Economic Diversity and the Resilience of Cities

Board of Governors of the Federal Reserve System International Finance Discussion Papers ISSN 1073-2500 (Print) ISSN 2767-4509 (Online) Number 1426 December 2025 Economic Diversity and the Resilience of Cities Franc¸ois de Soyres, Simon Fuchs, Illenin O. Kondo, and Helene Maghin Please cite this paper as: de Soyres, Franc¸ois, Simon Fuchs, Illenin O. Kondo, and Helene Maghin (2025). “Economic Diversity and the Resilience of Cities,” International Finance Discussion Papers 1426. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/IFDP.2025.1426. NOTE: International Finance Discussion Papers (IFDPs) 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 International Finance Discussion Papers Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at www.ssrn.com.

Economic Diversity and the Resilience of Cities∗ François de Soyres† Simon Fuchs‡ Federal Reserve Board Federal Reserve Bank of Atlanta Illenin O. Kondo§ Helene Maghin¶ Federal Reserve Bank of Minneapolis Analysis Group October 2025 Abstract We develop a framework to assess how economic shocks affect local labor markets and worker welfare, with a focus on city-level economic diversity. Using detailed worker flow data across cities, sectors, and occupations, we construct theory-consistent welfare measures. Our approach combines a dynamic discrete choice model with a dual representation that captures both direct effects and the insurancevalueoflocaleconomicdiversity.AppliedtoFrenchlabormarkets,wefindthatdiversification dampenstheeffectofnegativeshocks:bothjob-to-jobmovesandnetinflowsdeclinelessindiversecities thaninconcentratedones. Overall,wedocumentsizablewelfareinsurancegainsfromlocaleconomic diversity. Keywords: sufficientstatistic,laborflows,concentration,economicdiversity,welfare JELclassificationcodes: J61,J62,J21. ∗We thank Stefan Faridani, Kim Ruhl, Abbie Wozniak and seminar participants at the Urban Economics Association North Americameeting, theSouthernEconomicAssociationannualmeeting, andtheMid-AtlanticInternationalTradeconferenceat Dukeforveryhelpfulcomments.WealsothankKenCowlesforhisexcellentresearchassistance.Thisworkwascompletedwhile HeleneMaghinwasatKULeuven.TheviewsexpressedhereinarethoseoftheauthorsandnotnecessarilythoseoftheBoardof GovernorsoftheFederalReserveSystem,theFederalReserveBankofAtlanta,theFederalReserveBankofMinneapolis,orofany otherpersonassociatedwiththeFederalReserveSystem.Allerrorsareourown. †Email:francois.m.desoyres@frb.gov. ‡Email:sfuchs.de@gmail.com. §Email:kondo@illenin.com. ¶Email:helene.maghin@analysisgroup.com. 1

1 Introduction Areworkersinmoreeconomicallydiversecitiesbetterorworseoffwhentheirindustryoroccupationfaces a labor demand shock? We argue that answering this question requires accounting for the full range of options available to workers, including staying in the same job, switching locally to another occupationindustrypair,ortonon-employment,orrelocatingtoadifferentcity. Inthispaper,wedevelopanapproach to capture these effects by deriving sufficient statistics for a second-order approximation of local welfare changeswithinastandarddynamicdiscretechoice(DDC)framework. Usingrichworkerflowsdatafrom France,weshowthateconomicdiversityacrosscitiesgeneratessizablewelfareinsurancegainsforworkers. Figure1. LocalEconomicDiversityandNonspatialLaborReallocationinFrance (a)NonspatialMobilityandSector-OccupationHHI (b)LaborMarketConcentration(HHI)Map Notes:Theleftpanel,Figure1a,showsascatterplotofthesector-occupationHerfindahl-HirschmanIndex(HHI)ofacommuting zoneinFranceandameasureofnonspatiallaborreallocation:theshareoflocalemploymentthatswitchestoanotheroccupation, sectororbothwithoutmovingtoanewlocation.Therightpanel,Figure1b,showsamapofFrenchcommutingzonesandtheir sector-occupationHHI. Figure 1 illustrates the central idea of the paper: cities with more diverse local economies experience moreworkerreallocationwithinthecity. Panel1arelatessector-occupationconcentration,measuredusing theHerfindahl-HirschmanIndex(HHI),tonon-spatiallaborflows,definedastheshareoflocalemployment thatswitchesoccupation,sector,orbothwithoutmoving. CitieswithlowerHHI,thatis,moreeconomically diverse cities, exhibit higher local churn, meaning a larger share of workers change jobs within the same city. Panel 1b maps French commuting zones by their sector-occupation HHI, highlighting substantial variationinlocaleconomicdiversity. Building on this, we analyze how worker flows, both within and across cities, respond to local labor demand shocks, and how local economic diversity shapes this response. We then use these estimates as 2

theory-consistentsufficientstatisticstoevaluatethewelfarevalueofdiversity. Ourmainfindingisthatmore diversecitiesaremoreresilienttonegativelocallabordemandshocks,providingresidentswithsignificant insurancebenefits. To infer welfare, we build a revealed preference argument based on a standard DDC model: welfare changes can be approximated using observed baseline choice probabilities and their dynamic response to shocks,followingtheconvex-analyticapproachofChiong,GalichonandShum(2016).Ourmaintheoretical resultshowsthatwelfarechangesequalaweightedsumofchangesinchoiceprobabilitiestofirstorder,and, to second order, they depend on the covariance of choice adjustments and the cross-elasticities between alternatives. WeapplythisframeworktodataonFrenchworkers’employmenthistories,constructingflowsacrosslocation–industry–occupationcells. UsingBartik-styledemandshocksandlocalprojections-IV(Jordà,2005), wetracebothgrossandnetworkerflowsandseparatespatialfromnon-spatialadjustments. Wefindthatlocaleconomicdiversitysignificantlyshapeslabormarketresponses. Afternegativeshocks, diversified cities experience a smaller drop in within-city churn and less severe net inflow contractions, suggesting that diversity provides a form of insurance. Responses to positive shocks are more symmetric: churn reacts similarly across cities, while net inflows are only slightly larger in diversified cities. In short, concentratedcitiesfacedeeperdownturnsandonlymodestlystrongerexpansions. Finally,weuseourtheoreticalframeworktotranslatetheseflowresponsesintowelfaretermsandfind asizable“insurancevalue”oflocaleconomicdiversification,ascapturedbythesecond-orderterm. Overall,ourapproachhastwomainadvantages. (i)Itreliesonlyonobservedchoiceprobabilitiesand imposes less structure than a fully specified model, and is therefore less dependent on functional form assumptions. (ii) Its second-order approximation decomposes welfare changes into a part due to average utilitychangesandaninsurancetermduetocomovingreallocationacrosschoices. Thispapercontributestoseveralresearchstrands: First,ourpapercontributesmethodologicallytothe literature evaluating the welfare consequences of shocks using approximations and sufficient statistics.1 Closest to our paper, Kim and Vogel (2020) develop a methodology to approximate first-order welfare differencesinthecontextofastaticmodelusingasufficientstatisticapproachappliedtoU.S.commuting zones. Our paper builds on advances in the DDC literature from Chiong, Galichon and Shum (2016) to derivehigher-orderwelfarechangeapproximationsthatleveragethedynamicsandcompositionofworker choice probabilities across more adjustment margins. Relative to models of the labor market that fully specify the worker’s reallocation choice (e.g., Artuç, Chaudhuri and McLaren (2010); Caliendo, Dvorkin andParro(2019)),ourframeworkislighteronassumptions: itrecoverscontinuationvaluesdirectlyfrom conditional choice probabilities and delivers a transparent welfare decomposition into first- and second- 1See,forexample,Deaton(1989),KimandVogel(2020),Atkin,FaberandGonzalez-Navarro(2018),BaqaeeandBurstein (2023),Allenetal.(2022),BaqaeeandFarhi(2020),Kleinman,LiuandRedding(2021),Porto(2006),WolfandMcKay(2022), andBeraja(2023). 3

orderterms. Theflipsideisthatitdoesnotidentifyunderlyingfundamentalsorallowsforcounterfactual analysis. WeillustratetheapproachbycombiningLPIVestimatesfromrichworkerflowsdatainFranceto constructtheory-consistentwelfaremeasuresandtoderivetheinsurancevalueoflocaleconomicdiversity. Second, the paper contributes to an extensive literature on labor flows across cities and sectors in responsetoemploymentshocksfollowingthecanonicalstudybyBlanchardandKatz(1992).2 Studiessuch as Monte, Redding and Rossi-Hansberg (2018) and Marinescu and Rathelot (2018) highlight the heterogeneity in local employment elasticity and worker mobility preferences. The literature on adjustments to trade-inducedshocksalsohighlightstheroleofspatial,sectoralcompositionandoccupationaladjustments. See,forexample,Autoretal.(2014),Ebensteinetal.(2014),Fuchs(2021),Kondo(2018)andTraiberman (2019),amongothers. DisciplinedbytheDDCtheory,weuserichindustry-occupation-locationworkerflow datainFrancetodocumentsizableasymmetriesinthedynamicadjustmentpatternsofworkersacrosstheir spatial and non-spatial options and estimate significant heterogeneity across labor markets with different degrees of economic diversity. We combine these estimates through the lens of our theory to suggest a significantinsurancecomponentinthewelfarevalueofeconomicallydiverselabormarkets. The remainder of this paper is structured as follows: Section 2 introduces our approach to measure welfareimplicationsofshocksusingsufficientstatistics, Section3introducesthedata, providessummary statistics on worker flows in France, and covers our analysis of local projections on worker adjustments. Section4introducesourwelfareresults,andSection5concludes. 2 Theory Webeginfromthesimpleobservationthatworkersfacedynamicanddiscretechoices—staying,switching sectorswithinacity,ormoving—whereswitchingcostsandoptionvaluesmakedecisionsforward-looking. A dynamic discrete choice (DDC) framework is therefore natural for tracing how local shocks propagate intoreallocationandwelfare. DDCmodelsarewidelyusedinlaborandtradeeconomicsbecausetheylink observed worker flows to welfare in a theory-consistent way. We build on recent convex-analytic identification results for DDC models (Fosgerau et al., 2021; Chiong, Galichon and Shum, 2016) and measure welfareviathesocialsurplus,theexpectedvalueofthemenuofchoicesfacinganagent. Putplainly,given what a worker can do today and tomorrow, how do changes in local labor-market conditions shift overall welfare? We first lay out the general DDC environment and then present our key surplus approximation, whichweusetoquantifythevalueofeconomicdiversityacrosscities. 2.1 Welfare in Dynamic Discrete Choice Models: An Approximation We use a DDC model that describes how workers choose among discrete options over time, taking into accountcostsandbenefitsofeachchoice. 2SeeDao,FurceriandLoungani(2017)forarecentupdateofBlanchardandKatz(1992)fortheU.S. 4

