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Working Paper Series
Reputation, Career Concerns, and Job
Assignments
WP 06-01R
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/economic_
research/working_papers/index.cfm
Leonardo Martinez
Federal Reserve Bank of Richmond
Reputation, career concerns, and job assignments∗
Leonardo Martinez†
Federal Reserve Bank of Richmond Working Paper No. 06-01R
Abstract
Does a worker who had a successful career have stronger or weaker incentives to manipulate his reputation than a worker who performed poorly? This paper presents a tractable
model that allows us to study career concerns when the strength of a worker’s incentives depends on his employment history (the history of his past actions, jobs, and performances).
More specifically, the paper incorporates standard job assignments into the main model in
Holmstrom’s (1999) seminal paper on career concerns. Equilibrium wages, equilibrium job
assignments, and the strength of career-concern incentives are the same for all employment
histories that lead to the same worker’s reputation. (With reputation we refer to beliefs
about the worker’s future productivity.) We show that, typically, workers with a better
reputation have stronger incentives than workers with a worse reputation. Furthermore,
we show that when the strength of incentives depends on employment history, (i) a ratchet
effect may appear, (ii) in spite of this ratchet effect, incentives may be stronger, and (iii)
incentives may be stronger when beliefs about ability are more precise.
Journal of Economic Literature Classification Numbers: C73, D82, D83.
Keywords: career concerns, job assignments, reputation, agency, learning, dynamic
games.
∗
I thank the editor, Armin Schmutzler, and two anonymous referees for helpful comments. I benefited from
discussions of this paper and related work with Marina Azzimonti Renzo, John Duggan, Huberto Ennis, Mark
Fey, Juan Carlos Hatchondo, Hugo Hopenhayn, Arantxa Jarque, Per Krusell, Roger Lagunoff, Pierre Sarte, Uta
Schoenberg, Francesco Squintani, and participants at seminars and conferences. I also thank Elaine MandalerisPreddy for writing guidance. Remaining mistakes are mine. Any opinions expressed in this paper are those of
the author and do not necessarily reflect those of the Federal Reserve Bank of Richmond or the Federal Reserve
System.
†
Research Department. Federal Reserve Bank of Richmond. E-mail: leonardo.martinez@rich.frb.org.
1
1
Introduction
Fama (1980) suggests that workers are disciplined by opportunities in markets for their services,
that is, by their career concerns. These career concerns appear in situations where firms do not
know a worker’s future productivity, but they learn about it by observing his performance. In
general, a worker’s compensation is influenced by firms’ beliefs about his future productivity.
A worker may want to improve his performance in order to make firms believe that he is more
productive and thus obtain a better compensation.
Does a worker who had a successful career have stronger or weaker career-concern incentives
than a worker who performed poorly? For tractability reasons, previous work sidesteps the
relationship between employment history—i.e., the history of a worker’s past actions, jobs, and
performances—and the strength of career concerns.1 Earlier research does so by restricting the
analysis to environments in which there is only one period with career concerns (and, thus, no
relevant employment history), or to multi-period environments in which the strength of career
concerns does not depend on employment history.2 In contrast, we present a multi-period model
that allows us to study career concerns when the strength of a worker’s incentives to manipulate
his future reputation depends on his employment history (with reputation we refer to beliefs
about the worker’s future productivity).
We study a parsimonious departure from the main model in Holmstrom’s (1999) seminal
paper on career concerns. In that model, workers can only be employed in one type of job. In
contrast, we study an economy with N job types. Each period, a worker can be assigned to
any of the N jobs. The output produced by a worker depends on the quality of his labor input
(henceforth the quality) and on the job to which he is assigned.
1
See, for example, Holmstrom (1999), Gibbons and Murphy (1992), Besley and Case (1995), Jeon (1996),
Prendergast and Stole (1996), Meyer and Vickers (1997), Dewatripont, Jewitt, and Tirole (1999a, 1999b), Persson
and Tabellini (2000), Alesina and Tabellini (2003), De Motta (2003), Ortega (2003), Le Borgne and Lockwood
(2004), Eggertsson and Le Borgne (2005), and Shi and Svensson (2006).
2
For instance, when describing their model, Dewatripont, Jewitt, and Tirole (1999a) explain that “The model
is thus a ‘two-period’ model (effort in period 1, reward in period 2). Its extension to more than two periods is,
except in the additive normal case, a complex matter that is left for future research. In a multi-period context,
the agent acquires private information about the marginal productivity of her effort and the impact of effort on
future wages....” We study a model in which the agent may have private information about the impact of effort
on future wages.
2
Quality depends on the worker’s ability, his effort level, and luck. Firms do not observe the
worker’s ability, effort, and luck; instead they observe his performance. Good current performance
by the worker signals that he is capable of good performance in the future. The wage offered to
the worker increases if he is expected to perform better. Therefore, the worker exerts effort to
improve his current performance.
There are two properties that define a job. One is the fixed amount of output produced by
a worker independently of the quality of his labor input. The other is the marginal return to
quality. As is standard in the literature on job assignments, we assume that workers who are
expected to provide input of higher quality are also expected to be more productive in jobs with
higher marginal return to quality.3 We refer to jobs with higher marginal return to quality as
managerial jobs that are at higher levels in a firm’s hierarchy.4
We show that career concerns are typically stronger for workers with better reputation. A
worker’s reputation conveys all the payoff-relevant information. That is, for all employment
histories that lead to the same reputation at the beginning of a period, equilibrium wages,
equilibrium job assignments, and the strength of career-concern incentives are the same. In a
competitive labor market, the wage offered to a worker increases with the expected quality of his
labor input, and the change in the wage offered with respect to expected quality is increasing in
the marginal return to quality of the job to which the worker will be assigned. A worker with
a better current reputation is more likely to be assigned to jobs with a higher marginal return
to quality. As a result, he expects a larger increase in wages from making firms believe that his
labor input will be of better quality. Consequently, he chooses a higher effort level. This finding
is consistent with evidence from time-use surveys that shows that workers with more years of
education allocate more time for market work (see, for example, Aguiar and Hurst (2007)).5
3
This is the case, for example, in Bernhardt (1995) and Gibbons and Waldman (1999) (see also the references
therein). In these studies, workers do not decide their effort level.
4
For empirical evidence of the “demand for talent at the top,” see Cuñat and Guadalupe (2006) and the
references therein. Span-of-control models present a theory of why employees with higher ability are assigned to
higher levels in hierarchies (see, for example, Rosen, 1982).
5
Note that one way of exerting more effort on the job is working more hours. Like effort in agency models,
some working hours (for example, the time spent thinking about a problem) are not directly observed by the
employer. Furthermore, in general, workers with more years of education are more likely to work in managerial
jobs, where the marginal return to quality is higher. Thus, these workers could choose to work more hours (or
exert more effort) because they expect a larger benefit from making firms believe that the expected quality of
3
This paper highlights three other lessons from the study of a model of career concerns where
the strength of incentives depends on a worker’s employment history. First, the paper shows that
this dependence may lead to a ratchet effect in implicit incentives: a worker’s incentives to exert
effort in order to increase his next-period wage may be weakened by the effect of effort on his
wages in other future periods. On the one hand, a worker has incentives to exert effort in order
to increase his next-period wage: a worker’s effort positively affects firms’ current-period learning
about his ability and, thus, exerting effort increases a worker’s next-period wage. On the other
hand, a worker’s effort may negatively affect firms’ next-period learning about his ability. As a
result, exerting effort may decrease a worker’s wage after the next period. We show that the
latter (ratchet) effect appears because next-period equilibrium effort depends on employment
history.
