Long live the vacancy

Häfke, Christian; Reiter, Michael Working Paper Long live the vacancy IHS Working Paper, No. 22 Provided in Cooperation with: Institute for Advanced Studies (IHS), Vienna Suggested Citation: Häfke, Christian; Reiter, Michael (2020) : Long live the vacancy, IHS Working Paper, No. 22, Institut für Höhere Studien - Institute for Advanced Studies (IHS), Vienna This Version is available at: https://hdl.handle.net/10419/223416 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. 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Kunst Title Long Live the Vacancy Institut für Höhere Studien - Institute for Advanced Studies (IHS) Josefstädter Straße 39, A-1080 Wien T +43 1 59991-0 F +43 1 59991-555 www.ihs.ac.at ZVR: 066207973 License „Long Live the Vacancy“ by Christian Haefke, Michael Reiter is licensed under the Creative Commons: Attribution 4.0 License (http://creativecommons.org/licenses/by/4.0/) All contents are without guarantee. Any liability of the contributors of the IHS from the content of this work is excluded. All IHS Working Papers are available online: https://irihs.ihs.ac.at/view/ihs_series/ser=5Fihswps.html This paper is available for download without charge at: https://irihs.ihs.ac.at/id/eprint/5419/ Long Live the Vacancy† Christian Haefke ‡Michael Reiter§ September 2020 Abstract We reassess the role of vacancies in a Diamond-Mortensen-Pissarides style search and matching model. In the absence of free entry long lived vacancies and endogenous separations give rise to a vacancy depletion channel which we identify via joint unemployment and vacancy dynamics. We show conditions for constrained efficiency and discuss important implications of vacancy longevity for modeling and calibration, in particular regarding match cyclicality and wages. When calibrated to the postwar US economy, the model explains not only standard deviations and autocorrelations of labor market variables, but also their dynamic correlations with only one shock. JEL Codes: E24, E32, J63, J64 Keywords: Beveridge Curve, Business Cycles, Job Destruction, Random Matching, Separations, Unemployment Volatility, Wage Determination. †The authors wish to thank Almut Balleer, Melvyn Coles, Russell Cooper, Wouter den Haan, Jason Faberman, Shigeru Fujita, Andreas Gulyas, Johanna Haefke, John Haltiwanger, Ay¸se ˙ Imrohoro˘ glu, Leo Kaas, Philipp Kircher, Moritz Kuhn, Robert Kunst, John Leahy, Christian Merkl, Monika Merz, Jun Nie, Tamás Papp, Gilles St. Paul, Thijs van Rens, Etienne Wasmer, and participants of the 2017 Abu Dhabi Global Macro Workshop, 2017 Computing in Economics and Finance Conference, 2017 Labor Market Dynamics Workshop at the Board of Governors and the 11th Tsinghua Workshop in Macroeconomics 2018 for helpful discussions and valuable suggestions. ‡Social Science Division, New York University, Abu Dhabi, UAE. E-mail: [email protected] §Institute for Advanced Studies, Vienna, Austria and NYU Abu Dhabi. E-mail: [email protected] 1 Introduction In early 2017 mean vacancy duration ranged from fifteen days for small and medium enterprises to sixty five days for large enterprises1. Job search websites like indeed.com give guidelines on when and how to re-post jobs, resume-now.com provides strategies to job seekers who face reposted job ads. Clearly, vacancies have varying durations and are not necessarily destroyed at the end of a day/week/month if unfilled. In the absence of free entry for firm startups, e.g. as in Diamond (1982) or Melitz (2003), this observation implies that an economy’s stock of vacancies becomes a state variable and that in addition to the well studied job creation a second channel – vacancy2depletion – becomes relevant for labor market dynamics. Without explicitly labeling the channel, Coles and Moghaddasi Kelishomi (2018) illustrate the persistence and magnification of business cycle shocks that arise when a large number of newly unemployed job-seekers deplete the vacancy pool that then replenishes only gradually. We propose a model with endogenous separations that encompasses the well known Hall Milgrom model (Hall and Milgrom, 2008) and the model of Coles and Moghaddasi Kelishomi (2018). We identify the key parameter that governs the relationship between vacancy creation and vacancy depletion and show how it relates to the lead/lag structure of unemployment and vacancies. We establish that vacancies lead unemployment for a set of OECD countries with long available vacancy series and the U.S. since 1951 (cf. Figure 1), which indicates an overall dominating vacancy creation channel. Nevertheless, the quantitative importance of the vacancy depletion channel is substantial and the impact of long-lived vacancies on the model is far-reaching theoretically and empirically, as has been anticipated by Elsby, Michaels and Ratner (2015) who wrote “. . . areas in which additional research seems especially fruitful [. . . ] include the role of wage determination and entry into vacancy creation on the volatility and sluggishness of vacancy dynamics”. In addition to the important amplification of shocks generated by the depletion channel, the stock flow dynamics of vacancies generate highly persistent effects of temporary shocks (Fujita and Ramey, 2007) implying protracted deviations from steady state, in particular for unemployment. It is an empirical fact3that not every separation of an employment relationship leads to the destruction of the underlying job. Long lived vacancies naturally 1Based on data released by Steve Davis at https://www.dice.com/indicators/contents/data/2017/ 08/2017-08-DHI-HIring-Indicators-FINAL-1.xlsx. Mueller et al. (2020) report vacancy durations of comparable magnitude for Austrian linked employer-employee data. 2The analogous effect on the unemployment pool is widely acknowledged. Longer expansions typically lead to an exhaustion of the job-searcher pool, job-filling probabilities fall because it becomes increasingly harder to find qualified job applicants as expansions continue. 3On average the BLS reports 7.1 million destroyed jobs per quarter between 2001 and 2019 while an average of 13.7 million employment relationships were separated according to JOLTS. 