The Review of Economic Studies, Ltd.

Persistence of Employment Fluctuations: A Model of Recurring Job Loss Author(s): Michael J. Pries Reviewed work(s): Source: The Review of Economic Studies, Vol. 71, No. 1 (Jan., 2004), pp. 193-215 Published by: Oxford University Press Stable URL: http://www.jstor.org/stable/3700716 . Accessed: 21/05/2012 21:12 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

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Review of Economic Studies (2004) 71, 193-215 ? 2004 The Review of Economic Studies Limited

0034-6527/04/00090193$02.00

of

Persistence

Fluctuations: A Job

Employment Model of Recurring Loss

MICHAEL J. PRIES Universityof Maryland First version receivedDecember 2000; final version accepted July 2002 (Eds.) Standardmodels of employmentfluctuationscannotreconcile the unemploymentrate'sremarkable persistence with the high job-finding rates found in workerflows data. A matching model emphasizing high hazardrates among newly formed firm-workermatches can resolve this shortcoming.In the model, matchesareexperiencegoods; consequently,newly employedworkersface higherhazardrates.Following a job loss, workers may experience several short-livedjobs before finding stable employment. At an aggregatelevel, an initial burstof job loss precipitatesa steady flow of recurringjob loss. A simulation shows thatthis recurringjob loss can accountfor the fact thatthe unemploymentrateremainselevatedfor as much as 4 or 5 years following an initialjump.

1. INTRODUCTION Intuitively,the labourmarketfrictions associated with the difficultand time-consumingprocess of matchingunemployedworkerswith appropriatejobs representan appealingway to enhance conventionaldynamicgeneralequilibriummodels of economic fluctuationsthatgenerallyexhibit very little internalpropagationof business cycle shocks.1 Nevertheless, althoughconventional matching models, such as the canonical Mortensenand Pissarides (1994) model, successfully explain many other importantfeaturesof the labourmarket,they fail to explain the high degree of persistence observed in the U.S. unemploymentrate. The primaryreason for this failure is that dataon labourmarketflows show thatunemployedworkersactually findjobs quite quickly. Therefore,following a shock that triggers a burst of job loss, the high job-finding rate implies that the unemploymentrate in these models returnsquite rapidlyback towardits averagelevel. This representsa significantobstacle to the notion that frictionsin the labourmarketaccountfor much of the propagationof business cycle shocks. This paper asserts that existing matching models fail to capture an importantfeature of the labour marketthat can reconcile the dramaticpersistence of the unemploymentrate with the evidence of high job-finding rates. Specifically, although unemployed workers find jobs quickly,the newly foundjobs often last only a shorttime. After an initialjob loss, a workermay experience several short-livedjobs before settling into more stable employment.This recurring job loss slows the pace at which the unemploymentrate returnsto its normallevel following an adverseshock. This idea is formalized in a model that integratesa signal extractionproblem-similar in spiritto the one in Jovanovic(1979b)-into a relativelystandardmatchingenvironment.Whereas 1. Cogley and Nason (1995) and Rotembergand Woodford(1996), for example, emphasize the lack of internal propagationmechanisms in standardreal business cycle models, in the sense that the time-series properties of the importantmacro aggregatesin those models closely resemble the time-seriespropertiesof the shocks themselves. 193

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matchingmodels such as the Mortensenand Pissarides(1994) model emphasizethe uncertainty and learningaboutmatchqualitythatoccurs duringthe job-matchingprocess (as capturedin the matchingfunction),the model presentedhere also highlightsthe importanceof uncertaintyabout matchqualitythatremainsfollowing the creationof a worker-firmmatch.In particular,matches are experiencegoods: the matchingprocess that brings togetherworkersand employers fails to weed out all bad matches, so thatworkersand firmsmust resolve uncertaintyabouttheirmatch's qualityby observingits productivityover time. To understandthe process by which firms and workersin the model determinethe quality of their matches, imagine that the observed outputproducedby a match, yt, is the sum of two unobservedcomponents,y and Mtt.The firstcomponent,y, representsthe fundamentalqualityof the worker-firmmatch.Factorsthatdeterminey might include the degree to which the worker's skills mesh with the firm's particulartechnology, or the ability of the workerto get along with the firm'sother employees. The second component,ktt, reflects transitoryfactors that affect the match.For example, the workermay have an argumentwith the manageror the demandfor the match's outputmay dip. When one of these events occurs, the workerand firm cannot ascertain whetherits impactis temporaryor whetherthe event reveals somethingmore fundamentalabout the match'squality. Faced with a signal extractionproblemof this sort, the workerand firmuse each additional observationof Ytto update their belief about the true underlyingvalue of y. If the information from the observedvalues of Ytleads the workerand firmto believe that their match is bad, they choose to separateand searchfor a new match. Otherwise,they continueproducing. Because matchesdeterminedto be bad areterminated,the probabilitythata survivingmatch is actuallygood-and thus will not be terminated-increases with tenure.Put differently,hazard rates decline with tenure.This negative relationshipbetween tenure and workers' hazardrates, absent in standardmodels, has importantimplications for aggregate fluctuations.Because the employment-to-unemploymenthazardrate is high early in a job, workers who lose their jobs following an adverse shock to the economy may subsequentlypass throughseveral short-lived jobs before settlinginto more stableemployment.Consequently,the unemploymentratedeclines very slowly towardits averagelevel. The goal, then, is to explore whethera matchingmodel that incorporatesrecurringjob loss can generatemore realisticpersistenceof employmentfluctuations,in spite of a high job-finding rate. If so, then search and matching models of employment fluctuationscan indeed provide an importantinternal propagationmechanism-the economy's persistent response to adverse shocks can be explained within our models, and not simply as the result of persistence in the shocks themselves. A brief review of the related literatureis useful at this point. Cole and Rogerson (1999) provide a rigorous examination of the canonical Mortensen and Pissarides (1994) matching model and of its ability to explain key business-cycle facts. Their primaryconclusion is that the model performs best if the average duration of a non-employmentspell is 9 monthsin other words, if the monthly job-finding rate is far below the rate found in the data (0-24). From the perspectiveof the presentpaper,this is not surprisingsince the Mortensen-Pissarides model completely precludes the tenure effects that can generate episodes of recurringjob loss, due to the assumption that a match's productivityfollows a stochastic process that is entirely memoryless. Whereas Cole and Rogerson suggest that the lower job-finding rate can be justified if the definition of unemployment is broadened to include workers with lower degrees of labourmarketattachment,the presentpaper suggests that the lower job-finding rate needed in the Mortensen-Pissaridesmodel, which precludes recurringjob loss, is appropriate if it is instead interpretedas the rate at which workers sort themselves back into enduring matches.

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The appeal of labour market frictions as a primary propagationmechanism has been explored in the context of dynamic stochastic general equilibriummodels by Merz (1995), Andolfatto(1996), and den Haan,Ramey and Watson(2000). Both Merz (1995) and Andolfatto (1996) show that searchfrictions enhance propagationrelativeto standardRBC models, but an unrealisticallylow job-finding rate is needed to generatethe observed degree of persistence in aggregatefluctuations.In den Haan et al. (2000) the interactionof search frictions and costly capitaladjustmentis shown to producesignificantlymore propagation. With regard to different theories of why hazard rates decline with tenure, Jovanovic (1979a,b) proposes two different explanations-the learning mechanism considered here and match-specifichumancapital accumulation.Nagypial(2000) uses a worker-firmmatcheddataset to measurethe relativeimportanceof the two mechanisms,and findsthatthe dataindicatethat learningis more important.Pissarides(1994) examinesa model with tenureeffects in which only low-tenureworkersengage in on-the-jobsearch,due to job-specific humancapitalaccumulation. While this type of turnoveramong low-tenureworkersis consistent with the recurringjob loss emphasized in the present paper, it clearly cannot explain unemploymentrate dynamics since job-to-job transitionscompletely bypass unemployment. The idea thathigh turnoveramonglow-tenurejobs may constitutean importantpropagation mechanism was considered in a simple probabilisticmodel in Hall (1995). A recent paper by Chatterjeeand Sill (2001) has joined this paper in attemptingto more thoroughlyexamine the idea in the context of modem search and matching models. In their paper,high turnoverrates amonglow-tenureworkersarisebecause marketsare incompleteand workerscannotinsuretheir consumptionwell against unemploymentrisk. As a result, workersare more impatientin their searchand accept lower qualitymatches-which are thus less likely to endure-than they would be undercomplete markets. Pries and Rogerson (2001) examine how dismissal costs, unemploymentinsurance, and wage compressionpolicies can explain cross-countrydifferences in firms' hiring practices and in the importance of recurringjob loss in a country's overall labour market turnover.For example, in countriesin which it is expensive to dismiss workers,firms are less likely to engage in the type of "hire then screen" approachprevalentin the model of the present paper, but instead are likely to expend relatively greater resources on screening workers before hiring them. Consequently,countrieswith higher dismissal costs should display longer unemployment durations,higherunemploymentrates,fewer workersin low-tenurejobs, andlower employmentto-unemploymenthazard rates. More generous unemployment insurance and greater wage compressionhave similarimplications.Pries and Rogerson (2001) show that evidence from the U.S., Canada,and variousEuropeancountriesgenerally supportthese implications, suggesting that in less regulatedlabourmarketslike that of the U.S., high turnoveramong newly matched workersis indeed importantfor understandinglabourmarketoutcomes. The next section of this paperpresentsa simple frameworkfor understandingthe challenges of developing a matching model that exhibits significantunemploymentpersistence. It argues that a successful model must incorporaterecurringjob loss. Section 3 describes the model's economic environment.A simple learningprocess is achieved by assuming that there are only two types of matches, "good"and "bad",and by assuming that the noisy element of a match's output (Ltt above) is uniformly distributed.Moreover,in order to isolate the role of recurring job loss, the model is set up so that the job-finding rate remains constant in the face of the shocks thatcause fluctuationsin the unemploymentrate.These simplifying assumptionsprovide the model with a straightforwardequilibrium,which is examined in Section 4. The main result from Section 4 is a stochastic vector difference equation that describes the flows of workers between threelabourmarketstates:unemployment,employmentin a matchof unknowntype, or employmentin a match known to be good. From this, the model's unemploymentrate is shown

