Monitoring for Worker Quality:
Task Assignment, Job Ladders, Wages and
Mobility in Nonmanagerial Internal Labor Markets
Gautam Bose
School of Economics, University of New South Wales
g.bose@unsw.edu.au
Kevin Lang
Department of Economics, Boston University, and NBER/IZA
lang@bu.edu
July 12, 2015
Abstract
We analyze job-assignment and worker-monitoring when workers face occasional challenges. Good workers overcome challenges; bad workers fail. Firms
observe failures but must monitor workers to observe successes. Firms prefer
to assign good workers to a di¢ cult task and not to employ bad workers. If
monitoring costs are positive but small, monitoring is nonmonotonic in the
…rm’s belief about the probability a worker is good, even among those in the
low task. The model explains several empirical regularities regarding nonmanagerial internal labor markets: low use of performance pay, seniority pay, rare
demotions, wage ceilings within grade and wage jumps at promotion.
Key words: internal labor markets, personnel, worker evaluation
The research in this paper was supported in part by NSF grant SES-1260917. We are grateful to
Costas Cavounidis, Bob Gibbons, Sambuddha Ghosh, Eddie Lazear, Bart Lipman, Andy Newman,
Mike Waldman and participants at seminars, workshops and conferences at Boston University, MIT,
the NBER, SOLE, Tel Aviv University, UC Santa Barbara and the University of New South Wales
for helpful comments and suggestions. The usual caveat applies.
In this paper we present a model in which employers monitor workers to evaluate
their productive quality. The information is valuable because high-quality, but not
low-quality, workers produce greater pro…ts in high-level tasks. We propose a new
interpretation of worker quality and a corresponding structure of information about
quality. We derive the equilibrium monitoring rule and the resulting wage and promotion pro…le. The wage-pro…le predicted by our model is consistent with a number
of regularities observed in non-managerial labor markets. In particular, the model
predicts that workers’ pay will in large part be based on factors— such as age, experience and seniority— that may be related to productivity, but are not themselves
measures of productivity (Doeringer and Piore, 1971).
Our model makes three signi…cant departures from the current literature. First,
we assume that the primary purpose of monitoring is to evaluate the quality of the
worker and the quality of the match between the …rm and worker. Contrary to
standard theory, monitoring in this paper is not undertaken to prevent workers from
shirking on the job. In order to focus attention on the evaluative role of monitoring,
we abstract from moral hazard altogether. The validity of this assumption is addressed
in sections 1.1 and 1.2 below.
Second, in most jobs much of the work is routine, and workers of di¤erent quality
or ability are equally productive. Therefore most of the time there is little new
information about worker quality. Opportunities to observe worker quality arrive
randomly and discretely in the shape of nonroutine situations where the worker must
use her judgment to choose a corresponding nonroutine action. If the worker makes
an error, there is a signi…cant cost to the …rm. We term these situations “challenges.”
Good workers are likely to face challenges successfully, while bad workers are likely
to fail.
Third, failure and success in responding to challenges are observationally asymmetric. Failures are much more noticeable than successes, and are observed even if
the …rm does not actively monitor the worker. However, when a worker successfully
handles a challenge there is no disruption to the ‡ow of output; hence the …rm is less
likely to observe a success if it is not actively monitoring the worker. This technology
is, in many ways, similar to the Biais et al (2012) model of moral hazard in which very
costly challenges arrive infrequently. In their model, e¤ort reduces the probability of
failing a challenge. In our model, in contrast, high-ability workers are able to address
the challenge when it arises, perhaps because they have put in place an early and
1
critical measure that makes the challenge manageable with routine measures when it
arises.
The last two assumptions are consonant with an environment in which work is
mostly routine so that any worker with the appropriate training can do most of the
required work most of the time. The famous con-artist, Frank Abagnale, claims to
have worked for eleven months as chief resident pediatrician in a Georgia hospital
until he was faced with an oxygen-deprived baby and was almost exposed (Abagnale
1980). Though he had no medical training or tertiary education, he was able to deal
with routine tasks for eleven months, until he encountered an unusual challenge.
Consonant with our perspective, challenges may not always be apparent. A patient
comes in with what appears to be heartburn. The bad physician gives the patient an
antacid. The good physician thinks there is reason to be concerned that this could
be a myocardial infarction (MI) and also orders an electrocardiogram (EKG). Often
the EKG looks good; the patient has heartburn. Occasionally, the good physician
discovers an MI and prevents the patient’s death. Unless someone is monitoring
the physician’s behavior closely, the bad physician’s failure to note the additional
considerations that merited the EKG will go unnoticed. But eventually the failure
to note the need for an EKG will lead to a patient’s death, which will be observable
without monitoring.
It is mathematically and notationally convenient to assume that bad workers’failure to address a challenge is always observed. Perhaps more realistically, as in our
example above, bad workers engage in behaviors that sometimes lead to observably
bad outcomes. Workers assigned to watch a monitor go to sleep, wander o¤ or otherwise neglect their obligations. A salesperson working with a demanding and rude
customer responds inappropriately. Workers do not follow safety regulations. Generally in the absence of monitoring, these behaviors go unrecorded. Nothing serious
happens on the shift. The customer is annoyed but does not act on his annoyance. No
one is injured, etc. But occasionally the sleeping worker fails to catch an impending
disaster, a pilot reports a near miss due to a failure of tra¢ c control, a worker is seriously injured or the salesperson’s behavior leads to a successful civil rights claim or the
loss of a major client. Yet, with monitoring, all these behaviors would be observable.
The model can be adjusted with minor changes to address these examples.
We do not claim that this asymmetry between success and failure is present in all
jobs. A related paper (Cavounidis and Lang, 2015) examines a setting where both
2
successes and failures are observed only when workers are monitored. While that
paper has interesting implications for labor market equilibrium, it provides little or
no insight into internal labor markets. And we can certainly think of settings where
the opposite may be true. It may be easy to observe when a lawyer or consultant
has landed a big contract but not when she has “squandered the opportunity.” We
conjecture, but have not proved, that such settings are conducive to up or out contracts. But, we believe that the type of asymmetry we assume is more plausible in
lower-level jobs, which is why we apply it to nonmanagerial internal labor markets.
Thus our representation of productivity and information about productivity is
signi…cantly di¤erent from other prominent models in the literature. In Gibbons and
Waldman (1999a), for example, good workers are more productive ceteris paribus than
bad workers at all times, but information is noisy because productivity is a¤ected by
exogenous shocks. In the present model, good and bad workers di¤er only in their
ability to handle challenges which arrive randomly, and failures are publicly observed
while successes are not observed unless the worker is monitored.
The model has important implications for the sequencing of monitoring. The
…rm will not monitor workers who are either very unlikely or very likely to be good.
This result is quite general. However, in our context, since …rms acquire information
just by waiting, workers who are unlikely to be good will not be monitored initially
but only after they have been with the …rm for a while. Similarly, workers who are
not quite su¢ ciently likely to be good to be assigned to a more challenging task are
nevertheless not monitored because, with high probability, they will be promoted
shortly anyway. Only if the monitoring cost is not too high and the …rm’s assessment
of the worker is neither too high nor too low, is it e¢ cient to monitor.
Much of the theoretical literature on internal labor markets that incorporates monitoring focuses on moral hazard, although there are important exceptions. Therefore
in sections 1.1 and 1.2, we justify our alternative focus on learning about workers. We
show that our model can explain why, in many jobs, wages are largely determined by
initial human capital and seniority rather than by productivity. The model predicts,
or is consistent with, many of the regularities in the literature on nonmanagerial
internal labor markets described in section 1.3.
Our discussion of the theory then proceeds in two parts. In Sections 2 and 3
we discuss e¢ cient monitoring and task-assignment. In Section 4, we complete the
model by considering wage determination and turnover when ability is match-speci…c.
3
Section 5 relates our results to the regularities discussed in Section 1.3. Section 6
presents some extensions, and the …nal section concludes.
1
Monitoring, Learning and Internal Labor Markets
1.1
Monitoring
Payment on the basis of either measured output or subjective performance requires
some form of monitoring. An important element of our approach is that monitoring
is not designed to reduce or eliminate worker moral hazard. While we do not deny
that this is sometimes a motive for monitoring, we believe that focus on this aspect
has led theorists to ignore an equally important reason for monitoring, namely the
evaluation of worker quality. Practitioners are often surprised that the personnel
economics literature on monitoring almost universally assumes that its sole purpose
is to allow …rms to punish workers who shirk, cheat or steal and thus to create
incentives for e¤ort. This assumption generates empirical predictions and theoretical
results that are, at the least, problematic.
Theoretically, since detection is costly and deterrence depends on the probability
of detection multiplied by the cost of punishment, punishment for shirking should
be as large as possible and monitoring should be minimal. Dickens, Katz, Lang and
Summers (1989) refer to this as the monitoring puzzle. Akerlof and Katz (1986) show
that if backloading pay or otherwise requiring a bond is costly, the only solution to
models of this sort is the one derived by Becker and Stigler (1974): workers should
“buy” their jobs and have the purchase price returned to them when they retire. If
workers’ ability to purchase their job is limited, …rms may require them to engage
in rent-dissipating behavior (Murphy and Topel, 1990). Neither purchase of jobs
nor obviously rent-dissipating requirements are a common feature of job contracts.
Admittedly, the absence of bonds can be accounted for if there is also adverse selection in the labor market (MacLeod and Malcomson, 1988). In our model, we will
also abstract from adverse selection by assuming that workers do not have private
information about their type.
More general earnings pro…les are possible if bonding is costless, but the logic of
the argument requires that wages be less than the value of marginal product (VMP)
4
early in seniority and more than VMP later (Lazear, 1979, 1981). The e¢ cient
contract sets hours so that VMP equals the worker’s marginal value of leisure, while
workers would wish to choose to set the marginal value of leisure equal to their wage.
This implies that junior workers will want to work less than required by the optimal
contract while senior workers will want to work more. Kahn and Lang (1992, 1995)
show this is not consistent with the data on desired work hours while Shaw and
Lazear (2008) study workers whose physical output can be measured and show that
productivity rises faster than earnings.
In moral hazard models, in general, and tournament models, in particular, incentives will be stronger when outside competition is restricted. However, Bond (2012)
points out that 71 percent of UK workers are employed in …rms that do not favor
internal candidates when …lling vacancies.
