Declining Search Frictions, Unemployment, and Growth Revisited

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Declining Search Frictions, Unemployment, and Growth Revisited Juan Carlos C´ordoba, Anni T. Isoj¨arvi, Haoran Li 2025-098 Please cite this paper as: C´ordoba, Juan Carlos, Anni T. Isoj¨arvi, and Haoran Li (2025). “Declining Search Frictions, Unemployment, and Growth Revisited,” Finance and Economics Discussion Series 2025-098. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2025.098. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Declining Search Frictions, Unemployment, and Growth Revisited∗ Juan Carlos Co´rdoba† Anni T. Isoja¨rvi‡ Haoran Li§ October 21, 2025 Abstract Thispaperrevisitstheconditionsunderwhichsearchmodelsgeneratebalancedgrowthpaths (BGPs)—equilibria where unemployment, vacancies, and job flows remain steady as search frictionsdecline. MartelliniandMenzio(2020)claimthatsuchpathsexistonlywhenmatches are“inspectiongoods”andmatchqualityfollowsaParetodistribution. Weshowthatthese conditions are sufficient but not necessary. Their implementation assumes a strong form of stationarity—requiringtheendogenousdistributionofmatchqualitiestoremaininvariantunder proportional scaling. This restriction forces the reservation quality to grow at a constant, strictlypositiverate,mechanicallytyingdecliningfrictionstolong-termgrowthandyielding counterfactual implications of eliminating search frictions—persistent unemployment and infinitewelfaregains. Relaxingthisrestriction,balancedgrowthcanariseunderalternative forms of scaling, such as additive transformations that restore stationarity without Pareto tailsorinspection. Wefurthershowthatbiasedtechnologicalprogress,whenvacanciesand unemployedworkersarecomplementaryinputs,alsogenerateswell-behavedBGPswithfinite welfaregainsandvanishingunemploymentassearchfrictionsdisappear. Keywords: searchfrictions;balancedgrowth;inspectionmodels;Paretotails;biasedtechnological change JELCodes: E24;J64;O41 ∗We thank participants at the Federal Reserve Bank of Kansas City Workshop (October 2024), the 2025 SED Conference, the2025SAETConference, the2025Omaha MacroeconomicForum, Bancodela Repu´blica, Guido Menzio,DirkKrueger,andAndyGloverforhelpfulcommentsandsuggestions. †IowaStateUniversity,DepartmentofEconomics. E-mail: cordoba@iastate.edu. All errors our own. ‡BoardofGovernorsoftheFederalReserveSystem. E-mail: anni.t.isojaervi@frb.gov. Theviewsexpressedare thoseoftheauthorsandnotnecessarilythoseoftheFederalReserveBoardortheFederalReserveSystem. §SchoolofAppliedEconomics,RenminUniversityofChina. E-mail: haoranl@ruc.edu.cn. 1

”Ifsearchfrictionsinthelabormarkethavediminishedoverthelast90years,why do we not see a secular inward shift of the Beveridge curve, a secular negative trend in the unemployment rate, and a secular rise in the UE rate?... We seek a balanced growthpath(BGP)forthiseconomy,thatis,anequilibriumalongwhichunemployment, vacancies,UE,andEUratesareconstantovertime... ABGPexistiff(a)thequalityof afirm-workermatchisasample fromaParetodistributionwithsometailcoefficient... and(b)theworkers’benefitfromunemploymentandthefirms’costofmaintaininga vacancygrowatthesamerateasaverageproductivity. Theassumptionthatmatchesare inspectiongoodscouldbeconsideredthethirdconditionfortheexistenceofaBGP.” (MartelliniandMenzio,2020,pp.4392). 1. Introduction Despitedramaticimprovementsinjob-searchtechnologyoverthepastcentury—fromnewspaper classifieds to online platforms to algorithmic matching—aggregate labor market outcomes have remained remarkably stable. Unemployment rates show no secular decline, the Beveridge curve has not shifted inward over the long run, and job-finding and separation rates appear stationary. This apparent disconnectbetween technologicalprogressand stable labor-market outcomes posesa fundamental puzzle for macroeconomic theory: how can declining search frictions coexist with steadyemploymentoutcomes? Martellini andMenzio (2020) (hereafterMM) offer aninfluential answer. In the spirit ofKing, Plosser,andRebelo(1988),theyseeknecessaryandsufficientconditionsforbalancedgrowthin search-theoreticmodels ofunemployment. Theirsolutionisstrikinglysharp: abalancedgrowth path exists if and only if firm–worker matches are “inspection goods” and match quality follows a Pareto distribution.1 Under these assumptions, the long tail of the Pareto distribution induces increasinglyselective matchingthat offsetstheeffectsofdecliningfrictions,therebypreserving the stabilityoflabormarketoutcomeswhilegeneratinglong-rungrowth. This paper revisits and challenges that characterization. We show that MM’s conditions are sufficientbutnotnecessary. Thekeyliesintheirimplementationofthebalanced-growthconcept. In their definition—stated in the abstract and introduction—a BGP requires that aggregate labor variables such as unemployment, vacancies, and job flows remain constant over time, but says nothing about the underlying, unobservable distribution of match qualities. MM nevertheless impose astrong assumption on thislatent distribution: itmust remaininvariantunder proportional scaling. Thisrestriction—commoningrowthmodelsbutuntestedandempiricallyunmotivatedin 1Inspectionmeansthatwhenaworkerandavacancymeet,thematch-specificproductivityisrevealedandtheparties decidewhethertoformanemploymentrelationship. 2

labor-marketsettings—forcesboththesamplingandtheendogenousmatch-qualitydistributionsto beParetoandmechanicallyturnsdecliningsearchfrictionsintoasourceoflong-rungrowth. Yet proportionalscalingisnotrequiredforastationaryBGP.2 Oncescalingistreatedmoregenerally, theParetostructureisnolongernecessary,anddecliningfrictionsneednotaffecttheeconomy’s growthrate. Wedemonstratethatalternativetimetransformationscanrestorestationaritywithoutinvoking Pareto tails. For example, under exponential sampling of match quality, improvements in search efficiency cause the reservation quality to drift linearly over time, rather than multiplicative as in the Pareto case. This additive transformation of time ensures that the distribution of accepted matches remains stationary even as search efficiency improves exponentially. In equilibrium, self-selectionadjustsjustenoughtooffsetthedestabilizingeffectofimprovedmatching—restoring thetime-invarianceoflabormarketoutcomesbutwithoutgeneratinglong-rungrowth. The restrictive structure implied by multiplicative scaling also generates counterfactual implications. In the MM setup, unemployment persists even when posting vacancies is costless, and the welfare gains from eliminating search frictions are infinite. In contrast, the canonical Diamond–Mortensen–Pissarides(DMP)frameworkpredictsthatunemploymentvanishesasposting costsfallandthatwelfaregainsarefinite,boundedbytheforgoneoutputofunemployedworkers. Under exponential sampling, one core DMP prediction is restored—unemployment vanishes as postingcostsfall—but,asintheParetoinspectioncase,thewelfaregainsfromeliminatingfrictions remainunboundedbecauseever-easiermatchinginducesever-higherreservationqualitywhenmatch qualityisunboundedabove. Balancedgrowthcanalsoariseoutsidetheinspectionframework. Whentechnologicalprogress isinput-biasedandvacanciesandunemployedworkersarecomplementaryinputsinthematching function,theeconomyconvergestoawell-behavedBGPwithstationaryunemployment,tightness, and job-finding rates. In this setting, unemployment vanishes as frictions disappear only if technological progress is worker-augmenting. Unlike inspection models, the welfare gains from eliminatingfrictionsarefinite. However,theresultingBGPisnecessarilyinefficient: themarket equilibrium supports a stable labor market with declining frictions, but the planner’s allocation doesnot. Thisinefficiencyarisesbecausethebargainingweightinthemarketequilibriumisfixed, whereastheplanner’sshadowbargainingpowervarieswithtightness,preventingtheHosioscondition (Hosios, 1990)from beingsatisfied. Thus, while thebiased-technology DMPframework restores thecorequalitativepropertiesofDMP modelsandavoids theimplausiblewelfareimplicationsof inspection,itrevealsanintrinsicinefficiencybetweenmarketandplannerallocations. Together,theseresultsshowthatMM’scharacterizationoverstatestheconditionsrequiredfor 2MMdonotexplicitlyusethestationaryterminology. Weuseittoclarityourcontribution,asexplainedinSection 3.5. 3

balancedgrowth. Paretodistributionsandinspectionaresufficientunderaparticular(multiplicative) time transformation, but not necessary once alternative transformations are considered. In this article, we focus on the two most transparent cases—multiplicative and additive detrending—but the logic extends to more general transformations of time that can also restore stationarity. Our analysisclarifiesthelogicalstructureofbalancedgrowthinsearchmodelsandprovidesabroader foundationforunderstandinghowdecliningfrictionsinteractwithstationarityandgrowth. Therestofthepaperproceedsasfollows. Section2outlinesthemainmechanisms. Section3 revisitstheinspectionmodelfromtheplanner’sperspective,definestheBGP,stationaryBGP,and scale-invariantBGP,derivestheefficiencyconditionsfortheSI-BGP,andshowsthattheMMmodel is efficient under the Hosios condition and Cobb-Douglas matching. It further demonstrates that withexponentialsampling,aBGPexistsinwhichunemploymentandvacanciesremainstationary while declining frictions do not generate growth, and extends the analysis to more general cases. Section4analyzesaDMPmodelwithhomogeneousworkersandbiasedtechnologicalprogressin thematchingfunction,showingthatawell-behavedBGPariseswhenunemploymentandvacancies arecomplementaryinputs. Section5concludes. 2. Overview Becausethepaperistechnicallydetailed,itisusefultobeginwithanoverviewofthemainideas. MM analyze a two-stage matching process in the labor market. In the first stage, a worker meets a vacancy with arrival rate 𝐴 𝑝(𝜃 ), where 𝜃 is labor market tightness defined as vacancies over 𝑡 𝑡 𝑡 unemployedworkersand 𝐴 isatechnologicalparametergrowingexogenouslyattheconstantrate 𝑡 𝑔 𝐴 : 𝐴 𝑡 = 𝐴 0 𝑒𝑔 𝐴 𝑡. MM refer to this as the meeting rate. If a meeting occurs, the process moves to the second stage, where the productivity of the match is drawn from a distribution 𝐹(𝑧). The unemployment-employment(UE)rateisthus ℎ 𝑢𝑒,𝑡 = 𝐴 𝑡 𝑝(𝜃 𝑡 ) (1− 𝐹(𝑅 𝑡 )), (1) where 𝑅 isthereservationthresholdofmatchquality. 𝑡 MMseekabalancedgrowthpath(BGP)inwhichboth ℎ and 𝜃 remainconstant,consistent 𝑢𝑒 withtheempiricalevidenceofunemploymentandvacanciesdiscussedbyMM.Sincethematching technologyimprovesovertime,thereservationthresholdmustriseaccordinglytosatisfytheequation atalltimes. Time-differentiatingyields 𝐹′(𝑅 𝑡 ) · 𝑔 𝐴 = 𝑅 𝑡 , ∀𝑡, (2) 1− 𝐹(𝑅 ) 𝑡 andundertheassumptionthat 𝑅 growsataconstantrate,MM obtaintheirkeycondition(equation 𝑡 4

(10)),whichimpliesthat 𝐹 mustbePareto. Theyfurthershowthat 𝑅 indeedgrowsataconstant 𝑡 rateasanequilibriumoutcome,andthatincreasingselectivityaddstoeconomicgrowth. Ourpaperbuildsonthreeobservationsthatqualifyandextendthisframework. 1. Paretoisnot necessary. MM’s equation(10)is aspecialcase ofthemore generalcondition, equation(2). WhileParetosamplingproducesaconstantUEtransitionrate,otherdistributions,such asexponential, candoso aswellonce theadjustment ofreservationquality isspecifieddifferently. · · For instance, if 𝑅 is constant (rather than 𝑅/𝑅 ), the reservation quality increases linearly rather 𝑡 𝑡 thanexponentiallyovertime,andthesolutiontothedifferentialequation(2)isexponentialrather than Pareto. With exponential sampling, unemployment, tightness, and transition rates remain stationary,butoutputgrowthslowsdownandeventuallystops. Thus,decliningsearchfrictionsdo notdrivelong-rungrowth,andstationaritydoesnotuniquelyrequireaParetodistribution. 2. Inspectionmodelshavecounterfactualimplications. Introducinginspectionfundamentally changes the standard DMP logic. In the canonical model, eliminating search frictions (making vacancies costless) drives unemployment to zero and yields finite welfare gains. In inspection models,bycontrast,costlesspostingpushestightnessandreservationproductivitytoinfinity. With Paretosampling,bothforcescanceleachotheroutandunemploymentpersistsevenundercostless posting. Moreover, the welfare gains from eliminating frictions become infinite, a problematic predictionasitovershadowstheroleofanyotherfrictionsknownineconomics. 3. Technologicalprogressmaybebiased. MMassumeHicks-neutralprogress,with 𝐴 𝑡 scaling theentiremeetingrate. Amoregeneralformulationis ℎ 𝑢𝑒,𝑡 = 𝑝(𝜃 𝑡 , 𝐴 𝑡 )(1− 𝐹(𝑅 𝑡 )), where 𝐴 affectstheproductivityofunemploymentandvacanciesdifferently. Whentheseinputsare 𝑡 complements,biasedprogressinoneinputeventuallyfacesdiminishingreturns: improvementsin 𝐴 nolongerraisethemeetingratewithoutbound,andtheneedforever-risingreservationthresholds 𝑡 disappears. Inthelimit,searchefficiencygainsnolongerfuelgrowth,underminingMM’sclaim thatdecliningfrictionsnecessarilygeneratelong-runexpansion. 3. Inspection This section considers variants of the MM inspection model, with the aim of characterizing the necessary and sufficient conditions for the existence of a suitable balanced growth path. While the original framework analyzes a decentralized equilibrium, we study the corresponding social 5