Setup. Time is discrete and there are infinite periods. The state variable is x ∈ (cid:88) and agents choose betweenafinitenumberofactions y ∈(cid:89). Thesingleperiodutilityflowofchoosing yisgivenbyu¯ (x)+ϵ y y where ϵ denotes an i.i.d additive utility shock. The set of utility shocks is assumed to follow a joint y distribution function Q(·;x). Following Rust (1987), we assume conditional independence. Agents are dynamicoptimizersfacingthefollowingproblem: y ∈argmax (cid:166) u¯ ˜y (x)+ϵ ˜y +β(cid:69) x′,ϵ′ (cid:2) V¯(cid:0) x ′ ,ϵ′(cid:1)| x,˜y (cid:3)(cid:169) , ˜y∈(cid:89) whereβ isthediscountfactorandprimesdenotenextperiodvalues. NotethatV(x ′)istheexpectationof V¯(x ′ ,ϵ′): V(x ′) ≡ (cid:69) ϵ′V¯(x ′ ,ϵ′). We assume time-homogeneous dynamics (i.e. constant transition kernel Pr(x t+1 | x t ,y t )anddistributionQ),anddroppedtimesubscriptsforsimplicity. Wedefinechoice-specific continuationvaluesas: w y (x)≡u¯ y (x)+β(cid:69) x′ (cid:2) V (cid:0) x ′(cid:1)| x,y (cid:3) , whichcapturesthetotalexpectedvalueofselecting y today,includingfuturepayoffs.3 ApproximatingWelfare. Tosummarizeexpectedwelfareinthisenvironment,weusethesocialsurplus function4 ,whichistheexpectedmaximumutilityacrossallpossiblechoices: (cid:149) (cid:152) (cid:71)(w;x)=(cid:69) max (cid:0) w (x)+ϵ (cid:1)| x , (1) y y y∈Y wheretheexpectationistakenoverthedistributionofutilityshocks. Becausewetakeanexpectationover the shocks, this object summarizes ex-ante welfare and is a smooth function of w. Importantly, the social surplusfunctioniswell-behaved(finite,convex,differentiable),whichallowsustouseasecond-orderTaylorexpansiontoapproximatehowwelfarechangeswhencontinuationvaluesorchoiceprobabilitiesshift.5 Thefollowingpropositiongivesasecond-orderTaylorapproximationof(cid:71),showinghowwelfarechanges relatetochangesincontinuationvaluesandchoiceprobabilities. DetailedderivationsareinAppendixD.1. Proposition1(ApproximateSocialSurplus). Wehave, (cid:88) 1(cid:88)(cid:88) dlnG(w;x)= ω y (x)dlnw y + 2 ϱp y, ,w y′ ω y (x)dlnw y dlnw y′ +o(·) y y y′ where ω (x) = p y (x)w y (x) is a weight that measures the relative contribution of choice y to the expected y (cid:71)(w;x) 3Notethatw istheexantevalueofanoption y,thatisbeforetherealizationofthepreferenceshockϵ. y 4Foracomplementaryaggregatetreatmentthatalsodefinesaggregatewelfareviaasocialwelfarefunctionandderivesafirstorder decomposition—technology, dispersion of marginal utility, fiscal and technological externalities, and redistribution—see Donald,FukuiandMiyauchi(2025). Ourfocusislocalworkerwelfarerecoveredfromchoices(includingan“insurance”componentfromdiversification),whereastheirsisaggregateaccountingandconditionsunderwhichthetechnologytermalonecan characterizewelfarechanges. 5Assumingacontinuousshockdistributionwithfinitemomentsensuresthat(cid:71)(w;x)isfinite,convex,differentiable,andhas awell-definedconvexconjugate. 5

welfare and where ϱp,w ≡ ∂lnp y = ∂p y w y′ is the elasticity of the choice probability p with respect to y,y′ ∂lnw y′ ∂w y′ p y y changesinthechoice-specificcontinuationvalue w y′. Insights and Limitations. Proposition 1 implies that approximating welfare changes requires three ingredients: (i) changes in continuation values {dlnw }; (ii) the cross–elasticity matrix {ϱp,w}; and (iii) y y,y′ baselinewelfareweights{ω }. Thesecond–ordercomponentemphasizesthatitisthecovariancestructure y of option–specific changes in continuation values—interacting with cross–elasticities—that drives curvature. In more economically diverse locations, option payoffs tend to move less in lockstep, reducing that covarianceandattenuatingcurvature(i.e.,offeringinsurance). However,toimplementthisapproximation, twochallengesarise: continuationvaluesareunobserved,andcross-elasticitiesarehigh-dimensional. DualApproachviatheLegendre-FenchelConjugate. Tocircumventthesechallenges,weusetheLegendre- Fenchel conjugate of (cid:71). Intuitively, the Legendre-Fenchel conjugate offers a dual perspective: instead of asking“whatchoiceprobabilitiesfollowfromgivenutilities?”,itallowsustoask“whatutilitiesareconsistentwithobservedchoiceprobabilities?”. Chiong,GalichonandShum(2016)showhowthisinversionlets usrecovercontinuationvaluesnon-parametricallyfromobservedchoiceprobabilities,withoutimposinga specificfunctionalformonflowutilities. Theconjugateisdefinedasfollows: (cid:40) (cid:41) (cid:88) (cid:71)∗(p;x)= sup p (x)w (x)−(cid:71)(w;x) (2) y y w∈(cid:82)(cid:89) y∈(cid:89) where the vector p denotes the choice variable of the conjugate (cid:71)∗ . Therefore, w and p are connected through convex duality: p can be seen as the shadow prices associated with w, and vice versa. Moreover, since duality holds, we can approximate the convex conjugate for an equivalent welfare approximation, whichwesummarizeinthefollowingproposition:6 Proposition2(ApproximateConjugateSocialSurplus). Wehave, (cid:88) 1(cid:88)(cid:88) dln(cid:71)∗(p;x)= ω∗ y (x)dlnp y + 2 ϱw y, , y p ′ ω∗ y (x)dlnp y dlnp y′ +o(·) y y y′ whereω∗ (x)= w y (x)p y (x) isaweightthatmeasurestherelativecontributionofchoice ytotheconjugatesocial y (cid:71)∗(p;x) surplusfunction(cid:71)∗ andwhereϱw,p ≡ ∂lnw y′ = ∂w y′ p y istheinverseelasticityofthechoiceprobability p y,y′ ∂lnp y ∂p y w y′ y withregardtochangesinthechoice-specificcontinuationvalue w y′. Proposition 2 is an equivalent, but empirically more convenient approximation of the social surplus. First, it works with changes in choice probabilities (dlnp ), which are observed and estimated directly y 6Notethatwerequirestrictlypositiveconditionalchoiceprobabilityandcontinuationvalues,withGstrictlyconvexandtwice continuouslydifferentiable,sothatHessiansandcross-elasticitiesexist,asshowninChiong,GalichonandShum(2016).Moreover, innon-stationarysettings,themethodcanbeappliedperiodbyperiodusingthecorrespondingTaylorcoefficients,thoughatthe costofrecomputingsufficientstatisticseachperiod. 6

in our data, instead of changes in continuation values (dlnw ). Second, following Chiong, Galichon and y Shum (2016), the dual representation provides a simple “mass–transport” inversion that maps observed p into the continuation values w needed for the weights—no logit assumption required. Finally, because ourprimaryinterestisinunderstandinghoweconomicdiversityaffectswelfare,weusethisframeworkto deriveclosed-formexpressionsthatlinkwelfaregainsfromdiversificationdirectlytothelocalHHI.Aswe showinmoredetailinAppendixD.3,thefollowingcanbederived: Corollary 1 (Value of Insurance and Diversification). Consider a vector of shocks s ={s } ≡{dlnw } m m m m withmean(cid:69)[s]=0,andcovarianceΣ=(cid:69)[ss ⊤]. Assumethecross-elasticityofconditionalchoiceprobabilities (cid:128) ϱ k p , , m w |ℓ = ∂ ∂ l l n n w p m k| | ℓ ℓ (cid:138) areiso-elasticsuchthatϱ k p , , m w |ℓ ≡−γ ℓp m|ℓ +γ ℓ 1 {m=k}. Theexpectedwelfarechangefrom suchshockis (cid:20) (cid:21) 1 (cid:88) (cid:88) (cid:69)[dlnW(cid:67) ]= π ℓ γ ℓ λ m|ℓ Σ mm −λ⊤ ℓ Σλ ℓ , 2 ℓ∈(cid:67) m whereλ m|ℓ ≡ w (cid:71) m |ℓ p m|ℓ arewelfareshareweightsandπ ℓ arelocalpopulationshares. ℓ Furthermore,underaconstantsubstitutionelasticity(γ ℓ ≡γ),andequicorrelatedshockssuchthatΣ mm = σ2andΣ =νσ2form̸=k,theexpectedwelfaregainsareproportionaltotheshockvarianceanddecreasing mk inmarketconcentration: γ (cid:69)[dlnW(cid:67) ]= σ2(1−ν)(1−HHI(cid:67) ), 2 whereHHI(cid:67) ≡(cid:80) ℓ π ℓ (cid:88) λ2 m|ℓ isamarketconcentrationindex. m (cid:124) (cid:123)(cid:122) (cid:125) HHIℓ Motivated by this corollary, we use the HHI statistic as a proxy for economic concentration in the empiricalsectioninordertotraceandquantifydifferencesacrosslocationsinthisinsuranceterm.7 Taking Stock. Proposition 2 clarifies the methodological advantages of the mass-transport sufficientstatisticsapproachandprovidesanempiricallyconvenientrepresentationoftheworker’sconjugatesocial surplus. Thefirst-ordertermnowdependsonwelfareweights(ω∗ )andimpulseresponsesofchoiceproby abilities (dlnp ). The second-order term involves own and cross-price elasticities (ϱw,p ), which can be y y,y′ calibrated once an iso-elastic labor supply system is assumed, as discussed below. Our approach presents theadvantagethatitrecoverswelfareweights(ω∗ )non-parametricallyandusessimplereduced-formesy timationtoobtainchoiceprobabilityimpulseresponses. Theimpliedsecond-orderexpansionalsoisolates directeffectsfromtheinsurancevalueofeconomicdiversity.8 7Strictlyspeaking,theconcentrationindexHHI(cid:67) ≡=(cid:80) ℓ π ℓHHIℓ =(cid:80) ℓ π ℓ (cid:80) m λ2 m|ℓ inCorollary1isnottheHHIin(cid:67) using thelocalsharesπ ℓ,butanaverageofHHI-likeindexescomputedusingwelfareweightsλ m|ℓasshares. 8NotethatChiong,GalichonandShum(2016)introducetheconjugatedualityapproachweuse,whichmapsobservedchoice probabilitiestounobservedchoice-specificvalues. Thisavoidstheneedtocontrolforcontinuationvaluesorspecifytheshock distribution,unlikeArtuç,ChaudhuriandMcLaren(2010).Theyrecasttheproblemasamasstransportproblemanddevelopan estimator,the“masstransportapproach(MTA)”,torecoverareferencevector(cid:71)(w )≡0.Choice-specificvaluesaredetermined 0 onlyuptoaconstant,typicallyresolvedbysettingareferenceoptionwithoutlossofgeneralityinstaticsettings. 7

