Are Supply Networks Efficiently Resilient?

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Are Supply Networks Efficiently Resilient? Agostino Capponi, Chuan Du, Joseph E. Stiglitz 2024-031 Please cite this paper as: Capponi, Agostino, Chuan Du, and Joseph E. Stiglitz (2024). “Are Supply Networks EfficientlyResilient?,” FinanceandEconomicsDiscussionSeries2024-031. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2024.031. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Are supply networks efficiently resilient? Agostino Capponi Chuan Du Joseph E. Stiglitz * May 16, 2024 Abstract We show that supply networks are inefficiently, and insufficiently, resilient. Upstreamfirmscanexpandtheirproductioncapacitytohedgeagainst supply and demand shocks. The social benefits of such investments are not internalized, however, because of market power and market incompleteness. Upstreamfirmsunderinvestincapacityandresilience,passingonthecoststo downstreamfirms,anddrivetradeexcessivelytowardthespotmarkets. There isawedgebetweenthemarketsolutionandaconstrainedoptimalbenchmark, which persists even without rare and large shocks. Policies designed to incentivize capacity investment, reduce reliance on spot markets, and enhance competitionamelioratetheexternality. (JELD21,D24,D25,D43,D85,E23, L13) *Capponi:DepartmentofIndustrialEngineeringandOperationsResearch,ColumbiaUniversity, NewYork;email: ac3827@columbia.edu. Du: BoardofGovernorsoftheFederalReserveSystem, Washington D.C.; email: chuan.du@frb.gov. Stiglitz: Columbia Business School, Columbia University,NewYork;Email:jes322@columbia.edu.Theviewsinthispaperaresolelytheauthors’and shouldnotbeinterpretedasreflectingtheviewsoftheBoardofGovernorsoftheFederalReserve SystemorofanyotherpersonassociatedwiththeFederalReserveSystem. Asignificantpartofthis workwasundertakenwhileChuanDuwasemployedattheBankofEngland. Theviewsexpressed inthispaperdonotnecessarilyreflecttheviewsoftheBankofEnglandoranyofitscommittees.We thankourdiscussantsandconferenceparticipantsattheAlliedSocialScienceAssociationAnnual Meeting2023andtheInternationalEconomicAssociationWorldCongress2023,aswellasseminar participantsattheBankofEnglandandtheFederalReserveBoard,fortheirconstructivecomments andsuggestions. 1

The shortages and spikes in prices of certain intermediate goods during the COVID-19 pandemic demonstrated the fragility of supply chains. Prominent examples included a global shortage of semiconductors that led to a dramatic rise in the price of secondhand cars in the U.S. and an unprecedented demand for hand sanitizer and personal protective gear that triggered supply shortages in their respective,aswellasinterlinked,industries. Policymakersreactedstronglybytaking industry-specific actions to repair linkages and improve resilience. For example, the Biden-Harris Administration worked in partnership with Congress to provide new legislation to alleviate specific supply chain disruptions and promote greater resilience in future situations. Moreover, while the large and small supply chain disruptions during COVID-19 had propelled the issue into popular discourse, the cracks had been evident before the pandemic. Hanjin Shipping, a world’s top 10 containercarrier,filedforbankruptcyinSeptember2016becauseofsluggishfreight ratescausedbyweakdemandandsoaringglobalcapacity. Thebankruptcyaffected global supply chains, because half of Hanjin’s container ships were denied access toports. MajorU.S.retailers,suchasJ.C.PenneyandWalmart,begantodivertand switch carriers for their containers to other suppliers. Similarly, the failure of Carillion in January 2018, once the second-largest construction company in the U.K., broughtdownmanyofitssuppliers.1 These experiences with supply network disruptions left open the question: Had firms invested too little in resilience ex ante? The pandemic was an extreme event, and, in general firms should not be expected to anticipate and plan for every possible contingency. Doing so would almost surely be inefficient, entailing excessive focus on resilience. We show here, however, that given market power and market incompleteness,oneshouldexpectmarketstounderinvestinresiliencerelativetoa constrainedefficientbenchmark. We formulate a tractable theoretical model whereby a collection of intermedi- 1Fordetailedcoverageoftheseepisodes,seetheFinancialTimes:"Carchipshortageshineslight onfragilityofUSsupplychain"; CNN:"Distilleriesaremakinghandsanitizerwiththeirin-house alcohol and giving it out for free to combat coronavirus"; Biden-Harris Supply Chain Disruptions TaskForce;DeutscheWelle:"BankruptHanjinsparksshippingcrisis";andtheGuardian:"Carillion collapse: Twoyearson, thegovernmenthaslearnednothing". . Amoreacademicaccountcanbe foundinBaqaeeandFarhi(2022),Guerrierietal.(2022)andDiGiovannietal.(2022). 2

ateandfinalgoodsproducersformsupplylinkagestomeetuncertainconsumerdemandandaccommodatesupplyshocks. Eachfinalgoodsproducer(thedownstream firm) can source differentiated inputs from one or more suppliers of the intermediate goods (the upstream firms). Intermediate goods producers engage in price that is, Bertrand - competition with differentiated products, taking the prices set by competitorsasgiven. Loweringthepricechargedallowsanintermediategoodsproducer to increase demand on the extensive margin (by attracting more final goods producers).2 Intermediate goods producers face uncertainty in demand and supply conditions. They invest in non-scalable production capacity before the realization of shocks, reflecting that some factors of production cannot be readily adjusted at short notice.3 Given the structural frictions in the economy - namely, the lags in production and the uncertainty around future market conditions - over-investment in capacity can be just as inefficient as under-investment. A supply network that is efficiently resilient strikes the optimal balance on resilience, taking into account thesestructuralfrictions. Usingthemodel,wedemonstratetheexistenceofamarketfailureindecentralized supply networks, whereby upstream firms do not fully internalize the social benefits of building production capacity. When upstream firms over-invest in capacity, part of the cost savings are passed on to downstream firms via lower prices; but when firms underinvest, they can defend their profit margins despite mounting costs by charging higher prices. The shortages that result from underinvestment enhance market power, which the upstream firms rationally anticipate. As a result, upstreamfirmswillalwaysleantowardunderinvestment. This pecuniary externality is not internalized by the decentralized market because of a combination of (1) market power and (2) market incompleteness. First, upstream intermediate goods producers exhibit market power because (a) only a finite number of such firms exist, and (b) the intermediate goods they produce are imperfect substitutes of each other. Second, firms do not have access to the full set of Arrow-Debreu securities, and instead must trade either on the pre-order market, 2Inamoregeneralcase,loweringthepricemayalsoaffecttheintensivemargin. 3Semiconductors are an example of an important intermediate goods that requires significant capacityinvestmentupfront.IntheEuropeanUnion,theEuropeanChipsAct(2023)aimstoprovide additionalpublicandprivateinvestmentsofmorethanEUR15billion. 3

or on the spot market once the shocks have realized. The pre-order market offers partial insurance to both the upstream and downstream firms. For the upstream firms, pre-orders establish a minimum level of demand for their outputs, and help with their upfront non-scalable capacity investment decision. For the downstream finalgoodsproducers,apre-ordercontractlocksinanagreedpricefortheintermediary inputs in their production, shielding them from cost shocks in the upstream sector. Ifrealizeddemandforfinalgoodsexceedswhatcanbefulfilledthroughpreorders,thedownstreamfirmcanthensourcetheextrainputsrequiredfromthespot market. As we observe in practice, the spot and pre-order markets are insufficient todealwiththefullspectrumofpossibleshocks,andthusareunabletoprovidefull insuranceagainstsupplynetworkdisruptions.4 Asaconsequenceofthisexternality,weshowthatthemarket-basednetworkinvests too little in production capacity (K∗) relative to a constrained optimal benchmark (KSP) with a social planner facing the same informational and technological constraints as the private market. Even under the constrained benchmark, it is not optimal to build enough capacity to account for all contingencies. So there will be times when firms ex post have considerable market power, which, obviously, the social planner would not take advantage of but private firms would. In short, market-basedsupplynetworksareinefficientlyresilient: K∗ <KSP. Remarkably,thiswedgebetweenthedecentralizedandcentralizedsolutionarises evenwhenrarelargeshocksareabsent,andtheeconomyoperatesina“fullproduction” equilibrium whereby supply is sufficiently agile to accommodate all possible demand. Our results do not depend on an arbitrary specification of the distribution of shocks - for example, we do not require a threshold for the probability of large negative shocks. Nor do we need to impose a level of risk aversion on the part of private agents or the social planner. Capacity investment is suboptimally low, even wheneveryagent–includingtheconstrainedsocialplanner–isriskneutral. Extending the analysis to account for rare disasters (in the appendix), we show that the response of market-based supply networks to shocks can be highly nonlin- 4Itisobviousthatsuchfullinsurancedoesnotexist. Giventherangeofshocksthatcouldoccur –someofwhicharenownotevenreallyconceivable–theincompletenessofinsurancemarketsis inevitable. Theoriesofasymmetricinformationprovidefurtherexplanationsoftheabsenceofafull setofinsurancemarkets. SeeGreenwaldandStiglitz(1986)andStiglitz(1982). 4

ear. Private supply networks are seemingly resilient during normal times and can comfortably withstand small to moderate shocks, but they are fragile to rare large shocks, when real rigidities prevent suppliers from fully meeting the needs of the market.5 With a large enough shock, there is a transition from a monopolistically competitive regime to a local monopoly regime, whereby upstream firms are no longerpricingtocompeteandeachdownstreamfirmwillreceiveonlyonecredible offer for inputs. In other words, in a crisis, individual suppliers prioritize the needs oftheirlocalmarketbutwithincreasedmargins.6 Supplynetworkfragilitycanlead toanincreaseinmarketpower(inourmodel,reflectedinsuboptimalretrenchment inmarketcoverage),especiallywhendemandisatitsgreatest. The size of the wedge between the decentralized and centralized solution depends endogenously on firms’ reliance on the spot market, and exogenously on the structural parameters of the economy. An economy exhibiting greater scalability (production functions that rely less on non-scalable capacity investments), higher substitutability(intermediategoodsinputsthataremoreinterchangeable)andmore competition(moreupstreamfirms)willbemoreefficientlyresilient. Therefore,therearebroadlythreeavenuesfornarrowingthewedge. First,adirect governmental subsidy targeting investment in production capacity could serve as the most pragmatic remedy. Second, enhancing incentives for the use of preordermarketscanofferupstreamfirmstheassuranceofrecoupinginitialcosts. We show that an overreliance on the spot market contributes to fragility in the supply network.7 Third, the government can promote structural changes in the economy to enhance scalability, substitutability and competition. Enhancing competition is 5By “seemingly resilient,” we mean that demand can be fully met at some price. It is still the casethatthereistoolittlecapacity. 6ThesurgeindemandforCOVIDvaccinesin2021andthefranticpursuitofnaturalgasduring the European energy crisis in 2022 serve as illustrative examples. Global supply constraints often leadtoredirectiontowardwealthiernations,leavingless-affluentdevelopingmarketseconomically disadvantaged during challenging times. During the post-COVID recovery, evidence suggests a marked increase in market power (markups) associated with the supply chain interruptions. See KonczalandLusiani(2022). 7For instance, in 2021 and 2022, more than 30 energy companies in the U.K. failed as a result of to a rapid increase in wholesale natural gas prices and inadequate hedging through futures/forward contracts by the energy companies. For details, see https://www.forbes.com/uk/advisor/energy/failed-uk-energy-suppliers-update. 5

goodinitsownright,anddoublysowhenmakingsupplynetworksmoreefficiently resilient. 1 Related literature The literature on the resilience of supply networks to shocks can be roughly categorized into two branches. The first focuses on analyzing the mechanisms through which idiosyncratic shocks propagate and amplify within a fixed network of firms with pre-specified relationships. Acemoglu and Tahbaz-Salehi (2020) examine the effect of productivity shocks on the distribution of economic surplus, firm failures, and the amplification of shocks through disruptions. Acemoglu et al. (2012) propose a model that explains how micro shocks can be magnified into macro fluctuations through input-output linkages. Carvalho et al. (2021) use data from the 2011 Japanese earthquake to demonstrate the significant macroeconomic implications of idiosyncratic shocks. Barrot and Sauvagnat (2016) reveal evidence of fragility caused by the propagation of firm-specific shocks, using data on natural disasters. We refer to Carvalho (2014) and Carvalho and Tahbaz-Salehi (2019) for athoroughreviewofsuchmechanisms. That markets would not be prepared for every shock they confront is not a surprise. The analytically interesting question is the normative one: Relative to an appropriatebenchmark,dotheyadequatelyprepareforshocks? Thefailureofeach firm in a competitive environment to take account of how capacity decisions affect thedistributionofpricesinthespotmarketsisoneofthetwocentralmarketfailures thatweidentify. The second branch of literature focuses on firms’ strategic responses to mitigate the negative effects of supply chain disruptions. Birge et al. (2023) explore howfirmsinasupplychainnetworkstrategicallyreactpost-disruptionbyoptimally switching demand and rerouting supply from defaulted firms. Amelkin and Vohra (2020) examine the competing retailers’ decisionmaking process when selecting suppliers, taking into account factors such as prices and suppliers’ reliability as measuredbyyielduncertaintyandcongestion.8 8Afewotherstudiesfromtheoperationsmanagementliteratureanalyzethemechanismsthrough 6

