美联储-财经系列讨论 Are Supply Networks Effciently Resilient 2024

FinanceandEconomicsDiscussionSeries

FederalReserveBoard,Washington,D.C.

ISSN1936-2854(Print)

ISSN2767-3898(Online)

AreSupplyNetworksE伍cientlyResilient?

AgostinoCapponi,ChuanDu,JosephE.Stiglitz

2024-031

Pleasecitethispaperas:

Capponi,Agostino,ChuanDu,andJosephE.Stiglitz(2024).“AreSupplyNetworksE伍-cientlyResilient?,”FinanceandEconomicsDiscussionSeries2024-031.Washington:BoardofGovernorsoftheFederalReserveSystem,

/10.17016/FEDS.2024.031

.

NOTE:StafworkingpapersintheFinanceandEconomicsDiscussionSeries(FEDS)arepreliminarymaterialscirculatedtostimulatediscussionandcriticalcomment.TheanalysisandconclusionssetfortharethoseoftheauthorsanddonotindicateconcurrencebyothermembersoftheresearchstafortheBoardofGovernors.ReferencesinpublicationstotheFinanceandEconomicsDiscussionSeries(otherthanacknowledgement)shouldbeclearedwiththeauthor(s)toprotectthetentativecharacterofthesepapers.

1

Aresupplynetworksefficientlyresilient?

AgostinoCapponiChuanDuJosephE.Stiglitz*May16,2024

Abstract

Weshowthatsupplynetworksareinefficiently,andinsufficiently,re-silient.Upstreamfirmscanexpandtheirproductioncapacitytohedgeagainstsupplyanddemandshocks.Thesocialbenefitsofsuchinvestmentsarenotinternalized,however,becauseofmarketpowerandmarketincompleteness.Upstreamfirmsunderinvestincapacityandresilience,passingonthecoststodownstreamfirms,anddrivetradeexcessivelytowardthespotmarkets.Thereisawedgebetweenthemarketsolutionandaconstrainedoptimalbenchmark,whichpersistsevenwithoutrareandlargeshocks.Policiesdesignedtoin-centivizecapacityinvestment,reducerelianceonspotmarkets,andenhancecompetitionamelioratetheexternality.(JELD21,D24,D25,D43,D85,E23,L13)

*Capponi:DepartmentofIndustrialEngineeringandOperationsResearch,ColumbiaUniversity,NewYork;email:ac3827@.Du:BoardofGovernorsoftheFederalReserveSystem,WashingtonD.C.;email:chuan.du@.Stiglitz:ColumbiaBusinessSchool,ColumbiaUni-versity,NewYork;Email:jes322@.Theviewsinthispaperaresolelytheauthors’andshouldnotbeinterpretedasreflectingtheviewsoftheBoardofGovernorsoftheFederalReserveSystemorofanyotherpersonassociatedwiththeFederalReserveSystem.AsignificantpartofthisworkwasundertakenwhileChuanDuwasemployedattheBankofEngland.TheviewsexpressedinthispaperdonotnecessarilyreflecttheviewsoftheBankofEnglandoranyofitscommittees.WethankourdiscussantsandconferenceparticipantsattheAlliedSocialScienceAssociationAnnualMeeting2023andtheInternationalEconomicAssociationWorldCongress2023,aswellasseminarparticipantsattheBankofEnglandandtheFederalReserveBoard,fortheirconstructivecommentsandsuggestions.

