Endogenous Bargaining Power and Declining Labor Compensation Share

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Endogenous Bargaining Power and Declining Labor Compensation Share Juan C. Co´rdoba, Anni T. Isoj¨arvi, Haoran Li 2023-030 Please cite this paper as: C´ordoba, Juan C., Anni T. Isoj¨arvi, and Haoran Li (2023). “Endogenous Bargaining Power and Declining Labor Compensation Share,” Finance and Economics Discussion Series 2023-030. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2023.030. 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.

Endogenous Bargaining Power and Declining Labor Compensation Share Juan C. Córdoba∗ Anni T. Isojärvi† Haoran Li‡ Abstract: Workhorsesearchandmatchingmodelsassumeconstantbargainingweights,whilerecent evidence indicates that weights vary across time and in cross section. We endogenize bargaining weights in a life-cycle search and matching model by replacing a standard Cobb-Douglas(CD) matchingfunctionwitha generalconstantelasticityof substitution (CES) matching function and study the implications for the long-term labor share and bargainingpowerintheU.S.TheCESmodelexplains64percentofthereporteddecline inthelaborsharesince1980,whiletheCDmodelexplainsonly28percentofthedecline. We then use the model to recover changes in bargaining power and find that workers’ ∗IowaStateUniversity,DepartmentofEconomics. E-mail: cordoba@iastate.edu †BoardofGovernorsoftheFederalReserveSystem. Email: anni.t.isojaervi@frb.gov ‡SchoolofAppliedEconomics,RenminUniversityofChina. Email: haoranl@ruc.edu.cn Note: TheviewsexpressedarethoseoftheauthorsandnotnecessarilythoseoftheFederalReserveBoard ortheFederalReserveSystem. 1

bargainingpowerhasdeclined11percentbetween1980and2007becauseofadeclinein tightness. Laborshare,Endogenousbargainingpower,Searchandmatching,CESmatch- Keywords: ingfunction E25,J30,J50 JELCodes: 0

1 Introduction Both academics and the public have raised concernsabout workers’ declining bargaining powerintheUnitedStates. Evidencefromrecentdecadespointsinthatdirection: Labor’s shareofincomehasdeclined,medianwagegrowthhasbeensluggish,profitabilityoffirms hasrisen, andboth unionmembership andcoverage have droppeddrastically(Stansbury andSummers,2020). Whileworkhorsesearchandmatchingmodels,liketheDiamond- Mortensen-Pissarides(DMP)model,includebargainingbetweenfirmsandworkers,these models typically assume constant bargaining weights, making them unable to capture long-termchangesinworkers’bargainingpower. Weaddressthisdiscrepancybyendogenizingbargainingweightsinalife-cyclesearchand matchingmodel. InsteadofassumingthatthematchingfunctiontakesthestandardCobb- Douglas (CD) form, we rely on a more general constant elasticity of substitution (CES) matchingfunction. Thisapproachendogenizesthematchingelasticityand,conditional on the Hosios condition holding (Hosios, 1990), bargaining weights. We calibrate both modelsusingtheCurrentPopulationSurvey(CPS)dataandhistoricalvacancydatafrom Petrosky-Nadeau and Zhang (2021) to match job-finding rates, wages, and tightness rates. WefindthattheCESmodelexplainsabout64percentof theobserved4-percentdecline inthelaborshareintheU.S.betweentwobusinesscyclepeaks,1980and2007,whilethe CDmodelexplainsonly28percentofthedecline. Thus,theCESmodelwithendogenous matching elasticity and bargaining power performs drastically better at capturing the declineinthelaborshare,indicatingthatthebargainingpowerchannelisakeyingredient forthemodeltomatchthedata. Thisisthefirstmaincontributionofthispaper. 1

Theworkers’bargainingweightequalsthematchingelasticitywithrespecttojobseekers whentheHosiosconditionholds. UndertheCESmatchingfunctionandreasonableparameter values, the bargaining weight increases withlabor market tightness, indicating thata greaterdemandforlabortranslatesintohigherbargainingpowerofworkers. Intuitively, whenlabormarketsaretightandjobsare abundant,workershavemorebargainingpower overwagesandothertermsofemployment. Ourmodelthusimpliesthatchangesinmarket conditions,reflectedindeclinedtightness,cangenerateareductioninworkers’bargaining power—evenintheabsence ofinstitutionalchanges. Thisisthe secondmaincontribution ofthepaper. WeusetheCESmodeltorecoverworkers’efficientbargainingpowerandstudyhowithas changedovertime. Ourresultssuggestthattheaggregatebargainingpowerhasdeclined about 11 percent between 1980 and 2007 because of a decline in labor market tightness. Previousliterature hasfoundthat bargainingpowervaries amongdifferentdemographic groups1. So we also include disaggregated results of bargaining power trends for four groups: males and females with at least some college education, as well as males and females without a college education. We find that the bargaining power of males has decreasedmore than thatof females, leading to adecrease inthe gender bargaining power 1Recentliteraturehasdocumentedagenderbargainingpowergap,andthegapcanexplainafractionof thegenderwagegap(BiasiandSarsons,2021;BlauandKahn,2017;Cardetal.,2016;Hardingetal.,2003). Someliteraturealsousesbargainingpowerdifferencestoexplainawagegapbetweenolderworkersand youngworkers(FarmandandGhilarducci,2019andGloverandShort,2020). Theliteraturethussuggests thatassumingaconstantbargainingpoweracrossgroupsisproblematicandthataccountingforthenoted differencesinbargainingpowerisimportanttounderstandthedynamicsofthelabormarket. 2

gap. Thebargainingpowerofbothcollegeandnon-collegemaleshasdeclinedaround17 percent,whilethedeclineshavebeen1percentforcollege-educatedfemalesand6percent fornon-collegefemales. Lastly,whilethegenderbargainingpowergaphasdiminished, the opposite is true for the education gap—especially for females—as college-educated workers’bargainingpowerhasdecreasedlessrelativetonon-collegeworkers. WehighlightthattheCESmatchingfunctionhasdesirablepropertiesovertheCDmatching inourexerciseandingeneral. First,theCESfunctionistheoreticallysounder,astheCD introducesdiscontinuitiesandrequirestruncation. Second,itgeneratesintuitivelysensible matching elasticities. Specifically, when vacancies v and job seekers u are complements in a matching process, matching elasticity with respect to job seekers is increasing in tightness. Intuitively,thenumberofsuccessfulmatchesismoresensitivetothenumberof jobseekerswhentherearemanyavailablevacanciesrelativetojobseekers. Third,weshow thattheCESmatchingfunctionwithcomplementaritybetweenv anduisconsistentwith micro-evidencethatshowsthatgroupswithweakerlabormarkets(forexample,women) have lowerbargaining power. Fourth,theCES matching function is consistentwith casual evidence that workers’ bargainingpowerincreases with labor scarcity, such as during and aftertheCOVID-19pandemic. Asweshow,thesepropertiesturnouttobequantitatively importantinexplainingthelong-rundeclineinthelaborshare. Our model includes human capitalaccumulation through learning-by-doing and a nonparticipationstate. Weincludethesefeaturesinthemodelforthefollowingreasons. First, we areinterested instudying howbargainingpowerevolvesoverthelifecycle;bargaining powerlikelyevolveswithhumancapital—amoreexperienced,skilledworkerhashigher 3

bargaining power compared with a less experienced worker, all else being equal. And second,manystudies,suchasChoietal.(2015)andVeracierto(2008),havepointedout thatthenonparticipationstateisimportantinunderstandinglabormarketdynamics. Inour setting,nonparticipationmatters,especiallyforstudyinggenderdifferencesinbargaining power, as females are traditionally more likely toexperience nonparticipationperiods over thelifecycle. Forthatreason,weextendthemodeltoincludenonparticipation. AsourresultsrelyontheHosiosconditionholding,weshowthatitholdsinourextended model. Theoriginalconditionstatesthatanequilibriumallocationinasearchandmatching modelis constrained efficientwhen theworkers’bargainingweightequalstheelasticityof the vacancy-fillingratewith respecttolabormarkettightness. We show thatthe condition holds in our model despite the life-cycle dynamics, human capital accumulation, and nonparticipationwheneverthereisenoughsegmentationinlabormarkets.2 Finally,wedecomposethedeclineinbargainingpower. Weconcludethatanincreasein κ, therelativevacancy-postingcost, hasdriventhedecline intightness,and thusbargaining 2ThisresultcontrastswithLaureys(2021),whobuildsaDMPmodelwithsimilarhumancapitalaccumulation. Hermodelassumesintegratedlabormarketsforworkerswithdifferenthumancapitallevelsandthat thedecentralizedlabormarketisinefficientbecauseofalaborcompositionexternality. Weprovethatthere isamoregeneralizedHosiosconditionthatguaranteesefficiencyindecentralizedlabormarketswhenlabor marketsaresegmented. Moreover,weshowthattheHosiosconditionholdsendogenouslyinourmodelif wefollowthecompetitivesearchtheoryliterature(seeWrightetal.,2021)andassumethatfirmspostamenu ofbargainingpowersandthatworkerschoosetoapplytojobsthatofferbargainingpowerthatmaximizes theirutility. Inthatcase,bargainingpowerworksasapricedevicethatguaranteesthatthedecentralized allocationisconstrainedefficient. 4

power. Whileourcalibrationresultspointtoanimprovedmatchingefficiencyandhigher returns to experience, increasing tightness for all groups, we find that vacancy-posting costs have risen. This rise is necessary for the model to match the observed decline in tightnessalongwiththeobservedemploymentandwagetrends. . Onthetheoryside,themostrelatedpaperisManginandSedláček Relationtotheliterature (2018). They study business cycle fluctuations of the labor share by building a search andmatchingmodelwhere heterogeneousfirmscompeteoverworkers andinwhichthe divisionofoutputbetweenfirmsandworkersinendogenous. Specifically, atighterlabor marketincreaseslabor’sshareofoutput—amechanismthatisliketheoneinourmodel. However, Mangin and Sedláček (2018) focus on explaining the business cycle dynamics of the labor share, while our focus is on longer-term changes in bargaining power and the laborshare. WearenotthefirsttousetheCESmatchingfunctioninsearchandmatchingmodels—den Haanetal.(2000)are. TheyusetheCESmatchingfunctionandhighlightthepreferable propertiesofCESmatchingfunctionthatguaranteematchingprobabilitiesbetweenzero andone. ACESmatchingfunctionisalsoused,forexample,byHagedornandManovskii (2008)andPetrosky-Nadeauetal.(2018). Stevens(2007)microfoundsamatchingfunction byshowingthata"telephoneline"PoissonqueuingprocessimpliesaCESmatchingfunction. Recently,Bernsteinetal.(2022)studiedhowaCESmatchingfunctionandcyclicalityof matching efficiency affect nonlinear business cycle properties of search and matching models and foundquantitatively important effects. Whilethese papers allow variation in matchingelasticities,theyassumeconstantbargainingweights. 5

Our paper also relates to the literature that studies workers’ bargaining power—both the long-runtrendsandthedifferencesamongdifferent workergroups—andtherelationship betweenbargainingpowerandthelaborshare.3 Wecontributetotheeffortstomeasure changesinbargainingpowerbyindirectlyinferringchangesinefficientbargainingpower for different demographic groups using a general equilibrium model with endogenous bargainingpower. Consistentwiththepreviousliterature,wefindthatbargainingpower hasdeclinedinthepastfourdecadesandthattherearegapsinbargainingpoweracross gender,education,andage. Whilepreviousliteraturehasfocusedonstudyingadeclinebargainingpowerarisingfrom changes in labor market institutions (Stansburyand Summers, 2020 and Ratner and Sim, 2022), we focus on studying changes in bargaining power arising from labor demand. StansburyandSummers(2020)arguethatthreefactorshavecausedthedeclineinworker powerintheU.S.overrecentdecades: (1)institutionalchangeslikedecreasedunionism, (2) within-firm changes like an increase in shareholder power that has led to pressure tocutlaborcosts,and(3)changesineconomicconditions—likeincreasedglobalization and technology— that have improved employers’ outside options. While Stansbury and Summers(2020)focusonstudyingandpresentingsupportingevidenceforthefirsttwo factors,wecomplementtheirworkbyfocusingonthethird. Wethusbringanewangletotheliterature: Wearguethatbargainingpowerhasdecreased 3Seeforexample,BentalandDemougin(2010);BiasiandSarsons(2021);BlauandKahn(2017);Card etal.(2016);FarmandandGhilarducci(2019);GloverandShort(2020);RatnerandSim(2022);Roussille (2022);StansburyandSummers(2020). 6

asanequilibriumreactiontoadecreaseinlabordemand. Iflabordemandhasdecreased, workersfaceatrade-offbetweenchoosingalowerwage(andimplicitlylowerbargaining power)orahigherunemployment. Inaddition,thedecliningunionizationdocumented by Stansbury and Summers (2020) can also reflectthe described response. Workers may belessinclinedtojoinunionsforfearofjobsdisappearingquickly. Charlesetal.(2021)findasimilarresultwhenlookingatadeclineinunionization. They estimatethecausaleffectofincreasedimportcompetitionfromChinaontheaccelerateddeclineintherateofunionelectionsbetween1990and2007. Theyfindthatthe"Chinashock" contributed to 4.5 percent of the decline among workers in directly exposed industries, whiletheshockcontributedto8.8percentofthedeclineamongworkersindirectlyexposed through weaker local relative labor demand. In other words, workers in industries that werenotdirectlyexposedtotheChinashockunionizedlessbecausetheshockweakened theiroutsideemploymentoptionsinthefaceofajobloss. Wefindasimilarmechanismusingastructuralgeneralequilibriummodel,butwefocus onstudyingtheeffectonbargainingpower. Anincreasedvacancycost,whichcancapture the China shock, reduces rents from any match and leads to weaker labor demand via lowertightness. Thisincreasesthecostofjoblossbecauseworkers’job-findingrategoes down. Wealsoshowthatworkers’efficientbargainingpowerdecreases. Thedecreasesin thejob-findingrateandbargainingpowerthenleadtoadecreaseinthelaborshare. Therestofthepaperisorganizedasfollows. Section2summarizesthepropertiesoftheCD versusCESmatchingfunctions. Section3introducesthemodel,andSection4describes 7

