Search Frictions, Labor Supply, and the Asymmetric Business Cycle

Board of Governors of the Federal Reserve System International Finance Discussion Papers ISSN 1073-2500 (Print) ISSN 2767-4509 (Online) Number 1355 August Search Frictions, Labor Supply, and the Asymmetric Business Cycle Domenico Ferraro and Giuseppe Fiori Please cite this paper as: Ferraro, Domenico and Giuseppe Fiori (2022). “Search Frictions, Labor Supply, and the Asymmetric Business Cycle ,” International Finance Discussion Papers 1355. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/IFDP.2022.1355. NOTE: International Finance Discussion Papers (IFDPs) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the International Finance Discussion Papers Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at www.ssrn.com.

Search Frictions, Labor Supply, and the Asymmetric Business Cycle* Domenico Ferraro Giuseppe Fiori Arizona State University Federal Reserve Board August 3, 2022 Abstract We develop a business cycle model with search frictions in the labor market and a labor supply decision along the extensive margin that yields cyclical asymmetry betweenpeaksandtroughsoftheunemploymentrateandsymmetricfluctuationsofthe laborforceparticipationrateasintheU.S.data. Wecalibratethemodelandfindthat cyclical changes in the extent of search frictions are solely responsible for the peaktroughasymmetry. Participationdecisionsdonotgenerateasymmetrybutcontribute tothefluctuationsinsearchfrictionsbychangingthesizeandcompositionofthepool of job seekers, which in turn affects the tightness ratio and thereby slack in the labor market. The participation rate would be counterfactually asymmetric absent labor supplyresponsestoshocks. JELClassification: E24;E32;J63;J64. Keywords: Asymmetricbusinesscycles;Laborsupply;Searchfrictions;Employment; Unemploymentrate;Laborforceparticipationrate. *Ferraro: Department of Economics, W.P. Carey School of Business, Arizona State University, PO Box 879801,Tempe,AZ85287,UnitedStates(email: domenico.ferraro@asu.edu);Fiori: FederalReserveBoard, Division of International Finance, 20th and C Sts., NW, Washington, DC 20551, United States (email: giuseppe.fiori@frb.gov). First version: February 12, 2018. We thank Steven Davis, Maximiliano Dvorkin, NezihGuner,KyleHerkenhoff,CosminIlut,LoukasKarabarbounis,KurtMitman,ToshihikoMukoyama, PietroPeretto,José-VíctorRíos-Rull,RichardRogerson,RaülSantaeulàlia-Llopis,ToddSchoellman,Henry Siu, Nora Traum, Venky Venkateswaren, Ludo Visschers, Yaniv Yedid-Levi, and several conference and seminaraudiencesforhelpfulcommentsandsuggestions. Disclaimer:Theviewsexpressedinthispaperaresolelytheresponsibilityoftheauthorsandshouldnotbe interpretedasreflectingtheviewsoftheBoardofGovernorsoftheFederalReserveSystemorofanyother personassociatedwiththeFederalReserveSystem.

1 Introduction IntheUnitedStates,cyclicalfluctuationsintheemployment-to-populationratiodisplaya strikingasymmetry: deviationsbelowtrend(“troughs”)arelargerthandeviationsabove trend(“peaks”). Thisasymmetrybetweenpeaksandtroughsproducessignificanthigherorder moments, such as negative skewness in the distribution of the employment-topopulation ratio in deviations from trend. Sichel (1993) first refers to this phenomenon as “deepness,” which since then has become one of the stylized facts of the U.S. business cycle(McKayandReis,2008). The large and growing literature on the topic studies this phenomenon through the lensofaDiamond-Mortensen-Pissarides(DMP)modelinwhichsearchfrictionsgenerate unemployment(Diamond,1982;Mortensen,1982;Pissarides,1985). Thisapproachusesa two-staterepresentationofthelabormarket,whichabstractsfromparticipationdecisions altogether (Andolfatto, 1997; Abbritti and Fahr, 2013; Dupraz, Nakamura and Steinsson, 2020; Ferraro, 2017, 2018; Hairault, Langot and Osotimehin, 2010; Petrosky-Nadeau and Zhang,2013,2017;Pizzinelli,TheodoridisandZanetti,2020). In contrast, labor force participation decisions take center stage in this paper. We begin with documenting a new, overlooked fact: cyclical fluctuations in the labor force participation rate are symmetric around the trend, implying that deepness in the U.S. employment-to-populationratioisaccountedforsolelybytheunemploymentrate. Given these observations, one might conclude that participation decisions, and thus worker flows in and out of the labor force, are inconsequential for the study of cyclical asymmetry. There are, however, at least two reasons to be skeptical about this view; one is empirical, and the other is theoretical. First, worker flows in and out of the labor force account for around one-third of the cyclical volatility in the unemployment rate (Elsby, Hobijn and S¸ahin, 2015). Also, transition probabilities from nonparticipation to unemployment andfromunemploymenttononparticipationarecountercyclicalandpro-cyclical,respectively. Thus, during recessions, a larger share of individuals who would have left the labor force remains unemployed, and a larger share of nonparticipants who would have stayed out of the labor force enters the unemployment pool. These patterns exacerbate congestioninthelabormarketduringrecessions,contributingtogeneratingdeepness. Second, in the context of a DMP model extended to allow for an active participation margin,thelackofcyclicalasymmetryintheparticipationrateissomewhatpuzzling. Ina three-state model, an important object impinging on the decision to enter the labor force 1

is the probability of finding a job. In the data, cyclical fluctuations in such probability display significant deepness (Ferraro, 2018). Thus, a model that successfully reproduces deepness in the probability of finding a job would naturally generate a sharp fall in the individuals’willingnesstoenterthelaborforceduringrecessions. Atthesametime,such labor supply decisions change the size and the composition of the pool of job seekers competingforjobs,affectingvacancypostingandtheextentofslackinthelabormarket. To quantify these mechanisms, we develop a business cycle model that reconciles the asymmetry in the unemployment rate with the symmetric fluctuations in the labor force participation rate. The model combines search frictions with vacancy posting, as in the DMP model, with a labor supply decision with indivisible labor, as in Hansen (1985) and Rogerson (1988). In the model, workers are heterogeneous in home productivity, which changes over time because of persistent idiosyncratic shocks. Aggregate market productivityshocksgeneratebusinesscycles. Themodelembodiestwopropagationmechanismsofproductivityshocks: (i)shiftsin theindividuals’willingnesstoworkand(ii)fluctuationsintheextentoffrictions,i.e.,the speed at which job seekers meet employers posting vacancies, which depends on market tightness (the ratio of vacancies to job seekers). Unlike in the DMP model, in our setting, the market tightness ratio is determined in equilibrium by posted job vacancies (labor demand) jointly with participation decisions (desired labor supply). Actual and desired laborsupplydifferbecauseofsearchfrictions. First, a separation and a search cutoff on home productivity determine whether an individual is out of, attached, or not attached to the labor force. An individual attached to the labor force is either employed or unemployed and searching for a job, whereas a non-attachedindividualparticipatesinsofarasheorsheisemployed. Theresponseofthe twocutoffstoproductivityshocks,keepingthetightnessratioandsotheleveloffrictions fixed, is what we refer to as the “labor supply channel,” which captures the endogenous, yetpartialequilibrium,adjustmentofindividuallaborsupplytoproductivityshocks. Second, the market tightness ratio falls in response to a negative productivity shock, generatingslackinthelabormarket. Werefertothismechanismasthe“slacknesschannel,” whichcapturestheequilibriumfeedbackeffectbetweenvacancypostingandindividuals’ participation decisions. Notably, equilibrium vacancies are determined based on the size and composition of the pool of job seekers: the participation margin directly contributes to the cyclical movements in labor market frictions. This channel is absent in the DMP model,whereallindividualsareparticipantsatalltimes. 2

We calibrate the model to U.S. data and find that it accounts reasonably well for the deepness of the unemployment rate and the lack thereof in the participation rate. In addition, the model captures salient features of the cyclical movements in gross worker flows, a well-known challenge for existing three-state models of the labor market (see, e.g.,Shimer,2013;Tripier,2004;Veracierto,2008). To study the role of labor supply vis-à-vis search frictions, we propose a structural quantitative accounting exercise. Specifically, we generate two counterfactual time series for the unemployment rate and the labor force participation rate, keeping the same realization of productivity shocks. In the first counterfactual, we drop the indifference conditions determining the separation and search cutoffs, fix the two cutoffs on home productivity at their steady-state values, and let the tightness ratio vary in response to shocks as implied by the free-entry condition. In the second counterfactual, we drop the free-entry condition instead, fix the tightness ratio at its steady-state value, and let the cutoffsvary.1 We find that the slackness channel— and so fluctuations in the extent of frictions— is the key driving force of deepness in the employment rate, and that the participation margin per se does not generate cyclical asymmetry. In the model, the matching process between job seekers and vacancies is subject to congestion due to random search, implying that the probability that a job-seeker meets an employer falls more in response to adverse shocks than it rises in response to positive shocks. In other words, if the labor supply channel were the only driving force of fluctuations, we would observe symmetric fluctuations in employment, as in the frictionless real business cycle (RBC) model. To be sure, this is not to say that the labor force participation margin is inconsequential for cyclical asymmetry. On the contrary, in a three-state model like ours, individuals’ flows in and out of the labor force depend on market tightness, and they all contribute to the stocks of employment, unemployment, and nonparticipation— and so the mass of job seekers competing for jobs. During recessions, in the model, as in the data, unemployed individuals are less likely to drop out of the labor force, and individuals out of the labor force are more likely to enter the labor force as unemployed. Accounting for the cyclicality of these gross worker flows is critical for the model to generate the cyclical volatility andasymmetryinthedata. 1Such a decomposition cannot be implemented solely with data on labor market stocks and average transition probabilities as in, say, Elsby, Hobijn and S¸ahin (2015). The reason is that observed transition probabilitiesareequilibriumobjectsjointlydeterminedbytheindividuals’willingnesstoworkforagiven level of market tightness and the probability of finding a job, which in turn depends on the collection of individuals’participationdecisionsandjobvacancies. 3

Furthermore,absentthelaborsupplychannel,thelaborforceparticipationratewould be markedly asymmetric, mirroring the cyclical asymmetry in the probability of finding a job, which is at odds with the data. The lack of asymmetry in the participation rate is not hardwired into the model; rather, it is the result of equilibrium forces inherent to the jointdeterminationofmarkettightnessandthecutoffsonhomeproductivity. The rest of the paper is organized as follows. In Section 2, we discuss the related literature. Section 3 briefly presents the observations that motivate the paper. Section 4 presents the model. In Sections 5 and 6, we take the model to the data and study its quantitative properties. Finally, Section 7 concludes. Appendices A, B, and C contain datasources,derivations,andadditionalresults. 2 Related Literature This paper contributes to our understanding of business cycle asymmetry. In the RBC tradition,HansenandPrescott(2005)explainthenegativeskewnessinU.S.markethours worked (in deviations from trend) in the context of a neoclassical growth model with occasionally-bindingcapacityconstraints. VanNieuwerburghandVeldkamp(2006)study asymmetry in output growth rates using an RBC model augmented with learning about technology shocks. At the end of a boom, agents have accurate estimates of the state of technology so that a negative productivity shock prompts abrupt actions, leading to a sharp fall in investment and hours. Jovanovic (2006) explains the negative skewness in output growth rates through adopting technologies of uncertain skill requirements. Quadraticcostsinskillmismatchimplythatagoodmatchraisesoutputbylessthanabad match reduces it, such that output growth rates are negatively skewed. McKay and Reis (2008)showthatamodelwithasymmetricadjustmentcostsinemploymentandachoice of when to scrap old technologies reconciles the brevity and violence of the contractions inemploymentwiththenearlysymmetricfluctuationsinoutput. Ordonez(2013)singles out financial frictions as an explanation for the observation that cyclical asymmetry is morepronouncedincountrieswithlessdevelopedfinancialsystems. Usingasearch-theoreticmodel,Andolfatto(1997)arguesthatasymmetricfluctuations in the job destruction rate can qualitatively account for the fast rises and slow declines in the U.S. unemployment rate. Petrosky-Nadeau and Zhang (2013) argue that a DMP model,calibratedtomatchthecyclicalvolatilityintheunemploymentrate,producesthe asymmetry between peaks and troughs in the data. Building on this result, Petrosky- 4

