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- [email protected]). 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 But the increase in hiring also shortens unem- material is based upon work supported by the National 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 version of this paper, and to Mihai Manea, and especially 1 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 Ϫ0.89 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 Ϫ0.1 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 ϭ 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 matching 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 vv/uf 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 Ϫ0.894 Ϫ0.971 Ϫ0.949 0.709 Ϫ0.408 v — 1 0.975 0.897 Ϫ0.684 0.364 Correlation matrix v/u — — 1 0.948 Ϫ0.715 0.396 f —— — 1Ϫ0.574 0.396 s ————1Ϫ0.524 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 unemployment. Christopher Flinn and James Heckman (1983) show that unemployed workers are significantly 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 procyclical, 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 5 both direct and indirect evidence against this parameter 10 . hypothesis. As direct evidence, one would expect 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 decreasing in unemployment. I conclude that aggregate job search activity is positively correlated with unemployment.

B. Vacancies

The flip side of unemployment is job vacancies. 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 recruiting from outside of the establishment to averaged and seasonally adjusted. The dashed line shows 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 establishment 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 Ϫ0.89 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. 6 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 constructed by the Conference Board with normalization 1987 ϭ 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 seasonally adjusted monthly series and are expressed as devi- 5 standard deviation of the cyclical variation in ations from an HP filter with smoothing parameter 10 . unemployment and vacancies is almost identical, 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 ϵ m(u,v)/u ϭ m(1,␪), where level and short-term unemployment level. Let ␪ ϵ s v/u is the v-u ratio. The job-finding rate f ut 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 ϭ ͑ Ϫ ͒ ϩ s vacancy rates in the opposite direction. ut ϩ 1 ut 1 ft ut ϩ 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 Unemployment next month is the sum of the series and are expressed as deviations from an HP filter with 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 Ϫ s ut ϩ 1 ut ϩ 1 m(u,v). Data limitations force me to impose two (1) ft ϭ 1 Ϫ . ut restrictions on the estimated function. First, because 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 7 pute ft from 1951 to 2003. 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- ϭ 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) ϭ ␮u␣v1Ϫ␣, 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 unemployment. They suggest some methods of dealing with this 8 ϭ ␮ ϩ discontinuity. In this paper, I simply inflate short-term un- Consider the CES matching function log ft log ␳ ␣ ϩ Ϫ ␣ ␪ ␳ 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 ␣ ϭ 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 the unemployment level and mean unemploy- FIGURE 7. MONTHLY SEPARATION PROBABILITY FOR EMPLOYED WORKERS, 1951–2003 ment duration to obtain a weighted average of the job-finding rate in the unemployed popula- Notes: The separation rate is computed using equation (2), with employment, unemployment, and short-term unem- tion, with weights proportional to each individ- ployment data constructed and seasonally adjusted by the 9 ual’s unemployment duration. 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. The trend is an HP filter of the quarterly data with smooth- job-finding rate of workers with short unem- 5 ployment duration using the ratio of workers ing parameter 10 . 