(1) how tightness, the labor share, and other outcomes have evolved between 1980 and 2007;(2)howweparameterizethemodel;and(3)thecalibrationresults. Wethenmove on to reporting model-generated changes in efficient bargaining power (Section 5) and counterfactuals(Section6). Section7concludesthepaper. 2 ThePropertiesoftheCDandtheCESMatchingFunctions Thissectionhighlights somepropertiesofthe CESmatchingfunctionthat speaktoitsuse. First, under theCES matchingfunction, matchingprobabilitiesare between zeroand one (denHaanetal.,2000andPetrosky-Nadeauetal.,2018). DefinetheCESmatchingfunction as      A(αuρ +(1−α)vρ)1/ρ ifρ ≤ 1,ρ ̸= 0    M(u,v) = , (1)    Auαv(1−α) ifρ = 0    whereureferstojobseekers,v referstovacancies,α ∈ (0,1)istheshareparameter,andA isthematchingefficiency. Theelasticityofsubstitutionisσ ≡ 1 ∈ (0,∞). Thecaseofthe 1−ρ CD matching function with ρ = 0 implies that σ = 1. The value of ρ is negative when u andv arecomplements. Giventhematchingfunction,afirm’svacancy-fillingrateq(θ) = M(u,v)/v = M(1/θ,1)is givenby      A(αθ−ρ +(1−α))1/ρ ifρ ≤ 1,ρ ̸= 0    q(θ) = . (2)    Aθ−α ifρ = 0    8

Noticethatq′(θ) < 0. Furthermore,        0if ρ < 0         A(1−α)1/ρ ifρ < 0          q(0) = , q(∞) = A(1−α)1/ρ if ρ > 0 .    ∞ifρ ≥ 0               0ifρ = 0    Therefore,theprobabilityoffillingavacancyiswellbehavedwhenρ < 0if1 ≥ A(1−α)1/ρ. Whenthatisthecase,q(θ) ∈ [0,1]forallθ ≥ 0. Incontrast,q(θ)isnotwellbehavedwhen ρ ≥ 0,asq(0) = ∞. Inasimilarmanner,ajobseeker’sjob-findingratef(θ) = M(u,v)/u = M(1,θ)is      A(α+(1−α)θρ)1/ρ ifρ ≤ 1,ρ ̸= 0    f(θ) = . (3)    Aθ1−α ifρ = 0    Ajob-findingrateisincreasingintightness,f′(θ) > 0and,        Aα1/ρ ifρ > 0       Aα1/ρ if ρ < 0    f(0) = , f(∞) = .    ifρ ≤ 0       ∞ifρ ≥ 0    Therefore,thejob-findingprobabilityiswellbehavedwhenρ < 0if1 ≥ Aα1/ρ. Inthatcase, f(θ) ∈ [0,1]forallθ ≥ 0. Whenρ ≥ 0,f(θ)isnotwellbehaved,asf(∞) = ∞. To conclude, the CES matching function produces sensible job-finding and vacancy-filling probabilitieswhen1 ≥ max[A(1−α)1/ρ,Aα1/ρ]. ThisisnotthecasewiththeCDmatching function. Second,weshowhowtheCESmatchingfunctiongeneratesintuitivelyreasonablematching elasticities. Note first that with the CES matching function, q′(θ) = −Aα[αθ−ρ + (1 − 9

α)]ρ 1−1(θ−ρ−1). Therefore,wecan writethematchingelasticitywithrespect tojobseekers asafunctionofθ: u q′(θ)θ α M (u,v) = α(θ) = − = . (4) u M q(θ) α+(1−α)θ(x)ρ This expression includes the CD result with ρ = 0. As is well known, the CD matching functionimpliesaconstantmatchingelasticityα. Moreover,noticethatα′(θ) > 0whenever ρ < 0—thatis,thematchingelasticityisincreasingintightnesswhenv anduarecomplements in the matching process. Asα(θ) represents the elasticity of the matching function withrespecttou,alowerθ meansthattherearerelativelymorejobseekerscomparedwith vacancies. Thismeansthatthenumberofsuccessfulmatchesislesssensitivetothenumber ofjobseekers,aresultthathighlightsthecomplementarityofjobseekersandvacanciesin thematchingprocess. Third, we show that the CES matching function generates efficient bargaining power dynamics consistent with both micro-evidence and macro-evidence. The well-established Hosioscondition(Hosios,1990)statesthatthedecentralizedsolutionofthestandardDMP modelisconstrainedefficientaslongastheelasticityofthematchingfunctionwithrespect tothenumberofjobseekersequalsthebargainingweightϕoftheworker, α ϕ(θ) = α(θ) = . (5) α+(1−α)θ(x)ρ This simple formulation shows that the bargaining power of workers is increasing in α, butmoreimportantly,bargainingpowerisincreasingintheendogenoustightnessrateθ whenever ρ is negative. We now have efficient bargaining power that depends on labor markettightness. 10

Proposition1. UndertheCESmatchingfunctionM (u,v) = A(αuρ+(1−α)vρ)1/ρ,theefficient bargaining powerof workersdecreaseswithlabormarket tightnessifρ > 0;theefficientbargaining powerofworkersincreaseswithlabormarkettightnessifρ < 0. From the point of view of the social planner, proposition 1 means that the large relative numberofjobseekersreducesthepotentialforformingamatchbecauseofthecomplementarityofuandv. Toincreasethenumberofvacancies,itisoptimaltoreducethesurplus shareofworkerstospurvacancycreation. Itisalsoeasytoseethattheexpressionnests theCobb-Douglascase: Whenρ = 0,thebargainingpowerisexactlyα. TheCobb-Douglascasealsomeansthatthebargainingpowerof workersdoesnotdependonworkers’characteristicsorlabormarketconditions,contrasting withbothmicro-evidenceaswellascasualobservations,asnotedintheintroduction.4 Let’sfurtherderivethebargainingpowerelasticitywithrespecttotightnessθ: ∂ϕθ θ[α+(1−α)θρ] ε = = −α[α+(1−α)θρ]−2 ×[ρ(1−α)θρ−1]× ϕ,θ ∂θ ϕ α (6) (1−α)θρ = −[α+(1−α)θρ]−1 ×[ρ(1−α)θρ] = −ρ . α+(1−α)θρ Theaboveexpressionimpliesthattheelasticityofbargainingpowerwithrespecttotightness is positive whenever ρ < 0, and that bargaining power increases with tightness more 4Intuitively,whenρ<0,bargainingpowerrespondstolabordemandandsupply. Labormarkettightness reflectstherelativedemandforlabor. Asalargerθmeansashiftinthedemandcurvetotheright,thelabor marketendogenouslygivesalargerproductionsharetoworkersthroughlargerbargainingpowerϕ. This relationshipisalsoinlinewiththefindingofFortin(2006),whoshowsthatthecollegewagepremiumis negativelyrelatedtothesupplyofhighlyeducatedworkers. 11

wheneverρgetssmallerandthecomplementaritybetweenvacanciesandjobseekersin thematchingprocessgetshigher. Again, theaboveformulaincludestheCobb-Douglas case: Whenρ = 0,ε = 0. ϕ,θ Weusetheexpressionforbargainingpowertodiscussbargainingpowergapsdocumented in the literature. What can explain the lower bargaining power of females? Females generally have higherseparation rates comparedwith malesover the life cycle (Choiet al., 2015andCórdobaetal.,2021). Doesthatimplythatfemaleswillbeinaweakerposition when bargainingwith firms? Theanswer depends onthebargaining power elasticity. We derivetheeffectofaseparationrate,π ,ontheefficientbargainingpower: EN ∂ϕ ∂ϕ ∂θ −α(1−α)ρθρ−1 ∂θ ϕ ∂θ = = × = ε × × . ∂π ∂θ ∂π [α+(1−α)θρ]2 ∂π ϕ,θ θ ∂π EN EN EN EN The sign of ∂ϕ depends on the signs of ρ and ∂θ . ∂θ is negative, and the intuition ∂πEN ∂πEN ∂πEN for ∂θ beingnegativeisthathiringworkerswithhigherseparationrateswilllowerthe ∂πEN matchcontinuationvalue,andthelowercontinuationvalueneedstobecompensatedby a higher chance of successfully hiring such workers. Then the sign of ∂ϕ can be fully ∂πEN pinneddownbyρ.Whenρ < 0,groupswithhigherseparationrateshavelowerbargaining powercomparedwithgroupswithlowerseparationrates,allelseequal. To conclude, we argue that there are four reasons why the CES matching function with ρ < 0 is a sounder choice for a matching function: (i) it is theoretically sounder as the CDintroducesdiscontinuitiesandrequirestruncation;(ii)itgeneratesintuitivelysensible matchingelasticitiesandefficientbargainingpower;(iii)itisconsistentwithmicro-evidence that shows that groups with weaker labor markets (for example, women) have lower 12

bargainingpower;(iv)itisconsistentwithcasualevidence—forexampleduringandafter COVID-19pandemic—ofworkers’bargainingpowerincreasingwithlaborscarcity. 3 DynamicSearchandMatchingModel 3.1 BasicEnvironment The model presented here closely follows the life-cycle search and matching model presentedinCórdobaetal.(2021). Themodelfeaturesalifecyclewithafinitehorizon,human capitalaccumulation,andanonparticipationstate. Timeisdiscrete,andageisdenotedby a,where a ∈ A ≡ [a,a¯]. Themodelfocusesonworking years,which aretheyearsbetween one’seducationandretirement. Workers . Workers enter the labor market at age a, retire at age a , and die at age a¯, R where a¯ > a . At any point in time, a worker is either employed, E, unemployed, U, or R not participating in the labor force, N. Let s ∈ S ≡ (cid:8) E,U,N (cid:9) denote the labor market status of a worker. Workers enter the labor market without work experience and gain experience by working. Let e ∈ [0,a −a] denote the years of experience. Experience R increases by one unit during each period of employment, e = e +1, and we assume a+1 a that experience stays constant during nonemployment, e = e . Workers also differ in a+1 a termsoftheirdemographicslikegender(femaleormale)andlevelofschooling(college or non-college), which is captured by i. Let I denote the set of demographic groups. A workerisfullyidentifiedbyheryearsofexperience(e),age(a),labormarketstatus(s), and demographic group (i). Denote the state or the type of a worker by x = (e,a,s,i), wherex ∈ [0,a¯−a]×[a,a ]×S ×I,andletx′ = (e+1,a+1,s′,i). Thestateofaretireeis R definedasx = (cid:0) e,a ,N,i (cid:1). R R 13

Letm(x)bethemassofworkersoftypex. Theinitialmassdistribution,ms(0,a),istaken i as given for all s and i. Workers transition into unemployment and nonparticipation at exogenousratesπ (x),π (x),π (x),andπ (x),andintoemploymentatendogenous EU EN UN NU ratesf (cid:0) e,a,U ¯(cid:1)andf (cid:0) e,a,N ¯(cid:1) .Workersseektomaximizetheirexpectedpresentvalueof i i consumption. Theyareriskneutralanddiscountthefutureaccordingtothediscountfactor β ∈ (0,1). There are no savings. Wages of employed workers are determined by Nash bargainingbetweenworkersandfirms,whileconsumptionofnon-employedworkersand retireesisgivenbyc(x),anexogenousparametricform. Forsimplicity,wedonotexplicitly describethedomainofeachfunctionwheneveritisclear. Forexample,w(x)refersonlyto thewagesoftheemployedworkers,x = (cid:0) e,a,E ¯ ,i (cid:1). : Each worker is endowed with one unit of labor, but workers differ in Human Capital termsoftheirlaborproductivity. Werefertotheproductivityofaworkerashumancapital, h(x),and itisofthegeneraltype. Weassumethefollowingfunctional formfor thehuman capital: h(x) = y exp(r(x)e), (7) i where y is the baseline level of human capital that a member of a group i has when i enteringthelabormarket,andr(x)istype-specificreturnstoexperience. Bothy andr(x) i areexogenous. Weinterprety aseducation-relatedhumancapital—thehumancapitalof i anewworkerforwhome = 0. : Thecontinuum ofinfinitelylived firmsseekto maximizetheir Firmsand LaborMarkets expectedpresentvalueofprofitsnetofhiringcosts. Firmsareriskneutralanddiscount the future at the same rate as workers do. Labor markets are assumed to be perfectly 14

segmentedacrossworkertypes. Firmscanfreelyenteranysegmentedmarkets. Firmspost vacanciesforlong-term positionsatacostofκ(x)pervacancy, acostthatmaydepend on aworker’stype. Onceafirmismatchedwithaworker,aworkerproducesh(x)unitsof outputperperiod,whilegrossper-periodprofitsofthefirmareh(x)−w(x). Amatchis destroyedexogenouslyatarateofd(x) = π (x)+π (x). EU EN A worker and afirmwitha vacantposition are randomly matched Matching Technology: ineachsubmarketaccordingtothematchingtechnologyM (u(x),v(x);x),whereu(x)and v(x) are the masses of workers and firms, respectively, searching in a labor market. We assume that (1) all unemployed workers search for a job, (2) employed workers do not search,and(3)afractionψ(x) ≤ 1ofnonparticipantssearch. Thus,themassofworkers searchingatagivenemploymentstatuscanbedefinedasfollows:      m(x), ifs = U,    u(x) ≡ (8)    ψ(x)m(x), ifs = N.    Labormarkettightnessforeachmarketxisdefinedasθ(x) ≡ v(x)/u(x),thevacancy-filling rate as q(θ(x)) = M (u(x),v(x))/v(x), and the job-finding rate as f(θ(x)) = θ(x)q(θ(x)). Onceamatchisformed,thematchoutputisdistributedaccordingtoaNashbargaining solutioninwhichaworker’sbargainingpowerisϕ(x). LaborFlows. Giventheinitialdistributionofworkers,ms(0,a),andjob-findingratesf(x) i forallx,thesubsequentdistributionofworkersm(x)canbecalculatedassumingalawof largenumbers. SeedetailsinAppendixC. 15

3.2 ValueFunctionsofFirmsandWorkers Firms. LetV ¯ bethevalueofafirmwithoutaworkerandJ(x)bethevalueofafirmwitha workeroftypex = (cid:2) e,a,E,i (cid:3). Thevalueofafirmpostingavacancyinmarketxis (cid:8) (cid:2) ¯(cid:3) (cid:9) V (x) = max −κ(x)+β q(x)J (e,a+1)+(1−q(x))V ,0 . i The maximum value of posting a vacancy in any labor market is then given by V ¯ = max{V (x),0}. Free entry of firms into any labor market guarantees that the values of x unfilled vacancies must all be equal to zero: V (x) = 0 for all feasible x. As a result, the maximum value ofposting a vacancy mustbe zeroas well: V ¯ = 0. Active firmsare thus indifferenttowhichtypeofworkertohireandtowhichsegmentedmarketstheyoperate in. Theproblemofafirmwithaworkeristhen (cid:26) (cid:27) J(x) = h(x)−w(x)+β(1−d(x))J (x′) if a ≤ a < a −1 , (9) R whichstatesthat thevalueofafirm withaworker istheflowofgrossprofitsplusthe discountedcontinuationvalueofthematch. Thefirmvaluerightbeforeaworker’sretirement, a = a −1,isJ(x) = h(x)−w(x). R Thevalueoffirmsthatpostvacanciessimplifiesto κ(x) = βq(x)J (e,a+1) = βf (x)θ(x)−1J (e,a+1) fora ≤ a < a −1. (10) i i R The last equation states that the expected present value offilling a vacancy must be just enoughtorecoverthecostsofpostingthevacancy. 16

