Job Assignments under Moral Hazard - School of Economics and ...

Job Assignments under Moral Hazard
Alexander K. Koch a and Julia Nafziger b∗
a Aarhus University and IZA
b Aarhus University
November 29, 2010
Abstract
Inefficient job assignments are usually explained with incomplete information about
employees’ abilities or contractual imperfections. We show that inefficient assign-
ments arise even with full information about employees abilities and complete con-
tracts. Building on this result and allowing effort levels to vary across hierarchy
levels, we provide a new perspective on the Peter Principle: the output of an under-
qualified employee need not decline post promotion because he is pushed to exert
more effort. As a result, an employee moving up the career ladder may produce and
earn more only because he works harder and not because the job match is efficient.
JEL Classification: D82, D86, J33
Keywords: Moral Hazard; Information; Job Assignments; Peter Principle.
∗a School of Economics and Management, Aarhus University, Building 1322, 8000 Aarhus C, Den-
mark, Email: akoch@econ.au.dk, Tel.: +45 8942 2137. b School of Economics and Management, Aarhus
University, Building 1322, 8000 Aarhus C, Denmark, Email: jnafziger@econ.au.dk, Tel.: +45 8942
2048. We thank Mathias Dewatripont, Guido Friebel, Paul Heidhues, Ian Jewitt, Thomas Kittsteiner,
Matthias Kräkel, Johan Lagerlöf, Meg Meyer, David Myatt, Urs Schweizer, and John Sutton for helpful
comments. Nafziger is grateful to Nuffield College, University of Oxford for its kind hospitality.
1

1 Introduction
Organizations face the problem of how to match individuals with the job in which they
are most productive and to provide them with incentives. In practice one observes
that job assignments are often used to motivate employees – leading to an inefficient
allocation of employees to jobs. For example, people often are promoted to a job for
which they are less competent and then remain there, a phenomenon known as the Peter
Principle (Peter and Hull 1969). The appearance of such inefficiencies seems puzzling
to economists. Why do firms not separate job assignment policies from the provision of
incentives, leaving the latter to pay-for-performance schemes?
The literature on careers in organizations stresses the role of incomplete information
about employees’ abilities and contractual imperfections in accounting for inefficiencies
in job assignments (see the section on related literature). Our approach is complementary
to this. We start from a moral hazard model and show that job assignment decisions
cannot be separated from incentive provision even if the principal is able to write a
complete contract and knows about the agent’s productivity.
The driving force behind inefficient job assignments in our model is a static trade-off
between incentive provision and efficient allocation of agents to jobs. We develop this
trade-off in a moral hazard model, where a risk neutral principal assigns a risk neutral
and wealth constrained agent to one of two jobs (a high ability and a low ability one).
The probability with which the agent succeeds in producing a high output depends in
each job on his ability, and his effort. While the principal knows the ability of the agent,
she cannot observe his effort and thus designs an output-contingent reward scheme to
induce a certain effort. The cost of getting the agent to exert this effort level in a given
job depends on how likely it is that success in this job was generated by high and not
by low effort.
Because the job in which an agent of a certain ability produces the highest expected
output need not be the one in which it is cheapest for the principal to implement effort,
a trade-off between efficient job assignment and incentive provision arises – leading to an
inefficient assignment policy. 1 The necessary and sufficient condition that we derive for
such an inefficient outcome to occur depends on the relative informativeness of output
about effort in the two jobs, i.e., the ranking of likelihood ratios across jobs.
The rather technical condition becomes more intuitive once we interpret the high ability
job as a position higher up in the hierarchy than the low ability job. Jobs further down
1 One might suspect that the root of inefficiencies in job assignments are distortions in implemented
effort levels that arise in moral hazard problems. Our binary effort model shows that this is not the
case. It allows us to consider the job assignment problem for a parameter range where the principal does
not distort effort and implements the efficient level in both jobs. Our continuous effort model combines
the effects of assignments and distortions in implemented effort levels.
2

in the hierarchy often put low demands in terms of the inputs ability and effort. Hence,
in the low ability job success might be quite likely even without hard work (i.e., it is is
a relatively easy task). In the high ability job, if the agent does not work hard, there
is a bigger drop in expected output (i.e., it is a more difficult task for him). Seeing
such an agent succeed in what is for him a difficult job, thus, indicates that he must
have worked hard. So the high ability job poses a harder test of the agent’s effort than
the low ability job, and this makes it less costly to get the agent to exert effort. That
output in the higher level job is often more informative about the agent’s effort than
output in the lower level job is a view also found in the literature (Baker, Gibbons,
and Murphy 1994, Maskin, Qian, and Xu 2000, Ortega 2003). As we show, the reduced
cost of incentives (due to differences in informativeness about effort across hierarchy
levels) outweighs the negative effect of a lower success probability near the ability cutoff
level for efficient job assignment. Hence, the principal sets a lower ability threshold for
assignment to the more informative higher hierarchy level than is efficient.
The mechanism at work is a static one: the job assignment decision in one period affects
the contemporaneous cost of providing incentives. This static mechanism on its own
offers several new insights into the driving forces governing promotion decisions in firms.
Embedding our model in a stylized dynamic framework and extending it to allow for
continuous effort, we show that it may be optimal for a firm to promote an employee to
a higher level job even though it knows that he will be less productive than he could be
in his previously held position. This finding provides a new perspective on possible roots
of the Peter Principle: even in ideal circumstances where a firm is perfectly aware of its
employees’ abilities one may find employees in jobs for which they are not well suited.
Our analysis highlights that promoting an employee to a job for which he is less com-
petent need not lead to an observed decline in output – a narrow definition of the Peter
Principle often used in the literature (e.g. Lazear 2004). Output may actually rise be-
cause the promoted agent works harder – hiding the fact that he actually is untalented
for his new job. Existing theoretical treatments of promotions in firms have ignored
this effect of job assignments on implemented effort levels: they either abstract from
incentives altogether or assume that effort is constant across hierarchy levels. This effect
may, however, be an important driving force for promotion decisions, as our analysis
shows. We therefore suggest an extended definition of the Peter Principle that includes
both the possibility that a promoted agent works harder or less hard than he would if
he were reassigned to the previous job. Our setting with varying effort levels across hi-
erarchical levels allows to show that an employee moving up the career ladder may earn
more and produce more only because he has to work harder than he would in his old job.
The paper is organized as follows. Next we discuss the related literature. Section 2
3

introduces the static job assignment model with discrete effort. Section 3 extends the
model to a continuous effort setting to analyze promotions and explore the relation to
the Peter Principle. The final section discusses our findings and concludes. All proofs
are in the appendix.
Related Literature
The large literature on careers and incentives in organizations emphasizes two roles of
job assignment and promotion decisions. First, learning about the abilities of employees
drives the allocation policy. Firms initially place individuals in jobs where they can do
little harm. Those who prove to be able are then promoted to positions that require a
higher level of ability. Second, promotions are a source of incentives. 2
This literature identifies several reasons for distortions in job assignments. The principal
may create distortions to influence the information about employees’ abilities she herself
(e.g. Meyer 1991) or outsiders receive (e.g. Waldman 1984a, Bernhardt 1995), or what is
revealed to an employee about his own ability (Ishida 2006, Nafziger 2008). Restrictions
on incentive contracts may also lead to distortions. For example, a Lazear and Rosen
(1981)-style promotion tournament may select the agent with lower expected ability
when agents have private information about their type (Clark and Riis 2001). If job
assignment decisions need to be delegated, distortions may arise because of the additional
agency layer (Carmichael 1988, Baliga and Sjöström 2001, Fairburn and Malcomson 2001,
Carrillo 2003, Friebel and Raith 2004).
When asking about the reasons for distortions in job assignments, the literature tries to
address two puzzles. First, why can the role of matching individuals with jobs and that
of providing incentives not be separated; leaving the former to job assignment decisions
and the latter to pay-for-performance schemes (Baker et al. 1988)? Explanations for this
are based on contractual imperfections: for example, performance is not verifiable (e.g.
Malcomson 1984), or some aspects of performance – such as human capital accumulation
– are not contractible (e.g. Prendergast 1993, Kwon 2006).
Second, even though employees often turn out to be less productive in their job than
in their former position, why are they not demoted? Lazear’s (2004) model, based on
temporary shocks to productivity, offers an explanation for one interpretation of the so-
called Peter Principle. A promotion means that a worker has delivered a high measurable
output, which is the sum of permanent and transitory components. Because of regression
to the mean of the temporary productivity shocks, output declines on average after a
2 For an excellent survey of the literature see Gibbons and Waldman (1999a). Examples for the first
strand are: Prescott and Visscher (1980), O’Flaherty and Siow (1992), and Valsecchi (2003); examples
for the second strand are: Lazear and Rosen (1981), Malcomson (1984), Rosen (1986), and MacLeod
and Malcomson (1988).
4

