The Cyclical Behavior of Equilibrium Unemployment and ...kkasa/Shimer_AER05.pdfThe Cyclical Behavior of Equilibrium Unemployment and Vacancies By ROBERT SHIMER* This paper argues that

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, 1994;Pissarides, 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 insurance,firing 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 elements,unemployment and vacancies, which are bothhighly variable and strongly negatively corre-lated in U.S. 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 cheap.1 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 workers’ threat pointin wage bargaining, and therefore raising the

* Department of Economics, University of Chicago,1126 East 59th Street, Chicago IL 60637 (e-mail:[email protected]). A previous version of this paperwas entitled “Equilibrium Unemployment Fluctuations.” Ithank Daron Acemoglu, Robert Barro, Olivier Blanchard,V. V. Chari, Joao Gomes, Robert Hall, Dale Mortensen,Christopher 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 jobs.Higher 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 U.S. businesscycle facts: unemployment u is strongly coun-tercyclical, vacancies v are equally strongly pro-cyclical, and the correlation between the twovariables is �0.89 at business cycle frequen-cies. As a result, the v-u ratio is procyclical andvolatile, with a standard deviation around itstrend equal to 0.38 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(u,v), as inPissarides (1985), the job finding rate is f �m(u,v)/u, an increasing function of the v-u ratio.I use unemployment-duration data to measurethe job-finding rate directly. The standard devi-ation of fluctuations in the job-finding ratearound trend is 0.12 log points and the correla-tion with the v-u ratio is 0.95. Finally I look atthe two proposed impulses. The separation rateis less correlated with the cycle and moderatelyvolatile, with a standard deviation about trendequal to 0.08 log points. Average labor produc-tivity is weakly procyclical and even more sta-ble, with a standard deviation about trend of0.02 log points.

In Section II, I extend the Pissarides’s (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 equilibrium,there is only one real economic decision: firms’

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 “leisure” 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 utility.The separation rate has an even smaller impacton the v-u ratio, with an elasticity of �0.1according to the comparative statics. Moreover,while 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 U.S. data along as many dimensions aspossible, and Section II E presents the results.The 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 vacancies,again 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 workers’bargaining 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 0.01 log points aroundtrend.

Section III provides another angle fromwhich to view the model’s 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 workers’ 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 markets.Finally, 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 important—butstandard—feature of the search and matchingframework that I exploit throughout this paper:workers 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. Lucas,Jr., 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 inelastic.It 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. U.S. 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 2003,5.67 percent of the U.S. labor force was out ofwork, available for work, and actively seekingwork. This time series exhibits considerabletemporal variation, falling to as low as 2.6 per-cent in 1953 and 3.4 percent in 1968 and 1969,but reaching 10.8 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 quarterlydata.2 The difference between log unemploy-ment and its trend has a standard deviation of0.19, so unemployment is often as much as 38percent above or below trend. Detrended unem-ployment also exhibits considerable persistence,with quarterly autocorrelation 0.94.

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 conditions.Chinhui 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 activity,worsening 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 utilized.3 Indirect evidencecomes from estimates of matching functions,which universally find that an increase in un-employment is associated with an increase in

3 Shimer (2004b) discusses this evidence in detail.

TABLE 1—SUMMARY STATISTICS, QUARTERLY U.S. DATA, 1951–2003

u v v/u f s p

Standard deviation 0.190 0.202 0.382 0.118 0.075 0.020Quarterly autocorrelation 0.936 0.940 0.941 0.908 0.733 0.878

u 1 �0.894 �0.971 �0.949 0.709 �0.408v — 1 0.975 0.897 �0.684 0.364

Correlation matrix v/u — — 1 0.948 �0.715 0.396f — — — 1 �0.574 0.396s — — — — 1 �0.524p — — — — — 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 duration,all 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 sector,constructed 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 U.S. UNEMPLOYMENT (IN MILLIONS)AND TREND, 1951–2003

Notes: Unemployment is a quarterly average of the season-ally adjusted monthly series constructed by the BLS fromthe CPS, survey home page http://www.bls.gov/cps/. 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: “A 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 signs,accepting applications, or using similar meth-ods.”4 Unfortunately, JOLTS began only in De-cember 2000 and comparable data had neverpreviously been collected in the United States.Although 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 4.6 million job openings. Thisnumber fell rapidly during 2001 and averagedjust 2.9 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 newspapers.5 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 firms’ 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 trend.Notably, 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 curve.The correlation of the percentage deviation ofunemployment and vacancies from trend is�0.89 between 1951 and 2003.6 Moreover, the

4 This definition comes from the Bureau of Labor Sta-tistics news release, July 30, 2002, available at http://www.bls.gov/jlt/jlt_nr1.pdf.

