CoordinatingInvestment,ProductionandSubcontractingJanA.VanMieghem(vanm - PDF document

CoordinatingInvestment,ProductionandSubcontractingJanA.VanMieghem(vanmieghem@kellogg.nwu.edu)Nov10,1998(Extendedversionwithproofs;abridgedversionappearedinManagementScience45(7)954-971,July1999)AbstractWevaluetheoptionofsubcontractingtoimprovenancialperformanceandsystemcoordi-nationbyanalyzingacompetitivestochasticinvestmentgamewithrecourse.Themanufacturer theirproductionlevelsinstagetwo,constrainedbytheirearlierinvestmentdecisions.Subcontract-ingisviewedasatradeofthesupplier’sproductforthemanufacturer’smoney.Section2analyzestwoscenarios(thecentralizedrmvs.twoindependentrmswithoutanysubcontracting)forperformancereference.InSection3westudyprice-onlycontractswhereanex-antetransferpriceissetforeachunitsuppliedbythesubcontractor.Wecharacterizethesub-gameperfectin-vestmentstrategyandformulateanoutsourcingthresholdconditionintermsofthemanufacturer’sinvestmentcost.Ahighertransferpricemayincreasethemanufacturer’sprot.Thissuggeststhataprice-focusedstrategyformanagingsubcontractorscanbackreonthemanufacturer.Whilealowerpriceallowscheapsupply,itdoesnotguaranteeitsavailability.Ourmodelconrmsthatop-timalmanufacturerandsuppliercapacitylevelsareimperfectsubstituteswithrespecttocapacitycostsandcontributionmargins.Wealsoshowthatmanufacturerswillindeedsubcontractmorewhenthelevelofmarketuncertainty(risk)increasesandwhenmarketsaremorenegativelycorre-lated.Similartonancialoptions,thisincreasestheoptionvalueofsubcontracting(realassets).InSection4westudytwoothercontracttypes.Oneusestheincompletecontractingapproachnoexplicitcontractscanbemadeandbothpartiesnegotiateoverthesubcontractingtransfer.Thisallowsustoanalyzetheroleofthe“bargainingpower”ofthecontractoronoutsourcingdecisionsandsystemperformanceimprovement,whichmaybegreaterthanwithprice-onlycontracts.Thelattersuggeststhatsometimesrmsmaybebetteroleavingsomecontractparametersunspeci-edex-anteandagreeingtonegotiateafterdemandisobserved.Ourthirdcontracttypeconsistsstate-dependentprice-onlyandincompletecontractsforwhichweshowanequivalenceresult.Whilesubcontractingwiththesethreecontracttypescancoordinateproductiondecisionsinthesupplysystem,onlystate-dependentcontractscaneliminatealldecentralizationcostsandcoordi-natecapacityinvestmentdecisions.Wepresenttheminimallysucientprice-onlycontractthatachievescoordination.Section5closeswithadiscussionofmorecomplexcontractsintheliteratureandsuggestionsforfurtherwork.Manyliteraturesdiscussthecostsandbenetsofsubcontracting.Accordingtothestrategyliterature,subcontractingandoutsourcingoccurbecausearmmaynditlessprotableorinfea-sibletohaveallrequiredcapabilitiesinhouse:“armshouldconcentrateonitscorecompetenciesandstrategicallyoutsourceotheractivities[19]”and“notonecompanybuildsanentireightve-hicle,noteventhesimplestlightplane,becauseoftheexceptionalrangeofskillsandfacilitiesrequired[1]”.Subcontractingandoutsourcingmayalsobe“animpetusandagentforchange”and“mayimproveundulymilitantorchange-resisting”employeerelations[4].Thesebenetscomeatacostbyexposingthecontractortostrategicrisks,suchasdependenceonthesubcontractor(withitsinherentlossofcontrolandassociatedhold-uprisk)andvulnerability(e.g.,lowerbarrierstoentryandlossofcompetitiveedgeandcondentiality)[19].Theoperationsliteraturehighlightstheexibilitythatsubcontractingoerstoproductionandcapacityplanning.Likedemandandinventorymanagement,subcontractingallowsforshorttermcapacityadjustmentsinthefaceoftemporaldemandvariations.Subcontracting,however,hasthedistinguishingfeaturethatit“re-quiresagreementwithathirdpartywhomaybeacompetingrmwithconictinginterests[14].”(Theimplicationbeingthatanyreasonablemodelofsubcontractingmustincorporatemultipledecisionmakers.)Fromanancialperspective,themainreportedbenetsofsubcontractingandoutsourcingareloweroperatingcostsandlowerinvestmentrequirementsforthecontractor,andthespreadingofriskbetweenthetwoparties.Empiricalstudiesreportthatcosteciencyistheprimemotivationforoutsourcingmaintenance[4]andinformationsystems[16].Itisalsoarguedthatcon-tractors‘pushthehighrisk’ontosubcontractorsbyhavingthem“carryadisproportionateshareofmarketuncertainties[8].”Thenancialcostsofsubcontractingincludedecreasedscaleeconomiestothecontractor[10]andthetransactioncostsresultingfromtheinitiationandmanagementofthecontractingrelationship[19].Finally,anextensiveeconomicsliteraturediscussesourtopicwhen studyingverticalintegrationbutthatliteraturegenerallyignorescapacityconsiderations.Fewpapersexplicitlystudyananalyticmodelofsubcontracting.KamienandLi[14]presentamulti-period,gametheoreticaggregateplanningmodelwithgivencapacityconstraintsandshowthattheoptionofsubcontractingresultsinproductionsmoothing.Kamien,LiandSamet[15]studyBertrandpricecompetitionwithsubcontractinginadeterministicgamewithcapacityconstraintsimplicitintheirconvexcoststructure.Hanson[11]developsandempiricallytestsamodeloftheoptimalsharingoftheownershipofagiven,exogenouslydeterminednumberofunitsofanassetbetweenamanufacturerandasubcontractor.Tournas[20]capturesasymmetriesinin-houseinformationinaprincipal-agentmodelandcomparesthemwiththebargainingcostofacaptiveoutsidecontractor.BrownandLee[5]proposeaexiblereservationagreementinwhichamanufacturermayreservesuppliercapacityintheformofoptions.Finally,thereissignicantliteratureonoutsourcinginsupply-chains.CachonandLariviere[6]giveanoverviewofvariouscontracttypes,whichwillbediscussedinmoredetailinSection5.Ourmodelisdierentinthatthecapacityinvestmentlevelsofboththemanufacturerandthesubcontractoraredecisionvariables.Ourmulti-variate,multi-dimensionalcompetitivenewsvendorformulationisanextensionofunivariate,one-dimensionalsupplymodelsandoftheunivariatecompetitivenewsvendormodelsofLi[17]andLippmanandMcCardle[18].Themulti-variatedemanddistributionallowsustoinvestigatetheimportantroleofmarketdemandcorrelationandprovidesagraphicalinterpretationofthesolution.Ourmulti-dimensionalmodelallowsustostudytheimpactofsubcontractingonbothplayers’in-houseinvestmentlevelsandonthebuyer’soutsourcingdecision,whichispre-assumedincaptive-buyercaptive-suppliermodels.Weshowthatthehighercomplexityofsubcontracting(twocapacitydecisions)makescoordinationmorecultcomparedtotraditionaloutsourcingmodels(onlysuppliercapacity)insupplychains;weexplicitlydistinguishcoordinationofex-anteinvestmentdecisionsfromex-postproductionandsalesdecisionscoordination.Finally,wehavechosentomakebothmodelsessentiallysingle-periodandtopositnoinformationasymmetriesbetweenthetwoparties.Thereforeweshallnotdiscusshowsubcontractingcansmoothproductionplansovertime,createormitigateinformationasymmetryproblems,oraectthelong-runcompetitivepositionoftherms.2ASubcontractingModel2.1TheModelConsideratwo-stagestochasticmodeloftheinvestmentdecisionprocessoftworms.Instageonemarketdemandsareuncertainandbothrmsmustdecideseparately,yetsimultaneously,ontheircapacityinvestmentlevels.Atthebeginningofstagetwo,marketdemandsareobservedandbothrmsmustdecideontheirproductionlevelstosatisfyoptimallymarketdemands,constrainedbytheirearlierinvestmentdecisions.Atthisstage,bothrmshavetheoptiontoengageinatrade.Thesubcontractorcansupplythemanufactureratransferquantityinexchangeforapayment,asshowninFigure1.Beforeweexplainthespecicsofthesupplycontractinthenextsection,letusdiscussmodelfeatures,notationandtworeferencescenariosthatareusefulinevaluatingtheimpactofsubcontractingonrmperformance.Intherstreferencescenariobothrmsoperatecompletelyindependentofeachotherandsubcontractingisnotanoption(i.e.,transferquantity).Bothrmsgosoloandeachwillselltoitsownmarket.Forsimplicity,wewillassumethatbothrmshaveexclusiveaccesstotheirrespectivemarkets.Becausethesubcontractorlackstheassembly,marketingandsalescloutofthemanufacturer,shedoesnothavedirectaccesstomarket.Inpractice,however,themanufacturermayhaveaccesstomarketthroughwhollyownedupstreamsubsidiariesthatprovidethemand xM KM Market Market Manufacturer Figure1:TheSubcontractingModel.otherswithpartsorsubsystems.GeneralMotors,forexample,ownsDelphiAutomotivesthatsuppliesGMandotherautoassemblerswithbrakesystemsandotherparts.Atthesametime,GMmultisourcessomepartsfromoutside,independentsubcontractors.Thus,marketwouldrepre-senttheendmarketforcarsandmarkettheintermediatemarketforparts.GMcouldcompeteinmarketbutwewillabstractfromsuchcompetitiontohighlightthesubcontractingoption.Also,noticethatdirectsourcingfrommarketinsteadoffromthesubcontractorisnotanoptionforthemanufacturer.Thismodelingassumptionreectstherelationship-specicinformationtypicallypresentinsubcontractinganditimpliesthatwearenotdiscussingthepurchaseofstandardized,-the-shelfproductsincommoditymarkets.Thesecondreferencescenariorepresentstheotherextremeinwhichbothrmsareintegratedandcontrolledbyasingledecisionmaker.Inthiscentralizedscenariotheintegratedrmwillservebothmarkets.Subcontracting,then,istheintermediatescenarioinwhichbothrmsareindependentlyownedsothatwehavetwodecisionmakers,yettradingispossible.(Thusthesubcontractor’stechnologyissucientlyexiblethatitcanproducethesameproductasthemanufacturer’stechnology.)denote’scapacityinvestmentlevel,where.Firmisassumedtofaceaconstantmarginalinvestmentcost,sothatitscapacityinvestmentcostislinearintheinvestmentlevel.Productionlevelsandarelinearlyconstrainedbythecapacityinvestmentlevels:.Forsimplicity,weassumethatbothrmsmakeconstantcontributionmarginsperunitsoldinmarket.Stronger,wewillassumezeromarginalproductioncostssothatrepresentsthexedsalespriceinmarket.Toavoidtrivialsolutionsweassumethat.Letdenotetheproductdemandinmarket.LikeKamienandLi[14],weassumesymmetricinformationinthesensethateachrmhascompleteinformationabouttheother’scostandprotstructureandinvestmentlevel,andtheyshareidenticalbeliefsregardingfuturemarketdemands.Thesebeliefscanthenberepresentedbyasingle,multi-variateprobabilitymeasure.Forsimplicity,weassumethatmarketdemandsarenitewithprobabilityoneandthathasacontinuousdensityonthesamplespace.TheexpectationoperatorwillbedenotedbyE.Weassumezeroshortagecostsandzerosalvagevaluesforbothproductsandproductionassets.Finally,bothrmsareassumedtobeexpectedprotmaximizersandtheresearchquestioncanthusbeformulatedinthetworeferencescenariosasfollows. Relaxationoftheseassumptionstoincludeconvexinvestmentcosts,market-and-rmspecicunitcontributionmargins,shortagecostsandsalvagevalues,andnon-unitcapacityconsumptionratesisrelativelystraightforward(asshownin[12])attheexpenseofaddednotationalcomplexity. KS !2K+KM DS M!3!4!5!6!1!0 Xa DXb Figure2:Productiondecisionsandtotalmarketsupplyvector,representedbyarrows,dependonthedemandrealizationandthescenario.2.2Independents:GoingSoloWhenbothrmsdonotsubcontract,eachrmdecidesonitsproductionandsalesdecisionstagetwobymaximizingitsoperatingprosubjecttothecapacityconstraintandthedemandconstraint.This‘productmix’linearprogramhasoptimalactivitylevelsolo=min(withprosolosolo.Instage1,choosesitsoptimalinvestmentlevelsolosoastomaximizeitsexpectedrmvalue,denotedby,whichistheexpectedoperatingprotminusinvestmentcosts:solo=argmaxsolowheresolo)=EsoloK,D(1)Acriticalfractilenewsvendorsolutionisoptimal:solo,whereisthetaildistri-butionof.Tobuildsomeintuitionforthesolutiontechniquethatwillbeusedbelow,letussummarizebrieyhowthisfamiliarresultcanbederivedusingthemulti-dimensionalnewsvendormodelofHarrisonandVanMieghem[12].Itwillbeconvenienttoletcapacityvectorpartitionthedemandspaceinto7regions···,asinFigure2(whereweabbreviatedthesumofthecomponentsof).Therectangularregionistheca-pacityregionofthistwo-rmsupplysystemwithoutsubcontracting.Wheneverisoutsidethiscapacityregion,somedemandcannotbemetandtheoptimalmarketsupplyrepresentedbyanarrowemanatingfrom,willbeonthecapacityfrontier.LinearprogrammingtheoryyieldsthattheprotvectorsoloK,Disuniqueandconcavein.Thus,thelinearsuperpositionsoloK,Dandthussoloarealsoconcavesothattheorderconditionsof(1)aresucient: solosoloandsolosolowheresoloistheoptimalLagrangemultiplierofthenon-negativityconstraint.Invoking[12],gradientandexpectationcanbeinterchangedtoyieldsolosolowhere’scapacityshadowvalue: .Theshadowvaluewhichistheoptimaldualvariable’sproductionlinearprogram,equalsaconstantineachdomainofFigure2.Thus,theexpectedmarginalprotcanbeexpressedas .Tosimplifynotation,neamatrixwhose-thcolumnistheshadowvectorindomain.Similarly,neavectorwhose-thcoordinateistheprobabilityofdomain Whenbothrms“gosolo”themarginalvectorissolosolosolosolosoloBecausecontributionmarginsexceedinvestmentcosts()bothrmswillinvest(soloandtheoptimalityequationsdirectlyyieldthefamiliarnewsvendorsolutions.2.3CentralizationWhenbothrmsarecontrolledbyonecentraldecisionmaker,theoptimalproductionandsalesvectorinstagetwomaximizessystemoperatingprot,subjecttosystemcapacityanddemandconstraints.Transfersarepossibleandoptimalactivitylevelscenandprocenarethesolutionoftheproductmixlinearprogram:cen=max(2)s.t.Theoptimalinvestmentvectorcenmaximizesexpectedsystemvalue:cen=argmaxcenwherecen)=EcenK,D(3)Theoptionoftransfersenlargesthesupplysystem’scapacityregionto,orinshort.Usingthisshorthandnotation,if23456,demandexceedssupplyandtheoptimalsupplyvectorwillbeontheboundaryofthecapacityregion.Thelinearprogram(2)canbesolvedparametricallyintermsof(therebydirectlymanifestingthedomainsnedearlier).Ifmarketyieldshighermarginsthanmarket,itgetspriorityinthecapacityallocationdecisionyieldingmarketsupplyvectorinFigure2.OtherwisemarketgetspriorityyieldingvectorinFigure2.Asbefore,cenK,DisconcaveandtheshadowvectorK,Dconstantineachdomainsothattheoptimalcapacityvectorcensolvescencencenandcen,where00min(min(max(min((4)-capacityislessexpensivethan-capacity(),itisprotabletoinvestinbothtypesofcapacity(cen).Otherwise,itisoptimaltosupplybothmarketsusingonlythecheaper-capacity:cenand.IntheAppendixof[23]weshowthatcenisstrictconcaveatcensothattheoptimalinvestmentvectorisunique.Wenowhavecompletelycharacterizedtheoptimalinvestmentstrategiesinbothreferencescenarios.Clearly,systemvaluesundercentralization(weakly)dominatethosewhenbothplayersgosolo:censolosolosolo.Thevaluegapsolosolocapturesthecostsofdecentralization.Intheremainderofthisarticle,wewillinvestigatehowsubcontractingcandecreasethevaluegapandwhetheritcan“coordinate”thesupplynetwork(thatis,eliminatethevaluegap).3SubcontractingwithPrice-OnlyContractsprice-onlycontractspeciesex-antetobothpartiesthetransfer(or“wholesale”)pricethatthemanufacturermustpayforeachunitsuppliedbythesubcontractor.Becausethissimple