promotion. 3 Bernhardt (1995) formalizes an idea from Peter and Hull (1969): promotion
is based on the performance in one task (say as a laborer), but the skills required in the
new task (say as a manager) may be quite different. While on average a more able
laborer is also likely to be a better manager, such a laborer may turn out to be a
failure as a manager once promoted. If there is asymmetric information about employee
abilities, the employer may, however, refrain from demotions because she has to pay a
bonus to retain a demoted worker (Bernhardt 1995). Fairburn and Malcomson (2001)
include incentive considerations, and assume that workers can sway the performance
evaluation of their supervisors with bribes. This makes incentive pay ineffective for
the principal: supervisors and their workers would simply collude to extract high wages
without any effort in return. Making workers’ pay contingent on their job, and managers’
pay contingent on the firm’s profits, aligns managers’ interests more closely with those of
the firm’s owners: promoting a very untalented worker reduces profits, and thus reduces
the supervisor’s pay by more than what the supervisor could gain by extracting a bribe.
Still, some workers close to the efficient promotion threshold are promoted even though
they should not be.
Our approach is complementary to this. We start from assumptions that are most
favorable to avoiding inefficiencies in job assignment decisions: there is no uncertainty
about the agent’s ability and complete contracts are possible. We show that nevertheless
a principal would not adopt an efficient job assignment policy. Our results contribute to
understanding both puzzles referred to earlier: job assignment and incentive provision
need not be separable even with complete contracts; and inefficient promotion decisions
(as reflected in the Peter Principle) may result from an optimal policy of the principal.
From a theoretical point of view, our model is most closely related to Robbins and
Sarath (1998). They show in a setting with a risk averse agent that the principal may
not always want to choose from among two output systems the one that generates the
highest revenue. Different output systems vary in how informative they are about the
agent’s effort. While a more informative system corresponds to lower implementation
costs, it need not be the one that creates a higher revenue. Output systems in their
paper correspond in our setting to agents with different abilities in the two jobs. Our
model with a risk neutral agent who is protected by limited liability allows us to look at
a continuum of output systems. This enables us to show how inefficient job assignments
may emerge even in a complete contracting setting, and how this can help understand
the Peter Principle.
3 This mechanism has recently been tested using personnel records (Barmby, Eberth, and Ma 2007)
and lab experiments (Dickinson and Villeval 2007).
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2 The static job assignment model
2.1 The model
A risk neutral principal (‘she’) needs to assign an agent (‘he’) to one of two jobs, j ∈
{l, h}. He is protected by limited liability and has a reservation utility of zero. In each
job he produces a verifiable and observable output. Output can either be high (revenue
for the principal normalized to 1) – or low (revenue normalized to 0). The probability
of high output, fj(e, θ), depends on the job j ∈ {l, h}, the agent’s ability θ ∈ [θ, ¯ θ], and
his effort e ∈ {e, ē}. We assume that f(·, θ) is twice continuously differentiable in θ, and
that higher effort leads to a higher success probability, i.e., fj(ē, θ) > fj(e, θ) ∀ θ. The
utility cost of effort is c if the agent provides high effort, and zero otherwise.
Job l entails ‘safe’ tasks, where a low ability agent can do no harm because the success
probability is insensitive to ability. Hence, we refer to it as the low ability job hereafter
and simplify notation to fl(e) ≡ fl(e, θ). In the high ability job h, expected output
increases with higher ability, and the impact of ability on output also (weakly) increases
with effort (i.e., ability and effort are weak complements). The following sensitivity to
ability (SA) assumption captures this: 4
0 = fl θ(e, θ) = fl θ(ē, θ) < fh θ(e, θ) ≤ fh θ(ē, θ). (SA)
Additionally, we assume that for a given effort, an agent with the highest ability level
( ¯ θ) is most productive in job h, and an agent with the lowest ability level (θ) is most
productive in job l. The following crossing property (SP) expresses this formally:
fh(e, θ) < fl(e, θ) and fl(e, ¯ θ) < fh(e, ¯ θ) for e ∈ {e, ē}. (SP)
Together with Condition (SA) this implies that for each pair of equal effort levels in the
two jobs (e, e) ∈ {(e, e), (ē, ē)} there exists a unique interior threshold ˆ θ(e, e) such that
all types with θ ≥ ˆ θ(e, e) have a higher success probability in job h than in job l, and all
types with θ < ˆ θ(e, e) have a lower one. 5 Figure 1 illustrates this general property with
a linear example. It also shows that the success probability functions for the two jobs
need not cross for unequal effort levels: in the example with θ = 0, there is a crossing
point ˆ θ(ē, e) but no crossing point ˆ θ(e, ē) in the range θ ∈ [θ, ¯ θ].
The timing is as follows. First, the principal offers a contract, specifying the job j ∈ {l, h}
and an output-contingent wage scheme. Both the principal and the agent know the
4 The subscript θ denotes the partial derivative with respect to θ.
5 On its own, Condition (SP) guarantees that at least one such threshold exists (by the intermediate
value theorem). Condition (SA) gives a unique threshold: it implies that fh(e,θ)
fl(e,θ)
higher ability the agent becomes relatively more productive in job h.
6
is increasing in θ – with

Figure 1: Illustration of the single crossing property (SP) with a linear example
[fh(e, θ) = qh(e) + θ, qh(ē) = 0.4, qh(e) = 0; fl(e) = ql(e), ql(ē) = 0.7, ql(e) = 0.35]
agent’s ability θ ∈ [θ, ¯ θ]. The contract fixes a wage ¯wj(θ) after a high output and w j(θ)
after a low one. If the agent accepts the contract, he then provides unobservable effort
e ∈ {e, ē}. At the last stage, output and payoffs are realized: the principal receives the
revenue net of the wage to the agent; the agent receives utility equal to the wage less
his effort cost. If the agent rejects the contract, the game ends with zero payoffs for all
players.
2.2 Observable effort (first best)
In the benchmark case where the principal can observe the agent’s effort, she simply
compensates him for his effort cost. That is, given job j and type θ her expected profit
F B
F B
is either Πj (ē, θ) = fj(ē, θ) − c for high effort, or Πj (e, θ) = fj(e, θ) for low effort. To
make the problem interesting, we assume that first-best profits for high effort are larger
than the ones for low effort:
Assumption 1 High effort is efficient in jobs j = l, h: fj(ē, θ) − fj(e, θ) ≥ c ∀θ ∈ [θ, ¯ θ].
The optimal job assignment rule is that the principal places an agent with type θ in job
h if and only if the maximized profit in job h is higher than in job l, which is equivalent
to fh(ē, θ) ≥ fl(ē). So, unsurprisingly, the first-best job assignment rule places the agent
in the job where he is most productive: job h if θ ≥ θ F B = ˆ θ(ē, ē) and job l otherwise.
7