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 s ϭ ͑ Ϫ 1 ͒ month. Nevertheless, all three measures are ut ϩ 1 st et 1 ft . highly correlated, and so the choice of which 2 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 s first that whenever an employed worker loses u ϩ ϭ t 1 her job, she becomes unemployed. Then the (2) st . e ͑1 Ϫ 1 f ͒ separation rate could simply be computed as the t 2 t ratio of short-term unemployed workers next s month, utϩ1, 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 ␳ ϭ 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 ␳ ϭ 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 Ϫ 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 ϭ 100. The trend is an HP ﬁlter of the quarterly 5 Costs program. The v-u ratio and labor productivity are data with smoothing parameter 10 . 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, 10 increase in unemployment ut, so that the num- with a maximum correlation of 0.56. 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, Ϫ0.37 and Ϫ0.43, 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 Ͼ 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 measure of vacancies in the economy, and ␪(t) ϵ 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)) ϵ 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)) ϵ m(␪(t)Ϫ1,1) ϭ 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 (pЈ,sЈ) is drawn from a state de- bargaining solution. The worker can threaten to ޅ pendent distribution. Let p,sXpЈ,sЈ 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, conditional 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 ␤ ʦ (0, 1) of the surplus, her “bargain- entry implies the firm would be left with nothing 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 ޅ Ϫ fire the worker or the worker would prefer expected change in match value p,sVpЈ,sЈ 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 Ϫ ␤ 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, ␪ .12 Given the state-contingent v-u p,s (5) c ϭ q͑␪ ͒͑1 Ϫ ␤͒V . ratio, the unemployment rate evolves according p,s p,s to a standard backward-looking differential equation, Eliminating current and future values of Vp,s from (4) using (5) gives ͑ ͒ ϭ ͑ ͒͑ Ϫ ͑ ͒͒ Ϫ ͑␪ ͒ ͑ ͒ (3) u˙ t s t 1 u t f p͑t͒,s͑t͒ u t r ϩ s ϩ ␭ ϩ ␤␪ (6) ͑␪ ͒ p,s where (p(t),s(t)) is the aggregate state at time t. q p,s A flow s(t) of the 1 Ϫ u(t) employed workers ␪ p Ϫ z 1 become unemployed, while a flow f( )ofthe ϭ͑ Ϫ ␤͒ ϩ ␭ޅ 1 p,s ͑␪ ͒ u(t) unemployed workers find a job. An initial c q pЈ,sЈ 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 ϭ p Ϫ ͑z ϩ f͑␪ ͒␤V ͒ Ϫ sV p,s p,s p,s p,s the literature. ϩ ␭͑ޅ Ϫ ͒ p,s VpЈ,sЈ Vp,s . 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 of this model that there is no other equilibrium, e.g., one in 13 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. Ϫs ␭ ϭ . First, suppose there are no aggregate shocks, ͑r ϩ s͒͑1 Ϫ ␩͑␪ ͒͒ ϩ ␤f͑␪ ͒ 0.14 Then the state-contingent v-u ratio satisfies p,s p,s Substituting the same numbers into this expres- Ϫ r ϩ s p Ϫ z sion gives 0.10. Doubling the separation rate ϩ ␤␪ ϭ ͑ Ϫ ␤͒ would have a scarcely discernible impact on the ͑␪ ͒ p,s 1 . q p,s 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 Ϫ z is steady state, equation (3) holds with u˙ ϭ 0. If the matching function is Cobb-Douglas, ␣ 1Ϫ␣ ϩ ϩ ␤ ͑␪ ͒ m(u,v) ϭ ␮u v , this implies r s f p,s ͑ ϩ ͒͑ Ϫ ␩͑␪ ͒͒ ϩ ␤ ͑␪ ͒ r s 1 p,s f p,s s͑1 Ϫ u ͒ 1/͑1 Ϫ ␣͒ ϭ ͩ p,s ͪ vp,s ␮ ␣ . where ␩(␪) ʦ [0,1] is the elasticity of f(␪). This up,s 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 Ϫ ␩(␪), 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. ␤ ϭ 0.1, but only at ␤ ϭ 0 does the elasticity of I can perform similar analytic exercises by the v-u ratio with respect to p Ϫ 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(␪) ϭ ␮. 