The Cyclical Behavior of Equilibrium Unemployment...

The Cyclical Behavior of EquilibriumUnemployment and Vacancies

By ROBERT SHIMER

This paper argues that the textbook search and matching model cannot generate theobserved business-cycle-frequency fluctuations in unemployment and job vacanciesin response to shocks of a plausible magnitude In the United States the standarddeviation of the vacancy-unemployment ratio is almost 20 times as large as thestandard deviation of average labor productivity while the search model predictsthat the two variables should have nearly the same volatility A shock that changesaverage labor productivity primarily alters the present value of wages generatingonly a small movement along a downward-sloping Beveridge curve (unemployment-vacancy locus) A shock to the separation rate generates a counterfactually positivecorrelation between unemployment and vacancies In both cases the model exhibitsvirtually no propagation (JEL E24 E32 J41 J63 J64)

In recent years the Mortensen-Pissaridessearch and matching model has become thestandard theory of equilibrium unemployment(Dale Mortensen and Chris Pissarides 1994Pissarides 2000) The model is attractive for anumber of reasons it offers an appealing de-scription of how the labor market functions it isanalytically tractable it has rich and generallyintuitive comparative statics and it can easilybe adapted to study a number of labor marketpolicy issues such as unemployment insurancefiring restrictions and mandatory advanced no-tification of layoffs Given these successes onemight expect that there would be strong evi-dence that the model is consistent with keybusiness cycle facts On the contrary I argue inthis paper that the model cannot explain the

cyclical behavior of two of its central elementsunemployment and vacancies which are bothhighly variable and strongly negatively corre-lated in US data Equivalently the model can-not explain the strong procyclicality of the rateat which an unemployed worker finds a job

I focus on two sources of shocks changes inlabor productivity and changes in the separationrate of employed workers from their job In aone-sector model a change in labor productiv-ity is most easily interpreted as a technology orsupply shock But in a multi-sector model apreference or demand shock changes the rela-tive price of goods which induces a change inreal labor productivity as well Thus theseshocks represent a broad set of possibleimpulses

An increase in labor productivity relative tothe value of nonmarket activity and to the costof advertising a job vacancy makes unemploy-ment relatively expensive and vacancies rela-tively cheap1 The market substitutes towardvacancies and the increased job-finding ratepulls down the unemployment rate moving theeconomy along a downward sloping Beveridgecurve (vacancy-unemployment or v-u locus)But the increase in hiring also shortens unem-ployment duration raising workersrsquo threat pointin wage bargaining and therefore raising the

Department of Economics University of Chicago1126 East 59th Street Chicago IL 60637 (e-mailshimeruchicagoedu) A previous version of this paperwas entitled ldquoEquilibrium Unemployment Fluctuationsrdquo Ithank Daron Acemoglu Robert Barro Olivier BlanchardV V Chari Joao Gomes Robert Hall Dale MortensenChristopher Pissarides two anonymous referees the editorRichard Rogerson and numerous seminar participants forcomments that are incorporated throughout the paper Thismaterial is based upon work supported by the NationalScience Foundation under grants SES-0079345 and SES-0351352 I am grateful to the Alfred P Sloan Foundationfor financial support to the Federal Reserve Bank of Min-neapolis for its hospitality while I worked on an earlyversion of this paper and to Mihai Manea and especiallySebastian Ludmer for excellent research assistance

1 The interpretation in this paragraph and its sequelbuilds on discussions with Robert Hall

25

expected present value of wages in new jobsHigher wages absorb most of the productivityincrease eliminating the incentive for vacancycreation As a result fluctuations in labor pro-ductivity have little impact on the unemploy-ment vacancy and job-finding rates

An increase in the separation rate does notaffect the relative value of unemployment andvacancies and so leaves the v-u ratio essentiallyunchanged Since the increase in separationsreduces employment duration the unemploy-ment rate increases and so therefore must va-cancies As a result fluctuations in theseparation rate induce a counterfactually posi-tive correlation between unemployment andvacancies

Section I presents the relevant US businesscycle facts unemployment u is strongly coun-tercyclical vacancies v are equally strongly pro-cyclical and the correlation between the twovariables is 089 at business cycle frequen-cies As a result the v-u ratio is procyclical andvolatile with a standard deviation around itstrend equal to 038 log points To provide fur-ther evidence in support of this finding I exam-ine the rate at which unemployed workers findjobs If the process of pairing workers with jobsis well-described by an increasing constantreturns-to-scale matching function m(uv) as inPissarides (1985) the job finding rate is f m(uv)u an increasing function of the v-u ratioI use unemployment-duration data to measurethe job-finding rate directly The standard devi-ation of fluctuations in the job-finding ratearound trend is 012 log points and the correla-tion with the v-u ratio is 095 Finally I look atthe two proposed impulses The separation rateis less correlated with the cycle and moderatelyvolatile with a standard deviation about trendequal to 008 log points Average labor produc-tivity is weakly procyclical and even more sta-ble with a standard deviation about trend of002 log points

In Section II I extend the Pissaridesrsquos (1985)search and matching model to allow for aggre-gate fluctuations I introduce two types ofshocks labor productivity shocks raise outputin all matches but do not affect the rate at whichemployed workers lose their job and separationshocks raise the rate at which employed workersbecome unemployed but do not affect the pro-ductivity in surviving matches In equilibriumthere is only one real economic decision firmsrsquo

choice of whether to open a new vacancy Theequilibrium vacancy rate depends on the unem-ployment rate on labor market tightness and onthe expected present value of wages in newemployment relationships Wages in turn aredetermined by Nash bargaining at least in newmatches In principle the wage in old matchesmay be rebargained in the face of aggregateshocks or may be fixed by a long-term employ-ment contract Section II A describes the basicmodel while Section II B derives a forward-looking equation for the v-u ratio in terms ofmodel parameters

Section II C performs simple analytical com-parative statics in some special cases For ex-ample I show that the elasticity of the v-u ratiowith respect to the difference between laborproductivity and the value of nonmarket activityor ldquoleisurerdquo is barely in excess of 1 for reason-able parameter values To reconcile this withthe data one must assume that the value ofleisure is nearly equal to labor productivity somarket work provides little incremental utilityThe separation rate has an even smaller impacton the v-u ratio with an elasticity of 01according to the comparative statics Moreoverwhile shocks to labor productivity at least in-duce a negative correlation between unemploy-ment and vacancies separation shocks causeboth variables to increase which tends to gen-erate a positive correlation between the twovariables Similar results obtain in some otherspecial cases

Section II D calibrates the stochastic model tomatch US data along as many dimensions aspossible and Section II E presents the resultsThe exercise confirms the quantitative predic-tions of the comparative statics If the economyis hit only by productivity shocks it movesalong a downward-sloping Beveridge curve butempirically plausible movements in labor pro-ductivity result in tiny fluctuations in the v-uratio Moreover labor productivity is perfectlycorrelated with the v-u ratio indicating that themodel has almost no internal propagation mech-anism If the economy is hit only by separationshocks the v-u ratio is stable in the face oflarge unemployment fluctuations so vacanciesare countercyclical Equivalently the model-generated Beveridge curve is upward-sloping

Section II F explores the extent to which theNash bargaining solution is responsible forthese results First I examine the behavior of

26 THE AMERICAN ECONOMIC REVIEW MARCH 2005

wages in the face of labor productivity andseparation shocks An increase in labor produc-tivity encourages firms to create vacancies Theresulting increase in the job-finding rate putsupward pressure on wages soaking up virtuallyall of the shock A decrease in the separationrate also induces firms to create more vacanciesagain putting upward pressure on wages andminimizing the impact on the v-u ratio andjob-finding rate On the other hand I examine aversion of the model in which only workersrsquobargaining power is stochastic Small fluctua-tions in bargaining power generate realisticmovements in the v-u ratio while inducing onlya moderately countercyclical real wage with astandard deviation of 001 log points aroundtrend

Section III provides another angle fromwhich to view the modelrsquos basic shortcoming Iconsider a centralized economy in which a so-cial planner decides how many vacancies tocreate in order to maximize the present value ofmarket and nonmarket income net of vacancycreation costs The decentralized and central-ized economies behave identically if the match-ing function is Cobb-Douglas in unemploymentand vacancies and workersrsquo bargaining power isequal to the elasticity of the matching functionwith respect to the unemployment rate a gen-eralization of Arthur Hosios (1990) But ifunemployment and vacancies are more substi-tutable fluctuations are amplified in the central-ized economy essentially because the shadowwage is less procyclical Empirically it isdifficult to measure the substitutability of un-employment and vacancies in the matchingfunction and therefore difficult to tell whetherobserved fluctuations are optimal

Section IV reconciles this paper with a num-ber of existing studies that claim standardsearch and matching models are consistent withthe business cycle behavior of labor marketsFinally the paper concludes in Section V bysuggesting some modifications to the model thatmight deliver rigid wages and thereby do abetter job of matching the empirical evidence onvacancies and unemployment

It is worth emphasizing one importantmdashbutstandardmdashfeature of the search and matchingframework that I exploit throughout this paperworkers are risk-neutral and supply labor inelas-tically In the absence of search frictions em-ployment would be constant even in the face of

productivity shocks This distinguishes thepresent model from those based upon intertem-poral labor supply decisions (Robert E LucasJr and Leonard Rapping 1969) Thus this pa-per explores the extent to which a combinationof search frictions and aggregate shocks cangenerate plausible fluctuations in unemploy-ment and vacancies if labor supply is inelasticIt suggests that search frictions per se scarcelyamplify shocks The paper does not examinewhether a search model with an elastic laborsupply can provide a satisfactory explanationfor the observed fluctuations in these twovariables

I US Labor Market Facts

This section discusses the time series behav-ior of unemployment u vacancies v the jobfinding rate f the separation rate s and laborproductivity p in the United States Table 1summarizes the detrended data

A Unemployment

The unemployment rate is the most com-monly used cyclical indicator of job-search ac-tivity In an average month from 1951 to 2003567 percent of the US labor force was out ofwork available for work and actively seekingwork This time series exhibits considerabletemporal variation falling to as low as 26 per-cent in 1953 and 34 percent in 1968 and 1969but reaching 108 percent in 1982 and 1983(Figure 1) Some of these fluctuations are al-most certainly due to demographic and otherfactors unrelated to business cycles To high-light business-cycle-frequency fluctuations Itake the difference between the log of the un-employment level and an extremely low fre-quency trend a Hodrick-Prescott (HP) filterwith smoothing parameter 105 using quarterlydata2 The difference between log unemploy-ment and its trend has a standard deviation of019 so unemployment is often as much as 38percent above or below trend Detrended unem-ployment also exhibits considerable persistencewith quarterly autocorrelation 094

2 I use the level of unemployment rather than the rate tokeep the units comparable to those of vacancies A previousversion of this paper used the unemployment rate with noeffect on the conclusions

27VOL 95 NO 1 SHIMER UNEMPLOYMENT AND VACANCIES

There is some question as to whether unem-ployment or the employment-population ratio isa better indicator of job-search activity Advo-cates of the latter view for example HaroldCole and Richard Rogerson (1999) argue thatthe number of workers moving directly intoemployment from out-of-the-labor force is aslarge as the number who move from unemploy-ment to employment (Olivier Blanchard andPeter Diamond 1990) On the other hand thereis ample evidence that unemployment and non-

participation are distinct economic conditionsChinhui Juhn et al (1991) show that almost allof the cyclical volatility in prime-aged malenonemployment is accounted for by unemploy-ment Christopher Flinn and James Heckman(1983) show that unemployed workers are sig-nificantly more likely to find a job than nonpar-ticipants although Stephen Jones and CraigRiddell (1999) argue that other variables alsohelp to predict the likelihood of finding a job Inany case since labor force participation is pro-cyclical the employment-population ratio is amore cyclical measure of job-search activityworsening the problems highlighted in thispaper

It is also conceivable that when unemploy-ment rises the amount of job-search activity perunemployed worker declines so much that ag-gregate search activity actually falls There isboth direct and indirect evidence against thishypothesis As direct evidence one would ex-pect that a reduction in search intensity could beobserved as a decline in the number of job-search methods used or a switch toward lesstime-intensive methods An examination ofCurrent Population Survey (CPS) data indicatesno cyclical variation in the number or type ofjob-search methods utilized3 Indirect evidencecomes from estimates of matching functionswhich universally find that an increase in un-employment is associated with an increase in

3 Shimer (2004b) discusses this evidence in detail

TABLE 1mdashSUMMARY STATISTICS QUARTERLY US DATA 1951ndash2003

u v vu f s p

Standard deviation 0190 0202 0382 0118 0075 0020Quarterly autocorrelation 0936 0940 0941 0908 0733 0878

u 1 0894 0971 0949 0709 0408v mdash 1 0975 0897 0684 0364

Correlation matrix vu mdash mdash 1 0948 0715 0396f mdash mdash mdash 1 0574 0396s mdash mdash mdash mdash 1 0524p mdash mdash mdash mdash mdash 1

Notes Seasonally adjusted unemployment u is constructed by the BLS from the Current Population Survey (CPS) Theseasonally adjusted help-wanted advertising index v is constructed by the Conference Board The job-finding rate f andseparation rate s are constructed from seasonally adjusted employment unemployment and mean unemployment durationall computed by the BLS from the CPS as explained in equations (1) and (2) u v f and s are quarterly averages of monthlyseries Average labor productivity p is seasonally adjusted real average output per person in the non-farm business sectorconstructed by the Bureau of Labor Statistics (BLS) from the National Income and Product Accounts and the CurrentEmployment Statistics All variables are reported in logs as deviations from an HP trend with smoothing parameter 105

FIGURE 1 QUARTERLY US UNEMPLOYMENT (IN MILLIONS)AND TREND 1951ndash2003

Notes Unemployment is a quarterly average of the season-ally adjusted monthly series constructed by the BLS fromthe CPS survey home page httpwwwblsgovcps Thetrend is an HP filter of the quarterly data with smoothingparameter 105


the number of matches (Barbara Petrongolo andPissarides 2001) If job-search activity declinedsharply when unemployment increased thematching function would be measured as de-creasing in unemployment I conclude that ag-gregate job search activity is positivelycorrelated with unemployment

