Dynamic Beveridge Curve Accounting

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Dynamic Beveridge Curve Accounting Hie Joo Ahn and Leland D. Crane 2020-027 Please cite this paper as: Ahn, Hie Joo, and Leland D. Crane (2020). “Dynamic Beveridge Curve Accounting,” Finance and Economics Discussion Series 2020-027. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2020.027. 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.

∗ Dynamic Beveridge Curve Accounting HieJooAhn LelandD.Crane† March25,2020 Abstract WedevelopadynamicdecompositionoftheempiricalBeveridgecurve,i.e.,thelevelof vacanciesconditionalonunemployment. Usingastandardmodel, weshowthatthree factorscanshifttheBeveridgecurve: reduced-formmatchingefficiency,changesinthe jobseparationrate,andout-of-steady-statedynamics. WefindthattheshiftintheBeveridgecurveduringandaftertheGreatRecessionwasduetoallthreefactors,andeach factor taken separately had a large effect. Comparing the pre-2010 period to the post- 2010period,afallinmatchingefficiencyandout-of-steady-statedynamicsbothpushed thecurveupward,whilethechangesintheseparationratepushedthecurvedownward. Theneteffectwastheobservedupwardshiftinvacanciesgivenunemployment. Inpreviousrecessionschangesinmatchingefficiencywererelativelyunimportant,whiledynamicsandtheseparationratehadmoreimpact. Thus,theunusualfeatureoftheGreat Recessionwasthedeteriorationinmatchingefficiency,whileseparationsanddynamics have played significant, partially offsetting roles in most downturns. The importance of these latter two margins contrasts with much of the literature, which abstracts from one or both of them. We show that these factors affect the slope of the empirical Beveridgecurve,animportantfeatureinrecentwelfareanalysesestimatingthenaturalrate ofunemployment. ∗WethankKatharineAbraham,GianniAmisano,AndrewFigura,DavidRatner,andseminarparticipantsat the2019SOLEmeeting,theFall2018MidwestMacromeeting,andtheFederalReserveBoard. ViviGregorich provided excellent research assistance. Opinions expressed herein are those of the authors alone and do not necessarilyreflecttheviewsoftheFederalReserveSystemortheBoardofGovernors. †Ahn:FederalReserveBoardofGovernors,hiejoo.ahn@frb.gov.Crane:FederalReserveBoardofGovernors, leland.d.crane@frb.gov.

1 Introduction TheempiricalBeveridgecurve—thelevelofvacanciesconditionalonunemployment—has long been of interest to economists and policy makers. Interest intensified in the wake of theGreatRecession,asthecurveappearedtoshiftupwards(seeFigure1),fuelingconcerns about the functioning of the labor market. There is not currently consensus on the cause of this shift (or historical Beveridge curve shifts). Many papers have attributed the shift to falling matching efficiency (whether due to mismatch, duration dependence, recruiting intensity, heterogeneity, or other causes.) Other researchers have argued that mechanical out-of-steady state dynamics can account for the apparent shift. Finally, it has also been noted that variation in the employment separation rate can also produce shifts in the Beveridgecurve. Eachofthesethreadsoftheliteraturehastakenaslightlydifferentmodelling approach. Some authors use steady-state approximations, while others assume a constant jobseparationrate. Inthispaperweprovideanew,unifiedaccountingmodelfortheBeveridgecurveanda relateddecompositionmethod. Inourbaselinemodel,wherethelabor-forcestatusiseither employed or unemployed, there are three main factors that matter for the position of the Beveridge curve: (1) matching efficiency, (2) the job-separation probability, and (3) out-ofsteady-state dynamics. We analyze how much each of these factors shifted the Beveridge curve. The model allows us to estimate how the contribution of each factor changed in different recessionary and recovery episodes. We extend our model to include the laborforce participation margin, to see how important labor-supply factors are in the dynamics ofBeveridgecurve. We find that matching efficiency, job separations and out-of-steady-state dynamics are all important in understanding the shifts of the Beveridge curve over business cycles, particularly in the Great Recession. Out-of-steady state dynamics (defined below) produced a netupwardshiftintheBeveridgecurveduringandaftertheGreatRecession,assuggested 1

0.05 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 Unemployment Rate etaR seicnacaV 2019m11 NBER Peak: 2007m12 NBER Trough: 2009m6 Note:Monthlydata,2000-2019. Source:CurrentPopulationSurvey(U.S.CensusBureau),JobOpeningsandLaborTurnoverSurvey(U.S.Bureau ofLaborStatistics). Figure1: TheBeveridgecurve by Christiano et al. (2015) and Furlanetto and Groshenny (2016).1 Those papers assume a constant job separation rate (i.e., the rate at which workers lose their jobs and enter unemploymentisconstantoverthebusinesscycle),butwefindthatchangesinthejobseparation rate shifted the Beveridge curve sharply down on net in the later part of the Great Recession. This downward shift of the Beveridge curve partially offset the combined upward shifts from out-of-steady-state dynamics and matching efficiency. In fact, changes in the separationratewerethelargestsinglefactormovingtheBeveridgecurve. Our accounting exercise is conditional on the observed path of unemployment. When considering, say, acounterfactualpathformatchingefficiency, wecalculatethelevelofvacancies in each period that is consistent with the actual path of unemployment unemployment under counterfactual matching efficiency. Intuitively, conditional on a path for unemployment, lower matching efficiency require more vacancies to offset lower hiring rates 1SeealsoEichenbaum(2015)forrelateddiscussion. 2

conditional on tightness. Thus, lower matching efficiency shifts the level of vacancies (the Beveridgecurve)up. Perhaps less intuitively, a higher job separation rate will shift the Beveridge curve up. Higherseparationsimpliesmoreinflowstounemployment. Tokeepunemploymentatthe observed levels, vacancies must be higher, to absorb the extra workers. The job separation probabilitywashighinthedownswingoftheGreatRecession,anditlaterfellbacktomore normal levels in the recovery. This had the effect of shifting the Beveridge curve up in the downswing and down in upswing. Elsby et al. (2015) documented a similar point, though they did not quantify the extent of the shift or compare it to the other shifters.2 We also find that matching efficiency fell significantly during and after the Great Recession, which pushed the Beveridge curve up. This result is consistent with, e.g., Barnichon and Figura (2015).3 Analyseswhichignoreoneormoreoftheseshifterswilleitherfailtomatchthedataor will risk making mistaken inferences. This leads to several concrete conclusions and recommendations: First,theimportanceofout-of-steady-statedynamicsimpliesthattheusual flow steady-state approximations are not appropriate for studying the Great Recession, or similar periods of rapid change in the unemployment rate. Flow steady-state approximationshavebecomeafundamentaltoolforsimplifyingandunderstandingthelabormarket (see, for example, Fujita and Ramey (2009), Elsby et al. (2009), Shimer (2012), Barnichon et al. (2012), Elsby et al. (2015).) Unfortunately, in the Great Recession unemployment was consistently far from the steady-state value implied by inflows and outflows, thus the approximationispoorduringthisperiod. Similarly,Wefindalargeroleforout-of-steady-state dynamicsinsomepreviousrecessions. Second,time-variationinthejobseparationprobabilityiscriticalforunderstandingthe Beveridge curve, and indeed was the single largest shifter of the Beveridge curve in the 2HallandSchulhofer-Wohl(2018)alsonotedtheunemploymentinflowratecomplicatesthebehaviorofthe Beveridgecurve. 3SeealsoBarnichonandFigura(2010)andBarnichonetal.(2012)formoreonmatchingefficiency. 3

