The Cyclical Behavior of Equilibrium

Unemployment and Vacancies

By ROBERT SHIMER*

This paper argues that the textbook search and matching model cannot generate the
observed business-cycle-frequency fluctuations in unemployment and job vacancies
in response to shocks of a plausible magnitude. In the United States, the standard
deviation of the vacancy-unemployment ratio is almost 20 times as large as the
standard deviation of average labor productivity, while the search model predicts
that the two variables should have nearly the same volatility. A shock that changes
average labor productivity primarily alters the present value of wages, generating
only a small movement along a downward-sloping Beveridge curve (unemployment-
vacancy locus). A shock to the separation rate generates a counterfactually positive
correlation between unemployment and vacancies. In both cases, the model exhibits
virtually no propagation. (JEL E24, E32, J41, J63, J64)

In recent years, the Mortensen-Pissarides cyclical behavior of two of its central elements,
search and matching model has become the unemployment and vacancies, which are both
standard theory of equilibrium unemployment highly variable and strongly negatively corre-
(Dale Mortensen and Chris Pissarides, 1994; lated in U.S. data. Equivalently, the model can-
Pissarides, 2000). The model is attractive for a not explain the strong procyclicality of the rate
number of reasons: it offers an appealing de- at which an unemployed worker finds a job.
scription of how the labor market functions; it is I focus on two sources of shocks: changes in
analytically tractable; it has rich and generally labor productivity and changes in the separation
intuitive comparative statics; and it can easily rate of employed workers from their job. In a
be adapted to study a number of labor market one-sector model, a change in labor productiv-
policy issues, such as unemployment insurance, ity is most easily interpreted as a technology or
firing restrictions, and mandatory advanced no- supply shock. But in a multi-sector model, a
tification of layoffs. Given these successes, one preference or demand shock changes the rela-
might expect that there would be strong evi- tive price of goods, which induces a change in
dence that the model is consistent with key real labor productivity as well. Thus these
business cycle facts. On the contrary, I argue in shocks represent a broad set of possible
this paper that the model cannot explain the impulses.
An increase in labor productivity relative to
the value of nonmarket activity and to the cost
* Department of Economics, University of Chicago, of advertising a job vacancy makes unemploy-
1126 East 59th Street, Chicago IL 60637 (e-mail: ment relatively expensive and vacancies rela-
shimer@uchicago.edu). A previous version of this paper tively cheap.1 The market substitutes toward
was entitled “Equilibrium Unemployment Fluctuations.” I
thank Daron Acemoglu, Robert Barro, Olivier Blanchard, vacancies, and the increased job-finding rate
V. V. Chari, Joao Gomes, Robert Hall, Dale Mortensen, pulls down the unemployment rate, moving the
Christopher Pissarides, two anonymous referees, the editor economy along a downward sloping Beveridge
Richard Rogerson, and numerous seminar participants for curve (vacancy-unemployment or v-u locus).
comments that are incorporated throughout the paper. This
material is based upon work supported by the National
But the increase in hiring also shortens unem-
Science Foundation under grants SES-0079345 and SES- ployment duration, raising workers’ threat point
0351352. I am grateful to the Alfred P. Sloan Foundation in wage bargaining, and therefore raising the
for financial support, to the Federal Reserve Bank of Min-
neapolis for its hospitality while I worked on an early
1
version of this paper, and to Mihai Manea, and especially The interpretation in this paragraph and its sequel
Sebastian Ludmer, for excellent research assistance. builds on discussions with Robert Hall.
25
26 THE AMERICAN ECONOMIC REVIEW MARCH 2005

expected present value of wages in new jobs. choice of whether to open a new vacancy. The
Higher wages absorb most of the productivity equilibrium vacancy rate depends on the unem-
increase, eliminating the incentive for vacancy ployment rate, on labor market tightness, and on
creation. As a result, fluctuations in labor pro- the expected present value of wages in new
ductivity have little impact on the unemploy- employment relationships. Wages, in turn, are
ment, vacancy, and job-finding rates. determined by Nash bargaining, at least in new
An increase in the separation rate does not matches. In principle, the wage in old matches
affect the relative value of unemployment and may be rebargained in the face of aggregate
vacancies, and so leaves the v-u ratio essentially shocks or may be fixed by a long-term employ-
unchanged. Since the increase in separations ment contract. Section II A describes the basic
reduces employment duration, the unemploy- model, while Section II B derives a forward-
ment rate increases, and so therefore must va- looking equation for the v-u ratio in terms of
cancies. As a result, fluctuations in the model parameters.
separation rate induce a counterfactually posi- Section II C performs simple analytical com-
tive correlation between unemployment and parative statics in some special cases. For ex-
vacancies. ample, I show that the elasticity of the v-u ratio
Section I presents the relevant U.S. business with respect to the difference between labor
cycle facts: unemployment u is strongly coun- productivity and the value of nonmarket activity
tercyclical, vacancies v are equally strongly pro- or “leisure” is barely in excess of 1 for reason-
cyclical, and the correlation between the two able parameter values. To reconcile this with
variables is ⫺0.89 at business cycle frequen- the data, one must assume that the value of
cies. As a result, the v-u ratio is procyclical and leisure is nearly equal to labor productivity, so
volatile, with a standard deviation around its market work provides little incremental utility.
trend equal to 0.38 log points. To provide fur- The separation rate has an even smaller impact
ther evidence in support of this finding, I exam- on the v-u ratio, with an elasticity of ⫺0.1
ine the rate at which unemployed workers find according to the comparative statics. Moreover,
jobs. If the process of pairing workers with jobs while shocks to labor productivity at least in-
is well-described by an increasing, constant duce a negative correlation between unemploy-
returns-to-scale matching function m(u,v), as in ment and vacancies, separation shocks cause
Pissarides (1985), the job finding rate is f ⫽ both variables to increase, which tends to gen-
m(u,v)/u, an increasing function of the v-u ratio. erate a positive correlation between the two
I use unemployment-duration data to measure variables. Similar results obtain in some other
the job-finding rate directly. The standard devi- special cases.
ation of fluctuations in the job-finding rate Section II D calibrates the stochastic model to
around trend is 0.12 log points and the correla- match U.S. data along as many dimensions as
tion with the v-u ratio is 0.95. Finally I look at possible, and Section II E presents the results.
the two proposed impulses. The separation rate The exercise confirms the quantitative predic-
is less correlated with the cycle and moderately tions of the comparative statics. If the economy
volatile, with a standard deviation about trend is hit only by productivity shocks, it moves
equal to 0.08 log points. Average labor produc- along a downward-sloping Beveridge curve, but
tivity is weakly procyclical and even more sta- empirically plausible movements in labor pro-
ble, with a standard deviation about trend of ductivity result in tiny fluctuations in the v-u
0.02 log points. ratio. Moreover, labor productivity is perfectly
In Section II, I extend the Pissarides’s (1985) correlated with the v-u ratio, indicating that the
search and matching model to allow for aggre- model has almost no internal propagation mech-
gate fluctuations. I introduce two types of anism. If the economy is hit only by separation
shocks: labor productivity shocks raise output shocks, the v-u ratio is stable in the face of
in all matches but do not affect the rate at which large unemployment fluctuations, so vacancies
employed workers lose their job; and separation are countercyclical. Equivalently, the model-
shocks raise the rate at which employed workers generated Beveridge curve is upward-sloping.
become unemployed but do not affect the pro- Section II F explores the extent to which the
ductivity in surviving matches. In equilibrium, Nash bargaining solution is responsible for
there is only one real economic decision: firms’ these results. First I examine the behavior of
VOL. 95 NO. 1 SHIMER: UNEMPLOYMENT AND VACANCIES 27

wages in the face of labor productivity and productivity shocks. This distinguishes the
separation shocks. An increase in labor produc- present model from those based upon intertem-
tivity encourages firms to create vacancies. The poral labor supply decisions (Robert E. Lucas,
resulting increase in the job-finding rate puts Jr., and Leonard Rapping, 1969). Thus this pa-
upward pressure on wages, soaking up virtually per explores the extent to which a combination
all of the shock. A decrease in the separation of search frictions and aggregate shocks can
rate also induces firms to create more vacancies, generate plausible fluctuations in unemploy-
again putting upward pressure on wages and ment and vacancies if labor supply is inelastic.
minimizing the impact on the v-u ratio and It suggests that search frictions per se scarcely
job-finding rate. On the other hand, I examine a amplify shocks. The paper does not examine
version of the model in which only workers’ whether a search model with an elastic labor
bargaining power is stochastic. Small fluctua- supply can provide a satisfactory explanation
tions in bargaining power generate realistic for the observed fluctuations in these two
movements in the v-u ratio while inducing only variables.
a moderately countercyclical real wage, with a
standard deviation of 0.01 log points around I. U.S. Labor Market Facts
trend.
Section III provides another angle from This section discusses the time series behav-
which to view the model’s basic shortcoming. I ior of unemployment u, vacancies v, the job
consider a centralized economy in which a so- finding rate f, the separation rate s, and labor
cial planner decides how many vacancies to productivity p in the United States. Table 1
create in order to maximize the present value of summarizes the detrended data.
market and nonmarket income net of vacancy
creation costs. The decentralized and central- A. Unemployment
ized economies behave identically if the match-
ing function is Cobb-Douglas in unemployment The unemployment rate is the most com-
and vacancies and workers’ bargaining power is monly used cyclical indicator of job-search ac-
equal to the elasticity of the matching function tivity. In an average month from 1951 to 2003,
with respect to the unemployment rate, a gen- 5.67 percent of the U.S. labor force was out of
eralization of Arthur Hosios (1990). But if work, available for work, and actively seeking
unemployment and vacancies are more substi- work. This time series exhibits considerable
tutable, fluctuations are amplified in the central- temporal variation, falling to as low as 2.6 per-
ized economy, essentially because the shadow cent in 1953 and 3.4 percent in 1968 and 1969,
wage is less procyclical. Empirically it is but reaching 10.8 percent in 1982 and 1983
difficult to measure the substitutability of un- (Figure 1). Some of these fluctuations are al-
employment and vacancies in the matching most certainly due to demographic and other
function, and therefore difficult to tell whether factors unrelated to business cycles. To high-
observed fluctuations are optimal. light business-cycle-frequency fluctuations, I
Section IV reconciles this paper with a num- take the difference between the log of the un-
ber of existing studies that claim standard employment level and an extremely low fre-
search and matching models are consistent with quency trend, a Hodrick-Prescott (HP) filter
the business cycle behavior of labor markets. with smoothing parameter 105 using quarterly
Finally, the paper concludes in Section V by data.2 The difference between log unemploy-
suggesting some modifications to the model that ment and its trend has a standard deviation of
might deliver rigid wages and thereby do a 0.19, so unemployment is often as much as 38
better job of matching the empirical evidence on percent above or below trend. Detrended unem-
vacancies and unemployment. ployment also exhibits considerable persistence,
It is worth emphasizing one important— but with quarterly autocorrelation 0.94.
standard—feature of the search and matching
framework that I exploit throughout this paper: 2
I use the level of unemployment rather than the rate to
workers are risk-neutral and supply labor inelas- keep the units comparable to those of vacancies. A previous
tically. In the absence of search frictions, em- version of this paper used the unemployment rate, with no
ployment would be constant even in the face of effect on the conclusions.
28 THE AMERICAN ECONOMIC REVIEW MARCH 2005

