Cascading Failures in Production Networks

60 Pages·2015·0.7 MB·English

∗
Cascading Failures in Production Networks
†
DavidRezzaBaqaee
August24,2015
Abstract
I show how the extensive margin of firm entry and exit can greatly amplify idiosyncratic
shocksinaneconomywithaproductionnetwork. Ishowthatinput-outputmodelswithentry
andexitbehaveverydifferentlytomodelswithoutthismargin. Inparticular,insuchmodels,
salesprovideaverypoormeasureofthesystemicimportanceoffirmsorindustries. Iderivea
newnotionofsystemicinfluencecalledexitcentralitythatcaptureshowexitsinoneindustry
will affect equilibrium output. I show that exit centrality need not be monotonically related
to an industry’s sales, size, or prices. Unlike the relevant notions of centrality in standard
input-outputmodels,exitcentralitydependsontheindustry’sroleasbothasupplierandasa
consumerofinputs. Furthermore,Ishowthatgranularitiesinsystemicallyimportantindustries
cancauseonefailuretosnowballintoalarge-scaleavalancheoffailures. Inthissense,shocks
canbeamplifiedastheytravelthroughthenetwork,whereasinstandardinput-outputmodels
theycannot.
Keywords: Propagation,networks,macroeconomics,entryandexit
∗FirstVersion: September2013. IthankEmmanuelFarhi, MarcMelitz, AlpSimsek, andJohnCampbellfortheir
guidance. IthankDaronAcemoglu,PolAntras,NatalieBau,AubreyClark,BenGolub,GitaGopinath,AlirezaTahbaz-
Salehi,ThomasSteinke,OrenZiv,andDavidYang,aswellasmanyseminarparticipants,forhelpfulcomments. Ithank
LudvigSinanderforhisdetailedcommentsonanearlierdraft.
†LondonSchoolofEconomicsandPoliticalScience,[email protected].
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1 Introduction
Inthispaper,Iinvestigatehowtheentryandexitoffirmscanacttoamplifyandpropagateshocks
in production networks. I model cascades of failures among firms linked through a production
network,andshowwhichfirmsandindustriesaresystemicallyimportantforaggregateoutcomes.
Ishowthattheextensivemarginofentryisapowerfulpropagationmechanismofshocksthrough
productionnetworks.
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Theideathatkeyconsumersorkeysupplierscanhaveoutsizede ectsonaneconomyifthey
enter or exit has long been popular with policy makers and the public. Policies as diverse as
agricultural subsidies, industrial policy, and government bail-outs frequently have been justified
on the grounds of systemic importance. In academic economics, such ideas have been around
since at least the time of Hirschman (1958) and Rosenstein-Rodan (1943). However, despite their
popularitywithpolicymakers,theseideashavenotbeenasinfluentialinmacroeconomics. Results
like those of Hulten (1978) suggest that the systemic importance of firms and industries can be
approximatedbytheirsales,eveninthepresenceoflinkagesbetweenfirms,andthereforeappeals
tosystemicimportanceareatbestmisguidedandatworstdishonest.
However, in recent times, and motivated by current events, these ideas have experienced a
resurgence in macroeconomics. After the failure of Lehman Brothers in the United States, the
existence of systemically important financial institutions, or SIFIs, became well-accepted in the
financeandmacro-financeliteratures.
Furthermore,recentworkhasalsohighlightedthatsystemicimportanceneednotbelimitedto
financialrelationships. AmemorableexampleofthisiscaseoftheUSautomobilemanufacturing
industry, as discussed by Acemoglu et al. (2012). In the fall of 2008, the president of Ford Motor
Co., Alan Mulally, requested government support for General Motors and Chrysler, but not for
Ford. ThereasonMullalywantedgovernmentsupportforhiscompany’srivalswasthatthefailure
ofGMandChryslerwerepredictedtoresultinthefailureofmanyoftheirsuppliers. SinceFord
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relied on many of those same suppliers, Mulally feared that the knock-on e ect on its suppliers
could put Ford itself out of business. Mulally’s actions go against the standard expectation that
Fordwouldbenefitfromthefailureofitsfiercestrivals,andthesalesofGMdidnottellusthefull
storyaboutthepotentialconsequencesofGM’sfailure.
