DDynamic Beveridge Curve Accounting ∗ Hie Joo Ahn Leland D. Crane † March 3, 2020
Abstract
We develop a dynamic decomposition of the empirical Beveridge curve, i.e., the level ofvacancies conditional on unemployment. Using a standard model, we show that threefactors can shift the Beveridge curve: reduced-form matching efficiency, changes in thejob separation rate, and out-of-steady-state dynamics. We find that the shift in the Bev-eridge curve during and after the Great Recession was due to all three factors, and eachfactor taken separately had a large effect. Comparing the pre-2010 period to the post-2010 period, a fall in matching efficiency and out-of-steady-state dynamics both pushedthe curve upward, while the changes in the separation rate pushed the curve downward.The net effect was the observed upward shift in vacancies given unemployment. In pre-vious recessions changes in matching efficiency were relatively unimportant, while dy-namics and the separation rate had more impact. Thus, the unusual feature of the GreatRecession was the deterioration in matching efficiency, while separations and dynamicshave played significant, partially offsetting roles in most downturns. The importance ofthese latter two margins contrasts with much of the literature, which abstracts from oneor both of them. We show that these factors affect the slope of the empirical Beveridgecurve, an important quantity in recent welfare analyses estimating the natural rate ofunemployment. ∗ We thank Katharine Abraham, Andrew Figura, David Ratner, and seminar participants at the 2019 SOLEmeeting, the Fall 2018 Midwest Macro meeting, and the Federal Reserve Board. Vivi Gregorich provided excel-lent research assistance. Opinions expressed herein are those of the authors alone and do not necessarily reflectthe views of the Federal Reserve System or the Board of Governors. † Ahn: Federal Reserve Board of Governors, [email protected]. Crane: Federal Reserve Board of Governors,[email protected]. a r X i v : . [ ec on . GN ] F e b Introduction
The empirical Beveridge curve—the level of vacancies conditional on unemployment—haslong been of interest to economists and policy makers. Interest intensified in the wake ofthe Great Recession, as the curve appeared to shift upwards (see Figure 1), fueling concernsabout the functioning of the labor market. There is not currently consensus on the causeof this shift (or historical Beveridge curve shifts). Many papers have attributed the shiftto falling matching efficiency (whether due to mismatch, duration dependence, recruitingintensity, heterogeneity, or other causes.) Others researchers have argued that mechanicalout-of-steady state dynamics can account for the apparent shift. Finally, it has also beennoted that variation in the employment separation rate can also produce shifts in the Bev-eridge curve. Each of these threads of the literature has taken a slightly different modellingapproach as, some authors use steady-state approximations, while others assume a constantjob separation rate.
Unemployment Rate V a c an c i e s R a t e NBER Peak: 2007m12 NBER Trough: 2009m6 2019m11
Note:
Monthly data, 2000-2019.
Source:
CPS, JOLTS.
Figure 1: The Beveridge curve1n this paper we provide a new, unified accounting model for the Beveridge curve and arelated decomposition method. In our baseline model, where the labor-force status is eitheremployed or unemployed, there are three main factors that matter for the position of theBeveridge curve: (1) matching efficiency, (2) the job-separation probability, and (3) out-of-steady-state dynamics. We analyze how much each of these factors shifted the Beveridgecurve. The model allows us to estimate how the contribution of each factor changed indifferent recessionary and recovery episodes. We also extend our model to include the labor-force participation margin, to see how important labor-supply factors are in the dynamicsof Beveridge curve.We find that matching efficiency, job separations and out-of-steady-state dynamics are all important in understanding the shifts of the Beveridge curve over business cycles, par-ticularly in the Great Recession. Out-of-steady state dynamics (defined below) produced anet upward shift in the Beveridge curve during and after the Great Recession, as suggestedby Christiano et al. (2015) and Furlanetto and Groshenny (2016). Those papers assume aconstant job separation rates, but we find that changes in the job separation rate shifted theBeveridge curve sharply down on net around the Great Recession. This downward shiftof the Beveridge curve partially offset the combined upward shift from out-of-steady-statedynamics and matching efficiency. In fact, changes in the separation rate were the largestsingle factor moving the Beveridge curve. Separations can shift the Beveridge curve since,for a given path of unemployment, a higher separation rate implies that vacancies must alsobe higher, in order to maintain the net change in unemployment at the observed values. Thejob separation probability was high in the downswing of the Great Recession, and it laterfell back to more normal levels in the recovery. This had the effect of shifting the Beveridgecurve up in the downswing and down in upswing. Elsby et al. (2015) documented a similarpoint, though they did not quantify the extent of the shift or compare it to the other shifters. See also Eichenbaum (2015) for related discussion. Hall and Schulhofer-Wohl (2018) also noted the unemployment inflow rate complicates the behavior of theBeveridge curve.