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 0.020
Quarterly autocorrelation 0.936 0.940 0.941 0.908 0.733 0.878
u 1 ⫺0.894 ⫺0.971 ⫺0.949 0.709 ⫺0.408
v — 1 0.975 0.897 ⫺0.684 0.364
Correlation matrix v/u — — 1 0.948 ⫺0.715 0.396
f — — — 1 ⫺0.574 0.396
s — — — — 1 ⫺0.524
p — — — — — 1

Notes: Seasonally adjusted unemployment u is constructed by the BLS from the Current Population Survey (CPS). The
seasonally adjusted help-wanted advertising index v is constructed by the Conference Board. The job-finding rate f and
separation 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 monthly
series. 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 Current
Employment Statistics. All variables are reported in logs as deviations from an HP trend with smoothing parameter 105.

participation are distinct economic conditions.

Chinhui Juhn et al. (1991) show that almost all
of the cyclical volatility in prime-aged male
nonemployment 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 Craig
Riddell (1999) argue that other variables also
help to predict the likelihood of finding a job. In
any case, since labor force participation is pro-
cyclical, the employment-population ratio is a
more cyclical measure of job-search activity,
worsening the problems highlighted in this
FIGURE 1. QUARTERLY U.S. UNEMPLOYMENT (IN MILLIONS) paper.
AND TREND, 1951–2003 It is also conceivable that when unemploy-
Notes: Unemployment is a quarterly average of the season- ment rises, the amount of job-search activity per
ally adjusted monthly series constructed by the BLS from unemployed worker declines so much that ag-
the CPS, survey home page http://www.bls.gov/cps/. The gregate search activity actually falls. There is
trend is an HP filter of the quarterly data with smoothing both direct and indirect evidence against this
parameter 105.
hypothesis. As direct evidence, one would ex-
pect that a reduction in search intensity could be
observed as a decline in the number of job-
There is some question as to whether unem- search methods used or a switch toward less
ployment or the employment-population ratio is time-intensive methods. An examination of
a better indicator of job-search activity. Advo- Current Population Survey (CPS) data indicates
cates of the latter view, for example Harold no cyclical variation in the number or type of
Cole and Richard Rogerson (1999), argue that job-search methods utilized.3 Indirect evidence
the number of workers moving directly into comes from estimates of matching functions,
employment from out-of-the-labor force is as which universally find that an increase in un-
large as the number who move from unemploy- employment is associated with an increase in
ment to employment (Olivier Blanchard and
Peter Diamond, 1990). On the other hand, there
is ample evidence that unemployment and non- 3
Shimer (2004b) discusses this evidence in detail.
VOL. 95 NO. 1 SHIMER: UNEMPLOYMENT AND VACANCIES 29

the number of matches (Barbara Petrongolo and

Pissarides, 2001). If job-search activity declined
sharply when unemployment increased, the
matching function would be measured as de-
creasing in unemployment. I conclude that ag-
gregate job search activity is positively
correlated with unemployment.

B. Vacancies

The flip side of unemployment is job vacan-

cies. The Job Openings and Labor Turnover FIGURE 2. TWO MEASURES OF U.S. JOB VACANCIES,
Survey (JOLTS) provides an ideal empirical 2000Q4 –2003Q4
definition: “A job opening requires that 1) a Notes: The solid line shows the logarithm of the number of
specific position exists, 2) work could start job openings in millions, measured by the BLS from the
within 30 days, and 3) the employer is actively JOLTS, survey homepage http://www.bls.gov/jlt, quarterly
averaged and seasonally adjusted. The dashed line shows
recruiting from outside of the establishment to the deviation from trend of the quarterly averaged, season-
fill the position. Included are full-time, part- ally adjusted Conference Board help-wanted advertising
time, permanent, temporary, and short-term index.
openings. Active recruiting means that the es-
tablishment is engaged in current efforts to fill
the opening, such as advertising in newspapers that help-wanted advertising is subject to low-
or on the Internet, posting help-wanted signs, frequency fluctuations that are related only tan-
accepting applications, or using similar meth- gentially to the labor market. In recent years, the
ods.”4 Unfortunately, JOLTS began only in De- Internet may have reduced firms’ reliance on
cember 2000 and comparable data had never newspapers as a source of job advertising, while
previously been collected in the United States. in the 1960s, newspaper consolidation may
Although there are too few observations to look have increased advertising in surviving newspa-
systematically at this time series, its behavior pers and Equal Employment Opportunity laws
has been instructive. In the first month of the may have encouraged firms to advertise job
survey, the non-farm sector maintained a sea- openings more extensively. Fortunately, a low-
sonally adjusted 4.6 million job openings. This frequency trend should remove the effect of
number fell rapidly during 2001 and averaged these and other secular shifts. Figure 3 shows
just 2.9 million in 2002 and 2003. This decline the help-wanted advertising index and its trend.
in job openings, depicted by the solid line in Notably, the decline in the detrended help-
Figure 2, coincided with a period of rising un- wanted index closely tracks the decline in job
employment, suggesting that job vacancies are openings measured directly from JOLTS during
procyclical. the period when the latter time series is avail-
To obtain a longer time series, I use a stan- able (Figure 2).
dard proxy for vacancies, the Conference Board Figure 4 shows a scatter plot of the relation-
help-wanted advertising index, measured as the ship between the cyclical component of unem-
number of help-wanted advertisements in 51 ployment and vacancies, the Beveridge curve.
major newspapers.5 A potential shortcoming is The correlation of the percentage deviation of
unemployment 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://
6
www.bls.gov/jlt/jlt_nr1.pdf. Abraham and Katz (1986) and Blanchard and Diamond
5
Abraham (1987) discusses this measure in detail. From (1989) discuss the U.S. Beveridge curve. Abraham and Katz
1972 to 1981, Minnesota collected state-wide job vacancy (1986) argue that the negative correlation between unem-
data. Abraham compares this with Minnesota’s help-wanted ployment and vacancies is inconsistent with Lilien’s (1982)
advertising index and shows that the two series track each sectoral shifts hypothesis, and instead indicates that busi-
other very closely through two business cycles and ten ness cycles are driven by aggregate fluctuations. Blanchard
seasonal cycles. and Diamond (1989) conclude that at business cycle
30 THE AMERICAN ECONOMIC REVIEW MARCH 2005

FIGURE 3. QUARTERLY U.S. HELP-WANTED ADVERTISING

INDEX AND TREND, 1951–2003
Notes: The help-wanted advertising index is a quarterly
average of the seasonally adjusted monthly series con-
structed by the Conference Board with normalization
1987 ⫽ 100. The data were downloaded from the Federal FIGURE 4. QUARTERLY U.S. BEVERIDGE CURVE,
Reserve Bank of St. Louis database at http://research. 1951–2003
stlouisfed.org/fred2/data/helpwant.txt. The trend is an HP Notes: Unemployment is constructed by the BLS from the
filter of the quarterly data with smoothing parameter 105. CPS. The help-wanted advertising index is constructed by
the 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.
standard deviation of the cyclical variation in
unemployment and vacancies is almost identi-
cal, between 0.19 and 0.20, so the product of
unemployment and vacancies is nearly acyclic. Gross worker flow data can be used to mea-
The v-u ratio is therefore extremely procyclical, sure the job-finding rate directly, and indeed
with a standard deviation of 0.38 around its both the unemployment-to-employment and
trend. nonparticipation-to-employment transition rates
are strongly procyclical (Blanchard and Dia-
C. Job-Finding Rate mond, 1990; Hoyt Bleakley et al., 1999; Ka-
tharine Abraham and Shimer, 2001). There are
An implication of the procyclicality of the two drawbacks to this approach. First, the req-
v-u ratio is that the hazard rate for an unem- uisite public use dataset is available only since
ployed worker of finding a job, his job-finding 1976, and so using these data would require
rate, should be lower during a recession. As- throwing away half of the available time series.
sume that the number of newly hired workers is Second, measurement and classification error
given by an increasing and constant returns-to- lead a substantial overestimate of gross worker
scale matching function m(u,v), depending on flows (John Abowd and Arnold Zellner, 1985;
the number of unemployed workers u and the James Poterba and Lawrence Summers, 1986),
number of vacancies v. Then the probability that the magnitude of which cannot easily be com-
any individual unemployed worker finds a job, puted. Instead, I infer the job-finding rate from
the average transition rate from unemployment the dynamic behavior of the unemployment
to employment, is f ⬅ m(u,v)/u ⫽ m(1,␪), where level and short-term unemployment level. Let
␪ ⬅ v/u is the v-u ratio. The job-finding rate f uts denote the number of workers unemployed
should therefore move together with the v-u for less than one month in month t. Then as-
ratio. suming all unemployed workers find a job with
probability ft in month t and no unemployed
worker exits the labor force,
frequencies, shocks generally drive the unemployment and
vacancy rates in the opposite direction. u t ⫹ 1 ⫽ u t 共1 ⫺ f t 兲 ⫹ u ts ⫹ 1 .
VOL. 95 NO. 1 SHIMER: UNEMPLOYMENT AND VACANCIES 31

