BARGAINING OVER PUBLIC GOODS

Transcription

1 RGINING OVER PULIC GOODS JULIO DÁVIL Université catholiqe de Lovain and Paris School of Economics JN EECKHOUT Universit of Pennslvania and Universitat Pompe Fabra CÉSR MRTINELLI Centro de Investigación Económica, Institto Tecnológico tónomo de Méico bstract In a simple pblic good econom, we propose a natral bargaining procedre, the eqilibria of which converge to Lindahl allocations as the cost of bargaining vanishes. The procedre splits the decision over the allocation in a decision abot personalized prices and a decision abot otpt levels for the pblic good. Since this procedre does not assme price-taking behavior, it provides a strategic fondation for the personalized taes inherent in the Lindahl soltion to the pblic goods problem. 1. Introdction The private provision of pblic goods in general leads to inefficient allocations in a competitive market environment. This inefficienc is often attribted to a missing market. If personalized markets cold be created that individall price the pblic good for each agent, then a competitive eqilibrim cold implement an efficient allocation. For an econom with pblic goods, this otcome is known as a Lindahl eqilibrim Lindahl 1999). Tpicall, however, a Jlio Dávila, Université catholiqe de Lovain, CORE, Lovain-la-Neve, elgim and Paris School of Economics, Centre d Economie de la Sorbonne, olevard de l Hopital, Paris, France Jlio.davila@clovain.be). Jan Eeckhot, Department of Economics, Universit of Pennslvania, 160 McNeil ilding, 3718 Locst Walk Philadelphia, P , and Department of Economics, Universitat Pompe Fabra, Ramon Trias Fargas, 25-27, arcelona, Spain. César Martinelli, Centro de Investigación Económica, Institto Tecnológico tónomo de Méico, Camino Santa Teresa 930, Méico DF, Méico. Jlio Dávila gratefll acknowledges research fnding as Promoter d n M.I.S.- Mobilité Ulsse from the elgian F.R.S-FNRS. Received Jl 2, 2008; ccepted Jne 9, C 2009 Wile Periodicals, Inc. Jornal of Pblic Economic Theor, 11 6), 2009, pp

2 928 Jornal of Pblic Economic Theor Lindahl eqilibrim is deemed nrealistic becase of a serios shortcoming: in the personalized markets pon which it rests the agents are assmed to be price-takers. Unfortnatel, b the personalized natre of those markets, there is onl one single agent on the demand side in each of them, which makes price-taking behavior of this single agent an tterl nrealistic assmption. In contrast, in this paper we propose a bargaining procedre that leads to an otcome arbitraril close to a Lindahl allocation as the cost of bargaining vanishes, withot the need of assming price-taking behavior. s a matter of fact, the agents in or model have qite a lot of market power. In the case of a missing market, as is the case in the presence of a pblic good, one wa to allocate the srpls left nappropriated is throgh Coasian bargaining. s pointed ot b Coase, as long as there remain gains from trade the parties involved have incentives to get together and strike a deal. The main featre of sch bargaining is that it is decentralized no benevolent government mst intervene), and the etent to which the srpls can be allocated to the parties depends on the details of the bargaining protocol and on whether the bargaining is costl or not. The std of this tpe of bargaining in legislatres has alread been addressed in aron and Ferejohn 1989), where a model is presented in which one of the different possible was to divide a given pie is chosen b a vote according to the majorit rle. It trns ot that tpicall there eist man eqilibria for this procedre. anks and Dggan 2000) present a far more general model in which the space of alternatives is a compact, conve sbset of a mltidimensional Eclidean space. The consider arbitrar voting rles and prove the eistence of stationar eqilibria, the pper hemicontinit of eqilibrim proposals in strctral and preference parameters, and a core eqivalence reslt. While several of these bargaining setps encompass economies with pblic goods, the differ from or own setp in that ors implicitl imposes constraints on what the proposer is able to offer to other agents. We believe these constraints reflect in a natral wa the sharing of power in a bargaining sitation over both the otpt level and the financing of a pblic good. s a conseqence of imposing sch constraints on the offers, the bargaining protocol considered here attains in the limit the otcome that wold reslt from completing the missing markets. This holds withot resorting to the heroic assmption of price-taking behavior with respect to personalized prices. Specificall, we model the collective decision-making process with respect to the provision of pblic goods as a seqential bargaining game. The seqential natre of the game reflects a realistic featre of the power of setting the agenda of the negotiation. The proposal and acceptance decisions are endogenos. gents can reject a proposal and have it modified in their trn. ecase of the cost of an dela in reaching an agreement, the bargaining otcome will in general be inefficient, ths reflecting the power of setting the agenda: when there is impatience for reaching an agreement, the agent who makes an offer that is accepted will etract more rents at the cost of