5 Abraham (1987) discusses this measure in detail. From1972 to 1981, Minnesota collected state-wide job vacancydata. Abraham compares this with Minnesota’s 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 U.S. Beveridge curve. Abraham and Katz(1986) argue that the negative correlation between unem-ployment and vacancies is inconsistent with Lilien’s (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 U.S. JOB VACANCIES,2000Q4–2003Q4

Notes: The solid line shows the logarithm of the number ofjob openings in millions, measured by the BLS from theJOLTS, survey homepage http://www.bls.gov/jlt, 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 0.19 and 0.20, so the product ofunemployment and vacancies is nearly acyclic.The v-u ratio is therefore extremely procyclical,with a standard deviation of 0.38 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(u,v), depending onthe number of unemployed workers u and thenumber of vacancies v. Then the probability thatany individual unemployed worker finds a job,the average transition rate from unemploymentto employment, is f � m(u,v)/u � m(1,�), where� � v/u 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 series.Second, measurement and classification errorlead a substantial overestimate of gross workerflows (John Abowd and Arnold Zellner, 1985;James 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 U.S. HELP-WANTED ADVERTISING

INDEX AND TREND, 1951–2003

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 http://research.stlouisfed.org/fred2/data/helpwant.txt. The trend is an HPfilter of the quarterly data with smoothing parameter 105.

FIGURE 4. QUARTERLY U.S. BEVERIDGE CURVE,1951–2003

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 2003.7 Figure 5 shows theresults. The monthly hazard rate averaged 0.45from 1951 to 2003. After detrending with theusual low-frequency HP filter, the correlationbetween ft and �t at quarterly frequencies is0.95, 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 U.S. data.

One can use the measured job-finding rateand v-u ratio to estimate a matching functionm(u,v). 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(u,v) 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(u,v) � �u�v1��, for someunknown parameters � and �. Again, the dataare not very informative as to whether this is areasonable restriction.8 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, 1951–2003

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 http://www.bls.gov/cps/. 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 U.S. MATCHING FUNCTION,1951–2003

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 CPS.Both 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 0.70 and0.75. With a first-order autoregressive residual,I get � � 0.72 with a standard error of 0.01.

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-ual’s unemployment duration.9 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 0.48 per month, while unemployment dura-tion data yield a job-finding rate of 0.34 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, ut�1

s , to employed workers this month,et. 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, i.e.,during 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 0.034 from 1951to 2003, so jobs last on average for about 2.5years. Fluctuations in the deviation of the logseparation rate from trend are somewhat smallerthan in the hiring rate, with a standard deviationof 0.08, and separations are countercyclical, sothe correlation with the detrended v-u ratio is�0.72.

limiting case of � � 0. When I estimate the CES functionusing nonlinear least squares and correct for first-orderautocorrelation, I get a point estimate of � � 0.06 with astandard error of 0.38.

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, 1951–2003

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 http://www.bls.gov/cps/. It is expressed as a quarterly average of monthly data.The 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 Diamond’s (1990) conclusionthat “the amplitude in fluctuations in the flowout of employment is larger than that of the flowinto employment.” This is easily reconciled.Blanchard 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(u,v) , this isconsistent with the independent evidence thatvacancies are strongly procyclical.

E. Labor Productivity

The final important empirical observation isthe weak procyclicality of labor productivity,measured 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 models,which 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 productivity.There is a positive correlation between the twotime series and some evidence that labor pro-ductivity leads the v-u ratio by about one year,with a maximum correlation of 0.56.10 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 0.5 log points about its trend leveland six times has fallen by 0.5 log points belowtrend.