B Vacancies

The flip side of unemployment is job vacan-cies The Job Openings and Labor TurnoverSurvey (JOLTS) provides an ideal empiricaldefinition ldquoA job opening requires that 1) aspecific position exists 2) work could startwithin 30 days and 3) the employer is activelyrecruiting from outside of the establishment tofill the position Included are full-time part-time permanent temporary and short-termopenings Active recruiting means that the es-tablishment is engaged in current efforts to fillthe opening such as advertising in newspapersor on the Internet posting help-wanted signsaccepting applications or using similar meth-odsrdquo4 Unfortunately JOLTS began only in De-cember 2000 and comparable data had neverpreviously been collected in the United StatesAlthough there are too few observations to looksystematically at this time series its behaviorhas been instructive In the first month of thesurvey the non-farm sector maintained a sea-sonally adjusted 46 million job openings Thisnumber fell rapidly during 2001 and averagedjust 29 million in 2002 and 2003 This declinein job openings depicted by the solid line inFigure 2 coincided with a period of rising un-employment suggesting that job vacancies areprocyclical

To obtain a longer time series I use a stan-dard proxy for vacancies the Conference Boardhelp-wanted advertising index measured as thenumber of help-wanted advertisements in 51major newspapers5 A potential shortcoming is

that help-wanted advertising is subject to low-frequency fluctuations that are related only tan-gentially to the labor market In recent years theInternet may have reduced firmsrsquo reliance onnewspapers as a source of job advertising whilein the 1960s newspaper consolidation mayhave increased advertising in surviving newspa-pers and Equal Employment Opportunity lawsmay have encouraged firms to advertise jobopenings more extensively Fortunately a low-frequency trend should remove the effect ofthese and other secular shifts Figure 3 showsthe help-wanted advertising index and its trendNotably the decline in the detrended help-wanted index closely tracks the decline in jobopenings measured directly from JOLTS duringthe period when the latter time series is avail-able (Figure 2)

Figure 4 shows a scatter plot of the relation-ship between the cyclical component of unem-ployment and vacancies the Beveridge curveThe correlation of the percentage deviation ofunemployment and vacancies from trend is089 between 1951 and 20036 Moreover the

4 This definition comes from the Bureau of Labor Sta-tistics news release July 30 2002 available at httpwwwblsgovjltjlt_nr1pdf

5 Abraham (1987) discusses this measure in detail From1972 to 1981 Minnesota collected state-wide job vacancydata Abraham compares this with Minnesotarsquos help-wantedadvertising index and shows that the two series track eachother very closely through two business cycles and tenseasonal cycles

6 Abraham and Katz (1986) and Blanchard and Diamond(1989) discuss the US Beveridge curve Abraham and Katz(1986) argue that the negative correlation between unem-ployment and vacancies is inconsistent with Lilienrsquos (1982)sectoral shifts hypothesis and instead indicates that busi-ness cycles are driven by aggregate fluctuations Blanchardand Diamond (1989) conclude that at business cycle

FIGURE 2 TWO MEASURES OF US JOB VACANCIES2000Q4ndash2003Q4

Notes The solid line shows the logarithm of the number ofjob openings in millions measured by the BLS from theJOLTS survey homepage httpwwwblsgovjlt quarterlyaveraged and seasonally adjusted The dashed line showsthe deviation from trend of the quarterly averaged season-ally adjusted Conference Board help-wanted advertisingindex


standard deviation of the cyclical variation inunemployment and vacancies is almost identi-cal between 019 and 020 so the product ofunemployment and vacancies is nearly acyclicThe v-u ratio is therefore extremely procyclicalwith a standard deviation of 038 around itstrend

C Job-Finding Rate

An implication of the procyclicality of thev-u ratio is that the hazard rate for an unem-ployed worker of finding a job his job-findingrate should be lower during a recession As-sume that the number of newly hired workers isgiven by an increasing and constant returns-to-scale matching function m(uv) depending onthe number of unemployed workers u and thenumber of vacancies v Then the probability thatany individual unemployed worker finds a jobthe average transition rate from unemploymentto employment is f m(uv)u m(1) where vu is the v-u ratio The job-finding rate fshould therefore move together with the v-uratio

Gross worker flow data can be used to mea-sure the job-finding rate directly and indeedboth the unemployment-to-employment andnonparticipation-to-employment transition ratesare strongly procyclical (Blanchard and Dia-mond 1990 Hoyt Bleakley et al 1999 Ka-tharine Abraham and Shimer 2001) There aretwo drawbacks to this approach First the req-uisite public use dataset is available only since1976 and so using these data would requirethrowing away half of the available time seriesSecond measurement and classification errorlead a substantial overestimate of gross workerflows (John Abowd and Arnold Zellner 1985James Poterba and Lawrence Summers 1986)the magnitude of which cannot easily be com-puted Instead I infer the job-finding rate fromthe dynamic behavior of the unemploymentlevel and short-term unemployment level Letut

s denote the number of workers unemployedfor less than one month in month t Then as-suming all unemployed workers find a job withprobability ft in month t and no unemployedworker exits the labor force

ut 1 ut 1 ft ut 1s

frequencies shocks generally drive the unemployment andvacancy rates in the opposite direction

FIGURE 3 QUARTERLY US HELP-WANTED ADVERTISING

INDEX AND TREND 1951ndash2003

Notes The help-wanted advertising index is a quarterlyaverage of the seasonally adjusted monthly series con-structed by the Conference Board with normalization1987 100 The data were downloaded from the FederalReserve Bank of St Louis database at httpresearchstlouisfedorgfred2datahelpwanttxt The trend is an HPfilter of the quarterly data with smoothing parameter 105

FIGURE 4 QUARTERLY US BEVERIDGE CURVE1951ndash2003

Notes Unemployment is constructed by the BLS from theCPS The help-wanted advertising index is constructed bythe Conference Board Both are quarterly averages of sea-sonally adjusted monthly series and are expressed as devi-ations from an HP filter with smoothing parameter 105


Unemployment next month is the sum of thenumber of unemployed workers this month whofail to find a job and the number of newlyunemployed workers Equivalently

(1) ft 1 ut 1 ut 1

s

ut

I use the unemployment level and the number ofworkers unemployed for 0 to 4 weeks bothconstructed by the BLS from the CPS to com-pute ft from 1951 to 20037 Figure 5 shows theresults The monthly hazard rate averaged 045from 1951 to 2003 After detrending with theusual low-frequency HP filter the correlationbetween ft and t at quarterly frequencies is095 although the standard deviation of ft isabout 31 percent that of t Given that bothmeasures are crudely yet independently con-structed this correlation is remarkable andstrongly suggests that a matching function is auseful way to approach US data

One can use the measured job-finding rateand v-u ratio to estimate a matching functionm(uv) Data limitations force me to impose tworestrictions on the estimated function First be-cause unemployment and vacancies are stronglynegatively correlated it is difficult to tell em-pirically whether m(uv) exhibits constant in-creasing or decreasing returns to scale But intheir literature survey Petrongolo and Pissar-ides (2001) conclude that most estimates of thematching function cannot reject the null hypoth-esis of constant returns I therefore estimate ft f(t) consistent with a constant returns-to-scalematching function Figure 6 shows the raw datafor the job-finding rate ft and the v-u ratio t anearly linear relationship when both variablesare expressed as deviations from log trend Sec-ond I impose that the matching function isCobb-Douglas m(uv) uv1 for someunknown parameters and Again the dataare not very informative as to whether this is areasonable restriction8 I estimate the matching7 Abraham and Shimer (2001) argue that the redesign of

the CPS in January 1994 in particular the switch to depen-dent interviewing reduced measured short-term unemploy-ment They suggest some methods of dealing with thisdiscontinuity In this paper I simply inflate short-term un-employment by 10 percent after the redesign took effect

8 Consider the CES matching function log ft log 1 log ( (1 )t

) Cobb-Douglas corresponds to

FIGURE 5 MONTHLY JOB-FINDING PROBABILITY FOR

UNEMPLOYED WORKERS 1951ndash2003

Notes The job-finding rate is computed using equation (1)with unemployment and short-term unemployment dataconstructed and seasonally adjusted by the BLS from theCPS survey home page httpwwwblsgovcps It is ex-pressed as a quarterly average of monthly data The trendis an HP filter of the quarterly data with smoothing param-eter 105

FIGURE 6 MONTHLY US MATCHING FUNCTION1951ndash2003

Notes The v-u ratio is constructed by the BLS from theCPS and by the Conference Board The job-finding rate isconstructed using equation (1) and BLS data from the CPSBoth are quarterly averages of seasonally adjusted monthlyseries and are expressed as deviations from an HP filter withsmoothing parameter 105


function using detrended data on the job-findingrate and the v-u ratio Depending on exactlyhow I control for autocorrelation in the residu-als I estimate values of between 070 and075 With a first-order autoregressive residualI get 072 with a standard error of 001

One particularly crude aspect of this measureof the job-finding rate is the assumption that allworkers are equally likely to find a job Shimer(2004a) proves that when the unemployed areheterogeneous ft measures the mean job-findingrate in the unemployed population That paperalso compares my preferred measure of the job-finding rate with two alternatives The first usesthe unemployment level and mean unemploy-ment duration to obtain a weighted average ofthe job-finding rate in the unemployed popula-tion with weights proportional to each individ-ualrsquos unemployment duration9 The secondfollows Robert Hall (2004) and measures thejob-finding rate of workers with short unem-ployment duration using the ratio of workerswith 0 to 4 weeks of unemployment to workerswith 5 to 14 weeks of unemployment Since thejob-finding rate declines with unemploymentduration I find that my preferred measure of thejob-finding rate lies between these two alterna-tives Hall measures an average job-finding rateof 048 per month while unemployment dura-tion data yield a job-finding rate of 034 permonth Nevertheless all three measures arehighly correlated and so the choice of whichmeasure to use does not qualitatively affect theconclusions of this study

D Separation Rate

I can also deduce the behavior of the separa-tion rate from data on employment short-termunemployment and the hiring rate Supposefirst that whenever an employed worker losesher job she becomes unemployed Then theseparation rate could simply be computed as theratio of short-term unemployed workers nextmonth ut1

s to employed workers this monthet But this masks a significant time-aggregation

bias When a worker loses her job she has onaverage half a month to find a new job beforeshe is recorded as unemployed Accounting forthis the short-term unemployment rate the nextmonth is approximately equal to

ut 1s st et 1

12

ft

Ignoring the probability of finding another jobwithin the month leads one to understate theseparation rate This problem is particularlyacute when the job-finding rate is high ieduring expansions I therefore measure the sep-aration rate as

(2) st ut 1

s

et 1 12

ft

Figure 7 shows the monthly separation ratethus constructed It averaged 0034 from 1951to 2003 so jobs last on average for about 25years Fluctuations in the deviation of the logseparation rate from trend are somewhat smallerthan in the hiring rate with a standard deviationof 008 and separations are countercyclical sothe correlation with the detrended v-u ratio is072

limiting case of 0 When I estimate the CES functionusing nonlinear least squares and correct for first-orderautocorrelation I get a point estimate of 006 with astandard error of 038

9 A previous version of this paper relied on that measureof ft This had little effect on the results

FIGURE 7 MONTHLY SEPARATION PROBABILITY FOR

EMPLOYED WORKERS 1951ndash2003

Notes The separation rate is computed using equation (2)with employment unemployment and short-term unem-ployment data constructed and seasonally adjusted by theBLS from the CPS survey home page httpwwwblsgovcps It is expressed as a quarterly average of monthly dataThe trend is an HP filter of the quarterly data with smooth-ing parameter 105


The strong procyclicality of the job-findingrate and relatively weak countercyclicality ofthe separation rate might appear to contradictBlanchard and Diamondrsquos (1990) conclusionthat ldquothe amplitude in fluctuations in the flowout of employment is larger than that of the flowinto employmentrdquo This is easily reconciledBlanchard and Diamond look at the number ofpeople entering or exiting employment in agiven month ftut or stet while I focus on theprobability that an individual switches employ-ment states ft and st Although the probabilityof entering employment ft declines sharply inrecessions this is almost exactly offset by theincrease in unemployment ut so that the num-ber of people exiting unemployment is essen-tially acyclic Viewed through the lens of anincreasing matching function m(uv) this isconsistent with the independent evidence thatvacancies are strongly procyclical

E Labor Productivity

The final important empirical observation isthe weak procyclicality of labor productivitymeasured as real output per worker in the non-farm business sector The BLS constructs thisquarterly time series as part of its Major SectorProductivity and Costs program The outputmeasure is based on the National Income and

Product Accounts while employment is con-structed from the BLS establishment survey theCurrent Employment Statistics This series of-fers two advantages compared with total factorproductivity it is available quarterly since1948 and it better corresponds to the concept oflabor productivity in the subsequent modelswhich do not include capital

Figure 8 shows the behavior of labor produc-tivity and Figure 9 compares the cyclical com-ponents of the v-u ratio and labor productivityThere is a positive correlation between the twotime series and some evidence that labor pro-ductivity leads the v-u ratio by about one yearwith a maximum correlation of 05610 But themost important fact is that labor productivity isstable never deviating by more than 6 per-cent from trend In contrast the v-u ratio hastwice risen to 05 log points about its trend leveland six times has fallen by 05 log points belowtrend

10 From 1951 to 1985 the contemporaneous correlationbetween detrended labor productivity and the v-u ratio was057 and the peak correlation was 074 From 1986 to 2003however the contemporaneous and peak correlations arenegative 037 and 043 respectively This has beenparticularly noticeable during the last three years of data Anexploration of the cause of this change goes beyond thescope of this paper

FIGURE 8 QUARTERLY US AVERAGE LABOR

PRODUCTIVITY AND TREND 1951ndash2003

Notes Real output per person in the non-farm businesssector constructed by the BLS Major Sector Productivityand Costs program survey home page httpwwwblsgovlpc 1992 100 The trend is an HP filter of the quarterlydata with smoothing parameter 105

FIGURE 9 QUARTERLY US VACANCY-UNEMPLOYMENT

RATIO AND AVERAGE LABOR PRODUCTIVITY 1951ndash2003

Notes Unemployment is constructed by the BLS from theCPS The help-wanted advertising index is constructed bythe Conference Board Both are quarterly averages of sea-sonally adjusted monthly series Labor productivity is realaverage output per worker in the non-farm business sectorconstructed by the BLS Major Sector Productivity andCosts program The v-u ratio and labor productivity areexpressed as deviations from an HP filter with smoothingparameter 105