3 argaining Over Pblic Goods 929 inefficienc. s the impatience or cost of a dela vanishes, those rents disappear and the allocation is efficient and independent of who had the power of setting the agenda. We show that, despite the fact that otpt decisions for pblic goods and their mode of financing are often the reslt of a political process rife with opportnities for strategic behavior, the Lindahl allocations can be implemented withot assming price-taking behavior with respect to personalized prices. This is obtained precisel throgh strategic bargaining b the parties over the financing of the pblic good. 1 In order to show this, we take here a first step toward modeling the political process behind otpt and financing decisions with regard to pblic goods as a seqential bargaining game of complete information. We consider an econom with an finite nmber of pblic goods and private goods not necessaril the same nmber of each). There are two agents in this econom who take trns alternatingl as in Rbinstein 1982) to propose a maimm level of provision of each of the pblic goods and a wa to split between them the cost of financing an level of the pblic goods to be provided p to the proposed maima this amonts to proposing personalized prices or taes). The other agent can then either accept or reject the proposal. In case of acceptance, this other agent chooses the amont of each pblic good to be provided sbject to the maimm amont offered in the proposal). Each agent pas for the pblic goods according to the personalized prices agreed pon. The levels of pblic goods and their financing are fied thereafter so the game is effectivel over. If instead the other agent rejects the proposal, then it is his trn to make a proposal himself of new maimm amonts and personalized prices, and so on. Within this setp we show that, as the discont factors of each agent in the econom converge to one, the allocation of an stationar sbgame perfect eqilibrim converges to a Lindahl eqilibrim allocation. In particlar, we show first that, for infinitel patient agents i.e., for discont factors eqal to 1), the set of Lindahl allocations coincides with the set of stationar sbgame perfect eqilibrim allocations of the alternating-offer bargaining game described above. Then we establish the pper hemicontinit of the correspondence mapping, to each pair of discont factors, the set of stationar sbgame perfect eqilibrim allocations of the bargaining game, and in particlar its pper hemicontinit for discont factors δ = δ = 1. The conclsion then follows from these two reslts. 1 The mied competitive mechanisms proposed in Groves and Ledard 1977) obviosl can efficientl allocate private and pblic goods to coincide with the Lindahl allocation. However, those direct mechanisms both rel on a centralized mechanism designer and the ma involve complicated mechanisms. The virte here is that an efficient allocation of private and pblic goods is obtained throgh a decentralized bargaining procedre. It is simple: it relies eplicitl on personalized contribtions taes) and incorporates the notion that agenda setters have power to etract more rents when bargaining is costl.

