훐맺맜샭췸⁷睷桥浡湡来渠훐맺ퟮ듳뗄웳튵맜샭풴훐탄TheDesignofTransnationalPublicGoodMechanismsforDevelopingCountries∗Jean-JacquesLaffon†tandDavidMartimort‡RevisedVersion:October13,,:TransnationalPublicGoods,:H41,D82.∗.†UniversitdeToulouse,(IDEI,GREMAQ,ARQADE)andUniversityofSouthCaliforniaatLosAngeles.‡UniversitdeToulouse(IDEI,GREMAQ)
훐맺맜샭췸⁷睷桥浡湡来渠훐맺ퟮ듳뗄웳튵맜샭풴훐탄1IntroductionItiswellrecognizedbynow,bothamongpractitionersandscholars,%%.Despitethisimportance,,,CalderonandServen(2002)showthat,from1980to1997,theLatinAmericainfrastructuregaprelativetoEastAsiagrewby40%forroads,70%fortelecommunicationsandnearly90%%oftheLatinAmerica’,andgiventhescarcityofpublicfundsinLDCs,,,,-ernmentsorconcessionnaries,thoserenegotiationshaveoftencreatedpublicoppositiontowhatissometimespresented,forwaterconcessionsinparticular,,,suchasthoseofCentralAmericaoroftheMediter-aneanRim,,,Calderon,EasterlyandServen(2002)andCalderonandServen(2002)“Whentimesarehard,capitalspendingoninfrastruc-tureisthefirstitemtogo...Despitethelong-termeconomicscostsofslashinginfrastructurespending,governmentsfinditlesspoliticallycostlythanreducingpublicemploymentorwages.”WorldDevelopmentReport1994,,LaffontandStraub(2002).,thatdamcorrespondsto25%ofenergy2
훐맺맜샭췸⁷睷桥浡湡来渠훐맺ퟮ듳뗄웳튵맜샭풴훐탄,underasymmetricinformation,,,,,(ormorepreciselytheirgovernments)%,,,Morocco,Tunisia,LybiaandEgypttodealwithmarinepollution3
,withinagivencountry,,,,maybedesignedundertheaegisofaninternationalagency(thereafterIA),defineseachcountry’,thereislittlehopetoseetransnationalinfrastructuresbeingpricedassimple“localpublicgoods”,,,’,acountry’
’,,oratamoredisaggregatedlevel,-beingofthepoorestagents,,,,:ontheonehand,thetraditionalanalysisofpublicgoodmechanismsunderinformationalconstraints;ontheotherhand,(1973),(1977)(1976)andWalker(1980)’AspremontandGrard-Varet(1979)andArrow(1979)showedthat,byweakeningtheequilibriumconceptfromdom-inantstrategytoBayesian-Nash,,LaffontandMaskin(1979),someauthorshaveturnedtothesecond-bestproblemofdesigningacollectivedecisionmechanismwhichmaximizesexpectedsocialwelfareunderincentive,(1999),Rob(1989)andMailathandPostlewaite(1990).,transnational(ormoregenerallyglobal)publicgoodshavebeenanalyzedingametheoreticenvironmentsbyArceandSandler(2000)andSandler(1998,2000).ThefocusofthisliteratureisontheroleofIAsasmediatorswhoprovide5
communicationdevices,,,(i=1,2),acommonpowergrid,−1nature:,-tityxofprivateconsumption8andthecorrespondingpaymentmadetcanbewrittenas:U=θv(x)−t,(1)wherev(·)isincreasingandconcavewiththeInadaconditionsbeingsatisfied(v′>0,v′′<0,v′(0)=+∞,v′(+∞)=0).
Theparameterθrepresentstheintensityofanagent’,θmayreflecttheagent’,richerpeoplearealsothemosteagertousetheinfrastructure.θbelongstothesetΘ={θ,θ¯}(wedenoteby∆θ=θ¯−θthespreadofthedistribution)withrespectiveprobabilities1−νandν.Wewillassumethatθ>ν1−ν∆θ,itisusefultodefinethefirst-bestsurplusofanagentwithtypeθconsumingxunitsofservicesprovidedbytheinfrastructureasS(θ,x)=θv(x)−cx−,efficiencyischaracterizedbythefollowingfirst-bestconsumptionsx∗(θ)foranagentwithtypeθ:θv′(x∗(θ))=c.(2)GimaximizesaweightedsumofCiinhabitants’(θ)theutilityofanagentwithpreferencesθ,Gi’sobjectivefunctionis:αiνUi(θ¯)+(1−αiν)Ui(θ)αi<1isanon-negativeparameterrepresentingGi’αiissmaller,αi=0,αiiscloseto1,-offbetweenefficiencyandredistributiveconcernsofthegovernment,itisusefultorewritethisobjectivefunctionas:()νUi(θ¯)+(1−ν)Ui(θ)−ν(1−αi)Ui(θ¯)−Ui(θ).(3)Countriesmaydifferwithrespecttotheweightsαα{α,α¯}(where0≤α<α¯<1)withrespectiveprobabilities1−∆α=α¯−α,wewillassumethatα>q1−∆α,(1999)(1983).7
::Agentscannotbediscriminateddirectlyonthebasisoftheirtasteswhichistheirprivateinformation,’sconcernsforredistributionleadustodefineinaratherstandardway11thevirtualsurplusS˜(θ,x,αi)ofanagentwithtypeθconsumingxunitsasrespectivelyhistruefirst-bestsurplusifθ=θ¯andamodifiedsurplus()S˜(θ,x,αi)=θ−νi)∆θv(x)−1−(1−αcx−Fν2whenθ=θ.Notethatthevirtualsurplusdependsexplicitlyonααiwereabletoself-financetheinfrastructure,:()θ−ν1−(1−αi)∆θv′(x˜(θ,αi))=c,(4)νwhereasarichonewouldconsumethefirst-bestquantityx˜(θ¯,αi)=x∗(θ¯).Withinanygivencountry,,α¯α¯-countryfavorsalessegalitariandistributionofutilitiesthantheα:ThegovernmentandtheagentswithinagivencountryCihaveprivate2informationonα,,thegovernmentofagivencountrymaybemoreorlessbiasedtowardstherichdependingonitsdegreeofcorruptionorthelattergroup’,thelinearspecificationofthegovernment’spreferencescanbeviewedasatractablewayofintroducingatrade-off11SeeLedyardandPalfrey(1999)α,inequilibrium,theinformedprincipalproblemwouldnotbeanissuebecauseweareinacontextofprivatevalueswheretheprincipal’(1990),thesamecontractualoutcomeasundercompleteinformationonα
