On-the-job search and cyclical unemployment: Crowding out vs. vacancy effects

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On-the-job search and cyclical unemployment: crowding out vs. vacancy effects Daniel Martin, Olivier Pierrard

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Journal of Economic Dynamics & Control

Received date: 19 July 2012 Revised date: 7 September 2013 Accepted date: 25 April 2014 Cite this article as: Daniel Martin, Olivier Pierrard, On-the-job search and cyclical unemployment: crowding out vs. vacancy effects, Journal of Economic Dynamics & Control, http://dx.doi.org/10.1016/j.jedc.2014.05.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

On-the-job Search and Cyclical Unemployment: Crowding Out vs. Vacancy Effects Daniel Martina , Olivier Pierrardb,∗ a Clark

b Banque

University, Economics Department, 950 Main St., Worcester, MA 01610, USA centrale du Luxembourg, Research Department, 2 Bd. Royal, L-2983 Luxembourg

Abstract Incorporating on-the-job search (OTJS) into a real business cycle model has been shown to increase the cyclical volatility of unemployment. Using a particularly simple model of OTJS, we show that the increased search of employed workers during expansions induces firms to open more vacancies, but also crowds out unemployed workers in the job search, resulting in an ambiguous overall effect on unemployment volatility. We show analytically and numerically that the difference between the employer’s share of the match surplus with an employed versus an unemployed job seeker determines the degree to which OTJS increases unemployment volatility. We use this result to re-consider some related papers of OTJS and explain the amplification of volatility they obtain. Finally, we show that a plausible calibration of the OTJS model allows to reproduce most significant features of the US labor data. Keywords: on-the-job search, cyclical properties JEL classification: E24, E32, J64 1. Introduction As is well known, when a standard search-matching unemployment model such as Pissarides (2000) is embedded into a standard dynamic ∗ Corresponding

author. Phone: +352 4774 4449. E-mail: [email protected].

Preprint submitted to Journal of Economic Dynamics and Control

May 2, 2014

stochastic general equilibrium model of the macro-economy, it generates too little volatility over the business cycle in the key labor market variables of unemployment and job vacancies (Shimer, 2005). A number of fixes have been proposed for this problem. One is to introduce some form of wage rigidity, by assumption (Gertler and Trigari, 2009), by calibration of the parameters relevant to the wage bargain (Hagedorn and Manovskii, 2008) or by altering the bargaining mechanism (Hall and Milgrom, 2008). A second solution is the introduction of countercyclical vacancy costs as in Yashiv (2006) or Fujita and Ramey (2007). A third potential fix that has recently generated interest is to incorporate on-the-job search (OTJS) by currently employed workers for better jobs. While OTJS has been explored extensively in the partial equilibrium literature, to our knowledge just five papers to date have examined how OTJS increases the unemployment volatility in a DSGE context: Krause and Lubik (2010) and Van Zandweghe (2010) consider two-tier labor markets in which workers with bad jobs search for good jobs, Tasci (2007) and Nagypal (2007) construct models with a distribution of match quality, in which all employed workers search, but they only accept matches with a higher quality than the one they are currently in, and Menzio and Shi (2011) investigate OTJS in a model with directed (as opposed to random) search.1 These models of OTJS are quite different from one another and rely on specific assumptions as well as on different mechanisms to amplify the volatility of labor market variables. In this paper, we present a very simple 1 See

subsection 5.1 below for a review of this literature.

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model of OTJS, staying as close as possible to the textbook model of Pissarides (2000). First, we show that OTJS may increase but also may decrease the volatility of unemployment, depending on the difference between (the employer’s share of) the surplus of a match with a previously employed versus unemployed worker. Second, we adapt our model to reconsider some of the above-mentioned papers on OTJS and we explain the amplification of volatility they obtain in the light of this result. Third, we show that a plausible calibration of the OTJS model allows to reproduce most significant features of the US labor data. Incorporating OTJS is expected to increase the volatility of unemployment and vacancies over the business cycle through several mechanisms. First, even if employed workers search with less intensity than unemployed workers (hereafter experienced and inexperienced workers respectively), as is the case in this paper, OTJS smooths the number of potential hires businesses face over the course of the business cycle, leading firms to post more vacancies during expansions than they otherwise would. This results in more matches with inexperienced workers and thus to lower unemployment during expansions. This mechanism is common to all OTJS models. Second, if workers’ gains from finding a better job are procyclical, experienced workers will expend greater search effort during expansions than during recessions, which serves to accentuate this first effect. This is the mechanism explored in Krause and Lubik (KL hereafter) and in van Zandweghe, and is incorporated in this paper. Third, when firms prefer experienced to inexperienced workers, they will open more vacancies when more of their hires are expected to be experienced, i.e. dur-

3

ing expansions. In Nagypal and Tasci, firms prefer experienced workers because these matches are expected to last longer due to higher average match quality. In addition, in Tasci, matches of higher quality generate more per-period output. In this paper, we motivate firms’ preference for experienced workers through a one-time productivity boost conferred in the first period of a match. While this is a somewhat ad hoc mechanism for generating a preference for experienced workers, we motivate it with the argument that workers typically switch to jobs where they are more productive, and additionally inexperienced workers might be expected to be less productive due to skill loss during unemployment.2 Allowing workers to realize their productivity gains at the new job through a single lump-sum bonus eliminates the need to track the distribution of match quality, greatly simplifying the model. In addition, because wages are renegotiated each period as in the standard model, the non-convexity problem of Shimer (2006) does not apply. These issues are discussed further in subsection 5.1. Moreover, this simple mechanism allows us to easily adjust firms’ degree of preference for experienced workers, and to calibrate this to match observed wage gains by job switching workers.3 As would be expected, and as in KL, OTJS activity is procyclical in our 2 A counterargument to this would be that some workers switch jobs out of fear of losing their previous position, and indeed some job switchers accept wage losses–see Fujita (2011). While this point complicates our calibration, we note that the prevalence of this latter type of job switching is likely to be countercyclical. 3 Specifically, we calculate the wage gains workers would experience if their one-time hiring bonus were amortized over their expected duration of employment and compare this to empirical results for job switchers.

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model, inducing firms to open more vacancies during expansions when more of their hires will be experienced. We term this the vacancy effect. At the same time, because both experienced and inexperienced workers search in the same labor market, unlike in Krause and Lubik (2010); Van Zandweghe (2010); Menzio and Shi (2011), a larger fraction of the job matches formed during expansions will be with experienced workers. We term this the crowding out effect (i.e. experienced workers crowd out inexperienced workers), and it leads to a countervailing decrease in the cyclical volatility of unemployment. The overall effect of OTJS on unemployed workers’ job finding prospects and thus on the unemployment rate is therefore ambiguous. This is similar to the ambiguous overall effect of high-skilled workers’ job search on low-skilled workers’ job-finding prospects in Gautier (2002), and of the effect of commuters’ job search on residents’ job-finding in Pierrard (2008).4 This observation of competing vacancy and crowding out effects of OTJS on cyclical volatility appears to be new to the literature. In addition, we incorporate one other novel feature in our model: experienced workers negotiate the wages at their new job (specifically, their hiring bonus) using their previous job as a fallback. How wages are negotiated when switching jobs is a topic that has generated a great deal of interest (see, e.g., Cahuc et al. (2006)), with the default assumption (e.g. in each of the 4 One

potential objection to the notion of job switchers crowding out unemployed job seekers is that each job switch generates a new vacancy: the job the switcher left. Because of the free entry condition (equation (4)), however, firms open vacancies until their net value is zero (which is a standard in search models) and thus abandoned jobs have no impact on the overall number of vacancies in the economy.