2.3 Unobservable effort (second best)
As the principal cannot observe the effort choice of the agent, the problem she faces is
not only to assign the agent to a job but also to design an appropriate incentive scheme
that makes it optimal for the agent to provide the effort she desires. To solve this, we
apply for each job separately the usual two-step procedure for moral hazard problems.
First, for a given effort level e, we find the cost minimizing wage scheme that implements
e in job j. Second, given these wage schemes, we maximize profits with respect to effort.
In a third step, we then determine the profit maximizing job assignment rule.
2.3.1 The incentive scheme
Consider the first part of the problem. If the principal wants to implement high effort,
she minimizes the expected wage bill
fj(ē, θ) ¯wj(θ) + [1 − fj(ē, θ)] w j(θ), (1)
taking into account that w j(θ), ¯wj(θ) ≥ 0 (the limited liability constraint), that the
agent’s expected utility must be larger than his reservation utility of zero (the participa-
tion constraint), and that it must be optimal for him to provide high effort. The latter
is the case if the following incentive constraint is satisfied:
fj(ē, θ) − fj(e, θ) ¯wj(θ) − w j(θ) ≥ c. (2)
The limited liability and incentive constraint together imply that the participation con-
straint is satisfied. But then, as a positive wage after a failure would only decrease
incentives, the principal sets this wage equal to zero. The wage after a success is set to
make the incentive constraint bind, i.e., ¯wj(θ) =
c . The principal can get the
fj(ē,θ)−fj(e,θ)
agent to work hard only by leaving him a rent in addition to the compensation for his
effort cost c.
The expected profit from assigning an agent with type θ to job j and implementing high
effort is Π SB
j (ē, θ) = fj(ē, θ) − Cj(ē, θ), where
Cj(ē, θ) =
fj(ē, θ)
c (3)
fj(ē, θ) − fj(e, θ)
is the expected wage payment, referred to in the following as the implementation cost.
To implement e the principal sets w j(θ) = ¯wj(θ) = 0, resulting in expected profit
Π SB
j (e, θ) = fj(e, θ).
Everything is in place now to consider the second part of the principal’s problem, what
effort level to implement. Facing an agent of ability θ, it is optimal for the principal to
implement high effort in job j if and only if Π SB
j (ē, θ) ≥ Π SB
j (e, θ), or:
[fj(ē, θ) − fj(e, θ)] 2 ≥ fj(ē, θ) c. (Condition 1)
8

If this fails to hold the principal distorts downward the implemented effort to save on
implementation costs.
2.3.2 Job assignments given high effort in both jobs
In this section, we characterize the job assignment rule that the principal adopts if she
always implements high effort (i.e., if Condition 1 holds for both jobs and for all worker
types). This way, should we find that the job assignment rule under moral hazard is
inefficient, we can be sure that this result is not an artefact of inefficient effort choices:
we are looking at the parameter range where the second-best effort level is efficient.
The second-best job assignment rule can be described by ability level(s) θ SB for which
Π SB
h
ē, θ SB − Π SB
l
(ē) = 0 holds. To characterize these threshold(s) further, we first
have to look more closely at the behavior of the profit functions (and their difference) in
θ. While for job l this is clear – profits do not depend on θ – there are two potentially
opposing effects in job h. On the one hand, a higher ability level increases the expected
output. On the other hand, a higher ability level may have a negative effect on imple-
mentation costs. 6 The following lemma shows that the first effect dominates, and thus
profits in job h and the difference in profits across jobs increase with the agent’s ability:
Lemma 1 Suppose Condition 1 holds for θ ∈ [θ, ¯ θ], then
(a) d
d θ ΠSB
h (ē, θ) > 0,
(b) d
d θ
Π SB
h (ē, θ) − Π SB
l (e) > 0 for e ∈ {e, ē}.
From Lemma 1 it follows that there is a unique threshold θ SB . Does θ SB equal the
first-best threshold θ F B = ˆ θ(ē, ē)? This relies on whether the second-best profits in
the two jobs, Π SB
h (ē, ˆ θ(ē, ē)) and Π SB
l (ē), are equal at ˆ θ(ē, ē) or not. At ability level
θ = ˆ θ(ē, ē) the job assignment does not change the expected output, because the agent
is equally productive in both jobs if he works hard. Hence, the profit difference is
simply the difference in implementation costs, i.e., Π SB
h (ē, ˆ θ(ē, ē)) = Π SB
l (ē) if and only
if Ch(ē, ˆ θ(ē, ē)) = Cl(ē).
So the question is when these implementation costs differ. Without loss of generality,
let us consider the conditions under which Ch(ē, ˆ θ(ē, ē)) < Cl(ē) arises. Using the imple-
mentation cost function given in Equation (3), it follows that the implementation costs
are lower when assigning an agent with ability θ = ˆ θ(ē, ē) to job h rather than to job l
if and only if
fh(ē, ˆ θ(ē, ē))
fh(e, ˆ θ(ē, ē))
fl(ē)
> . (4)
fl(e)
6 As can be seen from the proof of Lemma 1, Ch θ(e, θ) ≷ 0 depending on whether fh(ē,θ)
fh(e,θ) ≷ fh θ(ē,θ)
fh θ(e,θ) .
9

This boils down to fh(e, ˆ θ(ē, ē)) < fl(e), at ˆ θ(ē, ē) because fh(ē, ˆ θ(ē, ē)) = fl(ē). 7
Each ratio reflects how likely it is after observing high output in a particular job that
the agent provided effort ē instead of e. According to Inequality (4), it is more likely
in job h than l that high output of an agent with ability θ = ˆ θ(ē, ē) was the result of
hard work. Thus, if this agent wanted to shirk, he would have a harder time hiding this
in job h than in the low ability job l. In that sense, the principal can put the agent to
a harder test by placing him in the more demanding job h: success in the high ability
job is a sign that he must have worked hard. And this makes it is less costly to provide
incentives in job h. 8
As Demougin and Fluet (1998) argue, such a comparison of likelihood ratios in the high
output state is an appropriate criterion for ranking different information systems for risk
neutral agents that are protected by limited liability. And thus – because an information
system corresponds here to an output-type-job combination – Inequality (4) translates
to job h being more informative about effort than job l for an agent with ability θ. 9 So
our model shows that the principal may place an agent into the job with the highest
level of “transparency” about effort rather than into the job with highest productivity
of effort.
Proposition 1 Suppose the second-best effort levels are efficient (i.e., Condition 1 holds
for jobs j = l, h and θ ∈ [θ, ¯ θ]). Then θ SB = θ F B if and only if fh(e, ˆ θ(ē, ē)) = fl(e),
where θ SB and θ F B = ˆ θ(ē, ē) are the thresholds for assignment to the high ability job h
in the principal’s second- and first-best job assignment rules, respectively.
Proposition 1 shows that the roots of inefficient job assignments may often lie deeper
than incomplete information about employees’ abilities and contractual imperfections:
we eliminate these ingredients and show that the job assignment threshold is nevertheless
inefficient, except for the degenerate case where fh(e, ˆ θ(ē, ē)) = fl(e). Perhaps less
surprisingly, in settings where distortions in effort levels do occur, these additionally
distort the job assignment rule, as shown in Appendix A.
7 If Inequality (4) held for all θ, output in job h would be globally more informative than output in
job l. For our purposes it is, however, not necessary to have such a global ranking of jobs in terms of
their likelihood ratios for all ability levels θ ∈ [θ, ¯ θ]. Whether the second-best job assignment rule is
biased depends only on a much weaker local condition: are the likelihood ratios equal to each other or
not at θ = ˆ θ(ē, ē)?
8 While the principal seems to design the reward scheme as if he needed to estimate the effort put in
by the agent, he of course knows the agent’s effort in equilibrium.
9 Note that this definition of informativeness differs slightly from those proposed for the case of risk
averse agents with unlimited liability, where the whole likelihood ratio distribution matters (Grossman
and Hart 1983, Kim 1995, Jewitt 2006). The difference stems from the fact that for risk neutral agents
rewards are concentrated in one state. Because the wage after low output is zero, there is no need to
‘estimate’ the agent’s effort in the failure state.
10