Workers. Now consider the value functions of an employed worker, E, an unemployed worker, U, a nonparticipating worker, N, and a retired worker, R. The expected present valueofconsumptionofanewlyretireesimplysatisfies (cid:88) a 1−βa−aR−1 R(x ) = βi−aRc(x ) = c(x ), (11) R R R 1−β i=aR wherec¯(x )is consumptionof a retiree oftypex . The corresponding valuefunctionsE, R R U,andN canthenbewrittenrecursivelyas       E(x) =   w(x)+β   π EU (x)U(x′)+π EN (x)N(x′)  ,ifa ≤ a < a −1   , (12) R       +(1−π EU (x)−π EN (x))E(x′)         U(x) =   c(x)+β   f(x)E i (e,a+1)+π UN (x)N i (e,a+1)  ,ifa ≤ a < a −1   , (13) R       +(1−f(x)−π UN (x)U i (e,a+1)         N(x) =   c(x)+β   f(x)E i (e,a+1)+π NU (x)U i (e,a+1)  ,ifa ≤ a < a −1   , (14) R       +(1−f(x)−π NU (x))N i (e,a+1)   while E(x) = w(x)+βR (cid:0) e+1,a ,N (cid:1), U(x) = c(x)+βR (cid:0) e,a ,N (cid:1), and N(x) = c(x)+ i R i R βR (e,a ifa = a −1. Anemployedworkerconsumesherwagew(x)eachperiod. Amatch i R R betweenaworkerandafirmcanbedestroyedintwoways: (1)withprobabilityπ (x), EU aworkerbecomesunemployed,and-(2)withprobabilityπ (x),theworkerbecomesa EN nonparticipant. The worker continues producing with probability 1−π (x)−π (x) EU EN and stays in the employment state. At the beginning of each period, an unemployed worker consumes c(x). During the next period, she finds a job with probability f(x) = f (e,a,U ¯ ),inwhichcaseshemovestotheemploymentstate. Aworkermayalsomoveto i thenonparticipationstatewithprobabilityπ (x);otherwise,shewillstayunemployed. UN Asimilarinterpretationholdsforthevaluefunctionofanonparticipatingworker. 17

3.3 NashBargaining 3.3.1 BargainingSolution Wages are negotiated through Nash bargaining. A firm and a worker share the match surplus S(x) = S (x)+J(x), s ∈ {E,U,N}, S (x) = E(x)−U(x) for an unemployed, Es EU and S (x) = E(x)−N(x) for a nonparticipant. Given the bargaining weights ϕ(x) for EN theworkerand1−ϕ(x)forthefirm,themaximizationproblemiswrittenas: max(S (x))ϕ(x)J (x)1−ϕ(x) subjecttoS(x), Es SEs,J andthesolutionforeachlabormarketsatisfies 1−ϕ(x) J(x) = Θ(x)×(S (x))whereΘ(x) = . (15) Es ϕ(x) 3.3.2 TheHosiosConditionundertheDynamicDMPModel We provethat the Hosioscondition holds inour life-cyclemodel with human Efficiency. capitalaccumulationandnonparticipation. Proposition 2. UndertheDMPmodelwithalifecycle,humancapitalaccumulation,andnonparticipation,ϕU¯ (e−1,a+1) = − q′(θ i U¯(e,a))θ i U¯(e,a)) andϕN¯ (e−1,a+1) = − q′(θ i N¯(e,a))θ i N¯(e,a)) ensure i q(θU¯(e,a)) i q(θN¯(e,a)) i i labormarketefficiency. SeeAppendixA. Proof. Inshort,weshowthatlabormarketsinthemodelareefficientwhenthestandardHosios conditionholdsiflabormarketsaresegmented. Notethatthebargainingpowerissetatthe 18

timeofthematch—oneperiodbeforetheproductionoccursandrentsareshared—which implies that both the worker’s age and experience have evolved by one unit by the time of the production. In the proof, we assume that workers’ experience depreciates while non-employed,buttheconditionalsoholdsifweassumethatexperiencestaysconstant or increases over time. For simplicity, we denote the efficient bargaining condition by ϕ(x′) = −q′(θ(x))θ(x)). q(θ(x)) InAppendixB,wefurthershowthattheHosiosconditionarisesendogenouslyifwefollow thecompetitivesearchtheoryliteratureandassumethatfirmspostamenuofbargaining powersandworkers choosetoapplyjobsthatofferthe bargainingpowerthatmaximizes theirutility. 3.4 CharacterizationoftheSolution The solution for wages, tightness rates, and job-finding rates using backward induction is characterized as in Córdoba et al. (2021). In particular, we first obtain closed-form solutions for the last period of working life, which we then use to find solutions for the previousperiods. Thesolutionsforperiodsa < a −1canbeexpressedintermsofworkers’ R surplusesandvaluechangesdefinedas S (x) ≡ E(x)−U(xU¯ (e,a)); S (x) ≡ E(x)−N(xN¯ (e,a)); (16) EU i EN i S (x) ≡ N(x)−U(xU¯ (e,a)); S (x) ≡ U(x)−N(xN¯ (e,a)); NU i UN i ∆U(x) ≡ U(x)−U (e−1,a); ∆N(x) = N(e,a)−N (e−1,a). i i 19

Thesolutionsforw(x),θ(x),andf (x),for0 ≤ a < a −1,thensatisfy R h(x)+Θ(x)[c¯(x)+βΩ(x)] w(x) = , (17) 1+Θ(x) κ(x) = βA(x)(αθ(x)−ρ +(1−α))1/ρJs(e,a+1),and (18) i f(x) = A(x)(α+(1−α)θ(x)ρ)1/ρ,where (19) Js(e,a+1) = Θ(x)Si (e,a+1), and (20) i Es       f i U¯ (e,a)S E i U (e,a+1)+π UN (x)S N i U (e,a+1)+                π EN (x)[S EN (x′)−S EU (x′)]−∆Ui(e+1,a+1),ifs = U    Ω(x) = . (21)     f i N¯ (e,a)S E i N (e,a+1)−π NU (x)S N i U (e,a+1)+                π EU (x)[S EU (x′)−S EN (x′)]−∆Ni(e+1,a+1),ifs = N    3.5 LaborShare Thelaborshareisdefinedasashareofoutputh(x)thatgoestotheworker,asmeasured bythewagew(x), h(x)+Θ(x)[c¯(x)+βΩ(x)] w(x) s (x) = = 1+Θ(x) (22) L h(x) h(x) ϕ(x)h(x)+(1−ϕ(x))[c¯(x)+βΩ(x)] = h(x) [c¯(x)+βΩ(x)] = ϕ(x)+(1−ϕ(x)) . h(x) Wefurtherstudyhowthederivedlaborsharerespondstochangesinlabortightness: (cid:26) (cid:20) (cid:21)(cid:27) ds (x) c¯(x)+βΩ(x) dϕ(x) β dΩ(x) L = 1− × +[1−ϕ(x)]× × . (23) dθ(x) h(x) dθ(x) h(x) dθ(x) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) bargainingchannel outsideoptionchannel 20

The above equation shows that the effect of tightness on the labor share can be divided into two parts: a bargaining channel and an outside option channel.5 The bargaining channelmeasurestheeffectoftightnessonthelaborsharethatrunsthroughthechangesin (cid:104) (cid:105) workers’bargainingpower,weightedby 1− c¯(x)+βΩ(x) . WiththeCDmatchingfunction, h(x) thebargaining channeldisappears, as dϕ(x) = 0. With theCESmatching functionandρ< dθ(x) 0, bargaining power increases withtightness, so the bargaining channel is positive if the weightispositive. The outsideoptionchannel measuresthechangesthat run throughthe changesin Ω(x). Intuitively,andasequation(21)shows,highertightnessincreasesworkers’outsideoptions by increasing their job-finding rate. Thus, tightness increases the labor share via the outsideoptionchannelbyincreasingtheoutsideoptionvalue,andthiseffectisweighted by[1−ϕ(x)]× β . h(x) 4 Parameterization 4.1 StylizedFacts Inthissection,weinvestigatetherelationshipbetweenthelong-termevolutionofthelabor share, wage growth, and labor market tightness in the U.S. Earlier literature has linked thelaborsharedeclinewithadeclineinbargainingpower(BentalandDemougin,2010; GloverandShort,2020;StansburyandSummers,2020). Iftightnessservesasaproxyfor workers’ bargainingpower,and changes inbargaining powerdrivethedeclinein the labor shareandwagegrowth,weexpecttheseseriestomoveintandem. 5AsnotedearlierinthisSection,weassumethatc¯(x)isexogenousandthuswillnotvarywithθ(x). 21

Labormarkettightnessandlaborcompensationshare,1976–2016 Figure1. Note: PanelAshowsthequarterly,seasonallyadjustedlaborshareforallemployedpersonsinthenonfarm businesssector,andpanelBshowsthequarterlylabormarkettightnessseries,normalizedto1in1979:Q1. Thedashedlinesplottherawseries,whilethepurplesolidlinesplottheHodrick-Prescott(HP)-filtered serieswithlambdasetto1,600. Bothfiguresincludealineartrendwith95percentconfidencebounds. Source: BureauofLaborStatistics;Authors’calculationsbasedonPetrosky-NadeauandZhang(2021)and IPUMS-CPS. PanelAinfigure1showstheevolutionofthequarterlylaborsharefortheU.S.nonfarm businesssectorbetween1976and2016(U.S.BureauofLaborStatistics,2022). Aspreviously documented, the U.S. labor share has declined in the long run: Between 1976 and 2016, there has been about a 7.9 percent decline in the Bureau of Labor Statistics (BLS) data. Betweentwofive-yearperiodsthatprecededtobusinesscyclepeaks,1976–80and2003–07, thedeclineis4.0percent. However,themajorityofthedeclinehasoccurredafter2000,and, overall,thelaborshareshowssignificantvariationovertimeandacrossbusinesscycles. Weusehistoricalvacancy-ratedatafromPetrosky-NadeauandZhang(2021)andunem- 22

ploymentandnonparticipationdatafromtheCPSfromtheIntegratedPublicUseMicrodata Series(IPUMS-CPS)between1976and2016toconstructaquarterlyrateoflabormarket tightness. Specifically, our tightness measure includes the number of unemployed workers andnonparticipantsbetweentheages of25 and64 inthedenominator.6 Theunderlying dataseriesarereported onamonthlybasis, sowecalculate quarterly ratesbyaveraging monthly values. Panel B in figure 1 shows the results. Similar to the labor share, labor markettightnesshastrendeddownward.7 Thedeclinebetweenthetwoperiods1976–80 and2003–07is21.5percent. Comparedwiththelaborshareseries,labormarkettightness shows stronger business cycle fluctuations, but otherwise the two series exhibit similar trends: Neithershowastrongdownwardtrendbetween1976and2000,sothedeclinein both occurs mostly after 2000. Also, both fluctuate similarly over time, with changes in tightnessseemingtoleadchangesinthelaborshare. PanelAinfigure2plotsthequarterlynominalwagegrowthofworkersbetween1983and 2016,whichiscalculatedbyaveragingtherawmonthlygrowthratesfromtheAtlantaFed’s WageGrowthTracker.8 Followingthetrendsinboththelaborshareandtightness,wage 6Werelyonthismeasurebecausethesamemeasureisusedinthemodelcalibration. Themeasureis alsoconsistentwithcurrentliteraturethatnotesthatvacanciesoverunemploymentmaynotbethebest approximationoftightnessbecauseitignoreslargeemploymentflowsfromnonparticipationandbetween jobs(see,forexample,Abrahametal.,2020andHallandSchulhofer-Wohl,2018). Ourtightnessmeasureis alsocloselypositivelycorrelatedwithatightnessrateconstructedusingasthedenominatortheHornstein- Kudlyak-LangeNonemploymentIndex(Hornstein-Kudlyak-LangeNon-EmploymentIndex,2023)fromthe RichmondFed. Thecorrelationcoefficientis.97whenusingdatafrom1994to2016. Indexstartsin1994. 7Hall(2017)alsodocumentsthedecliningtrendintherateoflabormarkettightness. 8TheAtlantaFed’sWageGrowthTrackerusesmicrodatafromtheCPS.Thewagegrowthrateiscalculated 23