While a government bailout of GM and Chrysler prevented a tide of failures in the United
States,1 a scenario similar to the one Mulally described did play out in Australia. In May 2013,
Ford Australia announced it they would stop manufacturing cars in 2017. Seven months later,
GMAustraliaannounceditwouldalsostopmanufacturingcarsin2017. Threemonthsafterthat,
in February 2014, Toyota Australia also announced that it would close its manufacturing plants
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at the same time. This e ectively ended automobile manufacturing in Australia. The Australian
government predicted that this would result in the loss of over 30,000 jobs, a figure they arrived
1SeeGoolsbeeandKrueger(2015).
2
atbyaddingthenumberofpeopledirectlyemployedbythethreeautomakersandtheAustralian
carpartsindustry.
EmpiricalworkbyCarvalhoetal.(2014)providesfurthersupportfortheideathatfirmentry
andexithaveimportantspill-oversonotherfirmsthroughsupplyanddemandchains. Carvalho
etal.(2014)instrumentforfirmexitsusingthe2011Tohokuearthquake,andthenshowthatfirm
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exits have important negative spill-over e ects on both upstream and downstream firms, even
whenthesefirmsarenotdirectlyconnectedtotheexitingfirm. Theyshowthatexitsupstreamor
downstreamfromafirmreducethefirm’sprofitsandmakethefirmmorelikelytoexit.
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Despitetheimportanceofthesespill-overe ects,standardmacroeconomicmodels,evenones
with production networks, do not allow for such forces. In this paper, I explicitly incorporate
the extensive margin of firm entry into an input-output model of production, and show that the
extensivemarginaltersthequantitativeandqualitativepropertiesofthemodel.
This paper contributes to the growing literature on the microeconomic origins of aggregate
fluctuations. Two recent strands of this literature are the granular origins hypothesis of Gabaix
(2011)andthenetworkoriginshypothesisofAcemogluetal.(2012). Theformerpositsthatshocks
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to large, or granular, firms can have non-trivial e ects on real GDP. The latter posits that shocks
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to highly-connected firms can have non-trivial e ects on real GDP. In practice, both theories
emphasizeafirmorindustry’ssize,asmeasuredbyitsshareofsales,asbeingtherelevantmeasure
of that player’s importance. The intuition here is that if a firm is systemically important, then in
equilibrium, it will have a large fraction of total sales. However, as discussed above, we might
expectinterconnectednesstomatterinwaysthatarenotcapturedbypricesortotalquantities. In
thispaper,Ishowthatallowingforentryandexitmeansthatthenetworkstructureofproduction
can amplify shocks and determine systemic importance in ways that are not captured by size.
Furthermore, by combining granularity, networks, and the extensive margin, we can produce
dramaticdominochainreactionsthatwouldnotbepresentintheabsenceofanyoneofthesethree
ingredients.
Toplacemyanalysisinthebroaderliterature,Ifirstshowthatstandardinput-outputmacroe-
conomic models that follow Long and Plosser (1983), like Acemoglu et al. (2012), Atalay (2013),
orBaqaee(2015)havethreecruciallimitations. First, Ishowthatthesemodelshavetheproperty
that their responses to idiosyncratic productivity shocks can be summarized in terms of exoge-
ffi ffi
nous su cient statistics. Once we compute the relevant su cient statistics, we can discard the
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network structure. So, for example, I show that there are disconnected economies, with di erent
householdstastes,thatbehavepreciselylikethenetworkmodels. Thisresultissimilarinspiritto
theirrelevanceresultofDupor(1999).