2.4 Discussion: Biases in hierarchical job assignments
Proposition 1 tells us that we can expect an inefficient job assignment threshold to be
the norm. The direction of the bias depends on which job is (locally) more informative
about effort for an agent who is equally productive in both jobs. Two cases therefore
may arise:
(i) If job h is the (locally) more informative job, employees with θ ∈ (θ SB , θ F B ) are
assigned to job h for which they are “underqualified”.
(ii) If job l is the (locally) more informative job, employees with θ ∈ (θ F B , θ SB ) are
passed over for assignment to job h, even though they are “overqualified” for the
job l in which they are placed.
The issue of underqualification has received considerable attention in the literature on
careers in organizations. So under what circumstances it is reasonable to apply our
model to the underqualification case? 10 One channel through which higher level jobs may
become more informative about effort is that higher level jobs may be more sensitive to
effort than lower level ones. Lower level jobs often entail ‘safe’ tasks that have relatively
low demands in terms of worker inputs ability and effort. Because workers cannot do
much harm here, firms assign to such jobs individuals with low expected ability (e.g.
Calvo and Wellisz 1979, Rosen 1982, Waldman 1984b). In contrast, higher level jobs
tend to be more sensitive to workers’ inputs, and firms therefore assign to such positions
only workers with sufficiently high ability.
In our model, Condition (SA) captures the notion that job h is more sensitive to ability
than job l. In the context of a hierarchy, job h would hence be at a higher level than
job l. The property that higher level jobs are more sensitive to effort can be captured
in our setting by fh(ē, θ) − fh(e, θ) > fl(ē) − fl(e). This says that high effort in job h
raises the success probability of the project more than in job l. This sensitivity to effort
condition implies that
fh(e, ˆ θ(ē, ē)) < fl(e), (Condition 2)
because fh(ē, ˆ θ(ē, ē)) = fl(ē). If this condition is met, a consequence of our earlier
analysis is that the principal sets the threshold for assignment to job h too low: θ SB <
θ F B . In other words, the principal biases the job assignment threshold in favor of the
higher level job h.
Condition 2 has an intuitive interpretation: An agent who is almost equally productive
in both hierarchy levels l and h if he works hard (i.e., with ability level close to the
10 That jobs further up in the hierarchy are often more informative about an employee’s inputs is a
view that can also be found in the literature (Baker, Gibbons, and Murphy 1994, Maskin, Qian, and
Xu 2000, Ortega 2003).
11

efficient threshold ˆ θ(ē, ē)) perceives the higher ability job h as more difficult. So the
principal can put the agent to a harder test by placing him in the more demanding job
h. Hence, it is less costly to provide incentives than in the lower level job l and the
principal distorts the assignment in favor of job h.
Another channel through which higher level jobs may become more informative about
effort is that more accurate performance measures are available, a notion for which
Baker et al. (1994, p. 1127) provide a succinct illustration: “[. . . ] market-adjusted
stock-price performance may be a useful measure of a CEO’s contribution but typically
is an extremely noisy measure of the contributions of lower-level employees.” With less
noise the performance measures become more informative about effort. This again would
suggest that assignments are distorted in favor of the more informative higher level job.
3 Implications for promotions
3.1 Extending the model
In the following we give our model a simple dynamic interpretation and extend it to
continuous effort to derive implications for promotion decisions.
3.1.1 A “dynamic” interpretation of the model
Assignment to a higher level job is in many cases tantamount to a promotion. We saw
in the previous section that the principal’s (static) job assignment decision interacts
with her problem of providing incentives at lowest cost. As the literature on careers in
organizations has not looked at this effect, we now ask what novel insights the trade-
off can deliver for understanding promotion policies. Our purpose is to stay as close
as possible to the static job assignment model from Section 2 and isolate these novel
insights by giving our model a simple dynamic interpretation. For this we add a first
period and the two features described in the following.
Workers do not start their careers at the top of the hierarchy: job l is the entry level job in
the two-period model (Feature 1). Distortions in the job assignment/promotion decision
arise in our model because the principal wants to change the agent’s incentives within
the second period; first-period incentives therefore do not drive inefficiency results. This
point can be made most clearly in a setting where the agent simply earns a fixed wage in
the first work period, during which the principal learns about the agent’s ability as, for
example, in Waldman (1984a) (Feature 2). With these features the promotion decision
in the second work period becomes a static job assignment problem.
12

3.1.2 The continuous effort model
Our setup links promotion decisions to the cost of providing incentives: The principal
can use a promotion as a tool to reduce implementation costs or to extract more effort
from the agent. In our two-effort model, however, we do not have much flexibility in
varying effort levels in and across jobs. To elaborate on the effects that varying effort
levels across hierarchy levels can have, we extend our model to allow for continuous effort.
Expected outputs in jobs l and h are fl(e, θ) = α e and fh(e, θ) = max {α e θ − k, 0},
respectively, with k ≥ 0, θ ∈ [θ, ¯ θ] ⊆ R+. The cost of effort is c(e) = e 2 /2. Further, we
impose the following technical conditions:
Assumption 2
(i) α ∈ (0, 1) and e ∈ [0, 1].
(ii) k/α < ¯ θ < 1/α.
(iii) θ < √ α 2 +2 k
α
< ¯ θ.
The first two parts guarantee proper probability distribution functions. Specifically, (i)
implies that fl(e, θ) ∈ (0, 1). And the upper bound in (ii) ensures that fh(e, θ) < 1 for all
θ ∈ [θ, ¯ θ]. The lower bound guarantees that there exist interior ability levels θ ∈ (θ, ¯ θ)
for which positive output in job h is possible (i.e., fh(e = 1, θ) > 0). Finally, (iii) ensures
that there exists an interior first-best job assignment threshold.
The ‘warm up’ cost k ≥ 0 for the more complex job h allows us to vary parametrically
the difference in informativeness of output across jobs. As argued in Section 2.4, a job
higher up in the hierarchy (job h) often is more informative about effort. Assuming k > 0
captures this. Analogous to the two-effort case, the continuous version of the likelihood
ratio says how informative output in job j ∈ {l, h} is about effort: rj(e, θ) = ∂
∂ e fj(e,θ)
fj(e,θ) .
In our setup, the inverse likelihood ratios across jobs have a simple relation:
1
rh(e, θ) =
1 k
−
rl(e, θ) α θ .
So at k = 0 both jobs are equally informative about effort. With increasing k, job h
becomes more informative about effort relative to job l. This can be seen at an intuitive
level by comparing what success in a given job tells the principal about the possible
effort levels that could have led to this outcome. In job l this could have been any effort
level e > 0. In contrast, success in job h is more informative about effort, because a
success indicates that the agent put in at least an effort of k/(α θ). Similarly, in job h
output becomes less informative about effort with a higher ability level: the more able
an agent the more likely it is that he succeeds even with little effort.
13