Our work is closely related to Elliot et al. (2022) and that by Grossman et al. (2023),which(also)examinesupplynetworkformationandfragility. Intheirmodels, downstream firms source customized inputs from upstream firms. To insure against possible supply disruptions, downstream firms strategically invest in relationships with multiple potential suppliers.9 One might infer from their analyses that systemic fragility should be reduced if inputs were more (albeit still imperfectly) substitutable, and there existed a common spot market for such inputs. We show that not only would such a spot market be insufficient to eliminate supply network fragility, but that market participants’ overreliance on spot market transactions would actually amplify the inherent externalities. In our model, fragility withinthesupplynetworkisnotaconsequenceofacatastrophicbreakdownofupstream suppliers or a failure in supplier diversification but due to a more structural combinationofmarketpowerandincompletemarkets. On the empirical side, Atalay et al. (2011) estimate a model of firms’ buyersupplierrelationshipsusingmicrodataonfirms’customers. Crosignanietal.(2019) investigate the consequences of supply shocks resulting from NotPetya, one of the mostseverecyberattacksinhistory. Theyobservethattheaffecteddownstreamcustomers were more inclined to establish new relationships with alternative suppliers while terminating existing relationships with the directly affected firms. Lastly, Baldwin and Freeman (2022) examines the cross-border dimensions of resilience inglobalsupplychains. Therestofthepaperisorganizedasfollows. Section2setsupthemodeleconomy. Section 3 constructs the social planner benchmarks, and characterizes the constrained-optimal level of capacity investment (KSP). Section 4 characterizes the decentralized equilibrium and the market solution for capacity (K∗). Section 5 presentsourcoreresultthatfirms’investmentincapacityisinsufficient-K∗<KSP -anddiscussespotentialpolicyinterventions. Section6concludeswithsuggestions for further research. Detailed derivations and proofs are available in the appendix, along with an extension of the analysis to rare large shocks pushing the economy whichmulti-sourcingstrategiesandsupplierselectioncanhelpmitigateriskinsupplychains. See AnupindiandAkella(1993),Tomlin(2006),Babichetal.(2012)andBabichetal.(2007). 9SeealsoElliottandGolub(2022)forasurveyonsupplychaindisruptionsandtheirmacroeconomicimplications. 7

awayfromfullproduction. 2 Model Consider an economy with two types of goods: final goods (the consumption numeraire) and intermediate goods used in the production of the final goods. There is a continuum of final goods producers (that is, downstream firms, indexed i ∈ I =[0,1])andn≥2intermediategoodsproducers(thatis,upstreamfirms,indexed j ∈J ={0,1,...,n−1}), all located around a circle with unit circumference. The positions of the intermediate goods producers around the circle are represented by nodes, which divide the continuum of final goods producers into n “market segments.” Figure 2.1 illustrates a simplified example of such an economy with n=3 intermediate goods producers. Distance is quantified along the circle’s circumference,ensuringthatthemaximumdistanceseparatinganytwopointsis 1. 2 Figure2.1: IllustrativeEconomy Consideranillustrativeeconomywiththreeintermediategoodsfirms(j∈{0,1,2}).Theintermediategoodsfirmsarelocated equi-distantfromeachother,separatingthecircleintothreeequalmarketsegments{I 0,I 1,I 2 }.Inatypicalequilibrium,firms j=0and j=1competeoverfinalgoodsfirmslocatedinthemarketsegmentI 0. (cid:8) (cid:9) Intermediate goods producers j ∈ J are price-setters. They set prices p to j compete over final goods producers in their two neighboring market segments.10 10It is possible for any particular intermediate goods producer to price so aggressively as to 8

Themassoffinalgoodsproducersineachmarketsegmentisdenotedas{m } . k k=0,...,n−1 To fulfill the endogenous demand for intermediate goods, each intermediate goods producer j operates a Cobb-Douglas production function with partial delay: Y = j,t L αjK 1−αj, where L denotes the scalable input factors in production with factor j,t j,t−1 j price w > 0, and K the non-scalable capacity investments that must be installed j j one period in advance at unit price r >0. The key distinction is that non-scalable j inputsK cannotbeadjustedintheshortrun.11 Theparameterα ∈(0,1),theexpoj j nent of L, measures the scalability of each sector j. Crucially, intermediate goods producers jmustdecideonthelevelofnon-scalablecapacityinvestmentsK before j therealizationofshockstotheeconomy. Aswewilldiscussingreaterdetailbelow, the intermediate goods producer’s capacity investment (K ), and pricing decisions j (onboththespotandfuturesmarket)formthecoreofourmodel. We model the final goods producers in a more reduced-form fashion. Specifically, final goods producers i∈I are price-takers. Each atomistic final goods producerifacesanexogenousdemandQ foritsoutput,valuedatunitpricev.12 These i producersconvertintermediategoodsintothefinalgoodsusingalinearproduction functionY˜ i =∑ j f(d( 1 i,j)) q ij , whereY˜ i denotes the final goods output of firm i, q ij is the quantity of intermediate goods input firm i sources from firm j, and f (d(i, j)) isapenaltyfunctionthatdependsonthedistance(d(i, j)∈ (cid:2) 0,1(cid:3) )betweenthetwo 2 firms. capture demand from market segments further afield. This possibility corresponds to the “supercompetitive”regionofthedemandcurveinacirculareconomy(seeSalop(1979)). Forthepurpose ofthepresentanalysis,ourclosed-formsolutionsfocusonasymmetricequilibriuminwhichallintermediategoodsproducersfinditoptimaltosetthesameprice,thusrulingoutcompetitionoutside oftheneighboringmarketsegments. 11For brevity, we will henceforth drop the time subscripts, and note simply that K must be precommittedinadvanceofproduction. 12Inourmodel,finalgoodsfirmsformexpectationsoverthelevelofdemandQ,takingtheprice i vasafixedconstant,whereas,moregenerally,shockstofinalgoodsdemandwouldaffectboth(their desired)equilibriumquantityQ andpricev. Wesimplifytheanalysisbytakingtheintegralover i i thedistributionofQ only,insteadofthejointdistributionoverbothQ andv. Thissimplification i i i offersgreateranalyticaltractabilityandhighlightsthecriticalmarketfailures,whilepreservingthe essentialeconomicsofresilience. Onecanthinkofthismodellingapproacheitheras: (1)astylized portrayal of final goods demand - a demand curve with demand equal to Q for price equal or less than v, and zero demand for price above v, or (2) a description of specific markets (like that for electricity)inwhichallfirmshavesignedcontractstodeliveroutputatpricevregardlessofthelevel ofdemandthatmaterializes. 9

One way to think about this distance-based penalty function is that for every unit of intermediate goods j purchased by i, only a fraction 1 is usable. The f(d(i,j)) (cid:16) (cid:17) remainder, 1− 1 ,“perishesintransit.” Asecondinterpretationof f (·)isa f(d(i,j)) valuation-basedpenaltyfunction. Foranygivenvaluationv,theeffectivevaluation of the final goods i that uses inputs j is given by v . Therefore, the function f(d(i,j)) f (·) can also account for heterogeneous valuations of final goods. Specifically, a finalgoodsfirmiproducingoutputsusingmore“distant”intermediategoodswould experienceadiminishedvaluationforitsoutput. Athirdinterpretation(andtheone we focus upon in the discussion below) is that the different intermediate goods are imperfectsubstitutesforeachother. Theproductionatanyplaceinthecircleisdesignedforacertaintypeofintermediategoodsbutcanuseotherintermediategoods, though they yield less output per unit of input. (Think of an oil refinery designed to refine oil of a specific gravity and sulfur content. It can refine oil with other characteristics, but less efficiently). For ease of exposition, we will refer to f (·) henceforth as the distance-based penalty function (distance, in this interpretation, referstodistanceintheproductspace).13 We assume that f (·) is an increasing function, normalized such that f (0)=1. This penalty function, f (·), combined with the starting distance between firms, d(i, j),capturestheextentofsubstitutabilityamongintermediategoods. Thegreater thedistanced(i, j)betweentwofirms,andthesteepertheslope f′(x)ofthepenalty function, the more inefficient it becomes for final goods producer i to source inputs from intermediate goods producer j. For brevity, let f := f (d(i, j)) and ij ′ f :=(f ,..., f ) be the corresponding n×1 column vector of penalties for fii i0 i,n−1 nalgoodsproduceri. Figure 2.2 summarizes the timeline of the model. At period 0, there is uncertainty around the demand and supply conditions that will prevail in period 1. Specifically, the uncertainty around the demand for final goods produced by firm i (cid:104) (cid:105) is captured by the random variable Q. Q is distributed between Q,Q¯ , with cui i i i mulative density function (c.d.f.) G (·) and associated probability density function i (cid:8) (cid:9) (p.d.f.) g (·). There is also uncertainty around w , the price of the scalable i j j∈J 13For a discussion of the measurement of distance in product space, see, for example, Stiglitz (1986). 10

Figure2.2: ModelTimeline Timelineofevents,decisionsandactionsundertakenbyintermediateandfinalgoodsproducers. input factor, which affects the supply of the intermediate goods j. w is distributed j (cid:2) (cid:3) between w ,w¯ , with c.d.f. H (·) and p.d.f. h (·).14 Supply shocks are assumed j j j j to be independent of demand shocks. In our formulation, there is no uncertainty aboutthepriceofthefinalgoods-itisthenumeraire. Inperiod0,tohedgeagainstthesedemandandsupplyshocks,eachfinalgoods producer i decides whether to enter into a supplier contract with each intermedi- (cid:104) (cid:105)′ pre pre pre pre ate goods producer j, placing pre-orders q := q ,...,q ,...,q . Each i i0 ij in−1 intermediate goods producer j sets pre-order price φ . Concurrently, firm j make j a cost-minimizing decision on the level of non-scalable capacity K , incurring asj sociated costs denoted by r K . The pre-order contracts between final goods and j j intermediategoodsproducersdefinetheendogenousnetworkformedinperiod0. In period 1, firms observe the realization of the demand and supply shocks. Fi- (cid:104) (cid:105)′ spot spot spot spot nalgoodsproducerisubmitsspot-marketordersq := q ,...,q ,...,q . i i0 ij in−1 (cid:2) pre spot(cid:3) Thetotalcostofpre-ordersandspot-marketordersforfirmiisgivenby φ·q +p·q , i i (cid:2) (cid:3)′ where φ := φ ,...,φ ,...φ denote the vector of pre-order prices, and p the 0 j n−1 vectorofspot-marketprices. Atperiod1,intermediategoodsproducer j takesprecommitted capacity K as given, solves for the cost-minimizing scalable input L , j j andsetsprices p tomaximizeprofits. Productionoccurs,andcontractsaresettled. j Theexcessofproductionoverthecontractedpre-ordersissoldonthespotmarket. 14Withoutlossofgenerality,let∞>Q¯ >Q >0,∀i∈[0,1];and∞>w¯ >w >0,∀j∈J. i i j j 11

In our model, the final goods producers can buy from any intermediate goods producer at the posted price. This flexibility stands in contrast to much of the network literature discussed in Section 1 (for example, Elliot et al. (2022)), where final goods producers can only buy from the firms with whom they have previous relations,soshockstothosefirmsobviouslygetpassedonstronglythroughthenetwork. Here,ineffect,theexanteandexpostnetworkscanbedifferent. Weassume thattherearenocoststoestablishinganewlinkexpost.15 For analytical tractability, we impose symmetry on the primitives of the model andderiveclosed-formsolutionsfortheresultingsymmetricequilibrium. AssumptionA1 [Symmetry]: α = α and r = r, ∀j ∈ J; Q = Q, ∀i ∈ I; w = j j i j w, ∀j∈J ;m = 1, ∀k∈{0,...,n−1} k n By assumption, all intermediate goods producers share a common Cobb-Douglas production function: α = α,∀j ∈ J; and face the same non-scalable input costs j in period 0: r = r, ∀j ∈ J. We also assume that the shocks to the economy are j symmetric and identical. The realization of final goods demand is the same for all finalgoodsfirms: Q =Q, ∀i∈I;andtherealizationofscalableinputcostisalsothe i sameforallintermediategoodsfirms: w =w, ∀j∈J.16 Thissymmetrycapturesan j economy that is subject to systemic, correlated shocks. For instance, a symmetric demand shock might resemble the surge in demand for vaccines amid a pandemic, whereas a symmetric supply shock could be akin to a military conflict causing a spike in energy prices that affects all manufacturing sectors. Lastly, m = 1, ∀k k n implies that the sizes of each market segment are equal. The intermediate goods producersareuniformlydistributedaroundtheunitcircleatequidistantintervals. It is important to note that fully symmetric shocks to final goods demand (Q) andintermediategoodssupply(w)donotimmediatelyimplyfullysymmetricequilibrium outcomes. For instance, final goods producers that are further away from intermediategoodssuppliernodes(thatis,thosewithlesssubstitutableinputs)will 15Ourresultmaybegeneralizedbyassumingeitherthatthereisafixedcosttogoingtothemarket ortobuyingfromanyspecificfirmwithwhomonedoesnothaveapreviousrelation. Theproblem wouldbecomeanalyticallymorechallenging,butthemaininsightswouldstayqualitativelythesame 16Thisisaslightabuseofnotation. WeuseQ andw torepresentboththerandomvariableand i j itsrealizedvalue. Theintendedmeaningshouldbeclearwithinthegivencontext. 12

needtoordermoreofagiveninput-comparedwithanotherfinalgoodsfirmthatis closer - to meet the same level of final goods demand. In practice, perfectly correlated shocks are the most challenging for resilience, which makes them a key “test case”toexamine. Before we dive into the formal equations that define the decentralized equilibrium, it is useful to first explore the more straightforward problems of an unconstrained and constrained social planner. The planner solutions will serve as our benchmarksforcomparison. 3 The social planner benchmarks Wecharacterizethesymmetricequilibriumoutcomesfortwoseparatebenchmarks. Inthefirst,thesocialplannercanperfectlyobservetherealizationofthestatevariables (Q,w) before committing to intermediate goods production across the network. The planner can therefore perfectly adjust both input factors (L,K) in line with market conditions. We call this unconstrained planner’s solution the first-best perfect foresight benchmark. We re-introduce the informational and technological constraintsfacedbyprivateagentsinthesecond-constrainedoptimal-socialplanner’s benchmark. Of the two, the constrained optimal benchmark provides a more appropriatebasisforcomparison. However,theperfectforesightbenchmarkserves avaluableroleinisolatingtheeffectsofreal-worldfrictions-suchasuncertainties aroundstatesandlimitationsinproductiontechnology-fromthoseassociatedwith thedistortionsthatariseasaresultofmarketexternalitiesandotherimperfections. There are two key distinctions between the social planner (under both benchmarks) and the decentralized market. First, a social planner can directly allocate (cid:8) (cid:9) order flows q without the need to use price signals (p,φ) as a coordinating ij mechanism. Second, a social planner maximizes the welfare of the economy as a whole, whereas individual private agents maximize their own profit or utilities. Thus,thesocialplannerinternalizesanyexternalitiesthatmayarise. Werestrictattentiontoafullproductionequilibrium,wherethetotaldemandfor finalgoodscanbemetinasociallyprofitableway-thatiswhere,atthemargin,the valueofthefinalgoodsexceedsthemarginalcostofproduction. Thissettingfurther 13

underscores that our core findings are not contingent on the occurrence of rare, large-scaleshocks. Formally,asymmetriceconomyE ={f (·),α,w,r,Q;v}admits a full production equilibrium if there exists an equilibrium wherebyY˜ (Q,w)=Q, i ∀i∈[0,1],and,forallstatesoftheworld(Q,w)⊂R2. + AssumptionA2 [Full production]: We provide conditions on the model primitives that ensure that a symmetric economy can achieve a full production equilibrium. More specifically, we assume that at every point around the circle(thatis,∀i∈I=[0,1]),themarginalbenefitsofproducingfinalgoodswill atleastmatchorexceedthemarginalcostsinallpossiblescenarios:  1−α v (cid:18) w¯ (cid:19)α(cid:18) r (cid:19)1−α w¯Q¯ α 1 f (cid:0) 1 (cid:1) ≥ α 1−α  (cid:104) 1 (cid:105) forn≥2 (3.1) 2n E wQα Assumption A2 states that, the marginal benefit of delivering intermediate goods to the final goods producer located farthest from the nearest node (at a distance of 1 )isweaklygreaterthanthemarginalcostofproducingtheintermediategoods 2n (cid:32) (cid:33)1−α (cid:0)w¯(cid:1)α(cid:0) r (cid:1)1−α w¯Q¯α 1 ,evenwhenthenegativesupplyshockisatitsmost α 1−α (cid:104) 1(cid:105) E wQα extreme (w=w¯), and demand is at its upper bound (Q=Q¯). For any given v, this assumption is equivalent to a restriction on the range of the demand and supply shocks. The assumption guarantees full production under the constrained optimal benchmark, where the social planner faces the same informational and technological constraints as the decentralized market.17 The corresponding condition for the perfect foresight benchmark is v ≥ (cid:0)w¯(cid:1)α(cid:0) r (cid:1)1−α for n ≥ 2, where the f(1) α 1−α 2n marginalcostofproductionislowerbecausethesocialplannercanfullyadjustboth (cid:104) (cid:105) inputsofproduction(KaswellasL)inresponsetoshocks(thatis,w¯Q¯ α 1 >E wQα 1 byconstruction). AssumptionA2isthereforeasufficientconditionforfullproductionunderbothsocialplannerbenchmarks. Onatechnicalnote,thefullproduction assumptionalsoenablesustoavoidproblemsofnon-differentiabilityinthedemand function.18 17SeeAppendixC.2fordetails. 18SeeAppendixDforamoredetaileddiscussion. 14