Second, we show that when employment history affects the equilibrium effort level, the effect of effort on next-period wage may be stronger. As a result, in spite of the ratchet effect
described above, career concerns may be stronger. Wages reflect expected productivity, which is
increasing in the expected effort level. If the equilibrium effort strategy is an increasing function
of reputation (as is typically the case), by exerting effort in the current period, a worker not
only improves his next-period expected productivity by increasing his expected ability but also
increases the effort firms expect him to exert (firms expect the worker to exert the equilibrium
effort level). Thus, a worker’s incentives to exert effort are strengthened by his desire to make
the market believe that he will exert more effort in the future.
Third, the paper shows that a worker’s career concerns may be stronger if the prior belief
about his ability is more precise. We show that with more precise priors, an improvement in a
worker’s reputation may imply a greater increase in the next-period effort level firms expect him
to exert, which in turn would imply a greater next-period wage increase. As a result, incentives
to exert effort may be stronger.
The remainder of this paper is organized as follows. Section 2 presents the framework, Section
their labor input is higher. Of course, other theories could also explain the relationship observed in the data.
For instance, if workers with more education are offered a higher market wage, they could choose to sell more
observable hours in the labor market. To distinguish between the two theories, one could look at whether workers
are paid by the hour and whether employers can observe working hours.
4
3 discusses results, and Section 4 concludes.
2
Framework
Following Holmstrom (1999), we study a competitive economy with many identical, risk-neutral
profit-maximizing firms in the market each period, with labor as the only factor of production.
There is a single output good, the price of which is normalized to one. We study a worker’s
incentives to exert effort in this economy. The number of employment periods is discrete and
indexed by t ∈ {1, 2, ..., T }.
The amount of output yt produced by the worker depends on the quality of his labor input
(quality) qt and the job in which he is employed jt ∈ {1, ..., N}. Quality qt is a stochastic function
of ability ηt , and effort at ≥ 0. In particular,
qt = at + ηt + εt ,
where εt is a normally distributed random variable with expected value 0 and precision hε (the
variance is
1
).
hε
The worker’s ability ηt evolves as a random walk. In particular, ηt+1 = ηt + ξt ,
where ξt is assumed to be normally distributed with mean 0 and precision hξ .
In any job j, the output produced by the worker would be a linear function of quality. In
particular, we make the following assumption.
Assumption 1: If the worker is employed in job j, his output is given by
yt = Aj + λj (at + ηt + εt ) .
(1)
Assumption 1 implies that job j can be characterized by two parameters: the amount of
output produced independently of quality, Aj , and the marginal return to quality, λj > 0.6
Without loss of generality, we make the following assumption.
Assumption 2: λ1 < λ2 < ... < λN .
6
The production function in equation (1) is particularly close to the one studied by Gibbons and Waldman
(1999) (who do not study the worker’s effort choice). As in previous studies on job assignments, we assume that
there is only one type of ability. If N = 1, λ1 = 1, and A1 = 0, the production function in equation (1) is exactly
the one in the main model studied by Holmstrom (1999).
5
We also assume that no job is fully dominated by the other jobs in the economy. That is, for
any job j there is a quality level such that job j is the most productive job for that quality level.
This allows us to restrict attention to jobs to which the worker could be assigned in equilibrium.
Neither firms nor the worker know the worker’s ability. Before the worker starts his career,
firms and the worker share the same belief about his ability. This belief is normally distributed
with mean m1 and precision hη .
Firms and the worker learn about the worker’s ability using Bayesian learning. From this
point forward, belief refers to belief about the worker’s ability, unless stated otherwise.
Depending on the precision of the shock that determines the evolution of the worker’s ability, hξ , the precision of beliefs may be increasing or decreasing with respect to the number of
observations of the worker’s performances (t) (see Holmstrom, 1999). The following assumption
guarantees that the precision of beliefs is constant.
Assumption 3:
h2η + hη hε
hξ =
.
hε
(2)
Assumption 3 simplifies the analysis by allowing us to abstract from a tenure effect on the
strength of career concerns discussed in previous studies (see, for example, Holmstrom, 1999).
We focus on the relationship between employment history and the strength of career concerns,
which previous studies left unexplored.
The timing of events within each period is as follows. First, each firm offers a wage to the
worker. Second, the worker decides where to work, his employer pays him his wage wt , and he
consumes it.7 Third, the firm that hired the worker decides in which job he will be employed.
Fourth, the worker decides on his effort level. Fifth, the luck component of quality εt and the
shock that determines the evolution of the worker’s ability ξt are realized, and output yt is
observed (in period 1, ability η1 and ε1 are realized before y1 is observed).
Ability ηt , the shock that determines the evolution of ability ξt , and the luck component of
quality εt are unobservable. Firms do not observe the worker’s effort level at (which is of course
known by the worker). Job assignments jt are public information.
7
As in previous studies of career concerns, this paper does not study the worker’s participation decision. It is
assumed that the worker always works.
6
There is a cost to exerting effort, c (at ), with c′ (at ) ≥ 0, c′′ (at ) > 0, and c′ (0) = 0. The
worker’s period-t utility equals wt − c(at ). Let δ ∈ (0, 1) denote the firms’ and the worker’s
discount factor.
3
Results
This section is organized as follows. First, we discuss equilibrium learning. Second, we describe
the main properties of the equilibrium when the worker can only be employed for three periods.
Finally, we discuss whether equilibrium properties change when other employment durations are
considered.
3.1
Equilibrium learning
Since beliefs are Gaussian and have a constant precision (because of Assumption 3), their evolution is described by the evolution of their mean. Let mf t and mwt denote the mean of the
firms’ and the worker’s beliefs at the beginning of period t (from here on, at period t). We refer
to a belief with mean m as belief m. At t, if the firms’ and the worker’s belief coincide, mt =
mf t = mwt denotes their belief.
Define st ≡ ηt + εt . We refer to st as the signal of the worker’s ability ηt . Bayes’ rule implies
that the mean of the beliefs at t + 1 is a weighted sum of the mean at t and the period-t signal.
Assumption 3 implies that the precision of the period-t + 1 beliefs about the signal st+1 is always
equal to the precision of the period-t beliefs about the signal st . This precision is given by
H≡
hη hε
.
hε + hη
(3)
Since beliefs about the signal are Gaussian and have a constant precision, the evolution of these
believes can also be summarized by the evolution of their mean, which is equal to the mean
of the beliefs about ability. Assumption 3 implies that the weight of period-t mean beliefs in
period-t + 1 mean beliefs does not depend on t. This weight is given by
µ=
hη
.
hη + hε
7
(4)
Observing output yt and job jt allows firms and the worker to compute the signal st using
their knowledge about the effort exerted by the worker and the production function. The worker
knows the effort he exerted. Therefore, he is able to compute the true signal st =
yt −Ajt
λjt
− at .
Thus, the worker’s belief at t + 1 is characterized by
mwt+1 = µmwt + (1 − µ) st .