Figure 2: Second-best versus efficient promotion threshold
The following additional assumptions simplify the analysis by eliminating corner solu-
tions for the second-best outcomes. Intuitively, they restrict the warm-up cost k to be
sufficiently small so that positive effort always pays off in job h.
Assumption 3
(iv) θ ≥ √ 3 k
α .
(v) k <
3 α2
4 .
Part (iv) guarantees that it is optimal for the principal to implement a positive second-
best effort level in job h for all θ ∈ [θ, ¯ θ]. And part (v) ensures an interior second-best
promotion threshold.
We focus here on the implications of the model as well as the links to previous results,
and relegate formal derivations to Appendix C. To generate the figures that illustrate
our results we use the following parametrization: θ ∈ 1
2 , 2 , e ∈ [0, 1], α = 1/2, and
0 < k < 1/16.
3.2 Results
3.2.1 Optimal effort levels and promotion thresholds
We first discuss the properties of the optimal effort levels. The first-best effort lev-
F B els are el B
= α and eFh (θ) = α θ, respectively. In each job, we observe the usual
downward distortion of effort in a moral hazard model. The second-best effort level
in job l is half of the first-best one: e SB
l
B = 0.5 eFl . Comparing this to job h, where
14

eSB h (θ) = 0.5
F B k eh (θ) + , one sees that the distortion relative to the first-best level is
α θ
smaller than in job l, and decreases with a more informative output (higher k). For k = 0
the relative distortion of effort is equal for both jobs. The closed form solutions for the
optimal effort levels allow us to derive the first- and second-best promotion thresholds:
θ F B √
α2 + 2 k
≡
α
and θ SB ≡ α + √ α2 + 4 k
.
2 α
Again, it is interesting to see how the difference between first- and second-best thresholds
varies with the informativeness parameter k. For k = 0 the second-best promotion
threshold is efficient: because both jobs are equally informative about effort there is
neither a relative distortion in the second-best implementation costs, nor in the second-
best effort. Hence, the principal cannot gain by moving away from the efficient promotion
threshold. This is similar to the case in the two-effort model where the agent works hard
in both jobs (effort is constant across jobs) and likelihood ratios are equal (both jobs are
equally informative). Once the jobs differ in informativeness, the picture changes: for
k > 0 we have θ SB < θ F B . The gap between first- and second-best promotion thresholds
increases in k as Figure 2 illustrates. The greater k, the greater the distortion in the
promotion threshold. The following proposition summarizes.
Proposition 2 The second-best promotion threshold is lower than the first-best one for
k > 0; the gap between the two increases in k.
3.2.2 Expected output
So far we have seen that the promotion threshold is set inefficiently low. What does
this imply for the agent’s output? An agent promoted to the higher level job produces
more than an agent who remains in job l, as illustrated in Figure 3. But this does not
mean that the agent is more productive in the new job. Figure 4 illustrates this by
comparing the output in job h with the output the agent would produce in his old job
l, when providing the same effort as he does now. One can see that for ability levels
close to the threshold θ SB , the agent would be more productive with his effort if he
remained in job l instead of being promoted to job h. The reason for higher output
thus is that the promoted agent works harder than he would work in his old job. The
patterns illustrated in the figures arise in general in our model and are summarized in
the following proposition.
Proposition 3
1. Agents promoted to job h work harder than those remaining in job l.
2. Expected output increases for promoted agents relative to those remaining in job l.
15

Figure 3: Expected output
Figure 4: The (extended) Peter Principle
16

3. In the lower ability range of job h, the expected output of an agent would be higher
in job l than in job h, if he worked as hard as he now does in job h.
What drives the inefficient job assignment? Because output in the higher level job is more
informative about the agent’s effort, the principal can save on costs for implementing a
given level of effort by lowering the promotion threshold relative to the first-best one.
These savings allow the principal to get the agent to exert extra effort, which drives
up expected output (similar to the case in the two-effort model where the principal
implements low effort in job l and high effort in job h).
3.2.3 Inefficient job assignments and the Peter Principle
Inefficiencies in promotion policies are often attributed to the Peter Principle. It says
that some employees are promoted to their “level of incompetence” and then remain at
that hierarchy level (Peter and Hull 1969). So it refers broadly to cases where, because
of his “incompetence” an employee produces less for the same inputs after a promotion
than he would be capable of producing in his former job.
Often, however, the Peter Principle is interpreted in a more restricted sense, as an ob-
served decline in output after a promotion (e.g. Lazear 2004). An important caveat
arising from our analysis is that such a comparison of actual outputs may be a mislead-
ing test for “incompetence”. Previous theoretical treatments of promotion policies and
the Peter Principle have sidestepped this issue by simply assuming that effort does not
vary across hierarchy levels (see Section 1). 11 Indeed, Proposition 3 shows that output
may rise, simply because a promoted agent works harder – hiding the fact that he is
actually less talented for his new job than he is for his former job.
To capture the possibility that a promoted agent works harder or less hard than he would
if he were reassigned to the previous job, we therefore propose an extended definition of
the Peter Principle.
Definition 1 The (extended) Peter Principle holds for an agent with ability θ who is
promoted to job h, in which he exerts effort eh, if fh(eh, θ) < fl(eh, θ).
That is, the Peter Principle is said to hold if the agent produces less after a promotion
than he would be capable of producing in his former job given the same inputs, i.e.,
holding fixed his effort eh at its post-promotion level.
Our setup links the Peter Principle to the cost of providing incentives: the principal
can use a promotion as a tool to reduce implementation costs or to extract more effort
11 Interestingly, Barmby, Eberth, and Ma (2007) recognize this problem in their empirical test of
Lazear’s (2004) model, and attempt to control for differences in incentives across hierarchy levels using
differences in mean income.
17

from the agent. According to Proposition 3 the (extended) Peter Principle thus holds
for those who are just above the promotion threshold to job h. As illustrated in Figure
4, such an agent would produce more in his old job if he worked as hard as he does now
after the promotion.
3.2.4 The profile of expected earnings and the agent’s well-being
Figure 5 illustrates the cost saving effect that supports the (extended) Peter Principle.
Panel (a) shows that an agent who is just at the promotion threshold (ability equal to
θ SB ) earns more than he would if he remained in the lower level job. If, however, the
principal wanted to have the agent work as hard in the lower level job as he does in the
new job h, she would need to pay him more, as shown in Panel (b). The gap in earnings
between job h and l widens the more informative job h is relative to job l (i.e., the higher
k). Intuitively, the better information in job h reduces the rent the principal has to leave
to the agent. This allows her to squeeze more effort out of him.
The higher earnings in job h relative to the lower level job thus are the result of harder
work. Panel (a) of Figure 6 illustrates this: for a fixed k a more able agent can expect
to earn more than a less able colleague, both within the higher level job and across
jobs. That is, the expected wage is always higher in job h than in job l. But, as a
consequence of the cost saving effect, an agent in job h with an ability level close to the
promotion threshold enjoys a lower utility than he would in job l (see Panel (b) of Figure
6). The expected utility increases with higher θ and soon exceeds that from working in
the lower level job. This is in line with the commonly expressed view that a promotion
is a desirable career move for most employees: expected utility increases for most ability
levels in job h. For such an agent a promotion leads to a better job match, where higher
earnings more than compensate for the increased workload. As the following proposition
shows, the patterns illustrated in the figures arise in general in our model.
Proposition 4
1. A promotion leads to a jump in expected earnings relative to those who remain in
the lower level job l. In the higher level job h expected earnings increase with θ.
2. The net effect on expected utility relative to non-promotion for the agents in job h
(a) is negative in the lower ability range; they are overpromoted, i.e., less happy
than if they were in their old job with lower earnings but less work;
(b) is positive for all others; for them the higher earnings compensate for the
increased workload.
The second result is driven by the principal’s willingness to forgo gains in expected
output in order to lower the limited liability rent she has to leave to the agent. This
18