planner’sallocationandshowthatthetwocoincideiftheHosiosconditionholdsandthematching function is Cobb-Douglas. Our formulation generalizes the baseline model to incorporate both endogenous and exogenous job destruction, nesting the canonical case without declining search frictionsasaspecialcase. 3.1. Environment Theeconomyispopulatedbyacontinuumofworkersofmeasureoneandacontinuumoffirmsof positivemeasure. Ateachdate𝑡,eachworkeriseitherunemployed,𝑢 ,oremployedinajobwith 𝑡 firm-specific productivity 𝑧. Let 𝑛 (𝑧) denote the measure of workers at time 𝑡 employed in jobs 𝑡 with productivity 𝑧. Employed workers with productivity 𝑧 produce 𝑦 𝑡 𝑧, where 𝑦 𝑡 = 𝑦 0 𝑒𝑔 𝑦 𝑡 is an aggregateproductivitytermcommontoalljobsandgrowsatrate 𝑔 ≥ 0. Jobsaredestroyedatan 𝑦 exogenously rate 𝛿 ≥ 0, or endogenously when either the worker or the firm chooses to separate. In the planner’s allocation, endogenous separations are characterized by a productivity cutoff 𝑅 : 𝑡 matches with 𝑧 < 𝑅 are terminated, while those with 𝑧 ≥ 𝑅 continue. Unemployed workers 𝑡 𝑡 produce 𝑏 . 𝑡 Unemployedworkerscanbeassignedtojobsacrossdifferentproductivitylevels,butdoingso requires vacancy creation. Let 𝑣 denote vacancies posted at cost 𝑘 units of output per vacancy 𝑡 𝑡 perperiod. Thesevacanciesgenerate 𝐴 𝑀(𝑢 ,𝑣 ) randommatchesbetweenunemployedworkers 𝑡 𝑡 𝑡 andvacant jobs,where 𝐴 measuressearch efficiency, and 𝑀 isa constant-returns-to-scale (CRS) 𝑡 matchingfunction. 𝑀 isincreasingineachargument,concave,andsatisfiesthe Inadaconditions. Let 𝑀 and 𝑀 denote the corresponding partial derivatives with respect to the first and second 1 2 arguments,respectively. Whenaworkerandfirmmeet,amatchproductivity 𝑧 isdrawnfromacumulativedistribution 𝐹(𝑧) withdensity 𝑓(𝑧) andsupport [𝑧 ,∞). Thelawofmotionof 𝑛 (𝑧) is: 𝑙 𝑡 · 𝑛 𝑡 (𝑧) = 𝐴 𝑡 𝑀(𝑢 𝑡 ,𝑣 𝑡 )𝑓(𝑧) −𝛿𝑛 𝑡 (𝑧) for 𝑧 ≥ 𝑅 𝑡 and𝑡 ≥ 0. 3.2. Planner’sProblem Givenaninitialdistributionofemployment, [𝑛 (𝑧)]∞,theplannersolvesthefollowingproblem: 0 0 ∫ ∞ (cid:20)∫ (cid:21) max 𝑒−𝑟𝑡 𝑦 𝑧𝑛 (𝑧)𝑑𝑧 +𝑢 𝑏 − 𝑘 𝑣 𝑑𝑡 subjectto: 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 {𝑣 𝑡 ,𝑢 𝑡 ,𝑅 𝑡 ,n𝑡}∞ 𝑡=0 𝑡=0 𝑅 𝑡 · 𝑛 𝑡 (𝑧) = 𝐴 𝑡 𝑀(𝑢 𝑡 ,𝑣 𝑡 )𝑓(𝑧) −𝛿𝑛 𝑡 (𝑧) for 𝑧 ≥ 𝑅 𝑡 for𝑡 ≥ 0,and (3) ∫ ∞ 𝑢 𝑡 = 1− 𝑛 𝑡 (𝑧)𝑑𝑧, 𝑅 𝑡 ≥ 𝑧 for𝑡 ≥ 0. (4) 𝑙 𝑅 𝑡 6

The planner selects vacancies, unemployment, and a productivity cutoff to maximize the discountedpresentvalueofnetoutputatdiscountrate𝑟,subjecttothelawofmotionforemployment acrossjobproductivitiesandthelabormarketresourceconstraint. To connect this formulation with the MM decentralized equilibrium, it is useful to define the meeting rates implied by the matching function. An unemployed worker meets a vacancy at rate 𝑚 𝑡 = 𝐴 𝑡 𝑝 (𝜃 𝑡 ) where 𝑝 (𝜃) ≡ 𝑀(1,𝜃) and 𝜃 𝑡 = 𝑣 𝑡 /𝑢 𝑡 is labor market tightness. Symmetrically, a vacancymeetsanunemployedworkeratrate 𝑠 𝑡 = 𝐴 𝑡 𝑞 (𝜃 𝑡 ), where 𝑞 (𝜃) = 𝑝 (𝜃)/𝜃. 3.3. OptimalityConditions Let 𝑒−𝑟𝑡𝜆 (𝑧) and 𝑒−𝑟𝑡𝜂 betheLagrange multipliersassociatedwithequation(3)andequation(4), 𝑡 𝑡 respectively,for𝑡 ≥ 0. Theconditionsassociatedwiththeoptimalchoicesof 𝑣 ,𝑢 , 𝑅 ,and 𝑛 (𝑧) 𝑡 𝑡 𝑡 𝑡 are: ∫ ∫ 𝑘 𝑡 = 𝐴 𝑡 𝑀 2 (1,𝜃) 𝜆 𝑡 (𝑧)𝑓(𝑧)𝑑𝑧 = 𝐴 𝑡 𝑞 𝑡 (1− 𝜇 𝑡 ) 𝜆 𝑡 (𝑧)𝑓(𝑧)𝑑𝑧, (5) 𝑅 𝑅 𝑡 𝑡 ∫ ∫ 𝜂 𝑡 = 𝑏 𝑡 + 𝐴 𝑡 𝑀 1 (1,𝜃) 𝜆 𝑡 (𝑧)𝑓(𝑧)𝑑𝑧 = 𝑏 𝑡 + 𝐴 𝑡 𝑝 𝑡 𝜇 𝑡 𝜆 𝑡 (𝑧)𝑓(𝑧)𝑑𝑧, (6) 𝑅 𝑅 𝑡 𝑡 𝜂 𝑡 = 𝑦 𝑡 𝑅 𝑡 ,and (7) · 𝑦 𝑡 𝑧 −𝜂 𝑡 −𝛿𝜆 𝑡 (𝑧) = 𝑟𝜆 𝑡 (𝑧) −𝜆 𝑡 (𝑧) for 𝑧 ≥ 𝑅 𝑡 , (8) where 𝜇 𝑡 ≡ 𝜕 𝜕 𝑀 𝑢 𝑡 𝑀 𝑢 𝑡 = 𝜇(𝜃 𝑡 ) istheelasticityofmatcheswithrespecttounemployment. 𝑡 𝑡 Equation(5)statesthattheoptimalmassofvacanciesequatesthemarginalcost𝑘 tothemarginal 𝑡 benefit: theadditionalmatchescreatedbyanextravacancy, 𝐴 𝑡 𝜕 𝜕 𝑀 𝑣 𝑡 = 𝐴 𝑡 𝑞 𝑡 (1− 𝜇 𝑡 ),multipliedby 𝑡 theexpectedshadowvalueofafilledjob. Equation(6)requiresthattheshadowflowvalueofanunemployedworker,𝜂 , equalstheflow 𝑡 ofoutputwhileunemployed, 𝑏 ,plustheadditionalmatchesgeneratedbyanunemployedworker, 𝑡 𝐴 𝑡 𝜕 𝜕 𝑀 𝑢 𝑡 = 𝐴 𝑡 𝑝 𝑡 𝜇 𝑡 ,timesexpectedshadowvalueofamatch. 𝑡 Equation(7)impliesthatthe optimalreservationproductivity 𝑅 issuchthat theproductionof 𝑡 themarginalworker, 𝑦 𝑅 , equalstheshadowflowvalueofunemployment,𝜂 . Finally,equation(8) 𝑡 𝑡 𝑡 characterizestheshadowvalue𝜆 (𝑧) ofafilledjobofquality 𝑧. 𝑡 We now rewrite these expressions in a form that will be useful later. Combining equation (6) andequation(7)yields: ∫ 𝑦 𝑡 𝑅 𝑡 −𝑏 𝑡 = 𝐴 𝑡 𝑝 𝑡 𝜇 𝑡 𝜆 𝑡 (𝑧)𝑓(𝑧)𝑑𝑧. 𝑅 𝑡 Substitutingequation(5)intothisexpressionleadstoourfirstkeyrelationship: 𝑦 𝑅 −𝑏 𝑝 𝜇 𝜇(𝜃 ) 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 = = 𝜃 𝑡 . (9) 𝑘 𝑞 (1− 𝜇 ) 1− 𝜇(𝜃 ) 𝑡 𝑡 𝑡 𝑡 7

Next,substitutingequation(7)intoequation(8)gives: · (𝑟 +𝛿)𝜆 𝑡 (𝑧) = 𝑦 𝑡 𝑧 − 𝑦 𝑡 𝑅 𝑡 +𝜆 𝑡 (𝑧) for 𝑧 ≥ 𝑅 𝑡 . (10) This is the familiar value function equation in which 𝜆 (𝑧) denotes the social value of a match. 𝑡 Solvingthisdifferentialequation(seeAppendix)yields: ∫ 𝑑 𝜆 𝑡 (𝑧) −𝑒−(𝑟+𝛿)𝑑𝜆 𝑡+𝑑 (𝑧) = 𝑒−(𝑟+𝛿)𝜏 (𝑦 𝑡+𝜏 𝑧 − 𝑦 𝑡+𝜏 𝑅 𝑡+𝜏 )𝑑𝜏. (11) 0 Thetransversalitycondition 𝑒−(𝑟+𝛿)𝑑𝜆 𝑡+𝑑 = 0must hold. If 𝑑 isfinite, thisimplies𝜆 𝑡+𝑑 = 0or, fromequation(10), 𝑧 = 𝑅 . (12) 𝑡+𝑑 𝑡(𝑧) Equation(12)defines 𝑑 (𝑧), theoptimallongevityofamatchwithproductivity 𝑧 attime𝑡,absent 𝑡 anexogenousdestructionshock. Substitutingthisintoequation(11)gives: ∫ 𝑑(𝑧,𝑡) 𝜆 𝑡 (𝑧) = 𝑒−(𝑟+𝛿)𝜏 (𝑦 𝑡+𝜏 𝑧 − 𝑦 𝑡+𝜏 𝑅 𝑡+𝜏 )𝑑𝜏. (13) 0 3.4. Aggregates Themeasureofaggregateemploymentwithmatchqualitybelow 𝑧 is: 𝑡 ∫ 𝑧 𝑡 𝑁 (𝑧 ) ≡ 𝑛 (𝑥)𝑑𝑥. (14) 𝑡 𝑡 𝑡 𝑅 𝑡 Totalemploymentisthen: 𝑁 𝑡 = 𝑁 𝑡 (∞) = 1−𝑢. Fromequation(3)andequation(14),weobtain: · · · ∫ 𝑧 𝑡 · 𝑁 𝑡 (𝑧 𝑡 ) = 𝑛 𝑡 (𝑧 𝑡 )𝑧 𝑡 −𝑛 𝑡 (𝑅 𝑡 )𝑅 𝑡 + 𝑛 𝑡 (𝑥)𝑑𝑥 (15) 𝑅 𝑡 · · ∫ 𝑧 𝑡 = 𝑛 𝑡 (𝑧 𝑡 )𝑧 𝑡 −𝑛 𝑡 (𝑅 𝑡 )𝑅 𝑡 + [𝐴 𝑡 𝑀(𝑢 𝑡 ,𝑣 𝑡 )𝑓(𝑥) −𝛿𝑛 𝑡 (𝑥)] 𝑑𝑥 𝑅 𝑡 · · = 𝑛 𝑡 (𝑧 𝑡 )𝑧 𝑡 −𝑛 𝑡 (𝑅 𝑡 )𝑅 𝑡 +𝑢 𝑡 𝐴 𝑡 𝑝 𝑡 (𝐹 (𝑧 𝑡 ) − 𝐹 (𝑅 𝑡 )) −𝛿𝑁 𝑡 (𝑧 𝑡 ). This expression decomposes employment below 𝑧 into net inflows from new matches and outflows 𝑡 fromjobdestructions. 8

Similarly,thelawofmotionfortotalemploymentcanbeexpressedas: · ∫ ∞ · · 𝑁 𝑡 = 𝑛 𝑡 (𝑧)𝑑𝑧 −𝑛 𝑡 (𝑅 𝑡 )𝑅 𝑡 (16) 𝑅 𝑡 · = 𝑢 𝑡 𝐴 𝑡 𝑝 𝑡 (1− 𝐹(𝑅 𝑡 )) −𝛿𝑁 𝑡 −𝑛 𝑡 (𝑅 𝑡 )𝑅 𝑡 . Here, total employment depends on the inflow from unemployed workers matched to jobs above 𝑅 𝑡 and theoutflow fromjob destruction. Finally, definethe cumulative distributionof matchqualities as: 𝑁 (𝑧) 𝑡 𝐺 (𝑧) ≡ . (17) 𝑡 𝑁 𝑡 Thisisthefractionofemployedworkersinmatcheswithproductivitybelow 𝑧. 3.5. BalancedGrowth 3.5.1. Definition Thedefinitionofabalancedgrowthpath(BGP)iscentral,asitimposesrestrictionsonendogenous variablesandfreesequationsthat helpidentifytheexogenousforces—particularlythedynamicsof 𝑘 , 𝑏 ,andthefunction 𝐹—neededtosustainthepath. Webeginwiththegeneraldefinition. 𝑡 𝑡 Definition(BGP):ABalancedGrowthPath(BGP)isaninitialstate𝐺 0 (𝑧) andanassociated efficient allocation such that unemployment, tightness, and the employment-to-unemployment (EU) and unemployment-to-employment (UE) rates remain constant over time, while aggregate productivityandsearchefficiencygrowatconstantrates 𝑔 ≥ 0and 𝑔 ≥ 0. 𝑦 𝐴 This is the definition that MM emphasize in the abstract and introduction of their paper. It guaranteesconstancyofaggregatelabor-marketvariablesbutimposesnorestrictionontheevolution ofthematch-qualitydistribution𝐺 (𝑧 ). Itisnatural,however,torequirethat𝐺 (𝑧 ) exhibitsome 𝑡 𝑡 𝑡 𝑡 formofstationarity. Definition(SBGP):AStationaryBalancedGrowthPath(SBGP)isaBGPsuchthat 𝐺 𝑡 (𝑇 𝑡 (𝑧)) = 𝐺 0 (𝑧) forall 𝑧 ≥ 𝑅 0 , where 𝑇 (𝑧) is a time transformation of 𝑧 capturing how the distribution evolves over time. The 𝑡 following,morerestrictiveversion,istheoneMMadopttoprovetheirmainresult,Theorem1. Definition(SI-BGP):AScale-InvariantBalancedGrowthPath(SI-BGP)isanSBGPforwhich 𝑇 𝑡 (𝑧) = 𝑧𝑒𝑔 𝑧 𝑡. AnSI-BGPisthereforeaspecialcaseofanSBGP—onethatimposesaproportional(multiplicative)transformationontheendogenousdistribution𝐺 . MMhighlightthebroaderBGPdefinitionin 𝑡 theirabstractandintroductionbutadoptthemorerestrictiveSI-BGPintheiranalyticalderivations. 9