It is possible that the measured cyclicality oflabor productivity is reduced by a compositionbias since less productive workers are morelikely to lose their jobs in recessions I offer tworesponses to this concern First there is a com-position bias that points in the opposite direc-tion labor productivity is higher in morecyclical sectors of the economy eg durablegoods manufacturing And second a large lit-erature on real wage cyclicality has reached amixed conclusion about the importance of com-position biases (Abraham and John Haltiwan-ger 1995) Gary Solon et al (1994) pro-vide perhaps the strongest evidence that laborforce composition is important for wage cycli-cality but even they argue that accountingfor this might double the measured variabilityof real wages This paper argues that thesearch and matching model cannot accountfor the cyclical behavior of vacancies and un-employment unless labor productivity is at leastten times as volatile as the data suggest socomposition bias is at best an incompleteexplanation

II Search and Matching Model

I now examine whether a standard search andmatching model can reconcile the strong procy-clicality of the v-u ratio and the job-finding ratewith the weak procyclicality of labor productiv-ity and countercyclicality of the separation rateThe model I consider is essentially an aggregatestochastic version of Pissarides (1985 or 2000Ch 1)

A Model

I start by describing the exogenous vari-ables that drive fluctuations Labor productiv-ity p and the separation rate s follow a first-order Markov process in continuous time Ashock hits the economy according to a Pois-son process with arrival rate at which pointa new pair (ps) is drawn from a state de-pendent distribution Let psXps denote theexpected value of an arbitrary variable Xfollowing the next aggregate shock condi-tional on the current state (ps) I assume thatthis conditional expectation is finite which isensured if the state space is compact At everypoint in time the current values of produc-

tivity and the separation rate are commonknowledge

Next I turn to the economic agents in theeconomy a measure 1 of risk-neutral infinitely-lived workers and a continuum of risk-neutralinfinitely-lived firms All agents discount futurepayoffs at rate r 0

Workers can either be unemployed or em-ployed An unemployed worker gets flow utilityz from non-market activity (ldquoleisurerdquo) andsearches for a job An employed worker earnsan endogenous wage but may not search Idiscuss wage determination shortly

Firms have a constant returns-to-scale pro-duction technology that uses only labor withlabor productivity at time t given by the sto-chastic realization p(t) In order to hire aworker a firm must maintain an open vacancyat flow cost c Free entry drives the expectedpresent value of an open vacancy to zero Aworker and a firm separate according to aPoisson process with arrival rate governedby the stochastic separation rate s(t) leavingthe worker unemployed and the firm withnothing

Let u(t) denote the endogenous unemploy-ment rate11 v(t) denote the endogenous mea-sure of vacancies in the economy and (t) v(t)u(t) denote the v-u ratio at time t The flowof matches is given by a constant returns-to-scale function m(u(t) v(t)) increasing in botharguments This implies that an unemployedworker finds a job according to a Poisson pro-cess with time-varying arrival rate f((t)) m(1(t)) and that a vacancy is filled accordingto a Poisson process with time-varying arrivalrate q((t)) m((t)11) f((t))(t)

I assume that in every state of the worldlabor productivity p(t) exceeds the value of lei-sure z so there are bilateral gains from match-ing There is no single compelling theory ofwage determination in such an environmentand so I follow the literature and assume thatwhen a worker and firm first meet the expectedgains from trade are split according to the Nashbargaining solution The worker can threaten tobecome unemployed and the firm can threatento end the job The present value of surplus

11 With the population of workers constant and normal-ized to one the unemployment rate and unemploymentlevel are identical in this model I therefore use these termsinterchangeably


beyond these threats is divided between theworker and firm with the worker keeping afraction (0 1) of the surplus her ldquobargain-ing powerrdquo I make almost no assumptionsabout what happens to wages after this initialagreement except that the worker and firmmanage to exploit all the joint gains from tradeFor example the wage may be re-bargainedwhenever the economy is hit with a shockAlternatively it may be fixed at its initial valueuntil such time as the firm would prefer tofire the worker or the worker would preferto quit whereupon the pair resets the wage soas to avoid an unnecessary and inefficientseparation

B Characterization of Equilibrium

I look for an equilibrium in which the v-uratio depends only on the current value of pand s ps

12 Given the state-contingent v-uratio the unemployment rate evolves accordingto a standard backward-looking differentialequation

(3) ut st1 ut fptst ut

where (p(t) s(t)) is the aggregate state at time tA flow s(t) of the 1 u(t) employed workersbecome unemployed while a flow f() of theu(t) unemployed workers find a job An initialcondition pins down the unemployment rate andthe aggregate state at some date t0

I characterize the v-u ratio using a recursiveequation for the joint value to a worker and firmof being matched in excess of breaking up as afunction of the current aggregate state Vps

(4) rVps p z fps Vps sVps

psVps Vps

Appendix A derives this equation from moreprimitive conditions The first two terms repre-sent the current flow surplus from matching If

the pair is matched they produce p units ofoutput If they were to break up the match freeentry implies the firm would be left with noth-ing while the worker would become unem-ployed getting flow utility from leisure z andfrom the probability f(ps) of contacting a firmin which event the worker would keep a fraction of the match value Vps Next there is a flowprobability s that the worker and firm separatedestroying the match value Finally an aggre-gate shock arrives at rate resulting in anexpected change in match value psVps Vps

Another critical equation for the match valuecomes from firmsrsquo free entry condition Theflow cost of a vacancy c must equal the flowprobability that the vacancy contacts a workertimes the resulting capital gain which by Nashbargaining is equal to a fraction 1 of thematch value Vps

(5) c qps 1 Vps

Eliminating current and future values of Vpsfrom (4) using (5) gives

(6)r s

qps ps

1 p z

c ps

1

qps

which implicitly defines the v-u ratio as a func-tion of the current state (ps)13 This equationcan easily be solved numerically even with alarge state vector This simple representation ofthe equilibrium of a stochastic version of thePissarides (1985) model appears to be new tothe literature

C Comparative Statics

In some special cases equation (6) canbe solved analytically to get a sense of the

12 It is straightforward to show in a deterministic versionof this model that there is no other equilibrium eg one inwhich also depends on the unemployment rate See Pis-sarides (1985)

13 A similar equation obtains in the presence of aggre-gate variation in the value of leisure z the cost of a vacancyc or workersrsquo bargaining power


quantitative results implied by this analysisFirst suppose there are no aggregate shocks 014 Then the state-contingent v-u ratio satisfies

r s

qps ps 1

p z

c

The elasticity of the v-u ratio with respect toldquonet labor productivityrdquo p z is

r s fps

r s1 ps fps

where () [01] is the elasticity of f() Thisis large only if workersrsquo bargaining power issmall and the elasticity is close to one Butwith reasonable parameter values it is close to1 For example think of a time period as equalto one month so the average job-finding rate isapproximately 045 (Section I C) the elasticity() is approximately 028 (Section I C again)the average separation probability is approxi-mately 0034 (Section I D) and the interest rateis about 0004 Then if workersrsquo bargainingpower is equal to 1 () the so-calledHosios (1990) condition for efficiency15 theelasticity of the v-u ratio with respect to netlabor productivity is 103 Lower values of yield a slightly higher elasticity say 115 when 01 but only at 0 does the elasticity ofthe v-u ratio with respect to p z rise appre-ciably to 139 It would take implausible pa-rameter values for this elasticity to exceed 2This implies that unless the value of leisure isclose to labor productivity the v-u ratio is likelyto be unresponsive to changes in the laborproductivity

I can similarly compute the elasticity of thev-u ratio with respect to the separation rate

s

r s1 ps fps

Substituting the same numbers into this expres-sion gives 010 Doubling the separation ratewould have a scarcely discernible impact on thev-u ratio

Finally one can examine the independentbehavior of vacancies and unemployment Insteady state equation (3) holds with u 0If the matching function is Cobb-Douglasm(uv) uv1 this implies

vps s1 ups

ups 11

For a given separation rate s this describes adecreasing relationship between unemploymentand vacancies consistent with the Beveridgecurve (Figure 4) An increase in labor produc-tivity raises the v-u ratio which lowers the un-employment rate and hence raises the vacancyrate Vacancies and unemployment shouldmove in opposite directions in response to suchshocks But an increase in the separation ratescarcely affects the v-u ratio Instead it tends toraise both the unemployment and vacancy ratesan effect that is likely to produce a counterfac-tually positive correlation between unemploy-ment and vacancies

I can perform similar analytic exercises bymaking other simplifying assumptions Supposethat each vacancy contacts an unemployedworker at a constant Poisson rate indepen-dent of the unemployment rate so q() Given the risk-neutrality assumptions this isequivalent to assuming that firms must pay afixed cost c in order to hire a worker Theneven with aggregate shocks equation (6) yieldsa static equation for the v-u ratio

r s

ps 1

p z

c

In this case the elasticity of the v-u ratio withrespect to net labor productivity is

r s

and the elasticity of the v-u ratio with respect to

14 Shimer (2003) performs comparative statics exercisesunder much weaker assumptions For example in that paperthe matching function can exhibit increasing or decreasingreturns to scale and there can be an arbitrary idiosyncraticprocess for productivity allowing for endogenous separa-tions (Mortensen and Pissarides 1994) I show that theresults presented here generalize to such an environment ifworkers and firms are sufficiently patient relative to thesearch frictions

15 Section III shows that the Hosios condition carriesover to the stochastic model


the separation rate is s Since f() one can again pin down all the parameter valuesexcept workersrsquo bargaining power Using thesame parameter values as above including 072 I obtain elasticities of 112 and 0105almost unchanged from the case with no shocksMore generally unless is nearly equal to zeroboth elasticities are very small

At the opposite extreme suppose that eachunemployed worker contacts a vacancy at aconstant Poisson rate independent of the va-cancy rate so f() and q() Alsoassume that the separation rate s is constant andaverage labor productivity p is a Martingalepp p With this matching function equation(6) is linear in current and future values of thev-u ratio

r s

p

1 p z

c

pp

It is straightforward to verify that the v-u ratio islinear in productivity and therefore pp p ie

r s

p 1

p z

c

so the elasticity of the v-u ratio with respect tonet labor productivity is 1 regardless of work-ersrsquo bargaining power I conclude that with awide range of parameterizations the v-u ratio should be approximately proportional to net la-bor productivity p z

D Calibration

This section parameterizes the model tomatch the time series behavior of the US un-employment rate The most important questionis the choice of the Markov process for laborproductivity and separations Appendix C de-velops a discrete state space model which buildson a simple Poisson process corresponding tothe theoretical analysis in Section II B I definean underlying variable y that lies on a finiteordered set of points When a Poisson shockhits y either moves up or down by one pointThe probability of moving up is itself decreas-ing in the current value of y which ensures that

y is mean reverting The stochastic variables arethen expressed as functions of y

Although I use the discrete state space modelin my simulations as well it is almost exactlycorrect and significantly easier to think aboutthe behavior of the extrinsic shocks by discuss-ing a related continuous state space model16 Iexpress the state variables as functions of anOrnstein-Uhlenbeck process (see Howard Tay-lor and Samuel Karlin 1998 Section 85) Let ysatisfy

dy ydt db

where b is a standard Brownian motion Here 0 is a measure of persistence with highervalues indicating faster mean reversion and 0 is the instantaneous standard deviationThis process has some convenient properties yis conditionally and unconditionally normal itis mean reverting with expected value converg-ing asymptotically to zero and asymptoticallyits variance converges to 22

I consider two different cases In the first theseparation rate is constant and productivity sat-isfies p z ey(p z) where y is anOrnstein-Uhlenbeck process with parameters and and p z is a measure of long-runaverage productivity Since ey 0 this ensuresp z In the second case productivity is con-stant and separations satisfy s eys whereagain y follows an Ornstein-Uhlenbeck processand now s 0 is a measure of the long-runaverage separation rate In both cases the sto-chastic process is reduced to three parameters and either p or s

I now proceed to explain the choice of theother parameters starting with the case of sto-chastic productivity I follow the literature andassume that the matching function is Cobb-Douglas

f q 1

This reduces the calibration to ten parameters

16 I work on a discrete grid with 2n 1 2001 pointswhich closely approximate Gaussian innovations This im-plies that Poisson arrival rate of shocks is n 4 timesper quarter in the model with labor productivity shocks and 220 in the model with (less persistent) separationshocks


the productivity parameter p the value of lei-sure z workersrsquo bargaining power the dis-count rate r the separation rate s the twomatching function parameters and the va-cancy cost c and the mean reversion and stan-dard deviation of the stochastic process and

Without loss of generality I normalize theproductivity parameter to p 1 I choose thestandard deviation and persistence of the pro-ductivity process to match the empirical behav-ior of labor productivity This requires setting 00165 and 0004 An increase in thevolatility of productivity has a nearly propor-tional effect on the volatility of other variableswhile the persistence of the stochastic process scarcely affects the reported results For exam-ple suppose I reduce to 0001 so productivityis more nearly a random walk Because it isdifficult to distinguish small values of in afinite dataset after HP filtering the model-generated data the persistence and magnitudeof the impulse is virtually unchanged comparedwith the baseline parameterization But reassur-ingly the detrended behavior of unemploymentand vacancies is also scarcely affected by in-creasing the persistence of labor productivity

I set the value of leisure to z 04 Sincemean labor income in the model is 0993 thislies at the upper end of the range of incomereplacement rates in the United States if inter-preted entirely as an unemployment benefit

I normalize a time period to be one quarterand therefore set the discount rate to r 0012equivalent to an annual discount factor of 0953The analysis in Section I D suggests a quarterlyseparation rate of s 010 so jobs last for about25 years on average This is comparable toAbowd and Zellnerrsquos (1985) finding that 342percent of employed workers exit employmentduring a typical month between 1972 and 1982after correcting for classification and measure-ment error It is also comparable to measuredturnover rates in the JOLTS although someseparations in that survey reflect job-to-job tran-sitions a possibility that is absent from thismodel

Using the matching function estimates fromSection I C I set the elasticity parameter to 072 This lies toward the upper end of the rangeof estimates that Petrongolo and Pissarides(2001) report I also set workersrsquo bargainingpower to the same value 072 Although the

reported results are insensitive to the value ofthat parameter I show in Section III that if the ldquoHosios (1990) Rulerdquo the decentralizedequilibrium maximizes a well-posed socialplannerrsquos problem

I use the final two parameters the matchingfunction constant and the vacancy cost c topin down the average job-finding rate and theaverage v-u ratio As reported in Section I C aworker finds a job with a 045 probability permonth so the flow arrival rate of job offers1 should average approximately 135 on aquarterly basis I do not have a long time serieswith the level of the v-u ratio but fortunatelythe model offers one more normalization Equa-tion (6) implies that doubling c and multiplying by a factor 21 divides the v-u ratio inhalf doubles the rate at which firms contactworkers q() but does not affect the rate atwhich workers find jobs In other words theaverage v-u ratio is intrinsically meaningless inthe model I choose to target a mean v-u ratio of 1which requires setting 1355 and c 0213