contractdoesnotspecifyatransferquantityoranyothermodelvariables,itcannotforceapartytoenterthesubcontractingrelationship.UsingCachonandLariviere’s[6]terminology,contractcomplianceisvoluntaryandbothpartieswillenterthesubcontractingrelationship(or“trade”)onlyifitbenetsthem.Asbefore,bothplayersmustdecideseparately,yetsimultaneously,ontheircapacityinvestmentsinstage1beforeuncertaintyisresolved.Theresultingcapacityvectorisobservableandbecomescommoninformation.Afterdemandisobserved,bothpartiesmaketheirindividualproduction-salesdecisionsinstage2wheretheyhavetheoptiontosubcontract.ThemanufacturerMcanaskasupplyfromthesubcontractorS,whohastheoptiontolltherequest.Thatis,sheoersaquantity,whichisacceptedbyMinexchangeforapaymentWhenmakingdecisions,eachplayeractsstrategicallyandtakesintoaccounttheotherplayer’sdecisions.Anycapacityvector(orproductionvector)withthepropertythatnoplayercanincreasermvaluebydeviatingunilaterallyfrom(or)isaNashequilibriuminpurestrategiesandiscalledsimplyanoptimalinvestment(production)vector.Itsresultingrmvalue(provectorisdenotedby(or).TheanalysisofoursubcontractingmodelinvolvesestablishingandcharacterizingtheexistenceofaNashequilibriuminthistwo-player,two-stagestochasticgame.3.1TheProduction-SubcontractingSubgameAswithanydynamicdecisionmodel,westartwithstage2andsolvetheproduction-subcontractingsubgameforanygivenpairK,D.Bothplayersdecidesequentiallyontheirproductionandtransferlevelsinordertomaximizetheirownoperatingpros.t.=min(andmax=min(Incentivestosubcontractdependonthetransferprice.First,Mwillonlysubcontractifotherwisetheindependentsolosolutionemerges.Thus,fortheremainderofthisarticlewewillassumesothatMwillalwaysprioritizehisinternalcapacityandwillaskStolltheremainingdemand:=min(andSecond,ShasanincentivetollM’sdemandif,whileshewillprefertollherownmarketdemandif.Thus,wemustdistinguishbetweentwocases:1.Hightransferprice:.SprefersllingM’srequesttothebestofhercapacity:=min(.Thesubcontractingtransferis=minwhichmaterial-izeswheneverMhasexcessdemand,thatisif13456.Swilluseanyremainingcapacitytoherownmarketdemand:=min(.(TheresultingmarketsupplyvectorinFigure2is2.Lowtransferprice:min(.Sprefersservingherownmarket:=min(AnyremainingcapacitycanllM’sdemand:=min(.Thesubcontractingtransfer=min,andsubcontractingwillmaterializewhenMhasexcessmarketdemandShaslowmarketdemand,thatisif156.(TheresultingmarketsupplyvectorinFigure2isInbothcases,theproductionvectorK,DformsauniqueNashequilibriumbecausenoplayerhasanincentivetodeviateunilaterally.Atanytransitionpointbetweenthetwocases(e.g.,),playersareindierentbecausetheyreceivethesameprotineithercase,andacontinuumofproductionvectorsareNashequilibria.Thisposesnoproblems,however,because solosolo p- ptGM( )cM pt sub cen solo solosolo GM( )cM pt sub cen solo Figure3:Theintersectionofthecapacityreactioncurves(bold)and(dashedbold)denestheoptimalinvestmentwhensubcontractingwithlow(left)andhigh(right).linearprogrammingtheoryyieldsthattheassociatedprotvectorK,Disuniqueandconcave,whichisallweneedtosolvetheinvestmentgameinstage1.3.2TheCapacityInvestmentGameTodemonstratetheexistenceofasubgameperfectNashequilibriuminpureinvestmentstrategies,wewillshowthatthecapacityreactioncurveshaveastableintersectionpoint.Firm’scapacityreactioncurvespeciesitsoptimalinvestmentlevelgivenhascapacity.Itisdenedpointwiseas)=argmax.Asbefore,andinheritconcavityfromK,Dsothattherstorderconditions(FOC)aresucient:andwheremin(FirmreactioncurveisfoundbysolvingFOCasafunctionof.TheAppendixof[23]showsthat andthataxiscrossingsandasymptotesareasshowninFigure3.Thus,thereactioncurveshaveanintersectionsubatwhichatleastonereactioncurvehasaslope Hence,isuniqueandstable(Nash).3.3ProductionversusInvestmentCoordinationSubcontractingwithprice-onlycontractscancoordinateproductiondecisionsif,becauseonlythenisthecontingentproductionvectorundersubcontractingK,DequaltotheproductioncenK,Dinthecentralizedscenario.(Incentiveincompatibilitiesariseand3456:undersubcontractingthesupplierwillprioritizeitsownmarket cMcSpMpSpt sub 236136 sub 236136 sub subsub236sub !pt!cM!KM sub sub subsub236sub !pt!cS!KS sub Table1:Sensitivityoftheoptimalinvestmentlevelssubandvaluesub,where.Tableentriesrepresentpartialderivatives:sub forexample.whereasthecentralizedsystemwouldprioritizemarketM.)Thiscontractarrangement,however,cannotcoordinateex-antecapacityinvestmentdecisionsoreliminatealldecentralizationcostsasmeasuredbythevaluegapcenMathematically,theoptimalcentralizedandsubcontractinginvestmentvectorsingeneraldierastheuniquesolutionstocencencenandsub,respectively,with.Hence,cenbecausethevaluefunctionsarestrictlyconcaveattheoptimalinvestmentvectors.Economically,oursingle-parameterprice-onlycontractisunabletoprovidesucientex-anteincentivesforbothplayersto‘build’cen.Asinmostrealisticmulti-playermodels,therst-bestsolutionisnotattainedanddecentralizationcomesatacost.Thesecontractsdo,however,mitigatedecentralizationcostsandimproveperformance.Com-paringthecapacityreactioncurveswiththeoptimalitycurvesthatdenetheoptimalcentralizedandsoloinvestment(thethinlinesinFigure3)directlyshowsthatsubcontractingdrivesinvestmenttowardsthecentralizedinvestmentcensubsoloandcensoloThisiswhatoneexpects:subcontractingallowsthemanufacturertodecreasehisinvestment.Theoptionofsubcontractingmeanspotentiallymorebusinessforthesupplierandthuswarrantsaddi-tional‘relationship-specic’investment.Next,weinvestigatehowothermodelprimitivesimpactthecoordinationimprovement.3.4SensitivityoftheInvestment-SubcontractingStrategiesThesensitivityoftheoptimalinvestmentstrategywithrespecttochangesincapacitycostscontributionmargins,andtransferpriceissummarizedinTable1.Asexpected,optimalmanufacturingandsuppliercapacitylevelsareimperfectsubstituteswithrespecttocapacitycostsandmargins.Indeed,strategicdecisionmakingcapturedbyourgame-theoreticmodelmakesoneparty’sinvestmentlevelandrmvaluedependentontheotherparty’scostandrevenuestructure.Whenthemanufacturerfaceshigherinvestmentcosts,forexample,hewilldecreasehisinvestmentlevel.ThesupplieranticipatesthatlowermanufacturingcapacitymostlikelywillleadtohighersupplyrequestsThisgivesthesupplieranincentivetoincreaseherinvestment,reectingtheexternalitiesinourmodel.Theincreaseinhowever,doesnotmakeupforthedecreaseinsub(becausetransfersareonlymadewithaprobabilitystrictlylessthanone).Asimilarreasoningappliestoachangeinmarginbutithasasmallerimpactthanacostchangesimplybecausethemargindependencyisstate-dependent.Forexample,anincreaseinonlywarrantsanincreaseinmanufacturingcapacityifdemandissucientlylarge(e.g.,).Anincrease,ontheotherhand,alwaysjustiesadecreaseinmanufacturingcapacity,regardlessofthedemandrealization.Thisresultisinstarkcontrasttodeterministicsystemsandoneexpectsthissensitivitydierentialtoincreaseintheamountofdemandvariability. 