【欧洲央行】能源危机期间的财政和宏观审慎政策-2025

EUROPEANCENTRALBANK

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WorkingPaperSeries

RomanosPriftis,RaphaelSchoenleFiscalandmacroprudentialpoliciesduringanenergycrisis

No3032

Disclaimer:ThispapershouldnotbereportedasrepresentingtheviewsoftheEuropeanCentralBank(ECB).TheviewsexpressedarethoseoftheauthorsanddonotnecessarilyreflectthoseoftheECB.

Abstract

WeconstructaNew-KeynesianE-DSGEmodelwithenergydisaggregationandfi-nancialintermediariestoshowhowenergy-relatedfiscalandmacroprudentialpoliciesinteractinaffectingtheeuroareamacroeconomyandcarbonemissions.Whenashocktothepriceoffossilresourcespropagatesthroughtheenergyandbankingsector,itleadstoasurgeininflationwhileloweringoutputandcarbonemissions,absentpolicyinterventions.Bycontrast,imposingenergyproductionsubsidiesreducesbothCPIandcoreinflationandincreasesaggregateoutput,whileenergyconsumptionsubsidiesonlylowerCPIinflationandreduceaggregateoutput.Carbonsubsidiesinsteadpro-duceanintermediateeffect.Giventhatbothenergysubsidiesraisecarbonemissionsanddelaythe“greentransition,”accompanyingthemwithparallelmacroprudentialpolicythattaxesdirtyenergyassetsinbankportfoliospromotes“green”investmentwhileenablingenergysubsidiestoeffectivelymitigatetheadverseeffectsofsupply-typeshocks,witnessedinrecentyearsintheEA.

Keywords:DSGEmodel,energysector,energysubsidies,financialfrictions,macropru-dentialpolicy.

JELclassification:E52,E62,H23,Q43,Q58.

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Non-technicalsummary

Thespikeinglobalfossilfuelpricesthatbeganinearly2022,drivenlargelybyRussia’sinvasionofUkraine,combinedwithongoingpandemic-relateddisruptionssuchasrisingfoodpricesandsupplychainbottlenecks,resultedinsignificantinflationarypressuresacrossEurope.InmanyEuropeancountries,inflationsurgedbyover10%,placingaconsiderableburdenonbothhouseholdsandfirms.Policymakersrespondedwithavarietyoffiscalandmonetarymeasurestoalleviatetheimpactofhigherenergycosts,aimingtosoftentheblowofrisingprices.However,theseinterventionsposeapotentialconflictwithclimategoals,particularlythosetiedtotheEuropeanCommission’sGreenDeal,whichaimstoreducecarbonemissionsandtransitiontowardsagreenereconomy.

Thispaperdevelopsadynamicstochasticgeneralequilibrium(DSGE)model,calibratedtotheEuroarea(EA),toprovideinsightsintothemacroeconomicimpactsofanincreaseinfossilfuelpricesandtheensuingpolicyresponses.Themodelincludesadetailedbreakdownoftheenergysectorandaccountsfortheroleoffinancialintermediaries,offeringaframeworktoevaluatepolicychoicesinthecontextofenergypriceshocks.TheanalysisfirstexploresthetransmissionchannelsthroughwhichanincreaseinfossilfuelpricesaffectstheEAmacroeconomy.Itthenexaminestheeffectivenessofvariousfiscalmeasures,suchasenergyproductionandconsumptionsubsidies,aswellascarbonsubsidies.

Akeyfindingofthepaperisthattheimpactofenergysubsidiesvariessignificantlydependingontheirdesign.Energyproductionsubsidies,whichlowerthecostofenergyforfirms,reduceboththeheadlineandcoreinflation,whileboostingoveralleconomicout-put.Incontrast,energyconsumptionsubsidies,whichdirectlylowerthepriceofenergyforhouseholds,reduceheadlineinflationbutalsoloweraggregateoutputbyincreasingthedemandforenergywithoutaddressingthesupply-sideconstraints.Carbonsubsidies,whichfocusonreducingthepriceofcarbon-intensiveenergy,produceanintermediateresult,strik-ingabalancebetweenloweringinflationandoutput,butwithlessimpactthanproductionsubsidies.

