The CyclicalBehavior ofEquilibrium and Vacancies Unemployment By ROBERT
SHIMER*
Thispaperarguesthatthetextbook searchand matching modelcannotgeneratethe observedbusiness-cycle-frequency in unemployment andjob vacancies fluctuations in responseto shocksofa plausiblemagnitude. In theUnitedStates,thestandard deviationof thevacancy-unemployment ratiois almost20 timesas large as the standarddeviationofaveragelaborproductivity, whilethesearchmodelpredicts thatthetwovariablesshouldhavenearlythesamevolatility. A shockthatchanges altersthepresentvalue ofwages,generating averagelaborproductivity primarily onlya smallmovement alonga downward-sloping Beveridgecurve(unemploymentvacancylocus).A shockto theseparationrategeneratesa counterfactually positive correlation betweenunemployment and vacancies.In bothcases, themodelexhibits no propagation.(JEL E24, E32, J41,J63,J64) virtually In recent years, the Mortensen-Pissarides cyclicalbehavioroftwoofitscentralelements, search and matchingmodel has become the unemployment and vacancies,whichare both standardtheoryof equilibriumunemployment highlyvariableand strongly negativelycorrelatedin U.S. data.Equivalently, themodelcan(Dale Mortensenand Chris Pissarides,1994; fora notexplainthestrongprocyclicality oftherate Pissarides,2000). The modelis attractive numberof reasons:it offersan appealingdeat whichan unemployed workerfindsa job. ofhowthelabormarket itis I focuson twosourcesof shocks:changesin functions; scription andchangesin theseparation tractable;it has richand generally laborproductivity analytically intuitive and it can rate of statics; comparative easily employedworkersfromtheirjob. In a be adaptedto studya numberof labormarket one-sector model,a changein laborproductivas a technology or insurance, ityis mosteasilyinterpreted policyissues,suchas unemployment and mandatory advancednorestrictions, firing model,a supplyshock.But in a multi-sector tification of layoffs.Giventhesesuccesses,one or demandshockchangestherelapreference mightexpectthattherewould be strongevi- tivepriceof goods,whichinducesa changein dence thatthe model is consistentwithkey real labor productivity as well. Thus these businesscyclefacts.On thecontrary, I arguein shocks representa broad set of possible this paper thatthe model cannotexplainthe impulses. An increasein laborproductivity relativeto thevalue of nonmarket and to thecost activity * of advertising a job vacancymakesunemployDepartmentof Economics,Universityof Chicago, 1126 East 59th Street, Chicago IL 60637 (e-mail: mentrelativelyexpensiveand vacanciesrelaA previousversionof thispaper
[email protected]). toward tivelycheap.' The marketsubstitutes was entitled"Equilibrium Fluctuations." I Unemployment rate thankDaron Acemoglu,RobertBarro,OlivierBlanchard, vacancies,and the increasedjob-finding V. V. Chari,Joao Gomes,RobertHall, Dale Mortensen, pullsdowntheunemployment rate,movingthe theeditor Pissarides,twoanonymous referees, Christopher economyalonga downwardslopingBeveridge RichardRogerson,and numerousseminarparticipants for curve (vacancy-unemployment or v-u locus). comments thatare incorporated thepaper.This throughout But the in increase also shortens unemhiring materialis based upon work supportedby the National duration, Science FoundationundergrantsSES-0079345 and SESployment raisingworkers'threat point 0351352. I am grateful to theAlfredP. Sloan Foundation in wage bargaining, and therefore raisingthe forfinancialsupport, to theFederalReserveBank of Minwhile I workedon an early neapolis for its hospitality versionof thispaper,and to Mihai Manea, and especially SebastianLudmer,forexcellentresearchassistance.
1 The interpretation in this paragraphand its sequel buildson discussionswithRobertHall.
25
26
THE AMERICANECONOMIC REVIEW
expectedpresentvalue of wages in new jobs. Higherwages absorbmostof theproductivity theincentiveforvacancy increase,eliminating in laborprocreation.As a result,fluctuations have littleimpacton the unemployductivity rates. ment,vacancy,andjob-finding An increasein the separationratedoes not and affecttherelativevalue of unemployment vacancies,andso leavesthev-uratioessentially unchanged.Since the increasein separations reducesemployment duration,the unemploymustvamentrateincreases,and so therefore cancies. As a result, fluctuationsin the posiseparationrateinducea counterfactually and tive correlationbetweenunemployment vacancies. SectionI presentstherelevantU.S. business u is strongly councyclefacts:unemployment vacanciesv areequallystrongly protercyclical, betweenthe two cyclical,and the correlation variablesis -0.89 at businesscycle frequenand cies. As a result,thev-uratiois procyclical volatile,witha standarddeviationaroundits trendequal to 0.38 log points.To providefurI examofthisfinding, therevidencein support workersfind ine therateat whichunemployed withjobs jobs. If theprocessofpairingworkers is well-describedby an increasing,constant function returns-to-scale m(u,v),as in matching Pissarides(1985), thejob findingrateis f = function ofthev-uratio. an increasing m(u,v)/u, I use unemployment-duration data to measure deviratedirectly. The standard thejob-finding in the job-findingrate ation of fluctuations aroundtrendis 0.12 log pointsand thecorrelationwiththev-uratiois 0.95. FinallyI look at rate thetwoproposedimpulses.The separation is less correlated withthecycleandmoderately volatile,witha standarddeviationabouttrend equal to 0.08 log points.Averagelaborproductivityis weaklyprocyclicaland evenmorestable, witha standarddeviationabout trendof 0.02 log points. In SectionII, I extendthePissarides's(1985) modelto allow foraggresearchand matching I introducetwo types of gate fluctuations. shocksraise output shocks:labor productivity in all matchesbutdo notaffect therateatwhich their lose workers job; andseparation employed shocksraisetherateatwhichemployedworkers butdo notaffecttheprobecomeunemployed in surviving matches.In equilibrium, ductivity thereis onlyone realeconomicdecision:firms'
MARCH2005
choiceof whether to opena new vacancy.The equilibrium vacancyratedependson theunemandon rate,on labormarket ployment tightness, the expectedpresentvalue of wages in new relationships. Wages, in turn,are employment at leastin new determined by Nashbargaining, thewage in old matches matches.In principle, may be rebargainedin the face of aggregate shocksor maybe fixedbya long-term employmentcontract. SectionII A describesthebasic model,while SectionII B derivesa forwardlookingequationforthe v-u ratioin termsof modelparameters. comSectionII C performs simpleanalytical parativestaticsin some special cases. For exofthev-uratio ample,I showthattheelasticity withrespectto the difference betweenlabor andthevalueofnonmarket productivity activity or "leisure"is barelyin excessof 1 forreasonable parameter values. To reconcilethiswith the data, one mustassume thatthe value of leisureis nearlyequal to laborproductivity, so marketworkprovideslittleincremental utility. The separation ratehas an evensmallerimpact on the v-u ratio,with an elasticityof -0.1 statics.Moreover, accordingto thecomparative while shocksto laborproductivity at least inbetweenunemployduce a negativecorrelation mentand vacancies,separationshocks cause bothvariablesto increase,whichtendsto generate a positivecorrelation betweenthe two variables.Similarresultsobtainin some other specialcases. thestochastic modelto SectionII D calibrates matchU.S. data along as manydimensionsas possible,and SectionII E presentstheresults. The exerciseconfirms the quantitative predictionsofthecomparative statics.Iftheeconomy is hit only by productivity shocks,it moves alonga downward-sloping Beveridgecurve,but in laborproempirically plausiblemovements in the v-u resultin tinyfluctuations ductivity ratio.Moreover,laborproductivity is perfectly thatthe correlated withthev-uratio,indicating mechmodelhas almostno internal propagation anism.If theeconomyis hitonlyby separation shocks,the v-u ratiois stablein the face of so vacancies fluctuations, largeunemployment are countercyclical. Equivalently,the modelgenerated Beveridgecurveis upward-sloping. SectionII F explorestheextentto whichthe Nash bargainingsolutionis responsiblefor theseresults.FirstI examinethe behaviorof
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and wages in the face of labor productivity shocks.An increasein laborproducseparation to createvacancies.The encouragesfirms tivity rateputs resultingincreasein thejob-finding upwardpressureon wages,soakingup virtually all of the shock.A decreasein the separation to createmorevacancies, ratealso inducesfirms again puttingupwardpressureon wages and minimizingthe impacton the v-u ratio and rate.On theotherhand,I examinea job-finding versionof the model in whichonly workers' bargainingpoweris stochastic.Small fluctuations in bargainingpower generaterealistic in thev-uratiowhileinducingonly movements real wage,witha a moderately countercyclical standarddeviationof 0.01 log pointsaround trend. Section III provides anotherangle from I whichto viewthemodel'sbasic shortcoming. considera centralized economyin whicha social plannerdecides how manyvacancies to createin orderto maximizethepresentvalueof incomenetof vacancy marketand nonmarket and centralcreationcosts. The decentralized ifthematchized economiesbehaveidentically is Cobb-Douglasin unemployment ingfunction andvacanciesandworkers'bargaining poweris of thematching function equal to theelasticity withrespectto theunemployment rate,a generalizationof ArthurHosios (1990). But if and vacanciesare moresubstiunemployment areamplified in thecentralfluctuations tutable, ized economy,essentiallybecause the shadow wage is less procyclical.Empiricallyit is of unto measurethe substitutability difficult employmentand vacancies in the matching difficult to tellwhether and therefore function, are optimal. observedfluctuations SectionIV reconcilesthispaperwitha number of existingstudies that claim standard with modelsareconsistent searchandmatching the businesscycle behaviorof labor markets. Finally,the paper concludesin SectionV by somemodifications tothemodelthat suggesting mightdeliverrigid wages and therebydo a theempiricalevidenceon better job ofmatching vacanciesand unemployment. one important-but It is worthemphasizing standard-feature of the searchand matching thispaper: framework thatI exploitthroughout workers arerisk-neutral andsupplylaborinelasemtically.In theabsenceof searchfrictions, wouldbe constant evenin thefaceof ployment
27
shocks. This distinguishesthe productivity presentmodelfromthosebased uponintertemporallaborsupplydecisions(RobertE. Lucas, Jr.,and LeonardRapping,1969). Thusthispaperexplorestheextentto whicha combination of searchfrictionsand aggregateshockscan in unemploygenerateplausible fluctuations mentand vacanciesif laborsupplyis inelastic. It suggeststhatsearchfrictions perse scarcely amplifyshocks.The paper does not examine whethera searchmodel withan elastic labor supplycan providea satisfactory explanation for the observed fluctuations in these two variables. I. U.S. Labor MarketFacts
This sectiondiscussesthetimeseriesbehavior of unemployment u, vacanciesv, thejob findingratef, the separationrates, and labor p in the UnitedStates.Table I productivity thedetrended data. summarizes A. Unemployment The unemployment rate is the most commonlyused cyclicalindicatorofjob-searchactivity.In an averagemonthfrom1951 to 2003, 5.67 percentof theU.S. laborforcewas outof work,availableforwork,and activelyseeking work. This time series exhibitsconsiderable temporalvariation, fallingto as low as 2.6 percentin 1953 and 3.4 percentin 1968 and 1969, but reaching10.8 percentin 1982 and 1983 are al(Figure1). Some of thesefluctuations most certainlydue to demographic and other factorsunrelatedto businesscycles. To highI fluctuations, light business-cycle-frequency take thedifference betweenthelog of theunlow frelevel and an extremely employment (HP) filter quency trend,a Hodrick-Prescott withsmoothing parameter105 usingquarterly data.2The difference betweenlog unemploymentand its trendhas a standarddeviationof is oftenas muchas 38 0.19, so unemployment unempercentaboveorbelowtrend.Detrended also exhibits considerable ployment persistence, withquarterly autocorrelation 0.94. level ofunemployment rather 2 use the thantherateto keeptheunitscomparabletothoseofvacancies.A previous versionof thispaperused theunemployment rate,withno effecton theconclusions.