Figure 5: Expected earnings and cost saving effect (θ = θ SB )
Figure 6: Expected wage and expected utility (k = 1/32)
19

eduction in the limited liability rent makes an overpromoted agent less happy than he
could be in his old job. However, as he still is earning a limited liability rent, he prefers
the promotion over being fired and forced to take the outside option.
Clearly, an agent who rejects a promotion will not necessarily be fired in real world em-
ployment relationships. But there are cases where turning down a promotion is costly,
as the following two examples illustrate. In a recent case, the US Federal Circuit up-
held the dismissal of a general attorney in the Los Angeles office of the Small Business
Administration for rejecting a promotion. 12 The FBI had a practice of cutting the pay
of agents who refuse to accept promotion to a supervisory position. 13 Even when the
penalties are less severe, people may accept their role for reasons of prestige or status,
as illustrated by the following case: “a man in his early fifties had been overpromoted,
but held on to his job because, as he put it, ‘of the status and authority of the role’”
(Erskine and Brook 1971, p.55). Enriching the model with concerns for status (see e.g.
the model of Auriol and Renault 2008) seems to be an interesting direction for future
research.
For the cases where an agent will not or cannot reject a promotion our model provides a
link between promotions and workplace stress. Those at the bottom end of the earnings
distribution in the higher level job are kept on toes because they are not sufficiently
talented for the new position: to make up for their (slight) lack in talent they work
harder than they would in their old job for the same expected earnings. Consistent with
this finding, “overpromotion” has been identified as one of the factors causing stress
at the workplace (e.g. Cooper and Marshall 1976, Murphy 1995). 14 In line with this,
Boyce and Oswald (2008) find that promotions significantly worsen the psychological
strain of private sector employees, controlling for endogeneity of health measures with
longitudinal data.
4 Concluding remarks
The literature on careers in organizations stresses the role of incomplete information
about employees’ abilities and contractual imperfections for inefficiencies in job assign-
ments. Our model strips away these presumed sources of inefficiencies: even though
there is no uncertainty about the agent’s ability and complete contracts are possible, a
static trade-off between incentive provision and efficient assignment generically pushes
the job assignment threshold away from its efficient level.
12 Robinson v. Small Business Administration, U.S.C.A.F.C. No. 06-3065, 5/4/06.
13 Los Angeles Times, FBI Agents Rebel Over Mandatory Transfers, May 22, 2006.
14 A person is overpromoted when he “has reached the peak of his abilities [...] and is given responsi-
bility exceeding his capacity” (Cooper and Marshall 1976, p.18).
20

Applying our simple model to a promotion context shows that an organization may find
it optimal to promote an employee to a job for which he is less talented than for the
previously held lower level position (the Peter Principle). Such a promotion decision
makes it cheaper to induce hard work: facing a more challenging job, the employee
has to make up for his lack in talent by exerting more effort than he would in his old
job for the same compensation. In other words, the promotion policy complements the
wage policy as a tool to reduce the rents that employees enjoy because of moral hazard
problems in the organization.
The mechanism at work is a static one: the job assignment decision in one period affects
the contemporaneous cost of providing incentives. This alone is sufficient to create
inefficiencies in the assignment policy. To not obscure this, we deliberately kept the
promotion model simple. Nevertheless, the framework delivers novel predictions in line
with several stylized facts. First, it offers an explanation for why organizations seem
to be content with having employees in positions where they are less productive than
they would be in their former job (the Peter Principle). Second, it shows that employees
moving up the career ladder may earn more only because they have to work harder than
they would in their old jobs for the same pay. Third, the setup captures the idea that a
promotion can be a good thing for most workers, but at the same time may reduce the
rents enjoyed by the least able workers in a hierarchy level. Because the latter have to
put in extra effort to outweigh their lack in talent, they are less happy than they would
be if they could stay in their old job – a phenomenon that the literature on workplace
stress refers to as overpromotion (e.g. Cooper and Marshall 1976, Murphy 1995).
While our setup goes beyond previous theoretical job assignment models by allowing
for implemented effort levels to vary across hierarchy levels, there are many facets of
careers in organizations that our model does not capture. Indeed, our purpose was to
show what additional insights a tractable model of the trade-off between job assignment
and incentives yields. Our framework can serve as a building block that could usefully
be added to an integrated promotion model in the spirit of Gibbons and Waldman
(1999b, 2006), who abstract from incentives. In such an enriched model, for example,
the effective ability of an agent may vary with tenure t in the organization: ρt(θ). Or the
expected output in a job may directly depend on the agent’s experience. Taken together,
the success probability may be written as fj(e, ρt(θ), t). This more general formulation
can capture learning by doing and human capital accumulation: one possibility is that
the productivity of an agent increases over time, fj(e, ρ, t+1) > fj(e, ρ, t); or ability may
increase with experience ρt+1(θ) > ρt(θ). The model then can also encompass learning
about the ability of an agent, if ρt(θ) is thought of as the expected ability given the
information available. 15 Developing an integrated promotion model along these lines is
15 Note that in our model we would have ρ0(θ) = E[θ] and ρt(θ) = θ.
21

an interesting direction for future research.
Appendix
A Job assignments given inefficient effort levels
This section builds intuition for additional effects that inefficient second-best effort levels
can have on the job assignment rule. To illustrate these effects, we assume that job h
is locally more informative about effort than job l (Condition 2). We discuss here the
case where the principal implements the inefficient low effort in job l (e SB
l
Condition 1 is violated for job l). The driving forces for the other case (e SB
l
= e, i.e.,
= ē) are
similar to those in Section 2, and we leave the details for the proof of Proposition 5.
Suppose first that in job h Condition 1 is also violated for all ability levels, i.e., e SB
l
e SB
h
(θ) = e. Then no incentives are needed because the principal implements low effort
in either job. Hence, the job assignment problem reduces to maximizing the expected
revenue fj(e, θ) over j, resulting in threshold θ SB = ˆ θ(e, e) > θ F B = ˆ θ(ē, ē). In other
words, the downward distortion of effort in both jobs pushes the job assignment threshold
above the first-best level. As we saw in Section 2, if second-best effort levels in both
jobs are efficient (i.e., Condition 1 holds for all ability levels in both jobs), the principal
distorts the job assignment threshold in the opposite direction: assignment to job h leads
to an implementation cost saving effect that pulls the threshold below the first-best level:
θ SB < ˆ θ(ē, ē).
We expect both forces to be at work whenever Condition 1 holds for job h, but not
for job l. To illustrate that this is indeed the case, and to see how these two forces
interact, suppose that for job h Condition 1 holds for all θ. 16 Because no incentives are
needed in job l, placing the agent into job h cannot lead to cost savings. But it may be
cheaper to implement high effort in job h than it would be in job l. For that reason,
assigning the agent to job h and then implementing high effort might be worthwhile for
some θ < ˆ θ(e, e). Indeed, it is even possible that θ SB is lower than θ F B . When does this
occur? Assigning the agent to the higher level job and implementing high effort increases
output by fh(ē, θ) − fl(e). This has to outweigh the increase in implementation costs of
Ch (ē, θ). Hence, if there is a strict gain in profit from assigning an agent with ability
θ = θ F B to job h we know that θ SB < θ F B (by continuity). The following proposition,
which we prove formally in Appendix B, summarizes our results.
16 Dropping the qualifier “for all θ”, implies that multiple thresholds are possible. Then θ SB in the
explanations and results below refers to the lowest threshold for assigning the agent to job h (see the
proof of Proposition 5).
22
=

