GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 353

A Tenure-Clock Problem

Chia-Hui Chen and Junichiro Ishida

January 2015

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, JapanA Tenure-Clock Problem
Chia-Hui Chen∗ and Junichiro Ishida†
January 6, 2015

Abstract
We consider a “tenure-clock problem” in which a principal may set a deadline by which
she needs to evaluate an agent’s ability and decides whether to promote him or not. We
embed this problem in a continuous-time model with both hidden action and hidden
information, where the principal must induce the agent to exert effort to facilitate herlearning process. The value of committing to a deadline is examined in this environment,
and factors that make the deadline more profitable are identified. Our simple framework
allows us to obtain a complete characterization of the equilibrium, both with and withoutcommitment, and provides insight into why up-or-out contracts are prevalent in someindustries while they are almost non-existent in others.

JEL Classification Number: D82, D83
Keywords: dynamic agency, deadlines, experimentation, standard promotion practices, upor-out contracts.
∗†

Institute of Economic Reserach, Kyoto University. Email: chchen@kier.kyoto-u.ac.jp
Institute of Social and Economic Research, Osaka University. Email: jishida@iser.osaka-u.ac.jp

1

Introduction

Suppose that an employer needs to hire a worker to carry out a project over time. The
project is ability-intensive in that the worker can successfully complete the project only if he issufficiently productive. As is often the case, however, the worker’s productivity is not directly
observable to the employer, who must instead make an inference about his productivity froma sequence of observed outputs. A number of issues arise in this dynamic environment. How
much time should the employer give the worker before she terminates the project? Should
the employer commit to a deadline at the outset? If so, under what conditions?
In this paper, we attempt to address these issues by constructing a stylized model of a“tenure-clock problem,” in which a principal hires an agent and evaluates his productivity overtime in an ongoing relationship. The problem is embedded in a continuous-time frameworkwith both hidden action and hidden information. At each instance, the agent privately
chooses how much effort to supply. The outcome is either a success or a failure, dependingon his effort choice as well as his productivity type. The game ends immediately when theagent achieves a success (or a “breakthrough”). The principal’s task in this environment isto determine when to terminate the project, conditional on no success having occurred.
Two distinct cases are analyzed within this environment to evaluate the value of setting anevaluation deadline. In one case, the principal sets a deadline and commits to it at the outset,
and the project is terminated automatically when the deadline is reached without attaining
a success (but never before). In the other, the principal makes no such commitment, thereby
retaining discretion to terminate the project at any instance with no specific deadline. The
principal simply terminates the project when the principal judges that the agent is not up
to the task. By directly comparing these two solutions, we evaluate the extent to which theprincipal benefits from committing to an evaluation deadline at the outset of the relationshipin this dynamic environment.
The driving force of our analysis is a dynamic strategic interaction between the agent’s
effort choices and the principal’s termination strategy. On one hand, the agent’s effort choices
depend clearly on how much time is left until the project is terminated: since the net valueof achieving a success is low when the project is still far from termination, the agent tends
to start off with low effort and gradually shift to higher effort as the expected termination
date approaches. On the other hand, the principal’s willingness to terminate the project
depends also on the agent’s effort choices: when the agent is less motivated and exerts low
effort, less information is revealed about his type, which makes the principal more reluctant
to terminate the project. This strategic interaction can generate a vicious cycle where the1principal’s reluctance to terminate the project diminishes the agent’s motivation, which inturn makes the principal even more reluctant. It is in general profitable to commit to adeadline when each player’s incentive to “procrastinate” is sufficiently strong.
We provide an analytical framework that can capture this dynamic interaction, arguably
in its simplest form. Our simple framework allows us to obtain a complete characterizationof the equilibrium, both with and without commitment. The case that is technically morechallenging is the no-commitment one where the principal’s termination strategy must be
sequentially rational along the way, and the belief off the equilibrium path plays a crucialrole. Even in this case, we show that a (pure-strategy) perfect Bayesian equilibrium always
exists and is generically unique for any given set of parameters. We then build on this result
to derive a necessary and sufficient condition under which the principal can strictly benefit
from committing to an evaluation deadline.
Among other things, an interesting, somewhat counterintuitive, property of our model isthat the average duration of the project can be either shorter or longer with commitment
than without. In other words, there exists an equilibrium in which the principal prematurely
terminates the project when she does not commit to a deadline. This result is somewhat
surprising, provided that the inefficiency of the problem stems partly from the principal’s
reluctance to terminate the project. The principal tends to terminate the project too early
when the agent’s productivity under low effort is sufficiently small. To see this possibility
most clearly, suppose that the agent’s productivity under low effort is arbitrarily close tozero, in which case the principal can learn very little from failures (during the phase where
the agent is supposed to exert low effort). The principal’s belief thus declines very slowly
over time, forcing her to incur a large amount of loss if she is to wait until the instantaneous
payoff equals zero. We show that if the expected loss is prohibitively large, the principal
chooses to terminate the project early even when the belief is still relatively high.
The most prominent example of evaluation schemes with deadlines is perhaps the “up-orout system,” which is widely observed in academia and professional service industries suchas law, accounting, and consulting. From a practical point of view, our analysis offers some
implications for when up-or-out contracts are more valuable by identifying several key factorsthat favor the use of an evaluation deadline: high ability intensity, stable job descriptions,similar jobs across ranks, and initial screening. Each of these factors intensifies either theagent’s incentive to delay exerting high effort for a given deadline or the principal’s incentive
to delay terminating the project for a given effort sequence (or both). We later argue thatthese factors are staple features of industries that are traditionally characterized by the upor-out system.

2

The paper is organized as follows. The literature review is provided in the remainderof this section. The model is presented in section 2 and analyzed in section 3, where wecharacterize both the commitment and no-commitment solutions. These solutions are thencompared in section 4, in order to derive a necessary and sufficient condition under which it is
optimal to commit to a deadline and conduct comparative statics exercises. Some extensionsof the baseline model are discussed in section 5, and concluding remarks are offered in section
6.
Related Literature: The current analysis is most closely related to the experimentation
literature in that the principal here attempts to uncover the agent’s type though a sequenceof experiments.1 From the principal’s point of view, our model can be seen as a variantof the canonical two-armed bandit problem with one safe arm (continuing the project) and
one risky arm (terminating the project). A crucial difference from the existing models isthat experiments in our context are “intermediated,” i.e., experiments are conducted not by
the principal herself but by an informed intermediary who may or may not behave as theprincipal wishes.2
Recently, there have been increasingly many works that explore the optimal provision of
incentives in bandit problems. Manso (2011) considers the classic two-armed bandit problem
and shows that the optimal contract in this context must tolerate, or even reward, early
failures in order to encourage exploratory activities. Bergemann and Hege (1997, 2005) andHörner and Samuelson (2013) analyze a financing problem of a venture capitalist where theprincipal provides funding to the agent who conducts experiments on a project of unknown
quality. Gerardi and Maestri (2012) consider a similar environment where an agent conducts
experiments but assume that the outcome of each experiment can only be observed by theagent. The principal must hence devise a contract not only to induce costly effort but also
to truthfully reveal the information. Halac et al. (2013) analyze a model of long-term contracting for experimentation with hidden information about the agent’s ability and dynamic
moral hazard and obtain an explicit characterization of optimal contracts. Aside from sometechnical differences,3 these previous works are primarily concerned with characterizing op1

An early economic application of the bandit problem can be found in Rothschild (1974). See Bergemann
and Välimäki (2008) for a survey.
2
There are now increasingly many works, often called strategic experimentation, where a group of individuals, rather than a single individual, face bandit problems (Bolton and Harris, 1999; Keller et al., 2005; Klein
and Rady, 2011; Bonatti and Hörner, 2011). There is some connection to this strand of literature in that theexperimentation process in the current setup involves decisions of multiple individuals. In a different vein,
Guo (2014) analyzes a model of “delegated experimentation” in which a principal delegates experimentation
to an agent and solves for the optimal delegation rule.
3
As a key technical difference, we consider a case where the agent knows his own productivity (or the
project quality), so that our model belongs to the class of dynamic signaling, rather than of experimentation,3

timal contracts. In contrast, the aim of this paper is to compare the allocations under fullcommitment and no commitment, while focusing on a narrower class of contracts, in orderto evaluate the extent to which the principal can gain from committing to a deadline at theoutset.4
It is well known that players often wait until the deadline to reach an agreement in finitehorizon models. This behavior, which is referred to as the deadline effect, is a topic of utmost
concern in many bargaining and war of attrition models (Hendricks et al., 1988; Spier, 1992;
Fershtman and Seidman, 1993; Hörner and Samuelson, 2011; Chen, 2012; Damiano et al.,2012; Fuchs and Skrzypacz, 2013). Some recent works also explore the role of deadlines indynamic problems with multiple agents. Bonatti and Hörner (2011) analyze a dynamic moralhazard problem with a team in which two agents work on a project of unknown quality andbriefly discuss the optimal deadline in this context. Campbell et al. (2013) consider a similar
environment where two agents work jointly on a project. They assume that there is only one
project type but assume that one’s own outcomes are his private information. Each agent
can exert effort to produce a breakthrough individually, and a successful agent can revealthat he has been successful. The focus of these works is placed on the interaction between theagents, especially the freerider problem, whereas ours is on the dynamic interaction between
the principal’s termination decisions and the agent’s effort choices.
The use of up-or-out is clearly intriguing because of its unique features, and has naturally
spurred a lot of interest among economists. Two features are particularly crucial as defining
characteristics of up-or-out: (i) there is an evaluation deadline by which a promotion mustoccur; (ii) a worker must leave the firm if he is not promoted by the deadline.5 Most previous
studies focus on the second feature (that unpromoted workers must leave their respective
firms) and rationalize the use of up-or-out from that perspective.6 In contrast, our analysis
from the agent’s point of view.
4
Mason and Välimäki (2014) consider a dynamic moral hazard problem and derives optimal wage contracts,
both with and without commitment on wage payments. However, their work assumes only one agent typewith an exclusive focus on moral hazard and thus does not consider hidden information about the agent’s
productivity.
5
In addition, Kahn and Huberman (1988) also argue that an up-or-out contract must specify a retention
wage, which is taken implicitly here.
6
Some of the influential works on this topic focus on the second feature and emphasize its role in inducing
human capital investments. Kahn and Huberman (1988) consider an environment where a worker invests in
firm-specific human capital but the investment level is not verifiable. The unverifiability invites a problem,
often referred to as double moral hazard, where the firm cannot credibly compensate the worker for making
the investment. They show that the firm can effectively solve this problem by the use of an up-or-out contract,
because the contract gives the firm an incentive to retain the worker if and only if his productivity is highenough. Similarly, building on the model of asymmetric learning, Waldman (1990) emphasizes the role ofsignaling in inducing general human capital investments. From a different perspective, O’Flaherty and Siow
(1992) consider an environment where junior-level jobs are used to predict a worker’s productivity at seniorlevel jobs and argue that up-or-out contracts prevail when junior-level job slots are scarce.