Twoimportantlimitationsremain. First,aswithothersufficient-statisticsmethods,theanalysisrecovers only relative welfare changes across locations and does not pin down absolute welfare levels. Second, while the framework provides a detailed account of workers’ adjustment choices within segmented labor markets, it abstracts from additional dynamic decisions such as intertemporal consumption and saving. Withthesestrengthsandcaveatsinview,thenextsectionappliestheframeworktoassesstheresilienceof Frenchlocallabormarketstodemandshocks. 3 Adjustment to Labor Demand Shocks Our approximation provides a practical strategy to quantify how labor demand shocks affect worker welfare. Guided by Proposition 2, we estimate worker flow responses to local shocks and examine how local sectoralandoccupationaldiversityshapestheseresponses. Usingalarge,representativesampleofFrench employmentspells,weconstructworkerflowsacrosslocations,industries,andoccupations. Wealsobuild a standard labor demand shock and estimate the impulse responses of spatial and nonspatial adjustment margins using local projection IVs (LPIV). We then document significant heterogeneity in worker flow responses,illustratinghowcity-leveleconomicdiversitypotentiallyshapeswelfare. 3.1 Data: Worker Flows in France Data. Our data on French workers’ employment histories comes from the DADS (Déclarations Annuelles deDonnéesSociales)administrativepanel,maintainedbyINSEE(theFrenchNationalInstituteofStatistics andEconomicStudies). TheDADSFichierPostescontainsmandatorySocialSecurityfilingsforallsalaried employeesinprivateandsemi-publicfirmsinFrance.9 TheDADSPanelfollowsworkersborninOctoberof evenyears,coveringabout4percentoftheprivate-sectorworkforce. Thispanelallowsustotrackindividual workersacrossallnon-public-sectoremploymentspellsfrom2005to2019.10 Eachrecordprovidesaworker’sestablishment,includingauniqueidentifier,4-digitindustry,andmunicipality. Thisallowsustoobservetransitionsacrosssectors,occupations,andgeographiclocations,which is crucial for our analysis. For each spell, we define the sector using the establishment’s 2-digit industry code,theoccupationusingthe2-digitINSEEoccupationcode,andthelocationasthe“Zoned’Emploi”(commutingzone)derivedfromtheestablishment’spostalcode. Thefinaldatasetcontains30occupations,90 sectors,and300geographicareas,whichwerefertoas“cities”forsimplicity. Constructing Worker Flows. For each quarter t, we assign every active worker to a unique city-sectoroccupation (i,s,o) cell, keeping the job with the longest duration.11 Quarters with no recorded activity 9DADSexcludestheself-employed,centralgovernmentemployees(FonctionPubliqued’État),anddomesticworkers. 10AsinTraiberman(2019),werestrictthesampletoindividualsaged23–64. 11Iftwojobshavethesameduration,weselecttheonewiththehighersalary;ifsalariesareequal,wechoosethejobwithmore hoursworked. 8

areclassifiedas“non-employment,”abroadcategorythatincludesstandardunemploymentandexitsfrom theprivate-sectorlaborforce. Wedefine“job-to-job”transitionsascaseswhereaworker’sassignedlabormarketin t+1differsfrom that in t. Following our DDC framework, we use directed gross flows rather than net flows, constructing both (i) the stock of workers in each labor market and (ii) the full matrix of worker flows between every pairofmarkets,includingtransitionstoandfromnon-employment.12 AnatomyofWorkerReallocationinFrance. Table1reportskeymomentsofcity-levelworkerflows. In themedianFrenchcity,4.5%ofworkersswitchlabor-marketaffiliationeachquarterinatleastonedimension: city(i),sector(s),oroccupation(o)—equivalenttoroughly18%peryear.13 Rows2–4disaggregate flows by dimension, showing that 65% of switches involve changing occupation, 60% involve changing sector,andnearlyhalfinvolvechangingcity. FigureA.1furtherdecomposesflowsbyallcombinationsofcity,sector,andoccupationchanges. Nonspatial moves (Figure A.1a) are more common than spatial ones (Figure A.1b), with occupation switches dominatingoverall.Nonspatialreallocation,particularlyoccupation-onlyswitches,variesmuchmoreacross citiesthanspatialreallocation. Table1. Quarterlyworkerreallocationacrossmarkets Median SD p5 p25 p75 p95 Flows 1. i,s,o 4.54% 0.98 3.21% 3.90% 5.35% 6.10% Emp i 2. Flows i→i′ 48.95% 9.79 34.32% 41.48% 56.14% 65.65% Flows i,s,o 3. Flows s→s′ 61.29% 4.34 53.58% 58.92% 63.82% 66.64% Flows i,s,o 4. Flows o→o′ 64.91% 4.74 57.36% 62.19% 67.10% 70.29% Flows i,s,o 5. Flows ne→e 3.22% 1.70 2.10% 2.70% 4.21% 7.62% Emp i Notes: Thedisplayedvaluesareexpressedforthecross-sectionoflocationsorcities. Flowsarecomputedatthe quarterlyfrequency. Sub-indicesaredefinedasfollows: idenotescities,ssectorsandooccupations. Row1shows theshareofworkersinitiallyinalocallabormarketswitchingjobsacrossanydimension. Rows2to4displaythe shareofswitchersthatchangeatleastinthesub-indiceddimension. Theydonotsumto100astheyallowforthe twootherdimensionstochangeaswell,therebycausingoverlap. Row5showstheshareofworkersleavingnonemployment.Itencompassesbothnewjob-seekersandthoseexitingthelaborforce. 12BecauseDADScoversonlytheprivatesector,flowsintoandoutofnon-employmentincludetransitionstoandfrompublicsectorjobsandself-employment. 13Ourestimate(18%)issomewhathigherthanthe13%reportedbyTraiberman(2019)forDenmark,whotrackoccupation andsectorbutnotgeographicmoves. 9

3.2 Estimation Methodology LPIV. We estimate average dynamic responses of city-level outcomes to labor demand shocks using the localprojectionmethodofJordà(2005)forh=0,...,20quarters: 8 8 8 (cid:88) (cid:88) (cid:88) ∆y i,t+h =αh+γ t +γ i +β h Shock i,t + γh m y i,t−m + ωh m Shock i,t−m + δh m Z i,t−m (3) m=1 m=1 m=1 where ∆y i,t+h = log (cid:0) y i,t+h (cid:1)−log (cid:0) y i,t−1 (cid:1) and y i,t+h is the cumulative value at time t+h since t−1.14 Shock isanexogenouslocalshockand Z denotescontrolsforvariouslocallabormarketvariables: the i,t i,t shareoftotalflowstolocalemployment,thelocallaborforce,andratiosofcity-levellocalchurn,inflows, and outflows relative to all flows. γ and γ are city- and time-fixed effects accounting for cross-sectional i t heterogeneity and common macroeconomic shocks. Standard errors are Driscoll-Kraay, and we allow for m=1,...,8quartersofauto-correlation–astandardtemporalcorrelationwindowoftwoyears. Constructing Labor Demand Shocks. We use a shift-share approach (Bartik) to generate plausibly exogenouslabordemandshocksatthecitylevel. Wecomputenationalemploymentgrowthsforeachsectoroccupationpair(s,o)andconstructthelabordemandshockincity i as: (cid:88) Bartik = share s,o ·gnational (4) i,t i,2004 −i,s,o,t s,o where share s i, , 2 o 004 = E i,s,o,t=2004 /(cid:80) s′,o′ E i,s′,o′,t=2004 are within-location (i) shares of sector-occupation (s,o)cellsand g−i,s,o,t areleave-one-outsector-occupationemploymentgrowthsin2004. FigureA.2shows thedistributionofsector-occupationshocks. Citiesareonaverageexposedtomorepositivethannegative shocks,andsomeofthemexperiencelargechangesofupto-5or+10%. 3.3 Beyond Net Employment: The Dynamics of Gross Flows Ourworkerflowsdataallowustounpackthenetemploymentresponsetolabordemandshocksbyrevealing the underlying gross flows. Examining net flows provides a clear lens into how employment adjusts acrosscities,sectors,andoccupations,beyondwhataggregateemploymentchangesalonecanshow. Citylevel adjustments are presented in Figure 2.15 The left panel (Figure 2a) shows average gross spatial flows—city inflows and outflows—while the right panel (Figure 2b) shows spatial and nonspatial flows. Nonspatial flows include all worker transitions that do not involve a change of city, whereas spatial net flows are defined as inflows minus outflows, capturing employment changes before accounting for local unemploymentorexitsfromprivatesectoremployment. 14Whenthe y outcomeconceptisastock(e.g. employmentlevels)ratherthanaflow(e.g. location-changingworkers), we simply take the log differences in the levels from t−1 to t. When the outcome concept is a flow, y t+h is constructed as the cumulativesumoftheperiod-specificflows. 15TableA.2reportsresultsforadditionalmarginslistedinTable1. 10

Followingalabordemandshock,inflowsriseimmediately,steadilyincreasingandremainingelevated throughout the horizon. Outflows respond with a lag of roughly two years, then rise sharply and persist.16 Nonspatialflowsinitiallymatchspatialflowsbutexceedthemafterafewquarters,highlightingthe importanceoflocaljobreallocation. Figure2. AverageResponsetoaBartikshock: SpatialandNonspatialFlows (a)GrossSpatialFlows (b)SpatialandNonspatialFlows Notes:Figure2ashowsthegrossflowofworkersthatmoveoutofacity(solidline)andthegrossflowofworkersthatmovein acity(dashedline). Figure2bshowsthenetflowofworkersthatmoveinacityminustheonesthatleave(solidline)andthe averageflowofworkersthatreallocatewithinacity(dashedline). 3.4 The Asymmetric Role of City-Level Economic Diversity Motivated by the insights from the second-order approximation in Section 2, we posit that city-level economic diversity plays an important role in shaping how worker flows respond to labor demand shocks.17 WeaugmentthebaselinedynamicLPIVwithasymmetricandheterogeneouseffects. Wemeasurecity-level economicconcentration(theoppositeofdiversity)usingthestandardHerfindahl-HirschmanIndex(HHI) definedoversectorsandoccupationsas (cid:88)(cid:128) (cid:138)2 HHI = shareso , (5) i i,t=2004 s,o whereshare s,o istheshareoflocality i’sworkersemployedinasector-occupationpair(s,o)in2004. i,t=2004 Lower HHI indicates a more diverse labor market, with employment spread across multiple sectors and occupations,whilehigherHHIreflectsamoreconcentratedeconomy 16Thedelayedpositiveoutflowresponsemayreflectincomersreplacingexistingworkersorunobservedshocksinneighboring cities. 17InSectionD.3intheappendix,weshowthat,undercertainorthogonalityconditionsinourtheoreticalframework,theHHI mediatestheinsurancevalueofcity-leveldiversity. 11