28
THE AMERICANECONOMIC REVIEW TABLE 1-SUMMARY
STATISTICS, QUARTERLY U.S.
u Standarddeviation autocorrelation Quarterly
Correlation matrix
u v v/u f
s p
0.190 0.936 1 -
-
v
v/u
MARCH 2005
DATA, 1951-2003
f
s
p
0.202 0.940
0.382 0.941
0.118 0.908
0.075 0.733
0.020 0.878
-0.894 1
-0.971 0.975 1
-0.949 0.897 0.948
0.709 -0.684 -0.715
-0.408 0.364 0.396
-
1
-
--
-0.574
1 -
0.396
-0.524 1
Notes: Seasonallyadjustedunemployment u is constructed by theBLS fromtheCurrentPopulationSurvey(CPS). The index v is constructed ratef and seasonallyadjustedhelp-wanted advertising by the ConferenceBoard. The job-finding fromseasonallyadjustedemployment, and meanunemployment duration, separationrates are constructed unemployment, all computedbytheBLS fromtheCPS, as explainedin equations(1) and (2). u, v,f, ands arequarterly averagesofmonthly series.Averagelaborproductivity businesssector, p is seasonallyadjustedreal averageoutputperpersonin thenon-farm constructed by the Bureau of Labor Statistics(BLS) fromthe NationalIncomeand ProductAccountsand the Current All variablesare reported in logs as deviationsfroman HP trendwithsmoothing Statistics. 105. Employment parameter
12
are distincteconomicconditions. participation ChinhuiJuhnet al. (1991) showthatalmostall 10 of the cyclicalvolatilityin prime-agedmale is accountedforby unemploynonemployment ment.Christopher Flinn and JamesHeckman workersare sig(1983) showthatunemployed more to find a nificantly likely job thannonparticipants,althoughStephenJones and Craig Riddell(1999) arguethatothervariablesalso a job. In helptopredictthelikelihoodoffinding 2 since labor force is proanycase, participation the ratio cyclical, employment-population is a more cyclicalmeasureof job-searchactivity, 2000 1990 1980 1970 1960 worseningthe problemshighlightedin this paper. FIGURE 1. QUARTERLY U.S. UNEMPLOYMENT (IN MILLIONS) It is also conceivablethatwhenunemployAND TREND, 1951-2003 ment rises,theamountofjob-searchactivity per Notes:Unemployment is a quarterly of the seasonaverage workerdeclinesso muchthatagseriesconstructed unemployed ally adjustedmonthly by theBLS from the CPS, surveyhome page http://www.bls.gov/cps/. The gregatesearchactivityactuallyfalls.Thereis trendis an HP filterof thequarterly data withsmoothing bothdirectand indirectevidence againstthis parameter105. As directevidence,one wouldexhypothesis. in searchintensity couldbe pectthata reduction observedas a declinein the numberof jobThereis somequestionas to whether unem- searchmethodsused or a switchtowardless ortheemployment-population ratiois time-intensive methods.An examinationof ployment a betterindicatorofjob-searchactivity. Advo- Current PopulationSurvey(CPS) dataindicates cates of the latterview, for exampleHarold no cyclicalvariationin thenumberor typeof Cole and RichardRogerson(1999), arguethat job-searchmethodsutilized.3Indirectevidence the numberof workersmovingdirectlyinto comes fromestimatesof matchingfunctions, fromout-of-the-labor forceis as whichuniversally findthatan increasein unemployment is associatedwithan increasein largeas thenumberwhomovefromunemploy- employment mentto employment (Olivier Blanchardand PeterDiamond,1990). On theotherhand,there 3 Shimer(2004b) discussesthisevidencein detail. andnonis ampleevidencethatunemployment
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and thenumberofmatches(BarbaraPetrongolo declined Pissarides,2001). Ifjob-searchactivity increased,the sharplywhen unemployment matchingfunctionwould be measuredas deI concludethatagcreasingin unemployment. gregate job search activity is positively withunemployment. correlated B. Vacancies
1.6. 1.3
014
29
.3 -
JOLTS
r
Conference Board
CD
.0
1.2
.1
1.1 10
-.2
0
-.3
.9
2001:1 2001:3 2002:1 2002:3 2003:1 2003:3 is job vacanThe flipside of unemployment cies. The Job Openingsand Labor Turnover FIGURE2. Two MEASURESOF U.S. JOBVACANCIES, 2000Q4-2003Q4 Survey(JOLTS) providesan ideal empirical "A job openingrequiresthat 1) a definition: of Notes:The solidlineshowsthelogarithm ofthenumber specificpositionexists,2) work could start job openingsin millions,measuredby the BLS fromthe quarterly is actively JOLTS, surveyhomepagehttp://www.bls.gov/jlt, within30 days,and 3) theemployer averagedand seasonallyadjusted.The dashedline shows of to from outside the establishment recruiting thedeviationfromtrendof thequarterly averaged,seasonfill the position.Includedare full-time, part- ally adjustedConferenceBoard help-wantedadvertising and short-term index. time, permanent,temporary, meansthattheesopenings.Activerecruiting efforts to fill is engagedin current tablishment is subjectto lowin newspapers thathelp-wanted theopening,suchas advertising advertising fluctuations that are relatedonlytanor on the Internet, postinghelp-wanted signs, frequency Inrecentyears,the tothelabormarket. acceptingapplications,or using similarmeth- gentially Internetmay have reducedfirms'relianceon JOLTSbeganonlyin Deods."4Unfortunately, as a sourceofjob advertising, while cember2000 and comparabledata had never newspapers in in the consolidation the United States. been collected 1960s, newspaper may previously in surviving to look haveincreasedadvertising therearetoofewobservations newspaAlthough laws at thistimeseries,its behavior pers and Equal Employment Opportunity systematically In the firstmonthof the may have encouragedfirmsto advertisejob has been instructive. a lowsectormaintained a seaopeningsmoreextensively. Fortunately, survey,thenon-farm trend should remove the effect of million This 4.6 job openings. frequency sonallyadjusted numberfellrapidlyduring2001 and averaged theseand othersecularshifts.Figure3 shows indexanditstrend. advertising just 2.9 millionin 2002 and 2003. This decline thehelp-wanted in job openings,depictedby the solid line in Notably,the decline in the detrendedhelpFigure2, coincidedwitha periodof risingun- wantedindexcloselytracksthe declinein job thatjob vacanciesare fromJOLTS during openingsmeasureddirectly suggesting employment, theperiodwhenthelattertimeseriesis availprocyclical. To obtaina longertimeseries,I use a stan- able (Figure2). Board dardproxyforvacancies,theConference Figure4 showsa scatterplotof therelationmeasured as the of unemindex, advertising shipbetweenthecyclicalcomponent help-wanted in 51 numberof help-wantedadvertisements ploymentand vacancies,the Beveridgecurve. A potentialshortcoming is The correlation of thepercentagedeviationof majornewspapers.5 and vacancies fromtrendis unemployment -0.89 between1951 and 2003.6Moreover,the 4 This definition comes fromtheBureauof Labor Statisticsnews release, July30, 2002, available at http:// www.bls.gov/jlt/jlt_nrl.pdf. (1987) discussesthismeasurein detail.From 5 Abraham 1972 to 1981,Minnesotacollectedstate-wide job vacancy data.AbrahamcomparesthiswithMinnesota'shelp-wanted indexand showsthatthetwo seriestrackeach advertising otherveryclosely throughtwo businesscycles and ten seasonalcycles.
6 AbrahamandKatz (1986) andBlanchardandDiamond (1989) discusstheU.S. Beveridgecurve.AbrahamandKatz betweenunem(1986) arguethatthenegativecorrelation andvacanciesis inconsistent withLilien's (1982) ployment sectoralshiftshypothesis, and insteadindicatesthatbusinesscyclesare drivenby aggregatefluctuations. Blanchard and Diamond (1989) conclude that at business cycle
30
THE AMERICANECONOMIC REVIEW
0.6
110 100
MARCH2005
..........
90 .
0.o2-------
70 60
0.4
50 4030
0
@i
1960
1970
1980
1990
2000
FIGURE3. QUARTERLYU.S. HELP-WANTEDADVERTISING
INDEXANDTREND,1951-2003
~d
_
-0.6 -0.6
-0.4
-0.2
0.2
0.0
0.4
0.6
indexis a quarterly Notes:The help-wanted advertising Unemployment seriesconadjustedmonthly averageof theseasonally Board with normalization structed the Conference by FIGURE 4. QUARTERLY U.S. BEVERIDGE CURVE, from theFederal 1987= 100.Thedataweredownloaded 1951-2003 ReserveBank of St. Louis databaseat http://research. Thetrend is an HP Notes:Unemployment is constructed the stlouisfed.org/fred2/data/helpwant.txt. bytheBLS from datawithsmoothing filter ofthequarterly CPS. Thehelp-wanted indexis constructed parameter 105. advertising by theConference Board.Botharequarterly ofseaaverages seriesandareexpressed as devisonally adjusted monthly ationsfrom an HP filter withsmoothing parameter 105. variationin standarddeviationof the
cyclical and vacanciesis almostidentiunemployment cal, between0.19 and 0.20, so theproductof Grossworkerflowdata can be used to meaand vacanciesis nearlyacyclic. unemployment rate directly,and indeed The v-uratiois therefore extremely procyclical, sure thejob-finding and with a standarddeviationof 0.38 aroundits both the unemployment-to-employment transition rates trend. nonparticipation-to-employment are stronglyprocyclical(Blanchardand DiaRate C Job-Finding mond,1990; HoytBleakleyet al., 1999; KatharineAbrahamand Shimer,2001). Thereare of the twodrawbacksto thisapproach.First,thereqAn implicationof the procyclicality v-u ratiois thatthe hazardrateforan unem- uisitepublicuse datasetis availableonlysince a job, hisjob-finding 1976, and so using these data would require ployedworkerof finding rate,shouldbe lowerduringa recession.Asthrowing awayhalfoftheavailabletimeseries. and classification error is sumethatthenumberofnewlyhiredworkers Second, measurement of worker lead a substantial overestimate and constant returns-toan gross givenby increasing flows(JohnAbowdand ArnoldZellner,1985; scale matchingfunction m(u,v),dependingon workersu and the JamesPoterbaand LawrenceSummers,1986), the numberof unemployed of whichcannoteasilybe comthat themagnitude number ofvacanciesv.Thentheprobability ratefrom finds a individual worker puted.Instead,I inferthejob-finding unemployed job, any ratefromunemployment the dynamicbehaviorof the unemployment theaveragetransition = m(1,0), where level and short-term level. Let isf m(u,v)/u toemployment, unemployment denote the number of workers ratef 0 v/uis thev-u ratio.The job-finding unemployed ut move togetherwiththe v-u forless thanone monthin montht. Then asshould therefore workersfinda job with ratio. sumingall unemployed in month t and no unemployed probability ft workerexitsthelaborforce, and shocksgenerallydrivetheunemployment frequencies, vacancyratesin theoppositedirection.