4

focuses exclusively on the first, i.e., the presence of an evaluation deadline, as the essential
feature of up-or-out.7 We focus on this feature primarily because of the recent trend observedin professional service industries: the way up-or-out is implemented in practice has become
diverse even within a specific industry, and there are now increasingly many instances ofup-or-out that do not necessarily force unpromoted workers to leave their respective firms.8
This recent trend away from the traditional sense of up-or-out suggests that the second
feature is not necessarily an essential aspect of up-or-out whereas the presence of a deadline
is clearly so, almost by definition. Moreover, by focusing on the optimal deadline problem,the current framework can be applied to a wide range of circumstances in which a principal(an evaluator) must assess an agent’s upside potential that is only gradually revealed in an
ongoing relationship, e.g., a manager who must evaluate subordinates, a professor who mustevaluate graduate students, a head coach in professional sports who must evaluate players,
and so on.
As another departure from the literature, it is also worth noting that our analysis directly evaluates the value of up-or-out by comparing the commitment and no-commitment
solutions. This aspect of our analysis stands in sharp contrast to the existing literature: as
pointed out by Ghosh and Waldman (2010), most previous studies analyze up-or-out contracts
independently, with no specific reference to “standard promotion practices,” thereby leaving
unresolved why up-or-out contracts are commonly instituted in some industries whereas they
are virtually nonexistent in others.9 Our analysis provides a new take on when up-or-out
contracts can be more valuable vis-a-vis other more standard promotion practices, from aperspective previously unexplored in the literature.

2

A tenure-clock problem

Environment: We employ a continuous-time model because of its greater tractability; adiscrete-time counterpart of the model is briefly discussed in Appendix B. Consider a situation where a principal (female) hires an agent (male) to complete a project. The game ends
either when the agent attains a success (a “breakthrough”) or when the principal terminates
the project. The agent is either good with prior probability p0 ∈ (0, 1) or mediocre with prob7

As such, we are not necessarily concerned about what happens after the deadline: the agent in our context
may or may not be fired after the deadline; all we need is that the agent must incur some cost, in whateverform, for not being promoted before the deadline.
8
For instance, many law firms now shift away from the traditional up-or-out model, known as the Cravathsystem, and increasingly adopt a two-tier partnership model where they retain those who fail to make partner
as permanent associates or non-equity partners.
9
As an exception, Ghosh and Waldman (2010) build on an asymmetric-learning model and compare theperformance of up-or-out contracts with that of standard promotion practices.

5ability 1−p0 . The ability type is the agent’s private information and is not directly observableto the principal who must instead evaluate it from a sequence of observed outcomes.
Production: The agent makes unobservable effort at ∈ {l, h} at each instance t. We interpret that low effort (at = l) refers to the minimum level of effort that can be induced
via input monitoring while high effort (at = h) refers to any part of effort that cannot bedirectly monitored by any means. The instantaneous cost of effort a is denoted by da where
dh = d > 0 and dl = 0. A success arrives stochastically, depending on the effort choice aswell as the agent’s type. If the good type chooses at = a over time [t, t + dt), he attains a
success with probability λa dt where λh > λl > 0. In contrast, the mediocre type can neversucceed with any effort level.10 Define ∆λ := λh − λl .
Payoffs: A success yields a net present value of y > 0 to the principal; otherwise, she receives
zero. If the agent attains a success, the principal pays a bonus b ≥ 0 to him. Aside from thisbonus, the principal also needs to pay a fixed wage w ≥ 0 at each instance as long as she hires
the agent. We call (b, w) ∈ (0, ∞)2 the wage scheme for clarity and assume for most part that
it is given exogenously: for a practical interpretation, w reflects the ongoing market wage ofentry-level workers whereas b reflects that of senior-level workers.11 The reservation payoff isassumed to be zero for both players. The common discount rate is denoted by r ∈ (0, ∞).
Remark on exogenous wages: For most part of the analysis, we restrict our attention
to the case where the principal can only set a deadline to motivate the agent while taking
the wage scheme as given; the case with endogenous wages is briefly discussed in section 5.2.
Although we make this assumption to clarify the role of deadlines, we also argue that thisconstitutes a nice starting point for several reasons. First, in many instances, equilibrium
wages are dictated by market forces. Second, there are also many cases in reality wherecontingent monetary transfers are neither feasible due to contractual incompleteness nor
desirable as in a professor-student relationship. Finally, the cost and benefit of achieving
a success, or simply staying in a particular job, may involve non-transferrable gains suchas prestige, authority, and the sense of achievement. Monetary transfers are of secondary
importance when this non-pecuniary motive is sufficiently large. In any of these cases, theprincipal has little control over what we loosely refer to as the wage scheme, whereas the10

Our model specification is thus the “breakthrough” type in which one success can resolve all the uncertainty
regarding the agent’s type – an assumption that is predominant in the experimentation literature. See, forinstance, Keller et al. (2005) and Bonatti and Hörner (2011, 2013). The analysis remains essentially thesame even when the mediocre type can succeed with some probability, as long as the success probabilities are
sufficiently small. See section 5.3 for details.
11
In a broader sense, while we call b a bonus for expositional clarity, it is meant to capture all the benefits
which accrue from a promotion, including non-transferrable gains such as prestige, authority, and job security;
similarly, w is also meant to capture all the flow benefit from remaining in the current job.

6

option of setting a deadline is almost always viable.
Contract: The only contractible decision for the principal in this environment is whether
to set a deadline, and if so, at what point; we thus do not consider the possibility of any
screening contract (no direct mechanism).12 An incentive scheme offered by the principal isthoroughly summarized by T ∈ R+ ∪ {∅}, while the wage scheme (b, w) is taken as given.
If T ∈ R+ , the principal terminates the project at time T (but never before) if the agent
has not attained a success up to that point. If T = ∅, the principal chooses not to set any
specific deadline, in which case she can terminate the project at any instance. In either case,both parties receive zero payoff once the project is terminated.

3

Analysis

3.1

Agent’s effort decision

The agent decides whether or not to exert high effort at each instance. To analyze thisproblem, it is important to note that the principal’s belief affects the agent’s payoff only
through her termination decision. The agent’s optimal effort choice thus depends only on
the remaining time to the termination date (hereafter, simply the remaining time), i.e., themaximum length of time for which the principal continues the project without attaining a
success. The remaining time is obvious when the principal sets a deadline at T , in whichcase the remaining time at time t is simply given by T − t. Even without such an explicit
commitment, however, the remaining time can be computed from the principal’s equilibrium
strategy in essentially the same manner as we shall discuss below. For now, we proceed withthe presumption that the remaining time exists and is well-defined at any given point in time.
The agent’s problem is rather straightforward since the mediocre type, knowing that his
marginal value of effort is zero, never exerts high effort. As such, we can focus on the goodtype whom we refer to simply as the agent in what follows. Denote by U (k) the agent’s valuefunction when the remaining time is k. Taking k > 0 as given, the value function can be
written as
U (k) = max (λa b − da + w)dt + e−rdt (1 − λa dt)U (k − dt) .
a∈{l,h}Taking limit dt → 0, we obtain the Bellman equation:rU (k) = max λa b − da + w − λa U (k) − U̇ (k) ,a∈ {l,h}12(1)

As we will discuss below, we only consider deterministic deadlines, in which case there is no feasible way
to screen out the mediocre type. If we allow for stochastic deadlines, on the other hand, it may be possible
for the principal to screen out the mediocre type by a menu of contracts. In this paper, however, we do notconsider this possibility for the reasons mentioned later in footnote 13.

7

with limk→0 U (0) = 0. It is clear from this that the agent chooses high effort if and only if∆λ b − U (k) ≥ d.

As usual in this type of setup (see, e.g., Bergemann and Hege, 2005), a success today
implies the loss of future opportunities to earn rent, which inevitably diminishes the agent’s
incentive to work hard today. This rent-dissipation effect is naturally more accentuated when
the agent has more time to prove himself. As such, the agent exerts high effort only when
the project is sufficiently close to termination. This is a manifestation of the deadline effectthat lies at the core of our entire analysis.
Proposition 1 If
λh b − d + w
d>b−,
λh + r∆λ(2)

there exists k∗ such that the good type exerts high effort if the remaining time is less than orequal to k∗ and low effort otherwise. The threshold k∗ is given by(λh +r)(b− ∆d ) 
if b > ∆dλ ,− λh1+r ln 1 − λh b−d+w λ∗k =0if ∆dλ ≥ b.

(3)

If (2) does not hold, the agent always exerts high effort.
Proof: See Appendix A.

When ∆λ b ≤ d, the static incentive is too weak for the agent to exert high effort for any
remaining time. In contrast, when (2) fails to hold, the static incentive is strong enoughto overcome the rent-dissipation effect, and the agent is willing to supply effort under any
circumstance. As these cases only result in trivial solutions and are clearly of less interestfor the question we pose here, we restrict our attention to the case where the strength of thestatic incentive lies in some intermediate range by making the following assumption.
> b − ∆dλ > 0.
Assumption 1 λhλb−d+w
h +r
Among other things, the assumption implies that the optimal threshold k∗ is bounded fromabove and away from zero, i.e., k∗ ∈ (0, ∞).

8

3.2

Principal’s termination decision

Let pt denote the principal’s belief that the agent is good at time t. Conditional on no successhaving occurred by time t, the updated belief evolves according toṗt = −λat pt (1 − pt ),(4)taking the agent’s effort choice as given. It is clear that the belief is strictly decreasing overtime for any effort choice (due to the fact that λl > 0) until the agent attains a success, inwhich case the belief immediately jumps up to one.
In the current setup, because the principal’s belief is strictly decreasing over time, there
is a one-to-one relationship between the belief and the time for any given effort sequence.
Since time is only an intermediate concept that by itself does not admit any economicallymeaningful interpretation, it is often more convenient to represent strategies in terms of
the belief rather than time. The principal’s problem can thus be defined as one whereinshe chooses a cutoff belief (in the commitment case) or a set of cutoff beliefs (in the nocommitment case), at which she terminates the project. Accordingly, we represent the agent’s
strategy as A : P → {l, h} where P ⊂ [0, 1] refers to the relevant domain of the belief (to be
defined more precisely in later subsections).
Let τ (p; A) denote the time at which the belief reaches p under a given strategy A, withτ (p0 ; A) = 0 and pτ (p;A) = p for any p. The principal’s continuation payoff at time t for sometermination belief q is obtained as
Z τ (q;A)Rsv(ps ; A)e τ (pt ;A)−(λau pu +r)duds,

τ (pt ;A)subject to (4), where v(p; A) is the instantaneous payoff given byv(p; A) = λA(p) py − w.
It is immediate to see that for a given effort level, there exists a threshold belief below
which the instantaneous payoff is negative for the principal. We denote by qa := λwa y the“break-even” belief at which the principal’s instantaneous payoff equals zero under effort a(whenever it is well defined). Let qa = 1 if λwa y > 1. Combined with the fact that the
belief is strictly decreasing over time for any effort sequence, qh then represents the absolute
lowerbound of the belief, as the principal clearly has no incentive to continue the project
once her belief dips below this level. Since the belief must reach this level sooner or later(due to the fact that λl > 0 and w > 0), the presence of such a lowerbound suggests that thegame must end in some finite time. This allows us to solve the game by backward induction,
analogously to the “gap case” of the durable-good monopoly problem.
9Finally, if the value of a success is too small, the model only admits a trivial solution
where the principal chooses to stop immediately (or not to hire the agent in the first place).
In what follows, therefore, we assume that the value of a success is large enough for the
principal to hire the agent at least for some positive duration.
Assumption 2 p0 > qh ⇔ λh p0 y > w.