Labormarkettightnessandwagegrowth,1983–2016 Figure2. Note: PanelAincludesthequarterlywagegrowthratefromtheAtlantaFed,andpanelBshowsthequarterly labormarkettightnessseries. Weplottheseriesstartingfrom1983,thefirstavailableyearoftheAtlantaFed WageGrowthTrackerdata. Thepurplesolidlinesplottherawseries. Bothfiguresincludealineartrend with95percentconfidencebounds. Source: FederalReserveBankofAtlanta;Authors’calculationsbasedonPetrosky-NadeauandZhang(2021) andIPUMS-CPS. growth rates have trended down. Moreover, wage growth follows closely the business cyclevariationinbothtightnessandthelaborshare. WefurtherreportthePearsoncross-correlationbetweentightnessandbothwagegrowth and the labor share. The correlation coefficient between the tightness rate and labor shareseriesis.502andisstatisticallysignificant,indicatingastrongpositivecorrelation betweentheseries. Thecorrelationbetweentightnessandwagegrowthseriesis.693,again asthemedianpercentchangeinthehourlywageofindividualsobserved12monthsapart. Seedetailsat https://www.atlantafed.org/chcs/wage-growth-tracker. 24

indicatingastrongpositivecorrelationbetweentheseries.9 Wechecktherobustnessofthetrendsandcorrelationsusingastandardmeasureoflabor markettightness. TheresultsarereportedinAppendixD.2.,andthoseconfirmthetrends andcorrelationsreportedinthissection. 4.2 Preliminaries We calibrate both the CD and CES models to the same targets and compare the models’ performance in generating a decline in the labor share. We calibrate the models using data from two periods, t ∈ {1976–80, 2003–07}. Both periods reflect the peak of the business cycle, and we use the data for five-year periods to have a sufficient sample to estimate life-cycle labor marketflows for our disaggregatedgroups. We refer to these two periods usingsimplytheyears1980and2007. 4.3 ConstantParameters Wecalibratethelife-cycletrendsataquarterlyfrequencyandsetthediscountrateβ equal to0.9902,whichimpliesthattherealinterestrateequals4percentannually. Weassumethat workersworkbetweentheagesof25and64. Afterthat,workersretireandliveuntiltheage of80. Thisgivesusa = 0(age25),a = 163(age65),anda¯ = 319(age80). Weassumethat R unemployed workers andretired workers consume afixed fraction oftheir human capital: 9Wealsolookatthecross-correlationsbetweenthelaborshareandtime-laggedtightnessseries. The correlationcoefficientincreaseswhenlookingatthecross-correlationbetweenthelaborshareandlagged labormarkettightness. Forexample,thevalueofthecorrelationcoefficientbetweenthelaborshareandthe tightnessratelaggedbythreequartersequals.557,indicatingthatthelaborsharerespondswithadelayto changesinlabormarkettightness. 25

c¯(x) = γ ·h (e,a),andc¯(x ) = γR ·h (e,a ),respectively. We setthereplacement-rate i R i R−1 parameters for unemployed and retired workers to γ = 0.35 and γR = 0.33. Under these parametervalues,inthemodel,theaverageconsumptionduringunemploymentisabout 40 percent of the average consumption of the employed, and the average consumption duringretirementisabout50percentoftheaveragehumancapitalattheretirementage. Theparameterα inboth the CDandCES modelsis0.5,followingdenHaanet al.(2000), and we set the matching elasticity ρ in the CES model to −0.3, which is close to values in Stevens (2007), with ρ of −0.3, Blanchard and Diamond (1989), with ρ of −0.35, and HagedornandManovskii(2008),withρof−0.4. 4.4 Time-SpecificParameters Life-CycleParameters. Weassumethatatage25theinitialmassoneofworkers,ms(i,a),is t dividedbetweenemployment,unemployment,andnonparticipationsuchthatthevalues matchtheaveragevaluesforeachgroupiobservedintheIPUMS-CPSdataineacht. Thecalibrationoflife-cycleoutcomesofworkersfollowscloselyCórdobaetal.(2021). We observeaveragelabormarketflowsandaveragewagesforeveryagebutnotforeverylevel of experience in the data. For that reason, we use the model’s analytical averages over experience to match the corresponding data. We directly estimate the exogenous flows (πt (i,a),πt (i,a),πt (i,a),πt (i,a))foreachi,a,andtusingIPUMS-CPSdata.10 EU EN UN NU We use the model solution, equations (17)–(21), to recover the matching efficiency and human capital parameters for each demographic group i. We assume that the human 10SeeAppendixD.1. fordetails. 26

capitalofaworkeriofageaandexperienceeish (e,a) = y eri(a)e. Specifically,wecalibrate i i theinitialhumancapitalyt tomatchthewagerateofiatage25. Thenreturnstoexperience i rt(a) are calibrated using equation(17) such that we exactly match the life-cycle profileof i wagesobservedinthe CenterforEconomicandPolicyResearch (CEPR)data(Centerfor EconomicandPolicyResearch,CenterforEconomicandPolicyResearch(CEPR))foreach i. WeallowdifferentmatchingefficiencyAt(a),At (a)forjob-seekersfromtheunemployment i i,s poolandthenonparticipationpool. Specifically,usingequation(19),werecoverthelifecycleprofilesofthematchingefficienciesfortheunemployedbymatchingtheirjob-finding ratesweestimate fromtheIPUMS-CPSdata. Then wecalibrateψt(a)suchthatwe match i theunexplaineddifferencesinjob-findingratesbetweenunemployedandnonparticipants ofotherwisesimilarworkers. Hence,At (a) = ψt(a)At(a). i,s i i We set the consumption for nonparticipants c(x) = γt(e,a) · h (e,a) and calibrate the i i replacementrateγt(e,a)foreachgroupsuchthatthewageratesforworkerscomingfrom i unemploymentversusnonparticipationarethesameinthemodel. Vacancy-Posting Cost: κt. Hall (2017) suggests that the cost of posting a new vacancy i is a constant share of the worker’s productivity. This insight reflects the idea that the investment neededto createjobs increaseswith potentialrevenues. WefollowHall (2017) andsetκt(e,a) = κ¯t ×ht(e,a). i i i We separately calibrate the values of κ¯t for each gender–education group. Specifically, we i settheκ¯tto0.33fornon-collegemalesbetween1976and1980,setκ¯1980 to0.33—astandard i MNC 27

valuein the literature—and setκ¯1980 for othergroupssuch that the average tightnessgaps i betweennon-collegemalesandothergroupsarematched. Wethencalibratetheκ¯t forall i groups such that the changes in tightness rates between 1980 and 2007 are matched. In particular,thetightnesstargetinthedatathatweuseforeachgroupisvacanciespergroup overthenon-employedbetweentheagesof25and64(unemployed+nonparticipants),a targetthatcaneasilybemappedtoourmodel. Constructingthe group-specificvacanciesisnotstraightforwardbecausevacancypostings are not targeted to specific demographic groups. We rely on a simple assumption that currentemploymentsharesofeachgroupin eachyearprovideanestimateof thenumber ofvacanciesavailableforeachgroup. Thus,wecalculategroup-specificvacanciesvt for i eachgroupbymultiplyingthenumberofvacancieswiththeemploymentshareofgroupi attimet. Thetightness-ratedenominatorsforeachgrouparesimplytheirunemployment levelsatt. ThesedataaredirectlyobservedinIPUMS-CPSdata. Group-specificmeasures oftightnessarethen st ×vt θt ≡ E,i , (24) i ut +nt i i wherevt isthenumberofvacancies,ut isthenumberofunemployed,nt isthenumberof i i i nonparticipants,andst istheshareofgroupiofthetotalemploymentatt. E,i Labormarkettightnesshasdecreasedforallgroups,butspecificallyformales(seetable3 fordetails). For bothcollege andnon-collegemales, tightnessin 2007was aboutone-third of the tightness in 1980. The decline for females was more subdued: The labor market tightnesshasdecreasedabout14percentfornon-collegefemalesandabout8percentfor college-educatedfemales. 28

Parameter values for 1976–80 and 2003–07 steady states—common parameters Table 1. acrosstheCDandCESmodels Parameter Explanation Value Source β Discountfactor .9902 Quarterlyrate γ Replacementrate .35 Averageconsumptionduringunemployment γR Replacementrate: retired .33 Averageconsumptionduringretirement α Matchingfunction: share .5 denHaanetal.(2000) ρ Matchingelasticity: CESmodel -.3 Stevens(2007) πt (i,a) Quarterlyseparationrate SeefigureD.4,AppD.4 Authors’estimationusingIPUMS-CPSdata EU πt (i,a) Quarterlyseparationrate SeefigureD.4,AppD.4 Authors’estimationusingIPUMS-CPSdata EN πt (i,a) Quarterlyflowrate: UtoN SeefigureD.5,AppD.4 Authors’estimationusingIPUMS-CPSdata UN πt (i,a) Quarterlyflowrate: NtoU SeefigureD.5,AppD.4 Authors’estimationusingIPUMS-CPSdata NU 4.5 Results Tables 1 and 2 sum up the model parameters and how they are set. We focus next on describingthechangesinthemainparameters: matchingefficiencies,returnstoexperience, nonparticipantconsumption,andvacancy-postingcosts. We find that life-cycle returns to experience have increased for all groups between 1980 and2007(figure3),andthetrendsaresimilarinboththeCDandtheCESmodels. The increaseinthereturnstoexperiencehasbeenmostpronouncedforfemalesaged35and older, which has led to a convergence between the returns to experience of males and femaleswithintheeducationgroups. Theonlygroupforwhichreturnstoexperiencehave 29

Calibratedparametervaluesfor1976–80and2003–07steadystates Table2. Parameter Explanation Value: CD Value: CES y Initialhumancapital 1.28 1.22 MC,80 y Initialhumancapital 1.27 1.29 MC,07 y Initialhumancapital .97 1.12 MNC,80 y Initialhumancapital .99 1.03 MNC,07 y Initialhumancapital 1.16 1.07 FC,80 y Initialhumancapital 1.17 1.18 FC,07 y Initialhumancapital .85 .94 FNC,80 y Initialhumancapital .90 .93 FNC,07 rt(a) Returnstoexperience Seefigure3 Seefigure3 i At(a) Matchingefficiency: U¯ Seefigure4 Seefigure4 i ψt(a) Searcheffort: N¯ SeefigureD.6,AppD.4 SeefigureD.6 i Source: Authors’estimations. declinedismalesyoungerthan35withoutacollegeeducation. Asshownintable1,the calibration results also indicate that the initial human capital of non-college males has declinedfrom1980to2007. Theseresultsreflecttheobservedchangesinwagetrendsduringthesameperiod. First, an increase in returns to experience reflects the fact that real wages have increased for allgroupsexceptnon-collegemales. Second,thedeclineinthegendergapinreturnsto experiencegoeshandinhandwiththesignificantdeclineinthegenderwagegapduring the same period, as documented in Blau and Kahn (2017) and shown in figure D.2 in 30

Workers’life-cyclereturnstoexperiencein1980and2007,bygenderandeducation Figure3. Note: Panels A and C plot the simulated life-cycle returns to experience for workers with and withoutacollegeeducation,respectively,bygenderintheCDmodel. PanelsBandDshowthe samesimulationresultsintheCESmodel. Source: Authors’estimations. AppendixD.4. Our calibration results show that for all groups matching efficiency is higher in 2007 than in 1980 (figure 4). Matching efficiencies are consistently higher in the CES model compared with the CD model, reflecting the differences in matching functions, but the trends align. Men have higher matching efficiency than women within each education group except for college-educated women, who had higher matching efficiency in 1980 31

compared withcollege-educated men. Likewise, younger workers have highermatching efficiencycomparedwitholderworkers. Avalidationofourcalibrationresultsformatchingefficiencybetweenthelate1970sand 2000scomesfromthecorrespondingshiftsintheBeveridgecurve.11 ShiftsintheBeveridge curve are typically interpreted as changes in matching efficiency, with outward shifts reflectingalowermatchingefficiency. MichaillatandSaez(2021)andDiamondandŞahin (2015)studylong-term movementsoftheBeveridgecurveintheU.S.,andtheirfindings show that theBeveridgecurve in the late 1970swas located totheright of the curve inthe late2000s. Thisresultsupportsthefindingthatmatchingefficiencywaslowerinthe1970s. Figure5showsthereplacementratesfornonparticipantsoverthelifecycle,γt(a). First,the i calibratedvaluesarehigherintheCESmodelcomparedwiththeCDmodel. Thebargaining power channel in the CES model amplifies the wage gap between the unemployed and nonparticipantswhenevertightnessratesaredifferentforthesegroups. Theconsumption ofnonparticipantsneedstobehigherforwagestobesetequalbetweenthesegroups. Thecalibrationresultsalsoindicatethatthereplacementrateshaveincreasedbetween1980 and 2007 for females and college males but stayed fairly constant for non-college males. For college males, thereis a cleardrop inthe replacementratein 1980between theages of 30 and 40, while the drop largely disappears by 2007. This could capture the higher likelihoodoffathersengaginginchild-rearingactivitiesinthelatterperiod. 11TheBeveridgecurvereflectsthenegativerelationshipbetweenunemploymentandvacancyratesover thebusinesscycle(Beveridge,1944). 32

Unemployedworkers’life-cyclematchingefficienciesin1980and2007,bygender Figure4. andeducation Note: Panels A and C plot the simulated life-cycle matching efficiencies for workers with and withoutacollegeeducation, respectively, intheCDmodelbygender. PanelsBandDshowthe samesimulationresultsintheCESmodel. Source: Authors’estimations. Allthecalibratedvaluesofκ¯t areshownintable3,alongwiththecalibrationtargets. First, i the results show that we can closely match the tightness targets from data. Second, we findthatvacancycostsvarybygenderandeducation,likelycapturingthedifferencesin representativeoccupationsandindustriesforeachgroup. Moreinterestingly,wefindthat κ¯ hasincreasedforeverygroupbetween1980and2007. First,wefindthatκ¯ forbothmale groupshasmorethandoubled,indicatingalargeincreaseinvacancycosts. Thisincrease 33

Replacement rates of nonparticipants over the life cycle in 1980 and 2007, by Figure 5. genderandeducation Notes: PanelsAandCplotthesimulatedlife-cyclereplacementratesfornonparticipantswithand withoutacollegeeducation, respectively, intheCDmodelbygender. PanelsBandDshowthe samesimulationresultsintheCESmodel. Source: Authors’estimations. isalsoreflectedinthedecreasedtightnessrates. Second,whileκ¯ fornon-collegefemales hasalmostdoubled, theincreasehas beenmore moderateforcollege-educated females. Theκ¯ forcollege-educatedfemaleshasincreasedbyafactorof1.5. What are the potential reasons for the increased κ¯, and why has this increase varied significantly between groups? We interpret the changes in the vacancy posting costs to 34

Table3. Calibratedκ¯ bygenderandeducation: 1980and2007steadystates 1980 2007 Group θ,data θ θ κ¯ κ¯ θ,data θ θ κ¯ κ¯ CD CES CD CES CD CES CD CES Male,college 2.37 2.37 2.36 .20 .16 .72 .72 .72 .66 .73 Female,college .35 .35 .36 .73 .90 .31 .32 .32 1.05 1.34 Male,non-college 1.00 1.00 1.00 .33 .33 .37 .36 .36 .86 1.10 Female,non-college .20 .20 .20 .78 .99 .15 .15 .15 1.56 2.21 Note: Wenormalizethetightnessrateofnon-collegemalesin1980to1andsettheirκ¯ to0.33. We thencalibratetheremainingκ¯ tomatchtherelativetightnessratesofothergroups. Source: Authors’estimations. broadlyreflectthechangesinrelativecostsofcreatingjobsforcertaingroups.12 Anyoutside factorthat raises therelative cost ofopeninga vacancy intheU.S.,given thevacancyvalue intheU.S.,willbecapturedbyκ¯. Hence,naturalcandidatesareautomation,increasedglobalization,andimportcompetition. Asdescribed inSection 4.1,a large shareof thedrop inboth thelabor shareandtightness occurred after 2000. The sluggish employment growth in the U.S. in the 2000s is tightly linkedtoincreasedimportcompetition(Acemogluetal.,2016;Charlesetal.,2019). While import competition has directly depressed employment in the most affected industries, these effects have transmitted to other industries through input-output and aggregate demand linkages, further elevating employmentlosses (Acemoglu et al., 2016; Autor et al., 12Anotherwaytointerpretthecostofpostingavacancyistointerpretitasafixedentrycost,eitherinthe unitsofcapitalorlabor,asinManginandSedláček(2018). 35