Ut+1
=
Ut(1 -ft)
+
Us+1.
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SHIMER: UNEMPLOYMENTAND VACANCIES
0.3
0.70 0.65
00
0 .2
0.60
i
0.55 0.50 0.45
D
0.00.
0.1
I
oo
9 o ........................ 0
o
0
o
0.40
I
0.0
0.35 0.30
31
-0.2 1960
1970
1980
1990
2000
FIGURE 5. MONTHLY JOB-FINDING PROBABILITY FOR
UNEMPLOYEDWORKERS,1951-2003
Notes:Thejob-finding rateis computedusingequation(1), with unemployment and short-term data unemployment and seasonallyadjustedby the BLS fromthe constructed It is exCPS, surveyhomepage http://www.bls.gov/cps/. data. The trend averageof monthly pressedas a quarterly of thequarterly is an HP filter datawithsmoothing parameter105.
nextmonthis the sum of the Unemployment ofunemployed number workers thismonthwho fail to finda job and the numberof newly workers.Equivalently, unemployed
-1.2
-0.8
-0.4
0.0
0.4
0.8
1.2
Ratio Vacancy-Un Vacancy-UnemploymentRatio
FIGURE 6. MONTHLY U.S.
MATCHING FUNCTION,
1951-2003
Notes: The v-u ratiois constructed by the BLS fromthe CPS and by theConference Board.The job-finding rateis constructed usingequation(1) and BLS datafromtheCPS. Bothare quarterly averagesof seasonallyadjustedmonthly seriesandareexpressedas deviations froman HP filter with smoothing parameter 105.
One can use the measuredjob-finding rate and v-u ratioto estimatea matchingfunction Ut+I - /t+ forceme to imposetwo m(u,v).Data limitations (1) = on restrictions the estimated function. f, 1First,beUt causeunemployment andvacanciesarestrongly I use theunemployment levelandthenumber of it is difficult to tell emcorrelated, negatively workersunemployedfor 0 to 4 weeks,both piricallywhetherm(u,v)exhibitsconstant,inconstructed to scale. But in by theBLS fromtheCPS, to com- creasing,or decreasingreturns putef from1951 to 2003.7 Figure5 showsthe theirliterature survey,Petrongoloand Pissarhazardrateaveraged0.45 results.The monthly ides (2001) concludethatmostestimates of the from1951 to 2003. Afterdetrending withthe matching function cannotrejectthenullhypothHP filter, usual low-frequency the correlation esis ofconstant I therefore estimate returns; f= betweenf, and 0, at quarterlyfrequenciesis consistent with a constant returns-to-scale f(O8), function. 0.95, althoughthe standarddeviationof ft is matching Figure6 showstherawdata about 31 percentthatof O,. Given thatboth forthejob-finding rateftand thev-uratio0,,a measuresare crudelyyet independently con- nearlylinearrelationship whenbothvariables areexpressedas deviationsfromlog trend.Secstructed,this correlationis remarkableand function is a ond, I impose thatthe matchingfunctionis strongly suggeststhata matching usefulway to approachU.S. data. Cobb-Douglas,m(u,v) = Ctu?v1-a,for some unknownparameters a and /,.Again,thedata are notveryinformative as to whether thisis a reasonablerestriction.B I estimatethematching 7 Abrahamand Shimer(2001) arguethattheredesignof theCPS in January theswitchto depen1994,in particular dentinterviewing, reducedmeasuredshort-term unemployment.They suggestsome methodsof dealing withthis In thispaper,I simplyinflateshort-term undiscontinuity. employment by 10 percentaftertheredesigntookeffect.
function 8 ConsidertheCES matching logf, = log Ii + to 1/plog (a + (1 - a)O,). Cobb-Douglascorresponds
32
THE AMERICANECONOMIC REVIEW
function dataon thejob-finding usingdetrended rate and the v-u ratio.Dependingon exactly in theresiduhow I controlforautocorrelation als, I estimatevalues of a between0.70 and 0.75. Witha first-order residual, autoregressive I get a = 0.72 witha standard errorof 0.01. crudeaspectofthismeasure One particularly thatall rateis theassumption ofthejob-finding workersare equallylikelyto finda job. Shimer are (2004a) provesthatwhentheunemployed heterogeneous, f,measuresthemeanjob-finding ratein theunemployed population.Thatpaper measureofthejobalso comparesmypreferred The first uses ratewithtwoalternatives. finding level and meanunemploytheunemployment mentdurationto obtaina weightedaverageof ratein theunemployed thejob-finding populato each individtion,withweightsproportional duration.9The second ual's unemployment followsRobertHall (2004) and measuresthe rate of workerswithshortunemjob-finding duration using the ratioof workers ployment to workers with0 to 4 weeksofunemployment Sincethe with5 to 14 weeksofunemployment. job-findingrate declines withunemployment I findthatmypreferred measureofthe duration, these twoalternarate lies between job-finding rate tives.Hall measuresan averagejob-finding duraof 0.48 per month,whileunemployment rate of 0.34 per tion data yield a job-finding month.Nevertheless,all threemeasuresare and so the choice of which highlycorrelated, affectthe measureto use does notqualitatively conclusionsof thisstudy.
MARCH2005
0.050
0.045.
0.035 0.030 0.025 0.020
.......... 1960
......... 1970
............. 1980 1990
2000
PROBABILITY FOR FIGURE 7. MONTHLY SEPARATION EMPLOYEDWORKERS, 1951-2003 Notes:The separation rateis computedusingequation(2), and short-term unemwithemployment, unemployment, and seasonallyadjustedby the ploymentdata constructed BLS fromtheCPS, surveyhomepage http://www.bls.gov/ data. averageof monthly cps/.It is expressedas a quarterly of thequarterly datawithsmoothThe trendis an HP filter ing parameter 105.
bias. Whena workerloses herjob, she has on averagehalfa monthto finda newjob before for she is recordedas unemployed. Accounting ratethenext this,theshort-term unemployment monthis approximately equal to ut+l
=
stet(1 -
ft).
another of finding job Ignoringtheprobability the withinthe monthleads one to understate D. SeparationRate separationrate. This problemis particularly rate is high,i.e., acute when the job-finding measurethesepI can also deducethebehaviorof thesepara- duringexpansions.I therefore short-term arationrateas tionratefromdata on employment, and the hiringrate. Suppose unemployment, firstthatwheneveran employedworkerloses s,= et(1ut+ 1 t her job, she becomes unemployed.Then the (2) ratecouldsimplybe computedas the separation ratio of short-term unemployedworkersnext month,us+1,to employedworkersthismonth, Figure 7 shows the monthlyseparationrate It averaged0.034 from1951 time-aggregation thusconstructed. e,.Butthismasksa significant to 2003, so jobs last on averageforabout2.5 in the deviationof the log years.Fluctuations ratefromtrendaresomewhatsmaller limitingcase of p = 0. WhenI estimatetheCES function separation using nonlinearleast squares and correctfor first-order thanin thehiring deviation rate,witha standard I get a pointestimateof p = 0.06 witha autocorrelation, so are of and 0.08, countercyclical, separations standarderrorof 0.38. withthe detrended v-u ratiois thecorrelation 9 A previousversionofthispaperreliedon thatmeasure -0.72. off, This had littleeffecton theresults.
st= tiIf
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SHIMER: UNEMPLOYMENTAND VACANCIES
33
1.2
.06
120
o 0.8
-------------------------------------- .04
110
rr(D
130
0.4
.02
100 -a E e(-0.4.
90 80
0
-.02 -0
8
70
> -1.2.0
60
-.06 -1.6 1960
50 40
Ratio Vacancy-Unemployment AverageLaborProductivity 1990 2000 1970 1980
08
FIGURE9. QUARTERLYU.S. VACANCY-UNEMPLOYMENT
RATIO AND AVERAGE LABOR PRODUCTIVITY, 1951-2003
Notes:Unemployment is constructed by theBLS fromthe CPS. The help-wanted indexis constructed by advertising theConference Board.Bothare quarterly averagesof seabusiness Notes: Real outputper personin the non-farm is real series.Labor productivity sonallyadjustedmonthly sector,constructed by theBLS Major SectorProductivity averageoutputperworkerin thenon-farm businesssector, and Costsprogram, surveyhomepage http://www.bls.gov/constructed and by the BLS Major Sector Productivity of thequarterly Costs are lpc/,1992 = 100. The trendis an HP filter program.The v-u ratioand labor productivity data withsmoothing parameter 105. expressedas deviationsfroman HP filterwithsmoothing parameter 105. FIGURE8. QUARTERLYU.S. AVERAGELABOR
ANDTREND,1951-2003 PRODUCTIVITY
The strongprocyclicality of thejob-finding of rate and relativelyweak countercyclicality the separationratemightappearto contradict Blanchardand Diamond's (1990) conclusion in the flow that"theamplitudein fluctuations outofemployment is largerthanthatoftheflow This is easily reconciled. into employment." Blanchardand Diamondlook at thenumberof in a people enteringor exitingemployment givenmonth, fu, or se,, whileI focuson the thatan individualswitchesemployprobability mentstates,f, and st.Althoughtheprobability of enteringemployment f, declinessharplyin recessions,thisis almostexactlyoffsetby the increasein unemployment ut,so thatthenumis essenber of people exitingunemployment tiallyacyclic.Viewed throughthe lens of an increasingmatchingfunctionm(u,v), this is evidencethat consistentwiththe independent vacanciesare strongly procyclical. E. Labor Productivity The finalimportant is empiricalobservation the weak procyclicality of labor productivity, measuredas real outputperworkerin thenonfarmbusinesssector.The BLS constructs this timeseriesas partof itsMajor Sector quarterly and Costs program.The output Productivity measureis based on theNationalIncomeand
is conProductAccounts,while employment fromtheBLS establishment the structed survey, Current Statistics.This seriesofEmployment ferstwoadvantagescomparedwithtotalfactor it is available quarterlysince productivity: totheconceptof 1948;anditbetter corresponds in the subsequentmodels, labor productivity whichdo notincludecapital. Figure8 showsthebehavioroflaborproductivityand Figure9 comparesthecyclicalcomponentsof thev-uratioand laborproductivity. Thereis a positivecorrelation betweenthetwo timeseriesand some evidencethatlaborproleads thev-u ratioby aboutone year, ductivity witha maximumcorrelation of 0.56.10But the mostimportant factis thatlaborproductivity is stable,never deviatingby more than 6 percentfromtrend.In contrast, the v-u ratiohas twicerisento 0.5 log pointsaboutitstrendlevel and six timeshas fallenby0.5 log pointsbelow trend. 10From1951 to correlation 1985,thecontemporaneous betweendetrended laborproductivity and thev-uratiowas 0.57 andthepeakcorrelation was 0.74. From1986to 2003, and peak correlations are however,the contemporaneous This has been negative,-0.37 and -0.43, respectively. noticeableduringthelastthreeyearsofdata.An particularly explorationof the cause of thischangegoes beyondthe scope of thispaper.