3.3

The commitment solution

The principal’s problem under commitment is to choose a termination belief q which maximizes her expected payoff at time 0. Throughout the analysis, we restrict attention to deterministic deadlines where the principal stops when the belief reaches q with probability one.13
As we have already seen, the high-ability agent exerts high effort only in [max{T − k∗ , 0}, T ]for any given deadline T . Let A∗q denote the agent’s best response, given the principal’s
strategy q. To represent the agent’s best response, it is convenient to define a backward
operator:φ(p) :=p.
p + (1 − p)e−λh k∗That is, φ(p) indicates the belief wherein starting with ps = φ(p), if the agent exerts higheffort for t ∈ [s, s + k∗ ], we end up with ps+k∗ = p. Given this definition, the agent’s best
response is given byA∗q (p) =(l for p0 ≥ p > φ(q),h for φ(q) ≥ p ≥ q,

if p0 > φ(q); otherwise, A∗q (p) = h for all p ∈ [q, p0 ]. In what follows, we abbreviate thesubscript q and denote the best response simply by A∗ to ease notation.
Let τ ∗ (p) := τ (p; A∗ ), σ ∗ (q) := max{τ ∗ (q) − k∗ , 0} and V ∗ (p; q) := V (p; q, A∗ ). The
principal’s continuation payoff at time 0 can then be written as
Z σ∗ (q)Rs∗(λl ps y − w)e− 0 (λl pu +r)du ds
V (p0 ; q) =0

Z τ ∗ (q)R ∗− 0σ (q) (λl pu +r)du+e

σ∗ (q)

13R− σs∗ (q) (λh pu +r)du(λh ps y − w)eds.

Our exclusive focus on deterministic deadlines is motivated by the fact that we almost never observe acontract in which the deadline is stochastic in nature. One possible reason for this is that a contract withstochastic deadlines inherently requires a credible randomization device and is thus hard to implement inpractice. For instance, consider a contract where the principal terminates the project at t1 or t2 , t1 < t2 ,with equal probability. At time t1 , however, the principal’s preference is generically strict, i.e., it is strictly
better either to stop or to continue, so that there is no incentive to randomize at that point. If it is strictly
better to stop at time t1 , for instance, the principal stops at time t1 with probability one but knowing that,the contract effectively has a deterministic deadline.

10With some computation, we obtain∗p0 e−(λl +r)σ (q) πh
p 0 πl∗∗∗(1 − e−(λl +r)σ (q) ) +(1 − e−(λh +r)(τ (q)−σ (q)) )V (p0 ; q) =λl + rλh + r(1 − p0 )w∗−(1 − e−rτ (q) ),r∗

where πa := λa y − w is the expected instantaneous payoff from the good type under effort a.
The first two terms represent the expected gain from the high-ability type whereas the lastterm represents the expected loss from the mediocre type.
The commitment solution, denoted by q C , is defined as
q C := argmaxq∈[0,p0 ] V ∗ (p0 ; q).
The continuation payoff is continuous but kinked at q = ψ0 := φ−1 (p0 ). For p0 > q ≥ ψ0 , thefirst-order condition is given by∗

p0 πh e−λh τ (q) − (1 − p0 )w = 0.

(5)

For ψ0 > q, we have
λl + r∗∗∗p0 e−(λl +r)σ (q) πl −πh (1 − e−(λh +r)k ) − (1 − p0 )we−rτ (q) = 0,
λh + rwhich can also be written as∗p0 ∆λ e−(λl +r)σ (q)∗∗∗∗p0 πh e−(λl σ (q)+λh k ) − (1 − p0 )w e−rτ (q) =w + ry + πh e−(λh +r)k .
λh + r(6)Note that the left-hand side of (6) is the expected instantaneous payoff for the principal at
the time of termination, whereas the right-hand side is the marginal cost of extending thedeadline. Since the right-hand side is strictly positive, the principal must set the deadline ata point where the expected instantaneous payoff is positive if ψ0 > q.
Proposition 2 For any given set of parameters Θ := (b, y, w, d, λl , λh , r, p0 ) satisfying Assumptions 1 and 2, there exists a unique commitment solution q C ∈ [qh , p0 ). If
λl + r∗πh (1 − e−(λh +r)k ) ≥ πl ,λh + rthe commitment solution is given by

q h
qC =
−λ k∗
 p0 e h−λ k∗1−p0 +p0 e

for φ(qh ) ≥ p0 > qh ,for 1 > p0 > φ(qh ).

h

11

(7)

If (7) fails to hold, the commitment solution is given by
for φ(qh ) ≥ p0 > qh ,q h∗p0 e−λh kCq =for Π ≥ p0 > φ(qh ),∗1−p0 +p0 e−λh kq̂
for 1 > p > Π,0where Π :=to (6).

we−rk

∗

λ +r
∗
πl − λ l +r πh (1−e−(λh +r)k )+we−rk∗h∈ (φ(qh ), 1), and q̂ ∈ (qh , 1) is the unique solution

Proof: See Appendix A.

3.4

The no-commitment solution

The situation becomes more complicated, and perhaps more intriguing, when the principal
makes no commitment at the outset. Formally, for this no-commitment case, we solve for a
perfect Bayesian equilibrium in Markov strategies (hereafter, simply an equilibrium) with thebelief as the state variable. The restriction to Markov strategies is without loss of generality inthis context because the only observable history to the principal is the sequence of outcomesthat consists only of failures at any continuation game (conditional on no breakthrough having
occurred). The principal’s termination decision can thus depend only on her belief. Given
this and the fact that effort is unobservable, the agent’s strategy must also take a Markovian
form that depends only on the principal’s belief.
We first note that the instantaneous payoff must be negative for any effort choice oncethe belief dips below qh . It is therefore a dominant strategy for the principal to stop forany p ≤ qh . This implies that the strategies beyond that point are totally irrelevant, and we
specify the strategies only over [qh , p0 ].14 The agent’s strategy is thus specified over this range
as A : [qh , p0 ] → {l, h}. The principal’s termination strategy is now specified in a different
way, as her strategy off the equilibrium path may matter. More precisely, the principal’s
strategy is given by a set of termination beliefs Q ⊂ [qh , p0 ] that may not be a singleton.
The agent’s best response is now denoted by A∗Q (or simply A∗ when it does not cause any
confusion). With slight abuse of notation, let V ∗ (p; Q) denote the continuation payoff underthe best response A∗ .
We denote the equilibrium strategy by Q∗ . One can then easily show that although Q∗
may or may not be a singleton, it is always finite.
Lemma 1 Q∗ is always finite.
14

To be more precise, the principal always terminates the project immediately for any p ≤ qh . Given this,the best response for the agent is to exert high effort for any p ≤ qh , but this has no consequence, even off theequilibrium path, because it is a dominant strategy to stop when p ≤ qh .

12

Proof:Since Q is defined on a compact set [qh , p0 ], it suffices to show that there is no continuousinterval L ⊂ [qh , p0 ] such that the principal stops for p ∈ L. To show this, suppose on thecontrary that there is such an interval. Since the remaining time is zero during this wholeinterval, we must have A∗ (p) = 1 for p ∈ L. However, this implies that the principal’s
expected payoff must be positive since p > qh . As this gives her an incentive to deviate,
there cannot be such an interval in equilibrium.
Given this result, we denote each element of Q by q i where q 1 < q 2 < · · · < q n . Accordingly, the game is divided into distinct segments P i , i = 1, 2, ..., n by termination beliefs,
where P i = [q i , q i+1 ) for i = 1, 2, ..., n − 1 and P n = [q n , p0 ]. Only the last segment P n
is actually played on the equilibrium path, as the game ends with probability one by thetime the belief reaches q n . Still, when the principal makes no commitment to a deadline, theagent’s strategy off the equilibrium path may play a role in constructing an equilibrium. For
expositional clarity, we let q N C = q n and call it the no-commitment solution.
When the principal makes no commitment, her termination (or continuation) decisionmust be sequentially rational at each instance, given the current belief. To be more precise,for some given strategy A, the principal’s continuation payoff at the time of termination mustbe zero, whereas it must be strictly positive elsewhere. Given that the agent adopts the best
response A∗Q , these restrictions amount to the following two equilibrium conditions which theprincipal’s termination strategy Q must satisfy.
Condition T: (i) q 1 = qh and (ii) if n > 1, for each i = 2, ..., n, limp→qi −0 V ∗ (p; Q) = 0.
Condition C: V ∗ (p; Q) > 0 for p ∈ [qh , p0 ] \ Q.
Condition T is the usual indifference condition that requires the principal to terminate theproject when the continuation payoff (when she deviates) is zero.15 This indifference condition
alone is in general not sufficient because the instantaneous payoff v may not be monotonicallydecreasing over time: in any segment P i , the agent may start off with low effort, during which
the instantaneous payoff could be so small that the principal is tempted to stop prematurely.
Condition C assures that the principal does not stop before the intended termination belief
q i is reached.
The next result establishes that there exists a perfect Bayesian equilibrium that is always
unique even under no commitment.
15

The right-hand limit of the continuation payoff must, by definition, be zero at the time of termination.
The principal has an incentive to terminate the project only if the left-hand limit is also zero. This condition
effectively implies that V ∗ (p; Q∗ ) is continuous for all p ∈ [qh , p0 ].

13

Proposition 3 For any given set of parameters Θ satisfying Assumptions 1 and 2, there
exists a unique equilibrium in which the principal stops at q N C ∈ [qh , min{p0 , ql }). The
principal’s equilibrium strategy is characterized by a set of n ≥ 1 distinct termination beliefs
where (i) q 1 = qh and (ii) if n > 1, q i+1 > φ(q i ) > q i for all i = 1, 2, ..., n − 1.
Proof: See Appendix A.
When λl is sufficiently high, the principal’s instantaneous payoff is positive even under
low effort. In this case, the principal is not tempted to stop prematurely, and only Condition
T is sufficient to pin down the equilibrium. Figure 1 depicts this situation: the agent starts
with low effort and switches to high effort when the remaining time is k∗ ; the principal stops
when the belief reaches the lowerbound qh .
[Figure 1 about here]
In contrast, the instantaneous payoff may become negative under low effort when λl isrelatively low, in which case we may have a situation where the principal’s continuation payoff
also becomes negative before the belief reaches qh . Figures 2 and 3 show the evolution of
the belief and the expected payoff while fixing the principal’s strategy at Q = {qh }. In the
figures, ql is so high that the instantaneous payoff is negative for the entire interval during
which the agent exerts low effort, and there exists a point q 2 such that the continuation payoff
is negative for p ∈ (q 2 , p0 ]. This implies that Q = {qh } does not constitute an equilibrium as
it violates Condition C. In this case, the game is divided into two segments, P 1 and P 2 , asillustrated in Figure 4. The pair of strategies now satisfies the equilibrium conditions since
the instantaneous payoff is always positive in P 2 .
[Figures 2-4 about here]

3.5

The cooperative solution: a benchmark

Before we move on to discuss when it is optimal to commit to a deadline, we briefly examine
as a benchmark the case where both the principal and the agent attempt to maximize the sumof their individual payoffs to derive the first-best allocation. More precisely, the instantaneous
payoff in the cooperative case, common to both players, is the difference between the expected
benefit from a success minus the effort cost, i.e., λa p(y + b) − da for a given effort a.
There is one important caveat for the analysis of the cooperative case. If the agent shared
the same objective as we assume here, his private information could in principle be induced
at no cost: given that the agent’s reservation payoff is zero, the mediocre type is indifferent
14between participating and dropping out immediately (before time 0), as he would obtain zero
payoff in either case for any given deadline. As it turns out, however, the optimal deadline isindependent of the initial belief p0 and is hence identical irrespective of whether or not the
principal can immediately screen out the mediocre type. To make the situation comparable tothe noncooperative case, we assume that for some reasons, no information can be exchanged
between the principal and the agent.16
The agent’s problem is essentially the same and only needs a slight modification. LettingŨ (k) denote the agent’s continuation payoff, the agent exerts high effort if and only if∆λ (y + b) − Ũ (k) ≥ d.