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five above OTJS papers) being that job switchers first renounce their previous position then negotiate new wages using unemployment as a fallback. Our wage assumptions of a hiring bonus upon job switching makes changing workers’ fallback position straightforward, and we find negotiating wages based on the previous job appealing both on theoretical and empirical grounds. (Regarding the latter, see e.g. Fujita (2011); Barlevy (2001).) However, in section 5 we relax this assumption, forcing experienced workers to negotiate wages with unemployment as a fallback. Since this increases the surplus of an experienced match relative to an inexperienced one, we find that this increases the volatility of unemployment relative to our benchmark. For our calibration, we follow Krause and Lubik (2010) as closely as possible, using their estimates for OTJS activity. We show that when the match surplus with an experienced worker is similar to or not much larger than that with an inexperienced worker because the productivity gain of job switchers is small, the crowding out effect dominates the vacancy effect, and the net effect of endogenous (procyclical) search by employed workers is to decrease unemployment volatility (i.e. unemployment varies less over the business cycle than in a model with constant, exogenous OTJS intensity). When this productivity boost is significant enough to make the surplus of an experienced match sufficiently exceed that of an inexperienced match, the vacancy effect overpowers the crowding out effect, and endogenous OTJS results in greater unemployment volatility. By experimenting with a number of alternative model specifications (unemployment as a fallback for wage negotiation, a one-time hiring cost,

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allowing workers to switch jobs only once coupled with a permanent increase in productivity when they do so, a larger firm share in the wage bargain, and an endogenous search intensity of the unemployed – see section 5), we confirm that the difference between the (employer’s share of the) match surplus of an experienced versus an inexperienced match largely determines the degree to which OTJS increases unemployment and vacancy volatility. The remainder of the paper is organized as follows. Section 2 details the model. Section 3 proves the uniqueness of the steady state and determines under which conditions an increase in on-the-job search reduces steady state unemployment. Section 4 calibrates the model and produces dynamic simulations. Section 5 compares our results to related literature and to empirical data. Section 6 concludes. 2. Model The model embeds the search and matching framework of Pissarides (2000) into a dynamic stochastic general equilibrium model. Unlike most of the related literature, we include on-the-job search. Both unemployed and employed workers search for jobs in the same labor market. All unemployed workers search for a job with an intensity normalized to 1 – we consider the effects of relaxing this assumption below in subsection 5.3 – whereas all employed workers search OTJ with an endogenous intensity et > 0. The basic unit of production is the job match, so firms can be thought of in the one-job-one-firm variety. A firm that opens a vacancy may there7

fore match with an experienced (employed in the previous period) or an inexperienced (unemployed in the previous period) worker. Although the firm may have a preference between the two types of workers since the firm surplus depends on the worker (see equations 5 and 6 below), it never turns down an application as long as the asset value of the job match is positive, as it is under assumption 1 (see subsection 3.1). An alternative would be for a firm to direct ex-ante its search toward a specific type of worker. In general though, the existence of vacancy and crowding out effects is a consequence of the unified labor market assumption that is central to this paper, since both effects reflect the external effects of some workers’ search on other, qualitatively different workers. We discuss “directed search” models in the literature review, subsection 5.1 below. We assume that job vacancies enjoy an initial productivity, i.e. the productivity during the first period following the match, normalized to 1 if filled with an inexperienced worker, and of x¯ ≥ 0 if filled with an experienced worker (subject to an aggregate productivity shock, discussed ¯ although we may inbelow). A priori, we do not restrict the value of x, tuitively expect x¯ > 1 because unemployment depreciates skills, and because workers switch to jobs where they are more productive than the one they left. From the second period of the match onwards, we drop all distinction between workers who were employed or not before they started their current job; the productivity is 1 for all workers.

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2.1. Labor market flows The size of the labor force is normalized to 1 and split between the employed nt and the unemployed ut : ut = 1 − nt .

(1)

Firms open vacancies vt . The number of new matches between job seekers and firms is generated by a standard Cobb-Douglas matching function:5 1− μ

¯ vt mt = m

( e t nt + ut ) μ ,

(2)

¯ > 0 and 0 < μ < 1. The probability for an unemployed job seeker where m to find a job is pt = mt /(et nt + ut ), the probability for an employed job seeker to find a job is pt et , and the probability for a firm to fill a vacancy is qt = mt /vt . Experienced and inexperienced matches dissolve at the same exogenous rate 0 < ρ < 1 and a new match becomes productive after one period.6 Employment therefore evolves according to : nt +1 = (1 − ρ ) ( p t ut + nt ) . 5 Cobb-Douglas

(3)

matching functions are standard in the literature, although in general they allow that probability of matching in a given period can be greater than 1. This is not the case in our calibration. 6 Following KL, on whom we base our calibration, we allow for newly created jobs to experience a negative shock and dissolve in their first period before producing output. This is not the universal convention in such models, but is a fairly innocuous assumption.

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2.2. Representative household As in Merz (1995) or Andolfatto (1996), we assume a representative household pooling income between employed and unemployment workers. This household also owns all firms and therefore receives their profits. Moreover, it also pays a lump-sum tax to finance unemployment benefits. It lives indefinitely and chooses the optimal consumption path, with preferences represented by a log-utility function as in Krause and Lubik (2010)). As a result, firms and workers discount returns in the subsequent period according to βCt /Ct+1 , where Ct is consumption and 0 < β < 1 is the household’s exogenous discount factor. 2.3. Firms A firm pays a cost c > 0 to open a vacancy and thus the free entry condition implies:  c = q t ( 1 − ρ ) Et

β Ct Ct+1



e t nt ut Jtn+1 + Jo e t nt + ut e t nt + ut t +1

 .

(4)

Jtn is the asset value of a new job with an experienced worker (productivity x¯ the first period), and Jto is the asset value a new job with an inexperienced worker or of an old job (productivity normalized to 1): 

Jtn Jto

 β Ct o , = + ( 1 − ρ ) ( 1 − p t e t ) Et J Ct+1 t+1   β Ct o o = Pt − wt + (1 − ρ) (1 − pt et ) Et J . Ct+1 t+1 ¯ t − wnt xP

(5) (6)

Wages for experienced and inexperienced workers are wnt and wot respectively, and Et denotes the expectation operator. Pt is a multiplicative economy10

wide productivity shock that affects experienced and inexperienced matches identically and that drives business cycle dynamics. 2.4. Workers On the one hand, a worker may have just switched from one job to a new one and thus be experienced. In this case, his asset value is Wtn . On the other hand, a worker may have just found a new job from unemployment and be inexperienced or be continuing in an old match. In this case, his asset value is Wto .  Wtn = max et

 Et

   β Ct  (1 − ρ) (1 − pt et )Wto+1 + pt et Wtn+1 + ρ Ut+1 Ct+1 

Wto = max et

 Et

wnt − S(et ) +  (7)

wot − S(et ) +

   β Ct  (1 − ρ) (1 − pt et )Wto+1 + pt et Wtn+1 + ρ Ut+1 Ct+1

 (8)

S(et ) is the on-the-job search cost, which we specify to be quadratic, that is S(et ) = e¯ e2t /2 with e¯ > 0.7 The asset value of an unemployed worker 7 Christensen

( 1+ γ ) et / (1

et al. (2005) propose a more general search cost function S(e t ) =

e¯ + γ) and estimate the parameter γ on the Danish Integrated Database for Labour Market Research. Depending on the estimation procedure, they obtain γ = 0.84 and γ = 0.89, which suggests a cost of effort function that is approximately quadratic. We nevertheless discuss this parameter further in section 4.1.