34
THE AMERICANECONOMIC REVIEW
It is possiblethatthemeasuredcyclicality of laborproductivity is reducedby a composition bias, since less productiveworkersare more I offer two likelyto lose theirjobs inrecessions. responsesto thisconcern.First,thereis a compositionbias thatpointsin theoppositedirection: labor productivity is higherin more cyclical sectorsof the economy,e.g., durable And second,a largelitgoods manufacturing. eratureon real wage cyclicality has reacheda mixedconclusionabouttheimportance ofcompositionbiases (Abrahamand JohnHaltiwanger, 1995). Gary Solon et al. (1994) provide perhapsthe strongest evidencethatlabor forcecomposition is important forwage cyclicality,but even they argue that accounting forthismightdoublethemeasuredvariability of real wages. This paper argues that the search and matchingmodel cannot account forthecyclicalbehaviorof vacanciesand ununlesslaborproductivity is at least employment ten timesas volatileas the data suggest,so compositionbias is at best an incomplete explanation. Model II. Searchand Matching I nowexaminewhether a standard searchand modelcan reconcilethestrong procymatching rate clicalityofthev-uratioandthejob-finding withtheweakprocyclicality oflaborproductivrate. oftheseparation ityandcountercyclicality an aggregate The modelI consideris essentially versionof Pissarides(1985, or 2000, stochastic Ch. 1). A. Model I startby describingthe exogenous variLabor productivables thatdrivefluctuations. the rate s followa firstand p separation ity orderMarkovprocess in continuoustime.A shock hitsthe economyaccordingto a Poisson processwitharrivalrateA,at whichpoint a new pair (p',s') is drawnfroma statedeLet denotethe pendentdistribution. Ep,X,p,s, expected value of an arbitraryvariable X followingthe next aggregateshock, conditionalon thecurrentstate(p,s). I assumethat thisconditionalexpectationis finite,whichis ensuredifthestatespace is compact.At every point in time,the currentvalues of produc-
MARCH2005
tivityand the separationrate are common knowledge. Next I turnto the economicagentsin the economy,a measure1 ofrisk-neutral, infinitelylivedworkersand a continuum of risk-neutral, firms. All agentsdiscountfuture infinitely-lived payoffsat rater > 0. Workerscan eitherbe unemployedor emworkergetsflowutility ployed.An unemployed z from non-market activity("leisure") and searchesfora job. An employedworkerearns an endogenouswage but may not search.I discusswage determination shortly. Firmshave a constantreturns-to-scale productiontechnology thatuses only labor,with labor productivity at timet givenby the stochastic realizationp(t). In order to hire a worker,a firmmustmaintainan openvacancy at flowcost c. Free entrydrivesthe expected presentvalue of an open vacancyto zero. A workerand a firmseparateaccordingto a Poisson process with arrivalrate governed by the stochasticseparationrate s(t), leaving the workerunemployedand the firmwith nothing. Let u(t) denotethe endogenousunemploymentrate,11v(t) denotethe endogenousmeasureof vacanciesin theeconomy,and 0(t) denotethev-uratioat timet. The flow v(t)/u(t) of matchesis givenby a constantreturns-toscale function m(u(t),v(t)), increasingin both This impliesthatan unemployed arguments. workerfindsa job accordingto a Poissonprocess withtime-varying arrivalratef(O(t)) m(1,0(t)) and thata vacancyis filledaccording arrival to a Poissonprocesswithtime-varying rateq(0(t)) - m(0(t)', 1) = f(0(t))/0(t). I assume thatin everystateof the world, laborproductivity p(t) exceedsthevalue of leisurez, so thereare bilateralgainsfrommatching. There is no singlecompellingtheoryof in such an environment, wage determination and assumethat and so I followthe literature whena workerandfirmfirst meet,theexpected gainsfromtradeare splitaccordingto theNash to solution.The workercan threaten bargaining and thefirmcan threaten becomeunemployed to end thejob. The presentvalue of surplus of workersconstantand normali1 Withthepopulation ized to one, the unemployment rate and unemployment use theseterms levelareidenticalin thismodel.I therefore interchangeably.
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VOL. 95 NO. 1
beyondthese threatsis divided betweenthe workerand firm,withthe workerkeepinga fraction f3E (0, 1) ofthesurplus,her"bargaining power." I make almost no assumptions aboutwhathappensto wages afterthisinitial agreement,except that the workerand firm manageto exploitall thejointgainsfromtrade. For example,the wage may be re-bargained wheneverthe economyis hit with a shock. itmaybe fixedat itsinitialvalue Alternatively, untilsuch time as the firmwould preferto firethe workeror the workerwould prefer to quit,whereupon thepairresetsthewage so as to avoid an unnecessaryand inefficient separation. B. Characterization ofEquilibrium I look foran equilibriumin whichthe v-u ratiodependsonly on the currentvalue of p
and
Giventhe
35
the pair is matched,theyproducep unitsof output.If theywereto breakup thematch,free entryimpliesthefirmwouldbe leftwithnothing, while the workerwould become unemployed,gettingflowutilityfromleisurez and fromtheprobability of a firm, f(Op,s) contacting inwhicheventtheworker wouldkeepa fraction 0 of thematchvalue Vp,s.Next,thereis a flow s thattheworkerand firmseparate, probability thematchvalue. Finally,an aggredestroying gate shock arrivesat rate A, resultingin an expectedchange in matchvalue p,sVp,s p,s.
Another criticalequationforthematchvalue comes fromfirms'freeentrycondition.The flowcost of a vacancyc mustequal the flow thatthevacancycontactsa worker probability timestheresulting capitalgain,whichby Nash is equal to a fraction1 - 3 of the bargaining matchvalue Vp,s:
v-u
s, state-contingent Op,s.12 c = q(Op,s)(1- ) Vp,s. rateevolvesaccording (5) ratio,theunemployment to a standardbackward-looking differential equation, Eliminatingcurrentand futurevalues of Vp,s from(4) using(5) gives (3)
ui(t) = s(t)(l
- u(t)) --f(Op(t),s(t))u(t)
r+s+A q(O~
+ + p,, where(p(t), s(t)) is theaggregate stateat timet. A flows(t) of the 1 - u(t) employedworkers 1 becomeunemployed, whilea flowf(O) of the = (1 - p) p-z + AE workersfinda job. An initial u(t) unemployed c ' q(Op,,) condition rateand pinsdowntheunemployment -theaggregatestateat somedateto. I characterize thev-u ratiousinga recursive whichimplicitly definesthev-uratioas a funcequationforthejointvalueto a workerandfirm tionof thecurrentstate(p,s).13 This equation of beingmatchedin excessof breakingup as a can easilybe solved numerically, even witha function of thecurrent state vector. This of state, aggregate large simplerepresentation Vp,s. the equilibriumof a stochasticversionof the Pissarides(1985) model appearsto be new to (4) rVp,s=p - (z + f(Op,,)PVp,s) sVp,s theliterature. (6)
+ (EsV,,,-V
AppendixA derivesthisequationfrommore conditions. The firsttwotermsrepreprimitive sentthecurrent flowsurplusfrommatching. If
C. Comparative Statics In some special cases, equation (6) can be solved analyticallyto get a sense of the
12
It is straightforward to showin a deterministic version of thismodelthatthereis no otherequilibrium, e.g.,one in which0 also dependson theunemployment rate.See Pissarides(1985).
13 A similarequationobtainsin thepresenceof aggrein thevalueofleisurez, thecostofa vacancy gatevariation c, or workers'bargaining powerP3.
36
THE AMERICANECONOMIC REVIEW
resultsimpliedby this analysis. quantitative First,supposethereareno aggregate shocks,A = v-uratiosatisfies 0.14 Thenthestate-contingent r+s
p,O,= (1 - p)
+ q(O,~,) q
p-z c
The elasticity of thev-uratio0 withrespectto "netlaborproductivity" p - z is r+ s + (r + s)(1
-
Of(Op,,,) + Of(Op,s)
r(p,,s))
where-r(0)E [0,1]is theelasticity off(0). This is largeonlyifworkers'bargaining powerP3is smalland theelasticityri is close to one. But withreasonableparameter values,it is close to 1. For example,thinkof a timeperiodas equal to one month,so theaveragejob-finding rateis I 0.45 the elasticity approximately (Section C), 0.28 (SectionI C again), rq(0)is approximately is approxithe averageseparationprobability rate mately0.034 (SectionI D), andtheinterest is about 0.004. Then if workers'bargaining power 0 is equal to 1 - Tr(0),the so-called Hosios (1990) conditionfor efficiency,15the elasticityof the v-u ratiowithrespectto net labor productivity is 1.03. Lower values of P yielda slightly higherelasticity, say 1.15 when = = but at 0 does the of 0.1, only elasticity /3 3 thev-u ratiowithrespectto p - z rise appreciably,to 1.39. It would take implausibleparametervalues forthiselasticityto exceed 2. This impliesthatunlessthevalue of leisureis thev-uratiois likely close tolaborproductivity, to be unresponsiveto changes in the labor productivity. I can similarlycomputetheelasticity of the v-u ratiowithrespectto theseparation rate:
14 Shimer(2003) performs staticsexercises comparative Forexample,inthatpaper undermuchweakerassumptions. function can exhibitincreasing or decreasing thematching returns to scale and therecan be an arbitrary idiosyncratic allowingforendogenousseparaprocessforproductivity, tions (Mortensenand Pissarides,1994). I show thatthe if resultspresented heregeneralizeto suchan environment workersand firmsare sufficiently patientrelativeto the searchfrictions. 15 Section III shows thatthe Hosios condition carries overto thestochasticmodel.