0 0 Transfer Price t cSpSpMcM solosolosolo optionFirmValues 0 solosolosoloCapacityLevels Transfer Price t cSpSpMcM KSM+ Figure4:Capacitylevels,theoptionvalueofsubcontractingoptionandthedecreasedvaluegapasafunctionofthetransferpricewhenmarketdemandsareuniformbutstronglynegativelycorrelated.Moreinterestingly,whilethesupplier’svaluesensitivitydirectlyreectstheexternalitiesinthemodel,themanufacturer’svalueisalittlemoreintricate.Clearly,anincreaseinsuppliercostsleadstoadecreaseintotalsystemcapacity,whichimpactsbothparties’valuenegatively.Anincreaseinmanufacturingcostbenetsthesupplierwhoincreaseshercapacityinanticipationofalargertotaldemand.Recallthatthecentralizedsystemwouldputallcapacitywiththesupplierif.Hence,anythingthatshiftscapacityfromthemanufacturertothesupplierwilltendtobenetthesystem.(Thestructureofthecapacityreactioncurvesshowsthatmovestowardcen–andthussubmovestowardcenincreases,evenif.)Thisectcandominatetoyieldtheunexpectedresultthatthemanufacturer’svaluecanbeincreasinginitsinvestmentcost.Themanufacturerenjoysspill-overbenetsfromincreasedsuppliercapacitythatmayoutweighhisincreasedinvestmentcosts.Similareectscanoccurwhenincreasingthetransferprice.Thetableshowsthatthishasasimilareectasasimultaneousincreaseinmarginsand.Theabsoluteeectoninvestmentlevelsandrmvaluesisambiguous.Anincreaseinmakessubcontractingmoreexpensiveforthemanufacturerrelativetointernalcapacityinvestment.Thisisreectedbyarightwardmoveofthemanufacturer’sreactioncurveinFigure3.Increasedtransferprices,however,givethesupplierahigherincentivetoincreaseher“relationship-specic”investment.Thus,whileweexpectsubtodecreaseandsubtoincrease,thesupplier’sreactioncurvecanmoveupwardmorethanmovesrightsothatincreasesanddecreases.Figure4illustratestheintricateexternalitiesthatcanoccurinstochasticgames.Contractdesign,orthechoiceoftheoptimalthusbecomesverycasespecicanddependsontheobjective.(Onecanmaximizemanufacturer,supplierorsystemprots,orsomecombination,dependingonhowthetransferpriceisset.Itmaybetheoutcomeofnegotiationbetweenthetwopartners,oritmaybeequaltoanexternalreferencepriceifanotherexternalsupplymarketexists.)Inallournumericaltestproblems,systemprotsweremaximizedatyieldingasubstantialimprovementinthevaluegap,whichisinagreementwitheconomictheorystatingthattransferpricesshouldbesetequaltooutsideopportunitycosts.Ifthemanufacturersetsthetransferprice,however,hedoesnotnecessarily setitat.Indeed,becauseofdemandvariability,atransferpricebelowmayyieldoptimalprotsforthemanufacturer.Figure4illustratesthispossibilitywhenmarketdemandsarestronglynegativelycorrelated().Asarguedearlier,thecapacitylevelsareimperfectsubstituteswhileTable1showsthattotalindustryinvestmentlevelsubisincreasingin.Thegurealsoshowsthatinthecontextofourmodelsubcontractingmayreduceorincreaseindustryinvestmentcomparedtothesoloorcentralizedsetting.(Whilethegureshowsthatsolo,thisisnottrueingeneraleither.)Interestingly,similartothedependencedescribedearlier,ahighertransferpricemayincreasethemanufacturer’sprot.Thissuggeststhataprice-focusedstrategyformanagingsubcontractorscanbackonthemanufacturer.Whilealowerpriceallowscheapsupply,itdoesnotguaranteeitsavailability.Finally,tostudytheeectofuncertaintyontheoptimalinvestmentstrategies,weconsideraprobabilitymeasurewithdensitythatisparameterizedbyanuncertaintymeasuresuchasanelementofthemeandemandvectororcorrelationmatrix.Formally,theimpactofchangesinontheoptimalinvestmentstrategycanbeexpressedas: subsubwhereistheJacobianoftheoptimalityequationssubsuband sub dz.Althoughthisexpressionisoflimitedpracticalvalue,itmaybeusefulforestimatingthesign .Theappendixof[23]showsthatand.Thus, suband submayhaveoppositesignssothattheoptimalmanufacturerandsupplierinvestmentlevelswouldrespondinoppositewaystochangesinthedemanddistribution,akintothesubstitutionectstatedearlier.ThiseectispresentforchangesinthestandarddeviationorcorrelationofmarketdemandsintheexampleshowninFigure5.Forsimplicity,weassumedidenticalmeanandstandarddeviationsforandAsshownintheleftgraphsofFigure5,optimalinvestmentlevelsaremonotoneinvariabilityasmeasuredbythestandarddeviation,buttheycanbeincreasingordecreasing.Thisissimilartothewell-knowneectinone-dimensionalnewsvendormodelswithsymmetricdemanddistributionswhereoptimalinvestmentincreases(decreases)invariabilityifthecriticalratio Moreimportantly,comparedtotheindependent“solo”setting,anincreaseinmarketriskdecreasesthemanufacturer’srelativeinvestmentifthereisasubcontractingoption.Thiscanbeparaphrasedbysayingthatthemanufacturerwillsubcontractmoreasmarketriskincreasesandthesubcon-tractor’sresponseistoinvestmore.Thepresenceofdemanduncertaintyisakeydriverintheoptionvalueofsubcontracting,whichisincreasinginvariability.Thus,similartomanynancialoptions,moreuncertaintyisgoodforthisrealoption.Inabsoluteterms,however,morevariabilityreducesrmvalues.ThegraphattherightinFigure5showsthatthemanufacturerwillsubcontractlessasmarketcorrelationincreasesIndeed,whenmarketdemandsarepositivelycorrelated,thesubcontractingoptionhaslessvaluesothattheoptimalfractionofcapacitythatissubcontracteddecreases.IntermsofourgraphicalsolutiontechniqueofFigure2,thetriangularoptionregiongetsmoreprobabilitymassascorrelationbecomesmorenegative. Thisexamplewasgeneratednumericallyusingatwo-dimensionaldemanddistributionparameterizedbycorre-lationandstandarddeviationinmarketdemand.Explicitexpressionsforthesedistributionswererstpresentedin[22,pp.75-77].Thesubcontractor’soptimalinvestmentlevelseemstobelesssensitivetorisk,whichmaybeexplainedbyriskpooling:thesupplier’seectivedemandpoolsoverbothmarketsandthereforeislessvariable. 