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is:  Ut = z + Et

  β Ct  o Wt+1 pt (1 − ρ) + Ut+1 (1 − pt (1 − ρ)) , Ct+1

(9)

where z > 0 represents unemployment benefits. The first order condition for optimal search intensity is: 

S ( e t ) = p t ( 1 − ρ ) Et



  β Ct  n o w t +1 − w t +1 . Ct+1

(10)

This equation means that in equilibrium, the marginal cost of OTJS is equal to the expected discounted marginal return. 2.5. Wages Workers and firms negotiate wages at the beginning of every period through a Nash (1950) bargain over the surplus resulting from the match. Since workers and firms do not commit to future wages, the non-convexity problem discussed in Shimer (2006) does not arise.8 If an OTJ searcher finds a new job, he negotiates with the new firm over the joint surplus (increase in asset values) of the match, with the surplus defined relative to his asset value in the previous job.9 If an unemployed worker finds a new 8 When

workers and firms commit to future wages, the payoff set for workers and firms is non-convex in the negotiated wage, thus violating the standard assumptions for Nash bargaining. The non-convexity comes from the fact that workers paid higher wages are expected to stay with their employer longer than workers paid lower wages, and thus the present discounted value of the total gains to be split is increasing in the wage. See also Krause and Lubik (2007), section 6.3, for a discussion. 9 In essence, we assume that if the worker failed to reach an agreement with the new firm, he would continue to work at his old job, though in practice they always reach an agreement. It is worth noting that we impose that the new bargained wage cannot be renegotiate until the next period. If not, the new employer would immediately renego-

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job, or if an OTJ searcher fails to find a new job and remains on the same job, they bargain the period wage wot with unemployment as the fallback position. This gives: η ( Jtn + Wtn − Wto ) = Wtn − Wto ,

(11)

η ( Jto + Wto − Ut ) = Wto − Ut ,

(12)

where 0 < η < 1 is the worker’s bargaining power. The average wage in the economy, a statistic we report below, is given by: wt nt = (wnt nnt + wot not ),

(13)

where we define nnt+1 = (1 − ρ)( pt et nt ) and not+1 = (1 − ρ)((1 − pt et )nt + p t u t ). 2.6. Closing the model Aggregate productivity in the economy is given by Pt , which evolves according to an autoregressive process with white noise errors as in standard RBC models: Pt = Pta−1 exp (ut )

(14)

where 0 < a < 1 and ut ∼ N (0, σ). Finally, the final goods equilibrium implies that consumption is equal tiate the wage once the worker breaks the relationship with the previous employer. We also impose that the previous employer cannot make a counteroffer in response to the new offer as in Cahuc et al. (2006). Most other papers, e.g. Krause and Lubik (2010), with OTJS and Nash bargaining use unemployment as the fallback wage. See the related literature section for more.

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to net output: yt − vt c − S(et ) nt = Ct

(15)

When et = 0 ∀t, the model simplifies to the benchmark search and matching framework. 3. Qualitative analysis In this section, we first derive the steady-state asset values, and describe the equilibrium solution to the model. We then show the uniqueness of a steady-state equilibrium under reasonable assumptions. Next, provide conditions under which endogenous search by employed workers leads to greater cyclical volatility of unemployment, that is where the vacancy effect dominates the crowding out effect. Finally, we set up the log-linearized dynamic system. 3.1. Steady state analysis In what follows, we drop the subscript t to indicate steady-state values. To begin, we make two assumptions, which we consider reasonable: Assumption 1. J o > 0 and J n > 0. The positive asset values, J o > 0 and J n > 0, lead W n − W o > 0 and W o − U > 0. This assumption therefore implies that the Nash bargains always reach agreements. It is usually met with any standard calibration (see subsection 4.1).

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Assumption 2. J o > eJ n . This assumption is necessary to insure the uniqueness of the equilibrium (see subsection 3.2). Given that J o and J n , the firm shares of surpluses of the different types of matches, are expected to be of similar magnitude and e, the search effort of employed workers relative to unemployed workers, is expected to be small (less than 10%, see calibration section), this assumption is met by any reasonable calibration. Now, we want to write the steady state equilibrium. In the standard Mortensen-Pissarides model, it is easy compute the equilibrium as the solution of one equation with one variable. In our setup with OTJS, we characterize the whole equilibrium as the solution of two equations with two variables. The two variables are pe, which represents the quit rate of the OTJ seekers, and θ, which we define as v/(en + u) and represents the labor market tightness. To do so, first, we compute the steady-state asset values, for both the employers and the employees. Equations (5) and (6) reduce to: 1 − wo , 1 − (1 − ρ)(1 − pe) β = ( x¯ − 1) − (wn − wo ).

Jo = Jn − Jo

We observe that the difference in the employers’ asset values depends positively on the difference in productivity and negatively on the difference in wages. Since optimal OTJ search effort is unaffected by the type of job

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a worker is in, equations (7), (8) and (9) reduce to: z + β(1 − ρ) pW o , 1 − (1 − p(1 − ρ)) β wo − S(e) − z + β(1 − ρ) pe(wn − wo ) o , W −U = 1 − (1 − ρ)(1 − pe) β W n − W o = wn − wo . U =

We see that the difference in workers’ asset values between a new and old match is simply equal to the difference in the period wage. We may now use the Nash bargaining equations (11) and (12), respectively for experienced and inexperienced wages to eliminate wn and wo . It yields:

(1 + β(1 − ρ)( x¯ − 1) η pe) − z − S(e) , 1 − β(1 − ρ) (1 − pe(1 − η )2 − η p) = (1 − η )( J o + ( x¯ − 1)).

J o = (1 − η )

(16)

Jn

(17)

Second, we see from equation (2) that the probability p to find a job is a function of θ. We also observe that the search effort S(e) can be rewritten as e¯( pe)2 /(2p2 ), that is S is a function of pe and θ. As a result, asset values (16) and (17) are themselves function of pe and θ. Moreover, we note, still from equation (2), that q is a function of θ, that wn − wo = η/(1 − η ) J n , and that the steady state of (1) and (3) gives ρn = pu = p(1 − n). Third, it is now straightforward to write the optimal search behavior of employed

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workers, equation (10), and free entry of firms, equation (4) as respectively: β pe η (1 − ρ) n J (θ, pe) = 0, (18) 2 (1 − η ) c(ρ + pe(1 − ρ)) − pe(1 − ρ) J n (θ, pe) − ρJ o (θ, pe) (19) I (θ, pe) = βq(θ )(1 − ρ) = 0.

H (θ, pe) = S(θ, pe) −

We refer to these as H and I respectively and they fully characterize equilibrium through two endogenous variables: labor market tightness, θ, and the likelihood of an employed worker locating a job, pe.10 3.2. Uniqueness of a steady state equilibrium In the benchmark search and matching model (without OTJS), e = 0 and the H equation is absent. As a result, the equilibrium is simply given by I (θ, 0) = I (θ ) = 0. To insure the existence and uniqueness of equilibrium, it is sufficient that I  (θ ) > 0, along with appropriate limiting values as θ → 0, ∞. OTJS adds the H equation and the variable pe. To check the existence and uniqueness of an equilibrium, we first notice that H uniquely defines pe as a (positive) function of θ, as a consequence of the fact that equation (18) can be rewritten as A(θ )( pe)2 + B(θ ) pe + C (θ ) = 0 with C (θ ) < 0 < A(θ ) for all θ. The implicit function theorem implies that dpe/dθ = − Hθ /H pe > 0. This is shown in Appendix B. It means that employed workers are more likely to find (new) jobs in a tighter labor market. A gives the partial derivatives of J o , J n , S and I. We use these derivatives in the next subsections. 10 Appendix

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The free-entry condition (19) can therefore be rewritten as I (θ, pe(θ )) = 0 and a sufficient conditions to ensure that there exists at most one equilibrium is dI/dθ > 0. This condition is stated in the next proposition: Proposition 1. Given assumptions 1 and 2, there will exist at most one equilibrium if I pe >