betweenefficiencyandredistributioninamodelwithquasilinearutilityfunctionsasarguedbyLedyardandPalfrey(1999).,,,weshowthatabenevolentgovernmentputtingapriorianequalweightonalldifferenttypesofagentsinitsobjectivefunctionwouldactuallybehaveasin(3)becauseofasymmetricinformationontheagents’αireflectstheshadowcostofthecountry’α,wewillthusassimilatearich()countryashavingpreferencesforredistributiongivenbyα¯(resp.α),aα¯-countryislessconcernedwithredistributionthanaα,agentsineachcountryCilearntheirindividualtastesθ.GovernmentGiandagentsinCilearnalsothepreferencesforredistributionα,,,(IA),thedegreeofinequalityaversionofthegovernmentgenerallydependsonthebudgetconstraintitfaces(seeMartimort(2001)foramodelalongtheselines).Wheninequalityaversiondecreaseswithwealth,()countrycanthusbeviewedasonehavingaparameterα(resp.α¯).14LiketheWorldBankanditsregionalcounterparts,theAsianDevelopmentBank,theInter-AmericanDevelopmentBank,
“mediator”inthesenseofMyersonandSattherwaite(1983).Itcollectscontributions,,16Accordingly,(αi=1),-tierasymmetricinformationpostulated(bothontastesbutalsoonthecountries’preferencesforredistribution),amechanismstipulatesfirsttheprobabilityp(·)ofbuildingtheinfrastructureandtheoverallcontributionofeachcountryTi(·)asafunctionofthereportsαˆ=(αˆ1,αˆ2),giventhosereports,themechanismstipulatesalsowhatshouldbethepricepaidti(·,αˆ)andtheconsumptionsxi(·,αˆ)ofeachagentinCiasafunctiono{fhisownreportθˆonhistast}(αˆ);Ti(αˆ);ti(θˆ,αˆ);xi(θˆ,αˆ).UsingtheRevelationPrinciple,,thepriceschargedtobothtypesofconsumersmustsatisfyforanyprofileαofpreferences:Ti(α)=p(α)E(ti(θ,α)−cxi(θ,α)).(5)θ:Giventhesymmetryofthemodel,wefocusonsymmetricmechanisms,omitindicesandsometimesdenoteoverallcontributionsasT¯=Ti(α¯,α¯),Tˆ1=T1(α¯,α)=T2(α,α¯),Tˆ2=T1(α,α¯)=T2(α¯,α)andT=Ti(α,α).′thefixed-costoftheproject,andbyKtheamountofexternalfundsprovided,ourmodelcouldaccountforthisextensionprovidedthatF′−(2002)stresstheroleofinternationalorganizationsininducingmorecooperativeoutcomesbyfosteringtrustandprovidingexpertiseonstate-of-the-arttechnology,αiisknownbyboththegovernmentGiandtheresidentsofCi,theIAcoulduseamorecomplexrevelationmechanismelicitingcostlesslythiscommonlyknownbutnon-verifiableinformation(seeMaskin(1999)).Toruleoutthosemechanisms,,TcouldbereplacedwithT′=
infrastructurearealsowrittenasp¯=p(α¯,α¯),pˆ=p(α¯,α)=p(α,α¯)andp=p(α,α).Finally,stillforasymmetricmechanism,wedenoteconsumptionsbyxi(θ,α)=x(θ,α).Similarconventionsholdforthepricesti(·)andtheutilitiesUi(·).Inwhatfollows,weassumethatfirst,countriesreporttheirpreferencesforredistribu-tion,,α=(α1,α2).193CommonKnowledgeonCountries’PreferencesLetusfirstsupposethattheIAhascompleteinformationonthecountries’profileofpreferencesforredistributionα=(α1,α2).20Withineachcountry,theagents’incentiveconstraintscanthusbeexpressedintermsoftheutilityUi(θ,α)theygetandtheirconsumptionsxi(θ,α).Thesecontraintsarerespectively()()Ui(θ¯,α)=p(α)θ¯v(xi(θ¯,α))−ti(θ¯,α)≥p(α)θ¯v(xi(θ,α))−ti(θ,α)=Ui(θ,α)+p(α)∆θv(xi(θ,α)).(6)and()Ui(θ,α)=p(α)(θv(xi(θ,α))−ti(θ,α))≥p(α)θv(xi(θ¯,α))−ti(θ¯,α)=Ui(θ¯,α)−p(α)∆θv(xi(θ¯,α)).(7)Asitisstandardintwo-typeadverseselectionmodels,21therelevant(binding)con-straintisthatoftherichagentθ¯,namely(6)whereas(7)(expected)costofbuildingtheprojectunderanyprofile19Thisisakintoadominantstrategyrequirementwhenwritingtheagents’,α−i,(1992),thereisneverthelessnolossofgeneralityinfocusingondominantstrategyimplementationaslongasthedecisionruleismonotonic,,thiscanbeviewedasasettingwheretheIAusesrevelationschemeslaMaskinmakingbotheachgovernmentGiandtheinhabitantsofCireportthecommonlyknownpieceofinformationα(2002,Chapter2).11
ofpreferences:∑2Ti(α)≥p(α)F.(8)i=1Werewritethisconstraintusing(5)andthedefinitionsofUi(θ,α)givenaboveas:∑2()p(α)E(S(θ,xi(θ,α))−Ui(θ,α)≥0.(9)θi=1Intuitively,thesumofexpectedsurplusesinbothcountriescomputedfortheconsumptionprofilexi(θ,α)α,theIAsolvesthefollowingproblem:∑2FB(TP):maxV˜i(α),{p(·);Ui(·),xi(·),V˜i(·)}i=1subjectto(6)-(7)-(9)andV˜i(α)=αiνUi(θ¯,α)+(1−αiν)Ui(θ,α).(10)IntheAppendix,weshowthatsolving(TP)FBamountsinfacttosolvingthefollowingsimplerproblem:{∑}2()(TPFB∗):maxp(α)ES˜(θ,xi(θ,α),αi).{p(·),xi(·)}i=1θThissimpleobjectiveaggregatestherichagents’incentiveconstraint(6)andthebudget-balancedconstraint(9),,weassumethatthefollowingconditionshold:(H1)E(S˜(θ,x˜(θ,α¯),α¯))+E(S˜(θ,x˜(θ,α),α))>0,θθ(H2)E(S˜(θ,x˜(θ,α),α))<0.θ