ThegeneralBGPconceptrequiresonlythataggregatelabor-marketvariablesremainstationaryand imposes no structure on 𝐺 𝑡 , consistent with MM’s stated objective: “We seek conditions for the existenceofabalancedgrowthpath(BGP),whereunemployment,vacancy,andworker’stransition ratesremainconstantinthefaceofimprovementsintheproductionandsearchtechnologies.” Thescale-invarianceassumptionon𝐺 followsrelatedworkinendogenousgrowththeory(e.g., 𝑡 PerlaandTonetti,2014;LucasandMoll,2014;BueraandOberfield,2020;Benhabibetal.,2021), whereitispurposefulbecauseitgeneratesgrowth. Inthepresentcontext—focusedonexplaining labor-marketstationarity—itinadvertentlyintroducesgrowtheventhoughitisnotrequiredforthe statedobjective. Oneimplicationis thatthereservationproductivity, thelowerboundofsupport, mustgrowatrate 𝑔 alonganSI-BGP: 𝑧 𝑅 𝑡 = 𝑅 0 𝑒𝑔 𝑧 𝑡. The broader BGP and SBGP definitions impose no such restriction. In our exponential example below,𝑇 𝑡 (𝑧) = 𝑧 +𝜙𝑡,sothereservationproductivityinsteadfollows 𝑅 𝑡 = 𝑅 0 +𝜙𝑡, 𝜙 > 0. 3.6. BGPSystemofEquations AlongaBGP,thefollowingversionsofequations(5),(9),(12),(13),(15),and(16)musthold. First, thefirstorderconditionwithrespecttovacanciesbecomes ∫ 𝑘 𝑡 = 𝐴 𝑡 𝑀 2 (1,𝜃) 𝜆 𝑡 (𝑧)𝑓(𝑧)𝑑𝑧, (18) 𝑅 𝑡 whilethereservationconditionreads 𝑦 𝑅 −𝑏 𝑀 (1,𝜃) 𝜇(𝜃) 𝑡 𝑡 𝑡 1 = = 𝜃. (19) 𝑘 𝑀 (1,𝜃) 1− 𝜇(𝜃) 𝑡 2 Matchlongevityisdefinedby 𝑧 = 𝑅 , (20) 𝑡+𝑑 𝑡(𝑧) andtheshadowvalueofafilledjobsatisfies ∫ 𝑑(𝑧) 𝜆 𝑡 (𝑧) = 𝑒−(𝑟+𝛿)𝜏 (𝑦 𝑡+𝜏 𝑧 − 𝑦 𝑡+𝜏 𝑅 𝑡+𝜏 )𝑑𝜏. (21) 0 10

Thedynamicsofemploymentbyproductivityaregivenby · · 𝑛 𝑡 (𝑧 𝑡 )𝑧 𝑡 +𝑢𝐴 𝑡 𝑝 (𝜃) (𝐹 (𝑧 𝑡 ) − 𝐹 (𝑅 𝑡 )) = 𝑛 𝑡 (𝑅 𝑡 )𝑅 𝑡 +𝛿𝑁 𝑡 (𝑧 𝑡 ). (22) Insteadystate,inflowsintounemploymentmustequaloutflows,implying 𝑢ℎ 𝑢𝑒 = (1−𝑢) ℎ 𝑒𝑢 , (23) wherethejob-findingandjob-destructionratesaregivenby ℎ ≡ 𝐴 𝑝 (𝜃) (1− 𝐹(𝑅 )),and (24) 𝑢𝑒 𝑡 𝑡 𝑛 𝑡 (𝑅 𝑡 ) · ℎ ≡ 𝛿+ 𝑅 , (25) 𝑒𝑢 𝑡 𝑁 respectively,withtotalemployment 𝑁 = 1−𝑢. Givenparameters,theseconditionscanbeusedtosolvefor 𝑑 (𝑧), 𝑅 , 𝜃,𝜆 (𝑧), 𝑛 (𝑧), ℎ ,and 𝑡 𝑡 𝑡 𝑡 𝑢𝑒 · ℎ 𝑒𝑢 . Equation(22)correspondstoequation(15)when 𝑁 𝑡 (𝑧 𝑡 ) = 0, andequation(23)corresponds · toequation(16)when 𝑁 𝑡 = 0. Finally,equation(24)andequation(25)provideexplicitdefinitions forthesteady-statejob-findingandjob-destructionrates. AsinMM,it isconvenientto workwith thedistribution𝐺 definedinequation(17). Toderive 𝑡 theimpliedrestrictionson𝐺,substituteequations(23)-(24)intoequation(22): · 𝑛 𝑡 (𝑧 𝑡 )𝑧 𝑡 +𝑢𝐴 𝑡 𝑝(𝜃) (𝐹 (𝑧 𝑡 ) − 𝐹 (𝑅 𝑡 )) = ℎ 𝑒𝑢 𝑁 −𝛿𝑁 +𝛿𝑁 𝑡 (𝑧 𝑡 ) = 𝑢ℎ 𝑢𝑒 +𝛿(𝑁 𝑡 (𝑧 𝑡 ) − 𝑁) = 𝑢𝐴 𝑡 𝑝(𝜃) (1− 𝐹(𝑅 𝑡 )) +𝛿(𝑁 𝑡 (𝑧 𝑡 ) − 𝑁). Simplifyingyields: (cid:18)𝑁 (𝑧 ) (cid:19) · 𝑡 𝑡 𝑛 𝑡 (𝑧 𝑡 )𝑧 𝑡 = 𝑢𝐴 𝑡 𝑝(𝜃) (1− 𝐹(𝑧 𝑡 )) +𝛿(1−𝑢) −1 . 𝑁 Fromequation(17)wehavethat(1−𝑢)𝐺′ 𝑡 (𝑧) = 𝑛 𝑡 (𝑧).Substitutingthisintothepreviousexpression gives: (1−𝑢) (cid:16) 𝐺′ 𝑡 (𝑧 𝑡 )𝑧 · 𝑡 +𝛿(1−𝐺 𝑡 (𝑧)) (cid:17) = 𝑢𝐴 𝑡 𝑝(𝜃) (1− 𝐹(𝑧 𝑡 )), (26) which provides the implied restriction on the evolution of the endogenous employment distribution. Moreover,thejobdestructionrate(25)becomes: · ℎ 𝑒𝑢 = 𝛿+𝐺′ 𝑡 (𝑅 𝑡 )𝑅 𝑡 . (27) 11

Theselasttwo equationscanreplace equation(22)andequation(25)inthe definitionofBGP. On ascaled invariantBGP (SI-BGP),productivity growsat aconstant rate 𝑧(cid:164) 𝑡 /𝑧 𝑡 = 𝑔 𝑧 . Substituting thisintoequation(26)andequation(27)yields (1−𝑢) (cid:0)𝐺′ 𝑡 (𝑧 𝑡 )𝑧 𝑡 𝑔 𝑧 +𝛿(1−𝐺 𝑡 (𝑧)) (cid:1) = 𝑢𝐴 𝑡 𝑝(𝜃) (1− 𝐹(𝑧 𝑡 )), (28) ℎ 𝑒𝑢 = 𝛿+𝐺′ 𝑡 (𝑅 𝑡 )𝑅 𝑡 𝑔 𝑧 . (29) Thesetwoconditionsreplaceequation(26)andequation(27)inthedefinitionofanSI-BGP. Finally,averagematchqualitysatisfies ∫ 𝑍 𝑡 = 𝑧𝐺′ 𝑡 (𝑧)𝑑𝑧, (30) 𝑅 𝑡 whichlinksthedistributionofemployedmatchestoaggregateproductivity. Decentralizationandefficiency. Adirectcomparisonshowstheconnectionbetweenthesocial planner’ssolutionandthe decentralizedequilibriuminMM. Oursystem ofequationsdefining an SI-BGPisidenticaltotheirsif 𝛿 = 0(i.e.,alljobseparationsareendogenouslydeterminedbythe reservationrule)andthematchingelasticity𝜇(𝜃)isreplacedbyaconstantworkers’bargainingpower 𝛾. Inotherwords,whentheHosiosconditionholdsandthematchingfunctionisCobb–Douglas, MM’sdecentralizedequilibriumisefficient. Proposition 1 (Efficiency of the MM Equilibrium). Consider the decentralized equilibrium in Martellini andMenzio (2020). If job separations areentirely endogenous (𝛿 = 0) andthe matching function is Cobb–Douglas with elasticity 𝜇 equal to the workers’ bargaining power 𝛾, then the decentralizedequilibriumcoincideswiththesocialplanner’sallocation. Undertheseconditions, theequilibriumsatisfiestheHosiosefficiencycriterion,andthescale-invariantbalancedgrowth path(SI-BGP)isefficient. 3.7. Characterization Wenowpresentthreemainresults. ThefirsttwoparallelMM’sLemma1andTheorem1,which establish necessaryand sufficient conditions for the existenceof a SI-BGP. Our resultsare novelin that they pertain to efficient rather than decentralized equilibrium allocations and allow for both endogenousandexogenousseparations,whereasMMconsideronlytheendogenousseparations. Thethird resultis acounterexample: aSBGP thatis notanSI-BGP, wherethe distribution 𝐹 is exponentialratherthanPareto,yetlabormarketstatisticsremainconstantovertime. Thisexample demonstratesthatsearchmodelsneednotbeconstrainedbythenarrowconditionsofMM’sLemma 1andTheorem1. 12

Lemma1(NecessaryconditionsforaSI-BGP). Let𝑔 𝑦 ≥ 0and𝑔 𝐴 > 0bearbitrarygrowthrates fortheproductionandsearchtechnologies. 1. ASI-BGPmayexistonlyif(a)thedistribution 𝐹 isParetowithanarbitrarycoefficient 𝛼; (b)thegrowthrateofthevacancycost, 𝑔 ,andthegrowthrateoftheunemploymentbenefit, 𝑘 𝑔 ,areequalto 𝑔 +𝑔 ;and(c)thediscountrate𝑟 isgreaterthan 𝑔 +𝑔 . 𝑏 𝑦 𝑧 𝑦 𝑧 2. InanySI-BGP,thegrowthrate 𝑔 ofthedistribution𝐺 isequalto 𝑔 /𝛼. 𝑧 𝑡 𝐴 The proof of Lemma 1 follows MM closely and is therefore omitted. The need for a Pareto distributionfollowsfromourdiscussioninSection2. Intuitively,aParetodistributionisrequiredto ensurescaleinvarianceofmatchquality,sothatthereservationcutoffcangrowproportionallyover time. Moreover,equation(12)inMM,neededfortheproof,isanalogoustoourequation(19),except thatourexpressioninvolvestheendogenouselasticity 𝜇(𝜃) ratherthantheexogenousbargaining weight 𝛾. Butsince 𝜃 isconstantalongaBGP,thisdifferencedoesnotaffecttheproofofLemma1. AssumingthattheconditionsofLemma1aresatisfied,thenextstepistoshowthataSI-BGP existsandisunique. Thisamountstoverifyingthat,underthoseconditions,thesystemofequations defining a SI-BGP can be solved and yields a unique set of functions and values for 𝑑 (𝑧), 𝑅 , 𝜃, 𝑡 𝑡 𝜆 (𝑧), 𝐺 (𝑧), ℎ ,and ℎ . FollowingMM’ssteps,thissystemcanbereducedtotwoequationsin 𝑡 𝑡 𝑢𝑒 𝑒𝑢 thetwounknowns 𝑅 and 𝜃: 0 𝑅 0 = (𝐴 0 𝑀 2 (1,𝜃)Φ1 𝑦 0 /𝑘 0 ) 1/(𝛼−1), and (31) 𝑏 𝑀 (1,𝜃) 𝑘 0 1 0 𝑅 0 = + , (32) 𝑦 𝑀 (1,𝜃) 𝑦 0 2 0 where 𝑧𝛼 Φ1 = (𝛼−1) (cid:0) (𝛼−1)𝑔 𝑙 + (cid:0)𝑟 +𝛿−𝑔 (cid:1)(cid:1) . (33) 𝑧 𝑦 Giventheassumedpropertiesof 𝑀,thissystemhasauniquesolution. Equation(31)isastrictly decreasingfunctionof𝜃,spanningfrom+∞to0,whileequation(32)isastrictlyincreasingfunction of𝜃,spanningfrom0to+∞.Bytheintermediatevaluetheorem,thetwocurvesintersectataunique 𝜃, whichbindsdownauniquesolutionfor 𝑅 0 and 𝜃. Thesolutionfor 𝑅 𝑡 isthen 𝑅 𝑡 = 𝑅 0 𝑒(𝑔 𝐴/𝛼)𝑡. A uniquesolutionfor𝐺 (𝑧) canthenbefound. 0 Giventhat 𝐹 isPareto,equation(28)reads: (cid:18)𝑧 (cid:19)𝛼 (1−𝑢) (cid:0)𝐺′ 𝑡 (𝑧)𝑧𝑔 𝑧 +𝛿(1−𝐺 𝑡 (𝑧)) (cid:1) = 𝑢𝐴 𝑡 𝑝 (𝜃) 𝑧 𝑙 . (cid:16) (cid:17)𝛼 Thisisadifferentialequationfor𝐺 𝑡 (𝑧). Anaturalguessforthesolutionis𝐺 𝑡 (𝑧) = 1− 𝑅 𝑧 𝑡 . 13