1 allow the modeling of this fact, thus providing an important building block in modeling job and worker flows jointly. In our model firms create jobs by paying a sunk cost to post a vacancy. This vacancy can either be filled or destroyed. Vacancies that are neither filled nor destroyed carry over as stock to the subsequent period. We call these vacancies long lived in contrast to the canonical approach of destroying unfilled vacancies with probability one at the end of a period. To our knowledge long lived vacancies (LLV) were first introduced to the literature by Fujita (2004) and Fujita and Ramey (2007) who documented the substantial improvement in the modelling of persistence. A key mechanism in their setup is the departure from free entry of vacancies towards a job creation process that Coles and Moghaddasi Kelishomi (2018) call Diamond entry after Diamond (1982). Effectively, entrepreneurs draw a onetime job/vacancy creation cost and will enter only if the cost is sufficiently low. It is the combination of Diamond entry and long lived vacancies that (i) makes vacancy stock an important state rather than jump variable in the model and (ii) generates a positive value for unfilled vacancies which will be important for wage determination and potentially separations. Coles and Moghaddasi Kelishomi (2018) extend Fujita and Ramey (2007) to allow for exogenous, time varying job destruction and document an important interaction effect. A large inflow of unemployed in response to a negative productivity shock will match and thus absorb an important fraction of the vacancy stock. In the absence of free entry a below steady state vacancy stock leads to substantial persistence in unemployment and vacancies, a substantial labor market tightness response and a pronounced Beveridge Curve relationship. Our model with fully endogenous separations maintains these desirable properties. The original insight of Shimer (2005) and Costain and Reiter (2008) that Nash bargained wages fluctuate more than empirically observed is even more important in the absence of free entry when vacancies are long lived. Vacancy depletion substantially depresses wages in recessions. The wage that would decentralize the planner allocation varies substantially more than productivity. Over the last decade the bargaining protocol proposed by Hall and Milgrom (2008) has turned out to be a useful and convenient way to model wage determination in a Diamond Mortensen Pissarides (DMP) style search and matching model. As a generalization of Nash bargaining it offers a path to insulating wage negotiations from current unemployment and can thus generate empirically observed unemployment and wage fluctuations. When Hall and Milgrom introduced their bargaining protocol they focused on driving a wedge between workers’ threat points and outside options. In our model their protocol applies both to the worker and the firm side. Our model — when calibrated to US data for 1951–2003 — successfully matches a broad set of first and second moments including the Beveridge Curve, vacancy, unemployment and job finding variability as well as all relevant dynamic correlations. While the 2 Hall Milgrom benchmark does equally well in matching unemployment, we substantially improve for job finding probabilities as well as GDP persistence by allowing for labor market adjustments via both the hiring and separation margin through endogenous separations and the vacancy depletion channel. Finally, we fit our model to US labor market data from 1951–2018 and illustrate how the combination of features implied by long lived vacancies manages to capture the data surprisingly well with only one exogenous shock. While our model is successful in replicating both average job destruction and separations, capturing a richer set of establishment level characteristics and worker characteristics as in Cooper, Haltiwanger and Willis (2007) would require firm heterogeneity. Then there would be room for job-to-job transitions (e.g. Menzio and Shi (2011)), giving rise to vacancy ladders, where each transition absorbs a vacancy and opens another one. The entry mechanism based on noisy signals of productivity explored by Pries (2016) can be seen as an interesting complement to Diamond entry. The importance of the dynamics of vacancies and their values has also been pointed out by Shao and Silos (2013) in a model with Diamond entry, constant separations and long lived vacancies. Leduc and Liu (2020) estimate a two-shock model of endogenous search and recruiting intensity with exogenously varying match destruction in the spirit of Coles and Moghaddasi Kelishomi (2018) for 1967–2017 U.S. data. They show that both vacancy longevity and departure from free entry are necessary for recruiting intensity to depend on labor market conditions. Discount factor effects in models with long lived vacancies are particularly powerful because a vacancy can be considered capital. We focus on one shock and constant intensities to highlight the contribution of endogenous separations and microfounded wage bargaining. However, adding the discount factor shock and the variable intensity would help in further improving our model-fit in the post great recession period. The remainder of this paper is organized as follows. In section 2 we summarize the data. Section 3 describes the model. Theoretical insights are presented in section 4, in particular how endogenous separations and the Beveridge Curve relationship inform estimates of the unemployment elasticity in the matching function. Section 5 presents the calibration strategy, main numerical results, extensions and robustness checks before section 6 concludes. 2 Data To facilitate comparability we study the same period as Shimer (2005) for the major part of this paper and refer to his work for a comprehensive description of the data. Results remain qualitatively similar for alternative subperiods. Table 1 reports the usual business 3 cycle statistics4. Table 1: Descriptive Statistics for Key Labor Market Variables, 1951–2003. u v M φwδGDP Mean 5.655.675.70 —2.452.462.47 44.9744.9845.00 3.383.393.40 — StDev 0.170.190.21 0.190.200.22 0.0750.0830.093 0.1080.1180.126 0.0640.0710.080 0.0240.0260.029 RStDev 6.797.287.70 7.107.778.24 2.883.163.51 4.184.534.72 2.452.703.02 1.001.001.00 AC 0.880.930.98 0.880.930.98 0.830.900.97 0.890.940.99 0.770.860.95 0.870.930.98 u1−.95 −.93−.92 0.920.930.95 −.97 −.96−.95 0.680.730.80 −.93 −.91−.89 v1−.85 −.80−.76 0.930.950.96 −.72 −.64−.58 0.820.840.88 M1−.86 −.81−.78 0.730.790.85 −.85 −.82−.75 φw1−.72 −.63−.58 0.880.900.93 δ1−.80 −.74−.68 GDP 1 Except for the means all statistics have been computed for detrended (HP, smoothing parameter 10e5) and seasonally adjusted logarithms of the respective quarterly time series. Unemployment, u, is the UNRATE series of the St. Louis Fred, vacancy, v, data has been provided by Regis Barnichon (Barnichon, 2010). Job finding, φw, and separation, δ, probabilities have been computed based on the methodology described in Shimer (2012). Matches, M, are computed by multiplying the unemployment rate (u)with the job finding probability (φw). GDP is real GDP as downloaded from FRED. 95% bootstrapped confidence bounds are reported in subscripts to the left and right of the respective statistic. The average unemployment rate over the sample period, as published by the BLS based on CPS data, was 5.67%. Unemployment, u, fluctuates strongly around its trend with a standard deviation of 0.19, which is more than seven times the variability of real per capita GDP. Regis Barnichon (2010) provides a methodology to combine data on the Help Wanted Index and the more recent JOLTS data. We use the vacancy series provided on his website5, which combines the advantage of a long series from the Help Wanted Index with the accurate measurement of JOLTS since 2000. In Table 1 and Figure 2 we see that the cyclical component of vacancies, v, fluctuates approximately as much as unemployment and is highly persistent, consistent with the findings of Christiano, Eichenbaum and Trabandt (2016); Davis, Faberman and Haltiwanger (2013); Fujita and Ramey (2007) for the US and a number of other OECD countries (Amaral and Tasci, 2016). The contemporaneous correlation with unemployment is strongly negative with -0.91, however, the highest correlation between unemployment and vacancies obtains when vacancies are lagged one period, i.e. vacancies lead unemployment by one quarter. Similar behavior prevails for all European countries with at least 50 years of data availability in the OECD database as illustrated in Figure 1. 