Great Recession. Thus, the common simplifying assumption of a constant separation rate (madein,e.g.,Christianoetal.(2015))isnotappropriatewhentryingtomodeltheBeveridge curve. In fact, we find that variation in the separation rate was an important shifter of the Beveridgecurveinmanypreviousrecessionsaswell,andthisvariationalsoaffectstheslope of the empirical curve. Our analysis does not speak directly to the debate over the relative importance of the separations versus the job findings for the evolution of unemployment (see, e.g., Fujita and Ramey (2009), Elsby et al. (2009), Shimer (2012), Ahn and Hamilton (2019)). Rather,wesimplypointoutthattheBeveridgecurvecannotbeproperlyunderstood withoutthisingredient. Third, we confirm that there was a clear fall in reduced-form matching efficiency in the Great Recession, as has been documented in several other papers (see Elsby et al. (2010), Barnichon and Figura (2015)). We show that this drop in matching efficiency shifted the Beveridge curve substantially and persistently upward in the Great Recession (though the other shifters partially obscure this effect.) In this paper we do not attempt to explain why matchingefficiencyfell, insteadweseektoquantifytheeffectsontheBeveridgecurveand theinteractionswithotherfactors.4 Though all three of these factors are crucial in understanding the Beveridge curve, we also find that the relative importance of each factor differed across recessionary episodes.5 We find that the 1990’s recession was similar to the Great Recession in that matching efficiency was the key factor to the persistent outward shift of Beveridge curve. However, in theotherrecessionsinthe1970’s,1980’sand2001,thejobseparationprobabilityandout-ofsteady-statedynamicsplayedmoreimportantrolesthanmatchingefficiency. In addition to clarifying the source of loops in the Beveridge curve, we show that these shifters affect the slope of the empirical Beveridge curve. This occurs because the curve is beingshiftedwhilelabormarketupswingsanddownswingsprogress,notjustatpeaksand 4Manypapershaveofferedexplanationsforthefallinreduced-formmatchingefficiencyamongthemDavis et al. (2013), Sahin et al. (2014), Elsby et al. (2015), Barnichon and Figura (2015), Kroft et al. (2016), Ahn and Hamilton(2019),andHallandSchulhofer-Wohl(2018). 5Dalyetal.(2011)andDiamondandSahin(2015)documenthistoricalBeveridgecurveshifts. 4

troughs. Thustheslopeofthesteady-stateBeveridgecurveunderconstantseparationsand constant matching efficiency is very different from the empirical slope. This has direct implicationsfortheworkofMichaillatandSaez(2019),whoexploittheslopeoftheBeveridge curve to estimate the efficient level of unemployment and the unemployment gap. A back of the envelope exercise shows that using an arguably more appropriate slope cuts the estimated unemployment gap in half, relative to Michaillat and Saez (2019). We view this as evidencethatmoreworkisneededtounderstandhowtime-varyingfactorsaffecttheslope oftheempiricalBeveridgecurve. Forourbaselineresults,weworkwithalog-linearizedBeveridgecurve,whichexpresses thevacancyratealinearfunctionofvariousfactors. Thisfirst-orderapproximationmatches the observed Beveridge curve quite well, and the factors and their associated coefficients are easily interpretable. This analytical tool makes it easy to trace out the contributions of factors to the shifts in the Beveridge curve, and trace out counterfactual curves that hold variousfactorsconstant. OnepossibleconcernisthatresultsbasedonaTaylorseriesapproximationcanbeinaccurate. In addition, under an approximate Beveridge curve the implied paths of vacancies will not be exactly consistent with the matching function and the law of motion for unemployment. To address this concern we perform similar decompositions, holding various factors constant, using the actual, non-linear Beveridge curve relation, and show that the results are nearly unchanged. Of course, when using the non-linear version the exact contributionsofeachmargindependontheorderingofthevariablesinthedecomposition. But theresultsarequalitativelyconsistentacrossallorderings. The next section introduces the basic model. Section 3 discusses the data. Section 4 linearizes the model and presents the results for the Great Recession. Previous recessions are covered in Section 5, and the results of a three-state model are discussed in Section 6. Section 7 concludes. Appendix A addresses the robustness of the linearized results by calculatingexactnon-lineardecompositions. 5

2 Model ThissectionderivesaversionofthesimpleBeveridgecurveframeworkusedinChristianoet al.(2015)(hereafterCET)andEichenbaum(2015),whichisnearlyidenticaltothatofElsbyet al.(2015). Wedonotclosethemodelbymakingassumptionsaboutthejobcreationprocess, wagedetermination,orotherfundamentals. Insteadwefocusonderivingconclusionsthat must hold for any general equilibrium model whose labor market is described by (1) the standardlawofmotionforunemploymentand(2)theusualmatchingfunctionrelationship. LetU betheunemploymentrateinmontht. Similarly,letV and H denotethevacancy t t t and hires rates both of which are normalized by the labor force. There is no on-the-job search, no participation margin, and the size of the labor force is constant and normalized tounity.Atthebeginningofeachperiodtheunemployedsearchforjobs,andthosethatfind matchesarehired. Attheendoftheperiodafractionofcontinuing(andnotnew)matches aredestroyed. Theflowofnewhiresinmontht, H ,isgivenbythestandardCobb-Douglas t matchingfunction: H = σU1−αVα (1) t t t t where α is the elasticity of the matching function and σ is matching efficiency, which can t varyovertime. Thenthejob-findingprobabilityisgivenby f = σ(V/U )α. (2) t t t t Thelawofmotionforunemploymentis U t+1 = s t (1−U t )− f t U t +U t (3) wheres isthejobseparationprobability. Substitutingequation(2)into(3)andrearranging t 6

wearriveat (cid:34) (cid:35)1/α V = s t (1−U t )−∆U t+1 (4) t σU1−α t t where ∆U t+1 = U t+1 −U t . ThisisaslightgeneralizationofCETequation5.2. WhereasCET assumethats andσ areconstants,wepermittime-variationintheseparameters. Notethat t t if s is set to its observed values and σ is chosen to verify equation (1), then equation (4) is t t anidentity. Equation (4) is at the core of our analysis. To understand it better, consider the case wheres ,σ andU areconstants: t t t (cid:20) s(1−U) (cid:21)1/α V = . (5) σU1−α This is the steady state Beveridge curve relationship at the core of textbook search models (seePissarides(2000)): asteady-statewithlowU musthavehighV,andvice-versa. Taking equation(5)asthereferencepoint,variationins t ,σ t and ∆U t+1 changesthelevelofV t given U . Thus,withaslightabuseofterminology,wewillrefertothesefactorsasshifters.6 t Given a path for unemployment and hypothesized, possibly counterfactual, values of the parameters (α,s ,σ), one can calculate the implied path of vacancies from equation (4) t t andcompareittothetruepathofvacancies. ThisistheessenceofourexercisesinSection4. 3 Data We require data on all the variables and parameters in equations (3) and (4). We use the standardapproaches, basedmostlyonShimer(2012)andBarnichonandFigura(2015). We setU asthenumberofunemployeddividedbythelaborforce,asmeasuredintheCurrent t PopulationSurvey(CPS).Weset V equaltothecountofvacanciesfromJobOpeningsand t 6WeuseshifterstomeanfactorsthatchangeVt givenUt. Notesandσalsoshiftthesteady-stateBeveridge curve(5),while∆U t+1 doesnot. Thedynamicscapturedby∆U t+1 produceloopsaroundthesteady-stateBeveridgecurve,butdonotchangethatmodel-basedrelationship. 7

