http://www.econometricsociety.org/ Econometrica,Vol.81,No.6(November,2013),2127–2204 ORGANIZINGTHEGLOBALVALUECHAIN POLANTRÀS HarvardUniversity,Cambridge,MA02138,U.S.A. DAVINCHOR NationalUniversityofSingapore,Singapore117570,Singapore The copyright to this Article is held by the Econometric Society. It may be downloaded, printed and reproduced only for educational or research purposes, including use in course packs. No downloading or copying may be done for any commercial purpose without the explicit permission of the Econometric Society. For such commercial purposes contact the Office of the Econometric Society (contact information may be found at the website http://www.econometricsociety.orgorinthebackcoverofEconometrica).Thisstatementmust be included on all copies of this Article that are made available electronically or in any other format. Econometrica,Vol.81,No.6(November,2013),2127–2204 ORGANIZINGTHEGLOBALVALUECHAIN BYPOLANTRÀSANDDAVINCHOR1 Wedevelopaproperty-rightsmodelofthefirminwhichproductionentailsacontin- uumofuniquelysequencedstages.Ineachstage,afinal-goodproducercontractswith adistinctsupplierfortheprocurementofacustomizedstage-specificcomponent.Our model yields a sharp characterization for the optimal allocation of ownership rights alongthevaluechain.Weshowthattheincentivetointegratesuppliersvariessystem- aticallywiththerelativeposition(upstreamversusdownstream)atwhichthesupplier enterstheproductionline.Furthermore,thenatureoftherelationshipbetweeninte- grationand“downstreamness”dependscruciallyontheelasticityofdemandfacedby thefinal-goodproducer.Ourmodelreadilyaccommodatesvarioussourcesofasymme- tryacrossfinal-goodproducersandacrosssupplierswithinaproductionline,andwe showhowitcanbetakentothedatawithinternationaltradestatistics.Combiningdata fromtheU.S.CensusBureau’sRelatedPartyTradedatabaseandestimatesofU.S.im- portdemandelasticitiesfromBrodaandWeinstein(2006),wefindempiricalevidence broadlysupportiveofourkeypredictions.Intheprocess,wedeveloptwonovelmea- suresoftheaveragepositionofanindustryinthevaluechain,whichweconstructusing U.S.Input–OutputTables. KEYWORDS: Property-rights theory, contractual frictions, sequential production, downstreamness,intrafirmtrade. 1. INTRODUCTION MOSTPRODUCTIONPROCESSESaresequentialinnature.Atabroadlevel,the process of manufacturing cannot commence until the efforts of R&D centers inthedevelopmentorimprovementofproductshaveproventobesuccessful, while the sales and distribution of manufactured goods cannot be carried out until their production has taken place. Even within manufacturing processes, thereisoftenanaturalsequencingofstages.First,rawmaterialsareconverted into basic components, which are next combined with other components to producemorecomplicatedinputs,beforethemselvesbeingassembledintofi- nal goods. This process very much resembles Henry Ford’s original Model T 1Wethanktheeditorandthreeanonymousrefereesfortheirhelpfulcommentsandsugges- tions. We are also grateful to Arnaud Costinot, Don Davis, Ron Findlay, Elhanan Helpman, KalaKrishna,MarcMelitz,EstebanRossi-Hansberg,DanielTrefler,JonathanVogel,andDavid Weinstein,aswellasaudiencesatBerkeleyHaas,Chicago,Columbia,Harvard,Northeastern, NotreDame,NYUStern,Princeton,Stanford,UCLA,UNH,Wisconsin,Kiel,Munich,Tübin- gen,Bonn,CityUniversityofHongKong,HKUST,NanyangTechnologicalUniversity,National University of Singapore, Singapore Management University, the Econometric Society World Congress(Shanghai),theSocietyfortheAdvancementofEconomicTheoryConference(Singa- pore),theAsiaPacificTradeSeminars(Honolulu),theAustralasianTradeWorkshop(UNSW), andtheNBERSummerInstitute.WethankNathanNunnformakingavailablehisdata.Ruiqing Cao,MiraFrick,GurmeetSinghGhumann,FrankSchilbach,andZhichengSongprovidedex- cellent research assistance.Chor acknowledges the hospitality of the International Economics SectionatPrinceton,aswellasresearchfundingprovidedbyaSingLunFellowshipatSingapore ManagementUniversity.Allerrorsareourown. ©2013TheEconometricSociety DOI:10.3982/ECTA10813 2128 P.ANTRÀSANDD.CHOR production assembly line, but recent revolutionary advances in information and communication technology, coupled with a gradual reduction in natural and man-made trade barriers, now allow such value chains to be “sliced up” intogeographicallyseparatedsteps. The