Moreover, while monitoring may serve primarily as a discipline device in some
settings, there are numerous contexts in which this assumption violates everyday
experience. For example, we do not check our research assistant’s work mainly to
deter him from shirking, nor do we …re him if we catch an error. We want to ascertain
that the RA is suited to the task he has been assigned. The frequency with which we
…nd errors will a¤ect our assessment of his ability, and perhaps our decision to rehire
him. Our assessment of his abilities may rise if we …nd he has successfully negotiated
a common pitfall, or constructed a clever proof. But in this sense monitoring is part
of our process of evaluating the research assistant. The threat that we may …re or
not rehire the assistant may also encourage the assistant to work harder in order to
reduce the frequency of errors, but our monitoring decision is primarily driven by
our need to catch errors and to know at what tasks the assistant is competent. As
we become more convinced that the assistant can handle the tasks to which he has
been assigned, we are likely to reduce our monitoring. In this respect we resemble
private sector managers who often monitor workers as part of an evaluation process
and review the work to determine that it has been done correctly. We revisit this
theme in Section 6.
The primarily evaluative role of monitoring is very evident in the case of highachieving university graduates who obtain attractive career jobs at leading accounting,
consulting and law …rms. It is well-known that in the …rst few years these new recruits work extraordinarily long hours and are watched very closely by their superiors.
After a length of time, some of the recruits are placed in responsible positions, often
5
handling important client accounts, while others are …red or given a clear signal that
they should look for work elsewhere. The close supervision in the initial period is not
primarily to motivate the candidates, who are intensely career-oriented and eager to
prove their capabilities, but to sift out the ones that are good matches in their current
…rm from the ones that are not as well-suited to it.1 To a lesser extent, extensive
initial monitoring is undertaken with a similar goal in the case of non-managerial
workers who will be placed in a positions where mistakes may be costly.
1.2
Learning
This paper is about using monitoring to learn about the ability of workers. There
is a modest literature on learning in the labor market. Farber and Gibbons (1996),
Altonji and Pierret (2001) and Gibbons and Waldman (1999a, 2006) present models
of wage determination in which learning is about general market skills, all information
is public, and new information arises continuously. Lange (2007) estimates that such
learning is very fast. Waldman (1984), Greenwald (1986), Gibbons and Katz (1991),
Acemoglu and Pischke (1998), Schoenberg (2007) and Li (2013) examine wage determination when only the current employer learns about a worker’s skills but such skills
are still general. Autor (2001) empirically investigates the hypothesis that employers
provide free general skills training to workers in order to obtain private information
about their abilities. Meyer (1994) investigates how task assignment may be manipulated to facilitate learning about individual abilities when production happens in
teams. Jovanovic (1979) considers how learning about match-speci…c productivity
a¤ects turnover but assumes that wage equals value of marginal product despite the
existence of bilateral monopoly.
There are at least two weaknesses in the existing literature. The …rst is that
in these models learning is passive when, in fact, …rms seem to invest signi…cant
resources in learning about (evaluating) workers. Our …rst objective is therefore to
develop a tractable model in which the primary purpose of monitoring is to evaluate
ability, guide task-assignment and inform retention decisions.
The second weakness is that the theory of wage determination is, at best, underdeveloped when learning is about match-speci…c productivity. When productivity
1
Landers, Rebitzer and Taylor (1996) model long hours for associates as part of employers’
strategy for learning when workers have private information about their marginal disutility of e¤ort
or marginal utility of leisure.
6
is match-speci…c, employers and workers …nd themselves in a setting with bilateral
monopoly. Much of the literature either avoids the di¢ cult issue of wage determination by studying turnover and assuming that it is e¢ cient, or assuming that workers
are paid their value of marginal product despite …rms’monopoly power. The related
problem of wage determination when …rms have private information about worker
quality has been solved but only for two-period models or, in one case, a three-period
model (cf. some of the papers discussed in the …rst paragraph of this subsection).
Below we develop an empirically relevant model of wage determination when there
is learning about match-speci…c productivity. An important consequence of the model
is that, despite important variation in productivity among workers, nonmanagerial
workers are paid on the basis of a predetermined scale and not on assessments of their
performance. Simultaneously, we attempt to remain consistent with other regularities
described below in Section 1.3.
1.3
Nonmanagerial Internal Labor Markets
In the last two decades, the Baker, Gibbs and Holmstrom (1994a&b, hereafter
BGH) analysis of the career patterns of managerial workers within a single service
sector …rm has justi…ably received a great deal of attention. In particular, Gibbons
and Waldman (1999a, 2006, see also 1999b) develop a simple and elegant model designed to address many of the features of the internal labor market described in BGH.
Many, albeit not all, of the subsequent studies of internal labor markets (Dohmen,
2004; Dohmen, Kriechel and Pfann, 2004; Flabbi and Ichino, 2001; Gibbs and Hendricks, 2004; Kwong, 2006; Treble et al, 2001) have extended the study to nonmanagerial workers. Pay and promotion patterns for nonmanagerial workers are often
quite di¤erent from those described in the managerial literature. In particular, many
nonmanagerial workers are paid an hourly or weekly wage that does not depend on
either subjective or objective performance measures. To some extent, we, too, will
seek to explain some of the regularities in the internal labor market studied by BGH,
but we also emphasize that the features of that internal labor market are by no means
universal.
Initially, we seek to explain the following characteristics of nonmanagerial internal
labor markets. The …rst three regularities are drawn from a sample of nonunion
establishments in the UK Workplace Employment Relations Survey (WERS). The
7
WERS asks managers to list the determinants of pay in the largest nonmanagerial
occupation (e.g. skilled workers, administrative and secretarial, technical) in their
establishment.
1. Many establishments have no variation in pay. Roughly 20 percent (employment
weighted) of establishments report no variation in pay within their largest class
of nonmanagerial workers except for hours, overtime and shift di¤erentials. A
further 11 percent di¤erentiate pay on the basis of skills/core competencies
and/or job grade but not factors such as seniority or performance evaluations.
Since the occupations considered are quite broad, such variation may re‡ect
only di¤erences between jobs (e.g. carpenter and electrician). Finally, 2 percent
report using only these factors and some form of objective pay for performance
such as piece rate or commission.
2. In …rms that di¤erentiate pay, individual wages are largely determined by objective factors such as seniority, experience, occupational grade, education and
other formal quali…cations which may be correlates of productivity but are not
themselves measures of productivity. Among those …rms where there is clear
di¤erentiation in pay within similar jobs, 64 percent determine pay at least in
part by age, experience, seniority and/or formal quali…cations such as education
but not on the basis of subjective performance measures. In contrast, only 5
percent use subjective performance evaluations but not the objective factors in
the previous sentence, and 31 percent use both. Fully 52 percent use neither
objective nor subjective performance measures.
These numbers should be treated with considerable caution. Of managers who
reported that performance appraisal or assessment was one of the factors that
explain the di¤erences in the level of pay of full-time workers in the largest occupation in their establishment, only about one-quarter also reported that merit
pay, de…ned as pay related to a subjective assessment of individual performance
by a supervisor or manager, was used anywhere in the establishment. Thus,
payment for subjective performance is not used for the largest nonmanagerial
group in somewhere between 64 and 90 percent of (size-weighted) nonunion UK
establishments that otherwise di¤erentiate pay among these workers.
3. Wages rise with seniority. In over half of the establishments that di¤erentiate
pay within their largest class of nonmanagerial workers, wages depend at least
in part on seniority. In 62 percent, wages depend on either age or seniority.
8
4. Many salary scales have a …xed number of steps. Workers at the top of the scale
receive no further individual pay increase unless they are promoted. Dohmen
(2004) describes one such salary scale for blue-collar workers.
5. Demotions are rare. This result has been established for managerial workers
(BGH). We are not aware of any single-…rm study that focuses on nonmanagerial
workers that establishes this result. However, demotions are su¢ ciently rare in
the broad population of workers (e.g. Kosteas, 2011) that it is safe to conclude
that it applies to nonmanagerial workers as well.
6. Wages jump at promotion. For samples of only or primarily nonmanagerial
workers, this result can be found in Grund (2005), Kwon (2006) and Dohmen
(2004). We note that this …nding is one that many learning models (e.g. Gibbons and Waldman, 1999a) have di¢ culty explaining although it is consistent
with tournament models (Lazear and Rosen, 1981). Waldman (1984) combines
the two to generate promotion e¤ects.
We note a seventh regularity: despite the link between wages and seniority, subjective performance measures rise little, if at all, with seniority within grade (Flabbi
and Ichino, 2001; Dohmen, 2004; also Medo¤ and Abraham, 1980&1981 for managerial workers). Our model leaves no role for supervisor evaluations. The essence of the
model is that most information comes in big chunks, which leaves no room for small
pieces of information. Thus, our model predicts this result but does so trivially.
In sum, it appears that individualized pay based on individual performance is far
from a universal phenomenon, at least at the nonmanagerial level. It is, however,
very clear that much of pay is determined by objective factors that may be correlates
of productivity but are certainly not measures of productivity. Almost half of establishments that di¤erentiate pay do so on the basis of seniority within the …rm and
about three-quarters do so on the basis of either seniority or experience.
One of the tasks before us is therefore to explain why …rms might rely on such
factors, particularly seniority, rather than on either actual productivity or subjective
performance measures.
2
Intuition
Our model of monitoring re‡ects our view that much of the work in most nonmanagerial jobs is routine for any trained worker, and …rms are concerned about the
9
worker’s ability to do the di¢ cult tasks that arise intermittently. We do not wish to
imply that there are no specialized skills. Many of us do not know how to change
the oil in a car, but it is usually straightforward for a trained mechanic. However,
when a nonroutine situation arises, it is costly if the worker does not have the requisite aptitude to identify it and respond to it. Only when the …rm has reasonable
con…dence about the worker’s ability to respond to such challenges will the worker
be assigned to high-level jobs where failure to handle challenges can be costly. But
it is also valuable to make full use of skilled workers who are capable of responding
to challenges, rather than to assign them to low-level jobs where their skill is less
valuable. Therefore the …rm wants to learn whether or not its workers are skilled.
In our speci…cation, the …rm can learn about the worker in two ways: by passively
updating its assessment of the likelihood that the worker is good or by actively monitoring the worker. If the …rm does not monitor the worker, it does not know when
a challenge arrives. However, when a worker fails a challenge, this becomes immediately known to the …rm. As time passes and the worker is not observed to fail, it
becomes more and more likely that the worker has addressed one or more challenges
successfully. Thus even without monitoring, the …rm must update its assessment
of the likelihood that the worker is good. Alternatively, the …rm can monitor the
worker, in which case it observes when a challenge arises. When it does, and the
worker handles it, the …rm knows with certainty that the worker is good.
