

4.3. A federal tax system under which the full federation forms The reason why the full federation K may not form in the above examples is strictly related to the status quo allocation implicitly considered in computing the cost to be imputed to each winning coalition Indeed, these examples show that the federal voting equilibrium may not be individually rational for countries’ citizens, in the sense that some of them may be better treated at the pre-federal equilibrium than at the ratio equilibrium of the federal economy. On the other hand, federal voting equilibria of K (the ratio equilibria of E) are always individually rational with respect to the initial endowment of the economy E. This observation motivates us to argue that if the federal voting procedure was reformulated in order to let decisions bear on variations in the production of the public goods with respect to the prefederal equilibrium, the full federation would be individually rational in the sense of Definition 6. We will show that not only this is true, but that for a relevant class of economies the full federation satisfies a stronger stability requirement based on coalitional rationality.

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4.3.1. Individual rationality of the full federation We start by defining the modified federal voting process bearing on variations of the  T ; WT ; qKT Þ be the voting game in which players and public goods production. Let GðB winning coalitions are defined as in G(BT,WT, qKT), but in which coalitional preferences and the set of feasible proposals for a coalition S2WT are redefined as a function of the  rÞ. status quo pre-federal equilibrium ðq; We say that coalition S2WT prefers q 0 T, p 0 T to (qT, pT) if there exists some private consumption allocation x 0 T such that XX k X 0 X xi ; pi ½Ck ðqk0 Þ K Ck ðq k Þ C xi % i2S k2T

X

i2S

i2S

pkh ½Ck ðqk0 Þ K Ck ðq k Þ Z xh ; for all h ;S

k2T

ui ðq 0 ; x 0 ÞO ui ðq; xi Þ; for all i 2S: The first two lines are the feasibility constraint for coalition S modified to account for the producing the vector q 0 T of public goods costs the amount P fact that 0 k Þ, i.e. the cost of passing from the status quo vector qT to the new k2T ½Ck ðqk ÞK Ck ðq  T ; WT ; qKT Þ each agent i is considered as being vector q 0 T. Note that in the game GðB  rÞ. endowed only with the private good xi he owes at the pre-federal equilibrium ðq; The minority protection rule is reformulated as follows: coalition S2WT can propose any pair (q 0 T,pT) in objection to a pair (qT,pT) currently on the floor under the constraints: XX 0 XX k pi ½Ck ðqk Þ K Ck ðqk ÞR pi ½Ck ðqk Þ K Ck ðqk Þ: (3) i2S k2T

i2S k2T

 xÞ  the economy obtained from E by considering the pre-federal Denoting by ET ðq;  xÞ  as the initial endowment of the economy, we obtain the cost function for equilibrium ðq;  xÞ  as: Eðq; C k ðqk Þ Z Ck ðqk Þ Z Ck ðqk Þ:  xÞ  under the We can establish existence of a ratio equilibrium of the economy Eðq; following additional assumption. Assumption 5. xi O 0 for all i2I.  xÞ.  Lemma 1. Under Assumptions 1–5 there exists a ratio equilibrium for Eðq;  xÞ  and the set of voting The equivalence result between the set of ratio equilibria of Eðq;  WÞ follows again by the results in Kaneko (1977a,b) and equilibria of the game GðI;  xÞ  is equivalent to an economy in Hirokawa (1992), once we remark that the economy Eðq;  6 In the same way as we which public goods are the variations of production with respect to q.  x;  qKT Þ. have done for the original economy, we also define the sub-economy ET ðq; 6

 xÞ.  Note that the assumption of zero marginal cost at zero production does not hold for the economy Eðq; However, that assumption was used here to rule out boundary cases of zero public good production, to which the proof in Hirokawa does not apply. Note, however, that the case of zero production in the economy Eðq;  xÞ  coincides with the case of no change from the autarchic equilibrium, at which we must have positive production of public goods by our assumptions on costs. This rules out the boundary problem, and allows us to apply the result to the economy Eðq;  xÞ. 