+ − We separate positive shocks (B ) and negative shocks (B ), as workers may respond differently dei,t i,t pending on the shock’s sign.18 Both variables only take non-negative values and are interacted with local HHI measures: i ∆y i,t+h =αh+γ t +γ i +β h −×B − i,t +ψ− h (cid:128) B − i,t ×HHI i (cid:138) +β h +×B + i,t +ψ+ h (cid:128) B + i,t ×HHI i (cid:138) (6) 8 8 8 (cid:88) (cid:88) (cid:88) + γh m y i,t−m + ωh m B i,t−m + δh m Z i,t−m . m=1 m=1 m=1 Thisspecificationshowshowworkerflowsresponddifferentlytopositiveandnegativeshocks(asymmetric) and how these responses vary with city-level economic diversity (heterogeneous).19 While both the HHI andtheshift-shareinstrumentusesectoralshares, FigureA.4showsthatthedistributionofBartikshocks isverysimilaracrosshighlyconcentrated(highHHI)andhighlydiversified(lowHHI)cities. Figure3. AsymmetricandHeterogeneousResponse: Positive/NegativeShocks×High/LowHHI (a)Nonspatial (b)Spatial Notes: Figure 3a shows the differential response of nonspatial flows to a negative/positive labor demand shock for low-HHI (dashedblue/greenlines)andhigh-HHIcities(solidblue/greenlines).Figure3bshowsthedifferentialresponseofspatialflows toanegative/positivelabordemandshockforlow-HHI(dashedblue/greenlines)andhigh-HHIcities(solidblue/greenlines). Figure3showsasymmetricresponsesforspatialandnonspatialflows,evaluatedatthe90th(highHHI) and10th(lowHHI)percentilesoftheHHIdistribution. Theresultsconfirmourtheoreticalpredictionthat localeconomicdiversityshapeshowlabormarketstransmitshocks. Inmorediverse(lowHHI)cities,negativelabordemandshocksarenoticeablydampened. Bothnonspatialchurn(workersswitchingjobswithin the same city) and net spatial inflows fall less compared with more concentrated cities, suggesting that 18Wetrimthetopandthebottom0.5%oftheHHIforourregressions. Trimmingbeyondthetopandbottom0.5%doesnot affectourresults.NotethatthistrimmingdoesnotleadustodropParis’commutingzone. 19MorettiandYi(2024)highlighttheinteractionofeducationandlabormarketsize. Wecontrolforlocallaborforcesizeand focus on illustrating our high-order welfare approximation using city-level diversity. We leave a richer set of interactions and heterogeneousresponsesforfutureapplications. 12

diversifiedlocallabormarketsprovideabufferagainstemploymentdisruptions. Thiseffectissubstantial, highlightingthatthesameshockproducesverydifferentlabormarketdynamicsdependingonthedegree oflocaleconomicdiversification. Bycontrast,positiveshocksdonotexhibitasimilarlyasymmetricpattern: moreconcentratedcitiesdo notexperienceaproportionallystrongerincreaseininflowsornonspatialreallocationrelativetodiversified cities. Thisasymmetryisconsistentwiththeideathateconomicdiversityfunctionsprimarilyasaformof insurance,limitingthedownsideofnegativeshocksratherthanamplifyingtheupsideofpositiveshocks. Importantly,ourapproachprovidesanewlensonlabormarketadjustment. CombiningtheHHImeasurewithgrossworkerflowsoffersaclearerviewoflabormarketadjustmentthannetemploymentchanges alone,revealinghowboththedirectionandscaleofflowsaresystematicallyshapedbylocaldiversity. 4 From Estimation to Welfare We now combine the approximation results from Section 2 with our estimated worker flow responses to quantifythewelfareeffectsoflocaleconomicdiversityunderlabordemandshocks. 4.1 Methodology Setup. WeadaptthedynamicdiscretechoiceframeworkfromSection2tomodelflowsacrosssegmented local labor markets. The choice set (cid:89) is defined as sector-occupation-location labor markets: (cid:89) ≡ (cid:83) × (cid:79) ×(cid:76),where(cid:83) ,(cid:79),and(cid:76) arethesetsofsectors,occupations,andlocations,respectively. Socialsurplus anditsconvexconjugateremainasinEquations(1)and(2). To construct welfare measures from the LPIV-estimated changes in choice probabilities (dlnp ), we y collapsethefullsetofsector-occupation-locationtransitions((cid:89) ×(cid:89))intocoarseradjustmentmargins: a “stay”marginforworkerswhodonotchangetheirsector-occupation-location,a“local”marginforworkers whochangesectororoccupationwithinthesamecity,anda“spatial”marginforworkerswhochangecity.20 Approximating Welfare Changes. To implement Proposition 2, three ingredients are required: (i) We useourLPIVestimatestoobtainchangesinconditionalchoiceprobabilities{dlnp y } y∈Y alongallmargins. (ii) We construct theory-consistent welfare weights ω∗(x)= w y (x)p y (x) via the Mass-Transport Approach y G∗(p;x) (MTA) of Chiong, Galichon and Shum (2016).21 (iii) To parsimoniously organize cross-elasticity terms in 20Thiscoarserpartitionassumesahomogeneousresponseofflowswithineachsubset:“stay”,“local”,or“spatial”. 21TheMTAsolvesalinearassignmentproblemthatcouplestheobservedCCPswithadiscretizedapproximationtotheshock distributionQandreturnsbothG∗(p;x)andasubgradientw∈∂G∗(p;x). Inourimplementation,Qisspecifiedflexiblyasa correlatedmultivariatedistribution(i.i.d.acrossobservationsandtime),discretizedonSsupportpoints;wefixthescaleofQby constructionandimposethesinglenormalizationG(w;x)=0⇔p·w=G∗(p;x)ineachbaselinecell.BecauseG∗ispositively homogeneous,thewelfareobjectsarehomogeneousofdegreezeroin(w,scale(Q)),sothisnormalizationiswithoutlossforour comparativestatics. Logit/Gumbelarisesasaspecialcase. AppendixEdetailsthediscretizationofQ,theLP,therecoveryofw, andnormalization. 13

the second-order expansion, we adopt a common elasticity γ that does not enter step (ii): the first-order piecedependsonlyon(p,w),whileγscalesthequadraticterms. Wethenobtain: dlnG∗(p;x) ≈ (cid:88) (cid:20) ω∗(x) dlnp y (x)(cid:21) (7) dlnz¯ y dlnz¯ y=stay(x), local(x),spatial(x) (cid:124) (cid:123)(cid:122) (cid:125) firstorderapproximationterm + 1 (cid:88) (cid:150) ω∗ y (x) (cid:18)dlnp y (x)(cid:19)2(cid:153) − 1 (cid:88) (cid:88) (cid:20)ω∗ y (x)dlnp y (x)dlnp y′ (x)(cid:21) γ y=stay(x), 1−p y (x) dlnz¯ 2γ y,y’=stay(x), y′̸=y p y′ (x) dlnz¯ dlnz¯ local(x),spatial(x) local(x),spatial(x) (cid:124) (cid:123)(cid:122) (cid:125) secondorderapproximationterm as derived in Appendix D.2. Intuitively, this expression shows that the welfare impact of a labor demand shock can be captured using only three key ingredients. First, the labor supply elasticity γ governs how stronglyworkersadjusttheirchoicesinresponsetolocalchanges. Second,thebaselinechoiceprobabilities (cid:8) p (x),p (x),p (x)(cid:9) reflect the relative importance of different adjustment margins—staying stay spatial local in the same job, switching sectors or occupations locally, or moving to a different city. Third, the impulse responsesoftheseprobabilitiesalongeachmarginindicatehowthelikelihoodofstaying,switchinglocally, ormovingspatiallyrespondstoashock: (cid:26)dlnp stay|x (x) , dlnp spatial (x) , dlnp local (x)(cid:27) dlnzℓ dlnzℓ dlnzℓ Thefirst-ordertermcapturesthedirecteffectoftheshockonwelfarethroughchangesinchoiceprobabilities weighted by their importance. The second-order term accounts for curvature effects, including interactions across margins and the moderating role of labor supply elasticity. Together, these terms provide a compact but accurate approximation of the welfare consequences of local labor demand shocks, relyingonlyonobservedworkerflowsandafewstructuralparameters. FromLocalProjectionstoWelfare. Followingtheliterature,weassumeγ=2andcomputethewelfare effectsoflocallabordemandshocksusingourestimationresults. Foreachcityℓ,weobserveitssequence ofannualshocksandbaselinesector-occupationeconomicdiversificationindexHHIℓ. UsingtheLPIVestimates,wepredicttheimpliedchangesinchoiceprobabilitiesforeachmarginovertheestimationhorizon, fortherealizedaverageshocks (cid:8) B˜ + ,B˜ −(cid:9) : (cid:166) ∆lnp stay,t+h ,∆lnp spatial,t+h ,∆lnplocal,t+h (cid:169) .22 Finally,wecut t ℓ,t ℓ,t ℓ,t h mulatewelfareeffectsovertimeusingadiscountfactorβ =.99. 22B˜+ andB˜− aresimpleaveragesof (cid:166) B+ (cid:169) and (cid:166) B− (cid:169) . ToreducetheinfluenceofHHIoutliers,weusedecile-averageHHI t t ℓ,t ℓ ℓ,t ℓ ratherthaneachcity’sownvalue. Duetodisclosureconstraints,theresponseoftheemploymentstockisusedtoapproximate ∆lnpstay,t+h. ℓ,t 14

Table2. WelfareEffectsandDecompositionbyEconomyDiversification HHI PositiveShocks NegativeShocks Deciles Firstorder Secondorder Total Firstorder Secondorder Total 1st 7.10 -3.25 3.85 -1.00 -0.06 -1.06 2nd 7.18 -2.49 4.68 -1.00 -0.09 -1.09 3rd 7.03 -2.57 4.46 -0.90 -0.15 -1.05 4th 7.39 -2.82 4.57 -1.43 -0.24 -1.66 5th 6.77 -1.74 5.03 -1.82 -0.30 -2.12 6th 6.49 -0.74 5.75 -2.29 -0.33 -2.62 7th 7.36 -4.02 3.34 -2.21 -0.57 -2.78 8th 9.41 -5.82 3.59 -1.75 -0.77 -2.52 9th 8.76 -5.24 3.51 -2.20 -0.97 -3.17 10th 12.75 -6.94 5.81 -2.38 -2.55 -4.93 Notes:Wesetγ=2anddiscountfuturecontinuationvaluesrelativetothebaseperiodusingβ=0.99:   ∆ln(cid:71) ℓ ∗≡ t= 20 2 (cid:88) 1 0 9 0 Q 6Q 4 1h (cid:88)2 = 0 0 βt+h−2006  lo y ca = (cid:88) l, s s t p a a y t , ial (cid:148)ω∗ y|ℓ ∆lnp ℓ y , , t t+h(cid:151)+ γ 1 lo y ca = (cid:88) l, s s t p a a y t , ial (cid:150) 1 ω − ∗ y p |ℓ y|ℓ (cid:128)∆lnp ℓ y , , t t+h(cid:138)2 (cid:153) − 2 1 γ l y o , c y a (cid:88) ′ l = ,sp st a a t y ia , l y (cid:88) ′̸=y (cid:150)ω pn ∗ y | | ℓ ℓ∆lnp ℓ n , , t t+h∆lnp ℓ y , , t t+h (cid:153)  . ThefirstcolumndenotesHHIdeciles. Thewelfarechangesareinpercentagechangesrelativetothebaselineperiodforthe averageshockineachHHIdecilebin,separatelyreportingtheeffectofpositiveandnegativeshocks. Thenextthreecolumns present,forpositiveshocks:theFOAtermandtheSOAtermofthewelfareformulaalongwiththeirsum.Thelastthreecolumns presentsimilartermsfornegativeshocks. 4.2 Results Our welfare results are summarized in Table 2. We separately report the effect of positive and negative shocksforallHHIdeciles.23 Thefirst-orderapproximationterm(FOA)capturesthedirectwelfareimpact ofshocks,whilethesecond-orderapproximationterm(SOA)reflectstheinsurancevaluethatcomesfrom acity’seconomicdiversity. Overall, ourfindingshighlightthateconomicdiversityisakeydeterminantof theheterogeneityinwelfareoutcomesobservedacrosscities.24 Focusingfirstonpositiveshocks,oursufficient-statisticframeworkrevealsthatmoreconcentratedcities (with largestHHI values, or inhigher deciles) exhibitlarger first-order gainsthan more diversifiedcities, hinting at possible gains from specialization. That said, more negative values for SOA terms significantly alter these gains. The SOA terms are in fact large enough to revert the rising FOA gains with HHI in the upper half of HHI deciles. These findings suggest that lack of economic diversification also comes with negativeinsurancevalue,especiallyforthemostspecializedcities. The effects of negative shocks strikingly confirm that economic diversity plays a substantial role in 23Weplot,inFigureA.6intheappendix,thecity-levelvaluesthatareaverageddecile-by-deciletocreatethedecile-levelvalues reportedinTableA.7. 24TableA.7providesadditionaldecompositionalresults,indicatingtowhatextentshockandHHIheterogeneitywithineach HHIdeciledriveswelfaredifferences. 15