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to follow approximatelyan ARMA(2,1) process, the coefficients of which are functions of the model's parameters. Section 5 quantitativelyexplores the model's success in generatingrealistic unemployment rate persistence. Simulations show that, for parametervalues that imply a realistic hazard function, the recurringjob loss introducedby the model significantlyincreases unemployment rate persistence relative to existing models. Section 6 examines the validity of additional implicationsof the model, such as the earningslosses experiencedby workersbefore and after primaryjob losses. 2. THE CHALLENGEOF PERSISTENCEIN MATCHINGMODELS A reduced-formdescription of the behaviourof worker flows in a typical matching model is useful for understandingwhy such models struggleto generateunemploymentrate persistence on par with what is observedin the data.As a startingpoint, considera simple matchingmodel in which unemployedworkersface a constantjob-finding rate, f, and employed workersface a constantseparationrate, s. In such a model, the change in the unemploymentrate is given by the fractionof employed workerswho lost jobs last period minus the unemployedworkerswho foundjobs: Ut

-

Ut-l

= S(l -

Ut-l)

- fut-l.

(1)

In the absence of any type of disturbance,the unemploymentrate convergesto u = s+f To see how the unemploymentrate would adjustfollowing a shock, rearrangeequation(1) and add an i.i.d. innovation6t, which representsa stochasticflow of separationsnot accounted for by the constantseparationrate s, to get Ut -U = (1 - s - f)(ut-i

- U) + t.

(2)

This equation suggests that the unemploymentrate is an AR(1) process with a coefficient of 1 - s - f. Following a positive shock Et, the unemploymentrate should decline each period 1 - s - f per cent of the way towardits long-runmean, u. Using gross workerflows data from the CurrentPopulationSurveys (CPS), Blanchardand Diamond (1990) estimate an averagemonthlyjob-findingrate, f, equal to 0.24. Their estimate for the employment-to-unemploymenthazardrate, s, is 0.013, implying, for this simple model, an averageunemploymentrateof u = 0.05. It follows thatthe estimatedAR(1) coefficient should be 1 - s - f = 0.747. However, the first autocorrelationof the monthly U.S. unemployment rate is in fact much higher-it exceeds 0-95 even after filteringout low-frequencymovements associated with changes in the naturalrate. Put differently,while the unemploymentrate in a simple model with constants and f has a half-life of just a few months, U.S. data indicatethat the unemploymentrate's half-life actually exceeds a year. Given the failure of this simple baseline case with a constantjob-findingrate and constant separationrate, what modificationscould make a matchingmodel accordbetter with the facts? One simplistic answer would be that shocks are in fact not i.i.d.-ut has a high autocorrelation because Ethas a high autocorrelation.However,the spiritof the approachtakenin this paperis to examine how much persistence can be endogenously explained by models of labourmarket frictions.While some persistencein economic fluctuationsmay indeed arise for reasons outside the labourmarket,to simply assume persistentshocks is tantamountto giving up on the notion thatlabourmarketfrictionsare an importantpropagationmechanism. Anotherpossibility is thatthe simple model above fails because it ignores workersclassified as out of the labourforce. Supposethe immediateimpactof an adverseshock were to cause many workersto leave the labourforce. As they graduallychose to seek work again, the flow from out

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of the labourforce into unemploymentwould slow the unemploymentrate's decline. Although logically sound, the data suggest thatthis mechanismis not of first-orderimportance.2 Alternatively,the failureof the simple model may lie in the fact thatthe transitionrates are not constantin reality.More specifically,the unemploymentrate's persistence could be greater in a matchingmodel with a procyclicaljob-findingrate. If the job-findingrate fell following an adverseshock, the unemploymentrate would returntowardnormallevels more graduallythanif the job-findingrateremainedfixed. Although models that exhibit this type of labour marketsluggishness-Pissarides (1992) and Shimer (1997) provide two examples-agree with the commonly held notion that jobs are significantly harderto find during recessions than in good times, evidence from worker flows data and job flows data seem to indicate that in fact job-finding rates are only slightly procyclical.For example, Blanchardand Diamond (1990) estimatethat the job-findingrate falls by an averageof only 0-4 percentagepoints duringthe first 6 months afteran adverseshock that raises the unemploymentrate by 0.3 percentagepoints. Based on their analysis of CPS worker flows data and LongitudinalResearchDatabasejob flows data, Blanchardand Diamond (1990, p. 104) conclude that "[t]he increase in the flow out of employment accounts for more of the contractionin employmentthan does the decrease in the flow into employment.This remains true throughoutthe adjustmentprocess".In other words, recessions are better characterizedas periodsof increasedlabour marketturnoverthan as periods when labourmarketsare sluggish. Although more evidence on the cyclical behaviourof job-findingrates is needed,3 the fact remains thatjob-finding rates are high (averageunemploymentspell durationsare short) at all phases of the business cycle and fluctuationsin the job-finding rate alone cannot come close to explaining the high degree of persistence found in the U.S. unemploymentrate. The reason behind the unemploymentrate's extremelyhigh autocorrelationmust lie elsewhere. This points in the directionof a model like the one presentedhere, in which fluctuationsin s are an important source of unemploymentratepersistence. According to Blanchardand Diamond's (1990) estimates, an adverse shock that increases the unemploymentrate by three percentage points causes the employment-to-unemployment hazard rate to rise during the first 6 months following the shock from its average value of 1.29 to 1.89% (a 30% increase). Hall (1995) provides further evidence that a shock to the economy precipitatesa protractedperiod of above normal separations.That paper shows that there is a strong distributedlag relationshipbetween job destructionin manufacturing,which serves as a proxy for primary separationsof the type initially caused by an adverse shock, and new unemploymentspells. The distributedlag relationshipindicates that an initial burst of job destructionleaves in its wake a continued flow of new unemploymentspells that lasts for seven quarters.This evidence supportsthe notion that a disruptionin the labour marketis 2. To see this, one can takea three-statemodel of workerflows (unemployment,employmentandout of the labour force) andcalibrateit with the mean transitionratesbetween the threestates-reported in BlanchardandDiamond(1990). Then, startingfrom the ergodic distribution,suppose a shock displaces employed workersto unemploymentand to "out of the labourforce"in equal numbers.Carryingout this exercise demonstratesthatthe speed at which the unemployment rate returnstowardits mean following the shock is initially slower than in the model without the third state. However, after 5 or 6 months, the rate of adjustmentis very similar to the two-state model, reflectingthe fact that Blanchardand Diamond (1990) estimate quite high re-entryrates. So while this mechanism can explain why the unemploymentrate adjustssomewhatmore slowly duringthe first6 months or so, it cannot explain why the unemploymentrate often takes 3 or 4 years to fully adjust. 3. Otherpapersthatlook at the behaviourof job-findingrates over the business cycle include Murphyand Topel (1987) and Hall (1991). Hall (1991) uses simple regressionsto show that the marginalimpactof the unemploymentrate on job creation in manufacturingrises with the unemploymentrate. This suggests that the job matching rate actually increases when unemploymentrises. Murphyand Topel (1987) use data from the CPS Annual DemographicFile to estimate monthly job-finding rates among men with strong labour marketattachment.According to their results, the unemploymentexit rate duringthe deep recession of the early 1980s only fell by 5-7 percentagepoints, from a peak of 22.4% in 1978 to a low of 16.7%in 1982.