4When series are available at higher frequencies, they are averaged to quarterly series. For all series we study log-deviations from a slow moving HP trend with smoothing parameter 105. A detailed report of all data sources and transformations is provided in Appendix A together with extra figures. 5https://sites.google.com/site/regisbarnichon/cv/HWI_index.txt 4 -1.00-0.50 0.00 0.50 1.00 -1.00-0.50 0.00 0.50 1.00 Correlation U(t),V(t+∆)) -20 -10 0 10 20 ∆ of Vacancy Stock AUT -1.00-0.50 0.00 0.50 1.00 -1.00-0.50 0.00 0.50 1.00 Correlation U(t),V(t+∆)) -20 -10 0 10 20 ∆ of Vacancy Stock DEU -1.00-0.50 0.00 0.50 1.00 -1.00-0.50 0.00 0.50 1.00 Correlation U(t),V(t+∆)) -20 -10 0 10 20 ∆ of Vacancy Stock CHE -1.00-0.50 0.00 0.50 1.00 -1.00-0.50 0.00 0.50 1.00 Correlation U(t),V(t+∆)) -20 -10 0 10 20 ∆ of Vacancy Stock NOR -1.00-0.50 0.00 0.50 1.00 -1.00-0.50 0.00 0.50 1.00 Correlation U(t),V(t+∆)) -20 -10 0 10 20 ∆ of Vacancy Stock GBR -1.00-0.50 0.00 0.50 1.00 -1.00-0.50 0.00 0.50 1.00 Correlation U(t),V(t+∆)) -20 -10 0 10 20 ∆ of Help Wanted Index USA For European Countries, Source: OECD, quarterly Data 1961--2012, For US: FRED and Regis Barnichon, quarterly Data 1951-2003, HP filtered λ=1e5 U-V Dynamic Correlations Figure 1: Dynamic Unemployment – Vacancy Correlations for USA and five European Countries. US Data: FRED and Barnichon (2010), 1951–2003. European Data: OECD, 1961–2012. ∆denotes the time shift in the vacancy series when computing the dynamic correlations. The red line indicates the largest negative correlation. Vacancies lead by one quarter in all countries except Switzerland, where they lead by four. 5 Definition 1 (Equilibrium).A symmetric equilibrium for the model economy consists of a sequence of thresholds ¯ κvt,¯ κjt,¯ κmt; a sequence of labor market stocks et,ut,vtand flows nt; and a sequence of wages wt,˜wf t,˜ww t; such that for any time period t the following hold: 1. Vacancy Creation: ¯ κvt=V˘ Vtand Equation 2; 2. Endogenous Separations: ¯ κjt=VJ tand ¯ κmt=VJ t−VV t; 3. Labor Market Transitions: Equations 10–12; 4. Wage Bargaining: The firm (Equation 16) and worker indifference (Equation 17) conditions; and wage aggregation by 18. With value functions V ˘ Vt,VV t,˜ VJ t(w),VJ t,VU t,˜ VE t(w),VE tdefined in Equations 4, 5, 13 – 15 and 19 – 20. 4 Analytical Results In this section, we study some basic implications of long-lived vacancies. We first show that the well known conditions for constrained efficiency in Mortensen/Pissarides (MP) models continue to hold with long-lived vacancies. Then we investigate under which conditions long-lived vacancies make an important difference for labor market fluctuations. Even though these results only hold in a special case of the model with Nash bargaining and exogenous (but time-varying) separations, they yield important insights for the general case which is analyzed numerically in Section 5. We conclude section 4 by deriving some comparative steady state results, which are natural generalizations of the MP model and provide important guidance for the calibration. 4.1 Planner Solution and Decentralized Equilibrium In the canonical labor market matching model, the decentralized equilibrium under Nash bargaining is constrained efficient if the worker’s bargaining power equals the elasticity of matches w.r.t unemployment. Long-lived vacancies and Diamond entry seem to change the bargaining situation, because firms are left with a valuable vacancy if the bargain breaks down, while workers are unemployed. Nevertheless, the efficiency of the decentralized equilibrium is preserved. However, constrained efficiency holds only in the case of exogenous separations; the way we have modeled endogenous separations generates a hold-up 12 problem that makes the equilibrium inefficient. Thus assume for this section that rather than being endogenously determined, match and job destruction follow a stochastic exogenous process. The planner solves max {nt,˘vt,vt,ut} ∞ ∑ t=1 βtE0(yt−κk)(1−ut)+but−κs˘vt−Vk(nt)(21) subject to vacancy and unemployment dynamics as given in (9 – 11) and the matching technology as represented in equation (6). Here Vk(n)denotes the total cost of creating nnew vacancies, which is derived from the vacancy posting cost distribution as Vk(n) = RV−1(n) 0xdV(x). Denoting by W˘ Vt,WV tand WS tthe Lagrange multipliers of constraints (9), (10) and (11), respectively, we can derive the optimality conditions: W˘ Vt=V0 k(nt),(22a) W˘ Vt=−κs+WV t+φf t(1−α)WS t−WV t,(22b) WV t=β(1−δv)EtW˘ Vt+1,(22c) WS t=yt−κk−b+Et(1−δt+1)βWS t+1+δmt+1βWV t+1−φw t+1αβWS t+1−WV t+1.(22d) Since firms unilaterally determine vacancies, it is necessary for optimality that the value of a vacancy coincides for the firm and the planner. It is also sufficient, because vacancy posting is the only decision in this version of the model. Therefore the planner’s Lagrange multipliers WV tand W˘ Vtmust equal the vacancy values VV tand V˘ Vtfor firms. This is the case if the firm surplus (VJ t−VV t)equals (1−α)(WS t−WV t). Similarly, the planner’s Lagrange multiplier of employment, WS t, equals total surplus (VJ+VE−VU)in the decentralized economy if (VE−VU) = α(WS−WV). Both conditions are satisfied if wages are determined by generalized Nash bargaining with worker bargaining power ω=α, which extends the Hosios condition to the case of long lived vacancies with exogenously time varying separations. The details of the derivation are in Appendix B.1. Result 1 (Constrained Efficiency).If wages are determined by Nash bargaining, and worker bargaining power ωequals the elasticity of matches with respect to unemployment α, then the Planner allocation (22) with exogenously time varying separations coincides with the solution to the decentralized economy. 