MARCH 2005
-s + + Pf(O3,s)" (r s)(1 -q(O,,s)) thesamenumbers intothisexpresSubstituting siongives -0.10. Doublingtheseparationrate wouldhavea scarcelydiscernible impacton the v-u ratio. Finally,one can examinethe independent behaviorof vacancies and unemployment. In steadystate,equation(3) holds witht = 0. If the matchingfunctionis Cobb-Douglas, m(u,v)= LU'v1-a , thisimplies VP'S
(S 1
-
Ups) 1/(1a)
pt
P'
OuPa"
For a givenseparationrates, thisdescribesa betweenunemployment decreasingrelationship and vacancies,consistentwiththe Beveridge curve(Figure4). An increasein laborproductivityraisesthev-u ratiowhichlowerstheunrateand henceraisesthevacancy employment rate. Vacancies and unemployment should movein oppositedirections in responseto such shocks.But an increasein the separationrate thev-uratio.Instead,ittendsto scarcelyaffects raiseboththeunemployment andvacancyrates, an effectthatis likelyto producea counterfacbetweenunemploytuallypositivecorrelation mentand vacancies. I can perform similaranalyticexercisesby makingothersimplifying assumptions. Suppose that each vacancy contactsan unemployed workerat a constantPoissonrate .t, independentof theunemployment rate,so q(O) = t . Given the risk-neutrality this is assumptions, equivalentto assumingthatfirmsmustpay a fixedcost c/4Lin orderto hirea worker.Then evenwithaggregateshocks,equation(6) yields a staticequationforthev-u ratio: r+s p-z --- + p = p,s (1 - 3) c /x
In thiscase, theelasticity of thev-u ratiowith is respectto netlaborproductivity r + s + (P30 andtheelasticity ofthev-uratiowithrespectto
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SHIMER: UNEMPLOYMENTAND VACANCIES
theseparation rateis -s/P3otO. Sincef(O)= /10, one can againpindownall theparameter values exceptworkers'bargaining powerP3.Usingthe = sameparameter valuesas above,including/3 obtain and elasticities of 1.12 0.72, I -0.105, fromthecase withno shocks. almostunchanged Moregenerally, unlessP is nearlyequal tozero, bothelasticitiesare verysmall. At the oppositeextreme,supposethateach unemployedworkercontactsa vacancy at a constant Poissonratept,independent of theva= = so and rate, f(O) pt q(O) p/0.Also cancy assumethattheseparation rates is constant and p is a Martingale, averagelabor productivity function, equation Epp' = p. Withthismatching and futurevaluesof the (6) is linearin current v-u ratio:
(r+s+A +[0 =(1 - 3)
p-z
A
c
E,.
Itis straightforward toverify thatthev-uratiois linearinproductivity, andtherefore = IEp, OP,i.e.,
r+ + s
\
HPIH 1J
p= =(I-
p -z p-z c
so theelasticity of thev-uratiowithrespectto netlaborproductivity is 1, regardlessof workers' bargainingpower.I concludethatwitha widerangeofparameterizations, thev-uratio0 shouldbe approximately to netlaproportional borproductivity p - z. D. Calibration
37
The stochastic variablesare y is meanreverting. thenexpressedas functions of y. I use thediscretestatespace model Although in mysimulations as well,it is almostexactly correctand significantly easier to thinkabout thebehavioroftheextrinsic shocksby discussa related continuous state ing space model.16I of an expressthe statevariablesas functions Ornstein-Uhlenbeck process(see HowardTaylorandSamuelKarlin,1998,Section8.5). Lety satisfy dy = -yydt+ rdb whereb is a standardBrownianmotion.Here withhigher y > 0 is a measureof persistence, values indicatingfastermean reversion,and a > 0 is theinstantaneous standarddeviation. Thisprocesshas someconvenient y properties: is conditionally and unconditionally normal;it is meanreverting, withexpectedvalueconvergto zero; and asymptotically ing asymptotically its varianceconvergesto o2/2y. I considertwodifferent cases. In thefirst, the rateis constantandproductivity satseparation isfiesp = z + eY(p* - z), wherey is an Ornstein-Uhlenbeck processwithparameters y and o, and p* > z is a measureof long-run Since ey> 0, thisensures averageproductivity. > In the second is conz. case, productivity p stantand separationssatisfys = eYs*,where againy followsan Ornstein-Uhlenbeck process and now s* > 0 is a measureof thelong-run averageseparationrate.In bothcases, thestochasticprocessis reducedto threeparameters, y, a, and eitherp* or s*. I now proceedto explainthe choice of the otherparameters, withthecase of stostarting chasticproductivity. I followtheliterature and assume thatthe matchingfunctionis CobbDouglas,
This section parameterizesthe model to matchthetimeseriesbehaviorof theU.S. unrate.The mostimportant employment question = = f(O) Oq(O) tO'1. is thechoice of the Markovprocessforlabor and separations.AppendixC deproductivity to tenparameters: velopsa discretestatespacemodelwhichbuilds This reducesthecalibration on a simplePoisson processcorresponding to thetheoretical analysisin SectionII B. I define 16 I workon a discretegridwith2n + 1 = 2001 points, an underlying variabley thatlies on a finite which Gaussianinnovations. This imcloselyapproximate orderedset of points.When a Poisson shock thatPoissonarrivalrateofshocksis A = ny = 4 times plies hits,y eithermovesup or downby one point. perquarterin themodelwithlaborproductivity shocksand The probability of movingup is itselfdecreas- A = 220 in the model with (less persistent)separation valueofy,whichensuresthat shocks. ingin thecurrent
38
THE AMERICANECONOMIC REVIEW
theproductivity p*, thevalue of leiparameter surez, workers'bargaining power0, the discount rate r, the separationrate s, the two function a and p, thevamatching parameters and stancancycost c, and themeanreversion dard deviationof the stochasticprocess, y and o-. I normalizethe Withoutloss of generality, top* = 1. I choosethe productivity parameter standarddeviationand persistence of theproductivity processto matchtheempiricalbehavThis requiressetting ior of laborproductivity. = = An increasein the and 0.0165 0.004. y aofproductivity has volatility ao- a nearlyproportionaleffecton thevolatility of othervariables, whilethepersistence ofthestochastic processy results.For examscarcelyaffectsthereported ple,supposeI reduceyto 0.001,so productivity is morenearlya randomwalk. Because it is difficult to distinguish small values of y in a finitedataset,afterHP filtering the modeland the data, persistence magnitude generated oftheimpulseis virtually unchanged compared Butreassurwiththebaselineparameterization. behaviorofunemployment ingly,thedetrended and vacanciesis also scarcelyaffectedby inof laborproductivity. creasingthepersistence I set the value of leisureto z = 0.4. Since meanlaborincomein themodelis 0.993, this lies at the upperend of the rangeof income ratesin theUnitedStatesif interreplacement as an unemployment benefit. pretedentirely I normalizea timeperiodto be one quarter, setthediscountrateto r = 0.012, and therefore an to annualdiscountfactorof0.953. equivalent The analysisin SectionI D suggestsa quarterly rateofs = 0.10,sojobs lastforabout separation 2.5 years on average.This is comparableto Abowd and Zellner's(1985) findingthat3.42 percentof employedworkersexitemployment duringa typicalmonthbetween1972and 1982, aftercorrecting forclassification and measurementerror.It is also comparableto measured turnoverrates in the JOLTS, althoughsome inthatsurveyreflect job-to-jobtranseparations sitions,a possibilitythatis absentfromthis model. function estimatesfrom Using thematching to a = SectionI C, I settheelasticity parameter 0.72. Thislies towardtheupperendoftherange of estimatesthat Petrongoloand Pissarides (2001) report.I also set workers'bargaining power 3 to thesame value,0.72. Althoughthe
TABLE 2-PARAMETER
MARCH2005 VALUES IN SIMULATIONS OF THE MODEL
Sourceof shocks Parameter
Productivity Separation
p Productivity Separationrates Discountrater Value of leisurez q(O) Matchingfunction Bargainingpower 3 Cost of vacancyc Standarddeviationoy Autoregressive parameter Gridsize 2n + 1
stochastic 0.1 0.012 0.4
1 stochastic 0.012 0.4
0.72 0.213 0.0165 0.004 2001
0.72 0.213 0.0570 0.220 2001
1.3550-0.72
1.3550-0.72
Note:The textprovidesdetailson thestochastic processfor rate. and fortheseparation productivity
to thevalue of reportedresultsare insensitive I showin SectionIII thatifa = thatparameter, 3, the"Hosios (1990) Rule,"thedecentralized equilibriummaximizes a well-posed social planner'sproblem. I use thefinaltwoparameters, thematching function constantg and thevacancycost c, to rateand the pin downtheaveragejob-finding in I C, a v-u ratio. As Section average reported workerfindsa job witha 0.45 probability per month,so the flowarrivalrate of job offers -' 1.35 on a pt01 shouldaverageapproximately basis. I do not have a time series long quarterly withthelevel of thev-u ratio,but fortunately one morenormalization. themodeloffers Equation(6) impliesthatdoublingc and multiplying pkby a factor21-" dividesthev-u ratio0 in half,doubles the rate at whichfirmscontact workersq(O), but does not affectthe rate at whichworkersfindjobs. In otherwords,the in averagev-uratiois intrinsically meaningless a meanv-uratioof1, themodel.I choosetotarget whichrequires setting [L = 1.355andc = 0.213. In thecase of shocksto theseparation rate,I change only the stochasticprocess so as to resultsdiscussedin Section matchtheempirical I D. Productivity is constantand equal to 1, whilethemean separationrateis s* = 0.10. I set a = 0.057 and -y= 0.220, a muchless persistentstochasticprocess.This leaves the rate averagev-u ratioand averagejob-finding 2 the Table summarizes virtuallyunchanged. choicesin thetwosimulations. parameter I use equation(6) to findthestate-contingent v-uratioOand thensimulatethemodel.That
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VOL. 95 NO. 1
39
TABLE 3-LABOR PRODUCTIVITYSHOCKS
Standarddeviation autocorrelation Quarterly u
u
v
v/u
0.009 (0.001) 0.939 (0.018)
0.027 (0.004) 0.835 (0.045)
0.035 (0.005) 0.878 (0.035)
0.010 (0.001) 0.878 (0.035)
0.020 (0.003) 0.878 (0.035)
-0.927 (0.020) 1
-0.958 (0.012) 0.996 (0.001) 1
-0.958 (0.012) 0.996 (0.001) 1.000
-0.958 (0.012) 0.995 (0.001) 0.999
1
v matrix Correlation
v/u
(0.000) 1
f
p
f
-
-
-
-
p
(0.001) 0.999
(0.001) 1
in logs as deviations themodelwithstochastic All variablesarereported Notes:Resultsfromsimulating laborproductivity. standard errors-thestandard deviationacross10,000model froman HP trendwithsmoothing 105.Bootstrapped parameter in parentheses. The textprovidesdetailson thestochasticprocessforproductivity. simulations-arereported
withan initialunemployment rate is, starting and aggregatestateat time0, I use a pseudoto calculatethe arrandomnumbergenerator rivaltimeof thefirstPoissonshock.I compute theunemployment ratewhenthatshockarrives, stateusingthediscretea new generate aggregate stochasticprocess state-spacemean-reverting describedinAppendixC, andrepeat.Attheend I recordtheaggregate of each period(quarter), stateand theunemployment rate. I throwaway the first1,000 "quarters"of data.I thenuse themodelto generate212 data to quarterlydata from points,corresponding thelog ofthemodel1951to 2003,anddetrend HP an withtheusual data filter generated using smoothing parameter105. I repeatthis 10,000 times,givingme good estimatesof both the meanof themodel-generated dataandthestandarddeviationacrossmodel-generated observations. The latterprovides a sense of how preciselythemodelpredictsthevalue of a particularvariable.