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 xÞ  is weakly preferred by As noted above, each ratio equilibrium of the economy Eðq;  xÞ.  Since ratio equilibria each agent in I to the pre-federal equilibrium allocation ðq; characterize the set of federal equilibria of K, the following proposition follows. Proposition 2. Under Assumptions 1–5 the full federation K is individually rational.

4.3.2. Coalitional rationality Individual rationality refers to situations in which the pre-federal equilibrium is the only alternative to the formation of a federation. In this section we wish to explore the case in which countries may refuse to join a given federation in order to form a different one. The next definition of a coalitional rationality is aimed at accounting for such a possibility, under the simplifying assumption that only one federation can form in the economy. We will later discuss the role of this assumption and the parallel between coalitionally rational federations and the core of the gamma-game, studied by Chander and Tulkens (1997) for international pollution problems, and inspired by the gamma game of coalition formation studied by Hart and Kurz (1983). Definition 7. The federation T is coalitionally rational if there exists no set of countries Z4N such that ZhTs0, and no collection (Sj)j2Z of winning coalitions such that ui(q 0 ,x 0 )Oui(q*,x*) for all i2Sj, where (q*, x*) is the allocation induced by the federal equilibrium with respect to T, and (q 0 , x 0 ) is an allocation which can be induced (through private transfers among its members) by the collection (Sj)j2Z starting from the federal equilibrium with respect to Z. In order to obtain characterization results, we need to restrict the class of economies under consideration. In particular, we will make the following additional assumptions: Assumption 6. The public goods enter the P utility function in an linearly additive way: ui (q, xi) is of the form ui (Q, xi), where Q h k2K qk for all i 2I. Assumption 7. All countries have identical linear cost functions: Cj0 ðqj ÞZ Ck0 ðqk ÞZ c for all j, k2K. The class of preferences described by Assumption 6 covers interesting public good provision problems such as pollution abatement games, and has convenient properties in terms of the aggregate production of public good in the various equilibria considered in this paper. The following lemmata generalize a characterization results obtained by Chander and Tulkens (1997) for the subclass of quasilinear preferences.7 Proofs are contained in the appendix.

7

Chander and Tulkens (1997) do not assume linear and identical cost functions. They however make an assumption of preferences, restricting the magnitude of marginal rates of substitution at the efficient production levels with respect to the non cooperative lelvels. Although a similar assumption would do in the present context, we have chosen to impose a restriction directly in the primitives of the model. We remark, however, that the next characterization lemma does not rely on linearity of costs, but only on cost symmetry. Linearity is explicitely used in the Proof of Proposition 3.

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Lemma 2. Under Assumptions 1–7, if (Q*, x*) is a ratio equilibrium of the economy E and (Q, x) is a partial federal equilibrium with respect to T3K, then Q*OQ. Lemma 3. Let Assumptions 1–7 hold. At every partial federal equilibrium with respect to T3K, the aggregate produced public good is higher than at the pre-federal equilibrium; moreover, if T3K then the aggregate production of the countries outside T is lower that at the pre-federal equilibrium. We are now ready to state our main result: when imputed costs bear on variations of production levels, the full federation is preferred by all countries to any other federation and to the pre-federal equilibrium. Proposition 3. Under Assumptions 1–7 the full federation is coalitionally rational. Note that Proposition 3 does not rule out the emergence of other smaller coalitionally rational federations. Any such smaller federation should be payoff-equivalent, for the member countries, to the full federation. As we said, it is possible to interpret the above result as a core-stability property of the full federation in the sense of Chander and Tulkens (1997). In their paper, the formation of a coalition induces remaining players to split up into singletons, so that the induced allocation is the Nash equilibrium of a game played by the forming coalition and these remaining players. The result of Proposition 3 can be reinterpreted in terms of Chander and Tulkens’ gamma core as follows: for any profile of domestic ‘incumbent’ winning coalitions, each determining the preference relation of the represented country, the payoff allocation induced by the full federation is not improved upon by any set of countries, given that countries non involved in the defection remain isolated.