Relaxing Assumption A2 leads to cases where some segment of the economy (farthestawayfromtheintermediategoodsproducers)mightbecomeshutoutfrom the final goods market under adverse supply conditions. In such instances, intermediategoodssuppliersoperateaslocalizedmonopoliesratherthanasdirectcompetitors, each prioritizing the needs of their local markets (at higher margins) and leavingdemandfrommore“distant”firmsunfulfilled. Theemergenceoflocalmonopoliesintroducesanextralayerofdistortiontothedecentralizedmarketsolution, which further strengthens our core argument that there is insufficient investment in non-scalableproductioncapacity.19 3.1 The perfect foresight (PF) benchmark Considerthefirst-bestproblemforasocialplannerwithafullyscalableproduction functionandperfectforesight. ThesocialplanneroperatesastandardCobbDouglas production function for intermediate goods: Y = Lα K1−α.20 The planner can j,t j,t j,t (cid:8) (cid:9) (cid:8) (cid:9) also dictate input choices K ,L and order flows q for all firms j j j∈J ij i∈I,j∈J after observing the realization of final goods demand Q and scalable input cost w. Although production is delayed until period 1, there is no uncertainty. At period 0, firmsknowtherealizationoftheshocksthatarriveatperiod1. Mathematically,this is equivalent to all decisions being made in a single period optimization problem, wheretheobjectiveistomaximizethevalueofproductionnetofitscosts. 19WediscusstheconsequencesofrelaxingthisassumptioningreaterdetailinAppendixG. 20Wesuppressthet subscripthenceforthtosimplifythenotation. 15

[OptimizationProblemPF]: (cid:40) (cid:41) (cid:90) 1 W(Q,w)= max v (cid:2) min (cid:8) Q,Y˜(cid:9)(cid:3) di−∑ (cid:2) rK +wL (cid:3) i j j {K j } j∈J ,{L j } j∈J ,{q ij } i∈I,j∈J 0 j∈J (3.2) 1 s.t. Y˜ =∑ q [Productionfunctionforfinalgoodi] (3.3) i ij f j∈J ij Y =LαK1−α ∀j∈J [Productionfunctionforintermediategood j] j j j (3.4) (cid:90) 1 q di≤Y ∀j∈J [Feasibilityofintermediategoodsorderflow] (3.5) ij j 0 q ≥0 ∀i∈[0,1],∀j∈J [Nonnegativeinputs] (3.6) ij The solution is simple and intuitive. In the perfect foresight benchmark, the plannerwouldmeetfinalgoodsdemandbysourcingintermediategoodsinputsfrom the cheapest supplier and produce the required intermediate goods at minimal cost byoptimizingtheratiobetweenscalableandnon-scalableinputsineverystate. Proposition1. [Fullproductionsymmetricequilibriumunderperfectforesight] 1. The social planner allocates sufficient intermediate goods j to each final goods firm i to meet consumer demand Q, accounting for any imperfect substitutability f . The required intermediate goods inputs will be sourced from ij thelowesteffective-costsupplier(s)foreachi,wheneverthevalueofproductionvexceedsthemarginalcostofproduction:  f Q if j∈J(i) andv≥ f (cid:0)w(cid:1)α(cid:0) r (cid:1)1−α qPF(Q,w)= ij ij α 1−α (3.7) ij 0 otherwise (cid:110) (cid:111) where J(i):= j˜∈J|f (cid:0)w(cid:1)α(cid:0) r (cid:1)1−α ≤ f (cid:0)w(cid:1)α(cid:0) r (cid:1)1−α ∀j∈J is ij˜ α 1−α ij α 1−α thesetoflowesteffective-costsupplier(s). 2. The planner’s input choices in intermediate goods production satisfy the op- 16

timalitycondition: αrKPF(Q,w)=(1−α)wLPF(Q,w) (3.8) whichyieldstheexplicitsolution (cid:32) (cid:33) (cid:18) w(1−α) (cid:19)α (cid:90) 1 KPF(Q,w)= 2Q 2n f (i)di (3.9) r α 0 (cid:32) (cid:33) (cid:18) r α (cid:19)1−α (cid:90) 1 LPF(Q,w)= 2Q 2n f (i)di (3.10) w1−α 0 where f (i) = f := f (d(i,0)) is the shorthand for the distance penalty bei0 tween final goods firm i and intermediate goods firm 0, and, by symmetry, KPF =KPF andLPF =LPF forall j∈J. j j 3.2 The constrained optimal social planner (SP) benchmark Next, we consider the constrained optimal problem, whereby a social planner can (cid:8) (cid:9) (cid:8) (cid:9) dictateproductionchoices L ,K andorderflow q butissubjecttothe j j ij i∈I,j∈J sameinformationalandtechnologicallimitationsastheprivatesector. Wesolvethe constrainedoptimalproblemthroughbackwardinduction. In period 1, the social planner takes the pre-committed non-scalable capac- (cid:8) (cid:9) ity K = K, ∀j ∈ J as given and chooses the scalable input factor L and j j j∈J (cid:8) (cid:9) order flows q to maximize aggregate welfare for any given realization ij i∈I,j∈J of demand and supply conditions (Q,w). For given intermediate goods output Y = (cid:82)1q di, we can express the cost-minimizing level of the scalable factor as j 0 ij (cid:16) (cid:17) 1 − (1−αj ) L = (cid:82)1q di αj K αj . SubstitutingoutL andimposingsymmetry(Assumpj 0 ij j j tionA1),wecanexpresstheoptimizationproblem[SP1]intermsoftheorderflows (cid:8) (cid:9) q only: ij i∈I,j∈J [OptimizationproblemSP1]: 17

WSP(K|Q,w)= max (cid:40) v (cid:90) 1 (cid:32) ∑ 1 q ij (cid:33) di−∑ (cid:34) rK+w (cid:32) (cid:18)(cid:90) 1 q ij di (cid:19) α 1 K−(1− α α) (cid:33)(cid:35)(cid:41) {q ij } i∈I,j∈J 0 j∈J f ij j∈J 0 (3.11) 1 s.t. Q≥ ∑ q ∀i∈[0,1] [Demandcap] (3.12) ij f j∈J ij q ≥0 ∀i∈[0,1], j∈J [Nonnegativeinputs] (3.13) ij (cid:16) (cid:17) wherev (cid:82) 0 1 ∑ j∈J f 1 q ij diistheaggregatevaluederivedfromtheproductionoffiij (cid:20) (cid:18) (cid:16) (cid:17)1 (cid:19)(cid:21) nalgoodsand∑ j∈J rK+w (cid:82) 0 1q ij di α K−(1− α α) theaggregatecostofproducingthenecessaryintermediateinputs. Thedemandcapreflectsthatanyproduction inexcessoftherealizeddemandQwillbewasted.21 (cid:8) (cid:9) Back in period 0, the social planner chooses non-scalable inputs K to j j∈J maximize expected welfare in period 1, accounting for the probability distribution ofdemandandsupplyshocks(Q,w). [OptimizationproblemSP0]: (cid:104) (cid:105) WSP =maxE WSP(K|Q,w) K Thesolutionresemblesthatoftheperfectforesightscenario,butwithimportant distinctions,arisingfromthenecessityofcommittingtoaspecificlevelofcapacity investmentinperiod0,beforetherealizationofstatesinperiod1. Proposition 2. [Full production symmetric equilibrium in the constrained optimalbenchmark] 1. Inperiod1,thesocialplannerallocatessufficientintermediategoodstoeach finalgoodsfirmitomeetconsumerdemandQ,accountingforimperfectsubstitutability. Therequiredintermediategoodsinputswillbesourcedfromthe lowest effective-cost supplier(s) for each i, whenever the value of production 21In this analysis, we deliberately exclude the effect of inventory management because of the frame-work’sstatic,one-shotnature. SeeFerrari(2022)foranetworkmodelwithinventories. 18

vexceedsthemarginalcostofproduction:  qSP(Q,w)= f ij Q if j∈J(i) andv≥ f ij M(cid:103)C (3.14) ij 0 otherwise (cid:32) (cid:33)1−α where M(cid:103)C := (cid:0) α w(cid:1)α(cid:0) 1− r α (cid:1)1−α (cid:104) wQα 1 1(cid:105) is the marginal cost of pro- E wQα ducing the intermediate goods in the symmetric equilibrium and J(i) := (cid:110) (cid:111) j˜∈J|f ij˜ M(cid:103)C≤ f ij M(cid:103)C ∀j∈J isthesetoflowesteffective-costsupplier(s). Theoptimallevelofscalableinputisgivenby (cid:18)(cid:90) 1 (cid:19) α 1 (cid:16) (cid:17)−(1−α) LSP(Q,w)= qSP(Q,w)di KSP α (3.15) ij 0 2. In period 0, the optimal level of non-scalable production capacity KSP satisfiestheoptimalitycondition: (cid:104) (cid:105) αrKSP =(1−α)E wLSP (3.16) whichcanbesolvedexplicitlytogive (cid:32) (cid:33) (cid:18) 11−α (cid:19)α (cid:90) 1 (cid:16) (cid:104) (cid:105)(cid:17)α KSP = 2 2n f (i)di E wQα 1 (3.17) r α 0 3. Therelationshipbetweencapacityinvestmentacrossthetwobenchmarkscenarioscanbesummarizedasfollows:  (cid:104) 1 (cid:105)α E wQα KSP =KPF(Q,w)  (3.18) 1 wQα andbyJensen’sinequalitywehave KSP ≥E (cid:2) KPF(Q,w) (cid:3) (3.19) 19

The first part of the proposition relating to the optimal order flow (qSP) and ij the level of scalable capacity (LSP) is straight-forward. Here, we will focus discussions on the intuition behind the constrained optimal solution for capacity investment KSP. The choice of non-scalable capacity at period 0, KSP, influences aggregatewelfareinperiod1throughtwoprimarymechanisms. First,anyincrease in KSP generates a direct cost given by r. This cost, however, is partly offset by the resultant decrease in the scalable input LSP needed to achieve a given outputY, thus offering a direct benefit. Second, a rise in capacity KSP may increase aggregate intermediate goods productionY, and indirectly improve welfare through this output channel dY. However, in a full-production equilibrium where the demand dK for final goods is always met (that is, the demand cap is binding), there can be no further welfare gains from increasing aggregate intermediate goods production. Therefore,theindirecteffectofK onwelfareisexactlyzero.22 Weareleftwiththe familiaroptimalityconditionthatistypicalforCobb-Douglasproductionfunctions, αrKSP = (1−α)E (cid:2) wLSP(cid:3) , albeit with an expectation function to account for the exanteuncertainty. Finally, equation 3.18 illustrates the relationship between the level of capacity investment across the two benchmarks. Under the constrained optimal benchmark, the social planner must commit to a given level of capacity KSP before observing the shocks. Hence, capacity investment is lower than that in the perfect foresight case, KSP < KPF(Q,w), in states where marginal costs exceed expectations (whenw,Q,orbotharehigherthanexpected). Conversely,KSP>KPF(Q,w)when marginalcostsfallbelowexpectations. Importantly,thisresultimpliesthattheconstrained social planner recognizes that investing in a level of production capacity that accommodates all contingencies (KPF(cid:0) Q¯,w¯ (cid:1) ) would give rise to a supply network that is inefficiently resilient. Nevertheless, the constrained social planner in- 22Intheformalproof(seeAppendixC.2),weshowthattheoptimalityconditionfornon-scalable productioncapacityKSP (equation3.16)remainsunchangedwhenwerelaxthefullproductionassumption.WecansafelyignoretheindirecteffectsofKonwelfarethroughchangesinoutputY,becausetheseindirecteffectsaremultipliedbythedifferencebetweenthemarginalcostandmarginal benefit of production for the threshold buyer, which is equal to zero by construction. This result bearsresemblancetotheEnvelopeTheorem,inwhichthetotalderivativeofthevaluefunctionwith respecttotheparametersofthemodelisequaltoitspartialderivative.HereKisthechoicevariable, butthetotalderivativeofWSP(K|Q,w)withrespecttoK isalsoequaltoitspartialderivative. 20