2

TheshortagesandspikesinpricesofcertainintermediategoodsduringtheCOVID-19pandemicdemonstratedthefragilityofsupplychains.Prominentex-amplesincludedaglobalshortageofsemiconductorsthatledtoadramaticriseinthepriceofsecondhandcarsintheU.S.andanunprecedenteddemandforhandsanitizerandpersonalprotectivegearthattriggeredsupplyshortagesintheirre-spective,aswellasinterlinked,industries.Policymakersreactedstronglybytakingindustry-specificactionstorepairlinkagesandimproveresilience.Forexample,theBiden-HarrisAdministrationworkedinpartnershipwithCongresstoprovidenewlegislationtoalleviatespecificsupplychaindisruptionsandpromotegreaterresilienceinfuturesituations.Moreover,whilethelargeandsmallsupplychaindisruptionsduringCOVID-19hadpropelledtheissueintopopulardiscourse,thecrackshadbeenevidentbeforethepandemic.HanjinShipping,aworld’stop10containercarrier,filedforbankruptcyinSeptember2016becauseofsluggishfreightratescausedbyweakdemandandsoaringglobalcapacity.Thebankruptcyaffectedglobalsupplychains,becausehalfofHanjin’scontainershipsweredeniedaccesstoports.MajorU.S.retailers,suchasJ.C.PenneyandWalmart,begantodivertandswitchcarriersfortheircontainerstoothersuppliers.Similarly,thefailureofCar-illioninJanuary2018,oncethesecond-largestconstructioncompanyintheU.K.,broughtdownmanyofitssuppliers

.1

Theseexperienceswithsupplynetworkdisruptionsleftopenthequestion:Had

firmsinvestedtoolittleinresilienceexante?Thepandemicwasanextremeevent,and,ingeneralfirmsshouldnotbeexpectedtoanticipateandplanforeverypos-siblecontingency.Doingsowouldalmostsurelybeinefficient,entailingexcessivefocusonresilience.Weshowhere,however,thatgivenmarketpowerandmarketincompleteness,oneshouldexpectmarketstounderinvestinresiliencerelativetoaconstrainedefficientbenchmark.

Weformulateatractabletheoreticalmodelwherebyacollectionofintermedi-

1Fordetailedcoverageoftheseepisodes,see

theFinancialTimes:"Carchipshortageshineslight

onfragilityofUSsupplychain";

CNN:"Distilleriesaremakinghandsanitizerwiththeirin-house

alcoholandgivingitoutforfreetocombatcoronavirus";

Biden-HarrisSupplyChainDisruptions

TaskForce;

DeutscheWelle:"BankruptHanjinsparksshippingcrisis";and

theGuardian:"Carillion

collapse:Twoyearson,thegovernmenthaslearnednothing"..

Amoreacademicaccountcanbefoundin

BaqaeeandFarhi

(2022),

Guerrierietal.

(2022)and

DiGiovannietal.

(2022)

.

3

ateandfinalgoodsproducersformsupplylinkagestomeetuncertainconsumerde-mandandaccommodatesupplyshocks.Eachfinalgoodsproducer(thedownstreamfirm)cansourcedifferentiatedinputsfromoneormoresuppliersoftheintermedi-ategoods(theupstreamfirms).Intermediategoodsproducersengageinprice-thatis,Bertrand-competitionwithdifferentiatedproducts,takingthepricessetbycompetitorsasgiven.Loweringthepricechargedallowsanintermediategoodspro-ducertoincreasedemandontheextensivemargin(byattractingmorefinalgoodsproducers)

.2

Intermediategoodsproducersfaceuncertaintyindemandandsupplyconditions.Theyinvestinnon-scalableproductioncapacitybeforetherealizationofshocks,reflectingthatsomefactorsofproductioncannotbereadilyadjustedatshortnotice

.3

Giventhestructuralfrictionsintheeconomy-namely,thelagsinproductionandtheuncertaintyaroundfuturemarketconditions-over-investmentincapacitycanbejustasinefficientasunder-investment.Asupplynetworkthatisefficientlyresilientstrikestheoptimalbalanceonresilience,takingintoaccountthesestructuralfrictions.

Usingthemodel,wedemonstratetheexistenceofamarketfailureindecentral-izedsupplynetworks,wherebyupstreamfirmsdonotfullyinternalizethesocialbenefitsofbuildingproductioncapacity.Whenupstreamfirmsover-investinca-pacity,partofthecostsavingsarepassedontodownstreamfirmsvialowerprices;butwhenfirmsunderinvest,theycandefendtheirprofitmarginsdespitemountingcostsbycharginghigherprices.Theshortagesthatresultfromunderinvestmentenhancemarketpower,whichtheupstreamfirmsrationallyanticipate.Asaresult,upstreamfirmswillalwaysleantowardunderinvestment.

