(PDF) ON-THE-JOB SEARCH, PRODUCTIVITY SHOCKS, AND THE INDIVIDUAL EARNINGS PROCESS

INTERNATIONAL ECONOMIC REVIEWVol. 51, No. 3, August 2010

ON-THE-JOB SEARCH, PRODUCTIVITY SHOCKS, AND THE INDIVIDUALEARNINGS PROCESS∗

BY FABIEN POSTEL-VINAY AND HELENE TURON1

University of Bristol, U.K., Paris School of Economics, CEPR, and IZA;University of Bristol, U.K., and IZA

Individual labor earnings observed in worker panel data have complex, highly persistent dynamics. We investigatethe capacity of a structural job search model with on-the-job search, wage renegotiation by mutual consent, and i.i.d.productivity shocks to replicate salient properties of these dynamics, such as the covariance structure of earnings, theevolution of individual earnings mean, and variance with the duration of uninterrupted employment, or the distributionof year-to-year earnings changes. Structural estimation of our model on a 12-year panel of highly educated Britishworkers shows that our simple framework produces a dynamic earnings structure that is remarkably consistent with thedata.

1. INTRODUCTION

In this article, we aim to offer a theoretical representation of observed individual earningsdynamics by investigating the capacity of a structural model of job search with simple i.i.d. pro-ductivity shocks to capture the covariance patterns of observed earnings processes. Specifically,we show how the combined assumptions of on-the-job search (with search frictions) and wagerenegotiation by mutual consent can act as a realistic “internal propagation mechanism” of i.i.d.productivity shocks. This combination of assumptions, which we shall motivate momentarily,implies that purely transitory productivity shocks are translated into persistent wage shockswith a covariance structure that we find to be consistent with the data.

The intuitive mechanism at work is as follows. Consider firms and workers who are matchedin pairs, each match facing an idiosyncratic productivity (or “match quality”) shock in everyperiod. Also assume that, through on-the-job search, workers occasionally contact outside firmsthat then compete over their services with their current employer. Because of search frictions,worker–firm pairings produce a positive surplus that the wage rate splits into the worker’s valueand the employer’s profit. In this process, the maximum wage that the firm is willing to payleaves the firm with zero profit and follows productivity shocks. The minimum wage that theworker is willing to receive yields the worker her/his outside option value, which equals thevalue of unemployment except in periods when the worker raises an outside offer, in which caseit equals the value of this offer. Under a mutual-consent rule for renegotiation (meaning thatneither party can force the other to renegotiate against its will), three distinct situations arise.First, if the match receives a sufficiently adverse productivity shock to make it unprofitable forthe firm to keep employing the worker at her/his current wage, then the firm has a credible threat

∗ Manuscript received August 2006; revised July 2008.1 The authors thank Gadi Barlevy, Jeff Campbell, Morris Davis, Jean-Marc Robin, Rob Shimer, and Randy Wright

for their inspiring thoughts on this article and two anonymous referees for very constructive suggestions. Very usefulfeedback was also received from conference participants at the 2005 Essex Thanksgiving conference and the 2006SED meeting (Vancouver), and seminar audiences at Bristol, Paris I, CREST, UCL, Queen Mary, Kent, CarnegieMellon, Minnesota, the Chicago Fed, Toulouse, and Oxford. Postel-Vinay gratefully acknowledges financial supportfrom the ESRC (grant reference RES-063-27-0090). The usual disclaimer applies. Please address correspondence to:Fabien Postel-Vinay, Department of Economics, University of Bristol, 8 Woodland Road, Bristol BS8 1TN, U.K. Phone:+44(0)117-928-8431. Fax: +44(0)791-350-8660. E-mail: [emailprotected].

599C© (2010) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Socialand Economic Research Association

600 POSTEL-VINAY AND TURON

0 2 4 6 8 10 12 14 16 18 20Time

Outside offers raising the worker’s outside option.

Maximum wage paid by the firm(Yields zero profit to the firm

and follows productivity shocks.)

Worker’s outside option.(Corresponds to the value

of unemployment whenno outside offer is raised.)

Efficient separationfollowing

an outside offer.

Worker forces renegotiationfollowing an outside offer.

Prevailing wage.

Firm forces renegotiationfollowing an adverseproductivity shock.

FIGURE 1

THE WAGE PROCESS

to fire the worker, which it can use to renegotiate the wage downward. Second, whenever theworker receives an outside job offer paying a higher wage than her/his current wage, s/he cancredibly threaten to accept it in order to force her/his employer into upward wage renegotiation.This will lead to an efficient separation if the outside offer is greater than the maximum wagethe firm is able to pay. Finally, in any other event (i.e., no sufficiently adverse productivity shockand no sufficiently good outside job offer), neither party is in a position to force the other torenegotiate, and the wage remains unchanged.

The wage is only altered when one of those outside-option constraints becomes binding, inwhich case it is revised up or down by just enough to satisfy whichever constraint is binding.Indeed the pattern of wage dynamics implied by the model just sketched can be summarizedgraphically as in Figure 1 (which we adapt from MacLeod and Malcomson, 1993): Because ofsearch frictions and the rule of mutual consent, i.i.d. match quality shocks and outside jobs offersare only infrequently translated into wage shocks; hence wage shock persistence.

The intuitive idea that renegotiation by mutual consent causes some form of “price stickiness”has been around for a while (as studies surveyed by Malcomson, 1997, suggest). Yet our articleis, inasmuch as we know, the first to formalize it in the context of a structural job search modeland to provide a quantitative analysis of the resulting individual income dynamics.

We estimate our structural model on a sample of highly educated British workers taken fromthe British Household Panel Survey (BHPS) and provide an in-depth fit analysis of the model.In so doing we contribute to the growing body of research carrying out structural estimation ofvarious forms of search models, which have so far been essentially geared to the description ofcross-sectional wage dispersion. As a consequence, estimation of these models tends to mostlyrely on the cross-sectional dimension of the data, leaving aside the question of individual earningsdynamics. Yet search models are inherently dynamic and have strong predictions about theprocess followed by individual wages over time. What little attention has been paid to thosepredictions has led to the conclusion that, in the absence of individual-level shocks, job searchmodels fail to accommodate the observed downward wage flexibility.2 By contrast, we consider

2 See Eckstein and Van den Berg (2007) or Postel-Vinay and Robin (2006) for reviews of these arguments.

ON-THE-JOB SEARCH AND EARNINGS 601

individual-level shocks and exploit as much as possible the observed dynamics of individuallabor income as is allowed by the large longitudinal dimension of our panel and the dynamicpredictions of our model.

To our knowledge, our article is the first full-fledged analysis of individual wage dynamicswithin a structural job search model, perhaps with the exception of Flinn (1986).3 In that in-sightful paper, Flinn also offers a search-based interpretation of the observed patterns of wagecovariances by combining job search with Jovanovic’s (1979) model of learning about matchquality. Although Flinn’s empirical motivation is very much the same as ours, his approach isdifferent. In his model wages are assumed to equal match productivity, which follows an ex-ogenously specified process in any given match (essentially a match-specific fixed effect plus ani.i.d. shock). It is the workers’ endogenous turnover decisions (driven by the workers’ learningabout the fixed component of match quality) that “filter” the exogenously specified produc-tivity process into sequences of wages that are consistent with NLSY data. We, on the otherhand, completely abstract from learning and keep the modeling of turnover decisions to a min-imal level of simplicity. Our focus is on the division of match rents as a mechanism to explainwage dynamics, something that was left essentially unmodeled in Flinn’s paper (see Flinn, 1986,footnote 2). As such we view our contribution as complementary to Flinn’s.

Another and perhaps more general contribution of our article is to offer a structural counter-part to the large empirical literature on individual labor income processes (as Flinn, 1986, alsodoes), in which sophisticated stochastic processes (typically, but not exclusively, ARMA-typeprocesses) are fitted to longitudinal wage data taken from worker or household panels.4 Becauseit is primarily based on statistical models, that literature remains somewhat out of touch witheconomic theory. Although links to behavioral interpretations of the highlighted autocovariancepatterns are sometimes informally discussed, quantitative consistency with a formal behavioralmodel is typically not addressed.

It seems important, however, to understand the economic forces governing individual wagedynamics from a dual theoretical and quantitative standpoint.5 This article suggests that thecombination of on-the-job search and renegotiation by mutual agreement is a promising can-didate explanation of the widely documented persistence of earnings shocks. In particular, ourstructural approach highlights the interplay between job mobility and earnings dynamics: Themodel predicts that the individual probabilities of transitions between labor market states con-dition the individual earnings process in a way that is consistent with the data. More generally,our theory suggests that the income process should be thought of as following a particular accep-tance/rejection scheme of underlying i.i.d. productivity shocks, of which labor market transitionrates are a key determinant.

Finally, our theoretical model can be seen as a version of the matching model of labor mar-ket equilibrium, now routinely referred to as the Diamond–Mortensen–Pissarides, or “DMP”model,6 in which employed job search is allowed. Although virtually any wage formation mech-anism can be embedded into the DMP model, the typical (and by far dominant) practice is toassume a Nash-like sharing rule, whereby each party receives a given share of the match surplusat all times. Hidden underneath this constant-share feature is the assumption that wages are

3 We should also mention a related and independently written paper by Yamaguchi (2006), which we discuss inSection 6.

4 That literature is literally huge. A somewhat arbitrary selection includes the seminal papers by Lillard and Willis(1978), Lillard and Weiss (1979), MaCurdy (1982), and Abowd and Card (1989), and the comparative analyses of recentdevelopments by Baker (1997) or Alvarez et al. (2001). See also Blundell and Preston (1998) for an application to U.K.data and Meghir and Pistaferri (2004) as an example of a state-of-the-art paper in this field.

5 The following examples may illustrate that claim: A typical application of the empirical literature on wages is to usea particular permanent/transitory decomposition of incomes to test the permanent-income/life-cycle hypothesis. Nowsurely, as, e.g., Baker (1997) notes, such tests have “an obvious dependence” on the specific decomposition of income.Other frequent fields of application include the study of wage rigidity or that of wage growth over the working life.In both cases, the relevant policy implications vary quite a lot from one possible underlying theoretical framework toanother.

6 From Diamond (1982) and Mortensen and Pissarides (1994). For a complete exposition of the DMP model andmany extensions thereof, see Pissarides (2000).

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renegotiated at least every time the match is hit by a productivity shock. This assumption issomewhat arbitrary, as, in general, the occurrence of a shock to match productivity providesneither of the matched partners with a credible threat to force the other to renegotiate. Asadvocated by, e.g., Malcomson (1997, 1999), renegotiation by mutual consent is a more naturalassumption, at least for its consistency with a number of legal and/or economic facts.7

The rest of the article is organized as follows. In the next section, we pose the theoreticalmodel. In Section 3, we go on to derive the model’s solution in connection with our estimationprocedure, which is presented in Section 4, together with the data. Section 5 contains estimationresults and an analysis of the model’s performance at replicating some features of the earningsdata. Section 6 tackles the issue of persistent productivity shocks within our structural model.Finally, we conclude and discuss a number of potentially interesting extensions in Section 7.

2. THEORY

2.1. The Environment

2.1.1. Basics. We consider a labor market where a unit mass of workers face a continuum ofidentical firms producing a multi-purpose good sold in a perfectly competitive market. Workersand firms are infinitely lived, forward-looking, and risk-neutral and have a common exogenousper-period discount factor of β. Time is discrete, and the economy is at a steady state. Workersare either unemployed or matched with a firm. Firms operate constant-return technologies andare modeled as a collection of job slots that are either vacant and looking for a worker oroccupied and producing.

The output flow yt of a firm–worker match in period t is defined as

yt = p · εt .(1)

It is the product of a worker fixed-effect p and a transitory period- and match-specific shock εt .We should emphasize that because this shock is match specific, a realization of ε is not carriedover from one firm to the other in case the worker changes firms.

The population distribution of (log) worker fixed effects ln p is denoted as H(·). Identificationrequires normalization of one of the components of (1). We choose to normalize the mean valueof ln p at zero: EH(ln p) = 0.