Firms compute the signal using equilibrium effort levels. They are rational and understand
the game. In particular, firms know the worker’s strategy. They also know the history of outputs
y t−1 = (y1 , y2 , ..., yt−1 ) and the history of job assignments j t = (j1 , j2 , ..., jt ). Let af t (y t−1 , j t )
denote the equilibrium period-t effort level computed by firms.8 The period-t signal computed
by firms is given by
yt − Ajt
t−1 t
t−1 t
sf t y t , j t ≡
−
a
y
,
j
=
s
+
a
−
a
y
,
j
.
f
t
t
t
f
t
λj t
(5)
Thus, firms’ period-t + 1 belief is given by
mf t+1 = µmf t + (1 − µ) st + at − af t y t−1 , j t .
(6)
Recall that firms and the worker have the same period-1 belief. Moreover, in any period
in which the worker exerts the equilibrium effort level, firms and the worker compute the same
signal (see equation (5)). Consequently, the next lemma holds
Lemma 1 In equilibrium, the firms’ and the worker’s belief coincide (mf t = mwt ).
8
When the worker decides the effort he will exert in period t, his information is given by y t−1 , j t , at−1 where
at−1 = (a1 , a2 , ..., at−1 ) denotes the history of previous effort levels. For all t > 1, let awt y t−1 , j t , at−1 denote
the worker’s period-t optimal effort level.
Let aw1 (j1 ) denote the first-period equilibrium effort when the worker
is employed in job j1 . Using y t−1 , j t and the history of equilibrium effort levels, firms can compute the period-t
equilibrium effort level as
af t y t−1 , j t ≡ awt y t−1 , j t , at−1
y t−2 , j t−1 ,
f
where
at−1
y t−2 , j t−1 ≡ aw1 (j1 ) , aw2 (y1 , j1 , aw1 (j1 )) , ..., awt−1 y t−2 , j t−1 , at−2
y t−3 , j t−2
f
f
denotes the history of equilibrium effort levels computed using y t−2 , j t−1 .
8
3.2
A three-working-period version of the model
In this section, we characterize the equilibrium of an economy where the worker can only be
employed for three periods, i.e., t ∈ {1, 2, 3}. We solve for equilibrium using backward induction.
3.2.1
The last working period
In period 3, there is no future compensation that could be influenced by the worker. Therefore,
he always chooses a3 = 0.
The firm that hires the worker assigns him to the job for which his expected productivity is
higher. Since a3 = 0, the period-3 quality of the worker’s labor input expected by firms is given
by the ability expected by firms, mf 3 . Thus, if a firm assigns the worker to job j, it expects a
productivity Aj + λj mf 3 . Note that the higher the worker’s expected ability, the better it is to
assign him to a job with a higher marginal return to quality. Thus, we can present the optimal
job assignment rule using the following reputation thresholds: m13 = −∞, for all 2 ≤ j ≤ N,
mj3 =
Aj−1 − Aj
,
λj − λj−1
(7)
+1
and mN
= ∞. Equation (7) defines mj3 as the expected quality that makes a firm indifferent
3
between employing the worker in jobs j −1 and j, i.e., mj3 is such that Aj−1 +λj−1 mj3 = Aj +λj mj3 .
Thus, if mf 3 > mj3 , a firm would rather employ the worker in job j than in job j − 1 (recall
Assumption 2 states that λj > λj−1 ).
Since we assumed a competitive labor market, in equilibrium, the wage offered to the worker
is equal to his expected productivity, which is computed by firms using the equilibrium job
assignment rule. We assume that, when a firm is indifferent between employing the worker
in two jobs, it employs the worker in the job with the lowest marginal return to quality λj .
Following the discussion presented above, the next lemma characterizes equilibrium in the last
working period.
Lemma 2 In period 3, the worker does not exert effort in equilibrium (i.e., aw3 (y 2 , j 3 , a2 ) = 0
for any history (y 2 , j 3 , a2 )), he is assigned to job j if and only if mj3 < mf 3 ≤ mj+1
3 , and the
9
equilibrium wage is given by
ω3 (mf 3 ) = Aj + λj mf 3 ,
(8)
where j is such that mj3 < mf 3 ≤ mj+1
3 . Thus, the period-3 wage is the same for all histories
(y 2 , j 2 ) that imply the same firms’ belief mf 3 , and is an increasing function of mf 3 .
3.2.2
The second working period
In the remainder of the paper, the worker’s equilibrium effort strategy is characterized using
the first-order condition of his maximization problem for a given effort level used by firms to
compute the signal.9 Note that assuming concavity of the worker’s maximization problem for a
given effort level used by firms to compute the signal assures that, given that effort level, there
exists a unique optimal effort level. But in order to find the equilibrium effort strategy, we have
to solve a fixed-point problem: The effort level used by firms to compute the signal has to be
equal to the effort level the worker exerts in equilibrium. Concavity does not guarantee that the
equilibrium effort strategy exists and is unique. It could be that there is no equilibrium effort
level such that if firms use that effort level to compute the signal, the worker chooses that effort
level. Also, it could be that there exists more than one effort level such that if firms use that
effort level to compute the signal, it is optimal for the worker to choose that effort level.10
Let φ(s; m, H) denote the density function for a signal that is normally distributed with mean
m and precision H. For any j ≥ 2, and any firms’ period-two belief mf 2 , let
sj2
mj3 − µmf 2
(mf 2 ) ≡
1−µ
(9)
denote the smallest period-2 signal inferred by firms that, in period 3, would make assigning
the worker to job j better than assigning him to job j − 1. That is, sj2 (mf 2 ) is such that
9
In order to assure global concavity of the worker’s problem, it is sufficient to assume enough convexity in
c (a). For example, one could find an upper bound for the slope of the marginal benefit curve and assume that
the slope of the marginal cost curve is always higher. Another alternative is to assume the standard exponential
cost function, c(a) = an , and to assume that n is high enough. In particular, this makes the worker’s problem
globally concave in the examples discussed in this paper.
10
Dewatripont, Jewitt, and Tirole (1999b) present a model with one career-concern period in which there are
multiple equilibria. In a multi-period version of their environment, it could be possible to construct equilibria in
which the effect of employment histories on the strength of career-concern incentives goes beyond their effect on
beliefs.
10
µmf 2 + (1 − µ)sj2 (mf 2 ) = mj3 . The next proposition shows that a unique equilibrium effort
strategy exists (see Appendix A for the proof).
Proposition 1 In period 2, for any equilibrium belief m for the firms and the worker, there
exists a unique equilibrium effort level a∗2 (m) given by
" Z 2
Z s32 (m)
Z
s2 (m)
2
N
′ ∗
1
c (a2 (m)) = δ (1 − µ) λ
φ (s; m, H) ds + λ
φ (s; m, H) ds + ... + λ
s22 (m)
−∞
#
∞
φ (s; m, H) ds .
sN
2 (m)
(10)
Consequently, the equilibrium effort level is the same for all employment histories that imply the
same equilibrium belief m.
Equation (10) shows that the period-2 equilibrium effort strategy is such that the marginal
cost of exerting effort is equal to the expected marginal benefit of exerting effort. The worker
benefits from exerting effort in period 2 because period-2 effort increases the period-3 wage that
will be offered to him. As shown in equation (5), the signal computed by firms in period 2 is
increasing with respect to the worker’s period-2 effort level. Therefore, the worker’s period-3
ability expected by firms and thus the period-3 wage are increasing with respect to the period-2
effort level (see equations (6) and (8)). The increase in the period-3 wage implied by an increase
in the worker’s perceived ability is given by the marginal return to quality λj of the job in which
firms would employ the worker (see equation (8)). Consequently, when the worker decides his
period-2 effort level, he does not know how period-2 effort will affect his period-3 wage, because
he does not know to which job he will be assigned in period 3. In the expected marginal benefit
of exerting effort (right-hand side of equation (10)), the marginal return to quality of each job
j, λj , is multiplied by the probability of the worker being assigned to that job next period.