In the case of shocks to the separation rate Ichange only the stochastic process so as tomatch the empirical results discussed in SectionI D Productivity is constant and equal to 1while the mean separation rate is s 010 Iset 0057 and 0220 a much lesspersistent stochastic process This leaves theaverage v-u ratio and average job-finding ratevirtually unchanged Table 2 summarizes theparameter choices in the two simulations

I use equation (6) to find the state-contingentv-u ratio ps and then simulate the model That

TABLE 2mdashPARAMETER VALUES IN SIMULATIONS OF THE

MODEL

Parameter

Source of shocks

Productivity Separation

Productivity p stochastic 1Separation rate s 01 stochasticDiscount rate r 0012 0012Value of leisure z 04 04Matching function q() 1355072 1355072

Bargaining power 072 072Cost of vacancy c 0213 0213Standard deviation 00165 00570Autoregressive parameter 0004 0220Grid size 2n 1 2001 2001

Note The text provides details on the stochastic process forproductivity and for the separation rate


is starting with an initial unemployment rateand aggregate state at time 0 I use a pseudo-random number generator to calculate the ar-rival time of the first Poisson shock I computethe unemployment rate when that shock arrivesgenerate a new aggregate state using the discrete-state-space mean-reverting stochastic processdescribed in Appendix C and repeat At the endof each period (quarter) I record the aggregatestate and the unemployment rate

I throw away the first 1000 ldquoquartersrdquo ofdata I then use the model to generate 212 datapoints corresponding to quarterly data from1951 to 2003 and detrend the log of the model-generated data using an HP filter with the usualsmoothing parameter 105 I repeat this 10000times giving me good estimates of both themean of the model-generated data and the stan-dard deviation across model-generated observa-tions The latter provides a sense of howprecisely the model predicts the value of a par-ticular variable

E Results

Table 3 reports the results from simulationsof the model with labor productivity shocksAlong some dimensions notably the co-movement of unemployment and vacancies themodel performs remarkably well The empiricalcorrelation between these two variables is089 the Beveridge curve The model actuallyproduces a stronger negative correlation 093

although the difference is insignificant It isworth emphasizing that the negative correlationbetween unemployment and vacancies is a re-sult not a direct target of the calibration exer-cise The model also generates the correctautocorrelation for unemployment although thebehavior of vacancies is somewhat off target Inthe data vacancies are as persistent and volatileas unemployment while in the model the auto-correlation of vacancies is significantly lowerthan that of unemployment while the standarddeviation of vacancies is three times as large asthe standard deviation of unemployment fluctu-ations around trend It is likely that anythingthat makes vacancies a state variable such asplanning lags an adjustment cost or irrevers-ibility in vacancy creation would increase theirpersistence and reduce their volatility bringingthe model more in line with the data along thesedimensions Shigeru Fujita (2003) develops amodel that adds these realistic features

But the real problem with the model lies inthe volatility of vacancies and unemploymentor more succinctly in the volatility of the v-uratio and the job-finding rate f In a reasonablycalibrated model the v-u ratio is less than 10percent as volatile as in US data This is ex-actly the result predicted from the deterministiccomparative statics in Section II C A 1-percentincrease in labor productivity p from its averagevalue of 1 raises net labor productivity p z byabout 166 percent Using the deterministicmodel I argued before that the elasticity of the

TABLE 3mdashLABOR PRODUCTIVITY SHOCKS

u v vu f p

Standard deviation 0009 0027 0035 0010 0020(0001) (0004) (0005) (0001) (0003)

Quarterly autocorrelation 0939 0835 0878 0878 0878(0018) (0045) (0035) (0035) (0035)

u 1 0927 0958 0958 0958(0020) (0012) (0012) (0012)

v mdash 1 0996 0996 0995(0001) (0001) (0001)

Correlation matrix vu mdash mdash 1 1000 0999(0000) (0001)

f mdash mdash mdash 1 0999(0001)

p mdash mdash mdash mdash 1

Notes Results from simulating the model with stochastic labor productivity All variables are reported in logs as deviationsfrom an HP trend with smoothing parameter 105 Bootstrapped standard errorsmdashthe standard deviation across 10000 modelsimulationsmdashare reported in parentheses The text provides details on the stochastic process for productivity


v-u ratio with respect to net labor productivity isabout 103 with this choice of parameters giv-ing a total elasticity of with respect to p ofapproximately 166 103 171 percent Infact the standard deviation of log aroundtrend is 175 times as large as the standarddeviation of log p Similarly the job-findingrate is 12 times as volatile in the data as in themodel

Not only is there little amplification but thereis also no propagation of the labor productivityshock in the model The contemporaneous cor-relation between labor productivity the v-u ra-tio and the job-finding rate is 100 In the datathe contemporaneous correlation between thefirst two variables is 040 and the v-u ratio lagslabor productivity by about one year The em-pirical correlation between labor productivityand the job-finding rate is similar

Table 4 reports the results from the modelwith shocks to the separation rate These intro-duce an almost perfectly positive correlationbetween unemployment and vacancies an eventthat has essentially never been observed in theUnited States at business cycle frequencies (seeFigure 3) As a result separation shocks pro-duce almost no variability in the v-u ratio or thejob finding rate Again this is consistent withthe back-of-the-envelope calculations per-formed in Section II C where I argued that theelasticity of the v-u ratio with respect to theseparation rate should be approximately 010According to the model the ratio of the stan-

dard deviations is about 008 and the two vari-ables are strongly negatively correlated

One might be concerned that the disjointanalysis of labor productivity and separationshocks masks some important interaction be-tween the two impulses Modeling an endoge-nous increase in the separation rate due to lowlabor productivity as in Mortensen and Pissar-ides (1994) goes beyond the scope of this pa-per Instead I introduce perfectly negativelycorrelated labor productivity and separationshocks into the basic model More precisely Iassume p z ey(p z) and s esysboth nonlinear functions of the same latent vari-able y The parameter s 0 permits a differentvolatility for p and s

I start with the parameterization of the modelwith only labor productivity shocks and intro-duce volatility in the separation rate Table5 shows the results from a calibration with equalstandard deviations in the deviation from trendof the separation rate and labor productivity(s 1 z) The behavior of vacancies in themodel is now far from the data with an auto-correlation of 029 (compared to 094 empiri-cally) and a correlation with unemployment of043 (089) The difference between modeland data is highly significant both economicallyand statistically Moreover although cyclicalfluctuations in the separation rate boost the vol-atility of unemployment considerably theyhave a small effect on the cyclical volatility ofthe v-u ratio and job-finding rate which remain

TABLE 4mdashSEPARATION RATE SHOCKS

u v vu f s



u 1 0999 0906 0906 0908(0000) (0017) (0017) (0017)

v mdash 1 0887 0887 0888(0020) (0020) (0021)



s mdash mdash mdash mdash 1

Notes Results from simulating the model with a stochastic separation rate All variables are reported in logs as deviationsfrom an HP trend with smoothing parameter 105 Bootstrapped standard errorsmdashthe standard deviation across 10000 modelsimulationsmdashare reported in parentheses The text provides details on the stochastic process for the separation rate


at around 10 percent of their empirical valuesSmaller fluctuations in the separation rate natu-rally have a smaller effect while realisticallylarge fluctuations in the separation rate induce astrong positive correlation between unemploy-ment and vacancies even in the presence ofcorrelated productivity shocks

To summarize the stochastic version of thePissarides (1985) model confirms that separa-tion shocks induce a positive correlation be-tween unemployment and vacancies It alsoconfirms that while labor productivity shocksare qualitatively consistent with a downward-sloping Beveridge curve the search model doesnot substantially amplify the extrinsic shocksand so labor productivity shocks induce onlyvery small movements along the curve

F Wages

Until this point I have assumed that the sur-plus in new matches is divided according to ageneralized Nash bargaining solution but havemade no assumption about the division of sur-plus in old matches Although this is sufficientfor determining the response of unemploymentand vacancies to exogenous shocks it does notpin down the timing of wage payments In thissection I introduce an additional assumptionthat the surplus in all matches new or old isalways divided according to the Nash bargain-

ing solution as would be the case if wages wererenegotiated following each aggregate shockThis stronger restriction pins down the wage asa function of the aggregate state wps Thisfacilitates a more detailed discussion of wageswhich serves two purposes First modelingwages illustrates that flexibility of the presentvalue of wage payments is critical for many ofthe results emphasized in this paper And sec-ond it enables me to relate this paper to aliterature that examines whether search modelscan generate rigid wages Appendix B provesthat a continually renegotiated wage solves

(7) wps 1 z p cps

This generalizes equation (120) in Pissarides(2000) to a stochastic environment

Consider first the effect of a separation shockon the wage An increase in the separation rate sinduces a slight decline in the v-u ratio (see Table4) which in turn by equation (7) reduces wagesslightly Although the direct effect of the shocklowers firmsrsquo profits by shortening the duration ofmatches the resulting decline in wages partiallyoffsets this so the drop in the v-u ratio is small

Second consider a productivity shock A1-percent increase in net labor productivity p z raises the v-u ratio by about 1 percent (seeTable 3) Equation (7) then implies that thenet wage w z increases by about 1 percent

TABLE 5mdashLABOR PRODUCTIVITY AND SEPARATION RATE SHOCKS

u v vu f s p

Standard deviation 0031 0011 0037 0014 0020 0020(0005) (0001) (0006) (0002) (0003) (0003)

Quarterly autocorrelation 0933 0291 0878 0878 0878 0878(0020) (0085) (0035) (0035) (0035) (0035)

u 1 0427 0964 0964 0964 0964(0068) (0011) (0011) (0011) (0011)

v mdash 1 0650 0650 0649 0648(0042) (0042) (0042) (0042)

Correlation matrix vu mdash mdash 1 1000 1000 0999(0000) (0000) (0001)

f mdash mdash mdash 1 1000 0999(0000) (0001)

s mdash mdash mdash mdash 1 0999(0001)

p mdash mdash mdash mdash mdash 1

Notes Results from simulating the model with stochastic but perfectly correlated labor productivity and separations Allvariables are reported in logs as deviations from an HP trend with smoothing parameter 105 Bootstrapped standarderrorsmdashthe standard deviation across 10000 model simulationsmdashare reported in parentheses The text provides details on thestochastic process


soaking up most of the productivity shock andgiving firms little incentive to create new va-cancies Hence there is a modest increase invacancies and decrease in unemployment in re-sponse to a large productivity shock

To understand fully the importance of wagesfor the v-u ratio it is useful to consider a ver-sion of the model in which labor productivityand the separation rate are constant at p 1 ands 01 but workersrsquo bargaining power changes stochastically An increase in reducesthe profit from creating vacancies which putsdownward pressure on the v-u ratio It is diffi-cult to know exactly how much variability in is reasonable but one can ask how much wagevariability is required to generate the observedvolatility in the v-u ratio I assume is a func-tion of the latent variable y (y 1()) where is the cumulative standardnormal distribution If y were constant at zerothis implies but more generally issimply bounded between 0 and 1 I set thestandard deviation of y to 0099 and themean reversion parameter to 0004 Al-though this implies very modest fluctuations inwagesmdashthe standard deviation of detrended logwages computed as in equation (7) is just001mdashthe calibrated model generates the ob-served volatility in the v-u ratio with persis-tence similar to that in the model with laborproductivity shocks Table 6 shows the com-plete results Since bargaining power is the only

driving force wages are counterfactually coun-tercyclical (Abraham and Haltiwanger 1995)Nevertheless it seems plausible that a modelwith a combination of wage and labor produc-tivity shocks could generate the observed be-havior of unemployment vacancies and realwages Of course the unanswered question iswhat exactly a wage shock is

If wages are bargained in new matches butthen not continually renegotiated this analysisis inapplicable Nevertheless one can prove thatthe frequency of wage negotiation does not af-fect the expected present value of wage pay-ments in new matches but only changes thetiming of wage payments An increase in pro-ductivity or decrease in separations raises thepresent value of wage payments in new jobs andtherefore has little effect on the v-u ratio Anincreased workersrsquo bargaining power in a newemployment relationship induces a large reduc-tion in vacancies and in the v-u ratio

III Optimal V-U Fluctuations

Another way to highlight the role played bythe Nash bargaining assumption is to examine acentralized economy in which it is possible tosidestep the wage-setting issue entirely17 Con-

17 A number of papers examine a ldquocompetitive searcheconomyrdquo in which firms can commit to wages before

TABLE 6mdashBARGAINING POWER SHOCKS

u v vu f w



u 1 0915 0949 0949 0818(0045) (0032) (0032) (0112)

v mdash 1 0995 0995 0827(0001) (0001) (0128)



w mdash mdash mdash mdash 1

Notes Results from simulating the model with stochastic bargaining power All variables are reported in logs as deviationsfrom an HP trend with smoothing parameter 105 Bootstrapped standard errorsmdashthe standard deviation across 10000 modelsimulationsmdashare reported in parentheses The text provides details on the stochastic process for the workersrsquo bargainingpower


sider a hypothetical social planner who choosesa state-contingent v-u ratio in order to maximizethe present discounted value of output net ofvacancy creation costs The plannerrsquos problemis represented recursively as

rW p s u max

zu p1 u cu

Wup s us1 u uf

psWp s u Wp s u)

Instantaneous output is equal to z times theunemployment rate u plus p times the employ-ment rate minus c times the number of vacan-cies v u The value changes gradually as theunemployment rate adjusts with u s(1 u) uf() and suddenly when an aggregateshock changes the state from (ps) to (ps) atrate

It is straightforward to verify that the Bell-man value W is linear in the unemployment rateWu(psu) cf(ps) and the v-u ratio satis-fies

r s

fps ps1

fps

ps fps

p z

c ps 1

f ps

This implicitly defines the optimal ps indepen-dent of the unemployment rate

With a Cobb-Douglas matching functionm(uv) uv1 this reduces to

r s

qps ps

1 p z

c ps 1

qps

a special case of equation (6) with workersrsquobargaining power equal to the elasticity This generalizes the Hosios (1990) conditionfor efficiency of the decentralized equilibriumto an economy with stochastic productivity andseparation rates Since the numerical example inSection II E assumes a Cobb-Douglas matchingfunction with the equilibrium allocationdescribed in that section solves the social plan-nerrsquos problem Conversely if those parametervalues describe the US economy the observeddegree of wage rigidity is inconsistent with out-put maximization