implications of such sequential production for the workings of open- economy generalequilibrium modelshave beenwidely exploredin the litera- ture.Severalpapers,mostnotablyFindlay(1978),DixitandGrossman(1982), Sanyal(1983),Kremer(1993),Kohler(2004),andCostinot,Vogel,andWang (2013), have emphasized that the pattern of specialization along the value chain has implications for the world income distribution and for how shocks spreadacrosscountries.Others,includingYi(2003),Harms,Lorz,andUrban (2012), and Baldwin and Venables (2013), have unveiled interesting nonlin- earitiesintheresponseoftradeflowstochangesintradefrictionsinmodelsof productionwherevalueisaddedsequentiallyalonglocationsaroundtheglobe. The focus of our paper is different. Our aim is to understand how the se- quentiality of production shapes the contractual relationships between final- goodproducersandtheirvarioussuppliers,andhowtheallocationofcontrol rightsalongthevaluechaincanbedesignedinawaythatelicits(constrained) optimaleffortonthepartofsuppliers.Anobviouspremiseofourworkisthat, although absent from most general equilibrium models, contractual frictions arerelevantfortheefficiencywithwhichproductioniscarriedout,andalsofor the way in which production processes are organized across borders. We find this to be a natural premise particularly in international trade environments, inwhichdeterminingwhichcountry’slawsareapplicabletoparticularcontrac- tual disputes is often difficult. The detrimental effects of imperfect contract enforcementoninternationaltradeflowsareparticularlyacuteintransactions involving intermediate inputs, as these tend to be associated with longer lags betweenthetimeanorderisplaced(andthecontractissigned)andthetime thegoodsorservicesaredelivered(andthecontractisexecuted).Suchtransac- tions, moreover, often entail significant relationship-specific investments and other sources of lock-in on the part of both buyers and suppliers.2 The rele- vanceofcontractingfrictionsfortheorganizationofproductionalsonowrests onsolidempiricalunderpinnings.3 2Suppliers often customize their output to the needs of particular buyers and would find it hardtosellthosegoodstoalternativebuyers,shouldtheintendedbuyerdecidenottoabideby thetermsofthecontract.Similarly,buyersoftenundertakesignificantinvestmentswhosereturn canbeseverelydiminishedbyincompatibilities,productionlinedelays,orqualitydebasements associatedwithsuppliersnotgoingthroughwiththeircontractualobligations. 3Arecentliterature(see,forinstance,Nunn(2007)andLevchenko(2007))hasconvincingly documentedthatcontractinginstitutionsareanimportantdeterminantofinternationalspecial- ization.Anotherbranchofthetradeliterature,towhichourpaperwillcontribute,hasalsoshown thattheownershipdecisionsofmultinationalfirmsexhibitvariouspatternsthatareconsistent withGrossmanandHart’s(1986)incomplete-contracting,property-rightstheoryoffirmbound- aries(see,amongothers,Antràs(2003),NunnandTrefler(2008,2013),andBernard,Jensen, Redding,andSchott(2010)). ORGANIZINGTHEGLOBALVALUECHAIN 2129 In this paper, we develop a property-rights model of firm boundaries that permits an analysis of the optimal allocation of ownership rights in a setting where production is sequential in nature and contracts are incomplete. Our model builds on Acemoglu, Antràs, and Helpman (2007). Production of final goods entails a large number (formally, a continuum) of production stages. Each stage is performed by a different supplier, who needs to undertake a relationship-specific investment in order to produce components that will be compatiblewiththoseproducedbyothersuppliersinthevaluechain.Theser- vicesofthesecomponentsarecombinedaccordingtoaconstant-elasticity-of- substitution(CES)aggregatorbyafinal-goodproducerthatfacesanisoelastic demandcurve.Contractsbetweenfinal-goodproducersandtheirsuppliersare incompleteinthesensethatcontractscontingentonwhethercomponentsare compatibleornotcannotbeenforcedbythirdparties. ThekeyinnovationrelativetoAcemoglu,Antràs,andHelpman(2007)—and relative to the previous property-rights models of multinational firm bound- aries in Antràs (2003, 2005) and Antràs and Helpman (2004, 2008)—is that we introduce a natural (or technological) ordering of production stages, so that production at a stage cannot commence until the inputs