In either case, the …rm will know if the worker has failed a challenge. The advantage of monitoring is that if the worker succeeds, the …rm learns this immediately, and
because it now knows that the worker can handle challenges, can move him instantly
to a job where his skill is more valuable.
When should the …rm monitor the worker, and when should it simply wait? If
the …rm’s prior that the worker is good is very low, then monitoring has very little
expected bene…t since when a challenge arises, it is unlikely that the …rm will discover
that the worker is actually good. Therefore monitoring is unlikely to be pro…table.
Nor is it pro…table to monitor a worker about whom the …rm’s prior is very high.
Such a worker will be promoted soon provided that she does not fail a challenge in
the very brief period before promotion. By de…nition, ex ante the …rm expects to
make (almost) the same pro…t whether it places her in the high or low job, that is,
the expected ‡ow of revenue net of the cost of any challenges is nearly the same in the
two tasks. However, if the …rm monitors the worker, it will continue to monitor until
10
a challenge arises. The expected time until a challenge, and thus the expected cost
of monitoring until it arrives, is independent of the …rm’s current assessment of the
worker. Consequently the cost of monitoring the worker is bounded away from zero.
Put di¤erently, there is a chance that she will turn out to be bad, so if the …rm keeps
the worker in the low-task while monitoring her, it will save the cost of having her fail
to handle a challenge in the high-level job instead of the low-level job. But there is a
large probability that the worker will turn out to be good, and the …rm will have lost
the ‡ow of extra output from assigning her to the high task. These two considerations
o¤set each other more closely as the prior approaches the promotion cuto¤. Thus the
net gain must fall below the monitoring cost when the prior is su¢ ciently high.
So the …rm will not monitor workers who are either very unlikely to be good
or who are su¢ ciently likely to be good that they, with high probability, will be
promoted shortly anyway. However, if the monitoring cost is not too high and the
…rm’s assessment of the worker is neither too high nor too low, then it is e¢ cient to
monitor. The expected bene…t of learning that a worker is good is high when the
worker is otherwise not too close to promotion, since the good worker can be promoted
to the more productive task much more quickly. As we have already observed, the
expected cost of monitoring is independent of the …rm’s current assessment. Thus if
any workers are monitored, it will be those with an intermediate probability of being
good.
To be clear, the model does not imply that workers who are monitored are unlikely
to be promoted any time soon. Instead, it says that the workers who are monitored
are those who are some distance from being promoted in the absence of monitoring.
In the model, there are thus two potential paths to promotion. Workers who are
hired with su¢ ciently high expected productivity are never monitored. But workers
about whom expectations are su¢ ciently low when they are hired are monitored
during a period that, in equilibrium, just precedes promotion. Under at least one
interpretation of the model, this could be viewed as a training period which, upon
successful completion, leads to promotion.
We make this argument formally in the next section.
11
3
The Formal Model
We begin with the case where the …rm must choose either not to monitor, in
which case it never sees a worker successfully resolve a challenge, or to monitor the
worker fully so that it observes all challenges and their outcomes. We then extend the
model to the case of partial monitoring where the probability that the …rm observes
a worker when she successfully resolves a challenge is increasing in the expenditure
on monitoring.
3.1
Workers and jobs
An employer hires a worker whom he can put in a high task (H) or a low task (L).
The worker’s productivity in a given task depends on his type, which may be good
(G) or bad (B). Both types of worker produce a ‡ow output q per unit time in the
low task and g + q; g > 0 per unit time in the high task. denotes the …rm’s belief
that the worker is good. For the moment it is irrelevant whether type is general or
…rm-speci…c, but it may be helpful to think of it as …rm-speci…c. In the next section,
when we discuss internal labor markets, we will treat the value of when the worker
begins working for the …rm, denoted by e; as the probability that the worker is good
at a randomly chosen …rm. Thus a worker with e equal to 0:7 will be good at 70
percent of the …rms at which he may work and bad at the remaining 30 percent.
Challenges occur in both H and L tasks with a Poisson arrival rate . This
assumption ensures that task assignment is una¤ected by its impact on learning about
productivity. Assumptions of this nature are common in the literature on internal
labor markets (e.g. Gibbons and Waldman, 1999a). It might be more natural to
assume that challenges are more common in the high task, but it makes the algebra
more cumbersome, and provides little additional insight (see the end of Section 3.3).
Bad workers fail when a challenge arises. Failure generates negative output of cl
in the L-task and ch in the H-task, with ch > cl . If a worker is bad and a challenge
arises, then the failure is immediately observed, and the worker’s type is revealed.
Good workers resolve challenges when they occur, with no impact on productivity.
Thus if the worker is good, the occurrence of a challenge can be known only if the
worker is actively monitored, in which case the worker’s type is revealed. We assume
that g + q
ch < q
cl so that the expected ‡ow of output net of costs associated
with challenges is lower when a bad worker is placed in the H-task than when he is
12
placed in the L-task.
Time is continuous and the future is discounted at a rate r. Workers and …rms
live forever.
Under complete information, it is clear that good workers will be put in the Htask. We assume that if a worker is revealed to be bad, she separates from the …rm
immediately. This assumption is natural if type is at all …rm-speci…c. It also allows
us to ignore the worker’s outside option for the moment because worker turnover is
una¤ected by the monitoring choice.
3.2
Monitoring
The employer can use one of two procedures to assess workers: monitor (M) or
not-monitor (N). Under N, she assigns the worker to a task and does not monitor
him. Thus she only gets con…rmation of the worker’s type if the worker is bad and
a challenge arises, in which case the worker fails. If the worker is good, or until a
challenge arrives, the employer observes nothing, and can update her beliefs about
the worker as time passes.
Under M, the employer monitors the worker until a challenge arises, at which time
she learns the worker’s type. There is a ‡ow cost of b per unit time of monitoring,
and the cost must be borne until a challenge arises. Note that under the monitoring
procedure the employer’s beliefs remain unchanged until a challenge arises, at which
time the employer knows the worker’s type. Therefore, if it was e¢ cient to monitor
the worker, it will continue to be e¢ cient until the …rm observes a challenge.
3.3
E¢ cient promotion without monitoring
First consider a worker who is never monitored by the …rm. If, at any given time,
the worker’s probability of being good, 0 , is not too high, the employer will place
the worker in the L-task. If the worker fails at some time, his type is revealed to be
bad, and he will leave the …rm. If he has not failed until time t, the employer updates
her belief about the worker’s type to (t; 0 ). As time passes and (t; 0 ) becomes
su¢ ciently high, the employer may promote the worker to the H-task. Similarly, a
worker who comes in with a su¢ ciently high prior at time 0 will be placed immediately
in the H-task.
Given Poisson arrival, the density function for the arrival of the …rst challenge is
13
e t , hence the probability that the …rst challenge arrives by time is p( ) = 1 e .
Thus the probability that a bad worker does not fail by time is 1 p( ) = e .
Theorem 3.1. If the …rm does not monitor the worker, then it is e¢ cient to promote
the worker to the H-task when the …rm’s assessment of the probability that the worker
is good reaches
(ch cl ) g
(1)
=
(ch cl )
provided that the initial prior satis…ed
0
<
:
The present value of this strategy at the time when the prior is
U ( 0)
where
If
0
Q
=
0 g[
0
r ( + r)
1
q (1
)q
+
:
r
+r
0
(ch
g
cl )
g
]
r
(1
0 )cl
0
is given by
+r
+ Q for
0
(2)
, the …rm places the worker in the H task immediately.
(All proofs are in the appendix.)
Note that (1) has a natural interpretation. The worker is promoted when the
expected ‡ow payo¤s in the L and H tasks are equal, that is
q
(1
) cl = g + q
(1
) ch :
(3)
This follows from the assumption that learning about productivity is independent
of task assignment. Therefore the assignment decision is determined solely by the
e¤ect on expected output. It is plausible that the arrival rate of challenges would be
faster in the H-task. In this case, workers would be promoted later than implied by
equation (1). However, we have not obtained any additional insights from allowing
for di¤erent rates of arrival of challenges and therefore have not pursued this path.
3.4
Payo¤ with the monitoring strategy
When the employer monitors the worker, she knows when the …rst challenge arises,
and immediately identi…es the worker’s type. If the worker is good, it is e¢ cient to
14
immediately promote him; if he is bad, the worker separates from the …rm. Before
the arrival of the …rst challenge, no new information is generated, so there is no
continuous updating of beliefs.
At the time when the employer starts monitoring the worker, let 0 be the prior
that the worker is good. When the …rst challenge arrives, with probability 0 the
(good) worker resolves the challenge and is promoted to the H-task, with the complementary probability, he fails and leaves the …rm. In either case the employer ceases
to monitor him. Recall that monitoring has a ‡ow cost of b per unit time. We prove
in the appendix that
Theorem 3.2. The value of the monitoring strategy with e¢ cient promotion is given
by
cl
rb
0g
(1
+ Q:
(4)
U~ ( 0 ) =
0)
r( + r)
+r
We assume that
g
b
> 0:
(5)
r
If not, even if the …rm knew that the worker was good and even if the only way it
could assign the worker to the H-task was by monitoring and observing him solve a
challenge, it would prefer not to do so.
3.5
E¢ cient monitoring
Next we determine the e¢ cient time-pro…le of monitoring for the …rm. The …rm’s
action at any time depends on its current assessment of the worker’s quality. Given
the …rm can either decide to monitor immediately, or to not monitor until increases
to some higher value through updating, or to never monitor at all.2 We …nd that
there always exist ranges of values for in which the …rm does not monitor, and if
monitoring costs are not too high, there is a range of values in which the …rm monitors
immediately.
2
Formally the …rm’s strategy should specify, as a function of the history at each instant of time (or
other appropriate speci…cation), whether the worker should be monitored at that instant. However,
the …rm’s assessment of the worker is the only time-dependent variable in the model, and this
becomes static once the …rm starts monitoring and until a crisis arrives. Thus it cannot be optimal
for the …rm to switch to no-monitoring once it has optimally started monitoring, and it is su¢ cient
to consider at most a single switch from no-monitoring to monitoring.
15
Theorem 3.3. There is always a range [0;
e¢ cient not to monitor the worker.
a)
and a range ( b ;
] in which it is
Theorem 3.3 establishes that workers who are very unlikely to be good and workers
who are close to promotion will not be monitored, but it does not establish that …rms
will ever monitor workers in order to determine their quality.
Under what circumstances will the …rm engage in monitoring? The following
theorem addresses this question.
Theorem 3.4.