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 data
constructed and seasonally adjusted by the BLS from the
FIGURE 6. MONTHLY U.S. MATCHING FUNCTION,
CPS, survey home page http://www.bls.gov/cps/. It is ex-
1951–2003
pressed as a quarterly average of monthly data. The trend
is an HP filter of the quarterly data with smoothing param- Notes: The v-u ratio is constructed by the BLS from the
eter 105. CPS and by the Conference Board. The job-finding rate is
constructed using equation (1) and BLS data from the CPS.
Both are quarterly averages of seasonally adjusted monthly
series and are expressed as deviations from an HP filter with
Unemployment next month is the sum of the smoothing parameter 105.
number of unemployed workers this month who
fail to find a job and the number of newly
unemployed workers. Equivalently, One can use the measured job-finding rate
and v-u ratio to estimate a matching function
u t ⫹ 1 ⫺ u ts ⫹ 1 m(u,v). Data limitations force me to impose two
(1) ft ⫽ 1 ⫺ .
ut restrictions on the estimated function. First, be-
cause unemployment and vacancies are strongly
I use the unemployment level and the number of negatively correlated, it is difficult to tell em-
workers unemployed for 0 to 4 weeks, both pirically whether m(u,v) exhibits constant, in-
constructed by the BLS from the CPS, to com- creasing, or decreasing returns to scale. But in
pute ft from 1951 to 2003.7 Figure 5 shows the their literature survey, Petrongolo and Pissar-
results. The monthly hazard rate averaged 0.45 ides (2001) conclude that most estimates of the
from 1951 to 2003. After detrending with the matching function cannot reject the null hypoth-
usual low-frequency HP filter, the correlation esis of constant returns; I therefore estimate ft ⫽
between ft and ␪t at quarterly frequencies is f(␪t), consistent with a constant returns-to-scale
0.95, although the standard deviation of ft is matching function. Figure 6 shows the raw data
about 31 percent that of ␪t. Given that both for the job-finding rate ft and the v-u ratio ␪t, a
measures are crudely yet independently con- nearly linear relationship when both variables
structed, this correlation is remarkable and are expressed as deviations from log trend. Sec-
strongly suggests that a matching function is a ond, I impose that the matching function is
useful way to approach U.S. data. Cobb-Douglas, m(u,v) ⫽ ␮u␣v1 ⫺ ␣, for some
unknown parameters ␣ and ␮. Again, the data
are not very informative as to whether this is a
7
Abraham and Shimer (2001) argue that the redesign of reasonable restriction.8 I estimate the matching
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 this
discontinuity. In this paper, I simply inflate short-term un- 8
Consider the CES matching function log ft ⫽ log ␮ ⫹
employment by 10 percent after the redesign took effect. 1/␳ log (␣ ⫹ (1 ⫺ ␣)␪t␳). Cobb-Douglas corresponds to
32 THE AMERICAN ECONOMIC REVIEW MARCH 2005

function using detrended data on the job-finding

rate and the v-u ratio. Depending on exactly
how I control for autocorrelation in the residu-
als, I estimate values of ␣ between 0.70 and
0.75. With a first-order autoregressive residual,
I get ␣ ⫽ 0.72 with a standard error of 0.01.
One particularly crude aspect of this measure
of the job-finding rate is the assumption that all
workers are equally likely to find a job. Shimer
(2004a) proves that when the unemployed are
heterogeneous, ft measures the mean job-finding
rate in the unemployed population. That paper
also compares my preferred measure of the job-
finding rate with two alternatives. The first uses
FIGURE 7. MONTHLY SEPARATION PROBABILITY FOR
the unemployment level and mean unemploy- EMPLOYED WORKERS, 1951–2003
ment duration to obtain a weighted average of
Notes: The separation rate is computed using equation (2),
the job-finding rate in the unemployed popula- with employment, unemployment, and short-term unem-
tion, with weights proportional to each individ- ployment data constructed and seasonally adjusted by the
ual’s unemployment duration.9 The second BLS from the CPS, survey home page http://www.bls.gov/
follows Robert Hall (2004) and measures the cps/. It is expressed as a quarterly average of monthly data.
job-finding rate of workers with short unem- The trend is an HP filter of the quarterly data with smooth-
ing parameter 105.
ployment duration using the ratio of workers
with 0 to 4 weeks of unemployment to workers
with 5 to 14 weeks of unemployment. Since the
job-finding rate declines with unemployment bias. When a worker loses her job, she has on
duration, I find that my preferred measure of the average half a month to find a new job before
job-finding rate lies between these two alterna- she is recorded as unemployed. Accounting for
tives. Hall measures an average job-finding rate this, the short-term unemployment rate the next
of 0.48 per month, while unemployment dura- month is approximately equal to
tion data yield a job-finding rate of 0.34 per
month. Nevertheless, all three measures are u ts ⫹ 1 ⫽ s t e t 共1 ⫺ 12 f t 兲.
highly correlated, and so the choice of which
measure to use does not qualitatively affect the
conclusions of this study. Ignoring the probability of finding another job
within the month leads one to understate the
D. Separation Rate separation rate. This problem is particularly
acute when the job-finding rate is high, i.e.,
I can also deduce the behavior of the separa- during expansions. I therefore measure the sep-
tion rate from data on employment, short-term aration rate as
unemployment, and the hiring rate. Suppose
first that whenever an employed worker loses u ts ⫹ 1
her job, she becomes unemployed. Then the (2) st ⫽ .
separation rate could simply be computed as the e t 共1 ⫺ 12 f t 兲
ratio of short-term unemployed workers next
month, uts⫹ 1, to employed workers this month, Figure 7 shows the monthly separation rate
et. But this masks a significant time-aggregation thus constructed. It averaged 0.034 from 1951
to 2003, so jobs last on average for about 2.5
years. Fluctuations in the deviation of the log
limiting case of ␳ ⫽ 0. When I estimate the CES function separation rate from trend are somewhat smaller
using nonlinear least squares and correct for first-order than in the hiring rate, with a standard deviation
autocorrelation, I get a point estimate of ␳ ⫽ 0.06 with a
standard error of 0.38. of 0.08, and separations are countercyclical, so
9
A previous version of this paper relied on that measure the correlation with the detrended v-u ratio is
of ft. This had little effect on the results. ⫺0.72.
VOL. 95 NO. 1 SHIMER: UNEMPLOYMENT AND VACANCIES 33

FIGURE 9. QUARTERLY U.S. VACANCY-UNEMPLOYMENT

RATIO AND AVERAGE LABOR PRODUCTIVITY, 1951–2003
FIGURE 8. QUARTERLY U.S. AVERAGE LABOR Notes: Unemployment is constructed by the BLS from the
PRODUCTIVITY AND TREND, 1951–2003 CPS. The help-wanted advertising index is constructed by
the Conference Board. Both are quarterly averages of sea-
Notes: Real output per person in the non-farm business sonally adjusted monthly series. Labor productivity is real
sector, constructed by the BLS Major Sector Productivity average output per worker in the non-farm business sector,
and Costs program, survey home page http://www.bls.gov/ constructed by the BLS Major Sector Productivity and
lpc/, 1992 ⫽ 100. The trend is an HP filter of the quarterly Costs program. The v-u ratio and labor productivity are
data with smoothing parameter 105. expressed as deviations from an HP filter with smoothing
parameter 105.

The strong procyclicality of the job-finding Product Accounts, while employment is con-
rate and relatively weak countercyclicality of structed from the BLS establishment survey, the
the separation rate might appear to contradict Current Employment Statistics. This series of-
Blanchard and Diamond’s (1990) conclusion fers two advantages compared with total factor
that “the amplitude in fluctuations in the flow productivity: it is available quarterly since
out of employment is larger than that of the flow 1948; and it better corresponds to the concept of
into employment.” This is easily reconciled. labor productivity in the subsequent models,
Blanchard and Diamond look at the number of which do not include capital.
people entering or exiting employment in a Figure 8 shows the behavior of labor produc-
given month, ftut or stet, while I focus on the tivity and Figure 9 compares the cyclical com-
probability that an individual switches employ- ponents of the v-u ratio and labor productivity.
ment states, ft and st. Although the probability There is a positive correlation between the two
of entering employment ft declines sharply in time series and some evidence that labor pro-
recessions, this is almost exactly offset by the ductivity leads the v-u ratio by about one year,
increase in unemployment ut, so that the num- with a maximum correlation of 0.56.10 But the
ber of people exiting unemployment is essen- most important fact is that labor productivity is
tially acyclic. Viewed through the lens of an stable, never deviating by more than 6 per-
increasing matching function m(u,v), this is cent from trend. In contrast, the v-u ratio has
consistent with the independent evidence that twice risen to 0.5 log points about its trend level
vacancies are strongly procyclical. and six times has fallen by 0.5 log points below
trend.
E. Labor Productivity

The final important empirical observation is 10

From 1951 to 1985, the contemporaneous correlation
the weak procyclicality of labor productivity, between detrended labor productivity and the v-u ratio was
measured as real output per worker in the non- 0.57 and the peak correlation was 0.74. From 1986 to 2003,
farm business sector. The BLS constructs this however, the contemporaneous and peak correlations are
negative, ⫺0.37 and ⫺0.43, respectively. This has been
quarterly time series as part of its Major Sector particularly noticeable during the last three years of data. An
Productivity and Costs program. The output exploration of the cause of this change goes beyond the
measure is based on the National Income and scope of this paper.
34 THE AMERICAN ECONOMIC REVIEW MARCH 2005