As in the noncooperative case, there exists a threshold k̃∗ such that the good type exerts
high effort if and only if the remaining time is less than or equal to k̃∗ . Under the maintained
assumptions, one can show that the agent is generally more motivated in the cooperative
case than in the noncooperative case. Formally, we make the following statement.
Proposition 4 Under Assumption 1, ifdλh (y + b) − d>y+b−,
λh + r∆λ(8)

the threshold is given bydk̃∗ = −(λh + r)(y + b − ∆λ ) 1
ln 1 −.
λh + rλh (y + b) − d(9)

If (8) does not hold, the agent always exerts high effort. Moreover, k̃∗ > k ∗ .
Proof: See Appendix A.
The principal’s problem is, on the other hand, fundamentally different because in thissetup, there is no social cost of hiring the mediocre type who simply yields zero surplus
at every instance.17 To make the argument parallel to the noncooperative case, we denotethe principal’s continuation payoff in the cooperative case by Ṽ ∗ (p; q), which is simply a16
Since the mediocre type is indifferent, there is also an equilibrium where the mediocre type chooses toparticipate. Alternatively, one can think of this as a case where the mediocre type is selfish and cares about
his own payoff, as in the noncooperative case. In either event, however, the optimal deadline is exactly thesame; see the proof of Proposition 5 for details.
17
Of course, the current setup is a special case in that the net value of the mediocre type is precisely zero.
On the other hand, if there is a fixed cost of production, the net value of the mediocre type is strictly negative.
In this case, however, the mediocre type has a strict incentive to drop out immediately. The qualitative natureof the solution with a fixed cost remains the same: the principal never terminates the project until the agent
succeeds.

15

restatement of Ũ (k) in terms of beliefs. Further, define τ̃ ∗ (p) and σ̃ ∗ (p) := max{τ̃ ∗ (p)− k̃∗ , 0}analogously. We then obtainZ σ̃∗ (q)Rsλl ps (y + b)e− 0 (λl pu +r)du dsṼ ∗ (p0 ; q) =0

Z τ̃ ∗ (q)R ∗− 0σ̃ (q) (λl pu +r)du+eσ̃∗ (q)which can be written asṼ ∗ (p0 ; q) =

 − R s (λ p +r)duds,
ps λh (y + b) − d e σ̃∗ (q) h u

p0 λl (y + b)∗
(1 − e−(λl +r)σ̃ (q) )λl + r∗p0 e−(λl +r)σ̃ (q) (λh (y + b) − d)∗∗(1 − e−(λh +r)(τ̃ (q)−σ̃ (q)) ).
+λh + rThe cooperative solution, denoted by q̃, is defined asq̃ := argmaxq∈[0,p0] Ṽ ∗ (p0 ; q).
Since there is no social cost of hiring the mediocre type, the social surplus from hiring theagent is strictly positive for any belief level, and as such, it could be optimal for the principal
to hire the agent indefinitely. This is indeed what happens in the cooperative case: theprincipal never terminates the project even though the agent would never exert high effort
for any finite k̃∗ if the game were to continue indefinitely. The reason for this is simple. The
agent exerts low effort only when the continuation payoff, which coincides with the social
surplus by definition, is larger than the static gain. On the other hand, if it is ever optimal
to set a deadline to induce high effort, the deadline must be set at time k̃∗ , at which point
the continuation payoff must be lower than the static gain. As such, when the preferences
are aligned (and there is no fixed cost of production), the first-best allocation is to continue
the project indefinitely until the agent succeeds, although the agent may only exert low effort
all the way.
Proposition 5 In the cooperative case, the principal never terminates the project, and thegame continues indefinitely. The agent always exerts low effort if (8) is satisfied and higheffort otherwise.
Proof: See Appendix A.

4Up-or-out versus standard promotion practices

4.1

The value of up-or-out

The most prominent example of evaluation schemes with deadlines is arguably what is called
the “up-or-out system” – a promotion scheme that is widely observed in academia and profes16

sional service industries such as law, accounting, and consulting. Under a typical up-or-out
contract, the employer sets a deadline by which a promotion must occur, and the worker
must leave the firm if he is not promoted by the deadline. Our simple framework offers some
practical implications for up-or-out by precisely evaluating the value of committing to suchan evaluation deadline at the outset.
To clarify what we attempt to investigate at this juncture, it is important to note thatthe principal can gain no relevant information along the equilibrium path because the only
feasible history at any continuation game is the one consisting only of failures up to thatpoint. This immediately implies that the no-commitment solution can yield no higher payoffthan the commitment solution. Still, under some conditions, the no-commitment solution
can replicate the commitment solution, in which case the no-commitment solution becomes
optimal if the principal faces a (possibly very small) commitment cost.18 The main purposeof this section is to clarify this condition, i.e., clarify when the absence of commitment entails
no additional cost, under the presumption that the commitment solution is generally morecostly to enforce than the no-commitment solution.
Given that the no-commitment solution can at best replicate the allocation of the commitment solution, one can easily see that setting a deadline strictly benefits the principal
if and only if q C 6= q N C . On one hand, it is clear that the no-commitment solution gives
the principal a different, and necessarily lower, payoff when q C 6= q N C . On the other hand,when q C = q N C , the equilibrium allocation must be exactly the same because Proposition
1 ensures the unique equilibrium given the same horizon. Using this fact, we can obtain anexact necessary and sufficient condition under which setting a rigid deadline yields a strictly
positive value for the principal.
Proposition 6 The commitment solution generically yields a higher expected payoff for the
principal than the no-commitment solution if and only if p0 > φ(qh ),19 i.e.,dλ(λh + r)(b − ∆λ )  λh h+r
1 − p01 − qh 
qh
1
−
>
⇔
.
p0 >
∗
−λ
k
h
qh + (1 − qh )eqh
λh b − d + w
p0

(10)

Proof: The sufficiency is obvious from the argument we have seen thus far: if p0 > φ(qh ),the optimal condition for the commitment solution is given by (6), which clearly differs fromthat for the no-commitment solution. To show the necessity, note that if φ(qh ) ≥ p0 , the18
An obvious cost is that the principal must give up flexibility to adjust ex post to any uncertainty that may
resolve during the course of play although this aspect is assumed away in the current setup for clarity. The
commitment cost may also arise from the cost of writing and enforcing a formal contract.
19
To be more precise, the expected payoff of the no-commitment solution is always lower if p0 > φ(qh ) ≥ ql :in this case, q C > q NC = qh always holds. If p0 , ql > φ(qh ), we may have q NC > qh , and there could be some
non-generic cases where q C = q NC and the expected payoffs are identical.

17

agent exerts high effort from the beginning and the principal stops when the belief reaches
qh , which results in the same payoff for the principal, either with or without commitment.
The proposition states that the principal benefits from committing to a deadline if theinitial prior belief p0 exceeds a certain threshold φ(qh ). The value of up-or-out depends
crucially on each player’s propensity to “procrastinate.” In short, the threshold is lower andhence favors the commitment solution for a given initial belief either when the agent delays
exerting high effort for a given deadline or when the principal delays terminating the project
for a given effort sequence. Below, we briefly summarize how each player’s propensity toprocrastinate is determined:
1. The agent’s propensity to procrastinate is determined by the tradeoff between thecurrent gain of attaining a success and the potential loss of future payoffs. For a given
deadline, the agent tends to procrastinate more (a small k∗ ) when the potential loss of
future payoffs is large relative to the current gain.
2. The principal’s propensity to procrastinate depends on the break-even belief and thelikelihood of a success under high effort. The break-even belief is evidently the major
force determining how patient the principal can be, where the principal is more tempted
to wait for an eventual success when qh is low. Further, for a given qh , the principal
tends to wait longer when λh is small, because the information about the agent’s typeis revealed only slowly in that case.
Before we move on, it is important to note that our characterization result owes largely
to the fact that we consider a binary effort choice or, more importantly, that there is anupperbound for effort. In contrast, if the effort domain is unbounded with a convex effort cost,
the commitment solution in general yields a strictly higher payoff than the no-commitment
solution.20 Even in this case, however, the nature of the problem remains largely intact.
The intuition provided above carries over to this more general case as well because the valueof up-or-out is still determined in essentially the same way, although the analysis would besomewhat more complicated.
20

On the other hand, we can obtain essentially the same result even if the effort level is continuous as longas the feasible effort level is bounded from above.

18

4.2

When is up-or-out optimal?

Up-or-out contracts are quite common in academia and professional service industries (hereafter, we refer to them inclusively as professional service industries).21 In academia, it is often
the case that incoming assistant professors must be promoted within a certain period of time
(typically around six years). Outside of academia, a leading example of industries with upor-out is the legal service industry. Many law firms traditionally adopt a set of managerial
practices, known as the Cravath system, which include up-or-out (or “partnership track”)
where incoming associates must make partner within a certain period of time (roughly sevento ten years). This contrasts sharply with other typical firm organizations which almost never
specify deadlines by which workers must be promoted.
To understand why this is the case, note that these professional service industries areindeed unique in several regards. For instance, high knowledge intensity, combined with low
capital intensity, is an obvious hallmark of these industries (von Nordenflycht, 2010).22 Another distinctive feature is a professionalized workforce that builds on a particular knowledge
base (Torres, 1991; von Nordenflycht, 2010). This in turn creates well-defined job boundaries,
from which we derive three additional implications: stable job descriptions, job homogeneity
across different ranks, and initial screening. In what follows, we interpret our main result inlight of these characteristics to provide further insight into why up-or-out is more prevalent
in professional service industries.
Ability intensity: It is often argued that knowledge intensity is perhaps the most fundamental characteristic of professional service industries. Combined with the fact that they
are also less capital intensive,23 the productivity of an organization depends crucially on theextent of knowledge embodied in individuals. A likely consequence of this fact is that innate
ability matters and creates value that cannot be easily substituted by sheer effort, at least in
the short run – a feature which we call ability intensity for expositional clarity.
Since the mediocre type can never attain a success in the current setup, one way tomeasure the extent of ability intensity is by the (good type’s) success probabilities. We saythat the project is more ability-intensive when λl and λh are larger with a fixed ∆λ . Notethat an increase in λh , while fixing ∆λ , yields three conflicting effects. First, it raises theagent’s productivity at the margin, which in turn lowers the threshold qh . Second, it gives
21

The military is another example which is characterized by up-or-out. We do not consider this case because
the underlying structure appears to be quite different.
22
The discussion in von Nordenflycht (2010) is mainly about professional service industries, but these characteristics are clearly applicable to academia as well.
23
High knowledge intensity and low capital intensity are not equivalent in the strict sense of the word
because, as argued by von Nordenflycht (2010), an industry can be both knowledge- and capital-intensive at
the same time. For the purpose of this study, however, we do not make any distinction between them.