efficientprobabilitiesofrealizingtheproject,.,theoptimalprobabilitieswhenautil-itarianwelfaremaximizeraggregatesthecountries’welfarewhichtakeintoaccounttheconsumers’incentiveconstraints.(H1)and(H2)-tuningwillbethesourceofsomeproblemsunderasymmetricinformationontheα,ifitwasoptimaltodotheprojectwhateverthecountries’preferences,(somethinglessorequaltotheexpectedvirtualsurplusoftheα-one):Assumethatthepreferencesprofileαiscommonknowledgeandthatconditions(H1)and(H2),theoptimalmechanismforcollectivedecisionischaracterizedasfollows.•¯∗=pˆ∗=1andp∗=0.•Therichagents’incentiveconstraint(6)isalwaysbindingassoonastheprojectisdone.•Therichagentsconsumealwaysthefirst-bestamountx∗(θ¯)whereasthepooragentsinCiconsumethesecond-bestquantityx˜(θ,αi)aslongastheprojectisbuilt.•,,governmentsarenotefficiencymaximizers,,,,,satisfyingtheincentivecompatibilityconstraint(6)requirestogivemoreutilitytoaθ¯-agentthantoaθ,
decreasingtheconsumptionofthepoorasitcanbeseenfrom(6).22Sincepricingwithineachcountryonlydependsonthepreferenceforredistribution,thereisacompletedichotomybetweenpricingandthedecisiontobuildtheprojectornotwhich,instead,,’Preferences:ConstrainedEfficiencyWenowconsiderthecasewheretheIAisuninformedontheα,,’Bayesianincentiveconstraints:()Vi(α¯)=EE(Uαi(θ,α¯,α−i))−ν(1−α¯)(Ui(θ¯,α¯,α−i)−Ui(θ,α¯,α−i))−iθ()≥EE(Uαi(θ,α,α−i))−ν(1−α¯)(Ui(θ¯,α,α−i)−Ui(θ,α,α−i))−iθ=Vi(α)+ν∆αE(Uαi(θ¯,α,α−i)−Ui(θ,α,α−i)).(11)−i22Ofcourse,autilitariangovernment(correspondingtothelimitingcaseα¯=1),therichagents’incentiveconstraintbecomescostlessandthepoorconsumealsothefirst-bestconsumptionx∗(θ).14
and()Vi(α)=EE(Uαi(θ,α,α−i))−ν(1−α)(Ui(θ¯,α,α−i)−Ui(θ,α,α−i))−iθ()≥EE(Uαi(θ,α¯,α−i))−ν(1−α)(Ui(θ¯,α¯,α−i)−Ui(θ,α¯,α−i))−iθ=Vi(α¯)−ν∆αE(Uαi(θ¯,α¯,α−i)−Ui(θ,α¯,α−i)).(12)−iInwhatfollows,theonlyrelevant(binding)incentiveconstraintis(11).,,becauseofasymmetricinformationontastes,,,thelessegalitarianthedistributionofutilityinthepoorcountry,theharderitistosatisfytheincentiveconstraint(11).WhentheIAhasastrongabilitytoenforcethemechanism,thecountries’,6and7,-compatiblemechanismbeforeknowingtherealizationsoftheα(·)andthefactthattheIAmaximizesnowthesumofexpectedwelfaresinbothcountries,IA’sproblembecomes:∑2(TP)0:maxE(V{p(·);xi(·);Ui(·);Vi(·)}αi(αi)),ii=1subjectto(6)-(7)-(8)-(11)-(12).WealsoshowintheAppendixthat(TP)0canbeexpressedinamorecompactwayas:23{(∑)}2(TP∗)0:maxEp(α)E(S˜(θ,xi(θ,α),αi)){p(·),xi(·),Vi(·)}αi=1θsubjectto(11)-(12),and∑{(2∑)}2E(VE(S˜(θ,xi(θ,α),αi))(13)αi(αi))=Ep(α)iαi=1θi=,reciprocally,
Proposition2:Assumethatthepreferencesprofileαisprivateinformationofthecountriesandthatconditions(H1)and(H2),,namelyp¯∗=pˆ∗=1andp∗=’AspremontandGrard-Varet(1979)andArrow(1979).,ifitis,,,weshowthatthereexistsawholecontinuumofmechanisms(indexedbythelevelofaggregatewelfareinapoorcountry),,’Preferences:ConstrainedInefficiencyEventhoughtheresultsinProposition2seemattractiveonnormativegroundsandsug-gestthatasymmetricinformationmaynotbesuchanobstacletoinvestment,thepreviousmechanismssufferfromaseriousflawsincethecountries’,,eachcountrymustgetmorethanitspayoff(thatwehaveexogenouslynormalizedatzero)(TP)∗0definesonlytheexpectedwelfareqV(α¯)+(1−q)V(α),thereexists24Amongthesemechanisms,afocalonemightbetheso-calledpay-theexternalitymechanismstressedbyD’AspremontandGrard-Varet(1979)andArrow(1979).
awholerangeofpossiblevaluesof(V(α),V(α¯))whichsatisfytheincentiveconstraints(11)and(12),withinthisrange,thereexistsomepairs(Vi(α),Vi(α¯))satisfyingalsothefollowinginterimparticipationconstraintsofcountries:Vi(α¯)≥0,(14)Vi(α)≥0.(15)Infact,withthoseparticipationconstraints,:Theconstrainedefficientdecision-rule(p¯∗,pˆ∗,p∗)andthesecond-bestlevelsofconsumptionx˜(θ,αi)cannolongerbeimplementedwhenthecountries’interimparticipationconstraints(14)and(15)mustbesatisfiedifthefollowingconditionholds:()2q2E(S˜(θ,x˜(θ,α¯),α¯))+2q(1−q)E(S˜(θ,x˜(θ,α¯),α¯))+E(S˜(θ,x˜(θ,α),α))θθθ<2q2ν∆α∆θv(x˜(θ,α)).(16)Thisconditionsaysthatthesovereigntyofcountriesisasourceofinefficiencywhen-evertheaggregatedwelfarecomputedfortheconstrainedefficientoutcomeissmallerthanthecostofinducinginformationrevelationfromtheα¯−(16)isrelatedtoanearlierresultduetoLaffontandMaskin(1979)whoprovedthatBayesianincentivecompatibility,efficiency,budgetbalanceandindividualrationalitymaybein-compatible,andtoMakowskiandMezzetti(1994)andWilliams(1999)’spreferences,α¯-countryre-ceivesmorethantheα(11),theinducedinequalityisgreaterwhentheIAmaintainsanefficientprobabilityofbuildingtheinfrastructureandsetspˆ∗=,iftheconstrainedefficientoutcomewasimplemented,(11)couldbewrittenas:Vi(α¯)≥Vi(α)+νq∆α∆θv(x˜(θ,α))fori∈{1,2}.(17)Whenpˆ∗=1andthesecond-bestconsumptionx˜(θ,α)ismaintained,theright-handsideof(17)isratherlargemeaningthatVi(α)mustbesignificantlylowerthanVi(α¯)αα-countrycallsforreducingthe17