It is possible that the measured cyclicality of tivity and the separation rate are common
labor productivity is reduced by a composition knowledge.
bias, since less productive workers are more Next I turn to the economic agents in the
likely to lose their jobs in recessions. I offer two economy, a measure 1 of risk-neutral, infinitely-
responses to this concern. First, there is a com- lived workers and a continuum of risk-neutral,
position bias that points in the opposite direc- infinitely-lived firms. All agents discount future
tion: labor productivity is higher in more payoffs at rate r ⬎ 0.
cyclical sectors of the economy, e.g., durable Workers can either be unemployed or em-
goods manufacturing. And second, a large lit- ployed. An unemployed worker gets flow utility
erature on real wage cyclicality has reached a z from non-market activity (“leisure”) and
mixed conclusion about the importance of com- searches for a job. An employed worker earns
position biases (Abraham and John Haltiwan- an endogenous wage but may not search. I
ger, 1995). Gary Solon et al. (1994) pro- discuss wage determination shortly.
vide perhaps the strongest evidence that labor Firms have a constant returns-to-scale pro-
force composition is important for wage cycli- duction technology that uses only labor, with
cality, but even they argue that accounting labor productivity at time t given by the sto-
for this might double the measured variability chastic realization p(t). In order to hire a
of real wages. This paper argues that the worker, a firm must maintain an open vacancy
search and matching model cannot account at flow cost c. Free entry drives the expected
for the cyclical behavior of vacancies and un- present value of an open vacancy to zero. A
employment unless labor productivity is at least worker and a firm separate according to a
ten times as volatile as the data suggest, so Poisson process with arrival rate governed
composition bias is at best an incomplete by the stochastic separation rate s(t), leaving
explanation. the worker unemployed and the firm with
nothing.
Let u(t) denote the endogenous unemploy-
II. Search and Matching Model ment rate, 11 v(t) denote the endogenous mea-
sure of vacancies in the economy, and ␪(t) ⬅
I now examine whether a standard search and v(t)/u(t) denote the v-u ratio at time t. The flow
matching model can reconcile the strong procy- of matches is given by a constant returns-to-
clicality of the v-u ratio and the job-finding rate scale function m(u(t), v(t)), increasing in both
with the weak procyclicality of labor productiv- arguments. This implies that an unemployed
ity and countercyclicality of the separation rate. worker finds a job according to a Poisson pro-
The model I consider is essentially an aggregate cess with time-varying arrival rate f(␪(t)) ⬅
stochastic version of Pissarides (1985, or 2000, m(1,␪(t)) and that a vacancy is filled according
Ch. 1). to a Poisson process with time-varying arrival
rate q(␪(t)) ⬅ m(␪(t)⫺1,1) ⫽ f(␪(t))/␪(t).
A. Model I assume that in every state of the world,
labor productivity p(t) exceeds the value of lei-
I start by describing the exogenous vari- sure z, so there are bilateral gains from match-
ables that drive fluctuations. Labor productiv- ing. There is no single compelling theory of
ity p and the separation rate s follow a first- wage determination in such an environment,
order Markov process in continuous time. A and so I follow the literature and assume that
shock hits the economy according to a Pois- when a worker and firm first meet, the expected
son process with arrival rate ␭, at which point gains from trade are split according to the Nash
a new pair ( p⬘,s⬘) is drawn from a state de- bargaining solution. The worker can threaten to
pendent distribution. Let ⺕ p,sXp⬘,s⬘ denote the become unemployed and the firm can threaten
expected value of an arbitrary variable X to end the job. The present value of surplus
following the next aggregate shock, condi-
tional on the current state ( p,s). I assume that 11
With the population of workers constant and normal-
this conditional expectation is finite, which is ized to one, the unemployment rate and unemployment
ensured if the state space is compact. At every level are identical in this model. I therefore use these terms
point in time, the current values of produc- interchangeably.
VOL. 95 NO. 1 SHIMER: UNEMPLOYMENT AND VACANCIES 35

beyond these threats is divided between the the pair is matched, they produce p units of
worker and firm, with the worker keeping a output. If they were to break up the match, free
fraction ␤ 僆 (0, 1) of the surplus, her “bargain- entry implies the firm would be left with noth-
ing power.” I make almost no assumptions ing, while the worker would become unem-
about what happens to wages after this initial ployed, getting flow utility from leisure z and
agreement, except that the worker and firm from the probability f(␪p,s) of contacting a firm,
manage to exploit all the joint gains from trade. in which event the worker would keep a fraction
For example, the wage may be re-bargained ␤ of the match value Vp,s. Next, there is a flow
whenever the economy is hit with a shock. probability s that the worker and firm separate,
Alternatively, it may be fixed at its initial value destroying the match value. Finally, an aggre-
until such time as the firm would prefer to gate shock arrives at rate ␭, resulting in an
fire the worker or the worker would prefer expected change in match value ⺕ p,sVp⬘,s⬘ ⫺
to quit, whereupon the pair resets the wage so Vp,s.
as to avoid an unnecessary and inefficient Another critical equation for the match value
separation. comes from firms’ free entry condition. The
flow cost of a vacancy c must equal the flow
probability that the vacancy contacts a worker
B. Characterization of Equilibrium times the resulting capital gain, which by Nash
bargaining is equal to a fraction 1 ⫺ ␤ of the
I look for an equilibrium in which the v-u match value Vp,s:
ratio depends only on the current value of p
and s, ␪p,s.12 Given the state-contingent v-u
(5) c ⫽ q共 ␪ p,s 兲共1 ⫺ ␤ 兲V p,s .
ratio, the unemployment rate evolves according
to a standard backward-looking differential
equation, Eliminating current and future values of Vp,s
from (4) using (5) gives
(3) u̇共t兲 ⫽ s共t兲共1 ⫺ u共t兲兲 ⫺ f共 ␪ p共t兲,s共t兲 兲u共t兲
r⫹s⫹␭
(6) ⫹ ␤␪ p,s
where (p(t), s(t)) is the aggregate state at time t. q共 ␪ p,s 兲
A flow s(t) of the 1 ⫺ u(t) employed workers
become unemployed, while a flow f(␪) of the p⫺z 1
⫽ 共1 ⫺ ␤兲 ⫹ ␭⺕ p,s
u(t) unemployed workers find a job. An initial c q共␪p⬘,s⬘ 兲
condition pins down the unemployment rate and
the aggregate state at some date t0.
I characterize the v-u ratio using a recursive which implicitly defines the v-u ratio as a func-
equation for the joint value to a worker and firm tion of the current state (p,s).13 This equation
of being matched in excess of breaking up as a can easily be solved numerically, even with a
function of the current aggregate state, Vp,s. large state vector. This simple representation of
the equilibrium of a stochastic version of the
Pissarides (1985) model appears to be new to
(4) rV p,s ⫽ p ⫺ 共z ⫹ f共 ␪ p,s 兲 ␤ V p,s 兲 ⫺ sV p,s
the literature.
⫹ ␭共⺕
⺕p,s Vp⬘,s⬘ ⫺ Vp,s 兲.
C. Comparative Statics
Appendix A derives this equation from more
primitive conditions. The first two terms repre- In some special cases, equation (6) can
sent the current flow surplus from matching. If be solved analytically to get a sense of the

12
It is straightforward to show in a deterministic version
13
of this model that there is no other equilibrium, e.g., one in A similar equation obtains in the presence of aggre-
which ␪ also depends on the unemployment rate. See Pis- gate variation in the value of leisure z, the cost of a vacancy
sarides (1985). c, or workers’ bargaining power ␤.
36 THE AMERICAN ECONOMIC REVIEW MARCH 2005

quantitative results implied by this analysis. ⫺s

First, suppose there are no aggregate shocks, ␭ ⫽ 共r ⫹ s兲共1 ⫺ ␩共␪p,s 兲兲 ⫹ ␤f共␪p,s 兲
.
0.14 Then the state-contingent v-u ratio satisfies
Substituting the same numbers into this expres-
r⫹s p⫺z sion gives ⫺0.10. Doubling the separation rate
⫹ ␤␪ p,s ⫽ 共1 ⫺ ␤ 兲 . would have a scarcely discernible impact on the
q共 ␪ p,s 兲 c v-u ratio.
Finally, one can examine the independent
The elasticity of the v-u ratio ␪ with respect to behavior of vacancies and unemployment. In
“net labor productivity” p ⫺ z is steady state, equation (3) holds with u̇ ⫽ 0.
If the matching function is Cobb-Douglas,
r ⫹ s ⫹ ␤ f共 ␪ p,s 兲 m(u,v) ⫽ ␮u␣v1 ⫺ ␣, this implies
共r ⫹ s兲共1 ⫺ ␩ 共 ␪ p,s 兲兲 ⫹ ␤ f共 ␪ p,s 兲

where ␩(␪) 僆 [0,1] is the elasticity of f(␪). This

v p,s ⫽ 冉 s共1 ⫺ u p,s 兲
␣
␮ u p,s 冊 1/共1 ⫺ ␣ 兲
.

is large only if workers’ bargaining power ␤ is

small and the elasticity ␩ is close to one. But For a given separation rate s, this describes a
with reasonable parameter values, it is close to decreasing relationship between unemployment
1. For example, think of a time period as equal and vacancies, consistent with the Beveridge
to one month, so the average job-finding rate is curve (Figure 4). An increase in labor produc-
approximately 0.45 (Section I C), the elasticity tivity raises the v-u ratio which lowers the un-
␩(␪) is approximately 0.28 (Section I C again), employment rate and hence raises the vacancy
the average separation probability is approxi- rate. Vacancies and unemployment should
mately 0.034 (Section I D), and the interest rate move in opposite directions in response to such
is about 0.004. Then if workers’ bargaining shocks. But an increase in the separation rate
power ␤ is equal to 1 ⫺ ␩(␪), the so-called scarcely affects the v-u ratio. Instead, it tends to
Hosios (1990) condition for efficiency, 15 the raise both the unemployment and vacancy rates,
elasticity of the v-u ratio with respect to net an effect that is likely to produce a counterfac-
labor productivity is 1.03. Lower values of ␤ tually positive correlation between unemploy-
yield a slightly higher elasticity, say 1.15 when ment and vacancies.
␤ ⫽ 0.1, but only at ␤ ⫽ 0 does the elasticity of I can perform similar analytic exercises by
the v-u ratio with respect to p ⫺ z rise appre- making other simplifying assumptions. Suppose
ciably, to 1.39. It would take implausible pa- that each vacancy contacts an unemployed
rameter values for this elasticity to exceed 2. worker at a constant Poisson rate ␮, indepen-
This implies that unless the value of leisure is dent of the unemployment rate, so q(␪) ⫽ ␮.
close to labor productivity, the v-u ratio is likely Given the risk-neutrality assumptions, this is
to be unresponsive to changes in the labor equivalent to assuming that firms must pay a
productivity. fixed cost c/␮ in order to hire a worker. Then
I can similarly compute the elasticity of the even with aggregate shocks, equation (6) yields
v-u ratio with respect to the separation rate: a static equation for the v-u ratio:

r⫹s p⫺z
⫹ ␤␪ p,s ⫽ 共1 ⫺ ␤ 兲 .
14
␮ c
Shimer (2003) performs comparative statics exercises
under much weaker assumptions. For example, in that paper
the matching function can exhibit increasing or decreasing
In this case, the elasticity of the v-u ratio with
returns to scale and there can be an arbitrary idiosyncratic respect to net labor productivity is
process for productivity, allowing for endogenous separa-
tions (Mortensen and Pissarides, 1994). I show that the
results presented here generalize to such an environment if r ⫹ s ⫹ ␤␮␪
workers and firms are sufficiently patient relative to the ␤␮␪
search frictions.
15
Section III shows that the Hosios condition carries
over to the stochastic model. and the elasticity of the v-u ratio with respect to
VOL. 95 NO. 1 SHIMER: UNEMPLOYMENT AND VACANCIES 37