Labor Turnover Survey (JOLTS), divided by the size of the labor force. Figure 2 plots the twoseries. Unemployment and Vacancy Rates Unemployment Rate 0.1 Vacancy Rate 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 Jan2000 Jan2005 Jan2010 Jan2015 Jan2020 Note:Monthlydata,2000-2019.NBERrecessionsshadedingray. Source:CurrentPopulationSurvey(U.S.CensusBureau),JobOpeningsandLaborTurnoverSurvey(U.S.Bureau ofLaborStatistics). Figure2: UnemploymentandVacancyRates We set the monthly job-finding probability, f as in Shimer (2012), using data on the t number of short-term unemployed each month.7 We then choose s to satisfy the law of t motion(3)exactly.8 Figure 3 shows the job finding and separation probabilities. It is notable that the job findingprobabilityfellbyabout50percentintheGreatRecessionandtheseparationprobability increased by about 50 percent.9 This suggests that both margins may have played a inm 7T o h n a th tis t , + w 1 e . s T e h t u ft s = ft 1 is − th U e t+ p 1 U − ro t U b t s + a 1 b , i w lit h y e t r h e a U ta t s + w 1 i o s r t k h e e r n u u n m em be p r lo o y f e w d o i r n ke m rs on u t n h em tfi p n lo d y s e a d jo fo b r b le y ss t+ tha 1 n . I fi n v t e h w e e d e a k ta s itispossibleforsuchaworkertobothfindandloseajob(ormultiplejobs)beforet+1,butthediscrete-time modelweuserulesoutthispossibility. 8InbothoursetupandthecontinuoustimeformulationofShimer(2012),jobseparationflowsaresetsoas to make the observed sequence of stocks consistent with the flows. In the three-state model of Section 6 the transitionratesaretakendirectlyfromthedataandadjustedtobeconsistentwiththestocks. 9Christianoetal.(2015)notethatthejobseparationrate,asmeasuredbyJOLTS,fellintheGreatRecession. TheJOLTSseparationrateincludesjob-to-jobflows,whichareknowntobehighlyprocylical,aswellasflowsto nonparticipation.Theirmodel,likeours,doesnotallowforjob-to-jobflows.TheJOLTSseparationrateislikely 8

significantroleintheevolutionofunemployment. Wewillconfirmthisimpressioninwhat follows. Monthly Job Finding Probability 0.5 0.4 0.3 0.2 0.1 0 Jan2000 Jan2005 Jan2010 Jan2015 Jan2020 Monthly Job Separation Probability 0.03 0.02 0.01 0 Jan2000 Jan2005 Jan2010 Jan2015 Jan2020 Note:Monthlydata,2000-2019.NBERrecessionsshadedingray. Source:CurrentPopulationSurvey(U.S.CensusBureau),JobOpeningsandLaborTurnoverSurvey(U.S.Bureau ofLaborStatistics). Figure3: ObservedTransitionProbabilities Measurementofαandσ requireestimationofthematchingfunction. Weruntheusual t regression (cid:18) (cid:19) V ln f = lnσ+αln t +ε (6) t t U t whereε isthemean-zeroerrorterm,σ = σexp(ε )istime-varyingmatchingefficiency,and t t t σisinterpretedasaveragematchingefficiency. Figure4plotsthelogjobfindingprobabilityagainstthelogV-Uratio. Thedatafordifferentperiodsareplottedindifferentcolors. Itisevidentthatmatchingefficiencydeteriorated significantlypost-2008. Anychangeinthematchingelasticity α wasminorbycomparison, thecorrectmeasurewhenconsideringthefirm’sproblem,sinceitgivestheexpecteddurationofthematch.But whenconsideringtheevolutionofunemploymentitisbettertousetheinflowtounemployment,ratherthan includingjob-to-jobflows. 9

so we will continue assuming that α is a constant throughout the paper (as is standard in theliterature). -0.6 -0.8 -1 -1.2 -1.4 -1.6 -1.8 -2 -1.5 -1 -0.5 0 0.5 Log V-U ratio ytilibaborP gnidniF boJ goL Matching Function Estimation Pre-2008 data 2008 data Post-2008 data Predicted, estimated on pre-2008 sample Note:Monthlydata,2000-2019. Source: Current Population Survey (U.S. Census Bureau), Job Openings and Labor Turnover Survey (U.S. BureauofLaborStatistics). Figure4: MatchingFunctionEstimation We run equation (6) on a sample starting in 2000 (when the JOLTS series begins) and ending in 2007, a period where it is plausible that σ was indeed constant. We also run the regression on a post-2008 sample. Table 1 presents the results. The point estimates put α near 0.3, very similar to the estimates of Shimer (2005) and Barnichon and Figura (2015), who use longer time series. It is evident that average matching efficiency fell about 25% betweenthetwosamples. 4 Linearization and Results Inorder tosimplifythe discussion, we log-linearizeequation(4). Inparticular, we takethe first order Taylor approximation around a point (U t ,s t ,σ t , ∆U t+1 ) = (cid:0) U,s,σ,0 (cid:1) . The result isthefollowingexpression 10

(1) (2) Pre-2008Sample Post-2008Sample lnσ −0.77*** −1.00*** (0.02) (0.01) α 0.27*** 0.34*** (0.03) (0.01) Notes: OLS estimates of average matching efficiency (lnσ)andthematchingfunctionelasticity(α).*,**,and *** indicate statistical significance at the 10%, 5%, and 1%levels,respectively.Standarderrorsareinparentheses. Table1: MatchingFunctionEstimates (cid:32) (cid:33) U 1−α (cid:0) (cid:1) lnV ≈lnV− + lnU −lnU t (cid:0) (cid:1) t α 1−U α U 1 1 − ∆ lnU t+1 + (lns t −lns)− (lnσ t −lnσ) (7) αs(1−U) α(1−U) α (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)(cid:124) (cid:123)(cid:122) (cid:125) Shiftdueto Shiftdueto Shiftdueto MatchingEfficiency Dynamics Separations (cid:0) (cid:1) whereV isequation(4)evaluatedat U,s,σ,0 . The first line of equation (7) is the (approximate) steady-state Beveridge curve. The secondlinecontainsthe“shifters”. TreatinglnV asalinearfunctionoflnU , theseshifters t t move the y-intercept of the steady-state curve up and down. For example, we can see that when unemployment is rising( ∆ lnU t+1 > 0) thenlnV t will be lower thanthe steadystate curve. Thisisbecause,allelseequal,risingunemploymentimplieslowfindingandthuslow lnV,whichistheout-of-steady-statedynamicsmechanismoutlinedinPissarides(2000). t Whileincreasingin ∆ lnU t+1 shiftslnV t down, increasesinthejobseparationprobabilitys shiftthecurveup. Theintuitionisthatahigherjob-separationprobability,conditional t onafixedvalueof ∆ lnU t+1 ,requiresmoreequilibriumvacanciestoabsorbtheunemployment inflows. Increases in matching efficiency σ obviously shift the curve down, as fewer t vacanciesareneededtorationalizetheobservedvalueof ∆ lnU t+1 . 11