− Iθ /(dpe/dθ ). Proof. As discussed above, a sufficient condition for excluding multiple equilibria is that the total derivative of I with respect to θ is dI/dθ = Iθ + I pe (dpe/dθ ) be positive. Since dpe/dθ > 0 (see above), I pe > − Iθ /(dpe/dθ ) implies dI/dθ > 0. Appendix A gives the partial derivatives of I, from which it is clear that assumption 2 implies Iθ > 0. In words, proposition 1 means that there is at most one equilibrium if Ipe is not too negative. The intuition is the following. I reflects the net cost of opening a vacancy, which must be 0 in equilibrium. Holding pe constant, opening more vacancies has a direct effect on this cost (Iθ ) that is always positive because of assumption 2; but as firms open more vacancies, they also induce more OTJS (higher pe). Provided that matches with experienced workers have a larger surplus to split, Ipe can be negative. This is precisely the amplifying effect we’re interested in in this paper (see proposition 2). However, if this effect is too strong, it can lead to multiple equilibria because the value of hiring experienced workers too greatly exceeds that of hiring inexperienced workers. There will then exist a low-employment equilibrium where more hires are unemployed and the average value of matches is low, and a highemployment equilibrium where more hires are employed and average the 18

value of matches is consequently high (along with an intermediate unstable equilibrium). Note that requiring Ipe not to be too negative is equivalent to requiring that J n not be too large relative to Jo .11 In other words, Ipe cannot be too negative. In this subsection we have proven the uniqueness of a steady state equilibrium but not its existence. In fact, the existence of a solution depends on the parameter values in a complex way–for example, if ρ is large, no non-degenerate equilibrium exists–but our numerical results show that a solution does exist for our calibration. 3.3. Vacancy vs. crowding out effects We now drop the search-effort (H) equation, and take en to be exogenous in order to investigate the effect of search by employed workers, en, on unemployment, u. Specifically, we replace equation (18) with pe(θ, en) = en(ρ + (1 − ρ) p(θ ))/(1 − ρ). The amount of OTJS activity, en, has two effects on unemployment. On the one hand, it increases the competition for jobs, making it more difficult for the unemployed to find them (the crowding out effect). On the other hand, it stimulates the opening of additional vacancies, which makes it easier for the unemployed to find a job (the vacancy effect). The crowding out effect dominates when dθ/den < 0, that is when more search ρ ( 1− ρ )

o − helps to write I pe as −( pe(1 − ρ)(1 − η ) + ρ) J pe ( J n − J o ), where the ρ + pe (1− ρ ) first term reflects the decreased expected value of a match as pe increases because matches dissolve faster, and the second value reflects the increased expected value of a match as pe increases because a larger fraction of matches are with job switchers who are potentially more productive. Only through this second term being very large relative to the first, that is only when J n is very high with respect to J o , do you get I pe << 0 ⇒ dI/dθ < 0 and hence multiple equilibria. 11 It

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by the employed results in more vacancies per unemployed worker and thus a higher probability p(θ ) that an unemployed worker finds a job. Conversely, the vacancy effect dominates when dθ/den > 0. Proposition 2. Under the conditions of proposition 1, the crowding out effect dominates when I pe > 0, and the vacancy effect dominates when 0 > I pe > − Iθ /peθ . Proof. The implicit function theorem implies Iθ dθ + I pe peen den + I pe peθ dθ = 0. This gives dθ/den = − I pe peen /( Iθ + I pe peθ ). We have shown above that, under the relevant conditions, Iθ > 0; Appendix A shows that peθ > 0; and Appendix C shows that peen > 0. To better understand what represents Ipe , we can easily show that c o > (1 − ρ) J n + ( pe(1 − ρ)(1 − η ) + ρ) J pe βq   (1 − η ) β(ρ + pe(1 − ρ))(ρ + pe(1 − ρ)(1 − η )) o n ⇐⇒ J < 1 + J ρ − βρ(1 − ρ) (1 − pe(1 − η )2 − η p)



I pe > 0 ⇐⇒

≡α

⇐⇒ J n < αJ o

with α > 1. This means that the crowding out effect dominates when Jn is small. The vacancy effect dominates when J n becomes sufficiently larger than J o . The intuition is the following: I reflects the expected net cost of opening a vacancy. As pe increases, both J o and J n decrease (because of the higher chance of a worker leaving), which has a direct positive effect on the expected net cost I. At the same time though, as pe increases, the fraction of matches with experienced workers also increases. Through this 20

composition effect, I decreases if J n > J o and I increases if J n < J o . If J n < J o , the two effects go in the same direction, Ipe is positive, and crowding out dominates. If J n is slightly higher than J o , the two effects run contrary to one another but the first one dominates and Ipe is still positive. If J n is sufficiently higher than J o , i.e. if J n > αJ o , this will mean I pe < 0 and the vacancy effect dominates. When J n becomes too high, we know from proposition 1 that we may have multiple equilibria. In the section 4.1, we calibrate the model and adjust the value of the parameter x¯ to change the value of J n and illustrate these different cases. 4. Quantitative analysis 4.1. Calibration Our paper is methodological, and we want to demonstrate that OTJS has two counteracting effects. To highlight which effect dominates, we will adjust our parameter x¯ and use a fairly standard calibration for all the other parameters. For simplicity and comparability, we fully borrow this standard calibration – based on US quarterly data – from Krause and Lubik (2010). We fix μ = 0.4, ρ = 0.10, β = 0.99, η = 0.5, a = 0.90, u ta ∼ ¯ c, N (0, σ) and σ = 0.0049. Again as in Krause and Lubik, we determine m, e¯ and z to reproduce the steady states q = 0.70, c v/y = 0.05 (vacancy costs as a fraction of output), u = 0.12 and p e = 0.06 (probability to voluntary quit a job).12 12 The

average search intensity of the employed is not a standard element of most labor market models and so there is no consensus calibration. As with our other parame-

21

¯ However, to satisfy assumption 1, x¯ must Our only free parameter is x. not be so low as to generate a negative Jn . With our calibration, we meet this restriction when x¯ > 0.51. Moreover, according to assumption 2 and proposition 1, x¯ must also not be too high, so as to avoid Jn >> J o and problems with multiple equilibria. We meet this restriction with x¯ < 2.74. Finally, our calibration and proposition 2 imply that the crowding out effect dominates when 0.51 < x¯ < 1.44 whereas the vacancy effect dominates when 1.44 < x¯ < 2.74. In section 4.4, in light of these results, we pin down different values for x¯ and discuss the Shimer puzzle. Before going further, it may be worthwhile to discuss which values of x¯ are empirically justified. x¯ represents a hypothetical 1–period productivity boost standing in for the higher (permanent) productivity of job switchers, and so cannot be observed directly. Nevertheless, we can compute the implications of the productivity parameter in terms of wage gains. We define wage gains as ((1 − (1 − ρ)(1 − pe))wn /wo + (1 − ρ)(1 − pe) − 1) × 100.13 Figure 1 shows the results. For instance, x¯ = 0.51 implies a wage gain close to 0%, x¯ = 1.44 implies wage gains of 6.1% and x¯ = 2.74 implies a wage gain of 14.1%. Using US job-to-job flow statistics derived from the Longitudinal Employer-Household Dynamics, Hyatt and McEntarfer ters, we borrow the calibration of Krause and Lubik (2010) and refer readers there for a justification. Few papers have directly measured OTJS intensity itself. Fujita (2011), using UK data, reports that approximately 5.5% of workers report searching OTJ. Meisenhemier and Ilg (2000), using US Current Population Survey data, report similar values. Since some workers can be observed to switch jobs without having reported that they were searching OTJ, we consider our (and KL’s) slightly higher estimate for e, 0.06/0.88 = 6.8%, reasonable. 13 This refers to average wage gains over a spell of employment. The average wage of an experienced worker is w n the first period followed by wo until the job is destroyed.