0 solosubsubsoloStd. deviation Capacity 1 Correlationsolosubsubsolo Capacity Levels 0 censubsoloStd. deviation optionSystem 1 censubsolo option CorrelationSystem Figure5:Optimalinvestmentandtheoptionvalueofsubcontractingoptionasafunctionofstandarddeviationofdemandassumingwhenmarketdemandsareuncorrelated(left)andofcorrelationwhenmarketdemandsareuniform(right).3.5OutsourcingorCompleteSubcontracting:subThestructureofthecapacityreactioncurvesshowsthattheoptimalinvestmentstrategyhasoneoftwodistinctforms:eitherbothrmsinvestoronlythesupplierinvests.Inthelattercase,themanufacturerreliesforallsourcingontheoutsideparty.Onecanexpressanoutsourcingconditionintermsofathresholdonthemanufacturer’sinvestmentcostasfollows.Set=(0(0))anddenethethresholdcostsub,wheresubistherstrowofsub.ThenthemanufacturershouldoutsourceifandonlyifhisinvestmentcostexceedsthethresholdcostCoordinationofinvestmentdecisionswouldrequirethatbecausethecentralizedsys-temputsallcapacityatthesupplierifUnderprice-onlysubcontracting,however,thethresholdcostdependsnotonlyon,butalsoonthemargins,the“costtosubcontract”asexpressedbythetransferpriceandthejointdemanddistribution.Figure6illustratesthatthe‘outsourcingzone’ofthestrategyspaceissmallerthantheoutsourcingzoneundercentraliza-tion().Thisconrmsthatsubcontractingwithsimplepricecontractsimprovessystemperformanceascomparedtothesoloscenario(neveroutsourcing),yetitcannoteliminatethevaluegapingeneral. Transfer Price t 0 if # SubcontractOutsourceManufacturer’s Investment Cost Figure6:Thethresholdcostpartitionsthestrategyspace.(ShownforincreasinglevelsofuncertaintyasmeasuredbythestandarddeviationofdemandassumingIntheAppendixof[23]weshowthatforlowlevelsofdemanduncertainty,thethresholdlevelindependentofthedemanddistributionand (5)Thus,andwithlittledemanduncertainty()andlowtransferprices,nooutsourcingwillhappen.Indeed,inthiscaseMmuststillinvestinin-housecapacitybecauseoftwoeects,bothrelatedtothelowtransferpriceandlowuncertainty.First,Mcannotinducethesuppliertollhisrequests:with,Swillprioritizeherownmarket.Second,littleuncertaintyandlowtransferpricesgivethesupplierinsucientincentivetoinvestinextracapacitytoservethemanufacturer.Fortransferpriceshigherthanthesupplier’scapacitycost,outsourcingispossiblebecauseSnowhasanincentivetobuildextracapacity().Formediumtransferprices,thethresholdisdecreasinginsothatoutsourcingbecomesmorelikelywithhighertransferprices,reectingahigherincentiveforS.Whenthetransferpriceexceedsthesupplier’smargin,adiscontinuousdropinmakesoutsourcingevenmorelikely:ensuresMthatitsrequestswillnowgetprioritybyS.Asthetransferpriceincreases,however,subcontractingincreasinglybecomesmoreexpensiveforthemanufacturercomparedtoin-housecapacitysothatMhaslessincentivetooutsource.Whenthelevelofdemanduncertaintyrisesaboveacertainlevel(),thethresholdcostwilldecreaseforlowtomediumtransferprices()butincreaseforhightransferprices().Thus,forlowtomediumtransferprices,moreuncertaintymakeshighermanufacturingrequestsmorelikely,creatingastrongerincentiveforthesuppliertoinvestinextracapacity,whichmakesoutsourcingmorelikely.Forhightransferprices,ontheotherhand,moreuncertaintyincreasestheexpectedtotaltransfercosttothemanufacturerwhowillprefermorein-housecapacity,makingoutsourcinglesslikely. 4SubcontractingwithOtherContracts4.1IncompleteContracts:BargainingInsomesituations,ex-antecontractsmaybetooexpensiveorimpossibletospecifyorenforce.Start-upcompaniesandcompaniesindevelopingcountriesmayndittooexpensivetoenforceexecutionofacontract[11],whileinvestmentsbysuppliersinquality,informationsharingsys-tems,responsivenessandinnovationareoftennon-contractible.“Withouttheabilitytospecifycontractuallyinadvancethedivisionofsurplusfromnon-contractibleinvestments,thissurpluswillbedividedbasedontheex-postbargainingpowerofthepartiesinvolved[3].”Thisincom-pletecontractsapproachwasrstsuggestedbyGrossman,HartandMoore[9,13]tostudyverticalintegration.Thenegotiationonthesurplusdivisioncanbecastasbilateralbargaining.Manybargaininggamesarepossible(c.f.KamienandLi[14,p.1357]).Nashintroducedagamethatleadstosplittingthesurplusevenly.Rubinsteinpresentsasequentialgameinwhichplayergetsfraction ofthesurplus,whereisthe“impatience”ordiscountfactorofplayer,whichisex-anteobservable.Whicheverbilateralbargaininggameisused,themanufacturercanex-anteexpect(butnotcontractuallyspecify)toreceivefractionofthesurpluswhilethesupplierwillgetfraction.Onecanalsothinkofasthe‘bargainingpower’ofthemanufacturer.Theanalysisissimilartobeforeinthatbothrmshavetheoptiontoengageinatradeatthebeginningofstagetwo.Thermscandecidejointlyonproduction-salesdecisionssothattheresultingactivityvectorequalsthevectorchoseninthecentralizedscenario.Engaginginsubcon-tractingthusyieldsaprotsurplusK,DK,DsoloK,D,andbothpartiesthushaveanincentivetoimplementthecentralizedproductionvectorcenK,DbyengaginginthetradeK,D.Hence,productiondecisionsarealwayscoordinatedwithincompletecontracts(incontrasttoprice-onlycontractswherepoorproductiondecisionscanoccurifInvestmentcoordination,however,isnotachieved.Indeed,themanufacturer’soperatingprosolowhilethesupplier’sissolo.Because solo,thecapacityreactioncurvescanbeconstructedagainintermsofashadowmatrix:barcendiagsolo (pM0 (pM+(pSpMpM (pM+(pS0pSpS (pM+(pS (pM (pS¸M!pS· (pM0pMpMpMpM0pSpSpS (pM Bothcurveshaveaunique,stableintersectionthatdenestheoptimalinvestmentvectorbarthatingeneraldiersfrombecausebarcen.Thus,thedivisionoftheex-postsurplusgivesthesupplieranincentivetomakearelationship-specicinvestment,yetinsucienttoimplementcen.AsshowninFigure7,reductioninthevaluegapandtheoptionvalueofsubcontractingwithincompletecontractsismaximalwhensurplusisdividednottoounevenly(butitneednotbeafairsplit).Moreimportantly,incompletecontractsarenotinferiortoexplicitprice-onlycontracts.Forexample,comparingFigure7withcorrespondingFigures4and6showsthattheoptionvaluecanbelargerandthatoutsourcingismorelikely.Thehigheroptionvaluemayreectthefactthatwithincompletecontractsproductioncoordinationisalwaysachieved.Italsosuggeststhatsometimesrmsmaybebetteroleavingsomecontractparametersunspeciedex-anteandagreeingtonegotiateafterdemandisobserved. Obviously,ifex-antenegotiationsareallowed,bothpartieshaveanincentivetoimplementcenandinvestmentcoordinationwouldbeachieved. 