We are interested in approximating the Beveridge curve around the Great Recession. To that end, we center the Taylor approximation around post-2007 averages. This yields U = 0.068, s = 0.020, and σ = 0.359 . We set ∆ lnU t+1 = 0 at the approximation point, whichisclosetoitspost-2007averageanyway. 4.1 Results Figure 5 plots the (log) observed Beveridge curve, the first order approximation, and the steady-state Beveridge curve. The approximate Beveridge curve, which includes all the (firstorder)effectsoftheshifters,followstheactualcurveclosely,asidefromabriefperiod near the trough of the Great Recession. Most importantly, the approximate curve shows nearly the same shift (between recession downswing and recovery) as the observed curve. The good fit of the linearized curve gives us confidence that our decomposition of the linearizedcurvewillalsobeaccuratefortheexactcurve. AppendixAaddressesanylingering concerns about the accuracy of the linearized results by calculating a series of nonlinear decompositionsontheexactBeveridgecurve. BoththeactualBeveridgecurveandtheapproximatecurvearesignificantlyflatterthan the steady state curve. In log space, the slope of the steady state curve is roughly −1−α = α −2.66, while the slope of the empirical curve is near unity. The difference in slopes is due to slow variation in the shifters, which pushed vacancies up as the Great Recession took hold,andthenpushedvacanciesdownintherecovery. Figure6plotsthetimepathsofthe threeshiftertermsinequation(7),alongwiththenetshift(theblackline),allnormalizedto be zero in April 2007. The blue line shows the shift in the Beveridge curve attributable to out-of-steady-statedynamics(thatis,− αs(1 U −U) ∆ lnU t+1 .) Theredandyellowlinessimilarly showtheshiftsduetoseparationsandmatchingefficiency. Relative to the pre-Great Recession period (say, 2007), the net effect of the shifters was to move vacancies sharply upward during the recession. This effect then dissipated very slowly, with the shifters returning to their pre-recession net value only in 2017. This com- 12

-2.8 -3 -3.2 -3.4 -3.6 -3.8 -4 -4.2 -3.4 -3.2 -3 -2.8 -2.6 -2.4 -2.2 Log Unemployment seicnacaV goL Observed 1st order approximation Steady State Beveridge Curve Note:3monthmovingaveragesofmonthlydata,2000-2019. Source: Current Population Survey (U.S. Census Bureau), Job Openings and Labor Turnover Survey (U.S. BureauofLaborStatistics). Figure5: BeveridgeCurves binedeffectexplainswhytheslopeoftheempiricalBeveridgecurveissomuchflatterthan thesteadystatecurve. WereturntothispointinSection4.2. Turning to each shifter separately, contribution of each factor is complicated and timevarying. Out-of-steady-statedynamicspushedtheBeveridgecurveinterceptsharplydown in the recession, and modestly up in the recovery, more or less the way Pissarides (2000) describes. The contribution of separations is roughly the opposite, raising the intercept sharply, especially late in the recession, and then eventually pushing the intercept down. Finally, the deterioration in matching efficiency raised the intercept during and after the recession. Figure 6 cannot clearly tell us which factors are responsible for the shift in the empirical Beveridge curve between the downswing and the upswing of the Great Recession. To understand that, we need to condition on a level of unemployment and examine the vertical shiftevidentinFigure5. 13

2 1.5 1 0.5 0 -0.5 -1 -1.5 Jan2006 Jan2008 Jan2010 Jan2012 Jan2014 Jan2016 Jan2018 Jan2020 seicnacaV goL Dynamics Separations Matching Efficiency Net Shift (relative to steady-state curve) Period of Peak Unemployment Note: 3 month moving averages of monthly data, 2000-2019. NBER recessions shaded in gray. Shifters are relativetoApril2007values. Source: Current Population Survey (U.S. Census Bureau), Job Openings and Labor Turnover Survey (U.S. BureauofLaborStatistics). Figure6: ShiftersoftheApproximateBeveridgeCurve Say that there were two months, t and t(cid:48), where observed unemployment rates were exactly equal, U t = U t(cid:48). Then using equation (7) we could decompose the (approximate) differenceinvacancies,lnV t(cid:48) −lnV t ,asfollows: lnV t(cid:48) −lnV t ≈ U 1 1 − (∆ lnU t(cid:48)+1 −∆ lnU t+1 )+ (lns t(cid:48) −lns t )− (lnσ t(cid:48) −lnσ t ) (8) αs(1−U) α(1−U) α (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)(cid:124) (cid:123)(cid:122) (cid:125) Shiftdueto Shiftdueto Shiftdueto MatchingEfficiency Dynamics Separations Equation (8) provides an additive decomposition of the vertical shift in the Beveridge curve. The portion of lnV t(cid:48) −lnV t due to, say, differences in matching efficiency between t and t(cid:48) is just the log difference in matching efficiency, lnσ t(cid:48) −lnσ t , multipled by 1/α. The shiftsduetodynamicsandseparationsaresimilar. Theonlywrinkleinimplementingequation(8)isthatweneverobservetwomonthswithexactlythesameunemploymentrate,so 14

-2.8 -3 -3.2 -3.4 -3.6 -3.8 -4 -4.2 -3.4 -3.2 -3 -2.8 -2.6 -2.4 -2.2 Log Unemployment seicnacaV goL Approximate Beveridge Curve Downswing Sample: April 2007 - June 2009 Upswing Sample: April 2010 - June 2017 Note:3monthmovingaveragesofmonthlydata,2000-2019. Source: Current Population Survey (U.S. Census Bureau), Job Openings and Labor Turnover Survey (U.S. BureauofLaborStatistics). Figure7: DownswingandUpswingSamples welinearlyinterpolateallrelevantseries. Asthereferencepoints,weselecttheunemploymentratesobservedbetweenApril2007 andJune2009. ThesearehighlightedinredinFigure7(the“downswingsample”). Wecomparethedownswingsampletotheupswingsample,whichbeginsinApril2010(highlighted in blue). For each of the downswing points, we calculate the vertical distance between observed vacancies and the (linearly interpolated) upswing vacancy levels. We also calculate eachofthetermsinequation(8). The result is Figure 8. The x-axis is the unemployment rate. For each unemployment rate, the black line shows the vertical distance between the upswing and downswing samples, as measured in log vacancies. This is the shift in the Beveridge curve we are trying to explain. The black line is the sum of the other three lines, which are the contributions inequation(8). Thereareseveralstrikingresults. First, thejob-separationprobabilityisresponsibleforalargeshiftdownintheBeveridgecurve. Thisisbecauseseparationsroseearly 15

