Long Live the Vacancy

Long Live the Vacancy Christian Haefke Michael Reiter February 15, 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 C. 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 12 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 ))Vt+1 V +M ( κ mt+1 )Vt+1 J 3.3 Workers κmt+1 (5) ] κ m dm (κ m ). 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 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 m 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 purpose of this section is to illustrate in a simple, approximate but analytical way 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 in this mechanical analysis. We provide an approximate AR(1) representation for the log-deviation of vacancies from their steady state, driven by exogenous fluctuations in labor productivity y t. Write this process as v t = γ 1 v t 1 + γ 0 η V J y y t. (17) If y itself is an AR(1) process, with persistence ρ y >> γ 1, then it is a good approximation to consider a flow equilibrium in v t, given y t : v t = γ 0 1 γ 1 η V J y y t. (18) Amplification is then given by γ 0 1 γ 1. In the other extreme, if y is i.i.d., we have The amplification factor is then std(v) = γ 0 η V J 1 γ 2 y std(y). (19) 1 γ 0 1 γ 2 1, which is much smaller. Similar to Shimer (2005b), we find that the flow-equilibrium approximation (18) is a better approximation than (19). Why do long-lived vacancies help explaining vacancy fluctuations? Let us start with the dynamic equations for vacancies and unemployment, somewhat simplified by ignoring the terms (1 δ m ) and (1 δ): 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. Using the matching function (8) and, purely mechanical, δ mt = δ m + mu t and δ t = δ+ u t with positive parameters and m, we get the log-linear approximation 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 ( m δ m )û t 1, (20) û t = ( δ)û t 1 (1 α) φ w ˆv t 1 + (1 α φ w )û t 1. (21) where we have used v u = φ w as well as the definitions + 2 ū and φ f m m + 2 mū. 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 ]. (22) 9

Putting (22) into (20), we get The Beveridge Curve ˆv t = (1 δ v ) ( (1 (1 α) φ f ) ˆv t 1 αφ ) f û t 1 + n [ ] v ξ η VV φ α( ˆv f t û t ) + η VV y ŷ t φ + f φ w ( m δ m )û t 1. (23) 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 can be derived assuming a flow equilibrium in Equ. (21), meaning that u t = u t 1. This yields û t (1 α) = ˆv t α + δ. (24) φ w Labor market tightness is then given by (1 α) ˆv t û t = 1 + φ α + δ ˆv t = w + δ φ αφ w + δ ˆv t, (25) w and the log-deviation of matches are (1 α) δ ˆM t = (1 α) ˆv t α α + δ ˆv t = (1 α) φ αφ w + δ ˆv t. (26) w One sees from (26) that matches are procyclical (positively correlated with vacancies) if δ >. The cyclicality of matches will play a crucial role in the dynamics of vacancies. The coefficient can be estimated as corr(δ,u) std(logδ) δ It is bigger than δ if corr(δ,u) std(logδ) 1 std(logu) std(logu) u. u > 1. In the data, this number is 0.74 2.74 7.27 0.0546 5.1. So it appears that matches should be clearly counter-cyclical, and in fact they are. Counter-cyclical matches are then a necessary consequence of a strong Beveridge curve and strongly countercyclical separations. If = 0, then (26) implies that total matches are procyclical, but fluctuate much less than vacancies, because δ/ φ w = ē/ū 18. If is substantially bigger than δ, as we find in the data, then matches turn counter-cyclical, and fluctuate more strongly. Using the Beveridge curve, we can eliminate the unemployment rate from (23), which becomes [ ˆv t = (1 δ v ) + φ f (1 α) αφ w + δ + n v ξ [ η VV φ f α ( (1 δv )( δ) ( m δ m ) )] ˆv t 1 φ w + δ αφ w + δ ˆv t + η VV y ŷ t ]. (27) Next we need to derive an expression for the value of of vacancies. 10

Job value and vacancy value Next we 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]. (28) and [ ] Vt V = κ i + βe t φt f Vt+1 J + β(1 φt f )(1 δ v )Vt+1 V. (29) The equilibrium value of vacancies is Putting this into (28) we get V V = βφ f V J κ i 1 β(1 φ f t )(1 δ v ). (30) 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 ). (31) The last term in (31) 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. (32) 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. (33) This elasticity is a key characteristic of the model. Equ. (33) 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 (33). Case 1: ξ = and flow vacancies In the limit ξ, since ˆn is finite, (22) implies that ˆv t û t = ηvv y ŷ t η VV α. (34) φ f 11