19the agent a stronger incentive to procrastinate (a lower k∗ ) as it raises his expected future
payoff under high effort. Finally, it also facilitates the principal’s learning as each failure
reveals more information. The first two effects raises the value of up-or-out while the last
lowers it, such that the overall impact is not immediately clear.
One can readily show, however, that there exists some threshold λ̄ such that it is optimal
to set a deadline, thereby favoring the use of up-or-out, for λh > λ̄. To see this, we rewrite(10) as
qh
w
p0
>
∗ =
∗.
−λ
k
h
1 − p0(1 − qh )e(λh y − w)e−λh k(11)Under the maintained assumptions, λh > λ := max{ pw0 y , ∆dλ r(b − ∆dλ ) + d − w }. If ∆dλ r(b −wd∗∆λ ) + d − w > p0 y , we have limλh →λ k = ∞, and hence, (11) is never satisfied if λh is small.

In contrast, we haveqh
∗ = 0,
λh →∞ (1 − qh )e−λh klim

implying that (11) must hold if λh is sufficiently large.
Stable job descriptions: Due to various professional requirements and accreditation processes, workers in professional service industries are typically responsible only for a narrowand clear set of tasks, compared to workers in other industries. This implies that the natureof tasks that they are expected to carry out is very stable over time, and the production
environment is relatively immune to stochastic shocks. For instance, a demand or technology
shock that entirely changes the job description of a lawyer or a college professor is highly
unlikely.
We can interpret r as the rate at which the project is terminated for exogenous reasons,e.g., the arrival of a stochastic shock that makes the project completely worthless.24 Definer̄ such thatdλh b − d + w=b−.
λh + r̄∆λAssumption 1 then implies that r must be bounded between (0, r̄). It is clearly not profitable
to commit to a deadline if r is sufficiently close to r̄ because limr→r̄ k∗ = ∞. In contrast,
there exists some r such that it is profitable to commit to a deadline for r < r if

24

λ1 (b − ∆dλ )  1 − p0
1 − qh 1−>.
qh
λh b − d + w
p0Here, we allow the principal to terminate the project when the shock arrives; see section 5.1 for the casewhere the principal must abide by the deadline under any circumstance.

20

It can be seen from this argument that a decrease in r (a more stable relationship) in general
raises the value of up-or-out because the agent can be more forward-looking and tends toprocrastinate more, which in turn makes the principal unable to stop at the right time.
Job similarity across ranks: Another immediate consequence of stable job descriptions
is that jobs necessarily become similar across different ranks.25 In the current setup, this
aspect may be captured by y, which can be interpreted as the expected payoff of promoting
a high-ability agent. If entry- and senior-level jobs are similar, y tends to be larger because
a success at the entry level is a reliable predictor of productivity at the senior level. This is
not necessarily the case in typical firm organizations where jobs across ranks can differ to aconsiderable extent, as often discussed in the context of the Peter Principle.26
The effect of a change in y is fairly straightforward, as it only affects the break-even belief
with no impact on the agent’s behavior. A large value of y implies a small value of qh which
gives the principal a stronger incentive to wait for a breakthrough. Given this, the agent
also has a stronger incentive to procrastinate, hoping to achieve a breakthrough with low
effort. The principal can then unambiguously benefit from committing to a deadline when
y is sufficiently large, implying that the value of up-or-out is higher in industries where jobs
are similar and there is a strong correlation between performances at different ranks.
Initial screening: When the principal holds a very optimistic belief to begin with, theincentive to wait for a success is stronger, making it harder to stop at the right time. We
argue that this is generally the case in professional service industries, for reasons that pertain
to their professional nature and well-defined job boundaries. First, job candidates are all
trained specialists who have already gone through some selection processes. Second, it isrelatively easy to assess an individual’s upside potential even at the entry level because
there are readily available signals. Moreover, combined with the fact that these industries
are typically vertically differentiated, the job market is largely characterized by assortative
matching where top organizations attract top prospects: although different organizations
have different expectations for incoming workers, each hires a candidate only when there is areasonable chance for the candidate to clear the hurdle. All of these amount to the fact thatthe quality of workers is relatively high and homogenous in professional service industries,27
25
Ghosh and Waldman (2010) also raises job similarity as one of the distinctive characteristics of academia
and law. Job similarity is also crucial in Kahn and Huberman’s (1988) classic argument because if jobs
are sufficiently different, we may use promotions to solve the double moral-hazard problem as suggested byPrendergast (1993).
26
The Peter Principle states that every post tends to be occupied by an employee who is incompetent tocarry out its duties, because employees are promoted through positions where they have excelled until theyreach a level of incompetence (Peter and Hull, 1969). A premise of this argument is that jobs are inherently
different and become progressively harder as one climbs though ranks.
27
Abowd et al. (2012) show that most workers are of high ability and that technology favors positive21

which in turn raises the value of up-or-out.

4.3

Too early or too late?

In the absence of commitment, the principal often fails to stop at the right time because
she has a strong incentive to wait for a success. This reasoning seems to indicate that the
principal always terminates the project earlier with commitment than without, or equivalently, pC ≥ pN C for any given parameters Θ. As indicated in Figures 3 and 4, however,this is not necessarily true in the current setup. When the principal knows that she cannot
stop soon enough, and it is too costly for her, she may terminate the project even when the
instantaneous payoff is still strictly positive or, alternatively, when the belief is strictly higher
than the break-even point qh . This is generally the case when λl is low, in which case theprincipal must incur losses in the low-effort phase.
Proposition 7 pN C > pC for almost all p0 ∈ (φ(qh ), 1) as λl tends to zero.
Proof: It follows from Proposition 2 that if p0 > φ(qh ),∗p0 e−λh k≥ pC .
1 − p0 + p0 e−λh k∗
To prove the first part of the proposition, it thus suffices to show that∗pN C >

p0 e−λh k.
1 − p0 + p0 e−λh k∗(12)

We know from the proof of Proposition 3 that (12) holds if φ(q N C ) > p0 ≥ q N C : in thiscase, the agent starts with high effort and the principal stops when the belief reaches q N C
in equilibrium, which takes less than k∗ units of time by definition. Note that as λl → 0, we
have q N C → φ(q N C ), and thus, this holds for almost all p0 ∈ (φ(qh ), 1).
Two additional implications follow from this result. The first is concerned with the roleof λl . As in any moral-hazard environment, a low value of λl directly implies a high marginalvalue of effort, which tends to better motivate the agent given the same wage. Aside from thisconventional effect, our analysis also suggests an alternative logic by which the principal canbenefit from a low λl . Suppose that (10) holds, so that the principal has a strong incentive to
wait for a success. In fact, if λl is sufficiently large, the principal cannot help but wait until
the belief reaches qh , as there is no incentive to stop along the way. A lower value of λl canhelp in this context, because that gives the principal an incentive to stop before the belief
assortative matching in professional service industries.

22reaches qh , thereby prompting the agent to shift effort earlier. In other words, a low value of
λl may be beneficial for the principal because it works as a strong commitment device.
Second, the result also suggests that under no commitment, the average duration of the
project may not be monotonically increasing in the initial belief. When the initial belief isrelatively high, it takes long for the belief to reach the lowerbound qh , and the principal may
have to incur large losses if she is to wait until the moment at which the instantaneous payoff
equals zero. To avoid this situation and discipline the agent, the principal must act early, i.e.,
she may need to pull the trigger earlier for ex ante more promising agents in some cases.28

5Extensions

5.1

The case with uncertainty

An important cost of making commitment arises from the fact that the principal loses flexibility to adjust to future stochastic shocks. The baseline model does not capture this costbecause the principal can gain no relevant information on the equilibrium path, and there isthus no uncertainty regarding the termination date. As a consequence, it is weakly optimal
to commit to a deadline in the baseline model. This is apparently unsatisfactory, provided
that we live in a world filled with uncertainty where there is clearly value in being flexible.
While we focus on the simplest case to illuminate the key insight, it is of crucial interest
to examine whether and how our results can be extended to cases where the principal facessome uncertainty that can be resolved only gradually over time.
There are obviously many different ways to introduce uncertainty into the baseline model.
Among a plethora of possibilities, we here consider a case where a permanent productivity
shock may strike with some probability, which totally changes the nature of the task andsubsequently makes the agent unproductive, i.e., λh = λl = 0.29 For simplicity, we assume
that (i) the shock is permanent and arrives at most once, with a Poisson arrival rate β, and (ii)the arrival of the shock is publicly observable. Other cases, e.g., those with transitory shocks,
can in principle be analyzed in an analogous manner although it may be computationallymore cumbersome.
The agent: The agent’s problem requires a slight modification. The augmented Bellman
28

Formally, when the principal’s termination equilibrium strategy is characterized by multiple termination
beliefs, the average duration increases with the initial belief within each segment P i but jumps down to zero ateach pi . As is clear from this argument, the principal’s expected payoff, though continuous, is not monotonic
either when the principal’s strategy is characterized by multiple termination beliefs. The expected payoff issingle-peaked within each segment P i and equals zero at each pi .
29
Once the principal chooses to commit to a deadline, the principal must abide by it and is not allowed toterminate the project, unlike in the discussion in section 4.2.

23

equation is obtained as
rU j (k) = max (bλa − da + w) − (λa + β)U (k) + β Ũ j (k) − U̇ (k) , j = C, N C
a∈ {l,h}where Ũ j denotes the value function after the shock has arrived. Since the principal’s reaction
to the shock differs, Ũ j , and hence U j (k), depend on whether or not the principal makes
commitment. As in the baseline model, the incentive compatibility constraint (before theshock strikes) can be written as∆λ b − U j (k) ≥ d,although the value function U j (k) now differs. As such, the agent exerts high effort only when
the project is sufficiently close to termination. We let kC and kN C denote the threshold undercommitment and no commitment, respectively.
Proposition 8 For any Θ and β, kN C > k C . Both kN C and kC are strictly increasing in βdh −d+wif bλ
λh +β+r > b − ∆λ > 0.

Proof: See Appendix A.
It is interesting to note that the possibility of a negative shock generally lowers the
expected future payoff and hence induces the agent to exert high effort for longer, either withor without commitment. First, with commitment, the best the agent can do after the shock
strikes is to exert low effort and earn the flow payoff w, which is the lowest possible payoff
as long as the project survives. The situation is even worse without commitment, however,
because the principal can immediately terminate the project, thereby leaving no rent for theagent. The extent of the loss is therefore larger when the principal makes no commitment.
The agent is generally better motivated in the face of uncertainty, which in turn favors the
no-commitment solution.
The principal: We start with the case where the principal commits to a deadline at theoutset. Let V C (p0 ; q) denote the principal’s expected payoff at time 0, which can be written24as
V0C (p0 ; q) =Z σC (q)Rs(λl ps y − w)e− 0 (λl pu +β+r)du ds

0

Z τC
R C
− 0σ (q) (λl pu +β+r)duRs+e

σC (q)−βr

Z σC (q)(λh ps y − w)e− σC (λh pu +β+r)du ds
Rs

w(1 − e−r(τ −s) )e− 0 (λl pu +β+r)du ds

0

β R t̄
− e− 0 (λl pu +β+r)dur

Z τ C (q)R
− σsC (q) (λh pu +β+r)duw(1 − e−r(τ −s) )e

ds.

