enriching a theory of wage and promotion …web.mit.edu/rgibbons/www/gibbons_waldman_enriching...

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[ Journal of Labor Economics, 2006, vol. 24, no. 1]� 2006 by The University of Chicago. All rights reserved.0734-306X/2006/2401-0003$10.00

Enriching a Theory of Wage andPromotion Dynamics inside Firms

Robert Gibbons, Massachusetts Institute of Technology and

National Bureau of Economic Research

Michael Waldman, Cornell University

In previous work, we showed that a model that integrates job as-signment, human capital acquisition, and learning can explain severalempirical findings concerning wage and promotion dynamics insidefirms. In this article, we extend that model in two ways. First, weincorporate schooling and derive further testable implications thatwe then compare with the available empirical evidence. Second, andmore important, we show that introducing “task-specific” humancapital allows us to produce cohort effects. We further argue thattask-specific human capital is a realistic concept and may have manyimportant implications. We also discuss limitations of our (extended)approach.

I. Introduction

A standard theme in labor economics, industrial relations, and humanresource management has been that careers in organizations are incon-

We thank Orley Ashenfelter, George Baker, Dan Bernhardt, Gary Fields, PaulOyer, Jan Zabojnik, seminar participants at Cornell University, NorthwesternUniversity, Queen’s University, University of California, Los Angeles, and Wash-ington University in St. Louis, and conference participants at the 2002 Society ofLabor Economics conference and the 2002 Stanford Institute for Theoretical Eco-nomics and National Bureau of Economic Research conferences on personneleconomics, for helpful comments, and Ari Gerstle and Kameshwari Shankar, forresearch assistance. Contact the corresponding author, Michael Waldman, [email protected].

60 Gibbons/Waldman

sistent with standard economic theories of labor markets. For example,Doeringer and Piore (1971) described aspects of internal labor markets(such as ports of entry and wages attached to jobs) that depart fromstandard conceptions of competitive labor markets. More recently, Medoffand Abraham (1980, 1981), Lazear (1992), and Baker, Gibbs, and Holms-trom (1994a, 1994b) provided detailed empirical analyses of careers insidespecific firms and argued that many of their results again do not matchstandard conceptions of competitive labor markets. In particular, Bakeret al. (1994b) compared their empirical findings to two theories of wagedetermination—learning and incentives—and found their results to bequite different from the predictions of those theories.

In Gibbons and Waldman (1999b), we developed a theoretical modelthat integrates job assignment, human capital acquisition, and learning.Regarding job assignment, we followed Rosen (1982) and Waldman(1984b) by assuming that jobs are ranked in terms of the importance ofability, so it is efficient to assign higher-ability workers to higher levelsof the job ladder. Regarding human capital acquisition, we followed thespirit of Becker (1962) and Mincer (1974) by assuming that a worker’sproductivity rises during the worker’s career. (We also assumed that pro-ductivity rises faster for workers of higher innate ability.) Finally, re-garding learning, we followed Harris and Holmstrom (1982) and Farberand Gibbons (1996) by assuming that there is uncertainty about a worker’sability when she enters the labor force but that all firms learn about theworker’s ability from public output observations as the worker ages.

Our 1999b model captures many of the empirical results concerningwage and promotion dynamics found by Medoff and Abraham (1980,1981), Lazear (1992), Baker et al. (1994a, 1994b), and others.1 For example,Baker et al. (1994a, 1994b) find that (1) wage changes are serially cor-related, (2) promotions are associated with large wage increases, and (3)workers who receive large wage increases early in their stay at one levelof the job ladder are promoted quickly to the next level.2 Our model

1 Some of the other studies whose empirical findings are consistent with oneor more of our theoretical results include Rosenbaum (1984), Murphy (1985),Main, O’Reilly, and Wade (1993), McCue (1996), Baker (1997), and Podolny andBaron (1997). See Gibbons (1997) for a review of this literature through 1997.Also, see Lima and Pereira (2003) for a recent study testing our theory that usesa longitudinal matched employer-employee panel of 74 large manufacturing firmsin Portugal. Their results are in general supportive of our framework.

2 Our 1999b model also explains three other findings in the Baker et al. (1994a,1994b) study. The first is that real wage decreases are not rare but demotions are.The second is that, even though promotions are associated with large wage in-creases, wage increases at promotion are small relative to the difference in averagewages across levels of the job ladder. The third is that there are promotion fasttracks. Note that we provide an explanation for this third finding in our full-information model, while under symmetric learning the result holds under somebut not all parameterizations.

Wage and Promotion Dynamics inside Firms 61

captures the first and third of these findings because of the skill acquisitionprocess we assume: a worker who receives a large wage increase today islikely to be a worker with high ability, so the worker’s productivity willlikely grow quickly in the future, producing both a large wage increaseand an early promotion. Our model captures the second finding becauseof the learning process we assume: a promoted worker is likely to be aworker whose most recent output produced positive updating of thebeliefs concerning the worker’s ability, so the worker’s wage will likelyincrease significantly. Our 1999b model also explains Medoff and Abra-ham’s (1980, 1981) findings concerning performance evaluations, providedthat supervisors evaluate individuals relative to others with the same labormarket experience.

In this article, we extend our 1999b model in two directions. First, weaddress issues that we did not consider in our earlier work. In particular,we incorporate schooling into the model and derive a number of testableimplications that we then compare with the available empirical evidence.Second, and more important, we show that an important empirical findingthat is inconsistent with the Gibbons and Waldman (1999b) model be-comes consistent with our approach if we make a realistic new assumption.In particular, we incorporate task-specific human capital into the modeland thereby capture the Baker et al. (1994b) finding of cohort effects: acohort that enters a firm at a low wage will earn below average wagesyears later.

The first part of this article incorporates schooling into our previousmodel in two ways. First, schooling increases a worker’s starting level ofgeneral purpose human capital. Second, schooling is correlated with thespeed with which the worker accumulates human capital once she is onthe job. We show that this new model makes a number of predictionsconcerning how schooling affects wage and promotion dynamics. To be-gin, this extended model of course predicts a positive correlation betweenschooling and the starting wage, since more schooling means a higherstarting level of general human capital. But the model also makes otherpredictions, such as that schooling is positively related to promotion pros-pects (because a worker with more schooling accumulates human capitalmore quickly once she is on the job) and that schooling is positivelyrelated to the wage even after holding job level and experience fixed(because a worker with more schooling has a higher initial level of generalhuman capital). We compare these predictions with the available evidenceand find that, although the evidence is not clear concerning all of thepredictions, in general the predictions coincide well with the availableevidence.

The second part of this article considers cohort effects. Baker et al.(1994b) find that the average wage at which a cohort is hired is positivelycorrelated with the cohort’s average wage many years later, even after

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controlling for cohort differences in age, gender, race, and education. Inour earlier paper (Gibbons and Waldman 1999b), we argued that ourframework is inconsistent with this finding, because we believed the find-ing to be at odds with spot market contracting, which is a basic featureof our approach. Here we show that we were incorrect in that earlierdiscussion: introducing task-specific human capital into our frameworkproduces cohort effects even in the presence of spot market contracting.By task-specific human capital, we mean that some of the human capitalan individual acquires on the job is specific to the particular tasks beingperformed, as opposed to being specific to the firm. Hence, when a workeris promoted and given a new set of tasks to perform, some of that worker’sacquired human capital goes unutilized on the new job. We show that, iftask-specific human capital is an important part of the skill-acquisitionprocess, then cohort effects can arise.

Our view is that the first of these extensions, that concerning schooling,is quite straightforward but nonetheless advances the project we advocatedin our Handbook of Labor Economics chapter (Gibbons and Waldman1999a), namely, that theoretical models of wage determination shouldbegin to address broad patterns of evidence rather than isolated stylizedfacts. By incorporating schooling into our model, we add several newfindings to the (already fairly large) set of findings that we originally setout to explain.

In contrast to the first extension, we believe that the second extension,that concerning task-specific human capital, is more novel and potentiallyquite rich in its theoretical applications and empirical implications. SinceBecker’s (1962, 1964) seminal work, the literature has focused almostexclusively on general purpose and firm-specific human capital. We believethat task-specific human capital is potentially both as commonplace andas important as these better-known types. Clearly, when a worker per-forms a particular task, it is likely that some of the human capital sheacquires is useful for performing that task but of only limited use in theperformance of other tasks; that is, there may be task-specific learningby doing.3 By introducing task-specific human capital into our model inorder to analyze cohort effects, we hope to set the stage for a variety offuture theoretical and empirical exercises concerning issues such as jobdesign, job assignment, labor mobility, labor demand, and even businessstrategy.4

3 See Heckman, Lochner, and Cossa (2003) for a recent empirical study thatfinds evidence consistent with much of the human capital accumulation on jobsbeing due to learning by doing.

4 As far as we know, Kwon (forthcoming) was the first to analyze what we arecalling task-specific human capital. He considers this type of human capital in amodel similar to Prendergast’s (1993), where wage increases at promotion are usedto provide incentives for investment in human capital. Kwon’s analysis does not


Our idea of task-specific human capital is related to the recent skill-weights approach to firm-specific human capital articulated by Lazear(2003). In Lazear’s formulation, each firm’s production function is char-acterized by a firm-specific weighting across general skills. Each periodworkers must decide how much to invest in the accumulation of eachskill. Lazear shows that this approach not only provides a plausible storyfor why some human capital is firm specific (i.e., in this setting, humancapital can be firm specific because a firm places more weight on a par-ticular skill than do prospective employers) but also matches various as-pects of the available evidence quite well. Our emphasis on task-specifichuman capital has related implications. Lazear’s focus is on how humancapital accumulated at one firm will typically lose value at another firmthat has different skill weights. Our focus, which complements Lazear’s,is on how human capital accumulated in one job will typically lose valuein another job that requires a different mix of skills. At the end of SectionIII, we discuss how these two ideas can be integrated and what predictionsmight follow from such an integrated approach.

In sum, this article reinforces the contention in Gibbons and Waldman(1999b) that a framework that integrates job assignment, human capitalacquisition, and learning can explain a large number of empirical findingsconcerning wage and promotion dynamics inside firms.5 Our interpre-tation is not that these three model elements currently (or could ever)constitute a complete theory of wage and promotion dynamics insidefirms. Clearly, other elements—ranging from incentives to social norms—are important parts of such a theory. Rather, our interpretation is that,for a wide variety of firms, a complete theory of wage and promotiondynamics is very likely to include our three model elements as integralparts. Notice that we say a “wide variety” of firms as opposed to “all”firms. This is because we believe that our theory does not apply well to

consider our main focus here, namely, the implications of task-specific humancapital when cohorts vary in the mix of jobs at which they enter the firm. Ouridea of task-specific human capital is also similar in spirit to the idea of “vintagehuman capital,” recently explored in Violante (2002). In that analysis, a worker’shuman capital is specific to the machine the worker uses, which means that someof a worker’s accumulated human capital goes unutilized when a worker switchesmachines. Despite this similarity, however, the papers are quite different, becauseViolante’s paper ignores promotions and cohort effects and focuses instead onwage inequality. See also Chaudhury (2002) for another related analysis.

5 Other papers that integrate these three model elements include Bernhardt(1995) and Jovanovic and Nyarko (1997). In contrast to our approach, Bernhardt’sanalysis focuses on asymmetric learning, as in Waldman (1984a), Greenwald(1986), and Gibbons and Katz (1991), where a worker’s current employer receivesbetter information concerning the worker’s ability than prospective employersreceive. Jovanovic and Nyarko do not consider the ability of their model to capturea broad set of empirical findings, while Bernhardt does. See sec. 5 of Gibbonsand Waldman (1999b) for a detailed discussion of Bernhardt’s analysis.

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the type of internal labor market initially described by Doeringer andPiore (1971), one involving ports of entry, wages attached to jobs, andlittle turnover at higher levels of the job ladder. We return to this issuein Section IV, where we discuss limitations of our approach.

II. The Role of Schooling in Wage and Promotion Dynamics

In this section, we consider the role of schooling in wage and promotiondynamics inside firms. We begin by constructing a model that enrichesour 1999b model by incorporating schooling. We then analyze the modeland show that it produces several natural predictions concerning the roleof schooling in wage and promotion dynamics inside firms. We end thesection by discussing the empirical evidence on this topic, which is broadlyconsistent with the model’s predictions. Our main point in this sectionis not that this modest enrichment is important in its own right, but ratherthat enriching the model in this way adds several new findings to thebroad pattern of evidence that our approach can explain.