Assuming û t = ˆv t we have ˆv t = ηvv y 2α η VV φ f ŷ t. (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. 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 η VV φ f = 1, and η VV y = η V J y. Thus we can write ˆv t = 1 2α ηv J y ŷ t. (36) 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 (33). Case 2: ξ = and LLV We now consider the case where δ v is small compared to φ f. In this case, the value of a vacancy is approximately 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). Then and (34) gives Notice that φ f V V κ i η VV φ f = κ i φ f 2 ˆv t û t = 1 φ f V V 2α φ f V V = κ i φ f V, (38) V κ i η VV y ŷ t. (39) > 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 increase 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. 12

Case 3: finitely elastic vacancy creation and flow vacancies With flow vacancies n t = v t. We use again η VV φ f = 1 and η VV y = η V J y, and get from (27) that [ φ ˆv t = ξ α w + δ ] αφ w + δ ˆv t + η V J y ŷ t φ f (1 α)( m δ m ) ˆv αφ w t 1. (40) + ( δ ) In the special case where δ = and m = δ m, (40) reduces to ˆv t = ξ (1 + ξ) ηv J y ŷ t. (41) What this shows is that, unsurprisingly, a finite elasticity ξ dampens the vacancy reaction to a productivity shock. Letting ξ go to infinity, we are back in Case 1. Allowing stronger variations in separations with > δ, the denominator in the first fraction in (41) shrinks faster than the numerator, which increases the absolute value of this term, leading to even more dampening. Similarly in the second fraction in (41), where higher m and higher increase the absolute value of the term, and further dampen the effect. However, one should keep in mind that this was derived under the assumption of a strong vacancy curve, which is not guaranteed in a combination of endogenous separation and flow vacancies. However, models that do not generate a Beveridge curve are probably not interesting anyway. Case 4: finitely elastic vacancy creation and LLV We finally turn to the most interesting case, combining finitely elastic vacancy creation and LLV. With small δ v, we use again (38), and furthermore use n φ f v. Then we can write (27) as ( ˆv t = (1 δ v ) 1 (1 α) [ + φ f ξ κ i φ f V α V ) δ αφ w + δ φ w + δ ˆv t 1 α φ w + δ ˆv t + η VV y ŷ t ] φ f (1 α)( m δ m ) ˆv αφ w t 1. (42) + ( δ ) With finite ξ and short time periods, we can replace v t on the rhs of (42) by v t 1 (means that solving exactly for v t yields almost the same result) and write (42) in the form (17), with γ 0 = φ f ξ and γ 1 = (1 δ v )+ φ f [ (1 α) ((1 αφ δv w )( δ) ( m δ m ) ) ] α κ i + ξ + δ (1 α) V V ( φ w δ). (43) In the special case where δ = and m = δ m, we obtain γ 1 = 1 δ v ξ κ i. In this approximation, vacancies fluctuate much more than new vacancy postings. Putting it the other V V way 13

round, the value of vacancies, which determines the creation of new vacancies, has to fluctuate less than 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. This shows clearly the amplifying effect of LLV. Long live the vacancy! However, variations in δ v have little effect as long as it stays small compared to ξ κ i V V. Let us now look at variations in the separation rate. First notice that the denominator α φ w + δ is positive; this is a condition for the negative slope of the Beveridge curve, cf. (24). Then a more countercyclical separation rate (higher ) makes the numerator bigger and the denominator smaller, so that it has an unambiguous positive effect on γ 1. A higher m has a negative effect on γ 1, but it is clear that, which contains both job and match destructions, is larger than m, containing only match destructions. The total effect of variable separations on γ 1 is therefore clearly positive, therefore increasing amplification. The three effects captured in (43) are the following: The first, positive effect of comes from a counter-cyclical match formation. In a boom, the total number of matches goes down, so that fewer vacancies are depleted, which contributes to building up the vacancy stock. The negative effect of m arises because in a boom, fewer vacancies are created through separation that are match but not job destructions. The last positive effect of is more subtle. From the Beveridge curve (24) we see that a higher increases the ratio ût ˆv t, for a given parameter α. This dampens the negative effect of higher vacancies on the vacancy filling rate, and therefore on new vacancy creation. If we go a step further and take into account that variable separations affect the estimated α, we can take the ratio ût ˆv t as given (roughly equal to 1 in the data). In this case, (43) simplifies to γ 1 = (1 δ v ) + φ f [ ((1 δv φ w )( δ) ( m δ m ) ) + ξ α ] κ i 1 α V V ( φ w δ). (44) We confirm the conclusion that variable separations have a clearly positive effect on γ 1. 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 alternative 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 14