(13)

σC (q)

where τ C (q) is the time at which the belief reaches q under the agent’s best response, andσ C := max{τ C (q) − kC , 0}. The commitment solution is obtained asτ C = argmaxτ V0C (τ ).
As can be seen from (13), the marginal benefit of extending the deadline decreases with β,which prompts the principal to terminate earlier than in the case with no uncertainty.
In contrast, the problem is roughly the same as in the baseline model when the principal
makes no commitment because the principal only pays attention to the instantaneous payoff. As in the baseline model, the principal stops when the instantaneous payoff becomes
sufficiently low. To be more precise, define
φN C (p) :=p.
p + (1 − p)e−λh kNC

Given that no shock has arrived, if φN C (qh ) > ql , the principal stops when the belief reaches
qh . The only difference is that when the shock strikes, the instantaneous payoff falls to −w,
which prompts her to stop immediately. The principal’s expected payoff is given byVNC(p; q) =Z σNC (q)Rs(λl ps y − w)e− 0 (λl pu +β+r)du ds

0

R σ NC (q)+ e− 0(λl pu +β+r)duZ τ NCσNC (q)R− σsNC (q) (λh pu +β+r)du(λh ps y − w)ewhere τ N C and σ N C are defined analogously.

ds,

(14)

Note that for the same time horizon,

V N C (p0 ; q) > V0C (p0 ; q) for any q.
The optimal incentive scheme: In the baseline model with no uncertainty, any allocation
which can be realized without commitment can be realized with commitment as well. This
is no longer the case with uncertainty because the principal cannot foresee if and when theshock strikes, in which case she needs to immediately terminate the project.
25

Proposition 9 The principal strictly benefits from not committing to a deadline for any
β > 0 if φN C (qh ) ≥ p0 .
Proof: If φN C (qh ) ≥ p0 , the agent exerts high effort from the beginning under no commitment. The principal’s payoff is maximized, and the principal strictly benefits from making
no commitment in this case. To see this, observe that kN C > kC implies φN C (qh ) > φC (qh ).
If φN C (qh ) ≥ p0 > φC (qh ), the agent starts off with low effort, and the principal is forced to
set a deadline before the belief reaches qh . The profit is clearly lower than V0N C (τ N C ). On
the other hand, if φN C (qh ) > φC (qh ) ≥ p0 , the agent exerts high effort from the beginningeven under commitment. The profit is still lower, however, because the principal must incur
a flow loss when the shock strikes whereas she can immediately terminate the project if she
makes no commitment.
The presence of uncertainty lowers the value of up-or-out through two channels. First, by
making no commitment, the principal can retain the flexibility to adjust to the negative shockby immediately terminating the project whenever it strikes. Aside from this conventional
effect of flexibility, there also arises another force that favors the no-commitment solution:in the face of uncertainty, the agent is better motivated by the constant threat of projecttermination, and thus, he starts exerting high effort earlier. Since φN C (qh ) > φ(qh ) for any
β > 0 by Proposition 8, we could have φN C (qh ) ≥ p0 > φ(qh ), in which case the value of
up-or-out is strictly negative with uncertainty whereas it is strictly positive without.

5.2

Endogenous wages

We have thus far assumed that the principal has no control over wage payments and takes
the wage scheme (b, w) as given. While we argue that this covers a range of situations of
our interest, it is also important to understand how the wage scheme affects the equilibrium
allocation when the principal has some control over it. What is especially intriguing in thisrespect is the role of the flow wage w, as it can directly influence the principal’s termination
decision and hence can be used as a substitute for the deadline when the commitment costis prohibitively large.
Here, we briefly analyze the case where the principal can choose to pay any w ≥ 0 as
she likes. To analyze this problem, note that because the flow wage here is a pure transfer
payment with no incentive effect on the agent’s side, there is no reason for the principal tooffer any wage beyond the minimum level, which is normalized at zero, if she can stop at
the right time. However, as we have seen, this is not always true when the principal makes
no commitment to a deadline. As such, the main question we address here is whether there
26

are circumstances in which she offers any strictly positive w.30 Since only the case with nocommitment is relevant, we focus on this case by assuming that the commitment cost is solarge that it is never optimal to commit to a deadline.
In general, an increase in w above zero yields two opposing effects. On one hand, it raises
the critical value qh that may give the principal the incentive to stop earlier than otherwise.
On the other hand, it also raises the expected future payoff that in turn lowers k ∗ and gives
the agent the incentive to procrastinate even more. A necessary condition for w > 0 is thatthe first effect dominates the second, or alternatively that τ ∗ (q N C ) − k∗ decreases with w.31
Proposition 10 w∗ > 0 if λl is sufficiently small.
Proof: See Appendix A.
When w = 0, the instantaneous payoff is always strictly positive (as in the cooperativecase), no matter how unlikely the agent is to succeed. This is the most extreme case of theprincipal’s procrastination, as she can never terminate the project, and given that, the agent
never exerts high effort. The principal can alter this structure by raising w above zero because
the flow cost of employment is now positive, rendering the instantaneous payoff negative atsome point. This can be profit-enhancing for the principal because with the credible threatof termination, she can induce the agent to shift effort earlier, which is especially beneficial
when the success probability under low effort is low.

5.3

When the mediocre type too can succeed

We have thus far assumed that only the good type can succeed with positive probability,
which amounts to a breakthrough-type specification where one success can resolve all theuncertainty. As is clear from the argument, the assumption that the mediocre type can neversucceed is not crucial, as long as the model retains the breakthrough-type structure. Here,
we briefly make a remark on this assumption and show that the basic results are independentof the success probabilities of the mediocre type as long as they are sufficiently small.32
The success probabilities are now denoted by λja , where j = G, M indicates the agent’s
type (G for good and M for mediocre). Let ∆jλ := λjh − λjl . We now allow for the possibility
30

In other words, we assume that the minimum wage level is zero. On the other hand, if the principal can
set a negative w, she can immediately screen out the mediocre type by offering w < 0. This means that we
focus on the situation where the minimum wage is so high that even the mediocre type can earn a positive
rent by being employed.
31
Note that τ ∗ (q NC ) − k∗ is the interval during which the agent exerts low effort. If it is longer with asmaller k∗ , the principal’s expected payoff is necessarily lower.
32
In contrast, if the success probabilities are relatively large, the principal may not stop after one success,and the analysis would be much more complicated.

27

Mthat λMh > 0 and λl > 0, so that the mediocre type may achieve a success with positiveM
probability. We restrict attention to the case where λM
h and λl are both sufficiently small,G
M
M
such that λG
h > λl > λh > λl ≥ 0. When the mediocre type too can succeed, the principal’s

belief never hits the maximum, and hence some uncertainty always remains. We thus need
to formally address the optimal stopping problem for the principal, which we sidestepped in
the baseline model: after the agent achieves a success, the principal can either continue theproject in search of an additional success or to stop the process and “promote” the agent.33
Suppose that the principal earns payoffs of yG and −yM if she promotes the good type and the
mediocre type, respectively. The expected net payoff of promoting the agent at any instanceis now a function of the current belief pt and is given by B(pt ) = yG pt − yM (1 − pt ). Let s(p)denote the updated belief when a success occurs at belief p, which is given bys(p) =

p
λG
aG
t

λG
p + λM(1 − p)
aG
aM
t

,

t

where ajt denotes the effort choice of type j at time t.
In what follows, we solve for equilibria where the principal always promotes the agent
after one success; in other words, one success is still informative enough. Note first that theM
principal’s belief declines monotonically over time for any effort profile if λG
l > λh , whichholds by assumption. The agent’s best response thus remains the same, where there exists
some threshold k∗ such that the agent exerts high effort if and only if the remaining time isless than k∗ . With slight abuse of notation, let V ∗ (p, q) denote the principal’s value function
when the agent adopts the best response, and the principal always promotes the agent after
one success. For some given termination belief q, the principal promotes the agent after one
success if
B(s(p)) ≥ V ∗ (s(p), q).

(15)

There exists an equilibrium where the principal always promotes the agent after one success if
this condition holds for all p ∈ [q, p0 ] (on top of all the other equilibrium conditions). Clearly,
this is the case when λM
h is sufficiently small even if it is strictly bounded away from zero.

6

Conclusion

This paper presents a simple model of a tenure-clock problem with particular emphasis on therole of commitment to a deadline. We consider an environment where the principal chooses
33
In principle, the principal can promote the agent at any instance. We only consider the moment at which
a success occurs because it is never optimal to do so at any other instance (except at time 0).

28whether to commit to a deadline and if so, at what time. Within this framework, we establish
the generic uniqueness of the equilibrium and obtain a necessary and sufficient condition forcommitting to a deadline to be optimal. The simple framework also allows us to conductvarious comparative statics exercises to give insight into why up-or-out contracts are moreprevalent in some industries than others.
As a final note, since our model is highly stylized, there are several avenues to extend
the current analysis. Of particular interest is the case with symmetric information where theagent himself does not know his own type and needs to uncover it through experiments. In
this symmetric-information case, there is an additional effect that is absent in our asymmetricinformation setup: having observed no success, the agent gradually loses confidence in his own
ability and is eventually discouraged to work hard as time progresses. This seems to be an
important aspect of many professional industries and is worth a comprehensive investigation.
We leave this for the future.

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Appendix A: Proofs
Proof of Proposition 1: It is intuitively clear that the value function is strictly increasing
in k. Formally, defineUa (k) = (λa b − d + w) − λa Ua (k) − U̇a (k)(16)

as the value function when the effort level is fixed at a = l, h. Solving this differential equationand imposing the boundary condition U (0) = 0, we derive
Uh (k) =λh b − d + w(1 − e−(λh +r)k ).
λh + rIf ∆λ b > d, it is optimal for the agent to exert high effort as long as Uh (k) is sufficiently
small. Then, k∗ is obtained as a solution to Uh (k∗ ) = b − ∆dλ , which can be written as
λh b − d + w∗
d(1 − e−(λh +r)k ) = b −.
λh + r∆λNote that the agent always exerts high effort ifb−

λh b − d + w
d≥,∆λ
λh + r31(17)

while he never does so if ∆dλ ≥ b. For in-between cases, solving (17) yieldsk∗ = −

(λh + r)(b − ∆dλ ) 1,
ln 1 −λh + rλh b − d + w

so that the agent exerts high effort when the remaining time is less than or equal to k ∗ . For
k > k∗ , the agent exerts low effort because U (k) ≥ Uh (k) > b − ∆dλ .
Proof of Proposition 2: First, if φ(qh ) ≥ p0 > qh (the latter inequality holds by Assumption2), there exists a unique deadline T ∈ (0, k∗ ] that satisfies
p0 e−λh T= qh .
1 − p0 + p0 e−λh T
There is clearly no reason to wait beyond this point because the instantaneous payoff isstrictly negative for any effort choice. It is also not optimal to stop before qh because theinstantaneous payoff is still strictly positive. The optimal termination belief is then qh andit is unique in this range.
Second, suppose that p0 > φ(qh ). In this case, it is optimal to set a deadline at k∗ if∗limq→ψ0 −0 ∂V∂q ≤ 0. The condition for this can be written as
λl + r∗∗πh (1 − e−(λh +r)k ) ≤ (1 − p0 )we−rk ,p 0 πl −λh + rwhich can also be written as
λl + r∗∗∗we−rk ≥ p0 πl −πh (1 − e−(λh +r)k ) + we−rk .
λh + rThis condition holds for any p0 if