2e also find that matching efficiency fell significantly during and after the Great Recession,which pushed the Beveridge curve up. This result is consistent with, e.g., Barnichon andFigura (2015). Analyses which ignore one or more of these shifters will either fail to match the data orwill risk making mistaken inferences. This leads to several concrete conclusions and rec-ommendations: First, the importance of out-of-steady-state dynamics implies that the usualflow steady-state approximations are not appropriate for studying the Great Recession, orsimilar periods of rapid change in the unemployment rate. Flow steady-state approxima-tions have become a fundamental tool for simplifying and understanding the labor market(see, for example, Fujita and Ramey (2009), Elsby et al. (2009), Shimer (2012), Barnichon et al.(2012), Elsby et al. (2015).) Unfortunately, in the Great Recession unemployment was consis-tently far from the the steady steady-state value implied by inflows and outflows, thus theapproximation is poor during this period. We also find a large role for out-of-steady-statedynamics in some previous recessions.Second, time-variation in the job separation probability is critical for understanding theBeveridge curve, and indeed was the single largest shifter of the Beveridge curve in theGreat Recession. Thus, the common simplifying assumption of a constant separation rate(made in, e.g., Christiano et al. (2015)) is not appropriate when trying to model the Beveridgecurve. In fact, we find that variation in the separation rate was an important shifter of theBeveridge curve in many previous recessions as well, and this variation also affects the slopeof the empirical curve. Our analysis does not speak directly to the debate over the relativeimportance of the separations versus the job findings for the evolution of unemployment(see, e.g., Fujita and Ramey (2009), Elsby et al. (2009), Shimer (2012), Ahn and Hamilton(2019)). Rather, we simply point out that the Beveridge curve cannot be properly understoodwithout this ingredient.Third, we confirm that there was a clear fall in reduced-form matching efficiency in the See also Barnichon and Figura (2010) and Barnichon et al. (2012) for more on matching efficiency. why matching efficiency fell, instead we seek to quantify the effects on the Beveridge curve andthe interactions with other factors. Though all three of these factors are crucial in understanding the Beveridge curve, wealso find that the relative importance of each factor differed across recessionary episodes. We find that the 1990’s recession was similar to the Great Recession in that matching effi-ciency was the key factor to the persistent outward shift of Beveridge curve. However, inthe other recessions in the 1970’s, 1980’s and 2001, the job separation probability and out-of-steady-state dynamics played more important roles than matching efficiency.In addition to clarifying the source of loops in the Beveridge curve, we show that theseshifters affect the slope of the empirical Beveridge curve. This occurs because the curve isbeing shifted while labor market upswings and downswings progress, not just at peaks andtroughs. Thus the slope of the steady-state Beveridge curve under constant separations andconstant matching efficiency is very different from the empirical slope. This has direct im-plications for the work of Michaillat and Saez (2019), who exploit the slope of the Beveridgecurve to estimate the efficient level of unemployment and the unemployment gap. A backof the envelope exercise shows that using an arguably more appropriate slope cuts the es-timated unemployment gap in half, relative to Michaillat and Saez (2019). We view this asevidence that more work is needed to understand how time-varying factors affect the slopeof the empirical Beveridge curve.For our baseline results, we work with a log-linearized Beveridge curve, which expressesthe vacancy rate a linear function of various factors. This first-order approximation matches Many papers have offered explanations for the fall in reduced-form matching efficiency among them Daviset al. (2013), Sahin et al. (2014), Elsby et al. (2015), Barnichon and Figura (2015), Kroft et al. (2016), Ahn andHamilton (2019), and Hall and Schulhofer-Wohl (2018). Daly et al. (2011) and Diamond and Sahin (2015) document historical Beveridge curve shifts.
This section derives a version of the simple Beveridge curve framework used in Christiano etal. (2015) (hereafter CET) and Eichenbaum (2015), which is nearly identical to that of Elsby etal. (2015). We do not close the model by making assumptions about the job creation process,wage determination, or other fundamentals. Instead we focus on deriving conclusions thatmust hold for any general equilibrium model whose labor market is described by (1) thestandard law of motion for unemployment and (2) the usual matching function relationship.Let U t be the unemployment rate in month t , and let V t be the vacancy rate (i.e. vacanciesdivided by the labor force). There is no on-the-job search, no participation margin, and the5ize of the labor force is constant and normalized to unity. H t = σ t U − α t V α t (1)where α is the elasticity of the matching function and σ t is matching efficiency, which canvary over time. Then the job-finding probability is given by f t = σ t ( V t / U t ) α . (2)The law of motion for unemployment is U t + = s t ( − U t ) − f t U t + U t (3)where s t is the probability a job ends in a given month. We refer to s t as the “EU probability”,as it is the probability an employed worker transitions to unemployment in a given month.Substituting equation (2) into (3) and rearranging we arrive at V t = (cid:34) s t ( − U t ) − ∆ U t + σ t U − α t (cid:35) α (4)where ∆ U t + = U t + − U t . This is a slight generalization of CET equation 5.2. Whereas CETassume that s t and σ t are constants, we permit time-variation in these parameters. Note thatif s t is set to its observed values and σ t is chosen to verify equation (1), then equation (4) isan identity.Equation (4) is at the core of our analysis. To understand it better, consider the casewhere s t , σ t and U t are constants: V = (cid:20) s ( − U ) σ U − α (cid:21) α . (5)This is the steady state Beveridge curve relationship at the core of textbook search models6see Pissarides (2000)): a steady-state with low U must have high V , and vice-versa. Takingequation (5) as the reference point, variation in s t , σ t and ∆ U t + changes the level of V t given U t . Thus, with a slight abuse of terminology, we will refer to these factors as shifters . Given a path for unemployment and hypothesized, possibly counterfactual, values ofthe parameters ( α , s t , σ t ) , one can calculate the implied path of vacancies from equation (4)and compare it to the true path of vacancies. This is the essence of our exercises in Section 4. We require data on all the variables and parameters in equations (3) and (4). We use thestandard approaches, based mostly on Shimer (2012) and Barnichon and Figura (2015). Weset U t as the number of unemployed divided by the labor force, as measured in the CurrentPopulation Survey (CPS). We set V t equal to the count of vacancies from JOLTS divided bythe size of the labor force. Figure 2 plots the two series.We set the monthly job-finding probability, f t as in Shimer (2012), using data on thenumber of short-term unemployed each month. We then choose s t to satisfy the law ofmotion (3) exactly. Figure 3 shows the job finding and separation probabilities. It is notable that the jobfinding probability fell by about 50 percent in the Great Recession and the separation prob-ability increased by about 50 percent. This suggests that both margins may have played a We use shifters to mean factors that change V t given U t . Note s and σ also shift the steady-state Beveridgecurve (5), while ∆ U t + does not. The dynamics captured by ∆ U t + produce loops around the steady-state Bev-eridge curve, but do not change that model-based relationship. That is, we set f t = − U t + − U st + U t , where U st + is the number of workers unemployed for less than five weeksin month t +
1. Thus f t is the probability that a worker unemployed in month t finds a job by t +
1. In the datait is possible for such a worker to both find and lose a job (or multiple jobs) before t +
1, but the discrete-timemodel we use rules out this possibility. In both our setup and the continuous time formulation of Shimer (2012), EU flows are set so as to make theobserved sequence of stocks consistent with the flows. In the three-state model of Section 6 the transition ratesare taken directly from the data and raked for consistency with the stocks. Christiano et al. (2015) note that the job separation rate, as measured by JOLTS, fell in the Great Recession.The JOLTS separation rate includes job-to-job flows, which are known to be highly procylical, as well as flows tononemployment. Their model, like ours, does not allow for job-to-job flows. The JOLTS separation rate is likelythe correct measure when considering the firm’s problem, since it gives the expected duration of the match. But nemployment and Vacancy Rates Jan2000 Jan2005 Jan2010 Jan2015 Jan202000.010.020.030.040.050.060.070.080.090.1
Unemployment RateVacancy Rate
Note:
Monthly data, 2000-2019. NBER recessions shaded in gray.