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 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; 15

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. 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 16

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 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 5.1, 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 features are driving the results, we present in Section 5.2 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 5.3. Sections 5.4 and 5.5 provide further evidence on the empirical performance of the model. Finally, we show in Section 5.6 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. 17

5.1 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 5.6). The model replicates this fact. Table 4 provides the same information for the model of Hall and Milgrom (2008). Fol- 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 - lowing 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 5.2 for a further discussion of the calibration). We see that this model 18

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 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. 5.2 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. It is well known (Costain and Reiter, 2008; Hagedorn and Manovskii, 2008) and was derived above in Equ. (33), 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 "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 alternative 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 19

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. Table 8 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 the wage bargain from fluctuations in unemployment, thereby reduces the responsiveness of wages to productivity, and makes the incentive for vacancy creation more cyclical (Hall and Milgrom, 2008). We will now assess the following sequence of models according to whether they are compatible with a relatively high responsiveness of discounted wages to discounted productivity. The point estimate of this statistic in Haefke et al. (2013)is 0.79. Our benchmark model differs from the Hall/Milgrom model in four respects. First, vacancies are long lived. Second, new vacancy creation is not perfectly elastic, but is a positive function of the value of a vacancy, with finite elasticity, similar to Diamond (1982). Third, the separation rate is time variable and endogenous. Fourth, we calibrate the model to have a higher match surplus, both for the worker and for the firm. The first six lines of Table 8 show the results of a step-wise transformation of the Hall/Milgrom model into our benchmark. Our starting point, reported in the first line, is the Hall/Milgrom model. As explained above, our calibration uses an unemployment replacement rate of 0.71, and a firm profit of 1.055 percent of production. This calibration yields a value for worker utility under disagreement of b b = 0.45, which is about 70 percent of wages, again similar to Hall and Milgrom (2008). We have chosen firm profit so as to obtain δ b /δ = 4, the value used in Hall and Milgrom (2008). Reassuringly, the wage 20

responsiveness in the model, DWR = 0.79, is perfectly in line with the evidence in Haefke et al. (2013). Next we are going to increase the match surplus. The second line of the table refers to the same model with a lower unemployment replacement rate, RR=0.4 rather than 0.71. In the calibration, the outside option in bargaining b b is now slightly increased to 0.46, in order to obtain the same steady state wage with lower replacement rate, and the bargaining destruction rate also goes down. After recalibration, results are basically unchanged. This is not surprising: for a given b b the unemployment replacement rate has little effect on the wage bargain in the Hall/Milgrom model. In the third line, we increase the firm surplus to our benchmark value of 2.255 percent. This requires a larger change in the calibration. To explain labor market fluctuations, the bargaining destruction rate must be lowered, which substantially reduces wage responsiveness in the model. Additional results below will confirm this systematic effect of the firm surplus. In the fourth line we still stick to flow vacancies, but replace the free-entry condition, which amounts to the assumption of an infinitely elastic supply of vacancies, by a vacancy elasticity of one, which is our benchmark value. As we have expected from the discussion in Section 3.7, reducing the elasticity makes it harder for the model to match fluctuations. This version of the model needs substantial wage stickiness, which is only achieved by making the bargaining break-up rate smaller than the job separation rate. These results show that with a somewhat higher match surplus, and an arguably more plausible value of vacancy elasticity, the model with alternative-offer bargaining has a hard time explaining labor market fluctuations. We now introduce the two features that are the focus of our analysis, namely LLV and endogenous separation. The fifth line introduces LLV, which brings a dramatic improvement and allows to increase wage responsiveness to 0.7. This can be perfectly understood from our heuristic calculations in Section 3.7. Finally, in line 6 we make separations endogenous. Again in line with our findings from Section 3.7, the results show that endogenous separations bring a further substantial improvement. The benchmark model explains the data with a bargaining destruction ratio of 3.85, and a wage responsiveness of 0.78, very close to the empirical estimates. To confirm our findings, lines 7 to 9 go backwards from the benchmark to the Hall/Milgrom model, but on a different route. In line 7, we make vacancy creation very elastic, but maintain LLV. Again, this helps the model somewhat, increasing the wage responsiveness to 0.81. In line 8, we go back to flow vacancies, with a dramatic deterioration of the model. In line 9, we reduce worker and firm surplus to the values in the Hall/Milgrom model, which allows to increase wage responsiveness to 0.7. The only difference between line 9 and the original HM model (line 1) is endogenous versus constant separation. Here we learn that, 21