Proposition 5 Suppose job h is locally more informative than job l (Condition 2). In
case multiple cutoffs exist, θ SB refers to the lowest second-best threshold.
1. Suppose the second-best effort level in job l is inefficient (e SB
l
is violated for job l). Then,
= e, i.e., Condition 1
(a) θSB ≤ ˆ θ(e, e), and θSB is higher than it would be if eSB h (θ) = eSB l = ē ∀ θ;
(b) θ SB < θ F B if and only if
fh(ē, ˆ θ(ē, ē)) − fl(e) > Ch(ē, ˆ θ(ē, ē)). (Condition 3)
2. Suppose the second-best effort level in job l is efficient (e SB
l
holds for job l), then θ SB < θ F B .
B Proofs
Proof of Lemma 1.
= ē, i.e., Condition 1
d
d θ ΠSB h θ (ē, θ) = fh θ(ē, θ) − Ch θ(ē, θ) = fh θ(ē, θ) − fh(ē, θ) fh θ(e, θ) − fh(e, θ) fh θ(ē, θ)
[fh(ē, θ) − fh(e, θ)] 2 c
> fh θ(ē, θ) − fh(ē, θ) fh θ(e, θ)
c > 0,
[fh(ē, θ) − fh(e, θ)] 2
where the last inequality follows from fh θ(ē, θ) ≥ fh θ(e, θ) (Assumption SA) and
[fh(ē, θ) − fh(e, θ)] 2 > fh(ē, θ) c (Condition 1). This shows Part (a). Part (b) follows.
Proof of Proposition 1.
We prove one side of the inequality here, the reverse direction is analogous.
Π SB
h (ē, ˆ θ(ē, ē)) − Π SB
l (ē) > 0 ⇔ fh(ē, ˆ θ(ē, ē)) − Ch(ē, ˆ θ(ē, ē)) > fl(ē) − Cl(ē)
⇔ Ch(ē, ˆ θ(ē, ē)) < Cl(ē) ⇔ fh(ē, ˆ θ(ē, ē)) − fh(e, ˆ θ(ē, ē)) > fl(ē) − fl(e)
⇔ fh(e, ˆ θ(ē, ē)) < fl(e),
using repeatedly fh(ē, ˆ θ(ē, ē)) = fl(ē). θ SB < θ F B follows because θ F B = ˆ θ(ē, ē) and the
profit difference is strictly increasing in θ (Lemma 1).
Proofs of Propositions 2-4.
See Appendix C.
Proof of Proposition 5.
Part (1a): Denote by ˜ θ the lowest ability level where expected profits in jobs h and l
would be equalized if high effort was implemented in both jobs. That is, Π SB
h (ē, ˜ θ) =
23

Π SB
l (ē) (these are not the maximized second-best profit levels, as implementing high
effort is not optimal in job l, and may not be in job h either). From the proof of
Proposition 1 we know that ˜ θ < ˆ θ(ē, ē). Condition 2 implies that ˆ θ(ē, ē) < ˆ θ(e, e).
Denote by θ SB the lowest threshold for assigning the agent to the high ability job. We
now show that θ SB ∈ ( ˜ θ, ˆ θ(e, e)], for which we will divide the ability interval [θ, ¯ θ] into
subintervals over which Condition 1 may hold or fail for job h. Note that whether or
not Condition 1 holds for θ ≥ ˆ θ(e, e) or θ ≤ ˜ θ is irrelevant: all types above ˆ θ(e, e) are
assigned to job h, all below ˜ θ to job l.
Suppose first that Condition 1 also does not hold for any θ ≤ ˆ θ(e, e) in job h. Then
because ΠSB h (e, θ) = fh(e, θ) ≤ fl(e) = ΠSB l (e) for θ ≤ ˆ θ(e, e) and maxe ΠSB h (e, θ) ≥
fh(e, θ) > fl(e) = ΠSB l (e) for θ > ˆ θ(e, e) it follows that θSB = ˆ θ(e, e).
Suppose now that Condition 1 holds for θ ≥ ˜ θ in job h. Then, for these ability levels we
have Π SB
h
(ē, θ) ≥ ΠSB
h
(e, θ) and ΠSB h θ (ē, θ) > 0 (Lemma 1). Because Condition 1 does not
hold for job l, ΠSB l (e) > ΠSB l (ē) = ΠSB h (ē, ˜ θ), as ΠSB l (e) = fl(e) > ΠSB l (ē) = fl(ē)−Cl(ē).
Hence, it follows that θSB > ˜ θ. Moreover, ΠSB h (ē, ˆ θ(e, e)) > ΠSB h (e, ˆ θ(e, e)) = ΠSB l (e), so
θ SB < ˆ θ(e, e).
Finally, if Condition 1 holds only for some θ ∈
( ˜ θ, ˆ θ(e, e)].
˜θ, θ(e, ˆ e) , then a fortiori θSB ∈
Part (1b): To prove the if part suppose that fh(ē, ˆ θ(ē, ē)) − fl(e) > Ch(ē, ˆ θ(ē, ē)).
Together with Condition 2 this implies that fh(ē, ˆ θ(ē, ē)) − fh(e, ˆ θ(ē, ē)) > Ch(ē, ˆ θ(ē, ē)),
so Condition 1 holds at θ = ˆ θ(ē, ē). Thus, the maximized second-best profit in job h is
Π SB
h (ē, ˆ θ(ē, ē)) > Π SB
l (e). By continuity, θ SB < θ F B = ˆ θ(ē, ē).
To prove the only if part, suppose θ SB < θ F B = ˆ θ(ē, ē). By definition of the second-
best threshold Π SB
h (ē, θSB ) = fl(e). The next steps show that Π SB
h (ē, ˆ θ(ē, ē)) > fl(e)
follows from (Condition 3). Condition 2 implies that ˆ θ(ē, ē) < ˆ θ(e, e) so fl(e) > fh(e, θ)
for θ ∈ [θ SB , ˆ θ(ē, ē)]. From the previous step and Π SB
h (ē, θSB ) = fl(e) it follows that
Π SB
h (ē, θSB ) > fh(e, θ SB ), which is equivalent to Condition 1 holding at θ SB . By Lemma
1, Π SB
h θ (ē, θSB ) > 0. We now show that the profit never falls back to or below fl(e)
over the relevant range. By way of contradiction, suppose that there exist values
θn ∈ (θSB , ˆ θ(ē, ē)] for which ΠSB h (ē, θn) = fl(e), and let θ1 denote the smallest of these.
Because ΠSB h (ē, θSB ) = fl(e) and ΠSB h θ (ē, θSB ) > 0 it follows that ΠSB h θ (ē, θ1) < 0. This,
however, implies that Condition 1 is violated at θ1, i.e., ΠSB h (ē, θ1) < fh(e, θ1), which
leads to a contradiction because fl(e) > fh(e, θ1).
Part (2): Denote by ˜ θ the ability level where expected profits in jobs h and l would be
equalized if high effort was implemented in both jobs, as in the proof of Part (1a). We now
show that if Condition 1 holds for job l, then Condition 1 holds also for job h at θ = ˜ θ:
24