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 ϩ s p Ϫ z ϩ ␤␪ ϭ ͑1 Ϫ ␤͒ . ␮ p,s c 14 Shimer (2003) performs comparative statics exercises under much weaker assumptions. For example, in that paper In this case, the elasticity of the v-u ratio with the matching function can exhibit increasing or decreasing returns to scale and there can be an arbitrary idiosyncratic respect to net labor productivity is process for productivity, allowing for endogenous separations (Mortensen and Pissarides, 1994). I show that the ϩ ϩ ␤␮␪ 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 Ϫs/␤␮␪. Since f(␪) ϭ ␮␪, 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 ␤ ϭ in my simulations as well, it is almost exactly 0.72, I obtain elasticities of 1.12 and Ϫ0.105, 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(␪) ϭ ␮ and q(␪) ϭ ␮/␪. Also assume that the separation rate s is constant and dy ϭ Ϫ␥ydt ϩ ␴db average labor productivity p is a Martingale, ޅ Јϭ 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 ␥ Ͼ 0 is a measure of persistence, with higher v-u ratio: values indicating faster mean reversion, and ␴ Ͼ 0 is the instantaneous standard deviation. r ϩ s ϩ ␭ ͩ ϩ ␤ͪ␪ This process has some convenient properties: y ␮ p is conditionally and unconditionally normal; it is mean reverting, with expected value converg- p Ϫ z ␭ ing asymptotically to zero; and asymptotically 2 ϭ͑1 Ϫ ␤͒ ϩ ޅ ␪ Ј . ␴ ␥ c ␮ p p its variance converges to /2 . 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- ޅ ␪ ϭ ␪ ϭ ϩ y Ϫ linear in productivity, and therefore p pЈ p, i.e., isfies p z e (p* z), where y is an Ornstein-Uhlenbeck process with parameters ␥ r ϩ s p Ϫ z and ␴, and p* Ͼ z is a measure of long-run ͩ ϩ ␤ͪ␪ ϭ ͑1 Ϫ ␤͒ y Ͼ ␮ p c average productivity. Since e 0, this ensures p Ͼ z. In the second case, productivity is con- so the elasticity of the v-u ratio with respect to stant and separations satisfy s ϭ 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* Ͼ 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 Ϫ 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. unemployment rate. The most important question Ϫ ␣ f͑␪͒ ϭ ␪q͑␪͒ ϭ ␮␪1 . is the choice of the Markov process for labor productivity and separations. Appendix C develops 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 ϩ 1 ϭ 2001 points, ordered set of points. When a Poisson shock which closely approximate Gaussian innovations. This implies that Poisson arrival rate of shocks is ␭ ϭ n␥ ϭ 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- ␭ ϭ 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 vacancy 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 ϭ Value of leisure z 0.4 0.4 productivity parameter to p* 1. I choose the Matching function q(␪) 1.355␪Ϫ0.72 1.355␪Ϫ0.72 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 ␥ ␴ ϭ 0.0165 and ␥ ϭ 0.004. An increase in the Autoregressive parameter 0.004 0.220 Grid size 2n ϩ 1 2001 2001 volatility of productivity ␴ has a nearly proportional 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 example, 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 ␣ ϭ 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 ϭ 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 ␮␪1Ϫ␣ 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 ϭ 0.012, tion (6) implies that doubling c and multiplying equivalent to an annual discount factor of 0.953. ␮ by a factor 21Ϫ␣ 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 ϭ 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 ␮ ϭ 1.355 and c ϭ 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* ϭ 0.10. I model. set ␴ ϭ 0.057 and ␥ ϭ 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 ␣ ϭ 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 vv/uf 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 Ϫ0.927 Ϫ0.958 Ϫ0.958 Ϫ0.958 (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 Ϫ z by Ϫ0.89, the Beveridge curve. The model actually about 1.66 percent. Using the deterministic produces a stronger negative correlation, Ϫ0.93, model, I argued before that the elasticity of the 40 THE AMERICAN ECONOMIC REVIEW MARCH 2005

TABLE 4—SEPARATION RATE SHOCKS