10 From 1951 to 1985, the contemporaneous correlationbetween detrended labor productivity and the v-u ratio was0.57 and the peak correlation was 0.74. From 1986 to 2003,however, the contemporaneous and peak correlations arenegative, �0.37 and �0.43, 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 U.S. AVERAGE LABOR

PRODUCTIVITY AND TREND, 1951–2003

Notes: Real output per person in the non-farm businesssector, constructed by the BLS Major Sector Productivityand Costs program, survey home page http://www.bls.gov/lpc/, 1992 � 100. The trend is an HP filter of the quarterlydata with smoothing parameter 105.

FIGURE 9. QUARTERLY U.S. VACANCY-UNEMPLOYMENT

RATIO AND AVERAGE LABOR PRODUCTIVITY, 1951–2003

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 sector,constructed 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, e.g., 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 rate.The model I consider is essentially an aggregatestochastic version of Pissarides (1985, or 2000,Ch. 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 (p�,s�) is drawn from a state de-pendent distribution. Let ��p,sXp�,s� denote theexpected value of an arbitrary variable Xfollowing the next aggregate shock, condi-tional on the current state (p,s). 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-neutral,infinitely-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 (“leisure”) 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 rate,11 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)�1,1) � f(�(t))/�(t).

I assume that in every state of the world,labor 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 environment,and 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 “bargain-ing power.” 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 trade.For example, the wage may be re-bargainedwhenever the economy is hit with a shock.Alternatively, 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, �p,s.

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

(3) u̇�t� � s�t��1 � u�t�� f��p�t�,s�t� �u�t�

where (p(t) , s(t)) is the aggregate state at time t.A 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, Vp,s.

(4) rVp,s � p � �z � f��p,s �Vp,s � � sVp,s

� ��p,sVp�,s� � Vp,s�.

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(�p,s) of contacting a firm,in which event the worker would keep a fraction of the match value Vp,s. Next, there is a flowprobability s that the worker and firm separate,destroying the match value. Finally, an aggre-gate shock arrives at rate �, resulting in anexpected change in match value ��p,sVp�,s� �Vp,s.

Another critical equation for the match valuecomes from firms’ 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 Vp,s:

(5) c � q��p,s ��1 � �Vp,s .

Eliminating current and future values of Vp,sfrom (4) using (5) gives

(6)r � s � �

q��p,s �� p,s

� �1 � �p � z

c� ��p,s

1

q��p�,s� �

which implicitly defines the v-u ratio as a func-tion of the current state (p,s).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, e.g., 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 workers’ bargaining power .


quantitative results implied by this analysis.First, suppose there are no aggregate shocks, � �0.14 Then the state-contingent v-u ratio satisfies

r � s

q��p,s �� p,s � �1 � �

p � z

c.

The elasticity of the v-u ratio � with respect to“net labor productivity” p � z is

r � s � f��p,s �

�r � s��1 � ��p,s �� f��p,s �

where (�) � [0,1] is the elasticity of f(�). Thisis large only if workers’ 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 0.45 (Section I C), the elasticity(�) is approximately 0.28 (Section I C again),the average separation probability is approxi-mately 0.034 (Section I D), and the interest rateis about 0.004. Then if workers’ bargainingpower is equal to 1 � (�), the so-calledHosios (1990) condition for efficiency,15 theelasticity of the v-u ratio with respect to netlabor productivity is 1.03. Lower values of yield a slightly higher elasticity, say 1.15 when � 0.1, but only at � 0 does the elasticity ofthe v-u ratio with respect to p � z rise appre-ciably, to 1.39. It would take implausible pa-rameter values for this elasticity to exceed 2.This 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 � s��1 � ��p,s�� f��p,s�.

Substituting the same numbers into this expres-sion gives �0.10. 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 u̇ � 0.If the matching function is Cobb-Douglas,m(u,v) � �u�v1��, this implies

vp,s � �s�1 � up,s �

�up,s� � 1/�1 � ��

.

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 rates,an 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

�� p,s � �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 workers’ bargaining power . Using thesame parameter values as above, including �0.72, I obtain elasticities of 1.12 and �0.105,almost unchanged from the case with no shocks.More generally, unless is nearly equal to zero,both 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 Martingale,��pp� � 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�

�

��p�p� .

