Long Live the Vacancy

Long Live the Vacancy Christian Haefke Michael Reiter March 14, 2017 Preliminary Version Please do not circulate! Abstract We reassess the role of vacancies in a Diamond-Mortensen-Pissarides style search and matching model. Long-lived vacancies and endogenous job separations together with alternating offer bargaining greatly improve the ability of the model to replicate key stylized labor market facts. The model explains not only standard deviations and autocorrelations of labor market variables, but also their dynamic correlations. The model is consistent with a large surplus both on the worker and the firm side, and generates a wage response to productivity shocks that is in line with empirical evidence on the wage dynamics of new matches. With only one shock, the model captures the dynamics of the US labor market from 1951 to 2014 surprisingly well. Keywords: Beveridge Curve, Business Cycles, Job Destruction, Random Matching, Separations, Unemployment Volatility. JEL Codes: E24, E32, J63, J64 New York University, Abu Dhabi, UAE. E-mail: christian.haefke@nyu.edu Institute for Advanced Studies, Vienna, Austria. E-mail: michael.reiter@ihs.ac.at

1 Introduction The high persistence of vacancies is a pervasive empirical phenomenon. Amaral and Tasci (2016) corroborate high autocorrelations in vacancies for all OECD countries in their data set. For the United States both Fujita and Ramey (2007) and Christiano et al. (2016) document the sluggish response of vacancies to a productivity shock. However, the workhorse labor market matching model achieves simplicity by making strong assumptions about vacancies and job separations which imply rather low persistence of the vacancy stock. Vacancies are assumed to be perfectly elastic with respect to productivity and to perish after one period if not filled. Job separation is assumed constant in most versions of the model. These simplifications imply appealing analytical simplicity, and earlier attempts in the literature to depart from them did not lead to substantial improvements in the ability of the model to explain the data (Fujita and Ramey, 2007; Mortensen and Pissarides, 1994). In this paper, we make the point that the introduction of both long-lived vacancies (LLV) and endogenous job separation unambiguously helps to reconcile the labor market matching model with the data. It does so in two ways. Long lived vacancies allow the impact of increased vacancy formation to accumulate over time, so that the required variability of vacancy formation is much lower than the observed variability of the stock of vacancies. This reduces the need to generate a very volatile job surplus. A time-varying job separation rate provides a further source for unemployment fluctuations, again reducing the need to generate volatile vacancy creation. The two features interact in an interesting way: an active separation margin contributes to the variability of the job finding rate, because higher unemployment leads to a depletion of the vacancy stock and therefore a reduction in the job finding rate, as highlighted in Coles and Moghaddasi Kelishomi (2014). Although variable separations play an important role here, this mechanism is fully consistent with Robert Shimer s (2012) findings on the ins and outs of unemployment. The concept of LLV means that the creation of a vacancy implies a one-time cost, either in addition to or as a substitute for the flow-cost of a vacancy that is present in the standard model. One should think of the setup costs of creating a workplace, including capital expenditures as well as planning costs. Once this cost has been paid, the vacancy is present until it is either destroyed (becomes obsolete) or is filled with a worker. When a match is separated, this might or might not leave a vacancy that can be filled with a new worker, without paying the setup costs again. Vacancies can therefore accumulate or be depleted in different ways. Apart from making vacancies durable, we also relax the standard assumption of perfectly elastic supply of vacancies. While an infinite elasticity of vacancy supply helps the 1

model to explain labor market fluctuations, it is plainly unrealistic. Our model matches labor market fluctuations with an elasticity between one and two. The model with long lived vacancies robustly generates a strong negative correlation of unemployment and vacancies (Beveridge curve) even in the presence of endogenous separations, which is considered a litmus test of modern unemployment theory. The creation cost model of Fujita and Ramey (2007) captures the sluggishness in vacancy dynamics, which they call the propagation mechanism. However, their model does not show any additional amplification compared to the standard search and matching model. In contrast to their results, we show both numerically and analytically that LLV generate substantial amplification as well as propagation. The interplay between LLV and timevarying separations has previously been explored in Coles and Moghaddasi Kelishomi (2014), who take job separations as an exogenous driving force that is negatively correlated with TFP. In our model we explain both match formation and match separation endogenously, and show that a single shock is sufficient to drive almost all labor market fluctuations including the great recession. 2 Data If not otherwise stated we study quarterly US data for 1951 2003. When series are available at higher frequencies, they are averaged to quarterly series. For easier comparability we decided to use precisely the same period studied by Shimer (2005a), results remain qualitatively similar for alternative subperiods. For seasonally unadjusted series we remove the high frequencies of two to four quarters with a bandpass filter. For all series we study log-deviations from a slow moving HP trend with smoothing parameter 10 5. We report the usual business cycle statistics in Table 1 and show all series and their long run trends in Figure 1. We provide a detailed report of all data sources and transformations in Appendix D. 2.1 Unemployment We measure the unemployment rate, u, with the series UNRATE published by the St. Louis FED data archive. The average unemployment rate over the sample period was 5.67. Unemployment fluctuates strongly around its trend with a standard deviation of 0.19. This is more than seven times the variability of real per capita GDP. 2