u vv/uf 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 Ϫ0.906 Ϫ0.906 0.908 (0.000) (0.017) (0.017) (0.017) v —1 Ϫ0.887 Ϫ0.887 0.888 (0.020) (0.020) (0.021) Correlation matrix v/u — — 1 1.000 Ϫ0.999 (0.000) (0.000) f —— —1 Ϫ0.999 (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 ϫ 1.03 ϭ 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 Ϫ␴ relation between labor productivity, the v-u ra- assume p ϭ z ϩ ey(p* Ϫ z) and s ϭ e sys*, tio, and the job-finding rate is 1.00. In the data, both nonlinear functions of the same latent vari- ␴ Ͼ 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 ␴ ϭ Ϫ 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 Ϫ0.43 (Ϫ0.89). 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 Ϫ0.10. 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 vv/uf 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 Ϫ0.427 Ϫ0.964 Ϫ0.964 0.964 Ϫ0.964 (0.068) (0.011) (0.011) (0.011) (0.011) v — 1 0.650 0.650 Ϫ0.649 0.648 (0.042) (0.042) (0.042) (0.042) Correlation matrix v/u — — 1 1.000 Ϫ1.000 0.999 (0.000) (0.000) (0.001) f —— —1Ϫ1.000 0.999 (0.000) (0.001) s ————1Ϫ0.999 (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 section 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 ϭ ͑ Ϫ ␤͒ ϩ ␤͑ ϩ ␪ ͒ not substantially amplify the extrinsic shocks (7) wp,s 1 z p c p,s . 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 Ϫ 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 Ϫ z increases by about 1 percent, 42 THE AMERICAN ECONOMIC REVIEW MARCH 2005

TABLE 6—BARGAINING POWER SHOCKS

u vv/uf 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 Ϫ0.915 Ϫ0.949 Ϫ0.949 0.818 (0.045) (0.032) (0.032) (0.112) v — 1 0.995 0.995 Ϫ0.827 (0.001) (0.001) (0.128) Correlation matrix v/u — — 1 1.000 Ϫ0.838 (0.000) (0.124) f —— —1Ϫ0.838 (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 ϭ 1 and If wages are bargained in new matches but s ϭ 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, ␤ ϭ⌽(y ϩ therefore has little effect on the v-u ratio. An ⌽Ϫ1(␣)), 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 ␤ ϭ ␣, 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 ␴ ϭ 0.099 and the III. Optimal V-U Fluctuations mean reversion parameter to ␥ ϭ 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 persistence 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 rW͑ p, s, u͒ ϭ max͑zu ϩ p͑1 Ϫ u͒ Ϫ cu␪ Section II E assumes a Cobb-Douglas matching ␪ function with ␣ ϭ ␤, the equilibrium allocation described in that section solves the social plan- ϩ Wu ͑p, s, u͒͑s͑1 Ϫ u͒ Ϫ uf͑␪͒͒ ner’s problem. Conversely, if those parameter values describe the U.S. economy, the observed ϩ ␭ޅ ͑ ͑ Ј Ј ͒ Ϫ ͑ ͒͒ p,s W p , s , u W p, s, u ). degree of wage rigidity is inconsistent with output 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 ϵ u␪. The value changes gradually as the broken. At one extreme, if unemployment and unemployment rate adjusts, with u˙ ϭ s(1 Ϫ vacancies are perfect substitutes, i.e., f(␪) ϭ Ϫ ␪ ␣ ϩ ␣ ␪ u) uf( ), and suddenly when an aggregate u v , then the output-maximizing v-u ratio Ј Ј ␣ Ϫ Ͼ ϩ ϩ ␣ 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 ϭϪ Ј ␪ Wu(p,s,u) c/f ( p,s), and the v-u ratio satis- the other hand, if unemployment and vacancies ␪ ϭ ͗␣ ␣ ␪͘ fies are perfect complements, f( ) min u, v , the v-u ratio never strays from the efficient ratio ␣ /␣ . With imperfect complements, the impact r ϩ s ϩ ␭ f͑␪ ͒ u v Ϫ ␪ ͩ Ϫ p,s ͪ of productivity shocks on the v-u ratio is muf- Ј͑␪ ͒ p,s 1 ␪ Ј͑␪ ͒ f p,s p,s f p,s fled but not eliminated. The economics behind these theoretical find- p Ϫ z 1 ϭ ϩ ␭ޅ ͩ ͪ ings is simple. An increase in labor productivity p,s Ј͑␪ ͒ . c f pЈ,sЈ 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) ϭ ␮u␣v1Ϫ␣, this reduces to and vacancies are in the matching function. If they are strong complements, substitution is r ϩ s ϩ ␭ nearly impossible and the v-u ratio barely ϩ ␣␪ p,s changes. If they are strong substitutes, substitu- q͑␪p s ͒ , tion is nearly costless, and the v-u ratio is highly p Ϫ z 1 procyclical. ϭ͑ Ϫ ␣͒ ϩ ␭ޅ ͩ ͪ 1 p,s ͑␪ ͒ In the decentralized economy, the extent of c q pЈ,sЈ substitution between unemployment and vacancies is governed not only by the matching function 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 maximizes 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(␪)isex- (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 Ϫ 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 Ϫ0.26 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 Ϫ0.88. 