the separation rate is ⫺s/␤␮␪. Since f(␪) ⫽ ␮␪, y is mean reverting. The stochastic variables are
one can again pin down all the parameter values then expressed as functions of y.
except workers’ bargaining power ␤. Using the Although I use the discrete state space model
same parameter values as above, including ␤ ⫽ in my simulations as well, it is almost exactly
0.72, I obtain elasticities of 1.12 and ⫺0.105, correct and significantly easier to think about
almost unchanged from the case with no shocks. the behavior of the extrinsic shocks by discuss-
More generally, unless ␤ is nearly equal to zero, ing a related continuous state space model.16 I
both elasticities are very small. express the state variables as functions of an
At the opposite extreme, suppose that each Ornstein-Uhlenbeck process (see Howard Tay-
unemployed worker contacts a vacancy at a lor and Samuel Karlin, 1998, Section 8.5). Let y
constant Poisson rate ␮, independent of the va- satisfy
cancy rate, so f(␪) ⫽ ␮ and q(␪) ⫽ ␮/␪. Also
assume that the separation rate s is constant and dy ⫽ ⫺␥ydt ⫹ ␴db
average labor productivity p is a Martingale,
⺕ pp⬘ ⫽ p. With this matching function, equation where b is a standard Brownian motion. Here
(6) is linear in current and future values of the ␥ ⬎ 0 is a measure of persistence, with higher
v-u ratio: values indicating faster mean reversion, and
␴ ⬎ 0 is the instantaneous standard deviation.
冉 r⫹s⫹␭
␮
⫹ ␤ ␪p 冊 This process has some convenient properties: y
is conditionally and unconditionally normal; it
is mean reverting, with expected value converg-
p⫺z ␭ ing asymptotically to zero; and asymptotically
⫽ 共1 ⫺ ␤兲 ⫹ ⺕ p␪p⬘ . its variance converges to ␴2/2␥.
c ␮
I consider two different cases. In the first, the
It is straightforward to verify that the v-u ratio is separation rate is constant and productivity sat-
linear in productivity, and therefore ⺕p␪p⬘ ⫽ ␪p, i.e., isfies p ⫽ z ⫹ ey(p* ⫺ z), where y is an
Ornstein-Uhlenbeck process with parameters ␥

冉 r⫹s
␮ 冊
⫹ ␤ ␪ p ⫽ 共1 ⫺ ␤ 兲
p⫺z
c
and ␴, and p* ⬎ z is a measure of long-run
average productivity. Since ey ⬎ 0, this ensures
p ⬎ z. In the second case, productivity is con-
so the elasticity of the v-u ratio with respect to stant and separations satisfy s ⫽ eys*, where
net labor productivity is 1, regardless of work- again y follows an Ornstein-Uhlenbeck process
ers’ bargaining power. I conclude that with a and now s* ⬎ 0 is a measure of the long-run
wide range of parameterizations, the v-u ratio ␪ average separation rate. In both cases, the sto-
should be approximately proportional to net la- chastic process is reduced to three parameters,
bor productivity p ⫺ z. ␥, ␴, and either p* or s*.
I now proceed to explain the choice of the
other parameters, starting with the case of sto-
D. Calibration chastic productivity. I follow the literature and
assume that the matching function is Cobb-
This section parameterizes the model to Douglas,
match the time series behavior of the U.S. un-
employment rate. The most important question
f共 ␪ 兲 ⫽ ␪ q共 ␪ 兲 ⫽ ␮␪ 1 ⫺ ␣ .
is the choice of the Markov process for labor
productivity and separations. Appendix C de-
velops a discrete state space model which builds This reduces the calibration to ten parameters:
on a simple Poisson process corresponding to
the theoretical analysis in Section II B. I define
an underlying variable y that lies on a finite 16
I work on a discrete grid with 2n ⫹ 1 ⫽ 2001 points,
ordered set of points. When a Poisson shock which closely approximate Gaussian innovations. This im-
plies that Poisson arrival rate of shocks is ␭ ⫽ n␥ ⫽ 4 times
hits, y either moves up or down by one point. per quarter in the model with labor productivity shocks and
The probability of moving up is itself decreas- ␭ ⫽ 220 in the model with (less persistent) separation
ing in the current value of y, which ensures that shocks.
38 THE AMERICAN ECONOMIC REVIEW MARCH 2005

the productivity parameter p*, the value of lei- TABLE 2—PARAMETER VALUES IN SIMULATIONS OF THE

sure z, workers’ bargaining power ␤, the dis- MODEL

count rate r, the separation rate s, the two Source of shocks
matching function parameters ␣ and ␮, the va-
cancy cost c, and the mean reversion and stan- Parameter Productivity Separation
dard deviation of the stochastic process, ␥ Productivity p stochastic 1
and ␴. Separation rate s 0.1 stochastic
Without loss of generality, I normalize the Discount rate r 0.012 0.012
Value of leisure z 0.4 0.4
productivity parameter to p* ⫽ 1. I choose the Matching function q(␪) 1.355␪⫺0.72 1.355␪⫺0.72
standard deviation and persistence of the pro- Bargaining power ␤ 0.72 0.72
ductivity process to match the empirical behav- Cost of vacancy c 0.213 0.213
ior of labor productivity. This requires setting Standard deviation ␴ 0.0165 0.0570
Autoregressive parameter ␥
␴ ⫽ 0.0165 and ␥ ⫽ 0.004. An increase in the Grid size 2n ⫹ 1
0.004
2001
0.220
2001
volatility of productivity ␴ has a nearly propor-
tional effect on the volatility of other variables, Note: The text provides details on the stochastic process for
while the persistence of the stochastic process ␥ productivity and for the separation rate.
scarcely affects the reported results. For exam-
ple, suppose I reduce ␥ to 0.001, so productivity
is more nearly a random walk. Because it is reported results are insensitive to the value of
difficult to distinguish small values of ␥ in a that parameter, I show in Section III that if ␣ ⫽
finite dataset, after HP filtering the model- ␤, the “Hosios (1990) Rule,” the decentralized
generated data, the persistence and magnitude equilibrium maximizes a well-posed social
of the impulse is virtually unchanged compared planner’s problem.
with the baseline parameterization. But reassur- I use the final two parameters, the matching
ingly, the detrended behavior of unemployment function constant ␮ and the vacancy cost c, to
and vacancies is also scarcely affected by in- pin down the average job-finding rate and the
creasing the persistence of labor productivity. average v-u ratio. As reported in Section I C, a
I set the value of leisure to z ⫽ 0.4. Since worker finds a job with a 0.45 probability per
mean labor income in the model is 0.993, this month, so the flow arrival rate of job offers
lies at the upper end of the range of income ␮␪1 ⫺ ␣ should average approximately 1.35 on a
replacement rates in the United States if inter- quarterly basis. I do not have a long time series
preted entirely as an unemployment benefit. with the level of the v-u ratio, but fortunately
I normalize a time period to be one quarter, the model offers one more normalization. Equa-
and therefore set the discount rate to r ⫽ 0.012, tion (6) implies that doubling c and multiplying
equivalent to an annual discount factor of 0.953. ␮ by a factor 21 ⫺ ␣ divides the v-u ratio ␪ in
The analysis in Section I D suggests a quarterly half, doubles the rate at which firms contact
separation rate of s ⫽ 0.10, so jobs last for about workers q(␪), but does not affect the rate at
2.5 years on average. This is comparable to which workers find jobs. In other words, the
Abowd and Zellner’s (1985) finding that 3.42 average v-u ratio is intrinsically meaningless in
percent of employed workers exit employment the model. I choose to target a mean v-u ratio of 1,
during a typical month between 1972 and 1982, which requires setting ␮ ⫽ 1.355 and c ⫽ 0.213.
after correcting for classification and measure- In the case of shocks to the separation rate, I
ment error. It is also comparable to measured change only the stochastic process so as to
turnover rates in the JOLTS, although some match the empirical results discussed in Section
separations in that survey reflect job-to-job tran- I D. Productivity is constant and equal to 1,
sitions, a possibility that is absent from this while the mean separation rate is s* ⫽ 0.10. I
model. set ␴ ⫽ 0.057 and ␥ ⫽ 0.220, a much less
Using the matching function estimates from persistent stochastic process. This leaves the
Section I C, I set the elasticity parameter to ␣ ⫽ average v-u ratio and average job-finding rate
0.72. This lies toward the upper end of the range virtually unchanged. Table 2 summarizes the
of estimates that Petrongolo and Pissarides parameter choices in the two simulations.
(2001) report. I also set workers’ bargaining I use equation (6) to find the state-contingent
power ␤ to the same value, 0.72. Although the v-u ratio ␪p,s and then simulate the model. That
VOL. 95 NO. 1 SHIMER: UNEMPLOYMENT AND VACANCIES 39

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 deviations
from an HP trend with smoothing parameter 105. Bootstrapped standard errors—the standard deviation across 10,000 model
simulations—are reported in parentheses. The text provides details on the stochastic process for productivity.

is, starting with an initial unemployment rate although the difference is insignificant. It is
and aggregate state at time 0, I use a pseudo- worth emphasizing that the negative correlation
random number generator to calculate the ar- between unemployment and vacancies is a re-
rival time of the first Poisson shock. I compute sult, not a direct target of the calibration exer-
the unemployment rate when that shock arrives, cise. The model also generates the correct
generate a new aggregate state using the discrete- autocorrelation for unemployment, although the
state-space mean-reverting stochastic process behavior of vacancies is somewhat off target. In
described in Appendix C, and repeat. At the end the data, vacancies are as persistent and volatile
of each period (quarter), I record the aggregate as unemployment, while in the model the auto-
state and the unemployment rate. correlation of vacancies is significantly lower
I throw away the first 1,000 “quarters” of than that of unemployment, while the standard
data. I then use the model to generate 212 data deviation of vacancies is three times as large as
points, corresponding to quarterly data from the standard deviation of unemployment fluctu-
1951 to 2003, and detrend the log of the model- ations around trend. It is likely that anything
generated data using an HP filter with the usual that makes vacancies a state variable, such as
smoothing parameter 105. I repeat this 10,000 planning lags, an adjustment cost, or irrevers-
times, giving me good estimates of both the ibility in vacancy creation, would increase their
mean of the model-generated data and the stan- persistence and reduce their volatility, bringing
dard deviation across model-generated observa- the model more in line with the data along these
tions. The latter provides a sense of how dimensions. Shigeru Fujita (2003) develops a
precisely the model predicts the value of a par- model that adds these realistic features.
ticular variable. But the real problem with the model lies in
the volatility of vacancies and unemployment
E. Results or, more succinctly, in the volatility of the v-u
ratio ␪ and the job-finding rate f. In a reasonably
Table 3 reports the results from simulations calibrated model, the v-u ratio is less than 10
of the model with labor productivity shocks. percent as volatile as in U.S. data. This is ex-
Along some dimensions, notably the co- actly the result predicted from the deterministic
movement of unemployment and vacancies, the comparative statics in Section II C. A 1-percent
model performs remarkably well. The empirical increase in labor productivity p from its average
correlation between these two variables is value of 1 raises net labor productivity p ⫺ z by
⫺0.89, the Beveridge curve. The model actually about 1.66 percent. Using the deterministic
produces a stronger negative correlation, ⫺0.93, model, I argued before that the elasticity of the
40 THE AMERICAN ECONOMIC REVIEW MARCH 2005