4.2 When does vacancy longevity matter? Diamond entry and vacancy longevity are two important departures from the canonical model. Each on their own does not have a substantial impact, whereas their interaction 13 does. It can be easily seen from (22) that short lived vacancies, i.e. δv=1, are equivalent to a standard model with time varying search costs. Zero vacancy creation costs κvare a rather trivial special case where vacancy creation is determined by search costs κs. Equivalence with the canonical model and thus irrelevance of longevity then immediately holds because destroyed vacancies are worthless. More generally, with constant marginal entry costs, the impact of vacancy longevity is substantially diminished. First, by fixing the value of a vacancy, entry costs stabilize the capital cost related to vacancy destruction. The second role is probably more important: if vacancies can be created at a constant marginal cost, the vacancy stock adjusts to the value of a vacancy with infinite elasticity, eliminating the sluggishness of the stock-flow dynamics. The relationship between short and long lived vacancies is captured more formally in the following: Result 2 (Equivalence Short and Long Lived Vacancies).Assume that V0 k(nt) = κvand the job destruction rate δjt=δt−δmtare constant over time. We consider two economies that differ only in the vacancy destruction rate δv, vacancy creation cost κv, the search cost κs, and the capital cost κk. Index the two economies with S and L, so that economy S is characterized by the set of parameters {δs v,κs v,κs s,κs k}and economy L is characterized by the set of parameters {δl v,κl v,κl s,κl k}. Given any {δs v,κs v,κs s,κs k}and {δl v,κl v}, by setting κl k=κs k−[1−β(1−δj)]β1−δl vκl v+[1−β(1−δj)]β(1−δs v)κs v,(23) κl s=κs s+[1−β(1−δs v)]κs v−h1−β1−δl viκl v,(24) the economies S and L have the same total surplus Σt, the same market tightness θt, and therefore also the same unemployment rate ut. In the decentralized equilibrium with Nash bargaining, this also implies the same wage rate wtand values of employment and unemployment. To derive result 2, define total surplus Σas Σt= (WS t−WV t), and use (22c) to rewrite (22b) as κs+W˘ Vt−β(1−δv)EW˘ Vt+1= (1−α)φf tΣt.(25) Furthermore, rearrange (22d) to write the surplus as Σt=yt−κk−b+βEt1−δt+1−αφw t+1Σt+1+βWV t+11−δjt+1−WV t.(26) The left side of (25) can be interpreted as the user cost of an unfilled vacancy. Next to the flow cost κs, it captures the effect of discounting β, the depreciation rate δvand the expected capital gain EtW˘ Vt+1−W˘ Vt. In the special case where vacancy creation costs are constant (V0 k(n) = κv), (22c) shows that the value of a vacancy is also a constant, given by WV t=β(1−δv)κv, and the user cost of vacancies becomes κh≡κs+ (1−β(1−δv))κv. 14 Then (23) says that the user cost of unfilled vacancies must be identical across the two calibrations. The effect of vacancies on the surplus is given by the bracketed term on the right side of (26), which can be interpreted as the user cost of a filled vacancy. If, in addition, job destruction δjt=δt−δmtis constant, this term is also constant and given by −β[1−β(1−δj)](1−δv)κv. Then (24) says that the constant term in the surplus equation, given by c=−[κk+b−β[1−β(1−δj)](1−δv)κv]must be identical across the two calibrations. The details of the recalibration are in Appendix B.2. Under these special assumptions, we get the following system of two dynamic equations, where the exogenous process ytdrives the dynamics of the surplus Σtand market tightness θt: Σt=yt−c+βE1−δt−αθ1−α t+1Σt+1(27) κh= (1−α)θ−α tΣt.(28) In the general case, the user cost of a filled vacancy can be written as βEtWV t+11−δjt−WV t=Et(WV t+1−WV t)−WV t+1(1−β)−βWV t+1δjt.(29) It includes three components: the capital gain EtWV t+1−WV t, the interest cost (1−β)EtWV t+1, and the expected loss through job destruction βEtWV t+1δjt+1. From this and the dynamics of vacancies, we see that LLV affect labor market dynamics in three ways. First, they affect the formation of new vacancies through the user cost of unfilled vacancies. Second, they affect the total surplus of a filled job through the user cost of filled vacancies. Third, they introduce stock-flow dynamics into vacancy creation. If vacancies are very persistent, the stock reacts sluggishly to the creation of new vacancies, which is underlying the mechanism in Coles and Moghaddasi Kelishomi (2018). Equation 25 can be interpreted as a generalized job creation condition. The share of the present value of the surplus going to the firm has to equal the user cost of an unfilled vacancy. 4.3 Steady State Responses In this section we consider the deterministic steady state of the model with Nash bargaining. Throughout this section, variables without time index denote steady state values, hats denote log deviations from steady state, and ηx ydenotes the elasticity of any variable xwith respect to exogenous productivity y. Similar to Shimer (2005), we find that the flow-equilibrium approximation is good and focus on it in this section. Hence any variable xcan then be considered a function of productivity yonly. 15 4.3.1 The Mechanics of the Beveridge Curve With Time-Varying Separations We start by deriving a few relationships that just follow from the mechanics of unemployment and vacancy dynamics, and are independent of the economics of vacancy creation and job separations. In the standard MP model, it is well known that procyclical job destruction tends to make the correlation of unemployment and vacancies positive, From the steady state relationship u=δ δ+φwand the matching relationships φw∝ θ1−αand ˘v=θuwe obtain ˆu=(1−−(1−α)uφw δ)ˆ θ+(1−u)ˆ δ= (1−u)ˆ δ−(1−α)ˆ θ,and (30) ˆ ˘v=1−(1−α)uφw δˆ θ+(1−u)ˆ δ=ˆ θ+(1−u)ˆ δ−(1−α)ˆ θ.(31) From this we can infer that time varying separations, endogenous or exogenous, leave the Beveridge Curve (opposite sign of duand d˘v) intact as long as the tightness response is sufficiently strong. This is formalized in Result 3: Result 3 (The Beveridge Curve).Let us assume that both tightness and the job destruction rate are a function of labor productivity. Across steady states, unemployment and vacancies go in opposite directions if −ηδ y<1 (1−u)−(1−α)ηθ y.(32) In the data, the correlation between unemployment and vacancies is not just negative, but close to -1, and the fluctuations in the two variables are of about equal size, which implies ˆut ˆ ˘vt≈ −1. Using this in (30) and (31) yields the following Result 4 (Calibration of α). α=1−1 2(1−u)−ηδ y ηθ y .(33) It follows from (30) and (31) that countercyclical separations (negative ηδ y) tend to increase the fluctuations of unemployment relative to those of vacancies. When the Beveridge Curve relationship is imposed, the elasticity of separations determines the cyclicality of matches. We have the following result about the log deviation from steady state of matches out of unemployment ˆ M: Result 5 (Cyclicality of Matches:).Since ˆ M=αˆu+ (1−α)ˆ ˘v, we get from (30) and (31) that ˆ M= (1−α)uˆ θ+(1−u)ˆ δ. Using (33), this gives ˆ M= u 2(1−u)+ηδ y ηθ y!ˆ θ.(34) Since tightness is procyclical, so are matches in the model, if separations are constant (i.e. ηδ y=0). Countercyclical matches, as observed in the data by Blanchard and Diamond (1990) or Mortensen (1994), are consistent with a strong Beveridge curve only if separations are sufficiently countercyclical (ηδ y<0). 16 4.3.2 Tightness With Time-Varying Separations We now analyze the effect of changes in labor productivity yand the total separation rate δ, which is considered here as an exogenous parameter. As in the standard MP model (Costain and Reiter, 2008; Hagedorn and Manovskii, 2008), the volatility of the labor market in this model is strongly affected by the size of the dynamic surplus in steady state, Σ, which we can derive from (26) as Σ=y−κk−b+βWV(1−δj)−WV 1−β(1−δ−αθ1−α).