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made persistentby the turbulentemploymentexperiences of the workersinitially displaced by the disruption.The model presentedin the following section formalizesthis idea. 3. THE ENVIRONMENT This section describes the importantelements of the model. Time continues forever in discrete intervals.Workersexpend time and resources searchingfor an appropriateemployer, and vice versa. The search process is imperfect,however, and many of the matches that are established are of poor quality. The firm and worker do not directly observe their match's quality; they attemptto determinethat quality by observing the efficiency with which they producetogether. For appropriatelychosen parametervalues, the model's equilibrium,discussed in Section 4, has the featurethatif it becomes apparentto the firmand workerthattheirmatchis poor, they agree to separateand resume search.Otherwise,the matchis preserved. 3.1. Firms and workers

Thereis a continuumof firms,each consisting of one job, which is either filled or vacant.There is no cost for opening or closing a vacancy; however, firms must pay a cost c each period to maintaina posted vacancy. There is also a continuum of workers, who are either employed or searching for employment.The size (measure)of the labourforce is constantand is normalizedto one. Thus, the model abstractsfrom movements into and out of the labour market.While unemployed, workers receive a constant pay-off of Yu per period. Workers search for jobs only while unemployed-there is no on-the-jobsearch. Both firms and workers are risk-neutral.Thus, the objective of workers and firms is to maximize the present discounted value of income. Futurevalues are discounted at a constant rate; f denotes the discountfactor. 3.2. The matching process

When a workerand a firm with a vacancy make contact, they evaluate the potential quality of their match before deciding to form a productionunit. However,they have limited information about how well the worker's attributes(specific skills, reliability,intelligence, etc.) mesh with the firm'sattributes(flexibilityof hours,technology used, location, etc.). Based on what they can ascertain,they decide whetherto establish an employmentrelationshipor to continue searching. There is an aggregate matching function m(v, u) that gives the number of workerfirm matches that emerge from the search process each period with sufficiently optimistic prospectsthatthey actuallyform an employmentrelationship.The function'stwo argumentsare, respectively,the numberof vacancies and the numberof unemployed workers. The matching function is assumed to be non-negative,increasing in both arguments,concave, less than the minimum of its arguments,and homogeneous of degree one. The rate at which a vacancy is matched with a workercan be expressed as q(u) = m(l, U) = m(v, u)/v, with q(v) < 0. Job-seekersfindjobs at the rate f(v) = uq(u) = m(v, u)/u. For ease of exposition, 0 is used to denote the ratio v, which is a measureof labourmarket"tightness". 3.3. The learning process

Once a worker and firm establish a match, they attemptto resolve the remaining uncertainty about their match's quality. They cannot directly determinetheir match's quality, or average

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productivity,y, because their observed outputin each period, Yt,is contaminatedwith a matchspecific random component, utt, so that yt = y + lt. The random component /t represents transitoryfactors that are independentof the match quality, y. The workerand firm struggle to determinewhether a low observed yt reflects something about the match's overall quality or whetherit simply reflects a temporaryspell of bad fortune. Given knowledge of the distributionof ,ut, a sequence of observationsof yt allows the workerand firm to learn aboutthe quality of their match, y. Although a continuumof potential values for y is perhapsmost realistic, for simplicity it is assumed here that there are only two values: yg for "good" matches, and yb for "bad"matches. Furthermore,ptt is assumed to be distributeduniformlyon [-co, co].Thus, the outputof a bad match is drawneach period from a uniformdistributionover [Yb- -, Yb + w], whereas the outputof a good match is drawnfrom a uniform distribution over [yg -

, yg + co]. Finally, Yb and yg satisfy Yb - ( < yg - c <

yb + w) < yg + C.

These assumptionsgive rise to an "all-or-nothing"learningprocess thatsimplifiesthe model considerably.4If a firm and workerobserve a yt less than yg - w, they learn with certaintythat their match type is bad. Conversely,a yt greaterthan Yb + w reveals that the match is good. An observedvalue of yt between yg - C and Yb+ w fails to reveal a match'stype. Moreover,it does not change the firm'sand worker'sbeliefs aboutthe probabilityof being a good match since the probabilityof observinga yt between yg - c and Yb+ w, given thatthe matchis bad, equals the probabilityof observingthatvalue of Yt,conditionalon the matchbeing good (i.e. the likelihood ratio is one). In an equilibriumin which matches discoveredto be bad are terminated(parametervalues that guaranteesuch an equilibriumare discussed in Section 4), all workerswill eventuallysettle into matches known to be good unless there is another source of separations.To avoid this, it is assumed that in any given period a matched firm and worker receive an "exogenous" idiosyncratic shock with probabilityxt. The nature of the shock is such that the worker and firm learn of some event that has adversely affected one of the many factors that determinea match's fundamentalquality,y. This shock is exogenous in the sense that the informationis not revealedendogenouslythroughobservationsof yt. An exogenous shock can lead to two different outcomes. In one case, occurring with probability 1 - a, the worker and firm know immediately that the adverse event has rendered theirmatch'sfundamentalqualitybad. In the othercase, occurringwith probabilitya, the worker and firmare not certainwhetherthe adverseevent has renderedtheirmatchbad. As a result,they simply restartthe learningprocess (drawa new y). The notion that the impact of the exogenous shock on the match's quality is known in some cases and unknown in others is analogous to the idea that duringthe searchprocess workersand firms have informationthat allows them to weed out some, but not all, bad matches. Section 5 discusses how this assumptionaffects the simulationresults. Because there is a continuum of workers, xt denotes both the probabilitythat a worker receives an exogenous shock and the fraction of workers that receive such a shock in a given period.The stochasticprocess thatdeterminesxt is given by xt = x + et, where et E [-x, 1 - x] is some i.i.d., zero-meanprocess and x > 0. Thus, an adverse shock to the economy takes the form of a positive realizationof Et.This type of shock representsany event that adverselyaffects the viability of large numbersof existing employmentrelationships.Reallocativeshocks-such 4. A frameworkthat specified that Itt were, say, normallydistributed,or that allowed a continuumof values for y, would greatly complicate the state space relevantfor the workers' and firms' decision problems.This would clearly necessitate a much more complex computationalapproachto solving the model, while qualitativelythe results would not be different.Pries (1999, Chapter3) computes the solution of a similar model, with normallydistributednoise, to examine a slightly differentset of issues.

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as an oil shock that affects the quality of matches using an energy-intensivetechnology, or a shock that shifts demand away from a certain sector, like military supplies-provide perhaps the most obvious example.5In response to such a shock, some matches shut down immediately, while others scrambleto discernthe impact of the shock. 3.4. Timingof events withina period Within each period, the timing of events for workers and firms in established matches is as follows. First,the firmand workerbargainover a wage to be paid by the firmto the worker,in the mannerdescribedbelow. Second, the firm and workerproduce,which results in an observation Yt.Third,with probabilityxt the matchreceives an exogenous shock. Finally,based on what the firm and workerhave found out about their match's type (if anything),they either preservethe matchor separateand searchfor a differentpartnerin the next period. Meanwhile, among the workersand firmsengaged in search,firmspost vacancies to recruit unemployedworkers,with the numberof vacancies being determinedoptimally by firmsbased on the unemploymentrate at the beginning of the period. Matches that result from the search process must wait until the next periodto produce. 3.5. Wagedetermination All separations in the model are privately efficient but may be socially inefficient, due to search externalities(see Hosios, 1990). Privateefficiency arises because workersand firms are symmetrically informed and each period they can renegotiate the wage paid to the worker. Following convention in this class of models, the negotiated wage is determinedby a Nash bargainingrule that gives a fraction 7r of the match's expected joint surplus to the worker. Whenever this expected joint surplus is negative, the worker and firm agree to separate and returnto search. 4. EQUILIBRIUM This section analyses the model's equilibrium. The first subsection describes the Bellman equations that implicitly characterizethe optimal decisions made by firms and workers. The second subsection shows that because firms can freely open and close vacancies, the model's equilibriumis characterizedby a constantjob-findingrate.The thirdsubsectionformallydefines a search equilibrium of the model. Finally, the fourth subsection examines the equilibrium stochasticprocess that characterizesthe unemploymentrate. It is shown that the unemployment rate is essentially an ARMA(2,1) process whose coefficients dependon the model's parameters. 4.1. Bellman equations For the purposeof analysingworkers'and firms' optimal decisions regardingwhetherto accept or terminatematches,considerfirstthe value of searchto workers,denotedby U. An unemployed workerreceives a currentpay-off of Yu.For a given 0, the workerwill, with probability1 - f(0), still be unemployedand searchingin the following period,yielding a value of U. Withprobability 5. This type of shock is distinct from traditionalaggregate shocks-such as "demand"shocks or technology shocks-that affect the profitabilityof all productionunits. Pries (1999, Chapter3) considers fluctuationsdrivenboth by reallocative shocks like the ones consideredhere and by traditionalaggregateshocks. Restrictingattentionhere to reallocative shocks simplifies the equilibriumconsiderablyand has the additionalbenefit of implying a constantjobfindingrate and thus focusing attentionon the role of recurringjob loss.