2016). Moreover, there is no strong evidence of offsetting employment gains in other industries in the long-term: Out-migration from the most affected local labor markets has been modest, manufacturing job losses have translated to declines in employment-topopulationratios,andthenegativeeffectsoftheChinashockhavepersisteduntilthelate 2010s(Autoretal.,2021). Inourmodel,thesenegativeemploymenteffects arecaptured byκ¯,leadingtolowervacancypostingandadropinlabordemand. Moreover, evidence shows that the described negative employment effects from rising import competition have had heterogeneous effects on different gender and education groups,potentiallyexplainingtheheterogeneouschangesinthecalibratedvacancycosts. First,whilebothfemale-andmale-dominatedmanufacturingindustrieshavefacednegative employment and wage consequences from import competition, a larger share of males work in manufacturing, leading to a larger effect on men (Autor et al., 2019). Second, the negative effect of trade exposure on employment have concentrated in local labor marketswithasmallershareofcollege-educatedworkers(Autoretal.,2021). Thelower declineincollege-educatedfemales’ κ¯ couldarisefromthefactthat,comparedwithmales, college-educated females are moreoften working onhealth-care- and education-related occupations,whicharelessaffectedbyimportcompetition. Finally, the increase in the cost of creating a new job also lines up with the findings of Wolcott(2020). SheconcludesthatthedeclineinemploymentratesofU.S.maleworkers, especially those without a college education, since the late 1970s is driven by demand factorsratherthansupplyfactors. 36

Tosumup,wefindthatinordertomatchtheobservedwagetrends(increasesforother groups except non-college males), the increases in job-finding rates for females and the decreasesformales,andthedecreasesintightnessrates,theremustbecounteractingforces thatcanjointlygeneratethesetrends. First,increasedrealwageratesindicateanincrease in the productivitycapturedbyhuman capital parameters. Second, while tightness rates havedecreased,job-findingrateseitherhavedecreasedlessorhaveincreased,meaning thatmatchingefficienciesmusthaveincreased. Bothfactorsincreasedemandforworkers byincreasingthevalueofopeningavacancy,leadingtohighertightnessrates. Tomatch the declines in tightness rates, vacancy costs have grown, capturing the fact that while thevacancyvaluehasalsogrown,therehasbeenacounterforcethathasloweredlabor demand. We find that the CES Changes in the Labor Share in the CD and the CES Models. model generatesa 2.6 percent decline inthe labor share between1980 and 2007, while the CD model generates a notably smaller decline of 1.1 percent (table 4). The labor share has dropped 4.0 percent during the same period.13 The CES model thus explains about two-thirds (65 percent) of the decline in the labor share, while the CD model explains only28percentofthedecline.14 Thisresultimpliesthatthebargainingpowerchannelis quantitativelyimportantingeneratingthelaborsharedeclineinthemodel.15 13Thedeclineiscalculatedbycomparingtheaveragelaborsharesintheperiodsof1976–80and2003–07. 14Notethatwedonottargetthelaborsharedeclineinourcalibration. 15Wefurtherreportmodel-generatedchangesinthelaborsharesfordifferentgroupsinAppendixD.5. 37

Decreaseinthelaborsharefrom1980to2007 Table4. Data CDmodel CESmodel Percentdeclineinlaborshare 4.0 1.1 2.6 Percentofdeclineindata 100.0 27.5 65.0 Note: Table4presentsthedecreaseinthelaborshareinthedata,intheCDmodel,andintheCES model. Source: BureauofLaborStatistics;Authors’estimations. 5 ChangesinEfficientBargainingPower We study changes in efficient bargaining power of workers between 1980 and 2007 by gender and education (table 5). Our calibration results suggest a decline in bargaining power, consistentwith previousempiricalfindings. First, we findthataveragebargaining power over the life cycle has decreased for all groups. The decrease has been larger for malesandforworkerswithoutacollegeeducation. Bargainingpowerhasdroppedby16.8 percentforcollege-educatedmalesand1.2percentforcollege-educatedfemales. Atthe sametime,bargainingpowerhasdecreasedby17percentfornon-collegemalesandby5.5 percentfornon-collegefemales. Usingemploymentshares ofeachgroup asweights, we findthataggregatebargainingpowerhasdeclinedby11.1percent. Relyingonthelife-cyclefeatureofthemodel,wefurtherinvestigatethelife-cycletrendsof bargainingpower. Figure6plots thelife-cyclebargainingpowerpatters forallthegroups between the ages of 25 and 64. When looking at male groups, bargaining power slowly declinesoverthelifecycle. Thelife-cycletrendsinbargainingpoweraresomewhatdifferent 38

Model-implied efficient bargaining power ofworkersby gender andeducation: Table 5. 1980and2007steadystates Group 1980 2007 Percentchange Male,college .563 .467 -16.8 Female,college .403 .398 -1.2 Male,non-college .501 .415 -17.0 Female,non-college .365 .345 -5.5 Weightedaverage .467 .415 -11.1 Source: Authors’estimations. forwomencomparedwithmen,followinganS-shape. Bargainingpowerstartsatahigher level, decreasesatafaster paceuptoawoman’smid-30s, increasesagainuntilage45, and then decreases until retirement. However, the life-cycle levels and trends of males and femaleshaveconvergedovertime,withthetrendsoffemalesmorecloselyreflectingthose ofmales duringthelatter period. This isexpected,given the convergence inother labor marketoutcomes,likewagesandlabormarketflows. By using equation (5), the calibrated parameter values for α and ρ, and observed time seriesofaggregatetightness,weconstructanaggregatebargainingpowerseriesbetween 1976and2016. Basedontheaggregatedata,wefindthatthebargainingpowerofworkers has decreased around 8 percent between the two business cycle peaks of 1979 and 2007 (panelBinfigure7). Thedeclineinbargainingpowerissmallerthantheweightedaverage fromtable5, indicatingthatrelyingonaggregatetightnesscan leadtounderestimating theoveralldecline. 39

Workers’efficientlife-cyclebargainingpowerin1980and2007, bygender and Figure6. education Note: PanelsAandBplotthesimulatedlife-cyclebargainingpowerofworkerswithandwithouta collegeeducation,respectively,bygender. Source: Authors’estimations. Whenlookingatthevariationinbargainingpowerbetween1976and2016,wefindthat bargaining power decreased notably between 2000 and 2010, consistent with the declines in tightness and the labor share. However, bargaining power recovered strongly after 2010, whilethelaborshareincreasedslightly. 40

Efficientbargaining powerhasdeclinedbetween 1979and2007, alongwiththe Figure7. laborshare Note: PanelAincludesthequarterly,seasonallyadjustedlaborshareforallemployedpersonsinthe nonfarmbusinesssector,andpanelBshowsthequarterlybargainingpowerseriesnormalizedto1in 1979:Q1. Thedashedlinesplottherawseries,whilethepurplesolidlinesplottheHP-filteredseries withlambdasetto1,600. Bothfiguresalsoincludealineartrendlinewith95percentconfidence bounds. Source: BureauofLaborStatistics;Authors’estimations. 6 CounterfactualAnalysis We assess the effect of each exogenous parameter on the model-generated changes in bargainingpower. Wedothatbygivingeachparameterits1980valueoneatatimeandby keepingalltheotherparametersattheir2007levels. Table6showstheresults. 41

Wefindthatthreeparametershavedrivenchangesinbargainingpowerthroughchanges in tightness. First, the increased vacancy-posting cost κ¯ can explain the majority of the bargainingpowerdeclineforallgroups. Second,improvedmatchingefficiencyandhigher returns to experience have mitigated the decline in the bargaining power of all groups. Third,alowerπ hasmitigatedthedecline forfemales,whilea higherπ hasincreased EN EN thedeclineformales. Intuitively, better matching efficiencies and higher returns to experience increase labor market tightness by increasing a vacancy value. To match the observed in the decline tightness,alongwiththeobservedwagesandjob-findingrates,themodelpredictsthatthe vacancy-postingcosthasincreased. Anincreaseintherelativecostofopeningavacancy hasdecreaseddemandforlaborandthenumberofvacancies. To summarize our findings, we find that κ, a proxy for a decline in labor demand, has driventhedeclineintightness,andthusbargainingpower,between1980and2007. 7 Conclusion In this paper, we used a life-cycle DMP search and matching model with endogenous bargaining power to study how the labor share and workers’ bargaining power have changed in the past four decades. Specifically, we assumed that the matching function takes the CES form, which implies that bargaining power increases with labor market tightness whenever the Hosios condition holds and there is enough complementarity betweenvacanciesandjobseekersinthematchingprocess. First, we find that a DMP model with endogenous bargaining power generates a signif- 42

Counterfactuals—efficientbargainingpower Table6. Male, Female, Male, Female, College College Non-college Non-college Totalchange—model -16.8 -1.2 -17.0 -5.5 Parameter—percentcontributiontothechange y 0.1 -21.2 4.6 3.5 A -7.6 -26.2 -11.0 -58.5 ψ 1.0 4.3 -1.4 -25.9 r -4.5 -64.5 -9.3 -90.3 π 0.7 7.5 1.0 5.3 EU π 7.8 -78.0 14.4 -52.2 EN π 2.9 -8.6 6.6 -3.1 UN π 2.2 -1.2 3.2 -3.1 NU κ¯ 91.8 218.4 91.3 293.4 γ 3.8 60.8 -1.4 27.7 N Startingmasses,ms(i,a) 1.8 8.6 1.9 3.1 t Totalcontribution 100.0 100.0 100.0 100.0 Note: Thecontributionofeachcounterfactualisscaledsuchthatthesumofindividualcounterfactualsequals100percent. Source: Authors’estimations. icantly larger drop in the labor share compared with the model with fixed bargaining power. Second, our calibration results suggest that workers’ efficient bargaining power hasdecreasedabout11percentbetween1980and2007. Thiscanbeattributedtoahigher vacancy-posting cost, which has driven down the labor demand. We also find that the declineinbargainingpowerhasbeenlargerformenandworkerswithoutacollegeeducation. Overall,the declineinbargainingpowerbasedon ourmodelhasbeenmodest. However, 43

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AppendixA:A SocialPlanner’sProblemandHosios ConditioninaLife-CycleDMP ModelwithHumanCapitalAccumulation,Nonparticipation,andSegmentedMarkets AppendixA.1: SocialPlanner’sProblem Theplannermaximizesthesumofflowsofmarketandhomeproductionsnetofsearch costs. Inthelife-cycleframework,theseflowsareconsideredoverallagesaanddifferent timeperiods. ThereareaR overlappinggenerationsofdifferentagesaworkinginagiven periodt+a,wheretisthedateofbirthofagivencohort. Moreover,workersgainexperience whenemployedandmaybesubjecttoexperiencedepreciationwhenoutofwork. Thus, theflowsarealsoconsideredoverallexperiencelevels. (cid:110) (cid:111) Theplannerchoosestheoptimalfullsequencesofvacancies vU¯,t(e,a),vN¯,t(e,a) ,andemi i ployment,unemployment,andnonparticipationmasses{ME¯,t+1(e,a),MU¯,t+1(e,a),MN¯,t+1(e,a)}, i i i givena ∈ A ≡ [a,a ]ande ∈ A ≡ [e,a ]. Denotethecontrolvariablesby R R (cid:110) (cid:111) X = vU¯,t(e,a),vN¯,t(e,a),ME¯,t+1(e,a),MU¯,t+1(e,a),MN¯,t+1(e,a) . We assume that lai i i i i bor markets are segmented for each i, e, and a, which means that the planner faces a different matchingtechnology for workers fromdifferent statuses. This alsosimplifiesthe planner’sproblem: Itcantreateachproblemasaseparateoptimizationproblem. Moreover, the planner also observes the current labor market status of a job-seeker, U ¯ and N ¯. The socialplanner’sproblemis 49

∞ (cid:34) aR−1aR−1 max (cid:88) βt (cid:88) (cid:88) h (e,a)ME¯,t(e,a)+c¯U¯ (e,a)MU¯,t(e,a)+c¯N¯ (e,a)MN¯,t(e,a)− i i i i i i X t=0 e=1 a=a (cid:35) κU¯ (e,a)vU¯,t(e,a)−κN¯ (e,a)vN¯,t(e,a) (25) i i i i subjectto ME¯,t+1(e,a) = (cid:2) 1−πi (e−1,a−1)−πi (e−1,a−1) (cid:3) ME¯,t(e−1,a−1)+ i EU EN i (cid:32) (cid:33) (cid:32) (cid:33) vU¯,t(e+1,a−1) vN¯,t(e+1,a−1) q i vU¯,t(e+1,a−1)+q i vN¯,t(e+1,a−1),(26) MU¯,t(e+1,a−1) i (cid:98) MN¯,t(e+1,a−1) i i i MU¯,t+1(e,a) = MU¯,t(e+1,a−1)+πi (e−1,a−1)ME¯,t(e−1,a−1)− i i EU i (cid:32) (cid:33) vU¯,t(e+1,a−1) q i vU¯,t(e+1,a−1)−πi (e+1,a−1)MU¯,t(e+1,a−1)+ MU¯,t(e+1,a−1) i UN i i πi (e+1,a−1)MN¯,t(e+1,a−1), (27) NU i MN¯,t+1(e,a) = MN¯,t(e+1,a−1)+πi (e−1,a−1)ME¯,t(e−1,a−1)− i i EN i (cid:32) (cid:33) vN¯,t(e+1,a−1) q i vN¯,t(e+1,a−1)−πi (e+1,a−1)MU¯,t(e+1,a−1)+ (cid:98) MN¯,t(e+1,a−1) i UN i i πi (e+1,a−1)MN¯,t(e+1,a−1), (28) NU i fora = a,...,aR −1, e = 1,...,aR −1, andt = 0,...,∞, and given initial masses ME¯,t+1(1,a), MU¯,t+1(1,a), and MN¯,t+1(1,a) and terminal coni i i ditions ME¯,t+1(cid:0) e,aR (cid:1) = 0, MU¯,t+1(cid:0) e,aR (cid:1) = 0, and MN¯,t+1(cid:0) e,aR (cid:1) = 1. The terminal i i i conditionscapturetheassumptionthatworkersretireatageaR andmovetononparticipation, which also implies that πi (e,aR−1) = 1, πi (e,aR−1) = 0, πi (e,aR−1) = 1, and EN EU UN πi (e,aR−1) = 0. NU 50