5. Conclusions In this paper we have studied the problem faced by a group of countries deciding whether to form a federation. Incentives to merge are here provided by the externalities generated by the domestic production of international public goods. We have assumed that countries’ decisions are taken by parliamentary voting, and that, if formed, a federation delegates the legislative power to a centralized committee. This federal committee is made of the union of all domestic committee (think, for instance, of the European Parliament). We have shown that if federal decisions take the status quo pre-federal allocation as a reference point, all countries have the political incentives to form the full federation, in the sense that all domestic winning coalitions vote in favour of the formation of the full federation. This has been shown to hold, for economies with symmetric linear costs and in which a single international public good is produced in all countries, even when sub-coalitions of countries may form smaller federations.

S. Currarini / Research in Economics 59 (2005) 1–21

13

Acknowledgements I wish to thank Henry Tulkens for his constant support and advice.

Appendix A Proof of Proposition 1. It will be convenient to denote the set of ratio equilibria RðEK ðqKk ÞÞ as a function also of the distributive vector rKk. We therefore write RðEK ðqKk ; rKk ÞÞ. We denote by rk(qKk, rKk) the projection of the set RðEK ðqKk ; rKk ÞÞ on the space RC !Dnk of pairs (qk, rk). We also denote by r(q, r) the m-th product of the sets rk(qKk, rKk) for kZ 1,.,m. A fixed point of the map r(q, r) is a pair ðq ; r  ÞZ ðq ; r1 ; .; rm Þ such that (q*, r*)2r(q*, r*). By definition, a fixed point of r is therefore a pre-federal equilibrium. By Kakutani’s theorem, such a fixed point exists if the map r (q, r) is nonempty, upper-hemicontinuous and convex-valued for all (q, r), and if it is defined on a nonempty, compact and convex set. Since the product maintains these properties, it will be sufficient to check these conditions on the projection of each map rk(qKk, rKk). Let us first assume that q is chosen in the compact set m jqk % T; c k 2K with T !Ng: QTm Z fq 2RC

(A1)

Non-emptiness of rk(qKk, rKk) follows from Theorem 1 in Kaneko (1977a). Convexity of rk(qKk, rKk) is easily proved once we have shown that, under normality of the public goods, the economy EðqKk Þ admits a unique equilibrium ratio. To show this, let ri ðq^ k Þ denote the ratio for which agent i demands level q^ k of public good: ri ðq^ k Þ hri : q^ k Z argmax ui ðq; ui K ri Ck ðqk ÞÞ:

(A2)

Note that ri(qk) is defined by ri ðqk ÞCk0 ðqk Þ

vui ðqk ; qKk ; ui K ri ðqk ÞCk ðqk ÞÞ vqk Z : vui ðqk ; qKk ; ui K ri ðqk ÞCk ðqk ÞÞ vxi

(A3)

Since the LHS is increasing in ri(qk) while the RHS is decreasing in ri(qk) (by normality), ri(qk) is uniquely determined for any qkR0. Also, it is immediate to check that ri(qk) is increasing in qk: vui v vq k Ck0 ðqk Þ K vri vui vri ðqk Þ vxi ZK R 0: vui vqk v vqk ri Ck0 ðqk Þ K vqk vui vxi

(A4)

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S. Currarini / Research in Economics 59 (2005) 1–21

Suppose now that there exist two ratio equilibria, (qk, xk, rk) and (q 0 k, x 0 k, r 0 k). Suppose that qkOq 0 k. We already know that rk(qk)Rri(qk) for all i2Bk. Since both rk and r 0 k belong to the simplex, it must be that rkZr 0 k. To show that uniqueness of the equilibrium ratio implies convexity of the set rk(qKk, rKk), let a2[0,1] and consider the pair ðqk ; xk Þa haðqk ; xk ÞC ð1K aÞðq^ k ; x^k Þ, where (qk, xk) and ðq^k ; x^k Þ are two ratio equilibrium allocations. It is immediate to check that (qk, xk)a is feasible for each i in Bk given the (unique) equilibrium ratio rk. In fact, by the definition of a ratio equilibrium, we have that both pairs (qk, xk) and ðq^ k ; x^k Þ solve for all i in Bk the maximization problem max ui ðq; xi Þ

(A5)

st ui R xi C ri Ck ðqk Þ:  By strict monotonicity of preferences we We can then write ui ðqk ; xi ÞZ ui ðq^ k ; x^i ÞZ u. can write ui Z xi C ri Ck ðqk Þ Z x^i C ri Ck ðq^k Þ