is insignificant. It is althoughthe difference worthemphasizing thatthenegativecorrelation betweenunemployment and vacanciesis a reexersult,nota directtargetof thecalibration cise. The model also generatesthe correct forunemployment, autocorrelation the although In behaviorofvacanciesis somewhatofftarget. thedata,vacanciesareas persistent andvolatile as unemployment, whilein themodeltheautocorrelation of vacanciesis significantly lower thanthatof unemployment, whilethestandard deviationof vacanciesis threetimesas largeas thestandard deviationof unemployment fluctuationsaroundtrend.It is likelythatanything thatmakesvacanciesa statevariable,such as cost,or irreversplanninglags, an adjustment wouldincreasetheir ibilityin vacancycreation, and reducetheirvolatility, persistence bringing themodelmorein linewiththedataalongthese dimensions.ShigeruFujita (2003) developsa modelthatadds theserealisticfeatures. But thereal problemwiththemodellies in the volatilityof vacanciesand unemployment E. Results in thevolatility of thev-u or,moresuccinctly, ratio0 andthejob-finding ratefIn a reasonably Table 3 reportstheresultsfromsimulations calibratedmodel,the v-u ratiois less than10 of the model withlabor productivity shocks. percentas volatileas in U.S. data. This is exfromthedeterministic Along some dimensions,notably the coactlytheresultpredicted movement ofunemployment andvacancies,the comparative staticsin SectionII C. A 1-percent modelperforms well.The empirical increasein laborproductivity remarkably p fromitsaverage correlationbetween these two variables is valueof 1 raisesnetlaborproductivity p - z by -0.89, theBeveridgecurve.The modelactually about 1.66 percent.Using the deterministic of the -0.93, model,I arguedbeforethattheelasticity producesa stronger negativecorrelation,
40
THE AMERICANECONOMIC REVIEW
MARCH2005
TABLE 4---SEPARATION RATE SHOCKS
Standarddeviation autocorrelation Quarterly u
u
v
v/u
0.065 (0.007) 0.864 (0.026)
0.059 (0.006) 0.862 (0.026)
0.006 (0.001) 0.732 (0.048)
0.002 (0.000) 0.732 (0.048)
0.075 (0.007) 0.733 (0.048)
0.999 (0.000) 1
-0.906 (0.017) -0.887 (0.020) 1
-0.906 (0.017) -0.887 (0.020) 1.000
0.908 (0.017) 0.888 (0.021) -0.999
1
v Correlation matrix
v/u
--
f
-
-
-
s
-
-
-
s
f
(0.000) 1
-
(0.000)
-0.999
(0.000) 1
Notes:Resultsfromsimulating themodelwitha stochasticseparation rate.All variablesare reported in logs as deviations froman HP trendwithsmoothing standard errors-thestandard deviationacross10,000model parameter 105.Bootstrapped in parentheses. simulations-arereported The textprovidesdetailson thestochastic rate. processfortheseparation
v-uratiowithrespecttonetlaborproductivity is about1.03 withthischoiceof parameters, giving a totalelasticityof 0 withrespectto p of 1.66 X 1.03 = 1.71 percent.In approximately fact,the standarddeviationof log 0 around trendis 1.75 timesas large as the standard deviationof log p. Similarly,thejob-finding rateis 12 timesas volatilein thedataas in the model. Notonlyis therelittleamplification, butthere is also no propagation of thelaborproductivity shockin themodel.The contemporaneous correlationbetweenlaborproductivity, thev-urarateis 1.00. In thedata, tio,andthejob-finding the contemporaneous correlation betweenthe first twovariablesis 0.40 andthev-uratiolags laborproductivity by aboutone year.The embetweenlabor productivity pirical correlation and thejob-finding rateis similar. Table 4 reportsthe resultsfromthe model withshocksto theseparation rate.Theseintroduce an almostperfectly positivecorrelation betweenunemployment andvacancies,an event thathas essentiallyneverbeen observedin the UnitedStatesat businesscyclefrequencies (see Figure3). As a result,separationshocksproduce almostno variability in thev-uratioorthe with job findingrate.Again,thisis consistent the back-of-the-envelope calculations performedin SectionII C, whereI arguedthatthe elasticityof the v-u ratiowithrespectto the rateshouldbe approximately -0.10. separation Accordingto themodel,theratioof the stan-
darddeviationsis about0.08 and thetwo variables are strongly correlated. negatively One mightbe concernedthatthe disjoint and separation analysisof labor productivity shocksmasks some important interaction betweenthetwo impulses.Modelingan endogenousincreasein theseparation ratedue to low laborproductivity, as in Mortensen and Pissarides (1994), goes beyondthescope of thispaper. Instead,I introduceperfectly negatively correlatedlabor productivity and separation shocksintothebasic model.More precisely,I assume p = z + eY(p* - z) and s = e-"Ys*,
bothnonlinear functions ofthesamelatentvariabley. Theparameter > 0 permits a different os for and s. volatility p I startwiththeparameterization ofthemodel withonlylaborproductivity shocksand introduce volatilityin the separationrate. Table 5 showstheresultsfroma calibration withequal standard deviationsin thedeviationfromtrend of the separationrate and labor productivity (o-s= 1 - z). The behaviorofvacanciesin the modelis now farfromthedata,withan autoof 0.29 (comparedto 0.94 empiricorrelation and a correlation withunemployment of cally) -0.43 (-0.89). The difference betweenmodel anddatais highlysignificant botheconomically and statistically. Moreover,althoughcyclical in theseparation fluctuations rateboostthevolof atility unemployment considerably,they have a smalleffecton thecyclicalvolatility of thev-uratioandjob-finding rate,whichremain
TABLE 5-LABOR
Standarddeviation autocorrelation Quarterly u v matrix Correlation
41
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VOL. 95 NO. 1
PRODUCTIVITY AND SEPARATION RATE SHOCKS
s
u
v
v/u
0.031 (0.005) 0.933 (0.020)
0.011 (0.001) 0.291 (0.085)
0.037 (0.006) 0.878 (0.035)
0.014 (0.002) 0.878 (0.035)
0.020 (0.003) 0.878 (0.035)
0.020 (0.003) 0.878 (0.035)
-0.427 (0.068) 1
-0.964 (0.011) 0.650 (0.042) 1
-0.964 (0.011) 0.650 (0.042) 1.000
0.964 (0.011) -0.649 (0.042) - 1.000
-0.964 (0.011) 0.648 (0.042) 0.999
-1.000 (0.000) 1
0.999 (0.001) -0.999
1 -
v/u
f
(0.000)
f
-
s
-
-
-
-
p
-
-
-
-
1
(0.000)
-
p
(0.001)
(0.001) 1
laborproductivity and separations. All themodelwithstochasticbutperfectly Notes: Resultsfromsimulating correlated variablesare reportedin logs as deviationsfroman HP trendwith smoothingparameter105. Bootstrappedstandard in parentheses. The textprovidesdetailson the errors-thestandard deviationacross10,000modelsimulations-arereported stochastic process.
at around10 percentof theirempiricalvalues. in theseparation ratenatuSmallerfluctuations rallyhave a smallereffect,while realistically rateinducea in theseparation largefluctuations between correlation unemploystrongpositive mentand vacancies,even in the presenceof correlated shocks. productivity To summarize, thestochasticversionof the thatseparaPissarides(1985) model confirms tion shocksinduce a positivecorrelation beand vacancies. It also tween unemployment confirms shocks that,whilelaborproductivity are qualitatively consistentwitha downwardslopingBeveridgecurve,thesearchmodeldoes not substantially amplifythe extrinsicshocks and so labor productivity shocksinduceonly verysmallmovements alongthecurve.
as wouldbe thecase ifwageswere ingsolution, renegotiated followingeach aggregateshock. This stronger restriction pinsdownthewage as a functionof the aggregatestate,w,s,. This facilitates a moredetaileddiscussionof wages, which serves two purposes.First,modeling thatflexibility of thepresent wages illustrates is criticalformanyof value of wage payments theresultsemphasizedin thispaper.And second, it enables me to relatethis paper to a literature thatexamineswhether searchmodels can generaterigidwages. AppendixB proves thata continually renegotiated wage solves
(7)
+ + w,,,= (1 - P)z 3(p cO,,s).
This generalizesequation(1.20) in Pissarides environment. (2000) to a stochastic F. Wages shock Considerfirst theeffectof a separation on thewage.An increasein theseparation rates Untilthispoint,I have assumedthatthesur- inducesa slightdeclinein thev-uratio(see Table 4), whichin turn, byequation(7), reduceswages plus in new matchesis dividedaccordingto a Nash but solution have thedirecteffectof theshock generalized bargaining slightly. Although aboutthedivisionof sur- lowersfirms' made no assumption of theduration profits byshortening theresulting declinein wagespartially plus in old matches.Althoughthisis sufficient matches, fordetermining theresponseof unemployment offsets this,so thedropin thev-uratiois small. and vacanciesto exogenousshocks,it does not shock. A Second, considera productivity the of In in labor this increase down net p1-percent productivity pin timing wage payments. section,I introducean additionalassumption, z raises the v-u ratioby about 1 percent(see thatthe surplusin all matches,new or old, is Table 3). Equation (7) thenimplies thatthe alwaysdividedaccordingto theNash bargain- netwage w - z increasesby about 1 percent,
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TABLE 6-BARGAINING POWERSHOCKS
Standarddeviation autocorrelation Quarterly u
Correlation matrix
u
v
v/u
f
w
0.091 (0.018) 0.940 (0.023)
0.294 (0.086) 0.837 (0.046)
0.379 (0.099) 0.878 (0.036)
0.106 (0.028) 0.878 (0.036)
0.011 (0.015) 0.864 (0.047)
-0.915 (0.045) 1
-0.949 (0.032) 0.995 (0.001) 1
-0.949 (0.032) 0.995 (0.001) 1.000 (0.000) 1
0.818 (0.112) -0.827 (0.128) -0.838 (0.124) -0.838 (0.124)
1
v
-
v/u
-
-
f
-
-
-
-
-
-
w
-
1
Notes: Resultsfromsimulating themodelwithstochastic in logs as deviations bargaining power.All variablesarereported froman HP trendwithsmoothing standard deviationacross10,000model errors-thestandard parameter 105.Bootstrapped simulations-arereportedin parentheses. The textprovidesdetailson the stochasticprocessfortheworkers'bargaining power.