Whileenergysubsidiescanhelpmitigateinflationarypressuresintheshortterm,theymaycomeatthecostoftheunintendedconsequenceofslowingdownthetransitiontoalow-carboneconomybyraisingtheproductionofenergy.Forinstance,whileenergyproductionsubsidieshelplowercostsforfirmsandboostoutput,theyalsoslowthesubstitutiontowardscleanerenergybyreducingtheincentiveforenergyproducerstoadoptgreenertechnologies.Similarly,carbonsubsidies,whilebalancedintheirimpactoninflationandoutput,leadtohigheremissionsbymakingdirtyenergymoreaffordable.

Recognizingthetrade-offsbetweenshort-terminflationstabilizationandlong-termenvi-ronmentalgoals,theframeworkintroducestargetedmacroprudentialpoliciestocomplementfiscalmeasures.Inparticular,ataxondirtyenergyassetsinbankportfolioscantiltinvest-mentawayfromcarbon-intensivesectorsandtowardsgreenerindustries.This“emissions-prudential”policyhelpsreducebanks’exposuretodirtyenergy,supportingthefinancial

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sector’sstabilitywhilealsoincentivizinggreeninvestment.Bytaxingdirtyenergyassets,policymakerscanmitigatetheincreaseincarbonemissionsassociatedwithenergysubsidiesandacceleratethetransitiontowardsalow-carboneconomy.Overall,theframeworkun-derscorestheimportanceofconsideringthesectoralandfinancialtransmissionmechanismswhendesigningpolicyresponsesduringanenergycrisis.

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1Introduction

Globalfossilfuelpricesstartedtosoarinearly2022uponRussia’sinvasionofUkraine.CombinedwiththerepercussionsfromtheCOVID-19pandemicrelatedtoariseinfoodpricesandsupplybottlenecks,overallconsumerpricessurged,withinflationratesincreasingbymorethan10%inmanyEuropeancountries.Fiscalandmonetarypolicymakersintheeuroarea(EA)haverespondedwithabroadrangeofreliefmeasurestoeasetheburdenofinflationandespeciallyhigherenergycostsonhouseholdsandfirms.Yet,anunintendedconsequenceofsuchmeasuresisthattheycancounteractanotherimportantpolicygoal,thatis,thereductionofcarbonemissionsasoutlinedforexampleintheEuropeanCommission’sGreenDeal

.1

Tocomprehensivelyevaluatepolicychoiceinthecontextofanexogenousfossilpriceincrease,thispaperpresentsadynamicstochasticgeneralequilibrium(DSGE)modelwithadisaggregatedenergysectorandfinancialintermediaries,andprovidesseveralinsights.OuranalysisfirstexposesthechannelsthroughwhichanexogenousincreasesinthepriceoffossilresourcesaffectstheEAmacroeconomy,tothenconsidertheeffectivenessofabroadsetofenergy-relatedfiscalinstruments–someutilizedover2021-2023byEApolicymakers–towardsmitigatingtheinflationaryandcontractionaryeffectsofincreasesinfossilprices.Inparticular,imposingenergyproductionsubsidiesreducesbothCPIandcoreinflationandincreaseaggregateoutput,whileenergyconsumptionsubsidiesonlylowerCPIinflationandalsoreduceaggregateoutput.Carbonsubsidiesinsteadproduceanintermediateeffect.Recognizingthatsuchschemesmaytemporarilyslowdownthepathtowardsdecarboni-sation,ouranalysisshowsthatinfactaparallelsetofmacroprudentialpoliciesaimedattaxingdirtyenergyassetsinbankportfolioscaneffectivelyincentivize“greeninvestment”andspeedupthe“greentransition.”

OurstartingpointisthestandardNew-Keynesianmodelofe.g.,

Christianoetal.

(2005);

SmetsandWouters

(2007)withfinancialintermediariesasin

GertlerandKaradi

(2011),ex

-tendedtoincludedisaggregatedenergy-productionandenergy-consumptionsectors.Energyentersourframeworkinadisaggregatedfashionbothontheconsumptionandproductionside

.2

Thisdetailedsetupisparticularlyusefulallowingustoexploretheeffectivenessofdifferenttypesoffiscalinterventions,suchassubsidiesontheproductionandconsump-tionofenergyutilizedbyintermediategoodfirmsforproductionandonconsumptionofhouseholds,butalsocarbonsubsidies,whichtargetthecarbon-intensivenatureofenergy

1FormoredetailsontheEuropeanUnion’sFit-for-55package,seehttps://www.consilium.europa.eu/en/policies/green-deal/fit-for-55/.