When a worker and a firm meet, the idiosyncratic component of (potential) match qualityis drawn from a distribution M(·) with support [εmin, εmax]. Every ongoing firm–worker matchdraws a new value of εt at the beginning of each period t from that same distribution M(·).Depending on the realized value of εt , the match can go on under the same wage contract orunder a renegotiated contract. The precise cut-off values of the transitory shock under which acontract is renegotiated are determined below.

7 Mutual agreement is indeed a prerequisite to wage renegotiation under English law, which is relevant to the datawe use in the latter part of this article. In the United States, although the employment-at-will doctrine would in principleleave scope for more responsiveness of wages to productivity shocks, the empirical evidence reviewed in Malcomson(1997, 1999) reveals that wage changes occur much less frequently than would be consistent with a strict applicationof the employment-at-will rule, suggesting that mutual consent, although not an explicit legal provision in the UnitedStates, may nonetheless be common practice.

On the theory side, Mortensen and Pissarides (2003, footnote 4) recognize, without actually using it for their purposes,that the assumption of renegotiation by mutual consent “may well generate more realistic wage dynamics.” Fella (2004)does in turn implement this type of negotiation within the standard DMP model (without on-the-job search). However,ignoring on-the-job search leads to the counterfactual prediction that wage profiles unambiguously (stochastically)decline over the job spell. On the other hand, existing versions of the DMP model with on-the-job search (Pissarides,2000, chapter 4; Shimer, 2006), mostly shut down between-employer competition by assuming that the worker’s outsideoption is always unemployment, even when s/he winds up with an outside job offer. This, combined with the assumptionthat wages are renegotiated every time a shock hits the match, implies that individual wages fluctuate along with matchproductivity.

ON-THE-JOB SEARCH AND EARNINGS 603

Finally, throughout most of this article we assume that transitory shocks εt are uncorrelatedover time or across matches, implying that any source of persistence in a worker’s productivityeither over time or across jobs is picked up by the worker fixed effect, p. We opt for this admit-tedly disputable assumption based on the following two arguments. First, it greatly simplifies theanalytical characterization of wage dynamics. Indeed our relatively simple, closed-form char-acterization of wage dispersion and wage dynamics (see Subsection 3.3), which we view as anattraction of our approach, becomes impossible under any realistic pattern of serially correlatedshocks, with otherwise little gain in terms of new qualitative insights for our purposes.8 Second,commonly available worker- or household-level data of the type routinely used in the analysisof individual earnings dynamics typically do not convey any direct information on productivity,which poses difficulties in the identification of the productivity process. Although a combina-tion of the model’s structure and the wage information contained in the data would in principleensure identifiability of some measure of persistence in productivity, we will argue later in thearticle that this is not the case in practice, at least with the BHPS data that we are using. Wedefer the discussion of all those issues until Section 6, where we offer a simple way to relax theassumption of i.i.d. shocks and attempt to estimate the resulting slightly generalized version ofour model.

2.1.2. Unemployment income. In any given period, an unemployed worker with perma-nent productivity component p (henceforth a “type-p worker”) receives a flow income ofb · p, b > 0. This contains the assumption that unemployment income depends on the perma-nent individual productivity parameter in the same way (i.e., multiplicatively) as productivity ina match with a firm. This assumption is inessential—although not quantitatively innocuous—andagain is made because it simplifies the formal model somewhat.

2.1.3. Surpluses. Consider a match between a firm and a type-p worker, with current pro-ductivity ε. Denote the current wage in this match by φ.9

We denote the worker’s valuation of this match by V(φ, p), and the firm’s valuation by�(ε, φ, p). V(·) and �(·) are present discounted sums of future expected income or profit flows.We assume from the outset that V(·) is increasing in φ and is independent of ε, whereas �(·)is increasing in ε and is decreasing in φ. The consistency of these assumptions will be verifiedlater on. We further assume that a vacant job slot is worth 0 to the firm (as naturally results fromfree entry and exit of vacant jobs on the search market), and we denote the lifetime value ofunemployment by V0(p).

We define total match surplus as the value of the match net of the combined values of a vacantjob and an unemployed worker

S(ε, p) = [V(φ, p) − V0(p)] + [�(ε, φ, p) − 0].(2)

We shall start working under the provisional assumption that S(·) is independent of any wagevalue. This will be shown later to be a consistent assumption given risk-neutrality of workersand firms and given our (privately efficient) surplus-sharing mechanism.10 Moreover, total matchsurplus only depends on the determinants of current and future match output flows. Given the

8 Arguably the simplest way to increase persistence (while preserving serial independence) of match quality shockswould be to model their occurrence as a jump process similar to Mortensen and Pissarides (1994). Again, this wouldadd great complexity in the formalization of the wage process. Yamaguchi (2006) takes this route in closely related(and independently conducted) research and finds it intractable to solve the model explicitly. He thus resorts to indirectinference for estimation.

9 We omit period subscripts t when they are not strictly necessary.10 Intuitively, total match surplus involves the present discounted sum of expected future flow values of match surplus,

which in turn are the sum of wage flows net of forgone unemployment income flows (φ − b · p) plus net profit flows(p · ε − φ). As the φ terms cancel when net income and profit flows are added, surplus flows are independent of anywage value.

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assumed jump process for transitory shocks ε, those only include the permanent component ofmatch productivity, p, and the current value of its transitory component, ε.

2.1.4. Job search, match formation and match dissolution. The labor market is affected byjob search frictions: Firms and workers are brought together in pairs through random search.Specifically, any unemployed worker has a per-period probability λ0 of meeting a firm. We alsoallow employed workers to raise job offers and assume that they have a per-period probability λ1

of meeting a potential alternative employer. Note that we only allow workers (in any employmentstate) to contact at most one firm per period. Moreover, we assume that contacts made in earlierperiods cannot be recalled.

Not all firm–worker contacts are a priori conducive to an actual job move: For employedjob seekers, the decision of whether or not to quit an ongoing match for a new one involves acomparison of match surpluses that will be carried out in full detail in the next subsection.

Finally, all matches have a common, exogenous breakup probability of δ per period. Matchbreakup is thus formally disconnected from idiosyncratic shocks to ε. This assumption callsfor the following comments. The formation or continuation of any firm–worker match is sub-jected to the minimal requirement that total match surplus be nonnegative. Thus, implicit inthe specifications detailed above is the assumption S(εmin, p) ≥ 0 for all p, which amounts totruncating the “true” underlying distribution of productivity shocks from below.11 However,an alternative, probably more general view on the essence of the assumption of an exogenousjob destruction rate is that transitory shocks to match quality—that potentially cause individualincome fluctuations—are of a fundamentally different nature than shocks leading to a job loss.Although our formal setup clearly takes it that shocks to ε are match specific, we do not give anyspecific interpretation of the random event causing match destruction, which can reflect adverseshocks to any combination of the match, the individual, the market, or the firm.

2.2. Wage Determination

2.2.1. The wage rule. Wage contracts stipulate a constant wage and are only renegotiableby mutual consent in continuing matches. In other words, no firm or worker can force their matchpartner to revise the wage against the latter’s interest, unless the former has a credible threatto leave the match.12 The implications of this wage rule for wage dynamics (of which we gavean informal account in Figure 1 in the introduction) were analyzed theoretically by MacLeodand Malcomson (1993): In continuing matches, if one party has a credible threat to dissolve thematch, i.e., if the value of her/his outside option exceeds the value s/he gets from the existingrelationship, the other party consents to wage renegotiation up or down to the point wherethis outside option is matched.13 The existing match only survives if the surplus it generates isgreater than the sum of surpluses generated by the outside options (i.e., a vacant job and analternative worker–firm match or an unemployed worker). In case neither party has a crediblethreat to leave the match, there is no mutual consent to revise the wage and the current termsof employment continue to apply.14

11 In fact, the condition S(εmin, p) ≥ 0 will turn out not to depend on p, as the surplus will be shown to be proportionalto p—see below. Let us then define r0 by S(r0, p) = 0 and assume that there is an underlying latent sampling distributionof potential match qualities, say M0(·), with support (−∞, εmax). Then, any draw of a productivity shock above r0 yieldsa positive potential match surplus, whereas any draw falling short of r0 entails a negative surplus and causes matchdissolution. Hence the match destruction rate δ can be seen as equating M0(r0), whereas M(·) simply coincides withM0(·) truncated below at r0.

12 See Malcomson (1997, 1999) for a motivation of this principle.13 That is, when renegotiation occurs, outside options act as bounds on the parties’ payoffs. This is known in the

bargaining literature as the outside option principle (see, e.g., Sutton, 1986, or Binmore et al., 1989).14 Even though we appeal to MacLeod and Malcomson (1993) as a theoretical foundation of our wage setting

mechanism, alternative justifications exist. Models of self-enforcing wage contracts designed to allocate risk betweena risk-neutral employer and a risk-averse employee faced with uncertainty about match productivity and/or market

ON-THE-JOB SEARCH AND EARNINGS 605

In newly created matches, there are no pre-existing terms of the (potential) employmentrelationship, and a start-up wage has to be determined. Here we follow the approach of Postel-Vinay and Robin (2002a,b) and assume that firms make take-it-or-leave-it offers to the workers,so starting wages with new employers give workers the value of their outside option.15 The latterequals V0(p) for a worker hired from unemployment or the maximum value the worker couldextract from her/his previous employer if the new match follows a job-to-job quit. Thus in thelatter case, as explained in Postel-Vinay and Robin (2002a,b), we effectively let the incumbentand the outside employer Bertrand-compete for the worker’s services.

2.2.2. Negotiation baselines. It is useful to introduce at this stage the following conventionfor the description of all wages. At any time, the wage that the worker receives, φ, can beexpressed as a function of her/his type p and a value of match-specific productivity r (whichwill not in general be equal to match-specific productivity in the current match, ε), defined asfollows:

φ = φ(r, p) ⇔ V(φ, p) = V0(p) + S(r, p) ⇔ �(ε, φ, p) = S(ε, p) − S(r, p),(3)

where r gives a measure of the surplus that the worker enjoys over and above the value of beingunemployed, V0(p). Alternatively, r can be seen as the quality of the match from which theworker was last able to extract the whole surplus. For reasons that will become clear shortly, wewill term r the worker’s negotiation baseline.

The negotiation baseline is defined formally in the next three paragraphs. Before we proceed,however, it is important to note that φ(r, p) is a strictly increasing function of r. This flowsdirectly from the monotonicity properties of V(·) and S(·), which, together with (3), imply∂φ

∂r = ( ∂S∂r )/( ∂V

∂φ) > 0.

2.2.3. Starting wages. First consider an unemployed, type-p worker meeting a job-advertising firm. Given the assumptions just discussed and a current match quality of ε, thepotential match yields positive surplus, and a starting wage contract must be signed. As men-tioned above, we assume that, in a newly created match, the employer extracts all the matchrent by offering the worker her/his reservation value. This case is encompassed by Equation (3)and entails a starting wage equal to φ(r0, p), and a negotiation baseline of r0 such that

V(φ(r0, p), p) = V0(p) ⇔ S(r0, p) = 0.(4)

A formal definition of r0 based on (4) will be given below, where we will establish that r0

is independent of p. For now we should note that, in any generic match with quality ε andnegotiation baseline r, ε ≥ r ≥ r0 necessarily holds, otherwise either the firm would earn negativeprofits and fire the worker or the worker would find it preferable to quit into unemployment.

2.2.4. Outside offers. We now examine the situation that arises when an already employedworker with current match productivity p · ε and current negotiation baseline r meets another

opportunities deliver a similar wage rule (Harris and Holmstrom, 1982; Thomas and Worral, 1988). More generally,some degree of worker risk aversion (coupled with an inability to transfer wealth across time) can be appealed tojustify the focus on constant-wage contracts. Finally, Hall (2005) also considers this wage setting mechanism, which heinterprets as a social norm favoring wage rigidity while restricting wages to lie within the bargaining set (thus avoidinginefficiencies in the allocation of labor).

15 In terms of a Nash bargaining approach, we thus assume that the worker has zero bargaining power in newly formedmatches. Extending the model to allow for positive worker bargaining power is of potential quantitative importance(see Dey and Flinn, 2005, and Cahuc et al., 2006), yet it complicates the writing of the model somewhat. We leave thisextension for later work. At the other extreme, Flinn (1986) assumes a worker bargaining power of 1, whereby s/he getspaid the full marginal product at all times.

