Public Good Provision and Cooperation Among Representative Democracies1 Antoine Loeper2 June 2015
1 This
research bene…ted from the …nancial support of the grant ECO 2013-42710-P from the
Spanish “Ministerio de Ciencia e Innovacion”. 2 Universidad Carlos III de Madrid. Email:
[email protected]
Abstract This paper analyzes the provision of international public goods by sovereign democracies. The contribution of each country is decided by a representative elected by majority rule at the country level. We compare the case in which representatives choose their contributions noncooperatively with the case in which they choose it through Coasian cooperation. Cooperation among representatives internalize cross-border externalities but induces strategic voters to elect representatives who care less about the public good, so as to reduce their share of the cost of provision. The former e¤ect increases public good provision while the latter reduces it. We show that once voters’ incentives are taken into account, whether cooperation increases public good provision depends neither on voters’preferences, nor on the magnitude of spillovers, nor on the size, bargaining power, or e¢ ciency of each country. Instead, it depends only on the elasticity of the demand for the public good, and the degree of substitutability between countries’contributions. Hence, the desirability of international cooperation depends mostly on the type of public good considered. JEL Classi…cation Numbers: D62, D72, F55, H41, H77 Keywords: International Cooperation; Bargaining; Federalism; Public Good; Election; Strategic Delegation.
1
Introduction
Cross border externalities and transnational public goods lead to ine¢ ciencies and collective action failure when countries set their policies noncooperatively. In the absence of overarching political institutions, observers often call for greater coordination between national policy makers to internalize these externalities. However, despite the multiplication of international negotiations and summits, the supposed gains from international cooperation have arguably not fully materialized. Many global public goods such as the reduction of greenhouse gas emission, political asylum, disease eradication, …sh stocks, or …scal stimulus are still underprovided. Since Coase (1960), economists have invoked transaction costs of various sorts to explain the inability of bargaining parties to reach mutually bene…cial arrangements.1 This strand of literature focuses on the bargaining process and does not take into account the speci…cities of the decision process within each country. Others have argued that international policy coordination can exacerbate ine¢ ciencies in national politics.2 This paper assumes away any ine¢ ciencies in national politics or in the bargaining process, and focuses instead on the interaction between elections at the national level and cooperation at the international level. In modern democracies, most decision are taken not by the voters, but by political representatives appointed by the voters. As Persson and Tabellini (1992) …rst pointed out, even if one abstracts away from political agency issues, this distinction has important consequences, because sophisticated voters can use elections as a strategic delegation mechanism. Several papers have shown that once voters’ incentives are taken into account, the impact of international cooperation on public good provision is ambiguous (Segendor¤ 1998, Gradstein 2004, Buchholz et al. 2005, Kempf and Rossignol 2013). On the one hand, cooperation helps national policy makers internalize cross-border spillovers. This direct e¤ect increases public good provision. On the other hand, more public good requires greater contributions from 1
Among others, commitment and enforcement problems (Williamson 1985, North 1990, or Acemoglu
2003) or imperfect information (Mailath and Postlewaite 1990, Harstad 2007) can lead to ine¢ ciencies. 2 International policy coordination can exacerbate political agency problems (Brennan and Buchanan 1980, Buchanan and Faith 1987, Tabellini 1990, Persson and Tabellini 1995) or dynamic commitment problems between voters and politicians (Rogo¤ 1985, Kehoe 1989).
1
participating countries. Therefore, cooperation induce strategic voters to elect representatives who care less about the public good, so as to decrease their relative contribution. This electoral e¤ect decreases public good provision. In this paper, we determine the main drivers of the magnitude of this electoral e¤ect, and characterize the conditions under which it only mitigates, or completely o¤sets the direct e¤ect of cooperation. We consider a global public good model with two countries populated by a continuum of heterogeneous voters. The preferences of a given voter or candidate are characterized by a type that determines her trade-o¤ between public good and private good consumption. Alternatively, this type can be interpreted as her “tax price”of the public good. Countries’ contributions are determined by a two-stage game. In the …rst stage— the electoral stage— each country elects its representative by majority rule. In the second stage— the policymaking stage— the elected representatives choose, cooperatively or noncooperatively, their contributions to the global public good. In the cooperative scenario, the representatives implement the generalized Nash bargaining solution with the noncooperative outcome as the bargaining default. The main result is that whether cooperation increases public good provision in equilibrium depends neither on the distribution of voters’ preferences, nor on the magnitude of cross-border spillovers, nor on the relative size, e¢ ciency, or bargaining power of each country. Instead, it depends only on the curvature of the demand for the public good, and on the degree of substitutability between countries’ contributions. In the canonical case of a pure public good whose demand has a constant tax-price elasticity, cooperation increases the level of public good if and only if this elasticity is greater than two. The model further reveals that whether cooperation increases public good provision does not depend on how severe the underprovision of the public good is in the noncooperative equilibrium. For instance, greater cross-border externalities, or allowing for transfers between countries moves the e¢ ciency frontier farther away from the noncooperative equilibrium. However, once voters’incentives are taken into account, greater spillovers or monetary compensations do not make cooperation more likely to be bene…cial. To understand the intuition behind these results, consider voters’incentives in the electoral stage. Voters trade o¤ electing a public good lover to increase public good provision, 2
versus electing a public good skeptic to shift the cost of provision on the other country. As the elasticity of the public good demand decreases, the cooperative level of public good becomes less sensitive to the representatives’type, so the latter a¤ect mostly how the cost of provision is shared among the two countries. As a result, voters put a greater weight on reducing their share of the cost of provision, relative to increasing the level of public good, and thus appoint a representative with a lower demand for the public good. This explains why a more inelastic public good demand makes cooperation less likely to be bene…cial. Likewise, greater spillovers increase the gains from cooperation in the policy-making stage, but require greater contributions from countries, which increase voters’incentives to elect a low demand representative and shift the cost of provision on the other country. The latter, electoral e¤ect o¤sets the former, direct e¤ect, which explains why greater spillovers do not make cooperation more likely to be bene…cial. Our results can be related to the empirical literature on the demand for public goods. Interestingly, existing estimates of the tax-price elasticity vary greatly between public goods (see, e.g., Feldstein 1975, Brooks 2007). Hence, our results imply that the e¢ ciency of interjurisdictional cooperation can di¤er importantly across types of public goods. Moreover, estimated elasticities are typically smaller than two (Wildasin 1989, Auten et. al. 2002). Therefore, this model suggests that for plausible speci…cations, strategic voting can severely o¤set the gains from Coasian cooperation between sovereign democracies. Hence, stronger forms of cooperation are required, such as pooling of sovereignty, or explicit cost sharing. Our results also shed light on the debate over the structure of federal systems. Several competing principles have been invoked to determine the optimal allocation of policy responsibilities between central and local governments. One of them, “cooperative federalism,” states that federal policies should be negotiated by and “agreed to unanimously by the elected representatives from each of the lower tier governments” (Inman and Rubinfeld 1979). Our analysis suggests that cooperative federalism is not an adequate solution for the provision of local public goods with spillovers, and more coercive forms of centralization are needed. Moreover, contrary to received wisdom (Oates 1972), whether cooperative federalism dominates decentralization depends neither on the heterogeneity of local preferences nor on the magnitude of externalities, but on the elasticity of the demand for the public good. 3
The paper is organized as follows. Section 2 reviews the literature. Section 3 describes the pure public good model. Section 4 derives the main results. Section 5 analyzes the case of local public goods with spillovers. Section 6 considers the robustness of the results to a number of extensions, and Section 7 relates the results to the empirical literature on the demand for public goods.