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Second,Ishowthattherelevantsu cientstatisticsarecloselyrelatedtotheequilibriumsizeof
firms. Thismeansthat,withouttheextensivemargin,itisnottheinterconnectionsperse,buthow
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thoseinterconnectionsa ectafirm’ssizethatdeterminesafirm’ssystemicimportance. Ifwecan
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ffi ff
arrive at the same su cient statistics using a di erent (perhaps degenerate) network-structure,
the equilibrium responses will be the same. This fact explains why the theoretical implications
of the granular hypothesis of Gabaix (2011), where business cycles are driven by large firms, are
observationallyequivalenttothetheoreticalimplicationsofthenetworkhypothesisofAcemoglu
etal.(2012),wherebusinesscyclesaredrivenbywell-connectedfirms.
Third, I show that in canonical input-output models, systemic importance depends only on a
firm’sroleasasupplier. Inotherwords,aslongasfirmsiand jhavethesamestrengthconnections
to the same customers, then their systemic influence will be the same, regardless of what i and
j’s own supply chains look like. This contrasts with our starting anecdote about the American
automobileindustrybecauseitwasnotGM’sroleasasupplierofcars,butitsroleasaconsumerof
carparts, thatmadeGMsystemicallyimportant. Canonicalinput-outputmodelscannotbeused
tounderstandepisodeswhereaproductivityshocktravelsfromconsumersupstreamtosuppliers
andthentravelsbackdownstreamtootherconsumers.
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Whenweallowforentryandexit,allthreeoftheselimitationsdisappear. First,thesu cient-
statisticapproachbreaksdown. Thisisbecausetheextensivemarginmakes“systemicimportance”
anendogenouslydeterminedvaluethatisnotwell-approximatedbyequilibriumsizeorprices. A
firmthatmayseemlikeasmallplayer,whenmeasuredbysales,canhavepotentiallylargeimpacts
on aggregate outcomes. On the other hand, a firm that may seem like a key player, as measured
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by sales, can have relatively minor e ects on the equilibrium. Furthermore, the endogenously-
determined measures of systemic influence depend on a firm’s role as both a supplier and as
a consumer, as well as on how many close substitutes there are for the firm. The fact that
systemicimportanceisunrelatedtosalescontrastssharplywiththeneoclassicalworldofHulten’s
theorem. Hulten (1978) shows that with arbitrary constant-returns production functions and
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arbitrarynetworkstructures,thee ectonoutputofamarginalTFPshocktoanindustrydepends
onlyonthatindustry’ssales. Thisresultisnolongertruewhenwehaveameaningfulentry-exit
margin.
ThemodelpresentedinthispaperalsoallowsustocombinethegranularhypothesisofGabaix
(2011) and the network hypothesis of Acemoglu et al. (2012) in a new and interesting way. In
particular, when firms are not infinitesimal, extensive margin shocks can be locally amplified via
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interconnections. In other words, when a large and well-connected firm exits, it can set o an
avalancheoffirmfailuresthatactuallygetslargerasitgatherssteam. Thistypeofamplificationis
impossibleincanonicalmodels,whereshockscanonlydecayastheytravelthroughthenetwork.
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Thispaperalsocontributestothewiderliteratureondi usioninnetworktheory,bybridging
the gap between two alternative modelling traditions. Loosely speaking, there are two popular
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approaches to modelling di usion on social networks. First, there are continuous models of
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di usion like Katz (1953). Here, nodes influence each other in continuous ways – shocks travel
awayfromtheirsourcelikewavesandslowlydieout. Thestrengthoftheconnectionsbetweenthe
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nodescontrolstherateofdecay. Suchshocks,sometimescalledpulseprocesses,arecharacterized
bygeometricsums. Itiswell-knownthatsuchmodelsareincapableoflocalamplification: ashock
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to one node will always have its largest e ect at its source, and the shock will decay as it travels
throughtheconnections.2 ThesestructureswerefirststudiedbyLeontief(1936)inhisinput-output
modeloftheeconomy.