If
+r
r
g ( + r)
g
br
then there is an interval [ a ; b ] =
6 ;, with
monitor a worker if and only if 2 [ a ; b ].
<
a
=
(ch
b( +r)
,
(g+b)
cl )
b
g
;
(6)
such that it is e¢ cient to
Remark 1. The proof of theorem 3.4 shows that for any worker who joins the …rm
with e < a it is e¢ cient to begin monitoring the worker only when reaches a ,
provided that at a the payo¤ to monitoring is higher than the payo¤ to never monitoring.
Remark 2. Note that, except in a knife-edge case, it is not e¢ cient to begin monitoring at the where the value of monitoring …rst exceeds the value of never monitoring.
Condition (6) is not particularly informative. We can derive somewhat more
informative conditions. Recall that, by assumption, both sides of the inequality are
positive.
As rb ! g , the left-hand side goes to in…nity while the right-hand side remains
…nite. Thus when monitoring costs are high, not surprisingly it is never e¢ cient to
monitor. On the other hand, when monitoring costs are su¢ ciently low, there is a
range in which monitoring is e¢ cient.
The gain from monitoring is that the …rm is assured that it never places a bad
worker in the H-task. The ‡ow cost of doing so is given by (ch cl ). Again not
surprisingly, as this term gets large, there is always a range in which monitoring is
e¢ cient. When it gets small, or equivalently when the bene…t from placing a good
worker in the H-task gets small, monitoring is never e¢ cient.
16
An increase in the frequency of challenges, , lowers the left-hand-side and increases the right-hand side of inequality (6). Thus more frequent arrival of challenges
is associated with a larger range of other parameters consistent with monitoring.
Conversely, a higher rate of time discounting is in many ways similar to a lower rate
of arrival of challenges and thus is associated with a more restricted set of parameters
consistent with some monitoring.
3.6
Alternative Interpretation I: Formal Testing
We have assumed that the …rm can observe challenges if it monitors the worker.
The arrival of challenges that allow the …rm to observe successes is stochastic. It is
natural to ask whether, when learning is important, the …rm could simulate challenges.
This may be di¢ cult in some settings but simple in others. This subsection analyzes
the case where …rms can administer some form of costly test to determine whether or
not the worker is good. It could, for example, send the worker to a form of “training”
program where he faces manufactured challenges. While this is not isomorphic to the
model in the previous sections, the two resemble each other closely.
If the …rm is fairly sure that the worker is not good, it will not send him to the
program because the cost outweighs the expected bene…t. Similarly, for any …xed
positive training cost, there will be a < ; su¢ ciently close to , such that it will
not be worth spending that amount on the training program. But, if the training
program is not too expensive, then for intermediate values of ; it will be worthwhile.
Suppose that the test costs B. At 0 the value of administering the test is
UT =
0 (g
+ q)
r
(7)
B:
Compare this with the value of monitoring in the main model. It will be clear that
the …rm will administer the test at a if
a (g
+ q)
r
B
or
ag
rb
r( + r)
b
(1
(1
a ) (q
a)
cl
(1
aq
a) q
+
+
+r
r
+r
cl ) +
ag
(8)
Ba :
(9)
+r
The …rst two terms in the numerator represent the ‡ow cost of monitoring net of the
expected output from bad workers. The last term is the added value of output from
B
17
good workers since the test is immediate. All of this is adjusted for discounting and
the expected time at which a challenge arrives in the base model.
If B = Ba ; then the value of testing at b ; the upper limit of the monitoring zone,
is
U T ( b ) U~ ( b ) = ( b
cl )
a ) (g + q
which is zero only in the special case g + q
cl is zero. If it is positive, it may be
pro…table to test the worker even when the …rm’s assessment of his ability is too high
for monitoring to be worthwhile, and the converse is true if it is negative. Thus the
two models are not isomorphic. Nevertheless, they are similar in their interpretation
and predictions.3
3.7
Alternative Interpretation II: Increasing Risk of Failure
Some of the examples in the introduction and, indeed most of the examples that
we have come up with involve workers engaging in behaviors that create the possibility
of what we have called challenges. Workers fail to notice potential problems, do not
follow the prescribed procedures (either through lack of knowledge or moral hazard)
or otherwise put the …rm at risk. This model is isomorphic to the one we have
presented. In this case
would be the arrival rate of the opportunity to notice
potential problems. The cost of failures would be ! cl where ! is the proportion of
such events that lead to noticeably adverse outcomes. Replacing !cl with cl gives
our model.
3.8
Partial Monitoring
The preceding results do not depend critically on the assumption that monitoring
is a 0 1 variable. In this section we maintain the other assumptions of the model
but assume that the …rm can vary the e¤ort with which it monitors the worker. The
…rm can choose the e¤ort and the corresponding ‡ow cost b of monitoring. If the
worker resolves a challenge, the …rm observes the success with probability p = p(b).
We write the inverse function b = b(p) and assume that b0 0 and b00 0.
3
If both testing and monitoring are available options, then for given b and B we can calculate
~ and U T , and determine ranges in which it is respectively optimal to update, test and monitor.
U ; U
However, the intuition is su¢ ciently clear that the full exercise seems unwarranted.
18
In the appendix we derive a result (theorem A.1) that parallels theorem (3.3),
which is restated below:
Theorem 3.5. If b0 (0) > 0 and b00 (p) > 0 8p 2 [0; 1], there is always a range
[0; a ) and a range ( b ; ] in which it is e¢ cient not to monitor the worker. Further,
if b0 (0) is su¢ ciently small, then there is a non-empty interval [ a ; b ] in which a
positive amount of monitoring is e¢ cient.
Remark 3. If b0 (1) is su¢ ciently small, there will also be a range with complete
monitoring. Depending on the shape of the b(p) function, the solution can be bangbang as in our base model.
The more interesting case is when monitoring starts at a and increases smoothly
between a and some A at which p equals 1 (full monitoring). It remains at 1 for
[ A ; B ] and then decreases smoothly between B and b : Then if workers are hired
with e < a ; as in the baseline case, they will not be monitored, but, unless the worker
fails to resolve a challenge, the …rm’s assessment of will rise continuously until it
reaches a : Thereafter the …rm partially monitors the worker and continues to update
: If no challenge is observed, rises towards A :4
But the …rm may observe the worker resolving a challenge, in which case she is
immediately promoted. In the region between a and A ; the probability of promotion
is strictly increasing in since both the probability of being good and the probability
of being observed solving a challenge rise with :
If the worker is hired with e such that a < e < A ; the situation is similar except
that she never experiences the no monitoring regime.
If the worker is hired with A < e < B ; the …rm does not update except
simultaneously with a separation or promotion.
If B < e < b , the …rm continuously updates and gradually reduces monitoring.
It is clear that for close to b ; the probability of promotion must be lower than for
close to B ; but we have not been able to establish whether the relation between the
probability of promotion and is monotonic in this range and expect that it need not
be. Finally we note that if b < e < ; the worker is not monitored. In the absence
of a failed challenge, is updated continuously until it reaches
and the worker is
promoted to the high job.
4
It appears to us that
will reach
A
only asymptotically, but we have not proved this.
19
As in the base model, there are no promotions from the L-task to the H-task
originating at < a or b < < , and all promotions are from one task to the
other:
4
Wage Pro…les
To address the implications of the model for internal labor markets and wage
pro…les, we must model turnover and wage determination. We return to the case of
a dichotomous choice between monitoring and no monitoring.
This section is largely technical. Intuitively, we would expect the wage to increase
as the expected ‡ow of output increases or, possibly, as increases, which is not quite
the same. Readers who are happy to assume that the wage is increasing in expected
output and are uninterested in the details of the bargaining model can skip to section
5.
Because modeling bargaining with asymmetric information is problematic, with no
widely accepted solution, we restrict attention to settings in which there is symmetric
information although we will suggest very informally that under some conditions,
asymmetric information would not alter our conclusions.
While the assumption of symmetric information is not fully natural, neither is it
unnatural. It is most natural when monitoring takes the form of some sort of external
testing as in section 3.6 above. Rather than waiting for a challenge to arise, the
monitoring agent may be able to simulate challenges and assess whether the worker
solves them. Such “training” programs are undoubtedly designed to increase the
worker’s ability to handle a challenge successfully, but they also serve an evaluation
role.
We assume that when the …rm and worker meet, they Nash bargain over a fully
contingent contract, specifying monitoring, task assignment and wages conditional
on the history. The number of agreements consistent with Nash bargaining is large.
To ensure uniqueness, we impose a form of renegotiation-proofness. We require that
if negotiation were exogenously reopened, the continuation of the original agreement
would be the outcome of the Nash bargaining over future wages, task assignment and
monitoring. In the web appendix, we develop a Rubinstein bargaining model with
similar, but not identical results. But in that setting, ruling out implausible ine¢ cient
equilibria requires strong assumptions. It adds considerably to the mathematical
20
complexity with no gain in insight.
4.1
Turnover
Our model of turnover is simple. We interpret e, the value of when the worker
…rst arrives at the …rm, as the ex ante probability that the worker will be good at a
randomly selected job (possibly from within a class of jobs). Any information about
the ability of the worker to resolve challenges is match-speci…c. Therefore if the
updated assessment falls to < e; it is e¢ cient for the worker and …rm to separate,
and they do so. Otherwise it is e¢ cient for the worker and …rm to continue their
relationship, and there is no separation. Thus in the base model, the worker remains
with the …rm unless he fails to resolve a challenge in which case the worker quits or
is …red. Note that we do not really require that success or failure at the current …rm
provide no information about productivity elsewhere, only that it is more informative
about productivity at this …rm.5
It is also important to note that since turnover occurs if and only if the worker
is observed to fail and since observation of failure is una¤ected by the monitoring
decision, it was unnecessary in our earlier discussion to take account of the worker’s
outside option. This will not be true when we discuss wage determination.
4.2
The Nash Bargaining Solution
Since Nash bargaining is e¢ cient, monitoring and task assignment will be as
derived in the previous section. We focus on wage determination.
De…ne Wt ( t ) as the expected (future) discounted value of wage payments and
Vt ( t ) as the expected discounted value of output net of monitoring at this …rm going
forward from time t when is believed to be t and de…ne W as the value of moving
to another …rm (including possible additional moves). We denote the ‡ow value of
W and V by wt and vt : We treat W as constant, implying that success or failure
at this …rm does not provide additional information about the probability of success
at other …rms. Letting performance at this …rm be somewhat informative about the
probability of success at other …rms changes the details but not the fundamentals of
5
We abstract from unemployment, which might cause workers with a low probability of being
good at any …rm from pursuing a new match even if they know they are poorly matched with their
current …rm.