2.2 Vacancies 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 website 1, which combines the advantage of a long series from the Help Wanted Index with the accurate measurement of JOLTS since 2000. In Table 1 we see that the cyclical component of vacancies, v, fluctuates approximately as much as unemployment and is highly persistent, consistent with previous evidence for the U.S. (Christiano et al., 2016; Davis et al., 2013; Fujita and Ramey, 2007) and a number of other OECD countries (Amaral and Tasci, 2016). Figure 2 illustrates the strong negative correlation of -.91 with unemployment for 1951 2003. 2.3 Job Finding Probabilities To compute the job finding probabilities, φ w, we follow the methodology suggested by Shimer (2005a). The average monthly 2 job-finding probability over the sample period is 44.4%. The cyclical component of the job finding probability is substantially less volatile than the unemployment rate and fluctuates approximately four and a half times as much as GDP. Job finding probabilities are highly negatively correlated with unemployment. 2.4 New Matches Following Shimer s assumption that all hires go through unemployment we compute the number of matches, M, as product of the unemployment rate and job finding probabilities. The cyclical component of new hires is substantially less volatile than unemployment and fluctuates approximately three times as much as GDP. New Matches fluctuate about half as much as unemployment and are strongly positively correlated with unemployment. 3 2.5 Job Separation Probabilities To compute separation probabilities, δ, we follow the methodology suggested by Shimer (2005a). The average monthly separation probability is 3.3% over the sample period. Separations probabilities fluctuate substantially less than unemployment and vacancies, approximately 3 times as much as GPD. Autocorrelation is substantially lower. Separation probabilities spike in recessions (see Figure 3) which leads to a comparatively low autocorrelation of 0.8. In Table 13 and Figure 4 we see that the separation probability is leading all the other variables. 1 https://sites.google.com/site/regisbarnichon/cv/hwi_index.txt?attredirects=0 2 The literature typically reports monthly probabilities, so we do the same. However, all other statistics are based on logarithms of quarterly averages. 3 These observations are consistent with CPS micro-data on worker flows for the period 1984 onwards. 3

Table 1: Descriptive Statistics for Key Labor Market Variables, 1951 2003. u v M φ w δ GDP Mean 5.655.67 5.70 2.422.43 2.44 44.42 44.44 44.46 3.33 3.34 3.35 StDev 0.170.19 0.21 0.18 0.19 0.21 0.077 0.086 0.096 0.108 0.118 0.126 0.068 0.075 0.085 0.024 0.026 0.029 RStDev 6.777.27 7.70 6.87 7.39 7.87 2.96 3.29 3.65 4.16 4.51 4.73 2.59 2.87 3.23 1.00 1.00 1.00 AC 0.870.93 0.98 0.89 0.94 0.98 0.79 0.88 0.97 0.87 0.92 0.98 0.72 0.82 0.91 0.87 0.93 0.98 u 1.93.91.88 0.90 0.92 0.94.97.96.94 0.65 0.71 0.78.93.91.89 v 1.82.77.72 0.89 0.92 0.93.74.68.63 0.82 0.85 0.88 M 1.83.77.72 0.67 0.75 0.81.84.80.75 φ w 1.70.62.55 0.87 0.89 0.92 δ 1.77.70.64 GDP 1 Except for the means all statistics have been computed for detrended (HP, smoothing parameter 10e5) and seasonally adjusted logarithms of the respective time series. u is the UNRATE series of the St. Louis Fred, v has been provided by Regis Barnichon (Barnichon, 2010), φ w and δ have been computed based on the methodology described in Shimer (2012). Matches are computed by multiplying the unemployment rate (u) with the job finding probability (φ w ). GDP is real GDP as downloaded from FRED. 95% Bootstrapped confidence bounds in subscripts to the left and right of the respective statistic. 4

3 Model 3.1 Timeline 1. Aggregate shock and individual shocks are realized 2. Separations take place 3. Wages are negotiated 4. Vacancy destruction takes place Define Value functions 5. vacancy investment is made Measure Labor Market Stocks: e,u,v, 6. Production takes place 7. Matching of unemployed and vacancies 3.2 Firms and Vacancies Our model extends Hall and Milgrom (2008) in two dimensions. First, we allow for long lived vacancies that are in finitely elastic supply. Secondly, all separations are endogenous. There is one aggregate shock to labor productivity that generates business cycles, lny t = ρ y lny t 1 + σ y ε t. (1) Every period there is a unit mass of potential vacancies. Analogously to Fujita and Ramey (2007) and Coles and Moghaddasi Kelishomi (2014) firms that would like to open such a vacancy pay a one-time stochastic vacancy posting cost κ v V. Hence the vacancy posting decision will follow a threshold rule and firms with a draw κ v <= κ vt pay, so that the flow of new vacancies in period t is given by: n t = V ( κ vt ). (2) A vacancy remains open until exogenously destroyed with probability δ v, or filled with filling probability φt f. Let Vt C denote the continuation value of being matched with a worker at the end of period t. The value of a vacancy can then be written as V V t = κ i + φ f t Vt C + (1 φ f t )βe t (1 δv )V V t+1, (3) 5

where E t < > denotes expectations conditional on information available at time t. β denotes the discount factor that is common for firms and workers, and κ i denotes a cost of idle capital that has to be paid whenever a firm is not producing. When matched, a worker-firm pair produces output with a linear technology so that firm level output in period t is simply labor productivity y t. Every period that a firm produces, firms pay a wage ω t to workers and a capital cost κ k, so that the value of a filled job, V J, is given by: V J t = y t ω t κ k +V C t. (4) Upon separation of an employment relationship, the job can either survive or be destroyed. We allow either of these to happen and call the two types of separation match destruction and job destruction, respectively. Both are modeled in a similar fashion. While previous models of endogenous separation (den Haan et al., 2000; Mortensen and Pissarides, 1994) introduced idiosyncratic productivity fluctuations, we keep firms completely homogeneous. Occasional costs need to be paid in order to maintain an employment relationship. A job destruction event occurs with probability λ j. Upon arrival of a job destruction event, firms draw a job maintenance cost κ j from distribution J. For all costs κ j <= κ jt firms will pay the cost and continue the relationship, otherwise the match dissolves and the job/vacancy is destroyed. Hence, a job is destroyed with probability δ j = λ j J ( κ jt ). Similarly, a match destruction event occurs with probability 4 λ m. Upon arrival of a match destruction event, firms draw a match maintaince cost κ m from distribution M. For all costs κ m <= κ mt firms will pay the cost and continue the relationship, otherwise the match dissolves and the vacancy enters the existing stock of vacancies to be refilled. Hence, a match is destroyed with probability δ m = (1 λ j )λ m M ( κ mt ). The overall separation rate is consequently given by δ t = δ mt + δ jt. We can now state the firm-continuation value of an employment relationship: V C t = βe t (1 λ j )(1 λ m )V J + λ j [J ( ) κ jt+1 V J t+1 t+1 κ jt+1 ] κ j dj (κ j ) + (1 λ j )λ m [(1 M ( κ mt+1 ))(1 δ mt+1 )Vt+1 V +M ( κ mt+1 )Vt+1 J 3.3 Workers κmt+1 The worker side is completely standard. Every worker who is employed in period t receives wage ω t, every searcher receives b. Searchers find jobs with probability φ w t (5) ] κ m dm (κ m ). so that we can 4 For simplicity we assume that a firm will only be hit by at most one destruction event. Hence, a match destruction event can only hit if the firm has not been hit by a job destruction event already. 6