(A6)

from which ri ½Ck ðq^ k Þ K Ck ðqk Þ Z xi Z x^i :

(A7)

Consider now the feasibility condition for (qk, xk)a: ri Ck ðaqk C ð1 K aÞq^ k Þ C axi C ð1 K aÞx^i % ui :

(A8)

Using (A7) we can rewrite the above as ri Ck ðaqk C ð1 K aÞq^ k Þ C ari ½Ck ðq^ k Þ K Ck ðqk Þ C ri Ck ðq^ k Þ K ri Ck ðq^k Þ C x^i % ui (A9) or ri Ck ðaqk C ð1 K aÞq^ k Þ C ri ½aCk ðqk Þ C ð1 K aÞCk ðq^k Þ C ri Ck ðq^ k Þ C x^i % ui :

(A10)

By convexity of Ck and (A7), inequality (A10) holds. Quasiconcavity of ui can now be used to write:  ui ððqk ; xk Þa ÞR minfui ðqk ; xk Þ; ui ðqk0 ; xk0 Þg Z u;

(A11)

implying the result. We finally show upper-hemicontinuity of rk(qKk,rKk). By continuity of ui and compactness of QTm , the image of the map rk(qKk) is compact. It is then sufficient to show n n that the correspondence rk(qKk, rKk) is closed. Consider a sequence fqKk ; rKk g/ ðqKk ; rKk Þ Q T n n in QmK1 ! jsk Dnj and a sequence fqk ; rk g/ ðqk ; rk Þ in ½0; T !Dnk , such that fqnk ; rkn g 2 n n rk ðqKk ; rKk Þ for all n. We prove that (qk, rk)2rk(qKk, rKk). Suppose that, contrary to our statement, (qk, rk);rk(qKk, rKk). In this case, there exists agent h2Bk and level of public good q^ k 2½0; T such that: uh ðq^ k ; qKk ; uh K rh Ck ðq^k ÞÞ K uh ðqk ; qKk ; uh K rh Ck ðqk ÞÞO 0:

(A12)

S. Currarini / Research in Economics 59 (2005) 1–21

15

By continuity of ui, there exists a positive integer N such that for all nRN: n n uh ðq^ k ; qKk ; uh K rh Ck ðq^k ÞÞ K uh ðqk ; qKk ; uh K rh Ck ðqk ÞÞO 0:

(A13)

Again, by continuity of ui and of Ck, there exists a positive integer N 0 such that for all nRmax {N, N 0 }: n n uh ðq^ k ; qKk ; uh K rh Ck ðq^k ÞÞ K uh ðqnk ; qKk ; uh K rh Ck ðqnk ÞÞO 0:

(A14)

Once more, by continuity of ui, there exists a positive integer N 00 such that for all nRmax {N,N 0 ,N 00 }: n n uh ðq^ k ; qKk ; uh K rhn Ck ðq^ k ÞÞ K uh ðqnk ; qKk ; uh K rhn Ck ðqnk ÞÞO 0:

(A15)

n ðqnk ; rkn Þ 2rk ðqKk ; rkn Þ

The last inequality contradicts the assumption that for all n. We still have to show that for every fixed point (q*, r*) there does not exist any  ðq1 ; .; qk ; .; qm Þ such that qk O T and such that uh ðq;  uh K rh Ck ðqk ÞÞO vector qZ    uh ðq ; uh K rh Ck ðqk ÞÞ for some h2Bk. We omit the proof of this part, which is a direct application of the last part of the Proof of Theorem 1 in Kaneko (1977a). ,  xÞ  is defined as a triple Proof of Lemma 1. A ratio equilibrium for the economy Eðq; ^ such that each agent i maximizes its utility ui(q, xi) under the following constraint: ^ x; ^ pÞ ðq; xi R xi C