2Inparticular,householdsconsumeacompositefinalconsumptiongood,whichconsistsofenergyandaconsumptiongoodexcludingenergy.Anintermediategoodisproducedbymonopolisticallycompetitivefirmsusingvalueadded(acapital-labourbundle)andanenergycomposite.Energyinturnisproducedbyperfectlycompetitiveenergyproviders,whobundlecleananddirtyenergyinputsproducedbymonop-olisticallycompetitivedirtyandcleanenergyproducers,respectively.Eachofthedirtyandcleanenergyfirmsinturnproducetheirinputsbycombiningsector-specificvalueaddedwithnaturalresources(fossilsvs.renewables).Asaby-productofproductioninthedirtyenergysector,carbonemissionsarereleasedintotheatmosphere.

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productiondirectly.Thedisaggregatedmulti-sectorsetupalsoallowsforthepossibilityofsector-specificmacroprudentialpolicywherethecentralbanksetstaxesorsubsidiesonsec-toralbankassetstoaltertheirweightinbankportfoliosandalterthedegreeofsector-specificfinancialfrictions.

Inlinewithanadversesupplyshock,ourmodelsuggeststhatanincreaseinthepriceoffossilresourcesproducesinflationaryandcontractionaryeffectsthatareamplifiedastheshockpropagatesthroughthefinancialsector.Theinflationaryeffectsunfoldasfollows:Becauseenergyisdirectlyusedinconsumption,theincreaseinthefossilresourcepricefeedsdirectlythroughtothepriceofthefinalconsumptiongood,causingCPIinflationtosurge.Coreinflation(i.e.inflationexcludingenergy)alsorisesasintermediategoodproducers,whoutilizeabundleofcleananddirtyenergyforproductionoftheintermediategood,experienceanincreaseintheirmarginalcosts.Thecontractionaryrealeffectsentailafallincurrentandexpectedfuturerealincomeandprofitability.Theseeffectscausehouseholdstocutbackonconsumptionandnon-financialfirmstocutbackoninvestment.

Severalstraight-forwardimplicationsfortheenergymixinproductionandinvestmentarise.Atthesectorallevel,followinganincreaseinthepriceofdirtyenergy,energyproducerssubstituteawayfromutilizingmorecostlydirtyenergyandintoutilizingnowrelativelylesscostlycleanenergyfortheproductionoftheaggregateenergygoodusedinintermediategoodproduction.Giventheimperfectsubstitutabilityacrossenergytypes,theloweruseofdirtyenergyforproductioncontributestoloweringcarbonemissionsinthemediumterm.Atthesametime,sincefirmsineachofthedirty,clean,andintermediategoodsectorsutilizesector-specificcapital(andlabour)forproduction,sectoralinvestmentfollowssimilarpatterns:investmentinthedirtyenergysectordeclineswhile“greeninvestment”increases.

Becauseourframeworkfeaturesfinancialfrictions,thecontractionaryeffectsoffossilpriceshocksareamplifiedaccordingtotheusualfinancialacceleratorlogic.Inlinewith

GertlerandKaradi

(2011),oursetupintroducesfinancialfrictionsinastandardway.

Pro-ductioninallsectorsrequiresfinancingfrombankstopurchasethesector-specificcapitalinput.Asaresult,firmsissuesharestobanks,intheformofequity.Bankscollectliabilitiesintheformofdepositsfromhouseholds,andgivenamotivationformaturitytransformationandsubjecttoan(occasionally-binding)leverageconstraint,attachapositive,sector-specificspreadontheirloanstofirms.Sincenon-financialfirmsinthemodelrequirefinancingfrombanksforpurchasingsector-specificcapital,changesinassetpricesasaresultofchangesinfirmprofitabilityimpactbankbalancesheetsandamplifytheeffectsontherealeconomythroughafinancialacceleratorchannel.

Whathappensasaresultsofanadversefossilpriceshock?Whendirtyenergyproduc-ersreducetheirproductionofdirtyenergyinputs,pricesofassetsusedtofinancecapitalinthedirtyenergysectordecline.Analogously,theincreaseinthedemandforcleanen-ergyproductionputsupwardpressureonthepriceofcleanenergyassets.Inequilibrium,theformereffectdominates,loweringbankequityandconsequentlycausingtheleverageconstraintoffinancialintermediariestobecomebinding.Thetighteningofleveragecon-

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straintscontributestoincreasinglendingspreadsacrossassets,andfurtheramplifyingthereductioninthedemandforfinancingcapitalonbehalfofnon-financialfirms.Overall,thebankingsectorgeneratesanegativefeedbackloop,whichfurthercontractsinvestmentandassetpricesmagnifyingtheeffectsoffossilpriceshocks.