Thispecuniaryexternalityisnotinternalizedbythedecentralizedmarketbe-causeofacombinationof(1)marketpowerand(2)marketincompleteness.First,upstreamintermediategoodsproducersexhibitmarketpowerbecause(a)onlyafinitenumberofsuchfirmsexist,and(b)theintermediategoodstheyproduceareimperfectsubstitutesofeachother.Second,firmsdonothaveaccesstothefullset

ofArrow-Debreusecurities,andinsteadmusttradeeitheronthepre-ordermarket,

2Inamoregeneralcase,loweringthepricemayalsoaffecttheintensivemargin.

3Semiconductorsareanexampleofanimportantintermediategoodsthatrequiressignificantcapacityinvestmentupfront.IntheEuropeanUnion,

theEuropeanChipsAct

(2023)aimstoprovideadditionalpublicandprivateinvestmentsofmorethanEUR15billion.

4

oronthespotmarketoncetheshockshaverealized.Thepre-ordermarketofferspartialinsurancetoboththeupstreamanddownstreamfirms.Fortheupstreamfirms,pre-ordersestablishaminimumlevelofdemandfortheiroutputs,andhelpwiththeirupfrontnon-scalablecapacityinvestmentdecision.Forthedownstreamfinalgoodsproducers,apre-ordercontractlocksinanagreedpricefortheinterme-diaryinputsintheirproduction,shieldingthemfromcostshocksintheupstreamsector.Ifrealizeddemandforfinalgoodsexceedswhatcanbefulfilledthroughpre-orders,thedownstreamfirmcanthensourcetheextrainputsrequiredfromthespotmarket.Asweobserveinpractice,thespotandpre-ordermarketsareinsufficienttodealwiththefullspectrumofpossibleshocks,andthusareunabletoprovidefullinsuranceagainstsupplynetworkdisruptions

.4

Asaconsequenceofthisexternality,weshowthatthemarket-basednetworkin-veststoolittleinproductioncapacity(K*)relativetoaconstrainedoptimalbench-mark(KSP)withasocialplannerfacingthesameinformationalandtechnologicalconstraintsastheprivatemarket.Evenundertheconstrainedbenchmark,itisnotoptimaltobuildenoughcapacitytoaccountforallcontingencies.Sotherewillbetimeswhenfirmsexposthaveconsiderablemarketpower,which,obviously,thesocialplannerwouldnottakeadvantageofbutprivatefirmswould.Inshort,market-basedsupplynetworksareinefficientlyresilient:K*<KSP.

Remarkably,thiswedgebetweenthedecentralizedandcentralizedsolutionarisesevenwhenrarelargeshocksareabsent,andtheeconomyoperatesina“fullproduc-tion”equilibriumwherebysupplyissufficientlyagiletoaccommodateallpossibledemand.Ourresultsdonotdependonanarbitraryspecificationofthedistributionofshocks-forexample,wedonotrequireathresholdfortheprobabilityoflargenegativeshocks.Nordoweneedtoimposealevelofriskaversiononthepartofprivateagentsorthesocialplanner.Capacityinvestmentissuboptimallylow,evenwheneveryagent–includingtheconstrainedsocialplanner–isriskneutral.

Extendingtheanalysistoaccountforraredisasters(intheappendix),weshow

thattheresponseofmarket-basedsupplynetworkstoshockscanbehighlynonlin-

4Itisobviousthatsuchfullinsurancedoesnotexist.Giventherangeofshocksthatcouldoccur–someofwhicharenownotevenreallyconceivable–theincompletenessofinsurancemarketsisinevitable.Theoriesofasymmetricinformationprovidefurtherexplanationsoftheabsenceofafullsetofinsurancemarkets.See

GreenwaldandStiglitz

(1986)and

Stiglitz

(1982)

.