With other matching functions the link be-tween the equilibrium with wage bargainingand the solution to the plannerrsquos problem isbroken At one extreme if unemployment andvacancies are perfect substitutes ie f() u v then the output-maximizing v-u ratiois infinite whenever v(p z) c(r s u)and is zero if the inequality is reversed With near-perfect substitutability the output-maximizing v-uratio is very sensitive to current productivity Onthe other hand if unemployment and vacanciesare perfect complements f() minuv thev-u ratio never strays from the efficient ratiou v With imperfect complements the impactof productivity shocks on the v-u ratio is muf-fled but not eliminated

The economics behind these theoretical find-ings is simple An increase in labor productivityrelative to the value of non-market activity andthe cost of advertising a vacancy induces aswitch away from the expensive activity unem-ployment and toward the relatively cheap ac-tivity vacancies The magnitude of the switchdepends on how substitutable unemploymentand vacancies are in the matching function Ifthey are strong complements substitution isnearly impossible and the v-u ratio barelychanges If they are strong substitutes substitu-tion is nearly costless and the v-u ratio is highlyprocyclical

In the decentralized economy the extent ofsubstitution between unemployment and vacan-cies is governed not only by the matching func-tion but also by the bargaining solution asshown by the comparative statics exercises inSection II C The Nash bargaining solution ef-fectively corresponds to a moderate degree ofsubstitutability the Cobb-Douglas case Ifwages were more rigid an increase in produc-tivity would induce more vacancy creation and

hiring workers and can increase their hiring rate by prom-ising higher wages (Peters 1991 Montgomery 1991Moen 1997 Shimer 1996 Burdett et al 2001) It is bynow well-known that a competitive search equilibrium max-imizes output essentially by creating a market for jobapplications This discussion of output-maximizing searchbehavior therefore also pertains to these models


less unemployment analogous to a centralizedenvironment with a high elasticity of substitu-tion in the matching function

The substitutability of unemployment and va-cancies is an empirical issue Blanchard andDiamond (1989) use nonlinear least squares toestimate a Constant Elasticity of Substitution(CES) matching function on US data Theirpoint estimate for the elasticity of substitution is074 ie slightly less substitutable than theCobb-Douglas case although they cannot rejectthe Cobb-Douglas elasticity of 1 As footnote 8describes my data suggest an elasticity slightlyin excess of 1 although my point estimate isimprecise Whether the observed movements inunemployment and vacancies are optimal whenviewed through the lens of the textbook searchand matching model therefore remains an openquestion

IV Related Literature

There is a large literature that exploreswhether the search model is consistent with thecyclical behavior of labor markets Some paperslook at the implications of the model for thebehavior of various stocks and flows includingthe unemployment and vacancy rates but do notexamine the implicit magnitude of the exogenousimpulses Others assume that business cycles aredriven by fluctuations in the separation rate sThese papers either impose exogenously or derivewithin the model a counterfactually constant v-uratio A third group of papers has tried but failedto reconcile the procyclicality of the v-u ratio withextrinsic shocks of a plausible magnitude

Papers by Abraham and Lawrence Katz(1986) Blanchard and Diamond (1989) andCole and Rogerson (1999) fit into the first cat-egory matching the behavior of labor marketstocks and flows by sidestepping the magnitudeof impulses For example Abraham and Katz(1986) argue that the downward-sloping Bever-idge curve is inconsistent with models in whichunemployment is driven by fluctuations in theseparation rate notably David Lilienrsquos (1982)sectoral shifts model That leads them to advo-cate an alternative in which unemploymentfluctuations are driven by aggregate distur-bances eg productivity shocks Unfortu-nately they fail to examine the magnitude ofshocks needed to deliver the observed shiftsalong the Beveridge curve Blanchard and Dia-

mond (1989) also focus on the negative corre-lation between unemployment and vacanciesbut they do not model the supply of jobs andhence do not explain why there are so fewvacancies during recessions Instead they as-sume the total stock of jobs follows an exoge-nous stochastic process This paper pushes thecyclicality of the v-u ratio to the front of thepicture Likewise Cole and Rogerson (1999)argue that the Mortensen and Pissarides (1994)model can match a variety of business cyclefacts but they do so in a reduced form modelthat treats fluctuations in the job-finding rate andhence implicitly in the v-u ratio as exogenous

The second group of papers including workby Michael Pries (2004) Gary Ramey and JoelWatson (1997) Wouter Den Haan et al (2000)and Joao Gomes et al (2001) assumes thatemployment fluctuations are largely due totime-variation in the separation rate minimiz-ing the role played by the observed cyclicalityof the v-u ratio These papers typically deliverrigid wages from a search model consistentwith the findings in Section II E Building onthe ideas in Hall (1995) Pries (2004) shows thata brief adverse shock that destroys some oldemployment relationships can generate a longtransition period of high unemployment as thedisplaced workers move through a number ofshort-term jobs before eventually finding theirway back into long-term relationships Duringthis transition process the v-u ratio remainsconstant since aggregate economic conditionshave returned to normal Equivalently theeconomy moves along an upward-sloping Bev-eridge curve during the transition period incontradiction to the evidence Ramey andWatson (1997) argue that two-sided asymmetricinformation generates rigid wages in a searchmodel But in their model shocks to the sepa-ration rate are the only source of fluctuations inunemployment The job-finding rate f() is ex-ogenous and constant which is equivalent toassuming that vacancies are proportional to un-employment This is probably an important partof the explanation for why their model producesrigid wages Den Haan et al (2000) show thatfluctuations in the separation rate amplify pro-ductivity shocks in a model similar to the oneexamined here however they do not discussthe cyclical behavior of the v-u ratio SimilarlyGomes et al (2001) sidestep the v-u issue bylooking at a model in which the job-finding rate


is exogenous and constant ie vacancies areproportional to unemployment Again thishelps keep wages relatively rigid in their model

Mortensen and Pissarides (1994) have prob-ably the best known paper in this literature Intheir three-state ldquoillustrative simulationrdquo theauthors introduce without comment enormousproductivity or leisure shocks into their modelAverage labor productivity minus the value ofleisure p z is approximately three times ashigh in the good state as in the bad state18 Thispaper confirms that in response to such largeshocks the v-u ratio should also be about threetimes as large in the good state as in the badstate but argues that there is no evidence forthese large shocks in the data Even if oneaccepts the magnitude of the implied impulsesMortensen and Pissarides (1994) still deliveronly a correlation of 026 between unemploy-ment and vacancies far lower than the empiri-cal value of 088 This is probably because ofthe tension between productivity shocks whichput the economy on a downward-sloping Bev-eridge curve and endogenous movements in theseparation rate which have the opposite effectMonika Merz (1995) and David Andolfatto(1996) both put the standard search model intoa real business cycle framework with intertem-poral substitution of leisure capital accumula-tion and other extensions Neither paper canmatch the negative correlation between unem-ployment and vacancies and both papers gen-erate real wages that are too flexible in responseto productivity shocks Thus these papers en-counter the problem I highlight in this paperalthough they do not emphasize this shortcom-ing of the search model Finally Hall (2003)building on an earlier version of this paperdiscusses some of the same issues Hall (2005)proposes one possible solution real wages aredetermined by a social norm that does notchange over the business cycle

V Conclusion

I have argued in this paper that a search andmatching model in which wages are determined

by Nash bargaining cannot generate substantialmovements along a downward-sloping Bever-idge curve in response to shocks of a plausiblemagnitude A labor productivity shock resultsprimarily in higher wages with little effect onthe v-u ratio A separation shock generates anincrease in both unemployment and vacanciesIt is important to stress that this is not an attackon the search approach to labor markets butrather a critique of the commonly-used Nashbargaining assumption for wage determinationAn alternative wage determination mechanismthat generates more rigid wages in new jobsmeasured in present value terms will amplifythe effect of productivity shocks on the v-uratio helping to reconcile the evidence and the-ory Countercyclical movements in workersrsquobargaining power provide one such mechanismat least in a reduced-form sense

If the matching function is Cobb-Douglasthe observed behavior of the v-u ratio is notsocially optimal for plausible parameterizationsof the model but this conclusion could be over-turned if the elasticity of substitution betweenunemployment and vacancies in the matchingfunction is sufficiently large Estimates of aCES matching function are imprecise so it isunclear whether observed wages are ldquotoo rigidrdquo

One way to generate more rigid wages in atheoretical model is to introduce considerationswhereby wages affect the worker turnover rateFor example in the Burdett and Mortensen(1998) model of on-the-job search firms havean incentive to offer high wages in order toattract workers away from competitors and toreduce employeesrsquo quit rate The distribution ofproductivity affects an individual firmrsquos wageoffer and vacancy creation decisions in complexways breaking the simple link between averagelabor productivity and the v-u ratio in the Pis-sarides (1985) model In particular a shift in theproductivity distribution that leaves average la-bor productivity unchanged may appreciablyaffect average wages and hence the equilibriumv-u ratio

Another possibility is to drop some of theinformational assumptions in the standardsearch model19 Suppose that when a worker

18 This calculation would be easy in the absence ofheterogeneity ie if their parameter were equal to zeroThen p z would take on three possible values 00220075 and 0128 for a six-fold difference in p z betweenthe high and low states

19 Ramey and Watson (1997) develop a search modelwith two-sided asymmetric information Because they as-sume workersrsquo job finding rate is exogenous and acyclic


and firm meet they draw an idiosyncraticmatch-specific productivity level from somedistribution F Workers and firms know aboutaggregate variables including the unemploy-ment rate and the distribution F but only thefirm knows the realized productivity level Bar-gaining proceeds as follows with probability (01) a worker makes a take-it-or-leave-itwage demand and otherwise the firm makes atake-it-or-leave-it offer Obviously the firm ex-tracts all the rents from the employment rela-tionship when it makes an offer But if theuninformed worker makes the offer she faces atradeoff between demanding a higher wage andreducing her risk of unemployment so the wagedepends on the hazard rate of the distribution FThis again breaks the link between average la-bor productivity and the equilibrium v-u ratioExploring whether either of these models orsome related model deliver substantial fluctua-tions in the v-u ratio in response to plausibleimpulses remains a topic for future research

APPENDIX A DERIVATION OF THE EQUATION FOR

SURPLUS (4)

For notational simplicity alone assume thewage payment depends only on the aggregatestate wps not on the history of the match Ireturn to this issue at the end of this sectionDefine Ups Eps and Jps to be the state-contingent present value of an unemployedworker employed worker and filled job re-spectively They are linked recursively by

(8) rUps z fps Eps Ups

psUps Ups

(9) rEps wps sEps Ups

psEps Eps

(10) rJps p wps sJps

psJps Jps

Equation (8) states that the flow value of anunemployed worker is equal to her value of

leisure z plus the probability she finds a jobf(ps) times the resulting capital gain E Uplus the probability of an aggregate shock timesthat capital gain Equation (9) expresses a sim-ilar idea for an employed worker who receivesa wage payment wps but loses her job at rate sEquation (10) provides an analogous recursiveformulation for the value of a filled job Notethat a firm is left with nothing when a filled jobends

Sum equations (9) and (10) and then subtractequation (8) defining Vps Jps Eps Ups

(11) rVps p z fps Eps Ups

sVps psVps Vps

In addition the Nash bargaining solution im-plies that the wage is set so as to maximize theNash product (Eps Ups)

Jps1 which gives

(12)Eps Ups

Vps

Jps

1

Substituting for E U in equation (11) yieldsequation (4)

If I allow wages to depend in an arbitrarymanner on the history of the match this wouldaffect the Bellman values E and J however thewage and therefore the history-dependencewould drop out when summing the Bellmanequations for E and J In other words the matchsurplus V is unaffected by the frequency ofwage renegotiation

APPENDIX B DERIVATION OF THE WAGE

EQUATION

Assume that wages are continually renegoti-ated so the wage depends only on the currentaggregate state (ps) Eliminate current and fu-ture values of J from equation (10) using equa-tion (12)

wps p r s 1 Vps

ps1 Vps

Similarly eliminate current and future values ofV using (5)

wps p r s c

qps ps

c

qps their results are not directly applicable to this analysis

although their methodology may prove useful


Finally replace the last two terms using equa-tion (6) to get equation (7)

APPENDIX C THE STOCHASTIC PROCESS

The text describes a continuous state spaceapproximation to the discrete state space modelused in both the theory and simulations Here Idescribe the discrete state space model andshow that it asymptotes to an Ornstein-Uhlenbeck process

Consider a random variable y that is hit withshocks according to a Poisson process with ar-rival rate The initial value of y lies on adiscrete grid

y Y 13n n 1

0 n 1 n

where 0 is the step size and 2n 1 13 3 isthe number of grid points When a shock hitsthe new value y either moves up or down byone grid point

y y y with probability

1

2 1 y

n1

2 1 y

n

Note that although the step size is constantthe probability that y y is smaller wheny is larger falling from 1 at y n to zero aty n

It is trivial to confirm that y Y so the statespace is discrete To proceed further define n and For any fixed y(t) I examinethe behavior of y(t h) over an arbitrarily shorttime period h For sufficiently short h the prob-ability that two Poisson shocks arrive is negli-gible and so y(t h) is equal to y(t) withprobability 1 h has increased by withprobability h(1 y(t)n)2 and has de-creased by with probability h (1 y(t)n)2 Adding this together shows

yt h ytyt h

nyt hyt

Next the conditional variance of y(t h) y(t)can be decomposed into

Varyt h ytyt

[(y(th)y(t))2y(t)]

yt h ytyt2

The first term evaluates to h2 over a suffi-ciently short time interval h since it is equal to2 if a shock positive or negative arrives andzero otherwise The second term is (hy(t))2and so is negligible over a short time interval hThus

Varyt h ytyt h2 h2

Putting this together we can represent the sto-chastic process for y as

dy ydt dx

where for t 0 the expected value of x(t) givenx(0) is x(0) and the conditional variance is tThis is similar to a Brownian motion exceptthat the innovations in x are not Gaussian sincey is constrained to lie on a discrete grid

Now suppose one changes the three parame-ters of the stochastic process the step sizearrival rate of shocks and number of steps from(n) to ( n) for any 0 It iseasy to verify that this does not change eitherthe autocorrelation parameter n or theinstantaneous variance But as3 0 the distribution of the innovation processx converges to a normal by the Central LimitTheorem Equivalently y converges to anOrnstein-Uhlenbeck process20 This observa-tion is also useful for computation It is possibleto find a solution on a coarse grid and then torefine the grid by decreasing without substan-tially changing the results