21
the results.
Theorem 4.1. Let
be the worker’s Nash bargaining weight. Then the unique
renegotiation-proof solution to the Nash bargaining problem is
wt = vt + (1
) rW :
(10)
In other words, in each period, workers get a share of the ‡ow of output, net of
any monitoring and expected costs of mistakes and, in addition, receive part of the
‡ow value of their outside option.
While we do not discuss asymmetric information in detail, it is clear that there are
settings in which our results will carry over when information is asymmetric. First, if
only the …rm recognizes when the monitored worker successfully handles a challenge,
it can appropriate the worker’s share of the expected loss from failing to handle a
challenge, but it foregoes its share of the gain in productivity and the savings from
ending monitoring. Thus, if its share is su¢ ciently high, it will want to reveal its
information by assigning the worker to the high task. In the special case of = :5;
this condition becomes g + b > (1
t ) cl :
On the other hand, suppose the worker observes that she has responded successfully to a challenge even when not monitored. There are always uninformative
equilibria in which the …rm ignores workers’claims to have resolved a challenge and
workers’strategies are independent of their information. Because promotion is more
valuable for workers who know they are good, there is also always a money-burning
equilibrium in which informed workers engage in wasteful activities to separate themselves from the uninformed. However, we note that as in Gul and Sonnenschein (1988)
such money burning equilibria may not exist if money burning takes time and can be
stopped as soon as it is recognized.
Based on this assessment, we view equilibria with full information or isomorphic
to the equilibrium with full information as the most plausible. In the next section,
we restrict our attention to these equilibria.
5
Implications for Nonmanagerial ILMs
The Internal Wage Pro…le: We again focus on the simpler case where there is only a
binary choice about monitoring. Extension to the partial monitoring case is relatively
22
straightforward. Recall that there are four ranges in the data. For < a ; the worker
is assigned to the L-task and is not monitored. For a < < b ; the worker is
assigned to the L-task and monitored. For b < < ; the worker is assigned to the
L-task and not monitored. For > ; the worker is assigned to the H-task and not
monitored. The wage pro…le for workers who remain with the …rm therefore depends
on e; the value of when the worker is hired:
1. Workers with a low e will be placed into the low no-monitoring range. If they
remain with the …rm, the …rm gradually increases its assessment of ; and
therefore the wage, until = a : At this point, the …rm begins monitoring the
worker, and there is no updating of until a challenge comes along. Wages
remain …xed until the worker either leaves the …rm or is promoted to the high
task.
2. Workers with a somewhat higher initial will be placed immediately into the
monitoring range. Although their wage will generally be higher than w( a ) since
most will have e > a ; in other respects they are similar to workers who started
at a lower e and rose to = a :
e<
3. Workers with a yet higher b
remain in the L-task and receive continuous wage increases until = , at which point they are promoted to the
H-task and continue to receive wage increases that are asymptotic to the wage
associated with = 1:
4. Finally, any worker hired with a high e
is placed directly into the Htask and receives continuous wage increases in a manner analogous to those
promoted from the upper no-monitoring range.
The Hierarchical Structure: Heretofore we have referred to tasks rather than to jobs.
Yet in many organizations, collections of tasks that appear quite similar have di¤erent
job titles (secretary I and II, tenured associate and full professor). In the empirical
literature, hierarchies are sometimes determined by transition patterns across occupational titles as in BGH. In our model, it is natural to de…ne three occupation titles:
LT 1 (“low task 1”), consisting of workers in the low no-monitoring and monitoring
zones, i.e., 2 [0; b ], LT 2, consisting of the high no-monitoring zone 2 ( b ; ], and
HT where > . Although LT 2 is higher paid than LT 1, workers do not transition
from LT 1 to LT 2, or conversely. Instead both feed into HT . So LT 1 and LT 2 appear
to share a location at the bottom of the hierarchy below HT .
We return now to the regularities described in section 1.3.
23
1. Many establishments have no variation in pay. We view the model as consistent
with but not predictive of this regularity. If the assessments e of workers at the
point of entry fall within a narrow band in the monitoring range (between a and
b ), then the wage would be approximately equal for all workers in LT 1. There
is no strong reason to believe that this would happen frequently. However, if
the …rm typically hires workers with a standard set of quali…cations at an entry
level, and initial assessment is largely determined by quali…cations, then the
range of e may well be narrow.
2. In …rms that di¤erentiate pay, individual wages are largely determined by objective factors such as seniority, experience, occupational grade, education and
other formal quali…cations which may be correlates of productivity but are not
themselves measures of productivity. In our model, for workers who are in LT 1
and LT 2 at any given time, the wage is fully explained by e and seniority, though
it is not strictly increasing with seniority at all times. To the extent that the
…rm’s initial assessment e is largely determined by education and experience,
the model is consistent with this regularity. However, some workers in the monitoring zone of LT 1 are periodically promoted to the top of the hierarchy and
therefore supersede seniority.
3. Wages rise with seniority. Except for workers who are being monitored or at
the top of the pay scale, wages increase with seniority at the …rm.
4. Many salary scales have a …xed number of steps. Workers at the top of the scale
receive no further individual pay increase unless they are promoted. The model
is set in continuous time and therefore does not actually predict a series of steps.
However, it does predict that workers hired into LT 1 will hit the top of the scale
when they begin to be monitored and that the wages of workers in HT who
have not been promoted from the monitoring zone will increase asymptotically
to :5 (g + q).
5. Demotions are rare. In the base model, demotions are nonexistent. It is important to note, however, that the mechanism is quite di¤erent from that in
Gibbons and Waldman (1999a). There, worker productivity increases continuously so that only a large adverse information shock can cause demotions.
In our model, large adverse information shocks need not be rare but generate
separations rather than demotions. We revisit this point in Section 6.
6. Wages jump at promotion. In our model wages jump at promotion for those
24
workers who are promoted from LT 1 but not from LT 2. As noted previously,
wage jumps at promotion do not arise naturally in standard learning models
although they do so in tournament models, including those with learning.
In sum, our model is able to capture many of the regularities that we observe in
nonmanagerial internal labor markets.
The model also has some empirical weaknesses. The most important issue is that,
when monitoring is a binary choice, the model implies that the wage falls when the
worker is …rst monitored. This is a less obvious problem when monitoring can be
partial and thus begin slowly; nevertheless, it is somewhat disturbing. Wage cuts at
promotion are not unheard of, but they are not the norm.
We make three observations in this regard. First, under the alternative interpretation of section 3.6, the worker and …rm might share the cost of an external training
program so that the nominal wage would not drop. Secondly, while our model is
set in continuous time, in reality, wage adjustments, and frequently wage scales, are
discrete. If the requisite drop in pay when monitoring begins is su¢ ciently modest
relative to the growth in over the period leading up to the monitoring, then no wage
decrease may be observed. Thirdly, we recognize that it is well known that nominal
wage cuts are rare (Bewley, 1999), and this phenomenon is poorly understood. It is
therefore not surprising that our model cannot explain their absence.
We also note that the model with only a binary monitoring decision counterfactually predicts that promotions are concentrated at the top of the scale of each of the
lower level jobs in the hierarchy and thus come from only two places in the wage distribution. Yet, the evidence strongly suggests that promotions come from most parts
of the wage distribution within a level of the hierarchy. This di¢ culty is resolved by
allowing for partial monitoring.
6
Extensions
Throughout this paper we have maintained the assumption that failures and observed successes are fully informative. A worker who is seen to succeed in a challenge
is de…nitely good. As a consequence, our model predicts that all promotions end
up at either at the top of the upper level of the hierarchy (observed success) or at
its bottom (updating from the upper no-monitoring zone with no observed failure).
Further, when a failure is observed, it is known with certainty that the worker is
25
bad at this …rm, which results in immediate separation. In particular, there are no
demotions. It would seem natural to relax this assumption. We brie‡y discuss two
alternatives.
Consider what happens if monitoring can produce false positives, that is the worker
can appear to have resolved a challenge when none existed. Then if a success is
observed, the …rm will not increase all the way to 1: Speci…cally, let be the arrival
rate of false positives and let = = ( + ) be the proportion of apparent challenges
that are not really challenges. Then if the …rm observes that a worker with = 0
has “responded to a challenge,”the …rm’s assessment of will be updated to
=
0
(1
)
0
+
:
(11)
This would lead to di¤erent critical values ( a ; b ; ) for the monitoring zone and
task-assignment. However, it is both intuitive and straightforward to show that if,
when is 0; there is an interval of for which the …rm monitors the worker, then for
su¢ ciently small there is still an interval for which the …rm monitors the worker and
if the worker appears to resolve a challenge he is promoted to the H-task although
the updated is less than one.
Similarly, if there are false negatives, i.e., even good workers may fail in a challenge
with some probability < 1, the …rm will revise downwards following a failure to
resolve a challenge, but it will not revise to 0. This results in a separation only if
the revised falls below e.
If the …rm’s assessment of prior to the challenge was su¢ ciently greater than
e, then even following failure the revised will exceed e. This, in turn, requires
the worker to have spent su¢ cient time with the …rm without being monitored, or,
if there are false positives, to have been observed to have had some successes. In
particular we can show that, for a worker who has not been previously monitored, a
su¢ cient condition for a failure to result in a separation is that the worker’s tenure
at the …rm satis…es
1
1
:
(12)
t<
ln
2
Thus we predict that demotions will be positively related to the worker’s time
with the …rm. Bad information about a low tenure worker results in a separation,
not a demotion. This is in contrast with Gibbons and Waldman who predict that
26
demotions, while rare, will be most likely among the recently promoted. We know of
no data on this issue.
Finally, although our initial discussion underlined the role of monitoring to correct
mistakes, we have largely ignored this role in our analysis so far. We note …rst that
our model can easily be reinterpreted as one in which the goal of monitoring is at
least partially to correct mistakes. In our base model, we assume that monitored
workers are in the L-task. Suppose that they are instead in the H-task. Rede…ne b
as b
g where b is the full cost of monitoring so that b is the cost of monitoring in
the H-task net of the increased output. De…ne cm as the cost of a failure to resolve
a challenge when the worker is monitored, and therefore the supervisor can reduce,
but not eliminate, this cost. In the special case where cm = cl ; the two models are
isomorphic. Allowing cm 6= cl would change very little. Although the parameter
values would change, for < a and for b
< ; the worker would be assigned to
the L-task and not monitored. For
; she would be assigned to the H-task and
not monitored. For a
< b ; she would be monitored. The sole change is that for
this last range, she would be assigned to the H-task rather than the L-task.