5

ear.Privatesupplynetworksareseeminglyresilientduringnormaltimesandcan

comfortablywithstandsmalltomoderateshocks,buttheyarefragiletorarelargeshocks,whenrealrigiditiespreventsuppliersfromfullymeetingtheneedsofthemarket

.5

Withalargeenoughshock,thereisatransitionfromamonopolisticallycompetitiveregimetoalocalmonopolyregime,wherebyupstreamfirmsarenolongerpricingtocompeteandeachdownstreamfirmwillreceiveonlyonecredibleofferforinputs.Inotherwords,inacrisis,individualsuppliersprioritizetheneedsoftheirlocalmarketbutwithincreasedmargins

.6

Supplynetworkfragilitycanleadtoanincreaseinmarketpower(inourmodel,reflectedinsuboptimalretrenchmentinmarketcoverage),especiallywhendemandisatitsgreatest.

Thesizeofthewedgebetweenthedecentralizedandcentralizedsolutionde-pendsendogenouslyonfirms’relianceonthespotmarket,andexogenouslyonthestructuralparametersoftheeconomy.Aneconomyexhibitinggreaterscalability(productionfunctionsthatrelylessonnon-scalablecapacityinvestments),highersubstitutability(intermediategoodsinputsthataremoreinterchangeable)andmorecompetition(moreupstreamfirms)willbemoreefficientlyresilient.

Therefore,therearebroadlythreeavenuesfornarrowingthewedge.First,adi-rectgovernmentalsubsidytargetinginvestmentinproductioncapacitycouldserveasthemostpragmaticremedy.Second,enhancingincentivesfortheuseofpre-ordermarketscanofferupstreamfirmstheassuranceofrecoupinginitialcosts.Weshowthatanoverrelianceonthespotmarketcontributestofragilityinthesupplynetwork

.7

Third,thegovernmentcanpromotestructuralchangesintheeconomytoenhancescalability,substitutabilityandcompetition.Enhancingcompetitionis

5By“seeminglyresilient,”wemeanthatdemandcanbefullymetatsomeprice.Itisstillthecasethatthereistoolittlecapacity.

6ThesurgeindemandforCOVIDvaccinesin2021andthefranticpursuitofnaturalgasduringtheEuropeanenergycrisisin2022serveasillustrativeexamples.Globalsupplyconstraintsoftenleadtoredirectiontowardwealthiernations,leavingless-affluentdevelopingmarketseconomicallydisadvantagedduringchallengingtimes.Duringthepost-COVIDrecovery,evidencesuggestsamarkedincreaseinmarketpower(markups)associatedwiththesupplychaininterruptions.See

KonczalandLusiani

(2022)

.

7Forinstance,in2021and2022,morethan30energycompaniesintheU.K.failedasaresultoftoarapidincreaseinwholesalenaturalgaspricesandinade-quatehedgingthroughfutures/forwardcontractsbytheenergycompanies.Fordetails,see

/uk/advisor/energy/failed-uk-energy-suppliers-update

.

6

goodinitsownright,anddoublysowhenmakingsupplynetworksmoreefficientlyresilient.

1Relatedliterature

Theliteratureontheresilienceofsupplynetworkstoshockscanberoughlycate-gorizedintotwobranches.Thefirstfocusesonanalyzingthemechanismsthroughwhichidiosyncraticshockspropagateandamplifywithinafixednetworkoffirmswithpre-specifiedrelationships.

AcemogluandTahbaz-Salehi

(2020)examinethe

effectofproductivityshocksonthedistributionofeconomicsurplus,firmfailures,andtheamplificationofshocksthroughdisruptions.

Acemogluetal.

(2012)pro

-poseamodelthatexplainshowmicroshockscanbemagnifiedintomacrofluc-tuationsthroughinput-outputlinkages.

Carvalhoetal.

(2021)usedatafromthe

2011Japaneseearthquaketodemonstratethesignificantmacroeconomicimpli-cationsofidiosyncraticshocks.

BarrotandSauvagnat

(2016)revealevidenceof

fragilitycausedbythepropagationoffirm-specificshocks,usingdataonnaturaldisasters.Wereferto

Carvalho

(2014)and

CarvalhoandTahbaz-Salehi

(2019)for

athoroughreviewofsuchmechanisms.