TABLE 4—SEPARATION RATE SHOCKS

u v v/u f s
Standard deviation 0.065 0.059 0.006 0.002 0.075
(0.007) (0.006) (0.001) (0.000) (0.007)
Quarterly autocorrelation 0.864 0.862 0.732 0.732 0.733
(0.026) (0.026) (0.048) (0.048) (0.048)
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 deviations
from an HP trend with smoothing parameter 105. Bootstrapped standard errors—the standard deviation across 10,000 model
simulations—are reported in parentheses. The text provides details on the stochastic process for the separation rate.

v-u ratio with respect to net labor productivity is dard deviations is about 0.08 and the two vari-
about 1.03 with this choice of parameters, giv- ables are strongly negatively correlated.
ing a total elasticity of ␪ with respect to p of One might be concerned that the disjoint
approximately 1.66 ⫻ 1.03 ⫽ 1.71 percent. In analysis of labor productivity and separation
fact, the standard deviation of log ␪ around shocks masks some important interaction be-
trend is 1.75 times as large as the standard tween the two impulses. Modeling an endoge-
deviation of log p. Similarly, the job-finding nous increase in the separation rate due to low
rate is 12 times as volatile in the data as in the labor productivity, as in Mortensen and Pissar-
model. ides (1994), goes beyond the scope of this pa-
Not only is there little amplification, but there per. Instead, I introduce perfectly negatively
is also no propagation of the labor productivity correlated labor productivity and separation
shock in the model. The contemporaneous cor- shocks into the basic model. More precisely, I
relation between labor productivity, the v-u ra- assume p ⫽ z ⫹ ey(p* ⫺ z) and s ⫽ e⫺␴sys*,
tio, and the job-finding rate is 1.00. In the data, both nonlinear functions of the same latent vari-
the contemporaneous correlation between the able y. The parameter ␴s ⬎ 0 permits a different
first two variables is 0.40 and the v-u ratio lags volatility for p and s.
labor productivity by about one year. The em- I start with the parameterization of the model
pirical correlation between labor productivity with only labor productivity shocks and intro-
and the job-finding rate is similar. duce volatility in the separation rate. Table
Table 4 reports the results from the model 5 shows the results from a calibration with equal
with shocks to the separation rate. These intro- standard deviations in the deviation from trend
duce an almost perfectly positive correlation of the separation rate and labor productivity
between unemployment and vacancies, an event (␴s ⫽ 1 ⫺ z). The behavior of vacancies in the
that has essentially never been observed in the model is now far from the data, with an auto-
United States at business cycle frequencies (see correlation of 0.29 (compared to 0.94 empiri-
Figure 3). As a result, separation shocks pro- cally) and a correlation with unemployment of
duce almost no variability in the v-u ratio or the ⫺0.43 (⫺0.89). The difference between model
job finding rate. Again, this is consistent with and data is highly significant both economically
the back-of-the-envelope calculations per- and statistically. Moreover, although cyclical
formed in Section II C, where I argued that the fluctuations in the separation rate boost the vol-
elasticity of the v-u ratio with respect to the atility of unemployment considerably, they
separation rate should be approximately ⫺0.10. have a small effect on the cyclical volatility of
According to the model, the ratio of the stan- the v-u ratio and job-finding rate, which remain
VOL. 95 NO. 1 SHIMER: UNEMPLOYMENT AND VACANCIES 41

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. All
variables are reported in logs as deviations from an HP trend with smoothing parameter 105. Bootstrapped standard
errors—the standard deviation across 10,000 model simulations—are reported in parentheses. The text provides details on the
stochastic process.

at around 10 percent of their empirical values. ing solution, as would be the case if wages were
Smaller fluctuations in the separation rate natu- renegotiated following each aggregate shock.
rally have a smaller effect, while realistically This stronger restriction pins down the wage as
large fluctuations in the separation rate induce a a function of the aggregate state, wp,s. This
strong positive correlation between unemploy- facilitates a more detailed discussion of wages,
ment and vacancies, even in the presence of which serves two purposes. First, modeling
correlated productivity shocks. wages illustrates that flexibility of the present
To summarize, the stochastic version of the value of wage payments is critical for many of
Pissarides (1985) model confirms that separa- the results emphasized in this paper. And sec-
tion shocks induce a positive correlation be- ond, it enables me to relate this paper to a
tween unemployment and vacancies. It also literature that examines whether search models
confirms that, while labor productivity shocks can generate rigid wages. Appendix B proves
are qualitatively consistent with a downward- that a continually renegotiated wage solves
sloping Beveridge curve, the search model does
not substantially amplify the extrinsic shocks (7) w p,s ⫽ 共1 ⫺ ␤ 兲z ⫹ ␤ 共p ⫹ c ␪ p,s 兲.
and so labor productivity shocks induce only
very small movements along the curve. This generalizes equation (1.20) in Pissarides
(2000) to a stochastic environment.
F. Wages Consider first the effect of a separation shock
on the wage. An increase in the separation rate s
Until this point, I have assumed that the sur- induces a slight decline in the v-u ratio (see Table
plus in new matches is divided according to a 4), which in turn, by equation (7), reduces wages
generalized Nash bargaining solution but have slightly. Although the direct effect of the shock
made no assumption about the division of sur- lowers firms’ profits by shortening the duration of
plus in old matches. Although this is sufficient matches, the resulting decline in wages partially
for determining the response of unemployment offsets this, so the drop in the v-u ratio is small.
and vacancies to exogenous shocks, it does not Second, consider a productivity shock. A
pin down the timing of wage payments. In this 1-percent increase in net labor productivity p ⫺
section, I introduce an additional assumption, z raises the v-u ratio by about 1 percent (see
that the surplus in all matches, new or old, is Table 3). Equation (7) then implies that the
always divided according to the Nash bargain- net wage w ⫺ z increases by about 1 percent,
42 THE AMERICAN ECONOMIC REVIEW MARCH 2005

TABLE 6—BARGAINING POWER SHOCKS

u v v/u f w
Standard deviation 0.091 0.294 0.379 0.106 0.011
(0.018) (0.086) (0.099) (0.028) (0.015)
Quarterly autocorrelation 0.940 0.837 0.878 0.878 0.864
(0.023) (0.046) (0.036) (0.036) (0.047)
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 deviations
from an HP trend with smoothing parameter 105. Bootstrapped standard errors—the standard deviation across 10,000 model
simulations—are reported in parentheses. The text provides details on the stochastic process for the workers’ bargaining
power.

soaking up most of the productivity shock and driving force, wages are counterfactually coun-
giving firms little incentive to create new va- tercyclical (Abraham and Haltiwanger, 1995).
cancies. Hence there is a modest increase in Nevertheless, it seems plausible that a model
vacancies and decrease in unemployment in re- with a combination of wage and labor produc-
sponse to a large productivity shock. tivity shocks could generate the observed be-
To understand fully the importance of wages havior of unemployment, vacancies, and real
for the v-u ratio, it is useful to consider a ver- wages. Of course, the unanswered question is
sion of the model in which labor productivity what exactly a wage shock is.
and the separation rate are constant at p ⫽ 1 and If wages are bargained in new matches but
s ⫽ 0.1, but workers’ bargaining power ␤ then not continually renegotiated, this analysis
changes stochastically. An increase in ␤ reduces is inapplicable. Nevertheless, one can prove that
the profit from creating vacancies, which puts the frequency of wage negotiation does not af-
downward pressure on the v-u ratio. It is diffi- fect the expected present value of wage pay-
cult to know exactly how much variability in ␤ ments in new matches, but only changes the
is reasonable, but one can ask how much wage timing of wage payments. An increase in pro-
variability is required to generate the observed ductivity or decrease in separations raises the
volatility in the v-u ratio. I assume ␤ is a func- present value of wage payments in new jobs and
tion of the latent variable y, ␤ ⫽ ⌽(y ⫹ therefore has little effect on the v-u ratio. An
⌽⫺1(␣)), where ⌽ is the cumulative standard increased workers’ bargaining power in a new
normal distribution. If y were constant at zero, employment relationship induces a large reduc-
this implies ␤ ⫽ ␣, but more generally ␤ is tion in vacancies and in the v-u ratio.
simply bounded between 0 and 1. I set the
standard deviation of y to ␴ ⫽ 0.099 and the III. Optimal V-U Fluctuations
mean reversion parameter to ␥ ⫽ 0.004. Al-
though this implies very modest fluctuations in Another way to highlight the role played by
wages—the standard deviation of detrended log the Nash bargaining assumption is to examine a
wages, computed as in equation (7), is just centralized economy in which it is possible to
0.01—the calibrated model generates the ob- sidestep the wage-setting issue entirely.17 Con-
served volatility in the v-u ratio, with persis-
tence similar to that in the model with labor
productivity shocks. Table 6 shows the com- 17
A number of papers examine a “competitive search
plete results. Since bargaining power is the only economy,” in which firms can commit to wages before
VOL. 95 NO. 1 SHIMER: UNEMPLOYMENT AND VACANCIES 43