Proposition 1 shows that employment history only affects the strength of a worker’s incentives
through his reputation. The key insight for understanding why reputation conveys all the payoffrelevant information is as follows. Aside from determining beliefs, histories of outputs and job
assignments may only affect the strength of the worker’s incentives through the effort level
af 2 (y1 , j 1 ) used by firms to compute the period-2 signal sf 2 (y1 , j 1 ) (see the worker’s maximization
problem in Appendix A). In equilibrium, the effort level exerted by the worker is the effort level
11
used by firms to compute the signal. Therefore, the signal computed by firms is the true signal
(sf 2 (y1 , j 1 ) = s2 ), which does not depend on the history (y1 , j 1 ). Thus, the equilibrium effort
strategy only depends on the history (y1 , j 1 ) through beliefs.
Note that proposition 1 shows that the equilibrium effort level does not depend on the job in
which the worker is currently employed. This is the case because we assumed that the worker’s
capability of influencing the signal computed by firms is independent of the job in which he is
employed (see equation (5)). Assumption 1 implies that, in any job, the worker can use one unit
of effort to substitute one unit of ability.
Reputation and the strength of career concerns. Next, we show that equilibrium effort
is increasing with respect to reputation. Recall that, in the expected marginal benefit of exerting
effort, the marginal return to quality of each job is multiplied by the probability of the worker
being assigned to that job next period. Thus, if the worker believes it is more likely that next
period he will be assigned to jobs with a higher marginal return to quality, the expected marginal
benefit of exerting effort is higher. The better the worker’s reputation, the more likely he will be
assigned to such jobs. Consequently, equilibrium effort is increasing with respect to reputation.11
The following proposition states this result (see Appendix B for the proof).
Proposition 2 If N > 1, the equilibrium effort strategy a∗2 (m) is an increasing function of the
worker’s reputation m.
Even though in equilibrium the firms’ and the worker’s beliefs coincide, the role of each of
these beliefs on the worker’s decision can be studied separately. In equation (10), the firms’ belief
determines the minimum signal required for working in a certain job next period, sj2 (m). As
shown in equation (9), this signal is a decreasing function of the worker’s reputation. For example,
if at the beginning of the period firms believe that the worker is very talented, in the next period
he may be assigned to a managerial job even if the current-period signal is low (because the
11
Milbourn (2003) presents a similar result. He studies a model in which (i) the worker may lose his job, (ii) the
worker is more likely to stay on the job if his reputation is better, (iii) the worker’s effort has greater influence on
the value of the firm when he is more likely to stay on the job, and (iv) the worker’s compensation is increasing
with respect to the value of the firm. Consequently, incentives to exert effort are stronger when the worker’s
reputation is better.
12
firms’ next-period belief may still be good enough). That is, if m is higher, sj2 (m) is lower, and
the worker knows that he is more likely to work in a managerial job in the future. Therefore, if
the worker’s reputation is better, his expected gain from making firms believe that he has higher
ability is greater, and his equilibrium effort level is higher. In equation (10), the worker’s belief
determines the signal density function he uses for evaluating his problem, φ (s; m, H). Loosely
speaking, it determines how likely he thinks it is that a certain signal is realized. If the worker
believes he is better and, therefore, he believes a higher signal is more likely, he believes that it is
more likely that he will work in a managerial job next period. Consequently, the expected gain
from making firms believe that he has higher ability is greater, and he chooses to exert more
effort.
Note that if there is only one job in the economy (i.e., if N = 1), the equilibrium effort level
in equation (10) does not depend on the worker’s reputation. Recall that in models of career
concerns, a worker exerts effort to improve the quality of labor input he is expected to provide in
future periods. If N = 1, the worker’s compensation is a linear function of the expected quality
of his labor input, as in the main model in Holmstrom (1999). Therefore, the expected derivative
of the compensation with respect to expected quality does not depend on reputation, and thus
the strength of incentives to exert effort does not depend on reputation. In contrast, if N > 1,
compensation is a piece-wise convex function of expected quality.12
Equilibrium wages and job assignments. As in period 3, in period 2, the firm that hires
the worker assigns him to the job for which his expected productivity is higher. The period-2
quality of the worker’s labor input expected by firms is given by a∗2 (mj2 )+mj2 . Using this expected
quality, we can define reputation thresholds for job assignments, as we did for period 3. As in
period 3, the period-2 wage offered to the worker is equal to his expected productivity. The
following proposition characterizes the equilibrium job assignment rule and the equilibrium wage
function for period 2 (see Appendix C for the proof).
Proposition 3 There exists a unique set of threshold reputation levels mj2 such that, in equilib12
Holmstrom (1999) suggests that expanding his main model to consider multiple jobs would result in a convex
compensation function. Rosen (1982) shows that in a span-of-control economy, the compensation function is
piece-wise convex.
13
rium, the worker is assigned to job j in period 2 if and only if mj2 < mf 2 ≤ mj+1
2 . The threshold
reputations are such that
a∗2 (mj2 ) + mj2 = mj3 .
Therefore, mj2 < mj3 . The period-2 equilibrium wage is given by
ω2 (mf 2 ) = Aj + λj [a∗2 (mf 2 ) + mf 2 ] ,
where j is such that mj2 < mf 2 ≤ mj+1
2 . Thus, the period-2 equilibrium wage is the same for all
histories that imply the same firms’ belief mf 2 , and is an increasing function of mf 2 .
Proposition 3 shows that workers with better reputation will work in higher levels in the
hierarchies. Thus, even though workers’ equilibrium strategies do not depend on their current
job assignment, workers in higher levels in the hierarchies exert more effort.
3.2.3
The first working period
In this section, we describe equilibrium in period 1. We show that understanding the worker’s
interest in influencing the next-period effort level expected by firms is crucial for understanding
his incentives.
A ratchet effect. First, we describe how a ratchet effect weakens the worker’s incentives to
exert effort in period 1. This ratchet effect appears because of a negative effect of period-1 effort
on the worker’s period-3 wage.13
By exerting effort in period 1 the worker improves his period-2 reputation (see equation (6)).
On the one hand, the worker benefits from this because it increases the wage he will receive in
period 2 (recall that the equilibrium period-2 wage presented in proposition 3 is an increasing
function of the worker’s reputation).
13
The ratchet effect in implicit incentives presented in this paper is similar to that in explicit incentives highlighted in previous studies where current performance affects the terms of future explicit incentive contracts (see,
for example, Freixas, Guesnerie, and Tirole, 1985, and Meyer and Vickers, 1997). But the mechanism in our
paper is different from the one in these studies. Here, the ratchet effect appears through the effect of current
effort on future learning and not through the effect of current effort on future contracts.