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f(0), the workerfinds a job. In this case, the worst the workercan do is to reject the job, which yields a value of U. If the worker accepts the job, the pay-off is U plus a fraction rt of the expectedjoint surplus.Formally,this can be expressedas U = Yu+ P{f(0) max[7r(L- U - V) + U, U] + (1 - f(0))U},

(3)

where L is the expectedjoint value of a match in which the workerand firm are still trying to learnthe match'stype and V is the expected value of a vacancy,so that L - U - V is the expected joint surplusof a new match. Similarly,for a given 0, the value of searchto firmsis definedby V = -c + B{q(0) max[(l - r)(L - U - V) + V, V] + (1 - q(0))V}.

(4)

Optimal decisions by matched workers and firms are implicitly defined by the Bellman equationsthat characterize,for a given 0, the expectedjoint value of matches of unknowntype, L, matchesknown to be good, G, and matchesknown to be bad, B.6 The expectedjoint value of a matchwhen its type is unknownis definedby L = Yl+ {[a(l - p)(l - x) + (1 - a)x] max[B, U + V] + [1 - a(l - x) - (1 - a)x] max[L, U + V] + ap(l - x) max[G, U + V]}.

(5)

The parametera = Yg Yb denotes the probabilitythat an outputobservationreveals a match's type and p is the probabilitythatthe match'stype is actuallygood. Recall thatx = E(xt) denotes the (expected)probabilitythatthe matchwill receive an exogenous shock at the end of the current period, and 1 - a is the probabilitythatthe exogenous shock reveals that the match is bad. The value L in (5) consists of two parts,the expectedcurrentpay-off of a matchof unknown type, given by Yl = pyg + (1 - p)Yb, and the match's expected value at the beginning of the next period,which is discountedby the factor B. To understandthe firsttermof the continuation value, note that with probabilitya(1 - p)(l - x) the workerand firm observe a value of Ytthat reveals their match is bad and do not receive an exogenous shock at the end of the period;with probability(1 - a)x they receive an exogenous shock that tells them their match type is bad. When either event occurs, the firm and worker must decide whether the expected joint value of keeping the bad match, B, exceeds the returnfrom separating,given by U + V (parameter values are chosen below so that in equilibriummatches discovered to be bad are terminated, i.e. B < U + V). The second term of the continuationvalue representsthe possibility that the match's type is still unknownat the end of the period. This occurs unless either Yt reveals the match'stype and thereis no subsequentexogenous shock (this occurs with probabilitya(1 - x)) or there is an exogenous shock at the end of the period and it reveals that the match is bad (this occurs with probability(1 - a)x). The last term representsthe returnfrom finding out that the matchis good, and not sufferingan exogenous shock at the end of the period. The expectedjoint value of a matchknown by the workerand firmto be good is G = yg + {(1 - x) max[G, U + V] + ax max[L, U + V] + (1 -a)x max[B, U + V]}.

(6)

6. Because the separationdecision hinges on a comparisonof the joint value of a match with the sum of the workerand firm's alternativeopportunities,it is convenientand parsimoniousto work with the Bellman equationsthat describethe expectedjoint value of matchesof bad, unknown,and good type, ratherthanwith pairs of separate Bellman equationsfor the workerand firm.Withthis approach,wage determinationis not explicitly treated.Ratherit is implicitly representedin the Bellman equationsby use of the bargainingparameterJr. In the background,associated with each of the threetypes of matches,is a correspondingwage that achieves the Nash bargainingoutcome.

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In the currentperiod, the expected pay-off is yg. With probability 1 - x, the match receives no exogenous shock and remains a good match. With probability ax the match receives an exogenous shock that makes the match's quality uncertainto the worker and firm. With probability(1 - a)x, an exogenous shock tells the firm and workerthattheirmatch is bad. The expectedjoint value of a matchknown to be bad is B = Yb+

{(1 - a)

max[B, U + V] + ax max[L, U + V]}.

(7)

4.2. Free entry and exit by firms

Because firms can open and close vacancies freely, the value of a vacancy, V, must always be zero in equilibrium.After ruling out trivial equilibriain which L < U + V, equation (4) then becomes c

-( = P(1 - r)(L - U).

(8)

This equationjust states that the firm'sexpected cost of searchequals the firm'sexpected return from search. It has a unique solution for 0. Given the propertiesof the matching function, as 0 approacheszero, q(0) becomes large and the left side of (8) goes to zero. Moreover, as 0 approacheszero, so does the job-findingrate, f(0), and consequentlythe returnfromjob search, U, declines relative to the value of being in a match of unknowntype, L. Thus, the right side increases as 0 falls. As 0 becomes large, the opposite results hold. By the IntermediateValue Theorem,thereis a uniquevalue of 0 thatsatisfies(8). Because 0 is constant,so is thejob-finding rate f = f(0). This result, standardin matching models in which there are no fluctuationsin the pay-off of productionrelative to non-production,stems from the assumptionof a constant returnsto scale matchingfunction (see Mortensenand Pissarides, 1994). Of course, the evidence of movements along a negatively sloped Beveridge curve-as in Blanchardand Diamond (1989), for example-suggests that indeed 0 does vary somewhat with the business cycle. These movements are associated with the small fluctuationsin f that Blanchardand Diamond find in their subsequentpaper (1990). Although the fact that neither0 nor f fluctuatein the model presentedhere is not perfectlyconsistentwith thataspect of the data, the role of recurringjob loss in generatingunemploymentratepersistenceis more transparentif the small role of movementsin f is turnedoff. It is worthnoting here thatthere is a one-to-one mappingfrom c to 0 (and thus to f(0)). A higherrecruitingcost, c, reduces the numberof vacancies per unemployedworker-i.e. reduces 0. Consequently,when choosing parametersfor the model in Section 5 below, the job-finding rate, f, can be set to its empirical value by simply assuming that c takes on the appropriate value. 4.3. Equilibrium defined

Because the free-entrycondition implies that 0 must be constant in equilibrium,workers and firms' decisions do not depend on the distributionof workersacross labourmarketstates. As a result, an equilibriumof the model can be definedvery simply in the following way: Definition 1.

A search equilibrium is a list of real numbers U, V, L, G, and B, and a

positive real number0 such that: (i) (Optimization)Given 0, the values U, V, L, G, and B satisfy the Bellman equations(3), (4), (5), (6), and (7).

(ii) (Free entry)Given L and U, (8) determines0.

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4.4. Unemploymentrate dynamics The equilibriumdecision rules, togetherwith the stochasticprocesses that affect matches,imply a law of motion that characterizesthe evolution of the distributionof workersacross potential employment states-unemployment, employmentin a match known to be bad, employmentin a match of unknowntype, or employmentin a matchknown to be good. Of course, the model's parametersdeterminethe nature of the decision rules-for example, they determine whether B < U + V, so thatworkersand firmsterminatematches discoveredto be bad. Severaltypes of equilibriaarepossible. Of course,the only interestingone is an equilibrium in which workers'andfirms' decision rules dictatethatmatchesrevealedto be bad areterminated and all other matches are preserved.The pay-off received by unemployed workers, Yu,is the criticalparameterfor determiningwhetherbad matchesare destroyedand whethergood matches and matchesof uncertaintype arepreserved.All else equal, if Yuis too small, the value of search will be so low that even a firm and a workerin a bad match would optimally choose to remain in thatbad matchratherthanpassing througha phase of unemploymentto try for a bettermatch. This would be an uninterestingequilibriumbecause it would renderthe learning aspect of the model irrelevant.Conversely,if yu is too high, unemploymentwould become preferableto a matchof uncertaintype or even to a matchknown to be good. In this case, the stochasticprocess for the distributionof workerswould become degenerate,with all workersin unemploymentand no firmsposting vacancies. Intuitionsuggests thatthere is an intermediaterange of values for yu such thatbad matches are destroyedand othermatchesare preserved.The following propositionaffirmsthis intuition: Proposition 1. If f(0)n > ax, then any value of yu in the interval (Yb,Yl) assures that B < V + U, L > V + U, and G > V + U. Proof: See the Appendix.