(35) It differs from the standard formula by the capital costs related to vacancies, the term in parentheses in the numerator, which reduces the surplus, everything else being equal. For brevity, write this term as ¯c−βWVδjwhere ¯c=βWV−WV−κk−b.Use (35) in (28) and rearrange to obtain: κhθα 1−α=y+¯c−βWVδj+β1−δ−αθ1−ακhθα 1−α.(36) Total differentiation gives [1−β(1−δ)]αΣdθ θ=dy −βWVdδj−βακh 1−αdθ=dy −βWVdδj−βΣφwdθ θ.(37) Since the steady state level of yis normalized to unity ˆ θ=ˆy−βWVδjˆ δj 1−β(1−δ−φw)·1 αΣ.(38) Naturally, higher labor productivity tends to decrease the job destruction rate, therefore ˆ δj<0. From (38) we see that countercyclically varying separations allow to match a given magnitude of tightness fluctuations with either a larger surplus or lower exogenous productivity fluctuations. 4.3.3 Wages It is straightforward to write the worker surplus in steady state as: VE−VU=wt−b 1−β[1−φw−δ].(39) Since in a standard calibration the job finding rate is much higher than the separation and the discount rate, this can be approximated as VE−VU≈wt−b φw.(40) This gives w−b≈φwΣ=θφfΣ∝θ (41) 17 because φfΣis a constant in the case of constant vacancy creation costs. In percentage deviations, this implies ˆw=ˆ θw−b w.(42) In the data, tightness fluctuates about 13 times as much as output and almost 20 times as much as wages. If the model is suitable to explain the fluctuations in unemployment, and therefore in tightness, there are basically three alternatives: 1. average surplus w−b wis very small, as in Hagedorn and Manovskii (2008); 2. the link between tightness and the wage is weakened, e.g. by a different bargaining scheme as in Hall and Milgrom (2008); 3. wages fluctuate much more than output, which is what happens in Coles and Moghaddasi Kelishomi (2018). Since the third alternative is clearly at odds with the data and we want to show that the model can explain unemployment fluctuations even with a big worker surplus, we assume a wage setting mechanism that dampens wage fluctuations. Nevertheless, our wages fluctuate about as much as labor productivity, and are not “rigid” by conventional criteria. 5 Numerical results After discussing our baseline calibration in Section 5.1, we show in Section 5.2, that the model is very successful in matching the usual labor market statistics. As a benchmark, we compare our model to the well-established Hall Milgrom (HM) model. In Section 5.3 we investigate the main mechanisms in the model and explain the vacancy creation versus the vacancy depletion channel. As we have already discussed in Section 4.2, LLV matter little if new vacancies are extremely elastic with respect to the value of a new vacancy. Identifying this elasticity is therefore a key task in the empirical validation of the model, and we explain in Section 5.4 which aspects of the data can be used for identification. Section 5.5 contains a robustness check, showing that our model is able to explain labor market fluctuations even if workers’ outside option is procyclical. Finally, in Section 5.6 we show that our model with only one shock goes a long way in explaining the whole postwar US labor market history. 5.1 Calibration 5.1.1 Calibration of the baseline model We have chosen the time period as the 60th part of a quarter, which we refer to as a workday. We follow in this respect Christiano, Eichenbaum and Trabandt (2016), who subdivide their 18 Table 2: Calibrated Parameters: Values and Targets Interpretation Parameter Value Fixed Exogenously Discount factor β0.961/240 Utility of non-employment rtp b0.4544 Worker flow utility in disagreement rtp bb0.4544 Worker Probability of making an offer ω0.6491 Firm Bargaining cost in disagreement rtp κb0.1698 Firm Flow Search Cost rtp κs0.1024 Arrival Prob of Job Dest. Event λj0.0023 Arrival Prob of Match Dest. Event λm0.0011 Arrival Prob of Vacancy Opportunity λv1.0000 For Averages Capital Cost κk0.0000 Mean of Vacancy Creation Cost µv2.4382 Mean of match destruction shock µm0.0631 Mean of job destruction shock µj2.4977 Efficiency of Matching Function A0.0403 For Second Moments Expected quarterly labor productivity 1 AR parameter for labor productivity ρy0.9987 Standard Deviation of labor productivity innovation σy0.0009 Elasticity of new vacancies ξ15.8780 Elasticity of matches wrt unemployment α0.6491 Breakdown of bargaining while disagreeing δb4.93¯ δ Std.Dev of Match destruction cost σm0.2003 Std.dev Job Destruction cost σj0.1758 rtp denotes relative to productivity. 19 quarterly period into 60 subperiods when computing the bargaining outcome. We set the discount factor βto the conventional value of 0.99 quarterly. We think it is important that the model features a high worker surplus, that means a substantial difference between the wage and the unemployment benefit. In the benchmark, we set the unemployment benefit to 71 percent of the steady state wage, following Hall and Milgrom (2008). In a robustness check we also consider a replacement rate of 40 percent as in Shimer (2005). Several parameters are chosen so as to match steady state values. The matching productivity A=0.0403 and an average vacancy cost of µv=2.4382 are chosen so as to achieve a steady state unemployment rate of 5.67 percent, and a filling rate φfof one third per week. The latter value was taken from Fujita and Ramey (2007) and Coles and Moghaddasi Kelishomi (2018). We target a steady state wage of 64 percent of production, conforming to a 36 percent gross capital share. The wage is strongly influenced by the bargaining position of both sides, mainly the worker utility during disagreement, bb, and the capital costs while in disagreement, κb. Since we do not have strong evidence on these parameters, we normalize bb=b, and fix κbso as to obtain the 64 percent labor share. Out of the capital share, all the costs of maintaining the match must be covered. These include the capital costs κk, the expected job maintenance and match maintenance costs, as well as the expected costs of filling a vacancy. As we have explained above, κkshould measure the cost of capital that is not tied to the specific job but can be hired flexibly. Since we do not have any direct observation of the split of capital into general and job-specific, we make the extreme assumption that all capital is job-specific and therefore set κk=0, which leads to a very high average value of a vacancy. Following Silva and Toledo (2009), we set the search cost parameter κs so that the average cost per hire equals 4 percent of the quarterly wage. We estimate the six parameters ρy,α,ξ,σj,σm, and δb, so as to match six second moments in the data, namely the quarterly autocorrelation of GDP and of vacancies, the variances of unemployment and of vacancies, and the variances of job destruction and of match destruction. Although the effects of all these parameters are inter-dependent, there is a clear interpretation of which parameter is driving which moment: • We choose the autocorrelation coefficient of labor productivity so as to match the quarterly autocorrelation of GDP. This results in ρy=0.99893, which conforms to a quarterly value of 0.938. • We choose the