Source:
CPS, JOLTS.
Figure 2: Unemployment and Vacancy Ratessignificant role in the evolution of unemployment. We will confirm this impression in whatfollows.Measurement of α and σ t require estimation of the matching function. We run the usualregression ln f t = ln σ + α ln (cid:18) V t U t (cid:19) + ε t (6)where ε t is the mean-zero error term, σ t = σ exp ( ε t ) is time-varying matching efficiency, and σ is interpreted as average matching efficiency.Figure 4 plots the log job finding probability against the log V-U ratio. The data for differ-ent periods are plotted in different colors. It is evident that matching efficiency deterioratedsignificantly post-2008. Any change in the matching elasticity α was minor by comparison,so we will continue assuming that α is a constant throughout the paper (as is standard inthe literature). when considering the evolution of unemployment it is better to use the inflow to unemployment, rather thanincluding job-to-job flows. onthly Job Finding Probability Jan2000 Jan2005 Jan2010 Jan2015 Jan202000.10.20.30.40.5
Monthly Job Separation Probability
Jan2000 Jan2005 Jan2010 Jan2015 Jan202000.010.020.03
Note:
Monthly data, 2000-2019. NBER recessions shaded in gray.
Source:
CPS, JOLTS.
Figure 3: Observed Transition Probabilities -2 -1.5 -1 -0.5 0 0.5
Log V-U ratio -1.8-1.6-1.4-1.2-1-0.8-0.6
Log J ob F i nd i ng P r obab ili t y Matching Function Estimation
Pre-2008 data2008 dataPost-2008 dataPredicted, estimated onpre-2008 sample
Note:
Monthly data, 2000-2019.
Source:
CPS, JOLTS.
Figure 4: Matching Function Estimation91) (2)Pre-2008 Sample Post-2008 Sampleln σ − − ( ) ( ) α ( ) ( ) Notes : OLS estimates of average matching efficiency(ln σ ) and the matching function elasticity ( α ). *, **, and*** indicate statistical significance at the 10%, 5%, and1% levels, respectively. Standard errors are in parenthe-ses. Table 1: Matching Function EstimatesWe run equation (6) on a sample starting in 2000 (when the JOLTS series begins) andending in 2007, a period where it is plausible that σ was indeed constant. We also run theregression on a post-2008 sample. Table 1 presents the results. The point estimates put α near 0.3, very similar to the estimates of Shimer (2005) and Barnichon and Figura (2015),who use longer time series. It is evident that average matching efficiency fell about 25%between the two samples. In order to simplify the discussion, we log-linearize equation (4). In particular, we take thefirst order Taylor approximation around a point ( U t , s t , σ t , ∆ U t + ) = (cid:0) U , s , σ , 0 (cid:1) . The resultis the following expressionln V t ≈ ln V − (cid:32) U α (cid:0) − U (cid:1) + − αα (cid:33) (cid:0) ln U t − ln U (cid:1) − U α s ( − U ) ∆ ln U t + (cid:124) (cid:123)(cid:122) (cid:125) Shift due toDynamics + α ( − U ) ( ln s t − ln s ) (cid:124) (cid:123)(cid:122) (cid:125) Shift due toSeparations − α ( ln σ t − ln σ ) (cid:124) (cid:123)(cid:122) (cid:125) Shift due toMatching Efficiency (7)where V is equation (4) evaluated at (cid:0) U , s , σ , 0 (cid:1) .10he first line of equation (7) is the (approximate) steady-state Beveridge curve. Thesecond line contains the “shifters”. Treating ln V t as a linear function of ln U t , these shiftersmove the y-intercept of the steady-state curve up and down. For example, we can see thatwhen unemployment is rising ( ∆ ln U t + >
0) then ln V t will be lower than the steady statecurve. This is because, all else equal, rising unemployment implies low finding and thus lowln V t , which is the out-of-steady-state dynamics mechanism outlined in Pissarides (2000).While increasing in ∆ ln U t + shifts ln V t down, increases in the job separation probabil-ity s t shift the curve up. The intuition is that a higher job-separation probability, conditionalon a fixed value of ∆ ln U t + , requires more equilibrium vacancies to absorb the unemploy-ment inflows. Increases in matching efficiency σ t obviously shift the curve down, as fewervacancies are needed to rationalize the observed value of ∆ ln U t + .We are interested in approximating the Beveridge curve around the Great Recession.To that end, we center the Taylor approximation around post-2007 averages. This yields U = s = σ = ∆ ln U t + = Figure 5 plots the (log) observed Beveridge curve, the first order approximation, and thesteady-state Beveridge curve. The approximate Beveridge curve, which includes all the(first order) effects of the shifters, follows the actual curve closely, aside from a brief periodnear the trough of the Great Recession. Most importantly, the approximate curve showsnearly the same shift (between recession downswing and recovery) as the observed curve.The good fit of the linearized curve gives us confidence that our decomposition of the lin-earized curve will also be accurate for the exact curve. Appendix A addresses any lingeringconcerns about the accuracy of the linearized results by calculating a series of nonlineardecompositions on the exact Beveridge curve.Both the actual Beveridge curve and the approximate curve are significantly flatter than11
Log Unemployment -4.2-4-3.8-3.6-3.4-3.2-3-2.8
Log V a c an c i e s Observed1st order approximationSteady State Beveridge Curve
Note:
Source:
CPS, JOLTS.