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 labor 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៮ Ϫ 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៮ Ϫ 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 ␪ Ϫ 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 ␤ ʦ (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 ϵ ϩ Ϫ 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 ϭ Ϫ Ϫ ͑␪ ͒͑ Ϫ ͒ tradeoff between demanding a higher wage and (11) rVp,s p z f p,s Ep,s Up,s reducing her risk of unemployment, so the wage Ϫ ϩ ␭͑ޅ Ϫ ͒ depends on the hazard rate of the distribution F. sVp,s p,s VpЈ,sЈ Vp,s . This again breaks the link between average labor 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 Ϫ ␤ 1Ϫ␤ some related model, deliver substantial fluctua- Nash product (Ep,s Up,s) Jp,s , which gives tions in the v-u ratio in response to plausible Ϫ impulses remains a topic for future research. Ep,s Up,s Jp,s (12) ϭ V ϭ . ␤ p,s 1 Ϫ ␤ APPENDIX A: DERIVATION OF THE EQUATION FOR SURPLUS (4) Substituting for E Ϫ 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. ϭ ϩ ͑␪ ͒͑ Ϫ ͒ (8) rUp,s z f p,s Ep,s Up,s APPENDIX B: DERIVATION OF THE WAGE EQUATION ϩ ␭͑ޅ Ϫ ͒ p,s UpЈ,sЈ Up,s Assume that wages are continually renegoti- ϭ Ϫ ͑ Ϫ ͒ (9) rEp,s wp,s s Ep,s Up,s ated, so the wage depends only on the current aggregate state (p,s). Eliminate current and fu- ϩ ␭͑ޅ Ϫ ͒ p,s EpЈ,sЈ Ep,s ture values of J from equation (10) using equation (12): (10) rJ ϭ p Ϫ w Ϫ sJ p,s p,s p,s ϭ Ϫ ͑ ϩ ϩ ␭͒͑ Ϫ ␤͒ wp,s p r s 1 Vp,s ϩ ␭͑ޅ J Ј Ј Ϫ J ͒. p,s p ,s p,s ϩ ␭ޅ ͑ Ϫ ␤͒ p,s 1 VpЈ,sЈ . 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): ͑r ϩ s ϩ ␭͒c c their results are not directly applicable to this analysis, ϭ Ϫ ϩ ␭ޅ wp,s p ͑␪ ͒ p,s ͑␪ ͒ . although their methodology may prove useful. q p,s q pЈ,sЈ VOL. 95 NO. 1 SHIMER: UNEMPLOYMENT AND VACANCIES 47

Finally, replace the last two terms using equa- Var͓y͑t ϩ h͒ Ϫ y͑t͉͒y͑t͔͒ tion (6) to get equation (7). ϭޅ[(y(tϩh)Ϫy(t))2͉y(t)] APPENDIX C: THE STOCHASTIC PROCESS Ϫ͑ޅ͓y͑t ϩ h͒ Ϫ y͑t͉͒y͑t͔͒͒2. 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, Var͓y͑t ϩ h͒ Ϫ y͑t͉͒y͑t͔͒ ϭ h␭⌬2 ϭ h␴2. y ʦ Y ϵ ͕Ϫn⌬, Ϫ͑n Ϫ 1͒⌬, ... , Putting this together, we can represent the sto- 0, ... , ͑n Ϫ 1͒⌬, n⌬} chastic process for y as where ⌬Ͼ0 is the step size and 2n ϩ 1 Ն 3is dy ϭ Ϫ␥ydt ϩ ␴dx the number of grid points. When a shock hits, the new value yЈ either moves up or down by one grid point: where for t Ͼ 0, the expected value of x(t) given x(0) is x(0) and the conditional variance is t. 1 y This is similar to a Brownian motion, except ͩ1 Ϫ ͪ y ϩ ⌬ n⌬ that the innovations in x are not Gaussian, since Ј ϭ ͭ 2 y y Ϫ ⌬ with probability Ά1 y . y is constrained to lie on a discrete grid. ϩ ͩ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(⌬͌␧, ␭/␧, n/␧) for any ␧ Ͼ 0. It is the probability that yЈϭy ϩ⌬is smaller when easy to verify that this does not change either y is larger, falling from 1 at y ϭϪn⌬ to zero at the autocorrelation parameter ␥ ϭ ␭/n or the y ϭ n⌬. instantaneous variance ␴ ϭ ͌␭⌬. But as It is trivial to confirm that yЈ ʦ Y, so the state ␧ 3 0, the distribution of the innovation process space is discrete. To proceed further, define ␥ ϵ x converges to a normal by the Central Limit ␭/n and ␴ ϵ ͌␭⌬. For any fixed y(t), I examine Theorem. Equivalently, y converges to an the behavior of y(t ϩ 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 ϩ h) is equal to y(t) with refine the grid by decreasing ␧ without substan- probability 1 Ϫ h␭, has increased by ⌬ with tially changing the results. probability h␭(1 Ϫ y(t)/n⌬)/2, and has de- creased by ⌬ with probability h␭ (1 ϩ 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 Ϯn⌬. The h␭ unconditional distribution of the state variable is approxi- ޅ͓ y͑t ϩ h͒ Ϫ y͑t͉͒y͑t͔͒ ϭ Ϫ y͑t͒ ϭ Ϫh␥y͑t͒. mately normal with mean zero and standard deviation n ␴/͌2␥ ϭ⌬͌n/2. The limiting values of the state variables therefore lie ͌2n standard deviations above and below the mean. If n ϭ 1000, as is the case in the simulations, one Next, the conditional variance of y(t ϩ h) Ϫ 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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