providing insurance against adverse economic shocks. Indeed, we observe negative second-order welfare effects (compared to the magnitude of the FOA terms) that increasing more strongly in magnitude with HHI. In other words, more diversified cities are associated with significantly smaller welfare losses from negativeshocks,duetosmallerchangefromtheSOAtermmeasuring“insurancevalue”. Thisresultechoes ourempiricalfindingsinsection3.4whereweshowedthat,inresponsetonegativeshocks,localnon-spatial churnandnetspatialinflowsfalllessinmorediversifiedlabormarkets. Ourresultsunderscoretheimportanceofeconomicdiversificationasastrategyforenhancingeconomic resilience: Amorediversifiedeconomymayfacilitatelaborreallocation,reducecostlyspatialmobility,and allowformorestableemploymentoutcomesinthefaceofsector-oroccupation-specificdownturns.25 Perhapsmoreimportantly,ourresultsillustratetheapplicabilityandusefulnessofthehigher-ordersufficientstatisticapproachusinggranularworkerflowsthatweputforwardinthispaper. 5 Conclusion This paper develops a second-order sufficient-statistics framework to analyze local shocks affect workers’ welfareandappliesittotheroleoflocaleconomicdiversityusingrichworkerflowsdatafromFrance. We findsubstantialwelfareinsurancegainsfromdiversityinthefaceoflocallabordemandshocks. Atafirst-orderapproximation,workersinmorespecializedcitiesexperiencelargerwelfaregainsfrom positive shocks, highlighting potential benefits of specialization. However, the second-order term, which capturestheinsurancevalueofdiversity,showsthathigheconomicconcentrationentailslargerlossesfrom negative shocks. This underscores the crucial role of economic diversity in buffering local labor markets againstadverseshocks. Our approach and results provide a framework for future empirical and quantitative studies of local labormarketresponses,offeringnewinsightsintohowcity-levelcharacteristicsshapetheimpactofnational economicfluctuations. 25The spatial variation in these welfare effects is depicted in Figures A.7 and A.8 in the appendix. These figures show the distributionofwelfareoutcomesunderboththe“first-orderonly”approachandthewelfareoutcomeswhenaccountingforthe insurancevaluecapturedbytheSOAterm. 16

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A Additional Figures (a)Spatialmobility (b)Nonspatialmobility FigureA.1. Mobilityofworkersacrosscities,sectorsandoccupations Notes: Figure (a) illustrates the distribution (density) of workers who change cities (spatial mobility). The lines decompose thismobilityintodifferentcombinationsoftransitionsacrosssectorsandoccupations. Specifically, i representsmovesthatare purelygeographic,isincludesbothgeographicandsectoralchanges,iorepresentsgeographicandoccupationalchanges,andiso captureschangesacrossallthreedimensions:geographic,sectoral,andoccupational.Figure(b)displaysthedensityofworkers whoremaininthesamecity(nonspatialmobility).Thelinesbreakdownnon-geographictransitions,withsrepresentingsectoronly changes, o representing occupation-only changes, and so for workers who switch both sectors and occupations without changinglocation. A1

FigureA.2. DistributionofBartikshocks Notes: This figure shows the distribution of labor demand shocks as constructedinEquation(4). Thedistributioncapturestherangeofpositiveand negativeshocksaffectingcities,highlightingtheheterogeneityinlocallabor demandchangesacrossdifferentregions. FigureA.3. DistributionofHHIoverthetimeseries Notes:ThisfigurepresentsthedistributionoftheHerfindahl-HirschmanIndex (HHI)forlocallabormarketsovertime.Ithighlightskeypercentiles,including the95th(p95),90th(p90),mean,10th(p10),and5th(p5)values,offering insightintothevariationineconomicconcentrationacrosscitiesthroughout thepanelperiod. A2

FigureA.4. DensityofBartikshockbyHHITopandBottomQuartile Notes: ThisfigureshowsthedistributionofBartikshocks,asconstructedin Equation(4),categorizedbythebottomandtopquartilesoftheHerfindahl- HirschmanIndex(HHI)distribution. Itcompareshowlabordemandshocks aredistributedacrosscitieswithlowandhigheconomicconcentration. (a)HHIandnonspatialflows (b)HHIandspatialflows FigureA.5. HHIandflows Notes:Figure(a)displaysthecorrelationbetweentheHerfindahl-HirschmanIndex(HHI)andtheshareoflocalemploymentthat transitionstoanotheroccupation,sector,orboth,withoutleavingthelocality(nonspatialflows).Figure(b)showsthecorrelation betweentheHHIandtheshareoflocalemploymentthatinvolvesgeographicmobility(spatialflows),reflectinghoweconomic concentrationinfluencesworkerrelocationacrosscities. A3

FigureA.6. Welfarecomponents: FOA,SOA,HHI Notes: Thisfigurepresentsscatterplotsillustratingtherelationshipbetweenthefirst-orderapproximation(FOA)andsecondorder approximation (SOA) terms, as constructed in Section 4. The FOA terms represent the direct welfare impact of labor demandshocks,capturingtheimmediateeffectsonlocallabormarkets. TheSOAtermsmeasurethe"insurancevalue"derived fromeconomicdiversification,reflectinghowcitieswithdifferentlevelsofsectoralandoccupationaldiversitybuffertheeffectsof theseshocks. A4

FigureA.7. HeterogeneousWelfareEffects(Negative) FigureA.8. HeterogeneousInsurance(Positive) A5

B Additional Tables TableA.1. Quarterlyworkerreallocationacrossmarkets Median SD p5 p25 p75 p95 Emp 693 6,396 170 380 1,633 5,252 i ∆Emp 0.32% 0.26 -0.11% 0.16% 0.47% 0.75% i Flow i′→i 1.67% 0.59 1.04% 1.32% 2.08% 2.89% Emp i Flow i→i′ 1.63% 0.58 1.01% 1.32% 2.00% 2.91% Emp i Flow s,o 2.96% 0.66 2.11% 2.57% 3.48% 4.17% Emp i Flow i′→i −Flow i→i′ 0.03% 0.13 -0.14% -0.05% 0.12% 0.26% Emp Flow ne→e −F i low e→ne 0.32% 0.32 -0.11% 0.16% 0.50% 0.86% Emp i Notes:Thedisplayedstatisticsarecomputedforthecross-sectionofFrenchcities.Thefirstrowdisplaystheaverage distributionofemploymentsizeacrosslocallabormarkets.Thesecondrowdisplaystheaveragequarterlygrowthrate ofemployment. Thefollowingthreerowsshowtheshareofspatialinflows,spatialoutflowsandnon-spatialflows. Thelasttermdoesnotdisplayanyflowdirectionsinceitdoesnotinvolveanyspatialcomponent.Hence,non-spatial outflowsareequivalenttonon-spatialinflows. Finally,thelasttworowsdisplaythespatialreallocation,i.e. spatial inflow minus spatial outflow, and the participation reallocation, i.e. employment inflow minus non-employment inflow. TableA.2. Responsetoalabordemandshock Emp (cid:80) Flow i′→i (cid:80) Flow i→i′ (cid:80) Flow i′→i −(cid:80) Flow i→i′ (cid:80) Flow s,o (cid:80) Flow s,o −(cid:2)(cid:80) Flow i′→i −(cid:80) Flow i→i′ (cid:3) 0.40*** 0.37*** 0.19** 0.18*** 0.40*** 0.23** (0.10) (0.11) (0.09) (0.06) (0.09) (0.09) Notes:Notes:LPIVregressionsofdifferentoutcomevariablesonshift-shareinstrumentatthecommutingzonelevel (Equation(3)).Allcolumnsindicatethecumulativeresponsesatyear5orquarter20.Thefirstcolumnprovidesthe responseoftheemploymentstock. Thesecondcolumnprovidesthespatialinflow. Thethirdcolumnprovidesthe impactonthespatialoutflow.Thefourthcolumnprovidestheimpactonthenetflow.Thefifthcolumnprovidesthe impactonallnon-spatialadjustments(reallocationacrosssectorsoroccupationslocally).Thelastcolumnprovides thedifferencebetweennon-spatialadjustmentsandnetflows.Robuststandarderrorsinparentheses. A6