2
Related Literature
Since Olson (1965), the literature on collective action and public good provision has shown that the ine¢ ciency of noncooperative behavior is more severe— and thus the potential gains from cooperation are greater— when spillovers are large (Oates 1972, Sandler 1998), when countries’ contributions are closer substitutes (Hirshleifer 1983), or when preferences are homogeneous (Cornes 1993). In contrast, in our model, the distribution of spillovers and preferences do not matter, and cooperation is more likely to be bene…cial when countries’ contributions are closer complements. This paper di¤ers from that literature in that the latter focuses on the coordination failure between policy makers while this paper assumes that policy makers cooperate e¢ ciently, and focuses instead on the coordination failure between voters of di¤erent jurisdictions. A number of papers have shown that strategic delegation in elections can distort the outcome of Coasian cooperation between jurisdictions. Brueckner (2000) shows that one can restore e¢ ciency by appropriately allocating the proposal power. Harstad (2008) focuses on the impact of transfers on electoral incentives (the role of transfers and bargaining power in this model is analyzed in Section 6.1). Kempf and Rossignol (2013) investigate the impact of equity considerations in international agreements on electoral incentives, but do not compare electoral equilibria with and without cooperation.3 In Section 6.2, we show that our results hold unchanged when representatives use the egalitarian bargaining solution as in Kempf and 3
Kempf and Rossignol (2013) compare the cooperative and noncooperative policy outcome assuming that
the representatives elected in the cooperative scenario are also the policy makers in the noncooperative scenario. However, this comparison cannot be used to assess the impact of cooperation under strategic voting because in equilibrium, voters elect di¤erent representatives depending on whether they expect them to cooperate or not.
4
Rossignol (2013). Most closely related to this paper, Segendor¤ (1998), Gradstein (2004), and Buchholz et al. (2005) also investigate whether strategic delegation o¤sets the gains from international cooperation. Segendor¤ (1998) and Gradstein (2004) consider a symmetric, pure public good model, and a unilateral externality model with transfers, respectively. Our results show that their conclusion are driven by their particular speci…cation for the cost structure of the public good (see the discussion of these papers in Sections 5 and 6.1). Buchholz et. al (2005) consider a symmetric environment with public bads and transfers, and show that strategic voting leads to an in…nite amount of public bad as spillovers become global. The discrepancy between our results and theirs comes from the fact that they focus on symmetric equilibria, which exists only when spillovers are su¢ ciently small, so their result cannot be applied to the case of global spillovers. In Section 6.4, we show that once asymmetric equilibria are considered, their model yields the same results as ours. This paper departs from the aforementioned contributions in that we focus on the empirically more relevant case in which international cooperation is carried out without monetary transfers, and political leaders stay in place in case of negotiation failure. More importantly, we consider a large class of public good environments, and characterize the conditions under which strategic voting makes cooperation detrimental. Our analysis singles out the elasticity of the demand for the public good as the main driver of the cost of strategic voting. Even though this parameter has received a lot of attention in the empirical literature (see Section 7), its central role has been overlooked by the theoretical literature on strategic delegation. Some papers analyze the impact of strategic delegation on more institutionalized forms of cooperation (Chari et al. 1997, Cheikbossian 2000, Besley and Coate 2003, Redoano and Scharf 2004, Dur and Roelfsema 2005, Harstad 2010, Christiansen 2013). These models di¤er from ours in that cooperation is carried out through federal institutions, …scal arrangements, and/or majoritarian decision making. In contrast to the case of voluntary bargaining, these forms of cooperation generate electoral incentives that lead to overprovision. Common …scal pools induce voters to elect public good lovers to increase central spending in their preferred public good. Majoritarian decision making leads to expropriation by the winning coalition, and thus induces voters to elect representatives that are biased in favor of the public project so as to be included in the winning majority. 5
3
A Pure Public Good Model
We consider an economy composed of two countries. Each country c 2 f1; 2g is inhabited by a continuum of residents indexed by a set Ic . A voter i 2 Ic in country c is denoted by (i; c) : Each country c chooses the level of provision xc
0 of a public good, and …nance this
provision by national taxation. For expositional purpose, in this section, we …rst present the model in the canonical case of a pure, global public good: for a given vector of provision x = (x1 ; x2 ), the level of public good consumed by the residents of either country is g = x1 + x2 .
3.1
Preferences and the Demand for the PublIC Good
The preferences of each voter (i; c) are parametrized by a type pi;c
0 and are derived from
the following utility function: Uc (pi;c ; x) = G (x1 + x2 )
(1)
pi;c xc :
The term G (x1 + x2 ) in (1) corresponds to the utility she derives from public good consumption, while the term
pi;c xc captures the utility loss due to the extra tax she pays to
…nance her country’s provision xc : We assume that G is twice continuously di¤erentiable on R++ , and for all g > 0, G0 (g) > 0; G00 (g) < 0, limx!0 G0 (x) = +1, and limx!1 G0 (x) = 0. Note that the parameter pi;c allows for several sources of heterogeneity. It can capture how much voter (i; c) enjoys public good consumption relative to what she consumes with her after tax income, but also how much she contributes to the cost of public good provision relative to other tax payers in country c, or how e¢ cient her country c is in collecting taxes and using the receipts to provide the public good. The representative of each country c is selected among its population, and its type is denoted by prc . The distribution of types within each country (pi;c )i2Ic is assumed to have full support on R+ , and the national median type pm c is positive. If country c was the sole provider of the public good, and if voter (i; c) could decide its policy, the resulting level of public good would be (G0 )
1
(pi;c ). Thus, one can interpret pi;c
as the tax price of voter (i; c) for the public good, and the function D = (G0 )
1
as the public
good demand function. Accordingly, for any pro…le of representatives’types pr , we refer to 6
the representative who has the smaller (greater) prc as the high (low) demand representative, and likewise with pm c for the median voters. To illustrate our results, we occasionally use the following speci…cation for G: for all g > 0 and all " > 0 such that " 6= 1; G (g) =
" "
1
g
" 1 "
(2)
;
and G = ln when " = 1. The public good demand induced by (2) is D (p) = p " . Hence, its tax price-elasticity is constant and equal to
".
Some remarks about the speci…cation in (1) are in order. It is general in some respects and restrictive in others. It is general in that it places minimal assumptions on the public good technology G: More precisely, it allows for any continuous and decreasing public good demand function D. As we shall see, this degree of freedom turns out to be the crucial one in our model. This speci…cation also allows for any degree of preferences heterogeneity within and across countries via the type distribution. Moreover, di¤erences in aggregate tax price R p across countries can re‡ect di¤erences in population sizes, or in taxation and public i2Ic i;c
good provision e¢ ciency.
The speci…cation in (1) is restrictive in two main respects. First, the term G (x1 + x2 ) in (1) assumes that countries make perfectly substitutable contributions to a pure public good, and this public good induces the same demand function in both countries. These assumptions are relaxed in Section 5. Second, the term
i;c xc
in (1) implicitly assumes that
preferences are quasi-linear in after-tax income and that the tax price of each individual is constant over the relevant range of public good levels. This assumption can be viewed as a …rst order approximation which is justi…ed if the cost of provision of the public good under consideration is a small share of the total budget of each country. Also, this assumption makes our results more comparable with the existing literature, since all papers on strategic delegation and public good provision consider quasi linear environments. Nevertheless, to investigate the impact of tax distortions or preferences that are nonlinear in after tax income, we relax that assumption in Section 9.