Substitutingthisintothedifferentialequationyields (cid:18)𝑅 (cid:19)𝛼 (cid:18)𝑧 (cid:19)𝛼 𝑡 𝑙 (1−𝑢) (𝛼𝑔 𝑧 +𝛿) = 𝑢𝐴 𝑡 𝑝 (𝜃) . 𝑧 𝑧 Thisequationissatisfiedif (cid:16)𝑧 𝑙(cid:17)𝛼 (1−𝑢) (𝛼𝑔 𝑧 +𝛿) = 𝑢𝐴 𝑡 𝑝 (𝜃) = 𝑢𝐴 𝑡 𝑝 (𝜃) (1− 𝐹 (𝑅)) = 𝑢ℎ 𝑢𝑒 , 𝑅 or 𝛼𝑔 +𝛿 𝑔 +𝛿 𝑧 𝐴 𝑢 = = . 𝛼𝑔 +𝛿+ ℎ 𝑔 +𝛿+ ℎ 𝑧 𝑢𝑒 𝐴 𝑢𝑒 This condition is consistent with equation (23) and equation (29), together with the guessed solution for 𝐺 (𝑧), thereby confirming its validity. Hence, an efficient SI-BGP exists, is unique 𝑡 and—importantly—is continuous at 𝑔 𝐴 = 0 if 𝛿 > 0. This continuity is possible because 𝛿 > 0 allows for exogenous job destruction. By contrast, if 𝛿 = 0 and 𝑔 𝐴 = 0, there would be no job destruction along a SI-BGP making full employment an absorbing state. The following theorem collectstheseresults. Theorem1(ExistenceandpropertiesofaSI-BGP). Let𝑔 𝐴 > 0,𝑔 𝑦 ≥ 0,and𝛿 > 0.Anefficient SI-BGP exists if and only if (a) 𝐹 is Pareto with coefficient 𝛼 > 1; (b) 𝑔 and 𝑔 satisfy 𝑏 𝑘 𝑔 𝑏 = 𝑔 𝑦 +𝑔 𝐴 /𝛼 and 𝑔 𝑘 = 𝑔 𝑦 +𝑔 𝐴 /𝛼;and(c)𝑟 > 𝑔 𝑦 +𝑔 𝐴 /𝛼. If an efficient SI-BGP exists, it is unique, continuous at 𝑔 𝐴 = 0, and has the following properties: (i)𝑢, 𝜃, ℎ 𝑢𝑒 ,and ℎ 𝑒𝑢 areconstant,with ℎ 𝑢𝑒 = 𝐴 0 𝑝(𝜃) (1− 𝐹 (𝑅 0 )), ℎ 𝑒𝑢 = 𝑔 𝐴 +𝛿; (cid:16) (cid:17)𝛼 (ii)𝐺 𝑡 (𝑧𝑒𝑔 𝑧 𝑡) = 𝐺 0 (𝑧) with 𝑔 𝑧 = 𝑔 𝐴 /𝛼 and𝐺 0 (𝑧) = 1− 𝑅 𝑧 0 ;and (iii)laborproductivitygrowsattherate 𝑔 +𝑔 /𝛼. 𝑦 𝐴 The theorem confirms MM’s findings for the planner. Under DMP decentralization, efficient andmarketallocationsgenerallydiffer,exceptwhentheHosiosconditionholdsandthematching functionisCobb-Douglas. IncontrasttoMM,ourSI-BGPiscontinuousat 𝑔 𝐴 = 0. We now present the main result of this subsection: an example of an SBGP that is not scaleinvariant(SI-BGP).WerefertothiscaseasanAI-BGP(“AdditiveInvariant”). AsshowninSection 2, when the sampling distribution is exponential rather than Pareto, maintaining a constant UE raterequiresthereservationproductivitytoriselinearlyratherthanexponentiallyovertime. This linear drift preserves the stationarity of aggregate labor-market variables and prevents declining searchfrictionsfromgeneratinglong-rungrowth. Thelimitofourexamplethereforecorrespondsto anSI-BGP withoutgrowth. The following propositionestablishesthat suchan equilibriumarises 14

endogenously when unemployment benefits increase linearly over time while the cost of posting vacanciesremainsconstant. Proposition2(AI-BGP). Let 𝑦 = 1, 𝑔 𝐴 > 0, 𝐹(𝑧) = 1−𝑒−𝜈𝑧, 𝑘 𝑡 = 𝑘,and 𝑏 𝑡 = 𝑏 + (𝑔 𝐴 /𝜈)𝑡. Then thereexistsauniqueefficientBGPsuchthat: (i)𝑢, 𝜃, ℎ 𝑢𝑒,and ℎ 𝑒𝑢 areconstant,with ℎ 𝑢𝑒 = 𝐴 0 𝑝(𝜃) (1− 𝐹 (𝑅 0 )), ℎ 𝑒𝑢 = 𝑔 𝐴 +𝛿; (ii) 𝑅 𝑡 = 𝑅 0 + (𝑔 𝐴 /𝜈)𝑡; (iii)𝐺 𝑡 (𝑧 + (𝑔 𝐴 /𝜈)𝑡) = 𝐺 0 (𝑧) = 1−𝑒−𝜈(𝑧−𝑅 0); (iv)averageproductivitysatisfies 𝑍 𝑡 = 1/𝜈 + 𝑅 𝑡. Hence 𝑍(cid:164) 𝑔 𝑡 𝐴 = −→ 0 as𝑡 → ∞. 𝑍 1+𝜈𝑅 +𝑔 𝑡 𝑡 0 𝐴 Proof. FollowingMM’ssteps,thesystemreducestotwoequationsinthetwounknowns 𝑅 0 and 𝜃. Equation(24)becomes ℎ 𝑢𝑒 = 𝐴 0 𝑒𝑔 𝐴 𝑡𝑝 (𝜃)𝑒−𝜈𝑅 𝑡 = 𝐴 0 𝑝 (𝜃)𝑒−𝜈𝑅 0. Fromthisexpression,itfollowsthat 𝑔 𝐴 𝑅 𝑡 = 𝑅 0 + 𝑡. (34) 𝜈 Substitutingintoequation(19)yieldsthefirstequationintwounknowns: 𝑅 −𝑏 𝑅 −𝑏 𝑀 (1,𝜃) 𝑡 𝑡 0 1 = = . (35) 𝑘 𝑘 𝑀 (1,𝜃) 2 Next,usingequation(34)inequation(20)gives 𝑑(𝑧) = (𝑧 − 𝑅 𝑡 )/(𝑔 𝐴 /𝜈). Theappendixshowsthat ∫ 𝜆 𝑡 (𝑧)𝑓(𝑧)𝑑𝑧 = Φ2 𝑒−𝜈𝑅 𝑡, (36) 𝑅 𝑡 where Φ2 = 𝑟+ 1 𝛿 1 𝜈 (cid:104) 𝑟+ 𝑟 𝛿 + − 𝛿 𝑔 𝐴 + 𝑟+ 1 𝛿𝑟+𝛿 𝑔 + 2 𝐴 𝑔 𝐴 (cid:105) . Plugging this and equation (34) into equation (18) gives thesecondequation: 𝑘 = 𝐴 𝑡 𝑒−𝜈𝑅 𝑡𝑀 2 (1,𝜃)Φ2 = 𝐴 0 𝑒−𝜈𝑅 0𝑀 2 (1,𝜃)Φ2 . (37) Equations(35)and(37)determine𝜃 and 𝑅 . Giventheassumedpropertiesof 𝑀,thissystemadmits 0 auniquesolution. Nowconsiderequation(26): (1−𝑢) (cid:16) 𝐺′ 𝑡 (𝑧) 𝑔 𝜈 𝐴 +𝛿(1−𝐺 𝑡 (𝑧)) (cid:17) = 𝑢𝐴 𝑡 𝑝 (𝜃)𝑒−𝜈𝑧. Thisisadifferentialequationfor𝐺 (𝑧).Giventheexponentialformontheright-handside,anatural 𝑡 15

conjectureis 𝐺 𝑡 (𝑧) = 1−𝑒−𝜈(𝑧−𝑅 𝑡). Substitutingyields (1−𝑢)𝑒−𝜈(𝑧−𝑅 𝑡) (𝑔 𝐴 +𝛿) = 𝑢𝐴 𝑡 𝑝 (𝜃)𝑒−𝜈𝑧. Thisholdsifandonlyif (1−𝑢) (𝑔 𝐴 +𝛿) = 𝑢𝐴 𝑡 𝑝 (𝜃)𝑒−𝜈𝑅 𝑡 = 𝑢𝐴 𝑡 𝑝 (𝜃) (1− 𝐹 (𝑅 𝑡 )) = 𝑢ℎ 𝑢𝑒 , orequivalently, 𝑔 +𝛿 𝑔 +𝛿 𝐴 𝐴 𝑢 = = , 𝑔 𝐴 +𝛿+ ℎ 𝑢𝑒 𝑔 𝐴 +𝛿+ 𝐴 𝑡 𝑝 (𝜃)𝑒−𝜈𝑅 𝑡 which follows from equations (23), (24), (27), and the conjecture for 𝐺. Thus, the conjecture is verified. Finally, average match quality—equal to average labor productivity since 𝑦 = 1— defined inequation(30)satisfies 1 𝑍 𝑡 = + 𝑅 𝑡 . 𝜈 Hence, · · 𝑍 𝑅 𝑔 𝑡 𝑡 𝐴 = = → 0. 𝑍 𝑍 1+𝜈𝑅 +𝑔 𝑡 𝑡 𝑡 0 𝐴 □ Comparing Theorem 1 and Proposition 1, we find that if the sampling distribution is Pareto, a SI-BGP requires both the cost of posting a vacancy and unemployment compensation to rise exponentiallyovertime. Bycontrast,ifthesamplingdistributionisexponential,thecostofposting a vacancy can remain constant, while unemployment compensation needs to grow only linearly. The underlying reason is that under an exponential distribution, small changes in the reservation threshold have large effects on the probability of drawing a high-quality match. Consequently, parametersdonotneedtoadjustasmuchtosustainaBGP. Insummary,MMimposesanunnecessaryextraassumption. TheyassumethatinanyBGP,the reservation productivity must grow at a constant rate. Proposition (2) removes this assumption. While it is true that an increase in reservation productivity is necessary to offset the persistent improvementsinsearchtechnology,thisincreasedoesnotneedtobeexponentialovertime. 3.8. TheRoleofSearchFrictions We now turn to the implications of eliminating search frictions. In the canonical DMP model, removing frictions—by making vacancy posting costless—drives unemployment to zero. In the MM inspection framework, by contrast, unemployment does not vanish: although jobs become 16

easytofind,workerssimultaneouslybecomeexcessivelyselective,exactlyoffsettingtheimproved matching prospects. In our exponential benchmark, however, unemployment does vanish, since self-selectionistooweaktoneutralizetheeaseoffindingwork. The welfare predictions differ just as sharply. In the DMP model, the gains from eliminating frictions are finite, bounded by the additional output that unemployed workers could contribute. In both the MM model and our exponential inspection variant, however, the welfare gains are unbounded. Asmatchingbecomesarbitrarilyeasy,itisoptimalforworkerstoholdoutforever-better offers, and since match quality is unbounded above, expected gains diverge. This implausible implicationhighlightsacoreweaknessofinspectionframeworks. Thepresentsectionillustrates theseweaknesses,whileSection4developsanalternativemechanismthatavoidsthem. For this analysis, we adopt the canonical Cobb-Douglas (CD) matching function, 𝑀(𝑣,𝑢) = 𝑢𝛾𝑣1−𝛾,tofacilitatecomparisonwithMM’sformulas. 3.8.1. TheUnderlyingSourceofUnemployment Inthetwomodelsweareconsidering,theunemploymentratesatisfies 𝑔 +𝛿 𝐴 𝑢 = , (38) 𝑔 +𝛿+ ℎ 𝐴 𝑢𝑒 where ℎ satisfies equation (24). In this section, we are interested in the limit of this expres- 𝑢𝑒 sion as 𝑘 → 0. In the DMP model, free vacancy posting would drive vacancies, tightness, 𝜃, workers’matchingprobability,andtheUEtransitionprobability, ℎ ,toinfinite,thuseliminating 𝑢𝑒 unemployment. 3.8.2. MM’sModel IntheMMmodel,theUEtransitionsatisfies ℎ 𝑢𝑒 ≡ 𝐴 0 𝑝 (𝜃) (1− 𝐹(𝑅 0 )) = 𝐴 0 𝜃1−𝛾𝑧 𝑙 𝛼𝑅 0 −𝛼. (39) Inordertocharacterize ℎ ,weneedtocharacterize 𝜃 and 𝑅 . 𝑢𝑒 0 Usingtheassumedmatchingfunction,equation(31)andequation(32)simplifyto: (cid:32) (cid:33)1/(𝛼−1) 𝑦 𝑅 0 = 𝐴 0 (1−𝛾)Φ1 0 (𝜃𝑘 0 )−𝛾/(𝛼−1),and (40) 𝑘1−𝛾 0 𝑏 𝛾 𝜃𝑘 0 0 𝑅 0 = + . (41) 𝑦 1−𝛾 𝑦 0 0 17