It is straightforward to verify that the v-u ratio islinear in productivity, and therefore ��p�p� � �p, i.e.,

�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-ers’ 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 U.S. 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 point.The 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 model.16 Iexpress the state variables as functions of anOrnstein-Uhlenbeck process (see Howard Tay-lor and Samuel Karlin, 1998, Section 8.5). 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 deviation.This 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 �2/2�.

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 points,which 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, workers’ 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� � 0.0165 and � � 0.004. An increase in thevolatility of productivity � has a nearly propor-tional effect on the volatility of other variables,while the persistence of the stochastic process �scarcely affects the reported results. For exam-ple, suppose I reduce � to 0.001, 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 � 0.4. Sincemean labor income in the model is 0.993, 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 quarter,and therefore set the discount rate to r � 0.012,equivalent to an annual discount factor of 0.953.The analysis in Section I D suggests a quarterlyseparation rate of s � 0.10, so jobs last for about2.5 years on average. This is comparable toAbowd and Zellner’s (1985) finding that 3.42percent of employed workers exit employmentduring a typical month between 1972 and 1982,after 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 � �0.72. This lies toward the upper end of the rangeof estimates that Petrongolo and Pissarides(2001) report. I also set workers’ bargainingpower to the same value, 0.72. Although the

reported results are insensitive to the value ofthat parameter, I show in Section III that if � �, the “Hosios (1990) Rule,” the decentralizedequilibrium maximizes a well-posed socialplanner’s 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 0.45 probability permonth, so the flow arrival rate of job offers��1�� should average approximately 1.35 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 1,which requires setting � � 1.355 and c � 0.213.

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 1,while the mean separation rate is s* � 0.10. Iset � � 0.057 and � � 0.220, 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 �p,s and then simulate the model. That

TABLE 2—PARAMETER VALUES IN SIMULATIONS OF THE

MODEL

Parameter

Source of shocks

Productivity Separation

Productivity p stochastic 1Separation rate s 0.1 stochasticDiscount rate r 0.012 0.012Value of leisure z 0.4 0.4Matching function q(�) 1.355��0.72 1.355��0.72

Bargaining power 0.72 0.72Cost of vacancy c 0.213 0.213Standard deviation � 0.0165 0.0570Autoregressive parameter � 0.004 0.220Grid 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 arrives,generate 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 1,000 “quarters” 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 10,000times, 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 shocks.Along some dimensions, notably the co-movement of unemployment and vacancies, themodel performs remarkably well. The empiricalcorrelation between these two variables is�0.89, the Beveridge curve. The model actuallyproduces a stronger negative correlation, �0.93,

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 U.S. 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 1.66 percent. Using the deterministicmodel, I argued before that the elasticity of the

TABLE 3—LABOR PRODUCTIVITY SHOCKS

u v v/u f p

Standard deviation 0.009 0.027 0.035 0.010 0.020(0.001) (0.004) (0.005) (0.001) (0.003)

Quarterly autocorrelation 0.939 0.835 0.878 0.878 0.878(0.018) (0.045) (0.035) (0.035) (0.035)

u 1 �0.927 �0.958 �0.958 �0.958(0.020) (0.012) (0.012) (0.012)

v — 1 0.996 0.996 0.995(0.001) (0.001) (0.001)

Correlation matrix v/u — — 1 1.000 0.999(0.000) (0.001)

f — — — 1 0.999(0.001)

p — — — — 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 errors—the standard deviation across 10,000 modelsimulations—are reported in parentheses. The text provides details on the stochastic process for productivity.


v-u ratio with respect to net labor productivity isabout 1.03 with this choice of parameters, giv-ing a total elasticity of � with respect to p ofapproximately 1.66 1.03 � 1.71 percent. Infact, the standard deviation of log � aroundtrend is 1.75 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 1.00. In the data,the contemporaneous correlation between thefirst two variables is 0.40 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 �0.10.According to the model, the ratio of the stan-

dard deviations is about 0.08 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 � e��sys*,both 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 0.29 (compared to 0.94 empiri-cally) and a correlation with unemployment of�0.43 (�0.89). 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 4—SEPARATION RATE SHOCKS

u v v/u f s



u 1 0.999 �0.906 �0.906 0.908(0.000) (0.017) (0.017) (0.017)

v — 1 �0.887 �0.887 0.888(0.020) (0.020) (0.021)

Correlation matrix v/u — — 1 1.000 �0.999(0.000) (0.000)

f — — — 1 �0.999(0.000)

s — — — — 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 errors—the standard deviation across 10,000 modelsimulations—are reported in parentheses. The text provides details on the stochastic process for the separation rate.


at around 10 percent of their empirical values.Smaller 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 assumption,that 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 shock.This stronger restriction pins down the wage asa function of the aggregate state, wp,s. Thisfacilitates a more detailed discussion of wages,which 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) wp,s � �1 � �z � �p � c�p,s �.