α¯-countrymoreeagertomimictheα-one,hardeningtherebyitsincentiveconstraint(11).Incentivecompatibilityatthecountrylevelisthuseasiertoachieveifpˆisdistorteddownwardsandif,whenthetwocountrieshavedifferentpreferencesforredistribution,theconsumptionofthepooragentswithinaα’sproblembecomesnow:∑2SB(TP):maxE(Vi(αi)){p(·);xi(·),Ui(·),Vi(·)}αi=1isubjectto(6)-(7)-(9)-(11)-(12)-(14)-(15).WeshowintheAppendixthatthisproblemcanbesimplifiedto:{(∑)}2SB∗(TP):maxEp(α)E(S˜(θ,xi(θ,α),αi)){p(·),xi(·)}αi=1θsubjectto{(∑)}2∑2{}Ep(α)E(S˜(θ,xi(θ,α),αi))−νq∆α∆θE(p(α,αα−i)v(xi(θ,α,α−i))≥0.αθ−ii=1i=1(18)Thislatterconstraintaggregatestheexpostbudget-balancedconstraint(9),therele-vant(binding)incentiveconstraint(11)oftheα¯-countriesandtheparticipationconstraint(15)ofaα(18)simplymeansthattheaggregatewelfareoverbothcountriesshouldcovertheinformationalcostofinducinginformationrevelationbytheα¯,allocativedistortionsareneededtosatisfy(18)
wherex∞(θ¯,α)=x∗(θ¯)andx∞(θ,α)isdefinedby:(θ−ν(1−α)1−∆θ−νq∆)αv′(x∞(θ,α))=c.(20)ν(1−ν)(1−∆θq)(19)strengthenscondition(16),theredoesnotexistanymodificationofthesecond-bestconsumptionx˜(θ,α)which,alone,couldensurethatthefeasibilitycondition(18):Assumethatcountriescanoptoutofthemechanismiftheydonotgetanon-negativeexpectedwelfareandthatcondition(19).•Theα-countrygetszeroexpectedwelfare(itsparticipationconstraint(15)isbind-ing).•Thedecisiontobuildtheinfrastructureisdistortedwiththeprojectbeingrealizedlessoftenthanwhencountries’,whencountriesareasymmetric,theprobabilityofbuildingtheprojectispositivebutalwayslessthanone:p¯SB=p¯∗=1,pˆSB∈]0,1[,pSB=p∗=0.•Consumptionsintheα¯-countriesarestillsecond-best;xSB(θ¯,α¯,α−i)=x∗(θ¯)andxSB(θ,α¯,α−i)=x˜(θ,α¯)forallα−i.•Thereisanextradownwarddistortionoftheconsumptionofthepoorintheα-countries:xSB(θ¯,α,α¯)=x∗(θ¯)andxSB(θ,α,α¯)<x˜(θ,α).Denotingbyλ>0,theshadowcostofthefeasibilityconstraint(18),wehave:()θ−ν(1−α)−λνq∆α1−∆θ∆θv′(xSB(θ,α,α¯))=c.(21)ν(1+λ)(1−ν)(1−q)•Pricinginapoorcountrydependsontheshadowcostλα¯-countryandtheparticipationconstraintofaα,theproject26When(19)doesnotholdbutstill(16)holds,weareincasesofintermediateinefficiencieswhere,dependingonthefunctionalforms,,
,,theIAmustmakethedistributionofutilitieswithintheα-countrylessattractivetoaα¯α¯-country,inequalityislesscostlythaninaαα-country,aα¯α-countryapricingschemeinducingaveryegalitariandistributionofutilitieswithinthecountry,itreducesalsotheincentivesofaα¯-countrytomimicaαα(21),everythinghappensthusasiftheα-countryhasnowapositivestrongervirtualaversiontoinequalityα˜definedas:α˜=α−λqα<α.(1+λ)(1−∆q)Thismodificationoftheredistributiveconcernswithintheαα,byintroducingaparticipationconstraintatthecountrylevel,,thefactthatthenestedinformationstructureofourmodelleadstoextradistortionsawayfrom(constrained),,,eventhoughourinformationstructureisnestedjustasin(mostof)thisliterature,(2000),,enforcingarandommechanismisamoredifficulttaskthanenforcingadeterministiccontractbuttheIA’,MookherjeeandReichelstein(1995),McAfeeandMcMillan(1995)andFaure-Grimaud,LaffontandMartimort(2003)
Interestingly,theconsumptionxSB(θ,α,α¯)isalwaysstrictlyabovex∞(θ,α)whichisobtainedbysettingλ=+∞into(21).,iftheprojectissometimescanceled,,(18),λofthefeasibilityconstraint(18),whencondition(16)isalmostanequality,.,forafixed-costwhichisnottoolarge,31λissmall,theprobabilitypˆ,theIAwasmodeledasabenevolentmaximizerofthesumofbothcountries’ex-pectedwelfaresjustinlinewithMyersonandSattherwaite(1983)(constrained),.,betweencountries,:WenowassumethattheIAwantstomaximizethefollowingweightedsumoftheaggregatewelfareinbothcountries:∑2∑2()βqVi(α¯))+(1−βq)Vi(α)=E(Vαi(αi))−q(1−β)(Vi(α¯)−Vi(α))i=1i=1i31Ofcourse,thisfixedcostmustbelargeenoughtoensurethattwoα-countrieswouldnotbuildtheprojectundercompleteinformationsothat(H2)