Equation(40)depicts 𝑅 asadecreasingfunctionof 𝜃𝑘 whileequation(41)depicts 𝑅 asan 0 0 0 increasingfunctionof 𝜃𝑘 . Auniquesolutionfor 𝑅 and 𝜃𝑘 exists. Moreover,as 𝑘 decreases,the 0 0 0 0 curvefromequation(40)shiftsupwardsandthatfromequation(41)remainsunchanged,pushing the solution toward higher 𝑅 and 𝜃𝑘 , but also higher 𝜃. As 𝑘 → 0 both 𝑅 and 𝜃 diverge to ∞. 0 0 0 For 𝑘 sufficientlylow,theconstantterm 𝑏 0 inequation(41)becomesnegligiblesothat 𝑦 0 𝛾 𝜃𝑘 0 𝑅 ≈ . 0 1−𝛾 𝑦 0 Usingthisandequation(40),closedformsolutionfor 𝑅 and 𝜃 canbefoundas: 0 − 𝛼 𝜃 𝑘→0 = Ω1 𝑘 0 𝛼−(1−𝛾), and (42) 𝑅 𝑘→0 = 1− 𝛾 𝛾 Ω 𝑦 1 𝑘 0 − 𝛼− 1 ( − 1 𝛾 −𝛾), (43) 0 (cid:104) (cid:105) 1 whereΩ1 = 1− 𝛾 𝛾 (𝐴 0 (1−𝛾)Φ1 ) 1/(𝛼−1) 𝑦 0 𝛼/(𝛼−1) 1+𝛾/(𝛼−1) .Thesesolutionsarevalidinthelimit,but if 𝑏 0 = 0,theyarealsovalidforany 𝑘. Substitutingthesesolutionsintoequation(39),thesolution for ℎ isobtained. Thesolutionthencanbesubstitutedintoequation(38)tofindunemployment. 𝑢𝑒 Thefollowingpropositionsummarizesthemainresults. Proposition3. InaSI-BGPwith 𝑀(𝑢,𝑣) = 𝑢𝛾𝑣1−𝛾 ,and 𝑘 → 0, (cid:18)𝛼−1 (cid:19) ℎ 𝑢𝑒 → ℎ 𝑢 ∗ 𝑒 = (𝛼−1) 𝛼 𝑔 𝐴 +𝑟 +𝛿−𝑔 𝑦 /𝛾, 𝑔 +𝛿 𝑢∗ = 𝐴 > 0. 𝑔 +𝛿+ ℎ∗ 𝐴 𝑢𝑒 Proof. SeeAppendix. □ Discussion. Unlike the DMP model, unemployment persists in MM’s model even when posting vacancies is costless. Firms post infinitely many vacancies, making matches certain, but workers become increasingly selective and wait for the best possible draws. Consequently, unemployment remains strictly positive and independent of level parameters such as 𝐴 or 𝑧 . 0 𝑙 Moreover,unemploymentdecreaseswiththeParetotailparameter: thinnertailsinduceworkersto accept jobsmore readily. The keydriver oflong run unemployment in MM’smodel isthe random natureofmatchquality. If 𝛼 = ∞,sothatmatchqualityisdeterministically 𝑧 𝑙 ,thenunemployment wouldfalltozerowhen 𝑘 = 0. 18

3.8.3. SBGPwithExponentialSamplingDistribution TheoutcomechangeswhenthesamplingdistributionisexponentialratherthanPareto. Becausethe exponentialdistributionhasmuchthinnertails,theselectionmotiveistooweaktosustainpositive unemployment in the presence of abundant job offers. As a result, unemployment vanishes once vacanciesbecomecostless,asweshow. UndertheexponentialdistributionandCDmatching,theUEtransitionrateis ℎ 𝑢𝑒 = 𝐴 0 𝑝 (𝜃)𝑒−𝜈𝑅 0 = 𝐴 0 𝜃1−𝛾𝑒−𝜈𝑅 0. To determine ℎ , we solve for (𝑅 ,𝜃) using the efficient conditions (35) and (37). Under CD 𝑢𝑒 0 matching,theybecome: 𝑘 = 𝐴 0 𝑒−𝜈𝑅 0 (1−𝛾)𝜃−𝛾 Φ2 ,and (44) 1−𝛾 𝑅 −𝑏 0 0 𝜃 = . (45) 𝛾 𝑘 Substitutingequation(45)intoequation(44)givesasingleequationin 𝑅 : 0 𝑒−𝜈𝑅 0 𝑘1−𝛾 𝐿𝐻𝑆(𝑅 0 ) := (𝑅 0 −𝑏 0 ) 𝛾 = 𝐴 0 (1−𝛾) 1−𝛾 𝛾𝛾Φ2 . (46) Equation (46) admits a unique solution since 𝐿𝐻𝑆(𝑏 0 ) = ∞, 𝐿𝐻𝑆(∞) = 0, and 𝐿𝐻𝑆′(𝑅 0 ) < 0. Moreover,as 𝑘 → 0,𝑅 → ∞. Once 𝑅 isdetermined, 𝜃 followsfromequation(45). 0 0 Rewritingequation(44), 𝑘 = 𝐴 0 𝑒−𝜈𝑅 0 (1−𝛾)𝜃−𝛾 Φ2 = ℎ 𝑢𝑒 (1−𝛾)𝜃−1 Φ2 , whichimplies,usingequation(45), 𝑘𝜃 𝑘1− 𝛾 𝛾 𝑅 0− 𝑘 𝑏 0 𝑅 0 −𝑏 0 ℎ 𝑢𝑒 = = = . (1−𝛾)Φ2 (1−𝛾)Φ2 𝛾Φ2 ThustheUErateisincreasinginthereservationquality 𝑅 0 . Sincelim 𝑘→0 𝑅 0 = ∞, itfollowsthat ℎ → ∞. Consequently,unemploymentconvergestozero. 𝑢𝑒 19

3.8.4. WelfareCostofSearchFrictions We now highlight a problematic feature of inspection models: eliminating search frictions by makingvacancypostingcostlessleadstounboundedproductivityandinfinitewelfaregains. Using equation(35)andCDmatching,socialwelfarecanbewrittenas ∫ ∞ (cid:20)∫ (cid:21) 𝑊 = 𝑒−𝑟𝑡 𝑦 𝑡 𝑧𝑛 𝑡 (𝑧)𝑑𝑧 +𝑢𝑏 𝑡 − 𝑘 𝑡 𝑣 𝑑𝑡 (47) 𝑡=0 𝑅 𝑡 ∫ ∞ (cid:20) ∫ (cid:21) = 𝑒−𝑟𝑡 (1−𝑢) 𝑦 𝑡 𝑧 𝑡 𝑔 𝑡 (𝑧)𝑑𝑧 +𝑢𝑏 𝑡 − 𝑘 𝑡 𝜃𝑢 𝑑𝑡 𝑡=0 𝑅 𝑡 ∫ ∞ (cid:20) 1−𝛾 (cid:21) = 𝑒−𝑟𝑡 (1−𝑢) 𝑦 𝑡 𝑍 𝑡 +𝑢𝑏 𝑡 − (𝑦 𝑡 𝑅 𝑡 −𝑏 𝑡 )𝑢 𝑑𝑡, 𝛾 𝑡=0 where 𝑔 (𝑧) isthedensitycorrespondingtothedistributionfunction𝐺 (𝑧). 𝑡 𝑡 IntheMMmodel,Lemma1andProposition1yieldtheclosed-formexpression ∫ ∞ (cid:20) 𝛼 1−𝛾 (cid:21) 𝑊 = 𝑒−(𝑟−𝑔 𝑦−𝑔 𝑧 )𝑡 (1−𝑢) 𝑦 0 𝑅 0 +𝑢𝑏 0 − (𝑦 0 𝑅 0 −𝑏 0 )𝑢 𝑑𝑡 𝛼−1 𝛾 𝑡=0 (cid:104) (cid:105) (1−𝑢) 𝛼 − 1−𝛾𝑢 𝑦 𝑅 + 1𝑢𝑏 𝛼−1 𝛾 0 0 𝛾 0 = . (48) 𝑟 −𝑔 −𝑔 𝑦 𝑧 Since 𝑅 → ∞as 𝑘 → 0, welfaredivergesto+∞providedthecoefficienton 𝑦 𝑅 ispositive— 0 0 0 thatis,whenunemploymentisnottoohigh. Conditions(a)and(c)inTheorem1guaranteethatitis, infact,thecase. Proposition4. InanSI-BGPwith 𝑀(𝑢,𝑣) = 𝑢𝛾𝑣1−𝛾 ,lim 𝑘→0 𝑊 = ∞. Proof. SeeAppendix. □ Theimplication ofProposition 4is stark: eliminatingsearch frictions delivers infinitewelfare gains. In our AI-BGP variant, unemployment vanishes as 𝑘 → 0, so welfare given in equation (47) simplifiesto ∫ ∞ ∫ ∞ (cid:20)1 (cid:21) lim𝑊 = 𝑒−𝑟𝑡𝑍 𝑡 𝑑𝑡 = 𝑒−𝑟𝑡 + 𝑅 0 + (𝑔 𝐴 /𝜈)𝑡 𝑑𝑡 𝑘→0 𝑡=0 𝑡=0 𝜈 1 (cid:18)1 (cid:19) 𝑔 /𝜈 𝐴 = + 𝑅 0 + . 𝑟 𝜈 𝑟2 Since 𝑅 → ∞, welfareagaindivergesto+∞. 0 20

Conclusion. In both the Pareto and exponential inspection frameworks, eliminating search frictions leads to unbounded welfare gains. This outcome stands in sharp contrast to the DMP model,wherewelfaregainsarefinite,andhighlightsafundamentalweaknessofinspectionmodels: theypredictimplausiblylargebenefitsfromremovingfrictions. 3.9. PossibleGeneralizations AnSBGPrequiresthat𝐺 𝑡 (𝑇 𝑡 (𝑧)) = 𝐺 0 (𝑧). Sofar,wehavefocusedontwocases: theParetocase,in which𝑇 𝑡 (𝑧) = 𝑧𝑒𝑔 𝑧 𝑡,andtheexponentialcase,inwhich𝑇 𝑡 (𝑧) = 𝑧+𝜙𝑡. Werefertothecorresponding SBGPsastheSI-BGP(Scale-Invariant)andtheAI-BGP(Additive-Invariant),respectively. Wenow outlineageneral methodologyfordetermining thetime-transformation function𝑇 (𝑧) 𝑡 foragivensamplingdistribution 𝐹(𝑧). Notefirstthatbyconstruction, 𝑅 𝑡 =𝑇 𝑡 (𝑅 0 ). AlongaBGP, equation(1)canbewrittenas 𝑅 𝑡 = 𝐻−1(cid:0)𝜙 1 𝑒−𝑔 𝐴 𝑡(cid:1), where 𝐻(𝑅 𝑡 ) = 1− 𝐹(𝑅 𝑡 ) is the survival function and 𝜙 1 is a constant. Because 𝜙 1 = 𝐻(𝑅 0 ), it followsthat 𝑅 𝑡 =𝑇 𝑡 (𝑅 0 ) where 𝑇 𝑡 (𝑧) ≡ 𝐻−1(cid:0)𝐻(𝑧)𝑒−𝑔 𝐴 𝑡(cid:1) . Afewillustrativecasesinclude: 1. Pareto: 𝐹(𝑧) = 1− (cid:16) 𝑧 𝑧 𝑙 (cid:17)𝛼 , 𝐻(𝑅) = (cid:0)𝑧 𝑅 𝑙 (cid:1)𝛼 , 𝐻−1(𝑥) = 𝑧 𝑙 𝑥−1/𝛼,𝑇 𝑡 (𝑧) = 𝑧𝑒𝑔 𝐴 𝑡/𝛼. 2. Exponential: 𝐹(𝑧) = 1−𝑒−𝜈𝑅, 𝐻(𝑅) = 𝑒−𝜈𝑅, 𝐻−1(𝑥) = − 1 𝜈 ln𝑥,𝑇 𝑡 (𝑧) = 𝑧 + 𝑔 𝜈 𝐴 𝑡. (cid:16) (cid:17) 3. Gompertz: 𝐹(𝑧) = 1 − 𝑒−𝜏(𝑒𝜈𝑧−1), 𝐻(𝑅) = 𝑒−𝜏(𝑒𝜈𝑅−1), 𝐻−1(𝑥) = 1 𝜈 ln 1− 1 𝜏 ln𝑥 , 𝑇 𝑡 (𝑧) = (cid:16) (cid:17) 1 ln 𝑒𝜈𝑧 + 𝑔 𝐴 𝑡 . 𝜈 𝜏 4. Weibull: 𝐹(𝑧) = 1−𝑒−(𝜈𝑅)𝜏,𝐻(𝑅) = 𝑒−(𝜈𝑅)𝜏,𝐻−1(𝑥) = 1 𝜈 (−ln𝑥)1/𝜏,𝑇 𝑡 (𝑧) = 1 𝜈 ((𝜈𝑧)𝜏 +𝑔 𝐴 𝑡) 1/𝜏. Theseexamplesillustratethatthedetrendingrequiredtopreservestationaritydependssensitively on the underlying sampling distribution. Once𝑇 (𝑧) is determined, equation (19) can be used to 𝑡 backouttheimpliedparameterrestrictionsfor 𝑏 and 𝑘 : 𝑡 𝑡 𝑀 (1,𝜃) 1 𝑏 𝑡 = 𝑦 𝑡 𝑇 𝑡 (𝑅 0 ) − 𝑘 𝑡 . 𝑀 (1,𝜃) 2 Here, 𝑅 and 𝜃 areendogenousbutconstant. 0 Theprocedureaboveprovidescandidatesforbalancedgrowthpaths. Acompletecharacterization, however, requires verifyingthat all BGP-defining equationsare jointly satisfied. Because someof 21

the necessaryintegrals may not have closed-form solutions, numerical methods may be neededto verifyexistenceandfullycharacterizethecorrespondingSBGP. 4. Biased Technological Change Thissectiondevelops analternative totheinspectionframework: aDMPmodelwith homogeneous workersandbiasedtechnologicalprogressinthematchingfunction. Weshowthatwhentechnological changeisbiasedandtheinputsinthematchingfunctionarecomplements,awell-behavedlimiting BGPexists. ThisBGPpreservesthecentralpropertiesofthestandardDMPmodelunderworkeraugmentingtechnologicalprogress: welfaregainsfromeliminatingsearchfrictionsarefinite,and unemploymentvanisheswhenvacanciesarecostless. The section proceeds in two parts. The first subsection formalizes the notion of biased technological progress in the matching function and provides a sharp characterization for the constantelasticityofsubstitution(CES)case. Weshowthat,contrarytoMM’sclaim,theirresults are not robust to the introduction of biased progress. In particular, the limiting growth rate of matchesmayconvergetozerodespiteongoingtechnologicalimprovements,renderingtheirmain theoreminapplicableinsuchcases. Thisanalysisassumesastationarytightnessrate. ThesecondsubsectionembedsbiasedtechnologicalprogressintothefullDMPmodel. Weshow thataBGPwithconstanttightnessemergesendogenouslyasageneralequilibriumoutcome—our maincontribution. Akeyfeatureofthisequilibriumisthatitisnecessarilyinefficient,highlighting asharpcontrastbetweenplannerandmarketallocations. 4.1. BiasedTechnologicalChangeintheMatchingFunction Section 2assumed amatching functionof the form 𝐴 𝑀(𝑢 ,𝑣 ). Inthis formulation,technological 𝑡 𝑡 𝑡 progressisHicks-neutral. Wenowconsideramoregeneralspecification, 𝑀 (𝐴 𝑢 ,𝐵 𝑣 ), 𝑡 𝑡 𝑡 𝑡 where 𝐴 and 𝐵 represent unemployment- and vacancy-augmenting technologies, growing at 𝑡 𝑡 constantexogenousrates 𝑔 ≥ 0and 𝑔 ≥ 0,respectively. 𝐴 𝐵 Thejob-findingrateisdefinedas 𝑀 (𝐴 𝑢 ,𝐵 𝑣 ) 𝑡 𝑡 𝑡 𝑡 𝑚 𝑡 ≡ = 𝑀 (𝐴 𝑡 ,𝐵 𝑡 𝜃 𝑡 ) =: 𝑚 𝑡 (𝜃 𝑡 ). (49) 𝑢 𝑡 where 𝜃 ≡ 𝑣 isthemarkettightness. Forlaterpurposes,itisconvenienttodefineeffectivetightness 𝑢 22