7
Discussion and Conclusion
If we allow for partial monitoring, our model permits the following monitoring
stages as a function of : No Monitoring, Partial Monitoring, Full Monitoring, Partial Monitoring, No Monitoring, High Task. Not all stages need exist. For the no
monitoring range to exist, we require that b0 (0) > 0; so that is that it is costly to
do even a little monitoring. For partial monitoring, we require b00 > 0; and for the
existence of full monitoring, we require that b0 (1) be su¢ ciently small. Therefore, the
precise nature of the internal labor market depends on the monitoring technology.
We should not be surprised by variation in internal labor markets across companies
and types of workers.
Despite this variation and dependence on parameters, there are some striking regularities in the literature on nonmanagerial internal labor markets that are consistent
with our model. If monitoring is very expensive, wages are likely to be determined
largely by observable proxies for productivity such as education and seniority. If monitoring is inexpensive and challenges are very informative, there is likely to be little
wage growth within job assignment. At intermediate monitoring costs, wages may
27
rise formulaically within some job assignment until some maximum wage. With partial monitoring, they climb formulaically except for “fast-trackers” who get a boost
from resolving a challenge.
Our model can be contrasted with that of Gibbons and Waldman (1999a, 2006).
The major di¤erence is that in our model, learning comes in large chunks and when
it does not come, either there is no learning or updating depends only on the passage
of time. In contrast, in Gibbons and Waldman, …rms continuously receive information about workers which allows them to distinguish among them. We do not view
these approaches as strict alternatives. Clearly, information can come in both forms.
However, it is important to note that there are testable di¤erences in the predictions
derived from the two models. We have already noted that the models make di¤erent
predictions about the types of workers who are demoted. In addition, our model
suggests that large wage increases at promotion should be associated with settings in
which there is a ceiling to the wage-scale. With continuous learning there is no such
discontinuity and therefore no prediction of jumps. In principle, it should be possible
to look at di¤erent jobs in a hierarchy feeding into similar jobs at a higher level in
the hierarchy. When wages tend to stall at some value in a job at the lower level, we
should be more likely to see large increases at promotion. In contrast, if wages rise
continuously within the lower level job, we should be less likely to see wages jump at
promotion. We are not aware of any studies directly related to these predictions.
Our approach has both advantages and disadvantages. On the positive side, as
discussed above, in many jobs wages are determined solely by objective measures such
as tenure and education that are only very imperfectly related to productivity. In
our model, wages are explained perfectly by e; task assignment and seniority. If we
consider education and experience to be imperfect proxies for e; the model is strongly
consistent with this regularity. However, on the negative side, it is too strong. There
are many settings in which wages are determined in part by subjective performance
evaluations even though much of the variation in wages is explained by education,
seniority and tier in the hierarchy. This is particularly true in managerial internal
labor markets. Admittedly, the distance from the assumption that information comes
only in fully informative chunks to the conclusion that wages are independent of
performance is not far. Still, we believe that the broader predictive power of the
model suggests that viewing information as “chunky”has value.
Our model is also consistent with both the steady increase in wages (at least up to
28
some maximum) that often accompanies seniority and the large jumps in wages often
associated with promotions. The strong association between large wage increases and
promotions does not arise naturally in Gibbons/Waldman.
In both models demotions are rare, albeit for di¤erent reasons. In Gibbons and
Waldman workers acquire human capital over time. The worker will be demoted only
if new information is su¢ ciently negative to outweigh the growth in human capital and
if the worker’s productivity previously placed him just above the cuto¤ between two
levels of the hierarchy. In contrast, demotions are rare in our model because negative
information usually causes a separation. In Gibbons/Waldman demotions should be
concentrated among recently promoted workers. In our model, demotions should be
particularly rare among workers with low seniority. We know of no empirical …ndings
on these two issues.
In contrast, the Gibbons/Waldman model is better able to explain the frequency
of real wage decreases. In our model, in the absence of macroeconomic shocks, real
wage decreases are like demotions. Bad news is infrequent and generally results in
a separation, not in the worker remaining with the …rm but with a lower estimate
of : In our model, small real wage decreases happen only when there are negative
macroeconomic shocks although, as discussed above, allowing false negatives would
result in some real wage decreases.
Finally, we note that technology has made monitoring easier. In almost any model
including this one, this will make pay-for-performance more common. Consistent with
this expectation, the proportion of British workers receiving performance pay rose
from 16 to 32 percent of workers between 1988 and 1994 (Manning and Saidi, 2008).
But our model suggests some less obvious e¤ects. Reducing the cost of monitoring
could shift the nature of the hierarchy. When monitoring is relatively expensive, as
discussed above, we can have two apparent jobs at the low task, one comprised of
workers in or below the full monitoring range and one comprised of workers above the
full monitoring range, with both jobs leading directly to the high task and relatively
little “lateral”movement. When monitoring becomes less expensive, particularly if it
becomes easier to observe less informative challenges, there will be more movement
from the lower range of the low task into the upper range of the low task so that the
low task now appears more like a single job in the hierarchy. Thus we believe the
model could be used to help explain how hierarchical structures change over time.
29
8
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33
A
A.1
Proofs
Proof of theorem 3.1
Suppose a worker with prior 0 has been put in a job at time 0 and has not failed until
time t. If the worker is good, then non-failure occurs with probability 1, and if he
is bad then the probability of non-failure is equal to the probability that a challenge
has not occurred by time t. Thus the employer’s updated belief about the worker’s
type (i.e., the updated probability that the worker is good) is:
(t;
0)
0
=
0
+ [1
p(t)](1
0)
(13)
which for future reference we rearrange as
1
p(t) =
0 [1
(t)]
(t)[1
0]
(14)
Let be the threshold such that a worker who was initially placed in an L job is
promoted to the H job when (t; 0 )
. We will show below that is independent
of 0 . De…ne t( 0 ) such that (t( 0 ); 0 ) = . Below we will suppress the arguments
in t(:), (:) etc.
If 0 < , then the employer puts the worker in the L job, and promotes him if
he has not failed by time t. Thus a good worker produces q between times 0 and t,
and thereafter produces a ‡ow output of g + q. A bad worker fails before promotion
with probability p(t). With probability [1 p(t)] he produces q until t, and thereafter
produces g + q until the …rst challenge arrives, at which time he produces ch and is
…red. Hence the expected payo¤ from the N-strategy with prior 0 and threshold is
34
U ( 0 ; [1
p(t)])
=
(g + q)
0
Z
1
rt
e
dt + q
t
+(1
0 )[1
+(1
0)
0q
r
+(1
Note that e
U ( 0 ; [1
rt
= e[
p(t)])
=
t] r
= [1
1
[1
r
rt
+e
Z
rt g
+q
+r
t
( +r)t
e
0)
q
dt
dt
cl
ch
Z
Z
1
e
(t t)
t
t
e
( +r)t
1
[1
+r
r
p(t)](1
0) (
e
rt
dt
!
(15)
dt
0
0g
f
rt
e
0
=
!
t
0
p(t)] e
q
Z
(ch
cl )
g)g
cl
+r
(16)
r
p(t)] , which substituted in (16) yields
p(t)]
r
(17)
0g
1
[1 p(t)]
+r
(1
0q
0 )q
+
+
r
+r
+r
(1
0) (
(ch
cl )
g)
(1
0 )cl
+r
The employer maximizes this payo¤ by choosing , or equivalently t or p(t). Maximizing U ( 0 ; [1 p(t)]) in (17) with respect to [1 p(t)] we obtain the …rst order
condition:
r
1r
[1 p(t)] 1 0 g
r
cl ) g)
0 g = (1
0 ) ( (ch
0
[1
p(t)]
1
=
1
+r
+r
= ( (ch
=
cl )
g)
(ch cl ) g
(ch cl )
35
r
p(t)] (1
0) (
(ch
cl )
g)
(18)
Let (18) be solved at t = t , and correspondingly
simpli…es to
g
[1
[1
=
]
etc. Using (14), (18)
It can be checked that the second derivative of U ( 0 ; [1
negative at the solution, as follows:
@ 2U
=
@[1 p(t)]2
=
r
2
p](
[1
r
r
2)
r
0g
1)
0g
2 (1
p](
0 )[1
+ r [1
1
p]
0g
r
1)
(1
( (ch
0) (
cl )
g)
(ch
cl )
g)
0g
=
<
p](
[1
2
p(t)]) in (17) is strictly
[1
p]
0
so this is indeed a strict maximum.
Note also that the optimal threshold is independent of the prior 0 , from which
it follows that a worker with prior 0
will be placed directly in the H job. At
the optimum, the employer’s expected payo¤ from a new worker with prior 0
can be obtained by making the appropriate substitutions in (17) to give:
U ( 0) =
r ( + r)
0 g[
0
1
(ch
0
g
cl )
r
g
It follows directly that U is increasing in
check that the expected payo¤ is then
U ( 0) =
A.2
1
r
0
(g + q) +
(1
] + (1
0.
For
0 ) (g
+q
+r
0)
cl
0q
+
+r
r
for
0
(19)
. It is straightforward to
0
ch )
q
> U ( ) for
0
>
Proof of Theorem 3.2
When the …rst challenge arises, the …rm gets
counting is
Z 1
e rt e t dt =
0
36
g+q
0 r
+r
:
(1
0 ) cl
: Expected dis-
Expected discounted monitoring costs are
b
Z
1
e
rt
t
e
dt =
0
e ( 0) =
U
b
+r
0
+r
g+q
r
(1
0 ) cl
+
q
b
:
+r
Rearranging terms yields (4)
A.3
Proof of Theorems 3.3 and 3.4
A.3.1
Preliminaries:
Given a prior , it is better to monitor the worker than never monitor if
1
[
r( + r)
)
g 1
(
g
rb] +
(1
r
1
) (
(ch
e( )
U
U ( ))
)q
q
+
+r
r
r
g
)
cl ) g
1
r( + r)
g
1
(ch
g
cl )
r
g
+
A.3.2
g 1
(
r
) (
1
(ch
g
cl )
g
)
r
(21)
Theorem 3.3
If = 0 or = , Z ( ) = 0; which proves the existence of the lower and upper
no-monitoring ranges.