2 1.5 1 0.5 0 -0.5 -1 -1.5 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 Unemployment Rate seicnacaV goL Dynamics Separations Matching Efficiency Net Shift (between downswing and upswing) Note:3monthmovingaveragesofmonthlydata,2000-2019. Source: Current Population Survey (U.S. Census Bureau), Job Openings and Labor Turnover Survey (U.S. BureauofLaborStatistics). Figure8: AccountingfortheVerticalShift 16

in the recession. Our accounting exercise is conditional on the path of unemployment, so risingseparationsimpliesahigherlevelofvacanciestokeepunemploymentatitsobserved values. Separations later fell, resulting in a lower path of vacancies during the recovery. This net shift is offset by the combined effects of dynamics and matching efficiency, which bothpushedthecurveup(onnet.) Interestingly,out-of-steady-statedynamicsplayedaprominentrole,withacontribution largerthanthatofmatchingefficiencyovermuchoftherange. ThisspecificresultisconsistentwithChristianoetal.(2015)’sargumentthat,becausetheGreatRecessionwassolarge and so sudden, dynamics can produce a realistic loop in the Beveridge curve. However, their analysis ignores the separation probability and matching efficiency, which are at least as important for understanding what happened. In particular, matching efficiency more than accounts for the net shift across most of the range, so without a change in matching efficiencytheBeveridgecurvewouldhaveshifteddown,notup. To summarize, all three of the factors we consider shifted the Beveridge curve in nontrivial ways. The vertical shift in the empirical Beveridge curve is the net result of out-ofsteady state dynamics and matching efficiency both shifting the curve up, an effect which ispartiallyoffsetbyalargenegativecontributionfromtheseparationprobability. Thetime paths of these shifters are complicated and non-monotonic, leading the slope of the empirical Beveridge curve to differ from the model-implied steady-state curve. We now turn to thisresultinmoredetail. 4.2 TheSlopeoftheBeveridgeCurve RecentinnovativeworkbyMichaillatandSaez(2019)(MS)hasemphasizedtheimportance of the Beveridge curve slope for welfare and the natural rate of unemployment. In this sectionweshowhowourmeasurementmethodsrelatetotheirresults. In many models with a matching function (e.g., Shimer (2005)), the Beveridge curve describes the possible steady-state values of vacancies and unemployment. In short, an 17

economy that sustains a lower level of unemployment must have more vacancies in equilibrium, and vice versa. MS point out that this relationship can be used to estimate the welfare-maximizinglevelofunemploymentinaparticularlysimpleandgeneralway. They notethatasocialplannerwillseektoequalizethecostsofadditionalvacanciestothecosts of additional unemployment. In other words, the social planner will seek the location on theBeveridgecurvewherethemarginalcostofadditionalunemploymentequalsthesocial valueoftheresultingreductioninvacancies. Thispointthendefinesthenaturalrateofunemployment, and the difference between observed unemployment and natural rate is the unemploymentgap. MSuseestimatesofthecostsofvacancies,thecostsofunemployment, andtheslopeoftheBeveridgecurvetomaketheircalculations. MS measure the slope of the Beveridge curve by estimating regressions of V on U in t t periods where the Beveridge curve appeared stable (dropping the troughs of recessions, for example.) As we show above, these observed slopes reflect both (1) movements along a stable Beveridge curve (changes in U for fixed separations, matching efficiency and dyt namics) and (2) time variation in the shifters. This second factor can distort the empirical Beveridgecurverelativetotheplanner-relevant, steady-statecurve. Forexample, consider a bare bones model where the separation probability and matching efficiency are exogenous processes, possibly correlated with the aggregate productivity shock. Such a model fitsinourframework(andthatofMS),andcouldproducetheobserveddata,includingthe empirical Beveridge curve and the paths of the shifts. However, a planner, facing such an economy, would not look to the empirical Beveridge curve to estimate the unemploymentvacancy tradeoff. The reason is that the empirical curve include the effects of the (purely cyclical)shifters,whiletheplannerisinterestedinlong-run,steadystaterelationships. The correctslopefortheplannercomesfromthelinearizedcurve(7),whichtreatstheshiftersas fixed: 18

(cid:32) (cid:33) U 1−α 1−α − + ≈ − (9) (cid:0) (cid:1) α 1−U α α and is determined by the shape of the matching function. The planner would make decisions based on the steady-state curve in Figure 5, not the empirical curve. Thus, in this toy example the empirical Beveridge curve does not directly give us the planner-relevant, long-runrelationshipweseek. The key question is whether the planner should incorporate the effects of the shifters when making a choice about the long-run level of unemployment. Clearly, out-of-steadystate dynamics are fundamentally transitory, so the planner should always purge the Beveridge curve of their effect. However, it is possible that the separation probability and matching efficiency are, to some extent, functions of the long-run level of unemployment (unlike in the toy example above). In this case the planner should not remove (all of) their influencewhencalculatingthevacancy-unemploymenttradeoff. Determining the exact nature of the variation in separations and matching efficiency is well beyond the scope of this paper. Instead, we provide an example to demonstrate that these issues can have an economically meaningful impact on welfare calculations. From equation (9), the slope of the steady-state curve (treating the shifters as fixed) is very close to −1−α. Averagingtogetherthetwoestimatesof α inTable1, wesetset α = 0.3, implying α a Beveridge curve slope of −2.33. This is far steeper than the estimates of MS, which are around−0.9forthesameperiod. Wecancalculatetheefficientlevelsofunemploymentusingequation(5)fromMS,based on our two estimates of the Beveridge slope (−2.33 and −0.9). In both cases we use MS’s preferred values for the costs of vacancies and unemployment. Figure 9 shows the results (this figure is comparable to Figure 3 Panel D in Michaillat and Saez (2019).) The blue line is the actual unemployment rate. The red line shows the efficient level of unemployment according to MS’s calibration, with a Beveridge slope of −0.9. The black line shows the 19

0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 Jan2000 Jan2005 Jan2010 Jan2015 Jan2020 tnemyolpmenU Actual Efficient, Beveridge slope=-0.9 Efficient, Beveridge slope=-2.33 Note:Calculationsbasedon3monthmovingaveragesofmonthlydata,2000-2019. Source: Current Population Survey (U.S. Census Bureau), Job Openings and Labor Turnover Survey (U.S. BureauofLaborStatistics). Figure9: EfficientUnemploymentBasedontheBeveridgeTradeoff efficientlevelofunemploymentusingourpreferredBeveridgeslopeof −2.33. Itisevident thatthesteeperBeveridgecurvesignificantlyraisestheefficientlevelofunemployment,as reducingunemploymentwithasteepBeveridgecurveismorecostlyintermsofvacancies. In our calibration the natural rate of unemployment fluctuates between about 4 percent and 6percent, nearthe rangeof other estimates including theCongressional BudgetOffice (CBO)’s short-term natural rate of unemployment. Notably, our calibrated estimate moves verysimilarlytotheCBO’sestimateduringthepostGreat-recessionperiod. Our results suggest that careful work is needed to disentangle which features of the Beveridge curve the planner should care about. These choices have real consequences for the measurement of efficiency, as Figure 9 shows. One approach is to specify a more complete model, which explicitly links separations and matching efficiency to the rest of the economy. Withsuchamodelinhand, onecoulddeterminetheplanner-relevantBeveridge curveslope. 20