λl + r∗πh (1 − e−(λh +r)k ) ≥ πl .
λh + r(18)

If this holds, it is optimal to set the deadline at T = k ∗ . The optimal termination belief isgiven by∗qC =p0 e−λh k,
1 − p0 + p0 e−λh k∗which is again unique in this range.
Finally, if (18) fails to hold, there exists some Π ∈ (φ(qh ), 1) such that∗
λl + r∗∗we−rk = Π πl −πh (1 − e−(λh +r)k ) + we−rk .
λh + r

For p0 > Π, we can find a unique q̂ that solves∗∗
λl + r∗πh (1 − e−(λh +r)k ) = (1 − p0 )we−rτ (q̂) .
p0 e−(λl +r)σ (q̂) πl −λh + r32

This is the unique termination belief when p0 > Π.
Proof of Proposition 3: We first establish the following two facts that will be useful forthe subsequent argument.
Lemma 2 In any equilibrium, qh ∈ Q∗ and A∗ (p) = h for φ(qh ) ≥ p ≥ qh .
Proof: Suppose otherwise, i.e., that the agent chooses low effort for some interval [t + dt)
when pt ≤ φ(qh ). To satisfy the incentive condition, the agent must exert high effort at least
for [t + dt, t + k∗ + dt), but then the belief necessarily falls below qh at time t + k∗ + dt,
at which point the principal’s instantaneous payoff is negative for any effort choice. This istherefore a contradiction.
The lemma indicates that the agent always exerts high effort once the belief dips below
φ(qh ). Note that this result holds both on and off the equilibrium path. Of course, thisdoes not rule out the possibility that the agent exerts high effort for p > φ(q1 ); as we willsee below, there exists an equilibrium in which the agent exerts high effort before the belief
reaches φ(qh ).
Lemma 3 For each q i ∈ Q∗ , q i ∈ {qh } ∪ (φ(qh ), ql ].
Proof: We have already argued that it is a dominant strategy to terminate the project forany pt < qh because the instantaneous payoff is negative for any effort choice. Likewise, it isa dominated strategy to stop at any pt > ql because the instantaneous payoff is positive forany effort choice, which shows that q i ∈ [qh , ql ]. Note also that the instantaneous payoff under high effort is strictly positive for all p ∈ (qh , φ(qh )), such that Condition T is not satisfied.
Given these results, we show the existence and uniqueness of the equilibrium by construction. There are three cases we need to consider, depending on the initial belief p0 .
Case 1: φ(qh ) ≥ p0
Lemmas 2 and 3 directly imply Q∗ = {qh } in equilibrium. In the unique equilibrium, theagent always exerts high effort on the equilibrium path and the principal stops at pt = qh .
The no-commitment solution coincides with the commitment solution and is characterizedby q N C = qh .
Case 2: p0 > φ(qh ) ≥ ql

33

This is the case where the equilibrium may involve a phase during which the agent exerts
low effort on the equilibrium path. When φ(qh ) ≥ ql , however, we can show the existence
and uniqueness of the equilibrium in essentially the same manner as in Case 1.
An obvious candidate strategy in this case is that the agent exerts low effort until the
belief reaches φ(qh ) and then switches to high effort from that point on. To check whether
this strategy constitutes an equilibrium, note that once the belief reaches φ(qh ), the uniquecontinuation equilibrium is that the agent exerts high effort and the principal stops when
the belief reaches qh , as shown by Lemmas 2 and 3. Moving backwards, it is optimal for the
principal to exert low effort before the belief reaches φ(qh ). It follows from Lemma 3 that the
principal never stops before the belief reaches φ(qh ) because φ(qh ) > ql . The no-commitment
solution is then given by q N C = qh as above. It is also clear from this argument that theconstructed equilibrium must be unique: on one hand, there is only one equilibrium once thebelief reaches φ(qh ); on the other hand, it is never optimal to stop before the belief reaches
φ(qh ) because φ(qh ) > ql .
Case 3: p0 > φ(qh ) and ql > φ(qh )This is the case where if the agent starts out with low effort, the belief reaches ql before
switching to high effort. This can entirely change the structure of the equilibrium as theprincipal may be tempted to stop when the belief reaches ql ; in other words, Condition C
may now bind.
We show the existence and uniqueness of the equilibrium by checking Condition C sequentially until it is satisfied. We start with Q1 = {qh } and the corresponding best response
A∗Q1 such thatA∗Q1 (p) =(lhfor p0 ≥ p > φ(q),for φ(q) ≥ p ≥ qh .

(19)This pair of strategies constitute an equilibrium if V ∗ (p; Q1 ) > 0 for all p ∈ (qh , p0 ]. It is
obvious that this holds for p ∈ (qh , φ(qh )] during which the agent exerts high effort. It thus
suffices to show that V ∗ (p; Q1 ) > 0 for all p ∈ (φ(qh ), p0 ]. The following fact is convenient tothis end.
Lemma 4 Fix the principal’s strategy Q1 = {qh } and the agent’s best response A∗Q1 . Then,V ∗ (p; Q1 ) > 0 for all p ∈ (φ(qh ), p0 ] if and only if V ∗ (min{p0 , ql }; Q1 ) > 0.
Proof: Since the necessity is evident by definition, we only prove the sufficiency part. Suppose first that p0 ≥ ql . In this case, V ∗ (ql ; Q1 ) > 0 implies V ∗ (p; Q1 ) > 0 for all p > ql
because the instantaneous payoff is strictly positive for any effort choice when the belief34

is above ql . Given this, it suffices to show that V ∗ (p; Q1 ) > 0 for all p ∈ (φ(qh ), ql ) if
V ∗ (ql ; Q1 ) > 0. To show this, suppose on the contrary that V ∗ (ql ; Q1 ) > 0 but V ∗ (p; Q1 ) ≤ 0
for some p ∈ (φ(qh ), ql ). If this is the case, there must exist some p̂ ∈ (φ(qh ), ql ) such thatV ∗ (p̂; Q1 ) = 0. Since
V ∗ (p; Q1 ) = (1 − p)w

∗ ′
∗
∗ ′
∗
p(λl y − w)
1 − e−(λl +r)(τ (p )−τ (p)) −1 − e−r(τ (p )−τ (p))λl + rr−λl (τ ∗ (p′ )−τ ∗ (p)) −r(τ ∗ (p′ )−τ ∗ (p)) ∗ ′+ (1 − p + pe)e
V (p ; Q1 ),for any p > p′ > φ(qh ), due to the recursive structure of the expected payoff, we have
 (1 − ql )w
ql (λl y − w)∗∗∗∗
1 − e−(λl +r)(τ (p̂)−τ (ql )) −1 − e−r(τ (p̂)−τ (ql ))λl + rr−λl (τ ∗ (p̂)−τ ∗ (ql )) −r(τ ∗ (p̂)−τ ∗ (ql )) ∗+ (1 − ql + ql e)e
V (p̂; Q1 )
 (1 − ql )w

∗
∗
∗
∗
ql (λl y − w)
1 − e−(λl +r)(τ (p̂)−τ (ql )) −1 − e−r(τ (p̂)−τ (ql )) ,=λl + rr

V ∗ (ql ; Q1 ) =which is strictly negative by definition, and therefore a contradiction.

Now suppose that ql > p0 . As above, it suffices to show that V ∗ (p; Q1 ) > 0 for allp ∈ (φ(qh ), p0 ) if V ∗ (p0 ; Q1 ) > 0. We can apply the same reasoning to show that this holds.
The lemma shows that we only need to check the continuation payoff at x := min{p0 , ql }to see whether a given strategy is sequentially rational. We start with the candidate strategy
considered above, in which the agent exerts low effort until the belief reaches φ(qh ) andswitches to high effort from that point on (until the belief reaches qh ). The principal has no
incentive to deviate and stop if the expected payoff at x is positive, i.e.,Z σ∗ (qh )R(λ p +r)du− sds(λl ps y − w)e τ ∗ (x) l u
V ∗ (x; Q1 ) =τ ∗ (x)Z τ ∗ (qh )R σ ∗ (q )− τ ∗ (x)h (λl pu +r)du+e

σ∗ (qh )−(λh ps y − w)eRs

σ ∗ (qh )(λh pu +r)duds > 0.

Letting ∆τ = τ ∗ (φ(qh )) − τ ∗ (x), the condition can now be written as r(λh y − w) −(λ +r)∆τ∗r(λl y − w)e l
1 − e−(λh +r)k1 − e−(λl +r)∆τ +λl + rλh + r∗(1 − r)w
1 − e−r(∆τ +k ) .
>
r

(20)

If (20) holds, we can apply the same argument as in Case 2 to show the existence anduniqueness of the equilibrium.
The situation becomes more complicated when (20) fails to hold, in which case the principal has an incentive to deviate and stop at some point, and the candidate strategy no longer
35

constitutes an equilibrium. In this case, there instead exists a critical belief that satisfies(20) with equality, because V ∗ (p; Q1 ) is continuous in p with V ∗ (φ(qh ); Q1 ) > 0 > V ∗ (r; Q1 ).
Define q 2 such that q 2 := min{q ∈ (φ(qh ), x] : V ∗ (q; Q1 ) = 0} under the candidate strategy
A∗Q1 (p) in (19). Given the critical belief, we can then consider Q2 = {q 1 , q 2 }, where q 1 = qh ,and the best response


0



1
A∗Q2 (p) =
0



1

for p0 ≥ p > min{p0 , φ(q 2 )},for min{p0 , φ(q 2 )} ≥ p ≥ q2 ,for q 2 > p > φ(qh ),for φ(qh ) > p ≥ q1 = qh .

Note that if q 2 exists, q 2 > φ(q 1 ) > q 1 = qh .
The game is now divided into two segments (P 1 , P 2 ). First, we can show that theequilibrium is unique in P 1 (as done in Case 2). For P 2 , we can apply the same procedure
to the truncated interval [q 2 , p0 ] to see whether Q2 can be an equilibrium strategy. Moreprecisely, we need to check if V ∗ (p; Q2 ) > 0 for all p ∈ [p0 , φ(q 2 )). If this condition holds,
Q∗ = Q2 . If not, we can find yet another critical belief q 3 ∈ (φ(q 2 ), x] and repeat the sameprocess until we find a strategy that can satisfy the equilibrium conditions. The existence ofan equilibrium is assured because q i+1 > φ(q i ) > q i for all i = 1, 2, ..., n − 1, so that we arebound to have q i at some point that satisfies either (i) φ(q i ) ≥ p0 ≥ q i or (ii) p0 > φ(q i ) ≥ ql .
The uniqueness is also evident in this setup: (i) the equilibrium is unique in P 1 ; (ii) moving
backwards, we can also show that the equilibrium in each P i is unique for i = 2, 3, ..., n.
Proof of Proposition 4: The agent always exerts high effort in the noncooperative case if(2) fails to hold. An analogous condition for the cooperative case is given by
λh (y + b) − d
d
>y+b−
.
λh + r∆λApplying the same argument as in Proposition 1, we obtaindk̃∗ = −(λh + r)(y + b − ∆λ ) 1
ln 1 −.
λh + rλh (y + b) − dObserve that k̃∗ > 0 under Assumption 1. It follows from this that k̃∗ > k ∗ if
(λh + r)(y + b − ∆dλ ) (λh + r)(b − ∆dλ ) 11
ln 1 −
ln 1 −>−,−λh + rλh (y + b) − d
λh + rλh b − d + w

which is reduced toy + b − ∆dλλh (y + b) − d>36

b − Rdλλh b − d + w.