where0<β<-offfacedbytheIAbetweenlookingfora(constrained)-anismiftheywishso,theIAisconstrainedbythesamebudget-balanced,,thereducedformoftheIA’sproblemwritesas:{(∑)}2(TP∗)RC:maxEp(α)E(S˜(θ,xi(θ,α),αi)){p(·),xi(·)}αi=1θ∑2−νq(1−β)∆α∆θE(p(α,αα−i)v(xi(θ,α,α−i))))−ii=1subjectto(18).Proposition5:Assumethatcountriescanoptoutofthemechanism,thattheIAhassomeredistributiveconcernsbetweencountriesandthatconditions(H1),(H2)and(19),itentails:•Adistortioninthedecisiontobuildtheinfrastructurewhencountrieshaveasym-metricpreferences:p¯RC=1,pˆRC∈]0,1[,pRC=0.•Adownwarddistortionfortheconsumptionofthepoorintheα-country;xRC(θ¯,α,α¯)=x∗(θ¯)andxRC(θ,α,α¯)<x˜(θ,α)with()θ−ν1−(1−α)∆θ−(λ+1−β)ν∆α∆θv′(xRC(θ,α,α¯ν(1+))=c(22)λ)(1−ν)(1−q)whereλistheshadowcostofthefeasibilityconstraint(18).Iftheshadowcostλwasthesameforproblems(TP)SB∗and(TP)∗RC,,asinSection5,onlyindirectlybecauseofthepresenceoftheα-country’(α¯)
:LetuslookatthecasewheretheIAhassomeconcernsforpovertyattheindividuallevelandputsanextraexogenouspositiveweightµ,theIAnowmaximizes:(∑)(2∑)2EV)+µEαi(αi(Ui(θ,α))iαi=1i=1whereµ>’incentiveconstraintswithineachcountryarebinding,wehave:E(Uααi(θ,i,α−i))=Vi(αi)−αiν∆θE(p(α)v(x(θ,α))).(23)−iα−iWecanfinallyrewritetheIA’sobjectivefunctionas:()EVαi(αi)−αiµνi1+∆θE(p(α)v(x(θ,α))).µα−iBecauseµ>0,,thereducedformoftheIA’sproblemcanbewrittenas:{(∑))2(TP∗)RA:maxEp(α)E(S˜(θ,xi(θ,α),αi))−µθv(xi(θ,α)){p(·),xi(·)}αi=1θ1+ναµi∆subjectto(18).Weneedagaintodescribeapairofconditionssimilarto(H1)and(H2):((()))((()))(H1’)Eα¯α¯S˜θ,x˜θ,,+EααS˜θ,x˜θ,,>0,θ(((1+µ)1+µ))θ1+µ1+µα(H2’)EαS˜θ,x˜θ,0.θ1+,µ1+<µProposition6:Assumethatcountriescanoptoutofthemechanism,thattheIAhassomeredistributiveconcernsandcaresaboutpovertywithincountriesandthatconditions(H1’),(H2’)and(19),itentails:23
•Adistortioninthedecisiontobuildtheinfrastructurewhencountrieshaveasym-metricpreferences:p¯RA=1,pˆRA∈]0,1[,pRA=0.•Strongdownwarddistortionsfortheconsumptionofthepoorinbothaα¯−andaα-countryiftheprojectisrealized:(())θ−ν1−α¯∆θv′(xRA(θ,α¯,α¯))=c(24)1−ν1+µ(())λνq∆αθ−νv′(xRA(θ,α,α¯))=c1−1−αν1+∆θ−µ(1+λ)(1−ν)(1−∆θq)(25)whereλistheshadowcostofthefeasibilityconstraint(18).TheIA’,,everythinghappensasif,apriori,thepreferencesforredis-tributioncouldbecharacterizedbyanewparameterβ=α1+µ<α’,notethat,foranyprobabilityofmakingtheprojectandtheoverallcontributionmadebyacountry,consumptionsreflectnowonlythepreferencesinthiscountry,and24
theoptimalsecond-bestprofileofconsumptionsspecifictoeachcountryx˜(θ,αi),{p(αˆ);Ti(αˆ)}.LetusfirstredefinetheexpectedwelfareincountryCias:()FVi(αi)=E−Tαi(α)+p(α)E(S˜(θ,x˜(θ,αi),αi)+).(26)−iθ2Thatdefinitionalreadyincorporatesthefactthateachgovernmentchoosespricingac-cordingtoitspreferencesonlyandthatthecorrespondingconsumptionsarex˜(θ,αi).Wecanrewritethecountries’incentiveconstraintsas:()Vi(α¯)≥Vi(α)+E(p(α,αα−i))E(S˜(θ,x˜(θ,α¯),α¯)−E(S˜(θ,x˜(θ,α),α))(27)−iθθand()Vi(α)≥Vi(α¯)−E(p(α¯,αα−i))E(S˜(θ,x˜(θ,α¯),α¯)−E(S˜(θ,x˜(θ,α),α))(28)−iθθNotethatS˜(θ,x˜(θ,α¯),α¯)>E(S˜(θ,x˜(θ,α),α))because∆α>0.θTakingintoaccountthecountries’participationconstraints,theIA’sproblemcannowbewrittenas:∑2(TP)L:maxE(Vi(αi)){p(·),Vi(·)}αi=1isubjectto(8)-(14)-(15)-(26)-(27)-(28).Ofcourse,constrainedefficiencymaystillbeachievedevenwiththeparticipationconstraint(15),letusassumethatthefollowingconditionholds:()2q2E(S˜(θ,x˜(θ,α¯),α¯))+2q(1−q)E(S˜(θ,x˜(θ,α¯),α¯))+E(S˜(θ,x˜(θ,α),α))θθθ()<2q2E(S˜(θ,x˜(θ,α¯),α¯))−E(S˜(θ,x˜(θ,α),α)).(29)θθCondition(29)issimilarto(19)(H1)and(H2),wewillalsoassumethat,hadtheIAbeenawareofthepreferencesinbothcountries,itwouldhavebeenoptimaltobuildtheprojectifandonlyifatleastonecountryhadpreferencesα¯.Thisyieldstheconditions:()(H1”)2q2E(S˜(θ,x˜(θ,α¯),α¯)+2q(1−q)E(S˜(θ,x˜(θ,α¯),α¯))+E(S˜(θ,x˜(θ,α),α))>0,θθθ(H2”)2E(S˜(θ,x˜(θ,α),α))<0.θ25
Whenboth(H1”)and(H2”)hold,thedecisiontobuildornottheprojectiscase-sensitive,:Assumethatgovernmentsineachcountrykeepcontrolofpricingandthatconditions(H1”),(H2”)and(29).•Theincentiveconstraintofaα¯-country(27)andtheparticipationconstraintofaα-one(15)arebothbinding.•Thedecisiontobuildtheinfrastructureisdistortedwiththeprojectbeingrealizedlessoftenthanwhencountries’,whencountriesareasymmetric,theprobabilityofbuildingtheprojectispositivebutalwayslessthanone:p¯L=1,pˆL∈]0,1[andpL=0.•Thesecond-bestlevelsofconsumptionx˜(θ,αi)arealwayschoseninbothcountries.•Bydefinition,’,namelypˆ.Asaresult,weexpectgreaterdistortionsinthedecisiontorealizetheprojectwhenpricingoftheinfrastructureisoutoftheIA’,wehavetocomparetwothird-bestpoliciesand,asusual,,:Assumethat∆θissmallenough,thenpˆL<pˆ,
8ExtensionsandConclusionInthispaper,’concernsforredistributionmightbeexacerbatedbytheexternalconstraintsimposedbytheIA’’sconcernsforredistributioneitheracrosscountriesoracrossindividualswithinthesamecountryspillovertothelocallevel,reinforcethegovernments’ownconcernsforredistribution,(eitherbecauseitcannotforceacceptancebycountriesorbecauseitcannotcontrolprices),,,:,,,,(atleastpartially)