II

This proposition provides a set of sufficient conditions for the interesting type of equilibrium.7The remainderof this paperassumesparametervalues thatsatisfy these conditions. As a result, at any given time workers are distributedamong one of only three different employment states: unemployment,employment in a match of unknowntype, or employment in a match known to be good. The vector zt = [ut, lt, gt]' denotes this distributionof workers. The transitionrates between ut, It, and gt are easily gleaned from the coefficients on the max operatorsin the Bellman equationsthat define U, L, and G. Figure 1 demonstratesthe rates of each of the flows. From Figure 1, it is apparentthat the distributionof workers evolves according to zt = Tt-lzt1_, where the transitionmatrixTt-1 is given by a(l - p)(l - xt-l)

1-f

1 - a(l - xt-)

f

+

1-) ( )xt

a)xt-

- (1 - a)xt-

ap(l - xt-1)

0

.

axt-

(9)

1 - xt-

This transitionmatrix varies because of shocks to xt. If x + Et-l is substitutedfor xt-1, the evolutionof zt over time is describedby the following stochasticvector differenceequation ut

1 -f

It =

f

gt

0

a(l - p)(l - x) + (1- a)

-a(l

-x)-(1 ap(l -

-a)x )

(1 -

)x

ax 1 -x 5

ut-i

It_gt-1

Ytlt

+

(1 - yt)t

(10)

--7t

7. The intervalof admissible values for Yumay be larger.Moreover,even if f(0)nr < ax, there is nevertheless still an intervalof values for yu-a smallerinterval-that leads to the desirableequilibrium.

204

REVIEW OF ECONOMIC STUDIES unemployment,u -paM-)(l- t) +(1 - a)xt

_(1-)/ta(

o

f - xt) ap(1 1

matches known to be good, g

caxt

learning match type, I

FIGURE1 Equilibriumrates of transitionbetween employmentstates

where it =

t_l(aplt_1

+ gt-1)

and Yt =

[(-1 )

+(1oa(1-p)]t-l apit-l+gt-1 )gt-1.

A positive

shock Et-1

induces an extra flow rlt of workers out of good matches. A fraction Yt of those workers becomes unemployed, while the other 1 - Yt workers flow to It and attempt to learn the quality of their matches. Following the shock, the coefficient matrix in (10) determines the path along which ut, It, and gt return (absent a new shock to xt) toward their average levels. Note that rit and yt are dated time t even though they depend on time t - 1 variables. This is appropriate because even though the shock et-I occurs during period t - 1, the innovation in the distribution of workers that it induces occurs in the following period, t. Put differently, et-1 is uncorrelated with the time t - 1 distribution of workers, zt-1. From (10), it can be shown that the response of the unemployment rate to a shock Et can be expressed as Ut -U

= 0l(Ut-

-

U) +

2(Ut-2

- U) + Vt-lt-l

+ Ytt,

(11)

where 01 = (1 - x)(l - a) + 1 - f - (1 - a)x 02 = -(1 - X)[(l - a)(l - x) + (1 - ap)(ax - f)] ft-1 = a(l - p)(l - x) - yt-i(l - ap)(I - x). Notice that ft-l depends on yt-i. If Yt- is approximately constant, however, then so is ft-l and the unemployment rate is essentially an ARMA(2, 1) process with coefficients that depend on the model's parameters. To understand why the process for ut has two autoregressive terms rather than one, note that the unemployment rate two periods ago affects the current unemployment rate because an innovation in Ut-2 leads to an innovation in lt-l and gt-,l which in turn feeds back into ut in the form of separations during period t. To summarize, there is an equilibrium in which workers are either unemployed, employed in a match of unknown type, or employed in a match known to be good.8 The addition of a third 8. It is worth nothing that the model here is isomorphic to a model in which negative durationdependence and recurringjob loss result entirely from "technological"assumptions,ratherthan from a signal extractionproblem. Specifically,supposematchproductivitywere a three-stateprocess, with matchesstartingin the middle state, and always separatingwhen the poor stateoccurs. In such a model, for an appropriatelychosen transitionmatrix,new matcheswould be more likely to suffer a separation,workerswould exhibit recurringjob loss, and the resultingworkerflows would be identical. Similarly Jovanovic (1979b) could have explained the set of facts that motivatedthat paper with a purely "technological"model. However,because learningdoes likely accountfor much of the negativedurationdependenceas work by Nagypal (2000) suggests-the present model, like Jovanovic (1979b), endogenously explains it by way of the signal extractionproblemratherthan exogenously imposing it.

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205

state (relativeto a simple model in which workersare eitherunemployedor employed) in which workers are uncertainabout their match's quality makes the unemploymentrate adjust more slowly following a burstof job loss. After a shock that displaces workersfrom employmentstate g into u and 1, the displaced workersmay cycle throughu andI several times before returning to g. The extent to which the addition of this third state increases the autocorrelationof the unemploymentrate dependson the parametersthatdeterminethe transitionratesbetween states. These parameters-a, p, f, a, and x--determine the coefficients on the stochasticprocess for the unemploymentrate given in (11).

5. SIMULATINGTHE MODEL This section provides a simulationof the model in orderto demonstratethe role that recurring job loss plays in generatingpersistence in the unemploymentrate. After selecting parameter values, the response of the unemploymentrate to a shock is comparedwith the response of an ARMA model of the U.S. unemploymentrate and to the response of a reducedform model like the one in Section 2. The results show that recurringjob loss can successfully explain why the unemploymentrateroutinelytakes more than4 years to returnto its pre-shocklevel. Selecting a value for the model's job-findingrate, f, is straightforward.Section 4 showed that f is constantandthatthereis a one-to-one mappingbetween its equilibriumvalue and firms' recruitingcost, c. Therefore,c is assumedto take on whatevervalue yields an equilibriumvalue for f of 0.24, the meanjob-findingrate estimatedby Blanchardand Diamond (1990). For given values of the otherfour parameters,x determinesthe averageamountof turnover in the model and thus determinesthe mean unemploymentrate. Accordingly, given values for the other four parameters,the average probabilityof an exogenous shock, x, is set so that the model's mean unemploymentrate equals the mean of the U.S. unemploymentrate during the period 1948:01-2000:06, i.e. 5.7%. This yields a value of 0.01 for x. While neithera nor p (respectively,the probabilitythat a match's type is revealedand the probabilitythat the type is good) can be directly observed, their values can be chosen based on the fact that they determinethe rate at which workersand firms learn their match type and thus they determinethe shape of the hazardfunction.For a given p, a higher value of a raises hazard rates at low tenurelevels but decreasesthe time until the hazardfunctionflattensout. For a given a, a highervalue of p decreaseshazardrates at low tenurelevels and decreasesthe time until the hazardfunction flattensout. The choice of values for a and p is especially importantbecause if values are chosen such that low-tenurejobs are more likely to be terminatedthan they are in the data, the model will give an exaggeratedpictureof the role thatrecurringjob loss plays in explainingunemployment rate persistence.Thus, in orderto choose values for a and p, an estimateof the empiricalhazard functionis needed. Tenure supplements conducted every 4 years as part of the CPS provide one means of characterizingthe empiricalhazardfunction. Hall (1982) and Ureta (1992), among others,have used this data to estimate hazard functions. The data from the tenure supplementsreveal the distributionof workers across tenure categories at a particularpoint in time. Hazardrates can be calculated from these data by treatingthe distributionof workers across tenure categories at a particularpoint in time as if it were the distributionthat would result from observing a cohort of workers who startedtheir jobs at the same time and recording the number who remained on that job through each of the tenure levels (i.e. the survival function). With this cross-sectional or "snap-shot"approach,if N(r) is the number of workers with r years of tenure, then the hazard function is simply the rate of decline of N(r), or N(r)-N(r+1) N(r)

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206

-

20 1-

cP_

10

-

0

\

'

,

0-6 -

\5

\ 15 ~ \-

\-

a\ 5\^K -

_

5 10 Years of Tenure

X 0 -2

_

?'~ ~~~~~~?2

15

0 ?0

10 5 Years of Tenure

15

FIGURE 2

Leftpanel: Fit of estimatedtenuredistribution.Rightpanel: Estimatedhazardfunction (solid) and the model's hazard function (dashed)