A. The Model

All firms are identical, and the only input is labor. A worker’s careerlasts T periods, . Worker i enters the labor market with a schoolingT ≥ 3level, denoted Si, which can take on any integer value between 1 and N.We assume that there is a positive number of workers at each value of S.One simple interpretation is that a worker’s schooling level is the numberof years the worker spent in school prior to entering the labor market.6

Greater schooling makes a worker more productive, but so does greaterexperience. This distinction between schooling and experience is standardin the literature, but we consider the microfoundations of the effects ofschooling and experience in more than the usual level of detail, as follows.Let hit denote the worker’s “on-the-job human capital,” where

h p v f(x ). (1)it i it

In this specification of on-the-job human capital, vi is the worker’s abilityto learn on the job and xit is the worker’s labor market experience priorto period t (i.e., for a worker in her first period in the labor market, priorexperience xit equals zero). In this specification, ) is the speed with′v f (x )i it

which on-the-job human capital grows in period t, where ′f(0) 1 0, f 1

6 With this interpretation, it would probably be more accurate to assume thatworker i’s career lasts Ti periods, where But since nothing of sig-T p A � S .i i

nificance would be changed by assuming this, for simplicity of exposition, weassume that the career lasts T periods for all workers.


and 7 Thus, a worker with more ability to learn on the job (i.e.,′′0, f ! 0.higher v) accumulates on-the-job human capital more quickly. We assumethat a worker with schooling level S has an ability to learn on the jobequal to with probability p and an ability to learn on the jobf � B(S)H

equal to with probability , where and ′f � B(S) (1 � p) f 1 f B 1 0.L H L

That is, schooling is positively correlated with a worker’s ability to learnon the job (either because schooling is useful for learning on the job orbecause there is a positive relationship between schooling and innate abil-ity to learn on the job).

A firm consists of two different jobs, denoted 1 and 2. In Gibbons andWaldman (1999b), we allowed for three job levels, but little of the evidencediscussed below relies on three job levels, so here we assume only twolevels to simplify the analysis. If worker i is assigned to job j in periodt, then the worker produces

y p d � G(S ) � c (h � � ), (2)ijt j i j it ijt

where and are constants known to all labor market participants,d cj j

and and �ijt is a noise term drawn from a normal distribution′ ′′G 1 0 G ! 0,with mean zero and variance j2.8 Let h′ denote the amount of on-the-jobhuman capital at which a worker is equally productive at jobs 1 and 2.That is, h′ solves . We assume andd � c h p d � c h c 1 c 1 0 0 !1 1 2 2 2 1

, so the efficient assignment for a worker with is job 2, while′d ! d h 1 h2 1

the efficient assignment for a worker with is job 1. Figure 1 depicts′h ! h

expected output as a function of on-the-job human capital for each pos-sible job assignment, given a fixed level of schooling. Varying the levelof schooling varies the intercept of each output schedule in the same wayand so simply causes the two curves to shift in parallel.

We assume that each worker’s schooling level is known to all labormarket participants when the worker enters the labor market. In contrast,each worker’s value for vi is not known by either the firms or the worker,although the probability p and ability levels and aref � B(S) f � B(S)L H

common knowledge. Learning about vi takes place at the end of eachperiod when the realization of the worker’s output for that period is

7 We assume rather than One interpretation is that this modelf(0) 1 0 f(0) p 0.is a simplified version of a continuous-time model, where production in the firstperiod represents production in the early part of the worker’s career. Since, onaverage, during this early part of the career, the worker has a positive amount oflabor market experience, it is natural to assume rather thanf(0) 1 0 f(0) p 0.

8 We could have assumed that the extra productivity due to a worker’s startinglevel of general purpose human capital depends on the job the worker is assignedto. That is, rather than this productivity being equal to we could haveG(S ),i

assumed that it equals for a worker assigned to job j. Given the additionalG (S )j i

assumptions for all Si and for all Si, all the qual-′ ′G (S ) ≥ G (S ) G (S ) ≥ G (S )2 i 1 i 2 i 1 i

itative results that follow would still hold.

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Fig. 1.—Expected output as a function of on-the-job human capital and job assignment,given a fixed level of schooling.

publicly observed. The presence of the noise term �ijt in (2) implies thatlearning occurs gradually, as follows.

Let .z p [(y � d � G(S ))/c ] � B(S )f(x ) p h � B(S )f(x ) � �it ijt j i j i it it i it ijt

That is, zit is the signal about the worker’s on-the-job human capital aftersubtracting off the human capital due to the schooling term that theB(S )imarket extracts from observing the worker’s output in period t. We referto zit as the worker’s normalized output from period t (i.e., normalized toabstract from job assignment and schooling) and to ( ) as thez , … , zit�x it�1

worker’s normalized output history at date t. Because the signal zit is in-dependent of job assignment and schooling, there is no difference in therate of learning across jobs and schooling levels. Let denote worker i’sevit

expected ability to learn on the job at date t: .ev p E(vFS , z , … , z )it i i it�x it�1

From we can compute the expected on-the-job human capital of workerevit

i in period t:

e eh p v f(x ). (3)it it it

For example, if worker i enters the labor market in period t with schooling, then eS h p [ pf � (1 � p)f � B(S )]f(0).i it H L i

Workers and firms are risk neutral and have a discount rate of zero.There is no cost to workers from changing firms or to firms from hiringor firing workers. Under these assumptions, there are no benefits to long-


term contracts, so we assume that wages are determined by spot marketcontracting. Finally, to ease the comparison of the model with the em-pirical evidence, we restrict attention to wages that are paid in advanceof production, as opposed to one-period piece rate contracts.

There is free entry into production. At the beginning of each period,all firms simultaneously offer each worker a wage for that period. Theworker then works for the firm that offers the highest wage. If there aremultiple firms tied at the highest wage, the worker chooses randomlyamong these firms (unless one of these tied firms was the worker’s em-ployer in the previous period, in which case the worker remains with thatfirm). This tie-breaking rule is equivalent to assuming a moving cost thatis infinitesimally small; as a result, there is no turnover in equilibrium.

To reduce the number of cases that need to be considered, we restrictthe analysis to parameterizations that satisfy the following conditions.First, so that it is efficient for each′[ pf � (1 � p)f � B(N)] f(0) ! h ,H L

worker to be assigned to job 1 in the first period of the worker’s career.Second, [ for all S, so that′f � B(S)] f(T � 1) 1 h 1 [f � B(S)] f(T � 1)H L

for each schooling level it is efficient for workers with the highest abilityto learn on the job to be on job 2 by the end of their careers, but forworkers with the lowest ability it is efficient to remain on job 1.

As a final point, there are a number of ways this model could be mademore realistic. For example, we could start with workers having someknowledge of their own innate abilities and incorporate a cost of schoolingand then have the schooling distribution be derived endogenously, as inSpence (1973). Or we could assume that a worker’s schooling level ispositively related to the extra productivity due to a worker’s starting levelof general purpose human capital , but not in a deterministic way.G(S )iWe believe that our main results are robust to both changes. The reasonwhy we consider the particular specification above is that it allows us topresent our main results in a tractable analysis in which the intuition forour results is easy to follow.

B. Analysis

We begin with a general result concerning how workers are assignedto jobs and how they are paid. We then turn our attention to how school-ing affects wage and promotion dynamics. The general result is that, asin Gibbons and Waldman (1999b), in each period each worker is assignedto the job that maximizes her expected output in that period and paid awage equal to that expected output.9 Let wit denote the equilibrium wagepaid to worker i in period t.

9 Throughout the analysis, we assume that, if a firm is indifferent betweenassigning a worker to either of the two jobs, then the firm assigns the worker tothe high-level job. This assumption is not crucial for the analysis, but it simplifiesthe statements of some results.

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Proposition 1. In equilibrium, job assignments and wages are givenby i and ii.

i) If , then worker i is assigned to job 1 in period t and paide ′h ! hitew p d � G(S ) � c h .it 1 i 1 it

ii) If , then worker i is assigned to job 2 in period t and paide ′h x hit

.ew p d � G(S ) � c hit 2 i 2 it

Proof. See the appendix.Proposition 1 tells us that the assignment of workers to jobs follows

a simple rule: a worker is assigned to the high-level job if and only if herexpected on-the-job human capital is greater than or equal to the criticalvalue . Only the expected on-the-job human capital matters, rather than′h

other moments of the distribution, because output in each job is a linearfunction of on-the-job human capital and the rate of learning is inde-pendent of job assignment.

This model is a simple extension of the Gibbons and Waldman (1999b)model, so it, of course, reproduces most of the Baker et al. (1994a, 1994b)findings captured by our original model. Three such findings follow. First,this model captures the finding of serially correlated wage increases. Thereason is that a large wage increase today likely means that the workerhas high ability to learn on the job, which is positively correlated with alarge wage increase tomorrow. Second, this model captures the findingthat a large wage increase early in a worker’s stay at one level of the jobladder is positively correlated with quick promotion to the next level.The logic here is that, similar to the argument above, a large wage increaseearly in a worker’s stay at job 1 likely means that the worker has highability to learn on the job, which implies that on-the-job human capitalwill likely grow quickly and induce promotion. Third, this model alsocaptures the finding of large wage increases at promotion. As in our 1999bmodel, this result follows from a selection effect: a promoted worker willlikely be a worker for whom learning resulted in a large increase in ex-pected on-the-job human capital.10

This model also explains a Baker et al. (1994a, 1994b) finding that ourprevious model did not capture. Baker et al. found that adjacent levels ofthe job ladder had overlapping wage distributions: the highest-paid work-ers at level L made more than the lowest-paid workers at level . InL � 1our earlier model, there were no schooling levels and all workers wereex ante identical, so a worker’s wage and job assignment depended solelyon ; therefore, the highest-paid worker at level L was paid strictly lessehit

than the lowest-paid worker at level . In this section’s model, how-L � 1

10 All of these results follow easily from results in our earlier (1999b) paper,given that in each case we are referring to workers of a given schooling level. Webelieve that each of the results also holds without controlling for workers’ school-ing levels, but we have not formally shown this.


ever, a worker’s wage depends on both and the worker’s schoolingehit

level. Consequently, in contrast to our earlier analysis, this model exhibitsoverlapping wage distributions: the highest-paid workers assigned to job1 are paid more than the lowest-paid workers assigned to job 2. Fur-thermore, our theory not only explains overlapping wage distributionsbut does so in a testable way: if we consider pairs of workers on adjacentlevels, where the worker assigned to the lower level is paid more thanthe worker assigned to the higher one, the worker on the lower levelshould typically have more schooling.11

We now turn our attention to predictions concerning the relationshipbetween schooling and wage and promotion dynamics. Corollary 1 statesour (unsurprising) first prediction: schooling is positively related to thestarting wage.

Corollary 1. The wage a worker receives in her first period in thelabor market is a strictly increasing function of the worker’s schoolinglevel.

Proof. See the appendix.In this model, a worker’s starting wage is an increasing function of her

schooling level for two reasons. The first is that more schooling meansthat a worker enters the labor force with more general purpose humancapital and so is paid more. The second is that, because , a worker′B 1 0with more schooling has a higher expected starting level of on-the-jobhuman capital, (see n. 7).E(v f(0))i

The second prediction is that schooling is positively related to job level.Corollary 2 develops this prediction. Let denote the proportionp(X, S)of workers of experience X and schooling level S who are assigned to thehigh-level job.

Corollary 2. Given any pair, where and(X, S) 1 ≤ X ≤ T � 1. Also, the inequality is strict if X isS ≤ N � 1, p(X, S � 1) ≥ p(X, S)

sufficiently close to .T � 1Proof. See the appendix.The logic for this result follows from the way that schooling is cor-

related with the speed of human capital acquisition. Holding experience

11 In Gibbons and Waldman (1999b), we discussed a different extension of ouroriginal analysis that captures overlapping wage distributions (see n. 13 of thatpaper). In particular, we argued that overlapping wage distributions can be ex-plained by each firm having multiple job ladders, where promotions only occurwithin ladders and not across. Our argument was that, if the wage distributionsfor one ladder are shifted relative to another, then an analysis of the aggregatejob ladder would yield overlapping wage distributions. We believe that the school-ing argument above is a superior argument. The reason is that, although we arenot familiar with any evidence on the subject, we suspect that, when a firm hasmultiple job ladders, there are overlapping wage distributions for each job ladderthat makes up the aggregate job ladder, and the schooling argument presentedhere is consistent with such a finding.

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constant, workers with more schooling will likely have a higher abilityto learn on the job. This means that, if we consider any experience, X,and any critical value, , workers with schooling level will have a�h S � 1higher proportion whose expected on-the-job human capital exceeds �h

than will workers with schooling level S. Since any worker whose on-the-job human capital exceeds h′ will be assigned to the high-level job,we now have that, at any experience level, the proportion assigned to thehigh-level job will be increasing in the schooling level.

Notice that, although the corollary is directly a statement about jobassignments, it easily translates into a statement about promotions. Thatis, since all workers in our model start their careers in the low-level job,the corollary tells us that the promotion rate is higher for workers withhigher schooling levels; that is, is increasing in S. Ad-p(T � 1, S)/(T � 1)ditionally, although we do not show it formally, a refinement of thisprediction is that, in empirical analyses of promotion rates, the relation-ship between schooling and promotions will be weaker when other mea-sures of worker ability to learn on the job are included as explanatoryvariables. The logic here is that, in our model, schooling affects the prob-ability of promotion through its correlation with the worker’s ability tolearn on the job. Hence, if one included a noisy measure of this abilityin a regression analysis of promotion rates, the prediction is that themeasured impact of schooling on promotions should fall.