22

(2012) show that median earnings changes from direct job-to-job is on average around 6%. Using the UK labor force survey, Fujita (2011) reports that when unsatisfied with their current jobs, workers enjoy wage gains between 6% and 10% following a job-to-job transition. This suggests an empirically justified x¯ in the region where the “vacancy effect” starts to dominate. 4.2. Sensitivity analysis The KL calibration p e = 0.06 implies in our model that employer-toemployer transitions represent 35% of all separations. Using the Current Population Survey data, Nagypal (2008) estimates this number to be as high as 49% in the US labor market over the past decade. As a sensitivity ¯ as well as the x¯ which exactly exercise, we recompute the admissible x, offsets the crowding out and vacancy effects, for different values of p e. Figure 2 displays the results. First, a higher p e reduces Jn because the probability of a job separation is higher. As a result, we need a higher x¯ to meet assumption 1. Second, a higher p e reduces all wages but the reduction in wn is more important, because wn also uses wo as fallback position. As a result, J n − J o increases and assumption 2 and proposition 1 are al¯ By implication, for sufficiently high p e (above ready violated for lower x. 0.4 in figure 2), there is no vacancy effect area. The main conclusion of this exercise, however, is that for a large set of job-to-job transition calibrations (for instance the 49% of job-to-job transitions from Nagypal, which implies p e = 0.11), the admissible values for x¯ are still large and can generate both crowding out and vacancy effects. We also consider three further elements of the calibration from KL. 23

First, reducing the bargaining power η of workers, following Hagedorn ¯ and Manovskii (2008), only marginally changes the admissible values of x, but lowers the cut-off value where crowding out and vacancy effects just offset one other. Thus, it is easier to generate the vacancy effect. Second, having a more linear search cost function (see footnote 7) leaves figure 2 almost unchanged, with little change in the cut-off between crowding out and vacancy; but makes search effort more responsive and thus increases the magnitude of the crowding out effects in the crowding out area and the vacancy effects in the vacancy area. Third, according to Petrongolo and Pissarides (2001), the estimations of matching elasticity with respect to job seekers are biased upwards when OTJ seekers are omitted, as a direct implication of the procyclicality of employed job search. To take this into account, we reduce the elasticity μ from 0.4 to 0.3. Similarly to the linear search function, it leaves figure 2 unaffected although it increases the magnitude of the affects. 4.3. Dynamic equilibrium To compute the dynamic equilibrium, we must approximate equations. To do so, we choose to log-linearize the model and we define a “hat” variable as the proportionate deviation of that variable from its steady state level. Xst = [nˆ nt nˆ ot Aˆ t ] is a vector collecting the 3 state variables of the model, Xct = [Cˆ t Jˆtn Jˆto ] is a vector collecting the 3 control variables of the model and Yt is a vector collecting all the other hat variables. We can write

24

the log-linearized model as: ⎡

⎡

⎤

Xs ⎣ t +1 ⎦ Et Xct+1

= Ψ

⎤

Xs ⎣ t⎦ Xct

⎤

⎡ Yt = Ξ

+ Γ uta+1

Xs ⎣ t⎦ Xct

where Ψ, Γ and Ξ are matrices. We decompose Ψ = QΛQ −1 where Q is a matrix of eigenvectors and Λ is a diagonal matrix of eigenvalues. The dynamic solution of the model is unique if the number of unstable eigenvalues is exactly equal to the number of control variables. This is the case with fair calibrations and hence in the subsequent analysis. 4.4. Dynamic simulations In this section, economic fluctuations are driven by the productivity shock defined in section 2.14 We look at the cyclical properties of the model for different values of x¯ when (i) the number of on-the-job seekers et nt is constant15 and when (ii) OTJS et nt is endogenous. A positive productivity shock obviously increases vacancies and tightness, and decreases unemployment. When OTJS is endogenous, et nt also increases and stimulates 14 As in any log-linearized DSGE model, there is the possibility that some variables with

limited ranges, especially probabilities like p and q, which are restricted to be between 0 and 1, will take on unreasonable values outside the steady state that might affect the results. We confirm empirically that this does not occur. 15 Alternatively, we could assume that the fraction of on-the-job seekers is constant, without changing our qualitative conclusions.

25

further the opening of vacancies. Is this opening of vacancies sufficient to counteract the crowding out effect? Figure 3 shows that when x¯ < 1.44, the volatility of unemployment with exogenous et nt is higher than with endogenous et nt . As a result, the crowding out effect dominates. When x¯ > 1.44, the volatility of unemployment with exogenous et nt is lower than with endogenous et nt and the vacancy effect dominates. We are therefore able to generalize proposition 2 to the dynamic setup. In the Real Business Cycle literature, the volatility of unemployment is usually normalized with respect to the volatility of net output. Figure 4 shows that endogenous OTJS combined with a sufficiently high x¯ increases the relative volatility of unemployment bringing it closer to real data. Table 1 shows selected statistics when x¯ = 1.0 and x¯ = 2.4 (with exogenous vs. endogenous on the job search). We also compare these statistics to US data, and to those obtained from the standard search and matching model (with the same calibration). The main conclusion is therefore that OTJS may help generate realistic unemployment and vacancy volatility, provided that Jtn is high enough, i.e. provided that OTJ searchers are enticing enough to firms. In our model, this is achieved through a suffi¯ 16 ciently high x. 16 We

focus on second moments and correlations of unemployment and vacancies because this is what we are primarily interested in explaining. In section 5.4, we extend the analysis to other statistics and check how far the OTJS model may globally reproduce US data.

26

5. Discussion In the next subsections, we briefly review some related OTJS papers and we use our model to understand the mechanisms at work. Our main contribution is to show that in previous studies, models are built or calibrated to always fall in the vacancy effect area, and that the more to the right they are in this area, the higher is the volatility of unemployment. We also discuss endogenous search intensity for the unemployed, looking again at the role of x¯ and hence the difference between J n and J o . Finally, we show that a plausible calibration of the OTJS model allows to reproduce most significant features of the US labor data. 5.1. Related literature OTJS has been proposed as a possible solution to the Shimer (2005) puzzle of insufficient volatility of unemployment and vacancies in RBC models. A central problem with this work is that there must be some reason for workers to switch between jobs, which necessitates some form of heterogeneity between jobs. The distribution of workers over these different job types then becomes part of the model’s state space, which is potentially of high dimensionality. Our solution to this problem is to grant workers a one-time payment for switching jobs, so that they become indistinguishable from other workers in subsequent periods. The other models presented each handle this problem differently Tasci (2007) constructs a model with a distribution of match quality following Pries and Rogerson (2005), where wages are determined each period as a simple split of the expected match surplus, with unemploy-

27

ment as the fallback even when switching jobs. All workers engage in costless OTJS with the same intensity, but an employed worker will only (and always) accept a new job if it has higher estimated productivity than his current job. This implies that firms strictly prefer experienced to inexperienced workers when filling a vacancy because the former have higher estimated productivity on average, which both directly increases output and implies the worker is expected to remain with the job longer. This increases the value of a vacancy during expansions because more of the contacted workers will be currently employed, and leads to more cyclical volatility in job openings and by extension unemployment. Tasci uses the numerical approximation methods of Krusell and Anthony A. Smith (1998) to track the distribution of job types in the economy. Nagypal (2007)’s model is similar, except that all workers have the same productivity, and jobs only differ by a randomly drawn worker contentment reflecting such things as satisfaction with coworkers and commutes. (Additionally, Nagypal allows variable search effort by both employed and unemployed workers and shocks to the job destruction rate.) Wages do not vary across jobs by assumption, and firms prefer experienced hires because they are expected to reject more future offers to change employers and thus remain longer on the job. A fixed hiring cost is responsible for much of the increased volatility. Because workers have identical productivity, the endogenous distribution of workers in existing matches has more modest effects on vacancy creation than in Tasci’s model, and Nagypal appears to assume a constant distribution over the business cycle for these purposes (i.e. the steady state distribution).

28

Krause and Lubik (2010), assume a two-tier labor market for good and bad jobs in which good jobs are strictly preferred to bad jobs and so workers with bad jobs engage in OTJS for good jobs. Unemployed workers must choose whether to search for a good or bad job, and in equilibrium are indifferent between the two; thus, the expected duration of unemployment is higher if searching for a good job than if searching for a bad job. Wages are renegotiated each period with unemployment as the fallback position, even for job switchers. OTJS is costly, and since the wage difference between good and bad jobs is procyclical, so is search effort, and thus so is vacancy creation. By using a two-tier labor market, KL reduce the distribution of workers across job types to a two-dimensional state space.17 Menzio and Shi (2011) investigate OTJS in a model of directed, rather than random search, i.e. with a large number of different job types, where workers in existing jobs of a particular type search for jobs of just one other type. Because of this, workers of different types (including unemployed workers) do not crowd each other out, or induce additional vacancies to be opened that would affect the job search of workers of other types, the effects considered in this paper. This serves to eliminate the endogenous economy-wide distribution of workers over job types from the value functions of workers and firms, and makes the model solvable out of steady state. Because of this, and in contrast to the present paper and others considered in this section, the MS model exhibits little amplification of productivity shocks on labor market variables when job matches 17 Van Zandweghe (2010) also looks at OTJS in a two-tier labor market in a DSGE model

in order to study inflation propagation.