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potential employer through on-the-job search. We denote the match-specific component ofproductivity in the outside firm as η.

Consistent with the assumptions listed above, we let the incumbent employer and the“poacher” Bertrand-compete for the worker’s services. The worker extracts the whole surplusfrom the less productive of the two potential matches, which translates into a new negotiationbaseline of min{ε, η}. If ε < η, the poacher profitably attracts the worker with a wage offer ofφ(ε, p) (plus one cent)—an offer that the incumbent employer is unable to match without in-curring losses. Alternatively, if ε ≥ η, then the incumbent can profitably retain the worker bymatching the poacher’s maximal wage offer of φ(η, p). Here, as a result of the outside offer, theworker stays in her/his current job but can force wage renegotiation up to her/his new outsideoption, φ(η, p). In this latter case, however, renegotiation only takes place if the worker gainsfrom it, i.e., if η ≥ r (otherwise we assume that the worker always has the option to conceal theoutside offer s/he has received from the poacher).

We can summarize the possible outcomes of an outside offer received by the worker as follows:

η > ε Worker quits, mobility wage φ(ε, p), new negotiation baseline ε,

ε > η > r Worker stays, renegotiated wage φ(η, p), new negotiation baseline η,

r > η Offer is discarded, nothing changes.

2.2.5. Productivity shocks. A last potential cause of wage change is the occurrence of aproductivity shock. Consider a match with productivity p · ε and current negotiation baseline rand assume a new transitory shock value of ε′ is drawn. One of three situations can arise.16

A first, simple case is ε′ ≥ ε. In this case, the worker would like to capture some of the extrasurplus brought by the gain in match productivity through a wage increase. But the worker’s onlyoutside option is to resign and become unemployed, thus achieving a value of V0(p), equivalentto a negotiation baseline of r0. This is never preferable to keeping the existing contract, whichhas a negotiation baseline of r ≥ r0. In other words, the worker cannot force the firm to raisethe wage. The match thus goes on with an unchanged wage after such a productivity gain.

In the second case, ε > ε′ ≥ r , the match has undergone a loss of productivity and the firm’sprofit has decreased from S(ε, p) − S(r, p) to S(ε′, p) − S(r, p). The firm would thus want toshare some of this loss with the worker by lowering the wage. But as long as ε′ ≥ r , profitsremain positive at the current wage φ(r, p). At that point the firm’s only outside option is tofire the worker, thus ending up with a vacant job worth 0, whereas carrying on with the existingcontract still gives it a positive profit. Hence the firm cannot force the worker to accept a wagecut, and the match again goes on with an unchanged wage.

The third, more complicated case is when r > ε′ ≥ εmin. In this case, since ε′ ≥ εmin, and since byassumption εmin ≥ r0, the match is still viable, meaning that a mutually beneficial contract exists.However, keeping the existing wage φ(r, p) would imply a negative profit of S(ε′, p) − S(r, p).Here the firm is better off firing the worker than maintaining the match under the existingcontract and so has a credible threat that it can use to force the worker into renegotiation. Ourassumed wage rule then implies a wage cut down to the point where the firm enjoys in thecontinuing match the same value as in its outside option, here equal to zero with a vacant job.This leaves the worker with a wage value of φ(ε′, p) in the continuing match. Her/his negotiationbaseline has thus been updated to ε′.

2.2.6. Within-period timing of events. The last thing that requires further specificationbefore we can solve the model is the sequence of random events affecting firm–worker matcheswithin each period. We simply assume that all of these random events are realized simultaneously

16 For simplicity of exposition, we describe here the case where no outside offers are raised by the worker. The factthat productivity shocks and outside offers can occur simultaneously will naturally be taken into account in the followingsections.

ON-THE-JOB SEARCH AND EARNINGS 607

at the beginning of each period. The list of such events is the following: match destruction shocks(with probability δ, any given match is dissolved), firm–worker contacts (any unemployed workermeets a potential employer with probability λ0, and any employed worker meets a potentialalternative employer with probability λ1), and draws of transitory match productivity shocks ε inincumbent matches and in potential matches. We assume that job destruction shocks and outsideoffers cannot occur simultaneously so that the probability that neither occurs is 1 − λ1 − δ. Thenwage contracts are negotiated and signed, wages are paid, and production takes place.

2.3. Value Functions and Wages

2.3.1. Unemployment value V0(p). In any given period, an unemployed worker receivesa flow income of b · p. In the following period, that same worker can either fail to meet a firm(an event of probability 1 − λ0), in which case s/he stays unemployed and gets a continuationvalue of V0(p), or s/he can meet a firm and be hired (probability λ0). In this latter case, ourassumption that firms are able to extract the entire surplus from newly formed matches impliesthat the worker’s continuation value from finding a job is again equal to V0(p). Hence, given theworker’s discount factor of β, the value of unemployment is simply defined by

V0(p) = b · p + β · V0(p) ⇔ V0(p) = b · p1 − β

.(5)

2.3.2. Total match surplus S(ε, p). As we saw earlier (footnote 9), in any given period,the flow surplus from a match between a firm and a type-p worker with current match-specificparameter ε is p · (ε − b). It does not depend on any wage value.

If the match is dissolved in the following period, then the worker becomes unemployed andreceives a continuation value of V0(p), whereas the employer is left with the option of openinga vacant job slot, which is worth zero. Hence the continuation surplus of the firm–worker pairis zero in this case.

If the match is not dissolved, given the new value of the transitory component of productiv-ity ε′, the continuation surplus associated with the incumbent match is S(ε′, p). However, withprobability λ1, the worker meets a potential alternative employer with match quality η′ (drawnfrom M(·)) and associated potential match surplus S(η′, p). As described above, two configura-tions can arise. Either ε′ > η′, in which case the worker stays with the incumbent firm (possiblyunder a renegotiated contract) with a continuation surplus equal to S(ε′, p), or η′ > ε′ and theworker joins the poaching firm. In this latter case, the incumbent firm is left with a value of 0whereas Bertrand competition between the two employers implies that the worker extracts allthe surplus from the incumbent match, i.e., S(ε′, p). All this implies that the sum of the worker’scontinuation value and the incumbent firm’s continuation profits is equal to S(ε′, p), whetheran outside offer was raised by the worker or not.

Summing up, given a common discount factor of β for the worker and the employer, totalmatch surplus S(ε, p) is defined recursively by

S(ε, p) = p · (ε − b) + β(1 − δ) ·∫ εmax

εmin

S(ε′, p)dM(ε′),(6)

where ε = EM(ε). S(·) is therefore proportional to p and independent of any wage value.Before going any further, it is worth looking at the negotiation baseline r0 that workers start

with when they leave unemployment. By its definition (4), r0 satisfies S(r0, p) = 0 and thus

r0 = b − β(1 − δ)1 − β(1 − δ)

(ε − b).(7)

608 POSTEL-VINAY AND TURON

Note that r0 < b provided that ε > b, which is a necessary condition for any trade at all to takeplace in the labor market.

2.3.3. Worker values V(φ(r, p), p) and wages φ(r, p). The current-period flow earnings ofan employed worker are simply her/his current wage φ(r, p). When the next period begins, withprobability δ the match is hit by a dissolution shock. In this case the worker becomes unemployedand therefore receives a continuation value of V0(p). With probability 1 − δ, the worker staysemployed and her/his continuation value depends on the new value of her/his wage and henceon the new value of her/his negotiation baseline r ′. Formally

V(φ(r, p), p)

= φ(r, p) + β · [δV0(p) + (1 − δ)E(V(φ(r ′, p), p) | r, continuing employment)],

(8)

The definition (3) also states that V(φ(r, p), p) = V0(p) + S(r, p), which implies

φ(r, p) = V0(p) + S(r, p) − β · [V0(p) + (1 − δ)E(S(r ′, p) | r, continuing employment)]

= p · [r − β(1 − δ)(E(r ′ | r, continuing employment) − ε)]

def.= p · ϕ(r),

(9)

where the second equality uses (5), (6), and (7) to substitute for V0(·), S(·), and r0, respectively.The term E(r ′ | r, continuing employment) appearing in (9) hinges on the period-to-period evo-lution of r, which will be analyzed in the next section. However, what is most important about(9) is that it establishes that all wages are multiplicatively separable in the worker’s type p and afunction of the negotiation baseline, ϕ(r). A closed-form expression of ϕ(·) can be derived (seebelow footnote 25); however, the only property of ϕ(·) that really matters for what follows isthat it is a strictly increasing function.

3. MODEL SOLUTION AND ECONOMETRIC INFERENCE

Our aim is to estimate the model parameters with a standard and readily available panel ofindividual data on income and labor force transitions (in this application a subsample of theBHPS). In this section, we thus derive some of the model’s implications that are potentiallyuseful for econometric inference.

3.1. Worker Turnover. Our model’s very simple structure makes it particularly easy toestimate the set of transition parameters in a first stage, independently of the rest of the model.The key property that facilitates this decomposition is that the only dimension of heterogeneitythat impacts worker turnover is ε, the transitory productivity shock, which is i.i.d. across periodsand matches, and as such can very easily be integrated out of the likelihood of labor marketspell durations.17 First looking at job-to-job mobility, we have Pr{job-to-job move | ε} = λ1 M(ε),implying that the unconditional probability of a job-to-job mobility equals λ1

2 .18 Next turning

17 Although analytically convenient for our estimation purposes, we should acknowledge that this property has sometheoretically unappealing consequences. One is that the hazard rates of job destruction, job finding, and job-to-jobmove are constant with respect to spell duration. This counterfactual prediction could be improved upon by introducingsome worker heterogeneity in these transition probabilities. Another counterfactual prediction of our simple modelis that the instantaneous job separation rate is independent of the current wage paid in the job. Again, heterogeneityin the λ1’s (in a form that would be correlated with productive heterogeneity in the p’s) would potentially correctthis unfortunate disconnect. Another possibility would be to allow for persistent productivity shocks. We discuss thesepossible extensions in Section 7.

18 Throughout this article, a bar over a cdf will be used to denote the survivor function.

ON-THE-JOB SEARCH AND EARNINGS 609

to transitions in and out of employment, the probability of observing a worker moving fromemployment into unemployment is δ, independently of the worker’s type p or the particularvalue of ε in the worker’s initial match. In the opposite direction, the unemployment exit rateis λ0 for all workers. Incidentally, this implies a steady-state unemployment rate of u = δ

δ + λ0,

obtained from the flow-balance condition ensuring the constancy of the unemployment rate:λ0u = δ(1 − u). This latter condition will be used at various points below.

Most importantly, we see that all the transition probabilities can be retrieved by maximizationof the likelihood of observed job and unemployment spell durations.

3.2. Wage Distributions. The model laid out in Section 2 implies that all wages have amultiplicative form φ(r, p) = p · ϕ(r). It thus predicts that log-wages are additively separableinto a worker fixed effect ln p and a transitory/persistent match-specific component v = ln ϕ(r).Idiosyncratic shocks to the negotiation baseline r therefore only impact wages through themonotonically increasing transformation r �→ v = ln ϕ(r). So knowledge of the sampling andpopulation distributions of v, denoted F(·) and G(·), respectively, is sufficient to characterizeor simulate individual labor market trajectories. Indeed it will prove more convenient to workwith the transformed negotiation baseline and match productivity shock—ln ϕ(r) and ln ϕ(ε)—than with the underlying r and ε. Specifically, as the sampling distribution of ε is M(ε), thecorresponding distribution for v = ln ϕ(ε) is simply F(v) = M[ϕ−1(ev)].