vests in more capacity than its counterpart with perfect foresight does on average, KSP ≥E (cid:2) KPF(Q,w) (cid:3) , as a way to insure against uncertainty. A formal exposition oftheseresultscanbefoundinAppendixC.3. 4 The decentralized solution: Equilibrium in the spot and pre-order markets In the decentralized market equilibrium, firms adjust production in response to prices in both the pre-order and spot markets. We solve the model through backwardinduction. 4.1 Period 1 equilibrium in the spot market In period 1, each final goods producer can turn to the spot market to acquire additional intermediate goods beyond those that have been pre-ordered. Formally, each finalgoodsproduceritakesrealizeddemandforfinalgoodsQ,priorcommitments i q pre , pre-order and spot intermediate goods prices (φ,p) as given23, and purchases i spot intermediate goods q from intermediate goods producers on the spot market in i ordertomaximizeprofit: Π (cid:0) q pre ,φ,p (cid:1) =max (cid:8) vmin (cid:8) Q,Y˜(cid:9) −C˜ (cid:0) q pre ,q,φ,p (cid:1)(cid:9) (4.1) i i i i i i i spot q i spot s.t. q ≥0 [No-defaultconstraint] (4.2) i wherefinalgoodsproductionandtotalcostsaregivenby 1 (cid:16) (cid:17) Y˜ = ∑ q spot +q pre (4.3) i f ij ij j∈J ij C˜ (cid:0) q pre ,q,φ,p (cid:1) =φ·q pre +p·q spot (4.4) i i i i i 23AsisconventionalintheliteratureonBertrandequilibria,eachfirmassumesitcanbuyasmuch onthespotmarketasitwishes. Thisassumptionisparticularlyimportantfortheanalysisoffirms’ decisionmakingattime0. 21

spot We interpret q ≥ 0 as a “no-default constraint” because it implies that the i total volume of intermediate goods orders will never fall below the pre-ordered spot pre pre amount: q :=q +q ≥q . The final goods producers cannot renege on the i i i i promises made in period 0. In principle, a firm could also resell its pre-order to some other firm, so that the level of input could be less than the pre-ordered level. Inasymmetricequilibrium,however,thatneveroccurs.24 Ifthespotmarketwereperfectlycompetitive,eachintermediatesupplierwould produceuptothepointwherethepriceoftheintermediategoods(onthespotmarket)wereequaltothemarginalcostofproduction,andthedemandforintermediate goods would be determined in the usual way, with equilibrium in the spot market occurring at the price where demand equals supply. Instead, this is a highly differentiated market for intermediate goods, and each intermediate goods producer acts in a monopolistically competitive way, setting a spot price p and taking its j non-scalable production capacity K , and the price of its competitors p as given. j −j (cid:110) (cid:111) pre Pre-ordercontracts q arehonoredattheagreedpriceφ . Theprofitoffirm ij j i∈I j is given by its pre-order revenue plus spot-market revenue, minus the total costs ofproduction: (cid:16) (cid:16) (cid:17) (cid:17) (cid:110)(cid:104) (cid:105) (cid:104) (cid:105) (cid:111) Π K , φ ,q pre ,p =max φ Y pre + p Y spot − (cid:2) w L∗+r K (cid:3) (4.5) j j j j −j j j j j j j j j p j where (cid:90) 1 pre pre Y := q di (4.6) j ij 0 (cid:90) 1 spot spot Y := q di (4.7) j ij 0 are the level of intermediate goods production required to meet pre-order demands andspotmarket-demands,respectively. 24Conceptually,wecouldimagineanequilibriumwhere,say,insomestates,thoseinonesetof locations sold excess pre-orders to those in another set of locations. Our assumption of perfectly correlated shocks is what rules out this scenario. Alternatively, even with imperfectly correlated shocks,resellingexcessorderscanbeassumedaway,forexample,becausetherearesome(notfully specified here) adaptations of production to each producer, which make such sales impossible. In practiceresaleofpre-orderedinputsdooccur,thoughtheyarelikelylimitedinscale. 22

Similar to our treatment of the social planner benchmarks, we restrict attention toafullproductionsymmetricequilibriumforanalyticaltractability. Inasymmetric setting, all intermediate goods firms j ∈ J share the same characteristics α = α, j r =randw =w;andeveryfinalgoodsfirmi∈[0,1]willfacethesameexogenous j j demandQ =Q. Inequilibrium,inputchoiceswillbethesameacrossintermediate i goods firms: K =K and L =L, ∀j∈J; and final goods firms will fulfill the same j j proportion of the realized demand for final goods through the pre-order market: Q pre :=∑ 1 q pre =Qpre ∀i∈[0,1]. i j f ij ij pre ItisimportanttonotethatQ denotestheleveloffinalgoodsdemandfulfilled i pre throughpre-ordersandnotthequantityofintermediategoodspre-orderedq . The i linkbetweenthetwoisgivenbyQ pre :=∑ 1 q pre ,where 1 accountsforimperfect i j f ij f ij ij substitutability. Later,inSection4.2,weshowthatQ pre =Qpre ∀i∈[0,1]isindeed i an optimal equilibrium strategy in period 0, but this strategy implies pre-orders for pre intermediategoodsq arenotequalizedacrossi’s. ij Proposition 3. [Full Production Symmetric Equilibrium in the spot market] In period1,takingperiod0choices (cid:0)(cid:8) q pre,∗(cid:9) ,K∗,φ∗(cid:1) asgiven: i 1. Final goods firms order intermediate goods on the spot market from the sup- (cid:110) (cid:111) plierofferingthelowesteffective-prices j∈J(i;p):= j˜∈J : f p =min{f ◦p} : ij˜ j˜ i  (cid:0) pre(cid:1) pre f Q−Q ifQ≥Q , j∈J(i;p), andv≥ f p spot,∗ ij i i ij j q = ∀i∈[0,1] ij 0 otherwise (4.8) 2. Intermediategoodsfirms: • purchasethecost-minimizinglevelofscalableinputs: L∗ =L∗ = (cid:0) Ypre,∗+Yspot,∗(cid:1) α 1 (K∗) −1− α α ∀j∈J (4.9) j • setspot-marketpricesatamarkupovermarginalcosts: p∗ = p∗ = (1+µ)MC ∀j∈J (4.10) j (cid:124) (cid:123)(cid:122) (cid:125) mark-up≥1 23

where – Ypre,∗:= (cid:82)1q pre,∗ diandYspot,∗:= (cid:82)1q spot,∗ diarethelevelofinter- 0 ij 0 ij mediate goods production required to meet equilibrium pre-order demandandspotmarketdemand,respectively; – µ istheproportionalmark-upovermarginalcostsgivenby: 2f′(cid:0) 1 (cid:1)(cid:82) 2 1 n f (i)di µ := 2n 0 (4.11) (cid:0) f (cid:0) 1 (cid:1)(cid:1)2 −2f′ (cid:0) 1 (cid:1)(cid:82) 2 1 n f (i)di 2n 2n 0 – MC isthemarginalcostfacedbyintermediategoodssuppliers: w (cid:18) Ypre,∗+Yspot,∗(cid:19)1− α α MC= (4.12) α K∗ Intheperiod1equilibrium,eachfinalgoodsproducerfirstevaluateswhetherits pre-committedordersforintermediategoodswillbeadequatetosatisfytheexisting demand for final goods - that is, whether Q i pre :=∑ j f 1 q i p j re ≥Q i . Should the preij orders prove sufficient, the final goods producer i will eschew the spot market, spot setting q = 0. Otherwise, additional intermediate goods will be purchased on i the spot market to meet realized demand, provided that the cost of doing so is less than the value of the output v. Spot-market purchases are made from the cheapest intermediate goods producer, adjusting for the distance-based penalties (equation 4.8). For intermediate goods producers, L∗ is the cost-minimizing choice for given capacity investment K∗ (equation 4.9). Equation 4.10 characterizes the optimal spot-marketpricing. Intermediategoodsproducers engageinmonopolisticcompetitionandchargeamark-upovermarginalcosts. Thismark-upishigherwhensubstitutabilityispoorforthemarginalbuyer(thatis,when f ′(cid:0) 1 (cid:1) ishigh);andlower 2n whencompetitionisfierce(thatis,whennislarge). Inthelimit,asnapproachesinfinity-suchthatthedistancebetweennodesshrinkstozeroandintermediategoods becomeperfectsubstitutes-equation4.10simplifiesdowntopriceequalsmarginal cost(perfectcompetition). Weexplicitlyassumethatintermediategoodsproducers cannotengageinpricediscrimination,chargingthoseatagreaterdistancelessthan 24

those nearby. This assumption is natural in this context: intermediate goods producers may not fully observe the characteristics of the firms that seek to buy from them. From equations 4.10 and 4.12, we see that non-scalable capacity K plays a key role through the marginal cost function. Higher capacity investments by any firm j inperiod0reduceitsmarginalcostofproductionineverystateinperiod1(though moresoinsomestatesthaninothers). However,thisdecreaseinmarginalcostdoes not directly translate into proportionate increases in profit, especially if competing firms also expand their capacities, which would drive down the equilibrium spot priceandpassongainstofinalgoodsproducers. Thispriceresponsehasimportant implicationsforinvestmentincapacity,asthenextsectionshows. 4.2 Period 0 equilibrium in the pre-order market Inperiod0,thefinalgoodsproducerstakepre-orderpricesφ asgiven,formexpectations over the state contingent distribution of spot prices at period 1, and submit pre pre-ordersforintermediategoodsq tomaximizetheirexpectedprofit: i maxE (cid:2) Π (cid:0) q pre ;φ,p∗(cid:1)(cid:3) i i pre q i =vE[Q]−Pr (cid:0) Q>Q pre(cid:1) E (cid:2) p∗(Q,w)·q spot,∗ (Q,w)|Q>Q pre(cid:3) −φ·q pre (4.13) i i i i (cid:16) (cid:17)′ pre pre pre pre whereΠ istheprofitoffirmiinperiod1(eqn. 4.1),q := q ,q ,...,q i i i0 i1 i,n−1 ′ is the vector of pre-orders for intermediate goods, φ := (φ ,φ ,...,φ ) is the 0 1 n−1 menuofpre-orderprices,andQ pre :=∑ 1 q pre isthevolumeoffinalgoodsdemand i j f ij ij that can be met given the pre-orders and the linear production function for final goods. Thefinalgoodsproduceranticipatesthattherealizeddemandforfinalgoods pre Q may fall short of what could be produced from pre-orders Q with probability i (cid:0) (cid:0) pre(cid:1)(cid:1) 1−Pr Q>Q . In such a scenario, the final goods producer will eschew the i spotmarketinperiod1,andnotincur anyadditionalcostsbeyondthoseassociated with the pre-orders.25 With complement probability Pr (cid:0) Q>Q pre(cid:1) , the final goods i 25As discussed in the previous section, we have imposed a constraint q spot ≥0 ruling out the i resaleofpre-orderedintermediategoods. 25

producerwillneedtopurchaseadditionalintermediateinputsonthespotmarketat expectedcostE (cid:2) p∗(Q,w)·q spot,∗ (Q,w)|Q>Q pre(cid:3) . i i Simultaneously,eachintermediategoodsproducer j setspre-orderpriceφ takj ingitscompetitors’pricesφ asgivenandcommitstoalevelofnon-scalableinput −j factorK inordertomaximizeexpectedprofitinperiod1, j (cid:104) (cid:16) (cid:110) (cid:111) (cid:17)(cid:105) (cid:104) (cid:105) (cid:104) (cid:105) maxE Π K , φ ,φ ,q pre,∗ = φ Y pre,∗ −rK +E p∗Y spot,∗ −wL∗ j j j −j j j j j j j j K j ,φj (4.14) pre,∗ spot,∗ whereY andY aretheintermediategoodsoutputrequiredtomeetequij j libriumpre-ordersandspot-marketordersrespectively(definedinequations4.6and 4.7). Weshowthatinafull-productionsymmetricequilibrium,theoptimalpre-order price is equal to the unconditional expectation of spot-market prices. Without a discount over expected spot market prices, final goods firms pre-order only what is necessarytocoverthelowestrealizationofdemand. Therestraineddemandforpreorders affects the intermediate goods producer’s incentive to invest in non-scalable productioncapacity. Proposition 4. [Full production symmetric equilibrium in the pre-order market] Inperiod0, 1. Each final goods producer i pre-orders only what is necessary to cover the lowest realization of final goods demand from its nearest intermediate goods supplier:  f Q if j∈J(i;φ), andv≥ f φ pre,∗ ij ij j q = ∀i∈[0,1], (4.15) ij 0 otherwise (cid:110) (cid:111) where J(i;φ) := j˜∈J : f φ =min{f ◦φ} denote the set of suppliers ij˜ j˜ i that provides the lowest effective pre-order price for i, which is equivalent undersymmetrytothesetofthenearestsuppliers. 2. Eachintermediategoodsproducer j: 26