4 930 Jornal of Pblic Economic Theor In this simple setp, nanimit plas an important role in or modeling. In effect, a ta proposal can be adopted onl if it is acceptable to ever agent. This has the virte of conferring a Wicksellian character to the Lindahl soltion to the pblic goods problem. In a classical contribtion to the theor of pblic finance, Wicksell 1896) proposed nanimit as the criterion for jst taation. The game we propose incorporates the need for achieving consenss to effect ta proposals that to some etent ma characterize bdgetar procedres in political regimes with mltiple checks and balances or in parliamentar democracies withot a majorit part. Inefficiencies associated with bargaining are shown to disappear as agents become increasingl patient. Ths, a Wicksellian procedre trns ot to be consistent with a Lindahl reslt. Of corse, this reslt depends on the assmption of complete information. The reslts of this paper parallel reslts obtained in Dávila and Eeckhot 2008) for economies withot pblic goods. In that paper, the athors provide a bargaining fondation for Walrasian eqilibria in a two-agent echange econom in which agents are not price-takers. The strateg of the proofs here is similar, althogh the presence of pblic goods in coeistence with private goods introdces significant technical differences for the proofs provided in Dávila and Eeckhot 2008) not to go throgh straightforwardl. In particlar, the pblic goods problem agments the private goods problem in two was. First, in the private goods econom, all prices are common across agents and consmptions for each good are agent specific. In the pblic goods econom, we add to the previos setp prices that are agent specific for each pblic good, while the consmption of each pblic good is common across agents. Second, the presence of pblic goods natrall introdces prodction in the econom. In this sense, the reslts provided here do not follow as a corollar from those in Dávila and Eeckhot 2008). More importantl, we think that looking for a bargaining fondation for the Lindahl eqilibrim is an even more natral qestion to address given that the decision abot the provision of pblic goods tpicall takes place within legislatres. From this perspective, we think that the proposed setp captres well the problem faced b two parties with known preferences that tr to reach an agreement on the provision of some pblic goods. Moreover, this bargaining fondation addresses one of the main critiqes of the Lindahl eqilibrim notion, namel the inevitable market power that the assmption of personalized markets for the pblic goods entails. We discss possible etensions of or reslts in the final remarks. 2. The Model We consider a pblic good econom consisting of two agents and, and an arbitrar nmber n + m of goods: n private goods 1,..., n and m pblic goods 1,..., m. The agents are infinitel lived and time is discrete. For each agent i =,, let i R n + be i s consmption of private goods, and let Rm +

5 argaining Over Pblic Goods 931 be their common consmption of pblic goods. The agents have preferences over the two tpes of goods represented b standard tilit fnctions, ) and, ), that is to sa nonnegative, C 2, monotone, differentiabl strictl qasi-concave, 2 and well-behaved at the bondar of R n+m +. 3 This good behavior at the bondar garantees the possibilit of generating a srpls b the prodction of pblic goods, i.e., at a Lindahl eqilibrim necessaril > 0. The agents are endowed with amonts e i with total endowment e = e + e ) of the private goods. s a normalization, we assme the initial amont of the pblic goods is zero. linear technolog M R n m allows to prodce each pblic good j b means of the private goods, reqiring m ij nits of private good i for each nit of j, for all i = 1,...,n. We consider an alternating-offers bargaining game. In an given period prior to an agreement, an agent i makes an offer consisting of a vector p i = p i, p i ) of prices for the other agent in terms of, sa, the private good 1 ) and an pper bond q i on the other agent s contribtion of private goods to the provision of pblic goods or, eqivalentl, to the provision of pblic goods itself. fter receiving an offer, the other agent can either accept it or reject it. In case of acceptance, the accepting agent chooses his consmption of private goods and the qantities of pblic goods to be provided sbject to the accepted prices p i and pper bond on trades q i. This procedre is repeated ntil a proposal is accepted. The tilit of each agent i is disconted in each iteration b a positive discont factor δ i not bigger than 1. The tilit of never reaching an agreement is The Stationar Sbgame Perfect Eqilibrim llocations We consider first the stationar sbgame perfect eqilibria withot dela of the bargaining game, that is the stationar sbgame perfect SSP) eqilibria in which no agent has incentives to reject the offer received. Indeed, as arged in Section 6, there does not eist an SSP eqilibrim with dela if the agents are impatient, that is to sa if δ,δ < 1. The SSP eqilibria with no dela are characterized b a pair of offers p, q ) and p, q ), consisting each of a vector of prices and a maimm amont for the contribtions of private goods, sch that p, q ) maimizes the tilit that obtains from s immediate acceptance, sbject to the constraint that it is indeed in s interest to accept s offer, and similarl for p, q ). Formall, p, q ) solves ma e p, q ) M ỹ p, q ), ỹ p, q ) ) 1) 2 That is to sa, for all i =,, for all, ) R n+m, ++ D2 i, ) is negative definite in the space orthogonal to D i, ). 3 That is to sa, for all i =,, and all j, k, it holds tre that lim h 0 D h i, ) =, and lim k 0 D k i, ) =.