The third prediction is that wage increases are positively related toschooling. Corollary 3 develops this prediction. Let denote theW(X, S)average wage paid to workers of experience X who have schooling levelS.

Corollary 3. For any parameterization, if demotions are suffi-ciently rare, then given any X, 0 ≤ X ≤ T � 2, W(X � 1, S) � W(X, S)is a strictly increasing function of S.

Proof. See the appendix.The logic for this result is straightforward if we restrict attention to

values of X such that all workers are assigned to the low-level job atexperience levels X and , because the argument is then similar toX � 1why workers with more schooling receive more promotions. As statedabove, at any given experience level, workers with more schooling will,on average, have a higher ability to learn on the job. This means that, asthe workers gain experience, those with more schooling will, on average,experience faster growth in on-the-job human capital. In turn, since aworker assigned to the low-level job in adjacent periods will experiencea wage change equal to c1 multiplied by the change in the worker’s on-the-job human capital, we have that those with more schooling will, onaverage, experience faster wage growth.

The result still holds, but the argument becomes more complicatedwhen it is possible that a worker will be on different jobs at experience


levels X and , as long as demotions are rare (which is typically theX � 1case).12 For example, suppose that all workers are assigned to the low-level job when experience equals X but some workers are assigned to thehigh-level job when experience equals . Now there are two reasonsX � 1why schooling is positively related to wage increases. The first is thatthose with more schooling typically experience higher growth in on-the-job human capital. The second reason concerns the fact that, in contrastto what was true before, a worker’s wage change does not always equalc1 multiplied by the change in on-the-job human capital. Rather, for thosepromoted, the wage change equals a convex combination of c1 and c2

multiplied by the change in on-the-job human capital, where . Thisc 1 c2 1

results in a second reason that schooling leads to larger wage increases,because, from corollary 2, we know that higher schooling means thatmore workers are promoted.

Our last prediction is that schooling is positively related to wages evenafter controlling for experience and job assignment. Let be theW(X, S)j

average wage paid to workers of experience level X and schooling levelS who are assigned to job j. Also, to generate this prediction, we nowassume that the extra productivity due directly to schooling is given by

where , and .′ ′′G(S) p lg(S), l 1 0, g 1 0 g ! 0Corollary 4. Holding all other parameters fixed, if l is sufficiently

large, then for all triplets, whereW(X, S � 1) � W(X, S) 1 0 (X, S, j) 0 ≤j j

and .X ≤ T � 1, S ≤ N � 1, j p 1, 2Proof. See the appendix.This result follows because of the effect that a worker’s starting level

of general purpose human capital has on productivity. Since this effectdoes not vary across jobs, it does not affect job assignment, but it doesaffect productivity and thus wages. In turn, since higher schooling meanshigher productivity due to the higher starting level of general purposehuman capital, the presence of this factor serves to make the wage apositive function of schooling, even after controlling for experience andjob assignment.13 Note that there is a complicating factor, because a

12 The role of demotions being rare is as follows. When a worker is demoted,instead of the worker’s wage decrease being c2 multiplied by the decrease in thethe worker’s expected on-the-job human capital, the wage decrease is a convexcombination of c1 and c2 multiplied by the decrease in the worker’s expected on-the-job human capital, i.e., the wage decrease is smaller. The result is that, if thereis a much higher frequency of demotions at a lower schooling level, then it ispossible that the lower schooling level will have a higher return to experience.

13 Although the argument is a bit more complicated, the result also holds if, aswe briefly discussed in n. 8, the extra productivity due to a worker’s starting levelof general purpose human capital were allowed to vary with job assignment. Thereason is that, since this extra productivity would still increase with schooling, asufficiently high value of l is enough to ensure that the wage increases withschooling, even after holding experience and job assignment fixed.

72 Gibbons/Waldman

worker’s schooling level affects the underlying distribution of true abilityto learn on the job, so extra schooling will affect (and possibly reduce)the expected ability to learn on the job of workers with a given experienceand job assignment. The restriction that l is large means that the firstfactor dominates and therefore that the wage increases with schooling,even after controlling for experience and job assignment.

C. Empirical Evidence

We now consider the extent to which the available empirical evidenceconcerning schooling and wage growth supports our theoretical predic-tions. Unfortunately, most of the vast literature concerning schooling andwage growth does not speak to any of our predictions. That is, becausethe goal of previous studies was not to test the theory developed above,much of the existing evidence cannot be used either to refute or to supportour predictions. Nevertheless, there are a few studies that have reportedresults that do address our predictions, and, although the evidence is abit mixed, we feel that, overall, the results support our theoreticalpredictions.

We begin with Baker et al. (1994a). In Gibbons and Waldman (1999b),we discussed a variety of findings from the Baker et al. study, but therewere two findings concerning schooling that we did not address. First,Baker et al. find that the average schooling of workers promoted into ajob level increases with the level the worker is promoted into. Althoughthis is not exactly what corollary 2 states, it would be easy to extend themodel to additional job levels to capture exactly this prediction. Second,Baker et al. run a wage regression similar to the standard Mincerian humancapital earnings function discussed below, but they also include theworker’s job level. Consistent with our model (corollary 4), they findthat higher values for schooling are associated with higher wages, evenafter controlling for experience and job level.

There are a number of other studies that find evidence consistent withschooling being positively related to promotions and with schooling beingpositively related to the wage, even after controlling for experience andjob level. For example, similar to Baker et al. (1994a), Medoff and Abra-ham (1980, 1981) run Mincerian regressions that include job level, andthey find that schooling is positively related to earnings even after onecontrols for experience and job level. They also study the factors thataffect promotion, and they find that schooling is positively related topromotion probabilities. Similarly, McCue (1996) finds (using evidencefrom the Panel Study of Income Dynamics [PSID]) that promotion prob-abilities are positively related to schooling level, although this finding isstatistically significant only for white men; for other demographic groups,the coefficients typically are positive but not statistically significant.


In a recent study using German data, Lluis (2005) also looks at theeffect of schooling on promotion rates. She first shows that schooling issignificantly positively related to promotions. She then incorporates alagged value for each worker’s wage growth into her empirical analysisand shows that this weakens the correlation between schooling and pro-motions. Notice that, consistent with our discussion following corollary2, this is exactly what our model predicts should happen. The lagged valueof a worker’s wage growth is a noisy measure of the worker’s ability tolearn on the job, and our model thus predicts that the correlation betweenschooling and promotions should be weaker when lagged wage growthis included as an explanatory variable.

We now turn to the vast empirical literature concerning schooling andearnings that employs the earnings function first explored in Mincer(1974); see Willis (1986) for an early survey of this literature and Card(1999) for a more recent survey. As is well known, Mincer estimates alog wage equation:

2log y p b � b S � b X � b X � e, (4)1 2 3 4

where y is earnings, S is the number of years of schooling, X is labormarket experience, and e is an error term. Mincer’s finding that isb 1 02

consistent with our first prediction: initial compensation (i.e., the wagereceived when entering the labor market) is increasing in the schoolinglevel. This finding is also consistent with our second prediction: the ab-solute (rather than percentage) return to an extra year of experience isincreasing in schooling. The reason is that, given the log specification thatMincer studied, means that the percentage return to schooling isb 1 02

constant with experience, implying that the absolute return to schoolingis increasing with experience.

But just because (4) is the specification that Mincer and others havestudied, it is not necessarily the correct specification. For example, in hisoriginal study, Mincer added to (4) an experience-schooling interactionterm and found a significant negative coefficient on this term. This findingintroduces the possibility that the absolute return to an extra year ofexperience is not positively related to the schooling level but rather iseither independent or negatively related.

A recent study concerning schooling and wage rates that does not takethe standard Mincerian approach is Farber and Gibbons (1996). (Altonjiand Pierret [2001] is related but takes an approach more similar toMincer’s.) In testing a pure-learning model of wage dynamics, Farber andGibbons estimate the wage level (as opposed to log-wage) specificationgiven in equation (5):

2y p b � b S � b X � b X � b (S # X)1 2 3 4 5

� b U � b (U # X) � e, (5)6 7

74 Gibbons/Waldman

where U is a variable correlated with ability but unobserved by employers(such as a test score). The pure-learning theory that Farber and Gibbonsconsider predicts and : the absolute return to an extra yearb p 0 b 1 05 7

of schooling is independent of experience, whereas the absolute return toa higher value for U is increasing with experience.

Farber and Gibbons estimate regression (5) with and without the Uand terms. When they include controls for changes in the returnU # Xto education over time, they find results consistent with their predictionthat , both when they include the U and terms and whenb p 0 U # X5

they do not. When the U and terms are included, their findingU # Xthat is consistent with our model.14 But when the U andb p 0 U #5

terms are not included, our model predicts , so these Farber-X b 1 05

Gibbons results are problematic for our theory. But Farber and Gibbonsalso find that, when the controls for changes in the return to educationover time are not included, then when the U and terms areb 1 0 U # X5

not included, as our theory predicts. It is possible that data limitationsin the Farber-Gibbons sample created a strong relationship between aworker’s schooling level and the date at which the worker entered theirsample, so that the true effect of schooling on the return to experiencein this data set is better estimated without controls for changes in thereturn to education over time.15

More recently, Rubinstein and Weiss (2005) have also considered howschooling affects the return to labor market experience. They employthree data sets—the Current Population Survey (CPS) short panel, thePanel Survey of Income Dynamics (PSID), and the National LongitudinalSurvey of Youth (NLSY)—and directly consider how the percentage

14 The logic here is as follows. Our model assumes a correlation between school-ing (Si) and ability to learn on the job (vi), captured by the term In ourB(S ).i

model, this correlation with vi is the only way that schooling might have a mea-sured effect on the return to experience. Thus, if variables correlated with abilityto learn on the job are included in the regression specification and there is noremaining correlation between schooling and ability once those variables are in-cluded, then the prediction of our model is what Farber and Gibbons found:

andb p 0 b 1 0.5 715 Their sample consisted of workers in the NLSY between the ages of 14 and

21 on January 1, 1979, and data for these workers between 1979 and 1991, wherea worker typically entered the sample when he finished his schooling. Given thissample, there is likely a strong relationship between a worker’s schooling leveland the year when he enters the sample, i.e., those with high values for schoolingwill typically enter later. This causes two potential problems for finding the trueimpact that schooling has on the return to experience. First, there are likely tobe few data points for many of the workers whose schooling levels are high.Second, because many of the workers with high schooling levels enter the samplelater, some of the impact that education has on the return to experience might bepicked up by the controls Farber and Gibbons included for changes in the returnto education over time.


growth in earnings over different career intervals varies with schoolinglevel. For each of their three data sets, Rubinstein and Weiss find that, atleast for the first decade of experience, the return to labor market ex-perience increases with the schooling level. In particular, they find thatwage growth during this first decade is about 50% for high school grad-uates compared to 80% for workers who have 4 or more years of college.(Note that translating these figures into absolute changes rather than per-centage changes would increase the difference, since workers with 4 ormore years of college have higher starting salaries.) Although Rubinsteinand Weiss do not control for changes in the return to education overtime, as Farber and Gibbons do, the magnitude of these differences sug-gests that, in their samples, more education would increase the return tolabor market experience early in careers even if such controls wereincluded.16

In summary, although there is not much direct evidence concerningour theoretical predictions, on three of the predictions—schooling beingpositively related to promotions, the correlation between schooling andpromotions being weaker when a noisy measure of a worker’s ability tolearn on the job is included, and schooling being related to the wage evenafter controlling for experience and job level—the available evidence issupportive. For the remaining prediction—schooling being positively re-lated to the return to experience—the evidence is mixed. (We take as giventhat the evidence is consistent with our prediction that schooling posi-tively affects the starting wage.) Hence, all told, we feel that the availableevidence supports our theoretical model, but clearly more evidence isneeded before reaching any final conclusions.

III. An Explanation of Cohort Effects

In this section, we provide an explanation for why wage and promotiondynamics inside firms might exhibit cohort effects. We begin by discussingthe recent empirical literature on cohort effects and providing a basic in-tuition that explains how task-specific human capital can produce cohorteffects, even in the spot market framework that we analyze. We then presenta variant of the model from Section II, show that it exhibits cohort effects,and derive some other possible implications of task-specific human capitalin this model. Finally, we discuss possible results from integrating our ideaof task-specific human capital with Lazear’s skill-weights approach.

Our idea of task-specific human capital is closely related to occupation-

16 See Habermalz (2003) for a recent study using the CPS that allows for non-linearity concerning how the return to education varies with experience and findsresults that suggest that education positively affects the return to experience overa substantial part of workers’ careers.

76 Gibbons/Waldman

and industry-specific human capital.17 In each case, human capital is spe-cific to the nature of the work, not specific to the firm. Hence, whencapital is accumulated, multiple firms will typically value the capital, somost (or even all) of the value of the capital will be reflected in the worker’swage.18 The main difference between our idea of task-specific humancapital and occupation- and industry-specific human capital is how theidea is applied. For example, the specific issue we address here is cohorteffects, while in Gibbons and Waldman (2004) and in the conclusion ofthis article, we discuss how the idea can be applied to issues of job design.Because of these and other potential applications, we believe that task-specific human capital has much wider applicability than suggested (sofar) by the occupation- and industry-specific human capital literatures.