ihhdnayrtemmysa :noitacollaerrekrownokitraBfotceffE .3.AelbaT 5raeY 4raeY 3raeY 2raeY 1raeY 3Q 2Q 1Q ′i→i wolF− i→′i wolF− i pmE ∗∗∗28.0 ∗∗∗67.0 ∗∗∗66.0 ∗∗∗96.0 ∗∗64.0 ∗∗∗6.0 ∗∗∗9.0 ∗∗∗17.0 + kitraB 71.0 41.0 31.0 31.0 2.0 2.0 11.0 21.0 ∗∗∗29.0− ∗∗∗80.1− ∗∗∗22.1− ∗∗∗76.0− ∗∗∗49.0− ∗∗∗57.0− ∗∗∗27.0− ∗∗∗63.0− − kitraB 71.0 41.0 41.0 91.0 22.0 41.0 51.0 1.0 ∗∗80.0− 40.0− 10.0 30.0 40.0− 40.0− ∗∗∗90.0− 20.0− IHH×+ kitraB os 40.0 30.0 40.0 40.0 40.0 40.0 30.0 40.0 ∗∗1.0− ∗∗∗71.0− ∗∗∗51.0− 90.0− ∗∗∗11.0− ∗∗∗90.0− ∗∗80.0− 50.0− IHH×− kitraB os 40.0 50.0 30.0 70.0 30.0 20.0 40.0 30.0 ′i→i wolF ∗∗62.0 ∗∗∗63.0 ∗42.0 ∗∗∗53.0 71.0 61.0 ∗∗∗43.0 ∗∗∗3.0 + kitraB 11.0 21.0 31.0 21.0 11.0 21.0 70.0 90.0 ∗∗∗24.0− ∗∗∗92.0− ∗∗∗83.0− 41.0− ∗92.0− 40.0 60.0 ∗∗∗91.0 − kitraB 1.0 11.0 90.0 90.0 71.0 90.0 60.0 70.0 40.0− ∗∗40.0− 10.0− 20.0− ∗∗40.0− 10.0− ∗∗∗50.0− 10.0− IHH×+ kitraB os 20.0 20.0 20.0 20.0 20.0 10.0 20.0 20.0 ∗∗∗80.0− ∗∗80.0− ∗∗∗80.0− ∗∗80.0− 40.0− 0.0 0.0− 0.0− IHH×− kitraB os 30.0 40.0 20.0 30.0 30.0 20.0 20.0 10.0 wolF o,s ∗∗72.0 ∗∗23.0 ∗∗73.0 ∗∗63.0 1.0 71.0 51.0 ∗∗91.0 + kitraB 31.0 31.0 71.0 81.0 11.0 11.0 1.0 80.0 ∗∗∗93.0− ∗∗∗63.0− ∗∗∗54.0− ∗∗∗23.0− ∗∗∗84.0− ∗∗∗54.0− ∗∗∗64.0− ∗∗∗91.0− − kitraB 80.0 60.0 90.0 80.0 11.0 11.0 11.0 50.0 10.0 20.0 10.0 20.0 ∗40.0 40.0 40.0 30.0 IHH×+ kitraB os 20.0 20.0 20.0 20.0 20.0 30.0 30.0 30.0 ∗∗∗50.0− ∗∗∗60.0− ∗∗∗60.0− ∗∗∗70.0− ∗∗∗70.0− ∗∗∗80.0− ∗∗∗70.0− ∗∗60.0− IHH×− kitraB os 10.0 20.0 20.0 30.0 30.0 30.0 30.0 30.0 pmE i ∗∗∗35.0 ∗∗∗5.0 ∗∗∗64.0 ∗∗∗55.0 ∗∗∗95.0 ∗∗∗65.0 ∗∗∗24.0 ∗∗∗34.0 + kitraB 41.0 11.0 31.0 12.0 61.0 11.0 1.0 11.0 ∗∗∗35.0− ∗∗∗57.0− ∗∗∗36.0− ∗∗84.0− ∗∗∗6.0− ∗∗∗26.0− ∗∗∗23.0− ∗∗2.0− − kitraB 11.0 90.0 80.0 91.0 21.0 80.0 60.0 90.0 30.0− 20.0− 0.0 0.0− 50.0− 20.0− 20.0 10.0 IHH×+ kitraB os 20.0 30.0 30.0 60.0 30.0 20.0 20.0 20.0 ∗∗30.0− ∗∗∗80.0− ∗∗40.0− 20.0 ∗∗∗70.0− ∗∗∗70.0− 10.0 40.0 IHH×− kitraB os 10.0 10.0 20.0 50.0 20.0 10.0 20.0 30.0 0219 06201 00411 04521 08631 56931 05241 53541 snoitavresbO foesnopserehtsedivorpnoitcestsrfiehT .))6(noitauqE(levelenozgnitummocehttatnemurtsnierahs-tfihsnoselbairavemoctuotnereffidfosnoissergerVIPL :setoN :setoN tsuboR.stnemtsujdalaitapsdnalaitaps-nonneewtebecnereffidehtsedivorpnoitcestsalehT.swofltenlaitapsfoesnopserehtsedivorpnoitcestxenehT.stnemtsujdalaitaps-non .sesehtnerapnisrorredradnats A7

C Additional Results Tocomplementtheanalysisinthemaintext,wealsointroduceanLPIVthatidentifiestheheterogeneous response,butwithoutfocusingontheasymmetricresponse. Theestimatingequationisspecifiedasfollows: 8 8 8 ∆y i,t+h =αh+γ t +γ i +β h Shock i,t +ψ h (cid:0) Shock i,t ×HHI i (cid:1)+ (cid:88) γh m y i,t−m + (cid:88) ωh m Shock i,t−m + (cid:88) δh m Z i,t−m m=1 m=1 m=1 In this equation, ∆y i,t+h represents the change in the outcome variable of interest for location i at horizon h, allowing us to model the response over a specified time horizon. The intercept αh is a timespecific constant that adjusts for general trends in the data, while the fixed effects γ and γ control for t i temporalandspatialheterogeneity. Thesefixedeffectsareessentialtoensurethattheestimatedresponse totheshockisnotbiasedbytime-invariantlocation-specificfactorsorbroadermacroeconomictrends. The key term, β Shock , captures the direct effect of the shock at time t on the outcome at time t+h. The h i,t shockShock isanexogenousdisturbanceaffectinglocationiandisassumedtovaryacrosslocationsand i,t time. Thetermisfurthermoreinteractedwiththelocal HHI asintroducedinthemaintext. i Toaccountforpersistenceintheoutcomeandthepotentialforshockstohavelastingeffects,themodel includes lags of both the outcome variable and the shock itself. The terms (cid:80)8 m=1 γh m y i,t−m represent the influenceofpastvaluesoftheoutcomeoncurrentchanges,whiletheterms (cid:80)8 m=1 ωh m Shock i,t−m capture thedynamiceffectsofpastshocks. Theselaggedtermsallowustomodelthefulldynamicresponseofthe outcometoshocks,recognizingthattheeffectofashockmaypersistovermultipleperiods. A8

TableA.4. EffectofBartikonworkerreallocation Q1 Q2 Q3 Year1 Year2 Year3 Year4 Year5 Emp i Bartik 0.30*** 0.37*** 0.61*** 0.61*** 0.53*** 0.43** 0.60** 0.65*** (0.09) (0.07) (0.09) (0.14) (0.13) (0.16) (0.24) (0.17) Bartik×HHI 0.02 0.01 -0.05*** -0.06** -0.03* -0.00 -0.05 -0.05* so (0.02) (0.01) (0.02) (0.02) (0.02) (0.03) (0.04) (0.03) Flow s,o Bartik 0.22*** 0.35*** 0.36*** 0.34*** 0.35*** 0.34** 0.35** 0.43*** (0.06) (0.10) (0.10) (0.10) (0.11) (0.13) (0.15) (0.16) Bartik×HHI -0.02 -0.03 -0.03* -0.03*** -0.02* -0.01 -0.00 -0.02 so (0.02) (0.02) (0.02) (0.01) (0.01) (0.02) (0.02) (0.02) Flow i′→i Bartik 0.07 0.12** 0.35*** 0.43*** 0.37*** 0.26* 0.41*** 0.55*** (0.05) (0.06) (0.12) (0.15) (0.12) (0.14) (0.14) (0.18) Bartik×HHI 0.02*** 0.04*** -0.02 -0.05** -0.02 0.00 -0.02 -0.06** so (0.01) (0.01) (0.02) (0.02) (0.02) (0.02) (0.02) (0.03) Flow i→i′ Bartik 0.03 0.12** 0.05 0.23** 0.36*** 0.37** 0.38*** 0.56*** (0.04) (0.05) (0.07) (0.10) (0.10) (0.18) (0.14) (0.12) Bartik×HHI -0.00 -0.02** 0.00 -0.04* -0.06*** -0.04 -0.04* -0.09*** so (0.01) (0.01) (0.01) (0.02) (0.01) (0.03) (0.02) (0.02) Observations 14,535 14,250 13,965 13,680 12,540 11,400 10,260 9,120 Notes:Notes:LPIVregressionsofdifferentoutcomevariablesonshift-shareinstrumentatthecommutingzonelevel (Equation(C))interactedwiththeHHI.Thefirstsectionprovidestheresponseoftheemploymentstock.Thesecond sectionprovidestheresponseofnon-spatialadjustments. Thenextsectionprovidestheresponseforoutflows. The lastsectionprovidestheresponseoftheinflowmargin.Robuststandarderrorsinparentheses. A9

TableA.5. EffectofBartikonworkerreallocation Q1 Q2 Q3 Year1 Year2 Year3 Year4 Year5 Flow i′→i −Flow i→i′ Bartik 0.03 0.02 0.32** 0.22** -0.01 -0.13* 0.01 -0.04 (0.05) (0.06) (0.14) (0.09) (0.09) (0.07) (0.08) (0.06) Bartik×HHI 0.03** 0.05*** -0.02 -0.01 0.05*** 0.05*** 0.02 0.04** so (0.09) (0.01) (0.02) (0.01) (0.02) (0.01) (0.02) (0.01) Flow s,o −[Flow i′→i −Flow i→i′ ] Bartik 0.20*** 0.37*** 0.06 0.14 0.36*** 0.48*** 0.35** 0.47*** (0.07) (0.09) (0.10) (0.08) (0.10) (0.17) (0.17) (0.11) Bartik×HHI -0.05** -0.08*** -0.01 -0.02 -0.07*** -0.06** -0.02 -0.05** so (0.02) (0.02) (0.02) (0.02) (0.02) (0.02) (0.04) (0.02) Flow i,s,o Bartik 0.33*** 0.58*** 0.76*** 1.00*** 1.07*** 0.96** 1.13*** 1.51*** (0.10) (0.14) (0.19) (0.31) (0.26) (0.42) (0.39) (0.44) Bartik×HHI 0.00 -0.00 -0.05* -0.11** -0.10*** -0.04 -0.06 -0.16** so (0.03) (0.03) (0.03) (0.05) (0.04) (0.06) (0.05) (0.06) Flow ne→e −Flow e→ne Bartik 0.30*** 0.43*** 0.31** 0.37*** 0.50*** 0.52*** 0.54** 0.68*** (0.08) (0.07) (0.14) (0.13) (0.09) (0.17) (0.12) (0.09) Bartik×HHI -0.01 -0.05*** -0.04 -0.05** -0.07*** -0.04 -0.07* -0.09*** so (0.01) (0.02) (0.03) (0.02) (0.02) (0.03) (0.03) (0.02) Emp i −Flow i′→i −Flow i→i′ Bartik 0.53*** 0.80*** 0.70*** 0.74*** 0.88*** 0.89*** 0.93** 1.12*** (0.12) (0.13) (0.15) (0.19) (0.18) (0.27) (0.35) (0.26) Bartik×HHI -0.03 -0.08*** -0.07** -0.08*** -0.10*** -0.06** -0.07 -0.11*** so (0.03) (0.02) (0.03) (0.03) (0.03) (0.03) (0.05) (0.03) Observations 14,535 14,250 13,965 13,680 12,540 11,400 10,260 9,120 A10