K X

p^ ki ½Ck ðqk Þ K Ck ðq k Þ

(A16)

kZ1

or xi C

K X

p^ ki Ck ðq k ÞR xi C

kZ1

K X

p^ ki Ck ðqk Þ

(A17)

kZ1

Note first that the LHS of (A17) is strictly positive by Assumption 5. Therefore, the economy in which the endowments of agent i are given by the LHS of (A17) respects all requirements in Kaneko (1977a) for the existence of a ratio equilibrium. We conclude that  rÞ admits a ratio equilibrium. , also the economy Eðq; Proof of Lemma 2. Let IT denote the set of agents living in the set of countries T. Conditions for efficiency of (Q*, x*) and for a partial federal equilibrium w.r.t. T imply the following equalities: X uiq ðQ ; xi Þ Z c; (A18) ui ðQ ; xi Þ i2I x X uiq ðQ; xi Þ i2IT

uix ðQ; xi Þ

X uiq ðQ; xi Þ i2Bj

uix ðQ; xi Þ

Z c:

Zc

c j 2KnT:

(A19)

16

S. Currarini / Research in Economics 59 (2005) 1–21

Since cO0 and since marginal rates of substitution are always non negative we have: X uiq ðQ; xi Þ i2IT

uix ðQ; xi Þ

R

X uiq ðQ ; xi Þ i2IT

uix ðQ ; xi Þ

:

(A20)

Suppose Q*!Q. Then, by normality of the public good and by strict concavity of preferences w.r.t. the private good, for all h2IT (there must exist at least one) for which: uhq ðQ; xh Þ uhq ðQ ; xh Þ O : uhx ðQ; xh Þ uhx ðQ ; xh Þ

(A21)

it must be that: xh ! xh :

(A22)

Using the expressions for xh and xh, if condition (A22) becomes: uhq ðQ; xh Þ X uhq ðQ ; xh Þ X   ðq K q Þ! ðqk K q k Þ: k k uhx ðQ; xh Þ k2T uhx ðQ ; xh Þ k2K

(A23)

Using (A21) we can write: uhq ðQ; xh Þ X uhq ðQ; xh Þ X   ðq K q Þ K ðqk K q k Þ! 0: k k uhx ðQ; xh Þ k2T uhx ðQ; xh Þ k2K

(A24)

But this implies that for all i2IT: uiq ðQ; xi Þ X uiq ðQ; xi Þ X  ðqk K qk Þ K i ðq K q k Þ! 0: i ux ðQ; xi Þ k2T ux ðQ; xi Þ k2K k

(A25)

In particular, for all i for which (A21) does not hold, we can write: uiq ðQ; xi Þ X uiq ðQ * ; xi* Þ X * ðqk K qk Þ K i ðq K qk Þ! 0; i ux ðQ; xi Þ k2T ux ðQ * ; xi* Þ k2K k implying that xi K xi ! 0. Therefore, we obtain X  X xi ! xi : i2IT

(A26)

(A27)

i2IT

The same argument can be used for those players i in I\IT. By (A18) we write for all j2K\T: X uiq ðQ ; xi Þ i2Bj

uix ðQ ; xi Þ

%

X uiq ðQ; xi Þ i2Bj

uix ðQ; xi Þ

:

(A28)

If Q*!Q, conditions (A21) and (A22) can be applied to some h2Bj, for all j2K\T. Condition (A23) becomes:

S. Currarini / Research in Economics 59 (2005) 1–21

uhq ðQ; xh Þ X uhq ðQ ; x Þ ðqk K qk Þ! h  h ðqj K qj Þ: h ux ðQ; xh Þ k2T ux ðQ ; xh Þ

17

(A29)

The same passages used in conditions (A24)–(A26) can be used here to conclude that for all j2K\T: X X xi ! xi : (A30) i2Bj

i2Bj

Conditions (A27) and (A30) imply: X  X xi ! xi : i2I

(A31)

i2I

Since by assumption Q*!Q, we obtain a contradiction of the assumption that (Q*, x*) is an efficient allocation for the economy. This directly obtains by strict monotonicity of preferences with respect to the private good and monotonicity with respect to the public good. , Proof of Lemma 3. We proceed in two steps. Let Q and Q be the pre-federal equilibrium and the Partial federal equilibrium w.r.t. T aggregate quantities of public goods, and let q j and qj denote the respective quantities produced by country j in the two regimes. Step 1  Step 2 shows that QR Q implies q j R qj for all j2K\T. shows that QR Q:  then q k % qk for all k2K, Step 1: we proceed by showing that if, on the contrary, Q! Q, implying a contradiction. ðaÞ