This approachhas some potential shortcomings. First, if the tenure distributionis not stable over time, the cross-sectional approachis inappropriate.However, comparisons of the distributionin four different years show that it is roughly stable. Second, differences in the relationship between tenure and job-to-job transitions and the relationship between tenure and employment-to-unemploymenttransitionscan introduce biases into the resulting hazard functionestimate.Third,the hazardfunctionthatresultsfrom this approachcan also sufferfrom heterogeneitybias. These potential shortcomings notwithstanding,this approachis used here to obtain an estimate of the hazard function that can guide the selection of parametervalues.9 Due to choppiness in the tenure distributiondata, a three-parameterdurationdistribution(a Gamma mixture of Weibull distributions,known as a Burr distribution)is used to provide a smooth approximation.After finding the parameter values that achieve the maximum fit of that distribution'ssurvivalfunctionto the empiricaltenuredistribution,the hazardfunctionassociated with the distributioncan then be evaluatedat those parametervalues. The left panel of Figure 2 shows the resulting fit of the parameterizedmodel to the 1995 tenure distribution.10The solid line in the right panel shows the estimated hazard function, which exhibits strong negative durationdependence. The dashed line in the right panel shows the model's hazard function for the chosen set of parametervalues (although the model is simulatedat a monthly frequency,the hazardrates in the figure have been annualized).Given the potentialshortcomingsof the hazardfunction estimation,the values for a and p--013 and 0.4, respectively-were chosen to err on the conservativeside. As the figure shows, the model's hazardratesstartat a slightly lower level thanthe estimatedones anddropmorequickly,implying less turnoverin the model (relativeto the estimatedhazardfunction)among low-tenurejobs. The value for a remains to be chosen. Recall that a is the fraction of matches that are affectedby the exogenous shock, but do not sufferan immediateseparation.Effectively,it serves to prolong slightly the initial impact of a shock. Interpretedin the context of the model, in 9. A more ideal approach would require a longitudinal data-set both with highly accurate information on employment histories and with a sample large enough to include a significant number of labour market transitions. Farber(1994) attemptsto use the NLSY data-setto estimatethe hazardfunctionandhe finds, aftercontrollingfor worker heterogeneity,strongevidence of negative durationdependence(althoughthe hazardraterises in the firstfew months). 10. The data was provided by Diebold, Neumark and Polsky (1997), and reflects adjustmentsto account for "heaping"and roundingproblems,as describedin theirpaper.

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207

which workershave a difficult time knowing their match quality,if something occurs that has the potentialto affect that quality,then it may take time to figure out whetherit has in fact been affected. Understoodmore broadly, this feature representsthe many frictions and adjustment costs thatpreventfirmsfrom immediatelydismissing workersupon experiencinga shock. As a matter of worker-flowsaccounting, any model that is calibratedto a roughly 25% monthlyjob-finding rate will experience a dramaticdrop in the the unemploymentrate during the firstfew monthsafterthe shock, since workersobviously have to leave unemploymentbefore they can experience anotherjob loss. Consequently,some mechanism (here provided by the parametera) is needed thatwill prolongby a few monthsthe burstsof primaryjob loss thatdrive the unemploymentrate fluctuations.In this sense, the recurringjob loss model of this paper is best suited to explain why the unemploymentrate takes more than 3 or 4 years to fully adjust, ratherthanjust 6 monthsor so. The value chosen for a is 0 8. This means thatthe impactof the adverseshock is completely understoodright away by 20% of the affected matches. Moreover,for the chosen values of a and p, well over two-thirdsof the workerswho will suffer a primaryjob loss due to the shock will have already done so by the 6th month following the shock. This is the sense in which the role of recurringjob loss should not be understoodin terms of its ability to explain what occurs in the first few months following a shock, but ratheron the model's ability to explain why the unemploymentrate remainselevated after 3, 4, or even 5 years. The simulationresults below make clear precisely what portionof the model's unemploymentrate persistencecan be attributedto recurringjob loss. To providea point of comparisonfor the model, monthly datafor the periodJanuary1948June 2001 are used to estimate a time-series model of the unemploymentrate in the U.S. To remove fluctuationsmore likely associated with low-frequencyfluctuationsin the naturalrate than with fluctuationsat business cycle frequencies, a time-varyingestimate of the NAIRUcalculated using the method in Staiger, Stock and Watson (1997)-is removed.1l Standard Box-Jenkinsidentificationtechniquesreveal thatthe following ARMA(1,5) model describesthe detrendedunemploymentrate series very well: Ut = 0.957ut_1 + 0.171t_5

+ 0-131rt_4 + 0.1526t_3 + 0-2426t_2 + 0-0226t_1 + 6t.

Figure 3 compares,for four differentmodels, the dynamic response of the unemployment rateto a shock (the magnitudeof the shocks were chosen such thateach of the responsespeaked at the same unemploymentrate, about8%).The solid line shows the responseof the ARMA(1,5) model of the U.S. data. The dashed line shows the response, as given by equation (11), for the model of recurringjob loss with the parametervalues indicatedabove. At the time thatthe shock hits (month zero), the distributionof workersacross the three labour marketstates is given by its long-runaverage.The dash-dottedline shows the response of the simple reducedform model of labourmarketflows with a constantjob-findingrate of 0-24 and a constantseparationrate of 0.015. To put the comparisonof this simple AR(1) model on equal footing with the recurringjob loss model, the flows of primaryseparationsin the recurringjob loss model, which are persistent due to the fact thatthe parametera drawsout the initialimpactof the shock, areused as impulses for the AR(1) model. The figure demonstratesthat indeed the recurringjob loss in the model generatessignificantlymore unemploymentratepersistencethan a simple matchingmodel with 11. Fluctuations in the natural rate can be traced to various factors, such as changes in the demographic composition of the workforce,changes in unemploymentinsurancepolicy, or changes in searchcosts. One way to deal with demographicchanges, at least partially,would be to use the unemploymentratefor prime age males. This approach resultedin even more persistencein the detrendedseries, suggestingthatthe othersources of naturalratemovementsare important.Detrendingwith the Hodrick-Prescottfilter or with a quartictime trendproducedresults very similarto the NAIRU approach.

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Months FIGURE 3

Dynamic responses of an ARMA(1,5) model of the U.S. unemploymentrate (solid line), the baseline parameterization of the recurringjob loss model (dashedline), the recurringjob loss model with a = 0 (dottedline), and a simple reduced form model with constantseparationrate andjob-findingrate (dash-dottedline)

just two labourmarketstates.In spite of the conservativeparameterizationof the hazardfunction, the rate at which the model's unemploymentrate revertstowardits mean is similar to the very gradualrate of reversionfound in the U.S. data. The dottedline in Figure3, which shows the responsefor the recurringjob loss model when a = 0 (and with x reset to 0.0062 to achieve the same mean unemploymentrate), demonstrates that recurringjob loss alone cannot account for the dynamic behaviourof the unemployment rate;some initial persistencein the primaryseparationsis needed.Witha = 0, so thatthereis no persistencein primaryseparations,the unemploymentratefalls rapidlyafteran initial spike as the displacedworkersfind theirway back into employmentat rate f = 0.24. Because workersmust firstfindjobs beforethey can lose them again,recurringjob loss cannotslow the adjustmentof the unemploymentrateuntil severalmonthsafterthe initial spike.As the dashedline shows, however, when the recurringjob loss kicks in, aftera few months,it slows the adjustmentsignificantly. One can assess the extent to which the recurringjob loss and the persistencein the primary separationsaccountfor the overallunemploymentratedynamicsin the baseline parameterization of the recurringjob loss model by comparingthe responseof thatmodel with eitherthe recurring job loss model with a = 0 (which reflects no persistence in primaryseparations)or with the simple AR(1) model (which reflects no recurringjob loss). Figure 4 provides anotherway of understandingthe role of the initial persistencein primaryseparationsthatresults from allowing a to be positive. Recall that among the extra Et matches affected by a reallocative shock only a fraction 1 - a become unemployedimmediately;the remaininga matches simply restartthe learning process (i.e. flow to it). Consequently,a shock leads to an increase both in the pool of unemployed workersand in the stock It of workerstrying to determinetheir match quality. Figure4 shows the deviationof ut, lt, and gt from theirlong-runaveragesfollowing the shock Et thatgeneratedthe model's responsein Figure3. In the monthsimmediatelyfollowing a shock, the shift in the distributionof workerstowardIt, where separationsare more likely, and away from gt, where separationsare less likely, results in a continuedflow of workersinto unemployment and keeps the unemploymentrate from falling precipitously.After a few periods, these primary separationsbecome less importantand give way to the recurringjob loss that takes place as the workersdisplacedby the shock begin to find new jobs, and then lose them. Figure5 providesa more directpictureof the relativeimportanceof primaryand secondary separations.The solid line on top gives the total numberof separations.Thereis a spike in period

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PERSISTENCEOF EMPLOYMENTFLUCTUATIONS

209

01.