NadeauandZhang(2017)showthatafirst-orderapproximationoftheDMPequilibrium dynamics neglects nonlinearities in the propagation of shocks. Ferraro (2018) develops a search-and-matching model with heterogeneous workers in skills that reconciles the cyclical asymmetry in the unemployment rate with the nearly symmetric fluctuations in output. Abbritti and Fahr (2013) and Dupraz, Nakamura and Steinsson (2020) study cyclical asymmetry in a DMP model with downward nominal wage rigidity. This body ofworkabstractsfromparticipationdecisionsaltogether. Ourworkalsorelatestotheliteraturethatstudiestheaggregateimplicationsofthreestate models of the labor market. The bulk of this body of work considers steady-state outcomes only (Garibaldi and Wasmer, 2005; Krusell et al., 2008, 2010, 2011; Pries and Rogerson, 2009). However, a few papers have confronted these models with the cyclical properties of labor market outcomes, too (Cairó, Fujita and Morales-Jiménez, 2019; Shimer, 2013; Tripier, 2004; Veracierto, 2008). Only recently, Krusell et al. (2017) show that a model with idiosyncratic risk, incomplete markets, and labor market frictions can account for the cyclical volatility and co-movement of U.S. gross worker flows. In their setting, job-finding rates are exogenous stochastic processes. By contrast, in our setting, job-finding rates are endogenously determined as an equilibrium outcome, based on the individuals’participationdecisionsandthefree-entryconditionforvacancyposting. This equilibrium property is instrumental in quantifying the role of search frictions as the sourceofcyclicalasymmetry. Ourcontributiontotheliteratureistwofold. First,weformulateandquantifyathreestate model that accounts for the deepness asymmetry in the unemployment rate and the symmetric fluctuations in the labor force participation rate, alongside key features of gross worker flows. Second, we quantify the importance of labor supply vis-à-vis search frictionsforthecyclicalvolatilityandasymmetryintheemployment-to-populationratio, aquestionthatpreviousstudieshavenotaddressed. 3 Motivating Facts In this section, we detail the empirical observations that motivate our work. Based on Sichel (1993), we measure cyclical asymmetry with the third standardized central mo- 5

ment,orskewness,ofthecyclicalcomponent xˆ ofthetimeseries x : t t (cid:104) (cid:105) E (xˆ −E[xˆ ])3 t t skew(xˆ ) = , t σ3 xˆ E where denotesthemathematicalexpectationoperatorand σ thestandarddeviationof xˆ thecyclicalcomponent xˆ expressedinpercentdeviationsfromtrend. Asiscustomaryin t the literature, fluctuations at the business cycle frequency are identified as occurring between 2 and 32 quarters. Also, since there is no firm consensus on the filtering approach, we report skewness statistics based on two alternative bandpass methods due to Baxter andKing(1999)andChristianoandFitzgerald(2003),aswellastheprocedureinHodrick and Prescott (1997). To test for asymmetry against the null hypothesis of symmetry, we usethetestdevelopedbyBaiandNg(2005).2 Table1reportsskewnessstatistics,withassociated p-values,fortheU.S.employmentto-population ratio, the employment rate (one minus the unemployment rate), and the laborforceparticipationrateinthepostwarperiod1948-2016. Tointerprettheresults,we considerthefollowingdecompositionoftheemployment-to-populationratio: (cid:18) (cid:19) (cid:18) (cid:19) emp unemp emp+unemp = 1− × . pop emp+unemp pop (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) employmentrate participationrate This decomposition shows that employment as a fraction of the working-age population equals the employment rate (fraction of employed workers in the labor force, one minus the unemployment rate) times the participation rate (fraction of the population in the labor force). Hence, in an accounting sense, cyclical asymmetry in the employment-topopulationratiomayresultfromtheunemploymentrate,theparticipationrate,orboth. The results in Table 1 establish that cyclical fluctuations in labor force participation are virtually symmetric, which leaves the unemployment rate as the key driving force of asymmetry in the employment-to-population ratio. Specifically, the cyclical component of the employment-to-population ratio displays significant negative skewness. Note that this negative skewness remains significant and of similar magnitude also in the pre-1980 period. Thus, cyclical asymmetry is not driven by the so-called jobless recoveries of the 1990s, the 2000s, or the Great Recession of 2007-2009; rather, it is a systematic feature of 2SeeAppendixAfordetailsondatasources. 6

theU.S.labormarketovertheentirepost-warperiod. Table1:SkewnessintheU.S.LaborMarket Skewness Baxter- Christiano- Hodrick- King Fitzgerald Prescott A.Sampleperiod:1948:Q1-2016:Q4 Employment-to-populationratio −0.44 −0.29 −0.32 (0.02) (0.08) (0.03) Employmentrate −0.85 −0.51 −0.70 (0.00) (0.00) (0.00) Participationrate 0.09 0.05 0.05 (0.38) (0.44) (0.38) B.Sampleperiod:1948:Q1-1980:Q4 Employment-to-populationratio −0.42 −0.34 −0.43 (0.03) (0.05) (0.02) Employmentrate −0.80 −0.62 −0.76 (0.00) (0.01) (0.00) Participationrate 0.12 0.03 −0.06 (0.34) (0.45) (0.41) Notes: ForBaxter-KingandChristiano-Fitzgerald,weconsiderfrequenciesbetween2 and32quarters. TheorderofthemovingaveragefortheBaxter-Kingfilterissetto8 quarters. ThesmoothingparameterfortheHodrick-Prescottfilteris1,600. Variables areexpressedinlog-deviationsfromtrend.P-values(one-sidedtest)inparentheses. TounderstandwhatarethemechanismsthatshapethecyclicalasymmetryintheU.S. unemploymentrateandthelackthereofintheparticipationrate,webuildaquantitative modelthatincorporatesemployment,unemployment,andnonparticipationanduseitas alaboratorytocarryoutcounterfactualanalysis. Weturntotheseissuesnext. 4 Model 4.1 Environment Time is discrete and continues forever, indexed by t = 0,1,2,..., ∞ . The economy is inhabited by two types of agents: individuals and employers. Both agents are infinitely lived, are risk neutral, and discount future values at the same rate β ∈ (0,1). The mass 7

ofindividualsisnormalizedtoone. Anindividualisendowedwithoneunitoftimethat canbeallocatedtothreeuses: marketwork,jobsearch,andnonmarketwork(e.g.,leisure andhomeproduction). Marketworkandjobsearcharemutuallyexclusiveactivities. An employer is either matched with an individual and producing output or unmatched and postingjobvacancies. Themassofemployersisdeterminedinfree-entryequilibrium. Preferences and budget constraints. We assume that an individual has linear utility over consumption, c , and maximizes E ∑∞ βtc , subject to the following flow budget t 0 t=0 t constraints: consumption is equal to the wage, c = w , if he or she is employed; cont t sumption is equal to unemployment insurance (UI) benefits, c = b, if the individual is t unemployed; and consumption is equal to home production, c = yh, if he or she is a t t nonparticipant, where yh depends on idiosyncratic home productivity, x , and aggregate t t marketproductivity, y ,inawaythatwemakepreciselater. t Heterogeneityandhomeproductivity. AsinGaribaldiandWasmer(2005),individuals are heterogeneous in home productivity, x .3 The value of x may change over time with t t probability λ. In that event, the new value x for the next period is drawn from a t+1 probabilitydistributionfunction f(x),takentobelognormalwithparameters µ and σ , x x (cid:2) (cid:3) and defined over the bounded support x ∈ xmin,xmax . With probability 1−λ, home t productivity maintains its current value into the next period. Hence, at the individual level, home productivity is persistent. But conditional on a switch, its current value does notaffectitsnext-periodrealization.4 Aggregateproductivityshock. Productionrequiresamatchbetweenoneemployerand oneindividual. Whenajob-seekerandanemployermeetandagreetocreateamatch(or, equivalently, a job), they produce output, y , which evolves stochastically over time act cordingtoafirst-orderautoregressive(AR(1))processinlogs: log(y ) = (1−ρ )ln(y¯)+ t+1 y iid ρ log(y )+σ ϵ , where y¯ is the unconditional mean of output and ϵ ∼ N(0,1) are iny t y t+1 t novations to the (log) output of a job. The parameters ρ and σ control the persistence y y 3WeextendtheworkofGaribaldiandWasmer(2005)alongtwoimportantdimensions.First,weamend their model to allow for worker flows from nonparticipation to employment, which are both large and highly volatile in the data (see Krusell et al., 2017). This modification implies that the composition of thepoolofjobseekerscontributestodetermininglabormarkettightness. Second, wefocusontransition dynamicstriggeredbybusinesscycleshocks,ratherthanjustfocusingonsteady-stateoutcomes. 4Everythingelsebeingequal, persistenceinhomeproductivityasgovernedby λ allowsthemodelto generaterealisticpersistenceinthetransitionprobabilitiesinandoutofthelaborforce. 8