TableA.6. WelfareEffectsandDecompositionbyEconomyDiversification HHI PositiveShocks NegativeShocks Deciles Firstorder Secondorder Total Firstorder Secondorder Total 1st 3.54 -1.94 1.60 -0.44 -0.03 -0.46 2nd 3.70 -1.50 2.20 -0.42 -0.04 -0.46 3rd 3.71 -1.55 2.15 -0.35 -0.07 -0.42 4th 3.86 -1.70 2.16 -0.56 -0.10 -0.66 5th 3.64 -1.06 2.58 -0.69 -0.13 -0.82 6th 3.50 -0.46 3.04 -0.90 -0.14 -1.04 7th 3.91 -2.38 1.53 -0.85 -0.23 -1.08 8th 5.02 -3.41 1.61 -0.70 -0.31 -1.01 9th 4.75 -3.06 1.69 -0.84 -0.39 -1.23 10th 6.92 -3.97 2.95 -0.92 -0.99 -1.91 Notes:Wesetγ=2anddiscountfuturecontinuationvaluesrelativetothebaseperiodusingβ=0.9:   ∆ln(cid:71) ℓ ∗≡ t= 20 2 (cid:88) 1 0 9 0 Q 6Q 4 1h (cid:88)2 = 0 0 βt+h−2006  lo y ca = (cid:88) l, s s t p a a y t , ial (cid:148)ω∗ y|ℓ ∆lnp ℓ y , , t t+h(cid:151)+ γ 1 lo y ca = (cid:88) l, s s t p a a y t , ial (cid:150) 1 ω − ∗ y p |ℓ y|ℓ (cid:128)∆lnp ℓ y , , t t+h(cid:138)2 (cid:153) − 2 1 γ l y o , c y a (cid:88) ′ l = ,sp st a a t y ia , l y (cid:88) ′̸=y (cid:150)ω pn ∗ y | | ℓ ℓ∆lnp ℓ n , , t t+h∆lnp ℓ y , , t t+h (cid:153)  . ThefirstcolumndenotesHHIdeciles. Thewelfarechangesareinpercentagechangesrelativetothebaselineperiodforthe averageshockineachHHIdecilebin,separatelyreportingtheeffectofpositiveandnegativeshocks. Thenextthreecolumns present,forpositiveshocks:theFOAtermandtheSOAtermofthewelfareformulaalongwiththeirsum.Thelastthreecolumns presentsimilartermsfornegativeshocks. TableA.7. ExtendedWelfareEffectsandDecompositionbyEconomyDiversification HHI PositiveShocks NegativeShocks Deciles FOA(Avg) FOA(Obs) ∆Exposure SOA Total FOA(Avg) FOA(Obs) ∆Exposure SOA Total 1st 7.03 7.10 0.07 -3.25 3.85 -1.53 -1.00 0.54 -0.06 -1.06 2nd 7.20 7.18 -0.02 -2.49 4.68 -1.56 -1.00 0.56 -0.09 -1.09 3rd 7.34 7.03 -0.31 -2.57 4.46 -1.58 -0.90 0.68 -0.15 -1.05 4th 7.49 7.39 -0.11 -2.82 4.57 -1.61 -1.43 0.18 -0.24 -1.66 5th 7.68 6.77 -0.91 -1.74 5.03 -1.64 -1.82 -0.19 -0.30 -2.12 6th 7.82 6.49 -1.33 -0.74 5.75 -1.66 -2.29 -0.63 -0.33 -2.62 7th 7.96 7.36 -0.60 -4.02 3.34 -1.68 -2.21 -0.54 -0.57 -2.78 8th 8.19 9.41 1.22 -5.82 3.59 -1.71 -1.75 -0.04 -0.77 -2.52 9th 8.53 8.76 0.22 -5.24 3.51 -1.76 -2.20 -0.44 -0.97 -3.17 10th 10.28 12.75 2.47 -6.94 5.81 -2.03 -2.38 -0.35 -2.55 -4.93 Notes:Wesetγ=2anddiscountfuturecontinuationvaluesrelativetothebaseperiodusingβ=0.99: ∆ln(cid:71) ℓ ∗≡ t= 20 2 (cid:88) 1 0 9 0 Q 6Q 4 1h (cid:88)2 = 0 0 βt+h−2006 (cid:26) y∈{stay, (cid:88) local,spatial} ω∗ y|ℓ ∆lnp ℓ y , , t t+h+ γ 1(cid:88) y 1 ω − ∗ y p |ℓ y|ℓ (cid:128)∆lnp ℓ y , , t t+h(cid:138)2− 2 1 γ y (cid:88) ̸=y′ ω py ∗ y ′| | ℓ ℓ∆lnp ℓ y , ′ t ,t+h∆lnp ℓ y , , t t+h (cid:27) . (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) FOA SOA(own) SOA(cross) ThefirstcolumnreportsHHIdeciles. Weseparatelyreportpositiveandnegativeshocks. Foreachshocksign,thefivecolumns are:(i)FOA(Avg):first-orderwelfarecomputedusingacommonshockpathanddecile-meanHHI(holdsexposureheterogeneity fixed);(ii)FOA(Obs):first-orderwelfareusingeachdecile’sobservedHHIandexposure;(iii)∆Exposure≡FOA(Obs)−FOA(Avg), whichisolatesthecontributionofexposureheterogeneity;(iv)SOA:second-orderterm(ownandcrosscomponentscombined); (v)Total=FOA(Obs)+SOA.Allentriesarepercentagechangesrelativetothebaselineperiod;rowsaredecile-binaverages. A11

D Additional Derivations This appendix section provides additional derivations for the results in Section 2 and 4 as well as a motivationfortheutilizationofHHIindices. D.1 Proof of Proposition 1 and 2 (Section 2) Inthissubsection,weprovideadditionalderivationsfortheresultsinSection2. Specifically,wepresentthe derivationsfortheapproximationresults. Tobegin,itwillbehelpfultoreiteratekeyresultsfromChiong, GalichonandShum(2016),specifically,theirTheorem1,whichrelatesconditionalchoiceprobabilitiesand continuation values to the subdifferential of the (cid:71) function and the convex conjugate of the (cid:71) function, i.e. p∈∂(cid:71)(w) and, w∈∂(cid:71)∗(p) where the former equation is a generalization of the Williams-Daly-Zachery (WDZ) theorem as discussedinChiong,GalichonandShum(2016)andthelattergeneralizesresultsinHotzandMiller(1993). Wethenstartbyapproximatingthe(cid:71)∗ functionaround w , 0 (cid:88)∂(cid:71)(w) (cid:128) (cid:138) 1(cid:88)(cid:88) ∂2(cid:71)(w) (cid:128) (cid:138)(cid:128) (cid:138) (cid:71)(w)=(cid:71)(w0)+ y ∂w y w y −w0 y + 2 y y′ ∂w y ∂w y′ w y −w0 y w y′ −w0 y′ +o(·) (cid:88)∂(cid:71)(w) 1(cid:88)(cid:88) ∂2(cid:71)(w) d(cid:71)(w)= y ∂w y dw y + 2 y y′ ∂w y ∂w y′ dw y dw y′ +o(·) andsimilarly,fortheconvexconjugatefunction, (cid:71)∗(p)=(cid:71)∗(p0)+ (cid:88) y ∂(cid:71) ∂p ∗( y p) (cid:128) p y −p0 y (cid:138) + 1 2 (cid:88) y (cid:88) y′ ∂ ∂2 p (cid:71) y ∂ ∗( p p y ) ′ (cid:128) p y −p0 y (cid:138)(cid:128) p y′ −p0 y′ (cid:138) +o(·) (cid:88)∂(cid:71)∗(p) 1(cid:88)(cid:88) ∂2(cid:71)∗(p) d(cid:71)∗(p)= y ∂p y dp y + 2 y y′ ∂p y ∂p y′ dp y dp y′ +o(·) Intermsoflogchangeswehave, dln(cid:71)(w)= (cid:88) ∂ ∂ (cid:71) (cid:71) w ( ( w y w ) w ) y dlnw y + 1 2 (cid:88)(cid:88) ∂ ∂ w 2 y (cid:71) ∂ ( (cid:71) w w) y ( ′ w w ) y w y′ dlnw y dlnw y′ +o(·) y y y′ ∂(cid:71)∗(p) ∂2(cid:71)∗(p) dln(cid:71)∗(p)= (cid:88) (cid:71) ∂p ∗ y (p p ) y dlnp y + 1 2 (cid:88)(cid:88) ∂p y ∂ (cid:71) p ∗ y ( ′ p p ) y p y′ dlnp y dlnp y′ +o(·) y y y′ ApplyingTheorem1fromChiong,GalichonandShum(2016),weobtain, dln(cid:71)(w)= (cid:88) (cid:71) p y ( w w y ) dlnw y + 1 2 (cid:88)(cid:88) ∂ ∂ w p y y ′ (cid:71) w p ( y y w ′ p ) y w y dlnw y dlnw y′ +o(·) y y y′ A12

dln(cid:71)∗(p)= (cid:88) (cid:71) w ∗ y ( p p y ) dlnp y + 1 2 (cid:88)(cid:88) ∂ ∂ p w y y ′ (cid:71) p w ∗ y y ( ′ w p) y p y dlnp y dlnp y′ +o(·) y y y′ Definingthecross-elasticity,ϱp,w ≡ ∂lnp y = ∂p y w y′ andϱw,p ≡ ∂lnw y = ∂w y p y′ , y,y′ ∂lnw y′ ∂w y′ p y y,y′ ∂lnp y′ ∂p y′ w y dln(cid:71)(w)= (cid:88) (cid:71) p y ( w w y ) dlnw y + 1 2 (cid:88)(cid:88) ϱp y, ,w y′(cid:71) p y ( w w y ) dlnw y dlnw y′ +o(·) y y y′ dln(cid:71)∗(p)= (cid:88) (cid:71) w ∗ y ( p p y ) dlnp y + 1 2 (cid:88)(cid:88) ϱw y, , y p ′(cid:71) w ∗ y ( p p y ) dlnp y dlnp y′ +o(·) y y y′ Rewriteintermsofgenericweights,ω ≡ w y p y andω∗ ≡ w y p y, y (cid:71)(w) y (cid:71)∗(p) (cid:88) 1(cid:88)(cid:88) dln(cid:71)(w)= ω y dlnw y + 2 ϱp y, ,w y′ ω y dlnw y dlnw y′ +o(·) y y y′ (cid:88) 1(cid:88)(cid:88) dln(cid:71)∗(p)= ω∗ y dlnp y + 2 ϱw y, , y p ′ ω∗ y dlnp y dlnp y′ +o(·) y y y′ whichisthestatedresultandgivesusasecond-ordercharacterizationofthe(conjugate)socialsurplus.1 D.2 Derivations for Section 4 Turningtowardsanisoelasticlaborsupplysystem,i.e. γ w p = y y (cid:80) γ w y y Wecantotallydifferentiateandobtain, ϵw y, , y p ′ ≡ ∂ ∂ l l n nw p y y ′ = (cid:18) ∂ ∂ w p y y ′ w p y y ′ (cid:19)−1 =(cid:0)−p y′ γ(cid:1)−1 if y = y ′ ϵw,p ≡ ∂ lnw y = (cid:18) ∂p y w y′ (cid:19)−1 =(cid:0)γ−p γ(cid:1)−1 if y ̸= y ′ y,y′ ∂ lnp y′ ∂w y′ p y y Substituting,weobtain, ω∗ ω∗ dln(cid:71)∗(p)= (cid:88) y ω∗ y dlnp y + γ 1(cid:88) y 1− y p y (cid:0) dlnp y (cid:1)2− 2 1 γ (cid:88) y y (cid:88) ′̸=y p y y ′ dlnp y dlnp y′ +o(·) Assuming, y ∈(cid:89) =[Stay,Local,Spatial],weobtain, 1Throughoutthetext,whatwecalla“second-orderapproximationofthesocialsurplus”isformallyafirst-orderlog-linearization ofafullsecond-orderTaylorexpansioninlevelsofG(orG∗). A13