 Q! Q0 q j % qj

for all j 2knT:

Suppose qj ! qj . Then Cj ðqj Þ% Cj ðqj Þ and, by convexity of Cj ; Cj0 ðqj Þ% Cj0 ðqj Þ. By partial equilibrium and Nash equilibrium conditions we obtain  xi Þ uiq ðQ; xi Þ uiq ðQ;  xi Þ O uix ðQ; xi Þ : uix ðQ;

(A32)

Note that the assumption that Q! Q and concavity of ui together imply:  xi Þ: uiq ðQ; xi0 ÞR uiq ðQ;

(A33)

Also, Ci ðqi Þ% Ci ðq i Þ implies xi R xi , which, together with normality of the public good, imply:  xi ÞR uiq ðQ;  xi Þ: uiq ðQ; Conditions (A32) and (A33) imply:  xi Þ: uiq ðQ; xi ÞR uiq ðQ;

(A34)

Using again concavity and normality of ui we write:  xi ÞR uix ðQ;  xi ÞR uix ðQ; xi Þ: uix ðQ;

(A35)

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S. Currarini / Research in Economics 59 (2005) 1–21

Conditions (A34) and (A35) yield:  xi Þ uiq ðQ; xi Þ uiq ðQ; i  xi Þ % uix ðQ; xi Þ ; ux ðQ;

(A36)

a contradiction. ðbÞ

 Q! Q0 q k % qk

for all k 2T:

By step 1 we know that QT ! Q T . By first order conditions for a Nash and a Partial Nash  xÞ  must satisfy equilibrium, (Q, x) and ðQ; X X uiq ðQ; xi Þ j2T i2Bj

uix ðQ; xi Þ

Z C 0 ðqk Þ c k 2T;

(A37)

 xi Þ X uiq ðQ; Z C 0 ðq i Þ c k 2T: i   ux ðQ; xi Þ i2Bk

Since utility levels are assumed to be non negative, the conditions (A37) imply that for all k2T:  xi Þ X uiq ðQ; xi Þ X uiq ðQ; 0 K C ðq Þ% K C 0 ðqk Þ: k i 0 i  u ðQ ; x Þ  u ð Q; x Þ x i i x i2B i2B k

(A38)

k

Suppose now that qk ! qk for all j2T. Then strict convexity of costs would imply that for all k2T X uiq ðQ; xi Þ i2Bk

uix ðQ; xi Þ

!

 xi Þ X uiq ðQ;  xi Þ : uix ðQ;

(A39)

i2Bk

 the above condition holds for all k2T only if xi ! xi for all i2gk2TBk. This Since Q! Q, would violate (local) efficiency of (QT, xT). So it must be that qj O qj for some j2T. However, in this case, identical cost functions imply that qk O q k for all k2T, the desired result. Step 2. We show that if Q! Q then qj R qj for all j2K\T. Note first that by concavity of ui,  xi Þ: uiq ðQ; xi Þ% uiq ðQ;

(A40)

Suppose now that qj ! qj . Increasing costs imply xi O xi and, by normality of the public good,  xi Þ% uiq ðQ;  xi Þ: uiq ðQ;

(A41)

Conditions (A40) and (A41) imply:  xi Þ: uiq ðQ; xi Þ% uiq ðQ;

(A42)

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19

Note now that xi O xi together with concavity and normality of ui, imply:  xi Þ: uix ðQ; xi ÞR uix ðQ; xi ÞR uix ðQ;

(A43)

Conditions (A42) and (A43) yield:  xi Þ uiq ðQ; xi Þ uiq ðQ; : % i  i ux ðQ; xi Þ ux ðQ; xi Þ

(A44)

Since in any Nash equilibrium and any Partial equilibrium we have for all j2K\T: uiq ðQ; xi Þ Z C 0 ðqj Þ; uix ðQ; xi Þ

 xi Þ uiq ðQ; 0  xi Þ Z C ðq j Þ; uix ðQ;