20 Notably for large n it is extraordinarily unlikely thatthe state variable reaches its limiting values of n Theunconditional distribution of the state variable is approxi-mately normal with mean zero and standard deviation2 n2 The limiting values of the state variablestherefore lie 2n standard deviations above and below themean If n 1000 as is the case in the simulations oneshould expect to observe such values approximately once in10436 periods


REFERENCES

Abowd John M and Zellner Arnold ldquoEstimat-ing Gross Labor-Force Flowsrdquo Journal ofBusiness and Economic Statistics 1985 3(3)pp 254ndash83

Abraham Katharine G and Katz Lawrence FldquoCyclical Unemployment Sectoral Shifts orAggregate Disturbancesrdquo Journal of Politi-cal Economy 1986 94(3) pp 507ndash22

Abraham Katherine G ldquoHelp-Wanted Advertis-ing Job Vacancies and UnemploymentrdquoBrookings Papers on Economic Activity1987 0(1) pp 207ndash43

Abraham Katharine G and Haltiwanger JohnC ldquoReal Wages and the Business CyclerdquoJournal of Economic Literature 1995 33(3)pp 1215ndash64

Abraham Katharine G and Shimer RobertldquoChanges in Unemployment Duration andLabor-Force Attachmentrdquo in Alan BKrueger and Robert M Solow eds Theroaring nineties Can full employment besustained New York Russell Sage Founda-tion 2001 pp 367ndash420

Andolfatto David ldquoBusiness Cycles and Labor-Market Searchrdquo American Economic Re-view 1996 86(1) pp 112ndash32

Blanchard Olivier Jean and Diamond PeterldquoThe Beveridge Curverdquo Brookings Paperson Economic Activity 1989 0(1) pp 1ndash60

Blanchard Olivier Jean and Diamond PeterldquoThe Cyclical Behavior of the Gross Flows ofUS Workersrdquo Brookings Papers on Eco-nomic Activity 1990 0(2) pp 85ndash143

Bleakley Hoyt Ferris Ann E and Fuhrer Jef-frey C ldquoNew Data on Worker Flows duringBusiness Cyclesrdquo New England EconomicReview 1999 0(0) pp 49ndash76

Burdett Kenneth and Mortensen Dale ldquoEquilib-rium Wage Differentials and Employer SizerdquoInternational Economic Review 1998 39(2)pp 257ndash74

Burdett Kenneth Shouyong Shi and WrightRandall ldquoPricing and Matching with Fric-tionsrdquo Journal of Political Economy 2001105(5) pp 1060ndash85

Cole Harold L and Rogerson Richard ldquoCan theMortensen-Pissarides Matching ModelMatch the Business-Cycle Factsrdquo Interna-tional Economic Review 1999 40(4) pp933ndash59

den Haan Wouter J Ramey Garey and Watson

Joel ldquoJob Destruction and Propagation ofShocksrdquo American Economic Review 200090(3) pp 482ndash98

Flinn Christopher J and Heckman James JldquoAre Unemployment and Out of the LaborForce Behaviorally Distinct Labor ForceStatesrdquo Journal of Labor Economics 19831(1) pp 28ndash42

Fujita Shigeru ldquoThe Beveridge Curve Job Cre-ation and the Propagation of Shocksrdquo Un-published Paper 2003

Gomes Joao Greenwood Jeremy and RebeloSergio ldquoEquilibrium Unemploymentrdquo Jour-nal of Monetary Economics 2001 48(1) pp109ndash52

Hall Robert E ldquoLost Jobsrdquo Brookings Paperson Economic Activity 1995 0(1) pp 221ndash56

Hall Robert E ldquoModern Theory of Unemploy-ment Fluctuations Empirics and Policy Ap-plicationsrdquo American Economic Review2003 93(2) pp 145ndash50

Hall Robert E ldquoEmployment Efficiency andSticky Wages Evidence from Flows in theLabor Marketrdquo Unpublished Paper 2004

Hall Robert E ldquoEmployment Fluctuations withEquilibrium Wage Stickinessrdquo AmericanEconomic Review 2005 95(1) pp 53ndash69

Hosios Arthur J ldquoOn the Efficiency of Match-ing and Related Models of Search and Un-employmentrdquo Review of Economic Studies1990 57(2) pp 279ndash98

Jones Stephen R G and Riddell W Craig ldquoTheMeasurement of Unemployment An Empir-ical Approachrdquo Econometrica 1999 67(1)pp 147ndash61

Juhn Chinhui Murphy Kevin M and TopelRobert H ldquoWhy Has the Natural Rate ofUnemployment Increased over TimerdquoBrookings Papers on Economic Activity1991 0(2) pp 75ndash126

Lilien David M ldquoSectoral Shifts and CyclicalUnemploymentrdquo Journal of Political Econ-omy 1982 90(4) pp 777ndash93

Lucas Robert E Jr and Rapping Leonard AldquoReal Wages Employment and InflationrdquoJournal of Political Economy 1969 77(5)pp 721ndash54

Merz Monika ldquoSearch in the Labor Market andthe Real Business Cyclerdquo Journal of Mone-tary Economics 1995 36(2) pp 269ndash300

Moen Espen R ldquoCompetitive Search Equilib-riumrdquo Journal of Political Economy 1997105(2) pp 385ndash411


Montgomery James ldquoEquilibrium Wage Dis-persion and Interindustry Wage Differen-tialsrdquo Quarterly Journal of Economics 1991106(1) pp 163ndash79

Mortensen Dale T and Pissarides ChristopherA ldquoJob Creation and Job Destruction in theTheory of Unemploymentrdquo Review of Eco-nomic Studies 1994 61(3) pp 397ndash415

Peters Michael ldquoEx Ante Price Offers inMatching Games Non-Steady StatesrdquoEconometrica 1991 59(5) pp 1425ndash54

Petrongolo Barbara and Pissarides ChristopherA ldquoLooking into the Black Box A Survey ofthe Matching Functionrdquo Journal of Eco-nomic Literature 2001 39(2) pp 390ndash431

Pissarides Christopher A ldquoShort-Run Equilib-rium Dynamics of Unemployment Vacan-cies and Real Wagesrdquo American EconomicReview 1985 75(4) pp 676ndash90

Pissarides Christopher A Equilibrium unem-ployment theory 2nd ed Cambridge MAMIT Press 2001

Poterba James M and Summers Lawrence HldquoReporting Errors and Labor Market Dynam-icsrdquo Econometrica 1986 54(6) pp 1319ndash38

Pries Michael J ldquoPersistence of EmploymentFluctuations A Model of Recurring JobLossrdquo Review of Economic Studies 200471(1) pp 193ndash215

Ramey Garey and Watson Joel ldquoContractualFragility Job Destruction and Business Cy-clesrdquo Quarterly Journal of Economics 1997112(3) pp 873ndash911

Shimer Robert ldquoContracts in a Frictional LaborMarketrdquo Unpublished Paper 1996

Shimer Robert ldquoThe Cyclical Behavior ofEquilibrium Unemployment and VacanciesEvidence and Theoryrdquo National Bureau ofEconomic Research Inc NBER WorkingPapers 9536 2003

Shimer Robert ldquoSearch Intensityrdquo Unpub-lished Paper 2004

Solon Gary Barsky Robert and ParkerJonathan A ldquoMeasuring the Cyclicality ofReal Wages How Important Is CompositionBiasrdquo Quarterly Journal of Economics1994 109(1) pp 1ndash25

Taylor Howard and Samuel Karlin An intro-duction to stochastic modeling San DiegoAcademic Press 3rd ed 1998


expected present value of wages in new jobsHigher wages absorb most of the productivityincrease eliminating the incentive for vacancycreation As a result fluctuations in labor pro-ductivity have little impact on the unemploy-ment vacancy and job-finding rates

An increase in the separation rate does notaffect the relative value of unemployment andvacancies and so leaves the v-u ratio essentiallyunchanged Since the increase in separationsreduces employment duration the unemploy-ment rate increases and so therefore must va-cancies As a result fluctuations in theseparation rate induce a counterfactually posi-tive correlation between unemployment andvacancies

Section I presents the relevant US businesscycle facts unemployment u is strongly coun-tercyclical vacancies v are equally strongly pro-cyclical and the correlation between the twovariables is 089 at business cycle frequen-cies As a result the v-u ratio is procyclical andvolatile with a standard deviation around itstrend equal to 038 log points To provide fur-ther evidence in support of this finding I exam-ine the rate at which unemployed workers findjobs If the process of pairing workers with jobsis well-described by an increasing constantreturns-to-scale matching function m(uv) as inPissarides (1985) the job finding rate is f m(uv)u an increasing function of the v-u ratioI use unemployment-duration data to measurethe job-finding rate directly The standard devi-ation of fluctuations in the job-finding ratearound trend is 012 log points and the correla-tion with the v-u ratio is 095 Finally I look atthe two proposed impulses The separation rateis less correlated with the cycle and moderatelyvolatile with a standard deviation about trendequal to 008 log points Average labor produc-tivity is weakly procyclical and even more sta-ble with a standard deviation about trend of002 log points

In Section II I extend the Pissaridesrsquos (1985)search and matching model to allow for aggre-gate fluctuations I introduce two types ofshocks labor productivity shocks raise outputin all matches but do not affect the rate at whichemployed workers lose their job and separationshocks raise the rate at which employed workersbecome unemployed but do not affect the pro-ductivity in surviving matches In equilibriumthere is only one real economic decision firmsrsquo

choice of whether to open a new vacancy Theequilibrium vacancy rate depends on the unem-ployment rate on labor market tightness and onthe expected present value of wages in newemployment relationships Wages in turn aredetermined by Nash bargaining at least in newmatches In principle the wage in old matchesmay be rebargained in the face of aggregateshocks or may be fixed by a long-term employ-ment contract Section II A describes the basicmodel while Section II B derives a forward-looking equation for the v-u ratio in terms ofmodel parameters

Section II C performs simple analytical com-parative statics in some special cases For ex-ample I show that the elasticity of the v-u ratiowith respect to the difference between laborproductivity and the value of nonmarket activityor ldquoleisurerdquo is barely in excess of 1 for reason-able parameter values To reconcile this withthe data one must assume that the value ofleisure is nearly equal to labor productivity somarket work provides little incremental utilityThe separation rate has an even smaller impacton the v-u ratio with an elasticity of 01according to the comparative statics Moreoverwhile shocks to labor productivity at least in-duce a negative correlation between unemploy-ment and vacancies separation shocks causeboth variables to increase which tends to gen-erate a positive correlation between the twovariables Similar results obtain in some otherspecial cases

Section II D calibrates the stochastic model tomatch US data along as many dimensions aspossible and Section II E presents the resultsThe exercise confirms the quantitative predic-tions of the comparative statics If the economyis hit only by productivity shocks it movesalong a downward-sloping Beveridge curve butempirically plausible movements in labor pro-ductivity result in tiny fluctuations in the v-uratio Moreover labor productivity is perfectlycorrelated with the v-u ratio indicating that themodel has almost no internal propagation mech-anism If the economy is hit only by separationshocks the v-u ratio is stable in the face oflarge unemployment fluctuations so vacanciesare countercyclical Equivalently the model-generated Beveridge curve is upward-sloping

Section II F explores the extent to which theNash bargaining solution is responsible forthese results First I examine the behavior of









A Unemployment









u v vu f s p


u 1 0894 0971 0949 0709 0408v mdash 1 0975 0897 0684 0364







B Vacancies












C Job-Finding Rate




ut 1 ut 1 ft ut 1s









(1) ft 1 ut 1 ut 1

s

ut














D Separation Rate




ut 1s st et 1

12

ft


(2) st ut 1

s

et 1 12

ft
























A Model


















psVps Vps




(5) c qps 1 Vps


(6)r s

qps ps

1 p z

c ps

1

qps








r s

qps ps 1

p z

c


r s fps

r s1 ps fps



s

r s1 ps fps



vps s1 ups

ups 11



r s

ps 1

p z

c


r s







r s

p

1 p z

c

pp


r s

p 1

p z

c


D Calibration




dy ydt db




f q 1














MODEL

Parameter

Source of shocks








E Results





u v vu f p



u 1 0927 0958 0958 0958(0020) (0012) (0012) (0012)

v mdash 1 0996 0996 0995(0001) (0001) (0001)













u v vu f s



u 1 0999 0906 0906 0908(0000) (0017) (0017) (0017)

v mdash 1 0887 0887 0888(0020) (0020) (0021)








F Wages



(7) wps 1 z p cps





u v vu f s p



u 1 0427 0964 0964 0964 0964(0068) (0011) (0011) (0011) (0011)

v mdash 1 0650 0650 0649 0648(0042) (0042) (0042) (0042)















u v vu f w



u 1 0915 0949 0949 0818(0045) (0032) (0032) (0112)

v mdash 1 0995 0995 0827(0001) (0001) (0128)







rW p s u max

zu p1 u cu

Wup s us1 u uf

psWp s u Wp s u)



r s

fps ps1

fps

ps fps

p z

c ps 1

f ps



r s

qps ps

1 p z

c ps 1

qps

















V Conclusion











SURPLUS (4)



psUps Ups


psEps Eps


psJps Jps





sVps psVps Vps


Jps1 which gives

(12)Eps Ups

Vps

Jps

1




EQUATION


wps p r s 1 Vps

ps1 Vps


wps p r s c

qps ps

c








y Y 13n n 1

0 n 1 n



1

2 1 y

n1

2 1 y

n



yt h ytyt h

nyt hyt


Varyt h ytyt

[(y(th)y(t))2y(t)]

yt h ytyt2


Varyt h ytyt h2 h2


dy ydt dx





REFERENCES




















































A Unemployment









u v vu f s p


u 1 0894 0971 0949 0709 0408v mdash 1 0975 0897 0684 0364







B Vacancies












C Job-Finding Rate




ut 1 ut 1 ft ut 1s









(1) ft 1 ut 1 ut 1

s

ut














D Separation Rate




ut 1s st et 1

12

ft


(2) st ut 1

s

et 1 12

ft
























A Model


















psVps Vps




(5) c qps 1 Vps


(6)r s

qps ps

1 p z

c ps

1

qps








r s

qps ps 1

p z

c


r s fps

r s1 ps fps



s

r s1 ps fps



vps s1 ups

ups 11



r s

ps 1

p z

c


r s







r s

p

1 p z

c

pp


r s

p 1

p z

c


D Calibration




dy ydt db




f q 1














MODEL

Parameter

Source of shocks








E Results





u v vu f p



u 1 0927 0958 0958 0958(0020) (0012) (0012) (0012)

v mdash 1 0996 0996 0995(0001) (0001) (0001)













u v vu f s



u 1 0999 0906 0906 0908(0000) (0017) (0017) (0017)

v mdash 1 0887 0887 0888(0020) (0020) (0021)