Inthisdisaggregatedproductioneconomywithfinancialfrictions,ouranalysisalsofo-cusesontheroleoffiscalpolicy.Inlinewithcurrentpolicypractices,forexamplefiscalauthoritiesintheEU27havepredominantlyutilizeduntargetedpricemeasuresintheformofexciseduties(orVATchanges)(see,e.g.,

Sgaravattietal.

(2023)),thefiscalpolicies

employedtomitigatetheinflationaryconsequenceoffossilresourcepriceshockshavetakentheformofenergysubsidies.Inourmodel,wedistinguishbetweensubsidiestofirmsandsubsidiestohouseholds(thatisasubsidytotheproductionofenergy,andasubsidytoitsconsumption).Wealsoconsiderthecaseofacarbonsubsidy(i.e.theinverseofacarbontax),wherethepriceofdirtyenergyusedforproductionisdirectlytargeted.

Whilealltypesofenergysubsidiesoperatebyaimingtoreducethepriceofenergy,anuancedinsightfromourframeworksisthatthetransmissiononCPIandcoreinflation,aswellasonaggregateoutput,differsmeaningfullyacrosssubsidytypes.Thisisadirectresultofthedifferentialsectoraleffectsthatmaterializeinadisaggregatedsetup.Sinceeachoftheinputswhicharesubsidizedareaggregatedtogetherwithotherfactorsofproductionusingconstantelasticityofsubstitution(CES)aggregators,eachsubsidyoperatesbyaffectingsomecombinationofrelativeprices;however,thesetsofrelativepricesaffectedeachtime,aswellastheverticalpositionofeachinputintheproductionorconsumptionstructure,producesdifferentimplicationsforthesignandmagnitudeoftheresponsesofaggregateoutputandinflationrates.

Specifically,ouranalysisshowsthatanenergyproductionsubsidycanlowerbothcoreinflationandCPIinflationandincreaseaggregateoutput,whileanenergyconsumptionsubsidycanonlylowerCPIinflation,whileincreasingcoreinflationandreducingaggregateoutput.Inturn,acarbonsubsidyliesinthemiddle,loweringbothCPIandcoreinflationandincreasingaggregateoutput,butlessthananenergy-productionsubsidy.Quantitatively,wefindthatthepresentvaluemultiplierfortheenergyproductionsubsidyconvergesto1.7inthelongrun,whiletheenergyconsumptionsubsidyproducesanegativepresentvaluemultiplierthroughoutthehorizonandconvergingto-0.26inthelongrun.

Thesedifferencesariseduetoadifferentialimpactofsubsidiesonmarginalcostsofproductionandspecificallyonthepost-subsidyenergypricefacedbyhouseholdsandfirms.Theenergyproductionsubsidylowerstheenergypricefacedbyintermediategoodproducersresultinginasignificantfallintheirmarginalcosts.LowermarginalcostsaresubsequentlypassedthroughtofinalgoodfirmsresultinginareductionincoreinflationandCPIinflation.Bycontrast,anenergyconsumptionsubsidylowersCPIinflationthroughitsdirectimpactonthepriceofenergyusedforconsumption,butinequilibrium,raisesthedemandforenergyusedforconsumption.Dirtyandcleanenergyproducersthenincreasetheirproductionofenergytomeettheadditionaldemand,resultinginanincreaseofthepost-subsidypriceof

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energyfacedbyfirms.

Finally,ourframeworkdeliverstwoinsightsinthecontextofcarbonemissions:First,giventhatemissionsareabyproductofproductioninthedirtyenergysector,asproductionofdirtyenergyrespondsfollowingchangesinenergy-relatedsubsidies,emissionssimplyrespondaccordingly.Acarbonsubsidy,whichrewardstheproductionofdirtyenergybyloweringitsprice,althoughbalancedintermsofitsimpactoninflationandoutput,leadstothelargestincreaseincarbonemissionsrelativetothecasewithoutsubsidies.Bycontrast,energyconsumptionandproductionsubsidiesonlymodestlyincreasesemissions,astheydonotsignificantlyaffecttherelativepricesofdirtyandcleanenergyinputs,utilizedfortheproductionofaggregateenergy.Clearly,energyproductionsubsidiesarebettersuitedtomitigatethecontractionaryeffectsoffossilpriceshocksinthemediumterm,whileonlymodestlyslowingdownthetransitiontoalow-carboneconomy.