shockand soakingup mostof theproductivity givingfirmslittleincentiveto createnew vacancies. Hence thereis a modestincreasein vacanciesand decreasein unemployment in reshock. sponseto a largeproductivity To understand of wages fullytheimportance forthev-u ratio,it is usefulto considera version of the modelin whichlaborproductivity andtheseparation rateareconstant atp = 1 and = but workers' s 0.1, bargainingpower /3 An increasein 0 reduces changesstochastically. the profitfromcreatingvacancies,whichputs downwardpressureon thev-u ratio.It is difficultto knowexactlyhow muchvariability in 01 is reasonable,butone can ask how muchwage is requiredto generatetheobserved variability in thev-uratio.I assume 3 is a funcvolatility tion of the latentvariabley, 03 = (y + D is where the cumulative standard S'-l(a)), normaldistribution. If y wereconstantat zero, this implies 3 = a, but moregenerally0 is simplyboundedbetween0 and 1. I set the standarddeviationof y to o- = 0.099 and the mean reversionparameter to -y= 0.004. Alin thoughthisimpliesverymodestfluctuations deviationofdetrended wages-the standard log wages, computedas in equation(7), is just 0.01-the calibratedmodel generatesthe observedvolatilityin the v-u ratio,withpersistence similarto thatin the model withlabor shocks.Table 6 shows the comproductivity pleteresults.Sincebargaining poweris theonly
coundrivingforce,wagesarecounterfactually and (Abraham tercyclical 1995). Haltiwanger, it seems plausiblethata model Nevertheless, witha combination of wage and laborproductivityshockscould generatethe observedbehaviorof unemployment, vacancies,and real Of the unanswered course, wages. questionis whatexactlya wage shockis. If wages are bargainedin new matchesbut thennotcontinually thisanalysis renegotiated, is inapplicable. onecanprovethat Nevertheless, thefrequency of wage negotiation does notaffectthe expectedpresentvalue of wage paymentsin new matches,but only changesthe An increasein protimingof wage payments. or decreasein separationsraisesthe ductivity valueofwagepayments innewjobs and present has littleeffecton the v-u ratio.An therefore increasedworkers'bargaining powerin a new induces a largereducemployment relationship tionin vacanciesand in thev-u ratio. III. OptimalV-UFluctuations Anotherway to highlight theroleplayedby theNashbargaining is to examinea assumption centralized economyin whichit is possibleto issue entirely.17Considestepthewage-setting 17 A
numberof papersexaminea "competitive search economy,"in whichfirmscan committo wages before
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43
a special case of equation(6), withworkers' bargainingpower P equal to the elasticitya. This generalizesthe Hosios (1990) condition forefficiency of the decentralized equilibrium and to an economywithstochastic productivity rates.Sincethenumerical examplein separation SectionII E assumesa Cobb-Douglasmatching rW(p, s, u) = max(zu + p(l - u) - cuO allocation function witha = /, theequilibrium 0 describedin thatsectionsolvesthesocial planner's problem.Conversely,if thoseparameter + Wu(p,s, u)(s(l u) uf(O)) valuesdescribetheU.S. economy,theobserved is inconsistent without+ AEP,S(W(p', degreeofwagerigidity s,u))). s',u) W(p, maximization. put the linkbeWithothermatchingfunctions, Instantaneous outputis equal to z timesthe rateu plusp timestheemploy- tween the equilibriumwith wage bargaining unemployment mentrateminusc timesthenumberof vacan- and the solutionto the planner'sproblemis if unemployment and as the broken.At one extreme, cies v uO.The valuechangesgradually = vacancies are with rate substitutes, i.e., s(1 f(O) = perfect unemployment adjusts, tu v-uratio u) - uf(O),and suddenlywhen an aggregate a, + aO, thentheoutput-maximizing is infinite whenevera,(p - z) > c(r + s + shockchangesthestatefrom(p,s) to (p',s') at au) Withnearandis zeroiftheinequality is reversed. rateA. theoutput-maximizing v-u to verifythattheBellIt is straightforward perfect substitutability, to current On manvalueWis linearintheunemployment rate, ratiois verysensitive productivity. = if and vacancies and the v-u ratio satisthe other hand, unemployment W,(p,s,u) -c/f'(Op,s), areperfect fies f(O) = min(au,aO), the complements, ratio v-u rationeverstraysfromthe efficient the Withimperfect complements, impact r+ s + A a,/av. f( o,s) of productivity shockson thev-u ratiois muf, ( f' fledbutnoteliminated. , Opsf' Os) (Op,s) O findThe economicsbehindthesetheoretical p +-z Is ingsis simple.An increasein laborproductivity relativeto thevalue of non-market and c activity ,s,,) f'p~(6, the cost of advertisinga vacancy induces a unemswitchawayfromtheexpensiveactivity, and towardtherelatively acdefinestheoptimal indepen- ployment, Thisimplicitly cheap vacancies.The magnitude of theswitch rate. Op,s, dentof theunemployment tivity, With a Cobb-Douglas matchingfunction dependson how substitutable unemployment If and vacanciesare in thematchingfunction. m(u,v)= -I, thisreducesto uAuvI is substitution they are strongcomplements, + p+s + nearly impossibleand the v-u ratio barely r+s+A + substituchanges.If theyare strongsubstitutes, q(O,s) 'ps tionis nearlycostless,andthev-uratiois highly procyclical. In thedecentralized economy,theextentof = - ) c + AEs andvacansubstitution between q(Op,,S,) unemployment funccies is governednotonlybythematching tion but also by the bargainingsolution,as shownby the comparativestaticsexercisesin hiringworkersand can increasetheirhiringrateby promising higherwages (Peters, 1991; Montgomery,1991; SectionII C. The Nash bargaining solutionefMoen, 1997; Shimer,1996; Burdettet al., 2001). It is by to a moderatedegreeof fectively corresponds maxsearchequilibrium nowwell-known thata competitive the Cobb-Douglas case. If substitutability, imizes output,essentiallyby creatinga marketforjob were more of search discussion wages rigid,an increasein producThis output-maximizing applications. also pertainsto thesemodels. behaviortherefore tivitywouldinducemorevacancycreationand
socialplannerwhochooses sidera hypothetical v-uratioin ordertomaximize a state-contingent the presentdiscountedvalue of outputnet of vacancycreationcosts.The planner'sproblem as is represented recursively
44
THE AMERICANECONOMIC REVIEW
less unemployment, analogousto a centralized environment witha highelasticityof substitufunction. tionin thematching The substitutability ofunemployment andvacancies is an empiricalissue. Blanchardand Diamond(1989) use nonlinearleastsquaresto estimatea ConstantElasticityof Substitution (CES) matchingfunctionon U.S. data. Their ofsubstitution is pointestimatefortheelasticity thanthe 0.74, i.e., slightlyless substitutable theycannotreject Cobb-Douglascase, although of 1. As footnote theCobb-Douglaselasticity 8 describes,mydatasuggestan elasticity slightly in excess of 1, althoughmy pointestimateis in theobservedmovements imprecise.Whether and vacanciesareoptimalwhen unemployment thelens of thetextbooksearch viewedthrough andmatching remainsan open model,therefore, question. IV. RelatedLiterature There is a large literaturethat explores withthe whether thesearchmodelis consistent Somepapers cyclicalbehavioroflabormarkets. of the model forthe look at the implications behaviorof variousstocksand flows,including andvacancyrates,butdo not theunemployment oftheexogenous examinetheimplicit magnitude impulses.Othersassumethatbusinesscyclesare in the separation rates. drivenby fluctuations orderive Thesepaperseither imposeexogenously v-u constant withinthemodela counterfactually ratio0. A third groupofpapershastriedbutfailed ofthev-uratiowith toreconcile theprocyclicality extrinsic shocksof a plausiblemagnitude. Papers by Abraham and Lawrence Katz (1986), Blanchardand Diamond (1989), and Cole and Rogerson(1999) fitintothefirstcategory,matchingthe behaviorof labormarket themagnitude stocksand flowsby sidestepping of impulses.For example,Abrahamand Katz Bever(1986) arguethatthedownward-sloping withmodelsin which idgecurveis inconsistent in the is drivenby fluctuations unemployment David Lilien's (1982) separationrate,notably sectoralshiftsmodel.Thatleads themto advocate an alternativein which unemployment are drivenby aggregatedisturfluctuations shocks. Unfortubances, e.g., productivity nately,theyfail to examinethe magnitudeof shocks needed to deliverthe observedshifts alongtheBeveridgecurve.Blanchardand Dia-
MARCH2005
mond(1989) also focuson thenegativecorrelationbetweenunemployment and vacancies, buttheydo notmodelthe supplyof jobs and hence do not explain why thereare so few vacanciesduringrecessions.Instead,theyassumethetotalstockofjobs followsan exogenous stochasticprocess.This paperpushesthe cyclicalityof the v-u ratioto the frontof the picture.Likewise,Cole and Rogerson(1999) and Pissarides(1994) arguethattheMortensen model can matcha varietyof businesscycle facts,buttheydo so in a reducedformmodel in thejob-finding thattreats fluctuations rate,and henceimplicitly in thev-uratio,as exogenous. The secondgroupof papers,includingwork by MichaelPries(2004), GaryRameyand Joel Watson(1997),WouterDen Haan et al. (2000), and Joao Gomes et al. (2001), assumes that are largely due to employmentfluctuations in the separationrate,minimiztime-variation ing theroleplayedby theobservedcyclicality of thev-u ratio.These paperstypically deliver from a search consistent model, rigidwages withthefindings in SectionII E. Buildingon theideasinHall (1995),Pries(2004) showsthat a briefadverseshock thatdestroyssome old can generatea long employment relationships of as the transition period highunemployment, displacedworkersmove througha numberof short-term jobs beforeeventuallyfindingtheir way back intolong-term relationships. During this transition process,the v-u ratioremains constant,since aggregateeconomicconditions have returnedto normal. Equivalently,the Beveconomymovesalongan upward-sloping the transition curve eridge during period,in contradictionto the evidence. Ramey and Watson(1997) arguethattwo-sided asymmetric information generatesrigidwages in a search model.But in theirmodel,shocksto thesepain rationratearetheonlysourceof fluctuations Thejob-finding ratef(O) is exunemployment. ogenousand constant,whichis equivalentto to unassumingthatvacanciesareproportional Thisis probablyan important employment. part oftheexplanation forwhytheirmodelproduces rigidwages. Den Haan et al. (2000) showthat rateamplifyproin theseparation fluctuations shocksin a modelsimilarto theone ductivity examinedhere;however,theydo not discuss thecyclicalbehaviorofthev-uratio.Similarly, Gomes et al. (2001) sidestepthe v-u issue by rate lookingat a modelin whichthejob-finding
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45
is exogenousand constant,i.e., vacanciesare Again, this proportionalto unemployment. in rigid theirmodel. helpskeepwagesrelatively Mortensenand Pissarides(1994) have probIn ablythebestknownpaperin thisliterature. theirthree-state "illustrativesimulation,"the enormous authorsintroduce, without comment, or leisureshocksintotheirmodel. productivity minusthevalue of Averagelaborproductivity leisurep - z is approximately threetimesas in in the state as the bad state.'8This high good in confirms that to paper response such large shocks,thev-uratioshouldalso be aboutthree timesas largein the good stateas in thebad state,but arguesthatthereis no evidencefor these large shocks in the data. Even if one of theimpliedimpulses, acceptsthemagnitude Mortensenand Pissarides(1994) still deliver of -0.26 betweenunemployonlya correlation mentand vacancies,farlowerthantheempirical valueof -0.88. Thisis probablybecauseof thetensionbetweenproductivity shocks,which Bevputtheeconomyon a downward-sloping inthe eridgecurve,andendogenousmovements rate,whichhave theoppositeeffect. separation Monika Merz (1995) and David Andolfatto (1996) bothputthestandardsearchmodelinto a realbusinesscycleframework withintertemof leisure,capitalaccumulaporal substitution tion,and otherextensions.Neitherpaper can matchthenegativecorrelation betweenunemand and both vacancies, papersgenployment eraterealwagesthataretooflexiblein response to productivity shocks.Thus thesepapersenin thispaper, counterthe problemI highlight do not this shortcomalthoughthey emphasize ing of the searchmodel.Finally,Hall (2003), buildingon an earlierversionof this paper, discussessomeof thesameissues.Hall (2005) proposesone possiblesolution:real wages are determinedby a social norm that does not changeoverthebusinesscycle.