or components fromallupstreamstageshavebeendelivered.Absentabindinginitial(exante) agreement,thefirmanditssuppliersarelefttosequentiallybargainoverhow thesurplusassociatedwithaparticularstageistobedividedbetweenthefirm and the particular stage supplier. As in Grossman and Hart (1986), in this incomplete-contracting environment, owning a supplier is a source of power for the firm because the residual rights of control associated with ownership allow the firm to take actions (or make threats) that enhance their bargain- ingpowervis-à-visthesupplier.However,theoptimalallocationofownership rightsdoesnotalwaysentailallproductionstagesbeingintegratedbecause,by reducingthebargainingpowerofsuppliers,integrationreducestheincentives ofsupplierstoinvestintherelationship.4 We begin in Section 2 by developing a benchmark model of firm behav- ior that isolates the role of the degree of “downstreamness” of a supplier in shaping organizational decisions. A key feature of our analysis is that the relationship-specific investments made by suppliers in upstream stages affect the incentives to invest of suppliers in downstream stages. The nature of this dependenceisshaped,inturn,bywhethersuppliers’investmentsaresequential complements orsequential substitutes,according to whether higher investment levelsbypriorsuppliersincreaseordecreasethevalueofthemarginalproduct ofaparticularsupplier’sinvestment.Eventhough,fromastricttechnological pointofview(i.e.,inlightoftheCESaggregatorofinputs),inputsarealways 4ZhangandZhang(2008,2011)introducedsequentialelementsinastandardGrossmanand Hart (1986) model, but focused on one-supplier environments in which either the firm or the supplierhasafirst-moveradvantage.Otherpapersthathavestudiedoptimalincentiveprovision insequentialproductionprocessesincludeWinter(2006)andKimandShin(2012). 2130 P.ANTRÀSANDD.CHOR complements,suppliers’investmentscanstillprovetobesequentialsubstitutes when the price elasticity of demand faced by the final-good producer is suffi- ciently low, since in such cases, the value of the marginal product of supplier investmentsfallsparticularlyquicklyalongthevaluechain.Whetherinputsare sequential complements or sequential substitutes turns out to be determined onlybywhether theelasticity offinal-gooddemandis(respectively)higher or lowerthantheelasticityofsubstitutionacrosstheservicesprovidedbythedif- ferentsuppliers’investments. The central result of our model is that the optimal pattern of ownership along the value chain depends critically on whether production stages are se- quential complements or substitutes. When the demand faced by the final- goodproduceris sufficiently elastic, then there exists a unique cutoff produc- tionstagesuchthatallstagespriortothiscutoffareoutsourced,whileallstages (if any) after that threshold are integrated. Intuitively, when inputs are se- quentialcomplements,thefirmchoosestoforgocontrolrightsoverupstream suppliers in order to incentivize their investment effort, since this generates positivespilloversontheinvestmentdecisionstobemadebydownstreamsup- pliers.Whendemandis,instead,sufficientlyinelastic,theconverseprediction holds:itisoptimaltointegraterelativelyupstreamstages,andifoutsourcingis observedalongthevaluechain,itnecessarilyoccursrelativelydownstream. In Section 3, we show that these results are robust to alternative contract- ing and bargaining assumptions, and stem mainly from the sequential nature of production rather than the sequential nature of bargaining. Furthermore, we show that our framework can easily accommodate a hybrid of sequential andmodularproductionprocesses(or“snakes”and“spiders”intheterminol- ogy of Baldwin and Venables (2013)) as well as several other features which have been built into the recent models of global sourcing cited earlier. These include (headquarter) investments by the final-good producer, productivity heterogeneity across final-good producers, productivity and cost differences across suppliers within a production chain, and partial contractibility. These extensionsproveusefulinguidingourempiricalanalysis. In Sections 4 and 5, we develop an empirical test of the main predictions of our framework. The nature and scope of our test are shaped in significant ways by data availability. Although our model does not