Another argument closely related to ours is the classic argument madeby Adam Smith in the Wealth of Nations concerning returns to special-ization. Smith’s (1776) argument was that, due to learning by doing atthe level of the task, productivity can be enhanced by having each jobentail fewer tasks. We believe that Smith was correct in focusing on learn-ing by doing at the level of the task as an important idea for thinkingabout organizations, and part of the goal of this section is to begin aninvestigation of other implications of this idea for the design and operationof organizations.

17 For a formal analysis of occupation-specific human capital, see Weiss (1971).18 Interestingly, recent empirical papers such as Neal (1995), Parent (2000), Kam-

bourov and Manovskii (2002), and Kwon and Meyersson Milgrom (2003) arguethat firm-specific human capital is less important than traditionally thought, whileindustry- and occupation-specific human capital are more important. For example,in an analysis of displaced workers, Neal shows that taking into account industryexperience yields that employer tenure is unimportant in wage determination,while industry experience is strongly significant; similarly, Kambourov and Man-ovskii employ the PSID and show that including occupational experience in theirregression analysis causes both employer tenure and industry experience to be-come unimportant, while occupational experience is strongly significant. Ourfocus on task-specific human capital is similar in spirit to these findings. It isworth noting, however, that, in some cases, task-specific human capital can alsobe firm specific. An interesting recent paper related to this point is Huckman andPisano (2003). In this study, a surgeon’s performance at a given hospital is sig-nificantly positively related to the surgeon’s annual volume of procedures at thathospital, but the surgeon’s performance at the given hospital is not significantlyimproved by increases in the surgeon’s annual volume of procedures at otherhospitals. A plausible interpretation is that, in this case, human capital is taskspecific but the team of individuals the surgeon works with is important. Hence,to the extent that the other team members are not mobile, this is a case in whichtask-specific human capital is also firm specific. Note also that the discussion ofLazear (2003) later in this section is related to this issue.


A. Empirical Evidence and Basic Argument

There are two well-known recent studies of cohort effects: Baker,Gibbs, and Holmstrom (1994b) and Beaudry and DiNardo (1991). Bakeret al. study the personnel records of managers in a single firm over a 20-year period. Regarding cohort effects, they find that a cohort’s averagewage upon entering the firm is an important determinant of the cohort’saverage wage years after entry. Of course, this would not be surprisingif a cohort’s average wage upon entry reflected differences in the com-position (and thus average productivity) of a cohort. But Baker et al. findno evidence that composition differences in race, sex, age, or educationcan explain a cohort’s average wage years after entry.

Beaudry and DiNardo (1991) find cohort effects in panel data coveringa large cross section of occupations and industries. Their initial findingis that the unemployment rate in the year a worker enters the firm affectswages years later (a similar result is found in Akerlof, Rose, and Yellen[1990]). Beaudry and DiNardo conduct further analyses that allow boththe current unemployment rate and the lowest unemployment rate sincethe entry year to affect current wages. They find that each of these threeregressors (entry unemployment, current unemployment, and lowest un-employment) is significant on its own, but that the lowest unemploymentrate since entry is the only significant regressor when all three are included.These Beaudry-DiNardo findings suggest alternative interpretations ofthe Baker et al. cohort effect. For example, the apparent influence of acohort’s entry wage on its current wage might disappear if one controlledfor the highest wage paid to any entering cohort since the first cohortentered.19

In Gibbons and Waldman (1999b), we argued that the spot marketcontracting framework we considered in that paper (and we are consid-ering in this article) is inconsistent with the cohort effects found by Bakeret al. (1994b) and with the range of findings by Beaudry and DiNardo(1991). Our argument was based on Beaudry and DiNardo’s interpre-tation of their own results in terms of long-term insurance contracts, akinto those analyzed by Harris and Holmstrom (1982). Since our 1999b

19 MacLeod and Malcomson (1993) put forth a model that can explain the cohorteffects found by Beaudry and DiNardo (1991). In their analysis, fixed-wage con-tracts are employed to induce efficient general investments by the firm (i.e., in-vestments that are not specific to a particular worker). These contracts are suchthat the wage moves only when one of the outside options changes, so that therelationship will be terminated if the wage remains unchanged (even if it is efficientfor the relationship to continue). Although such contracts can explain the Beaudryand DiNardo finding that the best labor market conditions since the start of aworker’s job have a significant positive effect on the worker’s current wage, weare skeptical concerning the ability of such contracts to match other importantfindings in the empirical literature concerning careers.

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model assumes that wages are determined each period by competitionbetween a worker’s current employer and prospective employers (andthat all parties are risk neutral), long-term insurance contracts are neithernecessary nor feasible, so we concluded that cohort effects were incon-sistent with our model. In this section, we show that we were incorrectin our previous conclusion: a competitive framework is potentially con-sistent with cohort effects, even after controlling for composition differ-ences, because a cohort’s prior experiences can have a significant effecton actual or perceived productivity. In the next subsection, we developan argument along this line and show how it can explain the Baker et al.(1994b) finding that a cohort’s entry wage is correlated with the cohort’swage years later.

There are two key aspects to this new model: (i) the state of the economycan be either good or bad; and (ii) human capital acquisition is taskspecific, so some of a worker’s acquired human capital goes unutilizedwhen a worker is promoted and is assigned a new set of tasks. When theeconomy is in the bad state, a higher proportion of young workers enterthe firm at the low-level job. Since human capital is task specific, someof a worker’s acquired human capital goes unutilized when she is pro-moted, so a cohort hired in the bad state has a low average wage yearslater. To capture the Baker et al. (1994b) finding that low wages at entrypredict low wages later, we require that the cohort’s entry wage be lowin the bad state. If the two states are sufficiently different, this occurs inour model because the bad state has both more entry workers on thelow-level job and a lower entry wage for any young worker assigned tothe high-level job in both states of the world.20

20 Although we focus on the Baker et al. (1994b) finding that a cohort’s entrywage is correlated with the cohort’s average wage years later, we believe that ourbasic argument can also capture the Beaudry and DiNardo findings. As discussedabove, they find that when entry unemployment, current unemployment, andlowest unemployment since entry are all included as explanatory variables in awage regression, it is only the lowest unemployment rate since entry that matters.Our approach will capture this finding (or at least that the lowest unemploymentrate since entry matters more) if the lowest unemployment rate since entry ismore highly correlated with prior assignments to high-level jobs and thus morehighly correlated with the amount and quality of the cohort’s human capital inthe current period. We also believe that task-specific human capital can explainWelch’s (1979) findings concerning cohort size. Welch finds that being a memberof a large cohort has a significant negative effect on the worker’s wage when sheenters the labor force and a smaller, but still negative, effect later in the worker’scareer. He interprets this as limited substitutability across cohorts, where thislimited substitutability gets less important as the cohort ages. An alternative ex-planation is that human capital is task specific: when a large cohort enters thelabor market, because there are a limited number of high-level positions availableto entry-level workers, the cohort earns on average lower wages and works onaverage in lower-level jobs. In turn, if human capital is task specific, then there


One interesting aspect of this argument is that, although we explainthe Baker et al. (1994b) finding that a cohort’s entry wage is correlatedwith the cohort’s wage years later, this is not because the entry wage itselfmatters. Instead, what matters is the proportion of workers who start atthe low-level job, which affects the number and productivity of workersin the high-level job years later. This explanation for the Baker et al.cohort effects is consistent with further evidence reported by Baker etal.: although they find no evidence that composition differences in race,sex, age, and education across cohorts explain their cohort-effects finding,they do find that a higher proportion of new workers entered the firmat lower job levels during the period 1976–85, which is when the entrywage was low. This is exactly what this explanation predicts: there is alow wage upon entry when more workers enter the firm in the lower joblevels, and it is the low skill acquisition (or the lack of transferability ofskills acquired in the low job) that affects wages years later.21

Our argument here is similar in spirit to Ellwood’s (1982) finding thatunemployment spells when workers are young have long-lasting negativeeffects on wages. This finding can be explained by positing that humancapital is accumulated more quickly in employment than in unemploy-ment, so a spell of unemployment when a worker is young negativelyaffects future wages. We take this argument one step further by assumingthat the nature of human capital accumulation is affected not only bywhether a worker is employed or unemployed but also by the type ofjob the worker holds. In particular, if workers who enter the labor marketduring recessions enter, on average, in lower-level jobs, then the humancapital accumulated by such workers will be less valuable and so willreduce future wages (i.e., cause cohort effects), much like Ellwood’sfinding.22

It is important to note that, although our analysis focuses on how task-

is a subsequent negative effect on wages as the cohort ages, because the humancapital the cohort acquires while young is less valuable after promotions takeplace.

21 Note that what we call the “state of the economy” could just as well be thestate of the industry (or even the state of the firm, if we were to introduce somefirm-specific human capital into the model). These two reinterpretations are quiteplausible for the financial services firm that Baker et al. (1994a, 1994b) study.

22 Our argument can be similarly related to the hysterisis literature of the 1980s.This literature discussed and empirically analyzed the idea that the natural rateof unemployment is path dependent (e.g., higher unemployment in one periodincreases the natural rate of unemployment in future periods; see Cross [1988]for articles on the subject). We are similarly focused on why history matters, butour focus is on wages rather than on unemployment and our mechanism throughwhich history matters is human capital accumulation. See Cross (1990) for ahysteresis argument concerning unemployment that also depends on human cap-ital accumulation.

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specific human capital can explain cohort effects, the basic point is moregeneral: cohort effects can arise in the spot market contracting frameworkwe explored in our 1999b paper whenever a cohort’s past experiencesaffect subsequent actual and/or perceived productivity. For example, sup-pose that there is no task-specific human capital but that the rate at whichworkers acquire human capital depends on how “busy” the firm is (e.g.,workers learn more from “interesting” projects, but such projects arescarce in slack times). Then a cohort that enters the firm when it is notbusy may receive a low wage upon entry and also do poorly years laterbecause little was learned during the entry period.

Another argument one can construct in this vein builds on Rosen’s(1972) classic analysis of learning in the labor market.23 In Rosen’s analysis,jobs vary in terms of their learning opportunities (or, equivalently, in therate of on-the-job human capital accumulation) and workers are willingto work for less at jobs with better learning opportunities. Given this,suppose that, when an economy is in a recession, wages are low acrossthe board. Then it would not be surprising if during recessions youngworkers were willing to “purchase” less in terms of learning opportunities(i.e., accumulate less human capital). The result is that workers who enterthe labor market during a recessionary period would earn, on average,less when young and also less when old because of the smaller accu-mulation of human capital.24

B. Model

There are three main differences between the model explored here andthe one analyzed in Section II. The first difference concerns the idea thathuman capital is now task specific. But to keep the model simple, werestrict this effect to job 2. That is, we assume that job 1 is analogous toboth jobs in Section II (and all the jobs in our 1999b paper): past labor

23 We thank Derek Neal for suggesting the following argument to us.24 Related to this discussion, one prediction of the specific model we consider

is that young workers who start their careers on a high-level job during a recessionwill on average do better later in their careers than young workers who start theircareers on a high-level job during a nonrecessionary period. The reason is thatonly the very highest ability young workers start their careers on a high-leveljob during a recession. The reason we mention this point here is that, althoughthis is a prediction of the specific model we consider, it is not a prediction of thegeneral approach. That is, if all workers are less busy during recessions and thusaccumulate human capital more slowly during recessionary periods, then theyoung workers who start their careers in high-level jobs during a recession shouldactually earn less when they become old. Similarly, if young workers purchaseless learning during recessionary periods, as in our extension of Rosen’s argument,then, again, young workers who start their careers in high-level jobs during arecession should actually earn less when they become old. We thank Dan Bern-hardt for suggesting this issue to us.


market experience improves a worker’s current productivity on job 1,and the extent of this productivity improvement is independent of howmuch of the past experience was in which jobs. Formally, we assume that,if worker i has experience xit in period t, then this worker’s output in job1 in that period is

y p d � c [v f(x ) � � ], (6)i1t 1 1 i it i1t

which is similar to equation (2) in Section II. For job 2, however, wemake a different assumption. Now skill acquisition is task specific: pastexperience improves a worker’s current productivity on job 2, but ex-perience on job 2 creates a larger productivity improvement than doesexperience on job 1. Formally, if worker i has experience xit in period t,then this worker’s output in job 2 in that period is

y p d � c [v f(x � ax ) � � ], (7)i2t 2 2 i i2t i1t i2t

where xijt is the past experience that worker i has accumulated in job jprior to period t, so and is a parameter thatx � x p x a � (0, 1)i1t i2t it

reflects the extent to which experience on job 1 creates a productivityimprovement on job 2. That is, because acquired human capital is taskspecific in job 2, when a worker is promoted from job 1 to job 2, aprevious period that the worker spent in job 1 counts as only a of aperiod in the skill-acquisition function.25 Equations (6) and (7) no longerinclude the productivity term , because it is not needed for any ofG(S )ithe results below. We do, however, continue to assume heterogeneity inschooling, because of the role it plays in initial job assignments.