~ O

,-

0-05-

r. \

I

N

I

0

-

-0-05-0.1

/

010

20

40 60 Months

80

FIGURE 4

Deviations of ut (solid line), It (dash-dottedline), and gt (dashedline) from their steady-statevalues following a shock ,

0-025

002-

0.015--

001

N

20

60 40 Months

80

FIGURE 5

Separationrate following a shock. Total separations(area below solid line), primaryseparations(between dashed and dash-dottedlines), and recurringseparations(between solid and dashedlines)

zero when the shock hits (the spike is cut off in Figure5 so thatthe verticalscale allows the figure to be readmore easily). Absent anothershock, this line convergesback to the steady-statehazard rate. After the shock hits, the separationsare divided into three categories. The area below the dash-dottedline on the bottom shows the separationsamong matchesnot affected by the shock. The area between the dashed line and the dash-dottedline indicates the primary separations among the workersaffected by the shock. In the periods following the shock, these are the first separationssufferedby the workerswho areoriginallydisplacedto It andthen subsequentlylearn thattheirmatchis a poor one. As expected, these separationseventuallydisappear. The area between the dashed line and the solid line representsthe recurringjob loss-the secondary separationsof workers affected by the shock who have already lost their job once. Again, initially these separationsare rare because the number of workers who have suffered an initial separationand have already found a new job is very small in the periods right after the shock. When this recurringjob loss becomes more important,after a few months, it is the principalreason thatthe unemploymentrate declines so slowly. Flows between ut and It remain high as many workerspass throughseveral short-livedjobs. Only very graduallydoes the total

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REVIEWOF ECONOMICSTUDIES

separationratereturnto normal,as workerseventuallysettle backinto matchesknown to be good (flow from ut and It back to gt).

6. EARNINGSLOSSES FOLLOWINGPRIMARYSEPARATIONS To bolsterthe explanationfor unemploymentratepersistenceput forth in this paper,this section examines the validity of other implications of the model. In particular,the model has strong implicationsregardingthe behaviourof wages andearningsbefore andaftera primaryseparation, or displacement.Prior to suffering a primaryseparation,some workers-those who first pass through employment state i-experience a drop in wages. After a displacement, wages also typically remain low for an extended period of time as workers cycle through states u and I before ultimatelyattaininga good match.Togetherwith the higher incidence of unemployment, these lower wages mean that earningsremain stubbornlylow for an extendedperiod following the initial displacement. As a point of comparisonfor the model, Jacobson,LaLonde and Sullivan (1993) provide a detailed analysis of the earningsexperiences of workersbefore and after a job displacement. The top panel of Figure 6 reproduceshere Figure 2 from that paper. Based on results from administrativedata for Pennsylvaniafor the years 1974-1986, it shows the quarterlyearnings losses for workersin the years before and after a displacementassociated with a mass lay-off, after adjustingfor individuallevel heterogeneity.(The figure shows results from two different specifications,one with and one without individual-specifictrend growth rates.) When judged against a mean quarterlyearningsof about $6000, both specificationsreveal a drop in earnings prior to displacementof about 15-20%, a cumulative drop of nearly 50% by the quarterthat includes the displacement,and then a slow recoverytowarda level that still remainsabout25% below initial levels after6 years. This evidence is clearly at odds with standardmatchingmodels such as Mortensenand Pissarides (1994), which predictthat a newly hired worker's prospects are no differentfrom, or perhapsare even better,thanthose of workersin continuingmatches. The bottom panel of Figure 6 displays the impact of displacementon quarterlyearnings for the simulated model. The figure is generatedby simulating 10,000 workers' employment histories for several hundredperiods (beginning from the ergodic distribution),synchronizing the time period of the initial job displacements,aggregatingmonthly earnings up to quarterly earnings,and then averagingthe quarterlyearnings,across workers,for each period before and afterthe displacement.Of course, in orderto determinewages in typeI matches, wl, and type g matches, Wg,values must be assigned to severalparametersfor which values did not have to be specified in the preceding analysis-specifically,

f, JT, yg, Yb, and Yu. Given parameter values,

equilibriumwages are calculatedusing an iterativetwo-step process that requiresthe individual Bellman equationsfor both workersand firms, as opposed to the joint Bellman equationsgiven in Section 4. In the firststep, the Bellman equationsarejointly solved by value functioniteration taking as given guesses for the wl andwg. In the second step, the wages are adjustedin the directionneeded to satisfy the Nash Bargainingconditions for matches of typesI and g. These steps are repeateduntil the Nash Bargainingconditionsare satisfied. The parametervalues used for the simulationdisplayedin Figure6 are 3 = 0-995, n = 0.7, = yg 2200, Yb = 1000, and y, = 1000; all otherparametervalues remainthe same as above. The overall level of the productivityparameterswere chosen to generateaveragequarterlyearnings on parwith Jacobsonet al. (1993) ($6000). The differencesbetween yg, Yb, andy, togetherwith the value for r, determinethe other quantityof interest:the differencebetween the two wages. For the chosen parametervalues, the monthly wages are wl = 1608 and wg = 2112. The top panel of Figure 6 reveals that in the data workersbegin to experience a noticeable drop in earnings about 3 years prior to the displacement-an average loss of $1000 by the

PRIES

211

PERSISTENCEOF EMPLOYMENTFLUCTUATIONS

3: WithoutTrends Fi WithTrends

1000 a

-1500 0

=. i3

-2000 -2500 -3000 Years Since Displacement

-1000 E act e4 u 0

-2000 -3000

0. r-) 0

a.)

-4000

u

-5000 fnn\I f-1 vJUU

-5

-4

I

I

I

-3

-2

-1

I

I

I

I

I

2 1 0 Years Since Displacemerit

3

4

5

6

i

I

FIGURE6 Toppanel: Reproductionof Figure 2 from Jacobson et al. (1993). Bottom panel: Model simulation. Impact of job displacementon earnings

time the displacement occurs. This could be due either to reduced hours (or temporarylayoffs) or to reduced wages, but the available data do not allow for a decomposition of the two causes.12 The simulation generates similar earnings losses prior to displacement. Of course, reduced employment plays no role in the simulation;the model's earnings losses are entirely accounted for by the fact that many workers flow into I (because a > 0) and earn the lower wage wl, priorto displacement.Jacobsonet al. (1993) provide evidence that indeed the average wage losses may be drivenby a subset of the whole population.For example, workerssuffering non-mass-lay-offdisplacementsexhibit small pre-displacementearningslosses when compared 12. Huff Stevens (1997) uses the PSID to look at both earningsand wage declines before and afterdisplacements. She finds that both wage and earningsfall priorto the displacement,and both reducedwages and reducedemployment are importantfor understandingpost-displacementearningslosses. Ruhm (1991) has similarfindings.The focus here is on the results in Jacobsonet al. (1993) because the quarterlyfrequencyreducesthe problemof identifying the timing of the displacementthatexists with the PSID.

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with mass-lay-off displacements.They also find significantheterogeneityacross industriesin the pre-displacementlosses. AlthoughJacobsonet al. (1993) do not reportthe fractionof workersthatexperiencea wage loss upon re-employment,relative to their pre-displacementwages, it is of interest to examine this for the model. Nearly 70% of the displaced workers are in state 1 in the month prior to displacement,meaning that their wage upon re-employmentwill representno additionalwage loss. However, only 47% spend all 3 months priorto displacementin state 1, so if earningsare reportedat a quarterlyfrequency,the percentagewho would reportno additionalearningsloss is smaller.Moreover,for examining these types of statistics the model may be too restrictive, relativeto reality,in that it only allows two match-qualitystates, g and 1. In a model with more states, many workerswould suffer some wage losses prior to displacement,as the confidence in their match quality declined, but then would suffer additional earnings losses following a displacement. It is useful to note that the earnings losses prior to a displacementare entirely consistent with the model's source of persistence in primaryseparationsfollowing a reallocative shock. At the individual level, because an adverse shock does not always lead to an immediate separation,but merely begins the learningprocess anew, there is often deteriorationof earnings prior to a displacement; at the aggregate level, the delay between the impact of the shock and the separationscauses the full impact of a reallocative shock to be drawn out for several months. Otherways of modelling the persistence in primaryseparationswould not necessarily be consistent with the pre-displacementearnings losses. Suppose, for example, that xt itself exhibited persistence. In this type of model, separationsthat occurredwhen xt were at average levels would exhibitno pre-displacementwage loss, whereaswages would decline for all matches when xt were aboveits mean, since the lower probabilityof continuingin the matchwould reduce a match's surplus.Thus, matchesthat survivedan initial increasein xt, but which later ended in separationdue to the continuinghigher separationprobabilities,would exhibit a wage loss prior to the separation.However, matches that sufferedno separationwould also experience a wage loss and thereforeone would not expect the results found in Jacobson et al. (1993), since the pre-displacementwage losses in thatpaperare measuredrelativeto the controlgroupof workers who experienceno displacement. Turn now to the post-displacementearnings losses. Here again, the model is relatively successful in matchingthe data. Notice, however,that the initial drop in earningsis much larger for the simulation,with averageearningslosses duringthe quarterthatcontainsthe displacement representingabout90% of averagequarterlyearnings($6000). In contrast,in the datathe losses are only about 50%. The severe drop in the model results from the fact that only about 42% of workers find employment during the quarterthat includes the displacement.The smaller drop seen in the data almost certainlyresults, at least in large part,from the fact that Jacobsonet al. (1993) cannot identify precisely when in the quarterthe displacementoccurs, whereas in the simulationthe displacementis always assigned to the first month of the quarter.Consequently, in the dataaverageearningsfor the quarterwhen the displacementoccurs probablyinclude some earnings received prior to the displacement. Without these earnings, the initial earnings loss would be more severe.To the extentthatthis timingissue does not fully reconcile the discrepancy, it would appearthatin the Jacobsonet al. (1993) datasome workersmust findjobs at a ratefaster thanthe model's rateof f = 0-24. Aside from the slight difficulty that the model has in matching the first quarteror two of post-displacementearnings losses, it capturesquite well the extremely slow recovery that follows. As in the data, the model displays significantearningslosses as much as 6 years after the initial displacement,althoughthe model's losses are slightly smallerin magnitudeafterabout 2 years. In the model, about 60% of the earningslosses duringeach of the 6 years following a