The first constraint represents employment dynamics betweentwo periods, t and t−1 for eacheanda,whilethelasttwoconstraintsrepresenttheevolutionofunemploymentand nonparticipationmasses. Aslabormarkettightnessisdefinedasθs,t(e,a)= v i s,t(e,a) ,wheres ∈ U ¯ ,N ¯,theplanner’s i Ms,t(e,a) i problemcanbewrittenintermsoftightness. TheLagrangianbecomes ∞ (cid:26)aR−1aR−1 L =max (cid:88) βt (cid:88) (cid:88) h (e,a)ME¯,t(e,a)+MU¯,t(e,a) (cid:16) c¯U¯ (e,a)−κU¯ (e,a)θU¯,t(e,a) (cid:17) i i i i i i X t=0 e=1 a=a (cid:27) (cid:16) (cid:17) + MN¯,t(e,a) c¯N¯ (e,a)−κN¯ (e,a)θN¯,t(e,a) i i i i ∞ aR−1aR−1 (cid:26) + (cid:88) βt (cid:88) (cid:88) λt(e,a) (cid:0) 1−πi (e−1,a−1)−πi (e−1,a−1) (cid:1) ME¯,t(e−1,a−1) i EU EN i t=0 e=0 a=a (cid:27) (cid:16) (cid:17) (cid:16) (cid:17) + f θU¯,t(e+1,a−1) MU¯,t(e+1,a−1)+f(cid:98) θN¯,t(e+1,a−1) MN¯,t(e+1,a−1)−ME¯,t+1(e,a) i i i i i ∞ aR−1aR−1 (cid:26) + (cid:88) βt (cid:88) (cid:88) µt(e,a) (1−f (cid:16) θU¯,t(e+1,a−1) (cid:17) −πi (e+1,a−1))MU¯,t(e+1,a−1) i i UN i t=0 e=0 a=a (cid:27) + πi (e−1,a−1)ME¯,t(e−1,a−1)+πi (e+1,a−1)MN¯,t(e+1,a−1)−MU¯,t+1(e,a) EU i NU i i ∞ aR−1aR−1 (cid:26) + (cid:88) βt (cid:88) (cid:88) ηt(e,a) (1−f(cid:98) (cid:16) θN¯,t(e+1,a−1) (cid:17) −πi (e+1,a−1))MN¯,t(e+1,a−1) i i UN i t=0 e=0 a=a (cid:27) + πi (e−1,a−1)ME¯,t(e−1,a−1)+πi (e+1,a−1)MU¯,t(e+1,a−1)−MN¯,t+1(e,a) EN i UN i i +µ (0)[u (0)(1−f (θ (0)))−u (0)] 0 t t t+1 ThefirstorderconditionswithrespecttoME¯,t+1(e,a),MU¯,t+1(e,a),MN¯,t+1(e,a),θU¯,t(e,a) i i i i andθN¯,t(e,a)arewrittenasfollows: i ME¯,t+1(e,a) :βt+1h (e,a)−βtλt(e,a)+βt+1λt+1(e+1,a+1) (cid:0) 1−πi (e,a)−πi (e,a) (cid:1) i i i i EU EN +βt+1µt+1(e+1,a+1)πi (e,a)+βt+1ηi (e+1,a+1)πi (e,a) = 0 i EU t+1 EN 51

(cid:16) (cid:17) MU¯,t+1(e,a) : βt+1(c¯U¯ (e,a)−κU¯ (e,a)θU¯,t+1(e,a))+βt+1λt+1(e−1,a+1)f θU¯,t+1(e,a) i i i i i i (cid:104) (cid:16) (cid:17) (cid:105) +βt+1µt+1(e−1,a+1) 1−f θU¯,t+1(e,a) −πi (e,a) i i UN +βt+1ηt+1(e−1,a+1)πi (e,a)−βtµt(e,a) = 0 i UN i (cid:16) (cid:17) MN¯,t+1(e,a) : βt+1(c¯N¯ (e,a)−κN¯ (e,a)θN¯,t+1(e,a))+βt+1λt+1(e−1,a+1)f(cid:98) θN¯,t+1(e,a) i i i i i i (cid:104) (cid:16) (cid:17) (cid:105) +βt+1ηt+1(e−1,a+1) 1−f(cid:98) θN¯,t+1(e,a) −πi (e,a) i i NU +βt+1µt+1(e−1,a+1)πi (e,a)−βtηt(e,a) = 0 i NU i (cid:16) (cid:17) θU¯,t(e,a) :−βtMU¯,t(e,a)κU¯ (e,a)+βtλt(e−1,a+1)f′ θU¯,t(e,a) MU¯,t(e,a) i i i i i i (cid:16) (cid:17) −βtµt(e−1,a+1)f′ θU¯,t(e,a) MU¯,t(e,a) = 0 i i i (cid:16) (cid:17) θN¯,t(e,a) :−βtMN¯,t(e,a)κN¯ (e,a)+βtλt(e−1,a+1)f(cid:98)′ θN¯,t(e,a) MN¯,t(e,a) i i i i i i (cid:16) (cid:17) −βtηt(e−1,a+1)f(cid:98) ′ θN¯,t(e,a) MN¯,t(e,a) = 0. i i i Inthesteadystate,t = t+1forallt,andwecanreorganizeandwritethefollowing: λ (e,a) i =h (e,a)+λ (e+1,a+1) (cid:2) 1−πi (e,a)−πi (e,a) (cid:3) β i i EU EN +µ (e+1,a+1)πi (e,a)+η (e+1,a+1)πi (e,a); i EU i EN µ (e) (cid:16) (cid:17) i =(c¯U¯ (e,a)−κU¯ (e,a)θU¯ (e,a))+λ (e−1,a+1)f θU¯ (e,a) β i i i i i (cid:104) (cid:16) (cid:17) (cid:105) +µ (e−1,a+1) 1−f θU¯ (e,a) −πi (e,a) +η (e−1,a+1)πi (e,a); i i UN i UN 52

ηt(e,a) (cid:16) (cid:17) i =(c¯N¯ (e,a)−κN¯ (e,a)θN¯ (e,a))+λ (e−1,a+1)f(cid:98) θN¯ (e,a) β i i i i i (cid:104) (cid:16) (cid:17) (cid:105) +η (e−1,a+1) 1−f(cid:98) θN¯ (e,a) −πi (e,a) +µ (e−1,a+1)πi (e,a); i i UN i NU (cid:16) (cid:17) (cid:16) (cid:17) −κU¯ (e,a)+λ (e−1,a+1)f′ θU¯ (e,a) −µ (e−1,a+1)f′ θU¯ (e,a) = 0; i i i i i (cid:16) (cid:17) (cid:16) (cid:17) −κN¯ (e,a)+λ (e−1,a+1)f(cid:98) ′ θN¯ (e,a) −η (e−1,a+1)f(cid:98) ′ θN¯ (e,a) = 0. i i i i i Moreover, define SU¯∗(e,a) = (λ (e,a)−µ (e,a))/β,SN¯∗(e,a) = (λ (e,a)−η (e,a))/β, i i i i i i andwrite λ (e,a)/β = h (e,a)+λ (e+1,a+1)−(πi (e,a)+πi (e,a))βSU¯∗(e+1,a+1) i i i EU EN i −πi (e,a)[µ (e+1,a+1)−η (e+1,a+1)] EN i i = h (e,a)+λ (e+1,a+1)−(πi (e,a)+πi (e,a))[βSN¯∗(e+1,a+1)] i i EU EN i −πi (e,a)[η (e+1,a+1)−µ (e+1,a+1)]; EU i i (29) (cid:104) (cid:105) µ (e,a)/β = (c¯U¯ (e,a)−κU¯ (e,a)θU¯ (e,a))+βf θU¯ (e,a) SU¯∗(e−1,a+1)+µ (e−1,a+1) i i i i i i i +πi (e,a))[η (e−1,a+1)−µ (e−1,a+1)]; UN i i (30) (cid:16) (cid:17) η (e,a)/β = (c¯N¯ (e,a)−κN¯ (e,a)θN¯ (e,a))+βf(cid:98) θN¯ (e,a) SN¯∗(e−1,a+1)+η (e−1,a+1) i i i i i i i +πi (e,a)[µ (e−1,a+1)−η (e−1,a+1)]; NU i i (31) 53

(cid:16) (cid:17) κU¯ (e,a) = βf′ θU¯ (e,a) SU¯∗(e−1,a+1); (32) i i i (cid:16) (cid:17) κN¯ (e,a) = βf(cid:98) ′ θN¯ (e,a) SN¯∗(e−1,a+1). (33) i i i Then, subtractequation (30)from equation (29), and(31) from(29), andinsert (32)and (33): λ (e,a)−µ (e,a) SU¯∗(e,a) ≡ i i i β (cid:104) (cid:104) (cid:105) (cid:104) (cid:105)(cid:105) = h (e,a)−c¯U¯ (e,a)+βSU¯∗(e−1,a+1) θU¯ (e,a)f′ θU¯ (e,a) −f θU¯ (e,a) i i i i i i +µ (e+1,a+1)−µ (e−1,a+1)−πi (e,a)[η (e−1,a+1)−µ (e−1,a+1)] i i UN i i −πi (e,a)[µ (e+1,a+1)−η (e+1,a+1)]+β (cid:2) 1−πi (e,a)−πi (e,a) (cid:3) SU¯∗; EN i i EU EN i (34) λ (e,a)−η (e,a) SN¯∗(e,a) ≡ i i i β (cid:104) (cid:104) (cid:105) (cid:104) (cid:105)(cid:105) = h (e,a)−c¯N¯ (e,a)+βSN¯∗(e−1,a+1) θN¯ (e,a)f(cid:98) ′ θN¯ (e,a) −f(cid:98) θN¯ (e,a) i i i i i i +η (e+1,a+1)−η (e−1,a+1)−πi (e,a)[µ (e−1,a+1)−η (e−1,a+1)] i i NU i i −πi (e,a)[µ (e+1,a+1)−η (e+1,a+1)]+β (cid:2) 1−πi (e,a)−πi (e,a) (cid:3) SN¯∗; EU i i EU EN i (35) η (e,a)−µ (e,a) SU¯∗(e,a)−SN¯∗(e,a) = i i . (36) i i β 54

Thus,theplanner’ssolutionissummarizedbythefollowingsevenequations: λ (e,a)−µ (e,a) SU¯∗(e,a) ≡ i i i β (cid:104) (cid:104) (cid:105) (cid:104) (cid:105)(cid:105) = h (e,a)−c¯U¯ (e,a)+βSU¯∗(e−1,a+1) θU¯ (e,a)f′ θU¯ (e,a) −f θU¯ (e,a) i i i i i i +µ (e+1,a+1)−µ (e−1,a+1)−πi (e,a)[η (e−1,a+1)−µ (e−1,a+1)] i i UN i i −πi (e,a)[µ (e+1,a+1)−η (e+1,a+1)]+β (cid:2) 1−πi (e,a)−πi (e,a) (cid:3) SU¯∗; EN i i EU EN i (37) λ (e,a)−η (e,a) SN¯∗(e,a) ≡ i i i β (cid:104) (cid:104) (cid:105) (cid:104) (cid:105)(cid:105) = h (e,a)−c¯N¯ (e,a)+βSN¯∗(e−1,a+1) θN¯ (e,a)f(cid:98) ′ θN¯ (e,a) −f(cid:98) θN¯ (e,a) i i i i i i +η (e+1,a+1)−η (e−1,a+1)−πi (e,a)[µ (e−1,a+1)−η (e−1,a+1)] i i NU i i −πi (e,a)[µ (e+1,a+1)−η (e+1,a+1)]+β (cid:2) 1−πi (e,a)−πi (e,a) (cid:3) SN¯∗; EU i i EU EN i (38) η (e,a)−µ (e,a) SU¯∗(e,a)−SN¯∗(e,a) = i i (39) i i β (cid:104) (cid:105) µ (e,a)/β = (c¯U¯ (e,a)−κU¯ (e,a)θU¯ (e,a))+βf θU¯ (e,a) SU¯∗(e−1,a+1)+µ (e−1,a+1) i i i i i i i +πi (e,a))[η (e−1,a+1)−µ (e−1,a+1)]; UN i i (40) (cid:16) (cid:17) η (e,a)/β = (c¯N¯ (e,a)−κN¯ (e,a)θN¯ (e,a))+βf(cid:98) θN¯ (e,a) SN¯∗(e−1,a+1)+η (e−1,a+1) i i i i i i i +πi (e,a)[µ (e−1,a+1)−η (e−1,a+1)]; NU i i (41) (cid:16) (cid:17) κU¯ (e,a) = βf′ θU¯ (e,a) SU¯∗(e−1,a+1); (42) i i i 55

(cid:16) (cid:17) κN¯ (e,a) = βf(cid:98) ′ θN¯ (e,a) SN¯∗(e−1,a+1). (43) i i i AppendixA.2: DecentralizedProblem Next,wecharacterizethedecentralizedproblemanditssolution. Weassumemarketsare segmented,whichimpliesthatfirmscanchoosehowmanyvacanciestopostforeachtype ofworkeracrossi,e,a,ands. Workers’valuefunctionsarewrittenasfollows: AvalueofanemployedworkerfromtheunemploymentpoolisEU¯ andfromthenonpari ticipationpoolisEN¯: i EU¯ (e,a) = wU¯ (e,a)+β (cid:2) πi (e,a)U (e+1,a+1)+πi (e,a)N (e+1,a+1) i i EU i EN i (cid:105) + (cid:0) 1−πi (e,a)−πi (e,a) (cid:1) EU¯ (e+1,a+1) EU EN i (cid:104) = wU¯ (e,a)+β EU¯ (e+1,a+1)−(πi (e,a)+πi (e,a))DU¯ (e+1,a+1) i i EU EN i (cid:3) − πi (e,a)(U (e+1,a+1)−N (e+1,a+1)) ; EN i i (cid:104) EN¯ (e,a) = wN¯ (e,a)+β EN¯ (e+1,a+1)−(πi (e,a)+πi (e,a))DN¯ (e+1,a+1) i i i EU EN i (cid:3) + πi (e,a)(U (e+1,a+1)−N (e+1,a+1)) , EU i i whereDU¯ (e,a) = EU¯ (e,a)−U (e,a);DN¯ (e,a) = EN¯ (e,a)−N (e,a). i i i i i i 56