andvolatilityoftheinnovations, ϵ ,respectively.5 t Wagedetermination. AsinShimer(2004)andmanyothers,weassumeanadhocwage η rule relating the wage to labor productivity: w = w¯y , where w¯ is a constant and the t t parameterη governsthecyclicalsensitivityofthewagetolaborproductivity. Thebenefit ofthisparsimoniousspecificationistwofold. First,itconsiderablysimplifiesthesolution of the model. As the firm’s value of a job is independent of home productivity, only the share of unemployed and nonparticipant individuals (instead of the full cross-sectional distribution) is relevant for vacancy posting.6 Second, depending on the value of η, the wageruleaccommodatesdifferentdegreesofwageflexibility. Meeting technology. The matching process between searchers and employers posting vacancies is subject to a search friction. We assume a constant-returns-to-scale meeting technology: m = χsεv1−ε, where m denotes the number of meetings between searchers t t t t and vacancies and s and v are the mass of searchers and vacancies, respectively. The t t probabilitythatasearchermeetsavacancyis p(θ ) = θ1−ε,whereθ ≡ v /s isthemarket t t t t t tightnessratio. Similarly,theprobabilitythatavacancymeetsasearcheris q(θ ) = θ −ε. t t In our setting, unemployed individuals compete with a subset of nonparticipants for jobs. Unemployedindividualsareclassifiedas“active”searchers,collectUIbenefits,and meet a vacancy with probability p(θ ). Nonparticipants who are randomly drawn in the t pool of “passive” searchers, enjoy home production, do not collect UI benefits, and meet a vacancy with probability ϕp(θ ), where ϕ ∈ (0,1) is an exogenous constant.7 The main t advantageofhavinganotionofpassivesearchersinthemodelisthatitallowsforworker flows from nonparticipation to employment, that are both large and highly volatile over the business cycle (see Krusell et al., 2017).8 Note also that while ϕ is exogenous and 5Altug˘,AshleyandPatterson(1999)findnoevidencefornonlinearityintotalfactorproductivityusing aggregate-levelU.S.data.Ilut,KehrigandSchneider(2017)confirmthisfindinginestablishment-leveldata. 6WithNashbargaining,thevalueofajobwouldnaturallydependonhomeproductivityviathewage. Thehigherthevalueofhomeproductivity,thehighertheopportunitycostofemployment,thehigherthe wagewouldhavetobeforasearchertoacceptajob. 7OurclassificationofactivesearchersasunemployedandparticipantsandofpassivesearchersasnonparticipantsisconsistentwiththeapproachoftheBureauofLaborStatistics(seeJonesandRiddell,1999, forfurtherdiscussion). 8Weacknowledgethat,inthedata,someoftheobservedflowsfromnonparticipationtoemployment may be due to time aggregation. As labor market data are sampled at the monthly frequency, measured flows from nonparticipation to employment may be due to unmeasured flows from nonparticipation to unemployment and from unemployment to employment insofar as they occur within the month. Here, we follow Krusell et al. (2017) and introduce a constant exogenous probability of becoming a (passive) searcher. Nonetheless, the flows from nonparticipation to employment remain endogenous in the sense 9

constant, the decision of accepting a job offer upon meeting an employer, and the choice of whether to become an active searcher next period or to remain out of the labor force continuetobeendogenous. Timing of events. Within the period, events unfold as follows. At the beginning of the period, the aggregate (y ) and idiosyncratic (x ) states are realized. After these events, t t the period consists of two stages. In the first stage, separation, participation, and search decisions are made simultaneously. In the second stage, output is produced and wages arepaid. Inoursetting,thereisadistinctionbetween(i)ameetingbetweenavacancyand a job seeker and (ii) the creation of a job. Only if profitable for both parties, a meeting is convertedintoajob. Themodelusesthe“instantaneoushiring”view,inwhichnewhires begin working right away rather than with a one-period delay. As discussed in Davis, Faberman and Haltiwanger (2006), this timing describes the U.S. labor market flows at a quarterlyfrequency. 4.2 Individual Agents’ Problems Weformulatetheindividualagents’problemsinrecursiveformandwritevaluefunctions at the production stage when idiosyncratic and aggregate states have been realized and the agents’ current decisions of continuing, destroying, or creating a match have been made. 4.2.1 Individuals Atthebeginningofeachperiod,anemployeedecideswhethertoremaininthematchand receive the wage or separate. Conditional on separating, the individual has the option to become either unemployed or a nonparticipant, thus dropping out of the labor force. Similarly, a non-employed individual has the choice to search for a job or stay out of the laborforce. Again,conditionalonbeingoutofthelaborforce,anindividualcannotmeet a job vacancy unless he or she receives a random job offer. In that event, the individual chooses whether to accept the job offer or remain out of the labor force. All flows across thethreelabormarketstatesarethusendogenous. that the individuals optimally decide whether to accept a job, or remain out of the labor force, given the realizationsofthestatevariables. 10

Attached employed. At the production stage, the value of employment depends on whether the individual is attached or non-attached to the labor force. Let (cid:0) xv,x q(cid:1) denote t t the search and separation cutoffs, respectively, whose determination we describe later. Thevalueofemploymentforanindividualattachedtothelaborforceis Wa = w (instantaneousreturn) t t +β E (cid:8) (1−δ) (cid:2) 1−λ+λF(xv ) (cid:3)(cid:9) Wa (continuingasattachedemployed) t t+1 t+1 +β E (cid:8) δ (cid:2) 1−λ+λF(xv ) (cid:3) p(θ ) (cid:9) Wa (restartingasattachedemployed) t t+1 t+1 t+1 (cid:40) (cid:41) (cid:90) x q +β E [(1−δ)λ+δλϕp(θ )] t+1 Wna (x)dF(x) t t+1 t+1 xv t+1 (continuingorre-startingasnon-attachedemployed) +β E (cid:8) δ (cid:2) 1−λ+λF(xv ) (cid:3) (1− p(θ ))U (cid:9) (becomingunemployed) t t+1 t+1 t+1 (cid:40) (cid:41) (cid:90) x q +β E [δλ(1−ϕp(θ ))] t+1 Ha (x)dF(x) t t+1 t+1 xv t+1 (becominganattachednonparticipant) (cid:40) (cid:41) (cid:90) xmax +β E λ Hna (x)dF(x) , (becominganon-attachednonparticipant) t q t+1 x t+1 where (Wna(x),U ,Ha,Hna) are the value of employment for an individual not attached t t t t to the labor force, the value of unemployment, and the values of nonparticipation for an attachedandnon-attachedindividual,respectively. 11

Non-attachedemployed. Thevalueofemploymentforanindividualnotattachedtothe laborforceis Wna(x) = w (instantaneousreturn) t t +β E (cid:8)(cid:2) (1−δ)λF(xv )+δλF(xv )p(θ ) (cid:3) Wa (cid:9) t t+1 t+1 t+1 t+1 (continuingorrestartingasattachedemployed) +β E (cid:8) [(1−δ)(1−λ)+δ(1−λ)ϕp(θ )]Wna (x) (cid:9) t t+1 t+1 (continuingorrestartingasnon-attachedemployedw/same x) (cid:40) (cid:41) (cid:90) x q +β E [(1−δ)λ+δλϕp(θ )] t+1 Wna (x)f(x)dx t t+1 t+1 xv t+1 (continuingorrestartingasnon-attachedemployedw/new x) +β E (cid:8)(cid:2) δλF(xv )(1− p(θ )) (cid:3) U (cid:9) (becomingunemployed) t t+1 t+1 t+1 (cid:40) (cid:41) (cid:90) x q +β E [δλ(1−ϕp(θ ))+δ(1−λ)(1−ϕp(θ ))] t+1 Ha (x)dF(x) t t+1 t+1 t+1 xv t+1 (becominganattachednonparticipant) (cid:40) (cid:41) (cid:90) xmax +β E λ Hna (x)dF(x) . (becominganon-attachednonparticipant) t q t+1 x t+1 Unemployed. Thevalueofunemploymentis U = b (instantaneousreturn) t +β E (cid:8)(cid:2) 1−λ(1−F(xv )) (cid:3) p(θ )Wa (cid:9) (startingasattachedemployed) t t+1 t+1 t+1 (cid:40) (cid:41) (cid:90) x q +β E λϕp t+1 Wna (x)dF(x) (startingasnon-attachedemployed) t t+1 t+1 xv t+1 +β E (cid:8) δ[1−λ(1−F(xv ))](1− p )U (cid:9) (continuingasunemployed) t t+1 t+1 t+1 (cid:40) (cid:41) (cid:90) x q +β E λ(1−ϕp ) t+1 Ha (x)dF(x) (becominganattachednonparticipant) t t+1 t+1 xv t+1 (cid:40) (cid:41) (cid:90) xmax +β E λ Hna (x)f(x)dx . (becominganon-attachednonparticipant) t q t+1 x t+1 12

Attachednonparticipant. Thevalueofnonparticipationforanattachedindividualis Ha(x) = yh (instantaneousreturn) t t +β E (cid:8) λF(xv )p(θ )Wa (cid:9) (becomingattachedemployed) t t+1 t+1 t+1 +β E (cid:8) (1−λ)ϕp(θ )Wna (x) (cid:9) (becomingnon-attachedemployedw/same x) t t+1 t+1 (cid:40) (cid:41) (cid:90) x q +β E λϕp(θ ) t+1 Wna (x)dF(x) t t+1 t+1 xv t+1 (becomingnon-attachedemployedw/new x) +β E (cid:8) λF(xv )(1− p(θ ))U (cid:9) (becomingunemployed) t t+1 t+1 t+1 +β E (cid:8) (1−λ)(1−ϕp(θ ))Ha (x) (cid:9) t t+1 t+1 (remaininganattachednonparticipantw/same x) (cid:40) (cid:41) (cid:90) x q +β E λ(1−ϕp(θ )) t+1 Ha (x)dF(x) t t+1 t+1 xv t+1 (remaininganattachednonparticipantw/new x) (cid:40) (cid:41) (cid:90) xmax +β E λ Hna (x)dF(x) , (becominganon-attachednonparticipant) t q t+1 x t+1 wherethehomeproductiontechnologyisspecifiedas yh = x y /y¯.9 t t t 9Wescalehomeproductionbyy¯sothat,inthedeterministicsteadystateofthemodel,yh = x . t t 13

Non-attachednonparticipant. Thevalueofnonparticipationforanon-attachedindividualis Hna(x) = yh (instantaneousreturn) t t +β E (cid:8) λF(xv )p(θ )Wa (cid:9) (becomingattachedemployed) t t+1 t+1 t+1 +β E (cid:8) (1−λ)ϕp(θ )Wna (x) (cid:9) (becomingnon-attachedemployedw/same x) t t+1 t+1 (cid:40) (cid:41) (cid:90) x q +β E λϕp(θ ) t+1 Wna (x)dF(x) t t+1 t+1 xv t+1 (becomingnon-attachedemployedw/new x) +β E (cid:8)(cid:2) λF(xv )(1− p(θ )) (cid:3) U (cid:9) (becomingunemployed) t t+1 t+1 t+1 +β E (cid:8) (1−λ)(1−ϕp(θ ))Hna (x) (cid:9) t t+1 t+1 (remaininganon-attachednonparticipantw/same x) (cid:40) (cid:41) (cid:90) x q +β E λ(1−ϕp(θ )) t+1 Ha (x)dF(x) t t+1 t+1 xv t+1 (becominganattachednonparticipantw/new x) (cid:40) (cid:41) (cid:90) xmax +β E λ Hna (x)dF(x) . t q t+1 x t+1 (remaininganon-attachednonparticipantw/new x) 4.2.2 Employers Fromtheemployer’sperspective,thevalueofbeinginanemploymentrelationship(value of a job, for short) is always positive. This fact implies that employers never initiate job destruction. As argued earlier, however, individuals initiate job destruction when the valueofnonparticipationexceedsthevalueofemployment. Valueofajob. Attheproductionstage,thevalueofajobis J = y −w +β E [(1−d )J +d V ], (1) t t t t t+1 t+1 t+1 t+1 wheretheindividual’sdecisionofdestroyingthematchissubsumedintheindicator (cid:40) δ if H < W t t d = , (2) t 1 if H ≥ W t t 14