Figure 5: Beveridge Curvesthe steady state curve. In log space, the slope of the steady state curve is roughly − − αα = − − U α s ( − U ) ∆ ln U t + .) The red and orange lines similarlyshow the shifts due to separations and matching efficiency.Relative to the pre-Great Recession period (say, 2007), the net effect of the shifters wasto move vacancies sharply upward during the recession. This effect then dissipated veryslowly, with the shifters returning to their pre-recession net value only in 2017. This com-bined effect explains why the slope of the empirical Beveridge curve is so much flatter thanthe steady state curve. We return to this point in Section 4.2.Turning to each shifter separately, contribution of each factor is complicated and time-12 an2006 Jan2008 Jan2010 Jan2012 Jan2014 Jan2016 Jan2018 Jan2020-1.5-1-0.500.511.52 Log V a c an c i e s DynamicsSeparationsMatching EfficiencyNet Shift (relative to steady-state curve)Period of Peak Unemployment
Note:
Source:
CPS, JOLTS.
Figure 6: Shifters of the Approximate Beveridge Curvevarying. Out-of-steady-state dynamics pushed the Beveridge curve intercept sharply downin the recession, and modestly up in the recovery, more or less the way Pissarides (2000)describes. The contribution of separations is roughly the opposite, raising the interceptsharply, especially late in the recession, and then eventually pushing the intercept down.Finally, the deterioration in matching efficiency raised the intercept during and after therecession.Figure 6 cannot clearly tell us which factors are responsible for the shift in the empirical
Beveridge curve between the downswing and the upswing of the Great Recession. To un-derstand that, we need to condition on a level of unemployment and examine the verticalshift evident in Figure 5.Say that there were two months, t and t (cid:48) , where observed unemployment rates wereexactly equal, U t = U t (cid:48) . Then using equation (7) we could decompose the (approximate)13ifference in vacancies, ln V t (cid:48) − ln V t , as follows:ln V t (cid:48) − ln V t ≈− U α s ( − U ) ( ∆ ln U t (cid:48) + − ∆ ln U t + ) (cid:124) (cid:123)(cid:122) (cid:125) Shift due toDynamics + α ( − U ) ( ln s t (cid:48) − ln s t ) (cid:124) (cid:123)(cid:122) (cid:125) Shift due toSeparations − α ( ln σ t (cid:48) − ln σ t ) (cid:124) (cid:123)(cid:122) (cid:125) Shift due toMatching Efficiency (8)Equation (8) provides an additive decomposition of the vertical shift in the Beveridgecurve. The portion of ln V t (cid:48) − ln V t due to, say, differences in matching efficiency between t and t (cid:48) is just the log difference in matching efficiency, ln σ t (cid:48) − ln σ t , multipled by 1/ α . Theshifts due to dynamics and separations are similar. The only wrinkle in implementing equa-tion (8) is that we never observe two months with exactly the same unemployment rate, sowe linearly interpolate all relevant series.As the reference points, we select the unemployment rates observed between April 2007and June 2009. These are highlighted in red in Figure 7 (the “downswing sample”). We com-pare the downswing sample to the upswing sample, which begins in April 2010 (highlightedin blue). For each of the downswing points, we calculate the vertical distance between ob-served vacancies and the (linearly interpolated) upswing vacancy levels. We also calculateeach of the terms in equation (8).The result is Figure 8. The x-axis is the unemployment rate. For each unemploymentrate, the black line shows the vertical distance between the upswing and downswing sam-ples, as measured in log vacancies. This is the shift in the Beveridge curve we are tryingto explain. The black line is the sum of the other three lines, which are the contributionsin equation (8). There are several striking results. First, the job-separation probability isresponsible for a large shift down in the Beveridge curve. This is because separations roseearly in the recession, pushing up vacancies, and later fell, making upswing vacancies lower.This shift is offset by the combined effects of dynamics and matching efficiency, which bothpushed the curve up on net. 14 Log Unemployment -4.2-4-3.8-3.6-3.4-3.2-3-2.8
Log V a c an c i e s Approximate Beveridge CurveDownswing Sample: April 2007 - June 2009Upswing Sample: April 2010 - June 2017
Note:
Source:
CPS, JOLTS.