dlnG∗(p;x) ≈ (cid:88) (cid:20) ω∗ (x) dlnp y (x)(cid:21) + 1 (cid:88) (cid:150) ω∗ Y (x) (cid:18)dlnp y (x)(cid:19)2(cid:153) dlnx¯ Y dlnx¯ γ 1−p (x) dlnx¯ y=stay, y=stay, y local,spatial local,spatial − 1 (cid:88) (cid:88) (cid:20)ω∗ y (x) dlnp y (x) dlnp n (x)(cid:21) 2γ y=stay, y′̸=y p y′ (x) dlnx¯ dlnx¯ local,spatial asreportedinSection4. Note that, while assuming stationarity is convenient because it allows us to interpret our first- and second-order welfare approximations as steady-state elasticities and to obtain a clean mapping between continuationvaluesandConditionalChoiceProbability(CCP),asdiscussedinChiong,GalichonandShum (2016),itisnotstrictlynecessary. Indeed,theseapproximationscouldalsobecomputedperiodbyperiod inanon-stationarysetting. D.3 Proof of Corollary 1 (Section 2) Wenowconnectoursecond-orderwelfareapproximationtoanindexoflocallabormarketdiversity. Consider a city (cid:67) with local sub-population shares π ℓ. Starting from the general expression for welfare changes, (cid:88) dlnW(cid:67) = π ℓdln(cid:71) ℓ ℓ∈(cid:67) (cid:20) (cid:21) = ℓ (cid:88) ∈(cid:67) π ℓ (cid:88) m w (cid:71) m ℓ p m|ℓdlnw m + 1 2 (cid:88) m w (cid:71) m ℓ p m|ℓ γ ℓ (cid:0) dlnw m (cid:1)2− 1 2 (cid:88) m,k w (cid:71) m ℓ p m|ℓ γ ℓp k|ℓdlnw m dlnw k , (8) where p m|ℓ areconditionalchoiceprobabilitiesandγ ℓ governstheircrosssubstitutionelasticitiessuchthat ∂ ∂ l l n n w p m k| | ℓ ℓ ≡−γ ℓp m|ℓ +γ ℓ 1 {m=k}. Define the log-elasticity weights λ m|ℓ ≡ w (cid:71) m ℓ p m|ℓ (which satisfy (cid:80) m λ m|ℓ = 1) and let s m ≡ dlnw m denotethevectorofshocks. Then(8)becomes   (cid:88) (cid:88) 1(cid:88) 1(cid:88) dlnW(cid:67) = π ℓ λ m|ℓs m + 2 γ ℓ λ m|ℓs m 2 − 2 γ ℓ λ m|ℓ λ k|ℓs m s k. (9) ℓ m m m,k Takingexpectationswith(cid:69)[s]=0andCov(s)=Σeliminatesthelineartermandyields (cid:20) (cid:21) (cid:69)(cid:2) dlnW(cid:67) (cid:3)= 1(cid:88) π ℓ γ ℓ (cid:88) λ m|ℓ Σ mm −λ⊤ ℓ Σλ ℓ . (10) 2 ℓ m Under equicorrelation (Σ mm =σ2 and Σ mk =νσ2 for m̸= k) and common elasticities (γ ℓ =γ∀ℓ), this simplifiesto   (cid:69)(cid:2) dlnW(cid:67) (cid:3)= 1 2 σ2(1−ν)γ    1− (cid:88) π ℓ (cid:88) λ2 m|ℓ    , (11)  ℓ m  (cid:124) (cid:123)(cid:122) (cid:125) HHIℓ A14

linkingtheexpectedwelfarechangetothecomplementofaconcentrationindexakintoHerfindahl–Hirschman index(HHI). Thisformalizestheintuitionthatthe“insurancevalue”ofdiversityisproportionaltotheshockvariance (σ2)andinverselyrelatedtomarketconcentration(HHI(cid:67) ≡(cid:80) ℓ π ℓHHIℓ). A15

E Computational Details Thissection explainshowwerecover welfare-consistentchoicevaluesfrom observedprobabilitiesviathe mass-transportapproachandusetheminourempiricalwelfareaccounting. ThefirstsubsectionformallysummarizestheChiong,GalichonandShum(2016)setup–convexsurplus, its conjugate G ∗(p), and the optimal-transport formulation whose dual delivers continuation values and, afterasinglenormalization,welfareweights. The second subsection details how we deploy their approach and combine it with estimated dynamic responses (including sign asymmetry and HHI heterogeneity) to derive first- and second-order welfare effects. Thefinalsubsectionprovidesapseudo-codeasanoverviewofourapproach. E.1 Mass-Transport Approach (Chiong, Galichon and Shum, 2016) LetY beafinitesetofactionsand,forstate x,letw=(w y ) y∈Y denotechoice-specificcontinuationvalues andϵ=(ϵ y ) y∈Y ∼Q(·| x)idiosyncraticshocks. Thesocialsurplus (cid:149) (cid:12) (cid:152) (cid:12) G(w;x)=(cid:69) max{w +ϵ } (cid:12)x y∈Y y y (cid:12) isconvex,and(underregularity)mapsvaluestoCCPsviatheWilliams–Daly–Zacharyrelationp=∇G(w). Itsconvexconjugate G ∗(p;x)=sup{p·w−G(w;x)} w satisfies G(w;x)+G ∗(p;x)=p·wwith p∈∂G(w;x) ⇐⇒ w∈∂G ∗(p;x). Hence,givenobservedCCPs ∗ p,onecaninverttorationalizingvalues wusingthesubgradientof G . Mass-transportformulation(CGS). Chiong,GalichonandShum(2016)showG ∗(p;x)equalsthevalue ofanoptimal-transportproblemthatcouplesthemultinomialpoverY withtheshockdistributionQ,under cost c(y,ϵ) = −ϵ . With a discrete approximation Q ≈ {ϵ(s)}S (with weights 1/S per Kennan, 2006), y s=1 thisbecomesthelinearprogram (cid:88) (cid:88) (cid:88) max ϵ(s)π s.t. π =p ∀y, π = 1 ∀s, π ≥0 y ys ys y ys S ys y,s s y whose objective equals G ∗(p;x) (up to sign). The associated dual variables deliver a subgradient w ∈ ∂G ∗(p;x),i.e.,thecontinuationvalues(uptoanadditiveconstant). Thismass-transportapproachaccommodatesgeneral(includingcorrelated)shockdistributionsandprovidesanumericallyconvenientinversion fromCCPstovalues,whichweuseasthebasisforourwelfareobjectsandweights. E.2 Implementation Overview Welfare weights. For each city–year cell, we start from observed conditional choice probabilities over three margins (stayers, spatial moves, non-spatial switches). We approximate the distribution of choice shockswithafinitesupportandsolvetheassignmentlinearprogramwhosevalueequalstheconvexconjugate G ∗(p) of the social surplus. The program’s dual variables deliver a subgradient of G ∗ , which we interpret as choice–specific continuation values, unique up to an additive constant. We impose a single normalizationsothatthedotproductofthesevalueswithobservedprobabilitiesequalsG ∗(p)inthatcell. Theresultingtheory–consistentwelfareweightsare ω∗ =(w p )/G ∗(p).2 y y y 2We set S=1000 support points for the shock discretization, with a positive covariance across margins (diagonals around onehalf, moderateoff–diagonalsinthecovariancematrixasinthebaselinecodeinChiong, GalichonandShum,2016). The A16

Impulse responses, shocks, and welfare effects. We then combine these weights with dynamic responsesofprobabilitiesthatweestimateusinglocalprojectionswithexternalshifters. Theempiricalspecificationallows(i)separatepositiveandnegativeshocks,(ii)dynamiceffectsovermultiplehorizonsincluding the contemporaneous term, and (iii) heterogeneity by market concentration via interactions with a Herfindahlindex. Fromtheseestimatedimpulseresponses,weformpredictedpathsforthethreemargins under three counterfactual scenarios of concentration and shock realization: holding both concentration andshiftersattheiracross–cellaverages,holdingshiftersattheiraveragewhilelettingconcentrationvary bycell,andallowingbothtovaryatobservedvalues. Perperiodandcell,first–orderwelfareistheinnerproductofweightsandpredictedprobabilitychanges. Second–order terms expand welfare around the baseline using the multinomial geometry: own–margin quadratic components enter with w /(1−p ) and cross–margin components enter with the appropriate y y negative terms scaled by w y /p y′. We discount over horizons and sum across time. For reporting, we aggregate by concentration quantiles, produce scatter and binscatter diagnostics against log concentration, andcompiletablesthatseparatepositiveandnegativecontributionsanddecomposesecond–ordereffects intodiagonalandoff–diagonalparts. assignmentproblemissolvedwithastandardLPsolver. A17

E.3 Pseudo-code Implementation Pseudo-code—MTAInversionandWelfare 1. Inputs(perbaselinecell). Choiceprobabilitiesp;discretizationsizeS;discountfactorδ;elasticityγ;impulse-responsecoefficientswithsignasymmetryandHHIinteractions;counterfactual pathofshocks. 2. Discretizeshocks. Draw{ϵ(s)}S fromacorrelatedmultivariatedistribution;set w =1/S. s=1 s 3. Mass-transportprogram(conjugateside). Solve (cid:88) (cid:88) (cid:88) max ϵ(s)π s.t. π =p , π =w . π ≥0 y ys ys y ys s ys y,s s y 4. Recover continuation values and normalize. Construct a subgradient w ∈ ∂G ∗(p) from the duals and impose the single normalization (cid:80) w p = G ∗(p). Form welfare weights ω∗ = y y y y (w p )/G ∗(p). y y 5. Build probability paths. From the estimated responses, create ∆p (h) over horizons h = y 0,...,4 with separate positive/negative components and HHI interactions; generate variants withaveragedvs.observedHHIandshifters. 6. First-orderwelfare(FOA).Foreachcellandhorizon, (cid:88) ∆W (1)(h)= ω∗ ∆p (h) y y y 7. Second-orderwelfare(SOA).Foreachcellandhorizon, ω∗ ω∗ ∆W (2)(h)= γ 1(cid:88) y 1− y p y (cid:2)∆p y (h)(cid:3)2− 2 1 γ (cid:88) y y (cid:88) ̸=y′ p y y ′ ∆p y (h)∆p y′ (h), 8. Discountedwelfareeffects. Sumoverhorizonshforcell-leveltotals∆W(h)=δh(cid:2)∆W (1)(h)+ 1∆W (2)(h)(cid:3) ;aggregateoverallandbyHHIquantiles;reportFOAvs.SOA,positivevs.negative, γ anddiagonalvs.off-diagonalcomponents. A18