write the period t values for being employed (Vt E ) and unemployed (Vt U ) as: 3.4 Transitions V E t = ω t + βe t V E t+1 δ t+1 ( V E t+1 V U t+1), (6) V U t = b + βe t V U t+1 + φ w t+1 (1 δ t+1 ) ( V E t+1 V U t+1). (7) The stock of unemployed, u t, and the stock of vacancies, v t, are matched using a constant returns to scale matching technology, where α denotes the elasticity of matches with respect to unemployment and A the constant matching productivity. The workers job finding probability φt w and the firms job filling probability φt f immediately follow: M (u t,v t ) = Au α t v 1 α t, (8) φ w t = M (u t,v t ) u t, (9) φ f t = M (u t,v t ) v t. (10) We assume that every worker is either employed or unemployed and normalize population to unity. Worker stocks are thus governed by: u t = δ t e t 1 + ( 1 φ w t 1(1 δ t ) ) u t 1, (11) and e t = 1 u t. (12) Vacancies in period t are the surviving 5 vacancies of the previous period less successful matches (those vacancies that were matched and not immediately separated by match destruction). The inflow into the vacancy stock comes from from newly formed vacancies, n, and those employment relationships separated by match destruction: [( ] v t = n t + (1 δ v ) 1 φ f t 1 (1 δ mt) )v t 1 + δ mt e t 1. (13) 3.5 Wages Wages are bargained via an alternating offer protocol. We follow Hall and Milgrom (2008) in assuming that the threatpoint in wage bargaining is not outright separation. Instead, when disagreeing, workers receive b b, while firms have to pay the cost of idle capital κ i. Recent work by Chodorow-Reich and Karabarbounis (2016) has challenged the assumption of acyclical opportunity costs of employment. In section 5.7 we will allow b and b b to 5 It would be straightforward to endogenize vacancy destruction. However, we see from equation (13) that the lion share of vacancy outflow is attributable to new matches, which are at least one order of magnitude larger. 7

be time-varying and cyclical. Results are robust. With probability δ b negotiations break down, the worker returns to the unemployment pool and the vacancy to the vacancy pool. If the worker makes the first offer, he will offer ω t which gives rise to the value Ṽt E employed and having made the wage offer, of being Ṽ E t = ω t + βe t V E t+1 δ t+1 ( V E t+1 V U t+1). (14) In equilibrium the firm will be indifferent between accepting wage offer ω t and producing, and rejecting the wage offer in order to offer ω t+1 in the subsequent period: y t ω t κ k +Vt C = δ b (1 δ v )Vt V + (1 δ b ) ( κ i + βe t Vt+1) J. (15) If the firm gets to offer first, they offer ω t, giving rise to Vt E and the worker has to be indifferent between accepting ω t and getting to make the next offer in the subsequent period: Vt E = δ b Vt U + (1 δ b ) ( b b + βe t Ṽt+1) E. (16) We assume that the firm always makes the first offer 6. 3.6 Equilibrium The aggregate state at the beginning of period t is summarized by the realization of aggregate labor productivity y t, and the size of the unemployment, u t 1, and vacancy pool v t 1 : Ω t = {y t,u t 1,v t 1 }. Definition 1 (Equilibrium). A symmetric equilibrium for the model economy consists of a sequence of wages ω t, ω t ; a sequence of thresholds κ vt, κ mt, κ jt ; and a sequence of labor market stocks and flows e t,u t,v t,n t ; such that for any time period t the following hold: 1. Vacancy Creation: κ vt = Vt V and Equation 2; 2. Endogenous Separations: κ jt = Vt J and κ mt = Vt J Vt V ; 3. Wage Bargaining: The worker (Equation 15) and firm indifference (Equation 16) conditions hold; 4. Labor Market Transitions: Equations 11 13. With value functions V J t,v C t,v V t,vt U,V E t,ṽ E t ; defined in Equations 3 7 and Equation 14. 6 Hall and Milgrom assume that the actual negotiated wage is an average of the case when the firm makes the first offer and the alternative case when the worker makes the first offer. In the calibration this impacts the calibrated value of b b but has no substantial effect on the results. 8