explicitly distinguish between domestic and offshore sourcing decisions of firms, data on domes- tic sourcing decisions are not publicly available. We therefore follow the bulk of the recent empirical literature on multinational firm boundaries in using U.S.Censusdataonintrafirmtradetomeasuretherelativeprevalenceofver- tical integration in particular industries.5 More specifically, we correlate the 5See, for example, Nunn and Trefler (2008, 2013), Bernard et al. (2010), and Díez (2010). Antràs(2013)containsacomprehensivesurveyofempiricalpapersusingotherdatasets,includ- ingseveralfirm-levelstudies,thathavesimilarlyusedtheintrafirmimportsharetocapturethe propensitytowardintegrationrelativetooutsourcing. ORGANIZINGTHEGLOBALVALUECHAIN 2131 shareofU.S.intrafirmimportsintotalU.S.importsreportedduringtheperiod 2000–2010withtheaveragedegreeof“downstreamness”ofthatindustry,and we study whether this dependence is qualitatively different for the sequential complements versus sequential substitutes cases. We propose two measures of downstreamness, both of which are constructed from the 2002U.S. Input– OutputTables.Ourfirstmeasureistheratiooftheaggregatedirectusetothe aggregate total use (DUse_TUse) of a particular industry i’s goods, where the directuseforapairofindustries(i(cid:2)j)isthevalueofgoodsfromindustryi di- rectlyusedbyfirmsinindustryj toproducegoodsforfinaluse,whilethetotal use for (i(cid:2)j) is the value of goods from industry i used either directly or indi- rectly(viapurchasesfromupstreamindustries)inproducingindustryj’soutput for final use. A high value of DUse_TUse thus suggests that most of the con- tribution of input i tends to occur at relatively downstream production stages thatarecloseto(onestageremovedfrom)finaldemand.Oursecondmeasure ofdownstreamness(DownMeasure)isaweightedindexoftheaverageposition inthevaluechainatwhichanindustry’soutputisused(i.e.,asfinalconsump- tion, as direct inputs to other industries, as direct inputs to industries serving asdirectinputstootherindustries,andsoon),withtheweightsbeinggivenby theratiooftheuseofthatindustry’soutputinthatpositionrelativetothetotal output of that industry. Although constructing such a measure would appear torequirecomputinganinfinitepowerseries,weshowthatDownMeasurecan be succinctly expressed as a simple function of the square of the Leontief in- versematrix.AsdiscussedinAntràs,Chor,Fally,andHillberry(2012),thereis acloseconnectionbetweenDownMeasureandthemeasureofdistancetofinal demandderivedindependentlybyFally(2012). Ourempiricaltestsalsocallonustodistinguishbetweenthecasesofsequen- tial complements and substitutes identified in the theory. For that purpose, weusetheU.S.importdemandelasticitiesestimatedbyBrodaandWeinstein (2006)anddataonU.S.Input–OutputTablestocomputeaweightedaverage of the demand elasticity faced by the buyers of goods from each particular industry i. The idea is that, for sufficiently high (respectively, low) values of this average demand elasticity, we can be relatively confident that input sub- stitutability is lower (respectively, higher) than the demand elasticity. Ideally, one would have used direct estimates of cross-input substitutability (and how they compare to demand elasticities) to distinguish between the two theoret- ical scenarios, but, unfortunately, these estimates are not readily available in theliterature. Figure 1 provides a preliminary illustration of our key empirical findings, which lend broad support for the theoretical implications of our model. As is apparent from the dark bins, for the subset of industries with above-median average buyer demand elasticities (labeled as “Complements”), the average U.S.intrafirmimportshare(fortheyear2005)risesaswemovefromthelowest tercileof DownMeasure tothehighest.Inthelightbins,thispatternisexactly reversedwhenconsideringthoseindustriesfacingbelow-medianaveragebuyer 2132 P.ANTRÀSANDD.CHOR FIGURE1.