Theothertraditionconsistsofmodelsthatbehavediscontinuously,astypifiedbyMorris(2000)
orElliottetal.(2012). Inthisclassofmodels,sometimescalledthresholdmodels,eachnodehasa
thresholdandiseitheractiveorinactive. Whenanodecrossesitsthresholditchangesstatesand,
by changing states, pushes its neighbors closer to their thresholds. Such models are frequently
used to study the spread of epidemics, products, or even ideas. One of the earliest and most
influential threshold models is the Schelling (1971) model of segregation. Threshold models do
not have wave-like properties since the rate at which shocks decay is not geometric. Crucially,
these models are capable of generating local amplification – that is, shocks can be amplified as
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theytravelthroughthenetwork. Unfortunately,thesemodelsarenotoriouslydi culttoanalyze.
In this paper, I consider a model that bridges the gap between the continuous and discrete
models. Specifically,Iexplicitlyaccountforthemassoffirmsinagivenindustry. Industrieswith
a continuum of firms behave continuously – the mass of firms responds continuously to shocks.
Ontheotherhand,lumpyindustries,withonlyafewfirms,behavediscontinuously. Forinstance,
anegativeshocktoanunconcentratedindustry,saybakeries,willresultinsomefractionofbakers
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exiting. The fraction exiting will be a continuous function of the size of the shock. The e ect
on a neighboring upstream industry, like flour mills, or downstream industry, like cafes, will be
attenuated by the strength of its connections to bakeries. However, a negative shock to a highly
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concentratedindustry,likeautomobilemanufacturing,willhavenoe ectonthenumberoffirms
unless it is large enough to force an exit. But once a large firm exits, it imparts an additional
impulsetothesizeoftheshock,whichcantriggeracascade. Becausethemodelisflexibleenough
to express both behaviors, I can provide conditions under which we would expect a continuous
approximationtoadiscontinuousmodeltoperformwell.
The idea of cascading – domino-like – chain reactions also appears outside of economics. In
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particular,modelsofcontagionanddi usionlikethethresholdmodelsconsideredbyKempeetal.
(2003) have these features. In these models, notions of connectedness play a key role since the
only way contagion can spread is via connections between nodes. An interesting implication of
embeddingacontagionmodelintoageneralequilibriumeconomyistherolepricesandaggregate
demand play – a role that does not have analogues in other threshold models. Typically, in a
threshold model, shocks can only travel along edges. Contagion can only spread to nodes who
are connected to an infected node. This important intuition breaks down in general equilibrium
models since all firms are linked together via aggregate supply and demand. This means that
2See,forexample,page253inBermanandPlemmons(1979)andalsoappendixD.
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general equilibrium forces can act like long-distance carriers of disease. Shocks in one fragile
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industry, like the financial industry, can jump via aggregate demand or supply, to a di erent
fragileindustrylikeautomobilemanufacturing,evenifthesetwoindustriesarenotconnected.
This paper is belongs to the literature seeking to derive aggregate fluctuations by combining
microeconomic shocks with local interactions. Empirical work on this front includes Foerster
et al. (2011), Di Giovanni et al. (2014), Barrot and Sauvagnat (2015), Carvalho et al. (2014), and
Acemoglu et al. (2015). Theoretical work can be roughly divided into two categories. The first
categoryarepapersthatbuildonthemulti-sectorneoclassicalmodelofLongandPlosser(1983),
like Horvath (2000), Acemoglu et al. (2012), Atalay (2013), and Jones (2011). These are log-linear
models of propagation, and follow the tradition of Leontief (1936) in having linear-geometric
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di usion. As shown by Hulten (1978), the propagation of TFP shocks in such models is, at
least locally, tied to the distribution of firm sizes. This means that in order for a firm to be
systemically important, it must have high sales. The second category are models that do not
satisfytherequirementsofHulten’stheorem. Thesemodels,likeJovanovic(1987),Durlauf(1993),
andScheinkmanandWoodford(1994),featurestrongnonlinearities,theirsalesdistributionisnot
ffi ff
a decisive su cient statistic, and they do not feature the kind of geometric patterns of di usion
found in input-output models. However, these models depart dramatically from the standard
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assumptions of macroeconomics and can be di cult to work with. This paper has elements in
commonwithbothoftheseliteratures. Theindividualingredientsinmymodelareveryfamiliarto
macroeconomists–monopolisticcompetition,free-entry,andinput-outputlinkagesviaconstant-
elasticitity-of-substitution production functions. However, the presence of all three ingredients
togetherresultsinstronglynonlinearinteractionsandanuncouplingofsystemicimportancefrom
thedistributionofsales.