7
3.2
The Elections
To focus on the role of elections as delegation mechanisms, and abstract away from other electoral e¤ects such electoral competition or political agency, we model national elections via a simpli…ed “citizen-candidate” model as in Persson and Tabellini (1992). Representatives are selected from the pool of voters, and any voter is willing to become the representative of her country if she receives a majority of votes. Candidates cannot make credible electoral promises (Alesina 1988). Therefore, once elected, policy makers maximize their own utility. Voters are consequentialist and rational: they care only about the policy outcome and foresee how the type of their representative a¤ect the policy outcome. To abstract away from the issue of voters’miscoordination, we assume that the election winner in a given country is the citizen who is preferred by a majority of voters to any other candidate.4 The following remark implies then that the national median voters are pivotal in all elections. Remark 1 Voters’ preferences within each country c satisfy the intermediate preferences property of Grandmont (1978),5 so the majority preferences in country c over any pair of policy vectors coincide with the preferences of its median type pm c . Depending on the nature of the public good, voting on the type of the representative can be interpreted as choosing a candidate with a particular belief about the intrinsic value of the public good (e.g., a global warming skeptic, or a monetarist in the case of …scal stimulus), a special inclination towards the polluting industry or the environmental lobbies, or a candidate who seek the support of voters from a particular income group, and thus from voters with a particular tax price. 4
One can model elections as a normal-form game in which the strategy of each voter is the individual for
whom she votes, and the outcome is the policy vector induced by the type of the plurality winner. This game typically has multiple Nash equilibria (e.g., all voters voting for a given type). However, whenever a type is a Condorcet winner, there is a strong Nash equilibrium in which a majority of voters vote for a candidate of that type, and all strong Nash equilibria are outcome equivalent, consistently with our assumption. 5 This property is satis…ed because the type pi;c appears linearly in (1) (see Grandmont 1978).
8
4
Public Good Provision by Elected Representatives
4.1
Public Good Provision without Cooperation
In this section, we consider the benchmark case in which elected representatives choose their contributions noncooperatively. The game proceeds as follows. In the …rst stage— the electoral stage— the voters of each country elect a representative among themselves by majority rule, taking the representative of the other country as given. In the second stage— the policy-making stage— elected representatives choose their policy noncooperatively, taking the policy of the other country as given. We …rst derive the subgame equilibrium in the policy-making stage. For any pro…le of representatives pr elected in the …rst stage, the noncooperative policy equilibrium is: 8 < (D (pr ) ; 0) if pr < pr ; 1 1 2 xN (pr ) = : (0; D (pr )) if pr > pr ; 2
1
(3)
2
and if pr1 = pr2 , there is a continuum of equilibria, but our results do not depend on which is selected in that case. Since public good contributions are perfect substitutes, (3) shows that the contribution of the high-demand representative completely crowds out the contribution of the low-demand representative, and the latter free rides in equilibrium. Solving by backward induction, the equilibrium in the electoral stage is de…ned as follows: De…nition 1 A vector of types pr is a noncooperative electoral equilibrium (henceforth NEE) if for all p1 2 R+ , xN (pr ) is majority preferred in country 1 to xN (p1 ; pr2 ), and for all p2 2 R+ , xN (pr ) is majority preferred in country 2 to xN (pr1 ; p2 ) : Remark 1 implies that a CEE is a Nash equilibrium of the game in which national median voters control the type of their representatives pr , and whose outcome is xN (pr ). Proposition 1 For any pm 2 > 0; there exists ; 2 R such that 0 < a If
pm 1 pm 2
2 =
;
0— that is, when B (g) = thus S (p) = p" — then S 0 (p)
"+1 " g " "+1
and
p3 = p"+2 which is increasing in p for all " > 0: Hence,
Proposition 9 implies that, contrary to the public good model, for the isoelastic speci…cation, cooperation is always detrimental. This model corresponds to the “global pollution” case in Buchholz et. al. (2005)— the limit case s ! 1 using their notations. These authors show that, unlike in this model, pollution can increase in…nitely as the public bad becomes pure. The discrepancy between our results and theirs comes from the fact that they focus on symmetric equilibria, which do not exist when spillovers are large (see Remark 3 in the appendix).12
6.5
Tax Distortions
The models analyzed in Sections 3 and 5 assume that for each voter (i; c), the tax price for the public good pi;c is independent of xc . This assumption could be violated if the tax levied to …nance the public good generates distortions which grow more than proportionally with the level of taxation. Increasing marginal tax distortions can be most easily introduced in the heterogeneous public good model by replacing the term
pi;c xd in (5) by
pi;c T (xc ) for some strictly
increasing and convex function T : Uc (pi;c ; x) = Gc (x1 ) +
c0 ;c Gc0
(xc0 )
pi;c T (xc ) ;
(9)
This speci…cation can be mapped into the original model by expressing countries’policies in terms of level of taxation instead of level of provision. Speci…cally, if we use the change of 12
Bucholtz et al. (2005) also di¤er from this paper in that they allow for monetary transfers. However, as
argued in Section 6.1, this assumptio does not change the results.
21
variable yc = T (xc ) for all c 2 fl; rg and xc that G is replaced by G T
1
0; then (9) becomes identical to (5) except
; which yields the following result:
Proposition 10 For all c 2 f1; 2g ; if DcT denotes the public good demand function derived
from the public good technology GTc $ Gc , and p!
T
1
, then for the speci…cation (9), for all pm ,
, the level of public good c is greater (smaller) in any CEE than in the NEE if 0
DcT (p)
p2 is decreasing (increasing).
Note that the demand function DcT has a di¤erent interpretation from Dc : DcT (p) is the most preferred level of provision of public good c for a voter with tax price p; whereas DcT (p) is the level of tax that she would pay at this level of provision. Intuitively, when tax payments increase more than linearly with the level of provision— that is, when T is convex instead of linear as in the basic model— DT should be less elastic, because as p decreases and as the level of provision increases, the price of an additional unit of public good increases more rapidly, so public good demand increases less rapidly. Thus, Corollary 2 suggests that increasing tax distortions should make cooperation less likely to be bene…cial. This intuition can be formalized in the following environment. Suppose that for all g
0; T (g) = g + g 2 ; so the case
= 0 corresponds to the basic model of Section 3.
Simple calculus shows (see Remark 4 in the appendix) that if Gc is such that (Dc )0 (p) is increasing, then
0
DcT (p)
cooperation is detrimental for
6.6
p2
p2 is also increasing. Therefore, from Proposition 10, if = 0, cooperation must also be detrimental for all
> 0:
Equilibrium Existence
Propositions 2 and 3 hold for any CEE, but do not prove the existence of such equilibria. Existence of an electoral equilibrium in the cooperative scenario is di¢ cult to ascertain because for the general speci…cation considered in (1) and (5), the Nash bargaining solution does not have a closed-form solution, which makes it di¢ cult to verify that voters’ payo¤ are quasi-concave in their representatives’types. Nevertheless, Proposition 4 shows that in the heterogeneous public good case, when G
ln— i.e., when the demand for public good
has a tax-price elasticity equal to 1— a CEE exists for any distribution of bargaining power, spillovers, and types. 22
For general speci…cations of the function G, one can prove the existence of a CEE in mixed strategies in the heterogeneous public good model. Suppose that voters are expected utility maximizer with Bernouilli utility function given by (5). A mixed CEE consists of two probability distributions
1
type prc in the support of probability distribution
and c
c0
2
over R+ such that for each c 2 f1; 2g ; any representative’s
is majority-preferred in country c to any other type, given the of the type prc0 elected in the other country. Hence, De…nition
2 corresponds to a mixed CEE with degenerate distributions
1
and
13 2.