The second difference between this section’s model and the model ofSection II is that now there are two states of the world, where the stateof the world affects relative productivities and hence job assignments. Thesimplest way to capture this possibility is to have the state of the worldaffect the productivity of the high-level job.26 In particular, we assumethat there are two intercepts for the high-level job. If the economy is inthe good state, then ; if it is in the bad state, then , whereG Bd p d d p d2 2 2 2

. We further assume that the probability that the economy is inG Bd 1 d2 2

the good state in any period is q, independent across periods, and thatthe realization of the state is determined and publicly observed at the

25 As indicated, to keep the model simple, we have allowed task-specific humancapital only in job 2. We could also allow task-specific human capital in job 1.If we then further restricted the analysis to parameterizations for which there arefew (if any) demotions (which is what most of the data show), then this alternativespecification would produce basically the same results as we derive.

26 We could assume that the state of the world affects productivity in both jobs.All of our qualitative results would still go through as long as the effect that thestate of the world has on the high-level job is larger than its effect on the low-level job.

82 Gibbons/Waldman

Fig. 2.—Expected output for a worker with one period of labor market experience inthe good state of the world as a function of ability to learn on the job and current andprevious job assignments.

beginning of each period (before wages and job assignments are deter-mined). When the economy is in the good state, holding all else equal,more workers will be assigned to the high-level job, because .G Bd 1 d2 2

Figure 2 depicts the expected output of a worker with one period of labormarket experience in the good state of the world as a function of the jobassignment, the number of prior periods the worker was in job 2, andthe worker’s expected ability to learn on the job. Note that changing thestate of the world to bad would cause a parallel shift downward of thecurves depicting assignment to job 2.27

The third difference between this section’s model and the model ofSection II is that, because we want to explore how the history of jobassignments for young workers affects job assignments and wages whenworkers become old, we no longer restrict the analysis to parameteri-zations for which all entry workers are assigned to the low-level job. Tothe contrary, we now restrict the analysis to parameterizations where someentry workers are assigned to the high-level job. In particular, we assumethat a worker with the highest schooling level (N) is assigned to the high-level job in the worker’s first period in the labor market if the state of

27 We have drawn the figure so that a worker who was in job 1 in the previousperiod produces more in job 2 than in job 1 if the worker’s ability to learn onthe job is sufficiently high. This condition is not required.


the world is good and that a worker with the lowest schooling level (1)is assigned to the low-level job in the worker’s first period in the labormarket if the state of the world is bad.28

Although the model in this section seems closely related to the modelin Section II and the main model of our 1999b paper, the model in thissection is in fact much more challenging analytically. In Section II andour earlier work, a worker’s output in any period was independent ofthe worker’s job assignment history. As a result, in each period, the as-signment of workers to jobs was simple: a worker is always assigned tothe job that maximizes that period’s expected output. But that assignmentrule is not optimal here. Instead, the current job assignment is now partlyan investment in the worker’s expected future productivity, so theworker’s optimal assignment may not maximize current productivity. Be-cause of this, to keep the intuition behind cohort effects clear, we beginby assuming that careers last two periods (i.e., ). Note that, for thisT p 2part of the analysis, we use the terms “young” and “old” to refer toworkers in their first and second periods in the labor market, respec-tively.29 We then consider what happens if careers last for more than twoperiods.

C. Analysis

As indicated, we start by considering the case . We begin ourT p 2analysis of this case by considering how workers are assigned to jobs.Consider, first, what happens when workers are old. Since there are nosubsequent productivities to be concerned with, when a worker is old,she will be assigned to the job that maximizes her expected output thatperiod. But the solution to this second-period assignment problem de-pends on the worker’s job assignment when young. In particular, becausesome human capital goes unutilized when a worker is promoted, moreold workers will be assigned to the high-level job this period if moreyoung workers were assigned to the high-level job last period.

Now consider assignments when workers are young. In contrast towhat happens when workers are old, the optimal job assignments whenworkers are young do not simply maximize expected productivity in thecurrent period. Rather, the assignments now take into account the effectthat this period’s assignment has on next period’s optimized productivity.

28 We also restrict the analysis to parameterizations such that at least one school-ing group in its first period in the labor market is assigned to the high-level jobin the good state of the world and to the low-level job in the bad state.

29 In this case, we assume that a is sufficiently close to 1 and/or f(1) is sufficientlysmall that some promotions occur when a worker becomes old. If a were suf-ficiently small and f(1) were sufficiently large, there would be no promotions inequilibrium because the underutilization of acquired human capital associated witha promotion would make all promotions unprofitable.

84 Gibbons/Waldman

In particular, because assigning a young worker to the high-level jobincreases the worker’s output in the following period if she is again as-signed to the high-level job, assignments will be biased toward the high-level job relative to assignments that optimize today’s productivity. Theother important aspect of assignments (both when workers are youngand when they are old) is that more workers are assigned to the high-level job in the good state of the world than in the bad state. This followsbecause the high-level job is even more productive relative to the low-level job in the good state of the world.

We formalize the above discussion in proposition 2. Let denote theKvY

critical value of such that young worker i in period t is assigned to theevit

high-level job if and is assigned to the low-level job ife K e Kv ≥ v v ! v ,it Y it Y

where is the state of the world in period t. Similarly, let K,JK p G, B vO

denote the critical value of such that old worker i in period t is assignedevit

to the high-level job when and is assigned to the low-level jobe K,Jv ≥ vit O

when , where ,B is again the state of the world in periode K,Jv ! v K p Git O

t and denotes the worker’s job assignment in period . Finally,J p 1, 2 t � 1let denote the critical value of if the assignment of a young workerK ev * vY it

in state K were to maximize current productivity, and let denote theK,Jv *O

critical value of if the assignment of an old worker in state K were toevit

maximize current productivity, given that the worker was assigned to joblevel J in the previous period.

Proposition 2. If , then job assignments satisfy i–iv:T p 2i) for .K Kv ! v * K p G, BY Y

ii) for all pairs, and .K,J K,Jv p v * (K, J) K p G, B J p 1, 2O O

iii) and for .G B G,J B,Jv ! v v ! v J p 1, 2Y Y O O

iv) for .K,2 K,1v * ! v * K p G, BO O

Proof. See the appendix.We now turn our attention to cohort effects. Let denote the averageKWY

wage paid to young workers in state of the world K and denote′K,KWO

the average wage paid to old workers when the current state of the worldis K and the state of the world in the previous period was K′.

Proposition 3. Suppose that . Then, holding all the otherT p 2parameters fixed, there exists a value such that, if then wages′ ′G G Gd d 1 d ,2 2 2

satisfy i and ii:i) G BW 1 W .Y Y

ii) forK,G K,BW 1 W K p G, B.O O

Proof. See the appendix.Proposition 3 tells us that, if is sufficiently large, then the modelGd2

exhibits a cohort effect like that found by Baker et al. (1994b): a cohortthat enters a firm with a high entry wage will earn more when the cohortbecomes old. To see this, consider, first, what happens when a cohortbecomes old. Holding constant this period’s state of the world, the oldcohort’s average pay will be higher when the previous period’s state was


good rather than bad. This result occurs because more young workersare assigned to the high-level job in the good state, and being assignedto the high-level job when young increases productivity in the high-leveljob when old.

Now consider what happens when a cohort enters the labor market.As just stated, relative to entry in the bad state, more workers are assignedto the high-level job in the good state. In turn, there are two factorsaffecting whether the average wage for young workers is higher in thegood state or the bad state. First, because the high-level job is moreproductive in the good state, the average wage is higher in the good state,since those assigned to the high-level job in both states are paid more inthe good state. But, second, again because the high-level job is moreproductive in the good state, some workers are assigned to the high-leveljob only in the good state, and these workers may earn on average eithermore or less in the good state than in the bad state. They can earn morebecause the high-level job is more productive in the good state, but theycan earn less because a young worker is willing to work in the high-leveljob for less given that human capital accumulated on that job is on averagemore valuable when the worker becomes old. The assumption that isGd2

sufficiently large ensures that the first of these factors is the dominantone and, thus, that the model exhibits cohort effects in the sense that agood state of the world when a cohort is young translates into high wagesboth when the cohort is young and when it is old.

As discussed earlier, one interesting aspect of our explanation for theBaker et al. (1994b) finding is that, although a cohort’s average wage whenyoung is correlated with its average wage when old, the wage when youngis only a proxy for the cohort’s job assignments when young (and con-sequently for the amount and quality of human capital that the cohortacquires when it is young). This is important for two reasons. First, froma competitive standpoint, it would be surprising for the average wage acohort earns when young to affect the average wage the cohort earnswhen it ages, but it is not surprising for the amount and quality of thehuman capital that a cohort accumulates when it is young to affect thecohort’s average wage when it ages. Second, as discussed earlier, this aspectof our explanation for the Baker et al. cohort effects finding is consistentwith another aspect of their empirical analysis: in their data, when theentry wage was low, more workers entered the firm at lower job levels.

There are two other pieces of empirical evidence consistent with theabove analysis that are worth mentioning. First, Kahn (2004) uses theNLSY to study the effect that the national unemployment rate in the yeara worker graduates from college has on the worker’s earnings over hisor her career. Consistent with propostion 3, Kahn finds that graduatingfrom college in a bad year has large negative effects on wages, both in

86 Gibbons/Waldman

the year of graduation and much later in the career.30 Second, Devereux(2002) uses the PSID to show that the average education level of newhires within occupations is higher when the economy is contracting ratherthan expanding. This is consistent with our model in the sense that prop-osition 2 tells us that, for both the low-level and high-level jobs, theaverage education level for new hires is higher when the state of theeconomy is bad rather than good.31

We now consider what happens when In this analysis, we con-T 1 2.sider both cohort effects and other aspects of wage and promotion dy-namics when there is task-specific human capital.32

Proposition 4. If then i–iii are satisfied:T 1 2,i) Holding all other parameters fixed, there exists a value suchG′d2

that, if then there is a positive correlation between aG G′d 1 d ,2 2

cohort’s average wage when young and its expected compensationover the last periods that the cohort is in the labor market.T � 1

ii) Hold all other parameters fixed other than a and let a′ be thehighest value for a such that Gd � c vf(T � 1) ≥ d �1 1 2

for all Such a value′c vf(a (T � 1)) f � B(0) ≤ v ≤ f � B(N).2 L H

will necessarily exist if f(0) is sufficiently small. If then′a ! a ,there exists a smallest value such that any worker′ ′t , t ≤ T � 1,

30 Neal (1999) finds a significant rate of career changes early in careers, evenfor the college educated. On the one hand, this finding throws some doubt onour model being the correct explanation for Kahn’s result. The reason is that, ifmany, and possibly most, older workers are in different industries and occupationsthan they were when young, then task-specific human capital acquired on theirinitial jobs should matter relatively little in terms of their wages when old. Onthe other hand, a variant of the model that we consider can explain Kahn’s resultand be consistent with Neal’s finding. Consider a model with multiple industriesand occupations and task-specific human capital. Further, suppose that, during arecession, a young worker is less likely to initially find a job in his preferredoccupation and industry, so that career changes are more likely for workers whoenter the labor force during recessions. In such a model, there would be cohorteffects both for the reason we identify above and, possibly more importantly,because workers who enter the labor force during a recession would earn lesswhen old due to a higher number of career changes. Given this possibility, itwould be interesting to know how career changes for young workers vary withthe business cycle.

31 Kwon and Meyersson Milgrom (2004) find preliminary evidence that cohorteffects are driven by promotions. In particular, young workers who enter thelabor market at a particular job level during a recession are promoted more slowlythan similar workers who enter the labor market at the same job level during aboom. This finding would not be consistent with the precise model analyzedabove, but it would be consistent with the related model mentioned briefly above,in which the rate of accumulation of human capital depends on how busy thefirm is.

32 As for our analysis of we continue to restrict the analysis to param-T p 2,eterizations such that there are some promotions in equilibrium (see n. 29).


who spends her first t′ periods in job 1 is not promoted in anyof her last periods in the labor market.′T � t

iii) There are parameterizations in which some promotions are as-sociated with wage decreases.

Proof. See the appendix.Proposition 4 tells us that there are three results that characterize the

case .33 The first, captured in i, is that, if is sufficiently large, thenGT 1 2 d2

the model continues to exhibit cohort effects: the average wage paid toa cohort in its first period in the labor market is correlated with averagewages paid to the cohort later. As above, this occurs because acquiredhuman capital is task specific, so the state of the world in the worker’sfirst period in the labor market influences not only job assignments andwages in that first period but also, consequently, job assignments andwages later in these careers.