—Downstreamnessandtheshareofintrafirmtrade. demand elasticities (labeled as “Substitutes”), with the intrafirm trade share steadilyfallingacrosstercilesofDownMeasureinstead.6 Ourregressionanalysiswillconfirmthattheabovepatternsholdundermore formal testing. We uncover a positive and statistically significant relationship between each of the measures of downstreamness and the intrafirm import share in a given sector, with this relationship emerging only for high values of the demand elasticity faced by buyer industries (i.e., in the complements case). These findings hold when controlling for other determinants of the in- trafirm trade share raised in the literature, and which our theoretical exten- sions also indicate are important to explicitly consider. They are, moreover, robustinspecificationsthatfurtherexploitthecross-countrydimensionofthe intrafirmtradedata,whilecontrollingforunobservedvariationinfactorcosts withcountry-yearfixedeffects.Forawiderangeofspecifications,wewillalso reportasignificantnegativerelationshipbetweendownstreamnessandthein- trafirmimportshareforgoodswithlowaveragebuyerdemandelasticities(i.e., inthesubstitutescase),aspredictedbyourmodel. 6Itisnothardtofindexamplesoflargeindustriesthatexhibitsimilardegreesofdownstream- ness,butfaceverydifferentaveragebuyerdemandelasticitiesandalsoverydifferentintegra- tionpropensities.Forinstance,Women’sapparel(IO315230)andAutomobiles(IO336111)are amongthetenmostdownstreammanufacturingindustries,butbuyerstendtobemuchlessprice- sensitiveintheirdemandfortheformer(elasticity=4.90)thanforthelatter(elasticity=19.02). Thesetwoindustriesarethusclassifiedunderthesequentialsubstitutesandcomplementscases, respectively,and,consistentwithourmodel,theshareofintrafirmtradeislowinWomen’sap- parel (0.108) and very high in Automobiles (0.946). As we shall see later in our econometric analysis,thisbroadpatterncontinuestoholdwhencontrollingforotherindustrycharacteristics thatmightalsoaffectthepropensitytowardintrafirmtrade. ORGANIZINGTHEGLOBALVALUECHAIN 2133 Theremainderofthispaperisorganizedasfollows.InSection2,wedevelop our benchmark model of sequential production with incomplete contracting andstudytheoptimalownershipstructurealongthevaluechain.InSection3, wedevelopafewtheoreticalextensionsanddiscusshowweattempttotakethe modeltothedata.Wedescribeourdatasourcesandempiricalspecificationin Section4,andpresenttheresultsinSection5.Section6offerssomeconcluding remarks. All the proofs in the paper are relegated to the Appendix (and the SupplementalMaterial(AntràsandChor(2013))). 2. AMODELOFSEQUENTIALPRODUCTIONWITHINCOMPLETECONTRACTS Webeginbydevelopingabenchmarkmodeloffirmbehavioralongthelines ofAcemoglu,Antràs,andHelpman(2007),butextendedtoincorporateade- terministic sequencing of production stages. The model is stylized so as to emphasizethenewinsightsthatemergefromconsideringthesequentialityof production. We will later incorporate more realistic features and embed the frameworkinindustryequilibriumtoguidetheempiricalanalysis. 2.1. BenchmarkModel 2.1.1. SequentialProduction We consider the organizational problem of a firm producing a final good. Productionrequiresthecompletionofameasureoneofproductionstages.We indexthesestagesbyj∈[0(cid:2)1],withalargerj correspondingtostagesfurther downstream(closertothefinalendproduct),andweletx(j)betheservicesof compatibleintermediateinputsthatthesupplierofstagej deliverstothefirm. Thequality-adjustedvolumeoffinal-goodproductionisthengivenby (cid:2)(cid:3) (cid:4) 1 1/α (1) q=θ x(j)αI(j)dj (cid:2) 0 whereθisaproductivityparameter,α∈(0(cid:2)1)isaparameterthatcapturesthe (symmetric) degree of substitutability among the stage inputs, and I(j) is an indicatorfunctionsuchthat: ⎧ ⎨1(cid:2) ifinputj isproduced I(j)= afterallinputsj(cid:3)<j havebeenproduced(cid:2) ⎩ 0(cid:2) otherwise. We normalize x(j)= 0 if an incompatible input is delivered at stage j. Al- though production requires completion of all stages, note that α>0 ensures thatoutputremainspositiveevenwhensomestagesmightbecompletedwith incompatible inputs. In words, although all stages are essential from an engi- neeringpointofview,weallowsomesubstitutioninhowthecharacteristicsof 2134 P.ANTRÀSANDD.CHOR these inputs shape the quality-adjusted volume of final output. For example, producingacarrequiresfourwheels,twoheadlights,onesteeringwheel,and soon,butthevalueofthiscarforconsumerswilltypicallydependontheser- vicesobtainedfromthesedifferentcomponents,withahighqualityincertain partspartlymakingupforinferiorqualityinothers. Ourproductionfunctionin(1)resemblesaconventionalCESfunctionwith a continuum of inputs, but the indicator function I(j) makes the production technologyinherentlysequentialinthatdownstreamstagesareuselessunless the inputs from upstream stages have been delivered. In fact, the technology in(1)canbeexpressedindifferentialformbyapplyingLeibniz’sruleas 1 q(cid:3)(m)= θαx(m)αq(m)1−αI(m)(cid:2) α (cid:8) where q(m) = θ( mx(j)αI(j)dj)1/α. Thus, the marginal increase in output 0 brought about by the supplier at stage m is given by a simple Cobb–Douglas function of this supplier’s (compatible) input production and the quality- adjustedvolumeofproductiongenerateduptothatstage(whichcanbeviewed asanintermediateinputtothestage-mproductionprocess). 