as 𝐵 𝑣 𝐵 𝑡 𝑡 𝑡 (cid:98) 𝜃 𝑡 ≡ = 𝜃 𝑡 . 𝐴 𝑢 𝐴 𝑡 𝑡 𝑡 When 𝐴 𝑡 = 𝐵 𝑡 ,thefunctionsimplifiesto 𝐴 𝑡 𝑀 (𝑢 𝑡 ,𝑣 𝑡 ), theHicks-neutralcaseconsideredbyMM. MMarguethattheirresultsextendbeyondHicks-neutralprogress: ”In the case of input-augmenting search progress, the rate 𝑔 converges to some 𝑚 𝑔∗... Inthelimitas𝑔 → 𝑔∗,ourtheoremsholdwith𝑔∗ replacing𝑔 .”(MM,footnote 𝑚 𝑚 𝑚 𝑚 𝐴 10)3. However,fortheirresultstohold,itisessentialthat𝑔∗ > 0. Otherwise,theirmainresultsdonot 𝑚 apply. If 𝑔 𝑚 = 0, thenaParetodistributioncannotbederivedfromtheirequation(10). Moreover, with 𝑔 𝐴 = 0and their assumption 𝛿 = 0, job destruction disappears, and unemployment vanishes in thelimit. Toseewhy 𝑔∗ = 0maynaturallyariseunderbiasedtechnologicalprogress,considertheCES 𝑚 matchingfunction: (cid:40) (𝛼(𝐴𝑢) 𝜎 + (1−𝛼) (𝐵𝑣) 𝜎 ) 1/𝜎, 𝜎 ≦ 1,𝜎 ≠ 0. (cid:41) 𝑀 (𝐴𝑢,𝐵𝑣) = . (50) (𝐴𝑢) 𝛼 (𝐵𝑣) 1−𝛼 if 𝜎 = 0. This specification has a long tradition in the search-and-matching literature (e.g., Den Haan etal.,2000;HagedornandManovskii,2008;Petrosky-Nadeauetal.,2018). TheCobb-Douglascase correspondsto 𝜎 = 0. Workersandvacanciesarecomplementsif 𝜎 < 0andsubstitutes if 𝜎 > 0. Co´rdobaetal.(2024)discussesseveraladvantagesoftheCESfunctionwithcomplementarity. FortheCESfunction,thegrowthrateofmeetingsis (cid:16) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17) 𝑔 𝑚,𝑡 = 𝜇 (cid:98) 𝜃 𝑡 𝑔 𝐴 + 1− 𝜇 (cid:98) 𝜃 𝑡 𝑔 𝐵 , (51) where (cid:16) (cid:17) 𝛼 𝜇 (cid:98) 𝜃 = . (52) 𝛼+ (1−𝛼)𝜃𝜎 (cid:98) The formula confirms that when technological progress is Hicks-neutral (𝑔 𝐴 = 𝑔 𝐵 ), we obtain 𝑔 𝑚,𝑡 = 𝑔 𝐴 = 𝑔 𝐵 > 0. The next proposition characterizes the limit behavior when technological progressiseitherworkerorvacancyaugmenting. Proposition 5. Suppose 0 < 𝜃 < ∞, and either (i) 𝑔 𝐴 > 0 and 𝑔 𝐵 = 0; or (ii) 𝑔 𝐵 > 0 and 𝑔 𝐴 = 0. 3MMusethenotation𝑔 forthegrowthrateofmeetings;weuse𝑔 . 𝑝 𝑚 23

Then   max{𝑔 𝐴 ,𝑔 𝐵 } > 0if 𝜎 > 0     𝑔 𝑚 ∗ = lim 𝑔 𝑚,𝑡 =  𝛼𝑔 𝐴 + (1−𝛼)𝑔 𝐵 > 0if 𝜎 = 0  . 𝑡→∞     0if 𝜎 < 0     Proof. As𝑡 → ∞, effectivetightnesssatisfies (cid:40) (cid:41) 0, 𝑔 > 0, 𝐴 𝜃 → . (cid:98)𝑡 ∞, 𝑔 > 0 𝐵 Fromequation(52): 𝜎 > 0 𝜎 < 0     (cid:16) (cid:17)     lim 𝜇 (cid:98) 𝜃 𝑡 = 𝑔 𝐴 > 0 1 0 . 𝑡→∞    𝑔 > 0 0 1   𝐵    Substitutingintoequation(51)yields: 𝜎 > 0 𝜎 < 0         lim 𝑔 𝑚,𝑡 = 𝑔 𝐴 > 0 𝑔 𝐴 0 . 𝑡→∞    𝑔 > 0 𝑔 0   𝐵 𝐵    □ Thepropositionshowsthatwheninputsaresubstitutes, 𝑔∗ > 0,whichisnecessaryforMM’s 𝑚 results to hold. The more interesting cases are when inputs are complements (𝜎 < 0) and technologicalprogressisbiased,eitherworker-augmentingorvacancy-augmenting,inwhichcases 𝑔∗ = 0. In these cases, MM’s results do not hold. The reason is that biased progress runs into 𝑚 diminishingreturns: thenon-improvinginputbecomesabottleneckunderstrictcomplementarity, creatinganupperboundonmeetingseveninthepresenceofcontinuedtechnologicalprogress. WenowexploretheimplicationsofthislimitbehaviorwithintheDMPsearch-and-matching modelwithdecliningsearchfrictions. 4.2. DMPModelwithBiasedTechnologicalChange Thereisaunitoneofworkers,ofwhich𝑛 areemployedand𝑢 areunemployed. Employedworkers 𝑡 𝑡 produce 𝑦 𝑡 = 𝑦𝑒𝑔𝑡, theunemployedworkersproduce 𝑏 𝑡 = 𝑏𝑒𝑔𝑡, where 𝑦 > 𝑏,andvacancyposting 24

costs 𝑘 𝑡 = 𝑘𝑒𝑔𝑡 attime𝑡. Givenaninitialemploymentlevel 𝑛 0 ,thesocialplannersolves ∫ ∞ max 𝑒−(𝑟−𝑔)𝑡 (𝑛 𝑦 +𝑢 𝑏 − 𝑘𝑣 )𝑑𝑡 subjectto 𝑡 𝑡 𝑡 {𝑛 𝑡 ,𝑣 𝑡 ,𝑢 𝑡}∞ 𝑡=0 0 · 𝑛 𝑡 = 𝑀 (𝐴 𝑡 𝑢 𝑡 ,𝐵 𝑡 𝑣 𝑡 ) −𝛿𝑛 𝑡 ,∀𝑡 ≥ 0, (53a) 𝑢 𝑡 = 1−𝑛 𝑡 , ∀𝑡 ≥ 0, (53b) where 𝐴 𝑡 = 𝐴 0 𝑒𝑔 𝐴 𝑡 and 𝐵 𝑡 = 𝐵 0 𝑒𝑔 𝐵 𝑡,𝑟 > 𝑔, and 𝛿 > 0. Let 𝜌 ≡ 𝑟 −𝑔 denotetheeffectivediscount rate. Optimality conditions. Let 𝑒−𝜌𝑡𝜆 𝑡 and 𝑒−𝜌𝑡𝜂 𝑡 denote the Lagrange multipliers associated with constraints(53a)and(53b),respectively. Thefirst-orderconditionswithrespectto𝑣 , 𝑢 , and𝑛 are: 𝑡 𝑡 𝑡 𝜕𝑀 𝑡 𝑘 = 𝜆 𝑡 = 𝑠 𝑡 (1− 𝜇 𝑡 )𝜆 𝑡 , (54) 𝜕𝑣 𝑡 𝜕𝑀 𝑡 𝜂 𝑡 = 𝑏 + 𝜆 𝑡 = 𝑏 +𝑚 𝑡 𝜇 𝑡 𝜆 𝑡 , and (55) 𝜕𝑢 𝑡 · 𝑦 −𝜂 𝑡 −𝛿𝜆 𝑡 = 𝜌𝜆 𝑡 −𝜆 𝑡 . (56) Here, 𝑠 isthejob-fillingrateand 𝜇 theelasticityofthematchingfunctionwithrespectto”effective” 𝑡 𝑡 jobseekers: 𝑀 𝑡 𝑠 𝑡 ≡ = 𝑚 𝑡 (𝜃 𝑡 )/𝜃 𝑡 and (57) 𝑣 𝑡 𝜕𝑀 𝑡 𝐴 𝑡 𝑢 𝑡 (cid:16) (cid:17) 𝜇 𝑡 ≡ = 𝜇 (cid:98) 𝜃 𝑡 . 𝜕 (𝐴 𝑢 ) 𝑀 𝑡 𝑡 𝑡 Optimalityalsorequiresthetransversalityconditionlim 𝑡→∞ 𝑒−𝜌𝑡𝜆 𝑡 𝑛 𝑡 = 0. Equations(54)to (56)mirrorthose ofthecanonicalDMP framework. Fromequation(54),the efficientnumberofvacanciesequatesthemarginalcost 𝑘 withthemarginalnewmatchesassociated toavacancy, 𝜕𝑀 𝑡,multipliedbytheshadowvalueofamatch,𝜆 . Themarginalgain 𝜕𝑀 𝑡 equalsthe 𝜕𝑣 𝑡 𝜕𝑣 𝑡 𝑡 averagegain 𝑠 𝑡 = 𝑀 𝑣 𝑡, scaleddownbytheelasticity1− 𝜇 𝑡 . 𝑡 Equation (55) states that the shadow value flow of an unemployed worker, 𝜂 , is equal the 𝑡 worker’sownoutput, 𝑏,plustheexpectedcontributiontonewmatches, 𝜕𝑀 𝑡,weightedby𝜆 . Here, 𝜕𝑢 𝑡 𝑡 themarginalgain 𝜕𝑀 𝑡 equalstheaveragegain, 𝑚 ,scaledby 𝜇 . 𝜕𝑢 𝑡 𝑡 𝑡 Finally,combiningequation(55)andequation(56)yieldsthevalueofamatch: · 𝜌𝜆 𝑡 = 𝑦 −𝑏 − (𝛿+𝑚 𝑡 𝜇 𝑡 )𝜆 𝑡 +𝜆 𝑡 . (58) Thisexpressionshowsthatthenetreturnofamatchistheaddedoutput 𝑦 −𝑏 plusthethecapital 25

· gains, 𝜆 , offset by the effective depreciation rate. Depreciation includes both the exogenous job 𝑡 destructionrate, 𝛿, andtheendogenouseffect 𝑚 𝜇 , whichcapturesthefactthatasuccessfulmatch 𝑡 𝑡 reducesthepoolofjobseekersandtherebylowersfuturematchingopportunities. 4.2.1. BalancedGrowthCharacterization Considerbalancedgrowthpaths(BGPs)alongwhichvariablesgrowatconstantrates. Thefollowing proposition shows that—despite improvements in the matching technology—labor-market variables remainstationaryalonganyBGP. Proposition6. Alongabalancedgrowthpath,thegrowthratesof 𝑛,𝑢, 𝑚, 𝜇,𝜆, 𝑠,and 𝜃 arezero. Proof. Sincepopulationisconstant,bothemployment(𝑛)andunemployment(𝑢)mustbeconstant alongaBGP.Equation(53a)thenreducesto 𝛿𝑛 = 𝑚 𝑡 𝑢 = 𝑚𝑢. Thus 𝑚 𝑡 = 𝑚 along a BPG. Similarly, since 𝜇 𝑡 ∈ [0,1], we must have 𝜇 𝑡 = 𝜇. Substituting these resultsintoequation(58)andimposingthetransversalityconditionyields 𝑦 −𝑏 𝜆 𝑡 = 𝜆 = . 𝜌 +𝛿+𝑚𝜇 Substitutingintoequation(54)gives 𝑠 (1− 𝜇) 𝑡 𝑘 = (𝑦 −𝑏). (59) 𝜌 +𝛿+𝑚𝜇 Equation(59)impliesthat 𝑠 𝑡 = 𝑠,andsince 𝑚 = 𝑠𝜃 𝑡 then 𝜃 𝑡 = 𝜃 alongaBGP. □ Equation(59)determines 𝜃, andtheunemploymentratefollowsfrom 𝛿 𝑢 = . (60) 𝑚 +𝛿 Markets. Equation (59) parallels Pissarides (2000, Eq. 1.24) in a decentralized setting where firms post vacancies with success probability 𝑠∗, workers find jobs with probability 𝜃∗𝑠∗, firms captureafraction1−𝛾 ofthematchsurplus,andfreeentryholds. Inournotation: 𝑠∗ (1−𝛾) 𝑘 = (𝑦 −𝑏).4 (61) 𝜌 +𝛿+𝑚∗𝛾 4Pissarides(2000)assumes𝑔 =0,unlikehere. 26