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displacementis due to lower employmentand about40% is due to lower averagewages (a higher fraction of workers earning wl instead of wg). The model could, with parametervalues that produceda largerwedge between wl and Wg,replicate the earningslosses for years 3 through 6. However, the model would performworse, relative to the data, for the periods prior to and immediatelyfollowing displacement. The fact that the model's implicationsfor the earningslosses associatedwith displacement are broadlyconsistentwith the empiricalevidence is particularlyimportantbecause the adverse effects of a displacementare intimatelylinked to the way in which recurringjob loss translates into unemploymentrate persistence in the model. At the onset of an economic downturn,the model featuresa significantshift of workersfrom the upperend of the matchqualitydistribution (matchesknown to be good) towardthe lower end (matcheswith more tenuous prospects).The disruptionof the good matches,with the accompanyingdecline in wages and earnings,is crucial because if only the most tenuousmatches(i.e. new matchesof unknownquality)were destroyed, then when the affected workersfound new jobs they would be no more likely to lose theirjobs than they had been before the initial displacement;consequently,the aggregateseparationrate would be no higher and there would be no additionalpersistencedue to recurringjob loss. To put this in terms of an ongoing debate in the related literature,the model is at odds with the "cleansingeffect" view of recessions, which maintainsthat recessions have a positive impacton averageproductivityandwages by wiping out low-qualitymatches.Otherpapers,such as Barlevy (2002), have challenged the cleansing hypothesis on the groundsthat the evidence suggests thatin fact thereis a shift towardmore low-qualitymatchesduringdownturns.Because match quality is not directly observed in the data, the evidence on the wage and earnings losses associated with displacementsis a useful means of examining whether indeed primary separationsoften affect high-qualitymatches as well as low-qualitymatches. There is a wide variety of other evidence that lends support to the general view that downturnsreduce the number of stable, or high-quality,matches and increase the number of tenuous matches. Nagypal (2000) finds strong evidence, using a matchedworker-firmdata-set, that firmsthatexperiencenegativeshocks reducetheiremploymentof high-tenureworkersby as much as they reduce their employmentof low-tenureworkers.Barlevy (2000) finds, using data from the NationalLongitudinalSurvey of Youth,that recessions tend to give rise to, ratherthan cleanse, low-wage jobs. Evidence on the cyclicality of wages, such as Bils (1985), who findsthat wages are particularlyprocyclicalamong individualswho experiencejob changes, also suggests that downturnsinvolve a movement of workers from the upper end of the wage distribution towardjobs at the lower end of the distribution. 7. CONCLUDINGREMARKS Business cycle researchhas benefitedgreatly in recent years from the successful integrationof unemploymenttheory into formal models of economic fluctuations.The explicit inclusion of unemploymentin these models has provided for a more complete understandingof both the amplificationand the persistence of shocks to the economy and indeed has led to an important transformationin economists' understandingof the fundamentalnature of business cycles. Recessions are no longer narrowlyinterpretedas prolongedperiods of depresseddemand,or of technological retardation,or of increaseddesirabilityof leisure. That is, aggregatefluctuations are no longer seen as resultingfrom periodic ups and downs of some exogenously given "good times"indicator. Instead, the moder view argues that recessions arise when some shock disrupts the normal functioning of the economy. In the labour market, the disruptionleads to a surge in unemployment.For a variety of reasons, such as contractingproblems, the labour market's

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response may amplify the disruption considerably. The recovery from the shock is drawn out by the gradual process of matching the unemployed workers with new employers. Evidence that workers find new jobs relatively quickly, and thus that periods of aboveaverage unemployment should not last so long, has been a nagging obstacle to this view of business cycles. This paper removes that obstacle by demonstrating that the highly persistent fluctuations in the unemployment rate can be explained-despite the high job-finding rateby the recurring job loss that occurs as many of the workers displaced by the initial shock pass through several short-lived jobs before finally settling back into a more stable employment relationship. APPENDIX Proof of Proposition 1. (5) to get

Given the free-entryassumption,set V = 0. To show that L > U for Yu < Yl,use (3) and

- x) + (1 - a)x] max[B - U, 0] L - U y= - Yu+ p{[a(l - (1 + ap(l -x) max[G - U, 0] + [1 -a(l - x) - (1 - a)x - f(0)r]max[L - U, 0]}.

(A.1)

Suppose L < U. Thenthe rightside of (A. 1) is positive since l - Yu is positive. Thatis, L > U, which is a contradiction. Thus, L > U when yu < Yl. To show that B < U when Yb < Yu < Yl and f(O)7r > ax, use (3) and (7) to get

B - U = Yb- Yu + {(1- ax) max[B - U, 0] + [ax - f(0)7r] max[L - U, 0]}.

(A.2)

Suppose B > U. Then, from (A.2) it follows (since L > U when Yu < yl) that [1 - i(1 - ax)](B -U)

= Yb-u

+ [ax - f(O)](L

- U).

Because f (0)7 > ax, the R.H.S. is negative,which contradictsB > U. Thus, B < U. Finally, L > U implies G > U. Consider the value of Yu > y-

call it

u--that satisfies L = U. At Yu = yu,

equation(A. 1) becomes 0 = 1 - ' + ap(l - x)max[G - U, 0]. Since Yl Yu < 0, it follows that max[G - U, 0] > 0, or G > U. Because G is decreasingin Yu,G > U for all values of Yuless ththan ((and thus for all values less than Yl). I Acknowledgements. Bob Hall deserves special thanksfor the advice andencouragementhe gave me while writing this paper.I also thank, for their many helpful suggestions, Orazio Attanasio (the editor), three anonymousreferees, Pierre-OlivierGourinchas,Michael Horvath,ChadJones, Eva Nagypal, RichardRogerson,Tom Sargent,John Shea, and Akila Weerapana,as well as seminarparticipantsat StanfordUniversity,U.C. San Diego, the Universityof Maryland,the Universityof Pennsylvania,the Universityof Virginia,TexasA&M University,the FederalReserve Boardof Governors, the Federal Reserve Bank of Cleveland, and the 1998 NBER Summer Institute.Financial supportfrom the Alfred P. Sloan Foundationis gratefullyacknowledged. REFERENCES ANDOLFATTO,D. (1996), "BusinessCycles and Labor-MarketSearch",AmericanEconomicReview,86, 112-132. BARLEVY, G. (2000), "Is There a Cleansing Effect in Recessions? Evidence from Wage Data"(Mimeo, Northwestern University). BARLEVY,G. (2002), "TheSullying Effect of Recessions",Reviewof EconomicStudies,69, 65-96. BILS, M. J. (1985), "RealWages over the Business Cycle: Evidence from Panel Data",Journalof Political Economy,93, 666-689. BLANCHARD,O. J. andDIAMOND, P. (1989), "TheBeveridgeCurve",BrookingsPaperson EconomicActivity,No. 1, 1-76. BLANCHARD,O. J. andDIAMOND,P. (1990), "TheCyclical Behaviorof the GrossFlows of U.S. Workers", Brookings Paperson EconomicActivity,No. 2, 85-143. CHATTERJEE,S. and SILL, K. (2001), "Unemploymentand Labor Force Participationin a Model of Precautionary Savings"(Mimeo, FederalReserve Bank of Philadelphia).

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