3.7 The Role of Long-lived Vacancies The aim of this section is a heuristic analysis of the amplification mechanism coming from LLV, and its interaction with endogenous separation. Part of this analysis is reduced form, not structural, but the numerical results of the full model will bear out the analytical insights obtained here. We will derive an approximate AR(1) representation for the deviation of vacancies from their steady state, driven by exogenous fluctuations in labor productivity y t. Write this process as ˆv t = γ 1 ˆv t 1 + γ 0 η V J y ŷ t. (17) where η V J y denotes the elasticity of the value of a filled job w.r.t. labor productivity. The notation ˆx stands for the percentage deviation of any variable x from its steady state value. If ŷ itself is an AR(1) process, with persistence ρ y >> γ 1, then it is a good approximation to consider a flow equilibrium in ˆv t, given ŷ t : ˆv t = γ 0 1 γ 1 η V J y ŷ t. (18) Amplification is then given by γ 0 1 γ 1. In the other extreme, if ŷ is i.i.d., we have std( ˆv) = with the smaller amplification factor γ 0 η V J 1 γ 2 y std(ŷ). (19) 1 γ 0 1 γ 2 1. Similar to Shimer (2005b), we find that the flow-equilibrium approximation (18) is a better approximation than (19). In each case, amplification depends positively on both γ 0 and γ 1. Putting it the other way round, the value of vacancies, which determines the creation of new vacancies, does not have to fluctuate as much as the stock of vacancies. This makes it much easier to generate unemployment fluctuations. We see that this effect is the stronger the smaller is δ v, demonstrating the amplification effect of LLV. Why do long-lived vacancies help explaining vacancy fluctuations? Let us start with the dynamic equations for vacancies and unemployment, simplified by ignoring terms that are quadratic in transition rates such as φ f t 1 δ m and φt 1 w δ, which are very small in our highfrequency simulation: ( ) v t = (1 δ v ) 1 φ f t 1 v t 1 + n t + δ mt (1 u t 1 ), u t = δ t (1 u t 1 ) + ( 1 φ w t 1) ut 1. We approximate separations by the reduced form relationships δ t = δy ηδ y t and δ mt = δ m y m t. The parameters η δ y and m are positive, which reflects the fact that job and match separations are a decreasing function of job and match value, respectively, which in turn depend 9

positively on labor productivity. Using the matching function (8), we get the following linear approximation, expressed in percentage deviations around the steady state: ( ˆv t = (1 δ v ) (1 (1 α) φ f ) ˆv t 1 αφ ) f û t 1 + n v ˆn φ t f φ w δ m û t 1, mφ f ŷ t (20) û t = (1 α φ w δ)û t 1 φ w η δ yŷ t (1 α) φ w ˆv t 1 (21) where we have used ȳ = 1 and the first order approximation δ t (1 u t 1 ) δ(1 u) = δy ηδ y t (1 u t 1 ) δ(1 u) δû t 1 u δη δ yŷ t (1 u) = ( δût 1 + φ w η δ yŷ t )u (22) If we make the assumption that the value of vacancies depends only on the current level of productivity y and the rate φ f at which vacancies can be filled, we can write ˆn t = ξv ˆ [ ] V ˆf t ξ φt + η VV y ŷ t = ξ η VV φ f [ η VV φ f α( ˆv t û t ) + η VV y ŷ t ]. (23) where η VV and η φ f VV y denote the elasticity of the vacancy value w.r.t. the filling probability and labor productivity, respectively. The Beveridge Curve To reduce the two-dimensional dynamical system (20) and (21) to one dynamic equation, we use the "Beveridge curve", the strong negative correlation between unemployment and vacancies. This correlation is 0.91 in the data, and even stronger in most variants of our model. The Beveridge curve arises if unemployment adjusts quickly to a new flow equilibrium in reaction to relatively persistent changes in the level of vacancies; this requires, among other things, a high job finding probability φ w. Under this assumption, we can set u t = u t 1 in Equ. (21) to solve it for the flow equilibrium between u and Vac. If we further assume a stable relationship between productivity and vacancies, which we write as ˆv t = η v y, we get û t ˆv t = φ [ w (1 α) + η δ y/η v ] y α φ w + δ Notice that variable separations (positive η δ y) tend to increase the fluctuations of unemployment relative to those of vacancies. In the data, the fluctuations in the two variables are of about equal size, and the correlation is highly negative. Therefore (24) û t ˆv t 1 (25) is a good approximation. For the model to match this stylized fact, we need the right choice of α. Combining (24) and (25) we get α = φ ( w 1 + η δ y/η v y) δ 2φ w (26) 10

For constant separation rate (η δ δ y = 0), because of φ = u w e 0.06, we get α 0.47. If we approximate η δ y/η v y by the relative volatiliby of δ and y, we get η δ y/η v y = 2.74/7.27 = 0.377, so that α 0.66. Although we arrived at these values through rough approximations, they are close to the calibrated values we obtain in our model simulations, cf. Section 5.3. From (24), labor market tightness is given by [ φ [ w (1 α) + η δ y/η v ]] y ˆv t û t = 1 + αφ w + δ and the total number of matches is φ [ ˆM w (1 α) + η δ y/η v ] y t = (1 α) ˆv t α αφ w + δ ˆv t = φ ( w 1 + η δ y/η v y) + δ αφ w + δ ˆv t, (27) ˆv t = (1 α) δ αφ w η δ y/η v y αφ w + δ ˆv t. (28) Using the value of α calibrated in (26), Equ. (28) simplifies to ( ) ˆM t = φ δ w ηδ y/η v y ˆv t. (29) With constant separation rate (η δ y = 0), we see from (29) that matches are procyclical (positively correlated with vacancies). This turns around with time-varying separations: as explained above, in the data we have δ φ w 0.06. and η δ y/η v y 0.377. We will see in Section 5.4 that the cyclicality of matches plays a crucial role in the dynamics of vacancies. The analysis above shows that time-variable separations and the observed Beveridge curve mechanically require that the total number of matches is counter-cyclical. We compute total matches in the data as ˆM t = φ w t u t 1, which is consistent with our model. Then total matches are strongly counter-cyclical. There are other estimates of total matches, for example in JOLTS, which seem to be procyclical. These measures are more broadly defined, including both job-to-job transitions, and transitions from out of the labor force. In the CPS, the total number of flows from unemployment to employment are counter-cyclical, while flows from out of the labor force are procylical. It remains a task for future research to explain why those different measures of hiring activity behave differently over the cycle. Job value and vacancy value To analyze the dynamics of vacancies, we have to explain new vacancy formation, which requires expressions for the value of jobs and vacancies. We first define a simplified expression for the value of a filled job, abstracting from match and job maintenance costs, and from the effect of endogenous separation on firm values. This effect is small, because separations only happen when the surplus is zero. Combining (4) and (5) we obtain V J t = y t ω(y t ) κ k + βe t [ δ m V V t+1 + (1 δ m δ j )V J t+1]. (30) 11