14
On the other hand, improving his period-2 reputation may harm the worker because it worsens
the period-2 signal of his ability computed by firms and therefore may decrease his period-3
wage. Recall that the signal computed by firms is a decreasing function of the effort they use for
computing the signal (see equation (5)). Firms compute the period-2 signal using the equilibrium
effort level a∗2 (mf 2 ). Thus, since a∗2 (mf 2 ) is an increasing function of mf 2 , the period-2 signal
computed by firms is a decreasing function of mf 2 . If firms compute a lower period-2 signal, this
decreases mf 3 and, therefore, it decreases w3 (mf 3 ). This ratchet effect is not present in previous
studies of career concerns where equilibrium effort does not depend on employment history and,
therefore, a worker does not change the effort firms expect him to exert next-period by changing
his next-period reputation.14
The remainder of this paper focuses on environments in which the worker wants to exert effort
because of career concerns.15 That is, we focus on environments in which the ratchet effect is not
strong enough to prevent the worker from exerting effort. As indicated above, the importance
of the ratchet effect depends on the responsiveness of period-2 effort to reputation. A bound
to this responsiveness guarantees situations in which the worker wants to exert effort because
of career concerns. It is easy to limit this responsiveness. For example, one can assume a low
enough value for the precision in the signal distributions.16
14
For instance, Meyer and Vickers (1997) show that a ratchet effect does not appear in their managerial model
of career concerns. They also show that a ratchet effect appears in a career-concern model of regulation. But
the ratchet effect in this paper is different from the one in their paper. The ratchet effect in their paper does
not appear because of the effect of the current action on future learning. In their model of regulation, since a
higher perceived ability (or “intrinsic efficiency”) translates into a lower regulated price, the agent (the regulated
firm) is worse off if he is perceived to have higher ability. This makes their regulation model different from their
managerial model in which a worker benefits from improving his reputation (his wage is higher if he is perceived
to have higher ability).
15
We focus on situations in which the worker expects his continuation utility to increase if firms believe his
expected ability is higher. The worker’s effort at t increases mf t+1 (see equation (6)). Consequently, the worker
exerts effort if and only if he expects his continuation utility to be increasing with respect to mf t+1 .
16
As explained before, the strength of the incentives to exert effort in period 2 depends on the probability of
having a managerial job in period 3. Consequently, the derivative of the equilibrium effort strategy with respect to
the worker’s reputation can be made small by decreasing the derivative of the probability of having a managerial
job with respect to reputation (see Appendix B). This in turn can be achieved by assuming high variance in the
signal distributions (a low H). This is illustrated by the examples presented in Figures 1 and 2.
15
The equilibrium effort level. Let M (m, s) ≡ µm+(1 − µ) s denote the mean in the posterior
belief when m is the mean in the prior belief and s is the signal used to update the prior. Let
r1 (mf 2 ) ≡
dM (mf 2 , s + a2 − a∗2 (mf 2 ))
′
= µ − (1 − µ) a∗2 (mf 2 )
dmf 2
(11)
denote the derivative of firms’ period-3 belief with respect to their period-2 belief for a given
′
period-2 effort level a2 , where a∗t (m) denotes the derivative of a∗t (m) with respect to m. For all
j ≥ 2, let
sj1 ≡
mj2 − µm1
1−µ
denote the smallest period-1 signal inferred by firms that, in period 2, would make assigning the
worker to job j better than assigning him to job j − 1. That is, sj1 is such that µm1 + (1 − µ)sj1 =
mj2 . The next proposition presents the worker’s equilibrium effort level in his first period of
employment (see Appendix D for the proof).
Proposition 4 In the first period of employment, there exists a unique equilibrium effort level
a∗1 that satisfies
c′ (a∗1 ) = P 2G + P 3G
(12)
where
"
1
P 2G ≡ δ (1 − µ) λ
Z
s21
−∞
Z
i
h
′
1 + a∗2 (M (m1 , s)) φ (s; m1 , H) ds + ... + λN
∞
sN
1
h
i
′
1 + a∗2 (M (m1 , s)) φ (s; m1 , H) ds
and
P 3G ≡ δ
Z
∞
r1 (M (m1 , s)) c′ (a∗2 (M (m1 , s))) φ (s; m1 , H) ds.
−∞
Career concerns when the worker can affect the effort expected by firms. Equation
(12) shows that, as in period 2, in period 1, equilibrium effort is such that the marginal cost
of exerting effort is equal to the expected marginal benefit of exerting effort. The worker may
benefit from exerting effort in period 1 because period-1 effort may increase the wage that will
be offered to him in periods 2 and 3. In the marginal benefit of exerting effort, P 2G represents
the expected gain from increasing the period-2 wage, and P 3G represents the expected gain from
increasing the period-3 wage.
16
#
P 2G shows that when the worker can affect the next-period effort expected by firms, the
effect of effort on the next-period wage is stronger. The gain from increasing the next-period
wage represented in P 2G is similar to the gain from exerting effort in period 2 represented in the
right-hand side of equation (10). Exerting effort increases the next-period quality expected by
firms and, therefore, it increases the next-period wage. Period-2 effort only increases the nextperiod expected quality through its direct effect on the next-period expected ability. In contrast,
period-1 effort also increases the next-period expected quality through the indirect effect of the
next-period expected ability on the next-period equilibrium effort level, which is used by firms
to compute next-period expected quality (recall that the period-2 equilibrium effort level is a
function of expected ability and the period-3 equilibrium effort level is not). This is why the
derivative of the next-period effort level is part of P 2G but does not appear on the right-handside of equation (10). Note that when the worker can affect the next-period effort expected by
firms, since the effect of effort on the next-period wage is stronger, incentives to exert effort
because of career concerns may be stronger in spite of the ratchet effect described above.
The expected gain from increasing period-3 wage represented in P 3G describes the typical
intertemporal tradeoff in dynamic models. In order to influence his period-3 wage, the worker
can exert effort in periods 1 and 2. That is, in order to have any given expected period-3 wage,
he can choose to exert more effort in period 1 and less effort in period 2, or vice versa. Thus,
the worker compares the costs and the effectiveness of exerting effort in these two periods.
In P 3G, r1 represents the relative effectiveness in changing the period-3 wage of exerting
effort in period 1 (compared to exerting effort in period 2). Recall that in order to influence
his period-3 wage, the worker needs to affect mf 3 . The worker’s period-1 effort affects mf 2
directly, and affects mf 3 through mf 2 . His period-2 effort affects mf 3 directly. Thus, the relative
effectiveness is given by the derivative of mf 3 = M (mf 2 , s + a2 − a∗2 (mf 2 )) with respect to mf 2 .
If r1 is lower than one, it implies that a1 was less effective than a2 in changing mf 3 .
In the relative effectiveness, the ratchet effect described above is represented in the second
term of the right-hand side of equation (11). As explained before, this effect represents a negative
impact of a higher mf 2 on mf 3 because of the effect of mf 2 on the signal computed by firms in
period 2. In equation (11), this effect is weighted by 1−µ, the weight of the signal in the posterior
17
belief. The first term of the right-hand side of equation (11) represents a positive effect of mf 2
on mf 3 . A better prior implies a better posterior. The importance of this effect is represented
by the weight of the prior in the posterior, µ.