29

are inspection goods, i.e. when the quality of a match is observable before the match is formed, as is the case in e.g. Nagypal (2007) and Tasci (2007). (KL and the present paper lack an analogous distinction between the inspection and experience character of a match, but the effect of this is that the amplification mechanism in MS is not at work in these models.) Instead, significant volatility amplification arises only when matches are experience goods whose quality cannot be observed before matching. In this case, a (permanent) increase in aggregate productivity induces more than proportional increases in employment because job search is inherently a gamble, and a positive shock to productivity increases the payoff to a successful job search without much affecting the cost of search. We depart from the above models by motivating job search with a onetime lump sum payment negotiated using the prior job as the fallback position and perhaps reflecting a one-time productivity benefit intended to represent permanent higher productivity on the new job. This implies that experienced workers stay on their new jobs no longer than inexperienced workers, a potential defect of the model. In exchange, we are relieved of the need to track the endogenous distribution of job types over the business cycle because all workers look identical after their first period of employment. Unlike in KL or MS, all workers search in the same labor market, so that workers’ search behavior affects one another through vacancy and crowding out effects in the spirit of the original MP model. 5.2. Comparison of different mechanisms The benchmark model described in section 2 is simple and close to Pissarides (2000), chapter 4. In this section, we modify the model to intro30

duce some of the features described in Tasci (2007), Nagypal (2007) and Krause and Lubik (2010). We investigate how the changes affect J n , the wage gains, and the amplification mechanisms. Simulation (a) uses the benchmark model described in section 2 with the calibration from section 4.1 and x¯ = 1.44. We see that endogenous onthe-job search weakly amplifies the relative volatility of unemployment by 5% (see also figure 4). Simulation (b) introduces a hiring cost H > 0 as in   βC Nagypal (2007) and equation (4) becomes c = qt (1 − ρ) Et Ct+t1 et nettn+tut Jtn+1  ut o + et nt +ut Jt+1 − H . The ratio J n /J o is unchanged and the amplification of unemployment fluctuations is due to a sunk cost mechanism similar to Fujita and Ramey (2007). This is therefore not directly related to an OTJS mechanism in our model. Simulation (c) replaces equation (11) by η ( Jtn + Wtn − Ut ) = Wtn − Ut . We therefore have unemployment as fallback position, even when OTJS, as in Tasci (2007), Nagypal (2007) and Krause and Lubik (2010). Obviously, this reduces wage gains upon jobto-job transitions and therefore increases the difference between Jn and J o , which amplifies unemployment volatility by 16%. Simulation (d) is the benchmark model described in section 2 but with x¯ = 2.4. We increase wage gains but also J n /J o which leads to a huge increase (+90%) in unemployment volatility (see also figures 4 and 1, and table 1). Simulation (e) introduces good jobs and bad jobs as in Krause and Lubik (2010). More precisely, we relax our initial assumption that workers who switch jobs get a one-time productivity advantage on their new jobs, and we instead assume that they enjoy this advantage until the job is destroyed. As a result, workers who have switched jobs once do not search OTJ anymore.

31

This approach must be combined with using U as the fallback position for all workers when bargaining. We calibrate the permanent productivity advantage x¯ = 1.20 to obtain the same J n /J o ratio than in the simulation (d) and we see that, although the approach is somewhat different, we obtain very similar results and amplification. These exercises underline again that a high J n relative to J o magnifies the vacancy effects and is therefore the key ingredient to amplify unemployment volatility through OTJS. As a last illustration, assuming a lower bargaining power η for the workers would reduce wage gains, increase the difference in the Js and therefore amplify further the unemployment volatility–see simulation (f) in table 2. A combination of some of the mechanisms described above would obviously push the effects further up still. 5.3. Endogenous search intensity of the unemployed So far we have assumed a constant search intensity by the unemployed normalized to 1. This assumption is not entirely innocuous; however the cyclical behavior of search by the unemployed is theoretically ambiguous and empirical work shows little evidence for significant variation–see Shimer (2004); DeLoach and Kurt (2013). See also Merz (1995) for a model with endogenous search effort by the unemployed but without OTJS. Incorporating endogenous search effort by the unemployed into our model should increase the volatility of unemployment, but we expect that the effects on the volatility of vacancies will depend on the match surplus of an experienced versus an inexperienced match. To illustrate this, consider kt as the endogenous search intensity of an unemployed. Equa1− μ

¯ vt tion (2) becomes mt = m

(et nt + kt ut )μ and we also modify accord32

ingly the subsequent equations. We define the search cost as V (kt ) = ¯ k2 and the first order condition with respect to kt is V  (kt ) = pt (1 − k/2 t   ρ)Et β Ct /Ct+1 (Wto+1 − Ut+1 ) , where we choose k¯ to obtain k = 1 at the steady state. ¯ the volatility of unemployment Figure 5 shows that, for any value of x, is higher when the unemployed search intensity is endogenous. However, figure 6 shows that the effects of endogenous unemployed search intensity ¯ i.e depend again on the volatility of vacancies depend on the value of x, on the difference between J n and J o . When x¯ is low (resp. high), firms prefer unemployed (resp. employed) job seekers and endogenous search of the unemployed therefore does (resp. does not) give firms the incentive to open more vacancies. 5.4. Further empirical comparisons So far, our empirical discussion has focused almost exclusively on how OTJS may–or may not–increase the cyclical volatility of unemployment. In this section, we expand the analysis to include a number of other labor market variables in order to determine whether OTJS is important to understand significant features of the data. US data show that labor market variables (employment, unemployment, vacancies, etc.) are highly volatile, whereas wages are much less volatile and have only a weak correlation with GDP (see the “US data” rows in table 3). We know from Shimer (2005) that the basic search and matching model fails to reproduce most of these labor market statistics (see the “Mortensen-Pissarides, η = 0.5” rows in table 3). For instance, the volatility of employment in the standard model is 6 times lower than 33

in data, the volatility of labor market tightness (θ) in the model is 8 times lower than in data, and wages are almost perfectly correlated with GDP in the model, whereas the correlation is only 0.40 in the data. From Shimer (2005) onwards, there have been many suggestions to alter the standard model to better fit the data, but many of these attempts must use extreme assumptions or calibrations. For example, while reducing workers’ bargaining power helps improves almost all cyclical properties (rows “Mortensen-Pissarides, η = 0.4” in table 3), a reasonable fit would require a bargaining power close to 0, as in Hagedorn and Manovskii (2008). Since many empirical papers show that job-to-job transitions are large and economically important (see, for instance, Hyatt and McEntarfer (2012) for the US economy, or Fujita (2011) for the UK economy), OTJS seems a natural mechanism by which to better relate the model to empirical facts. We observe from rows “OTJS, η = 0.5, x¯ = 1” of table 3 that OTJS does not necessarily improve the statistical properties. In fact, as we’ve shown, what is needed is OTJS with a strong vacancy effect to generate realistic statistical properties (rows “OTJS, η = 0.5, x¯ = 2.4” in table 3). In this case, volatility of most variables are quite close to the data, although the–positive or negative–correlations with GDP are somewhat too strong. However, the main weakness is that a strong vacancy effect implies a strong and unrealistic wage gain (+12%) with job switching.18 Lowering 18 Hyatt

and McEntarfer (2012) show that median earning changes from direct job-tojob transition, that is without a spell of unemployment, decreased from 11% in 1998 to 7% in 2010 for a switch to new job in the same quarter, and from 6% in 1998 to 0% in 2010 for a switch to job in the following quarter. Giving a similar weight to both observations implies that average wage gains decreased from 8.5% in 1998 to 3.5% in 2010. Fujita (2011) computes similar values using UK data.