The second result, captured in ii, is that, if the task specificity of humancapital is sufficiently large (i.e., a is sufficiently small), then a worker whospends substantial time in job 1 at the beginning of his career can getstuck in that job with no possibility of a subsequent promotion. That is,it can be the case that a worker who spends his first periods in′t (! T)the low-level job has a zero probability of being promoted to the high-level job in any of the last periods. There are two factors leading′T � tto this result. First, as the number of periods in job 1 grows, the loss inlifetime productivity upon promotion becomes large (because some job1 human capital will be underutilized after promotion). Second, again asthe number of periods in job 1 grows, the gain in lifetime productivityupon promotion from human capital acquisition on job 2 becomes small,because fewer periods remain in the worker’s career.

This second result is consistent with the Baker et al. (1994a) evidenceconcerning promotion rates for workers currently in level 2 as a functionof the number of years the worker has been in level 2 or below. Extendingthe logic from the above discussion to the case of three job levels yieldsthat a worker who spent many years in level 2 or below would have littleor no probability of further promotion. The logic is that the large numberof years in level 2 or below means that much human capital would gounutilized with a promotion, and fewer years would exist after a pro-motion to build human capital. Baker et al. (1994a) find just such a result:

33 When workers are in the labor market for more than two periods, this modelis similar to the analysis in Gibbons and Waldman (1999b). Thus, as was true forthe model of Sec. II, this model should be consistent with a variety of findingsin Baker et al. (1994a, 1994b) and elsewhere, such as that wage increases are seriallycorrelated, real wage decreases are not rare but demotions are, and workers whoreceive large wage increases early in their stay at one level of the job ladder arepromoted quickly to the next. Here we focus on results other than those foundin our 1999b analysis.

88 Gibbons/Waldman

the promotion rate is less than 2% for workers who were at level 2 orbelow for at least 15 years (compared to an overall promotion rate forall workers currently at level 2 of 19%).34

The last result, captured in iii, is that there are parameterizations inwhich some (possibly many) promotions will be associated with wagedecreases. This result also follows from the presence of task-specific hu-man capital (and it differs from our 1999b model, which did not havetask-specific human capital). Because being in the high-level job increasesproductivity in the high-level job in subsequent periods as shown in figure2, in period t, a worker will sometimes prefer a lower wage on the high-level job to a higher wage on the low-level job. Extending this argumentto incorporate the worker’s wage on the low-level job in period t � 1yields that, if task-specific human capital is significant (i.e., a is small),some promotions may be associated with wage decreases.35

Notice that this argument is similar to a classic argument concerningwage profiles over the career (e.g., Rosen 1972; and Topel and Ward 1992).In that argument, there are no promotions and no turnover, but firmsvary in terms of the rate at which a worker in the firm accumulates humancapital over the worker’s career. As a result, young workers are willingto work at high-capital-accumulation firms for lower starting wages be-cause of the effect on subsequent productivity and subsequent wages. Theresult is that firms with fast accumulation of human capital have lowerstarting wages but higher wage growth over the career. Our logic here isthe same except that fast accumulation of human capital is associated withpromotions, so some promotions are likely to be associated with wagedecreases.

Given the above, the obvious question is, to what extent are somepromotions associated with wage decreases even though on average pro-motions are associated with large wage increases? Baker et al. (1994a,1994b) do not report the percentages of their promotions associated withwage increases and decreases. However, McCue (1996) also studies pro-

34 Our general approach also provides a second potential explanation for thisfinding: workers who work many years in each of the two lowest job levels mayhave sufficiently low ability to learn on the job that it is rarely optimal to promotesuch workers to level 3.

35 In our model, a worker’s wage each period equals that period’s expectedproductivity, so the idea that promotions are sometimes associated with wagedecreases means that the model also predicts that promotions will sometimes beassociated with productivity decreases. Task-specific human capital, therefore, isan additional explanation to the ones identified by Lazear (2004) for why pro-ductivity might sometimes fall after a promotion occurs. Further, the fact thatpromotions can be associated with wage decreases means that, in addition to theexplanation developed in Sec. II concerning schooling, task-specific human capitalserves as an explanation for why adjacent job levels can have overlapping wagedistributions.


motions and wage changes, and she does report information concerningthe percentages of promotions associated with wage increases and de-creases. Consistent with Baker et al. (1994a, 1994b) and many other stud-ies, McCue finds that promotions on average are associated with largewage increases. But she also finds that roughly 25% of promoted workersin her sample received real wage decreases.36 This finding is inconsistentwith our 1999b paper and the model of Section II, but it is consistentwith a world where task-specific human capital is important.

D. Integrating Task-Specific Human Capital with Lazear’sSkill-Weights Approach

In this concluding subsection, we briefly discuss what would happenif one were to integrate Lazear’s (2003) skill-weights approach with ourapproach emphasizing task-specific human capital. As discussed in theintroduction, in Lazear’s skill-weights approach, each firm is characterizedby a weighting of general skills, where the weights vary across firms. Inour formulation, in contrast, each job within a firm is characterized bya weighting of general skills, where the weights vary across jobs withina firm. In the particular model we consider, these job-specific weightsvary across jobs that are at different levels of the firm’s job ladder.

Now suppose that one were to consider a model that captures bothideas: human capital acquired at a particular job within a particular firmis differently valued if the worker is promoted and/or if the workerswitches firms. One advantage of such a model is that it would capturein a single framework both the empirical results that motivated our anal-ysis and the empirical results that motivated Lazear’s. That is, such amodel should be capable of capturing not only the various cohort-effectand related findings discussed above but also the various findings thatLazear mentions, such as that the wage loss associated with mass layoffsis greater than the wage loss associated with other types of turnover (seeJacobson, LaLonde, and Sullivan 1993).

In addition to capturing both sets of empirical findings with a singletheoretical framework, we believe that such a model would also likelymake novel predictions not captured by either Lazear’s model or ourmodel alone. For example, consider again the wage loss associated witha mass layoff. In Lazear’s model, where the skill weighting can be inter-preted as constant across all jobs within a firm, the prediction seems likelyto be that the increased wage loss should not differ much across workers.But by adding our idea that the value of accumulated human capital varies

36 McCue reports the real wage increase for promoted workers at the 25thpercentile broken down by race and gender. She finds that, for white men, thisvalue is negative but small, while for the other categories—white women, blackmen, and black women—the value is positive but small.

90 Gibbons/Waldman

across jobs within a firm, one gets a more nuanced prediction: now theincreased wage loss seems likely to be larger for laid-off workers for whommany workers in similar jobs were also laid off and for whom there arefewer similar jobs elsewhere.

IV. Limitations of Our Approach

We feel that, in Gibbons and Waldman (1999b) and here, we have madea strong case that our approach is able to capture a variety of facts con-cerning wage and promotion dynamics in a large class of settings. Butthere are some settings in which we believe our approach does not applyand some facts that we are not able to capture. In this section, we discusssome limitations of our approach.

We start with settings in which we believe our approach does not apply.There are two key interrelated aspects of our approach that are importantfor generating predictions but that also limit its applicability. First, thereis only spot market contracting; there are no long-term contracts. Second,there is no firm-specific human capital (and the value of task-specifichuman capital does not vary across employers). In combination, thesetwo assumptions imply that wages in every period are determined by thewage offers of prospective employers. In a setting in which each level ofa firm’s job ladder exhibits a significant proportion of workers both en-tering and exiting the firm (as is true for the Baker et al. [1994a, 1994b]firm), we believe that these aspects of our approach make sense and helpthe model better match the data. The fact that each level of the job ladderexhibits significant entry and exit suggests that firm-specific human capitalis limited. Further, since there is a significant probability of a workerleaving at every job level, the wage offers of prospective employers arelikely to be important in determining what a worker’s initial employer iswilling to pay.37

But there are other settings in which our approach is probably not agood one, exactly because of the two assumptions mentioned above. Forexample, consider a firm with an internal labor market like that describedby Doeringer and Piore (1971). Among other aspects, this firm is char-acterized by ports of entry (meaning little or no entry at higher levels ofthe job ladder) and little exit at higher levels of the job ladder. The factthat there is little turnover at higher levels of the job ladder suggests thatthe labor market practices of this firm may not be driven by the wage

37 We feel that turnover in the data makes it more likely that our model isapplicable, even though our model itself exhibits no turnover. In our model, ineach period, a worker is close to indifferent between moving and staying (oneinterpretation of our model is that there is an infinitesimal moving cost that resultsin no turnover but this means that each worker prefers staying only by an in-finitesimal amount). It would be easy to make small changes in the model thatresult in some turnover but cause few other changes.


offers of prospective employers. Rather, it may be that significant firm-specific capital insulates the firm’s workers from the outside market, sosome alternative theory of wage and promotion dynamics is needed tounderstand careers in this firm.

Our approach shares one feature with the pure-form internal labormarket (ILM) described by Doeringer and Piore (1971): an important rolefor job levels in wage determination. But in the pure-form ILM, wagesare attached to jobs (i.e., there is little or no wage variation within a job,except perhaps formulaic variation with seniority). In our approach, incontrast, a worker’s wage is determined by his productivity. Because ourmodel incorporates learning and heterogeneity in human capital acqui-sition, wage variation within a job cannot be explained solely by variationin seniority. Thus, within-job wage variation may be another way (inaddition to entry and exit at higher levels of the job ladder) to identifysettings in which our approach applies rather than the pure-form ILMdescribed by Doeringer and Piore (1971).

We now turn to facts that we are still not able to capture. In our earlierpaper (Gibbons and Waldman 1999b), we discussed three such facts, sohere we will focus on two new ones.38 One implication of our approach(in this article and in our earlier [1999b] paper) is that learning shouldbecome less important as labor market experience increases. That is, aftera number of periods of observing worker output, the worker’s ability tolearn on the job should be known relatively precisely, so additional outputobservations should have little effect on the market’s expectation of thisability. Therefore, results or predictions that depend heavily on the learn-ing aspects of the model should vary with the worker’s labor marketexperience. For example, one result in our earlier paper is that promotedworkers should receive large wage increases (relative to wage changes inthe absence of a promotion), and one of the reasons for this result is that,on average, promoted workers are those for whom learning was verypositive. In turn, since learning becomes less important for those withsubstantial labor market experience, the prediction is that wage increasesupon promotion (relative to wage changes in the absence of a promotion)should be smaller for those with more labor market experience.

But this prediction is problematic because, at least in the Baker et al.(1994a, 1994b) study, wage increases due to promotion tended to riserather than fall with hierarchical level. What we draw from this is that,as we briefly discussed in the introduction, our modeling approach seemsto capture enough stylized facts about career dynamics that the building

38 The three facts that we discussed in our earlier (1999b) paper were (i) nominalwage rigidity, (ii) cohort effects, and (iii) green card effects. In earlier sections,we showed that, in fact, in contrast to the discussion in our earlier paper, ourapproach is capable of capturing cohort effects.

92 Gibbons/Waldman

blocks of our approach—job assignment, human capital accumulation,and learning—are likely to be important elements for understanding whatactually occurs in many firms. But a complete theory would need toincorporate additional elements, such as incentives. For example, it is quitepossible that incorporating incentives, as in Lazear and Rosen’s (1981)classic contribution, would allow us to amend our framework so that itis consistent with wage increases upon promotion being a positive ratherthan a negative function of hierarchical level. The idea here would be that,because higher-level decisions have more important effects on firm prof-itability, at higher levels of the hierarchy, the wage increases upon pro-motion need to be higher at these higher levels (see Rosen [1986] for arelated analysis).

There is also a second prediction of our approach that is not consistentwith the empirical evidence, again due to the rate of learning going tozero as labor market experience gets large. Holding the current wage fixed,our approach predicts that, as labor market experience gets large, thevariance associated with period-to-period wage changes should go to zero.One way to enrich our approach that would make the models morerealistic and eliminate this result is to assume that a worker’s underlyingability to learn on the job changes over time in a stochastic manner. Thenthe rate of learning and the variance of period-to-period wage changeswould not approach zero as labor market experience gets large because,even for very old workers, new realizations of output would providevaluable information about recent changes in the worker’s underlyingability.39

V. Conclusion

A large number of theories have been put forth to explain wage andpromotion dynamics inside firms. We believe that an important goal insuch model building should be to explain broad patterns of evidence ratherthan isolated stylized facts. In Gibbons and Waldman (1999b), we showedthat a model that combines job assignment, human capital acquisition,and learning captures a number of findings concerning wage and pro-motion dynamics found in Baker et al. (1994a, 1994b) and elsewhere andthat our theory matches the evidence better than available alternatives do.In this article, we extend that earlier analysis in two ways. First, weincorporate schooling into the framework and develop several predictions,

39 An alternative but complementary way of enriching the model to addressthis issue is to assume that ability to learn on the job is multidimensional andjobs vary in terms of which dimensions are more important for the human capitalaccumulation process. Then the rate of learning would not go to zero as labormarket experience gets large in the sense that this rate would increase every timea worker is promoted.


most of which match well with available evidence concerning the role ofschooling in wage and promotion dynamics. Second, we show that theintroduction of task-specific human capital allows the model to captureBaker et al.’s (1994b) finding of cohort effects, which in our earlier paper(Gibbons and Waldman 1999b) we incorrectly argued was inconsistentwith our theoretical approach.