Thevaluefunctionsforunemployedandnonparticipantsarethefollowing: (cid:26) (cid:16) (cid:17) (cid:16) (cid:16) (cid:17) (cid:17) U (e,a) =cU¯ (e,a)+β f θU¯ (e,a) EU¯ (e−1,a+1)+ 1−f θU¯ (e,a) −πi (e,a) U (e−1,a+1) i i i i i UN i (cid:27) + πi (e,a)N (e−1,a+1) UN i (cid:26) (cid:16) (cid:17) =cU¯ (e,a)+β U (e−1,a+1)+f θU¯ (e,a) DU¯ (e−1,a+1) i i i i (cid:27) + πi (e,a)(N (e−1,a+1)−U (e−1,a+1)) . UN i i (cid:26) (cid:16) (cid:17) (cid:16) (cid:16) (cid:17) (cid:17) N (e,a) =cN¯ (e,a)+β f(cid:98) θN¯ (e,a) EN¯ (e−1,a+1)+ 1−f(cid:98) θN¯ (e,a) −πi (e,a) N (e−1,a+1) i i i i i UN i (cid:27) + πi (e,a)N (e−1,a+1) UN i (cid:26) (cid:16) (cid:17) =cN¯ (e,a)+β N (e−1,a+1)+f(cid:98) θN¯ (e,a) DN¯ (e−1,a+1) i i i i (cid:27) − πi (e,a)(N (e−1,a+1)−U (e−1,a+1)) . NU i i Togetworkersurpluses,subtractthevalueofunemployment(nonparticipation)fromEU¯ i (EN¯): i (cid:26) DU¯ (e,a) =wU¯ (e,a)−cU¯ (e,a)+β (1−πi (e,a)−πi (e,a))DU¯ (e+1,a+1)+U (e+1,a+1) i i i EU EN i i (cid:16) (cid:17) − f θU¯ (e,a) DU¯ (e−1,a+1)]−πi (e,a)(U (e+1,a+1)−N (e+1,a+1)) i i EN i i (cid:27) − U (e−1,a+1)−πi (e,a)(U (e−1,a+1)−N (e−1,a+1)) i UN i i (cid:26) DN¯ (e,a) =wN¯ (e,a)−cN¯ (e,a)+β (1−πi (e,a)−πi (e,a))DN¯ (e+1,a+1)+N (e+1,a+1) i i i EU EN i i (cid:16) (cid:17) − f(cid:98) θN¯ (e,a) DN¯ (e−1,a+1)]+πi (e,a)(U (e+1,a+1)−N (e+1,a+1)) i i EU i i (cid:27) − N (e−1,a+1)+πi (e,a)(N (e−1,a+1)−U (e−1,a+1)) . i NU i i 57

Regardingthefirms’problem,firms’valuefunctionscanbewrittenasfollows. First,the valueoffillingthevacancyfromtheunemploymentpoolis JU¯ (e,a) = h (e,a)−wU¯ (e,a)+β (cid:0) 1−πi (e,a)−πi (e,a) (cid:1) JU¯ (e+1,a+1), i i i EU EN i andfromthenonparticipantpoolis JN¯ (e,a) = h (e,a)−wN¯ (e,a)+β (cid:0) 1−πi (e,a)−πi (e,a) (cid:1) JN¯ (e+1,a+1). i i i EU EN i Thevaluesofunfilledvacanciesarewrittenas VU¯ (e,a) = max{−κU¯ (e,a)+β[q[θU¯ (e,a)]JU¯ (e−1,a+1)+(1−[q[θU¯ (e,a)])V ¯ ],0} and i i i i i VN¯ (e,a) = max{−κN¯ (e,a)+β[q[θN¯ (e,a)]JN¯ (e−1,a+1)+(1−[q[θN¯ (e,a)])V ¯ ],0}. i i (cid:98) i i (cid:98) i With free entry, the values of unfilled vacancies are zero, so the previous two equations simplifyto κU¯ (e,a) = βq[θU¯ (e,a)JU¯ (e−1,a+1) and i i i κN¯ (e,a) = βq[θN¯ (e,a)JN¯ (e−1,a+1). i (cid:98) i i WagesaredeterminedthroughNashbargaining. Thematchsurpluses,EU¯ (e,a)−U (e,a)+ i i JU¯ (e,a)andEN¯ (e,a)−N (e,a)+JN¯ (e,a),aresharedaccordingtotheNashproduct: i i i i max (E i U¯ −U i )ϕU i ¯(e,a)J i U¯1−ϕU i ¯(e,a) subjecttoS i U¯ = D i U¯ +J i U¯ and Ei−Ui,Ji max (E i N¯ −N i )ϕN i ¯(e,a)J i N¯1−ϕN i ¯(e,a) subjecttoS i N¯ = D i N¯ +J i N¯ . Ei−Ni,Ji 58

Thesolutionforunemployedsatisfies ϕU¯ (e,a) EU¯ (e,a)−U (e,a) i = i i or 1−ϕU¯(e,a) JU¯(e,a) i i (cid:16) (cid:17) EU¯ (e,a)−U (e,a) = ϕU¯ (e,a)SU¯ (e,a)andJU¯ (e,a) = 1−ϕU¯ (e,a) SU¯ (e,a), i i i i i i i andfornonparticipants ϕN¯ (e,a) EN¯ (e,a)−N (e,a) i = i i or 1−ϕN¯(e,a) JN¯(e,a) i i (cid:16) (cid:17) EN¯ (e,a)−N (e,a) = ϕN¯ (e,a)SN¯ (e,a)andJN¯ (e,a) = 1−ϕN¯ (e,a) SN¯ (e,a). i i i i i i i Thus,thedecentralizedsolutionisdefinedbythefollowingsixequations: SU¯ (e,a) ≡JU¯ +DU(e,a) i i i = h (e,a)−cU¯ (e,a)+β (cid:2) 1−πi (e,a)−πi (e,a) (cid:3) SU¯ (e+1,a+1) i i EU EN i (cid:26) (cid:104) (cid:105) −βf θU¯ (e,a) DU¯ (e−1,a+1)+β U (e+1,a+1)−U (e−1,a+1) i i i i (cid:27) − πi [U (e+1,a+1)−N (e+1,a+1)]−πi (e,a)[N (e−1,a+1)−U (e−1,a+1)] EN i i UN i i (44) SN¯ (e,a) ≡JN¯ +DN(e,a) i i i = h (e,a)−cN¯ (e,a)+β (cid:2) 1−πi (e,a)−πi (e,a) (cid:3) SN¯ (e+1,a+1) i i EU EN i (cid:26) (cid:104) (cid:105) −βf(cid:98) θN¯ (e,a) DN¯ (e−1,a+1)+β N (e+1,a+1)−N (e−1,a+1) i i i i (cid:27) + πi [U (e+1,a+1)−N (e+1,a+1)]+πi (e,a)[N (e−1,a+1)−U (e−1,a+1)] EU i i NU i i (45) (cid:26) (cid:16) (cid:17) U (e,a) =cU¯ (e,a)+β U (e−1,a+1)+f θU¯ (e,a) DU¯ (e−1,a+1) i i i i i (46) (cid:27) + πi (e,a)(N (e−1,a+1)−U (e−1,a+1)) UN i i 59

(cid:26) (cid:16) (cid:17) N (e,a) =cN¯ (e,a)+β N (e−1,a+1)+f(cid:98) θN¯ (e,a) DN¯ (e−1,a+1) i i i i i (47) (cid:27) − πi (e,a)(N (e−1,a+1)−U (e−1,a+1)) NU i i κU¯ (e,a) = βq[θU¯ (e,a)]JU¯ (e−1,a+1), and (48) i i i κN¯ (e,a) = βq[θN¯ (e,a)]JN¯ (e−1,a+1). (49) i (cid:98) i i AppendixA.3: ProofofProposition1 Let’s now compare the social planner’s solution and the decentralized solution. When comparingequations(37)–(43)and(44)–(49),weseethatthetwosystemsareequivalent when ϕU¯ (e − 1,a + 1) = − q′(θ i U¯(e,a))θ i U¯(e,a)) and ϕN¯ (e − 1,a + 1) = − q′(θ i N¯(e,a))θ i N¯(e,a)), when i q(θU¯(e,a)) i q(θN¯(e,a)) i i alsonotingthat µi(e,a) = U (e,a)and ηi(e,a) = N (e,a). β i β i Appendix B: Competitive Equilibrium with Bargaining Posting in DMP Model with a LifeCycle,HumanCapitalAccumulationandNonparticipation Thewagesettinginthemodel followstheonecommonlyusedinthecompetitivesearch theory: While competitive search theory assumes thatfirms directly post wage rates, we assumethatfirmspostbargainingweightsandworkersdirecttheirsearchtowardstheir utility-maximizingbargainingweight. As notedinWrightetal.(2021)on page 131,these approaches are fundamentally the same. Competitive search equilibrium implies that the match surplus shares are not constant but respond to market conditions. The only exception is the special case of the Cobb-Douglas matching function, which guarantees constantsurplusshares. Inourcase,explicitlyfocusingonbargainingpowerpostingallowsustousethemodelto 60

discusshowbargainingpowermighthavechangedinresponsetolabormarketconditions, given that bargaining power is not directly observed in the data. Once a firm and a worker arematched inasubmarketdeterminedbythebargainingweightϕ(x),the firm andthe workersharethematchsurplususingtheoptimalbargainingweight,asinNashbargaining. Firmsandworkersupdatewageseveryperiod. Weassumethatbargainingweightsrespond to current market conditions: Whenever market tightness changes, bargaining weights reactaccordingly,evenifaworkerandafirmarealreadymatched. Following Moen (1997) competitive search equilibrium, we show that the profit- and utility-maximizing behaviors offirms andjob seekers determinetheoptimalbargaining power. WhileMoen(1997)showshowfirmsandworkersoptimallychooseawagefrom a menu of wages, we assume instead that both parties choose their optimal bargaining weights. Thisassumptionleadstocompetitiveallocation,whichweconfirmcoincideswith thesociallyoptimalallocation. AssumeagainthesimilardynamicDMPmodelwithnonparticipation,lifecycle,andhuman capitalaccumulationasdiscussedbefore. Assumethatforeachlabormarketxthereisaset ofΦequilibriumsub-labormarkets,whereΦstandsforallthepossibleworkers’bargaining powers. Thenthevaluesofbeingeitherajobseekeroremployedinasubmarketwithϕ andstatexaregivenas Es(x;ϕ) = w(x;ϕ)+β[Es(x′;ϕ)−(π (x)+π (x))Ds(x′;ϕ)−π (x)(U(x′;ϕ)−N(x′;ϕ))]; EU EN EN (cid:104) (cid:105) U(x;ϕ) = c(x)+β U(x′;ϕ)+f (θ(x;ϕ))DU¯ (x′;ϕ)+π (x)(N(x′;ϕ)−U(x′;ϕ)) ; UN (cid:104) (cid:105) N(x;ϕ) = c(x)+β N(x′;ϕ)+f(cid:98)(θ(x;ϕ))DN¯ (x′;ϕ)−π (x)(N(x′;ϕ)−U(x′;ϕ)) . NU 61

Wecanalsodefinethevalueofpostingavacancyandthevalueofhavingavacancyfilled as: (cid:110) (cid:104) (cid:105) (cid:111) VU¯ (x,ϕ) = max −κ(x)+β q(θ(x;ϕ))JU¯ (x′;ϕ)+(1−q(θ(x;ϕ)))VU¯ (x;ϕ) ,0 ; (cid:110) (cid:104) (cid:105) (cid:111) VN¯ (x,ϕ) = max −κ(x)+β q(θ(x;ϕ))JN¯ (x′;ϕ)+(1−q(θ(x;ϕ)))VN¯ (x;ϕ) ,0 ; (cid:98) (cid:98) Js(x,ϕ) = h(x)−w(x;ϕ)+β{(ϕ (x)+ϕ (x))Vs(x;ϕ)+(1−ϕ (x)−ϕ (x))Js(x′;ϕ)}. EU EN EU EN Becauseofthefree-entryconditionandtheworkers’searchbehaviors,wehave Vs(x;ϕ) = 0∀ϕ∈ϕandforanyx, U(x,ϕ) = U(x)∀ϕ∈ϕandforanyx, N(x,ϕ) = N(x)∀ϕ∈ϕandforanyx. NowwedifferentiatethevaluefunctionsU(x;ϕ),N(x;ϕ),VU¯ (x;ϕ),andVN¯ (x;ϕ),andwe get dU(x;ϕ) dθ(x;ϕ) d(DU¯ (x′;ϕ)) = f′(θ(x;ϕ)) (DU¯ (x′;ϕ))+f(θ(ϕ)) ) = 0; (50) dϕ dϕ dϕ dN(x;ϕ) dθ(x;ϕ) d(DN¯ (x′;ϕ)) = f(cid:98) ′(θ(x;ϕ)) (DN¯ (x′;ϕ))+f(cid:98)(θ(ϕ)) ) = 0; (51) dϕ dϕ dϕ dVU¯ (x;ϕ) dθ(x;ϕ) dJU¯ (x′;ϕ) = q′(θ(x;ϕ)) (JU¯ (x′;ϕ))+q(θ(ϕ)) ) = 0; (52) dϕ dϕ dϕ dVN¯ (x;ϕ) dθ(x;ϕ) dJN¯ (x′;ϕ) = q′(θ(x;ϕ)) (JN¯ (x′;ϕ))+q(θ(ϕ)) ) = 0. (53) (cid:98) (cid:98) dϕ dϕ dϕ Let’ssolvetheoptimalbargainingpowerfortheunemployed. Thesamesolutionappliesto the optimal bargaining powerfornonparticipants. Rearrange equations(50)and equation (52)andget dθ(x;ϕ) dDU¯ (x′;ϕ) f′(θ(x;ϕ)) DU¯ (x′;ϕ) = −f(θ(ϕ)) ) dϕ dϕ 62

dθ(x;ϕ) dJ(x′;ϕ) q′(θ(x;ϕ)) (J(x′;ϕ)) = −q(θ(ϕ)) ) dϕ dϕ f′(θ(x;ϕ))DU¯ (x′;ϕ) f(θ(x;ϕ))dDU¯ (x′;ϕ)/dϕ ⇒ = . (54) q′(θ(x;ϕ)) J(x′;ϕ) q(θ(x;ϕ)) dJ(x′;ϕ)/dϕ Combining the equation (54) with the fact that a surplus is S(ϕ) = J(x;ϕ) + E(x;ϕ) − U(x;ϕ)16 and the sharing rule of surplus is E(x;ϕ) − U(x;ϕ) = ϕS(x;ϕ), J(x;ϕ) = (1 − ϕ)S(x;ϕ),weget f′(θ(x;ϕ)) ϕS(x′;ϕ) f(θ(x;ϕ)) dϕS(x′;ϕ) (cid:20) d(1−ϕ)S(x′;ϕ) (cid:21)−1 = × × q′(θ(x;ϕ)) (1−ϕ)S(x′;ϕ) q(θ(x;ϕ)) dϕ dϕ θq′(θ(ϕ))+q(θ(ϕ)) ϕ ⇒ = −θ(ϕ). q′(θ(ϕ)) 1−ϕ Whentheequationissimplified,thesolutionforthesystembecomes q′(θ(x;ϕ))θ(x;ϕ) ϕ = − . (55) q(θ(x;ϕ)) Thus,equation (55)isexactlythesociallyefficientconditionforbargainingpower,andthe efficientconditionforbargainingpowerholdsforanygivensetΦofsubmarketsthatexists intheequilibrium. Inotherwords,thebargainingpowerϕservesasapricedevicetoadjust the relative demand and supply of labor. In equilibrium, firms are picking the efficient submarketstopursuethehighestprofit,andsodotheutility-maximizingjobseekers. The “price”ofbargainingpowermustfollowtheefficientruleϕ = −q′(θ(ϕ))θ(ϕ). q(θ(ϕ)) The decentralized equilibrium isefficient. Equivalently, theefficient bargainingcondition arisesendogenously. 16Noteherethesurpluscreatedisirrelevantofthebargainingpower. Thereasonisthatoncetheworker enteredthebargainingprocess,thesurplusisfixed,thebargainingisjustdividingthissurplusbetweentwo parties. Becauseofthis,thesurplushasalreadybeenmaximizedbeforedeterminingtheshareofeachparty. Thus,ithasnoeffectonthesurplus. 63