F Wages



(7) wps 1 z p cps





u v vu f s p



u 1 0427 0964 0964 0964 0964(0068) (0011) (0011) (0011) (0011)

v mdash 1 0650 0650 0649 0648(0042) (0042) (0042) (0042)















u v vu f w



u 1 0915 0949 0949 0818(0045) (0032) (0032) (0112)

v mdash 1 0995 0995 0827(0001) (0001) (0128)







rW p s u max

zu p1 u cu

Wup s us1 u uf

psWp s u Wp s u)



r s

fps ps1

fps

ps fps

p z

c ps 1

f ps



r s

qps ps

1 p z

c ps 1

qps

















V Conclusion











SURPLUS (4)



psUps Ups


psEps Eps


psJps Jps





sVps psVps Vps


Jps1 which gives

(12)Eps Ups

Vps

Jps

1




EQUATION


wps p r s 1 Vps

ps1 Vps


wps p r s c

qps ps

c








y Y 13n n 1

0 n 1 n



1

2 1 y

n1

2 1 y

n



yt h ytyt h

nyt hyt


Varyt h ytyt

[(y(th)y(t))2y(t)]

yt h ytyt2


Varyt h ytyt h2 h2


dy ydt dx





REFERENCES


















































u v vu f s p


u 1 0894 0971 0949 0709 0408v mdash 1 0975 0897 0684 0364







B Vacancies












C Job-Finding Rate




ut 1 ut 1 ft ut 1s









(1) ft 1 ut 1 ut 1

s

ut














D Separation Rate




ut 1s st et 1

12

ft


(2) st ut 1

s

et 1 12

ft
























A Model


















psVps Vps




(5) c qps 1 Vps


(6)r s

qps ps

1 p z

c ps

1

qps








r s

qps ps 1

p z

c


r s fps

r s1 ps fps



s

r s1 ps fps



vps s1 ups

ups 11



r s

ps 1

p z

c


r s







r s

p

1 p z

c

pp


r s

p 1

p z

c


D Calibration




dy ydt db




f q 1














MODEL

Parameter

Source of shocks








E Results





u v vu f p



u 1 0927 0958 0958 0958(0020) (0012) (0012) (0012)

v mdash 1 0996 0996 0995(0001) (0001) (0001)













u v vu f s



u 1 0999 0906 0906 0908(0000) (0017) (0017) (0017)

v mdash 1 0887 0887 0888(0020) (0020) (0021)








F Wages



(7) wps 1 z p cps





u v vu f s p



u 1 0427 0964 0964 0964 0964(0068) (0011) (0011) (0011) (0011)

v mdash 1 0650 0650 0649 0648(0042) (0042) (0042) (0042)















u v vu f w



u 1 0915 0949 0949 0818(0045) (0032) (0032) (0112)

v mdash 1 0995 0995 0827(0001) (0001) (0128)







rW p s u max

zu p1 u cu

Wup s us1 u uf

psWp s u Wp s u)



r s

fps ps1

fps

ps fps

p z

c ps 1

f ps



r s

qps ps

1 p z

c ps 1

qps

















V Conclusion











SURPLUS (4)



psUps Ups


psEps Eps


psJps Jps





sVps psVps Vps


Jps1 which gives

(12)Eps Ups

Vps

Jps

1




EQUATION


wps p r s 1 Vps

ps1 Vps


wps p r s c

qps ps

c








y Y 13n n 1

0 n 1 n



1

2 1 y

n1

2 1 y

n



yt h ytyt h

nyt hyt


Varyt h ytyt

[(y(th)y(t))2y(t)]

yt h ytyt2


Varyt h ytyt h2 h2


dy ydt dx





REFERENCES














































B Vacancies












C Job-Finding Rate




ut 1 ut 1 ft ut 1s









(1) ft 1 ut 1 ut 1

s

ut














D Separation Rate




ut 1s st et 1

12

ft


(2) st ut 1

s

et 1 12

ft
























A Model


















psVps Vps




(5) c qps 1 Vps


(6)r s

qps ps

1 p z

c ps

1

qps








r s

qps ps 1

p z

c


r s fps

r s1 ps fps



s

r s1 ps fps



vps s1 ups

ups 11



r s

ps 1

p z

c


r s







r s

p

1 p z

c

pp


r s

p 1

p z

c


D Calibration




dy ydt db




f q 1














MODEL

Parameter

Source of shocks








E Results





u v vu f p



u 1 0927 0958 0958 0958(0020) (0012) (0012) (0012)

v mdash 1 0996 0996 0995(0001) (0001) (0001)













u v vu f s



u 1 0999 0906 0906 0908(0000) (0017) (0017) (0017)

v mdash 1 0887 0887 0888(0020) (0020) (0021)








F Wages



(7) wps 1 z p cps





u v vu f s p



u 1 0427 0964 0964 0964 0964(0068) (0011) (0011) (0011) (0011)

v mdash 1 0650 0650 0649 0648(0042) (0042) (0042) (0042)















u v vu f w



u 1 0915 0949 0949 0818(0045) (0032) (0032) (0112)

v mdash 1 0995 0995 0827(0001) (0001) (0128)







rW p s u max

zu p1 u cu

Wup s us1 u uf

psWp s u Wp s u)



r s

fps ps1

fps

ps fps

p z

c ps 1

f ps



r s

qps ps

1 p z

c ps 1

qps

















V Conclusion











SURPLUS (4)



psUps Ups


psEps Eps


psJps Jps





sVps psVps Vps


Jps1 which gives

(12)Eps Ups

Vps

Jps

1




EQUATION


wps p r s 1 Vps

ps1 Vps


wps p r s c

qps ps

c








y Y 13n n 1

0 n 1 n



1

2 1 y

n1

2 1 y

n



yt h ytyt h

nyt hyt


Varyt h ytyt

[(y(th)y(t))2y(t)]

yt h ytyt2


Varyt h ytyt h2 h2


dy ydt dx





REFERENCES














































C Job-Finding Rate




ut 1 ut 1 ft ut 1s









(1) ft 1 ut 1 ut 1

s

ut














D Separation Rate




ut 1s st et 1

12

ft


(2) st ut 1

s

et 1 12

ft
























A Model


















psVps Vps




(5) c qps 1 Vps


(6)r s

qps ps

1 p z

c ps

1

qps








r s

qps ps 1

p z

c


r s fps

r s1 ps fps



s

r s1 ps fps



vps s1 ups

ups 11



r s

ps 1

p z

c


r s







r s

p

1 p z

c

pp


r s

p 1

p z

c


D Calibration




dy ydt db




f q 1














MODEL

Parameter

Source of shocks








E Results





u v vu f p



u 1 0927 0958 0958 0958(0020) (0012) (0012) (0012)

v mdash 1 0996 0996 0995(0001) (0001) (0001)













u v vu f s



u 1 0999 0906 0906 0908(0000) (0017) (0017) (0017)

v mdash 1 0887 0887 0888(0020) (0020) (0021)








F Wages



(7) wps 1 z p cps





u v vu f s p



u 1 0427 0964 0964 0964 0964(0068) (0011) (0011) (0011) (0011)

v mdash 1 0650 0650 0649 0648(0042) (0042) (0042) (0042)















u v vu f w



u 1 0915 0949 0949 0818(0045) (0032) (0032) (0112)

v mdash 1 0995 0995 0827(0001) (0001) (0128)







rW p s u max

zu p1 u cu

Wup s us1 u uf

psWp s u Wp s u)



r s

fps ps1

fps

ps fps

p z

c ps 1

f ps



r s

qps ps

1 p z

c ps 1

qps

















V Conclusion











SURPLUS (4)



psUps Ups


psEps Eps


psJps Jps





sVps psVps Vps


Jps1 which gives

(12)Eps Ups

Vps

Jps

1




EQUATION


wps p r s 1 Vps

ps1 Vps


wps p r s c

qps ps

c








y Y 13n n 1

0 n 1 n



1

2 1 y

n1

2 1 y

n



yt h ytyt h

nyt hyt


Varyt h ytyt

[(y(th)y(t))2y(t)]

yt h ytyt2


Varyt h ytyt h2 h2


dy ydt dx





REFERENCES














































(1) ft 1 ut 1 ut 1

s

ut














D Separation Rate




ut 1s st et 1

12

ft


(2) st ut 1

s

et 1 12

ft
























A Model


















psVps Vps




(5) c qps 1 Vps


(6)r s

qps ps

1 p z

c ps

1

qps








r s

qps ps 1

p z

c


r s fps

r s1 ps fps



s

r s1 ps fps



vps s1 ups

ups 11



r s

ps 1

p z

c


r s







r s

p

1 p z

c

pp


r s

p 1

p z

c


D Calibration




dy ydt db




f q 1














MODEL

Parameter

Source of shocks








E Results





u v vu f p



u 1 0927 0958 0958 0958(0020) (0012) (0012) (0012)

v mdash 1 0996 0996 0995(0001) (0001) (0001)













u v vu f s



u 1 0999 0906 0906 0908(0000) (0017) (0017) (0017)

v mdash 1 0887 0887 0888(0020) (0020) (0021)








F Wages



(7) wps 1 z p cps





u v vu f s p



u 1 0427 0964 0964 0964 0964(0068) (0011) (0011) (0011) (0011)

v mdash 1 0650 0650 0649 0648(0042) (0042) (0042) (0042)















u v vu f w



u 1 0915 0949 0949 0818(0045) (0032) (0032) (0112)

v mdash 1 0995 0995 0827(0001) (0001) (0128)







rW p s u max

zu p1 u cu

Wup s us1 u uf

psWp s u Wp s u)



r s

fps ps1

fps

ps fps

p z

c ps 1

f ps



r s

qps ps

1 p z

c ps 1

qps

















V Conclusion











SURPLUS (4)



psUps Ups


psEps Eps


psJps Jps





sVps psVps Vps


Jps1 which gives

(12)Eps Ups

Vps

Jps

1




EQUATION


wps p r s 1 Vps

ps1 Vps


wps p r s c

qps ps

c








y Y 13n n 1

0 n 1 n



1

2 1 y

n1

2 1 y

n



yt h ytyt h

nyt hyt


Varyt h ytyt

[(y(th)y(t))2y(t)]

yt h ytyt2


Varyt h ytyt h2 h2


dy ydt dx





REFERENCES















































D Separation Rate




ut 1s st et 1

12

ft


(2) st ut 1

s

et 1 12

ft
























A Model


















psVps Vps




(5) c qps 1 Vps


(6)r s

qps ps

1 p z

c ps

1

qps








r s

qps ps 1

p z

c


r s fps

r s1 ps fps



s

r s1 ps fps



vps s1 ups

ups 11



r s

ps 1

p z

c


r s







r s

p

1 p z

c

pp


r s

p 1

p z

c


D Calibration




dy ydt db




f q 1














MODEL

Parameter

Source of shocks








E Results





u v vu f p



u 1 0927 0958 0958 0958(0020) (0012) (0012) (0012)

v mdash 1 0996 0996 0995(0001) (0001) (0001)













u v vu f s



u 1 0999 0906 0906 0908(0000) (0017) (0017) (0017)

v mdash 1 0887 0887 0888(0020) (0020) (0021)








F Wages



(7) wps 1 z p cps





u v vu f s p



u 1 0427 0964 0964 0964 0964(0068) (0011) (0011) (0011) (0011)

v mdash 1 0650 0650 0649 0648(0042) (0042) (0042) (0042)















u v vu f w



u 1 0915 0949 0949 0818(0045) (0032) (0032) (0112)

v mdash 1 0995 0995 0827(0001) (0001) (0128)







rW p s u max

zu p1 u cu

Wup s us1 u uf

psWp s u Wp s u)



r s

fps ps1

fps

ps fps

p z

c ps 1

f ps



r s

qps ps

1 p z

c ps 1

qps

















V Conclusion











SURPLUS (4)



psUps Ups


psEps Eps


psJps Jps





sVps psVps Vps


Jps1 which gives

(12)Eps Ups

Vps

Jps

1




EQUATION


wps p r s 1 Vps

ps1 Vps


wps p r s c

qps ps

c








y Y 13n n 1

0 n 1 n



1

2 1 y

n1

2 1 y

n



yt h ytyt h

nyt hyt


Varyt h ytyt

[(y(th)y(t))2y(t)]

yt h ytyt2


Varyt h ytyt h2 h2


dy ydt dx





REFERENCES





























































A Model


















psVps Vps




(5) c qps 1 Vps


(6)r s

qps ps

1 p z

c ps

1

qps








r s

qps ps 1

p z

c


r s fps

r s1 ps fps



s

r s1 ps fps



vps s1 ups

ups 11



r s

ps 1

p z

c


r s







r s

p

1 p z

c

pp


r s

p 1

p z

c


D Calibration




dy ydt db




f q 1














MODEL

Parameter

Source of shocks








E Results





u v vu f p



u 1 0927 0958 0958 0958(0020) (0012) (0012) (0012)

v mdash 1 0996 0996 0995(0001) (0001) (0001)













u v vu f s



u 1 0999 0906 0906 0908(0000) (0017) (0017) (0017)

v mdash 1 0887 0887 0888(0020) (0020) (0021)








F Wages



(7) wps 1 z p cps





u v vu f s p



u 1 0427 0964 0964 0964 0964(0068) (0011) (0011) (0011) (0011)

v mdash 1 0650 0650 0649 0648(0042) (0042) (0042) (0042)















u v vu f w



u 1 0915 0949 0949 0818(0045) (0032) (0032) (0112)

v mdash 1 0995 0995 0827(0001) (0001) (0128)







rW p s u max

zu p1 u cu

Wup s us1 u uf

psWp s u Wp s u)



r s

fps ps1

fps

ps fps

p z

c ps 1

f ps



r s

qps ps

1 p z

c ps 1

qps

















V Conclusion











SURPLUS (4)



psUps Ups


psEps Eps


psJps Jps





sVps psVps Vps


Jps1 which gives

(12)Eps Ups

Vps

Jps

1




EQUATION


wps p r s 1 Vps

ps1 Vps


wps p r s c

qps ps

c








y Y 13n n 1

0 n 1 n



1

2 1 y

n1

2 1 y

n



yt h ytyt h

nyt hyt


Varyt h ytyt

[(y(th)y(t))2y(t)]

yt h ytyt2


Varyt h ytyt h2 h2


dy ydt dx





REFERENCES





















































psVps Vps




(5) c qps 1 Vps


(6)r s

qps ps

1 p z

c ps

1

qps








r s

qps ps 1

p z

c


r s fps

r s1 ps fps



s

r s1 ps fps



vps s1 ups

ups 11



r s

ps 1

p z

c


r s







r s

p

1 p z

c

pp


r s

p 1

p z

c


D Calibration




dy ydt db




f q 1














MODEL

Parameter

Source of shocks








E Results





u v vu f p



u 1 0927 0958 0958 0958(0020) (0012) (0012) (0012)

v mdash 1 0996 0996 0995(0001) (0001) (0001)













u v vu f s



u 1 0999 0906 0906 0908(0000) (0017) (0017) (0017)

v mdash 1 0887 0887 0888(0020) (0020) (0021)