6 932 Jornal of Pblic Economic Theor sbject to the constraint p, q ), ỹ p, q )) δ e p, q ) M ỹ p, q ), ỹ p, q ) ), 2) where p, q ), ỹ p, q ) is the soltion to ma, ) p e, ) 0 e, ) q, 3) where stands for the Eclidean norm, 4 and smmetricall for p, q ). Note that an SSP eqilibrim withot dela can eqivalentl be characterized b the allocations effectivel offered b the agents. In effect, we establish in Lemma 1 in ppendi that, conditional on immediate acceptance, an offer b of p, q ) amonts to offering the bndle p, q ), ỹ p, q )), which is characterized b satisfing the condition, ),, ) ) ) e. 4) Conversel, an bndle, ) satisfing the previos ineqalit is a soltion to s problem above for some offer p, )b. s a conseqence, an SSP eqilibrim withot dela can also be characterized b allocations,, ) and,, ), proposed b and respectivel, sch that,, ) solves ma, ), ),, ) ) ) e, ) δ, ) + + M = e + e 5) 4 This choice is done for analtical convenience. s a matter of fact, being in a finitedimension vector space, all norms are eqivalent, so that the choice of the norm in Eqation 3) is inessential.

7 argaining Over Pblic Goods 933 given,, ) and,, ) solves ma, ), ),, ) ) ) e, ) δ, ) + + M = e + e 6) given,, ). t this point, it ma be worth to clarif the main difference between Rbinstein s setp and ors. Note that in choosing their offers the agents are constrained not onl to garantee each other the contination vale at each stage, bt also that this vale is compted onl for prices and qotas that are sbgame perfect. conseqence of introdcing this additional constraint is that, when the agents are impatient, there ma remain srpls still to be distribted at an SSP eqilibrim. In Rbinstein s setp this is rled ot since there, all the feasible tilit profiles are available for negotiation. Here the tilit profiles available are constrained to be those attainable as SSP eqilibrim otcomes of bargaining over prices and maimm provisions of pblic goods. In fact, withot constraint 4) in problem 5) and the analogos constraint in problem 6), each proposal wold be Pareto efficient like in Rbinstein. In Section 6, we show that sch inefficienc vanishes as the agents become arbitraril patient. 4. The Lindahl llocations Lindahl eqilibrim consists of a vector of private good prices p, a vector of personalized prices for the pblic goods p, p, and an allocation,, ) sch that 1) the allocation is feasible, i.e.,,, and + + M = e + e. 7) 2) each agent s consmption i, ), i =,, maimizes his tilit given p and p i, i.e., i i, ), i i, ) ) = λ i p, p i ), p i e i ) + p i = 0, 8) for some λ i > 0.

8 934 Jornal of Pblic Economic Theor Figre 1: Lindahl eqilibrim in Kolm s triangle. 3) the prodction of pblic goods maimizes profits, which implies, when 0, 5 p + p p M = 0. 9) Therefore, a Lindahl eqilibrim allocation,, ) is a feasible allocation sch that, ) and, ) are colinear, and it allocates to each agent i =, his demand i, ) at his personalized relative prices, implicitl eqal to the marginal rates of sbstittion determined b his marginal tilities at i, ), i.e., ) i i i, ), i e i i, )) = 0. 10) For the case of one private good and one pblic good prodced with a 1-nit linear technolog, Lindahl eqilibrim allocations can be represented in a Kolm triangle, the pblic goods eqivalent of the Edgeworth bo of a private goods echange econom see Figre 1; Thomson 1999, provides a sefl presentation and discssion of the Kolm triangle). The Kolm triangle assmes a linear prodction technolog reqiring one nit of the private good for each nit of the pblic good. The height of the triangle represents the initial total endowment of the private good. The orthogonal distance from an point within the triangle to each of its sides represents each agent s allocation of the private good and ), while the vertical distance to the base of the triangle represents their common consmption of the pblic good. Ths, at the initial endowment e, e ) there is no provision of the pblic good. 5 The prodction of an amont 0 for some pblic good cannot be an eqilibrim otcome given that the tilit fnctions behavior at the bondar garantees interior soltions for positive prices.