Free entry further implies that the expected net return of posting a vacancy, 𝑠∗ (𝑦 −𝑤∗)/𝑘 − 𝛿, equalstheeffectivemarketreturn 𝜌, where 𝑤 whichyieldsthewageequation: (𝜌 +𝛿) 𝑘 𝑤∗ = 𝑦 − . 𝑠∗ Themarketequilibriumisgenerallyinefficientbecausetheworker’sbargainingpowerisfixedata constant value 𝛾 > 0, while in the planner’s allocation the effective bargaining power is variable, (cid:16) (cid:17) 𝜇 𝜃 . ThispermitsthemarketequilibriumtosustainaBGP;thecorrespondingefficientallocation, (cid:98)𝑡 bycontrast,necessarilyrulesoneout. 4.2.2. Equilibrium Considerfirstthemarketsolution. Accordingtoequation(61),aBGPwithdecliningsearchcosts existsiftheydonotaffect 𝑚∗ or 𝑠∗ = 𝑚∗/𝜃∗. Takingtimederivativesofequation(49)yields · ∗ · · 𝑚 𝐴 𝐴 𝜃𝐵 𝐵 𝑡 = 𝑀 1 (𝐴 𝑡 ,𝐵 𝑡 𝜃∗) 𝑡 𝑡 + 𝑀 2 (𝐴 𝑡 ,𝐵 𝑡 𝜃∗) 𝑡 𝑡 (62) 𝑚∗ 𝑀 𝐴 𝑀 𝐵 𝑡 𝑡 𝑡 (cid:16) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17) = 𝜇 (cid:98) 𝜃 𝑡 ∗ 𝑔 𝐴 + 1− 𝜇 (cid:98) 𝜃∗ 𝑔 𝐵 . Alltermsinthisexpressionarenon-negative. Thefollowingpropositionfollowsnaturally: Proposition 7. There is no BGP in the market economy when 𝑔 𝐴 > 0 and 𝑔 𝐵 > 0. In particular, thereisnoBGPwithHicks-neutraltechnologicalprogressinthematchingfunction. This proposition confirms MM’s result for the Hicks-neutralcase, namely that there is no BGP in a model without inspection. The next lemma suggests that a BGP may exist when technological changeiseithervacancyorworkeraugmenting. 𝑚·∗ Lemma2 Suppose 𝜃 = 𝜃∗ ∈ (0,∞). For 𝑡 = 0, oneofthefollowingtwoconditionsmusthold: 𝑚∗ 𝑡 (i) 𝑔 𝐴 > 0, 𝑔 𝐵 = 0, and 𝜇(0) = 0, or (ii) 𝑔 𝐵 > 0, 𝑔 𝐴 = 0, and 𝜇(∞) = 1. (cid:16) (cid:17) 𝑚·∗ Proof. (i) If 𝑔 𝐴 > 0 and 𝑔 𝐵 = 0, then (cid:98) 𝜃 𝑡 ∗ → 0 and 𝜇 (cid:98) 𝜃 𝑡 ∗ → 𝜇(0) = 0. Hence, 𝑚∗ 𝑡 = 𝑡 (cid:16) (cid:17) (cid:16) (cid:17) 𝜇 (cid:98) 𝜃 𝑡 ∗ 𝑔 𝐴 → 0. (ii) If 𝑔 𝐴 = 0 and 𝑔 𝐵 > 1, then (cid:98) 𝜃 𝑡 ∗ → ∞ and 𝜇 (cid:98) 𝜃 𝑡 ∗ → 𝜇(∞) = 1. Hence, 𝑚 𝑚 · 𝑡 = (cid:16) 1− 𝜇 (cid:16) (cid:98) 𝜃 𝑡 ∗ (cid:17)(cid:17) 𝑔 𝐵 → 0. □ 𝑡 At this point, it is convenient to focus on the CES matching function given in equation (50). Applyingthelemma,wefindthatstrictcomplementarityisanecessaryconditionfortheexistence 27

(cid:16) (cid:17) of a BGP. In the CES case, the function 𝜇 𝜃 satisfies equation (52). When inputs are strict (cid:98) complements, 𝜎 < 0, we have 𝜇(0) = 0 and 𝜇(∞) = 1, exactly as required by the lemma. In contrast,when inputsaresubstitutes (𝜎 > 0),we obtain 𝜇(0) = 1and 𝜇(∞) = 0—theopposite of the condition required by the lemma. Therefore, strict complementarity is necessary, though not sufficient,fortheexistenceofaBGP. 4.3. CESMatching Proposition8. Suppose 𝑀 isaCESmatchingfunction. AninteriorBGPofthemarketeconomy existsinthefollowingtwocases: (i) 𝑔 𝐴 > 0, 𝑔 𝐵 = 0, 𝜎 < 0, 𝛾 > 0, and (1−𝛼) 1/𝜎 𝐵(1−𝛾) 𝑘 < (𝑦 −𝑏); (63) 𝜌 +𝛿 (ii) 𝑔 𝐴 = 0, 𝑔 𝐵 > 0, 𝜎 < 0, 𝛾 < 1, and 𝑦 > 𝑏. Proof. (i)Underthestatedconditions,thematchingfunctionconvergesto𝑀 (𝐴𝑢,𝐵𝑣) = (1−𝛼) 1/𝜎 𝐵𝑣. Hence, 𝑠∗ = (1−𝛼) 1/𝜎 𝐵, 𝑚∗ = (1−𝛼) 1/𝜎 𝐵𝜃∗, 𝑢∗ = 𝛿 , 1 1 1 1 𝑚∗+𝛿 1 (1−𝛼) 1/𝜎 𝐵(1−𝛾) (𝑦 −𝑏) − (𝜌 +𝛿) 𝑘 𝜃∗ = , and 1 (1−𝛼) 1/𝜎 𝐵𝛾𝑘 (𝜌 +𝛿) 𝑘 𝑤∗ = 𝑦 − . 1 (1−𝛼) 1/𝜎 𝐵 Condition (63) guarantees that an interior solution for 𝜃∗ exists. (ii) Under the stated conditions, the matching function converges to 𝑀 (𝐴𝑢,𝐵𝑣) = 𝛼1/𝜎𝐴𝑢. Thus, 𝑠∗ = 𝛼1/𝜎𝐴/𝜃∗, 𝑚∗ = 𝛼1/𝜎𝐴, 2 2 2 𝑢∗ = 𝛿 , 2 𝑚∗+𝛿 2 𝛼1/𝜎𝐴(1−𝛾) 𝑦 −𝑏 𝜃∗ = , and 2 𝜌 +𝛿+𝛼1/𝜎𝐴𝛾 𝑘 (𝜌 +𝛿) (1−𝛾) (𝑦 −𝑏) 𝑤∗ = 𝑦 − . 2 𝜌 +𝛿+𝛼1/𝜎𝐴𝛾 Aninteriorsolutionexistiff0 < 𝛾 < 1and 𝑦 > 𝑏. □ Discussion. ThelimitBGP characterizedinProposition 1emergesbecause theCESmatching functionconvergestoalineartechnologyinwhichthesoleeffectiveinputistheonenotexperiencing technological progress. With labor-augmenting progress, the matching function converges to a 28

linear function of effective vacancies. Conversely, with vacancy-augmenting progress, it converges toalinearfunctionofeffectiveunemployedworkers. Despitetheseasymptoticlinearities,theunemploymentrateremainswellbehaved. Forexample, increases in the vacancy posting cost, unemployment benefits, or workers’bargaining power reduce markettightnessandraiseunemploymentintheusualway. Proposition8providesacounterexampletoMM’sclaim—madeintheirfootnote10—thattheir model remains valid in the limit even under input-specific technological change. Not only the growthrateofthemeetingsrategoestozerointhesecases,butthecanonicalDMPmodeldeliversa well-definedlimitBGPwithoutrequiringheterogeneity,Paretodistributions,orinspection. Ofthe twocasesidentified inProposition 8,case (i)is theonlyonethat delivers thestandard resultthatunemploymentvanisheswhenvacancypostingiscostless. Furthermore,Co´rdobaetal. (2024)alsoshowthatlabor-augmentingtechnologicalprogressinthematchingfunctioncanaccount for a significant share of the decline in the labor share and the fall in market tightness observed between1980and2007. 4.3.1. WelfareCostofSearchFrictions AnimportantdistinctionbetweentheDMPmodelanalyzedinthissectionandtheinspectionmodels discussedpreviouslyliesinthepotential welfaregainsfrom eliminating searchfrictions. Ina BGP, socialwelfaresatisfies: 𝑛𝑦 +𝑢𝑏 − 𝑘𝑣 𝑦 −𝑢𝑦 +𝑢𝑏 − 𝑘𝜃𝑢 𝑊(𝑢) = = 𝑟 −𝑔 𝑟 −𝑔 (cid:16) (cid:17) 𝑦 − (𝑦 −𝑏) 𝜌+𝛿+𝑚 𝑢 𝜌+𝛿+𝑚𝜇 = (usingequations(57)and(61)) 𝑟 −𝑔 (cid:16) (cid:17) 1− (1−𝜑) 𝑢(𝜌+𝛿)+(1−𝑢)𝛿 𝑢 𝑢(𝜌+𝛿)+(1−𝑢)𝛿𝜇 = 𝑦 (usingequation(60)), 𝑟 −𝑔 where 𝜑 = 𝑏/𝑦. Therelativewelfarecostsofsearchfrictionscanthenbedefinedas: 𝑊(0) −𝑊(𝑢) (cid:18) 𝑢(𝜌 +𝛿) + (1−𝑢)𝛿 (cid:19) Ψ(𝑢) ≡ = (1−𝜑) 𝑢. 𝑊(0) 𝑢(𝜌 +𝛿) + (1−𝑢) 𝜇𝛿 Thismeasureisrelativetotheidealbenchmarkoffullemployment. Suchbenchmarkisachieved when 𝑘 = 0incase(i)ofProposition8butnotincase(ii). Thekeypointisthatthewelfarecostsof searchfrictions,or unemploymentforshort,isboundedabove by1−𝜑,whichoccurs when 𝜇 = 0 and 𝜌 = 0. 29

In practice, estimated welfare costs fall well below this upper bound. For example, under the parametrizationemployedbyShimer(2005),thewelfarecostis Ψ(𝑢) = 6.5%.5 5. Conclusion MartelliniandMenzio(2020)poseafundamentalpuzzle: howcantechnologicalprogressinthe matching function (“declining search frictions”) be reconciled with the empirical stationarity of unemployment, tightness, and the Beveridge curve? In the spirit of King et al. (1988), they seek necessaryandsufficientconditionsunderwhichbalancedgrowthcancoexistwiththosestationary labor-marketfacts. Thispaperoffersthreemainconclusions. First, MM’s characterization is too strong. Their conditions (inspection goods with Paretodistributed quality) are sufficient but not necessary. Balanced growth paths arise outside their framework. Second, the inspection approach has implausible implications in the cases we study. In the Paretoversion,unemploymentpersistsevenwhenvacanciesarefreetopost,andthewelfaregains from eliminating search frictions are unbounded. In the exponential version, unemployment does vanishwithcostlessposting,yetthewelfaregainsremaininfinitebecauseworkerskeepraisingtheir reservationstandardsasmatchingbecomesarbitrarilyeasyandthequalitysupportisunbounded. Third, a constructive alternative exists within a standard DMP environment once we allow for biased technological change and complementarity in matching. With complementary inputs, biased progress in one input makes the other input relatively scarcer, creating a bottleneck and hence diminishing returns to search improvements. The growth rate of meetings falls to zero, deliveringawell-behavedBGPwithstationaryunemployment,tightness,andtransitionrates. Inthis setting,unemploymentvanishesasfrictionsdisappearbutonlyifprogressisworker-augmenting, and—crucially—welfaregainsarefinite. However,theBGPisnecessarilyinefficient: themarket equilibrium admits a stationary path with declining frictions, whereas the planner’s allocation does not,reflectingthefailureoftheHosiosconditionwhenbargainingweightsarefixedbuttheplanner’s shadowelasticityvarieswithtightness. TheseresultsreframetheinterpretationsMMconsider. Theydonotsupporttheviewthatsearch frictions are irrelevant, nor that the historical decline in frictionshas been too small. Instead, they pointtoaspecificcountervailingmechanism—endogenousbottlenecksfromcomplementarityunder biased progress—that can neutralize the growth effects of improved matching while preserving stationarylabor-marketvariables. TheyalsoshowthatMM’ssufficiencyresultdoesnotpindown a unique path: stationarity can emerge without perpetual growth in reservation quality (as in the 5Weuse𝜑=0.4, 𝜌 =0.012, 𝜇 =0.72,𝛿 =0.1,and𝑢 =4%. 30

exponentialcase)andwithoutattributinggrowthtodecliningfrictions(asinthebiased-technology DMPcase). Finally, this agenda opens clear avenues for future work. Empirically, measuring the bias in the matching progress (worker- vs. vacancy-augmenting) and the degree of complementarity is central todistinguishing between inspection and bottleneck mechanismsand to conducting credible welfare assessments. Theoretically, exploring policy in environments with biased progress and complementarity—wheremarketBGPsareinefficient—canclarifytheroleofbargaininginstitutions aswellasvacancytaxesorsubsidies. Relatedly,Co´rdobaetal.(2024)showthatCESmatchingwith worker-augmentingprogresscanaccountforsecularmovementsinthelaborshareandtightness, underscoringtheempiricalrelevanceofbiasedtechnologicalchangeinmatching. References Benhabib, J., J. Perla, and C. Tonetti (2021). Reconciling models of diffusion and innovation: A theoryoftheproductivitydistributionandtechnologyfrontier. Econometrica89(5),2261–2301. Buera,F.J.andE.Oberfield(2020). Theglobaldiffusionofideas. Econometrica88(1),83–114. Co´rdoba,J.C.,A.T.Isoja¨rvi,andH.Li(2024). Endogenousbargainingpoweranddeclininglabor compensation share. Technical report, Opportunityand Inclusive GrowthInstitute, Minneapolis Fed,WorkingPaperNo.92. Den Haan, W. J., G. Ramey, and J. Watson (2000). Job destruction and propagation of shocks. AmericanEconomicReview90(3),482–498. Hagedorn,M.andI.Manovskii(2008). Thecyclicalbehaviorofequilibriumunemploymentand vacanciesrevisited. AmericanEconomicReview98(4),1692–1706. Hosios,A.J.(1990). Ontheefficiencyofmatchingandrelatedmodelsofsearchandunemployment. TheReviewofEconomicStudies57(2),279–298. King,R.G.,C.I.Plosser,andS.T.Rebelo(1988). Production,growthandbusinesscycles: I.the basicneoclassicalmodel. JournalofMonetaryEconomics21(2-3),195–232. Lucas,R.E.andB.Moll(2014). Knowledgegrowthandtheallocationoftime. JournalofPolitical Economy122(1),1–51. Martellini,P.andG.Menzio(2020). Decliningsearchfrictions,unemployment,andgrowth. Journal ofPoliticalEconomy128(12),4387–4437. 31