elasticity of matches with respect to unemployment, α, so as to match the ratio of unemployment volatility to vacancy volatility. This gives α=0.6491, close to the value used in Coles and Moghaddasi Kelishomi (2018), and in-between the most common value in the literature, which is 0.5, and the value estimated in 20 Shimer (2005), which is 0.72. Motivated by the equivalence in Result 1 we impose the Hosios condition ω=α. • The elasticity of vacancy creation ξis the key parameter that determines the strength of the vacancy creation versus the vacancy depletion channel, and therefore the phase shift between vacancies and unemployment. We target a lead of vacancies to unemployment of 0.18 quarters (cf. Section 5.4). This gives a value of ξ=15.8780 which substantially exceeds the estimate of ξ=1 in Fujita and Ramey (2007), and the estimate of ξ=0.26 in Coles and Moghaddasi Kelishomi (2018). One should notice that it is not easy to compare this value across models. Since it is an elasticity with respect to the value of a vacancy, it depends very much on the average value of a vacancy. Because of our choice κk=0, we calibrate a high vacancy creation cost and therefore a high equilibrium value. Increasing κkand lowering the vacancy creation cost, the model would generate similar equilibrium dynamics with a substantially lower elasticity ξ. • Mean µjand dispersion σjof the job maintenance cost drive mean and variance of job destruction; given job destruction, mean µmand dispersion σmof the match maintenance cost drive mean and variance of total separations. We observe an average monthly separation rate of 3.3 percent, and an average job destruction rate equalling 66.87 percent of all separations. It is difficult to obtain evidence on job-flows (rather than worker flows) for the entire sample period. However, based on the pioneering work of Davis and Haltiwanger (1992), quarterly aggregate numbers for total private sector establishment level job destruction are published by the BLS as the Business Dynamics Statistics of the US Census Bureau. For 1994–2014 the (imputed) average monthly probability of job destruction is 1.75%, i.e. approximately two thirds of total separations. The remaining separations are match destructions, each leaving an unfilled vacancy. We choose σjand σmso as to match the variance of the two destruction rates. Given any σjand σm,µjand µmcan be set in steady state so as to obtain the right mean. This results in σj=0.1758, σm=0.2003, µj=2.4977 and µm=0.0631. • The probability of break-up during disagreement, δb, determines how strongly the unemployment rate influences wages, and is therefore a key determinant of the variance of unemployment. Our calibrated value δb=4.93¯ δis higher than in Hall and Milgrom (2008), who set δb=4¯ δ. Endogenous separations make it easier for our model to match unemployment fluctuations. This leaves three parameters undetermined. Two of them can basically be considered as 21 increase, so that the stock of vacancies start rising. Both effects lead to a gradual increase in the job finding probability. The reduction in unemployment leads to a reduction in the number of job matches, which in in turn leads to a further increase in the vacancy stock. The increase in vacancies then leads to a further reduction in unemployment, and so on. All this is the consequence of the interplay between endogenous vacancy destruction and LLV, which has been described very insightfully in Coles and Moghaddasi Kelishomi (2018). Notice that the peak in vacancies slightly precedes the peak in unemployment. This coincides with the finding that vacancies lead unemployment in the phase analysis, and is markedly different from the Coles-Kelishomi model with ξ=0.265. Figure 5 shows that vacancy dynamics depend critically on the value of ξ, the elasticity of new vacancies with respect to vacancy value. Our calibration identifies a very high elasticity, ξ=15.8780. The bottom right panel displays the same decomposition, but now for the more conventional parameter value ξ=1. In this case, the reduction in total matches becomes the driving force for vacancy dynamics almost immediately after the shock hits. 5.4 Identifying ξ We have seen in Section 5.3 that a higher vacancy creation elasticity ξstrengthens the importance of vacancy creation versus vacancy depletion. Why does this matter? More elastic vacancy creation leads to a faster absorption of newly unemployed workers into employment. The lower left panel of Figure 5 confirms this with the following experiment. We start out of steady state, with an unemployment rate of 1 percentage point above steady state, while productivity is always at the steady state level. In the HM model, only 16 percent of the additional unemployment are left after 3 months, while 38 percent remain in our model. Roughly speaking, the return to normal is about twice as fast in the HM model. We perform the same experiment for a wide range of values of ξ. For very high values of ξ, our model converges to the HM model, but it makes little difference whether ξ=1 or ξ=30 in this respect. This is partly due to the fact that, for each value of ξ, we recalibrate the model so as to match the volatility of unemployment, vacancies, job destruction and match destruction. The difference to the case without recalibration is not big, however. How is the value of ξidentified? There are two dimensions that are strongly affected by ξ: the phase relationship between vacancies and other variables, and the persistence of vacancies. As explained in the calibration section, we choose ξso as to match the observed phase shift between unemployment and vacancies. Table 5 provides summary information on the phase relations between the four variables unemployment, vacancies, job finding and separation rates, for various values of ξ, again using the same recalibration strategy. A negative number means that the second variable is leading. For example, the number -0.18 in the first column means that unemployment (the first variable in u,v) has the strongest abso28 lute correlation with the second variable, v, if vis lagged by 0.18 quarters.15 The baseline calibration with ξ=15.8780 does a very good job in matching the dynamic correlations. It gets the sign of all the leads right, and it matches the length of the lead within about half a quarter in all cases. As expected, the higher ξ, the stronger is the lead of vacancies relative to unemployment and other variables. No matter which phase shift one is targeting, the data always indicate a value of ξbetween 5 and approximately 30. Although this is a huge numerical range, we have seen in Figure 5 that these values have about the same implications for the speed of absorption of unemployment. The shape of the correlation functions is shown in Figure 6. In each panel, we show the observed correlations, bootstrapped 95% confidence bounds of the data, as well as the simulated correlations of the baseline model calibration. Since our one-shock model always generates stronger correlations than what we find in the data, all the model-generated correlation functions