F Wages



(7) wps 1 z p cps





u v vu f s p



u 1 0427 0964 0964 0964 0964(0068) (0011) (0011) (0011) (0011)

v mdash 1 0650 0650 0649 0648(0042) (0042) (0042) (0042)















u v vu f w



u 1 0915 0949 0949 0818(0045) (0032) (0032) (0112)

v mdash 1 0995 0995 0827(0001) (0001) (0128)







rW p s u max

zu p1 u cu

Wup s us1 u uf

psWp s u Wp s u)



r s

fps ps1

fps

ps fps

p z

c ps 1

f ps



r s

qps ps

1 p z

c ps 1

qps

















V Conclusion











SURPLUS (4)



psUps Ups


psEps Eps


psJps Jps





sVps psVps Vps


Jps1 which gives

(12)Eps Ups

Vps

Jps

1




EQUATION


wps p r s 1 Vps

ps1 Vps


wps p r s c

qps ps

c








y Y 13n n 1

0 n 1 n



1

2 1 y

n1

2 1 y

n



yt h ytyt h

nyt hyt


Varyt h ytyt

[(y(th)y(t))2y(t)]

yt h ytyt2


Varyt h ytyt h2 h2


dy ydt dx





REFERENCES














































r s

qps ps 1

p z

c


r s fps

r s1 ps fps



s

r s1 ps fps



vps s1 ups

ups 11



r s

ps 1

p z

c


r s







r s

p

1 p z

c

pp


r s

p 1

p z

c


D Calibration




dy ydt db




f q 1














MODEL

Parameter

Source of shocks








E Results





u v vu f p



u 1 0927 0958 0958 0958(0020) (0012) (0012) (0012)

v mdash 1 0996 0996 0995(0001) (0001) (0001)













u v vu f s



u 1 0999 0906 0906 0908(0000) (0017) (0017) (0017)

v mdash 1 0887 0887 0888(0020) (0020) (0021)








F Wages



(7) wps 1 z p cps





u v vu f s p



u 1 0427 0964 0964 0964 0964(0068) (0011) (0011) (0011) (0011)

v mdash 1 0650 0650 0649 0648(0042) (0042) (0042) (0042)















u v vu f w



u 1 0915 0949 0949 0818(0045) (0032) (0032) (0112)

v mdash 1 0995 0995 0827(0001) (0001) (0128)







rW p s u max

zu p1 u cu

Wup s us1 u uf

psWp s u Wp s u)



r s

fps ps1

fps

ps fps

p z

c ps 1

f ps



r s

qps ps

1 p z

c ps 1

qps

















V Conclusion











SURPLUS (4)



psUps Ups


psEps Eps


psJps Jps





sVps psVps Vps


Jps1 which gives

(12)Eps Ups

Vps

Jps

1




EQUATION


wps p r s 1 Vps

ps1 Vps


wps p r s c

qps ps

c








y Y 13n n 1

0 n 1 n



1

2 1 y

n1

2 1 y

n



yt h ytyt h

nyt hyt


Varyt h ytyt

[(y(th)y(t))2y(t)]

yt h ytyt2


Varyt h ytyt h2 h2


dy ydt dx





REFERENCES















































r s

p

1 p z

c

pp


r s

p 1

p z

c


D Calibration




dy ydt db




f q 1














MODEL

Parameter

Source of shocks








E Results





u v vu f p



u 1 0927 0958 0958 0958(0020) (0012) (0012) (0012)

v mdash 1 0996 0996 0995(0001) (0001) (0001)













u v vu f s



u 1 0999 0906 0906 0908(0000) (0017) (0017) (0017)

v mdash 1 0887 0887 0888(0020) (0020) (0021)








F Wages



(7) wps 1 z p cps





u v vu f s p



u 1 0427 0964 0964 0964 0964(0068) (0011) (0011) (0011) (0011)

v mdash 1 0650 0650 0649 0648(0042) (0042) (0042) (0042)















u v vu f w



u 1 0915 0949 0949 0818(0045) (0032) (0032) (0112)

v mdash 1 0995 0995 0827(0001) (0001) (0128)







rW p s u max

zu p1 u cu

Wup s us1 u uf

psWp s u Wp s u)



r s

fps ps1

fps

ps fps

p z

c ps 1

f ps



r s

qps ps

1 p z

c ps 1

qps

















V Conclusion











SURPLUS (4)



psUps Ups


psEps Eps


psJps Jps





sVps psVps Vps


Jps1 which gives

(12)Eps Ups

Vps

Jps

1




EQUATION


wps p r s 1 Vps

ps1 Vps


wps p r s c

qps ps

c








y Y 13n n 1

0 n 1 n



1

2 1 y

n1

2 1 y

n



yt h ytyt h

nyt hyt


Varyt h ytyt

[(y(th)y(t))2y(t)]

yt h ytyt2


Varyt h ytyt h2 h2


dy ydt dx





REFERENCES























































MODEL

Parameter

Source of shocks








E Results





u v vu f p



u 1 0927 0958 0958 0958(0020) (0012) (0012) (0012)

v mdash 1 0996 0996 0995(0001) (0001) (0001)













u v vu f s



u 1 0999 0906 0906 0908(0000) (0017) (0017) (0017)

v mdash 1 0887 0887 0888(0020) (0020) (0021)








F Wages



(7) wps 1 z p cps





u v vu f s p



u 1 0427 0964 0964 0964 0964(0068) (0011) (0011) (0011) (0011)

v mdash 1 0650 0650 0649 0648(0042) (0042) (0042) (0042)















u v vu f w



u 1 0915 0949 0949 0818(0045) (0032) (0032) (0112)

v mdash 1 0995 0995 0827(0001) (0001) (0128)







rW p s u max

zu p1 u cu

Wup s us1 u uf

psWp s u Wp s u)



r s

fps ps1

fps

ps fps

p z

c ps 1

f ps



r s

qps ps

1 p z

c ps 1

qps

















V Conclusion











SURPLUS (4)



psUps Ups


psEps Eps


psJps Jps





sVps psVps Vps


Jps1 which gives

(12)Eps Ups

Vps

Jps

1




EQUATION


wps p r s 1 Vps

ps1 Vps


wps p r s c

qps ps

c








y Y 13n n 1

0 n 1 n



1

2 1 y

n1

2 1 y

n



yt h ytyt h

nyt hyt


Varyt h ytyt

[(y(th)y(t))2y(t)]

yt h ytyt2


Varyt h ytyt h2 h2


dy ydt dx





REFERENCES















































E Results





u v vu f p



u 1 0927 0958 0958 0958(0020) (0012) (0012) (0012)

v mdash 1 0996 0996 0995(0001) (0001) (0001)













u v vu f s



u 1 0999 0906 0906 0908(0000) (0017) (0017) (0017)

v mdash 1 0887 0887 0888(0020) (0020) (0021)








F Wages



(7) wps 1 z p cps





u v vu f s p



u 1 0427 0964 0964 0964 0964(0068) (0011) (0011) (0011) (0011)

v mdash 1 0650 0650 0649 0648(0042) (0042) (0042) (0042)















u v vu f w



u 1 0915 0949 0949 0818(0045) (0032) (0032) (0112)

v mdash 1 0995 0995 0827(0001) (0001) (0128)







rW p s u max

zu p1 u cu

Wup s us1 u uf

psWp s u Wp s u)



r s

fps ps1

fps

ps fps

p z

c ps 1

f ps



r s

qps ps

1 p z

c ps 1

qps

















V Conclusion











SURPLUS (4)



psUps Ups


psEps Eps


psJps Jps





sVps psVps Vps


Jps1 which gives

(12)Eps Ups

Vps

Jps

1




EQUATION


wps p r s 1 Vps

ps1 Vps


wps p r s c

qps ps

c








y Y 13n n 1

0 n 1 n



1

2 1 y

n1

2 1 y

n



yt h ytyt h

nyt hyt


Varyt h ytyt

[(y(th)y(t))2y(t)]

yt h ytyt2


Varyt h ytyt h2 h2


dy ydt dx





REFERENCES




















































u v vu f s



u 1 0999 0906 0906 0908(0000) (0017) (0017) (0017)

v mdash 1 0887 0887 0888(0020) (0020) (0021)








F Wages



(7) wps 1 z p cps





u v vu f s p



u 1 0427 0964 0964 0964 0964(0068) (0011) (0011) (0011) (0011)

v mdash 1 0650 0650 0649 0648(0042) (0042) (0042) (0042)















u v vu f w



u 1 0915 0949 0949 0818(0045) (0032) (0032) (0112)

v mdash 1 0995 0995 0827(0001) (0001) (0128)







rW p s u max

zu p1 u cu

Wup s us1 u uf

psWp s u Wp s u)



r s

fps ps1

fps

ps fps

p z

c ps 1

f ps



r s

qps ps

1 p z

c ps 1

qps

















V Conclusion











SURPLUS (4)



psUps Ups


psEps Eps


psJps Jps





sVps psVps Vps


Jps1 which gives

(12)Eps Ups

Vps

Jps

1




EQUATION


wps p r s 1 Vps

ps1 Vps


wps p r s c

qps ps

c








y Y 13n n 1

0 n 1 n



1

2 1 y

n1

2 1 y

n



yt h ytyt h

nyt hyt


Varyt h ytyt

[(y(th)y(t))2y(t)]

yt h ytyt2


Varyt h ytyt h2 h2


dy ydt dx





REFERENCES















































F Wages



(7) wps 1 z p cps





u v vu f s p



u 1 0427 0964 0964 0964 0964(0068) (0011) (0011) (0011) (0011)

v mdash 1 0650 0650 0649 0648(0042) (0042) (0042) (0042)















u v vu f w



u 1 0915 0949 0949 0818(0045) (0032) (0032) (0112)

v mdash 1 0995 0995 0827(0001) (0001) (0128)







rW p s u max

zu p1 u cu

Wup s us1 u uf

psWp s u Wp s u)



r s

fps ps1

fps

ps fps

p z

c ps 1

f ps



r s

qps ps

1 p z

c ps 1

qps

















V Conclusion











SURPLUS (4)



psUps Ups


psEps Eps


psJps Jps





sVps psVps Vps


Jps1 which gives

(12)Eps Ups

Vps

Jps

1




EQUATION


wps p r s 1 Vps

ps1 Vps


wps p r s c

qps ps

c








y Y 13n n 1

0 n 1 n



1

2 1 y

n1

2 1 y

n



yt h ytyt h

nyt hyt


Varyt h ytyt

[(y(th)y(t))2y(t)]

yt h ytyt2


Varyt h ytyt h2 h2


dy ydt dx





REFERENCES





















































u v vu f w



u 1 0915 0949 0949 0818(0045) (0032) (0032) (0112)

v mdash 1 0995 0995 0827(0001) (0001) (0128)







rW p s u max

zu p1 u cu

Wup s us1 u uf

psWp s u Wp s u)



r s

fps ps1

fps

ps fps

p z

c ps 1

f ps



r s

qps ps

1 p z

c ps 1

qps

















V Conclusion











SURPLUS (4)



psUps Ups


psEps Eps


psJps Jps





sVps psVps Vps


Jps1 which gives

(12)Eps Ups

Vps

Jps

1




EQUATION


wps p r s 1 Vps

ps1 Vps


wps p r s c

qps ps

c








y Y 13n n 1

0 n 1 n



1

2 1 y

n1

2 1 y

n



yt h ytyt h

nyt hyt


Varyt h ytyt

[(y(th)y(t))2y(t)]

yt h ytyt2


Varyt h ytyt h2 h2


dy ydt dx





REFERENCES














































rW p s u max

zu p1 u cu

Wup s us1 u uf

psWp s u Wp s u)



r s

fps ps1

fps

ps fps

p z

c ps 1

f ps



r s

qps ps

1 p z

c ps 1

qps

















V Conclusion











SURPLUS (4)



psUps Ups


psEps Eps


psJps Jps





sVps psVps Vps


Jps1 which gives

(12)Eps Ups

Vps

Jps

1




EQUATION


wps p r s 1 Vps

ps1 Vps


wps p r s c

qps ps

c








y Y 13n n 1

0 n 1 n



1

2 1 y

n1

2 1 y

n



yt h ytyt h

nyt hyt


Varyt h ytyt

[(y(th)y(t))2y(t)]

yt h ytyt2


Varyt h ytyt h2 h2


dy ydt dx





REFERENCES























































V Conclusion











SURPLUS (4)



psUps Ups


psEps Eps


psJps Jps





sVps psVps Vps


Jps1 which gives

(12)Eps Ups

Vps

Jps

1




EQUATION


wps p r s 1 Vps

ps1 Vps


wps p r s c

qps ps

c








y Y 13n n 1

0 n 1 n



1

2 1 y

n1

2 1 y

n



yt h ytyt h

nyt hyt


Varyt h ytyt

[(y(th)y(t))2y(t)]

yt h ytyt2


Varyt h ytyt h2 h2


dy ydt dx





REFERENCES















































SURPLUS (4)



psUps Ups


psEps Eps


psJps Jps





sVps psVps Vps


Jps1 which gives

(12)Eps Ups

Vps

Jps

1




EQUATION


wps p r s 1 Vps

ps1 Vps


wps p r s c

qps ps

c








y Y 13n n 1

0 n 1 n



1

2 1 y

n1

2 1 y

n



yt h ytyt h

nyt hyt


Varyt h ytyt

[(y(th)y(t))2y(t)]

yt h ytyt2


Varyt h ytyt h2 h2


dy ydt dx





REFERENCES

















































y Y 13n n 1

0 n 1 n



1

2 1 y

n1

2 1 y

n



yt h ytyt h

nyt hyt


Varyt h ytyt

[(y(th)y(t))2y(t)]

yt h ytyt2


Varyt h ytyt h2 h2


dy ydt dx





REFERENCES













































REFERENCES













































The Cyclical Behavior of Equilibrium Unemployment...

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