A.3.3
Theorem 3.4
The value of no monitoring until time
37
)q
q
+
+r
r
(20)
rb
Name the left-had-side of (20) Z( ):
Z( ) =
(1
and then monitoring is
0
q
Z
e
rt
dt + (1
0)
q
0
+
0e
0 qe
r
r
+ (1
q
r 0e r
r
q
r
dt
cl
0 ) (q
= [1
e
( +r)t
dt
+ (1
0) e
(r+ )
q
cl b
+r
(22)
:
cl ) e
( +r)
b
g
+
+ r r ( + r)
e
Z
0
b
g
+
+ r r ( + r)
or
Recall that e
e
( +r)t
0
Maximizing with respect to
0 =
Z
=
+ (r + ) (1
(r+ )
q
cl b
(23)
+r
g
0
rb
b(r + )(1
0) e
0)
p( )]. Hence by (14), we have
( ;
0)
=
b (r + )
(g + b)
which is independent of 0 . Call this a . For a worker with an initial below this
value, the …rm will not monitor the worker until such time as t reaches this value.
At this point monitoring is more pro…table than never monitoring if Z( a )
rb.
Substituting for a and rearranging terms yields condition 6.
e ( ) U ( ) > 0 at = a . We already know that the
Thus if (6) holds then U
e ( ) U ( ) is strictly
reverse holds at = . It is straightforward to show that U
concave. Thus there is a unique b 2 ( a ; such that monitoring is more pro…table
than not monitoring for 2 [ a ; b ], but less pro…table for higher .
A.4
Proof of Theorem 3.5
Preliminaries: We state without proof that b = 0 for > :
Let be a small interval of time, p the probability that a challenge that occurs during
this interval is detected (given the intensity of monitoring), and the change in the
prior that occurs over given that the worker neither fails a challenge nor observed
to solve one. Bayesian updating leads to the following backdating formula
38
=
t
1
(1
=
(1
(24)
(1 p) )
e
)
)
(25)
p)
(1
t
(1 p)
e
t (1
ln (1t (1t )( t t+
=
e
t
t
+
1
(1 p)
)
t) ( t
(26)
)
Discretize the problem so that the …rm must do the same level of monitoring over
some period ; and consider the case of a worker in the L-task. Then
U(
) = (q
t
+(
) 1
t
(1
Substitute for
U(
(
(1
)) 1
t
+ (1
(
U ( (27)
t)
)) e
t
g+q
r
e
cl
p
e
p
)e
t
(28)
using (25) and (26)
b (p))
+(
(
(
(29)
)
t
= (q
r
b (p)) + e
ln
(1
)
t
t (1
t+ )
(1 t )( t
)
t
(1
) 1
t
(
(1
t
t
+
t
t
)
(1 p)
t+
+ (1
p
(1 p)
)
t) ( t
(1
)
t) ( t
)
t
+
p
)
t (1
(1
(1
(1
t) ( t
)) 1
t
Maximizing U (
p)
+
r
(1 p)
)
(1
t
+
)
t) ( t
39
!
))
t
g+q
r
!
1
!
(1 p)
)
) wrt to p gives
(
cl
t
(1
(1
t
+
t) ( t
1
(1 p)
)
)
!
U ( t)
0 =
dU (
)
t
dp
b0
t (1
t+
ln
(1 p) (1
t) ( t
=
r+
1
2
1
(1 p)
t
t
(1
which for
0 =
p)
1
)
t
)
+ (q
(1
(1
t
+
t) ( t
ln (1t (1t )( t t+
)
)
p)
r+
(1 p)
ln
)
+r
r
)
(30)
2
(1
)
p
(1 p)
1
(
b)
t) ( t
!
r
t
(1
(1
+
t
t) ( t
( t
( 1 + t) (
)
U ( t)
)
r
(1 p)
1)
t
:
)
t
) (g + q)
r
t
2
( t
)) t (1
t+
2
(1
(1 p)
t) ( t
1
(1 p)
)
ln
)
(1
t
(1
t
+
)
t) ( t
)
cl
> 0; p < 1 gives
dU (
)
t
dp
b0
(q
+
(1 p)
(1
=
r+
1
2
1
(1 p)
t
(1
(1
t
b)
p)2
t
t
( 1+
(31)
+
t
(1
(1
t
+
r+
(1 p)
)
t) ( t
t) ( t
U ( t)
)
p
(1 p)
1
t
1
(
t
t
( 1+
t ( 1+ t )(
(1
+
t
1
ln
t
)
+r
r
)
!
r
t
( 1+
r
(1 p)
1
t
t) ( t
:
)
) (g + q)
p)2 r
( t
)) t (1
t+
2
(1
(1 p)
t) ( t
1
(1 p)
)
cl
)
Now take the limit
lim
!0
dU (
t
dp
)
=
b0 r
b0 rp
qr + br + U ( t ) r2 + U ( t ) r
(1 p)2 r
tg
tq
+ cl r
(32)
40
cl r
t
Then we have
(q
(1
t)
cl ) +
t
g+q
r
U ( t ) ( + r) = b + b0 (1
p) ;
0 < p( 0 ) < 1
(33)
(q
(1
t)
cl ) +
(q
(1
t)
cl ) +
g+q
r
g+q
t
r
t
U ( t ) ( + r)
b0 (0) ;
p( 0 ) = 0
(34)
U ( t ) ( + r)
b (1) ;
p( 0 ) = 1
(35)
The result we want is:
Theorem A.1. If b0 (p) > 0 8p 2 [0; 1], then there is a ; b with 0 <
such that
(i) p( ) = 0 in the interval [0; a )
(ii) p( ) = 0 in the interval ( b ; ].
(iii) If b0 (0) is not too large, then p( ) > 0 for some 2 [ a ; b ].
a
b
<
Proof of (i) and (ii): Suppose there is no monitoring at t : Then U ( t ) is given by (2)
if there is never monitoring and is greater if monitoring is e¢ cient at a later point,
and therefore the proof parallels theorem 3.3 with b0 (0) substituting for b: If there is
monitoring at t then the left-hand-side of (33) is less using U ( t ) than (2) and thus
will be less than b0 (0) whenever it is less than b0 (0) substituting (2) for U ( t ) :
Proof of (iii): If there were no monitoring range, U ( t ) would be given by (2) and the
left-hand-side of (33) would reduce to (21) which is strictly positive for all 2 (0; )
and therefore greater than b0 (0) for b0 (0) su¢ ciently small.
Theorem (3.5) in the text is a restatement of parts (i) and (ii) above.
A.5
Proof of Theorem 4.1
If a worker with = t works for the …rm, he gets total value Wt +Ua (1
t ) = ( + r)
since he only goes to the alternative job if he fails. The worker’s surplus from the
relation is therefore Wt + Ua (1
Ua or Wt Ua (r + t ) = ( + r)
t ) = ( + r)
while the …rm’s surplus is Vt Wt where Vt ( t ) is the expected present discounted
value of the worker’s output at the current …rm.
41
So we have the Nash bargaining maximand
M axWt (1
! (1
Wt =
) ln (Vt
) Wt
Vt + (1
Wt ) + ln Wt
r+ t
Ua
+r
r+ t
)
Ua :
+r
=
(Vt
r+ t
Ua
+r
(36)
Wt )
(37)
To see that (10) integrates to (37), note …rst that the ‡ow value of output, vt ,
by de…nition integrates to the expected present value of output, Vt . If workers receive (1
) rUa in each period they are employed then with probability t ; the
present value of what they receive is (1
) Ua and with probability (1
t ) it is
(1
) rUa = ( + r) : But
t
(1
) Ua + (1
t ) (1
) rUa = ( + r) = (1
)
r+ t
Ua
+r
(38)
which proves that 10) is a solution to the bargaining problem. Since, by the renegotiationproofness requirement, this condition must hold for all t; it is the unique solution.
42
APPENDIX WITH RUBINSTEIN BARGAINING
Not for Publication
We assume that wages are determined as the outcome of an alternating o¤ers
bargaining game. Our setting is somewhat nonstandard for two reasons. First, the
parties must bargain not only over wages but also over the rules governing task
assignment and monitoring status. Second, since information about changes over
time, it is possible that one party or the other would want to renegotiate any prior
agreement. We look for a Bayesian perfect equilibrium of the game in which at any
time either party may announce that it wants to abrogate the agreement and can
o¤er a new agreement after a one-period delay.
Since renegotiation is costly, it is natural to assume that the parties will make
o¤ers that do not result in renegotiation. However, this appears to be true in the
general case only if we assume that any renegotiation results in an o¤er that would
not, itself, result in renegotiation. Equilibria with frequent negotiation are implausible
because they are ine¢ cient. We therefore assume a form of renegotiation-proofness:
The bargaining parties only make o¤ers to which they would be willing to commit
if they could and have the property that, if bargaining were exogenously reopened, they
would accept the original agreement’s continuation if it were o¤ered.
We assume that e, the worker’s ex ante probability of being good at this …rm, and
the other parameters of the model are common knowledge. We also assume that the
worker can observe to which task he is assigned, whether he is being monitored and
if he is monitored, whether he has risen to a challenge. Hence, the worker and …rm
have the same Bayesian update of . Wages can be conditioned on job assignment,
monitoring status and but not productivity-irrelevant factors such as time.6
Let V denote the expected present discounted value of output net of any monitoring costs associated with a given o¤er. Let Wf and Ww denote the present discounted
value of wages made by the …rm and worker. Finally let W denote the worker’s
outside option if she separates from the …rm. Then we have
Theorem A.2. Let be the time between o¤ers. Both worker and …rm propose
e¢ cient task assignment and monitoring with expected discounted value of marginal
6
Time can, of course, be relevant through its e¤ect on but not independently. If there is no
monitoring, conditioning on time and e is equivalent to conditioning on :
43
product: The worker proposes
Ww =
1
1+e
r
Vt
r
e
e
r
(1
t)
W = ( + r)
(39)
W = ( + r) :
(40)
and the …rm proposes
Wf =
As
1
1+e
r
Vt
(1
t)
! 0; both parties o¤er
W = :5 Vt
(1
t)
W = ( + r) :
(41)
The theorem could be extended to the case where discount rates di¤er. However,
di¤erent discount rates would also mean that the worker and …rm value the future
di¤erently and thus would prefer monitoring at di¤erent times. Unless we are willing
to assume that discounting of time during bargaining is somehow di¤erent from discounting outside of bargaining, which we are reluctant to do, we are stuck with equal
sharing.
We also note that in contrast with the static Nash bargaining solution where the
wage is increasing in the outside option and the one-shot Rubinstein bargaining solution which is independent of the outside option, in this setting the wage is decreasing
in the worker’s outside option. This is because the worker bene…ts from the outside
option only when the employment relationship ends. Bargaining delays therefore also
delay the worker’s opportunity to try another job.