5 Previous Recessions WecanalsouseourframeworktoanalyzerecessionspriortotheGreatRecession. Interms of data, the only change is that up through 2016 we use the composite vacancy series from Barnichon (2010) instead of JOLTS. After 2016 we continue the series by splicing on the JOLTS series. For four historical labor market downturns, we calculate the log-linearized Beveridge curve, as in Section 4. For each episode the curve is linearized around the local mean, to ensure a good fit. Figure 10 compared the observed and linearized Beveridge curves. The fit is generally good, although some of the linearized Beveridge curves show lessofashift,orcounter-clockwiseloop,thantheirobservedcounterparts. Weviewthisas atopicforfurtherinvestigation With the linearized Beveridge curves in hand, we can read off the implied contribution ofeachfactortotheshiftinthecurveateverypointintime. Figure11isanalogoustoFigure 6foreachrecession: thenetshiftintheBeveridgecurve,andthecontributions,asfunctions oftime. ItisapparentthatineachrecessiontheBeveridgecurveinterceptbeganshiftingup at the onset of the recession, and slowly drifted down once unemployment began falling. Rising separations usually drove this upward shift, partially offset by out-of-steady-state dynamics. It can be seen that in all recessions, out-of-steady-state dynamics shifted the Beveridge curvesignificantlydownintheinitialstagesandgenerallyupintherecovery. Interestingly, thisshiftispartiallyoffsetbythecontributionofseparations, which(asintheGreatRecession) tend to push Beveridge curve sharply upward in the initial stages of a recession and more moderately upward afterward. Thus the changes in the job-separation probability tendtoflattentheobservedBeveridgecurve,andcanceloutsomeofthecounter-clockwise loopthatout-of-steady-statedynamicsinduce. In most previous recessions, changes in matching efficiency had little impact, and were swamped by changes in the other factors. The 1990 recession appears to be an exception here. Duringthe1990recessionandtherecoveryperiod,thedeteriorationinmatchingeffi- 21

-3.2 -3.3 -3.4 -3.5 -3.6 -3.7 -3.8 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 Log unemplyoment seicnacav goL 1973 Recession -3.3 Observed Tayor approximation -3.4 -3.5 -3.6 -3.7 -3.8 -3.9 -4 -2.55 -2.5 -2.45 -2.4 -2.35 -2.3 -2.25 -2.2 -2.15 -2.1 Log unemplyoment (a) seicnacav goL 1981 Recession Observed Tayor approximation (b) -3.3 -3.4 -3.5 -3.6 -3.7 -3.8 -3.9 -2.9 -2.85 -2.8 -2.75 -2.7 -2.65 -2.6 -2.55 -2.5 -2.45 Log unemplyoment seicnacav goL 1990 Recession Observed -3.3 Tayor approximation -3.4 -3.5 -3.6 -3.7 -3.8 -3.9 -4 -3.3 -3.2 -3.1 -3 -2.9 -2.8 -2.7 Log unemplyoment (c) seicnacav goL 2001 Recession Observed Tayor approximation (d) Note:ObservedandapproximatedBeveridgecurvesforhistoricaldownturns.Log-linearTaylorapproximations aretakenaboutaveragesfortheperiodsplotted. Source: CurrentPopulationSurvey(U.S.CensusBureau),Barnichon(2010),andauthors’calculations. Figure10: ObservedandApproximateBeveridgeCurves 22

1973 Recession, Contrib. to shift 1 0.5 0 -0.5 Jan1974 Jan1975 Jan1976 Jan1977 Jan1978 seicnacaV goL 1981 Recession, Contrib. to shift 0.8 0.6 0.4 0.2 0 -0.2 -0.4 Jan1981 Jan1982 Jan1983 Jan1984 Jan1985 Dynamics Separations Matching Efficiency Net Shift (a) seicnacaV goL Dynamics Separations Matching Efficiency Net Shift (b) 1990 Recession, Contrib. to shift 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 Jan1990 Jan1991 Jan1992 Jan1993 Jan1994 Jan1995 seicnacaV goL 2001 Recession, Contrib. to shift 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 Jan2000 Jul2002 Jan2005 Jul2007 Dynamics Separations Matching Efficiency Net Shift (c) seicnacaV goL Dynamics Separations Matching Efficiency Net Shift (d) Note:Solidblacklineshowsthetimepathofthenetshiftofthe(log-linearized)Beveridgecurveintercept.NBER recessionsshadedingray. Source: CurrentPopulationSurvey(U.S.CensusBureau),Barnichon(2010),andauthors’calculations. Figure11: DecompositionsofApproximateBeveridgeCurves 23

ciencycontinuedtopushtheBeveridgecurveup, whichisquitesimilartowhathappened in the Great Recession. In fact, the two recessions are similar to each other in a sense that long-term unemployment continued to increase substantially after the recession was over. This suggests that mismatch or related factors might have been an important driver in the riseoflong-termunemploymentinthetworecessionepisodes. Weviewthislineofreasoningasatopicforfutureresearch. ThetentativeconclusionisthattheGreatRecessionwasexceptional,insofarasthedrop in matching efficiency had first-order effects on the Beveridge curve (with the possible exception of the early 1990s recession). In previous recessions matching efficiency usually playedlittlerole. However,themodestcounter-clockwiseloopsinpreviousrecessionwere notsimplytheproductofmodestout-of-steady-statedynamics,butwerethenetresultdramatic dynamics being offset by large contributions from the separations margin. Out-ofsteady-state dynamics and the separations margin played critical roles in all the recessions examinedhere. 6 Three State Model Theresultssofarhaveassumedthatallworkersareeitheremployedorunemployed. This is a simplification, since empirically flows into and out of the labor force are important for understanding total hires and evolution of unemployment. In this section we add the participationmarginanddiscusstherobustnessofourresultsintheexpandedmodel. 6.1 Model The population is still normalized to unity, but we add a nonparticipation state. Let N t be the stock of nonparticipants, so that E +U + N = 1. Consider the law of motion for t t t 24