and [ ] Vt V = κ i + βe t φt f Vt+1 J + β(1 φt f )(1 δ v )Vt+1 V. (31) The equilibrium value of vacancies is Putting this into (30) we get V V = βφ f V J κ i 1 β(1 φ f t )(1 δ v ). (32) Vt J ( δ m + δ j + 1 β ) = y t ω(y t ) κ k + βδ m φ f V J t κ i 1 β(1 φ f t )(1 δ v ). (33) The last term in (33) measures the continuation value in case that a match is separated while the job is maintained as a vacancy. If the value of a vacancy is close to the value of a filled job, this term is similar to reducing the separation rate from δ m + δ j to δ j. For simplicity, we abstract from this term and write V J t = y t ω(y t ) κ k δ m + δ j + 1 β. (34) We interpret the reduced-form relationship ω(y t ) as capturing both the direct effect of productivity on wages and the indirect effect through higher job finding rate. Then we get η V J y t = 1 ω (y t ) y t ω(y t ) κ k. (35) This elasticity is a key characteristic of the model. Equ. (35) illustrates the two main determinants: the elasticity is high if wages are sticky (low value of ω (y t )), or if the surplus is small, as measured by the denominator in (35). The elasticiticies η VV and η V J φ f y look different in the case of flow vacancies rather than LLV. In the case of flow vacancies and free entry, the value of a vacancy has to be interpreted as after paying the flow cost; before paying the cost, the value of the vacancy is zero. This value is Vt V = βe t φt f Vt+1 J Since the effect of labor market conditions on the job value is assumed to be captured by the wage function, there is formally no effect of φ f on V J, and we have With LLV, δ v is small compared to φ f. In this case, the value of a vacancy is approximately FlowVac: η VV φ = 1, η f VV y = η V J y (36) V V φ f V J t κ i φ f + δ v + 1 β V J t κ i φ f (37) because φ f is the dominant term in the denominator of (37), and we have η VV φ f = κ i φ f 2 φ f = V V κ i φ f V V. For given φ f, the derivative of V V w.r.t. productivity is the same as that of V J ; 12

therefore the elasticities only differ by the relative size of V V and V J. For the purposes of our approximation, we can assume LLV: η VV φ κ i f φ f V, V ηvv y η V J y (38) Case 1: ξ = and flow vacancies In the limit ξ, since ˆn is finite, (23) implies that With the approximation (25), (39) implies ˆv t û t = ηvv y ŷ t η VV α. (39) φ f Using (38), this further simplifies to ˆv t = ηvv y 2α η VV φ f ŷ t. (40) ˆv t = 1 2α ηv J y ŷ t. (41) This is the well known case where the fluctuations of vacancies are basically proportional to the fluctuations of the surplus, which again is governed by wage responsiveness and the average surplus size, as shown in (35). In this case, past vacancies play no role, since new vacancies adjust flexibly such that the stock of vacancies has its optimal value. If vacancy creation is perfectly flexible, the stock of vacancies adjusts until the increase in the value of a vacancy, caused by higher productivity, is compensated by the lower filling probability. What is the effect of endogenous job separation in this case? As we have discussed above, variable job separations lead to a higher estimate of α, giving more weight to unemployment in match creation. According to (41), this dampens the response of vacancies to productivity in equilibrium, because any increase in vacancies has then a stronger negative effect on the vacancy filling rate, which follows from φ f = (v/u) α. This also reduces the fluctuations of unemployment, since (41) was derived under the assumption of an unchanged Beveridge relationship. This means that under the standard assumption of perfectly elastic flow vacancies, endogenous separations make it harder to explain unemployment fluctuations, so that a higher degree of wage rigidity is needed. Case 2: ξ = and LLV Using the LLV approximation (38), Equ. (39) becomes ˆv t û t = 1 φ f V V 2α κ i η VV y ŷ t. (42) 13

Notice that φ f V V κ i > 1, otherwise a vacancy would have a negative value. More interestingly, LLV insulate vacancy creation to some extent from the size of the surplus. A high surplus tends to reduce the elasticity of the job value and the vacancy value w.r.t. an increase in productivity, but if vacancies are long lived, it also reduces the elasticity of the vacancy value w.r.t. the filling rate. A lower filling rate increases the cost of the waiting time, but if the value of a job is big, this is a minor increase compared to the value of a vacancy. An increase in the job value of equal magnitude is then compatible with a larger increase in vacancies. Case 3: finitely elastic vacancy creation and flow vacancies With flow vacancies, n t = v t and δ v = 1. Plugging (23) into (20) we then get ˆv t = n [ v ξ φ η VV φ α( ˆv f t û t ) + η VV y ŷ t ]. f φ w δ m û t 1, mφ f ŷ t (43) Using (38) and (25), this becomes [ ] n ˆv t = v ξηvv y mφ f ŷ t 2ξ n v α κ i φ f V ˆv V t φ f φ w δ m ˆv t 1 (44) In the limit case ξ, (44) reduces to (40). With finite ξ, using the approximation n v φ f, (44) becomes With high frequency simulation, [ ˆv t 1 + 2ξα κ ] [ ] i V V = φ f ξη VV y mφ φ f ŷ t f φ w δ m ˆv t 1 (45) κ i V V is very small compared to 1, so that [ ] ˆv t = φ f ξη VV y mφ φ f ŷ t f φ w δ m ˆv t 1 (46) is a good approximation. Then (46) shows that a low elasticity ξ dampens the vacancy reaction to a productivity shock, and thereby reduces amplification in the model. What is the effect of variable separations? In (46), a positive m reduces the reaction of vacancies to productivity, because a lower match destruction rate reduces the number of vacancy repostings. It may be surprising that this is the only effect showing up in (46), but remember that this approximation was derived under the assumption that parameters are recalibrated so as to yield a Beveridge curve described by (25). Our numerical simulations below will focus on those calibrations. 14