0 1 0 M’s Bargaining Power solosolosolocen optionFirmValues 1 Manufacturer’s Investment Cost M’s Bargaining Power SubcontractOutsource S Figure7:Theoptionvalueofsubcontractingwithanincompletecontractanditsoutsourcingthreshold(withdashedbounds)asafunctionofthemanufacturer’sbargainingpowerforthesamemodelparametersasFigure4.Letushighlighttheroleofthebargainingpower,becausethesensitivityoftheinvestmentstrategytootherparametersissimilartothatunderprice-onlycontracts.Asearlier,wecanexpressanoutsourcingconditionintermsofathresholdonthemanufacturer’sinvestmentcostTheappendixof[23]derivesthefollowingboundsontheoutsourcingthreshold:min pS¶+¯(pM" cM"pM,¯(pM+( Thethresholdisdecreasing(almostlinearly)forsmallwhichimpliesthatoutsourcingismorelikelyformorepowerfulmanufacturers.Theargument,however,cannotbegeneralizedtoverypowerfulmanufacturers():thethresholdmaybeincreasingnearasshowninFigure7.Indeed,ifisnear,outsourcingislesslikelybecausethesubcontractorreceiveslessex-postsurplusandhaslessex-anteincentivetomakearelation-specicinvestment.Similarly,ifbargain-ingpowerisverysmall,mostsurplusgoestothesupplier.Aswithprice-onlycontracts, andtheoutsourcingzoneunderthiscontractisagainsmallerthanthezoneundercentralization:meresuppliercostadvantageofthesubcontractoriscientforthemanufacturertoout-sourcebecausethesurplusdivisionincentiveisinsucientforthesubcontractortoimplementthecentralizedcapacitylevel.4.2State-dependentPrice-OnlyContractsstate-dependentprice-onlycontractspeciesanex-antetransferpriceK,Dforeachpossiblecontingentstatevector.(Suchacontractrequiresthatcapacitylevelsarenotonlyobservablebythetwormsasassumedearlier,butalsoveriablebyathirdparty.)Notonlycanthesecontractsimproveperformancebecauseoftheirincreaseddegreesoffreedom,optimalstate-dependentprice-onlycontractdesigncancoordinateinvestmentdecisionsandeliminatealldecentralizationcostsIndeed,itisdirectlyveriedthatthesucientconditionsubissatisedifwithK,Dforand.Suchacontractachievesinvestmentcoordination(andproductioncoordinationbecause)byaligningincentives:subcontractingis costlessin(equaltoS’smarginalopportunitycostwhengoingsolo)givingMthecorrectincentivetoreduceitsinvestmenttocenandrelyonsubcontracting,whileatransferpriceinequaltoM’smarginalopportunityprogivesStherightincentivetoincreaseitsinvestmentcen.Similarly,ifcoordinationcallsforand.HigherincentivesarenownecessaryforStoprioritizemarketMaboveitsownmarket(asthecentralizedsystemwoulddo):transferpricesmustatleastequalitsownmargininandbehigherintoinduceproductionandinvestmentcoordination.Itissurprisingthatourmodelsetupallowsustocharacterizethesenecessaryandsucientconditionsforcoordinationthiseasily.Inaddition,noticethatthesesucientstate-dependentcontractsareactuallysimplerthantheirnamesuggests:oneonlymustspecifythetransferpricesundersixscenariosinourmodelandnotforeachstateState-dependentprice-onlycontractscanberelatedtoincompletecontractsasfollows.Theexecutionoftheinter-rmtransferbarbarandthesurplusdivisionisimplementedbyspec-ifyingthequantityK,Dtobeprovidedbythesubcontractorandtheunittransferpricebartobepaidbythemanufacturer.Thistransferpriceisdenedimplicitlyinthebargainingmodelinthatitguaranteesthecorrectdivisionofsurplus:barcenbarcen(recallthatbarcenandcensolo)andrearrangingtermsyieldsbarcensolocen(6)Thistransferpaymentbaristhecompositionoftwoterms:cenisthegrosssurplusderivedfromsubcontractingwhilesolocenisthesubcontractor’sopportunitycostortheproforgonebysubcontracting.Thegrosssurplusisreceivedbythemanufacturerwhopaysthesharecentothesubcontractor.ThesubcontractorbearstheopportunitycostandiscompensatedbythemanufacturerforthesharesoloSolving(6)ineachdomainyieldsthestate-dependenttransferprices:Ifbar156(itsvaluein0234isirrelevantbecausenotransferoccursthen).ThisbarK,Dcontractyieldsproductioncoordinationbecausebar.Thus,ifthisstate-dependentprice-onlycontractisequivalenttotheincompletecontractwithparameter:itimplementsidenticalcentralizedproductiondecisionsandtheparticularchoiceofK,Dguaranteesthatexpectedoperatingprotsequalthoseunderthebargainingmodelandhencetheirinvestmentvectorsareidentical.If,however,theexistenceofanequivalentstate-dependentprice-onlycontractisnotguaranteedingeneral.Solving(6)yieldsbarbarbar bar .Ingeneral,suchaprice-onlycontractdoesnotguaranteeproductioncoordination.If,however,Mhaslimitedbargainingpowersothat,thenK,Dandproductioncoordinationisguaranteedsothatthecontractyieldsthesameinvestmentvectorastheincompletecontract.4.3State-dependentIncompleteContractsAstate-dependentincompletecontractisanincompletecontractwithstate-dependentsurplusdi-vision(bargaining)parameterK,D.Giventheirequivalencewithstate-dependentprice-onlycontractsifsuppliermarginsarehigher(),itisnotsurprisingthatthesecontractsalsocoordinatethesupplysystem.Indeed,if,thesucientconditionforinvestmentcoordina-tionbarcenissatisedwith(Mreceivesallsurplus)inanyconstantand(Sreceivesallsurplus)in.(ThisK,Disalsofoundbyrequiringthattheequivalentbariscoordinating:barandbarSimilarly,with,equalityofbarandcenrequiresanyconstant,butnoconstantexiststoequalizebarandcen.Hence,wemust lookforavariablefunctionK,Dover.Asbefore,toachieveinvestmentcoordinationthisK,DmustsatisfythesucientFOCequalitybarcencen.Ifvariesoveradomain,however,themarginalprotvectorinthatdomainmustbeexpandedtobar,ibar KM," ,whereindomain.Hence,theFOCbecomeasystemofpartialdierentialequations:withK,Dmustsatisfy (pM+(pS K,D KM# (7)whereK,D.Luckily,avalidsolutionK,Disinspiredbytheequivalentbar .Recallthatastate-dependentprice-onlycontractrequirestoinduceinvestmentcoordination.SolvingbarforyieldsK,D (pM#pSM#KMS(KS#DS)=(pM#pSM#KM) K,D(8)whichindeedsatises(7).Hence,thisincompletecontractwithtrulystate-dependentK,Dcoordinatesthesupplysystemif5DiscussionandExtensionsInadditiontothethreecontractsstudiedhere,manyothercontractstructurescanbeusedtoregulatesubcontractingbyaddingmoreparameterstothecontractspecication.CachonandLariviere[6]giveanoverviewofmoresophisticatedcontractsusedintheliterature,whichtypicallyalsospecifysomeconditionsonthetransferquantityoronthemanufacturer’sliabilityofthesupplier’sexcesscapacity.CachonandLariviereshowthatthesemoreadvancedcontractscan,butdonotnecessarily,improvesystemcoordinationandhighlighttheroleoftheinformationstructureandtheveriability(andthusenforcement)oftheplayers’actions.Inthepresenceofinformationasymmetries,complexcontractsprovideforapowerfulsignalingdevicethatcanimproveperformance.Tsay[21]hasshownthatsomeprice-quantitycontractsalsoimprovesystemcoordination.Whileweanalyzedonlysimplecontracts,webelievethatmanyofthecharacteristicsofmorecomplexoutsourcingcontractswillcarryovertooursubcontractingmodel.Otherextensionssuchastheinclusionofspecictransactioncostsandmergingcostsarerela-tivelystraightforward.Wehaveassumedthattheinitiationandmanagementofthesubcontractingrelationshipwascostless.Apositivecostisdirectlyincorporatedsothatbothpartieswouldenterintotherelationshiponlyiftheex-postsurplusexceedsthetransactioncost.Similarly,onecanincludemergingcosts,whichwouldexplainwhybothpartiesdonotalwayschoosetomergeintoasingle,centralizedorganization.Anothervariationistomakebothrmsmoreequal‘partners’bydroppingthenon-negativityconstraintontoallowforbi-directionaltransfers.