unemploymentwhenworkerscanmoveintoandoutofthelaborforce: ∆U t+1 = E t +N t nu t −U t un t −U t ue t (10) Thetransitionratefromnonparticipationtounemploymentinmonthtisnu . Thetermsun t t and ue are similarly defined, with eu replacing s for consistency. The law of motion for t t t nonparticipationissymmetric: ∆N t+1 = E t en t +U t un t −N t ne t −N t nu t (11) Summing equations (10) and (11) yields an expression involving total hires (H = N ne + t t t U ue ) t t ∆U t+1 +∆N t+1 = E t eu t +E t en t −H t (12) wheretheflowsbetweenunemploymentandnonparticipationhavecanceled. Wecanwritethematchingfunctionas H = σ(U +ξNN )1−αVα (13) t t t t t t where ξN is the search effort of the nonparticipants relative to the unemployed. Thus the t effectivemassofsearchersisU +ξnN andσ continuestorepresentreduced-formmatching t t t t efficiency. Combiningequations(12)and(13),andassumingbalancedmatching(thatis,hiresfrom unemployment are a share Ut of total hires), we have the following expression for Ut +ξ t NNt vacancies: V = (cid:20) (1−U t −N t )(eu t +en t )−∆U t+1 −∆N t+1 (cid:21)1/α (14) t σ(U +ξNN )1−α t t t t When the non-employed can participate in job search, it is more sensible to think of a Bev- 25

eridge curve which relates vacancies to searchers (both unemployed and nonparticipants) insteadofunemployment. Tothisend,wedefinetwonewgroups. First,wedefinethepool ofsearchersS as t S = U +ξNN. (15) t t t t Second,wedefinethepoolof“truenonparticipants”as (cid:16) (cid:17) N˜ = 1−ξN N. (16) t t t Whilewetakenostandonwhether ξN isthefractionofnonparticipantswhosearchorthe t searcheffortofeachnonparticipantrelativetotheunemployed,theformerinterpretationis convenient here. Note that if ξN = 1 all the nonparticipants search and N˜ = 0. Using S t t t and N˜ ,wecanwrite(14)as t V = (cid:34)(cid:0) 1−S t −N˜ t (cid:1) x t −∆S t+1 −∆N˜ t+1 (cid:35) α 1 (17) t σS1−α t t where x = eu +en isthetotaljob-separationprobability. Log-linearizingyields t t t 1 lnV = − [lnσ −lnσ ] t t 0 α (cid:40) (cid:41) (1−α) 1 S − α + α (cid:0) 1−S 0 −N˜ (cid:1) [lnS t −lnS 0 ] 0 0 (cid:40) (cid:41) 1 S − α (cid:0) 1−S − 0 N˜ (cid:1) x [∆ lnS t+1 ] 0 0 0 (cid:40) (cid:41) − α 1 (cid:0) 1−S N˜ 0 −N˜ (cid:1) (cid:2) lnN˜ t −lnN˜ 0 (cid:3) 0 0 (cid:40) (cid:41) − α 1 (cid:0) 1−S N˜ − 0 N˜ (cid:1) x (cid:2)∆ lnN˜ t+1 (cid:3) 0 0 0 1 + [lnx −lnx ] (18) t 0 α 26

Likeequation(4),equation(17)canbeusedtoanalyzetheBeveridgecurve. Thisdecomposition, naturally, has more shifters than the two-state model. In this model movements alongtheBeveridgecurvearecapturedbythelnS −lnS term,sincethecurveisdefinedin t 0 termsofsearchers,notmerelytheunemployed. Theeffectsofmatchingefficiencyandseparationsstillappear,onthefirstandlastlinesofequation(17)respectively. Finally,thereare now two out-of-steady state terms, ∆ lnS t+1 and ∆ lnN˜ t+1 , as well as a term capturing the levelofnon-searchers,lnN˜ −lnN˜ . Notallofthesetermshaveatransparentinterpretation, t 0 butasweshallseebelow,manyofthemarenotquantitativelyimportanteither. 6.2 Data Toimplementthethreestatemodel,weneeddataonthetermsappearinginequation(14). We obtain the stocks of employed, unemployed, and nonparticipants from the CPS labor force status flows.10 We normalize these stocks to satisfy E +U + N = 1 in all periods. t t t The transition rates eu ,nu ,un ,ue are also taken from the labor force status flows. These t t t t transition rates are not exactly consistent with the stocks, due to missing month-to-month linkagesandsamplerotation. Weadjustfortheinconsistencybyrakingtheratesuntilthey areconsistentwiththestocks.11 Thisresultsinverysmalladjustmentstothetransitionrates. Undertheassumptionofbalancedmatching, ξN canbeidentifiedbytheratiooftransit tionratestoemployment: ne ξN = t t ue t Finally, αandσ canbeidentifiedbythematchingfunctionregression,usingU +ξNN t t t t asthepopulationofeffectivesearchers. 27

-3.6 -3.8 -4 -4.2 -4.4 -4.6 -4.8 -2.4 -2.3 -2.2 -2.1 -2 -1.9 -1.8 Log Searchers seicnacaV goL Observed Beveridge Curve Approximate Beveridge Curve Note:4monthmovingaveragesofmonthlydata,2000-2019.Morerecentmonthsareshadeddarker. Source: Current Population Survey (U.S. Census Bureau), Job Openings and Labor Turnover Survey (U.S. BureauofLaborStatistics). Figure12: ThreeStateApproximateBeveridgeCurve 6.3 Results Figure 12 shows that, as with the two state model, the three state approximate Beveridge curve is a good approximation of the observed curve. Here “searchers” are the pool of actively searching workers, U +ξNN. To show the direction of time, more recent periods t t t areshadeddarker. Figure 13 shows the shifters as a function of time, similar to Figure 6 the story is similar to the two-state model. Matching efficiency slowly and steadily pushed the Beveridge curveupwardsduringandaftertheGreatRecession. Theseparationprobability,x ,pushed t the Beveridge curve up during the recession, but this was short-lived. The out-of-steadystatedynamicsterms,onnet,pushedthecurvedown,thoughinterestinglythe ∆N˜ t+1 term partially offsets the ∆S˜ t+1 term. Strikingly, there is no shift in the Beveridge curve under 10Accessibleathttps://www.bls.gov/webapps/legacy/cpsflowstab.htm. 11Thisisalsocallediterativeproportionalfitting: alternatelyscalingeachrowandcolumnofthetransition matrixtomatchthestocksuntiltheyconverge. 28

0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 Jan2006 Jan2008 Jan2010 Jan2012 Jan2014 Jan2016 Jan2018 Jan2020 seicnacaV goL Note: 4 month moving averages of monthly data, 2007-2019. NBER recessions shaded in gray. Shifters are relativetoApril2007values. Source: Current Population Survey (U.S. Census Bureau), Job Openings and Labor Turnover Survey (U.S. BureauofLaborStatistics). Figure13: ThreeStateModel-ShiftersoftheApproximateBeveridgeCurve constantmatchingefficiency. Thisconfirmstheresultsfromthetwostatemodel(andmuch oftheliterature)thatthedeclineinmatchingefficiencywasanimportantcontributortothe loopintheBeveridgecurve. 7 Conclusion The empirical Beveridge curve is easy to calculate, as it only requires data on the stocks of unemployed workers and job openings. This ease of measurement may help explain the attentionithasreceived. Unfortunately,theBeveridgecurveis(eveninasimplemodel)the product of multiple factors, and can be difficult to interpret. Our hope is that our results helpclarifythebehavioroftheBeveridgecurveandreconcilesomeconflictingideasinthe literature. Wehaveshownthatreduced-formmatchingefficiency, changesintheseparationprobability,andout-of-steady-statedynamicsallplayedimportantrolesintherecentshiftofthe 29