fh(ē, ˜ θ)−Ch(ē, ˜ θ) = fl(ē)−Cl(ē) ⇔ fh(ē, ˜ θ)−fl(e)−Ch(ē, ˜ θ) = fl(ē)−fl(e)−Cl(ē). The
right-hand side is nonnegative because Condition 1 holds for job l. Using fh(e, ˜ θ) < fl(e)
therefore yields fh(ē, ˜ θ)−fh(e, ˜ θ) > Ch(ē, ˜ θ) (Condition 1). Thus, the maximized second-
best profit in job h at θ = ˜ θ is Π SB
h (ē, ˜ θ), and by the definition of ˜ θ it follows that
θ SB = ˜ θ. Furthermore, we know that θ SB = ˜ θ < θ F B = ˆ θ(ē, ē). We now show that
θ SB is the only job assignment threshold below θ F B , i.e., Π SB (ē, θ) never falls back to
or below fl(ē) − Cl(ē) for θ ∈ (θ SB , ˆ θ(ē, ē)]. By way of contradiction, suppose that
there exist values θn ∈ (θSB , ˆ θ(ē, ē)] for which ΠSB h (ē, θn) = fl(ē) − Cl(ē), and let θ1
denote the smallest of these. Because ΠSB h (ē, θSB ) = fl(ē) − Cl(ē) and ΠSB h θ (ē, θSB ) > 0
(by Lemma 1) it follows that ΠSB h θ (ē, θ1) < 0. This, however, implies that Condition 1
is violated at θ1, i.e., ΠSB h (ē, θ1) < fh(e, θ1), which leads to a contradiction because
fl(ē) − Cl(ē) ≥ fl(e) > fh(e, θ1).
C Continuous effort model
First best
F B
The efficient effort level given θ solves maxe fj(e, θ) − c(e). For job l this yields e
l
= α
F B
B
and for job h we have eh (θ) = α θ. Hence, the expected profits are ΠFl α2 = 2 and
F B Πh (θ) = α2θ2 2
B
− k, respectively. Setting profits equal defines the efficient cutoff θF
implicitly by α2θ2 − 2 k − α2 = 0. Solving for θF B yields
θ F B √
α2 + 2 k
≡ .
α
Second best
To implement effort e in job j the wage after success must be set at ¯wj(e, θ) = ce(e)
fj e(e,θ) ,
where subscript e denotes the partial derivative with respect to effort. That is,
¯wl(e) = e
α and ¯wh(e, θ) = e . Denote the expected cost of implementing effort in job j by
α θ
Cj(e, θ) = fj(e, θ) ¯wj(e, θ). Maximizing the principal’s expected profit fj(e, θ) − Cj(e, θ)
over e gives the second-best effort level for a type-θ agent. In job l this yields e SB
l
half the efficient effort level. In job h we have 17
e SB
h (θ) = 1
α θ +
2
k
.
α θ
α = 2 ,
17 SB
Note that an effort of zero yields an expected utility of zero. So for eh (θ) > 0 to be optimal the
incentive constraint requires that EUh(eSB h (θ), θ) ≥ 0. Equation (5) and the derivations below show
that EUh(eSB h (θ), θ) ≥ 0 ⇔ θ ≥ √ 3 k
α . This explains A4.
25

So if k = 0, the second-best effort is also exactly half the efficient effort level, and with
higher k the effort increases. Expected profits are:
Πl = fl(e SB
l ) − Cl(e SB
l ) = α2
4 ,
Πh(θ) = fh(e SB
h (θ), θ) − Ch(e SB
h (θ), θ) = (α2 θ2 − k) 2
4 α2 θ2 .
Setting profits equal defines the second-best cutoff θ SB implicitly, and solving for θ SB
yields 18
Proofs of Propositions 2-4.
θ SB ≡ α + √ α2 + 4 k
.
2 α
We now prove formally the remaining results discussed in the text.
Promotion threshold θ SB < θ FB for k > 0: By way of contradiction, suppose θ F B ≤
θ SB ⇔ 2 √ α 2 + 2 k ≤ α + √ α 2 + 4 k. Squaring both sides: 4 (α 2 + 2 k) ≤ 2 (α 2 +
α √ α 2 + 4 k + 2 k) ⇔ α 2 + 2 k ≤ α √ α 2 + 4 k. Squaring again both sides: α 4 + 4 k α 2 +
4 k 2 ≤ α 4 + 4 k α 2 , yielding a contradiction for k > 0. Taking derivatives shows that
|θ F B − θ SB | decreases in k.
Expected output: For an agent placed in job h, the expected output is higher than it
would be if he were instead assigned to job l (with the second-best effort level for the
respective job). The difference in expected output fh(eSB SB
h (θ), θ) − fl(el ) = α2 (θ2−1)−k , 2
is increasing in θ:
∂
fh(e
∂ θ
SB
h (θ), θ) − fl(e SB
l ) = α 2 θ > 0.
At θSB there is an upward jump in expected output:
fh(e SB
h (θ SB ), θ SB ) − fl(e SB
l ) = α (√ α 2 + 4 k − α)
4
As the output difference is increasing it follows that fh(eSB SB
h (θ), θ) > fl(el ) also for all
θ > θSB .
The Peter Principle: Keeping effort constant at the second-best level for job h, the
expected output for an agent with ability θ SB placed in job h is lower than it would be
18 Without A5 a corner solution would arise whenever α+ √ α 2 +4 k
2 α
> 0.
≤ √ 3 k
α . Then, θSB = θ. We focus
here on the interesting case where promotion is not automatic, but requires ability level of at least
θ SB > θ.
26

if he were instead assigned to job l:
fh(e SB
h (θ SB ), θ SB ) − fl(e SB
h (θ SB )) =
−k α
α + √ α 2 + 4 k
< 0.
Effort eSB h (θ) > eSB l for θ > θSB : At θSB there is an upward jump in effort:
e SB
h (θ SB ) − e SB
l =
The effort gap is increasing in θ: d
d 2
d θ2 eSB h
d θ eSB h
2 k
α + √ α 2 + 4 k .
(θ) = 1
2
(θ) > 0 and d
d θ eSB
h (θ) θ=θ SB > 0, as is easy to check.
k α − α θ2
SB > 0 for θ > θ , because
Expected earnings: For an agent assigned to job h, expected earnings (EE) are higher
than they would be if he remained in job l.
EEh(e SB
h (θ), θ) = fh(e SB
h (θ), θ) × ¯wh(e SB
h (θ), θ) = (α2 θ2 − k) (α2 θ2 + k)
4 (α2 θ2 ,
)
EEl(e SB
l ) = fl(e SB
l ) × ¯wl(e SB
l ) = α2
4 ,
EEh(e SB
h (θ), θ) − EEl(e SB
l ) = α4 θ2 (θ2 − 1) − k2 4 α2 θ2 .
At θ = θ SB there is an upward jump in the expected earnings: 19
EEh(e SB
h (θ SB ), θ SB ) − EEl(e SB
l ) =
Since the difference is increasing in θ,
d
d θ
α k
α + √ α 2 + 4 k
EEh(e SB
h (θ), θ) − EEl(e SB
l ) = α4 θ 4 + k 2
2 α 2 θ 3
> 0.
> 0,
we conclude that the expected earnings difference is positive for all θ > θ SB .
Expected utility: The expected utility (EU) in job j is given by the expected earnings,
calculated above, net of the effort cost c(e SB
j (θ)):
EUh(e SB
h (θ SB ), θ SB ) = (α2 θ2 − 3 k) (α2 θ2 + k)
8 (α2 θ2 , (5)
)
EUl(e SB
l ) = α2
8 .
19 Alternatively, this can be seen also from our earlier result that the success probability at second-
best effort is higher in job h, combined with the fact that the wage after success in job h is also higher:
¯wh(e SB
h (θSB ), θ SB ) − ¯wl(e SB
l ) = 8 k
(1+ √ 1+16 k) 2 > 0.
27

Close to the threshold θ SB , the agent has lower expected utility in job h than if he could
remain in job l:
EUh(e SB
h (θ SB ), θ SB ) − EUl(e SB
2 k
l ) = −
2
√ 2 < 0.
α + α2 + 4 k
The expected utility in job h is increasing in θ:
∂
∂ θ
EUh(e SB
h (θ), θ) − EUl(e SB
l ) = α4 θ 4 + 3 k 2
4 α 2 θ 3
> 0.
The gap thus narrows and is eventually closed for some ˇ θ < 1/α, as one can show. An
agent with θ > ˇ θ is better off in job h than in job l.
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