9 argaining Over Pblic Goods 935 The personalized prices p and p represent the terms of echange of one nit of pblic good for p nits of the private good from agent and p nits of private good from agent. balanced bdget where the total contribtions eqal the total cost of prodction of the pblic good implies that for all, I i=1 p i =. For a given price schedle p, the offer crve OC i gives the optimal amont of pblic good and private good demanded b agent i at those terms of trade. n intersection of the offer crves represents then optimal qantities of the pblic good and the private good consmed b the agents given a vector of personalized prices that balances the bdget. This corresponds to a Lindahl eqilibrim allocation. There are of corse other efficient allocation represented b the Pareto set P, bt the are not attainable b means of price schedles starting from the initial endowment e, e ). 5. SSP Eqilibrim llocations are Lindahl llocations when δ = δ = 1 First we show that for infinitel patient agents, that is, when the discont factors δ and δ are 1, the Lindahl eqilibrim allocations, and onl these allocations, are offered at a stationar sbgame perfect eqilibrim of the bargaining game above. For discont factors eqal to one, the onl SSP eqilibrim allocation is the Lindahl allocation. This is qite remarkable and in sharp contrast with the standard Rbinstein bargaining eqilibrim, where the entire Pareto frontier is an eqilibrim when there is no disconting. THEOREM 1: When δ = δ = 1, at ever SSP eqilibrim with immediate acceptance the agents offers lead to the same allocation. Moreover, this allocation is a Lindahl eqilibrim allocation. Conversel, ever Lindahl eqilibrim allocation is the allocation offered b the two agents at some SSP eqilibrim withot dela. Proof: Let,, ) and,, ) be the feasible allocations reslting from s resp. s) acceptance of s resp. s) offer of price and maimm provisions of pblic goods at an SSP eqilibrim with immediate acceptance for infinitel patient plaers. That is, let,, ) and,, ) be sch that,, ) arg ma, ),,, ),, ) ) ) e, ), ) 11) + + M = e + e given,, ),

10 936 Jornal of Pblic Economic Theor and,, ) arg ma, ),,, ),, ) ) ) e, ), ) + + M = e + e given,, ). 12) Then, from the from the first-order conditions, there eist mltipliers λ,µ,λ,µ and ν,ν sch that ), 0 ) + λ, 0 ), ), 0 ) ) + µ, +, e ) ) +, ) ), +, e ) ) +, e i n + νi e i = 0, 13) i=1 m t i where mi t is the ith row of M transposed as a colmn. Eqivalentl, eliminating the mltipliers νi, ) ), ) ) M t,, ) ) = λ, ), ) ) + µ, +, e ) ) + ), ) ), +, e ) ), + 14),

11 argaining Over Pblic Goods 937 and similarl ) ), ) ) M t,, ) ) = λ, ), ) ) + µ, +, e ) ) + ), ) ), +, e ) ). 15) +, ssme that,, ),, ). Since at an SSP eqilibrim with δ = δ = 1, the constraints, ), ), ), ) 16) are binding, 6 and hence both allocations are on the same indifference srface for both agents, 7 then none of these two allocations can be efficient while being different. Therefore the efficienc condition that ) ), ) and, ) 17), ) M t, ), ) are collinear cannot hold neither at,, ) nor at,, ). In particlar, there eist h, k sch that Sppose k, ) h, ) n m ik ) ) i, k, i=1 h, ). 18) n ) m k ik ) ), i, k, i=1 ) < ). 19) h, h, We claim that if Eqation 15) holds, then Eqation 19) cannot be satisfied for nonnegative mltipliers. similar argment shows that if 6 For instance, since,, ) satisfies D, ) e, ) agent will never choose at eqilibrim to let ask for a provision of pblic goods bigger than the one necessar to attain s demand at the implicit prices), then, ), ) holds as well. 7 For this step to hold tre, it is crcial that δ = δ = 1.