Beveridge curve. Comparing the pre-2010 period to the post-2010 period, out-of-steadystate dynamics and a fall in matching efficiency both pushed the curve upward, while the changes in the separation probability pushed the curve downward. The net effect was the observedupwardshiftintheempiricalBeveridgecurve. Ourresultsarelargelyunchanged when we include a nonparticipation margin. One area for more research is the effect of on-the-jobsearch,whichwouldaffectthemeasurementofmatchingefficiency. A realistic model of the Great Recession therefore needs, (1) a mechanism for reducedformmatchingefficiencytofallduringandaftertherecession,(2)anon-constantseparation probability, which can generate an increase in job losses towards the end of the recession. Furthermore,modelsshouldnotbeevaluatedusingsteady-stateapproximations,sincethe rapidchangesinthelabormarketaroundtheGreatRecessionmadeout-of-steady-statedynamicsafirst-orderissue. We reach similar conclusions regarding earlier recessions, though the role of matching efficiency is generally smaller. Importantly, the relatively small Beveridge curve loops in earlierrecessionsweretheproductofchangesintheseparationprobabilitynearlyoffsetting out-of-steady-statedynamics. WefindthattheseshiftersmovetheinterceptoftheBeveridge curve continuously, not just at business cycle peaks and troughs. As a result, the slope of theempiricalBeveridgecurveisdistinctfromtheslopeoftheimplied(constantseparation probability,constantmatchingefficiency)steady-statecurve. 30

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A Full Decompositions OnemaybeconcernedthatresultsbasedontheTaylorapproximationarenotrobust. While the fit of the approximate Beveridge curve is strikingly good, it is not perfect. Therefore, there is some room for non-linearities to affect the results. A related issues is that the loglinearizedBeveridgecurveisnotdynamicallyconsistent: Ifweplugimpliedvacanciesinto the matching function and the unemployment law of motion, we generally won’t get the observedU t+1 back. In this section we decompose the shift in the empirical Beveridge curve using the exact vacancy equation rather than the log-linearized version. Again, the goal is to measure the contributionstotheshiftdueduetoout-of-steady-statedynamics,changesintheseparation probability,andchangesinmatchingefficiency. The starting point of our decomposition is the standard, steady-state Beveridge curve, withconstantmatchingefficiencyandseparations: (cid:34) (cid:35)1/α Vs,σ,∆U = s(1−U t ) (19) t σU1−α t ThesteadystateBeveridgecurvesets ∆U t+1 = 0. Itthereforethelevelofvacanciesthat wouldprevailaftermanymonthsofconstantsandσ. Let tdown be a month from the downswing sample, and let tup be the corresponding (interpolated) period from the upswing with the same level of unemployment. Then the observed vertical shift in the Beveridge curve is V −V . The steady-state Beveridge up down curve(19)obviouslyentailsnoshift,soVs,σ,∆U −Vs,σ,∆U = 0. up down We can define other counterfactual vacancy series. We use superscripts with bars to denotethatthemarginisbeingheldconstant. Thus,forexample, 34

(cid:34) (cid:35)1/α Vσ = s t (1−U t )−∆U t+1 (20) t σU1−α t (cid:34) (cid:35)1/α Vσ,∆U = s t (1−U t ) (21) t σU1−α t withVs,Vs,σ,Vs,∆U ,andV ∆U definedsimilarly. t t t t Next,considertheaccountingidentity (cid:16) (cid:17) V −V = (cid:0) V −V (cid:1) − Vσ −Vσ up down up down up down (cid:16) (cid:17) (cid:16) (cid:17) + Vσ −Vσ − Vs,σ−Vs,σ up down up down (cid:16) (cid:17) (cid:16) (cid:17) + Vs,σ−Vs,σ − Vs,σ,∆U −Vs,σ,∆U . (22) up down up down This writes V −V as three double differences. The terms on the right hand side up down havethefollowinginterpretation: (cid:16) (cid:17) (cid:0) (cid:1) • V −V − Vσ −Vσ : TheshiftintheBeveridgecurveaccountedforbythe up down up down time-variation in matching efficiency, conditional on having s and ∆U at their obt t servedvalues. (cid:16) (cid:17) (cid:16) (cid:17) • Vσ −Vσ − Vs,σ−Vs,σ : Theshiftaccountedforbytime-variationinthesepup down up down aration probability, conditional on having ∆U at its observed values and σ held cont stant. (cid:16) (cid:17) (cid:16) (cid:17) • Vs,σ−Vs,σ − Vs,σ,∆U −Vs,σ,∆U : The shift accounted for by time-variation in up down up down ∆U t+1 ,conditionalonhavingmatchingefficiencyandtheseparationprobabilityheld constant. NotethatVs,σ,∆U −Vs,σ,∆U = 0byconstruction. up down Thus,wecaninterpretequation(22)asmovingusfromthesteady-stateBeveridgecurve 35

Ordering Dynamics Separations Matching ∆U t+1 ,s,σ 115.26 −177.18 161.92 ∆U t+1 ,σ,s 115.26 −444.82 429.55 s, ∆U t+1 ,σ 121.46 −183.38 161.92 s,σ, ∆U t+1 213.19 −183.38 70.19 σ, ∆U t+1 ,s 205.80 −444.82 339.01 σ,s, ∆U t+1 213.19 −452.21 339.01 Notes:PercentagepointcontributionstotheverticalshiftintheBeveridge curve, averaged over the “downswing” sample points discussedearlier.“Ordering”columnshowstheorderinwhichmargins are set to their observed values. For example, the row ∆U t+1 ,s,σ starts with the steady-state curve, then adds observed ∆U t+1 , then addsobservedst,andfinallyaddstheobservedσt. Table2: ContributionstotheShiftintheBeveridgeCurve (which cannot shift by construction) to the observed shift, by successively adding the observed time-variation in margins. Equation (22) first adds observed dynamics, then adds observed the separation probability, then adds observed matching efficiency. With three margins there are six possible orderings, and the results will, in general, depend on the ordering. Table2showstheresultsofallsixorderings. Theresultsareremarkablyconsistent. Inall versions, separations push the Beveridge curve down during the upswing period, relative tothedownswingperiod. Bothdynamicsandmatchingefficiencyhavetheoppositeeffect, contributing to the counter-clockwise loop in the observed Beveridge curve. Generally, the contributionofmatchingefficiencyislargerthanthatofdynamics,sometimesdramatically so. The only outlier is the fourth row. However, we believe that the first two rows are the most important, because they put ∆U t+1 first in the ordering, which ensures dynamic consistency. Nearly all of the contributions in Table 2 are well above 100 percent. This shows just howimportantallthreemarginsareinunderstandingtheshiftoftheBeveridgecurve. The shiftweobserveempiricallyisrelativelysmall,whencomparedtotheeffectsoftheshifters takenseparately. 36