where δ ∈ (0,1) isanexogenousrateofjobdestruction. Valueofavacancy. Thevalueofapostedvacancyis V = −k+q(θ )Ω J +(1−q(θ ))E V , (3) t t t t t t t+1 Ω where k istheper-periodunitcostofopeningandmaintainingavacancy. Inaddition, t in(3)isanequilibriumobjectthatcaptures“selection”intothepoolofjobsearchers(akin toaninverseMillsratio). Itmeasurestheshareofsearchersthatacceptsajoboffer: u +ϕna Ω ≡ t t , (4) t u +ϕ(na +nna) t t t where u is the stock of unemployed, na is the stock of nonparticipants attached to the t t labor force, and nna is the stock of nonparticipants non-attached.10 Active and passive t searchers (or unemployed and nonparticipant attached, respectively) accept job offers, while nonparticipant non-attached decline them. So, insofar as ϕ > 0, variation in u , na, t t and nna, induced by productivity shocks, leads to cyclical variation in Ω , which in turn t t affectsvacancypostingandtherebymarkettightness. Free-entrycondition. AsinPissarides(1985)andmanyotherstudiesafterthat,employerspostjobvacanciesuntilitisprofitabletodoso,whichyieldsthatthecostofpostinga vacancy equals its expected benefit at all times such that V = 0 for all realizations of the t aggregate shock y . As a result, the market tightness ratio, θ , is determined according to t t aforward-lookingequation: (cid:20) (cid:21) k k = y −w +β E (1−d ) . (5) q(θ )Ω t t t q(θ )Ω t+1 t t t+1 t+1 As in the standard DMP model, the probability that a vacancy is filled depends on the probability of meeting a searcher q(θ ); however, unlike in DMP, in our setting the jobt filling probability depends on the fraction of searchers who are willing to work and so accept a job offer as captured by Ω . Note that if ϕ = 0, then Ω = 1 at all times, which t t shutsdownthecompositionchannelaltogether,nestingthestandardfree-entrycondition inDMPmodels. 10All the stocks are computed at the beginning of the period after aggregate and idiosyncratic shocks havebeenrealizedbutbeforeoffersarereceivedandmatchesformed. 15

4.3 Equilibrium The equilibrium of the model is characterized by the solution to the value functions for individuals, (Wa,Wna(x),U ,Ha(x),Hna(x)), together with the free-entry condition (5), t t t t t whichyieldsthemarkettightnessratio,θ ,separationandsearchcutoffs, (cid:0) x q ,xv (cid:1) ,andthe t t t labor market stocks, (u ,na,nna). Unlike in the standard DMP model, one needs to solve t t t for the individual decision rules and the market tightness ratio jointly with the stocks of unemployment and nonparticipation. This property obtains because vacancy posting dependsonthestocksofunemployedandofattachedandnon-attachednonparticipants. Separationandsearchcutoff. Sincethevalueofnonparticipationisincreasingin x,itis possible to determine two threshold values that uniquely identify the separation cutoff, xq,andthesearchcutoff, xv ≤ xq. Anindividualseparatesfromamatchwhenthevalueofnonparticipationexceedsthe valueofworking. Theindifferenceconditionforseparation, Wna(x q ,y ) = Ha(x q ,y ) = Hna(x q ,y ), (6) t t t t t t q implicitly defines the cutoff value x . Intuitively, the worker compares the utility cost of t market work with the benefit of market work, which equals the wage plus the expected discounted value of continuing the employment relationship. Given that the value of nonparticipationisincreasingin x ,jobseparationsatisfiesthereservationproperty. That t q is, there exists a unique separation cutoff, x , so that all matches with individuals whose t q value of nonmarket work is x ≥ x are endogenously destroyed. Hence, aggregate t t shocksinducejobdestruction. Theindifferenceconditionforsearch, U(xv,y ) = H(xv,y ), (7) t t t t implicitly defines the cutoff value xv. The marginal individual weighs the utility cost t of job search against the benefit of job search, which equals the UI benefits, b, plus the expecteddiscountedvalueofenteringanemploymentrelationship. 16

Dynamics of labor market stocks. The stocks of employment (e ), unemployment (u ), t t andnonparticipation(n )evolveovertimeaccordingto t       e fee fue fne e t+1 t+1 t+1 t+1 t        u  =  feu fuu fnu × u , (8)  t+1   t+1 t+1 t+1   t  n fen fun fnn n t+1 t+1 t+1 t+1 t ij where f denotestheindividual’stransitionprobabilityfromthelabormarketstate i to j t attime t.11 Employed individuals separate from employers either exogenously with probability q q δ,orendogenouslywithprobability1−F(x ). Thus,δandtheseparationcutoffx jointly t t determine the workers’ transition probability from employment to unemployment, feu; t the workers’ transition probability from employment to nonparticipation, fen; and the t probabilityofremainingemployed, fee,withtherestrictionthat feu + fen + fee = 1. t t t t Unemployed individuals meet a posted vacancy with probability p(θ ). Time variat tion in p(θ ), resulting from changes in the tightness ratio, θ , captures endogenous fluct t tuations in the degree of labor market frictions. So the meeting probability, p(θ ), and t the separation, x q , and participation, xv, cutoffs jointly determine the workers’ transition t t probability from unemployment to employment, fue; the workers’ transition probability t from unemployment to nonparticipation, fun; and the probability of remaining unemt ployed, fuu,withtherestrictionthat fue + fun + fuu = 1. t t t t Finally, nonparticipant individuals meet a posted vacancy with probability ϕp(θ ). t So the meeting probability ϕp(θ ), and the separation, x q , and participation, xv, cutoffs t t t jointly determine the transition probabilities from nonparticipation to employment, fne; t the transition probabilities from nonparticipation to unemployment, fnu; and the probat bilityofremaininganonparticipant, fnn,sothat fne + fnu + fnn = 1. t t t t 4.4 Basic Properties of the Model To provide insight into the main forces at play in the model, here we discuss some basic properties of the deterministic steady state of the model, where the productivity shock is y = y¯ at all times and the stocks of employment, unemployment, and nonparticipation areconstant. 11SeeAppendixBfordetailsonthecalculationofthetransitionprobabilities. 17

4.4.1 SearchandSeparationCutoffs Figure 1 shows the cross-sectional distribution of home productivity alongside search and separation cutoffs for a calibrated version of the model, which we later use for our quantitativeanalysis. Figure1: Thefigureshowsthecross-sectionaldistributionofhomeproductivityinthedeterministicsteady stateofthemodel,wheretheproductivityshockisy = y¯. SeeSection5fordetailsontheparametrization ofthemodel. The steady state features three regions. First, individuals whose home productivity (or, equivalently, value of leisure) exceeds the separation cutoff, xq, are nonparticipants. Second, individuals with home productivity smaller than the search cutoff, xv, are either employed or unemployed. These individuals are attached to the labor force. Third, for home productivity between the search and separation cutoffs, individuals are employed ornonparticipants;theyareattachedtothelaborforce. 4.4.2 MarketTightness Size of the pool of job seekers. Participation decisions affect the size of the pool of job seekers in two ways. First, for a given pool of unemployed individuals, a fraction ϕ of nonparticipants is drawn into the pool of job seekers; these individuals are passive 18

searchers who congest the labor market, which reduces the probability that an unemployedindividualseekingworkfindsajob. Giventhatajobseeker’smeetingprobability isconcaveinthetightnessratio,thiscongestioneffectisrelativelymoreimportantduring recessions, when the incentives to post job vacancies are depressed. This channel becomesself-evidentifoneusesthedefinitionofthemarkettightnessratio,whichgivesjob vacancies as v = θ ×(u+ϕn), where u+ϕn is the size of the pool of job seekers. Hence, cyclical fluctuations in the measure of nonparticipants directly affect congestion insofar as ϕ > 0. In the calibrated model, as in the data, the participation rate is pro-cyclical, implyingthat n risesinrecessionsandfallsinexpansions,whichexacerbatescongestion. Second, we emphasize that in the model, the pool of unemployed individuals is itself affectedbyparticipationdecisions. Thissituationoccursbecauseduringrecessions,inthe model, as in the data, unemployed individuals tend to remain unemployed at a higher rate, and individuals out of the labor force are more likely to transition into unemployment. Overall, the effect of the flows from and to unemployment leans toward a higher congestion of the labor market during recessions. Everything else being equal, these two effectsincreasetheprobabilitythatanemployerpostingvacanciesmeetsajobseekerand depressestheprobabilitythatajobseekerfindsajob. Composition of the pool of job seekers. Participation decisions affect the composition of the pool of job seekers, too. To clarify this channel, using β = 1/(1+r), where r is the real interest rate, in the deterministic steady state we rewrite the free-entry condition (5) as k 1+r = (y¯−w¯y¯η), (9) q(θ)Ω r+δ where,again, Ω = (u+ϕna)/(u+ϕn)istheshareofjobseekerswhoarewillingtowork atthesteady-statewagew = w¯y¯η. UnlikeintheDMPmodel,inourmodel,anactivelabor supply decision implies that individual participation decisions endogenously determine the composition of the pool of job seekers. Whether composition amplifies or dampens Ω Ω the fluctuations in the tightness ratio θ depends on the cyclicality of . Notably, if is Ω pro-cyclical, fluctuations in θ are amplified; conversely, if is countercyclical, fluctua- Ω tions in θ are dampened. In the calibrated model, is countercyclical, thus contributing to dampening fluctuations in market tightness. We note that the countercyclicality of Ω is not hardwired into the model; rather, it critically depends on getting the right comovementbetweenunemploymentandnonparticipationwithoutput. 19

5 Parametrization To parametrize the model, we exogenously set the values of a subset of parameters and jointlycalibratetheremainingparametersusingthemethodofmoments. Thiscalibration exercise involves solving the model’s deterministic steady state and finding parameter valuessothatthemodelmatchesasetoftargetedmomentsinactualdata. Wearetoassignvaluesto15parametersrelatedtofrictionsinthelabormarket(η,k,ϕ, w¯, ε, δ, and χ), preferences (β), UI benefits (b), and idiosyncratic and aggregate stochastic processes (µ , σ , λ, y¯, σ , and ρ ). The length of a model period is set to one month, x x y y as crucial labor market targets are available at a monthly frequency, taken from Krusell et al. (2017). The sample period runs from 1978:M1 to 2012:M9. Table 2 summarizes the parametrizationofthemodel. 5.1 Exogenously Set Parameters We use standard values for the parameters β, η, b, ε, ρ , and σ based on commonly y y acceptedvaluesintheliterature. Thetimediscountfactor βissetto0.997sothattheannualrisk-freeinterestrateofour economy equals4%, a standard valuein the literature(see, e.g., Gomme, Ravikumarand Rupert, 2011; McGrattan and Prescott, 2003). We set the wage elasticity to labor productivity η = 0.7 to match microeconomic estimates of wage flexibility in Haefke, Sonntag andvanRens(2013). Wesetbtoobtaina50%replacementraterelativetothesteady-state wage,avalue consistentwiththegenerosityofthe unemploymentbenefitssysteminthe UnitedStates. Wesettheelasticityofmatchestojobseekersinthemeetingfunction, ε,to 0.6,themidpointoftheestimatesinPetrongoloandPissarides(2006). Transitory shocks to the productivity of a job are the source of aggregate fluctuations. Importantly, we assume that the stochastic process for log productivity follows an AR(1) process, such that it exhibits symmetric fluctuations around its steady-state value, y¯. We set the persistence of the log productivity, ρ , to be 0.975 and its conditional volatility, σ , y y to be 0.5% so that the model reproduces the cyclical properties of the quarterly series of labor productivity in the data. To be sure, other shocks may hit the economy at different times and with different intensities (see Ramey, 2016, for an overview). While our analysis can accommodate other real shocks, we target productivity shocks, as they have been thefocusofmuchofthebusinesscycleresearch. 20