Thatmarketswouldnotbepreparedforeveryshocktheyconfrontisnotasur-prise.Theanalyticallyinterestingquestionisthenormativeone:Relativetoanappropriatebenchmark,dotheyadequatelyprepareforshocks?Thefailureofeachfirminacompetitiveenvironmenttotakeaccountofhowcapacitydecisionsaffectthedistributionofpricesinthespotmarketsisoneofthetwocentralmarketfailuresthatweidentify.

Thesecondbranchofliteraturefocusesonfirms’strategicresponsestomit-igatethenegativeeffectsofsupplychaindisruptions.

Birgeetal.

(2023)explore

howfirmsinasupplychainnetworkstrategicallyreactpost-disruptionbyoptimallyswitchingdemandandreroutingsupplyfromdefaultedfirms.

AmelkinandVohra

(2020)examinethecompetingretailers’decisionmakingprocesswhenselecting

suppliers,takingintoaccountfactorssuchaspricesandsuppliers’reliabilityasmeasuredbyyielduncertaintyandcongestion

.8

8Afewotherstudiesfromtheoperationsmanagementliteratureanalyzethemechanismsthrough

7

Ourworkiscloselyrelatedto

Elliotetal.

(2022)andthatby

Grossmanetal.

(2023),which(also)examinesupplynetworkformationandfragility.

Intheirmod-els,downstreamfirmssourcecustomizedinputsfromupstreamfirms.Toinsureagainstpossiblesupplydisruptions,downstreamfirmsstrategicallyinvestinrela-tionshipswithmultiplepotentialsuppliers

.9

Onemightinferfromtheiranalysesthatsystemicfragilityshouldbereducedifinputsweremore(albeitstillimper-fectly)substitutable,andthereexistedacommonspotmarketforsuchinputs.Weshowthatnotonlywouldsuchaspotmarketbeinsufficienttoeliminatesupplynetworkfragility,butthatmarketparticipants’overrelianceonspotmarkettrans-actionswouldactuallyamplifytheinherentexternalities.Inourmodel,fragilitywithinthesupplynetworkisnotaconsequenceofacatastrophicbreakdownofup-streamsuppliersorafailureinsupplierdiversificationbutduetoamorestructuralcombinationofmarketpowerandincompletemarkets.

Ontheempiricalside,

Atalayetal.

(2011)estimateamodeloffirms’buyer

-supplierrelationshipsusingmicrodataonfirms’customers.

Crosignanietal.

(2019)

investigatetheconsequencesofsupplyshocksresultingfromNotPetya,oneofthemostseverecyberattacksinhistory.Theyobservethattheaffecteddownstreamcus-tomersweremoreinclinedtoestablishnewrelationshipswithalternativesupplierswhileterminatingexistingrelationshipswiththedirectlyaffectedfirms.Lastly,

BaldwinandFreeman

(2022)examinesthecross-borderdimensionsofresilience

inglobalsupplychains.

Therestofthepaperisorganizedasfollows.Section2setsupthemodelecon-

omy.Section3constructsthesocialplannerbenchmarks,andcharacterizestheconstrained-optimallevelofcapacityinvestment(KSP).Section4characterizesthedecentralizedequilibriumandthemarketsolutionforcapacity(K*).Section5presentsourcoreresultthatfirms’investmentincapacityisinsufficient-K*<KSP-anddiscussespotentialpolicyinterventions.Section6concludeswithsuggestionsforfurtherresearch.Detailedderivationsandproofsareavailableintheappendix,alongwithanextensionoftheanalysistorarelargeshockspushingtheeconomy

whichmulti-sourcingstrategiesandsupplierselectioncanhelpmitigateriskinsupplychains.See

AnupindiandAkella

(1993),

Tomlin

(2006),

Babichetal.

(2012)and

Babichetal.

(2007)

.

9Seealso

ElliottandGolub

(2022)forasurveyonsupplychaindisruptionsandtheirmacroeco

-nomicimplications.

8

awayfromfullproduction.