sider a hypothetical social planner who chooses a special case of equation (6), with workers’
a state-contingent v-u ratio in order to maximize bargaining power ␤ equal to the elasticity ␣.
the present discounted value of output net of This generalizes the Hosios (1990) condition
vacancy creation costs. The planner’s problem for efficiency of the decentralized equilibrium
is represented recursively as to an economy with stochastic productivity and
separation rates. Since the numerical example in
rW共 p, s, u兲 ⫽ max共zu ⫹ p共1 ⫺ u兲 ⫺ cu␪ Section II E assumes a Cobb-Douglas matching
␪ function with ␣ ⫽ ␤, the equilibrium allocation
described in that section solves the social plan-
⫹ Wu 共p, s, u兲共s共1 ⫺ u兲 ⫺ uf共␪兲兲 ner’s problem. Conversely, if those parameter
values describe the U.S. economy, the observed
⫹ ␭⺕ p,s 共W共p⬘, s⬘, u兲 ⫺ W共p, s, u兲兲). degree of wage rigidity is inconsistent with out-
put maximization.
Instantaneous output is equal to z times the With other matching functions, the link be-
unemployment rate u plus p times the employ- tween the equilibrium with wage bargaining
ment rate minus c times the number of vacan- and the solution to the planner’s problem is
cies v ⬅ u␪. The value changes gradually as the broken. At one extreme, if unemployment and
unemployment rate adjusts, with u̇ ⫽ s(1 ⫺ vacancies are perfect substitutes, i.e., f(␪) ⫽
u) ⫺ uf(␪), and suddenly when an aggregate ␣u ⫹ ␣v␪, then the output-maximizing v-u ratio
shock changes the state from (p,s) to (p⬘,s⬘) at is infinite whenever ␣v(p ⫺ z) ⬎ c(r ⫹ s ⫹ ␣u)
rate ␭. and is zero if the inequality is reversed. With near-
It is straightforward to verify that the Bell- perfect substitutability, the output-maximizing v-u
man value W is linear in the unemployment rate, ratio is very sensitive to current productivity. On
Wu(p,s,u) ⫽ ⫺c/f⬘(␪p,s), and the v-u ratio satis- the other hand, if unemployment and vacancies
fies are perfect complements, f(␪) ⫽ min具␣u,␣v␪典, the
v-u ratio never strays from the efficient ratio

冉 冊
␣u /␣v. With imperfect complements, the impact
r⫹s⫹␭ f共 ␪ p,s 兲
⫺ ␪ p,s 1 ⫺ of productivity shocks on the v-u ratio is muf-
f⬘共 ␪ p,s 兲 ␪ p,s f⬘共 ␪ p,s 兲 fled but not eliminated.

冉 冊
The economics behind these theoretical find-
p⫺z 1 ings is simple. An increase in labor productivity
⫽ ⫹ ␭⺕p,s .
c f⬘共␪p⬘,s⬘ 兲 relative to the value of non-market activity and
the cost of advertising a vacancy induces a
switch away from the expensive activity, unem-
This implicitly defines the optimal ␪p,s, indepen- ployment, and toward the relatively cheap ac-
dent of the unemployment rate. tivity, vacancies. The magnitude of the switch
With a Cobb-Douglas matching function depends on how substitutable unemployment
m(u,v) ⫽ ␮u␣v1 ⫺ ␣, this reduces to and vacancies are in the matching function. If
they are strong complements, substitution is
r⫹s⫹␭ nearly impossible and the v-u ratio barely
⫹ ␣␪ p,s changes. If they are strong substitutes, substitu-
q共 ␪ p,s 兲
tion is nearly costless, and the v-u ratio is highly

⫽ 共1 ⫺ ␣兲
p⫺z
c
⫹ ␭⺕ p,s 冉
1
q共␪p⬘,s⬘ 兲 冊 procyclical.
In the decentralized economy, the extent of
substitution between unemployment and vacan-
cies is governed not only by the matching func-
tion but also by the bargaining solution, as
hiring workers and can increase their hiring rate by prom- shown by the comparative statics exercises in
ising higher wages (Peters, 1991; Montgomery, 1991; Section II C. The Nash bargaining solution ef-
Moen, 1997; Shimer, 1996; Burdett et al., 2001). It is by fectively corresponds to a moderate degree of
now well-known that a competitive search equilibrium max-
imizes output, essentially by creating a market for job
substitutability, the Cobb-Douglas case. If
applications. This discussion of output-maximizing search wages were more rigid, an increase in produc-
behavior therefore also pertains to these models. tivity would induce more vacancy creation and
44 THE AMERICAN ECONOMIC REVIEW MARCH 2005

less unemployment, analogous to a centralized mond (1989) also focus on the negative corre-
environment with a high elasticity of substitu- lation between unemployment and vacancies,
tion in the matching function. but they do not model the supply of jobs and
The substitutability of unemployment and va- hence do not explain why there are so few
cancies is an empirical issue. Blanchard and vacancies during recessions. Instead, they as-
Diamond (1989) use nonlinear least squares to sume the total stock of jobs follows an exoge-
estimate a Constant Elasticity of Substitution nous stochastic process. This paper pushes the
(CES) matching function on U.S. data. Their cyclicality of the v-u ratio to the front of the
point estimate for the elasticity of substitution is picture. Likewise, Cole and Rogerson (1999)
0.74, i.e., slightly less substitutable than the argue that the Mortensen and Pissarides (1994)
Cobb-Douglas case, although they cannot reject model can match a variety of business cycle
the Cobb-Douglas elasticity of 1. As footnote 8 facts, but they do so in a reduced form model
describes, my data suggest an elasticity slightly that treats fluctuations in the job-finding rate, and
in excess of 1, although my point estimate is hence implicitly in the v-u ratio, as exogenous.
imprecise. Whether the observed movements in The second group of papers, including work
unemployment and vacancies are optimal when by Michael Pries (2004), Gary Ramey and Joel
viewed through the lens of the textbook search Watson (1997), Wouter Den Haan et al. (2000),
and matching model, therefore, remains an open and Joao Gomes et al. (2001), assumes that
question. employment fluctuations are largely due to
time-variation in the separation rate, minimiz-
IV. Related Literature ing the role played by the observed cyclicality
of the v-u ratio. These papers typically deliver
There is a large literature that explores rigid wages from a search model, consistent
whether the search model is consistent with the with the findings in Section II E. Building on
cyclical behavior of labor markets. Some papers the ideas in Hall (1995), Pries (2004) shows that
look at the implications of the model for the a brief adverse shock that destroys some old
behavior of various stocks and flows, including employment relationships can generate a long
the unemployment and vacancy rates, but do not transition period of high unemployment, as the
examine the implicit magnitude of the exogenous displaced workers move through a number of
impulses. Others assume that business cycles are short-term jobs before eventually finding their
driven by fluctuations in the separation rate s. way back into long-term relationships. During
These papers either impose exogenously or derive this transition process, the v-u ratio remains
within the model a counterfactually constant v-u constant, since aggregate economic conditions
ratio ␪. A third group of papers has tried but failed have returned to normal. Equivalently, the
to reconcile the procyclicality of the v-u ratio with economy moves along an upward-sloping Bev-
extrinsic shocks of a plausible magnitude. eridge curve during the transition period, in
Papers by Abraham and Lawrence Katz contradiction to the evidence. Ramey and
(1986), Blanchard and Diamond (1989), and Watson (1997) argue that two-sided asymmetric
Cole and Rogerson (1999) fit into the first cat- information generates rigid wages in a search
egory, matching the behavior of labor market model. But in their model, shocks to the sepa-
stocks and flows by sidestepping the magnitude ration rate are the only source of fluctuations in
of impulses. For example, Abraham and Katz unemployment. The job-finding rate f(␪) is ex-
(1986) argue that the downward-sloping Bever- ogenous and constant, which is equivalent to
idge curve is inconsistent with models in which assuming that vacancies are proportional to un-
unemployment is driven by fluctuations in the employment. This is probably an important part
separation rate, notably David Lilien’s (1982) of the explanation for why their model produces
sectoral shifts model. That leads them to advo- rigid wages. Den Haan et al. (2000) show that
cate an alternative in which unemployment fluctuations in the separation rate amplify pro-
fluctuations are driven by aggregate distur- ductivity shocks in a model similar to the one
bances, e.g., productivity shocks. Unfortu- examined here; however, they do not discuss
nately, they fail to examine the magnitude of the cyclical behavior of the v-u ratio. Similarly,
shocks needed to deliver the observed shifts Gomes et al. (2001) sidestep the v-u issue by
along the Beveridge curve. Blanchard and Dia- looking at a model in which the job-finding rate
VOL. 95 NO. 1 SHIMER: UNEMPLOYMENT AND VACANCIES 45

is exogenous and constant, i.e., vacancies are by Nash bargaining cannot generate substantial
proportional to unemployment. Again, this movements along a downward-sloping Bever-
helps keep wages relatively rigid in their model. idge curve in response to shocks of a plausible
Mortensen and Pissarides (1994) have prob- magnitude. A labor productivity shock results
ably the best known paper in this literature. In primarily in higher wages, with little effect on
their three-state “illustrative simulation,” the the v-u ratio. A separation shock generates an
authors introduce, without comment, enormous increase in both unemployment and vacancies.
productivity or leisure shocks into their model. It is important to stress that this is not an attack
Average labor productivity minus the value of on the search approach to labor markets, but
leisure p ⫺ z is approximately three times as rather a critique of the commonly-used Nash
high in the good state as in the bad state.18 This bargaining assumption for wage determination.
paper confirms that in response to such large An alternative wage determination mechanism
shocks, the v-u ratio should also be about three that generates more rigid wages in new jobs,
times as large in the good state as in the bad measured in present value terms, will amplify
state, but argues that there is no evidence for the effect of productivity shocks on the v-u
these large shocks in the data. Even if one ratio, helping to reconcile the evidence and the-
accepts the magnitude of the implied impulses, ory. Countercyclical movements in workers’
Mortensen and Pissarides (1994) still deliver bargaining power provide one such mechanism,
only a correlation of ⫺0.26 between unemploy- at least in a reduced-form sense.
ment and vacancies, far lower than the empiri- If the matching function is Cobb-Douglas,
cal value of ⫺0.88. This is probably because of the observed behavior of the v-u ratio is not
the tension between productivity shocks, which socially optimal for plausible parameterizations
put the economy on a downward-sloping Bev- of the model, but this conclusion could be over-
eridge curve, and endogenous movements in the turned if the elasticity of substitution between
separation rate, which have the opposite effect. unemployment and vacancies in the matching
Monika Merz (1995) and David Andolfatto function is sufficiently large. Estimates of a
(1996) both put the standard search model into CES matching function are imprecise, so it is
a real business cycle framework with intertem- unclear whether observed wages are “too rigid.”
poral substitution of leisure, capital accumula- One way to generate more rigid wages in a
tion, and other extensions. Neither paper can theoretical model is to introduce considerations
match the negative correlation between unem- whereby wages affect the worker turnover rate.
ployment and vacancies, and both papers gen- For example, in the Burdett and Mortensen
erate real wages that are too flexible in response (1998) model of on-the-job search, firms have
to productivity shocks. Thus these papers en- an incentive to offer high wages in order to
counter the problem I highlight in this paper, attract workers away from competitors and to
although they do not emphasize this shortcom- reduce employees’ quit rate. The distribution of
ing of the search model. Finally, Hall (2003), productivity affects an individual firm’s wage
building on an earlier version of this paper, offer and vacancy creation decisions in complex
discusses some of the same issues. Hall (2005) ways, breaking the simple link between average
proposes one possible solution: real wages are labor productivity and the v-u ratio in the Pis-
determined by a social norm that does not sarides (1985) model. In particular, a shift in the
change over the business cycle. productivity distribution that leaves average la-
bor productivity unchanged may appreciably
V. Conclusion affect average wages and hence the equilibrium
v-u ratio.
I have argued in this paper that a search and Another possibility is to drop some of the
matching model in which wages are determined informational assumptions in the standard
search model.19 Suppose that when a worker
18
This calculation would be easy in the absence of
heterogeneity, i.e., if their parameter ␴ were equal to zero.
Then p៮ ⫺ z would take on three possible values: 0.022, 19
Ramey and Watson (1997) develop a search model
0.075, and 0.128, for a six-fold difference in p៮ ⫺ z between with two-sided asymmetric information. Because they as-
the high and low states. sume workers’ job finding rate is exogenous and acyclic,
46 THE AMERICAN ECONOMIC REVIEW MARCH 2005