5.2 Calibrated Parameters The remaining parameters are calibrated to match targeted moments in U.S. data. While noneoftheparametershasaone-to-onerelationshiptoaspecificmoment,itisinstructive todescribethecalibrationasafewdistinctsteps. Table2:Parametrization Parameter Description Value Comments A.Labormarketfrictions η Wagefcn:elasticity 0.700 Haefke,SonntagandvanRens(2013) w¯ Wagefcn:scale 1.668 Methodofmoments ϵ Meetingfcn:elasticity 0.600 PetrongoloandPissarides(2006) χ Meetingfcn:scale 0.231 Methodofmoments κ Unitvacancycost 0.105 Steady-statetightness ϕ Prob.ofpassivesearching 0.521 Methodofmoments δ Exogenousseparationrate 0.022 Methodofmoments B.PreferencesandUnemploymentInsurance(UI)benefits β Timediscountfactor 0.997 Realinterestrate(4%) b UIbenefits 0.5w¯ Replacementrate(50%) C.First-orderautoregressive(AR(1))productivityshock y¯ AR(1):mean 1.715 Methodofmoments ρ AR(1):persistence 0.975 FittoAR(1),HP-filteredlaborprod. y σ AR(1):volatility 0.005 FittoAR(1),HP-filteredlaborprod. y D.Homeproductivityshock µ Log-normal:scale 0 Normalization x σ Log-normal:shape 1 Methodofmoments x λ Arrivalrate 0.032 Methodofmoments After the normalization of the steady-state value of the market tightness ratio, we jointly calibrate the following seven model objects using seven data moments: (1) the steady-state value of feu (0.014); (2) the steady-state value of fen (0.014); (3) the steadystate value of fnae (0.12); (4) the steady-state value of fue (0.23); (5) the steady-state labor forceparticipationrate(66.8%);(6)thesteady-stateshareofnonparticipantattached(8%); and(7)theelasticityof fue withrespecttolaborproductivity(3.09). AsinShimer(2005),thecostofpostingajobvacancy,k,issetto0.10sothatthemarket tightnessratioequals1inthedeterministicsteadystate,inwhichy = y¯. Wethencalibrate the arrival rate of the idiosyncratic shock, λ, the scale parameter of the meeting function, 21

χ, the probability that a nonparticipant is drawn in the pool of job seekers, ϕ, and the exogenous separation rate, δ, so that the deterministic steady state of the model jointly reproduces: (i) the average transition probability from employment to nonparticipation, f¯en; (ii) the average transition probability from unemployment to nonparticipation, f¯un; (iii) the probability that a nonparticipant attached becomes employed, as reported by Jones and Riddell (2019); and (iv) the average transition probability from employment to unemployment, f¯eu. To match a labor force participation rate of 66.8% and the 8% share of nonparticipant attached(BarnichonandFigura,2016),wesetthesteady-statevalueoflaborproductivity y¯ andthereal-wagescaleparameter w¯ to1.72and1.67,respectively. Finally, the distribution of idiosyncratic shocks captures unobserved heterogeneity in home production (or leisure values), which is an inherently latent object. To proceed, we assume that the idiosyncratic component of home production x is log-normally dist tributed with parameters µ (scale) and σ (shape). We normalize µ = 0, and we set x x x σ = 1 so that the model replicates the elasticity of the transition probability from unemx ploymenttoemploymentwithrespecttolaborproductivityof3.09.12 6 Quantification In this section, we study the quantitative properties of the calibrated model. To achieve this goal, we solve the deterministic steady state, and we compute the model’s dynamics in response to productivity shocks using an approximation of the model equilibrium conditionsaroundthedeterministicsteadystatethatisaccuratetothesecondorder.13 Operationally, we perform 200 simulations, each 870 periods long. We simulate the model at a monthly frequency and then construct quarterly series by averaging the data over three consecutive non-overlapping periods. We discard 40% of the initial simulated series, so we are left with 420 observations that, once aggregated at the quarterly frequency, match the length of the sample period in Krusell et al. (2017). For each simulation, we compute moments and report the median of those moments across the 200 simulations. 12Thelaggedelasticityofthetransitionprobabilityfromunemploymenttoemploymentwithrespectto labor productivity, η y f − ue 1 , is estimated by running the regression log(f t ue) = constant+η y f − ue 1 log(y t−1 )+u t onactualandartificialdatasimulatedfromthemodel. DataonoutputperworkerarefromHagedornand Manovskii(2011). 13We numerically solve the model by relying on a second-order approximation to the solution around thedeterministicsteadystate(see,e.g.,Schmitt-GrohéandUribe,2004). 22

Table3:BusinessCycleStatistics–LaborMarketStocks y θ v EPOP ER PR A.Standarddeviation Data 0.0225 24.01 13.15 0.99 0.90 0.26 Model:baseline 0.0225 8.21 7.59 0.40 0.34 0.07 Model:nolinkmarket-homeproductivity 0.0225 8.63 7.99 0.53 0.35 0.20 B.Correlationwithoutput Data 0.55 0.89 0.88 0.83 0.86 0.21 Model:baseline 0.99 0.97 0.95 0.97 0.96 0.86 Model:nolinkmarket-homeproductivity 0.98 0.95 0.92 0.93 0.97 0.75 C.Autocorrelation Data 0.75 0.92 0.91 0.92 0.93 0.69 Model:baseline 0.75 0.67 0.64 0.84 0.84 0.87 Model:nolinkmarket-homeproductivity 0.75 0.68 0.65 0.87 0.85 0.91 D.Beveridgecurve Data −0.92 Model:baseline −0.92 Model:nolinkmarket-homeproductivity −0.91 Notes: The variable y is labor productivity; θ is labor market tightness; v is vacancies; EPOP is the employment-to-populationratio;ERistheemploymentrate(oneminustheunemploymentrate);PRis theparticipationrate.Variablesarequarterlyaveragesofmonthlyseriesexpressedinlog-deviationsfrom theHodrick-Prescotttrendwithsmoothingparameter1,600.SeeAppendixAfordatasources. 6.1 Standard Business Cycle Moments We now turn to examining the time-series properties of the calibrated economy in terms of first- and second-order moments of labor market stocks and transition probabilities acrossthethreestatesofthelabormarketindeviationsfromtrend. 6.1.1 LaborMarketStocks Table 3 reports business cycle statistics calculated on artificial data simulated from the model, aggregated to a quarterly frequency, logged, and Hodrick-Prescott (HP)-filtered with a smoothing parameter of 1,600. First, the model generates 40% of the volatility of the employment-to-population ratio and 27% of the volatility of the participation rate in thedata. Notethatbyconstruction,themodelmatchesthevolatilityoflaborproductivity in the data. Also, the model reproduces approximately 34% and 58% of the volatility of thetightnessratioandjobvacancies,respectively,accountingforanontrivialshareofthe 23

Table4:BusinessCycleStatistics–TransitionProbabilities feu fen fue fun fne fnu A.Average Data:AZ-adjusted 0.014 0.014 0.228 0.135 0.022 0.021 Model:baseline 0.014 0.014 0.230 0.015 0.013 0.015 Model:nolinkmarket-homeproductivity 0.014 0.014 0.228 0.015 0.013 0.015 B.Standarddeviation Data:AZ-adjusted 0.089 0.083 0.088 0.106 0.103 0.072 Data:DeNUNified 0.069 0.036 0.076 0.066 0.041 0.063 Model:baseline 0.011 0.002 0.036 0.002 0.027 0.013 Model:nolinkmarket-homeproductivity 0.012 0.007 0.038 0.008 0.030 0.007 C.Correlationwithoutput Data:AZ-adjusted −0.630 0.430 0.760 0.610 0.520 −0.230 Data:DeNUNified −0.660 0.290 0.810 0.550 0.570 −0.560 Model:baseline −0.974 0.929 0.964 0.811 0.826 −0.982 Model:nolinkmarket-homeproductivity −0.950 −0.979 0.949 −0.961 0.825 −0.943 D.Autocorrelation Data:AZ-adjusted 0.590 0.290 0.750 0.620 0.380 0.300 Data:DeNUNified 0.700 0.220 0.850 0.580 0.480 0.570 Model:baseline 0.680 0.856 0.670 0.821 0.530 0.705 Model:nolinkmarket-homeproductivity 0.683 0.731 0.679 0.699 0.557 0.667 Notes: Thevariable fijisthetransitionprobabilityfromlabormarketstateitoj;eisemployment;uisunemployment;n = 1−e−uisnonparticipation;AZisAbowd-Zellner. Variablesarequarterlyaveragesofmonthlyseries expressedinlog-deviationsfromtheHodrick-Prescotttrendwithsmoothingparameter1,600. SeeAppendixAfor datasources. fluctuations in what the model identifies as determinants of search frictions. Here we stress that as a number of shocks of varying nature and magnitude hit the U.S. economy over time, it is not surprising that a model with only productivity shocks like ours does notaccountfortheentiretyofthecyclicalvolatilityinthedata.14 In light of these considerations, we compare some of the model’s predictions related totheelasticitiesoflabormarketstocksandworkers’transitionprobabilitieswithrespect to labor productivity with their empirical counterparts. In terms of labor market stocks, and focusing on Current Population Survey (CPS) data for the nonfarm business sector, we find that the contemporaneous and lagged estimated elasticities of the employment- 14Mortensen and Nagypal (2007) propose a similar argument in the context of the “unemployment volatility puzzle,” in reference to the inability of the DMP model to reproduce the cyclical volatility in theU.S.unemploymentrate. 24

to-populationratiotooutputperworkerare0.25and0.4,respectively. Runningthesame regressionsonartificialdatasimulatedfromthemodel,wefindelasticitiesof0.35thatare remarkably close to the untargeted estimates in actual data. The model also does reasonablywellinaccountingfortheelasticitiesofjobvacancies,tightness,andtheemployment ratetolaborproductivity,alluntargetedmoments.15 The model also accounts for the co-movement and persistence in the data, measured as the contemporaneous correlation with output and autocorrelation, respectively. Note that none of these moments are a target of our calibration; thus, one can assess how well the model does against a rich set of overidentifying restrictions. The positive and strong co-movementofjobvacancieswithoutputis,toalargedegree,notsurprising. Intuitively, in the model, shocks to the output of a job are the only source of aggregate fluctuations, so job vacancies are bound to be highly correlated with output. In this sense, a close matchwiththedataalongthatdimensioncannotbeviewedasasuccess. Bycontrast,the positive co-movement of the employment-to-population ratio is not hardwired into the model but crucially depends on the configuration of parameter values. Our calibrated model generates the strength of the comovement between the unemployment rate and outputinthedata,anditproducesacorrelationoftheparticipationratewithoutput. We stress that even accounting for the sign of the co-movement of both unemployment and participation rates has been a challenge for equilibrium models of the aggregate labor market(see,e.g.,Veracierto,2008;Shimer,2013). The model does reasonably well in accounting for the persistence of job vacancies. In the model, job vacancies have an autocorrelation of 0.64, which is comparable with the 0.91 in the data. The lack of persistence in vacancies is a well-known problem in searchand-matching models of the labor market. As shown by Fujita and Ramey (2006), one waytotacklethisshortcomingistoextendthemodelwithsunkcostsinvacancyposting. In our setting, though, the introduction of sunk costs in vacancy posting dramatically increasesthestatespaceofthemodelasthestocksofemployed(attachedandnon-attached, separately), unemployed, and nonparticipants become endogenous state variables, thus enormouslycomplicatingthecomputationoftheequilibrium. Finally, the model generates a downward-sloping Beveridge curve—i.e., the negative empirical relationship between job vacancies and unemployment—which is a wellknown challenge for three-state models of the labor market (see, e.g., Tripier, 2004; Veracierto, 2008). Our results along this dimension are in line with Arseneau and Chugh 15Table C.1 in Appendix C reports the estimated elasticities for labor market stocks, tightness, and job vacanciesinthemodelandinthedataforseverallaborproductivityseries. 25