(Thisalsoyieldsatwo-locationinventorymodelwithtransfersbetweenprotcenters.)Allowingfordemand-dependentsalesprices(andthusmargins)byincorporatingdownwardslopingdemandcurves(ourrmsareassumedtobepricetakers)wouldyieldaduopolymodelmorein-linewithtraditionaleconomics.Thisgeneralizationtoincorporatetacticalpricingdeci-sions,however,comesatconsiderablecost.Onenotonlylosestheconnectiontothetraditionalnewsvendormodelanditsintuitive,graphicalinterpretation,butthecompetitivepricingdecisionunderuncertaintygreatlyincreasesthecomplexityoftheanalysis.Allowingfornon-exclusive Allowingforinter-rmsubcontractingtransferswouldamounttoputtingyetanotherlayerofcomplexityonthecompetitiveinvestment-pricingmodelthatwestudiedin[24]. marketaccessisaneasierextensionthat,webelieve,willnotchangethequalitativeinsightsob-tainedhere.Finally,thetime-horizoncanbeextendedtoamulti-periodsettingtostudytheeectofpredictabletemporaldemandvariations,suchasoveraproductlifecycle(stochastictemporalvariationsmostlikelywillleadtoaproductionsmoothingeectasstudiedbyKamienandLi[14]).Acknowledgments:IamgratefultoSunilChopra,MaqboolDada,JimPatell,ScottSchaeferandseminarparticipantsatColumbiaUniversity,NorthwesternUniversityandStanfordUniversity.Ithanktheassociateeditorandtherefereesfortheirconstructivecommentsandinsightfulsuggestionsontwoearlierversions.References[1]Britannicaonline.Availableathttp://www.eb.com:180/cgi-bin/g?DocF=macro/5007/31/8.html.[December121996].[2]Merriam-Webster’sCollegiateDictionary.Merriam-Webster,Springeld,MA,tenthedition,1998.[3]J.Y.BakosandE.Brynjolfsson.Informationtechnology,incentivesandtheoptimalnumberofsuppliers.J.ofManagementInformationSystems,10(2):37—53,1993.[4]J.BensonandN.Ieronimo.Outsourcingdecisions:evidencefromaustralia-basedenterprises.Interna-tionalLabourReview,135(1):59—73,1996.[5]A.O.BrownandH.L.Lee.Optimal“paytodelay”capacityreservationwithapplicationtothesemiconductoreindustry.workingpaperStanfordUniversity,pages1—25,November8,1997.[6]G.P.CachonandM.A.Lariviere.Contractingtoassuresupplyorwhatdidthesupplierknowandwhendidheknowit?workingpaper,FuquaSchoolofBusiness,DukeUniversity:1—29,1997.[7]J.S.Day.SubcontractingPolicyintheAirframeIndustry.GraduateSchoolofBusinessAdministration,HarvardUniversity,Boston,MA,1956.[8]T.ElgerandC.Smith.GlobalJapanization:Convergenceandcompetitionintheorganizationofthelabourprocess.InT.ElgerandC.Smith,editors,GlobalJapanization:Thetransnationaltransformationofthelabourprocess,pages31—59.Routledge,London,1994.[9]S.J.GrossmanandO.D.Hart.Thecostsandbenetsofownership:Atheoryofverticalandlateralintegration.JournalofPoliticalEconomy,94:691—719,1986.[10]M.GuptaandD.Zhender.Outsourcinganditsimpactonoperationsstrategy.ProductionandInventoryManagementJournal,35(3):70—76,1994.[11]G.H.Hanson.Incompletecontracts,risk,andownership.InternationalEconomicReview,36(2):341—363,May1995.[12]J.M.HarrisonandJ.A.VanMieghem.Multi-resourceinvestmentstrategies:Operationalhedgingunderdemanduncertainty.EuropeanJ.ofOperationalResearch,113:17—29,1999.[13]O.D.HartandJ.Moore.Incompletecontractsandrenegotiation.Econometrica,56:755—785,1988.[14]M.I.KamienandL.Li.Subcontracting,coordination,exibility,andproductionsmoothinginaggregateplanning.ManagementScience,36(11):1352—1363,Nov.1990.[15]M.I.Kamien,L.Li,andD.Samet.Bertrandcompetitionwithsubcontracting.RandJournalofEconomics,20(4):553—567,1989.[16]M.C.LacityandR.Hirschheim.InformationSystemsOutsourcing.J.WileyandSons,WestSussex,England,1993.[17]L.Li.Theroleofinventoryindelivery-timecompetition.ManagementScience,38(2):182—197,Feb1992.[18]S.A.LippmanandK.F.McCardle.Thecompetitivenewsboy.OperationsResearch,45:54—65,1997. [19]J.B.QuinnandF.G.Hilmer.Strategicoutsourcing.SloanManagementReview,pages43—55,Summer1994.[20]Y.Tournas.ThreeessaystowardsageneraltheoryoftheDoctoralDissertation,NorthwesternUniversity,pages3—43,December1996.[21]A.A.Tsay.Thequantityexibilitycontractandsupplier-customerincentives.SantaClaraUniversityworkingpaper,1996.[22]J.A.VanMieghem.Multi-resourceinvestmentstrategiesunderuncertainty.DoctoralDissertation,GraduateSchoolofBusiness,StanfordUniversity,pages1—89,May1995.[23]J.A.VanMieghem.Coordinatinginvestment,production,andsubcontracting:Appendix.TechnicalReport1208,CenterforMathematicalStudiesinEconomicsandManagementScience,NorthwesternUniversity,November1998.Availableathttp://www.kellogg.nwu.edu/research/math.[24]J.A.VanMieghemandM.Dada.Priceversusproductionpostponement:Capacityandcompetition.ToappearinManagementScience,1999. 6AppendixAllrstorderconditions(FOC)foroptimalityareoftheform wherethematrixfunctiononlyof,whilethevector isfunctiononlyof(andofparametersintheprobabilitydistribution).ThestructureoftheFOCs(orcapacityreactioncurves)anduniquenessofanoptimalsolutionwillbeestablishedusingpartialderivativeswhicharefoundbyimplicitlydierentiatingoneorbothFOC.representacostormarginparameterofinterest.TotaldierentiationoftheFOCyields: $=d P=µ" "x#¶ P+J" istheJacobianmatrixoftheFOC: !Kj=!Vi ,whichcanbecalculatedexplicitly: !KM P! !KS P¤=#³'K wherethematrix canbeexpressedintermsofthelineintegralsoftheprobabilitydensityovertheboundarybetweendomainsandandij,kl Forexample:.Thus,alleortisreducedtoshowingthat invertiblewhichthenyields "xK=J"1 dxJ"1µ" "x#¶ (9)Thus,lettingwedirectlyhavethat !cMK! (10)andtheslopeof,theFOCforgiven,followsfromtotallydierentiatingthe’thFOC: !KiE$i +! !KjE$i =#!2 !Kj!KiVi !2 !K2iVi=#J 6.1CentralizedReferenceScenarioTheoptimalsolutionisattheintersectionofthetwoFOCcurves.WehavethatAllentriesinarenonpositivewithandsothat =#J Clearly,ifitisoptimaltoinvestonlyin-capacity:sothatcentcentcent.IfBecause,eitherand/orarepositivesothatFOCisstrictconcaveat),ergouniqueness.Ifandeitherand/orarepositive,againshowinguniqueness.Otherwise,if,weinvestinbothcapacitiesandatleastoneofthetermsinispositivesothatcenisstrictconcaveattheuniqueoptimal.Wecancomputesomepointsofthecentralizedcurves: ,thencentThus,cent(0)=soloand =½#pM"pS ,thensothatandsothat)=0 asituationthatremainsifdecreasesaslongas((0))=Clearly,thisminimalincreasesincorrelationandvariability.,thenand.Thus,cent cent thensothatand.Thus,cent solocent asituationthatremainsifdecreasesaslongas))=0Clearly,thisminimalincreasesinvariability.6.2SubcontractingwithPrice-OnlyContractsTheJacobianbecomesmin(6.2.1UniquenessofthesolutionsubAllentriesinarenonpositivewithandsothatmin( =#J Existenceofanintersectionfollowsfromtherelativepositionofaxiscrossingsandasymptotes:,then01356.Thus,(0)= solo(0) =½#pM"pt min( remainsincreaseswithlowuntilbecomespositive.Clearly,thismaximaldecreasesincorrelationandvariability.) ,thensothatand.Thus,if,wehave)=0,else soloand