Perla, J. and C. Tonetti (2014). Equilibrium imitation and growth. Journal of Political Economy122(1),52–76. Petrosky-Nadeau,N.,L.Zhang,andL.-A.Kuehn(2018). Endogenousdisasters. AmericanEconomic Review108(8),2212–45. Pissarides,C.A.(2000). Equilibriumunemploymenttheory. MITpress. Shimer, R.(2005). Thecyclicalbehavior ofequilibrium unemployment andvacancies. American EconomicReview95(1),25–49. Appendix A.1. Proofs of Equations and Propositions ProofofEquation(11): Tosolvethisdifferentialequation,writeitas (cid:104)· (cid:105) 𝑒−(𝑟+𝛿)𝜏 𝜆 𝑡+𝜏 (𝑧) − (𝑟 +𝛿)𝜆 𝑡+𝜏 (𝑧) = 𝑒−(𝑟+𝛿)𝜏 [𝑦 𝑡+𝜏 𝑅 𝑡+𝜏 − 𝑦 𝑡+𝜏 𝑧]. Integratingyields: ∫ 𝑑 ∫ 𝑑 (cid:104)· (cid:105) 𝑒−(𝑟+𝛿)𝜏 𝜆 𝑡+𝜏 (𝑧) − (𝑟 +𝛿)𝜆 𝑡+𝜏 (𝑧) 𝑑𝜏 = 𝑒−(𝑟+𝛿)𝜏 [𝑦 𝑡+𝜏 𝑅 𝑡+𝜏 − 𝑦 𝑡+𝜏 𝑧] 𝑑𝜏. 0 0 Theintegralontheleft-handsidesimplifiesto: (cid:2)𝑒−(𝑟+𝛿)𝜏𝜆 𝑡+𝜏 (𝑧) (cid:3) 0 𝑑 = 𝑒−(𝑟+𝛿)𝑑𝜆 𝑡+𝑑 (𝑧) −𝜆 𝑡 (𝑧). 32

ProofofEquation(36): Leta=: 𝑔 𝐴 /𝜈. Equation(13)becomes ∫ 𝑑(𝑧,𝑡) 𝜆 𝑡 (𝑧) = 𝑒−(𝑟+𝛿)𝜏 (𝑧 − 𝑅 𝑡 −𝑎𝜏)𝑑𝜏 0 ∫ 𝑑 𝑡(𝑧) ∫ 𝑑 𝑡(𝑧) = (𝑧 − 𝑅 𝑡 ) 𝑒−(𝑟+𝛿)𝜏𝑑𝜏 −𝑎 𝜏𝑒−(𝑟+𝛿)𝜏𝑑𝜏 0 0 (cid:20) 𝑒−(𝑟+𝛿)𝜏(cid:21)𝑑 𝑡(𝑧) (cid:20) (cid:18) 𝜏 1 (cid:19)(cid:21)𝑑 𝑡(𝑧) = (𝑧 − 𝑅 𝑡 ) − −𝑎 −𝑒−(𝑟+𝛿)𝜏 + 𝑟 +𝛿 0 𝑟 +𝛿 (𝑟 +𝑠) 2 0 𝑧 − 𝑅 𝑎 (cid:20) (cid:18) 1 (cid:19) 1 (cid:21) = 𝑡 (cid:2) −𝑒−(𝑟+𝛿)𝑑 𝑡(𝑧) +1(cid:3) − −𝑒−(𝑟+𝛿)𝑑 𝑡(𝑧) 𝑑 𝑡 (𝑧) + + 𝑟 +𝛿 𝑟 +𝛿 𝑟 +𝛿 𝑟 +𝛿 𝑎𝑑 (𝑧) 𝑎 (cid:20) (cid:18) 1 (cid:19) 1 (cid:21) = 𝑡 (cid:2)1−𝑒−(𝑟+𝛿)𝑑 𝑡(𝑧)(cid:3) + 𝑒−(𝑟+𝛿)𝑑 𝑡(𝑧) 𝑑 𝑡 (𝑧) + − 𝑟 +𝛿 𝑟 +𝛿 𝑟 +𝛿 𝑟 +𝛿 𝑎 (cid:20) (cid:18) 1 (cid:19) 1 (cid:21) = 𝑑 𝑡 (𝑧) (cid:2)1−𝑒−(𝑟+𝛿)𝑑 𝑡(𝑧)(cid:3) +𝑒−(𝑟+𝛿)𝑑 𝑡(𝑧) 𝑑 𝑡 (𝑧) + − 𝑟 +𝛿 𝑟 +𝛿 𝑟 +𝛿 𝑎 (cid:20) 1 1 (cid:21) = 𝑑 𝑡 (𝑧) (cid:2)1−𝑒−(𝑟+𝛿)𝑑 𝑡(𝑧)(cid:3) +𝑑 𝑡 (𝑧)𝑒−(𝑟+𝛿)𝑑 𝑡(𝑧) +𝑒−(𝑟+𝛿)𝑑 𝑡(𝑧) − 𝑟 +𝛿 𝑟 +𝛿 𝑟 +𝛿 𝑎 (cid:20) 𝑒−(𝑟+𝛿)𝑑 𝑡(𝑧) 1 (cid:21) = 𝑑 𝑡 (𝑧) + − . 𝑟 +𝛿 𝑟 +𝛿 𝑟 +𝛿 Wenextneedtocalculate ∫ 𝑎 ∫ (cid:20) 𝑒−(𝑟+𝛿)𝑑 𝑡(𝑧) 1 (cid:21) 𝜆 𝑡 (𝑧)𝑓(𝑧)𝑑𝑧 = 𝑑 𝑡 (𝑧) + − 𝑓(𝑧)𝑑𝑧 𝑟 +𝛿 𝑟 +𝛿 𝑟 +𝛿 𝑅 𝑅 𝑡 𝑡 = 𝑎 ∫ (cid:34) 𝑧 − 𝑅 𝑡 + 𝑒−(𝑟+𝛿) 𝑧− 𝑎 𝑅𝑡 − 1 (cid:35) 𝜈𝑒−𝜈𝑧𝑑𝑧 𝑟 +𝛿 𝑎 𝑟 +𝛿 𝑟 +𝛿 𝑅 𝑡 ∫ ∫ 𝑓(𝑧) 𝜆 𝑡 (𝑧)𝑓(𝑧)𝑑𝑧 = (1− 𝐹(𝑅 𝑡 )) 𝜆 𝑡 (𝑧) 𝑑𝑧 1− 𝐹(𝑅 ) 𝑅 𝑅 𝑡 𝑡 𝑡 = (1− 𝐹(𝑅 𝑡 ))𝐸 [𝜆 𝑡 (𝑧)|𝑧 > 𝑅] 𝑎 (cid:20) 𝑒−(𝑟+𝛿)𝑑 𝑡(𝑧) 1 (cid:21) = (1− 𝐹(𝑅 𝑡 )) 𝐸 𝑑 𝑡 (𝑧) + − |𝑧 > 𝑅 𝑟 +𝛿 𝑟 +𝛿 𝑟 +𝛿 𝑎 (cid:26) (cid:20)𝑒−(𝑟+𝛿)𝑑 𝑡(𝑧)(cid:21) 1 (cid:27) = (1− 𝐹(𝑅 𝑡 )) 𝐸 𝑧>𝑅 [𝑑 𝑡 (𝑧)] +𝐸 𝑧>𝑅 − . 𝑟 +𝛿 𝑟 +𝛿 𝑟 +𝛿 Now, (cid:20)𝑧 − 𝑅 (cid:21) 1 𝑅 1 (cid:18)1 (cid:19) 𝑅 1 𝑡 𝑡 𝑡 𝐸 𝑧>𝑅 [𝑑 𝑡 (𝑧)] = 𝐸 𝑧>𝑅 = 𝐸 𝑧>𝑅 𝑧 − = + 𝑅 𝑡 − = ;and 𝑎 𝑎 𝑡 𝑎 𝑎 𝜈 𝑎 𝑎𝜈 33

(cid:20)𝑒−(𝑟+𝛿)𝑑 𝑡(𝑧)(cid:21) 1 (cid:20)𝑒−(𝑟+𝛿)𝑑 𝑡(𝑧)(cid:21) 𝐸 𝑧>𝑅 = 𝐸 𝑧>𝑅 𝑟 +𝛿 𝑟 +𝛿 𝑟 +𝛿 1 (cid:104) 𝜈𝑎 (cid:105) = . 𝑟 +𝛿 𝑟 +𝛿+𝜈𝑎 Therefore, ∫ 𝑎 (cid:26) 1 1 𝜈𝑎 1 (cid:27) 𝜆 𝑡 (𝑧)𝑓(𝑧)𝑑𝑧 = (1− 𝐹(𝑅 𝑡 )) + − 𝑟 +𝛿 𝑎𝜈 𝑟 +𝛿𝑟 +𝛿+𝜈𝑎 𝑟 +𝛿 𝑅 𝑡 𝑎 (cid:20) 1 1 𝜈𝑎 1 (cid:21) = 𝑒−𝜈𝑅 𝑡 + − 𝑟 +𝛿 𝑎𝜈 𝑟 +𝛿𝑟 +𝛿+𝜈𝑎 𝑟 +𝛿 1 (cid:20) 1 (𝜈𝑎)2 𝑎𝜈 (cid:21) = 𝑒−𝜈𝑅 𝑡 1+ − (𝑟 +𝛿)𝜈 𝑟 +𝛿𝑟 +𝛿+𝜈𝑎 𝑟 +𝛿 1 (cid:34) 𝑟 +𝛿−𝑔 1 𝑔2 (cid:35) = 𝑒−𝜈𝑅 𝑡 𝐴 + 𝐴 . (64) (𝑟 +𝛿)𝜈 𝑟 +𝛿 𝑟 +𝛿𝑟 +𝛿+𝑔 𝐴 ProofofProposition3: Substitutingequation(42)andequation(43)intoequation(39): ℎ 𝑢 ∞ 𝑒 = 𝐴 0 𝑧 𝑙 𝛼 (cid:16) Ω1 𝑘 0 − 𝛼−( 𝛼 1−𝛾) (cid:17)1−𝛾 (cid:18) 1− 𝛾 𝛾 Ω 𝑦 1 𝑘 0 − 𝛼− 1 ( − 1 𝛾 −𝛾) (cid:19)−𝛼 0 𝛼(1−𝛾)−𝛼(1−𝛾) = Ω2 𝑘 0 𝛼−(1−𝛾) = Ω2 , where Ω2 = 𝐴 0 𝑧 𝑙 𝛼 Ω 1 1 −𝛾 (cid:18) 1− 𝛾 𝛾 Ω 𝑦 1 (cid:19)−𝛼 0 (cid:18) 𝛾 1 (cid:19)−𝛼 = 𝐴 0 𝑧 𝑙 𝛼 1−𝛾 𝑦 Ω 1 1 −𝛾−𝛼 0 = 𝐴 0 𝑧 𝑙 𝛼 (cid:18) 1− 𝛾 𝛾 𝑦 1 (cid:19)−𝛼 (cid:18)1− 𝛾 𝛾 (𝐴 0 (1−𝛾)Φ1 ) 1/(𝛼−1) 𝑦 0 𝛼 𝛼 −1 (cid:19) 1+ 1 𝛾 − / 𝛾 (𝛼 − − 𝛼 1) 0 1+ 1 1−𝛾−𝛼 𝛼+ 𝛼 1−𝛾−𝛼 = 𝐴 𝛼−11+𝛾/(𝛼−1)𝑦 𝛼−11+𝛾/(𝛼−1) 0 0 ×𝑧 𝑙 𝛼 (cid:18) 1− 𝛾 𝛾 (cid:19)−𝛼 (cid:18)1− 𝛾 𝛾 ((1−𝛾)Φ1 ) 1/(𝛼−1) (cid:19) 1+ 1 𝛾 − / 𝛾 (𝛼 − − 𝛼 1) (cid:18) 𝛾 (cid:19)−𝛼 (cid:16) (cid:17) 1−𝛾−𝛼 = 𝑧𝛼 𝛾−1 (1−𝛾) 𝛼/(𝛼−1) 1+𝛾/(𝛼−1) Φ−1 𝑙 1−𝛾 1 = 𝑧𝛼𝛾−𝛼− 1+ 1 𝛾 − / 𝛾 (𝛼 − − 𝛼 1) (1−𝛾) 𝛼 (1−𝛾)−𝛼 Φ−1 𝑙 = 𝑧𝛼𝛾−1 Φ−1. 𝑙 1 34

Therefore, 𝑧𝛼𝛾−1 ℎ 𝑈 ∞ 𝐸 = Ω2 = 𝑧 𝑙 𝛼𝛾−1 Φ− 1 1 = 𝑙 𝑧𝛼 𝑙 (𝛼−1)((𝛼−1)𝑔 𝑧+(𝑟+𝛿−𝑔 𝑦 )) (cid:18)𝛼−1 (cid:19) = (𝛼−1) 𝑔 𝐴 +𝑟 +𝛿−𝑔 𝑦 /𝛾. 𝛼 ProofofProposition4: Accordingtoequation(48),𝑊 → ∞as 𝑘 → ∞if (1−𝑢) 𝛼 > 1−𝛾𝑢 or 𝛼−1 𝛾 𝛼 𝛼 𝛼𝛾 𝛼−1 𝛼−1 = = 𝛼 + 1−𝛾 𝛼𝛾+(1−𝛾)(𝛼−1) 𝛼𝛾 + (1−𝛾) (𝛼−1) 𝛼−1 𝛾 (𝛼−1)𝛾 𝑔 +𝛿 > 𝑢∞ = 𝐴 . 𝑔 +𝛿+ ℎ 𝐴 𝑢𝑒 Thissimplifiesto: 𝛼 1−𝛾 ℎ∞ > (𝑔 +𝛿) 𝛼−1 𝑢𝑒 𝛾 𝐴 𝛼 𝛼 𝛼 𝛼−1 ℎ 𝑢 ∞ 𝑒 = 𝛾 (𝑟 +𝛿−𝑔 𝑦 +𝑔 𝐴 −𝑔 𝑧 ) = 𝛾 (𝑔 𝐴 +𝛿+𝑟 −𝑔 𝑦 −𝑔 𝑧 ). Aslongas𝑟 −𝑔 −𝑔 > 0,thenbecause 𝛼 > 1and1−𝛾 < 1 𝑦 𝑧 𝛼 𝛼 1−𝛾 𝛼−1 ℎ 𝑢 ∞ 𝑒 = 𝛾 (𝑔 𝐴 +𝛿+𝑟 −𝑔 𝑦 −𝑔 𝑧 ) > 𝛾 (𝑔 𝐴 +𝛿). 35