(2012). 6.1.2 TransitionProbabilities Table4showsaveragesandbusinesscyclestatisticsforthetransitionprobabilitiesacross employment, unemployment, and nonparticipation in the model and data. We report statisticsbasedondataadjustedforclassificationerrors,asinAbowdandZellner(1985), aswellas“deNUNified”data,asconstructedinElsby,HobijnandS¸ahin(2015). The model matches the calibration targets of the average transition probabilities from employmenttounemployment(f¯eu = 0.014),fromemploymenttononparticipation(f¯en = 0.014), and from unemployment to employment (f¯ue = 0.228). As a by product then, the model matches the average probability of staying employed (f¯ee = 1− f¯eu − f¯en = 1−0.014−0.014 = 0.972). Hence, in the model, as in the data, employment is a very persistent state. The model does reasonably well for other untargeted moments, too, such as the average probability from nonparticipation to employment (f¯ne = 0.013 in the model, compared with 0.022 in the data), and from nonparticipation to unemployment (f¯nu = 0.015inthemodel,comparedwith0.021inthedata). Our calibrated model accounts well for the cyclical properties of the workers’ transition probabilities across the three labor market states. Notably, it captures (i) the countercyclicality of the transition probabilities into unemployment (feu, fnu), (ii) the procyclicalityofthetransitionprobabilitiesoutofunemployment(fue, fun),and(iii)theprocyclicality of the transition probability from nonparticipation to employment (fne). The model is successful in reproducing the pro-cyclicality of the transition probability from employment to nonparticipation (fen) and that of the transition probability from unemployment to nonparticipation (fun), as in the data. This achievement is typically a challenge for three-state models of the labor market in which market productivity shocks are the only driving force of aggregate fluctuations. A positive productivity shock raises the match surplus across the board, so that individuals either continue working at a higher wage or continue seeking work at a higher expected value of future employment. Overall, these two forces induce countercyclical movements in both fen and fun. Indeed, a version of our model in which home productivity is not scaled by market productivity suffers from the same drawback, suggesting that the pro-cyclicality of the opportunity cost of employment is critical for the model to reproduce the pro-cyclicality of fen and fun.16 16Chodorow-Reich and Karabarbounis (2016) find that the opportunity cost of employment is pro- 26

Finally, by virtue of our calibration strategy, the model matches the lagged elasticity of the probability of finding a job fue with respect to labor productivity, a key model object determining the extent of slack in the labor market. The model does reasonably well in terms of the elasticity of fne to labor productivity, and it reproduces the negative signoftheestimatedelasticitiesof fen and fun;however,itgreatlyundershootsthestrong countercyclicalityofthetransitionprobabilityfromemploymenttounemployment, feu.17 6.2 Cyclical Skewness We now turn to evaluating the model’s ability to generate the cyclical asymmetry in the data, as measured by the skewness of a time series in deviations from trend, or “deepness”(Sichel,1993). Notethat,sincetheskewnessisnotatargetofourcalibration,aclose matchtothedataconstitutesanadditionalvalidationofthemodel. Table 5 reports skewness statistics for three different filtering or detrending methods: HP, Baxter-King (BK), and Christiano-Fitzgerald (CF) filters. Overall, the model is successful in reproducing the deepness in the employment-to-population ratio in the data and, crucially, the negative skewness in the employment rate and the lack of it in the laborforceparticipationrate. Focusing on the results based on the HP filter to streamline exposition, we find that the skewness in the artificial employment-to-population ratio generated by the model is −0.24, which is 75% of the skewness in the data. The model reproduces the disconnect between the asymmetry properties of unemployment and participation rates as well. In the model, cyclical fluctuations in the employment rate (one minus the unemployment rate)areleftskewed,withaskewnesscoefficientof−0.25,whereasthoseintheparticipationratearesymmetric,withaskewnesscoefficientofvirtuallyzero. Similarresultshold fortheBKandCFfilter. 6.3 Impulse Response Functions To illustrate the propagation mechanism of productivity shocks embodied in the model, inthissection,wediscussimpulseresponsefunctions(IRFs). Allresponsesareexpressed aslogdeviationsfromthedeterministicsteady-statelevels. cyclicalandvolatileoverthebusinesscycle. 17TableC.2inAppendixCreportstheestimatedelasticitiesforthetransitionprobabilitiesinthemodel andinthedataforseverallaborproductivityseries. 27

Table5:SkewnessofLaborMarketStocks EPOP ER PR A.Hodrick-Prescottfilter Data −0.32 −0.70 0.05 Model:baseline −0.24 −0.25 −0.07 B.Baxter-Kingfilter Data −0.44 −0.85 0.09 Model:baseline −0.25 −0.27 −0.09 C.Christiano-Fitzgeraldfilter Data −0.29 −0.51 0.05 Model:baseline −0.14 −0.14 −0.09 Notes: EPOPistheemployment-to-populationratio;ERisoneminustheunemploymentrate;PRistheparticipationrate. Variables arequarterlyaveragesofmonthlyseriesexpressedinlog-deviations fromtrend. ThesmoothingparameterfortheHodrick-Prescottfilteris1,600. FortheBaxter-KingandChristiano-Fitzgeraldfilters, weconsiderfrequenciesbetween2and32quarters. ThemovingaverageorderforBaxter-Kingissetto8quarters. SeeAppendixA fordatasources. IRFs: Searchandseparationcutoffs. Animportantpartofthepropagationmechanism ofshocksembodiedinthemodelishowlaborforceparticipationvariesoverthebusiness cycle. In the model, the participation margin of employment adjustment is described by theresponseofthesearchandseparationcutoffstoshocksandthemassofindividualsat thosecutoffs. Afterapositiveproductivityshock,laborsupplyisaffectedbytwocontrastingforces. On the one hand, high market productivity results in higher wages, increasing, ceteris paribus, the return of nonparticipant individuals to the labor force. Similarly, an individualmaypostponetheseparationdecisionandstayinthelaborforce. Inshort,individuals withhigherhomeproductivityareledintothelaborforce. Ontheotherhand,higherhome productivity increases the opportunity cost of market work, prompting nonparticipants to stay out of the labor force or employed individuals to drop out of it. Which of these twoforcesprevailsdependsontheparametrizationofthemodel. The IRFs in Figure 2 show that in the baseline model, in which home productivity is scaled by market productivity, the search (xv) and separation (xq) cutoffs fall in response to a technology shock that increases home and market productivity. By contrast, in the 28

case of a “pure” market productivity shock, in which home productivity is not scaled by market productivity, the cutoffs xv and xq move in opposite directions, which in turn makestheworkers’transitionprobabilities fen and fun fallinresponsetoapositivetechnology shock (see Figure C.1 in Appendix C). By assumption, home productivity is proportional to market productivity, implying that the opportunity cost of employment is pro-cyclical, consistent with the evidence in Chodorow-Reich and Karabarbounis (2016). Asitturnsout,thepro-cyclicalityoftheopportunitycostofemploymentiscriticalforthe modeltoreplicatethepro-cyclicalityof fen and fun inthedata. v q x x 0.8 Baseline 0.6 No link market-home prod. 0.7 0.5 0.6 0.4 0.5 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 -0.1 -0.1 -0.2 -0.2 10 20 30 40 10 20 30 40 Figure 2: The figure shows the impulse response function of the search (xv) and separation (xq) cutoffs to a productivity shock in the baseline model (solid line) and in a version of the model in which home productivity is not scaled by market productivity (dash-dotted line). All responses are expressed as log deviationsfromthedeterministicsteady-statelevels. SeeSection5fordetailsontheparametrizationofthe model. IRFs: Labormarketstocksandtransitionprobabilities. Figures3and4showtheIRFs of labor market stocks and workers’ transition probabilities, respectively. Note that by assumption, the productivity shock follows an AR(1) process so that its IRF features a jumponimpactandmonotonicreversiontowardtheunconditionalmean. Inresponseto 29

apositiveproductivityshock,labormarketstocksexhibithump-shapeddynamics. Productivity Output 1 1 0.5 0.5 10 20 30 40 10 20 30 40 EPOP ER 0.35 0.3 0.3 0.25 0.2 0.2 0.15 10 20 30 40 10 20 30 40 PR Tightness: 0.04 10 0.02 0 5 -0.02 10 20 30 40 10 20 30 40 Composition: Vacancies -0.1 10 -0.2 -0.3 -0.4 5 -0.5 10 20 30 40 10 20 30 40 Figure 3: The figure shows the impulse response function to a productivity shock. EPOP is the employment-to-population ratio; ER is the employment rate (one minus the unemployment rate); PR is the participation rate. Ω = u+ϕna is the fraction of job seekers that accepts a job offer. All responses u+ϕn are expressed as log deviations from the deterministic steady-state levels. See Section 5 for details on the parametrizationofthemodel. Jobvacanciesandthemarkettightnessratio(θ)riseonimpactandthenrevertbackto their steady-state values, mirroring the dynamics of the productivity shock. The fraction Ω of job seekers accepting a job offer ( ) instead displays hump-shaped dynamics; it falls inresponsetoaproductivityshock,slowlyrevertingtoitssteady-statelevel. 6.4 Role of Labor Supply and Search Frictions To gain further insight into the mechanism of fluctuations, we carry out a quantitative accountingexercisethatleveragesthestructureofthemodel. Labor supply versus slackness channel. To assess the importance of labor supply decisions vis-à-vis slack, we simulate two counterfactual economies in which, crucially, we keepthesameparametervaluesandthesamerealizationsofproductivityshocksasinthe 30

eu en f f -0.5 0.16 0.14 -1 0.12 10 20 30 40 10 20 30 40 ue un f f 4 0.1 3 0.05 2 0 10 20 30 40 10 20 30 40 ne nu f f 4 -0.5 -1 2 -1.5 10 20 30 40 10 20 30 40 Figure 4: The figure shows the impulse response function of the workers’ transition probabilities to a productivityshock.Allresponsesareexpressedaslogdeviationsfromthedeterministicsteady-statelevels. SeeSection5fordetailsontheparametrizationofthemodel. 31