(a) sets pre-order prices to the unconditional expectation of spot-market prices φ ∗ =E[p∗(Q,w)], (4.16) (b) invests in a level of non-scalable capacity K∗ given by the optimality condition:   αrK∗ =(1−α)E[wL∗]+E   ∂p∗ Yspot . (4.17) ∂K  (cid:124)(cid:123)(cid:122)(cid:125) <0 There is an important intermediate step to show why final goods firms find it optimal in equilibrium to pre-order only what is sufficient to meet the lowest realization of final goods demand. In Appendix F (Lemma 2), we characterize final goodsfirms’demandforpre-orders(Q pre,∗ :=∑ 1 q pre,∗ )intermsoftheequation: i j f ij ij φ =Pr (cid:0) Q>Q pre,∗(cid:1) E (cid:2) p∗(Q,w)|Q>Q pre,∗(cid:3) (4.18) i i (cid:8) pre,∗(cid:9) where Q>Q is the set of states in which the final goods firms need to puri chaseadditionalintermediategoodsfromthespotmarketinperiod1. Forevery(effective)unitofintermediategoodspre-orderedinperiod0,thefinalgoodsfirmwill pre,∗ needtoorderonefewerunitonthespotmarket,butonlyinstateswhereQ>Q . i Thus,foragivenpre-orderpriceφ,finalgoodsfirmswillpre-orderjustenoughintermediate goods such that the φ is equal to the expected marginal savings on the spot market, accounting for the fact that larger pre-orders reduce the probability thatspot-marketpurchaseswillberequired. The demand function for pre-orders (characterized by equation 4.18) has two immediate implications. First, aggregate pre-orders must be equalized across i in equilibrium (Q pre,∗ =Qpre,∗ for all i). Second, the maximum sustainable pre-order i price is the unconditional expectation of the spot market price φ =E[p∗]. As final goods firms are risk neutral, they will not pre-order if φ >E[p∗]. Likewise, intermediategoodsfirmsdonothaveincentivestoofferadiscountonpre-orders(thatis, pay a premium for insurance) by setting φ <E[p∗]. Intermediate goods producers 27

do not have incentives to reduce φ below E[p∗] to attract more pre-orders because they expect to make more marginal profit on the spot market. Critically, any extra marginal costs incurred from lower capacity investments can also be passed on to final goods firms on the spot market along with a mark-up. In fact, because the spotmarketmarkupisproportionaltomarginalcosts,andaggregateoutputremains unchanged in a full-production equilibrium, final goods firms’ profits measured in dollar terms are actually higher when there are symmetric and correlated negative supplyshocks. Withmarketpoweronthespotmarket,intermediategoodsfirmssee noneedtopromotepre-orderstoinsureagainstcorrelatedadversesupplyshocks. Inequilibrium,therefore,wehaveacornersolutionwithφ =E[p∗]andQpre,∗= Q. Intermediate goods firms set pre-order prices at the level that makes final goods firms indifferent between no pre-orders at all and pre-ordering only what is necessarytocoverthelowestrealizationofdemandQ. Inshort,intermediategoodsfirms sets the highest possible pre-order price that drives the final goods firms to their participationconstraint.26 Havingcharacterizedtheequilibriumquantityandpriceofpre-orders,theintermediate goods suppliers determine the amount of production required to meet preorders (Ypre,∗) and forecast expected prices (p∗) and production on the spot market (Yspot,∗). The intermediate goods suppliers then invest in a level of non-scalable capacity K∗ that minimizes expected costs for the anticipated level of production (equation 4.17). This optimality condition for K∗ is similar to its analogues under the social planner benchmarks (equations 3.8 and 3.16 for the unconstrained and (cid:104) (cid:105) constrainedcases,respectively),exceptfortheadditionofafinaltermE ∂p∗ Yspot . ∂K Thisfinaltermcapturesthepecuniaryexternalitythatarisesfromenhancedmarket powerandtheoverrelianceonspotmarkets. Itplaysanimportantroleinexplaining the wedge between the decentralized market solution and the constrained optimal benchmark. 26Both the full production and the symmetry assumption play an important role here. We no longerhaveφ =E[p∗]asanequilibriumconditionwhentheseassumptionsarerelaxed. Likewise, we will also move away from this corner solution if agents are risk-averse, though the presence and qualitative properties of the market failures we identify are likely to be the same. That is, while with risk aversion there is likely to be more investment in capacity (greater resilience) in themarketequilibrium,withmoreriskaverseagents,(constrained)Paretooptimalityalsorequires greaterresilience,andagapwillremainbetweenthetwo. 28

5 Decentralized solution versus constrained optimal benchmark We can now prove the core proposition of the paper. The level of investment in the non-scalable capacity in a decentralized market setting (K∗) is suboptimally low whencomparedwiththelevelintheconstrainedoptimalbenchmark(KSP). Proposition5. [Sub-optimalnon-scalablecapacityinvestment]K∗ <KSP. Proof. We prove K∗ < KSP by contradiction. This proof is instructive because it dp∗ highlights the importance of the pecuniary externality and the overreliance on dK thespotmarketYspot asthemaindriversbehindtheunderinvestmentincapacity. First,bythefull-productionassumption,weknowthatthelevelofintermediate goods production is the same under both the decentralized solution and the constrainedbenchmark-Y∗(Q,w)=YSP(Q,w)-inallstatesoftheworld(Q,w). The above equality implies that if K∗ = KSP, then L∗(Q,w) = LSP(Q,w) in everystate,leadingtoacontradiction: (cid:104) (cid:105) αrKSP = (1−α)E wLSP =(1−α)E[wL∗]   > (1−α)E[wL∗]+E   dp∗ Yspot =arK∗  dK  (cid:124)(cid:123)(cid:122)(cid:125) <0 IfinsteadK∗>KSP,thenL∗(Q,w)<LSP(Q,w)ineverystate,againgivingrise toacontradiction:   αrK∗ =E   dp∗ Yspot +(1−α)E[wL∗] dK∗  (cid:124)(cid:123)(cid:122)(cid:125) <0 (cid:104) (cid:105) <(1−α)E[wL∗]<(1−α)E wLSP =αrKSP 29

Thepropositionrevealsthatintermediategoodsproducersunderinvestincapacity upfront because they are unable to fully capture the cost savings generated by increased investment. Specifically, each dollar saved through efficiency gains from capacity investment does not yield a corresponding one-dollar increase in profits, because a part of these gains is transferred to final goods producers through lower (cid:104) (cid:105) spot-market prices. The key term of interest is E dp∗ Yspot , which captures the dK dp∗ interaction between the pecuniary externality ( the sensitivity of spot market dK pricestocapacityinvestment)andthedegreeofrelianceonthespotmarket(Yspot). dp∗ Focus first on the price sensitivity term and recall that equilibrium spot dK pricescanbeexpressedasaproportionalmark-upovermarginalcosts: p∗=(1+µ)MC. Allelsebeingequal,highercapacityinvestmentK,lowersthemarginalcost(MC= (cid:16) (cid:17)1−α w Y∗ α )ineverypossiblestateandthuslowersspotprices. Theextenttowhich α K∗ K matters depends on the scalability of the economy (α). As scalability improves andα →1,thelessimportantisK inproduction,andtheexternalityshrinks. 1 2f ′(1)(cid:82)2n f(i)di TheeffectofKonmarginalcostsisamplifiedbythemarkup(µ= 2n 0 ). 1 f(1)−2f′(1)(cid:82)2n f(i)di 2n 2n 0 The size of the mark-up depends on the substitutability between sectors, as measured by the distanced-based penalty function f (evaluated at the marginal buyer i = 1 ). Higher substitutability between sectors lowers mark-up and reduces the 2n wedge between the decentralized solution and the constrained optimal benchmark in equilibrium. Lastly, another important way to reduce the wedge is through enhanced competition (that is, a larger n), which also reduces the amplification of marginalcostchangesbyreducingmark-ups. Equally as important, the wedge results from an over-reliance on the spot market. UnlikeanArrow-Debreueconomy,inwhichagentscantradecontingentclaims for every conceivable state of the world, in our model - much like real-world conditions-thesetofcontractsthatcanfeasiblybewrittenandtradedismuchsmaller than the set of possible states. As a result, the pre-order, forwards, and futures markets will fall short of providing adequate risk insurance for intermediate goods producers. Downstream final goods producers fail to sufficiently compensate their suppliersforthepecuniaryexternalityarisingfromthebenefitsofincreasedcapital 30

investment.27 5.1 Policy interventions Using our model, it is possible to identify a number of ways to narrow the wedge between the supply network delivered by unfettered markets and the efficiently resilientnetworkcharacterizedunderaconstrainedoptimalbenchmark. First, the most straightforward strategy to address the externality in the model is to offer subsidies for capacity investments, thereby lowering the effective cost r incurred by intermediate goods producers for non-scalable capacity. Second, the government might extend tax benefits to downstream firms that engage in preorders or transact in the futures market or, alternatively, levy additional taxes on spot-market transactions. Futures markets facilitate greater risk sharing between upstream and downstream entities and diminish dependency on spot markets. A thirdavenueistoreducethesensitivityofspotpricestochangesincapacityinvestments. This approach could entail structural economic reforms such as lowering entrybarriers(includingtradebarriers),enactingstrongercompetitionpolicies,and enhancing the substitutability of intermediate products, all of which could reduce supplier markups. Similarly, technological advancements in production scalability couldshiftthefocustowardotherinputfactorsthatcanbemorereadilyadjustedon shortnotice. In practice, it may be harder to devise practical, implementable interventions. Directly subsidizing capacity investments offers a straightforward strategy, yet distinguishing such investments from other types of capital expenditure can be difficult, particularly in certain sectors. The government may want to intervene only in certain critical industries - for example computer chip production, where downstream externalities are especially significant and resilience is more important - by for instance, offering lower taxes for firms operating with excess capacity. While tax incentives for spot and pre-order markets can be effective in sectors like electricity, with its well-defined spot and futures markets, this approach becomes less 27Inasense,thispecuniaryexternalityisaspecialcaseofthegeneralpecuniaryexternalityarising in economies without a complete set of AD securities analyzed by Greenwald and Stiglitz (1986) andfirstdiscussedinStiglitz(1982). 31

straightforwardinindustrieswheremarketboundariesaremoreblurred. 6 Concluding remarks Especially since the pandemic and post-pandemic supply chain interruptions, the question of resilience has moved to the fore. Of course, we do not expect markets to be prepared for every shock, regardless of size, as doing so would be extraordinarilyexpensive. Thequestionis,dotheymakeappropriatepreparations,measured againstanappropriatebenchmark? Therearemanyreasonstothinkthattheymight not,Criticsofthemarket,forinstance,complainabout“short-termism.” Weexaminethenormativequestionofresilience,however,inaworldwithfully rationalexpectationsandinwhichfirmsdonotsufferfromshort-termism,showing that,nonetheless,thereisabiastowardexcessivevulnerabilityduetoinsufficientex antecapacityinvestmentsbyupstreamintermediategoodsproducers. Thisshortfall arises because these producers cannot fully capture the returns on their capacity investments: A portion of the economic gains is transferred downstream to final goodsproducersthroughreducedspot-marketprices. We believe that our study is the first to incorporate interactions in both the spot and futures markets in such a normative analysis of supply networks, which is essential for addressing the question at hand. Performing this analysis in the context ofdifferentiatedcompetitionnecessarilyentailsacertaindegreeofcomplexity. For tractabilityandeaseofexposition,wehaveintroducedanumberofsimplifications, however, in online Appendix G, we show how the results hold under significantly moregeneralconditions. Mostnotably,weshowthatifthereareverylargeshocks, suchthatthecostofmeetingthemarketdemandissohighthatthereare“unserved” customers(thatis,AssumptionA2FullProductionisnotsatisfied),thentheanalysisstillholds. Finally,wenotethatincertainindustries,forcesmaybepushingintheotherdirection. Somefirmsmaychoosetoholdexcessproductioncapacitypurelyasaway to deter prospective entrants, thereby reducing competition. Risk aversion on the partofintermediateandfinalgoodsproducers(andconsumers,translatedintomore profitable contracts signed with firms that have greater resilience) may also result 32

ingreaterresiliencethansuggestedbythismodel. Moreover,wehaveassumedthat market power resides in the upstream firms. Especially in more oligopolistic contexts,downstreamfirmsmayengageinsupplychaindiversificationandhigherlevels of pre-ordering, generating higher levels of capital investment in the upstream industries and greater market resilience, explicitly to limit the ability of the upstreamfirmstoexercisemarketpowerinthemannerillustratedhere. Theoneresult thatwebelieveisresilientisthatthereislikelytobeadisparitybetweenthemarket andtheconstrainedoptimallevelofresilience. The events of the past few years have made it clear that economists have paid insufficient attention to resilience. This paper is intended as a contribution to the nascent literature attempting to understand better why markets may have underinvestedinresilience. References Acemoglu, D., V. Carvalho, A. Ozdaglar, and A. Tahbaz-Salehi (2012). The networkoriginsofaggregatefluctuations. Econometrica80(5),1977–2016. Acemoglu, D. and A. Tahbaz-Salehi (2020). Firms, failures, and fluctuations: The macroeconomicsofsupplychaindisruptions. NBERWorkingPaper (27565). Amelkin, V. and R. Vohra (2020). Strategic formation and reliability of supply chainnetworks. WorkingPaper. Anupindi,R.andR.Akella(1993). Diversificationundersupplyuncertainty. ManagementScience39(8),944–963. Atalay, E., A. Hortacsu, J. Roberts, and C. Syverson (2011). Network structure of production. Proceedings of the National Academy of Sciences 108(13), 5199– 5202. Babich,V., G.Aydın, P.-Y.Brunet, J.Keppo, andR. Saigal(2012). Risk, financing andtheoptimalnumberofsuppliers. InSupplyChainDisruptions,pp.195–240. Springer. 33

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For Online Publication: Appendix A Cost functions for Cobb-Douglas production function A.1 Standard Cobb-Douglas Production ThestandardcostminimizationproblemwithaCobb-Douglasproductionfunction isgivenby: minC=wL−rK (A.1) L,K s.t. LαK1−α ≥Y SettinguptheLagrangianandcomputingthenecessaryfirstorderconditionsyields thefamiliaroptimalitycondition: (1−α)wL=αrK (A.2) Substitutingtheoptimalityconditionintotheproductionfunctionyieldstheoptimal inputchoicesK=Y (cid:0)w(cid:1)α(cid:0)1−α(cid:1)α ,andL=Y (cid:0)r(cid:1)1−α(cid:0) α (cid:1)1−α . Thecostfunction r α w 1−α isthereforegivenby: (cid:16)w(cid:17)α (cid:18) r (cid:19)1−α C(Y)=Y (A.3) α 1−α withconstantmarginalcost: (cid:16)w(cid:17)α (cid:18) r (cid:19)1−α ′ MC:=C (Y)= (A.4) α 1−α 36

A.2 Cobb-Douglas Production with Partial Delay With partial delay, the intermediate goods producer takes K as given in its period 1 costminimizationproblem: minC˜(K)=wL+rK L s.t.LαK1−α ≥Y TheoptimalL isgivensimplybytheminimumamountnecessarytoproduceY: L=Yα 1 K−(1− α α) (A.5) ThecostfunctionthereforedependsonboththedesiredoutputY andthecapacityK reservedexante: C˜(Y;K)=wYα 1 K−(1− α α) +rK (A.6) withamarginalcostofproductionthatdependsontheoutput-capacityratio (cid:0)Y(cid:1) : K dC˜(Y;K) w (cid:18) Y (cid:19)1− α α M(cid:103)C:= = (A.7) dY α K Notethattheimpactofcapacityontotalcostisgivenby: dC˜(Y;K) (1−α) (cid:18) Y (cid:19) α 1 =−w +r dK α K (cid:18) (cid:19) Y =r−(1−α) M(cid:103)C (A.8) K which is lower than r, because of the additional indirect cost savings on scalable inputcapacity. 37