34

slightly the worker’s bargaining power–but still keeping it within a reasonable range–allows us to reduce the productivity gain of job switching ¯ and thus also the wage gain, to a more realistic value, without destroyx, ing the improvement of the statistical properties (see rows “OTJS, η = 0.4, x¯ = 2.1” in table 3). Finally, while this model generates globally realistic cyclical properties, we nevertheless observe that the quit rate and and vacancies are too volatile and procyclical, whereas the unemployment rate is not volatile enough. Moreover, wages are also to procyclical. Introducing an endogenous search intensity of the unemployed as in section 5.3 helps improve the results along all theses dimensions (rows “OTJS + endo u search, η = 0.4, x¯ = 2.1” in table 3). This last version of the model gives overall a pretty good account of the data and allows us to conclude that OTJS is indeed important if we want to understand real data.19 6. Conclusion We present a very simple model of on-the-job search and show that unemployment volatility may increase or decrease, depending on the calibration of a single parameter. This parameter governs the difference between the match surplus of an experienced vs. an inexperienced match. Then we extend the model along several dimensions to reproduce the main features of related papers with OTJS, and confirm that the match surplus difference is the key ingredient to understand their results. 19 In

table 3, we set in bold the best fit to the real data for each variable.

35

We briefly comment here on one other mechanism, not explored in this or any other paper to our knowledge, by which OTJS might increase cyclical unemployment volatility. In a model with wage rigidity, OTJS reduces the ability of firms to hire workers at a discount during recessions relative to benchmark models, because such workers would be more likely to leave during subsequent expansions. In addition to the negative effect on vacancies during recessions, this would reinforce the procyclicality of OTJS intensity. While wage rigidity as a mechanism to increase cyclical volatility of unemployment is well-studied (see section 1); in models without OTJS, what is important is the responsiveness of the wages of newly formed matches (rather than of existing matches) (Bodart et al., 2006). Including OTJS potentially allows for wage rigidity only in the case of existing matches, and not of new matches, to have a significant impact on cyclical unemployment volatility. Introducing wage rigidity into models with OTJS, however, raises additional difficulties (see, e.g. Shimer (2006)) and often requires wage posting. Acknowledgment This paper should not be reported as representing the views of the BCL or the Eurosystem. The views expressed are those of the authors and may not be shared by other research staff or policymakers in the BCL or the Eurosystem. Daniel Martin thanks the Henry J. Leir Luxembourg ProgramClark University, The Leir Charitable Foundations, and the Banque centrale du Luxembourg for its hospitality.

36

Appendix A. Partial derivatives of J o, J n , S and I

β (1 − ρ ) η p  ( J o − eJ n ) 1 − β(1 − ρ) (1 − pe(1 − η )2 − η p(θ )) (1 − η ) β (1 − ρ ) = − Jo 1 − β(1 − ρ) (1 − pe(1 − η )2 − η p(θ )) = (1 − η ) Jθo

Jθo = − o J pe

Jθn

n o J pe = (1 − η ) J pe

2p S βη (1 − ρ) p e n J = − p 1−η 2S βη (1 − ρ) n = = J pe 1−η c(ρ + pe(1 − ρ))q − ( pe(1 − ρ)(1 − η ) + ρ) Jθo = − βq2 (1 − ρ) c o = − (1 − ρ) J n − ( pe(1 − ρ)(1 − η ) + ρ) J pe βq

Sθ = − S pe Iθ I pe

Appendix B. Sign of dpe/dθ Restricting 0 < θ, pe < ∞, rewrite equation (19) as: 0 = e¯(1 − η ) pe − βη (1 − ρ) p2 J n    ¯ −1 ( pe)2 0 = 1 + 2β(1 − ρ)(1 − η )2 ep  + −2β2 (1 − ρ)2 η ( x¯ − 1) P(1 − η )2 p

 +2β(1 − ρ)η (e¯ − P( x¯ − 1)) + 2e¯(1 − β(1 − ρ)) p−1 pe    + −2β(1 − ρ)η β(1 − ρ)η ( x¯ − 1) Pp2 + (1 − β(1 − ρ))( x¯ − 1) Pp + (1 − η )( P − z) As noted in the text, this is equivalent to A(θ )( pe)2 + B(θ ) pe + C (θ ) = 37

0 with A > 0 and C < 0 for all θ > 0, implying the existence of a unique positive value for pe(θ ). dpe/dθ is given by: dpe/dθ = βη (1 − ρ) p

2p J n + pJθn n e¯(1 − η ) − βη (1 − ρ) p2 J pe

n < 0. The numerator is too: The denominator is positive since Jpe

2p J n + pJθn

= 2(1 − β(1 − ρ)(1 − pe(1 − η )2 − η p)) p J n + p(1 − η ) β(1 − ρ)η p ( J o − eJ n ) = (2(1 − β(1 − ρ)) + β(1 − ρ)(2pe(1 − η )2 + 2η p)) J n + pβ(1 − ρ)η (1 − η ) J o − peβ(1 − ρ)η (1 − η ) J n = (2(1 − β(1 − ρ)) + β(1 − ρ)( pe(2 − η )(1 − η ) + 2η p)) J n + pβ(1 − ρ)η (1 − η ) J o Appendix C. Partial derivatives of pe

p (1 − ρ) pe ρ + (1 − ρ ) p ρ + (1 − ρ ) p = 1−ρ

peθ = peen

38

Andolfatto, D., 1996. Business cycles and labor-market search. American Economic Review 86 (1), 112–132. Barlevy, G., 2001. Why are the wages of job changers so procyclical. Journal of Labor Economics 19 (4), 837–878. Bodart, V., Pierrard, O., Sneessens, H., 2006. Calvo wages in a search unemployment model, IZA Discussion Paper 2521. Cahuc, P., Postel-Vinay, F., Robin, J., 2006. Wage bargaining with on-thejob search: Theory and evidence. Econometrica 4 (2), 323–364. Christensen, B., Lenz, R., Mortensen, D., Neumann, G., Werwatz, A., 2005. On-the job search and the wage distribution. Journal of Labor Economics 23 (1), 31–58. DeLoach, S. B., Kurt, M. R., 2013. Discouraging workers: Estimating the impacts of macroeconomic shocks on the search of the unemployed. Journal of Labor Research 34 (4), 433–454. Fujita, S., 2011. Reality of on-the-job search, the Federal Reserve Bank of Philadelphia, WP 10-34/R. Fujita, S., Ramey, G., 2007. Job matching and propagation. Journal of Economic Dynamics and Control 31 (11), 3671–3698. Gautier, P., 2002. Unemployment and search externalities in a model with heterogeneous jobs and heterogeneous workers. Economica 69, 21–40. Gertler, M., Trigari, A., 2009. Unemployment fluctuations with staggered nash wage bargaining. Journal of Political Economy 117 (1), 38–86. 39

Hagedorn, M., Manovskii, I., 2008. The cyclical behavior of equilibrium unemployment and vacancies revisited. American Economic Review 98 (4), 1692–1706. Hall, R., Milgrom, P., 2008. The limited influence of unemployment on the wage bargain. American Economic Review 98 (4), 1653–1674. Hyatt, H., McEntarfer, E., 2012. Job to job flows and the business cycle, CES WP 12-04, U.S. Census Bureau. Krause, M., Lubik, T., 2007. On-the-job search and the cyclical dynamics of the labor market, deutsche Bundesbank WP 15-2007. Krause, M., Lubik, T., 2010. On-the-job search and the cyclical dynamics of the labor market, the Federal Reserve Bank of Richmond, WP 10-12. Krusell, P., Anthony A. Smith, J., 1998. Income and wealth heterogeneity in the macroeconomy. Journal of Political Economy 106 (5), 867–896. Meisenhemier, J. R., Ilg, R. E., 2000. Looking for a ’better’ job: job-search activity of the employed. Monthly Labor Review 123(9), 3–14. Menzio, G., Shi, S., 2011. Efficient search on the job and the business cycle. Journal of Political Economy 119, 468–510. Merz, M., 1995. Search in the labor market and the real business cycle. Journal of Monetary Economics 36, 269–300. Nagypal, E., 2007. Labor-market fluctuations and on-the-job search, mimeo. 40