The other main message of the current article is the potential significanceof task-specific human capital. Beginning with the seminal work of Becker,most of the literature concerning human capital has focused on the con-cepts of general purpose and firm-specific human capital. We believe,however, that task-specific human capital is potentially as prevalent andimportant as general purpose and firm-specific human capital. Clearly,much of human capital acquisition involves becoming more proficient atthe task or tasks being performed. Thus, rather than human capital goingunutilized when a worker switches firms (as is the case when humancapital is firm specific), human capital goes unutilized when a workerswitches jobs and is assigned a new set of tasks, whether the switch entailsstaying within the same firm or moving across firms. In this sense, ournotion of task-specific human capital is akin to occupation- or even in-dustry-specific human capital. In this article, we have begun to exploresome of the implications of task-specific human capital. We have shownthat this concept can explain cohort effects, but we conjecture that thereare a variety of other applications concerning issues such as job design,job assignment, labor mobility, labor demand, and even business strategy.

To give a sense of what we have in mind here, consider how a typicalfirm structures its job ladder. Most theoretical models have all workersat any particular job level doing the same set of tasks (to the extent thattasks are modeled at all). In reality, however, there is often significantheterogeneity across the tasks performed by different workers at the samejob level. For example, in the spirit of the Baker et al. (1999a, 1999b)data, two workers at an investment bank might work at the same joblevel, but one might specialize in analysis and the other in sales. Thequestion, then, is, how is the promotion process structured? The answeris that promotions are not moves from one job at a particular level to arandom new job at the next level. Rather, there are typically standardpromotion paths where, for example, the job of an analyst who getspromoted is to supervise a group of analysts doing tasks similar to thetasks he previously performed.

The idea of task-specific human capital makes what we believe to beplausible predictions concerning these promotion paths. The basic idea isthat promotion paths should be structured so as to minimize the under-utilization of human capital when promotions occur. For example, consis-tent with the analyst example above, one prediction from this approach isthat a worker promoted into a supervisory position should frequently be

94 Gibbons/Waldman

someone who previously performed tasks similar to the ones being per-formed by the workers now being supervised. In Gibbons and Waldman(2004), we also discuss related predictions concerning how tasks are assignedto both lower-level and higher-level jobs and how job rotation schemes canbe explained by task-specific human capital.

There are a number of directions in which the analysis in this articlecould be extended, including the following three. First, as just indicated,one could extend our analysis of task-specific human capital to a varietyof topics. Second, as also discussed earlier, one could incorporate into ourframework other modeling elements that have received significant theo-retical attention, such as incentives and asymmetric learning, to seewhether the addition of such elements makes the model a better matchwith the evidence.40 Third, along the lines of our discussion in SectionIII.D, one could integrate our focus on careers inside firms with theextensive theoretical literature that attempts to explain evidence concern-ing wage dynamics and mobility between firms (see, e.g., Burdett 1978;Mortensen 1978; Jovanovic 1979). Since promotion and turnover are al-ternative exit routes from a worker’s current job, there should be a singletheoretical framework that captures the empirical evidence on bothtopics.41

Appendix

Proofs of Propositions and Corollaries

Proof of proposition 1. This proof is similar to the proof of proposition2 in Gibbons and Waldman (1999b). Because learning is symmetric andthere is no firm-specific human capital, competition among firms eachperiod yields both efficient job assignment and wages equal to expectedoutput. Given this, we compute worker i’s expected on-the-job humancapital in period t, , in (3) and then the worker’s expected output inehit

job j as Note that the linearity of the productioneEy p d � G(S ) � c h .ijt j i j it

function in (2) is key here: without linearity, expected output would notequal the output of a worker known to have on-the-job human capitalequal to . We now have that, given efficient job assignment and wagesehit

equal to expected output, if , then worker i is assigned to job 1 ine ′h ! hit

40 One prominent approach to incentives in firms involves tournaments. Papersconcerning the tournament issue include Lazear and Rosen (1981) and Rosen(1986), while asymmetric learning is studied in Waldman (1984a, 1990), Greenwald(1986), and Gibbons and Katz (1991), among others. See Zabojnik and Bernhardt(2001) for an interesting recent paper that combines the tournament and asym-metric learning approaches.

41 See Ghosh (2002) for a start in this direction. Also, Munasinghe (2001) ex-plores a model that captures a number of findings concerning wages and turnover,although he does not explore the extent to which his model captures findingsconcerning careers inside firms.


period t and paid while if , then workere e ′w p d � G(S ) � c h , h ≥ hit 1 i 1 it it

i is assigned to job 2 in period t and paid (see n.ew p d � G(S ) � c hit 2 i 2 it

9). Note here the fact that (2) reads means thatd � G(S ) � c (h � � )j i j it ijt

the signal about ability that can be extracted from output does not varyin its signal-to-noise ratio as a function of j. Thus, there is no way to usejob assignment to change the speed of learning about ability, so job as-signment is determined by current productive efficiency (i.e., maximizingexpected output this period), which, in turn, is solely a function of theworker’s current expected on-the-job human capital. QED

Proof of corollary 1. Given proposition 1 and that the analysis isrestricted to parameterizations such that it is efficient for each workerto be assigned to job 1 in the first period of the worker’s career (i.e.,

the wage of worker i in her first′[ pf � (1 � p)f � B(N)] f(0) ! h ),H L

period in the labor market can be written as d � G(S ) � c [ pf �1 i 1 H

Given and , we immediately have that′ ′(1-p)f � B(S )] f(0). G 1 0 B 1 0L i

the worker’s wage is a strictly increasing function of the worker’sschooling level. QED

Proof of corollary 2. Because the signal zit is independent of jobassignment and schooling level, the amount in any period that has beenlearned about fi is independent of a worker’s schooling level and pastjob assignments. In turn, since a worker is assigned to the high-level jobwhenever and , we immediately have that, fore ′ ′[f � B(S)] f(x ) ≥ h B 1 0it it

any pair, where and(X, S) 1 ≤ X ≤ T � 1 S ≤ N � 1, p(X, S � 1) ≥In turn, since by assumption ′p(X, S). [f � B(S)] f(T � 1) ! h ! [f �L H

for all S, we have that this inequality must be strict if X isB(S)] f(T � 1)sufficiently close to QEDT � 1.

Proof of corollary 3. Consider first values for X and such thatX � 1all workers of those experience levels are assigned to job 1. Let f* p

Given that expectations each period must be correct, onpf � (1 � p)f .H L

average, and output in job 1 is linear in on-the-job human capital, wehave that is given byW(X, S)

W(X, S) p d � G(S ) � c [f* � B(S)] f(X). (A1)1 i 1

In turn, the return to experience for a fixed schooling level is given by

W(X � 1, S) � W(X, S) p c [f* � B(S)][ f(X � 1) � f(X)]. (A2)1

Given and , (A2) immediately yields that′ ′B 1 0 f 1 0 W(X � 1, S) �is strictly increasing in S. Note that the increase in the return toW(X, S)

an extra year of experience as schooling is increased from S to isS � 1given by

[B(S � 1) � B(S)]c [ f(X � 1) � f(X)]. (A3)1

Now consider values X and such that some workers may beX � 1assigned to job 2 but there are no demotions. Consider a worker with

96 Gibbons/Waldman

experience level X and schooling level S who is assigned to job 1 and forwhom there is an expectation concerning f given by f′. Given that ex-pectations each period must be correct, on average, and that output injob 1 is linear in on-the-job human capital, if the worker is assigned tojob 1 with probability 1 next period, then his expected wage increase isgiven by Similarly, consider a worker with′c [f � B(S)][ f(X � 1) � f(X)].1

experience level X and schooling level S assigned to job 2 for whom thereis an expectation concerning f given by . This worker’s expected wage′′f

increase is given by Now consider a′′c [f � B(S)][ f(X � 1) � f(X)].2

worker with experience level X and schooling level S initially assigned tojob 1 for whom there is an expectation concerning f given by , where′′′f

there is a strictly positive probability that the worker will be assigned tojob 2 in the following period. Because expectations are correct, on average,every period, the expected wage increase is given by ′′′rc [f �1

, where′′′B(S)][ f(X � 1) � f(X)] � (1 � r)c [f � B(S)][ f(X � 1) � f(X)]2

. These expressions yield that integrating the wage increases over0 ! r ! 1all workers of experience X and schooling S yields

W(X � 1, S) � W(X, S) p

Rc [f* � B(S)][ f(X � 1) � f(X)] (A4)1

� (1 � R)c [f* � B(S)][ f(X � 1) � f(X)],2

where 0 ≤ R ! 1.Now consider workers of the same experience level but with schooling

level . Because a worker whose initial expectation for f is given byS � 1may now be assigned to job 2 in either period, this worker’s expected′f

wage increase must be greater than or equal to ′c [f � B(S � 1)][ f(X �1

Similarly, a worker whose initial expectation for f is given by1) � f(X)].must now be assigned to job 2 in each period, so this worker’s expected′′f

wage increase is Now consider a′′c [f � B(S � 1)][ f(X � 1) � f(X)].2

worker whose initial expectation for f is . Because, for any realization′′′f

for the stochastic term in which the worker is assigned to job 2 givenand S, the worker is also assigned to job 2 given and ,X � 1 X � 1 S � 1

this worker’s expected wage increase must be greater than or equal to′′′ ′′′rc [f � B(S � 1)][ f(X � 1) � f(X)] � (1 � r)c [f � (S � 1)][ f(X � 1) �1 2

. Given these expressions, integrating the wage increases over allf(X)]workers of experience level X and schooling level yieldsS � 1

W(X � 1, S � 1) � W(X, S � 1) ≥Rc [f* � B(S � 1)][ f(X � 1) � ( f(X)] (A5)1

� (1 � R)c [f* � B(S � 1)][ f(X � 1) � f(X)].2

We now have that the increase in the return to an extra period of


experience due to the schooling level increasing from S to is greaterS � 1than or equal to the expression here:

[B(S � 1) � B(S)][Rc � (1 � R)c ][ f(X � 1) � f(X)]. (A6)1 2

Together, (A3) and (A6) yield that, given no demotions, W(X � 1, S) �is strictly increasing in S.W(X, S)

We now consider what happens when demotions are possible. Considervalues for X and such that workers of those experience levels canX � 1be assigned to either job and there are some demotions as experienceincreases from X to Consider a worker with experience level XX � 1.and schooling level S who is assigned to job 1 and for whom there is anexpectation concerning f given by . As before, if the worker is assigned′f

to job 1 with probability 1 next period, then his expected wage increaseis given by Consider a worker with ex-′c [f � B(S)][ f(X � 1) � f(X)].1

perience level X and schooling level S assigned to job 2 for whom thereis an expectation concerning f given by f′′, where there is a positiveprobability that the worker will be assigned to job 1 in the followingperiod. This worker’s expected wage increase is strictly less than

where m is the proba-′′c [f � B(S)][ f(X � 1) � f(X)] � mc Df(X � 1),2 2

bility of demotion and D is the absolute value of the average reductionin the expectation concerning f when the worker is demoted. Considera worker with experience level X and schooling level S assigned to job 1for whom there is an expectation concerning f given by , where there′′′f

is a strictly positive probability that the worker will be assigned to job2 in the following period. As before, the expected wage increase for thisworker is given by ′′′ ′′′rc [f � B(S)][ f(X � 1) � f(X)] � (1 � r)c [f �1 2

where . These expressions yield that in-B(S)][ f(X � 1) � f(X)], 0 ! r ! 1tegrating the wage increases over all workers of experience X and school-ing S yields

W(X � 1, S) � W(X, S) ! Rc [f* � B(S)][ f(X � 1) � f(X)]1

� (1 � R)c [f* � B(S)][ f(X � 1) � f(X)] (A7)2

� Mc [f � f ] f(X � 1),2 H L

where M is the overall probability of demotion for these workers and.0 ≤ R ! 1

Now consider workers of the same experience level but with schoolinglevel . Because a worker whose initial expectation for f is given byS � 1

may now be assigned to job 2 in either period, this worker’s expected′f

wage increase must be greater than or equal to ′c [f � B(S � 1)][ f(X �1

Because a worker whose initial expectation for f is may be′′1) � f(X)]. f

assigned to job 1 in the following period, this worker’s expected wageincrease must be greater than or equal to ′′c [f � B(S � 1)][ f(X � 1) �2

98 Gibbons/Waldman

Because a worker whose initial expectation for f is has a higher′′′f(X)]. f

probability of being assigned to job 2 in the following period when theschooling level is rather than S, this worker’s expected wage increaseS � 1must be greater than or equal to ′′′rc [f � B(S � 1)][ f(X � 1) � f(X)] �1

These expressions yield that in-′′′(1 � r)c [f � B(S � 1)][ f(X � 1) � f(X)].2

tegrating the wage increases over all workers of experience X and school-ing yieldsS � 1