12 938 Jornal of Pblic Economic Theor the ineqalit is reversed, then Eqation 14) cannot be satisfied for nonnegative mltipliers.) In effect, since λ,µ ) mst solve Eqation 15), then it mst also solve ) h, n m ik ) ) i, k, i=1 ) ) = λ h, ) k, ) ) + µ h, + h h, e ) ) + ) k h, ) ) k, + h k, e ) ). + k k, 20) In particlar, µ is eqal to ) ) h, h, ) n k, m ik ) ) i, k, i=1 ) ) ) h, h, + h h, e ) ) +. k h, ) ) ) k, k, + h k, e ) ) + k k, 21) Since µ, and the nmerator is strictl positive according to Eqation 19), so mst be the denominator. t the denominator is eqivalent to ) ) ) k, h, [ ) ) ) ) h, h h, ) )] h k, h e h ) + ) ), k, k h, k k, k 22) which is negative since the first scalar prodct is nll and the Hessian of at, ) is definite negative in the space orthogonal to the gradient of at, ) and hence also to an 0,...,0, h e h, 0,...,0, k, 0,...,0) orthogonal to, ),, )), i.e., for all h e h, k) sch that h, ) h e h ) + k, ) k = 0. 23) Note that in this case k, ), h, )) is collinear to h e h, k) p to a positive constant.) It follows that if δ = δ = 1, then at an SSP eqilibrim with immediate acceptance the two agents offer the same allocation.

13 argaining Over Pblic Goods 939 Let,, be the common allocation offered at an SSP eqilibrim when δ = δ = 1. This allocation is therefore sch that, ) and, ) are collinear, it is interior, in particlar 0, given the bondar behavior of the tilit fnctions i, and + + M = e + e, 24) and, ) arg ma e ˆ M ŷ, ŷ) ˆ, ŷ), ˆ, ŷ) ) ) ˆ e ŷ ˆ, ŷ) e M, ) given,,, ) arg ma e ˆ M ŷ, ŷ) ˆ, ŷ), ˆ, ŷ) ) ) ˆ e ŷ ˆ, ŷ) e M, ) given,. 25) 26) In effect, since δ = δ = 1, shold, ) and, ) not be collinear, then there wold remain enogh gains from trade to be eploited for an of the two agents to deviate making a Pareto-improving offer that the other agent wold accept, contradicting that it is a SSP eqilibrim allocation. Sppose also that, ),, ) ) ) e > 0. 27) Then, since both and are concave,, ) arg ma e ˆ M ŷ, ŷ) ˆ, ŷ) e M, ) given,. Therefore, there eists λ>0sch that ) ), ) = λ, ), ) M t, ), ) ) ) In 0, ) = λ, M t I m, ) 28) 29)

14 940 Jornal of Pblic Economic Theor or eqivalentl,, ),, ) ) = λ, ),, ) ) ) I n M. 30) 0 I m t from Eqations 24), 27), and 30) we get, ),, ) ) ) e < 0, 31) which contradicts Eqation 26). Therefore,, ),, ) ) ) e = 0, 32) and similarl, ),, ) ) ) e = 0. 33) That is, the allocation proposed b both agents at an SSP eqilibrim is on both agents offer crves and sch that, ) and, ) are collinear and, so that it is a Lindahl eqilibrim allocation. 8 Conversel, let,, be the allocation of a Lindahl eqilibrim, i.e., an allocation sch that + + M = e + e, 34) and Let and, ),, ) ) ) e = 0, 35), ),, ) ) ) e = 0. 36) p, q ) =, ),, ) ), ) 37) p, q ) =, ),, ) ), ), 38) 8 The additional condition for it to be a Lindahl eqilibrim allocation, namel that p + p p M = 0, follows from the satisfaction of the bdget constraints with the prices p =, ), p =, ), and p = µ, ), µ > 0 being the scalar for which, ) = µ, ) given their colinearit) and the fact that 0.