PoliticalEconomy:,’:,,,,localgovernmentswouldchoosepricesnon-cooperativelyandthiswouldleadtonotenoughconsumptionbecauseofa(non-internalized),,:,oneimportantlessonofadverseselectionmodelswithtype-dependentreservationpayoffsisthatthoseoutsideopportunities,whenbinding,,inthecaseofatelecommunicationsnetwork,
:Eventhough,wehadinmindspecificexamplesoftransnationalinfrastructurefordevelopingcountriesinwritingthispaper,itslessonsmayhavealsosomevaluetounderstandthegovernanceofmoregeneralglobalpublicgoods(orbads)likeglobalwarming,diseaseprevention,tradeagreements,etc...,,(2002),“TransnationalPublicGoods;StrategiesandInstitutions”,WorkingPaper,,K.(1979),“ThePropertyRightsDoctrineandDemandRevelationunderIncompleteInformation”,inEconomicsandHumanWelfare,,(2002),“TheOutputCostofLatinAmerica’sInfrastructureLag”,inTheMacroeconomicsofInfrastructureinLatinAmerica,,(eds.),TheWorldBank,,C.,(eds)(2002),TheMacroeconomicsofInfrastructureinLatinAmerica,TheWorldBank,’Aspremont,’-Varet(1979),“IncentivesandIncompleteInformation”,JournalofPublicEconomics,11,-Grimaud,A.,(2003),“CollusionandSupervisionwithSoftInformation”,ReviewofEconomicStudies70:,J.,(1977),“”,CahiersduSminaired’Econometrie,,(1977),“CharacterizationofSatisfactoryMecha-nismsfortheRevelationofPreferencesforPublicGoods”,Econometrica,45,,T.(1973),“IncentivesinTeams”,Econometrica,41,(2002,Chapter3).29
Guasch,L.,(2002),“RenegotiationofConcessionContractsinLatinAmerica”,,(1983),“EfficientandDurableDecisionRuleswithIncompleteInformation”,Econometrica,51,,(2002),TheTheoryofIncentives:ThePrincipal-AgentModel,,(1979),“ADifferentiableApproachtoExpectedUtilityMaximizingMechanisms”,inAggregationandRevelationofPref-erences,(ed.),,(1999),“ACharacterizationofInterimEfficiencywithPublicGoods”,Econometrica,64,,(1990),“AsymmetricInformationBargainingProblemswithManyAgents”,ReviewofEconomicStudies,57,,(1994),“BayesianandWeaklyRobustFirst-Bestmechanisms:Characterization”,JournalofEconomicTheory,64:,D.(2001),“OptimalTaxationandStrategicBudgetDeficitunderPoliticalRegimeSwitching”,ReviewofEconomicStudies,68,,E.(1999),“NashEquilibriumandWelfareOptimality”,ReviewofEconomicStudies,66,,(1990),“ThePrincipal-AgentRelationshipwithanInformedPrincipalI:Private-Values”,Econometrica,58,,(1995),“OrganizationalDiseconomiesofScope,”JournalofEconomicsandManagementStrategy4:,N.,(1995),“HierarchicalDe-centralizationofIncentiveContracts”,RandJournalofEconomics,26:,(1992),“DominantStrategyImplementa-tionofBayesianIncentiveCompatibleAllocationRules”,JournalofEco-nomicTheory,56:,(1983),“EfficientMechanismsforBilateralTrading”,JournalofEconomicTheory,28:,R.(1989),“PollutionClaimsSettlementswithPrivateInformation”,JournalofEconomicTheory,47,
Sandler,T.(1998),“GlobalandRegionalPublicGoods:APrognosisforCollectiveAction”,FiscalStudies,19:,T.(2001),“OnFinancingGlobalandInternationalPublicGoods”,(eds.)InternationalPublicGoods:Incentives,MeasurementsandFinancing,Dordecht,NL:,(2002),“RegionalCooperation,andtheRoleofInternationalOrganizationsandRegionalIntegration”,WorldBankPolicyResearchWorkingPaper2872,,M.(1980),“OntheInexistenceofaDominantStrategyMechanismforMakingOptimalPublicGoodDecisions”,Econometrica,48,,S.(1999),“ACharacterizationofEfficientBayesianIncentiveCom-patibleMechanims”,EconomicTheory,14:,InfrastructureforDevelopment,WorldBank,•EndogenizingGovernment’sPreferencesandAsymmetricInformation:Letussupposethatpricingisusedtocoverarandomdeficitκ˜∈{κ,κ¯}withrespectiveprobabilitiesqand1−qandκ¯>κ.Wesupposethatthegovernmentisbenevolentandmaximizesthesumofutilitiesofthedifferenttypesofagentssubjecttotheagents’(asusual)onlyontherichagent’sincentiveconstraintandthepooragent’sparticipationone,thegovernment’sproblemcanbewrittenas:maxνU(θ¯)+(1−ν)U(θ).{x(·),U(·)}subjecttoU(θ¯)−U(θ)≥∆θv(x(θ)),()U(θ)≥U0,()andνU(θ¯)+(1−ν)U(θ)+κ≤ν(θ¯v(x(θ¯))−cx(θ¯))+(1−ν)(θv(x(θ))−cx(θ)),()wherethelatterconstraintisthebudgetconstraintofthestatewhenthedeficitisκ.Ofcourse,