B Proof for Proposition 1 - Perfect Foresight benchmark We start with the optimization problem for the perfect foresight benchmark [PF] characterized in the main text (eqns 3.2 to 3.6). With perfect foresight, both L and K canbesetasafunctionoftherealizedstate(Q,w)inperiod1. Thisisequivalent to saying that both L and K can be adjusted flexibly and simultaneously as the need arise. We thus have a standard Cobb-Douglas production for intermediate goods, with: optimal input choices characterized by (1−α)wL=αrK (eqn A.2); cost functionC (cid:0) Y (cid:1) =Y (cid:0)w(cid:1)α(cid:0) r (cid:1)1−α (eqn A.3); and constant marginal cost of j j α 1−α production ∂C = ∂C = (cid:0)w(cid:1)α(cid:0) r (cid:1)1−α (eqnA.4).28 ∂Y j ∂q ij α 1−α Substituting the optimal input choices and the associated cost function into the original optimization problem [PF] reduces the dimension of the problem to one in (cid:8) (cid:9) orderflows q only: ij [OptimizationProblemPF*] (cid:40) (cid:34) (cid:35) (cid:34) (cid:35)(cid:41) (cid:90) 1 1 (cid:18)(cid:90) 1 (cid:19) (cid:16)w(cid:17)α (cid:18) r (cid:19)1−α W(Q,w)= max v ∑ q di−∑ q di ij ij {q ij } i∈I,j∈J 0 j∈J f ij j∈J 0 α 1−α 1 s.t. ∑ q ≤Q ∀i∈[0,1] [Demandcap] ij i f j∈J ij q ≥0 ∀i∈[0,1], j∈J [Non-negativeinputs] ij WecansetuptheKuhnTuckerLagrangianforProblemPF*as: (cid:34) (cid:35) (cid:34) (cid:35) (cid:34) (cid:35) (cid:90) 1 1 (cid:18)(cid:90) 1 (cid:19) (cid:16)w(cid:17)α (cid:18) r (cid:19)1−α 1 LPF =v ∑ q di−∑ q di −∑λ ∑ q −Q ij ij i ij f α 1−α f 0 j∈J ij j∈J 0 i∈I j∈J ij wherebysymmetrywehaveQ =Q, ∀i∈[0,1]. i Thefirst-orderconditions(FOCs)withthecorrespondingcomplementaryslacknessconditionsaregivenby: 28Thefirstequality ∂C = ∂C holdswhenthe“feasibilityofintermediategoodsorderflow”binds ∂Yj ∂qij withequalityinequilibrium. 38

(cid:32) (cid:33) ∂LPF v (cid:16)w(cid:17)α (cid:18) r (cid:19)1−α λ qPF =q − − i =0 ∀i∈I, j∈J (B.1) ij ∂q ij f α 1−α f ij ij ij (cid:34) (cid:35) ∂LPF 1 λ =λ ∑ q −Q =0 ∀i∈I (B.2) i i ij ∂λ f i j∈J ij We observe from the FOCs that for each final goods i, the corresponding Lagrangian multiplier λ, when strictly positive, is determined by the supplier j ∈ J i thatcanprovidetheinputsmostcheaplytoi: (cid:40) (cid:41) (cid:16)w(cid:17)α (cid:18) r (cid:19)1−α λ =v−min f i ij j∈J α 1−α (cid:16)w(cid:17)α (cid:18) r (cid:19)1−α =v− f (B.3) ij α 1−α (cid:110) (cid:111) where j(i) ∈ J(i) := j˜∈J|f (cid:0)w(cid:1)α(cid:0) r (cid:1)1−α ≤ f (cid:0)w(cid:1)α(cid:0) r (cid:1)1−α ∀j∈J . ij˜ α 1−α ij α 1−α (cid:26) (cid:27) (cid:16) w (cid:17)αj (cid:16) r (cid:17)1−αj Alternatively, if v < min f j j , then q = 0 for all j ∈ J, j∈J ij αj 1−αj ij Y˜ =0 and λ =0 (i.e. it is not efficient for firm i to produce at all). This latter case i i isruledoutbythefullproductionassumption(A2). Thus,combiningequations(B.2,B.3),whenλ >0finalfirmiwillbeallocated i sufficientintermediategoodsfromitscheapestsuppliertomeetfinaldemandQ:   1 f Q for j∈J(i) qPF = n(J(i)) ij ij 0 for j̸=J(i) where n(J(i)) is the cardinality of the set J(i). In a symmetric equilibrium, the cheapestsupplier(s)coincideswiththeclosestsupplier(s). Withintermediategoods firms located equidistant around the circle, there are at most two closest suppliers for each i (e.g. nodes 0 and 1 for i= 1 ). In such cases when there are two closest 2n suppliers,insteadoftie-breakingbydividingordervolumesinhalf,weassumeeach intermediate goods node j wins the tie-break to its right on the circle, but loses the 39

tie-break to its left. This is loosely equivalent to imposing n(J(i))=1,∀i∈[0,1]; aconventionwewilladopttosimplifyexpositionwithoutlossofgenerality. (cid:110) (cid:111) Havingsolvedfortheoptimalorderflow qPF ,wecannowderivetheaggreij gate output of intermediate goods. By symmetry every intermediate goods firm j will produce the same amountY =YPF, ∀j∈J. So we can computeYPF from the j perspective of firm j =0, who is able to capture the two equal market segments to itsleftandright-handside,i∈ (cid:2) 0, 1 (cid:3) and (cid:2) 1− 1 ,1 (cid:3) : 2n 2n (cid:90) 1 (cid:90) 1 YPF(Q,w)= qPFdi=2Q 2n f di (B.4) i0 i0 0 0 Finally, we can substitute the equilibrium intermediate goods production YPF intotheCobb-Douglasproductionfunction,combinedwiththeoptimalitycondition forinputs(eqnA.2)toderiveexplicitsolutionsforKPF andLPF: (cid:34) (cid:35) (cid:18) w(1−α) (cid:19)α (cid:90) 1 KPF(Q,w)= 2Q 2n f di i0 r α 0 (cid:34) (cid:35) (cid:18) r α (cid:19)1−α (cid:90) 1 LPF(Q,w)= 2Q 2n f di i0 w1−α 0 ThiscompletestheproofforProposition1. C Proof for Proposition 2 - Social Planner’s Constrained Optimal problem C.1 Period 1 optimization Taking a similar approach to the Perfect Foresight benchmark, we start by formingthecorrespondingKuhnTuckerLagrangianforthesocialplanner’sconstrained 40

optimalprobleminperiod1[SP1]: L = max v (cid:90) 1 (cid:32) ∑ 1 q ij (cid:33) di−∑ (cid:34) rK+w j (cid:18)(cid:90) 1 q ij di (cid:19) α 1 K−(1− α α) (cid:35) ... {q ij ∈R + } i,j 0 j∈J f ij j∈J 0 (cid:32) (cid:33) (cid:90) 1 1 − λ ∑ q −Q di i ij f 0 j∈J ij From the Lagrangian we obtain the first-order derivatives with complementary slacknessconditions:  (cid:32) (cid:33)1−α  ∂L v w (cid:82)1q di α λ q ij =q ij − 0 ij − i =0 ∀i∈[0,1],∀j∈J ∂q f α K f ij ij ij (cid:32) (cid:33) ∂L 1 λ =λ ∑ q −Q =0 ∀i∈[0,1] i i ij ∂λ f i j∈J ij (cid:18) (cid:19)1−α where v isthemarginalbenefitfromsupplyingifrom j(i.e.,q ),and w (cid:82) 0 1q ij di α f ij ij α K is the marginal cost. Later, in the final step of this proof, we will substitute out the (cid:18) (cid:19)1−α (cid:32) (cid:33)1−α endogenouslydeterminedKandq toshowthat w (cid:82) 0 1q ij di α = (cid:0)w(cid:1)α(cid:0) r (cid:1)1−α wQα 1 . ij α K α 1−α (cid:104) 1(cid:105) E wQα By the full-production assumption, we know that the marginal benefit will alwaysweaklyexceedthemarginalcost,sotheLagrangianmultiplierforfinalgoods firm i, λ, is given by the intermediate goods firm j that offers the lowest effective i cost:  (cid:32) (cid:33)1−α  w (cid:82)1q di α  λ =v−min f 0 ij i ij j∈J  α K  (cid:18) (cid:19)1−α w Y α =v− f ij α K 41

Definingthesetoflowesteffectivecostsuppliersas:  (cid:32) (cid:33)1−α (cid:32) (cid:33)1−α   w (cid:82)1q di α w (cid:82)1q di α  J(i):= j˜∈J|f 0 ij ≤ f 0 ij ∀j∈J ij˜ α K ij α K   wearriveatthefirstpartoftheproposition(eqn3.14):  (cid:16) (cid:17)1−α f Q if j∈J(i) andv≥ f w YSP α qSP(Q,w)= ij ijα KSP ij 0 otherwise whereYSP = (cid:82)1qSPdi 0 ij Next, from the cost-minimization problem for the Cobb-Douglas production withpartialdelay(eqnA.5)wehavethenextpartofthepropositionfortheoptimal choiceofthescalableinputfactorinperiod1: LSP(Q,w)=Yα 1 K−(1− α α) = (cid:18)(cid:90) 1 qS ij Pdi (cid:19) α 1 (cid:16) KSP (cid:17)−(1− α α) 0 C.2 Period 0 Optimization As discussed in the main body, we can show that the optimality condition for nonscalable production capacity KSP (equation 3.16) holds with or without the full production assumption. To elucidate this point, note that when the full production assumption is relaxed, there may exist states of the world (Q,w) where some final goods firms situated far from intermediate goods production firms do not find it optimal to produce at all. In other words, let i¯SP(cid:0) Q˜,w˜ (cid:1) represent a “thresh- 0 old” firm in the final goods sector. This firm is indifferent between sourcing inputs from intermediate goods firm j = 0 and opting out of production altogether (cid:32) (cid:33)1−α in state (cid:0) Q˜,w˜ (cid:1) : v = (cid:0)w˜(cid:1)α(cid:0) r (cid:1)1−α w˜Q˜α 1 . Hence there may exist f(i¯S 0 P) α 1−α E (cid:104) wQα 1(cid:105) states (cid:0) Q˜,w˜ (cid:1) whereby i¯SP(cid:0) Q˜,w˜ (cid:1) < 1 , and the market segment (cid:2) i¯SP(cid:0) Q˜,w˜ (cid:1) , 1 (cid:3) on 0 2n 0 2n thecircleproducesnofinalgoodsoutputsandexperiences“emptyshelves”. Atfirst glance,onemightexpectthatanex-anteincreaseinnon-scalablecapacityK would 42

positively impact welfare. This expectation arises from the fact that an increase in K would endogenously boost the production of intermediate goods, Y. However, theindirecteffectscapturedby dY arezeroinequilibrium. Wecansafelyignorethe dK indirect effects of K onY, because the indirect effects are multiplied by the differencebetweenthemarginalcostandmarginalbenefitofproductionforthethreshold buyer, which is equal to zero by construction.29 Therefore, irrespective of whether thefull-productionassumptionholds,onlythedirecteffectsofK matter. Formally,recallthattheperiod1valuefunctionforgivenK andrealizationofQ andwcanbeexpressedasthedifferencebetweenthevalueoffinalgoodsproduced andthecostoftherequiredintermediarygoods: (cid:32) (cid:33) WSP(K|Q,w)=v 2n (cid:90) min{ 2 1 n ,i¯S 0 P} Qdi −n (cid:16) rK+wYα 1 K−(1− α α)(cid:17) (C.1) 0 where iSP is the threshold buyer for intermediate goods 0, (implicitly) defined as 0 the final goods firm i for which the marginal benefit of sourcing inputs from j =0 equalsthemarginalcost: (cid:18) (cid:19)1−α v w Y α = (C.2) f α K i¯SP,0 0 The upper limit of integration, min (cid:8) 1 ,i¯SP(cid:9) , reflects the possibility of “regime 2n 0 switching” when the full production assumption is relaxed. When the economy operates at a full-production equilibrium, the relevant threshold buyer for intermediategoodsfirm j=0isgivenbyi= 1 ,thefinalgoodsfirmlocatedatthehalfway 2n point between j = 0 and j = 1. This is a competitive regime, where intermediate goods firms engage in monopolistic competition. But without the full-production assumption, there may arise states of the world whereby the threshold buyer for j=0 is closer: i.e. i¯SP < 1 . This is a local monopolies regime, characterized by a 0 2n gapinmarketcoveragebetweentwosuppliernodes(e.g. between j=0and j=1). 29This result bears resemblance to the Envelope Theorem, in which the total derivative of the valuefunctionwithrespecttotheparametersofthemodelisequaltoitspartialderivative. HereK isthechoicevariable,butthetotalderivativeofWSP(K|Q,w)withrespecttoK isalsoequaltoits partialderivative. 43

Thedemandforfinalgoodsisnotfullymetforfirmslocatedinthisgap,andwesee “empty shelves” in some segments of the market. We account for the possibility of “regime switching” between the competitive regime and the local monopolies regimeintheanalysesthatfollows. Totally differentiating the expectation of WSP with respect to K will yield the desired first-order optimality condition for non-scalable capacity in period 0. For ease of exposition, we proceed with the differentiation in parts. In particular, note thatthechangeinthefinaloutputineachmarketsegmentwithrespecttoK isgiven by: d (cid:90) min{1,i¯SP} dmin (cid:8) 1 ,i¯SP(cid:9) 2n 0 Qdi= 2n 0 Q dK dK 0 which depends on the derivative of the threshold buyer i¯SP with respect to K. 0 Strictlyspeaking,thefunctionmin (cid:8) 1 ,i¯SP(cid:9) isnotcontinuouslydifferentiablew.r.t. 2n 0 K duetothekinkwhere 1 =i¯SP. Withoutlossofgenerality,wewilllooselydefine 2n 0 dmin{1,i¯SP} 2n 0 usingitsright-handsidederivative: dK  dmin (cid:8) 1 ,i¯SP(cid:9) 0 when min (cid:8) 1 ,i¯SP(cid:9) = 1 2n 0 = 2n 0 2n dK  di¯S 0 P >0 when min (cid:8) 1 ,i¯SP(cid:9) < 1 dK 2n 0 2n to account for the fact that when i¯SP ≥ 1 , the presence of demand caps in over- 0 2n lapping market segments means that any further increases in capacity would not increaseaggregateoutput. Next,theincreaseinscalableinputcosts(wL)fromchangesinK canbebroken down into two components: the indirect costs of requiring more scalable inputs when total output increase following a rise in K; minus the direct cost savings of needinglessL whenK increasesforgivenoutputY: (cid:34) (cid:18) (cid:19)1−α (cid:18) (cid:19)1(cid:35) d (cid:16) wYα 1 K−(1− α α)(cid:17) =w 1 Y α dY − 1−α Y α dK α K dK α K SinceY =2Q (cid:82)min{ 2 1 n ,i¯S 0 P} f di,theendogenousincreaseinincreaseinoutputY 0 i0 44