15 argaining Over Pblic Goods 941 so that = p, q ), = p, q ), and ỹ p, q ) = = ỹ p, q ). It can be easil checked that the Lindahl allocation,, ) is the otcome of the following SSP eqilibrim profile of strategies: 1) offers p, q ) whenever he has the opportnit to do so, and accepts onl offers p, q ) sch that p, q ), ỹ p, q )) δ p, q ), ỹ p, q ) ). 39) 2) offers p, q ) whenever he has the opportnit to do so, and accepts onl offers p, q ) sch that p, q ), ỹ p, q )) δ p, q ), ỹ p, q ) ). 40) 6. Convergence of SSP Eqilibrim llocations to Lindahl llocations as δ,δ 1 In the relevant case in which bargaining entails some frictions, so that the factors b which the agents discont ftre tilities are strictl smaller than 1, an SSP eqilibrim allocation is still arbitraril close to a Lindahl allocation if agents are patient enogh. This is a conseqence of the fact that the correspondence of SSP eqilibrim allocations is pper hemicontinos with respect to the agents discont factors, as Theorem 2 net establishes. In the proof it sffices to consider onl SSP eqilibria in which the first offer gets accepted immediatel, since there does not eist an SSP eqilibrim with dela whenever δ,δ < 1. 9 THEOREM 2: as δ,δ 1. Ever SSP eqilibrim allocation converges to a Lindahl allocation Proof: Given that there is no SSP eqilibrim with dela if δ,δ < 1, it is sfficient to show the pper hemicontinit of the correspondence of allocations of SSP eqilibria with immediate acceptance. 9 Consider a candidate SSP eqilibrim p, q ) and p, q ) in which, for instance, rejects and accepts. Let,, ) be the feasible allocation reslting from s acceptance of s offer of price and maimm provisions of pblic goods. Sppose that,, ) is efficient. Then cold deviate offering himself s offer instead, since will accept it anwa later, saving the cost of dela in reaching an agreement. Since,, ) is efficient, there is a price vector sch that if offered this price and a nonbinding qota, wold choose this allocation. Sppose that,, ) is inefficient. Then there is room for deviating and making an offer that is efficient and Pareto improving with respect to,, ) and that wold accept.

16 942 Jornal of Pblic Economic Theor Consider then the correspondence defined as,,,,, ; δ,δ ) = arg ma 0,,, ) arg ma 0,,, ) ) e ) e D, ) D, ), ) δ, ), ) δ, ) + + M = e + e + + M = e + e given,, ) given,, ), 41) where D i i, ) stands for i i, ), i i, )), for all i =,. Note that, b the Theorem of the Maimm, arg ma 0,,, ) ) e D, ), ) δ, ) + + M = e + e given,, ), 42) is a compact-valed, pper hemicontinos correspondence that depends eplicitl on,, and δ bt also triviall on and,,,δ. 10 nd similarl for agent s problem. Therefore, is the Cartesian prodct of compact-valed, pper hemicontinos correspondences, and hence it is compact-valed and pper hemicontinos itself Since depends continosl on, and also triviall on,,,,,,δ,δ, and the correspondence defined b the constraints { ),,,,,,δ,δ ) =,, ) R 2n+m D e, ), ) δ, ) } + + M = e + e is continos and compact-valed. 11 See Lemma 1 in Dávila and Eeckhot 2008).