2Model

Consideraneconomywithtwotypesofgoods:finalgoods(theconsumptionnu-meraire)andintermediategoodsusedintheproductionofthefinalgoods.Thereisacontinuumoffinalgoodsproducers(thatis,downstreamfirms,indexedi∈I=[0,1])andn≥2intermediategoodsproducers(thatis,upstreamfirms,indexedj∈J={0,1,...,n−1}),alllocatedaroundacirclewithunitcircumference.Thepositionsoftheintermediategoodsproducersaroundthecirclearerepresentedbynodes,whichdividethecontinuumoffinalgoodsproducersinton“marketseg-ments.”Figure

2.1

illustratesasimplifiedexampleofsuchaneconomywithn=3intermediategoodsproducers.Distanceisquantifiedalongthecircle’scircumfer- .

Figure2.1:IllustrativeEconomy

Consideranillustrativeeconomywiththreeintermediategoodsfirms(j∈{0,1,2}).Theintermediategoodsfirmsarelocatedequi-distantfromeachother,separatingthecircleintothreeequalmarketsegments{I0,I1,I2}.Inatypicalequilibrium,firmsj=0andj=1competeoverfinalgoodsfirmslocatedinthemarketsegmentI0.

Intermediategoodsproducersj∈Jareprice-setters.Theysetprices{pj}tocompeteoverfinalgoodsproducersintheirtwoneighboringmarketsegments

.10

10Itispossibleforanyparticularintermediategoodsproducertopricesoaggressivelyasto

9

Themassoffinalgoodsproducersineachmarketsegmentisdenotedas{mk}k=0,...,n−1.

Tofulfilltheendogenousdemandforintermediategoods,eachintermediategoodsproducerjoperatesaCobb-Douglasproductionfunctionwithpartialdelay:Yj,t=

L,K,

pricewj>0,andKjthenon-scalablecapacityinvestmentsthatmustbeinstalledoneperiodinadvanceatunitpricerj>0.Thekeydistinctionisthatnon-scalableinputsKjcannotbeadjustedintheshortrun

.11

Theparameterαj∈(0,1),theexpo-nentofL,measuresthescalabilityofeachsectorj.Crucially,intermediategoodsproducersjmustdecideonthelevelofnon-scalablecapacityinvestmentsKjbeforetherealizationofshockstotheeconomy.Aswewilldiscussingreaterdetailbelow,theintermediategoodsproducer’scapacityinvestment(Kj),andpricingdecisions(onboththespotandfuturesmarket)formthecoreofourmodel.

Wemodelthefinalgoodsproducersinamorereduced-formfashion.Specifi-cally,finalgoodsproducersi∈Iareprice-takers.Eachatomisticfinalgoodspro-ducerifacesanexogenousdemandQiforitsoutput,valuedatunitpricev

.12

Theseproducersconvertintermediategoodsintothefinalgoodsusingalinearproduction

functioni=Σjqij,whereidenotesthefinalgoodsoutputoffirmi,qijis

thequantityofintermediategoodsinputfirmisourcesfromfirmj,andf(d(i,j))isapenaltyfunctionthatdependsonthedistance(d(i,j)∈[0,firms.

capturedemandfrommarketsegmentsfurtherafield.Thispossibilitycorrespondstothe“super-competitive”regionofthedemandcurveinacirculareconomy(see

Salop

(1979))

.Forthepurposeofthepresentanalysis,ourclosed-formsolutionsfocusonasymmetricequilibriuminwhichallin-termediategoodsproducersfinditoptimaltosetthesameprice,thusrulingoutcompetitionoutsideoftheneighboringmarketsegments.

11Forbrevity,wewillhenceforthdropthetimesubscripts,andnotesimplythatKmustbepre-committedinadvanceofproduction.

12Inourmodel,finalgoodsfirmsformexpectationsoverthelevelofdemandQi,takingthepricevasafixedconstant,whereas,moregenerally,shockstofinalgoodsdemandwouldaffectboth(theirdesired)equilibriumquantityQiandpricevi.WesimplifytheanalysisbytakingtheintegraloverthedistributionofQionly,insteadofthejointdistributionoverbothQiandvi.Thissimplificationoffersgreateranalyticaltractabilityandhighlightsthecriticalmarketfailures,whilepreservingtheessentialeconomicsofresilience.Onecanthinkofthismodellingapproacheitheras:(1)astylizedportrayaloffinalgoodsdemand-ademandcurvewithdemandequaltoQforpriceequalorlessthanv,andzerodemandforpriceabovev,or(2)adescriptionofspecificmarkets(likethatforelectricity)inwhichallfirmshavesignedcontractstodeliveroutputatpricevregardlessofthelevelofdemandthatmaterializes.