Career concerns and the degree of uncertainty about ability. In models of career concerns, an increase in uncertainty about ability (a decrease in hη ) brings about an increase in the
relative quality of the signal as an indication of ability and, therefore, an increase in the weight
of the current-period signal in the next-period belief, 1 − µ (see equation (4)). Since the worker
affects the signal computed by firms with his effort level, if the signal weight in the next-period
belief is higher, the worker has stronger incentives to exert effort (see the marginal benefit of
exerting effort in equations (10) and (12)). This insight explains the conventional wisdom in the
career-concern literature on the relationship between the degree of uncertainty about ability and
the strength of career concerns. For instance, Holmstrom (1999) explains that career-concern
incentives “will work more effectively if the ability process is more stochastic.” Dewatripont,
Jewitt, and Tirole (1999b) explain that “since effort in the career concern model is driven by
talent uncertainty, the principal faces a tradeoff between the riskiness of overall performance and
effort....” Similarly, Ortega (2003) explains that one can expect career-concern incentives to be
stronger for younger workers for whom beliefs about ability are less precise. De Motta (2003) uses
this insight to show that when corporate headquarters have more precise information, stand-alone
firms provide stronger career-concern incentives than multidivisional firms.
By analyzing a framework in which the strength of career concerns depends on a worker’s
employment history, we can identify a mechanism that challenges the conventional wisdom in
the literature. In the framework presented in this paper, the effect described above is important,
however, another effect appears. In particular, in our framework, a positive relationship between
equilibrium effort and hη may arise, as seen in the following example. Suppose there are two
jobs in the economy (N = 2), A1 = A2 = 0, λ1 = 0.5, λ2 = 2, m1 = 0, c(a) = a5 , δ = 0.9, and
hε = 1000. If hη = 1, the worker’s equilibrium effort level in his first working period is given
by a∗1 = 0.74. If hη = 1000, the worker’s equilibrium effort level in his first working period is
given by a∗1 = 0.85. That is, in the initial working period, incentives are stronger in the economy
18
with less uncertainty about ability. The intuition behind this example is as follows. Suppose
hη is lower. This implies that the distribution of the signal is less precise (H is lower). This in
turn implies that equilibrium effort is less responsive to changes in reputation (see the derivative
of equilibrium effort with respect to reputation in Appendix B and the examples presented in
Figures 1 and 2). Thus, it is more difficult for the worker to increase his next-period wage by
making firms believe that he will exert more effort next period. Therefore, incentives to exert
effort are weaker.
Equilibrium wages and job assignments. In order to complete the characterization of the
three-working-period economy, we present the equilibrium job assignment rule and the equilibrium wage function for the worker’s first employment period. As in periods 2 and 3, in period
1, the firm that hires the worker assigns him to the job for which his expected productivity is
higher, and the wage offered to the worker is equal to his expected productivity. The following
proposition characterizes the equilibrium job assignment rule and the equilibrium wage function for period 1 (the proof of this proposition parallels the proof of proposition 3 presented in
Appendix C).
Proposition 5 There exists a unique set of threshold reputation levels mj1 such that, in equilibrium, the worker is assigned to job j in period 1 if and only if mj1 < m1 ≤ mj+1
1 . The threshold
reputations are such that
a∗1 + mj1 = mj3 .
The equilibrium wage for an inexperienced worker is given by
ω1 = Aj + λj (a∗1 + m1 ) ,
where j is such that mj1 < m1 ≤ mj+1
1 .
3.3
A T-working-period version of the model
We now discuss whether the insights from the study of the three-working-period economy presented above apply to economies in which the worker can be employed for more than three
19
periods. In the three-working-period version of the model, the relationship between the strength
of career concerns and employment history can only be studied in the second employment period
(in the first period there is no employment history, and in the last period there are no career
concerns). Proposition 2 shows that in a three-working-period model, in the second employment period, career concerns are stronger when the worker’s reputation is better. That period
is special in that next-period effort does not depend on reputation. We will show that when
more than three employment periods are considered, local nonmonotonicities may appear in the
equilibrium effort strategy. This implies that the expected quality of a worker’s labor input may
be decreasing with respect to his reputation because a worker with higher expected ability may
be expected to exert less effort. Therefore, since managerial jobs are assigned to workers with
higher expected quality, there could be laborers with better reputations than managers.17
For the remainder of the paper, we discuss economies with monotone equilibrium assignment
rules in which workers with better reputations are assigned to jobs with a higher marginal return
to quality.18 This facilitates the analysis of economies in which the worker can be employed for
more than three periods. We will discuss how parameter values can be restricted in order to
assure that assignment rules are monotone. We will show that equilibrium job assignment rules
may be monotone even in economies that present nonmonotonicities in the equilibrium effort
strategies.
3.3.1
Equilibrium.
The next proposition characterizes economies with monotone assignment rules (the derivation
of equation (14) follows closely the derivation of equation (12) presented in Appendix D; the
17
Likewise, Ortega (2003) explains that a firm may want to give more power to workers with lower expected
ability when these workers are expected to choose a higher effort level. Ottaviani and Sφrensen (2001) show
that increasing the informational quality of a committee member can decrease the informational value of the
committee. Similarly, Ottaviani and Sφrensen (2006) show that an expert with a better reputation may be less
informative in equilibrium than an expert with a worse reputation. They remark that this would make it difficult
to endogenously derive a monotonic reputational payoff in a dynamic version of their model with more than two
periods. The mechanism through which better reputations weaken incentives in their models is different from the
one in our model. Here, concerns about the effect of current actions on future actions expected by the market
are crucial. In their two-period models, these concerns are not present.
18
Similarly, in Martinez (2009), I focus on economies with political career concerns in which the equilibrium
reelection rule is monotone.
20
derivation of equilibrium wages and job assignment rules parallels the derivations for the threeworking-period version of the model presented in Appendix C).
Proposition 6 In an economy in which a worker with higher expected ability is employed at
higher levels in the hierarchies, there exists a unique set of threshold reputation levels mjt such
that the worker is assigned to job j in period t if and only if mjt < mt ≤ mj+1
. The threshold
t
+1
reputations are such that m1t = −∞, mN
= ∞, and for all 2 ≤ j ≤ N,
t
a∗t (mjt ) + mjt =
Aj−1 − Aj
,
λj − λj−1
(13)
where a∗T (m) = 0, and for any t < T , there exists a unique equilibrium effort strategy a∗t (m) that
satisfies
c′ (a∗t (m)) = NP Gt (m) + OP Gt(m),
(14)
where
"
N P Gt (m) ≡ δ (1 − µ) λ
1
Z
s2
t (m)
−∞
h
1+
OP Gt(m) ≡ δ
′
a∗t+1
Z
i
(M (m, s)) φ (s; m, H) ds + ... + λN
∞
sN
t (m)
h
1+
′
a∗t+1
#
i
(M (m, s)) φ (s; m, H) ds ,
∞
Z
−∞
rt (M (m, s)) c′ a∗t+1 (M (m, s)) φ (s; m, H) ds,
′
rt (m) ≡ µ − (1 − µ) a∗t+1 (m) ,
and, for all j ≥ 2,
sjt (m)
mjt+1 − µm
.