The dynamics of v = ln ϕ(r) will be characterized in the next subsection. Focusing on steady-state cross-sectional distributions for now, we first notice that (thanks essentially to the propor-tionality of all income flows to p) the transitory component of wages v is independent of p in across-section of employed workers. Conditional on p, log-wages are thus distributed as v, andwe now seek to determine the steady-state population distribution of v, G(v). Let us considerflows in and out of the stock (1 − u)G(v) of employed workers with a (transformed) negotiationbaseline less than v = ln ϕ(r). Workers exit this pool either if their match has been dissolved(probability δ) or if their new negotiation baseline is greater than r (probability λ1 F(v)2).19 Twoflows of workers enter this pool: λ0u previously unemployed workers and employed workerswith a previous negotiation baseline greater than r with probability (1 − δ)F(v).20

We can thus now write the balance of flows in and out of the stock (1 − u)G(v):

(1 − u)G(v)(δ + λ1 F(v)2) = (1 − δ)F(v) · (1 − u)G(v) + λ0 · u

⇐⇒ G(v) = δ + (1 − δ)F(v)δ + (1 − δ)F(v) + λ1 F(v)2

(10)

Note the existence of a mass at v0 = ln ϕ(r0), G(v0) = δδ+λ1

due to the unemployed workers allbeing hired at the minimum negotiation baseline v0. More precisely, we can decompose G(·) as

G(v) = δ

δ + λ1· 1{v≥v0} +

[δ + (1 − δ)F(v)

δ + (1 − δ)F(v) + λ1 F(v)2− δ

δ + λ1

]· 1{v≥vmin},(11)

where vmin = ln ϕ(εmin) is the lower support of F(·). Also, as expected, G(·) is identically equalto 1 in the absence of on-the-job search (i.e., if λ1 = 0).21

19 As observing an increase in the negotiation baseline requires the worker to receive an outside offer, and productivitydraws that are above her/his current negotiation baseline at both the poaching and the incumbent firm.

20 As, conditional on remaining employed, workers will have a new negotiation baseline lower than r if their current-period draw of the idiosyncratic shock falls short of r, whether they raise an outside offer or not.

21 Incidentally we may emphasize that, contrary to most job search models that have a “job-ladder” design andno idiosyncratic productivity shocks, our model features a nondegenerate equilibrium wage distribution even if oneassumes away any risk of unemployment (i.e., if δ = 0). Absent productivity shocks, continuously employed workerswould gradually climb up the wage ladder as they receive outside job offers (at a speed that depends on the partic-ular assumptions on wage determination. See Burdett and Mortensen, 1998, for the canonical wage-posting model,

610 POSTEL-VINAY AND TURON

3.3. Wage Dynamics. Wage dynamics are driven by the combination of two distinct forces:job offers and idiosyncratic shocks to match productivity. Given knowledge of the process gov-erning the arrival of job offers (i.e., given knowledge of the arrival rate λ1 of job offers and jobdestruction shocks δ, which we saw are identified from job spell durations and job transitions),observed individual wage dynamics thus convey information about the distribution of matchproductivity shocks.

3.3.1. Dynamics over one period. At any period t, an employed worker earns a wage φt

such that ln φt = ln p + vt , and we are left to analyze the dynamics of vt = ln ϕ(rt ), the worker’scurrent (transformed) negotiation baseline rt .

When period t + 1 begins, with probability δ the match is hit by a dissolution shock. In thiscase the worker becomes unemployed and her/his income flow becomes equal to p · b. Withprobability 1 − δ, the worker stays employed and her/his continuation wage depends on thenew value of her/his negotiation baseline r ′. We thus now examine the value of r ′ in the variouspossible cases.

If the worker fails to find any outside job opportunity (probability (1 − δ − λ1)), the onlysource of randomness is the realization of εt+1. Our assumptions concerning the wage settingrules then imply the following. If εmin ≤ εt+1 < rt , then the match is maintained but under arenegotiated contract, and the new negotiation baseline is εt+1. Otherwise, if εt+1 ≥ rt , then noneof the parties can force the other to renegotiate, and the match goes on under an unchangedcontract, leaving both negotiation baseline and wage unchanged.

Next consider the situation in which the match continues and the worker manages to contacta poacher (probability λ1). The idiosyncratic productivity component ηt+1 of a potential matchwith the poacher is drawn at random from F(·). We scan over all possible values of the shocksεt+1 and ηt+1 and see what happens in each case.

First, if εmin ≤ εt+1 < rt then the worker has a choice between playing off the two firms againsteach other or simply discarding the poacher’s offer. The former option will yield the worker anew negotiation baseline of min{εt+1, ηt+1} and the latter a negotiation baseline of εt+1. Onethus sees that the worker’s optimal choice yields a continuation value of the negotiation baselineof εt+1.22

Second, if εt+1 ≥ rt , then playing off the two employers against each other againyields a new negotiation baseline of min{εt+1, ηt+1}, whereas ignoring the poacher’s offeramounts to continuing a relationship with the incumbent employer under unchanged terms,thus keeping a negotiation baseline at rt and consequently an unchanged wage. It fol-lows that the worker’s optimal choice yields a continuation negotiation baseline equal tomax〈rt , min{ηt+1, εt+1}〉.23

Summarizing the above, the conditional distribution of the continuing (log) negotiation base-line vt+1 | vt is as follows:

Postel-Vinay and Robin, 2002a, for an offer-matching model closer to the one of this article, Burdett and Coles, 2003, fora model of explicit wage-tenure contracts, and finally Mortensen, 2003, for an overview). Job loss then acts as a “resetbutton” for this process of wage progression, as workers who have experienced a spell of unemployment essentiallyhave to start over at the bottom of the wage ladder (e.g., in our model, they start over with a negotiation baseline ofv0). In the standard model, without this reset button, all workers would end up at the top of the wage ladder and onlyone wage would be observed in the long-run equilibrium of a hom*ogeneous labor market. Here we see that, by causingoccasional wage cuts, productivity shocks prevent this from happening.

22 The optimal choice is to let the firms compete whenever ηt+1 > εt+1. The outcome of the Bertrand game thustriggered is that the worker joins the poaching firm with a negotiation baseline of εt+1.

23 The optimal choice is to let the firms compete whenever ηt+1 > rt . The outcome of the Bertrand game is thenthat the worker joins the poaching firm if ηt+1 > εt+1, and stays with her/his incumbent employer, with a wage raise, ifεt+1 ≥ ηt+1 > rt .

ON-THE-JOB SEARCH AND EARNINGS 611

vt+1 | vt =

vt with probability (1 − δ)F(vt ) − λ1 F(vt )2,

v′ < vt with density (1 − δ) f (v′),

v′ > vt with density 2λ1 f (v′)F(v′),

(12)

whereas with probability δ the worker becomes unemployed and vt+1 is irrelevant.Conditional on individual fixed-effects p, we thus predict that wages follow a first-order,

nonlinear Markovian process based on a specific acceptance/rejection scheme of i.i.d. wage in-novations.24 We also predict that the rates of transition between labor market states (δ and λ1)are key determinants of the individual earnings process. This strong prediction of our structuralmodel highlights the interplay between job mobility and income dynamics: Job mobility reflectsthe intensity of labor market competition between employers (as measured by the frequencyat which employed workers raise outside job offers), which in turn conditions the observed(dynamic) behavior of wages. If validated empirically, that prediction may help with the inter-pretation of observed wage dynamics.

The empirical properties of the process in (12)—and its differences with the conventionallinear ARMA specification—will be analyzed in Section 5.2. For the time being, we derive thefollowing moment that will be useful for estimation. Integration of (12) implies that, conditionalon employment at two consecutive dates t and t + 1,

E(vt+1 | vt , employment at t, t + 1) = vt + λ1

1 − δ

∫ vmax

F(x)2dx −∫ vt

vmin

F(x)dx.

This expression shows that conditional expected wage growth—i.e., E(vt+1 − vt | vt , employmentat t, t + 1)—is the sum of a positive term reflecting the impact of outside job offers causing wageincreases and a negative term coming from adverse productivity shocks causing downward wagerenegotiation. As intuition suggests, the former dominates among workers with a relatively lowcurrent negotiation baseline vt (which translates into a relatively low wage conditional on theirtype p), whereas the latter dominates for workers with a high current negotiation baseline (whichhas little chance to be exceeded by the minimum of a pair of random draws from F(·)).25,26

3.3.2. Dynamics over s periods. We now consider the cross-sectional distribution of vt+s

conditional on employment at dates t, . . . , t + s, i.e., the distribution of negotiation base-lines conditional on at least s periods of continuous employment. Designating the cdf of thisdistribution by Gs(·), we show the following in the Appendix:

24 Incidentally, our model is not the only one suggesting that this type of acceptance/rejection scheme is the rightway to think about wage dynamics. The process in (12) is indeed formally reminiscent of predictions obtained by Harrisand Holmstrom (1982) and Thomas and Worral (1988) in models of self-enforcing wage contracts. See the discussion infootnote 13.

25 Note that, beyond means, higher-order moments of the conditional distribution of vt+1 | vt are functions of vt . Thisleaves scope for ARCH-type effects, as were detected in U.S. data by Meghir and Pistaferri (2004).

26 Incidentally, Equation (13) can be rewritten in terms of the initial negotiation baseline r = ϕ−1(ev) as follows:

E(r ′ | r, continuing employment) = ε −∫ εmax

(M(r ′) − λ1

1 − δM(r ′)2

)dr ′.

Going back to (9), this leads to the following closed-form expression of the function ϕ(r) and any wage φ(r, p):

φ(r, p) = p · ϕ(r) = p ·(

r + β(1 − δ)∫ εmax

(M(r ′) − λ1

1 − δM(r ′)2

)dr ′

612 POSTEL-VINAY AND TURON

Gs(v) =[

1 −(

1 − F(v) − λ1

1 − δF(v)2

)s]· G∞(v) +

(1 − F(v) − λ1

1 − δF(v)2

· G(v),

(13)

where

G∞(v) = F(v)

F(v) + λ1

1 − δF(v)2

Hence, as one conditions on more periods of continuous employment, the cross-sectional dis-tribution of negotiation baselines gradually shifts from G(v) to G∞(v). An interesting propertyof this shift (see the Appendix) is that it features a monotonically increasing mean, i.e., EGs (v)increases with s. Hence, from a cross-section perspective, our model predicts positive returnsto continuous employment in that the mean negotiation baseline increases with the durationof continuous employment. Intuitively, a worker’s wage increases in expected terms with thenumber of job offers received by that worker since s/he was last unemployed. As one looks atindividuals that have been continuously employed for longer periods, the average number ofjob offers received by these individuals since they got out of unemployment (and hence theaverage wage or negotiation baseline) gradually increases. This selection effect is the drivingforce behind the increase in the mean wage with s.

Based on the definition (13), we can then compute any set of model-predicted moments touse in the estimation. In practice, as we discuss in the next subsection, we shall use all first- andsecond-order moments of Gs(·).

The earnings autocovariance structure also conveys potentially useful information about earn-ings dynamics. We establish in the Appendix that27

Cov(ln φt , ln φt+s) = VarH(ln p) − CovG

(vt ,

∫ vmax

(1 − F(x) − λ1

1 − δF(x)2

dx)

.(14)

(Subscripts indicate the distribution with respect to which expectations are taken.) Again theseautocovariances are the sum of a constant term (the population variance of the fixed-effect ln p),and a term that decreases toward to zero as s goes to infinity. This reflects the limited persistenceof wage shocks in our model: The memory of the initial negotiation baseline vt gradually fadesout as workers are hit by productivity shocks and/or outside offers causing renegotiation.

A panel length of T (i.e., T different dates at which we observe a cross-section of individualwages) thus provides us with 3T − 1 moment conditions (T means and T variances from (13)and T − 1 covariances from (14)) on which to base an estimation of the F(·) distribution andthe variances of the measurement error and fixed-effect distributions.

4. DATA AND ESTIMATION PROCEDURE

4.1. Structure of the Analysis Sample. We use a subsample of the British Household PanelSurvey. The BHPS is a 13-wave (1991 to 2003) panel of household data, of which we use waves 2–13, thus following individuals for up to 12 years.28 The BHPS provides information on individuallabor market spell histories and precise spell durations (down to the month or the day whennot missing), together with records of individual earnings and working hours every 12 months.There is some attrition from and entry into the panel, both of which we assume exogenous.

27 We continue to work conditionally on continuous employment between dates t and t + s. However, to avoid anotational overload, we now keep this conditioning implicit.