baseline economy. In the first counterfactual (“Ctrfl 1”), we re-solve the model by droppingthefree-entrycondition(5)andfixthemarkettightnessratioatitssteady-statevalue inthebaselineeconomy;separationandsearchcutoffsareallowedtovaryinresponseto shocks as implied by the separation and search indifference conditions (6) and (7). This exerciseproducescounterfactualtimeseriesoflabormarketstocksandflowsinwhichall thevariationcomesfromtheresponseofthetwocutoffstoproductivityshocks—namely, the“laborsupplychannel.” Conversely, in the second counterfactual (“Ctrfl 2”), we re-solve the model by dropping the indifference conditions for separation and search and fix the values of the two cutoffs at their steady-state values in the baseline economy; the tightness ratio is allowed insteadtovaryasimpliedbythefree-entrycondition(5). Thiscounterfactualisolatesthe roleoffluctuationsinthetightnessratio—namely,the“slacknesschannel.”18 We stress that this exercise is neither a test of whether a two-state model is a better abstraction than a three-state model nor a way to discriminate between frictional and frictionless models of the labor market. The constructed counterfactual series for the unemployment rate and the participation rate are not the equilibrium outcome of nested economies. Specifically, fixing the cutoffs on home productivity at their steady-state values does not render a two-state model of the labor market. The counterfactual economy withfixedcutoffscontinuestodisplayflowsinandoutofthelaborforce. Inaddition,the transition probabilities from out of the labor force to either employment or unemployment, and the transition probability from unemployment to attached nonparticipation directly depend on the tightness ratio. Thus, movements in market tightness alone drive fluctuations not only in the flows between employment and unemployment, but also in thoseinandoutofthelaborforce.19 Similarly, fixing the market tightness ratio at its steady-state value does not render a frictionless economy. The reason is that while the extent of frictions is not allowed to vary in response to shocks, the counterfactual economy with fixed tightness continues to displayunemploymentandfluctuationsintheunemploymentrate.20 18For each counterfactual, we recompute the equilibrium of the model by relying on a second-order approximationtothesolutionaroundthedeterministicsteadystate. 19TablesC.3andC.4inAppendixCreportresultsfortwoadditionalexperiments,inwhichwefixone cutoffatatime. 20Wenotethatsincethemodelisnonlinear,theresultsofourquantitativeaccountingexercisearetobe viewedastheory-basedcounterfactuals,asopposedtoresultsofalinearadditivedecomposition. 32

Table6:LaborSupplyversusSlackness Model Ctrfl1 Ctrfl2 (fixedtightness) (fixedcutoffs) A.Standarddeviation Employment-to-populationratio 0.40 0.06 0.43 Employmentrate 0.34 0.01 0.35 Participationrate 0.07 0.07 0.09 B.Correlationwithoutput Employment-to-populationratio 0.97 −0.24 0.96 Employmentrate 0.96 0.90 0.97 Participationrate 0.86 −0.37 0.91 C.Skewness Employment-to-populationratio −0.24 −0.05 −0.24 Employmentrate −0.25 0.04 −0.25 Participationrate −0.07 −0.05 −0.18 Notes:“Ctrfl1”referstothecounterfactualexperimentwherethemodelissimulatedwiththe tightnessratiofixedatitssteady-statevalueandvaryingsearchandseparationcutoffs.“Ctrfl 2”referstothecounterfactualexperimentwherethemodelissimulatedwithcutoffsfixedat theirsteady-statevaluesandavaryingtightnessratio.Inallcounterfactuals,wekeepthesame realizationsofproductivityshocks. Cyclical volatility and co-movement. Panels A and B of Table 6 show the results of the two experiments for the cyclical volatility and co-movement of the labor market stocks with output. First, through the lens of the model, absent the response of the search and separation cutoffs to productivity shocks, fluctuations in the market tightness ratio accountforthebulkofthecyclicalvolatilityintheunemploymentrateandareanimportant driverofthefluctuationsintheparticipationrate,too. Second, the counterfactual with a fixed tightness ratio yields the wrong co-movement betweentheparticipationrateandoutput,whichispositiveinthedataandinthebaseline economy but negative in the counterfactual. That is, in the counterfactual economy with fixedtightness,duringarecessionthelaborforceparticipationraterises,insteadoffalling as in the data, which highlights the critical role of the fluctuations in the probability of 33

findingajobforthecyclicalityoftheparticipationrate.21 Cyclical skewness (deepness). Panel C of Table 6 shows results for deepness, again measured as skewness of a variable in deviation from trend. First, the slackness channel, captured by endogenous fluctuations in the market tightness ratio, accounts for virtually all the negative skewness in the employment rate. The participation margin per se does notgenerateskewness. Infact,fluctuationsinthecutoffsalonewouldgeneratesymmetric fluctuationsintheunemploymentrate,whichisstronglyatoddswiththedata. Second,thelackofskewnessintheparticipationrateistheresultofcompetingforces. Fluctuations in the tightness ratio alone generate negative skewness in the participation rate, while the labor supply channel counteracts that effect. Capturing the relative strength of these two channels is key for the model to replicate the observed disconnect betweentheasymmetrypropertiesofunemploymentandparticipationratesinthedata. 7 Conclusion In the United States cyclical fluctuations in the employment-to-population ratio exhibit “deepness,” which refers to the pattern that the deviations below the trend (troughs) are larger than those above (peaks). This phenomenon produces negative skewness in the distributionoftheemployment-to-populationratioindeviationsfromthetrend. Our analysis starts with documenting a related, yet overlooked fact: deepness in the employment-to-populationratioisaccountedforsolelybytheunemploymentrateinthat fluctuationsinthelaborforceparticipationratearesymmetric. Toexplainthesefacts,we formulate a business cycle model featuring frictional unemployment and a labor force participationdecision. Themodel,restrictedtofitkeyobservationsofU.S.data,accounts for the observed cyclical skewness in the unemployment rate and the lack thereof in the participationrate,aswellassalientpropertiesofgrossworkerflowsacrossemployment, unemployment,andnonparticipation. Through the lens of a host of quantitative experiments, we find that cyclical fluctuations in the extent of search frictions, as measured by the speed at which job seekers find job opportunities, account for the deepness of the employment-to-population ratio. Individuals’ participation decisions contribute by affecting the size and the composition 21Panels B and C of Table C.4 in Appendix C report the standard deviations and the correlation with outputoftheworkers’transitionprobabilitiesacrosscounterfactualexperiments. 34

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Appendix A Data Sources Data for the monthly and seasonally-adjusted unemployment rate (series LNS14000000) and participation rate (series LNS11300000) are from the Current Population Survey of the Bureau of Labor Statistics (BLS) and available on the BLS website at www.bls.gov. The employment-to-population ratio is obtained as one minus the unemployment rate times the participation rate. Data for monthly hazard rates across different states (employment, unemployment, and nonparticipation) are taken from Krusell et al. (2017). DataforjobvacanciesarethemonthlycompositeHelp-WantedIndex(HWI)constructed by Barnichon (2010) and available on the author’s website at https://sites.google.com/ site/regisbarnichon. Quarterlydataareobtainedbyaveragingnon-overlappingmonthly observations in a given quarter. Seasonally adjusted quarterly data for real output per worker in the nonfarm business sector are produced by the BLS and available on the LaborProductivityandCostshomepageathttp://www.bls.gov/lpc. B Derivations ij We use f to denote the worker’s transition probability from labor market state i to j at t timet,andthelabels“a”and“na”toindicate“attached”and“non-attached”individuals, respectively. The stocks of employment (e), unemployment (u), and nonparticipation (n) evolve overtimeaccordingto ea = f eaeaea + f enaeaena + f ueau + f naeana + f nnaeanna; (B.1) t+1 t+1 t t+1 t t+1 t t+1 t t+1 t e na = f eaenaea + f enaenaena + f uenau + f naenana + f nnaenanna; (B.2) t+1 t+1 t t+1 t t+1 t t+1 t t+1 t u = f eau ea + f enau ena + fuu u + f nau na + f nnau nna; (B.3) t+1 t+1 t t+1 t t+1 t t+1 t t+1 t n a = f eanmea + f enanaena + f unau + f nanana + f nnananna; (B.4) t+1 t+1 t t+1 t t+1 t t+1 t t+1 t nna = f eannea + f enannaena + f unnau + f nannana + f nnannanna. (B.5) t+1 t+1 t t+1 t t+1 t t+1 t t+1 t 41

Theworkers’transitionprobabilitiesarecalculatedas: f eaea = (1−δ) (cid:8) 1−λ (cid:2) 1−F (cid:0) xv (cid:1)(cid:3)(cid:9) +δ (cid:8) 1−λ (cid:2) 1−F (cid:0) xv (cid:1)(cid:3)(cid:9) p ; (B.6) t+1 t+1 t+1 t+1 f enaea = (1−δ)λF (cid:0) xv (cid:1) +δλF (cid:0) xv (cid:1) p ; (B.7) t+1 t+1 t+1 t+1 f uea = p (cid:8) 1−λ (cid:2) 1−F (cid:0) xv (cid:1)(cid:3)(cid:9) ; (B.8) t+1 t+1 t+1 f naea = p λF (cid:0) xv (cid:1) ; (B.9) t+1 t+1 t+1 f nnaea = p λF (cid:0) xv (cid:1) ; (B.10) t+1 t+1 t+1 f eaena = (1−δ)λ[F (cid:0) x q (cid:1) −F (cid:0) xv (cid:1) ]+δλ[F (cid:0) x q (cid:1) −F (cid:0) xv (cid:1) ]ϕp ; (B.11) t+1 t+1 t+1 t+1 t+1 t+1 f enaena = (1−δ) (cid:8) 1−λF (cid:0) xv (cid:1) −λ (cid:2) 1−F (cid:0) x q (cid:1)(cid:3)(cid:9) (B.12) t+1 t+1 t+1 +δ (cid:8) 1−λF (cid:0) xv (cid:1) −λ (cid:2) 1−F (cid:0) x q (cid:1)(cid:3)(cid:9) ϕp ; (B.13) t+1 t+1 t+1 f uena = ϕp λ (cid:2) F (cid:0) x q (cid:1) −F (cid:0) xv (cid:1)(cid:3) ; (B.14) t+1 t+1 t+1 t+1 f naena = ϕp (cid:8) 1−λ (cid:2) 1−F (cid:0) x q (cid:1)(cid:3) −λF (cid:0) xv (cid:1)(cid:9) ; (B.15) t+1 t+1 t+1 t+1 f nnaena = ϕp λ (cid:2) F (cid:0) x q (cid:1) −F (cid:0) xv (cid:1)(cid:3) ; (B.16) t+1 t+1 t+1 t+1 f eau = δ (cid:8) 1−λ (cid:2) 1−F (cid:0) xv (cid:1)(cid:3)(cid:9) (1− p ); (B.17) t+1 t+1 t+1 f enau = δλF (cid:0) xv (cid:1) (1− p ); (B.18) t+1 t+1 t+1 fuu = (1− p ) (cid:8) 1−λ (cid:2) 1−F (cid:0) xv (cid:1)(cid:3)(cid:9) ; (B.19) t+1 t+1 t+1 f nau = (1− p )λF (cid:0) xv (cid:1) ; (B.20) t+1 t+1 t+1 f nnau = (1− p )λF (cid:0) xv (cid:1) ; (B.21) t+1 t+1 t+1 f eana = δλ (cid:2) F (cid:0) x q (cid:1) −F (cid:0) xv (cid:1)(cid:3) (1−ϕp ); (B.22) t+1 t+1 t+1 t+1 f emna = δ (cid:8) 1−λF (cid:0) xv (cid:1) −λ (cid:2) 1−F (cid:0) x q (cid:1)(cid:3)(cid:9) (1−ϕp ); (B.23) t+1 t+1 t+1 t+1 f una = (1−ϕp )λ (cid:2) F(x q )−F(xv ) (cid:3) ; (B.24) t+1 t+1 t+1 t+1 f nana = (1−ϕp ) (cid:8) 1−λF (cid:0) xv (cid:1) −λ (cid:2) 1−F(x q ) (cid:3)(cid:9) ; (B.25) t+1 t+1 t+1 t+1 f nnana = (1−ϕp )λ (cid:2) F(x q )−F(xv ) (cid:3) ; (B.26) t+1 t t+1 t+1 f eanna = λ (cid:2) 1−F (cid:0) x q (cid:1)(cid:3) ; (B.27) t+1 t+1 f enanna = λ (cid:2) 1−F (cid:0) x q (cid:1)(cid:3) ; (B.28) t+1 t+1 f unna = λ (cid:2) 1−F (cid:0) x q (cid:1)(cid:3) ; (B.29) t+1 t+1 f nanna = λ (cid:2) 1−F (cid:0) x q (cid:1)(cid:3) ; (B.30) t+1 t+1 f nnanna = 1−λF (cid:0) x q (cid:1) . (B.31) t+1 t+1 42

C Additional Results TableC.1:ElasticityofLaborMarketStockstoLaborProductivity l θ v EPOP ER PR A.Data,contemporaneous ηl:CPS,non-farmbusiness 4.949 5.171 0.251 0.171 −0.016 y ηl:CPS,allprivate 6.501 6.923 0.281 0.214 −0.052 y ηl:LPC,non-farmbusiness 2.597 2.910 0.035 0.054 −0.050 y ηl:LPC,allprivate 4.244 4.723 0.112 0.108 −0.057 y B.Data,lagged ηl :CPS,non-farmbusiness 6.204 6.427 0.396 0.244 0.012 y−1 ηl :CPS,allprivate 8.349 8.769 0.488 0.315 −0.005 y−1 ηl :LPC,non-farmbusiness 4.570 4.873 0.223 0.159 −0.027 y−1 ηl :LPC,allprivate 6.484 6.942 0.330 0.225 −0.025 y−1 C.Model,contemporaneous ηl:baseline 7.506 6.812 0.350 0.342 0.008 y ηl:nolinkmarket-homeproductivity 8.244 7.181 0.674 0.375 0.301 y D.Model,lagged ηl :baseline 7.114 6.405 0.352 0.345 0.007 y−1 ηl :nolinkmarket-homeproductivity 7.855 6.765 0.686 0.379 0.309 y−1 Notes:Thevariablesηl andηl denotetheelasticityofl∈{θ,v,EPOP,ER,PR}tocontemporaney y−1 ous(y)andlagged(y−1)outputperworker,respectively.CPSisCurrentPopulationSurvey;LPCis LaborProductivityandCosts.Contemporaneousandlaggedelasticitiesareestimatedbyrunning theregressionslog(lt ) = constant+η y llog(yt )+ut andlog(lt ) = constant+η y l −1 log(yt−1 )+ut onactualandartificialdatasimulatedfromthemodel.DataonoutputperworkerarefromHagedornandManovskii(2011). 43

TableC.2:ElasticityofTransitionProbabilitiestoLaborProductivity l feu fen fue fun fne fnu A.Data,contemporaneous ηl:CPS,non-farmbusiness −4.005 2.032 2.144 2.626 2.891 −0.394 y ηl:CPS,allprivate −5.749 2.683 2.432 4.039 3.918 0.130 y ηl:LPC,non-farmbusiness −3.632 0.685 0.191 1.259 1.725 0.389 y ηl:LPC,allprivate −5.128 1.324 0.735 2.383 2.813 0.762 y B.Data,lagged ηl :CPS,non-farmbusiness −3.639 2.716 3.093 3.853 3.751 −0.272 y−1 ηl :CPS,allprivate −4.843 3.768 4.273 5.269 5.741 0.166 y−1 ηl :LPC,non-farmbusiness −3.444 2.042 1.548 2.904 2.834 0.177 y−1 ηl :LPC,allprivate −4.289 3.149 2.444 4.234 4.339 0.486 y−1 C.Model,contemporaneous ηl:baseline −1.115 0.219 3.257 0.164 1.804 −1.273 y ηl:nolinkmarket-homeproductivity −1.113 −0.764 3.604 −0.790 2.334 −0.653 y D.Model,lagged ηl :baseline −1.062 0.220 3.086 0.171 1.603 −1.219 y−1 ηl :nolinkmarket-homeproductivity −1.059 −0.740 3.434 −0.756 2.126 −0.620 y−1 Notes: The variables ηl and ηl denote the elasticity of l ∈ {feu,fen,fue,fun,fne,fnu} to contemporaneous y y−1 (y)andlagged(y−1)outputperworker,respectively. CPSisCurrentPopulationSurvey;LPCisLaborProductivityandCosts. Contemporaneousandlaggedelasticitiesareestimatedbyrunningtheregressionslog(lt ) = constant+η y llog(yt )+ut andlog(lt ) = constant+η y l −1 log(yt−1 )+ut onactualandartificialdatasimulated fromthemodel.DataonoutputperworkerarefromHagedornandManovskii(2011). 44

eu en f f -0.4 -0.6 -0.4 -0.8 -1 -0.6 -1.2 -1.4 10 20 30 40 10 20 30 40 ue un f f -0.4 4 -0.6 3 2 -0.8 10 20 30 40 10 20 30 40 ne nu f f 4 -0.4 -0.6 2 -0.8 10 20 30 40 10 20 30 40 Figure C.1: The figure shows the impulse response function of the workers’ transition probabilities to a productivityshockinaversionofthemodelinwhichhomeproductivityisnotscaledbymarketproductivity. Allresponsesareexpressedaslogdeviationsfromthedeterministicsteady-statelevels. SeeSection 5fordetailsontheparametrizationofthemodel. 45

TableC.3:BusinessCycleStatistics–LaborMarketStocks y θ v EPOP ER PR A.Standarddeviation Data 0.0225 24.01 13.15 0.99 0.90 0.26 Model:baseline 0.0225 8.21 7.59 0.40 0.34 0.07 Ctrfl: θfixed 0.0225 0.00 0.00 0.06 0.01 0.07 Ctrfl: xvandxqcutoffsfixed 0.0225 8.33 7.71 0.43 0.35 0.09 Ctrfl: xvcutofffixed 0.0225 8.24 7.63 0.41 0.34 0.07 Ctrfl: xqcutofffixed 0.0225 8.32 7.69 0.43 0.35 0.08 B.Correlationwithoutput Data 0.55 0.89 0.88 0.83 0.86 0.21 Model:baseline 0.99 0.97 0.95 0.97 0.96 0.86 Ctrfl: θfixed 1.00 0.95 0.72 −0.24 0.90 −0.37 Ctrfl: xvandxqcutoffsfixed 0.99 0.96 0.94 0.96 0.97 0.91 Ctrfl: xvcutofffixed 0.99 0.96 0.95 0.97 0.97 0.92 Ctrfl: xqcutofffixed 0.99 0.96 0.94 0.96 0.97 0.93 C.Autocorrelation Data 0.75 0.92 0.91 0.92 0.93 0.69 Model:baseline 0.75 0.67 0.64 0.84 0.84 0.87 Ctrfl: θfixed 0.75 0.75 0.43 0.94 0.86 0.93 Ctrfl: xvandxqcutoffsfixed 0.75 0.67 0.64 0.85 0.84 0.88 Ctrfl: xvcutofffixed 0.75 0.67 0.64 0.84 0.84 0.87 Ctrfl: xqcutofffixed 0.75 0.67 0.64 0.85 0.84 0.87 Notes:Thevariableyislaborproductivity;θislabormarkettightness;visvacancies;EPOP istheemployment-to-populationratio; ERistheemploymentrate(oneminustheunemploymentrate);PRistheparticipationrate.Variablesarequarterlyaveragesofmonthlyseriesexpressedinlog-deviationsfromtheHodrick-Prescotttrendwithsmoothingparameter 1,600.SeeAppendixAfordatasources. 46

TableC.4:BusinessCycleStatistics–TransitionProbabilities feu fen fue fun fne fnu A.Average Data:AZ-adjusted 0.014 0.014 0.228 0.135 0.022 0.021 Model:baseline 0.014 0.014 0.230 0.015 0.013 0.015 Ctrfl: θfixed 0.014 0.014 0.228 0.015 0.013 0.015 Ctrfl: xvandxqcutoffsfixed 0.014 0.014 0.229 0.015 0.014 0.015 Ctrfl: xvcutofffixed 0.014 0.014 0.229 0.015 0.014 0.015 Ctrfl: xqcutofffixed 0.014 0.014 0.229 0.015 0.013 0.015 B.Standarddeviation Data:AZ-adjusted 0.089 0.083 0.088 0.106 0.103 0.072 Data:DeNUNified 0.069 0.036 0.076 0.066 0.041 0.063 Model:baseline 0.011 0.002 0.036 0.002 0.027 0.013 Ctrfl: θfixed 0.001 0.003 0.000 0.006 0.002 0.004 Ctrfl: xvandxqcutoffsfixed 0.012 0.001 0.036 0.001 0.028 0.012 Ctrfl: xvcutofffixed 0.011 0.001 0.036 0.001 0.027 0.011 Ctrfl: xqcutofffixed 0.012 0.001 0.036 0.002 0.028 0.014 C.Correlationwithoutput Data:AZ-adjusted −0.630 0.430 0.760 0.610 0.520 −0.230 Data:DeNUNified −0.660 0.290 0.810 0.550 0.570 −0.560 Model:baseline −0.974 0.929 0.964 0.811 0.826 −0.982 Ctrfl: θfixed −0.362 0.939 −0.998 0.998 0.097 −0.998 Ctrfl: xvandxqcutoffsfixed 0.682 0.601 0.673 0.674 0.544 0.674 Ctrfl: xvcutofffixed 0.676 0.836 0.671 0.653 0.531 0.672 Ctrfl: xqcutofffixed −0.976 −0.518 0.961 0.796 0.861 −0.979 D.Autocorrelation Data:AZ-adjusted 0.590 0.290 0.750 0.620 0.380 0.300 Data:DeNUNified 0.700 0.220 0.850 0.580 0.480 0.570 Model:baseline 0.680 0.856 0.670 0.821 0.530 0.705 Ctrfl: θfixed 0.936 0.792 0.747 0.747 0.879 0.747 Ctrfl: xvandxqcutoffsfixed 0.682 0.601 0.673 0.674 0.544 0.674 Ctrfl: xvcutofffixed 0.676 0.836 0.671 0.653 0.531 0.672 Ctrfl: xqcutofffixed 0.688 0.718 0.672 0.807 0.550 0.707 Notes:Thevariable fijisthetransitionprobabilityfromlabormarketstateitoj;eisemployment; uisunemployment;n = 1−e−uisnonparticipation;AZisAbowd-Zellner. Variablesarequarterlyaveragesofmonthlyseriesexpressedinlog-deviationsfromtheHodrick-Prescotttrendwith smoothingparameter1,600.SeeAppendixAfordatasources. 47