and firm meet, they draw an idiosyncratic leisure z plus the probability she finds a job
match-specific productivity level from some f(␪p,s) times the resulting capital gain E ⫺ U
distribution F. Workers and firms know about plus the probability of an aggregate shock times
aggregate variables, including the unemploy- that capital gain. Equation (9) expresses a sim-
ment rate and the distribution F, but only the ilar idea for an employed worker, who receives
firm knows the realized productivity level. Bar- a wage payment wp,s but loses her job at rate s.
gaining proceeds as follows: with probability Equation (10) provides an analogous recursive
␤ 僆 (0,1), a worker makes a take-it-or-leave-it formulation for the value of a filled job. Note
wage demand, and otherwise the firm makes a that a firm is left with nothing when a filled job
take-it-or-leave-it offer. Obviously the firm ex- ends.
tracts all the rents from the employment rela- Sum equations (9) and (10) and then subtract
tionship when it makes an offer. But if the equation (8), defining Vp,s ⬅ Jp,s ⫹ Ep,s ⫺ Up,s:
uninformed worker makes the offer, she faces a
tradeoff between demanding a higher wage and (11) rV p,s ⫽ p ⫺ z ⫺ f共 ␪ p,s 兲共E p,s ⫺ U p,s 兲
reducing her risk of unemployment, so the wage
depends on the hazard rate of the distribution F. ⫺ sVp,s ⫹ ␭共⺕
⺕p,s Vp⬘,s⬘ ⫺ Vp,s 兲.
This again breaks the link between average la-
bor productivity and the equilibrium v-u ratio. In addition, the Nash bargaining solution im-
Exploring whether either of these models, or plies that the wage is set so as to maximize the
some related model, deliver substantial fluctua- Nash product (Ep,s ⫺ Up,s)␤Jp,s1⫺␤
, which gives
tions in the v-u ratio in response to plausible
impulses remains a topic for future research. E p,s ⫺ U p,s J p,s
(12) ⫽ V p,s ⫽ .
␤ 1⫺␤
APPENDIX A: DERIVATION OF THE EQUATION FOR
SURPLUS (4) Substituting for E ⫺ U in equation (11) yields
equation (4).
For notational simplicity alone, assume the If I allow wages to depend in an arbitrary
wage payment depends only on the aggregate manner on the history of the match, this would
state, wp,s, not on the history of the match. I affect the Bellman values E and J; however, the
return to this issue at the end of this section. wage, and therefore the history-dependence,
Define Up,s, Ep,s, and Jp,s to be the state- would drop out when summing the Bellman
contingent present value of an unemployed equations for E and J. In other words, the match
worker, employed worker, and filled job, re- surplus V is unaffected by the frequency of
spectively. They are linked recursively by: wage renegotiation.

(8) rU p,s ⫽ z ⫹ f共 ␪ p,s 兲共E p,s ⫺ U p,s 兲 APPENDIX B: DERIVATION OF THE WAGE
EQUATION
⫹ ␭共⺕
⺕p,s Up⬘,s⬘ ⫺ Up,s 兲
Assume that wages are continually renegoti-
(9) rE p,s ⫽ w p,s ⫺ s共E p,s ⫺ U p,s 兲 ated, so the wage depends only on the current
aggregate state (p,s). Eliminate current and fu-
⫹ ␭共⺕
⺕p,s Ep⬘,s⬘ ⫺ Ep,s 兲 ture values of J from equation (10) using equa-
tion (12):
(10) rJ p,s ⫽ p ⫺ w p,s ⫺ sJ p,s
w p,s ⫽ p ⫺ 共r ⫹ s ⫹ ␭ 兲共1 ⫺ ␤ 兲V p,s
⫹ ␭共⺕
⺕p,s Jp⬘,s⬘ ⫺ Jp,s 兲.
⫹ ␭⺕ p,s 共1 ⫺ ␤兲Vp⬘,s⬘ .
Equation (8) states that the flow value of an
unemployed worker is equal to her value of Similarly, eliminate current and future values of
V using (5):
共r ⫹ s ⫹ ␭ 兲c c
their results are not directly applicable to this analysis, w p,s ⫽ p ⫺ ⫹ ␭ ⺕ p,s .
although their methodology may prove useful. q共 ␪ p,s 兲 q共 ␪ p⬘,s⬘ 兲
VOL. 95 NO. 1 SHIMER: UNEMPLOYMENT AND VACANCIES 47

Finally, replace the last two terms using equa- Var关y共t ⫹ h兲 ⫺ y共t兲兩y共t兲兴
tion (6) to get equation (7).
⺕[(y(t⫹h)⫺y(t))2兩y(t)]
⫽⺕
APPENDIX C: THE STOCHASTIC PROCESS
⫺ 共⺕
⺕关y共t ⫹ h兲 ⫺ y共t兲兩y共t兲兴兲2.
The text describes a continuous state space
approximation to the discrete state space model
used in both the theory and simulations. Here I The first term evaluates to h␭⌬2 over a suffi-
describe the discrete state space model and ciently short time interval h, since it is equal to
show that it asymptotes to an Ornstein- ⌬2 if a shock, positive or negative, arrives and
Uhlenbeck process. zero otherwise. The second term is (h␥y(t))2,
Consider a random variable y that is hit with and so is negligible over a short time interval h.
shocks according to a Poisson process with ar- Thus
rival rate ␭. The initial value of y lies on a
discrete grid,
Var关y共t ⫹ h兲 ⫺ y共t兲兩y共t兲兴 ⫽ h␭⌬2 ⫽ h␴2.
y 僆 Y ⬅ 兵⫺n⌬, ⫺共n ⫺ 1兲⌬, ... ,
Putting this together, we can represent the sto-
0, ... , 共n ⫺ 1兲⌬, n⌬} chastic process for y as

where ⌬ ⬎ 0 is the step size and 2n ⫹ 1 ⱖ 3 is

dy ⫽ ⫺␥ydt ⫹ ␴dx
the number of grid points. When a shock hits,
the new value y⬘ either moves up or down by
one grid point: where for t ⬎ 0, the expected value of x(t) given
x(0) is x(0) and the conditional variance is t.

冉 冊
冦
1 y This is similar to a Brownian motion, except

再
1⫺ that the innovations in x are not Gaussian, since
y⫹⌬ 2 n⌬

冉 冊
y⬘ ⫽ y ⫺ ⌬ with probability 1 y . y is constrained to lie on a discrete grid.
1⫹ Now suppose one changes the three parame-
2 n⌬ ters of the stochastic process, the step size,
arrival rate of shocks, and number of steps, from
Note that although the step size is constant, (⌬,␭,n) to (⌬公␧, ␭/␧, n/␧) for any ␧ ⬎ 0. It is
the probability that y⬘ ⫽ y ⫹ ⌬ is smaller when easy to verify that this does not change either
y is larger, falling from 1 at y ⫽ ⫺n⌬ to zero at the autocorrelation parameter ␥ ⫽ ␭/n or the
y ⫽ n⌬. instantaneous variance ␴ ⫽ 公␭⌬. But as
It is trivial to confirm that y⬘ 僆 Y, so the state ␧ 3 0, the distribution of the innovation process
space is discrete. To proceed further, define ␥ ⬅ x converges to a normal by the Central Limit
␭/n and ␴ ⬅ 公␭⌬. For any fixed y(t), I examine Theorem. Equivalently, y converges to an
the behavior of y(t ⫹ h) over an arbitrarily short Ornstein-Uhlenbeck process.20 This observa-
time period h. For sufficiently short h, the prob- tion is also useful for computation. It is possible
ability that two Poisson shocks arrive is negli- to find a solution on a coarse grid and then to
gible, and so y(t ⫹ h) is equal to y(t) with refine the grid by decreasing ␧ without substan-
probability 1 ⫺ h␭, has increased by ⌬ with tially changing the results.
probability h␭(1 ⫺ y(t)/n⌬)/2, and has de-
creased by ⌬ with probability h␭ (1 ⫹ y(t)/
n⌬)/2. Adding this together shows 20
Notably, for large n it is extraordinarily unlikely that
the state variable reaches its limiting values of ⫾n⌬. The
h␭ unconditional distribution of the state variable is approxi-
⺕ 关 y共t ⫹ h兲 ⫺ y共t兲兩y共t兲兴 ⫽ ⫺ y共t兲 ⫽ ⫺h␥y共t兲. mately normal with mean zero and standard deviation
n ␴/公2␥ ⫽ ⌬公n/2. The limiting values of the state variables
therefore lie 公2n standard deviations above and below the
mean. If n ⫽ 1000, as is the case in the simulations, one
Next, the conditional variance of y(t ⫹ h) ⫺ y(t) should expect to observe such values approximately once in
can be decomposed into 10436 periods.
48 THE AMERICAN ECONOMIC REVIEW MARCH 2005

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