28 We discard wave 1 (1991) because of substantial coding differences between this and the subsequent waves.

ON-THE-JOB SEARCH AND EARNINGS 613

TABLE 1SAMPLE DESCRIPTIVE STATISTICS

Job Spell Sample: First Spell

Mean Duration % Job-Job % Job-Un.Initial State N. Obs. (Months) % Censored Transitions Transitions

Employed 659 61.6 42.4 33.8 23.8Unemployed 58 15.2 25.9 — —

Job Spell Sample: Numbers of Transitions

0 1 2 ≥3

Percent of sample 40.8 23.9 15.0 20.3

Income sample (1992 cross-section)

N. Obs. Mean Std. Dev.

677 1.90 0.51(£7.70 per hour) (£4.96 per hour)

Our working sample is obtained from the following selection of the raw data. First, we dropthe few individuals that have gaps in their records. We then use data on males and females withmore than 5 years’ potential experience and aged less than 60, thus cutting 5 years at both endsof the individuals’ working lives. We restrict our analysis to individuals with A-level education ormore, both for the sake of brevity and also because the individual-level wage-bargaining/offer-matching process described in the theoretical model is arguably more relevant in high-skilllabor markets. For similar reasons, we do not consider individuals observed as self-employed oremployed in the public sector in their initial year in the survey.29

Based on these selection rules, we then construct two separate samples. The first one drawsfrom records of individual labor market spell histories and will serve for the estimation oftransition parameters: We take all selected individuals at their first interview date, follow themthroughout the 12 waves, and record all their labor market spell durations and transition types(job-to-job or job-to-unemployment).

Our second sample is an income sample gathering the yearly observations of wages andworking hours: We compute hourly wages using data on (before-tax) labor income receivedin the last month and on worked hours. We then regress these wages on indicators of year,education, gender, ethnic background, and labor market cohort. We use the residuals from thislatter regression as our measure of individual earnings. We finally trim the data by droppingthe top and bottom 2.5 percent of earnings (residuals). This trimming is useful to stabilize ourempirical estimates of cross-sectional wage variances.

Our job spell sample comprises 659 initially employed individuals and 58 initially unemployedindividuals, whereas our income sample comprises 599 individuals with a valid initial wageobservation. Table 1 gives more detailed descriptive statistics for the two samples.

4.2. Estimation Procedure. Following the above developments, we carry out a two-stepestimation procedure. In the first step we use the data on labor market spells to estimatethe transition rates δ and λ1 using maximum likelihood on observed job spell durationsand job transitions. In the second step, we use our income data to estimate the remainingparameters—i.e., the sampling distribution F(·) of productivity shocks and the distribution ofperson fixed-effects H(·)—by matching a series of wage means and covariances and a series ofthird moments of the distribution of wage changes, as derived in Subsection 3.3.30 Specifically,

29 Ideally, we would have liked to work on a more hom*ogeneous set of workers. Yet, as we shall see below, given theresulting sample size, this is probably the finest stratification of the original BHPS data that we can reasonably envisage.

30 Note that we shall only estimate F(·)—the sampling distribution of transformed match quality shocks v = ln ϕ(ε)—not the underlying sampling distribution of match quality shocks M(·). As we already mentioned, knowledge of F(·) is

614 POSTEL-VINAY AND TURON

we match the following 3T − 1 moments (where T is the panel length in years), for s =0, 12, 24, . . . , 12(T − 1):31

E(ln φt+s) = EGs (v)

Var(ln φt+s) = VarH(ln p) + VarGs (v) + σ 2me

Cov(ln φt , ln φt+s) = VarH(ln p) − CovG

(vt ,

∫ vmax

(1 − F(x) − λ1

1 − δF(x)2

dx)

(15)

All these moments are conditional on continuous employment between times t and t + s—thatis, we will match those theoretical predictions of E(ln φt+s), Var(ln φt+s), and Cov(ln φt , ln φt+s)to the corresponding empirical moments computed on the subsample of individuals observed tobe continuously employed between t and t + s.32 Also, we set our period length to be one month,hence the series of leads being taken at multiples of 12 periods to match our yearly wage data.Finally note in the second line of (15) the addition of a term σ 2

me to the theoretical expression ofcross-sectional income variances. This accounts for the presence of classical measurement error(with variance σ 2

me) in hourly wages.We match these moments using Equally Weighted Minimum Distance (EWMD) estimation.33

For convenience, we choose to parameterize the population distribution of productivityshocks, G(·) instead of the sampling distribution F(·). The latter can then easily be retrievedfrom G(·) using Equation (10). As detailed in Equation (11) in the theoretical section, G(·) isthe sum of a mass point at v0 corresponding to entry wages for previously unemployed workerswho all start off their employment spell with a negotiation baseline of v0, and a transformationof the sampling distribution F(·) with lower support vmin. We thus parameterize G(·) as the sumof a mass point (at v0) and a normal distribution, truncated below at vmin. Note that becausematch surpluses are nonnegative for any realization of the productivity shock, it has to be thecase that vmin ≥ v0. In fact we assume vmin = v0. This assumption seems natural in that it meansthat jobs and job offers exist for values of the productivity shock down to a value leading to amatch surplus of zero (see the discussion in footnote 11).

As to the distribution of worker fixed effects H(·), we see that only its variance, VarH(ln p)appears in the series of moments we aim to match. The distribution H(·) can, however, beretrieved from our estimate of G(·) and the actual wage distribution by deconvolution. Yetknowledge of VarH(ln p) will be sufficient for all of our simulation exercises.

5. RESULTS

5.1. Parameter Estimates

5.1.1. Transition rates. The estimated arrival rates of outside offers and job destructionshocks are reported in Table 2, in monthly values. The probability of receiving an outside offer

sufficient to simulate wages from our model. This also explains why we will not need an estimate of the discount factor β.Indeed the wage process derived from our theory—see (12)—is independent of β, which only affects the function ϕ(·),i.e., the way in which match quality shocks translate into wage shocks. As our estimation procedure is based on wagedata only (and on the assumption that productivity shocks are i.i.d. when they occur), knowledge of β is unnecessary forthe identification of the parameters of our wage process. It would only matter if we wanted to recover the distributionof match quality shocks implied by our model and by the earnings dynamics observed in our data.

31 The first T moments (mean wages) use the normalization E(ln p) = 0.32 This conditioning leads us to discard the information brought by observations for individuals who experienced a

complete unemployment spell between t and t + s, and still have a wage record at both dates. It is possible to write downthe means and autocovariances in (15) conditional on employment at t and t + s only. However, the correspondingformulas are cumbersome and unemployment is a sufficiently rare event in our sample to make the loss of informationentailed in the more stringent conditioning inconsequential.

33 Optimal minimum distance (OMD) estimation results are available upon request. They are qualitatively verysimilar to those reported here. As suggested by a referee, we report EWMD results in the main text, as OMD estimatesare fraught with a well-known small-sample bias problem (Altonji and Segal, 1996).

ON-THE-JOB SEARCH AND EARNINGS 615

TABLE 2TRANSITIONS AND MEAN SPELL DURATIONS (MONTHS)

δ(×100) 1/δ λ1(×100) 1λ1

λ0(×100) 1/λ0

0.323(0.021)

309.7(20.22)

1.481(0.055)

67.52(2.484)

4.108(0.716)

24.34(4.241)

TABLE 3DISTRIBUTIONAL PARAMETERS

Parameters of G(.) Test of OI Restrictions:v0 = vmin µ σ Var(ln p) σ 2

me Test Statistic(df:p-value)

1.801(0.078)

2.219(0.122)

0.200(0.099)

0.063(0.008)

0.016(0.015)

22.23(30: 0.784) Vs. σ 2

me �= 0:

1.806(0.074)

2.188(0.176)

0.279(0.068)

0.062(0.007)

0.000(const.)

24.35(31: 0.725)

5.18(1: 0.023)

NOTE: G(v) is parameterized as the sum of a mass of δδ+λ1

at v0 and a normal distribution ofmean µ and standard deviation σ truncated at vmin. Bootstrap standard errors obtained with500 joint estimations of the transition parameters and the distribution parameters.

is about 1.5 percent per month. Because the probability of job switching is equal to λ1/2 (seeSubsection 3.1), this number says that about 1 in 130 employed workers from our high-skillsample switches jobs each month. With a job destruction rate of just under a third of one percent(implying an average waiting time of about 25 years between two spells of unemployment), theaverage duration of a job spell, (δ + λ1/2)−1, is in the order of 7.9 years.

Finally, turning to the unemployment exit rate λ0, we estimate it at 4.11 percent monthly,implying a mean unemployment duration of just over 2 years. Combined with the estimatedvalue of δ, it also implies an unemployment rate of 7.2 percent. Considering that we have asample of highly educated workers, this may sound like a long duration and a high rate. Thesenumbers, however, accommodate the unemployment spell durations observed in our BHPSsample.34

5.1.2. Distributions. Table 3 contains the EWMD estimates and standard errors of thevarious distributional parameters involved in our moment conditions (15).35

The first thing we notice from the first row of Table 3 is the relatively poor precision of ourestimate of σ 2

me, the variance of the measurement error. Equality to zero is only borderlinerejected (a Wald test produces a p-value of 0.023 against the alternative σ 2

me = 0.016, our pointestimate), although the point estimate of 0.015—or 10.8 percent of the cross-sectional wagevariance—is of reasonable magnitude. For comparison, in the bottom row of Table 3, we alsoreport parameter estimates obtained subject to the constraint σ 2

me = 0. Comparison of the tworows shows that the parameters affected by this constraint is σ , which is estimated higher in theabsence of measurement error. Removing the possibility of a measurement error leads us toestimate a variance of the negotiation baseline v to account for 56 percent of the wage variance,as opposed to 44 percent in the base estimation.

34 Unemployment notoriously appears to be highly persistent in the BHPS data (Stewart, 2007). Also, the meanunemployment duration implied by our estimated value of λ0 may seem at odds with the mean duration of unemploymentspells reported in Table 1. We should bear in mind that our estimated λ0 aims to fit both spell durations and the initial(un)employment rate observed in our sample.

35 The reported standard errors are based on 500 bootstrap replications of both stages of our estimation procedureon 500 resamples with replacement. As such they do account for the presence of nuisance parameters δ and λ1, whichappear in the theoretical moments and which we fix to their first-stage estimated value in our second estimation step.

616 POSTEL-VINAY AND TURON

The minimum negotiation baseline, v0 = vmin that workers start off with when first hiredfrom unemployment is estimated at 1.80, to be compared with a value of the mean negotiationbaseline (which equals the mean log wage residual under our normalization EH(ln p) = 0) overall employed workers of 2.18. Recalling that G(v) is parameterized as the sum of a mass of

δδ+λ1

at v0 and a normal distribution of mean µ and variance σ 2 truncated at vmin, our estimatedtransition rates δ and λ1 put the fraction of employed workers with a minimum negotiationbaseline at 17.9 percent.36

The variance of the worker fixed-effect distribution, VarH(ln p), is 0.063. Taking the varianceof the measurement error to equal 0.015, this suggests that the relative contributions of the per-manent component of earnings ln p, the transitory component of earnings vt , and measurementerror to the total cross-sectional earnings variance (which equals 0.145) are 45 percent for ln p,about 44 percent for vt , and 11 percent for the measurement error.

Finally, as we use 12 years of data, (15) defines a set of 3T − 1 = 35 moments to match inthe estimation for only 5 parameters to estimate. Table 3 reports the test statistic and p-valueof a Wald test of overidentifying restrictions. With a p-value of 0.78, the model passes thisspecification test.

5.2. Simulations and Fit Analysis

5.2.1. Matched moments. The first three panels of Figure 2 displays the moments listed in(15) that we match in the estimation, as they are predicted by our model with the estimatedparameters and as they are observed in our sample. In all panels the dotted lines materialize 95percent confidence bands around the empirical moments.

The top left panel shows the progression of the mean log-wage as one conditions on zeroto 12T (=132) months of continuous employment. In the data, this mean log-wage increasesfrom 2.17 for the whole employed population to 2.24 for the subset of workers with a durationof continuous employment of 11 years or more.37 This wage growth is well replicated by ourmodel.

The top right panel shows the evolution of the variance of the distribution of log-earningsconditional on zero to 12T months of continuous employment. We showed above that theunderlying conditional distribution, Gs(·), gradually shifts from the steady-state populationdistribution G(·) when we condition on a minimum length of continuous employment of zeroperiods to G∞(·) when s is increased indefinitely. The predicted variance of Gs(·) clearly declineswith the minimum duration of uninterrupted employment, s. In spite of the poor precision ofthe empirical counterparts of these conditional wage variances, the figure is also suggestive of adecline of the latter with s, at least up to about s = 96 months (8 years). Although staying wellwithin the confidence bands, the model still seems to have a tendency to over-predict this declineat long lags. Yet at this level of precision, we may say that the empirical pattern is consistentwith the model.