Figure 7: Downswing and Upswing Samples
Unemployment Rate -1.5-1-0.500.511.52
Log V a c an c i e s DynamicsSeparationsMatching EfficiencyNet Shift (between downswing and upswing)
Note:
Source:
CPS, JOLTS.
Figure 8: Accounting for the Vertical Shift15nterestingly, out-of-steady-state dynamics played a prominent role, with a contributionlarger than that of matching efficiency over much of the range. This specific result is consis-tent with Christiano et al. (2015)’s argument that, because the Great Recession was so largeand so sudden, dynamics can produce a realistic loop in the Beveridge curve. However,their analysis ignores the separation probability and matching efficiency, which are at leastas important for understanding what happened. In particular, matching efficiency morethan accounts for the net shift across most of the range, so without a change in matchingefficiency the Beveridge curve would have shifted down , not up.To summarize, all three of the factors we consider shifted the Beveridge curve in non-trivial ways. The vertical shift in the empirical Beveridge curve is the net result of out-of-steady state dynamics and matching efficiency both shifting the curve up, an effect whichis partially offset by a large negative contribution from the separation probability. The timepaths of these shifters are complicated and non-monotonic, leading the slope of the empir-ical Beveridge curve to differ from the model-implied steady-state curve. We now turn tothis result in more detail.
Recent innovative work by Michaillat and Saez (2019) (MS) has emphasized the importanceof the Beveridge curve slope for welfare and the natural rate of unemployment. In thissection we show how our measurement methods relate to their results.In many models with a matching function (e.g., Shimer (2005)), the Beveridge curvedescribes the possible steady-state values of vacancies and unemployment. In short, aneconomy that sustains a low level of unemployment must have more vacancies in equi-librium, and vice versa. MS point out that this relationship can be used to estimate thewelfare-maximizing level of unemployment in a particularly simple and general way. Theynote that a social planner will seek to equalize the costs of additional vacancies to the costsof additional unemployment. In other words, the social planner will seek the location on16he Beveridge curve where the marginal cost of additional unemployment equals the socialvalue of the resulting reduction in vacancies. This point then defines the natural rate of un-employment, and the difference between observed unemployment and natural rate is theunemployment gap. MS use estimates of the costs of vacancies, the costs of unemployment,and the slope of the Beveridge curve to make their calculations.MS measure the slope of the Beveridge curve by estimating regressions of V t on U t inperiods where the Beveridge curve appeared stable (dropping the troughs of recessions,for example.) As we show above, these observed slopes reflect both (1) movements alonga stable Beveridge curve (changes in U t for fixed separations, matching efficiency and dy-namics) and (2) time variation in the shifters. This second factor can distort the empiricalBeveridge curve relative to the planner-relevant, steady-state curve. For example, considera bare bones model where the separation probability and matching efficiency are exoge-nous processes, possibly correlated with the aggregate productivity shock. Such a modelfits in our framework (and that of MS), and could produce the observed data, including theempirical Beveridge curve and the paths of the shifts. However, a planner, facing such aneconomy, would not look to the empirical Beveridge curve to estimate the unemployment-vacancy tradeoff. The reason is that the empirical curve include the effects of the (purelycyclical) shifters, while the planner is interested in long-run, steady state relationships. Thecorrect slope for the planner comes from the linearized curve (7), which treats the shifters asfixed: − (cid:32) U α (cid:0) − U (cid:1) + − αα (cid:33) ≈ − − αα (9)and is determined by the shape of the matching function. The planner would make de-cisions based on the steady-state curve in Figure 5, not the empirical curve. Thus, in thistoy example the empirical Beveridge curve does not directly give us the planner-relevant,long-run relationship we seek. 17he key question is whether the planner should incorporate the effects of the shifterswhen making a choice about the long-run level of unemployment. Clearly, out-of-steady-state dynamics are fundamentally transitory, so the planner should always purge the Bev-eridge curve of their effect. However, it is possible that the separation probability andmatching efficiency are, to some extent, functions of the long-run level of unemployment(unlike in the toy example above). In this case the planner should not remove (all of) theirinfluence when calculating the vacancy-unemployment tradeoff.Determining the exact nature of the variation in separations and matching efficiency iswell beyond the scope of this paper. Instead, we provide an example to demonstrate thatthese issues can have an economically meaningful impact on welfare calculations. Fromequation (9), the slope of the steady-state curve (treating the shifters as fixed) is very closeto − − αα . Averaging together the two estimates of α in Table 1, we set set α = − − − − − − an2000 Jan2005 Jan2010 Jan2015 Jan202000.010.020.030.040.050.060.070.080.090.1 U ne m p l o y m en t ActualEfficient, Beveridge slope=-0.9Efficient, Beveridge slope=-2.33
Note:
Calculations based on 3 month moving averages of monthly data, 2000-2019.
Source:
CPS, JOLTS.
Figure 9: Efficient Unemployment Based on the Beveridge TradeoffOur results suggest that careful work is needed to disentangle which features of theBeveridge curve the planner should care about. These choices have real consequences forthe measurement of efficiency, as Figure 9 shows. One approach is to specify a more com-plete model, which explicitly links separations and matching efficiency to the rest of theeconomy. With such a model in hand, one could determine the planner-relevant Beveridgecurve slope.