The following
proposition extends Proposition 3 to mixed CEE. Proposition 11 A mixed CEE always exists. Let D denotes the public good demand function derived from G (see Section 3.1). For all pm and (increasing), then for all mixed CEE ( 1 ; there exists prc0 in the support of
c0
2 ),
, if jD0 (p)j
p2 is decreasing
all c 2 f1; 2g and all prc in the support of
c;
such that the level public good in country c is greater
(smaller) in xB (pr ) than in the NEE. In the homogeneous public good model, the equilibrium existence issue is further complicated by the discontinuity of the noncooperative equilibrium as given by (3) at any pr such that pr1 = pr2 : In the cooperative scenario, this discontinuity can give incentives to the voters of the country of the high demand representative to elect a type slightly higher than the other country so as to become the free-rider at the bargaining default, and thus decrease their contribution at the bargaining outcome. Intuitively, such deviations are pro…table only m when pr1 and pr2 are su¢ ciently close, and thus when pm 1 and p2 are su¢ ciently close. The 13
Since the electoral stage represents a game between two electorates rather than two unitary players, the
interpretation of mixed equilibria is not the same as in standard normal form games. Nevertheless, such mixed strategy equilibria can be puri…ed in the spirit of Harsanyi (1973) as follows: the payo¤ of a voter (i; c) from a pro…le of elected representative pr is Uc pi;c ; xB (pr ) + shock common to all voters in country c; and
1
and
2
c
(prc ) ; where ( c (prc ))prc 2R is a random
are independent. The realization of
c
(prc ) can be
interpreted as the intrinsic popularity in country c of the candidate with type prc : Since voters’ payo¤ are still linear in the type, Remark 1 still holds, and for any realization of
c
(prc ), the majority preferences in
country c on lotteries over pr coincide with the preferences of the median type pm c . One can then choose such that for almost all its realizations, there is a unique best response prc for pm c given the probability distribution of these best response induced by indi¤erent between all types prc in the support of
c,
c
is
c:
If (
1;
2)
c0 ;
c
and such that
is a CEE, then since pm c is
these popularity shocks can be chosen arbitrarily small.
23
following proposition con…rms this intuition when G (g)
p
g— i.e., when the public good
demand has an elasticity equal to 2. Proposition 12 When G (g)
7
p
g, there exists a CEE if and only if
pm 1 pm 2
2 =
1+ 3
1
; 1+3
2
:
Concluding Remarks
It is instructive to relate our results to the empirical literature on the demand for public goods. Most empirical studies have found a relatively small tax-price elasticity of the public good demand, typically lower than one (see, e.g., Wildasin 1989) but estimates greater than two have been found for some particular public goods (see DelRossi and Inman 1999). Together with Corollaries 1 and 2, these studies suggest that strategic voting can greatly undermine the gains from interjurisdictional Coasian cooperation, but its e¢ ciency can vary substantially across types of public goods. It should be noted that most of these estimates are for local public goods. Tax-price elasticities for global public goods are more di¢ cult to estimate because the e¤ect of institutional or cultural di¤erences across countries is harder to factor out from the variations of public good demands. For public goods of national of international scope, existing estimates are based on individual donations. They are not equivalent to tax-price elasticities, because charitable donations re‡ect a desire for warm-glow which is of smaller magnitude when individual contributions take the form of mandatory taxation. Nevertheless, it is interesting to note that, as for the case of local public goods, these estimates are typically around or below one (see, e.g., Auten et. al. 2002, and the references therein), and more importantly, they vary greatly between public goods (Feldstein 1975, Brooks 2007). When empirical estimates of tax-price elasticities are not available, our results suggest the following rule of thumb: the negative e¤ect of strategic voting on international cooperation is more severe for public goods whose marginal return decreases more rapidly. Policy expertise can then be used to assess this rate of decrease for a given public good, say fundamental research, disease eradication, or global warming.
24
8
Appendix
To keep the algebra tractable, it will be convenient to renormalize the utility function in (1) as follows: Ui;c (x) = where
i;c
i;c G (x1
+ x2 )
xc ;
= 1=pi;c : Since the Nash equilibrium concept and the Nash bargaining solution
used in this paper are invariant through a¢ ne transformation of the utility function, this renormalization is w.l.o.g. Throughout the appendix, we use the above speci…cation (except in the proof of Propositions 5 and 8, in which the cardinal representation of Ui;c matters), and the following notations. Notation 1 The median voter of any country c 2 f1; 2g is denoted by im c , and its type is denoted by For all
r
m c :
2 R2+ , all x 2 R2+ , and all c 2 f1; 2g,
c
( r ; x) = Uc ( rc ; x)
r N c; x
Uc
( r)
denotes the welfare gains for the representative of country c at x relative to the noncooperative equilibrium xN ( r ). For all p > 0; D (p) $ (G0 )
1
r
Proof of Proposition 1. Let that
r 1
=
r 2.
(t) $
(p) and for all t > 0,
1 D0 t3
1 t
r 1
6=
be a NEE. Necessarily, r c
From (3), for at least one country c, xN
: r 2.
To see this, suppose r m c , ic
> 0, and by lowering
keeps
the public good level unchanged, but lowers its contribution to 0, a pro…table deviation. r 1
Suppose now to …x ideas that
1 such that (0; r
m 1
m 2 .
Lemma 1 If
r 1
r 2
>
m 1
m 2 ,
m 2 ,
and since G0 D (p) = p,
m 1
m 2 .
< 1 such that there
r 2)
>
r 1
if and only if r
from (3), for all such equilibria, xN
m 1
=
= xN (
and m
).
> 0, the Nash bargaining solution xB ( r ) as de…ned in Section 4.2 is
1
r 1 2
m 2 ,
=
there exists
r 1
r r B such that xB 1 ( ) > 0, and x2 ( ) > 0. It is given by 8 r r r 1 < xB G D r 1 2) G D 1 ( ) = ( 2 1 + r
: xB ( r ) = ( 2
m 1
is a NEE if and only if
in which country 2 free rides (i.e.,
When furthermore
> 0:
m 2
which is positive. This shows that for a given
A symmetric argument shows that for a given exists a NEE
1
G D
m 1
r 2 1)
1
G D
1
+
1D
1
+
2
r 1
2
G D
r r 1+ 2
r 1
1
r r 1+ 2
+
1
D
1
2D
D
r r 1+ 2
r 1
1
:
r 1
(10)
The level of public good at xB ( r ) is r r B xB 1 ( ) + x2 ( ) = D
r 1
1 +
r 2
(11)
;
Using Notation 1, the marginal e¤ect of the representatives’ types on the welfare of their median voters is 8 < @U1 ( :
for some t 2 ( r1 ;
r 1
)
m B r 1 ;x ( ) m 1 @ r1 B r @U2 m 2 ;x ( ) @ r2
+
(
r 1)
=(
)
=(
m 2
(
r 1
+
r 2)
r 2) :
26
r 2)
(
r 1
( +
1 r 2)
( r1 ) + 1
(t)
(t))
2 r 2
r 2
:
(12)
Proof. From (4), B ( r ; x) is concave in x, so xB ( r ) is characterized by the F.O.C. @B=@x = 0, which can be rewritten as follows: for all c 2 f1; 2g, 1
(
r 1 1 r ; xB )
+
(
2
r 2 2 r ; xB )
B G0 xB 1 + x2 =
c
c r
( ; xB )
(13)
:
which implies that 1
Dividing (13) by
c=
r
c
(
1 r
; xB )
2 r
=
( ; xB )
2
(14)
:
; xB and substituting (14), we obtain (
r 1
+
r 0 2) G
B xB 1 + x2 = 1;
which implies (11). If
r 1
>
r 2,
then from (3), xN ( r ) = D
8
0; (D (1=u))0 = (G D (1=u))0 = (u), so di¤erentiating (10) and simplifying, we obtain 8 B < @x1r = 2 G D r 1 r G D 1r + ( r1 + r2 ) r1 + 1 ( r1 ) @ + 1
:
1
@xB 2 @ r2
=
1
2
G D
From the mean value theorem, G D
1
1
1
G D
r r 1+ 2
1
1
G D
r r 1+ 2
+ (
r 1
=
r 1
r 2
r 1
+
@
2
2
r B Using (11), the welfare of im c at x ( ) is
@Uc
m B c ;x @ rc
( r)
1
m c G
=
m c
Substituting (16) into (17), we obtain (12). 27
1
D
(
r 1
2
r r 1+ 2
+
r 2)
r 2
r r 2) 2
(t) for some t in ( r1 ;
Substituting the latter equality into the above two equations, we obtain 8 r r r r r r < @xB1r = 2 (t) 2 + ( 1 + 2 ) 1 + 1 ( 1 ) 2 @ 1 @xB : 2 (t) r + ( r + r ) r r = 1
(u) u and
r 1
+
r 2 ).