Thestructureofpaperisasfollows. Insection2,Isetupthemodelanddefineitsequilibrium.
In section 3, I characterize the equilibrium conditional on the mass of entrants in each industry
and define some key centrality measures. In section 4, I study how the model behaves when the
extensive margin of firm entry and exit is shut down. I prove results showing that the network
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structure can be summarized by su cient statistics related to size. I also show that we can think
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of these models as non-interconnected models with di erent parameters. In section 5, I allow
for firm entry and exit. First, I characterize the model’s responses to shocks in the limit where
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all firms are massless. I show that su cient statistics are no longer available, and I derive an
endogenous measure of influence called exit centrality. Then, I consider the case when firms can
have positive mass, and show that with atomistic firms, shocks can be locally amplified. I prove
an approximability result showing conditions under which we can approximate an intractable
discontinuous model using a tractable continuous model. Motivated by Ford’s request for a
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bailout of GM, I also consider whether e cient bail-outs can be identified by surveying firms.
Specifically,whencanwetrustonefirm’stestimonyaboutwhetherornotanotherfirmshouldbe
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bailedout. Iconcludeinsection6.
2 Model
In this section, I spell out the structure of the model and define the equilibrium. There are three
types of agents: households, firms, and a government. Each firm belongs to an industry, and
thereareNindustries. Therearetwoperiods: entrydecisionshappensimultaneouslyinperiod1,
productionandconsumptiontakeplaceinperiod2.
Household’sProblem
The households in the model are homogenous with a unit mass. The representative household
maximizesutility
  σ
U(c1,...,cN) = (cid:88)N βkσ1ckσ−σ1σ−1 ,
k=1
wherec representscompositeconsumptionofvarietiesfromindustrykandσ > 0istheelasticity
k
β ≥
of substitution across industries. The weights 0 determine household tastes for goods and
k
services from the different industries. The composite consumption good produced by industry k
isgivenby
ε
  k
ck = M−kϕk(cid:88)Nk ∆kc(k,i)εkε−k1εk−1 ,
i=1
where c(k,i) is household consumption from firm i in industry k and ε > 1 is the elasticity of
k
substitution across firms within industry k. Here, N is the number of firms active in industry k
k
and ∆ is the exogenous weight of each firm.3 The total mass of varieties in industry k is given
k
−ϕ
by M = N ∆ . Finally, the term M k controls the strength of the love-of-variety effect. If we let
k k k k
ϕ = /ε ϕ =
1 , then there is no love-of-varieties, and if we let 0, we get the standard Dixit and
k k k
Stiglitz(1977)demandsystem.4 Thehousehold’sbudgetisgivenby
(cid:88) (cid:88)
p(k,i)c(k,i)∆ = wl+ π(k,i)∆ −τ,
k k
k,i k,i
wherep(k,i)isthepriceoffirmiinindustrykandπ(k,i)isfirmiinindustryk’sprofits. Thewage
iswandlaborisinelasticallysuppliedatl. Fortherestofthepaper,andwithoutlossofgenerality,
3Ifwetakethelimit∆ →0,thissumconvergestoaRiemannintegral.
k
4See section 5.1 for more discussion about the significance of the love-of-varieties effect in this model. Love of
varietiescanbethoughtofasareducedformforamorecomplexmodelwhereincreasingproductvarietyimproves
thematchbetweenproductsandconsumers. Andersonetal.(1992)provideamicrofoundationforthisviaadiscrete
choicemodelwhereeachconsumerconsumesonlyasinglevariety,butthereexistsaCESrepresentativeconsumerfor
thepopulationofconsumers.