whenK increaseis: dY (cid:34) dmin (cid:8) 1 ,i¯SP(cid:9) (cid:35) =2Q 2n 0 f dK dK min{ 2 1 n ,i¯S 0 P},0 (cid:34) dmin (cid:8) 1 ,i¯SP(cid:9) (cid:35) =2Q 2n 0 f dK i¯S 0 P,0 Thesecondequalitysimplifiesthefirstbynotingthat dmin{ 2 1 n ,i¯S 0 P} =0⇔min (cid:8) 1 ,i¯SPS(cid:9) = dK 2n 0 1 . 2n Taken together, we have the following first-order optimality condition for the period0problem: dE (cid:2) WSP(cid:3) =2nvE (cid:34) dmin (cid:8) 2 1 n ,i¯ 0 (cid:9) Q (cid:35) −nr−nE (cid:34) w (cid:32) 1 (cid:18) Y (cid:19)1− α α dY − 1−α (cid:18) Y (cid:19) α 1(cid:33)(cid:35) =0 dK dK α K dK α K ⇔0=2vE (cid:34) dmin (cid:8) 2 1 n ,i¯ 0 (cid:9) Q (cid:35) −r−E (cid:34) w (cid:32) 1 (cid:18) Y (cid:19)1− α α dY − 1−α (cid:18) Y (cid:19) α 1(cid:33)(cid:35) dK α K dK α K (cid:34) (cid:35) (cid:34) (cid:18) (cid:19)1−α (cid:35) (cid:34) (cid:18) (cid:19)1(cid:35) 1 dY w Y α dY 1−α Y α ⇔0=vE −r−E + E w f dK α K dK α K i¯SP,0 (cid:34)(cid:32) (cid:18) (cid:19)1−α(cid:33) (cid:35) (cid:34) (cid:18) (cid:19)1(cid:35) v w Y α dY 1−α Y α ⇔r=E − + E w f α K dK α K i¯SP,0 0 (cid:20) (cid:21) (cid:20) (cid:21) dY 1−α wL ⇔r=E 0× + E dK α K (cid:104) (cid:105) ⇔αrKSP =(1−α)E wLSP where the penultimate line holds because v = w(cid:0)Y(cid:1)1− α α (marginal benefit f i¯SP,0 α K 0 = marginal cost) is the definition of i¯SP, and from the Cobb-Douglas production 0 function, we have (cid:0)Y(cid:1) α 1 = L. In other words, the indirect effects of raising K on K K aggregate output Y neatly cancels out, leaving us with the familiar Cobb-Douglas inputsoptimalityconditioninthefinalline. 45

1 − (1−αj ) Using the production function to substitute out LSP = YαjK αj and rearrangingyieldstheexplicitsolutionforKSP:   (cid:32) (cid:33)1α (cid:18) 1−α 1 (cid:19)α (cid:90) min{1,i¯SP} α KSP = Ew 2Q 2n 0 f i0 di  α r 0 asrequiredforpart2oftheproposition. Finally, to complete the proof, we want to show that this level of KSP indeed leads to a full production equilibrium under assumption A2. We do this by substitutingouttheexplicitexpressionforKSP inthemarginalcostfunctiontoshowthat inequilibriumthemarginalcostofproductionisalwaysbelowthevaluationforthe finalgoods(adjustedforthedistance-basedpenalty): w (cid:18) YSP(cid:19)1− α α M(cid:103)C(Q,w) := α KSP  1−α α w (cid:18) α (cid:19)1−α YSP = r   α (1−α)  (cid:104) 1 (cid:105)α E w(YSP)α  1−α (cid:16)w(cid:17)α (cid:18) r (cid:19)1−α wQα 1 =  (cid:104) (cid:105) α 1−α 1 E wQα  1−α (cid:18) w¯ (cid:19)α(cid:18) r (cid:19)1−α w¯Q¯ α 1 ≤  (cid:104) (cid:105) ∀w,Q α 1−α 1 E wQα v ≤ byassumptionA2 f (cid:0) 1 (cid:1) 2n 1 whereYSP(Q,w)=2Q (cid:82) 2n f (i)di. 0 ThiscompletestheproofforProposition2. 46

C.3 Relationship between KPF and KSP Recallfromequations3.9,3.17thatwehave: (cid:32) (cid:33) (cid:18) w(1−α) (cid:19)α (cid:90) 1 KPF(Q,w)= 2Q 2n f (i)di r α 0 (cid:32) (cid:33) (cid:18) 11−α (cid:19)α (cid:90) 1 (cid:16) (cid:104) (cid:105)(cid:17)α KSP = 2 2n f (i)di E wQα 1 r α 0 Somestraight-forwardalgebrashowsthat:  (cid:104) 1 (cid:105)α E wQα KSP =KPF(Q,w)  1 wQα Furthermore,takingtheexpectationofKPF over(Q,w),wehave (cid:32) (cid:33) E (cid:2) KPF(Q,w) (cid:3) = (cid:18) 1(1−α) (cid:19)α 2 (cid:90) 2 1 n f (i)di E[wαQ] r α 0 TakentogetherwiththeexpressionforKSP,wecanshow: (cid:16) (cid:104) (cid:105)(cid:17)α 1 KSP E wQα = E[KPF(Q,w)] E[wαQ] suchthat,byJensen’sinequalityandgiveng(x):=xα isconcaveforα ∈(0,1),we have: KSP ≥E (cid:2) KPF(Q,w) (cid:3) asrequired. 47

D Sufficient condition for full production symmetric equilibrium in the decentralized solution First, we establish the sufficient conditions for the existence of a full-production symmetricequilibrium. Lemma 1. [Existence of Full Production Symmetric Equilibrium]: For every configurationoftheprimitivesofthemodelwiththeexceptionofv,E ={f (·),α,w,r,Q}, −v there exist a v¯∈R such that the economies E (v)={f (·),α,w,r,Q,v≥v¯} ad- ++ mitsafullproductionsymmetricequilibrium. Intuitively, the marginal benefit of production is increasing in the valuation of the final goods v, but the marginal cost is non-increasing in v. So, for every parameterization of the model, we can find a large enough v¯ to guarantee full production inasymmetricequilibrium. Formally,whileassumptionA2establishesthesufficientconditionsforfullproduction under the social planner benchmarks, the corresponding full-production conditionforthedecentralizedcaseisgivenby:  1−α (cid:18) 1 (cid:19) (cid:18) 1 (cid:19) (cid:18) 1 (cid:19) w¯ Q¯(cid:82) 2 1 n f (i)di α v≥ f p∗ = f µ(n)M(cid:103)C= f µ(n)  0  2n 2n 2n α K∗ f(1) where p∗istheequilibriumpriceforintermediategoods,µ(n):= 2n f(1)−2 f′( 2 1 n ) (cid:82)2 1 n f(i)di 2n f(1) 0 2n is the mark-up over marginal costs, and K∗ is the equilibrium level of non-scalable capacity. We argue that for every possible parameterization of the other primitives, thereexistsav¯∈R thatguaranteesfullproduction. ++ Consider an arbitrary economy E (v˜) = {f (·),α,w,r,Q;v˜} with valuation v˜. We want to show that by varying v˜ we can always construct an economy E (v¯) = {f (·),α,w,r,Q;v¯} that supports a full production symmetric equilibrium holding all other primitives the same. To do this, we compute K∗(v˜), the associated equilibrium level of capacity investment assuming full production; and the correspond- 48

(cid:32) (cid:33)1−α ing MC(v˜) = f (cid:0) 1 (cid:1) µ(n)w¯ Q¯(cid:82) 0 2 1 n f(i)di α , the highest possible realization of 2n α K∗(v˜) marginal costs in that economy. Note that K∗(v) is a non-decreasing function of v andthereforeMC(v)isanon-increasingfunctionofv(i.e. themarginalcostofproductioninanyfullproductionequilibriumdoesnotincreasewhenthevaluationincreases). Thenifv˜≥MC(v˜),thentheeconomyE (v˜)admitsafullproductionsymmetricequilibriumcharacterizedbyK∗(v˜). Ifinsteadv˜<MC(v˜),letv¯=MC(v˜)> v˜. Thenv¯=MC(v˜)≥MC(v¯). AndeveryeconomyE (v)={f (·),α,w,r,Q;v≥v¯} admitsafullproductionsymmetricequilibriumasrequired. Second,we remarkthatthe fullproductionassumption alsoenablesus toavoid problemsofnon-differentiabilityinthedemandfunction. Inaclassicaltreatmentof thecirculareconomy,Salop(1979)segmentsthedemandfunctionforintermediate goods into three sections: a “monopoly” regime (whereby the firm acts as if it is a monopoly);a“competitive”regime(whereitengagesinBertrandcompetitionwith itsneighbors);anda“super-competitive”regime(whereitpricessoaggressivelyas to take over its neighbor’s native market). The demand function exhibits a kink at theintersectionbetweenthemonopolyandcompetitiveregime,andmakesadiscontinuous jump between the competitive and super-competitive regime. We can rule out equilibria falling under the super-competitive regime by setting a sufficiently steep distance-based penalty function; and for the purpose of the main analyses in section3and4,thefullproductionassumptionensuresthedemandfunctioniscontinuously differentiable. In Appendix G we relax the full production assumption to examinetheinterplaybetweenthecompetitiveandmonopolyregime. E Proof of Proposition 3: Full production symmetric equilibrium in the spot market spot,∗ In period 1, the equilibrium spot market orders q by final goods firms, and the ij purchase of scalable inputs L∗ by intermediate goods firms, take a similar form to theircorrespondingexpressionsundertheconstrainedoptimalbenchmark. Weskip their derivations to avoid repetition, and concentrate instead on the solution for the 49

spotmarketprice p∗,givenbyequation4.10. To solve for p∗, we will first need to derive the demand function facing the intermediate goods firm j = 0 on the spot market. For now, we will also need to conjecturethattheaggregatevolumeofpre-ordersmustbeequalizedacrossallfinal goodsfirmsinequilibrium: Q pre =Qpre,∀i∈I,aresultthatwewillproveformally i laterinappendixF. E.1 Finding the slope of the demand curve The demand curve facing each intermediate goods firm is piece-wise linear (when plotted against p , for given p ). To see this, note that the period 1 equilibrium j −j is governed by two indifference thresholds. First, for given price vector (p ,p ), 0 −0 the participation threshold for firm j =0, i¯ , is defined as the final goods firm that 0 is indifferent between buying inputs from intermediate goods firm j = 0 and not producingatall: (cid:34) (cid:35) v f (i¯ )p =v, ∀p ∈ ,v (E.1) 0 0 0 f (cid:0)1(cid:1) 2 Second, the competitive threshold i¯ is the marginal final goods producer that 0,1 isindifferentfrombuyingfromsuppliernode j=0and j=1: f (d(i¯ ,0))p = f (d(i¯ ,1))p (E.2) 0,1 0 0,1 −0 Hence the demand curve facing firm j = 0 depends on the lower envelope of the participationandcompetitivethresholdfunctions: (cid:90) i¯∗ Y spot =2 0 f (i)·(Q−Qpre)di (E.3) 0 0 where i¯∗ :=min (cid:8) i¯ (p ),i¯ (p ,p ) (cid:9) (E.4) 0 0 0 0,1 0 −0 When the slope of the demand curve is well-defined (i.e., away from the knife- 50

edgecasewheni¯ (p )=i¯ (p ,p )),itisgivenby: 0 0 0,1 0 −0 dY spot (cid:20) di¯∗ (cid:21) 0 =2 0 f (i¯∗)(Q−Qpre) (E.5) dp dp 0 0 0 Under a full production symmetric equilibrium we have i¯∗ =i¯ (p∗,p∗)= 1 , 0 0,1 2n and (cid:18) (cid:19) f(i¯ ) ∂ 0,1 p /∂p di¯∗(p = p∗,p = p∗) di¯ (p∗,p∗) f(1−i¯ ) 0 0 0 0 −0 = 0,1 =− n 0,1 (cid:18) (cid:19) dp 0 dp 0 ∂ f(i¯ 0,1 ) p /∂i¯ f(1−i¯ ) 0 0,1 n 0,1 1 =− (cid:18) (cid:19) f′(i¯ ) f′(1−i¯ ) 0,1 + n 0,1 p f(i¯ ) f(1−i¯ ) 0 0,1 n 0,1 f (cid:0) 1 (cid:1) =− 2n (E.6) 2f′ (cid:0) 1 (cid:1) p 2n 0 E.2 Solving for the optimal spot market price We can derive the following first-order condition with respect to p from the inter- 0 mediategoodsproducer j=0’soptimizationproblem(equation4.5): (cid:32) w (cid:18) Y (cid:19)1− α α(cid:33) (cid:18) dYspot(cid:19) p∗− =Yspot/ − (E.7) α K dp∗ where w(cid:0)Y(cid:1)1− α α is the marginal cost of production for intermediate goods; Y := α K Yspot +Ypre is the total amount of intermediate goods production; and dYspot is dp∗ the slope of the demand curve in the spot market. By imposing symmetry we get p∗ = p∗ = p∗ forall j∈J. 0 j Substituting equations E.5 and E.6 into equation E.7 gives the optimal spot- 51

marketpriceasrequired: 1 2 (cid:82) 2n f (i)(Q−Qpre)di (p∗−MC)= 0 (cid:20) (cid:21) −2 − f( 2 1 n ) f (cid:0) 1 (cid:1) (Q−Qpre) 2f′(1)p∗ 2n 2n   2f′(cid:0) 1 (cid:1)(cid:82) 2 1 n f (i)di ⇒ p∗ =1+ 2n 0 MC (cid:0) f (cid:0) 1 (cid:1)(cid:1)2 −2f′ (cid:0) 1 (cid:1)(cid:82) 2 1 n f (i)di 2n 2n 0 F Proof of proposition 4: Full Production Symmetric Equilibrium in the pre-order market F.1 Final goods producers in period 0 WewillstartbyverifyingthattheconjectureQ pre =Qpre,∀i∈I isindeedanequii libriumsolution. Lemma2. [OptimalPre-orders]Inafull-productionsymmetricequilibrium,each finalgoodsproduceriwill: 1. pre-orderfromtheintermediategoodsproducersthatsetsthelowesteffectivepricefori.  pre,∗ f Q if j∈J(i;φ), and f φ ≤v pre,∗ ij i ij j q = (F.1) ij 0 otherwise (cid:110) (cid:111) whereJ(i;φ):= j˜∈J : f φ =min{f ◦φ} denotethesetofsuppliersthat ij˜ j˜ i providesthelowesteffectivepricefori. pre,∗ 2. set the aggregate quantity of pre-orders Q such that the marginal cost of i pre-ordersisequaltoitsexpectedmarginalbenefit. (cid:104) (cid:105) f φ =Pr (cid:0) Q>Q pre,∗(cid:1) E f p∗(Q,w)|Q>Q pre,∗ , for j˜∈J(i;φ), jˆ∈J(i;p) ij˜ j˜ i ijˆ jˆ i (F.2) (cid:110) (cid:111) whereJ(i;p):= jˆ∈J : f p =min{f ◦p} ijˆ jˆ i 52