(16)
2
r xB c ( ), so
r @xB c ( ) : @ rc
(17)
r 1
Necessarily,
r 2;
6=
r
Let
Proof of Proposition 2.
be a CEE, and assume w.l.o.g. that r 1
because as explained in the proof of Proposition 1, if
=
r 2;
r 2:
r 1
one of the
two median voter can discontinously increase its payo¤ at the bargaining default, and thus at the bargaining outcome by slightly undercuting the type of the other country. Therefore, 1
>
r 2.
r 1
Moreover,
assumption on G,
> 0 because the best response to
= 0 is
m 1 .
Finally, under our
is continuous and positive on R+ , so (12) yields m B 2 ;x @ r2
@U2
lim r
2 !0
which implies that
r 2
r 2
( r)
m 2
=
( r1 ) > 0;
r r B > 0, so from Lemma 1, xB 1 ( ) > 0 and x2 ( ) > 0.
Under the assumption of Proposition 2, country 1 also elects the high demand representative in the NEE. So the public good level in the NEE and the CEE is D (1= and D (1= ( if
m 1
r 1
+
r 1
+
r 2 )) ;
m 1 )
respectively. Hence, the former is smaller than the latter if and only
r 2. r 1.
From Remark 1, im 1 is pivotal in country 1 when voting on m B c ;x
is a CEE, necessarily, @U1
r 1
( r ) =@ m 1
r 1 1
=
Since
r 1
> 0 and since
r
= 0. From (12), this implies that ( r1 ) + ( r1 +
2 r 2)
(t)
r 2;
so r 1
Since
+
r 2
is positive, we see that
m 1
(
=
r r 1+ 2
r 1
+
r 2)
( r1 ) ( r1 + r2 ) 1
is greater (smaller) than
hand-side of the above equation is positive (negative). Since is positive (negative) if
2
m 1 r 1
(t)
r 2
(18)
if the numerator of the right
, the high demand
m 2
r 1
implies
r 2.
>
Moreover, there also exists a NEE in which the high demand median voter elects the high m 1
demand representative. This is the case for all NEE and all CEE when m 2 .
small or large relative to r
Proof. Let r 1
that
>
r 2.
be a CEE. As argued in the proof of Proposition 2, w.l.o.g., we can assume
From (19), r 1 r 2 r 1
Substituting m 1 m 2
(
>
r 1
is su¢ ciently
r 2
>
+ (
=
( (
r 1
+
r 2)
+
1
(t)) m 1 r ( 1+
( r 2)
2 m 2
(t) +
1
( r1 ))
m 2
;
in the above equality, we obtain
r 2) + 2 r r 1 + 2)
(t) + 1 ( r1 ) >1+ + 1 (t)
( r1 ) ( r1 + r2 ) + 1
1
(t)
2
+ min 0;
1
:
1
The fraction on the right-hand-side of the above inequality is bounded away from 0 in a neighborhood of m 1
1=2,
>
m 2 ,
1
= 1=2. This implies that when
1
and
2
are su¢ ciently close to
and from Proposition 1, there exists a NEE in which country 1 elects the
high demand representative. When furthermore
m 1
is su¢ ciently large relative to
m 2 ,
from
proposition 1, in all NEE, country 1 elects the high demand representative. Proof of Proposition 3.
As in the pure public good model, we renormalize w.l.o.g.
the utility function in (5) as follows: Ui;c (x) = Step 1: For all For all
r 2
r 2
r 1
0; if
0, xB (0;
r 2)
Gc (xc ) +
i;c
r 1
m r 1 ; 2)
= xN (0;
to xN (
m r 1 ; 2 ).
= 0 cannot be a best response to Step 2: For all
xB 1
r r 1; 2
r r 1; 2
r 2
(xc0 )
is preferred by im 1 to any other r 2 ).
0; if
r 1
xc : r 1
given
m r 1 ; 2)
to xN (0;
r 2 ),
r 1
then r 1
> 0. r 2,
and
re-
and by de…nition of xN ,
B By transitivity, im 1 prefers x ( r 2
r 2,
N Since xN 1 and x2 depend only on
N spectively, by de…nition of xN , im 1 prefers x ( B im 1 prefers x (
c0 ;c Gc0
m r 1 ; 2)
to xB (0;
r 2) ;
so
for im 1 .
is majority preferred in country 1 to any other
is greater (smaller) than xN 1 (
m
) if
and x = xB ( r ) is positive (negative). 29
@2B @2B @x1 @x2 @x1 @ r1
@2B @2B r (@x1 )2 @x2 @ 1
r 1,
evaluated at
then r
=
Remark 1 still holds in this environment, so the majority preferences in country 1 coincide B r @ [U1 ( m r r r 1 ;x ( ))] . From step 1, > 0, so with the preferences of im r 1 1 1 ; 2 = 0, or equivalently @ 1
m 0 1 G1
r r 1; 2
xB 1
=1
@xB 2 @ r1
r r 1; 2
@xB 1 @ r1
r r 1; 2
m 0 1 2;1 G2
r r 1; 2
xB 2
: r r 1; 2
Comparing the above equation with (6), we see that since G01 is decreasing, xB 1 xN 1 (
m
) is of the same sign as r r 1; 2
this shows that xB 1 F.O.C.
r
@B @x
"
@xB 2 @ r1
xN 1 (
r r 1; 2 m
@xB 1
=@
r r 1; 2
r 1
r . Since xB 1 ( ) is increasing in
) is of the same sign as
@xB 2 @ r1
r r 1; 2
. Di¤erentiating the
, we obtain that at r ; xB ( r ) , # 2 @2B @2B @2B @2B @2B @xB 2 = : @x1 @x2 @ r1 @x1 @x2 @x1 @ r1 (@x1 )2 @x2 @ r1
; xB ( r ) = 0 w.r.t.
@2B @2B (@x1 )2 (@x2 )2
r 14 1,
r 1
(20)
Since B is strictly concave, the term in bracket on the left-hand side of (20) is positive, as needed. Step 3: for all
r
2 R2++ ,
@2B @2B @x1 @x2 @x1 @ r1
@2B @2B r (@x1 )2 @x2 @ 1
evaluated at
r
; xB ( r ) is positive
(negative) if u (u) is increasing (decreasing) in u: 0 Using Notation 1, for all x 2 RN + and c 2 f1; 2g, if c denote the other country, c
( r ; x) =
r c
Gc (xc ) + r c
c0 ;c Gc0
1
Gc Dc
r c
(xc0 ) +
xc
c0 ;c Gc0
Dc0
1 c0 ;c0
r c0
+ Dc
1 r c
;
so @ c = @xc and
8 < :
r 0 c Gc
(xc )
@B @x1
=
1
@B @x2
=
1
1 and
@ c = @xc0
r 0 1 G1 (x1 ) 1 r 1 ( ;x) r 0 G 1 2;1 2 (x2 ) r 1 ( ;x)
+ +
2
r c
0 c0 ;c Gc0
r 0 2 1;2 G1 (x1 ) r ( ;x) 2 r 0 G (x 2 2 2) 1 r 2 2 ( ;x)
Using (22), the F.O.C. @B=@x = 0 that characterizes xB ( r ) is: 8 r 0 1 r1 G01 (x1 ) 2 1;2 G1 (x1 ) < = r r 1 2 1 ( ;x) 2 ( ;x) : r 0 1 r2 G02 (x2 ) 1 2;1 G2 (x2 ) : = r r 1 2 1 ( ;x) 2 ( ;x) 14
(xc0 ) ,
(21)
.