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wetakelabortobethenumerairesothatw = 1,andfixthesupplyoflaborl = 1. Lumpsumtaxes
τ
bythegovernmentaredenoted .
Firms’Problem
Letfirmiinindustrykhaveprofits
(cid:88)N (cid:88)Nl
π(k,i) = p(k,i)y(k,i)− p(l,j)x(k,i,l,j)∆ −wh(k,i)−wf +τ ,
l k k
l=1 j
where p(k,i) is the price and y(k,i) is the output of the firm. Inputs from firm j in industry l are
x(k,i,l,j)andlaborinputsareh(k,i). Finally,inordertooperate,eachfirmmustpayafixedcost f
k
τ
in units of labor and the firm potentially receives a lump sum subsidy . The firm’s production
k
function(oncethefixedcosthasbeenpaid)isconstant-returns-to-scale
  σ
y(k,i) = αkσ1(zkl(k,i))σ−σ1 +(cid:88)N ωkσ1,lx(k,i,l)σ−σ1σ−1 .
l=1
σ > ω
Here, 0isagaintheelasticityofsubstitutionamonginputs,and istheshareparameterfor
kl
howintensivelyfirmsinindustrykusecompositeinputsfromindustryl. TheN×Nmatrixofω ,
kl
Ω
denotedby ,determinesthenetwork-structureofthiseconomy. Onecanthinkofthismatrixas
α >
theadjacencymatrixofaweighteddirectedgraph. Theparameter 0givestheintensitywith
k
whichfirmsinindustrykuselabor(netoffixedcosts).
Laborproductivityshocks,liketheonesconsideredbyAcemogluetal.(2012)andAtalay(2013)
are denoted by z . Note that when σ (cid:44) 1, a productivity shock z to industry k is equivalent to
k k
changing that industry’s labor intensity from α to α zσ−1. Therefore, as long as σ (cid:44) 1, we can
k k k
α
think of as including both the productivity shock and the labor intensity. This way we do not
k
need to directly make reference to the shocks z since they are just equivalent to changing α. This
equivalence breaks down when σ = 1, and in those cases, we shall have to work directly with z.
Forthemajorityofthispaper, Ifocusonthepropagationoftheseproductivityshocks. Thesame
methodscaneasilybeusedtostudyothershockslikefixed-costshocksordemandshocks.
Thecompositeintermediateinputfromindustrylusedbyfirmiinindustrykis
  εl
x(k,i,l) = Ml−ϕl(cid:88)Nl ∆lx(k,i,l,j)εlε−l1εl−1 ,
j=1
whereε istheelasticityofsubstitutionacrossdifferentfirmswithinindustryl. Asbefore,theterm
l
−ϕ
M l affects the strength of the love-of-varieties effect in industry l. Letting ϕ = 0 gives us the
l l
traditionalDixit-Stiglitzformulation. Notethattheelasticitiesofsubstitutionarethesameforall
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usersofanindustry’soutput.
Government’sProblem
Thegovernmentrunsabalancedbudgetsothat
(cid:88)
τ M = τ.
k k
k
For now, we assume that the government is completely passive. In section 5.5, we return to how
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thegovernmentcanvaryitstaxesandsubsidiestoa ecttheequilibrium.
Notation
Lete denotetheithstandardbasisvector. LetΩbetheN×Nmatrixwhoseijthelementisequalto
i
ω . LetαandβbetheN×1vectorsconsistingofαsandβs. LetMbetheN×Ndiagonalmatrix
ij i i
whose ith element is M, and let M˜ be the N ×N diagonal matrix whose ith diagonal element is
i
1−ϕiεi
equaltoM εi−1 . Let◦denotetheelement-wiseorHadamardproduct,anddiag : RN → RN2 bethe
i
operatorthatmapsavectortoadiagonalmatrix. Tosimplifythenotation,whenavariableappears
withoutasubscript,thisrepresentsthematrixofallsuchvariables,andtheobjectvs,wherevisa
vectorandsisascalar,shouldbeinterpretedasthevectorvraisedtothepowerofselementwise.