Case 4: finitely elastic vacancy creation and LLV We finally turn to our benchmark case, combining finitely elastic vacancy creation and LLV. Short time-horizon, small δ v, do not distinguish t and t-1: [ ˆv t = (1 δ v φ f ) ˆv t 1 + α φ f n v ξηvv φ ]( ˆv f t 1 û t 1 ) φ f φ w [ n δ m û t 1 + Using (24) and (25) as well as (38), we can write (47) in the form (17) with [ n γ 0 = + v ξηvv y mφ f ( γ 1 = 1 δ v φ f + 2α v ξηvv y ] mφ f ŷ t (47) ] ŷ t (48) φ f n v ) κ i φ + f V V φ f φ w δ m ˆv t 1 (49) It is clear from (49) that LLV (small δ v ) riases γ 1 and therefore increases amplification and propagation. To get an idea of the approximate size of γ 1, consider the benchmark α = 0.5 and m = 0. With the approximation n φ f v, (49) simplifies to γ 1 = 1 δ v κ i. Counting V V in monthly frequency, δ v is about 2 percent in our benchmark calibration. The ratio of the cost of idle capital κ i to the value of a vacancy depends ultimately on the size of the job surplus. In our calibration, it is almost 40 percent of monthly vacancy value, so that it is the dominating term in γ 1, which gives approximately γ 1 = 0.6 at monthly frequency. Of course, the first order autocorrelation is much higher, due to the positive autocorrelation of the driving force y. This also makes clear that variations in δ v have little effect as long as it stays small compared to ξ κ i V V. What is the effect of variable job separation on amplification? γ 1 is affected by variable separation through recalibration, specifically through an increase in the calibrated α. This has two effects: the first term in paranthesis in (49) reflects the number of matches. We have discussed above that variable separations make match formation counter-cyclical. In a boom, the total number of matches goes down, so that fewer vacancies are depleted, which contributes to building up the vacancy stock. This creates a positive effect on γ 1. The second effect, which is negative, reflects the fact that the negative effect of tightness on the vacancy filling rate is stronger with higher α. This makes new vacancy postings react more sluggishly to changes in the vacancy value. In our calibration, φ f is about four times as big as κ i V V, so that the positive effect is dominating under the benchmark ξ = 1. However, this would not be true for highly elastic vacancy formation. A further effect of variable separation appears in Equ. (48). A positive m means that higher productivity leads to fewer match destructions, thereby reducing the number of "repostings", the flowing back into the vacancy pool of matches that are destroyed without the job being destroyed. Whether this is an important effect depends on two things: the value 15

of m, which we will estimate rather small, as well as the elasticity of the vacancy value w.r.t. labor productivity. In Section 5.4 we will study the size of these different effects. Summary We can summarize the above discussion as follows: 1. LLV increase magnification and propagation of the model, both under infinite and finite elasticity of vacancy creation. 2. A lower elasticity of new vacancy creation reduces amplification. 3. Time-variable separation rates reduces amplification under flow vacancies.. 4. Time-variable separation rates increase propagation under LLV. 5. Time-variable separation rates increase the calibrated value of α, making the total number of matches counter-cyclical. 4 Calibration We have chosen the time period as the 60th part of a quarter, which we refer to as a workday. With this short time period, the bargaining advantage of the firm, which is allowed to make the first offer in the alternating offer game, is very small. We follow in this respect Christiano et al. (2016), who subdivide their quarterly period into 60 subperiods when computing the bargaining outcome. A few parameters are standard or taken from the literature. We set the discount factor β to 0.99 quarterly. We follow Fujita and Ramey (2007) and Coles and Moghaddasi Kelishomi (2014) in calibrating a vacancy filling rate φ f of one third per week. For the cost of an idle vacancy, we follow HM and set it to 23 percent of average production. Average productivity per week is normalized to one. We set the autocorrelation coefficient of labor productivity to ρ y = 0.92 1/60, which conforms to a quarterly value of 0.92. With this, we do a good job in matching the autocorrelation coefficients of almost all variables in the model, cf. Section 5. The rest of the parameters is chosen so as to match seven first moments (steady state values), and four second moments (relative variances). We will chose parameters so as to match the following first moments of the labor markets in our model. We match three longrun averages in the labor market: an average unemployment rate of 5.67 percent, 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 16

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. The variability of total separations and job destruction was approximately equal with a cyclical variability of 5.3. We pay special attention to the size of the match surplus for both sides of the bargain. We match a steady state wage of 64 percent of production, conforming to a 36 percent gross capital share. Out of the capital share, all the costs of maintaining the match must be covered. These include the average job maintenance costs κ j, the average match maintenance costs κ m, and the capital costs κ k and κ i. Following Shimer (2005a) and Christiano et al. (2016), we set the unemployment benefit of the worker to 40 percent of the steady state wage. This is at the upper end of the worker surplus that the literature has considered. For the firm surplus, we us as a benchmark and starting point the value of 2.255 percent, taken from Hagedorn and Manovskii (2008). Since they have argued for a low match surplus, we consider this to be towards the lower end of plausible values. We will show that our model can work well with a substantially larger firm surplus. The parameters to match those targets are 1. the capital cost κ k ; it mainly drives the firm surplus; 2. worker utility while unemployed b directly gives the replacement rate; 3. worker utility during disagreement, b b, is the main driver for the steady state wage; 4. mean µ j and dispersion σ j of the job maintenance cost drive mean and variance of job destruction; 5. given job destruction, mean µ m and dispersion σ m of the match maintenance cost drive mean and variance of total separations; 6. the probability of break-up during disagreement, δ b, determines how strongly the unemployment rate influences wages, and is therefore a key determinant of the variance of unemployment; 7. the elasticity of matches with respect to unemployment, α, determines the variability of vacancies; 8. the average cost of posting a vacancy µ v and the match productivity A jointly determines the total number of vacancies and the number of matches; this is used to match the average unemployment and vacancy filling rate. 17