≡
1−µ
Equilibrium wages are given by
ωt (mt ) = Aj + λj [a∗t (mt ) + mt ] ,
(15)
where j is such that mjt < mt ≤ mj+1
.
t
This proposition shows that, as in the first employment period of the three-working-period
version of the model, in any period t < T , the marginal gains from exerting effort are the expected
increase in the next-period wage (NP Gt (m)) and the expected gain from increasing wages in
21
every other future period (represented by the lower marginal cost of exerting effort next period
in OP Gt (m)). It also shows that for any T , the model can easily be solved backwards as we did
for T = 3. A unique period-t equilibrium effort strategy can easily be obtained from equation
(14) once the unique period-t + 1 equilibrium effort strategy is known. For example, Figure 1
shows the equilibrium effort strategies for T − 1, T − 2, T − 3, and the limit of the finite-horizon
solution when the worker is far enough from retirement for an economy with N = 2, A1 = A2 = 0,
λ1 = 0.5, λ2 = 2, m1 = 0, c(a) = a5 , δ = 0.9, and hε = hη = 1. For each period, after deriving
the equilibrium effort strategy for the period, equilibrium job assignment rules and wages can
easily be obtained using equations (13) and (15).
Figure 1: Equilibrium effort strategies (H = 0.5)
Recall proposition 1 shows that, when the worker can only be employed for three periods,
in the second employment period, employment history only affects the strength of career concerns through its effect on the worker’s reputation. Proposition 6 shows that, in any period of
the T -working-period version of the model, when employment histories may be more complex,
reputation (and t) also conveys all the payoff-relevant information. Beliefs are the relevant state
variables. The optimal effort level is the same for all histories (y t−1 , j t , at−1 ) that imply the same
beliefs mw and mf (see the discussion of the optimal off-equilibrium effort strategy in Appendix
22
D). Equilibrium wages and job assignments are the same for all histories (y t−1 , j t−1 ) that imply
the same belief mf .
The fact that equilibrium strategies can be written as functions of beliefs allows us to increase
the assumed maximum length of employment histories without increasing the dimensionality of
the state space. Therefore, it facilitates solving the model for any number of employment periods.
This insight can also facilitate solving other models of career concerns where employment history
affects the strength of incentives. For instance, the insight allows Martinez (2009) to solve an
extension of a standard career-concern model of political cycles in which past performances affect
the strength of incentives.
3.3.2
Employment history and the strength of career concerns
Next, we describe how the worker’s reputation affects the strength of his incentives when he
takes into account how his current decision affects the effort level firms expect him to exert. On
the one hand, the effect presented in proposition 2 is still important. A worker with a better
reputation is more likely to have a managerial job next period. Therefore, he expects a larger
increase in his next-period wage if he makes firms believe that he will provide labor input of
higher quality. This implies that workers with better reputation could have stronger incentives
to exert effort. Figure 1 illustrates how this effect may be dominant and the equilibrium effort
strategies may be increasing in reputation.
On the other hand, local nonmonotonicities in the equilibrium effort strategies may appear.
Recall that effort affects the next-period wage through the next-period effort level expected by
′
′
firms. In NP G, this is represented by a∗t+1 (M (m, s)). As illustrated by Figure 1, a∗t+1 (M (m, s))
may be decreasing. Thus, a worker with a better reputation may expect his effort will be less
effective in making firms believe that he will exert more effort next period. Therefore, he may
have weaker incentives to exert effort. For instance, let us consider the example presented in
Figure 1 but with hε = hη = 1000. Figure 2 shows that in this economy, for reputation levels
′
where a∗T −1 (m) is decreasing, a∗T −2 (m) may be decreasing.
As explained above, nonmonotonicities in the equilibrium effort strategies arise because the
23
Figure 2: Equilibrium effort strategies (H = 500)
derivative of next-period equilibrium effort with respect to reputation may be decreasing. Hence,
one could avoid these nonmonotonicities by limiting the responsiveness of this derivative to
changes in reputation. Recall that the strength of incentives to exert effort is related to the
probability of having a managerial job next period. Thus, both the derivative and the second
derivative of this probability can be decreased by decreasing the precision in the distribution
of signals, H. Consequently, one could restrict attention to economies with monotone effort
strategies by assuming low values of H. This is apparent from comparing the examples presented
in Figures 1 and 2—in Martinez (2009) it is shown that, in a political agency model of career
concerns, similar restrictions have to be imposed to assure that voters reelect an incumbent
policymaker if and only if his reputation is good enough.
Note that, as illustrated in Figure 2, even if the period-t equilibrium effort strategy presents a
nonmonotonicity, the period-t equilibrium job assignment rule may be monotone—in the example
presented in Figure 2, the worker is assigned to job 2 for all reputations such that the equilibrium
effort strategy is above the -45 degree line (see equation (13)). This example illustrates how
in economies with nonmonotonic effort strategies, it may be possible to find the equilibrium
24
following proposition 6.
The importance of a higher probability of having a managerial job next period is also illustrated in Figure 2. In general (asides from the nonmonotonic region), equilibrium effort is
increasing in reputation. For the more extreme reputation levels, incentives from changing the
effort level expected by firms are not important and the equilibrium effort level is determined by
the probability of having a managerial job next period.
4
Conclusions
We presented a tractable model of career concerns in which the strength of career concerns
depends on a worker’s employment history. The strength of career-concern incentives, equilibrium
wages, and equilibrium job assignments are the same for all employment histories that lead to the
same reputation. This property of the equilibrium facilitates computing the equilibrium for games
of any length. We showed that typically, workers with the better reputations exert more effort
than workers with the worse reputations. But local nonmonotonicities in the effort-reputation
relationship may appear. Furthermore, we showed that when the strength of incentives depends
on a worker’s employment history, (i) a ratchet effect in implicit incentives may appear, (ii) the
effect of current-period effort on next-period wage may be stronger, and (iii) incentives may be
stronger if the prior beliefs about ability are more precise.
Appendix A: proof of proposition 1
For any history (y1 , j 1 ), effort expected by firms af 2 (y1 , j 1 ), and belief m, the worker’s period2 maximization problem can be written as
R 2
s2 (m)−a+af 2 (y1 ,j 1 )
{A1 + λ1 [µm + (1 − µ) (s + a − af 2 (y1 , j 1 ))]} φ (s; m, H) ds+
−∞
R s3 (m)−a+af 2 (y1 ,j 1) 2
max δ 22
{A + λ2 [µm + (1 − µ) (s + a − af 2 (y1 , j 1 ))]} φ (s; m, H) ds + ...
a s2 (m)−a+af 2 (y1 ,j 1 )
R∞
+ sN (m)−a+af 2 (y1 ,j 1 ) AN + λN [µm + (1 − µ) (s + a − af 2 (y1 , j 1 ))] φ (s; m, H) ds
2
− c(a) .
The equilibrium effort level a∗2 (m) satisfies the first-order condition of this problem when firms
compute the signal using a∗2 (m) (i.e., evaluated at a = af 2 (y1 , j 1 ) = a∗2 (m)). This first-order
25
condition is given by equation (10). The right-hand side in equation (10) does not depend on
effort and is positive. Therefore, there exists a unique a∗2 (m) that satisfies equation (10).
Appendix B: proof of proposition 2
In equation (10), the marginal benefit of exerting effort is given by
" Z 2
Z s32 (m)
Z
s2 (m)
1
2
N
δ (1 − µ) λ
φ (s; m, H) ds + λ
φ (s; m, H) ds + ... + λ
s22 (m)
−∞
#
∞
φ (s; m, H) ds .
sN
2 (m)
The sign of the derivative of this marginal benefit with respect to reputation m is given by the
sign of
λ
∂
1
R s22 (m)
−∞
∂
φ (s; m, H) ds
+ λ2
∂m
R s32 (m)
s22 (m)
φ (s; m, H) ds
∂m
+ ... + λN
∂
R∞
sN
2 (m)
φ (s; m, H) ds
∂m
,
which is equal to
∂s22 (m)
∂s32 (m)
∂sN
2 (m)
+ λ2 − λ3 φ s32 (m) ; m, H
+ ... + (λN−1 − λN )φ(sN
+
λ1 − λ2 φ s22 (m) ; m, H
2 (m) ; m, H)
∂m
∂m
∂m
Z s32 (m)
Z ∞
Z s22 (m)
∂φ (s; m, H)
∂φ (s; m, H)
∂φ (s; m, H)
ds + λ2
ds + ... + λN
ds.
