Miguel A Sanchez Rafael HortalaVallve London School of Economics and Political Science The Toyota Centre Suntory and Toyota International Centres for Economics and Related Discipli ID: 290193
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Hierarchic contracting Miguel A. Sanchez & Rafael Hortala-Vallve London School of Economics and Political Science The Toyota Centre Suntory and Toyota International Centres for Economics and Related Disciplines DARP 73 London School of Economics September 2005 Houghton Street London WC2A 2A Telephone: (+44 020) 7955 6674 We thank Frank Cowell, Jordi Blanes, Matthias Dahm and seminar participants at the London School of Economics for helpful comments and discussions. The second author gratefully acknowledges financial support from the Fundación Rafael del Pino. 1Introduction1.1MotivationProbablyoneofthemostimportantissuesregardingeveryorga-nizationishowtostructureitsoperation.Thesolutiontothisquestionencompassessomanyfactorsthatathoroughanswerisalmostimpossible.Inthepresentpaperwefocusonjustoneas-pectofit,namely,thecontractingstructure.Certainlytheallocationofcontractingrightsisimportantbothinthepublicsector(e.g.decentralizationofpublicservices)andprivatesectors,withinandbetween…rms(e.g.teamproduction,producingwithinthe…rmoroutsourcing,franchising).Ourgoalistohighlighthowthecontractingstructurea¤ectsthee¢ciencyoftheorganizations.Ourmodelconsistsofaprincipalwhohiresagroupofagentsinordertojointlyproduceanoutputinasituationwherethisoutputistheonlycontractiblevariable.Thecombinationofthesetwoelementsgeneratestheessentialingredientinouranalysis:apositiveexternalitybetweenagents.Sinceoutputisjointlyproducedandwagesarecontingentonit,thee¤ortofanyagentincreasesnotonlyherownwagebutalsoeveryotheragent’swages.However,eachplayer’sdecisiontakesintoaccountonlyherprivatebene…ts(wage)andcosts(riskplusdisutilityofe¤ort),andtheseare,duetothepositiveexternality,strictlylowerthanthesocialbene…ts.Asaconsequence,eachagentexertstoolittlee¤ortcomparedtothee¢cientsituation.ThemodeldepartsfromHolmstrom’steamproductionmodel(Holmstrom(1982))duetoourassumptionthatoutputisnotdeterministic,giventhee¤ortexertedbyeachmemberoftheteam.Hencetheprincipalcannoto¤erabudget-breakingcon-tracttoachieve…rstbest.Instead,theprincipalfacestheclassi-caltrade-o¤betweenincentivesandinsurance.Whentheriskislow,theprincipalisabletoprovideclose-to-…rst-bestincentivestotheagents(i.e.,incentivessuchthatprivatebene…tsareclosetothesocialones)butastheriskinessoftheprojectincreases,theprincipalcannolongerprovidesuchincentivestotheagents(itistoocostly)andhencethemisalignmentofincentivesbecomes1 TheseminalreferenceonmoralhazardinteamsisHolmstrom(1982).Aswesaidabove,ouranalysisdepartsfromitbecauseoftheassumptionthatthetechnologyisstochasticinsteadofdeterministic.MostrelatedtoourworkisItoh(1991),thoughinhismodeldi¤erentagentsproducedi¤erentnon-deterministicoutputsandtheissueiswhethertheprincipalisbettero¤bypro-vidingincentivesforthemtocooperate.1Notethatthefactthatinoursettingoutputisjointlyproducedprecludes"relativeper-formance"incentiveschemes,theimplementationoftournaments(asinLazearandRosen(1981))orcollusionamongagents.TheclosestreferencestoourstudyarethoseofMacho-StadlerandPerez-Castrillo(1998)andJelovacandMacho-Stadler(2002).Theiranalysesconsistofcomparingdi¤erentcontractingstruc-turesinabinarye¤ortmodelbuttheirstudyisbasedonthedi¤erenttimingofeventsratherthanthee¤ectoftheexternalitybetweenagents.Moreovertheirresultsdivergefromours.They…ndthatthehierarchicstructuredoesnotalwaysperformbetterthanthecentralizedone.AdetailedcomparisonofourresultstotheirsisprovidedinSection4.Therestofthearticleisorganizedasfollows:InSection2wepresentthemodelandwesolveitunderthedi¤erentcontractingstructures:centralizedanddecentralized.InSection3wecom-parebothstructuresanddiscussthehigherperformanceofthehierarchicone.Section4considerstherobustnessoftheresultstoalternativespeci…cationsandexploressomeextensions.Finally,Section5concludes.2TheModelThemodelpresentedbelowisbasedonHolmstromandMilgrom(1987). 1Alternatively,Itoh(1994)assumesjointproductionbutconsiderswhetheritisbestfortheprincipaltodelegatethetaskspartiallyorcompletely.Inthelattercasehealsocomparesthecaseswheretasksareundertakenbythesameindividualorbydi¤erentones.Hence,hestudiesthee¤ectoftheallocationoftasksinthepresenceofexternalitiesanddi¤erentdegreesofcostssubstitutabilitiesratherthanthecontractingstructure.3 Noticealsothatbothagentsaretreatedequally,theirparticipationconstraintsarebindingandtheexpectedpro…toftheprincipalattheoptimum(=1)istwicethatofthesingleagent…rstbestproblem.2.2CentralizedSecondBestAstandardassumptioninmoralhazardsettingsisthattheprin-cipalisnotabletodirectlycontractone¤ort.Thisistypicallyassumedtobetheconsequenceofade…cientortoocostlymoni-toringtechnology.Whichevertherationale,whenwemovefromthe…rstbestsettingwherecontractscanbecontingentone¤ort,theprincipalfacesthecustomarytrade-o¤betweenincentivesandinsurance.We…rstconsiderthestandardsecondbestsituation,wheretheprincipalcontractsdirectlywithbothagents.Inthesequelwecallthatsituationthecentralizedsecondbest.Thenweanalysethehierarchicsecondbestsituation,wheretheprincipalcontractswithoneoftheagents,andthisagentsubsequentlycontractswiththeremainingone.Duetothenon-contractibilityofe¤ort,theprincipalcano¤ercontractscontingentonlyonoutput.Hence,shemustdesignthecontractsinsuchawaythattheyprovidetherightincentivesfortheagentstoexerte¤ort.Technically,thisimpliesaddingtothe…rstbestprogramtwonewconstraints,theincentivecompatibilityconstraints(denotedIC1andIC2hereafter).Theprogramoftheprincipalreadsnowasfollows:maxfw1(x);e1;w2(x);e2gExfV(x)gs.t.8���������&#x]TJ ;� -2;.51; Td;&#x[000;&#x]TJ ;� -2;.51; Td;&#x[000;&#x]TJ ;� -2;.51; Td;&#x[000;&#x]TJ ;� -2;.51; Td;&#x[000;&#x]TJ ;� -2;.51; Td;&#x[000;&#x]TJ ;� -2;.51; Td;&#x[000;&#x]TJ ;� -2;.51; Td;&#x[000;&#x]TJ ;� -2;.51; Td;&#x[000;&#x]TJ ;� -2;.51; Td;&#x[000;:ExfU(w1(x);e1)gU0ExfU(w2(x);e2)gU0e12argmax_e1ExfU(w1(x);_e1)ge22argmax_e2ExfU(w2(x);_e2)g6 lowergiventhattheexpectedoutputoftheprojectistwicethatinthesingleagentcase.Thishighlightstheessentialelementthatisincorporatedintotheproblemwhenwemovefromthesingle-agenttothemulti-agentcase:thereisnowapositiveexternalitybetweentheagents.Sinceoutputisproducedjointlyandistheonlycontractiblevari-able,thewageofeachagentisafunctionofthee¤ortsofalltheagentsinvolvedinproduction,notonlyherown.Whenanagentincreaseshere¤ort,sheincreasestheexpectedwagesofalltheagents.However,theprincipalisunabletointernalizethisexternalitywhencontractinginacentralizedmanner.2.3HierarchicSecondBestNeverthelesswecanimproveuponthesecond-bestsolutionsimplybymodifyingthecontractingstructureofthemodel.Themainintuitionisthatbytransferringcontractingrightsto,say,agent1(henceestablishingahierarchybetweentheagents),sheinter-nalizesthepositiveexternalityhere¤orthasonagent2.Thisissobecausesherealizesthatanincreaseinherowne¤ortdoesnotonlyincreaseherexpectedsalarybutalsoincreasesthesalaryofhersubordinate,thusrelaxingPC2.Wecanclearlyseethattheprivatebene…tsofagent1havenowincreasedwithrespecttotheonesshehadunderthecentralizedstructure.Consequently,sheiswillingtoexertmoree¤ortatnoextracostfortheprincipal.Ultimately,theprincipalextractsallthisextrasurplusfromagent1andhencethechangeinthecontractingstructureresultsinahigherexpectedsurplusfortheprincipal.4Underthehierarchicstructure,theprincipalcontractswithonlyoneagent,whothensubcontractswiththeremainingone.Thetimingofthegameisasfollows:theprincipalo¤ersacontractW(x)=A+Bxtoagent1andsubsequentlyagent1o¤ersacontractw2(x)=a2+b2x(s.t.a2Aandb2B)toagent2;thentheagentsdecidetheiroptimale¤ort(IC)and,iftheirPCs 4Notethatthehierarchicstructureinternalizesthepositiveexternalitythatagent1hasonagent2(henceequatingherprivatebene…tstohersocialbene…ts)butobviouslydoesnota¤ectagent2’sprivatebene…ts.8 powerofhercontract(e1=Bande2=b2).However,inthiscaseagent1exertsastrictlyhighere¤ortthanagent2(B�1+ 1+2 B).Notethatata…rstglancethisdoesnotseemoptimalfromtheprincipal’sperspective:giventheconvexdisutilityofe¤orttheoptimalallocationshould,apriori,involvebothagentsexertingexactlythesamelevelofe¤ort.ThederivativeoftheLagrangianofthepreviousprogramwithrespecttoe1showsclearlyhowagent1internalizesthepositiveex-ternalitysheexertsonagent2throughitsparticipationconstraint.TheLagrangianofthepreviousprogramis:L=(A�a2)+(B�b2)E� 2(B�b2)2�1 2e21++a2+b2E� 2b22�1 2e22+(e2�b2)anditsderivativewithrespecttoe1shouldbeequaltozeroattheoptimum:@L @e1=(B�b2)�e1+b2=0Theaboveconditiondi¤ersfromtheIC1underthecentralizedstructureduetothefactthatitincludestheterm(b2).Thattermcapturespreciselythepositiveexternalitythatagent1hasonplayer2.Givenagent1’soptimalbehaviourwecanmoveonestageback-wardsandconsidertheprincipal’sproblem.Herprogramreadsasfollows:maxfw1(x);e1;w2(x);e2gExfV(x)gs.t.8��������&#x]TJ ;� -2;.52;&#x Td[;&#x]TJ ;� -2;.52;&#x Td[;&#x]TJ ;� -2;.52;&#x Td[;&#x]TJ ;� -2;.52;&#x Td[;&#x]TJ ;� -2;.52;&#x Td[;&#x]TJ ;� -2;.52;&#x Td[;&#x]TJ ;� -2;.52;&#x Td[;&#x]TJ ;� -2;.52;&#x Td[;:ExfU(w1(x);e1)gU0e1=Be2=b2=1+ 1+2 Ba=ah(B)10 Proposition1Thehierarchicstructurealwaysyieldsahigherex-pectedpro…ttotheprincipalthanthecentralizedone.Therelativegainsarehigherthehighertheriskoftheprojectand/ortheindexofriskaversionoftheagents. Figure1:Expectedpro…tsTheproofisimmediatefromcomparing:=1 1+ andh=1 2(2+3 )2 (1+2 )(2+3 + 2)andshowingthath=isincreasingon .Figure1showsbothexpectedpro…tsintermsof .Proposition2Underthehierarchicstructure,agent1exertsmoree¤ortthanagent2.Moreover,bothagentsexertmoree¤ortthanunderthecentralizedstructure,i.e.eh1�eh2�e1=e28 �0:12 downe¤ectthatmultipliestheinitial…rst-ordere¤ectofanin-creaseinBhbyincreasingalsobh2.However,thissecond-ordere¤ectisalwayslowerthanoneanddecreaseswith .Thatis,wheneverweincrease agent1islesswillingtoinducehigherin-centivesonagent2(itbecomesmorecostly).Figure2showsthee¤ortsofbothagentsunderbothcontractingstructuresasfunc-tionsoftheparameter :Inparticular,wecanseethattheriskbornebytheagentsunderthehierarchicstructuretendstozeroas !1;whichisconsistentwiththederivativeabove.ItisalsoeasytoseethatadirectconsequenceofProposition2isthattheexpectedoutputunderthehierarchicstructureisgreaterthanunderthecentralizedone.Noticethattheslopesofthewagecontracts,besidesprovidinginformationontheoptimallyinvestede¤orts,indicatealsotheriskbornebytheagents.Giventhecertaintyequivalent,theriskbornebyagentiisequaltor 