We can also use our framework to analyze recessions prior to the Great Recession. In termsof data, the only change is that up through 2016 we use the composite vacancy series fromBarnichon (2010) instead of JOLTS. After 2016 we continue the series by splicing on theJOLTS series. For four historical labor market downturns, we calculate the log-linearizedBeveridge curve, as in Section 4. For each episode the curve is linearized around the localmean, to ensure a good fit. Figure 10 compared the observed and linearized Beveridge19
Log unemplyoment -3.8-3.7-3.6-3.5-3.4-3.3-3.2
Log v a c an c i e s ObservedTayor approximation (a) -2.55 -2.5 -2.45 -2.4 -2.35 -2.3 -2.25 -2.2 -2.15 -2.1
Log unemplyoment -4-3.9-3.8-3.7-3.6-3.5-3.4-3.3
Log v a c an c i e s ObservedTayor approximation (b) -2.9 -2.85 -2.8 -2.75 -2.7 -2.65 -2.6 -2.55 -2.5 -2.45
Log unemplyoment -3.9-3.8-3.7-3.6-3.5-3.4-3.3
Log v a c an c i e s ObservedTayor approximation (c) -3.3 -3.2 -3.1 -3 -2.9 -2.8 -2.7
Log unemplyoment -4-3.9-3.8-3.7-3.6-3.5-3.4-3.3
Log v a c an c i e s ObservedTayor approximation (d)
Note:
Observed and approximated Beveridge curves for historical downturns. Log-linear Taylor approximationsare taken about averages for the periods plotted.
Source:
CPS, Barnichon (2010), and authors’ calculations.
Figure 10: Observed and Approximate Beveridge Curves20urves. The fit is generally good, although some of the linearized Beveridge curves showless of a shift, or counter-clockwise loop, than their observed counterparts. We view this asa topic for further investigationWith the linearized Beveridge curves in hand, we can read off the implied contributionof each factor to the shift in the curve at every point in time. Figure 11 presents the historicalversions of Figure 6: the net shift in the Beveridge curve, and the contributions, as functionsof time. It is apparent that in each recession the Beveridge curve intercept began shifting upat the onset of the recession, and slowly drifted down once unemployment began falling.Rising separations usually drove this upward shift, partially offset by out-of-steady-statedynamics.It can be seen that in all recessions, out-of-steady-state dynamics shifted the Beveridgecurve significantly down in the initial stages (the light blue line is below zero) and generallyup in the recovery (the bold blue line is above zero). Interestingly, this shift is partiallyoffset by the contribution of separations, which (as in the Great Recession) tend to pushBeveridge curve sharply upward in the initial stages of a recession and more moderatelyupward afterward. Thus the changes in the job-separation probability tend to flatten theobserved Beveridge curve, and cancel out some of the counter-clockwise loop that out-of-steady-state dynamics induce.In most previous recessions, changes in matching efficiency had little impact, and wereswamped by changes in the other factors. The 1990 recession appears to be an exceptionhere. During the 1990 recession and the recovery period, the deterioration in matching effi-ciency continued to push the Beveridge curve up, which is quite similar to what happenedin the Great Recession. In fact, the two recessions are similar to each other in a sense thatlong-term unemployment continued to increase substantially after the recession was over.This suggests that mismatch or related factors might have been an important driver in therise of long-term unemployment in the two recession episodes. We view this line of reason-ing as a topic for future research. 21
973 Recession, Contrib. to shift
Jan1974 Jan1975 Jan1976 Jan1977 Jan1978-0.500.51
Log V a c an c i e s DynamicsSeparationsMatching EfficiencyNet Shift (a)
Jan1981 Jan1982 Jan1983 Jan1984 Jan1985-0.4-0.200.20.40.60.8
Log V a c an c i e s DynamicsSeparationsMatching EfficiencyNet Shift (b)
Jan1990 Jan1991 Jan1992 Jan1993 Jan1994 Jan1995-0.8-0.6-0.4-0.200.20.40.6
Log V a c an c i e s DynamicsSeparationsMatching EfficiencyNet Shift (c)
Jan2000 Jul2002 Jan2005 Jul2007-0.6-0.4-0.200.20.40.60.8
Log V a c an c i e s DynamicsSeparationsMatching EfficiencyNet Shift (d)
Note:
Solid black line shows the time path of the net shift of the (log-linearized) Beveridge curve intercept. NBERrecessions shaded in gray.
Source:
CPS, Barnichon (2010), and authors’ calculations.
Figure 11: Decompositions of Approximate Beveridge Curves22he tentative conclusion is that the Great Recession was exceptional, insofar as the dropin matching efficiency had first-order effects on the Beveridge curve (with the possible ex-ception of the early 1990s recession). In previous recessions matching efficiency usuallyplayed little role. However, the modest counter-clockwise loops in previous recession werenot simply the product of modest out-of-steady-state dynamics, but were the net result dra-matic dynamics being offset by large contributions from the separations margin. Out-of-steady-state dynamics and the separations margin played critical roles in all the recessionsexamined here.
The results so far have assumed that all workers are either employed or unemployed. Thisis a significant simplification, since empirically flows into and out of the labor force areimportant for understanding total hires and evolution of unemployment. In this section weadd a participation margin and discuss the robustness of our results in the expanded model.