The bottom left panel graphs wage autocovariances at 0 to 12T lags (again conditional on con-tinuous employment over the number of lags considered). As expected, these autocovariancesdecline as one looks at longer lags, in a way that the model captures well. Note in particular theconvergence of Cov(ln φt , ln φt+s) toward Var(ln p) as the number of lags increases.

5.2.2. Wage changes. The bottom right panel graphs the third moment of the distributionof wage changes over an increasing horizon, i.e., (ln φt+s − ln φt )3, with s = 12, 24, . . . , 12T. Ourknowledge of the distribution Gs (see Equation (13)) indeed allows us to predict any momentof this distribution. We show in the Appendix the detail of the derivation of the third moment

36 The model further has it that v0 should equal the mean log wage among workers who were hired out of unem-ployment within the last month. Unfortunately we can only obtain a very imprecise direct estimate of this mean, as thenumber of unemployment exits occurring within a month of an interview date—i.e., a date at which we have a wageobservation—in our sample is very small (27 observations in total). Interestingly, however, this direct estimate is 1.87(std. err. 0.11), which is very close to the estimate of v0 shown in Table 3.

37 These individuals account for 37.9 percent of our sample.

ON-THE-JOB SEARCH AND EARNINGS 617

0 2 4 6 8 102.14

2.16

2.18

2.2

2.22

2.24

2.26

2.28

2.3

2.32 Mean wage progression

Number of lags (years = s/12)

E[ln

φ t +s |

empl

. at t

, ...

,t+s]

PredictedObserved95% confidence bands

0 2 4 6 8 100.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16 Evolution of wage variance

Number of lags (years = s/12)

Var

[lnφ t +

s | em

pl. a

t t, .

.. ,t +

0 2 4 6 8 100.04

0.06

0.08

0.1

0.12

0.14

0.16 Wage autocovariances

Number of lags (years = s/12)

Cov

[lnφ t,ln

φ t+s |

empl

. at t

, ...

, t+s]

1 2 3 4 5 6 7 8 9 10−0.02

−0.01

0.01

0.02

0.03

0.04

0.05

0.06 Third moment of wage changes

E[(

lnφ t+

s−ln

φ t)3 | em

pl. a

t t, .

.. , t+

Number of lags (years = s/12)

FIGURE 2

FIRST-, SECOND-, AND THIRD-ORDER MOMENTS OF THE WAGE PROCESS

of wage changes. The graph shows that, although not matched in the estimation, this series ofmoments is well predicted by our model.

That having been said, our structural model permits a much finer analysis of predicted wagechanges. Using the parameter estimates obtained above, we can now create a sample of simulateddata that can then be compared to the real data along any dimension we like. This will allow usto assess the model’s capacity to replicate features of the data that were not directly matchedin the estimation. In the reported simulations we alternatively use the values of σ 2

me = 0 andσ 2

me = 0.015 for the variance of the measurement error.An intuitive way of looking at earnings dynamics beyond first- and second-order moments

(which, as we just saw, the model replicates well enough) is to consider the distribution of wagechanges from one year to another. In the language of our theory, this is the distribution ofvt+s − vt , with s some chosen horizon.

Figure 3 plots three such distributions of wage changes over 1 year (s = 12 months, left panel).The solid line is the empirical cdf of log wage changes. The dashed line is the model-predictedcounterpart of that cdf σ 2

me = 0.015. Finally, the dash-dot line with a mass at zero is the cdf ofsimulated wage changes if one removes measurement error from the model.

First comparing the “observed” and “predicted, with measurement error” distributions ofyearly wage changes, we observe a near perfect replication of the top half of the distribution (i.e.,the part of the distribution corresponding to wage increases) and a slightly larger discrepancyin the bottom half, with the observed distribution tending to dominate the simulated one at thefirst order. Although the overall look of the graph is sensitive to the particular calibration of themeasurement error, the robust conclusion is that the model tends to slightly over-predict wagecuts.

618 POSTEL-VINAY AND TURON

−1 −0.5 0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Wage changes over one year

Yearly log wage changes (vt+12

−vt)

ObservedPredicted, with m.e.Predicted, no m.e.

FIGURE 3

DISTRIBUTION OF YEAR-TO-YEAR WAGE CHANGES

The model also predicts that many of the observed wage changes are in fact artificial and onlyreflect measurement error: Once measurement error is removed from the model, the predicteddistribution of year-to-year wage changes has a mass of about 60 percent at zero. Interestingly,the distribution without measurement error also seems more skewed than the observed one,as the model-predicted share of “genuine” wage increases is roughly 14 percent, whereas thecorresponding figure for wage cuts is about a 27 percent.

5.2.3. Conditioning on job-to-job mobility. The model has very strong implications aboutearnings dynamics around a job-to-job transition. As mentioned above, the date-t negotiationbaseline of a worker changing jobs at date t is equal to the date-t idiosyncratic match productivityshock at the incumbent firm, which has to be less than productivity at the poaching firm if theworker has switched from the former to the latter. In other words, conditional on observing a job-to-job transition, the negotiation baseline is the minimum of two independent draws from F(·),hence a draw from 1 − F(·)2. Most importantly, this new negotiation baseline vt is independentof any previous negotiation baseline vt−s .

Are these predictions borne out by the data? A partial answer can be sought in the followingadditional moment conditions, which are implied by the above considerations:38

E(ln φt | job-to-job transition at t) = E1−F2 (v)

Var(ln φt | job-to-job transition at t) = VarH(ln p) + Var1−F2 (v) + σ 2me

Cov(ln φt , ln φt+s | job-to-job transition between t + 1 and t + s) = VarH(ln p).

(16)

38 Of course these considerations imply much more than these moment restrictions. Indeed they even offer a potentialsource of nonparameteric identification of our model (up to the measurement error). Consider a worker experiencing ajob-to-job transition for whom we have two wage observations, one on each side of the transition. Let ln φb = ln p + vb

denote the wage observed before the transition and ln φa = ln p + va the wage after the transition. From the above weknow that va is independent of vb. In principle we can thus retrieve H(·) and G(·) by nonparameteric deconvolution—and hence F(·) as well, using (10). (This is an application of a general identification theorem by Kotlarski, 1967.)Unfortunately, implementation of this method requires a sufficient number of independent observations of a workerwith a job-to-job mobility occurring between two valid wage observations. We only have in the order of 55 suchobservations in our sample.

ON-THE-JOB SEARCH AND EARNINGS 619

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Number of lags (years = s/12)

Cov

[ln φ

t,ln φ

t+s |

empl

. t, .

.. ,t+

s ; j

−j b

etw

. t+

1 &

t+s]

PredictedObserved95% confidence bands

FIGURE 4

WAGE AUTOCOVARIANCES CONDITIONAL ON JOB-TO-JOB MOBILITY

Computation of the first two moments in (16) has to rely on independent wage observationsthat coincide with the occurrence of a job-to-job mobility, i.e., on the subset of workers movingfrom job to job in a month preceding an interview date. Because we only have one interview—therefore at best one wage observation—each year, and because job-to-job transitions are infre-quent events, we only have very few (indeed 25) such independent coincidences in our data set.39

This does not allow for a very precise estimation of the mean and variance of wages conditionalon mobility: The empirical mean is 2.25 (standard error of 0.09), and the empirical varianceequals 0.18 (standard error of 0.05). Yet, the corresponding model predictions are 2.62 and 0.11.Hence, although our prediction of the conditional variance is still acceptable (albeit on the lowside), the model seems to overstate the mean wage of job-to-job movers somewhat.

Computation of the series of conditional covariances in (16) can rely on a slightly wider set ofobservations, as they only involve pairs of wage observations that are anywhere on either sideof a job-to-job mobility (as opposed to wage observations that exactly coincide with a job-to-jobmobility). Exploiting this, Figure 4 depicts the empirical covariance of wages at dates t = 1992(the initial year) and t + s (for s = 0, 12, 24, . . . , 12(T − 1) months) among workers who haveexperienced at least one job-to-job transition between t + 1 and t + s. For comparison withthe model’s prediction, a horizontal line at VarH(ln p) is also drawn. Finally, at s = 0 the fig-ure reports the observed and predicted conditional variance Var(φt | job-to-job transition at t).Dotted lines represent confidence bands around the empirical moments.

Again given the scarce numbers of observations upon which we have to base our computa-tions,40 this figure only paints an indicative picture of the covariance profile of individual incomearound a job-to-job transition. Yet it still suggests that this profile is both lower and markedly“flatter” than the corresponding unconditional covariance profile plotted in the bottom left panelof Figure 2. Both properties are in accordance with and, indeed, quantitatively well captured bythe model.

39 The measurement of these “mobility” wages also runs into another problem, which is that the date at which a jobtransition exactly occurs is certainly measured with error. For instance, some workers are known to report as havingstarted a new job spell, when they have really only accepted an offer for a job that is effectively to start at some (near)future date. In these cases, it is unclear whether the reported wage pertains to the new or to the old job. Because it likelyadds some “pre-mobility” wages into our sample of mobility wages, this type of measurement error tends to bias ourempirical estimate of the conditional mean E(ln φt | job-to-job transition at t) downward and that of the correspondingvariance upward.

40 These numbers range from 21 to 60, depending on s.

620 POSTEL-VINAY AND TURON

A related property of the wage process generated by our model is also worth pointing out.It can be shown that the wage earned by an employed worker is independent of the numberof outside offers received by the worker since the beginning of his/her employment spell, pro-vided that this number is at least equal to one.41 This, together with the fact that the transitorycomponent vt+1 of the new wage obtained by a job mover is independent of the past transitorycomponent vt in the previous job, has the particular implication that wage gains for job-to-jobchangers are independent of the number of offers raised in the past, or of the number of pastjob changes. Barlevy (2008) finds empirical support for such independence using NLSY data ina different (search) context.

5.2.4. Linear ARMA model. As we mentioned in the introduction, the literature has along tradition of fitting ARMA-type models to individual wage trajectories. In this subsectionwe take another look at the covariance structure of (observed and simulated) wages under thisalternative angle.

The following describes a canonical ARMA specification found in the literature:

ln φi,t = zi + a Pi,t + aT

i,t ,

a Pi,t = a P

i,t−1 + ζi,t , with ζi,t i.i.d.,

aTi,t =

q∑�=0

θ� ξi,t−�, with ξi,t i.i.d. and θ0 = 1.

(17)

Here log-earnings are modeled as the sum of an individual fixed-effect zi , a permanent earningsshock a P

i,t following a martingale process, and a transitory earnings shock aTi,t following an MA(q)

process. The order q of the latter MA process is to be determined empirically, along with theparameters of this process (the θ�’s) and the innovation variances, σ 2

ζ and σ 2ξ .

We fit model (17) separately to our income sample from the BHPS and to a sample of sim-ulated data based on the parameter estimates obtained above.42 The order of the MA processis determined by looking at the sequence of autocovariances of first-differenced log wages,Cov(� ln φi,t , � ln φi,t+s), which should equal zero for any s ≥ q + 2. The top panel of Table 4reports these autocovariances at lags of up to 4 years for both the BHPS and simulated sample.We first notice that the pattern of wage autocovariances is very similar in the real and in thesimulated data samples, which was expected given the good fit to these covariances illustrated inthe bottom left panel of Figure 2. Second, the magnitude of the covariance point estimates drops10-fold in both samples between the first and the second lag and remains very small thereafter.Both patterns square in well with earnings following an MA(1) process in growth rates, thusimplying that earnings levels can be described along the lines of model (17) as the sum of arandom walk component and a serially uncorrelated—or MA(0)—component. However, forcompleteness we estimate model (17) under both the MA(0) and MA(1) specifications.43

Results for both samples and both specifications are displayed in the bottom panel of Table 4.Looking at the variances of innovations for the permanent and transitory components of theearnings process, σ 2

ζ and σ 2ξ , we observe again much similarity between those obtained with

41 A straightforward proof of this claim relies on flow-balance equations similar to (10). Details are available onrequest.