2Var(wi(x))= 2b2i.Thus,agent2’sriskis 2�bh22;andagent1’sis 2�Bh�bh22.Notethatagent1’sriskdependsonhernetwage(W(x)�w2(x)=A�a2+(B�b2)x).However,herinvestede¤ortdependsonthegrosswagegiventhatsheinternalizesherpositiveexternalityonagent2.Proposition3Underthehierarchicstructure,agent1bearslessriskthanagent2.Moreover,thelatterbearsmoreriskunderthehierarchicstructurethanunderthecentralizedone.Formally,thepropositioncanbere-writtenasfollows:8:bH&#x]TJ ;� -2;.52;&#x Td[;BH�bH;8 0bH&#x]TJ ;� -2;.52;&#x Td[;b;8 &#x]TJ ;� -2;.52;&#x Td[;0andisdepictedinFigure3.6 6Aquestionthatbecomesrelevantonceweallowforheterogeneousagentsonthelevelofriskaversioniswhethertheprincipalshouldcontractwiththemostortheleastriskaverseagentunderthehierarchicstructure.GivenProposition3theanswerisimmediate:theleastriskaverseshouldendupatthebottomgiventhattheagentatthetopisassumingthelowerlevelofrisk.Itiseasytoprovethatinthiscasetheprincipalisstillbettero¤underthehierarchicstructurethanunderthecentralisedone.14 atedwithmorerisk.Onceagain,thereasonisthatthehierarchicstructureprovidesextraincentivestoagent1throughtheinternal-izationofthepositiveexternalityshehasontheremainingagent.Whatarethemechanismsthatmakehierarchiesmoree¢cientthancentralizedstructures?Imagine,forillustrativepurposes,thattheprincipalo¤erstoagent1,underthehierarchicstructure,theoptimalcentralizedaggregatewagecontract(A=2aandB=2b).Agent1’soptimale¤ortise1=Bandshecano¤eragent2thecentralizedsecondbestcontract.Inthatcaseagent2investsthesameef-fortasinthecentralizedcasebutnowPC2isnotbindinggiventhatagent1isexertingtwicethee¤ortofthatinthecentralizedcase.Hence,agent1optimallyadjuststhecontracto¤eredtoagent2untilPC2isbinding(i.e.increasingagent2’sincentivesanddecreasingherownrisk),thusrelaxingherownparticipationconstraint.Ultimately,theprincipalextractsalltheextrasurplusobtainedbyagent1bymakingPC1bindingagain.Henceweseethatagent1exertsane¤orthigherthane1sincesheinternal-izesthepositiveexternalityshehasonagent2.Asaresultofthat,wealsoobserveasecondordere¤ectonagent2’se¤ort(thederivative):giventhatherparticipationconstraintisnotbindinganymore,sheiso¤eredaslightlymodi…edcontractwheremoreincentivesandlessinsuranceareprovided.Atthehierarchicoptimumtheprincipaldoesevenbetterando¤ersadi¤erentcontractfromtheaggregatecentralizedone.4DiscussionInspiteoftheconvexityofthedisutilityofe¤ortandtheriskaver-sionofagentsthehierarchicstructuredistributesasymmetricallytheriskandthee¤ortamongtheagents.But,regardlessoftheine¢ciencyinherentinthat,theinternalizationoftheexternalitymakestheprincipalbettero¤overall.16 Figure4:Limitedliabilitythelimitedliabilityconstraintbites).InFigure4wehavedepictedthesethresholdsintermsoftheparameter .Notethatthecurvethatcorrespondstothehierarchicnetcontractofthe…rstagentisalwayslowerthanthetworemainingcurves.Thatis,thelimitedliabilityconstraintofagent1underthehierarchicstructurebitesatalowervalueofrealizedoutputthantheoneofagent2.Thisindicatesthatagent1isabletoensurethatshegetsanegativepaymentlessoftenthanagent2(thatisconsistentwithProposi-tion3-agent1bearslessrisk-andwiththefactthattheprincipalisabletoinduceahighlevelofe¤ortonagent1atalowlevelofrisk).Whencomparingthecentralizedandhierarchicstructures,wecanseethatforlowlevelsof thehierarchicstructureismorea¤ectedbythelimitedliabilityconstraint(inparticular,agent2’srealizationofwagesisnegativeforalargerrangeofoutputrealiza-tions).However,forlargevaluesof ,limitedliabilityconstrainsthecentralizedstructuremore.8 8Theheuristicalargumentthatlimitedliabilityconstraintsbitelessunderthehierarchicstructureforhighlevelsof ;togetherwiththefactthatthe18 onlyinternalizingtheexternalitybutisalsomodifyingthewageo¤eredtohersubordinatewithrespecttothesecondbestsitua-tion.Nevertheless,heroptimalchoiceofe¤ortdoesnotdependonthedistributionofthewagesanddependsonlyonthetotalwages,giventhatsheinternalizescompletelyherexternalityonagent2throughherparticipationconstraint.Whenweconsideramoregeneralsettingtheanalysisbecomesambiguouspreciselybecausetheprevioustwoobservationsnolongerhold.Thatimpliesthatwhenweanalysethe…rstagentprogramunderthehierarchicstructureothere¤ectsmayoutweighthepos-itivee¤ectsofinternalizingtheexternality.Suppose…rstthatthenewsettingisstillsuchthattherearenostrategiccomplemen-taritiesorsubstitutabilities.Inthatcase,agent1mayswitchthecontractingstructureinsuchawaythatsheinducesadif-ferentriskstructureonagent2withrespecttotheonethattheprincipalwouldo¤erunderacentralizedstructure.Thiswouldnotmatteriftheinternalizationoftheexternalitymadeagent1’soptimale¤ortonlycontingentonthetotalwages(W(x))butingeneralthatisnottrue.Onecouldimagineasituationwheretheincentivesofagent1onsettingagent2’scontractmaymaketheoveralle¤ectofthehierarchicstructureworsefortheprincipal.Thepresenceofstrategiccomplementaritiesseemstoreinforcethepre-eminenceofthehierarchicstructureoverthecentralizedone.Butstrategicsubstitutabilitiesworkagainsttheincentivesofagent1toinvestmoree¤ortunderthehierarchicstructureandhencemayoutweighinsomecasesthepositivee¤ectsthatresultfrominternalizingtheexternality.Lastly,therootofthebene…tsofthehierarchicstructureoverthecentralizedoneisthatagent1internalizestheexternality.Forthattobethecaseweneedtoassumethatagent1isabletohavea…rstmoveradvantage.Otherwise,shewouldchoosehere¤ortaccordingtoherownnetwageregardlessofthepositiveexternalityshehasonagent2(technically,shedoesnotchooseherowne¤ortsubjecttotheparticipationconstraintofagent2andthusshedoesnotinternalizethepositivee¤ectthatherowne¤orthasontheparticipationconstraintofagent2;thekeyisthatagent1cantakeadvantageoftheinteractionofhertwochoice-variables(e1andw2(x)).Webelievethatthisisthemostnaturalsituation:the20 andbearlessriskthanthoseinlowertiers.Thecrucialfactorthatexplainsthepre-eminenceofadecen-tralizedstructureoverthecentralizedoneisthatthe…rstonegivesagentsmoredecisivepower(contractingrights).Thisresultsintheinternalizationoftheexternalitybyagent1and,duetothealignmentofthelatter’sandtheprincipal’sincentives,inhigherpro…tsfortheprincipal.Nostructurewheretheprincipalretainsallcontractingrightscan,therefore,outperform(orevenmatchtheperformanceof)thehierarchicone:theexternalitywouldnotbeinternalised.Moreover,theseresultsholdinspiteofassuminghomogeneousagents.Summingup,theaboveanalysismeansthathierarchicstruc-turesshouldnotonlymatterforsupervisingande¢cientlyallo-catingtasksbutmayintroducegainssimplybecauseofthechangeinthecontractingstructure.Itsrelevanceseemstoariseinseveraldistinctbuildingblocksofeconomiclife,andsothepresentanaly-siscanbeeasilyappliedtoissuessuchasdecentralizationwithinthegovernment,privatizationofpublicservices,hierarchiesanddelegationwithinthe…rm,subcontracting,outsourcing,franchis-ingandsharecropping,amongothers.22 [13]Prendergast,C.(1995).ATheoryofResponsibilityinOrga-nizations.JournalofLaborEconomics13(3),387-400.24