The population is still normalized to unity, but we add a nonemployment state. Let N t is the stock of nonemployed, so that E t + U t + N t =
1. Consider the law of motion forunemployment when workers can move into and out of the labor force: ∆ U t + = E t eu t + N t nu t − U t un t − U t ue t (10)The transition rate from nonemployment to unemployment in month t is nu t . The terms un t and ue t are similarly defined, with eu t replacing s t for consistency. The law of motion fornonemployment is symmetric: ∆ N t + = E t en t + U t un t − N t ne t − U t un t (11)23umming equations (10) and (11) yields an expression involving total hires ( H t = N t ne t + U t ue t ) ∆ U t + + ∆ N t + = E t eu t + E t en t + H t (12)where the flows between unemployment and nonemployment have canceled.We can write the matching function as H t = σ t ( U t + ξ Nt N t ) − α V α t (13)where ξ Nt is the search effort of the nonemployed relative to the unemployed. Thus theeffective mass of searchers is U t + ξ nt N t and σ t continues to represent reduced-form matchingefficiency.Combining equations (12) and (13), and assuming balanced matching (that is, hires fromunemployment are a share U t U t + ξ Nt N t of total hires), we have the following expression forvacancies: V t = (cid:20) ( − U t − N t )( eu t + en t ) − ∆ U t + − ∆ N t + σ t ( U t + ξ Nt N t ) − α (cid:21) α (14)When the non-employed can participate in job search, it is more sensible to think of a Bev-eridge curve which relates vacancies to searchers (both unemployed and nonemployed) in-stead of unemployment. To this end, we make two substitutions. First, we define the poolof searchers S t as S t = U t + ξ Nt N t . (15)Second, we define the pool of “truly nonemployed” as˜ N t = (cid:16) − ξ Nt (cid:17) N t . (16)While we take no stand on whether ξ Nt is the fraction of nonemployed who search or thesearch effort of each nonemployed relative to the unemployed, the former interpretation is24onvenient here. Note that if ξ Nt = N t =
0. Using S t and˜ N t , we can write (14) as V t = (cid:34) (cid:0) − S t − ˜ N t (cid:1) x t − ∆ S t + − ∆ ˜ N t + σ t S − α t (cid:35) α (17)where x t = eu t + en t is the total job-separation probability. Log-linearizing yieldsln V t = − α [ ln σ t − ln σ ] − (cid:40) ( − α ) α + α S (cid:0) − S − ˜ N (cid:1) (cid:41) [ ln S t − ln S ] − (cid:40) α S (cid:0) − S − ˜ N (cid:1) x (cid:41) [ ∆ ln S t + ] − (cid:40) α ˜ N (cid:0) − S − ˜ N (cid:1) (cid:41) (cid:2) ln ˜ N t − ln ˜ N (cid:3) − (cid:40) α ˜ N (cid:0) − S − ˜ N (cid:1) x (cid:41) (cid:2) ∆ ln ˜ N t + (cid:3) + α [ ln x t − ln x ] (18)Like equation (4), equation (17) can be used to analyze the Beveridge curve. This decom-position, naturally, has more shifters than the two-state model. In this model movementsalong the Beveridge curve are captured by the ln S t − ln S term, since the curve is defined interms of searchers, not merely the unemployed. The effects of matching efficiency and sep-arations still appear, on the first and last lines of equation (17) respectively. Finally, there arenow two out-of-steady state terms, ∆ ln S t + and ∆ ln ˜ N t + , as well as a term capturing thelevel of non-searchers, ln ˜ N t − ln ˜ N . Not all of these terms have a transparent interpretation,but as we shall see below, many of them are not quantitatively important either.25 .2 Data To implement the three state model, we need data on the terms appearing in equation (14).We obtain the stocks of employed, unemployed, and nonemployed from the CPS labor forcestatus flows. We normalize these stocks to satisfy E t + U t + N t = eu t , nu t , un t , ue t are also taken from the labor force status flows. These transitionrates are not exactly consistent with the stocks, due to missing month-to-month linkagesand sample rotation. We iteratively rake the rates until they are consistent with the stocks.This results in very small adjustments to the transition rates.Under the assumption of balanced matching, ξ Nt can be identified by the ratio of transi-tion rates to employment: ξ Nt = ne t ue t Finally, α and σ t can be identified by the matching function regression, using U t + ξ Nt N t as the population of effective searchers. Figure 12 shows that, as with the two state model, the three state approximate Beveridgecurve is a good approximation of the observed curve. Here “searchers” are the pool ofactively searching workers, U t + ξ Nt N t . To show the direction of time, more recent periodsare shaded darker.Figure 13 shows the shifters as a function of time, similar to Figure 6 the story is simi-lar to the two-state model. Matching efficiency slowly and steadily pushed the Beveridgecurve upwards during and after the Great Recession. The separation probability, x t , pushedthe Beveridge curve up during the recession, but this was short-lived. The out-of-steady-state dynamics terms, on net, pushed the curve down, though interestingly the ∆ ˜ N t + term Accessible at . Log Searchers -4.8-4.6-4.4-4.2-4-3.8-3.6
Log V a c an c i e s Observed Beveridge CurveApproximate Beveridge Curve
Note:
Source:
CPS, JOLTS.
Figure 12: Three State Approximate Beveridge Curve
Jan2006 Jan2008 Jan2010 Jan2012 Jan2014 Jan2016 Jan2018 Jan2020-0.6-0.4-0.200.20.40.60.8
Log V a c an c i e s Note:
Source:
CPS, JOLTS.