asituationthatremainsifdecreasesaslongas))=Clearly,thisminimalincreasesincorrelationandvariability.Notethatiscontinuousinforisboundedfrombelowbyapositivenumberwithprobabilityone(thedemanddensityiszeroatsothat.Forhighwehavethatandbecause,wehavethat(0) Withsmallvariability,wehavethat(0)(exact:(0))= Indeed,if,wewouldhavethat3456(0),whichcannotsatisfyFOC.)Forlowwehavethat.Ifthen.Ifhaslowvariabilityinthesensethatsolo)))=0isdiscontinuousatandwehavethat(0)(exactly: and(0) soloandthus(0) hashighvariability, thensothatand.Thus, soloand asituationthatremainsifdecreasesaslongas))=0Clearly,thisminimalincreasesinvariability.Uniquenessofsubfollowsfrom atintersection(assumehigh,lowissimilar)andbecauseisacontinuousmeasurewehavethatthatisstrictconcaveattheoptimalandthusthereactioncurveisunique.Moreover "0 sothat.Againthereactioncurveisuniquebutnow,aslongas (and Attheintersectionsubwehavethat whichshowsuniqueness(indeedwouldimply3456,whichcannotbeasolutiontoFOCmin(Similarlyforrm2’sreactioncurves(),iffollowsthatBecauseandthussothatisstrictconcaveattheoptimalandthusthereactioncurveisunique.Moreover (and =#"2 Giventhatthetworeactioncurvesareuniquewith ,andtherelativeaxiscrossingsareasgivenhighertogetherwith atanysolutiontotheFOC,itfollowsthatheyhaveauniqueintersectionwhichisastable,andthusNash,equilibrium. 6.2.2SensitivityofTheintersectionpointistheuniquesolutiontotheFOCanditfollowsfromtheFOCthatonewillneverinvesttocoveralldemandwithprobability1.Inotherwords,ifandatleastoneofthetermsindet(ispositivesothatandisinvertible:min(Because ,wehavethatbothcapacitiesareimperfectsubstitutesw.r.t.themarginalcostvector.Partialsw.r.t.themarginsare 000000000""""0000003456min( 000000236000000"""234min( 000000000000min(While cannotbesignedingeneral,wedohavethat 6.2.3SensitivityofsubWehavethat !Vi !KM!KM !x+!Vi !KS!KS !x+!Vi ,where underoptimalinvestment.Thecross-partial !Kj=! !KjE!i i,jcanbecomputedasbeforebytheweightedaverageoftheconstanti,jineachdomainsub sub min(Denoting,wegetsub sub sub sub subAsexpected !cS,!VS arenegativeand ispositive,while cannotbesignedingeneral.Forpricesensitivityconsiderhightransferprices(theothercaseissimilarbutreplacethe,andsub sub sub sub sub sub sub sub sub sub !ptt. 6.3OutsourcingConditionsFirstnotethatfromthestructureofthemanufacturer’sreactioncurveitfollowsthatthethresholdcost,becauseanecessaryconditionforoutsourcingisthatsothat )=06.3.1Lowtransferprice:min(Because,wehavethat13456andtheoptimalityequationsyieldsothat pS"PcS ptpt+pM#pt pScS"¯cM=ptM#pt)Ppt+pM#pt Also,Noticethatwithlowlevelsofuncertainty,oneeitherhas(0)(exactly:(0)(exactly: Asuncertaintyincreases,decrease.Indeed,if,increasinguncertaintywilldecrease3456fromandincrease,buthaslowercoecientinthedenitionof.If,increasinguncertaintywilldecreaseandincrease.FromFOC2weseethatwilldecreasemorethanwillincrease();thuswillalsoincrease,butagainlessthanthedecreasein,sothatdecreasebecause6.3.2Hightransferprice:Because,wehavethat13456andtheoptimalityequationsyieldsothat ptpt"¯cM=ptM#pt)Ppt+pM#pt Again,withlimitedlevelsofuncertainty,onecanonlyhave(Asuncertaintyincreases,willdecreasefromandwillgrow,leadingtoanincreasebecause.Finally,noticethatisdiscontinuousat6.4IncompleteContracts(Bargaining)6.4.1FOCWithincompletecontractingtheplayers’revenuefunctionsare:solocensolosolosolocensolosolo Because solo,thecapacityreactioncurvescanbeconstructedintermsofourprimitiveshadowmatricessolodiagcensolodiagcensolocensolo (pM0 (pM+(pSpMpM (pM+(pS0pSpS (pM+(pS (pM (pS¸M· (pM0pMpMpMpM0pSpSpS (pM elsewhere.6.4.2OutsourcingCase1:Because,wehavethatand13456andtheoptimalityequations pMP1 pM+(pS)PpMPM,pSP3 pM+(pS)P4+ (pMP5+ sothat )+( pM+(pS)PpMP (pM+(pSP(pMP (pM+(¡cS#( (pM+(pS)P4# (pMP5+(pSP6¢+(pMP (pM+(¡cS#( (pM+(pS)P4# (pMP5+pMP(pSP6¢= (pM+((cS#(#(pM+(pS)P4+(pMP5+(pSP6)= (pM+(cS+(2M#pS)P4+pMP5+pSP6) (pM+(cS"¯cM= (pM+(cS+(2M#pS)P4+pMP5+pSP6)" (pM+(cS+(2 (cS= (pM+( Case2:wehavethatand13456andtheoptimalityequationsIffollowsdirectlyfromFOCandfromFOCsothat pScS Andhence: pScS¯cM#()pM+(pMP("#()pM+( "#(cS. 6.4.3State-DependentPrice-OnlyContractsandCoordinationItisdirectlyveriedthatthesucientconditionforcoordinationissatisedifK,D)=0and.Similarly,ifcoordinationcallsforandTheequivalenttransferpriceisdenedimplicitlyinthebargainingmodelinthatitguaranteesthecorrectdivisionofsurpluscencen(recallthatcen).Weknowthatsolocensolosolocencencensolosolocencensolo[becausesolosothatcensolocensolocensolocen cenCase1:.Weknowthatnotransfershappeninandintheotherdomains: cen solo x p !1 DM#KM DS DS ¯(pM ! S#DS DS DS ¯(pM Case2:.Weknowthatnotransfershappeninandintheotherdomains: cen solo x p !1 DM#KM DS DS ¯(pM !3 DM#KM KS K+#DM ¯(pM+(pS !4 KS KS 0 ¯(pM+(pS !5 KS DS 0 ¯(pM+(pSDS KS !6 DM#KM DS K+#DM ¯(pM+(pSD+"K+ DM"KM. 6.4.4State-DependentBargainingandCoordinationAssumeK,D.CoordinationwouldrequirethattheFOCsforareidenticalwiththoseofcen.Or,ineachdomain cen.Withincompletecontractingtheplayers’revenuefunctionsare:solocensolosolosolocensolosoloBecause solo,theFOCscanbeconstructedintermsofourprimitiveshadowmatricesandgradientofbar)+( KM,# K,DRequiringidenticalrst-orderconditionsnowyieldsasystemofpartialdierentialequations:cen isamatrixwithK,D Lemma1,state-dependentbargainingwithanyconstant234andcoordinatesinvestmentdecisions.Proof:Usingtheresultsofwithaandcalculatingineachdomainyields: (pM0pMpMpMpM0pSpSpS (pM )000cenCoordinationrequiresthatcen.TheappropriatesolutionforK,Disfounddirectlybyinspection:anyconstant234Toshowthatthisistheuniquesolution,wesolvethesystemofpartialdierentialequationsineachdomain:-Indomain "KM(#(=#"," withuniquesolution: isafunctionthatdoesnotdependon.Tosatisfytheboundaryconditions,wemusthaveandhencesothatthemanufacturergetsallthesurplusin-Indomainswehaveequality.(Therearenotransfers;centralizedproduction=soloproduction;henceanywilldo.-Indomains "KM((D+#K+)" TheuniquesolutionofthesecondPDEis:andtherstPDErequiresthat,sothatandallsurplusgoestothesupplier.Lemma2,state-dependentbargainingwithanyconstantand K,Dcoordinatesinvestmentdecisions.Proof:Calculationofineachdomain:Wehavethatindomain Combiningwiththeresultsof (pM0 (pM+(pSpMpM (pM+(pS0pSpS (pM+(pS (pM cenByinspectionweagainseethatweshouldhave.Domainismorediasnosingleconstantyieldsasolution.Hence,wemustlookforavariablefunction.Recallthatastate-dependentprice-onlycontractrequirestoinduceinvestmentcoordination.Nowusetheequivalent whichyieldsthefollowingcandidatesoluationfunctionK,D WenowverifythatthisfunctionsatisesthesucientFOC,whichsimplifytorequireindomain (pM+(pS K,D KM# TesttheFOCtoverifycorrectnessofthecandidatesolution FOC:[( "KM(pM#pSM#KM) (pM#pSM#KMS(KS#DS)S#pM)pS(KS#DS) (pM#pSM#KMS(KS#DS)S#pM)·"#(p1#p21#x) FOC "KS(pM#pSM#KM) (pM#pSM#KMS(KS#DS)=pS(pM#pSM#KM) Conclusion:thecandidatesolutionK,DsatisestheFOCandhenceourstate-dependentbargainingcontractinducesinvestmentcoordination.