(22)
(23)
Since it is quite intuitive, the formal proof of this comparative statics is omitted for the sake of brevity.
It is available from the author upon request.
30
Di¤erentiating (22) and using (21), we obtain r 0 1 2;1 G2
@2B = @x1 @x2
1
r 0 1 G1 2
(x2 ) (1
(
r
(x1 ))
r
Using (23), the above expressions evaluated at @2B = @x1 @x2
r 0 1 2;1 G2
1
@ @
0 2;1 G2
( r ; x)
1
1
Using (23) and r
(x2 ) (
1
( r ; x)
1
r@ 1@
2 r 1
1 r 1
2
1
r 0 1 G1 2
(x1 ))
( r ; x))
r 2 1;2 1
( r ; x)
1 r 1
2
r
( ; x))
;
r 0 2 G2
2
1
(24)
;
( r1 ), we obtain
(1
( r ; x))2 ( r ; x) = D1
(x2 ))
; xB ( r ) becomes
( r ; x) =
r@ 1@
r 0 2 G2 2
(x1 ) (1
(
(x2 ) (1
(
2
Di¤erentiating (22) and using (21) and @2B = @x2 @ r1
+
( ; x))
1
r 0 2 1;2 G1
(
(x2 )) 2
r 2 1;2 1 2
( r1 )
( r ; x))
x1 , the above expressions evaluated at
r 1
; xB ( r ) becomes 0 2;1 G2
@2B = @x2 @ r1
1
1
(x2 ) D1 (
1
x1
r 1
2 1
2
r
( ; x))
1
( r1 )
r r 0 0 1 2;1 G2 (x2 ) (1 1 G1 r 0 G1 (x1 ) ( 1 ( ; x))2
2
(x1 ))
(25)
:
From (25), 2 1 2
r 1 ) (1
(x1 )) = 2 1 ( ; x))
( G01 (x1 ) (
1
1(
2
r 0 1 G1 r
r
;x)
@ @
r 1
1
1 r 1
( r ; x) (1
(
1
r 0 1 G1
(x1 )) 1
( r ; x))2
(26)
Di¤erentiating (22) and using (21), we obtain ! 2 r 00 r 0 @2B (1 1 G1 (x1 ) 1 G1 (x1 )) = 1 + r (@x1 )2 ( 1 ( r ; x))2 1 ( ; x)
@2B = (@x1 )2 = Di¤erentiating (22) and using G01 (x1 ) 1
1
G001 (x1 ) 1 0 G1 (x1 ) 1 ( r ; x) G001 (x1 ) 1 0 G1 (x1 ) 1 ( r ; x) @ @
2 r 1
( r ; x) =
2
( r ; x) + (1 (
1
r 0 1 G1 2
1 r 1
(x1 )) @@
( r ; x))
31
1 r 1
r
!
; xB ( r ) becomes
(x1 ))2 2 1 ( ; x)) r 0 2 1 G1 (x1 ) @ B : 0 2;1 G2 (x2 ) @x1 @x2
(1 ( 2 1
r 2 1;2 1
2 r 0 2 1;2 G1 (x1 ) ( 2 ( r ; x))2
r 00 2 1;2 G1 (x1 ) r 2 ( ; x)
Using successively (23) and (24), the above expression evaluated at
@2B = @x1 @ r1
r 0 2 1 G1 (x1 ) @ B r r 0 1 2;1 G2 (x2 ) @x2 @ 1
r 0 1 G1 r
(27)
( r1 ), we get ( r ; x) +
r 0 2 1;2 G1 2
(
r 2 1;2 2 r
(x1 ) 2
( ; x))
( r1 )
r
Using successively (23) and (26), the above expression evaluated at G01 (x1 )
@2B = @x1 @ r1
1
( r ; x) + (1
1
1
1 1
r
( ; x)
r 1
( r ; x)
1 r 1
+
r 1
r
2 r 0 1 G1 (x1 )) 2 r 1 ( ; x))
( r1 ) (1 G01 (x1 ) (
2 1 2
1
Using (27) and (28), we get that at @2B @2B @x1 @x2 @x1 @ r1
(x1 )) @@
( r ; x)) r 0 2 1 G1 (x1 ) @ B r: 0 2;1 G (x2 ) @x2 @ 1
(
=
r 0 1 G1 2
; xB ( r ) becomes
(28)
; xB ( r ) ,
@2B @2B r = (@x1 )2 @x2 @ 1
00 1 G1 r
@ 2 B G01 (x1 ) @x1 @x2 G001 (x1 ) r1
(x1 ) 0 1 ( ; x) G1 (x1 )
@2B @x2 @ r1
:
If we substitute (24) and (25) on the right-hand side of the above equation, after simpli…cation, we obtain: @2B @2B @x1 @x2 @x1 @ r1
@2B @2B r (@x1 )2 @x2 @ 1
2 0 1 2;1 G2
r 0 1 G1 3
(x2 ) (1 (
2
r
( ; x))
1
(x1 )) (
(G01
G001 2
r
; xB ( r ) =
(x1 ))
(x1 ))
2
(G01 (x1 ))
Note that
=
G00 1 (x1 )
1 G01 (x1 )
1 G01 (x1 )
1
0
(29) x1
2
0 @ (G1
(x1 )) G001 (x1 )
2
D1
1
1
r 1 1 1
r 1
1 G01 (x1 )
, and since (D1 (1=u))0 =
r 1
1
( r1 )A :
(u) u, the mean value
theorem implies x1
D1
1 r 1
= D1 (G01 (x1 ))
D1
1
=
r 1
G01
1 (x1 )
r 1
1
(t) t;
for some t 2 ( r1 ; 1=G01 (x1 )). Therefore, (29) can be simpli…ed as follows: @2B @2B @x1 @x2 @x1 @ r1 2 0 1 2;1 G2 (x2 ) 2(
From (23),
@2B @2B 2 @x @ r (@x1 ) 2 1
(1
1(
r 0 1 G1 (x1 )
r
)(
G00 1 (x1 ) 2
)
=
G01
1 (x1 )
1
G01
1 (x1 )
2t 1
(t)
r 1 1 1
( r1 ) :
(30)
;x))3 (G01 (x1 ))
r 0 1G
r xB 1 ( ) < 1 and
1
r
r ; xB 1 ( ) > 0, so the denominator on the left-hand
side of (30) is positive. Therefore, the numerator has the same sign as the right-hand side of (30). Since evaluated at
1
+ r
2
= 1 and
r 1
< t < 1=G0 (x1 ), this implies that
; xB ( r ) is positive (negative) if u
alently, if jD10 j (p)
1
@2B @2B @x1 @x2 @x1 @ r1
(u) is increasing (decreasing), or equiv-
p2 is decreasing (increasing), which proves step 3.
Proposition 3 follows then immediately from steps 1 to 3 and De…nition 2. Proof of Proposition 4. 32
@2B @2B 2 @x @ r (@x1 ) 2 1
If G1 r
G2
; xB ( r ) ,
ln, then for all u > 0, u
@2B @2B @x1 @x2 @x1 @ r1
and xB 2 is constant in r xB 1 ( ) ! 0 (+1) as
r 1 r 1
is equal to
c
(u) = 1. From (30), this means that for all
@2B @2B r, (@x1 )2 @x2 @ 1
so from (20), for all
r 1
> 0,
@xB 2 @ r1
r 1
, at
( r ) = 0,
on R++ . By the maximum theorem, xB 1 is continuous in ! 0 (+1). Hence,
r
r 1,
and
r ! xB 1 ( 2 ) is onto on R+ . Therefore, voters
choosing their respective representative at the electoral stage is strategically equivalent to them choosing their respective policy directly. Together with Remark 1, this shows that
r
is a CEE if and only if xB ( r ) is a Nash equilibrium of a two person game in which the median voter of each country choose their respective policies. The equilibrium of that game is xN (
m
), as given by (6).