Figure 13: Three State Model - Shifters of the Approximate Beveridge Curve27artially offsets the ∆ ˜ S t + term. Strikingly, there is no shift in the Beveridge curve underconstant matching efficiency. This confirms the results from the two state model (and muchof the literature) that the decline in matching efficiency was an important contributor to theloop in the Beveridge curve. The empirical Beveridge curve is easy to calculate, as it only requires data on the stocks ofunemployed workers and job openings. This ease of measurement may help explain theattention it has received. Unfortunately, the Beveridge curve is (even in a simple model) theproduct of multiple factors, and can be difficult to interpret. Our hope is that our resultshelp clarify the behavior of the Beveridge curve and reconcile some conflicting ideas in theliterature.We have shown that reduced-form matching efficiency, changes in the separation prob-ability, and out-of-steady-state dynamics all played important roles in the recent shift of theBeveridge curve. Comparing the pre-2010 period to the post-2010 period, out-of-steady-state dynamics and a fall in matching efficiency both pushed the curve upward, while thechanges in the separation probability pushed the curve downward. The net effect was theobserved upward shift in the empirical Beveridge curve. Our results are largely unchangedwhen we include a nonparticipation margin. One area for more research is the effect ofon-the-job search, which would affect the measurement of matching efficiency.A realistic model of the Great Recession therefore needs, (1) a mechanism for reduced-form matching efficiency to fall during and after the recession, (2) a non-constant separationprobability, which can generate an increase in job losses towards the end of the recession.Furthermore, models should not be evaluated using steady-state approximations, since therapid changes in the labor market around the Great Recession made out-of-steady-state dy-namics a first-order issue.We reach similar conclusions regarding earlier recessions, though the role of matching28fficiency is generally smaller. Importantly, the relatively small Beveridge curve loops inearlier recessions were the product of changes in the separation probability nearly offsettingout-of-steady-state dynamics. We find that these shifters move the intercept of the Beveridgecurve continuously, not just at business cycle peaks and troughs. As a result, the slope ofthe empirical Beveridge curve is distinct from the slope of the implied (constant separationprobability, constant matching efficiency) steady-state curve.29 eferences
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One may be concerned that results based on the Taylor approximation are not robust. Whilethe fit of the approximate Beveridge curve is strikingly good, it is not perfect. Therefore,there is some room for non-linearities to affect the results. A related issues is that the log-linearized Beveridge curve is not dynamically consistent: If we plug implied vacancies intothe matching function and the unemployment law of motion, we generally won’t get theobserved U t + back.In this section we decompose the shift in the empirical Beveridge curve using the exactvacancy equation rather than the log-linearized version. Again, the goal is to measure thecontributions to the shift due due to out-of-steady-state dynamics, changes in the separationprobability, and changes in matching efficiency.The starting point of our decomposition is the standard, steady-state Beveridge curve,with constant matching efficiency and separations: V s , σ , ∆ Ut = (cid:34) s ( − U t ) σ U − α t (cid:35) α (19)The steady state Beveridge curve sets ∆ U t + =
0. It therefore the level of vacancies thatwould prevail after many months of constant s and σ .Let t down be a month from the downswing sample, and let t up be the corresponding(interpolated) period from the upswing with the same level of unemployment. Then theobserved vertical shift in the Beveridge curve is V up − V down . The steady-state Beveridgecurve (19) obviously entails no shift, so V s , σ , ∆ Uup − V s , σ , ∆ Udown = σ t = (cid:34) s t ( − U t ) − ∆ U t + σ U − α t (cid:35) α (20) V σ , ∆ Ut = (cid:34) s t ( − U t ) σ U − α t (cid:35) α (21)with V st , V s , σ t , V s , ∆ Ut , and V ∆ Ut defined similarly.Next, consider the accounting identity V up − V down = (cid:0) V up − V down (cid:1) − (cid:16) V σ up − V σ down (cid:17) + (cid:16) V σ up − V σ down (cid:17) − (cid:16) V s , σ up − V s , σ down (cid:17) + (cid:16) V s , σ up − V s , σ down (cid:17) − (cid:16) V s , σ , ∆ Uup − V s , σ , ∆ Udown (cid:17) . (22)This writes V up − V down as three double differences. The terms on the right hand sidehave the following interpretation: • (cid:0) V up − V down (cid:1) − (cid:16) V σ up − V σ down (cid:17) : The shift in the Beveridge curve accounted for by thetime-variation in matching efficiency, conditional on having s t and ∆ U t at their ob-served values. • (cid:16) V σ up − V σ down (cid:17) − (cid:16) V s , σ up − V s , σ down (cid:17) : The shift accounted for by time-variation in the sep-aration probability, conditional on having ∆ U t at its observed values and σ held con-stant. • (cid:16) V s , σ up − V s , σ down (cid:17) − (cid:16) V s , σ , ∆ Uup − V s , σ , ∆ Udown (cid:17) : The shift accounted for by time-variation in ∆ U t + , conditional on having matching efficiency and the separation probability heldconstant. Note that V s , σ , ∆ Uup − V s , σ , ∆ Udown = ∆ U t + , s , σ − ∆ U t + , σ , s − s , ∆ U t + , σ − s , σ , ∆ U t + − σ , ∆ U t + , s − σ , s , ∆ U t + − Notes : Percentage point contributions to the vertical shift in the Bev-eridge curve, averaged over the “downswing” sample points dis-cussed earlier. “Ordering” column shows the order in which marginsare set to their observed values. For example, the row ∆ U t + , s , σ starts with the steady-state curve, then adds observed ∆ U t + , thenadds observed s t , and finally adds the observed σ t . Table 2: Contributions to the Shift in the Beveridge Curve(which cannot shift by construction) to the observed shift, by successively adding the ob-served time-variation in margins. Equation (22) first adds observed dynamics, then addsobserved the separation probability, then adds observed matching efficiency. With threemargins there are six possible orderings, and the results will, in general, depend on theordering.Table 2 shows the results of all six orderings. The results are remarkably consistent. In allversions, separations push the Beveridge curve down during the upswing period, relativeto the downswing period. Both dynamics and matching efficiency have the opposite effect,contributing to the counter-clockwise loop in the observed Beveridge curve. Generally, thecontribution of matching efficiency is larger than that of dynamics, sometimes dramaticallyso. The only outlier is the fourth row. However, we believe that the first two rows arethe most important, because they put ∆ U t +1