Furthermore,imposingsymmetryimplies φ =Pr (cid:0) Q>Q pre,∗(cid:1) E (cid:2) p∗(Q,w)|Q>Q pre,∗(cid:3) ∀i∈[0,1] (F.3) i i sothattheaggregatevolumeofpre-ordersmustbeequalizedacrossallfinalgoods firms: Q pre,∗ =Qpre,∗ ∀i∈[0,1] (F.4) i Equation F.2 is the first-order condition of final goods producer i’s period 0 optimization problem. It gives an implicit expression for the equilibrium aggregate volume of pre-orders Q pre,∗ as a function of spot and pre-order prices (p∗,φ). i On the left hand side of the equation, f φ is the effective marginal cost of preij˜ j˜ orders. Ontherighthandsideistheexpectedmarginalbenefitofpre-orders,which is equal to the probability that the spot market order of i will be strictly positive (cid:0) pre,∗(cid:1) Pr Q>Q , multiplied by the conditional expectation of the lowest effective i (cid:104) (cid:105) spotprice,giveni’sspot-marketorderisstrictlypositiveE f p∗(Q,w)|Q>Q pre,∗ . ijˆ i Under symmetry, p∗ = p∗ and φ = φ for all j ∈ J; so the nearest intermediate j j goodsnodetoiwillalwaysprovidethelowesteffectivepriceonboththepre-order and spot markets: f = f . Equation F.2 can thus be simplified to equation F.3, ij˜ ijˆ pre,∗ which we can also interpret as the demand function for pre-orders Q for given i pre-orderpriceφ. EquationF.3hastwoimmediateimplications: (1)aggregatepreorders must be equalized across i (equation F.4); and (2) the highest sustainable pre-orderpriceisφ =E[p∗],inwhichcasethefinalgoodsproducerswillonlypreorder to satisfy the minimal possible realization of demand Qpre,∗ = Q. For any pre-order price greater than the unconditional expectation of the spot market price, the aggregate quantity of pre-order will be zero. So we can view equation F.3 also asaparticipationconstraintforfinalgoodsfirmsonthepre-ordermarket. F.2 Intermediate goods producers in period 0 Recalltheexpectedprofitfunctionforintermediategoodsproducers: 53

maxE (cid:2) Π (cid:3) =E (cid:2) p∗Yspot−wL∗(cid:3) +φYpre−rK j φ,K (cid:90) Q¯ (cid:90) = (cid:0) p∗Yspot(cid:1) h(w)g(Q)dwdQ... Qpre w (cid:18)(cid:90) Qpre(cid:90) (cid:90) Q¯ (cid:90) (cid:19) − (wL∗)h(w)g(Q)dwdQ+ (wL∗)h(w)g(Q)dwdQ ... Q w Qpre w +φYpre−rK (F.5) We note that in a symmetric full production equilibrium, the aggregate produc- 1 tionofintermediategoodsY :=Ypre+Yspot =2Q (cid:82) 2n f (i)diisexogenouslypinned 0 down by the realization of final goods demand Q, and the distance-based penalty function f. But the relative importance of the spot market and the pre-order market (Ypre and Yspot) depends on the aggregate volume of pre-orders Qpre, which is determined by the choice of the pre-order price φ. On the other hand, the level of non-scalable capacity investment K affects the period 1 equilibrium spot market price p∗(Q,w) and scalable input demand L∗(Q,w) in each possible state. We examinetheoptimalityconditionsforφ andK inturn. First we take the derivative of expected profits with respect to φ. With some algebra,wecanshowthat (cid:104) (cid:16) (cid:17)(cid:105) pre dE Π j K j ,φ j ,q j (cid:18) dYpre(cid:19) (cid:20) ∂L∗ (cid:21) dYpre =φ − −Pr(Q≤Qpre)E w |Q≤Qpre ... dφ dφ ∂Y dφ (cid:18) dYpre(cid:19) + Ypre+φ (F.6) dφ (cid:20) ∂L∗ (cid:21) dYpre =−Pr(Q≤Qpre)E w |Q≤Qpre +Ypre ∂Y dφ (cid:20) ∂L∗ (cid:21)(cid:18) dYpre(cid:19) =Pr(Q≤Qpre)E w |Q≤Qpre − +Ypre >0 ∂Y dφ (F.7) This imply that the equilibrium must be a corner solution. Intermediate goods producerswouldliketosetthehighestpossiblepre-orderpricesubjecttotheparticipa- 54

tionconstraintoffinalgoodsproducers(eqnF.3). Thus,fromLemma2,equilibrium pre-orders will equal to the lowest possible realization of final goods demand, and theequilibrium pre-orderprice willequal theunconditionalexpectation ofthe spot marketprice: Qpre,∗ =Q (F.8) φ ∗ =E[p∗(Q,w)] (F.9) NextwetakethederivativeoftheexpectedprofitwithrespecttoK: (cid:20) ∂p∗ (cid:21) (cid:20) ∂L∗(cid:21) E Yspot −E w −r=0 (F.10) ∂K ∂K whereL∗ =(Ypre+Yspot)α 1 (K) −1− α α ,so ∂L∗ =− (cid:18) 1−α (cid:19) (cid:0) Ypre+Yspot(cid:1) α 1 K− α 1 ∂K α (cid:18) 1−α (cid:19) L∗ =− α K Substituting ∂L∗ backintothefirst-orderconditiontogive ∂K (cid:20) ∂p∗ (cid:21) E Yspot +(1−α)E[wL∗]=αrK∗ (F.11) ∂K asrequired. G Partial Production and Local Monopolies Inthisappendix,wediscusstheimplicationsofrelaxingthefullproductionassumption. Relaxing the assumption allows for shocks that are severe enough to shut out some market segments of final goods producers from the spot market. Final goods producers that are further away from intermediate goods suppliers (i.e., those with lesssubstitutableinputs)willexperiencegreaterdifficultyadjustingtotheshocks. 55

To see this, note that the period 1 equilibrium is governed by two indifference thresholds (which may or may not be binding). First, for given price vector (p ,p ), where p = p ∀j ̸=0, the participation threshold i¯ is defined as the 0 −0 j −0 0 final goods firm that is indifferent between buying inputs from intermediate goods firm j=0andnotproducingatall: (cid:34) (cid:35) v f (i¯ )p =v, ∀p ∈ ,v (G.1) 0 0 0 f (cid:0)1(cid:1) 2 Second, the competitive threshold i¯ is the marginal final goods producer that 0,1 isindifferentbetweenbuyingfromsuppliernode j=0and j=1: f (d(i¯ ,0))p = f (d(i¯ ,1))p (G.2) 0,1 0 0,1 −0 As Figure G.1 illustrates, the participation threshold i¯ (and its counterparts for 0 j ̸= 0) can be visualized as the arms that reaches out from each supplier node. Theparticipationthresholdthereforerepresentsthepotentialmarketreachforeach intermediate goods supplier. As long as the market reach from two nearby supplier nodes overlap, the two suppliers engage in competition and the competitive threshold i¯ is the binding threshold for computing demand. Under this compet- 0,1 itive regime, the intermediate goods suppliers’ market reach covers every market segment on the circle. The aggregate demand for final goods is met and we see “full shelves”. The competitive regime always prevails under the full production assumption. We can show further that the market reach of each intermediate goods supplier is increasing in the level of non-scalable capacity installed (K), and decreasing in the cost of the scalable input (w). For given level of non-scalable capacity K, the marketreachofeachsuppliernodegetsshorterasthesizeofthenegativecostshock increases, until eventually the participation thresholds i¯ and i¯ no longer overlap 0 1 and the two neighboring suppliers (j =0,1) behave like local monopolies. Under this local monopolies regime, there is a gap in market coverage between the two suppliernodes,andwesee“emptyshelves”insomesegmentsofthemarket.30 30Athirdpossibleregimeariseswhenthemarketreachofoneintermediategoodssuppliergoes 56

FigureG.1: Regimeswitching: competitionvslocalmonopolies The optimal pricing strategy of intermediate goods suppliers therefore depend onwhethertheyareoperatingunderthecompetitiveorthelocalmonopoliesregime, which in turn depends on the realization of demand and supply shocks in period 1. We formally characterize the symmetric equilibrium spot-market pricing strategy under the assumption that the distance-based penalty function f (x) takes the form ofanexponentialfunction,withparameterβ. AssumptionA3 Exponential distance-based penalty function: f (d) = exp(βd), where β ∈(0,1] governs the degree of substitutability between different intermediategoods. Proposition6. [Optimalspot-marketpricingundersymmetricequilibrium] 1. Under the competitive regime, we have i¯ = 1 ≤ i¯ and the equilibrium 0,1 2n 0 pricefortheintermediategoodsisgivenby: (cid:16) (cid:17) β exp 2n p∗ =MC · (G.3) c c (cid:16) (cid:17) 2−exp β 2n past the node of another. This is the “super-competitive” regime, whereby one supplier prices so aggressivelyastocapturethehomemarketoftheirneighboringcompetitor. Allowingforthispossibilitywouldleadtoadiscontinuousjumpinthedemandfunctionforintermediategoods. Inthe interestoftractability,wecanruleoutthepossibilityofasuper-competitiveregimebymakingthe distance-basedpenaltyfunction f(·)sufficientlypunishing. 57

whereMC isthemarginalcostfacedbyintermediategoodssuppliers c w (cid:18) Ypre+Y spot(cid:19)1− α α c MC = (G.4) c α K 2. Under the local monopolies regime, we have i¯ <i¯ = 1 and the equilib- 0 0,1 2n riumpricefortheintermediategoodsisgivenby: (cid:112) p∗ = v·MC (G.5) m m whereMC isthemarginalcostfacedbyintermediategoodssuppliers m w (cid:18) Ypre+Y spot(cid:19)1− α α m MC = (G.6) m α K Intuitively, the first part of Proposition 6 shows that under a competitive regime, intermediate goods suppliers charge a mark-up over marginal costs.31 The markup is higher when substitutability is lower (i.e., when β, the parameter governing the distance-based penalty function, is closer to 1), and lower when competition is fiercer(i.e.,whennislarge). Inthelimit,asnapproachesinfinity-andthedistance between nodes shrinks to zero such that intermediate goods become perfectly substitutable - equation G.3 simplifies down to the familiar condition of price equals marginalcost. ThesecondpartofProposition6showsthatwhenintermediategoodssuppliers operateaslocalmonopolies,thepricetheychargeisequaltothegeometricaverage betweentheirmarginalcosts(MC )andthehighestpossibleprice(v,thevaluation m of the final goods output by end consumers). Unsurprisingly, whilst intermediate goods suppliers operates as local monopolies, the number of other firms n is irrelevant to their pricing decision. Any changes in n instead influences whether the economyswitchesbetweenthelocalmonopoliesregimeandthecompetitiveregime (i.e. whetheri¯ islessorgreaterthani¯ = 1 ).32 0 0,1 2n 31Thispartofthepropositionisjustare-writingofourearlierresultsforthisspecificparameterization. 32Clearly,thisneatcharacterizationofthemonopolypriceasageometricaveragewon’tholdin 58

Other factors that influence the market pricing regime that prevails in equilibrium include the level of non-scalable production capacity in place K, and the cost ofthescalableinputw. Proposition7. [Regimeswitching]i¯ ,theparticipationthreshold(i.emarketreach) 0 offirm j=0,isincreasinginK anddecreasinginw: di¯ 0 >0 (G.7) dK di¯ 0 <0 (G.8) dw Proposition 7 formalizes our earlier discussion that, for given non-scalable capacity K, larger negative supply shocks (larger w) increases the likelihood that the economy will end up in the local monopolies regime. Under the local monopolies regime,themarketsegment(i∈(i¯ ,i¯ ))thatliesin-betweenthemarket-reachofthe 0 1 two nearby supplier nodes will not be able to fulfill their realized demand for final goods, and we observe “empty shelves”. Intuitively, the proposition holds because ahigherK,andalowerw,reducesthemarginalcostofproduction,whichincreases themarketreachoftheintermediategoodssupplier.33 A key implication of Proposition 7 is that the response of final goods outputs to shocks is non-linear. Under normal or benign market conditions, the economy might be operating under the competitive regime which ensures that demand from every market segment is met. Market reach of neighboring suppliers overlap, and continues to overlap for small perturbations in supply and demand. Under these benign conditions, the supply network appears robust. But when negative supply shocks becomes sufficiently large, the economy suddenly switches from the competitive regime to the local monopolies regime. The critical role capacity plays, therefore, is that it prevents empty shelves for a larger range of shocks. A larger general(e.g.withouttheexponentialfunctionalformfor f(d)).Buttheotherpartoftheproposition, thatinthelocalmonopoliesregimethenumberofotherfirmsisirrelevant,ismoregeneral. Evenif otherfirmsexist,theysimplyaren’tsellingineachother’s“submarket”. 33Notethatthisanalysisisnotcomparativestaticsinthestrictsense: wisanexogenousvariable, but K is an endogenous variable. With regard to the latter, we are asking how firms’ endogenous choiceofcapacityinvestmentinperiod0affectsmarketreachandthenatureofcompetitiononthe spotmarketinperiod1. 59

K allows for a larger market-reach overlap for any given input cost w, making the entirenetworkmorerobust. Butsincethedegreeofoverlapisinofitselfirrelevant, surpluscapacityis“wasted”intheabsenceoflargenegativesupplyshocks. Relaxing the full production assumption therefore reinforces our central message that K∗ <KSP. This is intuitive, because the possibility of a large shock shiftingtheeconomytoalocalmonopoliesregimeaddsanotherdistortiontothesystem. ExantecapacityinvestmentK increasesnetworkresiliencebyensuringfullproduction for a wider range of shocks, but is undervalued by market participants under business-as-usual scenarios. Robustness becomes an externality that may not be fully internalized by individual intermediate goods suppliers in their capacity decisions in period 0. Worse still, in imperfectly competitive economies, some firms mayprofitfromtheartificialscarcitythatarisesfromalackofresilience. 60