From()and(),()implies:ν(θ¯v(x(θ¯))−cx(θ¯))+(1−ν)(θv(x(θ))−cx(θ))≥κ+ν∆θv(x(θ))+U0.()Whenκislargeenough(butnottoolargesothattheconstrainedsetremainsnon-empty),thisconstraintisclearlynolongersatisfiedbythefirst-bestoptimallevelsofconsumptionsx∗(θ)andx∗(θ¯).Then,()isbindingattheoptimum(andconsequently()and()arealsobinding).Wecanrewritethegovernment’sproblemasmaxν(θ¯v(x(θ¯))−cx(θ¯))+(1−ν)(θv(x(θ))−cx(θ)){x(·)}subjectto().Denotingbyµ(κ)thepositivemultiplierof(),thegovernmentmaximizes(1−ν)(θv(x(θ))−cx(θ))+ν(θ¯v(x(θ¯))−cx(θ¯))−νµ(κ)∆θv(x(θ))1+µ(κ)forsomeµ(κ)>0,whereµ(κ)µ(κ)isincreasinginκ.Denoting1−α¯=µ(κ¯)1+µ1−α=µ(κ)(κand)1+µeobser(κ,wvethateverything)happensasifthegovernmentmaximizesanobjectivefunctionofthetypeανU(θ¯)+(1−αν)U(θ).PrivateinformationontheparameterαcanthusbeviewedasareducedformforprivateinformationontheshockκhittingthebudgetconstraintoftheState.•Transformationof(TP)FBinto(TP)FB∗andProofofProposition1:First,werewritetheobjectivefunctionoftheIAwhichbecomes:∑2()νUi(θ¯,α)+(1−ν)Ui(θ,α)−ν(1−αi)(Ui(θ¯,α)−Ui(θ,α))i=1foragivenpreferencesprofileα=(α1,α2).Clearly,thisshowsthat(9)mustbebindingattheoptimumandwegetthus:∑{2∑}2V˜i(α)=p(α)E(S(θ,xi(θ,α)))i=1∑i=1θ2−ν(1−αi)(Ui(θ¯,α)−Ui(θ,α))).()i=1Sinceαi<1,thesecondtermontheright-handsideof()isminimizedwhen(6)(7)isslackassoonasxi(θ,α)<xi(θ¯,α),
Inserting(6)bindinginto()yieldsthemaximandof(TP)FB∗,namely:{∑}2p(α)E(S˜(θ,xi(θ,α),αi)).=1θiThesecond-bestconsumptionsx˜(θ,αi)˜(θ,αi)<x∗(θ¯)and(6)bindingimpliesthat(7)(H1)and(H2),wehavepˆ∗=1andp∗=,bydef-initionofx˜(θ,α)andthefactthatthegovernmentinthepoorcountryismoreaversetoinequalitythatintherichcountry,x˜(θ,α)<x˜(θ,α¯).Finally,E(S˜(θ,x˜(θ,α¯),α¯))>θE(S˜(θ,x˜(θ,α),α)).Hence,assumption(H1)impliesalsothatp¯∗=1.θ•Transformationof(TP)0into(TP)∗0,andProofofProposition2:First,weobservethat:()Vi(αi)=EE(Uαi(θ,α))−ν(1−αi)(Ui(θ¯,α)−Ui(θ,α))−iθ(())=Ep(α)E(S(θ,x(θ,α)))+F−ν(1−αi)(Ui(θ¯,α)−Ui(θ,α))−E(Tα−iθ2αi(α)).−iForasymmetricmechanism,E(Tαi(α¯,α−i))=qT¯+(1−q)Tˆ1andE(T(αiα,α−i))=−i−iqTˆ2+(1−q),stillusingthesymmetryofthemechanism,wehave:∑2((∑2E(Vp(α)E(S(θ,xi(θ,α))+F))αi(αi))=Eαi=1iθ2i=1∑(2∑)2−ν(1−αi)(Ui(θ¯,α)−Ui(θ,α))−ETαi(α).i=1i=1Maximizationofthisexpressionsubjecttotheexpostbudgetconstraints(8),asintheproofofProposition1,theright-handsideaboveismaximizedwhen(6)isbinding,.,Ui(θ¯,α)−Ui(θ,α)=p(α)∆θv(xi(θ,α)),foralli∈{1,2}.Again,thefactthatxi(θ,α)<xi(θ¯,α)forthesolutionensuresthat(7),weobtain:∑2{(∑)2}E(Vp(α)E(S˜(θ,xαi(θ,α),αi)),()αi(αi))=Ei=1ii=1θ33
.,themaximandof(TP0)∗withtheincentiveconstraintsofbothcountries(11)and(12).-iblepairs(V(α¯),V(α)).Thereisstillawholerangeofsuchsymmetricpairs(V(α¯),V(α))whichsatisfy()asanequalityandtheincentiveconstraints(11)and(12).Since()holdsasanequality,definingV(α)definesalsoV(α¯).AllpossiblevaluesofV(α)describetheinterval[Vm(α),VM(α)]where:Vm(α)=A−νq∆α∆θv(x˜(θ,α¯)),2VM(α)=A−νq2∆α∆θv(x˜(θ,α));2andAistheright-handsideof()computedfortheconstrainedefficientprobabilitiesp¯∗=pˆ∗=1,p∗=0andthesecond-bestconsumptionsx˜(θ,α).Wehavethus:()A=2q2E(S˜(θ,x˜(θ,α¯),α¯))+2q(1−q)E(S˜(θ,x˜(θ,α¯),α¯))+E(S˜(θ,x˜(θ,α),α)).θθθSinceq<1andx˜(θ,α)<x˜(θ,α¯),wehaveindeedVM(α)>Vm(α).ForanyV(α)in[Vm(α),VM(α)]andthecorrespondingvalueofV(α¯)obtainedwhen()isbinding,wecanfindthevaluesofthesymmetrictransfers(T¯,Tˆ1,Tˆ2,T)whichimplementtheseutilitylevelsassolutionstothefollowingsystem:()V(α)=qpˆ∗E(S˜(θ,xSB(θ,α,α¯)))+F−(qTˆ2+(1−q)T)()(θ2)()V∗B(α¯)=qp¯E(S˜(θ,xSB(θ,α¯,α¯)))+F+(1−q)pˆ∗E(S˜(θ,xS(θ,α¯,α)))+Fθ2θ2−(qT¯+(1−q)Tˆ1)()2T¯=p¯∗F,()Tˆ1+Tˆ2=pˆ∗F,()2T=p∗F,()wherep¯∗=pˆ∗=1andp∗=()and()yieldimmediatelyT¯=F2andT=,(Tˆ1,Tˆ2)isimmediatelyobtainedasasolutionto()and().•ProofofProposition3:ThesamedecisionruleandconsumptionsasinProposition2cannolongerbeobtainedwhenthepreferencesforredistributionareunknownifVM(α)<(16).34
•Transformationof(TP)SBinto(TP)SB∗andProofofProposition4:Since(16)holds,(17)isnotsatisfiedbythesolutionobtainedwhenthepreferencesprofileα=(α1,α2),weshouldlookforasolutionof(TP)SBsuchthat(11)and(15)(andthus(18))(15)and(11)aresatisfiedimply(forapositivex(θ,α,α¯))that(14)holdsstrictly,(12)(whichcanbecheckedexpost).Then,wecanrewrite(TP)SBinamorecompactwayas:{(∑)2∑}2(TPSB′):maxEp(α)E(S(θ,xi(θ,α),αi))−ν(1−αi)(Ui(θ¯,α)−Ui(θ,α)),{p(·),xi(·),Vi(·)}αθi=1i=1subjectto(6)-(7)-(11)-(15)and∑{(2∑)2∑}2E(V(α)E(S(θ,xi(θ,α),αi))−ν(1−αi)(Ui(θ¯,α)−Ui(θ,α)).αii))=Ep(αiαi=1i=1θi=1()Using(11)and(15),weget{(∑)2∑}2Ep(α)E(S(θ,xi(θ,α),αi))−ν(1−αi)(Ui(θ¯,α)−Ui(θ,α))=αi=1θi=1∑2∑2E(Vαi(αi))≥νq∆αE(Ui(θ¯,α,α−i)−Ui(θ,α,α−i)).()iα−ii=1i=1OptimizingfirstwithrespecttoUi(·),(6)isbindingtoincreasethemaximandin(TP)SB′andrelaxconstraint().(7)(θ¯,α)−Ui(θ,α)into()andthemaximandof(TP)SB′,weget(18)andtheexpressionofthemaximandofIA’sproblemas(TP)SB∗.Letdenotebyλthepositivemultiplierof(18)into(TP)SB∗.TheLagrangeanis:{(∑2)}∑2E(1+λ)p(α)E(S˜(θ,xi(θ,α),α))−λνq∆α∆θE(v(xαi(θ,α,α−i))).αi1θi=−ii=1Optimizingwithrespecttoxi(·)yieldsasymmetricsolutionsuchthat•Forarichcountry,θ¯v′(xSB(θ¯,α¯,α−i))=c,∀α−iandsoxSB(θ¯,α¯,α−i)=x∗(θ¯);(θ−ν(1−)α¯)θv′SB(x(θ,α¯,α−i))=c,∀α−i1−∆νandsoxSB(θ,α¯,α−i)=x˜(θ,α¯);35