AppendixC:DMPModelDetails LaborFlows. Giventheinitialdistributionofworkers,ms(0,a),andjob-findingratesf(x) i forallx,thesubsequentdistributionofworkersm(x)canbecalculatedassumingalawof largenumbers. Themassofindividualswithexperiencee ∈ [1,a]atagea ∈ [a,a −2]is R determinedby mE¯ (e,a+1) = (1−π (x)−π (x))×mE¯ (e−1,a)+f (cid:0) e,a,U ¯(cid:1) ×mU¯ (e,a) i EU EN i i i +f (cid:0) e,a,N ¯(cid:1) ×mN¯ (e,a); i i mU¯ (e,a+1) = (1−π (x)−f (cid:0) e,a,U ¯(cid:1) )×mU¯ (e,a)+π (x)×mN¯ (e,a) i UN i i NU i (56) +π (x)×mE¯ (e−1,a); EU i mN¯ (e,a+1) = (1−π (x)−f (cid:0) e,a,N ¯(cid:1) )×mN¯ (e,a)+π (x)×mU¯ (e,a) i NU i i UN i +πi (x)×mE¯ (e−1,a). EN i Theaboveequationsnesttheflowsforindividualswithoutexperiencewhenonesetse = 0 andmE¯ (0,a) = 0. i AppendixD:DataandDetailedCalibrationResults AppendixD.1: DescriptionofData WeusethebasicmonthlyCPSdatafrom1976to1980and2003to2007(CenterforEconomic andPolicyResearch,CenterforEconomicandPolicyResearch(CEPR)andFloodetal., 2020). Thedataincludebothfull-andpart-timeU.S.workers. Wedisaggregatethedata based on an individual’s gender (male or female) and education status (college or noncollege). Anindividualisassignedtothecollegegroupifshehascompletedatleastsome collegeandtothenon-collegegroup,ifherhighestlevelofcompletededucationishigh 64

school or less. We then calculate life-cycle trends of average wages, and employment, unemployment,andnon-participationratesforeachofthedescribeddemographicgroups. WerelyonthehourlywageratesobtainedfromtheCEPR(CenterforEconomicandPolicy Research (CEPR)), while the other data are obtained from the raw CPS data files from IPUMS(Floodetal.,2020). TheadvantageofusingtheCEPRwagedatainsteadoftheraw CPSdataisthattheCEPRadjuststherawCPSwagedatasuchthattheconstructedwage dataseriesareconsistentandcomparableovertimeandareespeciallysuitableforresearch uses.17 Wealsoestimatemonthly,age-specifictransitionprobabilitiesbetweenemployment(E ¯ ), unemployment (U ¯ ), and nonparticipation (N ¯ ) separately for each group, following the methodinChoietal.(2015). Inpractice,thetransitionprobabilityestimatesareweightedaverageflowsbetweenlabormarketstatesforeveryagewhencontrollingforbirthcohorts. Foragivencohortandsurveyyear,weobservethefractionofindividualsofagivenage that transfers from one labor market state to another. Denote this variable as π (a,c,t), ss′ wheress′ denotesthetransitionfromastatuss ∈ (cid:8) E ¯ ,U ¯ ,N ¯(cid:9)toastatuss′ ∈ (cid:8) E ¯′,U ¯′,N ¯′ (cid:9),a denotesanindividual’sage,cdenotesthecohort(thebirthyear)anindividualbelongsto, andtdenotesthesurveyyear. Weobtaintheestimatedtransitionprobabilitiesbyrunningseeminglyunrelatedregressions ofπ (a,c,t)againstagedummies. Thecoefficientforeachagedummyistheprobability ss′ thatatransitionhappensatagea. AlimitationoftheCPSdataisthatitdoesnotcontain 17Foradetaileddescription,pleaserefertotheCEPR-CPSdocumentationfoundathttps://ceprdata.org/cpsuniform-data-extracts/cps-basic-programs/. 65

a variable capturing the work experience of individuals. As a result, only average (over experience)transitionprobabilitiescanbeestimated. Wedenotetheseestimatedtransition probabilities,π (e,a,s,i) ≡ π (x),asπ (a,s,i). ss′ ss′ ss′ To remove high-frequency reversals of transitions between unemployment and nonparticipation,wefollowthemethodsuggestedbyElsby etal.(2015)called "deNUNification." Thekeyideaistocorrectforapossibleclassificationerrorofanindividual’slabormarket state: Anindividualwhomovesfromnonparticipationtounemploymentandbacktononparticipationwithinashortperiodoftimeislikelytobeanonparticipant—includingthese high-frequencytransitionsbetweenstatesmayleadtospurioustransitionestimates. The correctionmethodthusrecodesthehigh-frequencytransitions,NUN,asNNN.Thesame methodisappliedtohigh-frequencytransitionsfromunemploymenttonon-participation andback. The estimated flows between different labor market states are flow probabilities from employment to unemployment and to nonparticipation—π (a,s,i) and π (a,s,i), re- EU EN spectively;unemploymenttononparticipation,π (a,s,i);nonparticipationtounemploy- UN ment,π (a,s,i);andunemploymentandnonparticipationtoemployment—π (a,s,i) ≡ NU UE f (cid:0) e,a,U ¯(cid:1)andπ (a,s,i) ≡ f (cid:0) e,a,N ¯(cid:1),respectively. Astheperiodinourmodelcalibrai NE i tionwillbesettoaquarterinsteadofamonth,wecalculatequarterlytransitionprobability matrices,Λ (a,s,i),asΛ (a,s,i) = (Λ (a,s,i)∧3,whereΛ (a,s,i)equals Q Q M M   1−π (a,s,i)−π (a,s,i) π (a,s,i) π (a,s,i)  EU EN EU EN       π (a,s,i) 1−π (a,s,i)−π (a,s,i) π (a,s,i) .  UE UE UN UN      π (a,s,i) π (a,s,i) 1−π (a,s,i)−π (a,s,i) NE NU NE NU 66

Labormarkettightnessandlaborcompensationshare,1976–2016 FigureD.1. Note: PanelAincludesthequarterly,seasonallyadjustedlaborshareforallemployedpersonsinthenonfarm businesssector,andpanelBplotsthequarterlylabormarkettightnessusingthenumberofunemployed workersovertheageof16asthedenominator. Thedashedlinesplottherawseries,whilethepurplesolid linesplottheHP-filteredserieswithlambdasetto1,600. Bothfiguresalsoincludealineartrendwith95 percentconfidencebounds. Source: BureauofLaborStatistics;Authors’calculationsbasedonPetrosky-NadeauandZhang(2021)and IPUMS-CPS. AppendixD.2: StylizedFacts—Robustness Figure D.1 plots the labor share and the standard tightness rate (total vacancies/all unemployedpeopleover16years),andtableD.1showscorrelationcoefficientsbetweenthe laborshareandboththestandardandalternativetightnessrates. Theresultsconfirmthat thelaborshareand tightnessarepositivelycorrelated,althoughthecorrelation isweaker betweenthelaborshareandthestandardtightnessrate. 67

Correlationcoefficientsbetweenthelaborshareanddifferentmeasuresoflabor TableD.1. markettightness Standard Alternative1 θ .353∗∗∗ .502∗∗∗ θ .394∗∗∗ .520∗∗∗ −1 θ .432∗∗∗ .539∗∗∗ −2 θ .464∗∗∗ .557∗∗∗ −3 Note: ***p<0.001. Bothtightnessmeasuressharethesamenumerator—allvacanciesfromPetrosky- Nadeau and Zhang (2021). The standard measure of tightness (column 1) uses the number of all unemployed workers over the age of 16 as the denominator. Alternative 1 (column 2) uses thenumberofallunemployedworkersandnonparticipantsbetweentheagesof25and64asthe denominator. Tableshowscorrelationcoefficientsandrelatedp-valuesbetweenrawlaborshare (U.S.BureauofLaborStatistics,2022)andtightnessratesandtheirlags. Lagsarequarterlylags: forexample,-1representsatightnessseriesthatislaggedbyonequarter. Source: Authors’calculationsbasedondatafromBureauofLaborStatistics,Petrosky-Nadeauand Zhang(2021)andIPUMS-CPS. AppendixD.3: CalibrationAlgorithm Wesolvethemodelusingbackwardsinduction. Giventhesetvaluesofβ,γ¯,γ ,α,ρ,and R π (i,a),thecalibrationalgorithmtorecovery ,r (a),A (a),ψ (a),γ (e,a),κ¯ andbargaining ss′ i i i i i i weightsϕs(e,a)isthefollowing: i Step1: Makeareasonableguessofthebargainingweightsofworkersϕs(e,a)andvacancyi 68

postingcostsκ¯. i Step2: Atgivenbargainingweights,vacancy-postingcosts,andotherparametervalues, solve the model and use model solutions to reverse engineer the group-specific human capital parameters (y , r (a)) and matching efficiencies (A (a), ψ (a)) to fit the observed i i i i wage rate and job-finding rates. We obtain γ (e,a) by equalizing the wage rates for the i unemployedandnonparticipants. Step3: Weusegroup-specifictightnessrateswithProposition2(theHosioscondition)to updatetheguessofbargainingpowerandgroup-specifictightnesstoupdateguessesfor κ¯. i Step 4: We repeat steps 2 and step 3 until the bargaining power series converge and the tightnessrateshittheirtargets. AppendixD.4: CalibrationResults Thissectionincludestheremainingcalibrationtargetsandresults. 69

Life-cyclehourlywagesin1976–80and2003–07,bygenderandeducation FigureD.2. Note: PanelsAandBplotthelife-cyclewagesforworkerswithandwithoutacollegeeducation, respectively,bygender. Allwagesareshownrelativetothewagerateofnon-collegemalesatage 25in1976–80period. Source: Authors’calculationsbasedondatafromCEPR. 70

Life-cyclejob-findingratesin1976–80and2003–07,bygenderandeducation FigureD.3. Note: PanelsAandCplotthelife-cyclejob-findingratesfromunemploymentforworkerswithand withoutacollegeeducation,respectively,bygender. PanelsBandDplotthelife-cyclejob-finding rates from nonparticipation for workers with and without a college education, respectively, by gender. Source: Authors’estimationsbasedondatafromIPUMS-CPS. 71

Life-cyclejobseparationratesin1976–80and2003–07,bygenderandeducation FigureD.4. Note: PanelsAandBplotsthelife-cycleseparationratesintounemploymentforworkerswithand withoutacollegeeducation,respectively,bygender. PanelsCandDplotsthelife-cycleseparation ratesintononparticipationforworkerswithandwithoutacollegeeducation,respectively,bygender. Source: Authors’estimationsbasedondatafromIPUMS-CPS. 72

Life-cycle flowsbetweenunemploymentandnonparticipationin1976–80and FigureD.5. 2003–07,bygenderandeducation Note: Panels A and B plot the life-cycle flow rates from nonparticipation to unemployment for workerswithandwithoutacollegeeducation, respectively, bygender. PanelsCandDplotthe life-cycleflowratesfromunemploymenttononparticipationforworkerswithandwithoutacollege education,respectively,bygender. Source: Authors’estimationsbasedondatafromIPUMS-CPS. 73

The fraction of nonparticipants searching over the life cycle in 1976–80 and Figure D.6. 2003–07,bygenderandeducation Note: PanelsAandCshowsthefractionofnonparticipants—withandwithoutacollegeeducation, respectively—searchingoverthelifecycleintheCDmodel. PanelsBandDshowthesameresults intheCESmodel. Source: Authors’estimations. 74

Workers’life-cyclebargaining power in1976–80 and2003–07, by genderand Figure D.7. education Note: PanelsAandBplotthesimulatedlife-cyclebargainingpoweroffemaleandmaleworkers, respectively,byeducationgroup. Source: Authors’estimations. 75

AppendixD.5: ChangesintheDisaggregatedLaborShares TableD.2summarizesthemodel-generatedchangesinthelaborsharefordifferentgroups. First, we find that the labor share has declined for all groups in the CES model. Male workershavefacedthelargestdecline: Non-collegemalesexperienceda5.6percentdecline intheir laborshare, whilecollege-educatedmales experienceda4.2 percentdecline. The laborsharedecreasedby2.8percentforcollege-educatedfemalesandby2.1percentfor non-collegefemales. Consistentwiththeaggregateresults,thelaborshareshave declined lessintheCDmodel—orhaveevenincreased,asisthecaseforcollege-educatedwomen. Simulatedlaborcompensationshares: 1976–80and2003–07 TableD.2. Group Percentchange,CDmodel Percentchange,CESmodel Male,college -2.7 -4.2 Female,college 3.1 -2.8 Male,non-college -3.6 -5.6 Female,non-college .0 -2.1 Note: TableD.2presentsthesimulatedchangeinthelaborshareforeachgroupundertheCDand theCESmodels. Source: Authors’estimations. 76