Nagypal, E., 2008. Worker reallocation over the business cycle: The importance of employer-to-employer transitions, mimeo. Nash, J., 1950. The bargaining problem. Econometrica 18 (2), 155–162. Petrongolo, B., Pissarides, C., 2001. Looking back into the black box: a survey of the matching function. Journal of Economic Literature 39, 390– 431. Pierrard, O., 2008. Commuters, residents and job competition. Regional Science and Urban Economics 38 (6), 565–577. Pissarides, C., 2000. Equilibrium unemployment theory. MIT Press. Pries, M., Rogerson, R., 2005. Hiring policies, labor market institutions, and labor market flows. Journal of Political Economy 113 (4), 811–839. Shimer, R., 2004. Search intensity, mimeo. Shimer, R., 2005. The cyclical behavior of equilibrium unemployment and vacancies. The American Economic Review 95 (1), 25–49. Shimer, R., 2006. On-the-job search and strategic bargaining. European Economic Review 50, 811–830. Tasci, M., 2007. On-the-job search and labor market reallocation, the Federal Reserve Bank of Cleveland, WP 0725. Van Zandweghe, W., 2010. On-the-job search, sticky prices, and persistence. Journal of Economic Dynamics and Control 34, 437–455.

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Yashiv, E., 2006. Evaluating the performance of the search and matching model. European Economic Review 50 (4), 909–936.

42

14

12

10

8

6

4

2

0 0.5

1

1.5

2

2.5

xb

Figure 1: Wage gains, as defined in section 4, for job switchers (%) for different values of ¯ x.

0.5

Jn<0

0.45 0.4 0.35

pe

0.3 0.25

n o J >>J

0.2 0.15 0.1 0.05 0 0.5

crowding−out effect 1

vacancy effect 1.5

2

2.5

xb

Figure 2: Admissible values of x¯ and frontier between crowding out and vacancy effects, for different values of p e. See assumptions 1 and 2, and propositions 1 and 2 for definitions.

43

0.06 exo en endo en 0.05

0.04

0.03

0.02

0.01

0

1

1.5

2

2.5

xb

Figure 3: Absolute standard deviation of unemployment for fixed versus endogenous ¯ OTJS effort and different values of x.

5.5 exo en endo en

5 4.5 4 3.5 3 2.5 2 1.5 1

1

1.5

2

2.5

xb

Figure 4: Relative standard deviation of unemployment (w.r.t. the standard deviation of ¯ net output) for fixed versus endogenous OTJS effort and different values of x.

44

6 exo k endo k

5.5 5 4.5 4 3.5 3 2.5 2 1.5 1

1

1.5

2

2.5

xb

Figure 5: Relative standard deviation of unemployment (w.r.t. the standard deviation of net output) for fixed versus endogenous search by the unemployed and different values ¯ of x.

9 exo k endo k 8

7

6

5

4

3

2

1

1.5

2

2.5

xb

Figure 6: Relative standard deviation of vacancies (w.r.t. the standard deviation of net ¯ output) for fixed versus endogenous search by the unemployed and different values of x.

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u

US data standard MP exo OTJS, x¯ = 1 endo OTJS, x¯ = 1 exo OTJS, x¯ = 2.4 endo OTJS, x¯ = 2.4

v

stdv

corr

stdv

corr

8.71 1.00 1.18 1.01 2.71 5.06

-0.87 -0.76 -0.79 -0.78 -0.86 -0.94

8.39 1.40 1.89 2.50 3.82 8.90

0.88 0.87 0.91 0.95 0.98 0.99

Table 1: Relative standard deviation (w.r.t. the standard deviation of GDP) and correlation with GDP of unemployment (u) and vacancies (v), in the US (1950Q1-2009Q4) [from Krause and Lubik (2010)] and using simulations. Simulations include models without on-the-job search (MP), with on-the-job search of fixed intensity (exo OTJS), with on-thejob search intensity fluctuations over the business cycle (endo OTJS), and with different ¯ job switching productivity boosts x.

J n /J o

wage gains

stdv exo

stdv endo

Δ

(a) (b) (c) (d) (e)

Wn − Wo H = 0.20 Wn − U x¯ = 2.4 good/bad jobs

1.20 1.20 1.85 5.88 5.88

+6.1% +6.1% +3.5% +12.0% +12.2%

1.53 2.08 1.83 2.71 2.73

1.60 2.21 2.12 5.06 5.59

+5.0% +6.3% +16% +87% +105%

(f)

η = 0.2

2.34

+2.1%

4.12

4.42

+15%

Table 2: Alternative models and unemployment relative standard deviation (w.r.t. the standard deviation of net output): a) benchmark with x¯ = 1.44, b) hiring cost, c) unemployment as fallback for wage negotiation, d) high job switching productivity boost, e) max one job switch, f) high firm share. (J n /J o is relative firm share of match surplus for job searchers and others, wage gains are defined in section 4.1, stdv exo is the relative standard deviation when OTJS is fixed, stdv endo is the relative standard deviation when OTJS is endogenous, and Δ is the relative difference between stdv endo and stdv exo.)

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steady state

relative standard deviation (wrt. GDP) w

n

lp

u

v

θc

qr

wg

US data

0.65

0.88

0.80

8.71

8.39

15.99

10.06

[3.5%,8.5%]

Mortensen-Pissarides, η = 0.5 Mortensen-Pissarides, η = 0.4 OTJS, η = 0.5, x¯ = 1 OTJS, η = 0.5, x¯ = 2.4 OTJS, η = 0.4, x¯ = 2.1 OTJS + endo u search, η = 0.4, x¯ = 2.1

0.84 0.79 0.86 0.57 0.46 0.31

0.14 0.19 0.14 0.69 0.73 0.79

0.87 0.86 0.96 0.97 0.90 0.80

1.00 1.39 1.01 5.06 5.35 5.77

1.40 1.94 2.50 8.90 9.45 8.44

1.98 2.75 3.17 13.58 14.37 13.87

– – 3.68 11.50 12.21 10.73

– – 3.27% 12.0% 7.8% 7.8%

w

n

lp

u

v

θc

qr

US data

0.40

0.82

0.63

-0.87

0.88

0.88

0.88

Mortensen-Pissarides, η = 0.5 Mortensen-Pissarides, η = 0.4 OTJS, η = 0.5, x¯ = 1 OTJS, η = 0.5, x¯ = 2.4 OTJS, η = 0.4, x¯ = 2.1 OTJS + endo u search, η = 0.4, x¯ = 2.1

0.99 0.99 0.99 0.92 0.90 0.78

0.76 0.80 0.78 0.94 0.96 0.96

0.99 0.98 0.99 0.95 0.93 0.93

-0.76 -0.79 -0.78 -0.94 -0.96 -0.96

0.87 0.84 0.95 0.99 0.98 0.98

0.99 0.99 0.99 0.99 0.99 0.99

– – 0.99 0.99 0.99 0.99

cross-correlation (wrt. GDP)

Table 3: w, n, u, v are respectively wages, employment, unemployment and vacancies, as defined in the paper. l p is labor productivity and defined as gross output over employment. θ c is the standard labor market tightness defined as vacancies over unemployment. qr is the job-to-job quit rate defined as p × e. wg is the average wage gain for job switchers as defined in section 4. US cyclical properties are computed from quarterly HPfiltered statistics (1950Q1–2009Q4) with a smoothing parameter of 1600. We borrow them from Krause and Lubik (2010) [see their section 2 and table 1 for details]. We borrow the steady state US wage gain from job to job transition from Hyatt and McEntarfer (2012) [see footnote 18 for more details].

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