This leaves two important parameters open, on which there is no direct evidence, and which seem hard to calibrate. The first one is the dispersion of the vacancy distribution cost, σ v. This parameter determines the elasticity of new vacancy creation with respect to the value of a vacancy. Fujita and Ramey (2007) use an elasticity of 1; Coles and Moghaddasi Kelishomi (2014) consider values between 0.5 and 1. Lacking direct evidence, we find 1 a more natural benchmark. We will see in Section 5 that this elasticity has a subtle effect on the labor market variables in the model, mainly driving the lead-lag structure of vacancies, job separations and new matches. We will find that an elasticity higher than unity comes closest to matching these aspects of the data. The second open parameter is the rate of vacancy destruction. Fujita and Ramey (2007) set this parameter equal to the job destruction rate. We follow them and take this value as our benchmark, but we think it is quite possible that an unfilled vacancy disappears much faster than a filled job. We therefore consider higher values of vacancy destruction, up to a rate of 10 percent weekly. We find that vacancy elasticity and the rate of vacancy destruction interact in a way that make it difficult to identify both of them. This leaves two parameters undetermined, which can basically be considered as normalizations, namely the arrival rate of job and match destruction. The arrival rate of job and match maintenance shocks determines the average cost of maintaining the match, but we compensate variations in those costs by a change in capital costs κ k, so that the average firm surplus in steady state is unchanged. We set the arrival rates such that on average, half of those shocks lead to a separation. Reasonable variations in these arrival rates have a minimal influence on the fluctuations in the model. The results of the calibration are contained in Table 2. Some of those values deserve some further discussion. The elasticity of matches with respect to the number of unemployed lies between the most common value in the literature, which is 0.5, and the value estimated in Shimer (2005a), which is 0.72. The probability of break-up under disagreement is almost exactly four times the probability of job separation, which perfectly conforms to the original calibration of Hall and Milgrom (2008), although many other aspects of the calibration are different in our model. The worker utility under disagreement is substantially higher than the value in unemployment; this gives a very good bargaining position to the worker, and generates the high worker surplus that we target. 5 Results We organize the discussion of our results in the following way. In Section 5.2, we present the results of the benchmark calibration, and compare them to the results of the model in Hall and Milgrom (2008), which is nested in our general model specification. To find out which 18

Table 2: Calibrated benchmark parameter values Parameter Symbol Value capital cost κ k 0.005 utility unemployed b 0.004 utility disagreement b b 0.007 mean job maintenance µ j 0.267 dispersion job maintenance σ j 0.141 mean match maintenance µ m 0.062 dispersion match maintenance σ m 0.172 probability break-up δ b 0.006 elasticity matches w.r.t. U α 0.647 average vacancy cost µ v 94.886 matching productivity A 0.038 arrival prob. match cost λ m 0.001 arrival prob. job cost λ j 0.002 features are driving the results, we present in Section 5.3 a sequence of models. We start from Hall/Milgrom, and add one model feature at a time until we arrive at our benchmark model. We provide further checks in this section, which reveal a remarkable robustness of this model with respect to changes in the match surplus. To explain this robustness, we inspect the main mechanism in the model more closely in Section 5.4. Sections 5.5 and 5.6 provide further evidence on the empirical performance of the model. Finally, we show in Section 5.8 that our model, which is very simple in the sense of being driven by a single shock, can account remarkably well for the US labor market history from 1951 to 2014. 5.1 Measuring success of the model We will present many versions of our model, and all of them succeed in matching the standard labor market statistics such as volatility of unemployment, vacancies, and the Beveridge curve. The questions is then which criteria one should apply to distinguish between the different variants or calibrations of the model. It is well known (Costain and Reiter, 2008; Hagedorn and Manovskii, 2008) and was derived above in Equ. (35), that two simple model features can be used to generate large fluctuations of vacancies and unemployment in a labor market matching model. The first feature is a small match surplus, which makes that fluctuations in productivity have a "leveraged" effect on the surplus, and therefore vacancy creation. This aspect was again stressed and analyzed in more general terms in Ljungqvist and Sargent (2016). The second feature is 19