Definition 2.1. An economy E is defined by the tuple E = (β,Ω,α,ε,σ, f,∆,ϕ). The vector β
Ω α
contains household taste parameters, captures the input-output share parameters, contains
ε >
the industrial labor share parameters, 1 is the vector of industrial elasticities of substitution,
σ > 0isthecross-industryelasticityofsubstitution, f isthevectoroffixedcosts,∆isthevectorof
ϕ ff
massesoffirmsineachindustry,and istheparametercontrollingthelove-of-varietye ect.
Equilibrium
WestudythesubgameperfectNashequilibrium. Inperiod1,potentialentrantsmakesimultaneous
entry decisions. In period 2, firms play general equilibrium conditional on period 1’s entry
decisions. Define a markup function µ : RN ×RN → RN to be a function that maps number of
entrants{N } andproductivityshocksz tomarkupsforeachindustry. Nowwecandefinegeneral
k k k
equilibrium as follows. I assume that firms set their prices to equal the markup function times
theirmarginalcost.
Definition 2.2. A general equilibrium of economy E is a collection of prices p(i,k), wage w, and
input demands x(i,k,l,j), outputs y(i,k), consumptions c(i,k) and labor demands l(i,k) such that
fornumberofentrants{N }N ,vectorofproductivityshocksz,andmarkupfunctionµ(N,z):
k k=1
9
(i) Eachfirmminimizesitscostssubjecttodemandforitsgoodsgiventhemarkupfunction,
(ii) therepresentativehouseholdchoosesconsumptiontomaximizeutility,
(iii) thegovernmentrunsabalancedbudget,
(iv) marketsforeachgoodandlaborclear.
Notethatpricesandquantitiesinageneralequilibriumdependonthenumberofentrants{N },
k
µ
thevectorofproductivityshocks,andtheassumedindustry-levelmarkups . Conditions(ii)–(iv)
are standard, but condition (i) merits further discussion. Condition (i) is deliberately vague. It
simply requires that the firms minimize their costs, given the demand they face and leaves their
µ
choice over prices unspecified – it simply requires that there be some function that determines
theirmarkupsasafunctionofthenumberofentrants{N }andtheproductivitiesz.
k
µ
The exact nature of the markup function , which will depend on what we assume about
competition in each industry, will play an important role in our analysis. However, for now, we
µ
canleavethemarketstructuregeneral. Notethatbychoosing correctly,wecanrepresentmany
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di erentpopularmodelsofcompetitionincludingCournot,Dixit-Stiglitz,andperfectcompetition.
Let Π : RN+ × RN+ → RN be the function mapping the masses of entrants M and vector of
productivity shocks z to industrial profits assuming general equilibrium in period 2. In theorem
3.5,Ianalyticallycharacterizethisfunction.
Definition2.3. Avectorofintegers{N }N isanequilibriumnumberofentrantsif
k k=1
Πi(Mi,M−i,z) ≥ 0 > Πi(Mi+∆i,M−i,z) (i ∈ {1,2,...,N}),
whereM = N ∆ ,forall j.
j j j
Intuitively, a vector of integers is an equilibrium number of entrants if all firms make non-
negativeprofits,andtheentryofanadditionalfirminanyindustryresultsinfirmsinthatindustry
∆ →
making negative profits. Note that if we let 0, the entry inequalities become a zero-profit
i
equalitycondition.
Beforeanalyzingthemodel,ithelpstodefinesomekeystatistics. Thesearestandarddefinitions
fromtheliteratureonmonopolisticcompetition. See,forexample,BettendorfandHeijdra(2003).
Definition2.4. Thepriceindexforindustrykisgivenby
  1
pk = M−kϕkεk(cid:88)Nk ∆kp(k,i)1−εk1−εk ,
i
10