cannotgeneratesubstantial by Nash bargaining Bevermovements a along downward-sloping idge curvein responseto shocksof a plausible shockresults magnitude.A labor productivity in higherwages,withlittleeffecton primarily thev-u ratio.A separationshockgeneratesan increasein bothunemployment and vacancies. It is important to stressthatthisis notan attack on the searchapproachto labor markets,but rathera critiqueof the commonly-used Nash for determination. bargaining assumption wage An alternative mechanism wage determination thatgeneratesmorerigidwages in new jobs, measuredin presentvalue terms,will amplify the effectof productivity shockson the v-u to reconcile the evidenceandtheratio,helping movements in workers' ory. Countercyclical powerprovideone suchmechanism, bargaining at leastin a reduced-form sense. If the matchingfunctionis Cobb-Douglas, the observedbehaviorof the v-u ratiois not sociallyoptimalforplausibleparameterizations ofthemodel,butthisconclusioncouldbe overturnedif theelasticityof substitution between and vacanciesin the matching unemployment functionis sufficiently large. Estimatesof a are imprecise,so it is CES matchingfunction unclearwhether observedwagesare"toorigid." One way to generatemorerigidwages in a theoretical modelis to introduce considerations affect the worker turnover rate. wherebywages For example,in the Burdettand Mortensen search,firmshave (1998) modelof on-the-job an incentiveto offerhigh wages in orderto attractworkersaway fromcompetitors and to reduceemployees'quitrate.The distribution of affectsan individualfirm'swage productivity offerandvacancycreationdecisionsin complex thesimplelinkbetweenaverage ways,breaking laborproductivity and thev-u ratioin thePissarides(1985) model.In particular, a shiftin the thatleaves averageladistribution productivity bor productivity unchangedmay appreciably V. Conclusion affectaveragewagesandhencetheequilibrium v-u ratio. I have arguedin thispaperthata searchand Anotherpossibilityis to drop some of the modelin whichwagesaredetermined informationalassumptionsin the standard matching searchmodel.19Suppose thatwhen a worker 18This calculationwould be easy in the absence of o-wereequal to zero. i.e., if theirparameter heterogeneity, Thenp - z would take on threepossiblevalues: 0.022, inp - z between 0.075,and0.128,fora six-folddifference thehighand low states.
19 Rameyand Watson(1997) develop a searchmodel withtwo-sidedasymmetric information. Because theyassume workers'job findingrateis exogenousand acyclic,
THE AMERICANECONOMIC REVIEW
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MARCH2005
she findsa job and firmmeet, they draw an idiosyncratic leisurez plus the probability level from some times the resulting capitalgain E - U productivity match-specific f(Op,s) shocktimes ofan aggregate distribution F. Workersand firmsknowabout plustheprobability a simthe that variables, (9) Equation expresses including unemploycapital gain. aggregate whoreceives mentrateand the distribution F, but onlythe ilaridea foran employedworker, level.Bara wage payment butloses herjob at rates. firmknowstherealizedproductivity wp,, as follows: with probability Equation(10) providesan analogousrecursive gainingproceeds forthevalue of a filledjob. Note 3 E (0,1), a workermakesa take-it-or-leave-itformulation whena filledjob thefirmmakesa thata firmis leftwithnothing wage demand,and otherwise ends. take-it-or-leave-it offer.ObviouslythefirmexSumequations(9) and (10) andthensubtract relatractsall therentsfromtheemployment + if it an offer. But the - Up,,: when makes equation(8), defining tionship Vp,'s Jp,s Ep, workermakestheoffer, shefacesa uninformed - Up,,) a higherwage and tradeoff betweendemanding (11) rVp,,= p - z - f(Op,s)(Ep,s so thewage herriskofunemployment, reducing - sVp,, + X(EEV,,, - V,,s). F. dependson thehazardrateofthedistribution This againbreaksthelinkbetweenaveragelav-u ratio. In addition,the Nash bargainingsolutionimbor productivity and theequilibrium pliesthatthewage is set so as to maximizethe Exploringwhethereitherof thesemodels,or - Up,)p 1-, whichgives fluctua- Nash product somerelatedmodel,deliversubstantial (Ep,s in ratio in to tions the v-u response plausible research. impulsesremainsa topicforfuture Jp,s Ep,s Up, V (12) = p, FOR A: DERIVATION OF THEEQUATION APPENDIX
forE - U in equation(11) yields Substituting equation(4). If I allow wages to dependin an arbitrary For notationalsimplicity alone, assumethe on thehistory of thematch,thiswould on the manner aggregate wage paymentdependsonly theBellmanvaluesE andJ;however,the affect state,wp,s,not on the historyof the match.I the history-dependence, returnto thisissue at the end of thissection. wage, and therefore and to be the state- would drop out when summingthe Bellman Define Up,s, Ep,,, Jp,, contingentpresentvalue of an unemployed equationsforE andJ.In otherwords,thematch worker,employedworker,and filledjob, re- surplusV is unaffected by the frequencyof by: wage renegotiation. Theyare linkedrecursively spectively. SURPLUS (4)
(8)
= z + f(O,,,)(E,,,- U,,,) rUp,,
+Aps Up,-Up) (9)
- s(Ep,s- Up,s) rEp,s= Wp,s +
(10)
- Ep,s) A(EE-p,sEps,
= rJp,s p wp,s sJP,S +
-
OF THEWAGE B: DERIVATION APPENDIX EQUATION
Assumethatwages are continually renegotiated,so thewage dependsonlyon thecurrent and fuaggregatestate(p,s). Eliminatecurrent turevaluesofJ fromequation(10) usingequation(12): - (r + s + A)(1 w,s = p
A(Ep,sJp',s' Jps).
Equation(8) statesthatthe flowvalue of an unemployedworkeris equal to her value of theirresultsare not directlyapplicableto this analysis, mayproveuseful. althoughtheirmethodology
3)Vp,s
+ AEp,s(1 P)Vp,,s,.
eliminate current andfuture valuesof Similarly, V using(5): Wp,s=
p
(r + s + A)c q(Op,s)
c Ep,s
p,s
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SHIMER: UNEMPLOYMENTAND VACANCIES
Finally,replacethelast two termsusingequation(6) to getequation(7).
Var[y(t+ h) - y(t)ly(t)] = [E[(y(t+h)-y(t))2ly(t)]
APPENDIXC: THE STOCHASTICPROCESS
The textdescribesa continuousstatespace to thediscretestatespace model approximation used in boththetheoryand simulations. HereI describethe discretestate space model and show that it asymptotesto an OrnsteinUhlenbeckprocess. Considera randomvariabley thatis hitwith shocksaccordingto a Poissonprocesswitharrival rate A. The initialvalue of y lies on a discretegrid, y E Y =- -nA, -(n-
1)A...,
0,..., (n - 1)A,nA}
47
- (E[y(t+ h)- y(t)ly(t)])2. The firsttermevaluatesto hAA2over a sufficientlyshorttimeintervalh, sinceit is equal to A2 if a shock,positiveor negative,arrivesand zero otherwise.The second termis (hyy(t))2, and so is negligibleovera shorttimeinterval h. Thus
+ h) - y(t)ly(t)] = hAA2= ho. Var[y(t we can represent thestoPuttingthistogether, chasticprocessfory as
whereA > 0 is thestepsize and2n + 1 3 is - hits, thenumberof gridpoints.Whena shock thenew value y' eithermovesup or downby one gridpoint:
dy = - yydt + adx
wherefort > 0, theexpectedvalueofx(t)given x(O) is x(0) and the conditionalvarianceis t. This is similarto a Brownianmotion,except thattheinnovations inx arenotGaussian,since y withprobability2 to lie on a discretegrid. y is constrained Now supposeone changesthethreeparame1+Yny) ters of the stochasticprocess,the step size, arrivalrateofshocks,andnumber ofsteps,from Note thatalthoughthe step size is constant, (A,A,n)to (A\V/,A/e,n/e)foranye > 0. It is theprobability thaty' = y + A is smallerwhen easy to verifythatthisdoes notchangeeither is the autocorrelation y larger,fallingfrom1 aty = -nA to zeroat parametery = /n or the - = = nA. instantaneous variance y XA. But as It is trivialto confirm thaty' E Y,so thestate e -- 0, thedistribution oftheinnovation process definey x convergesto a normalby the CentralLimit spaceis discrete.To proceedfurther, A/nandar \/A. Foranyfixedy(t),I examine Theorem. Equivalently,y converges to an thebehaviorofy(t+ h) overan arbitrarily short Omstein-Uhlenbeck process.20This observatimeperiodh. For sufficiently shorth,theprob- tionis also usefulforcomputation. It is possible abilitythattwo Poissonshocksarriveis negli- to finda solutionon a coarsegridand thento substangible, and so y(t + h) is equal to y(t) with refinethegridbydecreasinge without probability1 - hA,has increasedby A with tiallychangingtheresults. and has deprobabilityhA(1 - y(t)/nA)/2, creasedby A withprobability hA (1 + y(t)/ shows nA)/2.Addingthistogether
(yh) Py=1
?2nA)
E[y(t+ h) - y(t)ly(t)]=
n
hA y(t)= -hyy(t).
varianceofy(t+ h) - y(t) Next,theconditional can be decomposedinto
20 Notably,forlargen it is extraordinarily unlikelythat the statevariablereachesits limitingvalues of +nh. The unconditional distribution of the statevariableis approximatel normalwith mean zero and standarddeviation valuesofthestatevariables ofV2y = A-n/2. The limiting therefore lie n standard deviationsabove andbelowthe mean.If n = 1000, as is thecase in the simulations, one shouldexpectto observesuchvaluesapproximately oncein 10436periods.
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