"wage stickiness" in the sense of a low responsiveness of wages to productivity. Notice that this feature does not require stickiness in a formal sense: Hall and Milgrom (2008) provide a microfoundation, based on an alternating offer bargaining scheme, that can deliver a low responsiveness of wages despite continuous renegotiation. Neither of the two features can be used arbitrarily. A too small match surplus, for the average worker, is rather implausible, since it makes fluctuations in unemployment irrelevant from a welfare point of view, and it contradicts the cross-country evidence provided in Costain and Reiter (2008). Too high wage stickiness is not compatible with the empirical evidence for the wages of new hires (Haefke et al., 2013), which is the wage series relevant for vacancy creation. An important criterion for the success of the model is therefore whether it can replicate labor market fluctuations with a realistic responsiveness of wages, and with a substantial surplus from a match, both on the worker and on the firm side. Our measure of wage responsiveness to technology shocks is the ratio of discounted wage changes to discounted productivity changes. The discount factor is the sum of the steady state interest rate and the steady state total match separation rate. The point estimate for this statistic in Haefke et al. (2013) is 0.79. For the match surplus, we look separately at the worker and the firm surplus. On the worker side, we report the steady state replacement rate. At the firm side, we define the surplus as the difference between steady state production and the sum of wages, capital costs, and all other costs to maintain the match. Shimer (2012) made the somewhat provocative point that unemployment fluctuations are primarily driven by fluctuations in the job finding rate, not the job separation rate. This is based on the following approximate decomposition. Starting from the definition of a labor market flow equilibrium we can define δ t u t δ t + φt w, (50) ut Find δ δ + φt w (51) as the hypothetical unemployment rate, which would prevail under the actual job finding rate and the steady state separation rate. The coefficient of a regression of u t on ut Find can then be interpreted as the contribution of the job finding rate to the unemployment fluctuations. Shimer finds that this contribution was 0.77 for the time period 1948 2010, and 0.90 for the period 1987 2010. For a model of endogenous separations to be plausible, it must be possible to generate Shimer s empirical finding from the model output. We compute this statistic for simulations of our model, and report the result under the name ShimerCF. 20

5.2 Benchmark model Table 3 shows the usual set of second moments for the key labor market statistics, for the benchmark calibration of the model (data moments in parenthesis). The model is able to Table 3: Benchmark results Urate Vacancies Finding rate Sep. rate real GDP Rel. stdev 7.27 (7.27) 7.39 (7.39) 5.18 (4.51) 2.87 (2.87) 1.00 (1.00) Autocor. 0.95 (0.94) 0.95 (0.94) 0.95 (0.93) 0.87 (0.82) 0.93 (0.94) Cor. GDP -0.97 (-0.91) 0.97 (0.85) 0.97 (0.89) -0.97 (-0.70) 1.00 (1.00) Cor. U 1.00 (1.00) -1.00 (-0.91) -1.00 (-0.96) 0.88 (0.71) -0.97 (-0.91) WResponse 0.78 ShimerCF 70.9 match the four second-moment targets, namely the standard deviations of unemployment, vacancies, total job separation and job destruction (not shown in the table). It somewhat exaggerates the variance of the job finding rate. The model is very successful in generating the high autocorrelation of the labor market variables. Although the exogenous driving force, TFP, has a theoretical autocorrelation coefficient of 0.92, and 0.89 after HP detrending, unemployment, vacancies and the job finding rate have an autocorrelation coefficient of 0.95 after detrending. In this sense, the model has a substantial propagation mechanism. Only job separations are less autocorrelated; they are characterized by sharp spikes (cf. Section 5.8). The model replicates this fact. Furthermore, the model does a very good job by the criteria that we have adopted in Section 5.1). The dynamic wage response equals 0.78, almost identical to the point estimate of 0.79 in Haefke et al. (2013). And the measured contribution of the job finding rate ot unemployment fluctuations, which Shimer finds to be 0.77 percent in the data, is estimated as 71 percent from the model output. Table 4 provides the same information for the model of Hall and Milgrom (2008). Following this paper, we use an unemployment replacement rate of 0.71. We estimate δ b and α so that the the model replicates the relative standard deviations of unemployment and vacancies. We choose the firm profit as 1.055 percent, so that this calibration yields δ b = 4δ, as in the paper. (cf. Section 5.3 for a further discussion of the calibration). We see that this model is quite successful in matching labor market fluctuations. There is one noticeable difference to our benchmark model: the Hall/Milgrom model has a weaker propagation mechanism, in the sense that all variables have lower autocorrelation than in the LLVE benchmark, using the same productivity process in both models. The difference is greatest for vacancies, where the correlation in the Hall/Milgrom models falls clearly short of the correlation in the 21

Table 4: Hall/Milgrom model Urate Vacancies Finding rate Sep. rate real GDP Rel. stdev 7.27 (7.27) 7.39 (7.39) 7.92 (4.51) 0.00 (2.87) 1.00 (1.00) Autocor. 0.92 (0.94) 0.80 (0.94) 0.89 (0.93) - (0.82) 0.91 (0.94) Cor. GDP -0.99 (-0.91) 0.94 (0.85) 1.00 (0.89) - (-0.70) 1.00 (1.00) Cor. U 1.00 (1.00) -0.89 (-0.91) -0.97 (-0.96) - (0.71) -0.99 (-0.91) WResponse 0.79 ShimerCF - data. The results of both models are surprisingly similar, after recalibration of some key parameters. We will discuss the recalibrations in the next subsection. We will also see that the mechanisms differ substantially. 5.3 A sequence of models Both our benchmark and the Hall/Milgrom model are successful in matching many aspects of the aggregate labor market. We next want to ask which features of these models are responsible for this success, and whether this success is achieved with a plausible set of parameter values. Table 9 reports calibration outcomes for a wide variety of models. The parameters to the left of the vertical line describe the variant of the model that we are looking at. To the right of the line we report parameters that were estimated so as to match labor market fluctuations. For any model with endogenous separations, we estimate the model so as to replicate the relative standard deviations of unemployment, vacancies, job and match destruction (cf. Section 4). In any model with constant separations, we estimate δ b and α so that the the model replicates the relative standard deviations of unemployment and vacancies. The first remarkable result in this table is that, in all the models with constant separation rate, the estimated value of α is around 0.45; In all the models with endogenous separation rate, this value is considerably higher, around 0.64. We have explained this difference in Section 3.7. Notice that our model-based estimate is close to the regression estimate of 0.72 in Shimer (2005a). In all models we estimate workers outside option in the bargaining process b b at around 60 to 70 percent of the real wage, higher than the unemployment benefit in the benchmark calibration. A large b b compensates for the fact that we always give the firm the right to make the first offer. The two last columns of the table give the key information about the calibration: the bargaining breakup rate δ b, and the discounted wage responsiveness, DWR. A small δ b isolates 22