Asecondinsightinthecontextofemissionsisthatacomprehensivesetofmacropru-dentialpoliciescanbeintroducedtocompensatefornewlyemergingtransitionrisks.Thispossibilityarisesbecauseenergysubsidiescontribute,albeittodifferentextents,towardstemporarilyslowingdownthe“greentransition”whenfossilpricesincrease.Ouranaly-sisshowsthatwhentheregulatoryauthority(thegovernmentinoursetup)hasaccesstomacroprudentialtaxesandsubsidies,whichcanbeimposedonsectoralbankassetsinbankportfolios,emissionincreasesfromenergysubsidiescanbemitigated.Specifically,amacro-prudentialtaxondirtyenergyassetsinbankbalancesheetstiltsbankportfoliosawayfromdirtyenergyassetsandintoassetsusedfortheproductionofcleanenergyandtheinter-mediategood.Giventhelargeshareofintermediate-goodscapitalinproduction,aggregateinvestmentandoutputfallbylesswhileemissionsdeclinebymore.Moreover,sincethebankingsectorbecomeslessexposedtothedirtyenergy,bankequitydropsbyless,while“greeninvestment”increases,highlightingthebenefitsofsuchan“emissions-prudential”policy.

Relatedliterature

OurworkisrelatedtothestillscarceliteratureassessingtheimplicationsoffiscalandmonetarypolicyinmitigatingincreasesinenergypricesinDSGEmodel-basedstudies.

Paperscloselyrelatedtooursarethosethatstudythesupply-sideeffectofenergypriceshocks.

BaqaeeandFarhi

(2019),

BaqaeeandFarhi

(2022)and

Bachmannetal.

(2022),

findthatrisesinenergypriceshaveaverylimitedeffectonGDP,givenrealisticsubstitutionelasticities.

KharroubiandSmets

(2023)analyzehownegativeenergysupplyshockscan

manifestasnegativedemandshocks,oraKeynesiansupplyshock.However,sincethesepapersabstractfromnominalrigidities,theydonotfeaturearoleformonetarypolicy.In-stead,

Bodensteinetal.

(2011)and

Pataracchiaetal.

(2023)incorporatearoleformonetary

policythroughnominalrigidities.

Morerecently,

Ercegetal.

(2024),explorehowenergysubsidiescanhelpaddressthe

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consequencesofenergypriceincreasesinclosedandopeneconomymodels.Similartous,theyfindthatanenergyconsumptionsubsidyislesspreferable,butintheiranalysisthemechanismhingesontheinternationalnatureofthepolicy(i.e.whetheritisappliedgloballyorinaclosedeconomy).Ourpaperiscomplementary,butemploysagranularsetupofenergyproduction,focusingonaframeworkwithfinancialfrictionsthatamplifyeffectsofshocksandfiscalandmacroprudentialpolicies.Additionally,ouranalysisconsiderstheirimpactoncarbonemissions.

Muchlessvast,butcloselyrelatedtoourpaper,istheliteraturefeaturingmicroeconomicheterogeneityandenergy-relatedissues.

K¨anzig

(2023)and

Auclertetal.

(2023)study

themacroeconomiceffectsofenergypriceshocksinenergy-importingeconomiesusingaheterogeneous-agentNewKeynesianmodel.Whileourpaperfocusesonsimilarquestionsitconsidersheterogeneityonthesupplysideoftheeconomycombinedwithabankingsector.

Anotherstreamofpapersstudyclimatemitigationpoliciesinmodelswithfinancialintermediaries(

BenmirandRoman

(2020);

Diluisoetal.

(2021);

NakovandThomas

(2023);

Bartoccietal.

(2022);

FerrariandNispi-Landi

(2022),amongothers)

.Relativetotheseworks,ourframeworkincludesarolefordisaggregatedenergyandafocusaroundtheabilityoffiscalandmacroprudentialpolicyinstrumentstomitigatetheeffectsofenergypriceincreases.