This generalizes equation (1.20) 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 firms’ 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 5—LABOR PRODUCTIVITY AND SEPARATION RATE SHOCKS

u v v/u f s p

Standard deviation 0.031 0.011 0.037 0.014 0.020 0.020(0.005) (0.001) (0.006) (0.002) (0.003) (0.003)

Quarterly autocorrelation 0.933 0.291 0.878 0.878 0.878 0.878(0.020) (0.085) (0.035) (0.035) (0.035) (0.035)

u 1 �0.427 �0.964 �0.964 0.964 �0.964(0.068) (0.011) (0.011) (0.011) (0.011)

v — 1 0.650 0.650 �0.649 0.648(0.042) (0.042) (0.042) (0.042)

Correlation matrix v/u — — 1 1.000 �1.000 0.999(0.000) (0.000) (0.001)

f — — — 1 �1.000 0.999(0.000) (0.001)

s — — — — 1 �0.999(0.001)

p — — — — — 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 standarderrors—the standard deviation across 10,000 model simulations—are 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 � 0.1, but workers’ 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 zero,this implies � �, but more generally issimply bounded between 0 and 1. I set thestandard deviation of y to � � 0.099 and themean reversion parameter to � � 0.004. Al-though this implies very modest fluctuations inwages—the standard deviation of detrended logwages, computed as in equation (7), is just0.01—the 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 workers’ 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 entirely.17 Con-

17 A number of papers examine a “competitive searcheconomy,” in which firms can commit to wages before

TABLE 6—BARGAINING POWER SHOCKS

u v v/u f w



u 1 �0.915 �0.949 �0.949 0.818(0.045) (0.032) (0.032) (0.112)

v — 1 0.995 0.995 �0.827(0.001) (0.001) (0.128)

Correlation matrix v/u — — 1 1.000 �0.838(0.000) (0.124)

f — — — 1 �0.838(0.124)

w — — — — 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 errors—the standard deviation across 10,000 modelsimulations—are reported in parentheses. The text provides details on the stochastic process for the workers’ 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 planner’s problemis represented recursively as

rW� p, s, u� � max�

�zu � p�1 � u� � cu�

� Wu�p, s, u��s�1 � u� � uf��

� ��p,s�W�p�, s�, u� � W�p, 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 u̇ � s(1 �u) � uf(�), and suddenly when an aggregateshock changes the state from (p,s) to (p�,s�) atrate �.

It is straightforward to verify that the Bell-man value W is linear in the unemployment rate,Wu(p,s,u) � �c/f�(�p,s), and the v-u ratio satis-fies

r � s � �

f��p,s �� p,s�1 �

f��p,s �

�p,s f��p,s ��

�p � z

c� ��p,s� 1

f ��p�,s� ��.

This implicitly defines the optimal �p,s, indepen-dent of the unemployment rate.

With a Cobb-Douglas matching functionm(u,v) � �u�v1��, this reduces to

r � s � �

q��p,s �� p,s

� �1 � ��p � z

c� ��p,s� 1

q��p�,s� ��

a special case of equation (6), with workers’bargaining 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-ner’s problem. Conversely, if those parametervalues describe the U.S. 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 planner’s problem isbroken. At one extreme, if unemployment andvacancies are perfect substitutes, i.e., 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(�) � min��u,�v��, thev-u ratio never strays from the efficient ratio�u /�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, 1991;Moen, 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 U.S. data. Theirpoint estimate for the elasticity of substitution is0.74, i.e., 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 s.These 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 Lilien’s (1982)sectoral shifts model. That leads them to advo-cate an alternative in which unemploymentfluctuations are driven by aggregate distur-bances, e.g., 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 vacancies,but 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. Similarly,Gomes et al. (2001) sidestep the v-u issue bylooking at a model in which the job-finding rate