W(X � 1, S � 1) � W(X, S � 1) 1

Rc [f* � B(S � 1)][ f(X � 1) � f(X)] (A8)1

� (1 � R)c [f* � B(S � 1)][ f(X � 1) � f(X)].2

Comparing (A7) and (A8) yields that, for any parameterization, if de-motions are sufficiently rare, that is, M is sufficiently small, then

is strictly increasing in S. QEDW(X � 1, S) � W(X, S)Proof of corollary 4. Let be the average wage paid to workerslW (X, S)j

of experience level X and schooling level S who are assigned to job j givena value for l. Because the productivity due to a worker’s starting levelof general purpose human capital, does not depend on the worker’slg(S),job assignment, varying l will not affect job assignments. Hence,

can be written as This, in turn, yields that forl 0W (X, S) W (X, S) � lg(S).j j

all and l0 ≤ X ≤ T � 1, S ≤ N � 1, j p 1, 2, W (X, S � 1) �j

Given ,l 0 0 ′W (X, S) p W (X, S � 1) � W (X, S) � l[g(S � 1) � g(S)]. g 1 0j j j

this expression is strictly positive given l sufficiently large. QEDProof of proposition 2. Consider first an old worker. Since this is the

last period of the worker’s career, competition across firms means thatthe worker will be assigned to the job that maximizes productivity thatperiod and will be paid a wage equal to that productivity. We also knowthat, since if, in state of the world K, an old worker who workedc 1 c ,2 1

in job J last period with expected ability to learn on the job equal to v

is efficiently assigned to job 2, then, in state of the world K, an old workerwho worked in job J last period with expected ability to learn on the jobequal to , is also efficiently assigned to job 2. This means that for′ ′v v 1 v,each pair, and there exists a value such that,K,J(K, J) K p G, B J p 1, 2, vO

if period t is characterized by state of the world K, then old worker ipreviously assigned to job J is assigned to job 2 if and is assignede K,Jv ≥ vit O

to job 1 if . Further, since assignments maximize that period’se K,Jv ! vit O

expected output, we also have for all pairs,K,J K,Jv p v * (K, J) K p G, BO O

and This proves ii.J p 1, 2.Consider again This is the value for such that, in state of theK,J ev *. vO it

world K, an old worker who was in job J in the previous period has thesame expected output in each of the two jobs. In other words, K,2v *O


satisfies (A9), while satisfies (A10).K,1v *O

K,2 K K,2d � c [v *f(1)] p d � c [v *f(1)]. (A9)1 1 O 2 2 O

K,1 K K,1d � c [v *f(1)] p d � c [v *f(a)]. (A10)1 1 O 2 2 O

Note that (A9) yields while (A10)K,2 Kv * p (d � d )/[c f(1) � c f(1)],O 1 2 2 1

yields A comparison of these expres-K,1 Kv * p (d � d )/[c f(a) � c f(1)].O 1 2 2 1

sions yields for Note that, ifK,2 K,1 K,2 K,1v * ! v * K p G, B. v * ! v * ≤O O O O

for or for thenK,2 K,1f � B(1) K p G, B f � B(N) ≤ v * ! v * K p G, B,L H O O

this inequality does not translate into a difference concerning assignmentsin equilibrium. This proves iv.

We now consider the assignment of young workers to jobs. is theKv *Y

value for such that a young worker is equally productive in the twoevit

jobs. That is, satisfies (A11).Kv *Y

K K Kd � c [v *f(0)] p d � c [v *f(0)]. (A11)1 1 Y 2 2 Y

Let be the expected productivity of an old worker who wasJ eE(W Fv )O it

assigned to job J when she was young and whose expected ability to learnon the job when young was (where this expectation is taken prior toevit

the realization of the state of the world when the worker is old). Letbe the value for such that the worker’s expected output over herK ev vY it

career is the same whether the worker is assigned to job 1 or job 2. Thatis, is given by (A12).KvY

K 1 K K K 2 Kd � c [v f(0)] � E(W Fv ) p d � c [v f(0)] � E(W Fv ). (A12)1 1 Y O Y 2 2 Y O Y

Equation (A11) yields while equation (A12)K Kv * p (d � d )/(c � c )f(0),Y 1 2 2 1

yields Because beingK K 1 K 2 Kv p [d � d � E(W Fv ) � E(W Fv )]/(c � c )f(0).Y 1 2 O Y O Y 2 1

assigned to job 1 when young only counts a of a period in the humancapital accumulation function if the worker is assigned to job 2 when old,we know that Hence, a comparison of these ex-2 K 1 KE(W Fv ) 1 E(W Fv ).O Y O Y

pressions yields for This proves i.K Kv ! v * K p G, B.Y Y

The earlier expressions for and yield forK,2 K,1 G,J B,Jv * v * v ! v J pO O O O

Further, given for , it can be shown thatK,2 K,11, 2. v ! v K p G, B,O O

rises more quickly with than does Given this, the2 K K 1 KE(W Fv ) v E(W Fv ).O Y Y O Y

above expression for yields This proves iii. QEDK G Bv v ! v .Y Y Y

Proof of proposition 3. Competition and no firm-specific human cap-ital means that a worker’s wage each period equals her expected outputin the job to which she is assigned. Let Zn denote the number of workersof schooling level n in each cohort, and let Z denote the total number ofworkers in each cohort. Given that a worker is always paid her expected

100 Gibbons/Waldman

output in the job to which she is assigned, we have that must satisfy.BWY

N

BW X Z max {d � c E(vFS p n)f(0),�Y n 1 1 i inp1

Bd � c E(vFS p n)f(0)}/Z. (A13)2 2 i i

Now consider the good state of the world. satisfiesGv * d �Y 1

Given sufficiently high values forG G Gc v *f(0) p d � c v *f(0). c 1 c ,1 Y 2 2 Y 2 1

yield for all S. But, since proposition 2 tells us thatG Gd v * ! f � B(S)2 Y L

we now have that for all S for sufficiently highG G Gv ! v *, v ! f � B(S)Y Y Y L

values for For such a value for , we can write asG G Gd . d W2 2 Y

N

G GW p Z (d � c E(vFS p n)f(0))/Z. (A14)�Y n 2 2 i inp1

Comparing (A13) and (A14), in turn, yields that for sufficientlyG BW 1 WY Y

high values for This proves i.Gd .2

We now turn our attention to the average wage of old workers in stateof the world K. Consider first a worker of schooling level such that′n

Given from proposition 2 that we know , when′ B G BE(vFS p n ) ≥ v . v ! vi i Y Y Y

she is young, this worker is assigned to the high-level job in both statesof the world. In turn, since job assignments and wages when a workeris old depend on the current state of the world, the previous job assign-ment, and the current expectation concerning the worker’s ability to learnon the job (and not directly on the previous period’s state of the world),the worker’s expected wage when she is old is independent of whetherthe state of the world when the worker was young was good or bad.Further, a similar argument yields that the expected wage when a workeris old for a worker of schooling level such that is′′ ′′ Gn E(vFS p n ) ! vi i Y

independent of whether the state of the world when the worker was youngwas good or bad.

Now consider a worker of schooling level such that′′′ Gn v ≤Y

When this worker is young, she will be assigned to′′′ BE(vFS p n ) ! v .i i Y

the high-level job in the good state of the world and the low-level job inthe bad state. In turn, if the state of the world when she was young wasgood rather than bad, then her expected wage when she is old is higher.The logic here is as follows. First, given fixed values for vi and � whenthe worker is young and given , if she is assigned to the high-levela ! 1job in both states of the world, she earns more when the previous period’sstate of the world was good. Second, given fixed values for and � whenvi

the worker is young and given that there is no task-specific human capitalon job 1, if she is assigned to the low-level job in both states of the world,her pay when old is independent of the previous period’s state of theworld. Third, given fixed values for vi and � when the worker is young


and since she is sometimes assigned to the high-level job whenK,2 K,1v ! v ,O O

the previous period was good but assigned to the low-level job when theprevious period was bad. Further, whenever this is the case, since K,2vO

satisfies and she is paid moreK,2 K K,2d � c v f(1) p d � c v f(1) c 1 c ,1 1 O 2 2 O 2 1

when the previous period’s state of the world was good. In summary, herexpected wage when old is higher when the previous period’s state of theworld was good since she always earns at least as much when the previousperiod was good rather than bad and she sometimes earns more.

We now have that a worker’s expected wage when she is old is inde-pendent of the state of the world in the previous period when

and while when , theB G G BE(vFS ) ≥ v E(vFS ) ! v , v ≤ E(vFS ) ! vi i Y i i Y Y i i Y

worker’s expected wage when she is old is higher when the state of theworld in the previous period was good. Given that there is at least oneschooling group such that falls in the latter interval (see n. 28),E(vFS )i i

we now have for This proves ii. QEDK,G K,BW 1 W K p G, B.O O

Proof of proposition 4. Using the same logic as in the proofs of prop-ositions 2 and 3, it must be the case that and that there exists aG Bv ! vY Y

value such that, if then the average wage for young workers′ ′G G Gd d 1 d ,2 2 2

is higher when the state of the world is good rather than bad. Nowconsider the assignment of workers to jobs and wages over the last

periods a cohort is in the labor market. Consider, first, a workerT � 1of schooling level such that Given that in′ ′ B G Bn E(vFS p n ) ≥ v . v ! v ,i i Y Y Y

her first period in the labor market, this worker is assigned to the high-level job in both states of the world. In turn, since the state of the worldwhen a worker is young does not directly enter into the productionfunction for either job in later periods, we have that the worker’s expectedcompensation in the next periods must be independent of the stateT � 1of the world when she was young. Further, a similar argument yields that,for a worker of schooling level such that the′′ ′′ Gn E(vFS p n ) ! v ,i i Y

worker’s expected compensation over her last periods in the laborT � 1market is independent of the state of the world when she was young.

Now consider a worker of schooling level such that′′′ Gn v ≤Y

When this worker is young, she will be assigned to′′′ BE(vFS p n ) ! v .i i Y

the high-level job in the good state of the world and the low-level job inthe bad state. In turn, if the state of the world when she was young wasgood rather than bad, then the worker’s expected compensation over herlast periods in the labor market must be higher because of theT � 1following. Suppose that the state of the world when the worker was youngwas good. In this case, for every sequence of realizations of the stochasticterms, the worker has the option in each period of choosing the same jobassignment that she chooses when the state of the world when she wasyoung was bad rather than good. Since, in turn, some of these job as-signments are to the high-level job (see n. 32) and because this seta ! 1,of choices must result in a higher value for the worker’s expected com-

102 Gibbons/Waldman

pensation over her last periods in the labor market than she achievesT � 1when the state of the world was bad when she was young. But, since heractual choices must do at least as well as this, we now have that theworker’s expected compensation over her last periods in the laborT � 1market must be higher when the state of the world when she was youngwas good rather than bad.

We now have that a worker’s expected compensation over her lastperiods in the labor market is independent of the state of the worldT � 1

when the worker was young when and while,B GE(vFS ) ≥ v E(vFS ) ! v ,i i Y i i Y

when , this expected compensation is higher when theG Bv ≤ E(vFS ) ! vY i i Y

state of the world when the worker was young was good rather than bad.Given that there is at least one schooling group such that falls inE(vFS )i i

the latter interval (see n. 28), we now have that a cohort’s average com-pensation over the last periods it is in the labor market is higherT � 1when the state of the world was good rather than bad in its first periodin the labor market. This proves i.

Suppose that and consider a worker who has spent her first′a ! a ,1 periods in the labor market in job 1. Let be her expected ability′T � v

to learn on the job. Since we know that′ ′a ! a , d � c v f(T � 1) 11 1

Thus, she will be assigned to the low-level job inG ′d � c v f(a(T � 1)).2 2

her last period in the labor market. In turn, this immediately yields thatthere must exist a smallest value such that a worker who′ ′t , t ≤ T � 1,spends her first periods in job 1 is not promoted in any of her last′t

periods in the labor market. This proves ii.′T � tWe demonstrate the third point by presenting an example in which

some promotions are associated with wage decreases. Suppose thatT p 3, d p 20, c p d, d p 10, c p 1, a p 1/2, f(0) p d, f(1/2) p1 1 2 2

and where d is infinitesimally small. Also, we consider a9, f(3/2) p 16,schooling group for which and , which,f � B(S) p d f � B(S) p 1L H

given and yields that a worker in this school-d p 20, d p 10, f(0) p d,1 2

ing group is assigned to the low-level job in her first period in the labormarket. Let Wt denote the worker’s wage in her tth period in the labormarket. We have since and Now suppose thatW ≈ 20 c p d d p 20.1 1 1

learning after the first period is such that the expectation in the secondperiod concerning the worker’s ability to learn on the job is arbitrarilyclose to If the worker remains in job 1 for the remaining twof � B(S).H

periods, then she earns over these last two pe-W � W ≈ 20 � 20 p 402 3

riods, since and If instead she is promoted, thenc p d d p 20. W �1 1 2

, since her expected ability to learn on the job is ar-W ≈ 19 � 26 p 453

bitrarily close to Hence, she will be promoted at the beginningf � B(S).H

of the second period, but her wage falls from approximately 20 to ap-proximately 19 when the promotion takes place. This proves iii. QED


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