(A45)

condition (A44) implies: C 0 ðqj Þ% Ci0 ðqj Þ:

(A46)

This, by convexity of costs, implies that qj % qj , a contradiction. , Proof of Proposition 3. We proceed by contradiction. Suppose coalition T of countries prefer the partial federal equilibrium (qT pT, qKT, rKT) w.r.t. T to the voting equilibrium (q*, p*) of the full federation, with x* denoting the private good allocation induced by the pair (q*, p*). This means that there exist winning coalitions (Sj)j2T and private good allocations ðxi Þi2Sj for all j 2 T such that: ui ðQT C QKnT ; xi ÞO ui ðQ ; xi Þ c i 2Sj ; c j 2T

(A47)

and XX

xi Z

j2T i2Sj

XX j2T i2Sj

" xi K

X

# pki ½cðqk

K q k Þ :

k2T

Lemma 3, stating that QKnT % Q KnT , together with monotonicity of preferences, imply that for all i2Sj and all j2T: ui ðQT C Q KnT ; xi ÞR ui ðQ ; xi Þ: We will show that (A48) implies a contradiction. The argument goes by first showing that for all i2gj2TSj: X k ÞO xi : pk xi K i ½cðqk K q

(A48)

(A49)

k2T

Suppose not, so that for some i2gj2TSj: X k ÞO xi xi K pk i ½cðqk K q

(A50)

k2T

Note that by the ratio equilibrium properties of (q*, p*) we have for all i2Sj and all j2T:

20

S. Currarini / Research in Economics 59 (2005) 1–21

ui ðQ ; xi ÞR ui QT C Q KnT ; xi K

X

!  k Þ : pk i ½cðqk K q

(A51)

k2T

By strict monotonicity of preferences and condition (A50), we write: ! X k Þ O ui ðQT C Q KnT ; xi Þ: ui QT C Q KnT ; xi K pk i ½cðqk K q

(A52)

k2T

This, together with (A51), implies: ui ðQ ; xi ÞO ui ðQT C Q KnT ; xi Þ; which contradicts (A48). We now sum up condition (A49) over all agents in the set gj2TSj to obtain: " # X X X k k ðpi K pi Þðcðqk K q k Þ R 0: j2T i2Sj

(A53)

(A54)

k2T

j Noting that the linear additive nature of public goods implies that for all pk i Z pi and for all k, j2K, we rewrite (A54) as follows: XX  X ðpi K pi Þc ðqk K q k ÞR 0: (A55)

pki Z pji

j2T i2Sj

k2T

Lemma 3 allows us to say that condition (A55), and therefore (A47), hold only if XX  ðpi K pi ÞR 0: (A56) j2T i2Sj

We will prove our result by showing that condition (A56) does not hold. We first rewrite equilibrium ratios as follows: pi Z

MRSi ; c

pi Z

(A57)

MRSi ; c

where we have used the following notation: MRSi Z

uiq ðQ; xi Þ ; uix ðQ; xi Þ

MRSi Z

(A58)

uiq ðQ ; xi Þ : uix ðQ ; xi Þ

Using (A58) we now express the private good consumption levels xi and xi as follows for i2gj2TBj:

S. Currarini / Research in Economics 59 (2005) 1–21

xi Z xi K

MRSi X ½cðqj K q j Þ c j2T

xi Z xi K

21

(A59)

MRSi X ½cðqj K q j Þ: c j2K

We then note that pi R pi 4

MRSi MRSi R / MRSi ðQ; xi ÞO MRSi c c

(A60)

where the last inequality comes from normality and strict concavity of the utility function in the public good and the fact that Q*OQ. Note now that, again by normality and concavity, the last inequality implies that xi R xi : This in turn implies (see (A59)): X  X MRSi ðqj K q j Þ% MRSi ðqj K qj Þ: (A61) j2K

j2T

Using the fact that the aggregate efficient production of public good is higher than the aggregate production at any partial federal equilibrium (Lemma 2), and that efficient production is strictly positive in each country thanks to symmetry of costs, we obtain that pi ! pi for all i22gTSj, contradicting (A56) and concluding the proof. ,

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