is exogenous and constant, i.e., 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 “illustrative simulation,” theauthors introduce, without comment, enormousproductivity or leisure shocks into their model.Average labor productivity minus the value ofleisure p � z is approximately three times ashigh in the good state as in the bad state.18 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 impulses,Mortensen and Pissarides (1994) still deliveronly a correlation of �0.26 between unemploy-ment and vacancies, far lower than the empiri-cal value of �0.88. 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 effect.Monika 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 paper,although they do not emphasize this shortcom-ing of the search model. Finally, Hall (2003),building on an earlier version of this paper,discusses 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 vacancies.It 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 determination.An alternative wage determination mechanismthat generates more rigid wages in new jobs,measured 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 workers’bargaining power provide one such mechanism,at least in a reduced-form sense.

If the matching function is Cobb-Douglas,the 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 “too rigid.”

One way to generate more rigid wages in atheoretical model is to introduce considerationswhereby wages affect the worker turnover rate.For 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 employees’ quit rate. The distribution ofproductivity affects an individual firm’s 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 model.19 Suppose that when a worker

18 This calculation would be easy in the absence ofheterogeneity, i.e., if their parameter � were equal to zero.Then p� � z would take on three possible values: 0.022,0.075, and 0.128, 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 workers’ 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 � (0,1), 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 F.This again breaks the link between average la-bor productivity and the equilibrium v-u ratio.Exploring 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, wp,s, not on the history of the match. Ireturn to this issue at the end of this section.Define Up,s, Ep,s, and Jp,s to be the state-contingent present value of an unemployedworker, employed worker, and filled job, re-spectively. They are linked recursively by:

(8) rUp,s � z � f��p,s ��Ep,s � Up,s �

� ��p,sUp�,s� � Up,s�

(9) rEp,s � wp,s � s�Ep,s � Up,s �

� ��p,sEp�,s� � Ep,s�

(10) rJp,s � p � wp,s � sJp,s

� ��p,sJp�,s� � Jp,s�.

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(�p,s) 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 wp,s but loses her job at rate s.Equation (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 Vp,s � Jp,s � Ep,s � Up,s:

(11) rVp,s � p � z � f��p,s ��Ep,s � Up,s �

� sVp,s � ��p,sVp�,s� � Vp,s�.

In addition, the Nash bargaining solution im-plies that the wage is set so as to maximize theNash product (Ep,s � Up,s)

Jp,s1�, which gives

(12)Ep,s � Up,s

� Vp,s �

Jp,s

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-dependence,would 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 (p,s). Eliminate current and fu-ture values of J from equation (10) using equa-tion (12):

wp,s � p � �r � s � ��1 � �Vp,s

� ��p,s�1 � �Vp�,s� .

Similarly, eliminate current and future values ofV using (5):

wp,s � p ��r � s � ��c

q��p,s �� p,s

c

q��p�,s� �.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 � �n�, ��n � 1��, ... ,

0, ... , �n � 1��, n�}

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

y� � �y � �y � � with probability �

1

2 �1 �y

n��1

2 �1 �y

n��.

Note that although the step size is constant,the 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

�� y�t � h� � y�t��y�t�� h�

ny�t� � �h�y�t�.

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

Var�y�t � h� � y�t��y�t��

��[(y(t�h)�y(t))2�y(t)]

� ��y�t � h� � y�t��y�t��2.

The first term evaluates to h��2 over a suffi-ciently short time interval h, since it is equal to�2 if a shock, positive or negative, arrives andzero otherwise. The second term is (h�y(t))2,and so is negligible over a short time interval h.Thus

Var�y�t � h� � y�t��y�t�� h��2 � h�2.

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 t.This 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 size,arrival 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 as�3 0, the distribution of the innovation processx converges to a normal by the Central LimitTheorem. Equivalently, y converges to anOrnstein-Uhlenbeck process.20 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 deviation�/�2� � ��n/2. 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.


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The Cyclical Behavior of Equilibrium Unemployment and ...kkasa/Shimer_AER05.pdfThe Cyclical Behavior of Equilibrium Unemployment and Vacancies By ROBERT SHIMER* This paper argues that