10

Onewaytothinkaboutthisdistance-basedpenaltyfunctionisthatforeveryunitofintermediategoodsjpurchasedby.The

remainder,,“perishesintransit.”Asecondinterpretationoff

valuation-basedpenaltyfunction.Foranygivenvaluationv,theeffectivevaluation

ofthefinalgoodsithatusesinputsjisgivenby.Therefore,thefunction

f(·)canalsoaccountforheterogeneousvaluationsoffinalgoods.Specifically,afinalgoodsfirmiproducingoutputsusingmore“distant”intermediategoodswouldexperienceadiminishedvaluationforitsoutput.Athirdinterpretation(andtheonewefocusuponinthediscussionbelow)isthatthedifferentintermediategoodsareimperfectsubstitutesforeachother.Theproductionatanyplaceinthecircleisde-signedforacertaintypeofintermediategoodsbutcanuseotherintermediategoods,thoughtheyyieldlessoutputperunitofinput.(Thinkofanoilrefinerydesignedtorefineoilofaspecificgravityandsulfurcontent.Itcanrefineoilwithothercharacteristics,butlessefficiently).Foreaseofexposition,wewillrefertof(·)henceforthasthedistance-basedpenaltyfunction(distance,inthisinterpretation,referstodistanceintheproductspace)

.13

Weassumethatf(·)isanincreasingfunction,normalizedsuchthatf(0)=1.Thispenaltyfunction,f(·),combinedwiththestartingdistancebetweenfirms,d(i,j),capturestheextentofsubstitutabilityamongintermediategoods.Thegreaterthedistanced(i,j)betweentwofirms,andthesteepertheslopef′(x)ofthepenaltyfunction,themoreinefficientitbecomesforfinalgoodsproduceritosourcein-putsfromintermediategoodsproducerj.Forbrevity,letfij:=f(d(i,j))andfi:=(fi0,...,fi,n−1)′bethecorrespondingn×1columnvectorofpenaltiesforfi-nalgoodsproduceri.

Figure

2.2

summarizesthetimelineofthemodel.Atperiod0,thereisun-certaintyaroundthedemandandsupplyconditionsthatwillprevailinperiod1.Specifically,theuncertaintyaroundthedemandforfinalgoodsproducedbyfirmi

iscapturedbytherandomvariableQi.Qiisdistributedbetween,withcu-

mulativedensityfunction(c.d.f.)Gi(·)andassociatedprobabilitydensityfunction(p.d.f.)gi(·).Thereisalsouncertaintyaround{wj}j∈J,thepriceofthescalable

13Foradiscussionofthemeasurementofdistanceinproductspace,see,forexample,

Stiglitz

(1986)

.

11

Figure2.2:ModelTimeline

Timelineofevents,decisionsandactionsundertakenbyintermediateandfinalgoodsproducers.

inputfactor,whichaffectsthesupplyoftheintermediategoodsj.wjisdistributed

between[wj,j],withc.d.f.Hj(·)andp.d.f.hj(·)

.14

Supplyshocksareassumed

tobeindependentofdemandshocks.Inourformulation,thereisnouncertaintyaboutthepriceofthefinalgoods-itisthenumeraire.

Inperiod0,tohedgeagainstthesedemandandsupplyshocks,eachfinalgoodsproduceridecideswhethertoenterintoasuppliercontractwitheachintermedi-

ategoodsproducerj,placingpre-ordersqre:=′.Each

intermediategoodsproducerjsetspre-orderpriceφj.Concurrently,firmjmakeacost-minimizingdecisiononthelevelofnon-scalablecapacityKj,incurringas-sociatedcostsdenotedbyrjKj.Thepre-ordercontractsbetweenfinalgoodsandintermediategoodsproducersdefinetheendogenous