42 We make the simulated sample of equal size to the observed one. An important detail to keep in mind is that, forthis paragraph, the time unit is taken to be 1 year (as opposed to 1 month, as was the case thus far), in accordance withthe yearly frequency of wage observations in the BHPS data, so our simulated sample is made up of yearly observationstaken from a monthly simulated dataset. Finally, we ignore the presence of measurement error in this comparativeexercise. The measurement error variance cannot be identified within the specification (17)—see Meghir and Pistaferri(2004).

43 Results from the literature conclude to the presence of either an MA(0) or an MA(1) component in the earningsprocess, but not usually to higher order MA components. For the estimation, we proceed by EWMD matching of theautocovariance structure of yearly wage growth. Details are available on request.

ON-THE-JOB SEARCH AND EARNINGS 621

TABLE 4FITTING AN ARMA PROCESS

Cov(� ln φi,t , � ln φi,t+s), s = . . .

0 1 2 3 4

BHPS 0.032(0.0007)

−0.010(−0.001)

0.000(0.001)

−0.000(−0.001)

0.000(0.001)

Simulated 0.040(0.001)

−0.012(−0.001)

−0.001(−0.001)

−0.000(−0.001)

−0.001(−0.001)

ARMA Parameters: Test of OI Rest.:

σ 2ζ σ 2

ξ θ1 Test Statistic(df:p-value)

BHPS 0.011(0.001)

0.010(0.001)

0.071(0.054)

55.21(63: 0.747) Vs. θ1 �= 0:

0.012(0.001)

0.009(0.001)

0.000(const.)

56.44(64: 0.738)

1.23(1: 0.267)

Simulated 0.012(0.001)

0.014(0.001)

0.044(0.042)

70.79(63: 0.234) Vs. θ1 �= 0:

0.012(0.001)

0.013(0.001)

0.000(const.)

71.76(64: 0.236)

0.97(1: 0.325)

the simulated sample and with the BHPS data.44 Most intriguingly, as commonly found in theliterature, we obtain a significant variance for the innovation of the permanent earnings shock,thus concluding to the presence of a random walk component of the individual earnings process,both in the real and in the simulated data. Yet we know that, at least for the simulated data,the fitted ARMA process is misspecified and the true DGP is stationary (according to ourtheoretical model, earnings have a steady-state distribution characterized in (10), which has afinite variance). This illustrates the difficulty of numerically distinguishing between a processtruly exhibiting a unit root and other forms of persistent, possibly nonlinear processes.

The following exercise provides further illustration of this latter point. The ARMA speci-fication (17) was estimated by fitting moments of the wage growth rates (i.e., by taking firstdifferences of the wage data). The results shown in Table 4 indicate that our structural modelcorrectly captures these latter moments, even though it was fitted to the data in levels. One alsomay ask the converse question of how good the ARMA model is at replicating the autocovari-ance structure of wage levels that was used in the estimation of the structural model. Under thesimple assumption of hom*oskedastic innovations, this structure is characterized as follows:

Var(ln φi,t+s) = Var(zi + a Pi,t ) + sσ 2

ζ + (1 + θ21 )σ 2

ξ ,

Cov(ln φi,t , ln φi,t+s) = Var(zi + a Pi,t ) + θ1σ

2ξ × 1{s=1}.

(18)

Figure 5 shows the fit of (18) to the data in the same way as Figure 2 did for the correspondingpredictions based on the structural model.45 The very poor fit seems to point to an inconsistency

44 Wald tests further indicate that both specifications are accepted based on either sample at the five percent level(last column of Table 4). Moreover, neither sample allows rejection of the MA(0) against the MA(1) specification atconventional levels.

45 We are only reporting second-order moments to save on space. For means, the simple ARMA decomposition (17)postulates that E(ln φt+s) is independent of s, which is a poor fit to the empirical pattern observed in the top left panelof Figure 2. Figure 5 shows predictions based on both assumptions (0 or 1) about the order of the MA process in (17).Also, we should mention that the intercept term in (18), Var(zi + a P

i,t ), was set to fit the cross-sectional wage varianceat the initial date t. Although this is not a very efficient way to estimate this parameter, its only impact is to shift the(co)variance/time profile up or down, thus changing very little to the overall fit.

622 POSTEL-VINAY AND TURON

0 2 4 6 8 100.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28 Evolution of wage variance

Number of lags (years = s/12)

Var

[lnφ t+

s | em

pl. a

t t, .

.. ,t+

s] MA(0)

MA(1)

0 2 4 6 8 100.04

0.06

0.08

0.1

0.12

0.14

0.16 Wage autocovariances

Number of lags (years = s/12)

Cov

[lnφ t,ln

φ t+s |

empl

. at t

, ...

,t +s]

MA(0)

MA(1)

FIGURE 5

THE WAGE PROCESS IN LEVELS AND THE ARMA DECOMPOSITION

between the ARMA representation of wage growth and what would be the ARMA represen-tation of wage levels.46 The point of all this is not to say that a more sophisticated ARMAdecomposition would not perform better,47 but rather to emphasize the difficulty of choosing aspecific statistical model of income dynamics without theoretical guidelines.

6. PRODUCTIVITY SHOCK PERSISTENCE

The above results show that the covariance pattern of wages—most notably the degree of wagepersistence—can be accounted for by the mechanisms embedded in our simple model. Thoseresults were obtained under the maintained assumption of i.i.d. productivity shocks, i.e., from amodel where wage persistence arises solely from the mutual consent rule in wage renegotiationand the (in)frequency of outside job offers.

Clearly a fuller empirical exercise would consist of decomposing the observed persistenceof wages into components of their dynamics arising from shocks to match productivity andjob mobility shocks (outside offers). As argued before, however, this idea is fraught with twomain problems. One is the general lack of analytical tractability of our model under non-i.i.d.productivity shocks (see the discussion below). The other and more substantive one is thatthe typical earnings data available to researchers (such as the BHPS data used in this article)conveys no direct information on productivity. It may then seem more natural, given this lack ofdirect information, to assume away persistence in productivity instead of estimating it throughits manifestation in wage persistence.

It is nonetheless true that our model remains well defined—if considerably morecumbersome—without the i.i.d. assumption on productivity shocks. Moreover, as we arguebelow, the model has a tight enough structure to theoretically allow identification of at leastsome simple measure of productivity persistence from observed wage processes. We in briefpursue these ideas in this section.

To that end, we must go some way toward relaxing our assumption of i.i.d. productiv-ity shocks. Instead of assuming that these shocks occur every period, we assume that the

46 This inconsistency was already noticed in U.S. data by Baker (1997). Interestingly, Baker advocates an alternativespecification of the wage process—which he calls the “profile-heterogeneity model”—where wages are linear functionsof experience with individual-specific intercepts and slopes. Even though our structural model is nonlinear and onlyfeatures endogenous (and nonsystematic) heterogeneity in wage/experience profiles, it is formally closer to Baker’spreferred profile-heterogeneity model than to the linear ARMA model (17).

47 For instance Meghir and Pistaferri (2004) show that the variance of innovations in a decomposition similar to (17)follow a relatively complex time pattern.

ON-THE-JOB SEARCH AND EARNINGS 623

match productivity is redrawn from distribution F(·) with probability τ and stays at its pre-vious value with probability 1 − τ . Pre- and post-shock values of match productivity are stillindependent.

In order to keep the model analytically tractable, we further assume that the arrival of anoutside job offer is always accompanied by a shock to idiosyncratic match quality. This is clearlyan ad hoc assumption, and we certainly do not defend it as particularly theoretically appealing.It has, however, the advantage of keeping the model tractable while at the same time introducingpersistent productivity shocks in a simple, if somewhat constrained, way. We view this simplefirst pass at tackling the issue of productivity persistence as informative at least in the followingway: If identification of the (scalar) measure of productivity persistence, i.e., 1 − τ , fails in thissimple case, then any hope of identifying more complex or theoretically appealing patterns ofpersistence from the kind of worker-level data that is typically used in the analysis of individualearnings dynamics should probably be abandoned.

The crucial theoretical simplification that comes with our assumption that outside offersare always accompanied by a shock to ε is that current match productivity does not becomean additional state variable in the worker’s choice problem (as it would under any differentassumption, making the model analytically intractable).48 Here, every time the worker is in aposition to compare surpluses between her/his incumbent employer and a potential poacher,the idiosyncratic quality of these two matches are two independent draws from the distributionM(·). As a consequence the worker’s value of employment continues to be independent ofthe current level of match productivity (as was obviously the case under i.i.d. productivityshocks), implying that there is no option value of being employed in a match with a higherproductivity.

Given this extended model, it is easy to see that data on job and unemployment spell durationsnow only convey information on λ0, δ, and τ × λ1: The unconditional probability of a job-to-job transition is now τλ1/2, and the rest of the analysis of Subsection 3.1 stays unaffected. Wenow argue, however, that separate identification of λ1 and τ is theoretically possible from acombination of the model’s structure and the use of some higher-order moments of the wagedata.

In our modified setting the one-period dynamics of the negotiation baseline are the following:

vt+1 | vt =

vt with probability (1 − δ)[1 − τ F(vt )] − τλ1 F(vt )2,

v′ < vt with density τ (1 − δ) f (v′),

v′ > vt with density 2τλ1 f (v′)F(v′).

(19)

The Appendix then extends the above characterization of one-period wage dynamics to deriveexpressions for the 3T − 1 moments listed in (15) and matched in the estimation proceduredescribed in Section 4.

In addition to the set of means, variances, and covariances described in (15), we now con-sider the third moment of the distribution of wage changes in the hope of disentangling thearrival rate of productivity shocks τ from the job mobility transition rate τλ1/2. Indeed, look-ing at the wage process (19), it appears that τ alone drives wage cuts whereas τλ1 driveswage increases. Thus the relative size of these arrival rates will impact the skewness of the

48 Analytically tractable models in Postel-Vinay and Robin (2002b), Dey and Flinn (2005), and Cahuc et al. (2006)have fixed match- or firm-specific heterogeneity and no transitory productivity shocks. However, those models predictimplausibly (downward-)rigid wages within a job spell. Match productivity in Flinn (1986) is modeled as the sum ofa match fixed effect plus an i.i.d. shock, yet the model assumes that wages equal productivity at all dates, so that thewage process is exogenously specified in that paper. Finally, Yamaguchi (2006) and Lise et al. (2008) introduce variousforms of persistent productivity processes in models very similar to ours, but have to resort to numerical solutions andsimulation-based techniques to estimate their model.

624 POSTEL-VINAY AND TURON

TABLE 5ESTIMATING THE FREQUENCY OF PRODUCTIVITY SHOCKS

Moments Matched 35 46 46Constraint τ = 1 τ = 1 -

vmin 1.801(0.078)

1.806(0.083)

1.848(0.075)

µ 2.219(0.122)

2.217(0.121)

2.166(0.151)

σ 0.200(0.099)

0.197(0.101)

0.265(0.076)

Var(ln p) 0.063(0.008)

0.064(0.009)

0.054(0.016)

σ 2me 0.016

(0.015)0.017(0.014)

0.019(0.012)

τ 1 1 0.282(0.386)

Test of OI Restictions:

Test Statistic 22.23 79.98 73.93(df: p-value) (30: 0.784) (41: 0145) (40: 0.262)

NOTE: Bootstrap standard errors obtained from 500 replications of the entire two-stage estima-tion procedure on 500 resamples with replacement.

distribution of wage changes.49 Specifically, we use the following additional moments in theestimation:50

E[(ln φt+s − ln φt )3] = E

[(vt+s − vt )3] + 6σ 2

me · E[vt+s − vt ].(20)

Details of the analytical derivation of that moment are reported in the Appendix.51

We then proceed to match the basic set of moments listed in (15) (duly amended to accountfor persistence in productivity shocks—see the Appendix), supplemented by the T − 1 momentsin (20) computed for s = 0, 12, 24, . . . , 12(T − 1) using EWMD. Results of this estimation arereported in the third column of Table 5. The first column in that table merely repeats parameterestimates obtained from the basic model and reported earlier in Table 3 for comparison, whereasthe second column reports estimates obtained by matching the full set of moments, includingthe third-order moment (20), yet constraining productivity shocks to be i.i.d. by fixing τ at 1.