Proof of Proposition 5. Suppose that the most e¢ cient country is country 1: When transfers are available, country 1 produces all the public good and is compensated by a transfer from country 2: Thus, the availability of transfers amounts to let country 2 use the technology of country 1 in the policy stage of the cooperation scenario. Thus, in terms of payo¤, it is equivalent to assume that transfers are not available, but in the cooperative scenario, the tax price of voters in country 2 are uniformly scaled down by p1 =p2 : This transformation of tax prices does not apply in the NEE and in the bargaining default of the CEE, but in these two cases, since country 1 provides all the public good (as in Proposition 2, we assume that the same country elects the high demand representative in the CEE and the NEE), the outcome does not depend on the tax prices in country 2. Therefore, the outcome of the NEE and the CEE is the same as if transfers were not available but the tax price of the voters of country 2 were scaled down by p1 =p2 in all cases. This implies that (19) is still valid, since
m 2
does not appear in it. As argued in the proof of Proposition 2,
(19) implies that public good provision is greater (smaller) in the CEE than in the NEE if jD0 (p)j p3 is decreasing (increasing).
The Nash bargaining solution xB ( r ) is Pareto e¢ cient.
Proof of Proposition 6. Moreover, from (14), when
1
=
2;
1
r
; xB ( r ) =
2
r
; xB ( r ) : Therefore, xB ( r )
coincides with xE ( ) for that particular . To conclude, note that Proposition 2 does not depend on :
33
Proof of Proposition 7. Maximizing (7) yields 8 1 < D ;0 r r 1 1+ 2 2 r U x ( )= : 0; D 2 r r 1 + 2 1
and a non unique solution for strictly increasing in in
r 1,
r 1.
1
2.
=
2
From (31), if
1
if
1
2
>
is constant in
2
(31)
;
while xU2 ( r ) is
r 1
r U Therefore, the welfare of im 1 induced by x ( ) is strictly increasing
m so im 1 does not have a best response. A symmetric argument holds for i2 when
1
pm 1 and p2 > p2 =
m1 m p . m2 1
Therefore,
pr1 pr2 pr1 +pr2
>
m1 pm ; m2 +m1 1
the proof, note that the function g ! (m1 + m2 ) G (g)
; and from Proposition 2,
and so g CEE < g : To conclude
m1 pm 1 g is concave, with a maximum
at g ; so S g N EE Q S g CEE if and only if g N EE Q g CEE . One can readily check that all the steps in the proof of
Proof of Proposition 9.
Lemma 1, and Propositions 1 and 2 hold unchanged with G replaced by B and D replaced by S: So if
r
is a CEE such that
r 1
r 2
(the proof in the other case is identical), the level
of public bad in the NEE and the CEE are S
1
latter is smaller than the former if and only if
r 1
However, since (smaller) than
+
r 2
r r 1+ 2
m 1 .
; respectively. Hence, the
The formula (18) is still valid.
; is negative, so (18) implies that ( 1t ))t3 if j j is increasing (decreasing), or equivalently if S 0 (p)
(t) = m 1
1
1
and S
m 1
B 00 ((B 0 )
1
r 1
+
r 2
is greater
p3 is decreasing
(increasing). The following remark shows that, as claimed in Section 6.4, the symmetric equilibria characterized in Buchholz et. al. (2005) typically do not exist when spillovers are large. 34
Remark 3 Consider the same model as in Section 6.4 with partial and symmetric spillovers and transfers, as in Buchholz et al. (2005). So the utility of a voter with type
i;c
is given
2:
Suppose
by: Uc ( where
i;c )
=
i;c B
(x1 + x2 )
xc + tc ;
2 (0; 1) ; tc is the transfer received by country c; with t1 +t2 = 0; and
furthermore that S $ (B 0 )
1
1
=
does not increase exponentially, that is, lim supp!1
S(2p) S(p)
< 1—
which is the case for instance if S is isoelastic or if B (x) is a polynomial of any order n > 1. Then there exists
< 1 such that for all
for the median voters (
m
;
m
; for every symmetric pro…le of preferences
) ; there exists no symmetric CEE.
Proof. Suppose by contradiction that Remark 3 is false. Then there exists a sequence n
! 1 such that for all n 2 N;
< 1 and there exists a symmetric CEE for
n
=
n,
which
we denote ( ( ) ; ( )). As shown in Buchholz et al (2005), the CEE ( ( ) ; ( )) must satisfy their …rst order condition (12). As shown in the proof of their Proposition 4, this implies that m
2
1
; and thus that
(
n)
( )
! 0: At a symmetric CEE, transfers are equal to 0 and
the policy vector must satisfy their …rst order condition (6), which yields xB ( ( ) ; ( )) =
1 1+
S
1 (1 + ) ( )
1 (1 + ) ( )
;S
:
Therefore, the welfare of im 1 in the CEE is U1
Let r 2
m
; xB ( ( ) ; ( ))
d
=
m
B S
m
B S
1 (1 + ) ( ) 1 : 2 ( )
> 0 and consider the deviation from
= ( ) : For
close to 1,
r 1
+
equilibrium with these representatives is given by xN
d
1 (1 + ) ( ) (32)
= ( ) to
( ) is negligible relative to
1 S 1+
d
r 1
=
d
for im 1 ; while keeping
; so the noncooperative policy
; ( ) = (0; S (1= ( ))) : Since
utils are transferable, the bargaining equilibrium with these representatives xB must maximizes the sum of the representatives’utility. That is, xB max
x1 0;x2 0
d
B (x1 + x2 )
d
; ( )
; ( ) must solve
( ) B (x2 + x1 ) + x1 + x2 : 35
d
Note that the solution xB to this program is unique, and is such that xB 1 and xB 2
d
S 1= d
transfers t1
d
= ; so xB d
; ( ) and t2
; ( ) is bounded as
(B 0 )
! 1: Since
1
1
=
1=
d
the
2;
; ( ) must equalize the gains from cooperation for the
representatives, so U1
d
= U2
d
; xB
d
t1
d
; xB
; ( ) ; the above equality implies that
d
= U1
d
d
B
d
B S
S
; ( ) 1 ( ) 1 ( )
; ( )
+ t1
d
; ( )
d
1 ( )
; ( )
; (
n)
; ( )
:
are bounded, and
; ( )
1 ( )
+ ( )B S S
d
( ) ; xN
U2
d
( ) ; xN
U2 ( ) ; xB
and U2
; ( )
d
; xN
; ( )
+ O (1) 1 S ( )
+ O (1)
+ O (1) :
d
; ( ) and t1
d
B S
d
; ( ) is such that
1 ( )
S
By comparing (32) with (33), we see that the deviation from d
; ( )
d
t2
d
+ t2
B Therefore, the welfare of im 1 induced by x m
d
; xN
; xB
; ( )
d
d
U1
d
=
U1
; ( )
d
; ( ) =
; ( )
d
+ t1
( ) ; xB
From what precedes, U1 since t1
; ( )
r
1 ( )
=( (
n) ;
+ O (1) : (
n ))
to
(33) r
=
is pro…table to im 1 for n su¢ ciently large if d
1
B S
lim inf n!1
(
m
n)
B S
+S
1 (
n)
1 2 ( n)
< 1:
(34)
Since limx!+1 B 0 (x) = +1; x is negligible relative to B (x) as x tends to 1: Since limt!0 S (1=t) = 1; this implies that the second term on the numerator of (34) is negligible relative to the …rst term as n ! 1: Therefore, if B is such that lim sup p!1
then (34) holds for some
d
> 0, so ( (
B (S (2p)) < 1; B (S (p))
n) ;
(
n ))
(35)
cannot be a CEE for n su¢ ciently large.