The o¤ers that satisfy (39) and (40) are not unique because reopening negotiations
takes time. Wages can ‡uctuate over very short periods. For example, when the
worker has been revealed to be good at this job, she could get g+q and 0 in alternating
periods rather than :5 (g + q) in all periods as period length goes to 0. Thus average
wages over short …nite intervals are …xed but the wage at any instant is not.
Theorem A.3. Let = ( ) with
average wage over a period of length
(0) = 0 ! and lim
tends to
wt = :5 vt
(1
t)
where vt is the expected ‡ow productivity at time t:
44
W
!0
( ) = = 1; then the
A.6
Proof of Theorem A.2
Lemma A.1. No o¤er is ever made that triggers renegotiation.
Proof. Suppose that one party makes an o¤er that, if accepted,would …rst trigger
renegotiation when the worker is believed to have probability
of being good: If
under the terms of the o¤er, the parties can never believe that = ; the lemma
holds trivially.
Suppose instead that the belief =
occurs with strictly positive probability.
Let A1 ( ) ; <
be the wage, monitoring and task assignment at each level of
below
in the original o¤er and let A2 ( ) ;
be the average values of these
variables as a function of that will prevail after renegotiation. For a period of at
least ; during renegotiation, both worker and …rm make 0; but the post-renegotiation
agreement must have positive expected value for both worker and …rm. Therefore an
o¤er of A1 ( ) ; < that continues immediately with A2 ( ) ;
; is preferred by
both parties and involves no (…rst) renegotiation at . Since this argument applies
for any …rst triggered renegotiation, the proof is complete.
Proof of theorem:
We …rst prove e¢ ciency. We require that the o¤ers satisfy
Wf + (1
t)
W = ( + r) = e
r
Ww + (1
t)
W = ( + r)
(42)
and
Vw
Ww = e
r
(Vf
(43)
Wf )
where Vw and Vf denote the expected present discounted value of output associated
with the o¤ers. Then
Wf + (1
t)
W = ( + r) = e
Vw
Ww = e
r
r
Ww + (1
(Vf
t)
W = ( + r)
(44)
(45)
Wf )
or
Wf =
Ww =
( + r) e
r
( + r) (Vw
e r Vf )
W (1 e r ) (1
( + r) (1 e 2r )
e r Vf )
W (1 e r ) e r (1
( + r) (1 e 2r )
(Vw
45
t)
t)
(46)
:
(47)
Both Wf and Ww are increasing in Vw : So the worker is always best o¤ when she
maximizes V in her proposal. Similarly
Vf
Wf =
Vw
Ww =
( + r) (Vf
r
( + r) e
e r Vw ) + W (1 e r ) (1
)
( + r) (1 e 2r )
(Vf Vw e r ) + W (1 e r ) e r (1
( + r) (1 e 2r )
(48)
)
(49)
which establishes that the …rm always want to maximize V:
Substituting V for Vf and Vw in (46) and (47) and rearranging terms establishes
the second part of the theorem (equations 39 and 40). Setting to 0 in (39) and (40)
completes the proof.
A.7
Proof of Theorem A.3
Consider the wage over a short period of time : We consider the case where the
worker makes the …rst o¤er. This is without consequence as ! 0. We begin with
the case where the worker is not in the monitoring region.
Then we know from (39) that
t
Z
t+
e
r(x t)
w (x) dx + (1
t)
t
+
=
where Vt
that
1
1+e
r
Z
t+
e
(r+ )(x t)
w (x) dx
t
t
+ (1
1
1+e
r
e
r
r
(1
t) W
( + r)
t) e
V ( t)
e
Wt+
(50)
V ( t ). The requirement that neither party wishes to renegotiate implies
Vt+
e
r
(1
t+ ) W
( + r)
(51)
Wt+
1
1+e
46
r
e
r
Vt+
(1
) W
( + r)
t+
and therefore that
1
1+e
Vt
r
t + (1
t
Z
t+
r(x
e
e
r
(1
t) W
( + r)
e r
)
e
Vt+
t
1+e r
Z
t)
w (x) dx + (1
t)
e
r
e
(r+ )(x t)
(1
t+ ) W
( + r)
(52)
t+
(53)
w (x) dx
t
t
1
1+e
t
e
Vt
r
+ (1
r
(1
t) W
( + r)
e r
e
1+e r
t) e
r
(1
Vt+
) W
( + r)
t+
:
(54)
We have
e r (1
Vt
1+e
w=
t
t ) Ua
+r
r
R t+
t
t
e
r(x t)
e r (1
t) W
+r
1+e r
Vt
+ (1
t) e
e
t) e
e
Let
= ( ) ; with (0) = 0; and lim !0
implies 0 (0) = 1:Find the limit of w as
inequality X: Then
@X
@
lim X =
t
Evaluated at
=
+ (1
t) e
X=
r
r
e
Vt+
t+ ) W
+r
r
(r+ )(x t)
e r
(1
1+e
w (x) dx
t+
+r
)
W
r
:
0 dX
d
+
(56)
0
re
(55)
r
(1
) W + Vt+ ( + r)
:
( + r) (1 + e r )2
t+
(57)
=0
@X
=
@
V ( t)
e
(1
Vt+
= 1: Note that this latter condition
! 0:Call the right-hand-side of the
!0
@X
=
@
r
1+e
R t+
t) t
w (x) dx + (1
t + (1
e
r
e r (1
t) W
+r
r
1+e
r
t
(1
+ (1
t) W
+ Vt ( + r)
4 ( + r)
t) e
47
e
r
Vt+
(58)
e r
(1
1+e
t+
+r
r
)
W
(59)
dX
d
=
(1
t) e
+r
+ (1
t
t
e
e
1+e
Evaluated at
=
+ (1
(1
) W
t+
+r
1+e
r
e
@ t+
@
r
W
+r
r
r
1+e
=0
t )) Vt
= :5 (r + (1
t ))
2
r
e
r
t) e
= :5 (r + (1
:5
(60)
@Vt+
@
r
e
t
) W
t+
r
1+e
Vt+
r
e
t) e
(1
+r
t) e
+ (1
r
e
Vt+
r
(1
t
(1
t)
:5
+r
t) W
+r
(1
Vt
W
@Vt
@
:5
+ :5 (vt
W @@ t
+r
(61)
(r + (1
t )) Vt )
t) W
+r
:5 (1
t) W
= :5vt
(62)
(63)
Now
@
t+
@
=
t
(1
(64)
t)
So we have
dX
d
= :5 (r + (1
t ))
= :5 (r + (1
t )) Vt
(1
Vt
t)
W
@Vt
:5
@
+r
:5
@Vt
@
:5 (1
2
t
:5
(1
t) W
+r
(65)
(66)
t) W
Finally, when there is no monitoring, we write
Vt =
t
Z
t+
e
r(x t)
vg (x) dx + (1
t
+e
t)
Z
t+
e
(r+ )(x t)
e
(r+ )(x t)
vb (x) dx
(67)
t
r
t
+ (1
t) e
Vt+ :
Solving for Vt+
Vt+ =
Vt
t
R t+
t
e
r(x t)
vg (x) dx (1
e r ( t + (1
48
R t+
t) t
t) e
)
vb (x) dx
(68)
So evaluated at
=0
0
=
Vt+
vt + (r + (1
(69)
t )) Vt
So that
r
lim X =
!0
(1
t )W +Vt (
+r)
4( +r)
+ :5 vt
(1
0
t) W
(70)
0
= :5 vt
(1
t) W
(71)
:
Consider the case with no monitoring and again consider the wage over a short
period of time : We examine the case where the worker makes the …rst o¤er. Then
we know from (39) that
t
Z
t+
t
+ (1
t)
e
Z
(r+ )(x t)
w (x) dx +
Z
t+
1
e
(x t)
e
r(x t)
w dx
t
t+
e
(r+ )(x t)
e
r
w (x) dx + e
( +r)
Wt+ +
t
1
e
W1
t
=
1
1+e
Vt
r
(1
t) W
( + r)
(72)
:
The requirement that neither party wishes to renegotiate implies that
1
1+e
r
Vt+
e
r
(1
t+ ) W
( + r)
(73)
Wt+
1
1+e
49
r
e
r
Vt+
(1
) W
( + r)
t+
From (72) and (73) we obtain
Z
t
t+
t
+ (1
t)
e
Z
(r+ )(x t)
w (x) dx +
Z
t+
1
(x t)
e
r(x t)
e
w dx
t
t+
e
(r+ )(x t)
e
r
w (x) dx
t
1
1+e
1
1+e
=
e
r
Vt
r
Vt
1
1+e
( +r)
r
e
r
(1
t) W
( + r)
(1
t) W
( + r)
e
r
But, under monitoring, Vt+ = Vt ;
t
Z
t)
e
Z
(r+ )(x t)
=
t+
t+
t
+ (1
(1
Vt+
w (x) dx +
Z
( +r)
e
Wt+
) W
( + r)
t+
t:
1
t
t
1
e
W1
r
e
e
W(74)
1
Therefore, we have
t+
1
(x t)
e
e
r(x t)
w dx
t
t+
e
(r+ )(x t)
e
r
(75)
w (x) dx
t
1
Vt
1+e r
e
t 1
e
r
(1
t) W
( + r)
W1
e
( +r)
1
1+e
e
r
r
Vt
(1
t)
W
( + r)
Dividing by
wt
1
1+e
r
Vt
e
r
(1 t ) W
( +r)
e
( +r)
1
1+e
e
r
r
Vt
(1
t) W
( +r)
t
1
e
e
Taking limits gives
(1
50
t)
W
( + r)
+
t W1
W1
:
(76)
wt = :5 ( + r) Vt
r
(77)
but
Vt =
t
Z
1
e
t
+ (1
t)
(r+ )(x t)
Z
(q
b) dx +
Z
1
1
e
(x t)
e
r(x t)
(q + g) dx (78)
t
1
e
(r+ )(x t)
(q
cl
q+g
r
b
cl ) dx
0
=
q
b
(1
+r
t)
+
t
and
W1 ! :5
q+g
+r
(79)
g+q
r
(80)
which gives
wt = :5 ( + r)
q
b
(1
+r
g+q
e r
t
r
1+e
r
= :5 q b (1
t ) cl
t)
(1
cl
q+g
r
+
t
t)
W :
q+g
+r
Remark 4. The same result can be proved by taking limits as …rst
to 0:
51
(1
t)
W
(81)
( + r)
(82)
and then
goes