are shifted so as to have the same extreme value as the data. Then the shifted model correlations are within the confidence bounds almost always. Table 6 presents autocorrelations implied by the model for various choices of ξ. As expected, the autocorrelation of vacancies (and most other variables) is higher the lower is ξ. Taking the more conventional value of ξ=1 from Fujita and Ramey (2007) results in vacancies that are much more persistent than in the data. Matching the autocorrelation of vacancies would result in a somewhat higher value of ξ, around ξ=26. All these values point to a range where the absorption of unemployment is much slower than in the HM model. To sum up, the evidence relating vacancies, unemployment and job finding rates strongly points towards a high elasticity of vacancy creation, such that vacancies lead unemployment over the cycle. We have seen in the international data that this is a robust stylized fact. Although the elasticity is high, the model still predicts a speed of unemployment absorption that is much lower than in the standard MP and HM models. 5.5 Robustness check: cyclical employment opportunity cost Chodorow-Reich and Karabarbounis (2016) provide empirical evidence that the opportunity costs of employment are procyclical, which makes the job surplus less procyclical and makes it harder for the model to explain unemployment fluctuations. In the HM model, there are two types of opportunity costs of employment: the utility of being unemployed, and the utility of disagreement in the bargaining process. It is mostly the latter which affects bargaining, and it is not clear to what extent the evidence in Chodorow-Reich and Karabarbounis (2016) applies to the disagreement situation. To make sure that cyclical opportunity 15Since we observe variables only at a quarterly frequency, we interpolate quarterly correlations by a cubic spline and report the phase shift at which the absolute values of this spline interpolation reach their maximum. 29 Correlation: ut,vt+∆Correlation: ut,φw t+∆ 432101234 0.9 0.8 0.7 0.6 0.5 0.4 0.3 Data Model, shifted 432101234 1.0 0.9 0.8 0.7 0.6 0.5 0.4 Correlation: vt,φw t+∆Correlation: ut,δt+∆ 432101234 0.3 0.4 0.5 0.6 0.7 0.8 0.9 432101234 0.0 0.2 0.4 0.6 0.8 Correlation: vt,δt+∆Correlation: φw t,δt+∆ 432101234 0.8 0.6 0.4 0.2 0.0 432101234 0.8 0.6 0.4 0.2 0.0 Figure 6: Dynamic Correlations. The horizontal axis in each panel depicts the time-shift ∆, the vertical axis the correlation coefficient. 30 Table 5: Dynamic Correlations. Model u,v u,φwv,φwu,δv,δ φw,δ Data -0.18 -0.22 0.06 -1.25 -1.31 1.41 Model, ξ=1 0.51 0.25 -0.24 -1.39 -2.00 1.71 Model, ξ=5 0.27 0.14 -0.12 -1.30 -1.61 1.46 Model, ξ=10 0.01 0.01 0.00 -1.17 -1.19 1.19 Baseline, ξ=15.8780 -0.18 -0.09 0.09 -1.06 -0.86 0.97 Model, ξ=30 -0.41 -0.21 0.17 -0.91 -0.41 0.65 Model, ξ=100 -0.64 -0.34 0.21 -0.69 -0.02 0.24 Model, ξ=1000 -0.76 -0.42 0.20 -0.50 0.14 0.04 Hall/Milgrom model -0.80 -0.44 0.20 — — — Quarterly correlations are interpolated by a cubic spline. We report the phase shift at which the absolute values of this spline interpolation reach their maximum. Thus −0.18 for (u,v)means unemployment needs to be lagged by 0.18 quarters (0.18 ·365/4=16.4 days) in order to obtain the maximal correlation between unemployment and vacancies — i.e. vacancies lead unemployment. Table 6: Autocorrelations. u v φwδM Y y w Data 0.938 0.943 0.935 0.821 0.889 0.937 0.888 0.801 Model, ξ=1 0.953 0.958 0.957 0.811 0.920 0.929 0.888 0.936 Model, ξ=5 0.954 0.957 0.956 0.844 0.941 0.930 0.888 0.933 Model, ξ=10 0.953 0.953 0.953 0.870 0.952 0.930 0.888 0.929 Baseline, ξ=15.8780 0.950 0.945 0.948 0.883 0.951 0.929 0.888 0.924 Model, ξ=30 0.944 0.927 0.938 0.892 0.934 0.927 0.888 0.916 Model, ξ=100 0.931 0.876 0.914 0.893 0.873 0.920 0.888 0.900 Model, ξ=1000 0.922 0.818 0.892 0.889 0.796 0.912 0.888 0.890 Hall/Milgrom model 0.921 0.805 0.889 - 0.330 0.910 0.888 0.889 31 Table 7: Cyclical Employment Opportunity Costs. u v φwδM Y y w Data StdevRel 7.28 7.77 4.53 2.70 3.16 1.0 0.77 0.70 Baseline, δb=4.93¯ δ StdevRel 4.92 5.24 3.56 1.82 1.36 1.00 0.72 0.99 StdevRelZ 6.80 7.25 4.93 2.52 1.88 1.38 1.00 1.37 More rigid wages, δb=2.20¯ δ StdevRel 7.26 7.75 5.26 2.66 2.01 1.00 0.59 0.75 StdevRelZ 12.28 13.11 8.91 4.50 3.40 1.69 1.00 1.27 costs have an effect, we implement them by making both band bbproportional to TFP. Does the model still succeed if we assume procyclical employment opportunity costs? In the part of Table 7 titled “Baseline, δb=4.93¯ δ”, we have left all other parameters unchanged. The cyclical variability of unemployment is in fact reduced, to 4.92 from 7.28. A large part of this comes from a reduction in the variability of job separations: the incentive to separate in a recession is mitigated if outside options have deteriorated as well. To generate a realistic degree of unemployment variability, it appears necessary to shield the wage bargain more strongly from unemployment fluctuations. The results reported in the second part of Table 7 assume a breakup rate under disagreement of 2.20 times the total separation rate (all other parameters unchanged). Similarly to the case of low replacement ratio (cf. Table 4), a reduction in the breakup rate brings the model in line with the data, including the volatility of wages, which is higher than that of productivity. Again, there is no indication of “wage rigidity” when the usual criteria are applied. 5.6 The US labor market history 1951–2018 We have found that our model is quite successful in matching the dynamic properties of the data, even along some dimensions that are usually not considered, such as the dynamic correlations between different variables. The success may come as a surprise, given that our model has only one shock, namely a technology shock. This is in stark contrast to the recent DSGE literature, which assumes up to 10 different shocks, and usually finds that technology shocks contribute little to labor market dynamics. To see whether the one-shock structure is an important constraint, we explore to what extent the model can fit the labor market history of the US in the postwar period. For this purpose, we construct the shock series such that the model matches the stock of vacancies perfectly in every quarter. Since the data are quarterly and the model period is daily, we 32 assume that the data are constant over a quarter when we fit the model to the data. We initialize the state variables of the year 1951 by their expected values, according to our model, conditional on the observed values of unemployment, vacancies, the job finding rate and the total separation rate. With these starting values for 1951, and using the estimated series of shocks, we then simulate the US labor market variables for the whole time period 1951-2018. What we report are not one-period forecasts, but one long simulation, without resetting the states. Figure 7 shows the data and our simulated series. We show the total separation rate, not the components job separation and match separation separately, since there are no long enough data series for those. The stock of