To conclude the proof, we show that (35) is satis…ed whenever S (p) does not increase exponentially. Note …rst that since S 0 is positive and increasing, Z Z p Z p p p p 0 0 (S (p) S (0)) = S (q) dq qS (q) dq p S 0 (q) dq = p (S (p) 2 2 0 0 0 36
S (0)) :
Since (B (S (p)))0 = pS 0 (p) ; B (S (p)) = B (0) + B (S (2p)) B (S (p))
Rp 0
qS 0 (q) dq; so
B (0) + 2p (S (2p) S (0)) : B (0) + p2 (S (p) S (0))
The right hand-side of the above inequality is bounded when S (2p) =S (p) is bounded, as 1
needed. Finally, note that if B (x) is a polynomial of order n > 1; then S (p)
p n 1 ; so
1
S (2p) =S (p) !p!1 2 n 1 ,and if S is isoelastic, then S (2p) =S (p) = 2" : Remark 4 Consider the model with tax distortion discussed in Section 6.5 in which for all g
0; T (g) = g + g 2 : If G is such that jD0 (p)j p is increasing in p, then GT $ G T
is such that
1
0
DT (p) p is increasing in p.
Proof. Since D = (G0 )
1
; simple calculus shows that p ! jD0 (p)j p is increasing if and
G00 = (G0 )2 is increasing if and only if 1=G0 is convex. Elementary algebra yields that p p for all t 0; T 1 (t) = 1 + 1 + 4 t =2 ; so GT (t) = G 1 + 1 + 4 t =2 and only if
therefore, G
T 00
(GT )0
(t)
(t)
From what precedes,
p 1+ 1+4 t 2 p 1+ 1+4 t 2
G00 2
= G0
2
+p
2 1 + 4 t G0
1 p 1+ 1+4 t 2
G00 = (G0 )2 is increasing, so the …rst term on the right-hand side of GT
the above equation is increasing in t. Thus,
00
=
GT
0 2
is increasing whenever the
second term is also increasing. Using the change of variable g = term is equal to
:
1 . (g+1=2 )G0 (g)
increasing. From what precedes, increasing, which means that
g g+1=2
Since 1 G0
1 gG0 (g)
p 1+ 1+4 t , 2
the second
is increasing, (g+1=21 )G0 (g) is increasing if
is convex, so the slope
1 g 0
1 G0 (x)
1 G0 (0)
1 gG0 (g)
is
must be
is increasing, as needed.
Proof of Proposition 11. Since the Bernouilli utility function of each voter, as given by (5), is linear in her type
i;c ,
the expected utility is also linear in
i;c ;
so Remark 1 still
holds for preferences over policy lotteries. Therefore, as for pure CEE, the mixed CEE are the mixed Nash equilibria of the two-player game in which the median voter of each country c controls the type
r c
of her representative and her payo¤ is Uc
m B c ;x
( r ) : One can clearly
restrict the strategy space of each median voter to a compact subset of R+ . Moreover, since Uc ( rc ; x) is strictly concave in x for all
r c,
xB ( r ) is unique, and from Berge’s theorem, it is 37
r
continuous in
: Existence of a (possibly mixed) Nash equilibrium of this delegation game
follows then from the Debreu-Fan-Glicksberg theorem (see, e.g., Fudenberg and Tirole 1991 p. 35). Let ( 1 ;
2)
be a mixed strategy CEE, and let r 1
im 1 for a given representative’s type r 1
type in country 2. For all Z
m 0 1 G1
r r 1; 2
xB 1
+
r r 1; 2
G01
r 1
If jD10 (p)j 3, @xB 2 =@
r 1
r 2
@xB 2 @ r1
1 m 1
G (D (u)) = 2u and
r
2;
r r 1; 2
d
2
( r2 ) = 0:
the integrand must be
r r 1; 2
@xB 1 @ r1
m 1 2;1
r r 1; 2
m 1
r r 1; 2
G02 xB 2
:
r r 1; 2
1
(1=
m 1 )
r r 1; 2
= xN 1
: The
p Suppose G (g) = 2 g. Using Notation 1, for all u > 0,
(u) = 2. Substituting these identities into Lemma 1, we obtain that
2 R2+ such that
8 < U 1 :
(G02 )
m 1 .
p2 is increasing is analogous.
r 1
r 2,
0,
r r 1; 2
xB 1
@ 1, @
m 0 1 2;1 G2
The above equation implies that for some nonpositive, and since @xB 1 =@
( r1 ;
and a given distribution
in the support of
@xB 2 @ r1
1
m B 1 ;x
U2
be a CEE such that
r 1
+
( r) = 2 m B 2 ;x r 1 r
> =
r 2 2)
m 1
2
r 2)
+ m 2
r 1
(
r 1
(
r 2 2)
+
+
r 2)
m 2 ;
1 1+
r 2 1) ( 2)
; (1 +
+ (1 +
(1 +
;
r 2 1) ( 2)
r 2 1) ( 2)
:
(36)
From (19), 1 1+
m 1
Substituting (37) into (36), we obtain 8 m B < U 1 1 ;x m : U ; xB 2
r 1
(
( r) = 2 r 2.
r 2 1) ( 2)
(1 +
r r
1
=(
m 2 1 )
=2
m m 1 2
38
+
1
1 1+ 1 1+
m 2
:
(
m 2 2 )
1
1
(
m 2 2 )
(37)
:
(38)
The above equation shows that the equilibrium payo¤ of countries 1 and 2 are decreasing and increasing in
1,
respectively, as needed.
The initial assumption
r 1
>
m 1 m 2
>
2 1+
follows, we assume that
r 2 1
r c
> r
, and determine under which condition r c
that if im c has a pro…table deviation r c
m 1 m 2
is satis…ed at (37) if and only if
such that
r c
r
=
c,
2 1+
1
. In what
is a CEE. Note
then for " > 0 su¢ ciently small,
" is also a pro…table deviation, because the discontinuity in the bargaining default at =
r c
leads to an upward jump in the payo¤ of the voters of country c. So we can restrict r c
attention to deviations such that
6=
From (36), for all c 2 f1; 2g, Uc
r
c.
m B c ;x
( r ) is concave in
r c
r
on
2 R2+ :
r 1
r 2
>
.
Therefore, the F.O.C. of the CEE (which yields (37)) is su¢ cient on that domain. That is, at
r
, there is no pro…table deviation
Given
r m 1 , i2
r 1
m ( r2 ) for im 1 (i2 ) such that
clearly has no incentive to choose a type
r 2
r 1
r 2
> r 2
such that
(
>
r 1
>
r 1,
because such
r 2 ).
a deviation lowers her payo¤ at the bargaining default (country 2 is not free-riding anymore) as compared to that of country 1, and thus lowers her payo¤ at the bargaining outcome. Consider now a deviation of im 1 such that outcome xB
r r 1; 2
2 1+
1
and (37), we have 1 1+
^r
1
2 21+
This shows that the supremum of U1
1 m B 1 ;x
m 2
=
2 1+
r r 1; 2
expression on the right-hand side of (39) evaluated at
r 2
r 2:
w.r.t.
r 1
2
r
=
on 0;
r r 2; 2
r 2
is equal to the
. Substituting the latter
expression and (37) into (39), we get m B 1 ;x
sup U1 r r 2 1 [0; 2 )
r r 1; 2
=
4 m 1 1+
m 2 1
2 (1 + 2 ) ( m 2 ) : (1 + 1 )2
The comparison of (38) and (40) reveals that im 1 has no pro…table deviation and only if 3
m 2 m 1
2
4 (1 +
1)
39
m 2 m 1
+ (1 +
2 1)
0:
(40) r 1
on 0;
r 2
if
m 2 m 1
The left-hand side of the above inequality is a polynomial of order two in are
1+ 3
1
and 1 +
1.
Therefore, under our assumption that
is satis…ed if and only if
m 2 m 1
1+ 3
1
m 2 m 1