λ1
∂m
∂m
∂m
N (m)
−∞
s2
(m)
s
2
2
Recall that λj > λj−1 (Assumption 2), and
∂sj2 (m)
∂m
< 0. Therefore, the first N − 1 terms in the
expression above are positive.
Let x be the job such that sx2 (m) ≤ m < sx+1
(m). We can write the last N terms in the
2
expression above as
1
λ
Z
s22 (m)
−∞
∂φ (s; m, H)
ds+...+λx
∂m
Z
m
sx
2 (m)
∂φ (s; m, H)
ds+λx
∂m
Z
sx+1
(m)
2
m
∂φ (s; m, H)
ds+...+λN
∂m
Z
∞
sN
2 (m)
∂φ (s; m, H)
ds.
∂m
Recall that φ (s; m, H) denotes the density function for a normally distributed random variable
with mean m. Consequently,
−λ
x
"Z
s2
2 (m)
−∞
∂φ (s; m, H)
ds + ... +
∂m
Z
m
sx
2 (m)
#
"Z x+1
#
Z m
s2
(m)
∂φ (s; m, H)
∂φ (s; m, H)
∂φ (s; m, H)
x
ds = λ
ds + ... +
ds > 0
∂m
∂m
∂m
m
sx
2 (m)
Furthermore,
x
−λ
"Z
s22 (m)
−∞
∂φ (s; m, H)
ds + ... +
∂m
Z
m
sx
2 (m)
#
Z s22 (m)
Z m
∂φ (s; m, H)
∂φ (s; m, H)
∂φ (s; m, H)
ds ≥ −λ1
ds−...−λx
ds,
x
∂m
∂m
∂m
−∞
s2 (m)
26
and
λx
"Z
sx+1
(m)
2
m
∂φ (s; m, H)
ds + ... +
∂m
Z
m
sx
2 (m)
#
Z sx+1
Z ∞
(m)
2
∂φ (s; m, H)
∂φ (s; m, H)
∂φ (s; m, H)
x
N
ds ≤ λ
ds...+λ
ds.
N
∂m
∂m
∂m
m
s2 (m)
Thus,
λ1
Z
s22 (m)
−∞
∂φ (s; m, H)
ds+...+λx
∂m
Z
m
sx
2 (m)
∂φ (s; m, H)
ds+λx
∂m
Z
sx+1
(m)
2
m
∂φ (s; m, H)
ds...+λN
∂m
Z
∞
sN
2 (m)
∂φ (s; m, H)
ds ≥ 0.
∂m
Consequently, the marginal benefit of exerting effort is increasing with respect to m. The proposition follows from c′′ (a) > 0.
Appendix C: proof of proposition 3
Expected quality thresholds for job assignments are given by mj3 . The period-2 quality
expected by firms is given by a∗2 (mf 2 ) + mf 2 .
Since a∗2 (mf 2 ) is increasing with respect to
mf 2 , we know that a∗2 (mf 2 ) + mf 2 is increasing with respect to mf 2 . Moreover, we know that
for any mj3 , there is a high enough mf 2 such that a∗2 (mf 2 ) + mf 2 ≥ mj3 ; and there is a low
enough mf 2 such that a∗2 (mf 2 ) + mf 2 ≤ mj3 . Consequently, there is a unique mf 2 such that
a∗2 (mf 2 )+mf 2 = mj3 . Considering that the worker is assigned to the position where he is expected
to be more productive, and that the equilibrium wage is equal to his expected productivity
completes the proof.
Appendix D: proof of proposition 4
For any history (y1 , j1 , a1 ) such that the worker’s and the firms’ beliefs are given by mw2 and
mf 2 , respectively, the worker’s period-2 maximization problem is given by
R s22 (mf 2 )−a+a∗2 (mf 2 ) 1
A + λ1 [µmf 2 + (1 − µ) (s + a − a∗2 (mf 2 ))] φ (s; mw2 , H) ds
−∞
R s3 (m )−a+a∗ (m )
f2
f2
2
2
∗
2
2
max δ
+ s22 (mf 2 )−a+a∗2 (mf 2 ) A + λ [µmf 2 + (1 − µ) (s + a − a2 (mf 2 ))] φ (s; mw2 , H) ds + ...
a
R∞
N
+ N
A + λN [µmf 2 + (1 − µ) (s + a − a∗ (mf 2 ))] φ (s; mw2 , H) ds
∗
2
s2 (mf 2 )−a+a2 (mf 2 )
27
− c(a) .
(16)
Consequently, the optimal period-2 effort level only depends on the worker’s employment
history through the worker’s belief and the firms’ belief. Therefore, at the beginning of period
2, the worker’s expected lifetime utility only depends on these beliefs. Let
W2 (mw2 , mf 2 ) =
ω2 (mf 2 )−c(â∗2 (mw2 , mf 2 ))+δ
Z
∞
−∞
ω3 (M (mf 2 , s + â∗2 (mw2 , mf 2 ) − a∗2 (mf 2 ))) φ (s; mw2 , H) ds
denote this utility, where â∗2 (mw2 , mf 2 ) denotes the optimal effort for mw2 and mf 2 . The worker’s
period-1 maximization problem is given by
R 2
s1 −a+a∗1
W2 (M (m1 , s) , M (m1 , s + a − a∗1 ))φ (s; m1 , H) ds
−∞
R 3
−a+a∗1
∗
max δ + ss21−a+a
∗ W2 (M (m1 , s) , M (m1 , s + a − a1 ))φ (s; m1 , H) ds + ...
a
1
1
R∞
+ sN −a+a∗ W2 (M (m1 , s) , M (m1 , s + a − a∗1 ))φ (s; m1 , H) ds
1
1
The first-order condition of this problem evaluated in equilibrium reads
c
′
(a∗1 )
= δ (1 − µ)
"Z
s21
−∞
∂W2
∂mf 2
φ (s; m1 , H) ds + ... +
Z
∞
sN
1
M(m1 ,s),M(m1 ,s)
∂W2
∂mf 2
− c(a) .
#
φ (s; m1 , H) ds .
M(m1 ,s),M(m1 ,s)
(17)
j
For any m such that mj2 < m ≤ mj+1
2 , let λ2 (m) ≡ λ . By the envelope theorem, for all
m 6∈ {m22 , m32 , ..., mN
2 },
∂W2
∂mf 2
m,m
" Z
h ′
i
= λ2 (m) a∗2 (m) + 1 + δr1 (m) λ1
s22 (m)
φ (s; m, H) ds + ... + λN
−∞
Z
∞
φ (s; m, H) ds .
sN
2 (m)
By equation (10),
∂W2
∂mf 2
m,m
h ′
i r (m)
1
∗
= λ2 (m) a2 (m) + 1 +
c′ (a∗2 (m)) .
1−µ
Substituting into equation (17) yields equation (12).
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