2.1.2. InputProduction There is a large number of profit-maximizing suppliers who can engage ei- therinintermediateinputproductionorinanalternativeactivitythatdelivers an outside option normalized to 0. We assume that each intermediate input must be produced by a different supplier with whom the firm needs to con- tract.Eachsuppliermustundertakearelationship-specificinvestmentinorder toproduceacompatibleinput.Forsimplicity,weassumethattheinputisfully customized to the final-good producer, so the value of this input for alterna- tivebuyersisequalto0.Tohighlighttheasymmetriesthatwillarisesolelyfrom thesequencingofproduction,weassumethatproductionstagesareotherwise symmetric: the marginal cost of investment is common for all suppliers and equal to c, and in all stages j ∈[0(cid:2)1], one unit of investment generates one unitofservicesofthestage-j compatibleinputwhencombinedwiththeinputs from upstream suppliers. (We will relax these symmetry assumptions later in Section 3.) Incompatible inputs can be produced by all agents (including the firm)atanegligiblemarginalcost,buttheyaddnovaluetofinal-goodproduc- tionapartfromallowingthecontinuationoftheproductionprocess. 2.1.3. Preferences The final good under study is differentiated in the eyes of consumers. The goodbelongstoanindustryinwhichfirmsproduceacontinuumofgoodsand consumers have preferences that feature a constant elasticity of substitution ORGANIZINGTHEGLOBALVALUECHAIN 2135 acrossthesevarieties.Morespecifically,denotingbyϕ(ω)thequalityofava- riety and by q˜(ω) its consumption in physical units, the sub-utility accruing fromthisindustryisgivenby (cid:2)(cid:3) (cid:4) (cid:9) (cid:10) 1/ρ (2) U = ϕ(ω)q˜(ω) ρdω with ρ∈(0(cid:2)1)(cid:2) ω∈Ω where Ω denotes the set of varieties. Note that these preferences feature di- minishing marginal utility with respect to not only the quantity but also the quality of the goods consumed. As a result, in our previous car example, fur- ther quality improvements on a high-end car would not add as much satis- faction to consumers as they would in a low-en(cid:8)d car. As is well known, when maximizing(2)subjecttothebudgetconstraint p(ω)q˜(ω)dω=E,where ω∈Ω E denotes expenditure, consumer demand for a particular variety features a constantpriceelasticityequalto1/(1−ρ).Furthermore,theimpliedrevenue function of a firm that sells variety ω is concave in quality-adjusted output q(ω)≡ϕ(ω)q˜(ω)withaconstantelasticityρ.Combiningthisfeaturewiththe production technology in (1), the revenue obtained by the final-good produc- ingfirmunderstudycanbewrittenas (cid:2)(cid:3) (cid:4) 1 ρ/α (3) r=A1−ρθρ x(j)αI(j)dj (cid:2) 0 where A>0 isanindustry-widedemandshifterthatthefirmtreatsasexoge- nous. 2.1.4. CompleteContracts Before discussing in detail our contracting assumptions, it is instructive to considerfirstthecaseofcompletecontractsinwhichthefirmhasfullcontrol overallinvestmentsandthusoverinputservicesatallstages.Insuchacase,the firmmakesacontractoffer[x(j)(cid:2)s(j)]foreveryinputj∈[0(cid:2)1],underwhicha supplierisobligedtosupply x(j) ofcompatibleinputservicesasstipulatedin thecontractinexchangeforthepayments(j).Itisclearthatthefirmwillhave an incentive to follow the natural sequencing of production, so that I(j)=1 for all j, and the optimal contract simply solves the following maximization program: (cid:2)(cid:3) (cid:4) (cid:3) 1 ρ/α 1 max A1−ρθρ x(j)αdj − s(j)dj {x(j)(cid:2)s(j)}j∈[0(cid:2)1] 0 0 s.t. s(j)−cx(j)≥0(cid:9) Solving this problem delivers a common investment level x = (ρA1−ρθρ/ c)1/(1−ρ) for all intermediate inputs and associated firm profits equal to π = (1−ρ)A(ρθ/c)ρ/(1−ρ), while leaving suppliers with a net payoff equal to their outsideoptionofzero(i.e.,s=cx).