The main conclusion that we draw from those estimates is that the arrival rate of productivityshocks cannot be precisely estimated with our data. Looking at the third column of Table 5,the point estimate of τ is 0.282 with a (bootstrap) standard error of 0.386. Although that pointestimate may appear markedly lower than unity, a Wald test of the hypothesis τ = 1 produces a p-value of 0.045, indicating borderline rejection (or acceptance) of the restricted model against theunrestricted one at the five percent level.52 It should further be noted that estimates of all otherparameters remain very stable across the columns of Table 5. All these findings concur to suggestthat the particular value assumed by τ makes little difference to our model’s ability to fit the set ofmoments described above.53 In other words, given the model’s structure, there does not seem to

49 The discussion in Subsection 5.2 gives us additional a priori reason to be hopeful that the (asymmetry of the)distribution of wage changes has some practical informative content, as we then observed that our simple model withi.i.d. shocks has trouble replicating the asymmetry of that distribution. Specifically, it tends to over-predict wage cuts,which may be caused by an excessively high frequency of productivity shocks.

50 Once again, this moment is meant conditional on continuous employment between times t and t + s, although wekeep the conditioning implicit.

51 Note that we assume that the distribution of the measurement error is symmetric, i.e., has zero skewness. Also,looking at wage changes means that the worker fixed effect p does not appear in this expression and that we do notrequire any assumptions regarding the skewness of the distribution H(p).

52 Wald tests of overidentifying restrictions reported at the bottom of Table 5 otherwise lead to accepting eithermodels against an arbitrary alternative.

53 We also experimented with larger sets of moments, going up to the sixth moment of the distribution of wagechanges. All specifications yielded equally imprecise estimates of τ .

ON-THE-JOB SEARCH AND EARNINGS 625

be enough information in our data to practically identify the degree of persistence of productivityshocks. We conjecture that this negative result would also obtain given a more sophisticateddescription of the productivity process or without the ad hoc simplifying assumption about theforced joint occurrence of job offers and productivity shocks made here for convenience andthat firm-level data conveying direct information on productivity are needed to credibly identifyparameters of the productivity process.54

7. CONCLUSION

Our concern in this article has been the ability of a simple structural model to replicate themain features of the dynamics of individual labor earnings observed in the data. Our proposedmodel belongs to the family of search models a la Diamond-Mortensen-Pissarides (DMP). Ourspecific assumptions are that we allow on-the-job search and assume that wages can only berenegotiated with mutual consent by the firm and the worker. We investigate whether such amodel can produce a quantitatively plausible “internal propagation mechanism” of i.i.d. pro-ductivity shocks into persistent wage shocks using a 12-year panel of highly educated Britishworkers. Our key contributions are the following.

First, we formalize the assumption of renegotiation by mutual consent in the context of ajob search model with idiosyncratic productivity shocks to match quality. We then scrutinizethe model outcomes in terms of individual earnings dynamics, whereas the existing job searchliterature usually focuses on cross-sectional wage dispersion. Because the mutual consent rulewill only allow the wage level to be altered when one party has a credible threat to leave thematch, wage dynamics generated by our model are more persistent than under the constant-share Nash surplus sharing rule often used for wage determination.

Second, we show that our model, when estimated with 12 waves of BHPS data, producesa dynamic earnings structure that is remarkably consistent with the data. The main featuresof individual earnings dynamics, such as the covariance structure of earnings, the evolution ofindividual earnings mean and variance with the duration of uninterrupted employment, or thedistribution of year-to-year earnings changes are very well matched by our model predictions.

Third, we offer a structural counterpart to the “reduced-form” literature on individual earn-ings processes and we establish that the combination of on-the-job search and renegotiation bymutual agreement is a promising candidate explanation of the widely documented persistenceof earnings shocks. Our theory suggests that wage dynamics should be thought of as the outcomeof a specific acceptance/rejection scheme of i.i.d. wage shocks, thus offering an alternative to theconventional linear ARMA-type approach. Moreover, it highlights the link between labor mar-ket competition (as measured by the probability of raising outside job offers when employed),worker mobility across jobs, and individual earnings dynamics.

There are several avenues for further work building on this article. We now briefly discusssome of these. A first, relatively straightforward extension would be to close our theoreticalmodel in the manner of the DMP framework to include the firm’s job creation decision. Wecould also allow for the job destruction to become endogenous by allowing the match surplus tobe negative over some of the support of the distribution of productivity shocks. Although sucha closed model would be difficult to estimate (it would be fraught with the well-known difficultyof estimating a matching function), a calibrated version of it could still be used to analyze theimpact of various labor market policies. We pursue those ideas in more recent work (Postel-Vinay and Turon, 2008), where we include firing costs and minimum wage regulations to thepresent setting. Our mutual consent assumption then gives rise to endogenous agreements onfirm–worker separations with severance packages. We use this setup to analyze (inter alia) theimpact of firing restrictions on wage inequality. Finally, closing the model would allow us to relate

54 Of course an alternative route would be to gain identification power by further tightening the model’s structureand assuming more specific and less flexible functional forms for some elements of the model. This is the route followedby Yamaguchi (2006), to which we refer the reader.

626 POSTEL-VINAY AND TURON

our framework to well-known empirical results of the contract literature, such as Beaudry andDiNardo’s (1991) striking observation that individuals’ current wages are more strongly affectedby the lowest unemployment rate since the start of their job than by the current unemploymentrate or the unemployment rate at the start of the job.55

A second possible extension would be to incorporate the impact of human capital accumula-tion into our model. Although we have ignored it completely in this article, it obviously lines upas a potential cause of the positive returns to continuous employment. A simple way of introduc-ing human capital into our framework would be to allow the individual-specific component ofmatch productivity to grow at a constant rate over time. Although this will make the derivationsof the predicted moments of interest much more cumbersome,56 it is likely to improve the fitof the model in terms of the frequency and size of wage cuts, which the present model tends tooverestimate.

A third avenue of potential improvement would be the addition of worker heterogeneity inthe transition rates, i.e., the hazards of job destruction, job finding, and outside offers. This wouldhelp with our model’s currently counterfactual implication that these three hazards are constantwith spell duration or independent of current wages.

Finally, our model can be enriched by a more careful depiction of the employer side and a moreprecise description of what we referred to as “match productivity” or “match quality” shocks interms of firm-specific and truly match-specific shocks. On the theoretical side, the introductionof permanent firm heterogeneity is a far-from-trivial extension of our model, as it complicatesthe derivation and comparison of the workers’ valuation of employment at different firms byan order of magnitude. On the empirical side, Section 6 has demonstrated the need for directevidence on productivity shocks to ensure practical identification of the model. Such evidencecan in principle be found in matched employer-employee data of the type used in Abowd et al.(1999), Cahuc et al. (2006), or Guiso et al. (2005). In currently ongoing research, Lise et al. (2008)extend the model we have offered in this article to allow for more sophisticated, firm-specificproductivity processes and plan to estimate their model on matched employer-employee data.

APPENDIX: DERIVATION OF THEORETICAL MOMENTS

All derivations in this appendix are made under the general specification of Section 6, whichallows for non-i.i.d. productivity shocks. This encompasses the main baseline case of Sections 2–5,which is obtained by setting τ = 1 in what follows.

Derivation of Gs(·). We construct Gs(·) by induction. Considering the conditional distribu-tion of vt+1 | vt in (19), one has for any s ≥ 1

gs+1(v) =[

1 − τ F(v) − τλ1

1 − δF(v)2

]· gs(v) −

[τ f (v) − 2τλ1

1 − δf (v)F(v)

]· Gs(v) + τ f (v),

(A.1)

which integrates as

Gs+1(v) =[

1 − τ F(v) − τλ1

1 − δF(v)2

]· Gs(v) + τ F(v).(A.2)

55 Even though these authors, along with much of the related contract literature, emphasize risk-sharing considera-tions as the main driving force behind individual earnings dynamics, our model still bears a close formal relationship totheirs. Malcomson (1997) indeed notes that the assumption of renegotiation by mutual consent squares well with theevidence documented in Beaudry and DiNardo (1991).

56 See also Yamaguchi (2006) and Bagger et al. (2006), whose focus is on the decomposition of young workers’ wagegrowth between human capital accumulation and job search.

ON-THE-JOB SEARCH AND EARNINGS 627

Given the initial condition G0(·) ≡ G(·), this difference equation solves as (13) in the main text.With the notation b(v) = 1 − τ F(v) − τλ1

1−δF(v)2, we can rewrite this equation as

Gs(v) = (1 − b(v)s) · G∞(v) + b(v)s · G(v).(A.3)

It is straightforward to check that ∀v, G∞(v) ≥ G(v), i.e., G∞(·) first-order stochasticallydominates G(·). Integration by parts further shows that

EGs (v) = v0 +∫ vmax

G∞(x)dx −∫ vmax

b(v)s · [G∞(x) − G(x)]dx,(A.4)

which establishes that EGs (v) monotonically increases with s.

Derivation of Cov(ln φt , ln φt+s | employment at t, . . . , t + s). We begin by the derivation ofthe conditional expectation E(vt+s | vt , employment at t, . . . , t + s). Even though this is notamong the set of moments we are eventually going to directly match in the estimation, itsderivation is a useful intermediate step.

It is straightforward to show using (12) that for any differentiable function ϕ(·),

E[ϕ(vt+1) | vt , employment at t, t + 1] = (1 − τ )ϕ(vmax) + τ EF [ϕ(v)] −∫ vmax

ϕ′(x) · b(x) dx.

(A.5)

Next, defining Ts(vt ) = E(vt+s | vt , employment at t, . . . , t + s), the conditional prediction ofthe negotiation baseline s periods ahead given vt and given continuous employment betweendates t and t + s, one notices that for any s ≥ 2

Ts+1(vt ) ≡ E(vt+s+1 | vt , employment at t, . . . , t + s + 1)

= E[E(vt+s+1 | vt+1, employment at t + 1, . . . , t + s + 1) | vt , employment at t, t + 1]

= E[Ts(vt+1) | vt , employment at t, t + 1].

(A.6)

Reasoning by induction and differentiating (A.5) shows that T′s (vt ) = b(v)s . Substituting back

into (A.5) and (A.6) establishes the following for Ts(vt )

Ts(vt ) = (1 − τ )Ts−1(vmax) + τ EF [Ts−1(v)] −∫ vmax

b(x)sdx.(A.7)

Note that EF [Ts−1(v)] is a deterministic term depending on the “prediction horizon” s only.Then from (A.7) we directly obtain57

Cov(ln φt , ln φt+s) = Var(ln p) + Cov(vt , vt+s)

= Var(ln p) + Cov(vt , E(vt+s | vt ))

= Var(ln p) − Cov(

vt ,

∫ vmax

b(x)sdx)

(A.8)

Note that the distributions with respect to which expectations should be taken in all thesemoments are distributions in the population of employed workers, meaning G(·) for the momentsinvolving vt and H(·) for those involving p.

57 We continue to work conditionally on continuous employment between dates t and t + s. However, to avoid anotational overload, we now keep this conditioning implicit.

628 POSTEL-VINAY AND TURON

Derivation of E[(ln φt+s − ln φt )3]. We can rewrite (20) as

E[(ln φt+s − ln φt )3]

= EG[E

[(vt+s − vt )3 + 6σ 2

me(vt+s − vt ) | vt , employment at t, . . . , t + s]]

= EG[Rs,3(vt ) − 3vt · Rs,2(vt ) + 3v2

t · Rs,1(vt ) − v3t

] + 6σ 2me EG[Rs,1(vt ) − vt ],

(A.9)

where Rs,n(vt ) = E(vnt+s | vt ) for n = 1, 2, 3. In the first equality, we integrate on the distribution

G, as conditioning on continuous employment between the dates t and t + s is independent ofvt –employment is continuous provided the match has not been hit by a destruction shock δ,which occurs independently of the individual wage.

As in Equations (A.6) to (A.7), we obtain the following recurrent definition for Rs,n:

Rs,n(vt ) = (1 − τ )Rs−1,n(vmax) + τ EF (Rs−1,n(v)) −∫ vmax

R′s−1,n(x) · b(x) dx(A.10)

and R′s,n(x) = nxn−1b(x)s .

Predictions of the model with respect to fourth and fifth moments of the distribution of wagechanges are derived in a similar manner when computing the fit in terms of 62 moments. Detailsof the derivation are available upon request.

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