17 argaining Over Pblic Goods 943 Consider the correspondence Ɣ of fied points of as a fnction of δ,δ, i.e., { Ɣδ,δ ) =,,,,, ) R 22n+m),,,,, ),,,,, ; δ,δ )}. 43) Since is compact-valed and pper hemicontinos, then the correspondence mapping the fied points of,,,,,,δ,δ ) to each pair δ,δ ) is pper hemicontinos. 12 Finall, note that Ɣ is the correspondence of SSP eqilibrim allocations withot dela). Since this correspondence is pper hemicontinos, in particlar at δ,δ ) = 1, 1) and, according to Theorem 1, Ɣ1, 1) is the set of Lindahl allocations, then the claim follows. 7. Final Remarks In this paper, we consider the collective decision problem faced b two agents who mst agree on the otpt level and financing of pblic goods, and we show how bargaining over ta schedles can provide a fondation for Lindahl eqilibrim allocations. We think that this reslt provides new insights into the long-standing problem of finding strategic fondations for the Lindahl eqilibrim. In particlar, it addresses the criticism that in the presence of personalized Lindahl prices agents cannot be price-takers, b eplicitl letting the agents offer prices. We limit the analsis to the two-agent, complete information case. We think that this case captres the essentials of the actal bargaining taking place in legislatres over the fnding of pblic goods. Eamples of interesting and important instances of two-part bargaining are negotiations in legislatres between government and opposition, between territorial entities in federal contries, or in bilateral international agreements. natral qestion is how to etend or bargaining game to a sitation with more agents, as in the models considered b Harrington 1989), aron and Ferejohn 1989), and anks and Dggan 2000). Consider an n 3 person societ in which the agents alternate in the role of proposer in a fied order. proposal consists as in this paper of a price vector and a qota, and each agent other than the proposer seqentiall decides whether to accept or not the proposal and in case of acceptance chooses a maimm acceptable level of pblic goods. We conjectre that a characterization of allocations corresponding to SSP eqilibria withot dela analogos to that given b 12 See Lemma 3 in Dávila and Eeckhot 2008).

18 944 Jornal of Pblic Economic Theor Eqations 5) and 6) can be obtained in this setting, so that Theorems 1 and 2 still hold. We leave this qestion open for ftre research. ppendi LEEM 1: If, ) solves ma, ) p e, ) 0 e, ) q, where stands for the Eclidean norm, then ) e D, ), 1) 2) and conversel, if satisfies Eqation 2), then there eist p,q for which, ) solves Eqation 1). Proof: ssme q > 0, otherwise Eqation 2) is triviall satisfied. Since, ) solves Eqation 1), then there eist λ, µ sch that D, ) = λp + µ e, ) λp e, ) = 0 µ[ e) t e) + t q 2 ] = 0. 3) Therefore, D, ) e, ) = λp e, ) + µ e, ) t e, ) = µ e, ) t e, ). 4) Conversel, if, satisf D, ) e, ) = 0, let λ = 1 µ = 0 p = D, ) q 2 = e, ) t e, ). 5)

19 argaining Over Pblic Goods 945 If, satisf D, ) e, ) > 0, let λ>0 D, ) e, ) µ = e, ) t e, ) p = 1 [ ] D, ) e, ) D, ) e, ) λ e, ) t e, ) q 2 = e, ) t e, ), 6) where the weak ineqalit follows from the fact that if a R n ++ and b / Rn + are sch that ab > 0, then 13 a a b b b b. References RON, D. P., and J.. FEREJOHN 1989) argaining in legislatres, merican Political Science Review 83, NKS, J., and J. DUGGN 2000) bargaining model of collective choice, merican Political Science Review 94, DÁVIL, J., and J. EECKHOUT 2008) Competitive bargaining eqilibrim, Jornal of Economic Theor 139, GROVES, T., and J. LEDYRD 1977) Optimal allocation of pblic goods: soltion to the Free Rider problem, Econometrica 45, HRRINGTON, J. 1989) The advantageos natre of risk aversion in a three-plaer bargaining game where acceptance of a proposal reqires a simple majorit, Economics Letters 30, LINDHL, E. 1919) Jst taation: positive soltion, reprinted in part in Classics in Theor of Pblic Finance, R. Msgrave and. Peacock, editors 1958), New York: McMillan, pp RUINSTEIN,. 1982) Perfect eqilibrim in a bargaining model, Econometrica 50, THOMSON, W. 1999) Economies with pblic goods: n elementar geometrical eposition, Jornal of Pblic Economic Theor 1, WICKSELL, K. 1896) new principle of jst taation, reprinted in part in Classics in Theor of Pblic Finance, R. Msgrave and. Peacock, editors 1958), New York: McMillan, pp See Lemma 2 in Dávila and Eeckhot 2008).