•Forapoorcountry,θ¯v′(xSB(θ¯,α,α−i))=c,∀α−iandsoxSB(θ¯,α,α−i)=x∗(θ¯);(θ−ν(1−)α)λqν∆α1−∆θ−∆θv′SB(x(x(θ,α,α−i))=c,∀α−iν(1+λ)(1−q)(1−ν)andsoxSB(θ,α,α−i)<x˜(θ,α)sinceλ>¯,pˆ:p¯SB=1⇔2(1+λ)q2E(S˜(θ,x˜(θ,α¯),α¯))>0()θwhichholdsfrom(H1),pˆSB∈[0,1]⇔()2(1+λ)q(1−q)E(S˜(θ,x˜(θ,α¯),α¯))+E(S˜(θ,xSB(θ,α,α¯),α))θθ=2λq2ν∆α∆θv(xSB(θ,α,α¯)),()pSB=0⇔2(1+λ)(1−q2)ESB(S˜(θ,x(θ,α,α),α))<2λq(1−q)ν∆α∆θv(xSB(θ,α,α)).()θThislatterinequalityholdssincexSB(θ,α,α)<x˜(θ,α)impliesthatE(S˜(θ,xSB(θ,α,α),α))<θE(S˜(θ,x˜(θ,α),α))<0fromcondition(H2).θFrom(18)binding,andtakingalsointoaccountthatp¯SB=1andpSB=0,wegetthatpˆSB,ifitbelongsto]0,1[,isthesolutiontothefollowingequation:()pˆSB(1−q)E(S˜(θ,x˜(θ,α¯),α¯))+E(S˜(θ,xSB(θ,α,α¯),α))θθ+qE(S˜(θ,x˜(θ,α¯),α¯))=qνpˆSB∆α∆θv(xSB(θ,α,α¯)).()θUsing()and(),weobtain:pˆSBνv(xSB(θ,α1+∆α∆θ,α¯))=E(S˜(θ,x˜(θ,α¯),α¯)).()λθTheright-handsideispositivebycondition(H1).Hence,pˆSB>ˆSB<1isnecessarywhencondition(19)∞(θ,α)definedby(20)maximizes−qν∆α∆θv(x)+(1−q)(1−ν)S˜(θ,x,α).Therefore,eventhestrongestpossibledistortiononx(θ,α,α¯)makesitimpossibletosatisfy(18).Adistortionofpˆ
Finally,tofindthevaluesofthetransfers(T¯,Tˆ1,Tˆ2,T)(α)whichisalwayszero.•ProofofProposition5:Theconsolidationofincentive,budget-balancedandpar-ticipationconstraintsinto(18)β<1,theα¯(TP)∗:p¯RC=1⇔2(1+λ)q2E(S˜(θ,x˜(θ,α¯),α¯))>0,()θwhichholdsfrom(H1).pˆRC∈[0,1]⇔()2(1+λ)q(1−q)E(S˜(θ,x˜(θ,α¯),α¯))+E(S˜(θ,xRC(θ,α,α¯),α))θθ=2(λ+1−β)q2ν∆α∆θv(xRC(θ,α,α¯)),()pRC=0⇔2(1+λ)(1−q2)E(S˜(θ,xRC(θ,α,α),α))<2(λ+1−β)q(1−q)ν∆α∆θv(xRC(θ,α,α)).θ()Thislatterinequalityholdsfromcondition(H2).From(18)binding,andtakingalsointoaccountthatp¯RC=1andpRC=0,wegetthatpˆSB,ifitbelongsto]0,1[,isthesolutiontothefollowingequation:()pˆRC(1−q)E(S˜(θ,x˜(θ,α¯),α¯))+E(S˜(θ,xRC(θ,α,α¯),α))θθ+qE(S˜(θ,x˜(θ,α¯),α¯))=qνpˆRC∆α∆θv(xRC(θ,α,α¯)).()θUsing()and(),weobtain:νβpˆRC∆α∆θv(xRC(θ,α,α¯))=E(S˜1+(θ,x˜(θ,α¯),α¯)).()λθTheright-handsideispositivebycondition(H1).Hence,pˆRC>,onecanshowthatpˆRC<1isnecessarywhencondition(19)holds.•ProofofProposition6:Theconsolidationofincentive,budget-balancedandpartic-ipationconstraintsinto(18)(19)holds,37
theα¯(TP)∗(H1’)and(H2’)yieldstheprobabilitiesandconsumptionsinthetext.•ProofofProposition7:When(H1”)holds,wehaveE(S˜(θ,x˜(θ,α¯),α¯))>θoptimalpolicywhenthepreferencesprofileα=(α1,α2)iscommonknowledgeisthusp¯∗=pˆ∗=1,p∗=(TP)Linto(TP)∗Lbelow:{(∑)}2P∗(T)L:maxEp(α)E(S˜(θ,x˜i(θ,αi),αi)),{p(·)}αθi=1subjectto{(∑)}2Ep(α)E(S˜(θ,x˜(θ,αi),αi))αiθ∑i=12()−qE(p(α,αα−i))E(S˜(θ,x˜(θ,α¯),α¯))−E(S˜(θ,x˜(θ,α),α))≥0.()iθθi=1Todoso,λLthepositivemultiplierof()andoptimizingthecorrespondingLagrangeanwithrespecttop¯,pˆandpwithin[0,1]yields:p¯L=1⇔2(1+λL)q2E(S˜(θ,x˜(θ,α¯),α¯))>0,()θ()pˆL∈[0,1]⇔2(1+λL)q(1−q)E(S˜(θ,x˜i(θ,α¯),α¯))+E(S˜(θ,x˜(θ,α),α))(θθ)=2λLq2E(S˜(θ,x˜(θ,α¯),α¯))−E(S˜(θ,x˜(θ,α),α)),()θθpL=0⇔2(1+λL)(1−q2)E(S˜(θ,x˜(θ,α),α))θ()<2λLq(1−q)E(S˜(θ,x˜(θ,α¯),α¯))−E(S˜(θ,x˜(θ,α),α))(.)θθOfcourse,p¯∗=1andp∗=0implythat()and()¯L=1,pL=0and()into()bindingyields:()E(S˜(θ,x˜(θ,α¯),α¯))=pˆLE(S˜(θ,x˜(θ,α¯),α¯))−E(S˜(θ,x˜(θ,α),α)),(A.θ1+28)λLθθandthuspˆL>()binding,wecanderivepˆLexplicitlyas:pˆL=(qE()S˜(θ,x˜(θ,α¯),(α¯)))qE(S˜(θ,x˜(θ,α¯),α¯))−E(S˜(θ,x˜(θ,α),α))−(1−q)E(S˜(θ,x˜(θ,α¯),α¯))+E(S˜(θ,x˜(θ,α),α))θθθθ38
FinallypˆL<1whencondition(29)holds.•ProofofProposition8:WederivefromtheproofofProposition4(equation())thatλisgivenbythefollowingexpression:()(1−q)E(S˜(θ,x˜(θ,α¯),α¯))+E(S˜(θ,xSB(θ,α,α¯),α))λθθ1+=λqν∆α∆.θv(xSB(θ,α,α¯))For∆θsmallenough,xSB(θ,α,α¯)differsfromx˜(θ,α)byatermoforder∆θ.Sincex˜(θ,α)maximizesS˜(θ,x,α),thenumeratorabovediffersofE(S˜(θ,x˜(θ,α¯),α¯))+E(S˜(θ,x˜(θ,α),α))byθθtermsoforder∆θ,wehaveuptotermsoforder∆θ2ontheright-handsidebelow()(1−q)E(S˜(θ,x˜(θ,α¯),α¯))+E(S˜λ∆(θ,x˜(θ,α),α))()θ=θθ)∆x1−v′(x˜(θ,α)1+λqν∆αv(x˜(θ,α))v(x˜(θ,α))where∆x=xSB(θ,α,α¯)−x˜(θ,α)<,wegetfromtheproofofProposition7thatthemultiplierλLisgivenby()(1−q)E(S˜(θ,x˜(θ,α¯),α¯))+E(S˜(θ,x˜(θ,α),α))λL=(θθ).1+λLqE(S˜(θ,x˜(θ,α¯),α¯))−E(S˜(θ,x˜(θ,α),α))θθWethusobservethat:λE(S˜(θ,x˜(θ,α¯),α¯))−E(S˜(θ,x˜(θ,α),α))Lθθ.1+≥λλ1+λLν∆α∆θv(x˜(θ,α))Thebracketedtermintheright-handsideaboveisgreaterthanonesincewehaveE(S˜(θ,x˜(θ,α¯),α¯))−E(S˜(θ,x˜(θ,α),α))−ν∆α∆θv(x˜(θ,α))=θθ(1−ν)(S˜(θ,x˜(θ,α¯),α¯))−S˜(θ,x˜(θ,α),α¯))>0bydefinitionofx˜(θ,α¯).Finally,wegetthatλ>λˆSBandpˆ(uptotermsoforder∆θ2)(1+λ)E(S˜(θ,x˜(θ,α¯),α¯))pˆSB=θ,ν∆θ∆αv(x˜(θ,α))and(1+λL)E(S˜(θ,x˜(θ,α¯),α¯))pˆ(θL=).E(S˜(θ,x˜(θ,α¯),α¯))−E(S˜(θ,x˜(θ,α),α))θθSinceλL<λandE(S˜(θ,x˜(θ,α¯),α¯))−E(S˜(θ,x˜(θ,α),α))>ν∆θ∆αv(x˜(θ,α)),wegetpˆL<θθpˆ