Finally,ourworkrelatestothebroaderliteratureassessingtheinteractionbetweencli-mateoutcomesandtheeconomy,whichalsofeatureanenergysectorofdifferentgranularityintotheframework.

Golosovetal.

(2014)isoneofthefirstcontributionstoaddfossilfuel

(oilandcoal)inputsinanotherwisestandardDSGEmodel.Foradditionalcontributions,seee.g.,

AnnicchiaricoandDiDio

(2015),

Airaudoetal.

(2022),

Vargaetal.

(2021),

Coenen

etal.

(2023),amongothers

.Thefocusofallthesestudies,relativetoours,however,liesontheimpactofcarbontaxationinaffectingmacroeconomicoutcomes.

Thispaperconsistsof5sections.Section

2

describesthemodelindetailandpresentsthecalibration.Section

3

presentsquantitativeanalysisdescribingthepropagationofashocktothepriceoffossilresources.Section

4

explorestheeffectivenessofasetofenergysubsidies,whilesection

5

illustrateshowmacroprudentialpoliciescanbeutilizedinparalleltoenergysubsidiestomitigatetheincreaseincarbonemissions.Finally,section

6

concludes.AnAppendixreportsadditionalexercisesandrobustnesschecks.

2Model

Ourframeworkconsistsofaclosed-economyNewKeynesianmodelwithfinancialinterme-diariesanddisaggregatedenergyproductionanduse

.3

Onthedemandside,householdsconsumeacompositefinalconsumptiongood,whichiscomposedofenergyandaconsump-

tiongoodexcludingenergy.Onthesupplyside,theintermediategoodisproducedby

3Aschematicillustrationofthemodel’sstructurecanbeseeninFigure

1.

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monopolisticallycompetitivefirmsusingvalueadded(acapital-labourbundle)andanen-ergycompositeaggregatedinCESfashion.Theythenselltheintermediategoodtoperfectlycompetitivefirmsforprivateconsumption,privateinvestmentandgovernmentconsumptionpurchases..Theenergycompositeisproducedbyenergyproducers,whobundlecleananddirtyenergyinputsproducedbymonopolisticallycompetitivedirtyandcleanenergypro-ducers,respectively.Eachofthedirtyandcleanenergyproducersproducetheirinputsbycombiningsector-specificvalueaddedwithnaturalresources(fossilsvs.renewables).

Animportantfeatureofourmodelliesinthepresenceoffinancialfrictions:Productioninallsectorsrequiresfinancingfrombankstopurchasethesector-specificcapitalinput.Asaresult,firmsissuesharestobanks,intheformofequity.Bankscollectliabilitiesintheformofdepositsfromhouseholds,andgivenamotivationformaturitytransformationandsubjecttoan(occasionally-binding)leverageconstraint,attachapositivespreadsontheirloanstofirms,whicharealsosector-specific.

Onthepolicyside,theframeworkisclosedasfollows:Thefiscalauthorityappliesdistortionarysubsidies(taxesifpositive)ontheproductionandconsumptionofenergy(atdifferentlevels,asshownbelow),whicharefinancedbylump-sumtransfersfromhouseholds.ThecentralbankimplementsaTaylorrulethatfeaturesinterestratesmoothing.

2.1Households

Thereisacontinuumofidenticalhouseholdsofmeasure1.Householdsconsumeabasketofenergyandnon-energygoods,workineachofthethreesectors,andinvesttheirsavingsintobankdepositsandshort-termgovernmentbonds.Laborisperfectlymobileacrosssectors.

Inanyperiodt,afractionfofhouseholdsarebankersandtheremainingfraction1−fareworkers.Withinthefamily,thereisperfectconsumptioninsurance.Workerssupplylabortoeachsectorandreturntheir(after-tax)wageincometothehousehold.Bankersmanagefinancialintermediariesandalsotransfertheirearningsbacktothehousehold.Householdssavebydepositingfundsintointermediariestheydonotown.

Bankershaveafinitelifetime.Ineveryperiodahouseholdstaysabankerwithprobabilityσandwithprobability(1−σ)itremainsaworker.Hence,ineveryperiodafraction(1−σ)fofhouseholdsbecomesaworker.Exitingbankersarereplacedbythesamenumberof(random)workers,keepingtherelativesharesofeachhouseholdtypefixed.Enteringbankersareassignedwithanamountofstart-upfunds.

Thediscountedutilityofthehouseholdisgivenby

E0Σiβt−Liφ]