vacancies is perfectly matched by construction. Perhaps surprisingly, both our model and the HM model match vacancies by using almost identical shock sizes. Conditional on matching vacancies, the fit for unemployment is very good for both models. This basically says that the matching function is a good approximation to reality. In terms of correlation, both models are very good at fitting the job finding probability, but HM overpredicts the amplitude. This results from the lower value of α, and ultimately comes from the fact that the HM model needs a more volatile job finding rate, because it has a constant separation rate (cf. our discussion in Section 5.2). Perhaps the biggest discrepancy between data and models is that both models predict a faster recovery of the unemployment rate in the great recession, given the observed vacancies. This would imply a reduction in matching efficiency in that period. Variations in advertisement intensity of vacancies, the mechanism stressed in Leduc and Liu (2020), might explain this behavior. For the separations, the correlation between model and data is 0.673, lower than for the other labor market variables, mostly due to a bad fit over the time period 1985-1995. Nevertheless, the model captures well the sharp spikes and lower autocorrelation of separations. These findings suggest that the labor market variables are closely tied together through the mechanisms of the separation and the vacancy posting process, so that the dynamics can be explained by a model with only one shock. One can interpret this shock as the “marginal profitability of employment”. In our model, the marginal profitability is closely linked to the average productivity of labor. This relationship can be tested in the data. The bottom right panel of Figure 7 displays our identified shock, together with measured labor productivity (productivity per person; productivity per hours would yield a similar picture), and an index of “financial tightness”, namely Shiller’s price-earnings-ratio (series “CAPE”, detrended and aggregated to quarterly frequency). We can see that measured productivity follows our shock reasonably well until around 1985. After that, the series often run in opposite directions, so that the correlation over the full sample is only 0.085. This insight is not new; the “vanishing procyclicality” of productivity has been the object of intensive research over the last two decades. After 1985, there is a pretty good correlation between 33 Unemployment Rate, uJob Finding Probability, φw -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 1950 1960 1970 1980 1990 2000 2010 2020 Data Baseline Hall/Milgrom -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 1950 1960 1970 1980 1990 2000 2010 2020 Correlation with data for baseline: 0.923; HM: 0.913. Correlation with data for baseline: 0.892; HM: 0.896. Separation Rate, δWages, w -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 1950 1960 1970 1980 1990 2000 2010 2020 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 1950 1960 1970 1980 1990 2000 2010 2020 Correlation with data for baseline: 0.673; HM— Correlation with data for baseline: 0.390; HM: 0.426. Labor Productivity, yShocks -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 1950 1960 1970 1980 1990 2000 2010 2020 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 1950 1960 1970 1980 1990 2000 2010 2020 Labor Prodtty Data ModelShock CAPE,scaled Figure 7: Key Labor Market Variables and Implied Productivity for Estimated Models. 34 our shock and the financial index. Although the underlying processes driving the incentives for job creation and separation are changing over time, the models still generate similar labor market dynamics. Identifying the structural shocks creating these incentives is still an open task. 6 Conclusions We distinguish two channels generating labor market fluctuations: the vacancy creation channel, which is active in every textbook model of the Mortensen/Pissarides type, and the vacancy depletion channel, which results from the interplay of long-lived vacancies, less than perfectly elastic vacancy creation, and time-varying job separation rates. This mechanism was described, if not named, in Coles and Moghaddasi Kelishomi (2018). We develop a model that includes long-lived vacancies and endogenous job separations and use it to identify the importance of the two channels. The most important information for this identification are the phase relationships between labor market variables. While the vacancy creation channel is, overall, the dominating one, the depletion channel is quantitatively relevant. This is important for the question of how fast the unemployed can be absorbed into employment. Our estimates imply that the speed of absorption is about half of what is implied by the textbook labor market models which is important in the Covid crisis if we assume that many of the job losses are actually caused by job rather than just match destruction. If the vacancy depletion channel is active, wages must fluctuate much more than productivity for the decentralized economy to achieve an efficient allocation of vacancies over the business cycle. Such a strong variation of wages is not observed in the data, not even in the wages for new hires. To bring the model in line with the data, we adopt the alternative-offerbargaining framework of Hall and Milgrom (2008). Although this framework is usually seen as a form of “rigid wages”, wages in our model still fluctuate more than productivity, which would probably not be considered rigid. Therefore, to assess whether wages react efficiently to changes in the economic situation, a unit elasticity with respect to productivity is not the appropriate benchmark. If one accepts the view that vacancies are long lived, future work should focus on modeling job separations. We find that the model has no problem explaining the fluctuations in unemployment, if it is able to explain the fluctuations in job separations. The task is therefore to reconcile a sizeable match surplus, not only with large fluctuations in job finding rates (Ljungqvist and Sargent, 2017), but also with large fluctuations in job separation rates. As in den Haan, Haefke and Ramey (2005) a larger surplus increases the resilience of matches and reduces separations. 35 Finally, LLV create a link between the labor market and physical investment, because LLV are a form of capital, namely capital that is tied to a specific job. This changes the way that financial markets can affect the labor market. 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B.2 Proof of Result 2 Based on equations 27 and 28 the values for capital cost and search cost need to be derived so that the user cost of an unfilled vacancy κsand the surplus term remain constant. Pick κl s such that κhremains constant, i.e. κh=κl s+1−β1−δl vκl v =κs s+(1−β(1−δs v))κs v κl s=κs s−1−β1−δl vκl v+(1−β(1−δs v))κs v. Pick κl ksuch that cremains constant. Recall WV=β(1−δv)κv. c+b= [β(1−δj)−1]β1−δl vκl v−κl k = [β(1−δj)−1]β(1−δs v)κs v−κs k κl k=κs k−[1−β(1−δj)]β1−δl vκl v+[1−β(1−δj)]β(1−δs v)κs v. 44