With some algebra, we obtain
λl dy+ w y + b − ∆dλ > 0,∆λwhich always holds.

Proof of Proposition 5: Given some threshold k̃∗ , the only possibility is to set a deadline
at time k̃∗ , so that the agent exerts high effort from the beginning. This is because any otherfinite deadline yields a strictly lower payoff. Let Ũ∞ := Ṽ ∗ (p; 0) denote the continuation
payoff when the project continues indefinitely. We show that we cannot have Ũ (k) > Ũ∞ for
any finite k. Since, by definition,λh (y + b) − dd∗∗= Ũ (k̃ ) =(1 − e−(λh +r)k̃ ).
y+b−∆λλh + rit is optimal to set a deadline at time k̃∗ if and only ify+b−d> Ũ∞ .
∆λThis is the case, however, where (8) fails to hold, and the agent always exerts high effort,
i.e., k̃∗ → ∞. On the other hand, if (8) holds, it is strictly better to let the project continue
indefinitely as Ũ∞ ≥ U (k̃∗ ). This shows that no finite deadline can improve the social
surplus in the cooperative case.
Proof of Proposition 8: We first consider the case where the principal commits to adeadline. Note that the principal cannot terminate the project even if the shock arrives before
the deadline. Since the agent has no incentive to exert high effort, we have Ũ C (k) = w(1−e
r

−rk )

.

Let UaC (k) denote the value function under commitment when the effort level is fixed at a.
We then obtain
UhC (k) =bλh − d + w
βw  1 − e−(λh +β+r)ke−rk (1 − e−(λh +β)k ) .
(1 − e−(λh +β+r)k ) +−λh + β + rr
λh + β + r
λh + βThe threshold under commitment is a solution to UhC (kC ) = b − ∆dλ .
The problem is much more straightforward when the principal makes no commitment. In
this case, the optimal choice for the principal is to terminate the project as soon as the shock
arrives, which implies Ũ N C (k) = 0 for all k. As above, define UaN C (k) as the value function
under no commitment with fixed effort a, which is given by
UhN C (k) =bλh − d + w(1 − e−(λh +β+r)k ).
λh + β + r
37d
N C given by
h −d+wIf bλ
λh +β+r > b − ∆λ > 0, there exists an interior threshold k

kN C = −(λh + β + r)(b − ∆dλ ) 1.
ln 1 −λh + β + r
bλh − d + w

It is evident that kN C > kC because the expected payoff must be higher under commitment for a given horizon. Formally, we need to show thate−rk (1 − e−(λh +β)k )1 − e−(λh +β+r)k>,λh + β + r
λh + βwhich can also be written as
Z k−(λh +β+r)s

e

ds >

Z k

e−(λh +β)s−rk ds.

00This condition holds becausee−(λh +β+r)s > e−(λh +β)s−rk ⇔ e−rs > e−rk ,for any s < k.
dh −d+w
Finally, note that kC , kN C ∈ (0, ∞) when bλλh +β+r > b − ∆λ > 0. We can then showthat kj , j = C, N C, is increasing in β if Uaj (k) is decreasing in β for k ≤ kj . Straightforward
computation shows that UhN C (k) is strictly decreasing in β. In contrast, it is a little more
involved to show the same for UhC (k). To see this, with some computation, we obtain
λh w  1 − e−(λh +β+r)ke−rk (1 − e−(λh +β)k ) bλh − d(1 − e−(λh +β+r)k ) −−λh + β + rr
λh + β + r
λh + β−rkw(1 − e )+r−rk )bλh − d − λh w(1−eλh we−rk  1 − e−(λh +β)k1 − e−(λh +β+r)k r=(1 − e−(λh +β+r)k ) +−λh + β + rr
λh + βλh + β + rw(1 − e−rk ).
+rUhC (k) =−αkFirst, it is easy to verify that 1−eαis strictly decreasing for any α > 0. Further, observethat the second term is strictly decreasing in β. These two facts imply that it suffices to showthatbλh − d > λhw(1 − e−rk ),rλhfor k ≤ kC . To show this, since bλh − d > bλh − ∆d ≥ λh UhC (k) for k ≤ kC , we haveλλh (UhC (k) − w(1−erUhC (k) >λh + β + rw(1 − e−rk )
+

r

−rk )

)

(1 − e−(λh +β+r)k ) +,38

λh we−rk  1 − e−(λh +β)k1 − e−(λh +β+r)k −r
λh + βλh + β + rfor k ≤ kC . This is further reduced tow(1 − e−rk )  λh we−rk  1 − e−(λh +β)k1 − e−(λh +β+r)k 
λh (1 − e−(λh +β+r)k )  C
Uh (k) −>,−1−
λh + β + rrr
λh + βλh + β + rwhich gives UhC (k) > w(1−e
r

−rk )

.

Proof of Proposition 10: As w → 0, we have qh → 0, which implies τ ∗ (q N C ) → ∞. The
expected payoff converges tolim V ∗ (p0 , q N C ) =w→0

p0 λl y,λl + r(21)

for any p0 . Letdµ(w) :=

λ(λh + r)(b − ∆λ )  λh h+r
λh y − w 1−.
w
λh b − d + w̃Since limw→0 µ(w) = ∞ and µ(λh y) = 0, there must exist at least one w̃ such thatµ(w̃) =

1 − p0.
p0

If we set w = w̃, then τ ∗ (q N C ) = k∗ > 0 by definition, and the expected payoff is
Z k∗
Rs
NC
V (p0 , q ) =(λh yps y − w)e− 0 (λh pu +r)du ds,0which is always strictly positive and larger than (21) if λl is sufficiently small.

Appendix B: A discrete-time model
Our continuous-time setup is particularly convenient as it allows us to obtain a completecharacterization of equilibrium in a very tractable manner. The nature of the equilibrium
remains the same, however, even when actions can be taken only at discrete points in time. In
this appendix, we provide a discrete counterpart of our model to show how we can construct
an equilibrium and also that the constructed equilibrium converges to the continuous-timeequilibrium as the time interval vanishes to zero. Since the commitment case is essentiallythe same, we only consider the no-commitment case in this appendix.
Setup: We consider the same setting as in the continuous-time case, except that actions
can now be taken only at a fixed interval ∆ > 0. We refer to each interval as a “period,”each denoted by an integer t = 1, 2, 3, .... The timing of events within each period goes asfollows. At the beginning of each period, the principal first chooses whether to continue the39

project. If the principal chooses to continue, the agent then chooses the effort level, and theoutcome is realized. The game ends if the agent attains a success. If not, the game proceeds
to the next period. The common discount factor is replaced by δ := e−r∆ . Further, define−λa ∆θa := 1−e∆as the success probability (per unit time) for a given effort level a = l, h.

Agent’s effort decision: Let K denote the number of remaining periods, which is ananalogous concept to the remaining time in the continuous-time case: at the beginning of
period t, if the principal is to terminate the project at the beginning of period T , K = T − t.
Let U (K) denote the agent’s value function when there are K remaining periods. Similarly
to the continuous-time case, we show that the good type exerts high effort only in the lastfew periods.
Proposition B.1 There exists K ∗ such that the good type exerts high effort if and only if
the number of remaining periods is fewer than or equal to K ∗ .
Proof: We first show that U (K) is strictly increasing in K. Note that
U (1) = max{θh b − d + w, θl b + w}∆ > U (0) = 0.
It follows from this that
U (2) = max{(θh b − d + w)∆ + (1 − θh ∆)δU (1), (θl b + w)∆ + (1 − θl ∆)δU (1)}
> U (1) = max{(θh b − d + w)∆, (θl b + w)∆}.
Applying the same argument backward, we obtain U (K) > U (K − 1).
Given that there are K remaining periods, the good type exerts high effort if(θh b − d + w)∆ + (1 − θh ∆)δU (K − 1) ≥ (θl b + w)∆ + (1 − θl ∆)δU (K − 1).
This holds when U (K) is small enough. Since U (K) increases with K, there exists K ∗ suchthat the good type exerts high effort if and only if the number of remaining periods is fewer
than or equal to K ∗ .34
Principal’s termination decision: Since the belief makes discrete jumps on the equilibrium path, it is more convenient to define the principal’s continuation payoff as a function of
the number of remaining periods K rather than of the termination beliefs as we did in thecontinuous-time case. The principal’s strategy is represented by a set of termination dates
34

When ∆ → 0, K ∗ → ∞, but K ∗ ∆ converges to k∗ .

40

S := {T i }ni=1 . Let Vt (p; S, A) denote the principal’s continuation payoff at the beginning of
period t under strategies A and S. Straightforward computation yields
Vt (pt ; S, A) =

t+K
t −1
X

δs−t

s=t

s−t−1
Y
u=0


(1 − pt+u θat+u ∆) (ps θas b − das + w)∆,where Kt is the number of remaining periods in period t, subject topt+1 = Dat (pt ) :=pt e−λat ∆pt (1 − θat ∆)
=
.
−λ
∆
a
t
1 − pt θat ∆1 − pt + pt e

We can characterize the principal’s termination strategy in essentially the same mannerwith some technical modifications. One difference is that since the belief now makes discrete
jumps, the termination belief in the discrete-time case is given by a set of continuous intervals
rather than of discrete points. As in the continuous-time case, it is a strictly dominatedstrategy to continue the project if pt ≤ qh . More precisely, the game must end if pt ∈ Θh :=(Dh (qh ), qh ], which is the lowerbound of the belief for the discrete-time case. Notice that thelowerbound degenerates to a single point qh , the lowerbound for the continuous-time case, as∆ → 0.
As in the continuous-time model, the principal may also need to stop before the belief
reaches Θh . The principal terminates the project at the beginning of period T i if the continuation payoff is weakly negative in period T i but is positive from the next period on. Moreprecisely, the principal terminates the project if
VT i (pT i ; S, A) ≤ 0 and Vt (pt ; S, A) > 0,
for all t = T i + 1, ..., T i+1 − 1.
Equilibrium construction: The fact that the principal’s termination strategy is characterized by intervals gives rise to a technical issue that would not arise in the continuous-timecase. To see this, suppose that there exists some T such that pT ∈ Θh , so that the principal
terminates the project at the beginning of period T . The agent then starts exerting higheffort from period T − K ∗ , and the principal has no incentive to terminate the project until
period T . Before period T − K ∗ , however, there is an interval during which the agent exerts
low effort and the principal’s continuation payoff may become negative at some point. Letting
K := T − t denote the number of remaining periods, consider the agent’s strategy Â given by(
h if K ∗ ≥ K,
at = Â(K) =l if K > K ∗ ,and define a backward operator BK (p), where pT −K = BK (p) for some p ∈ Θh under the
strategy Â. Let V̂ (p; K) := VT −K (p; S, Â).
41

To construct an equilibrium, it is convenient to define the following notations. First,
define νK := (BK (Dh (qh )), BK (qh )]: if a continuation game starts from some p ∈ νK , thebelief reaches the lowerbound Θh after K periods under Â. Note that νK ∩ νK−1 is generally
nonempty, and when the belief falls in this overlapped range, there are (at least) two ways toreach the lowerbound Θh . Second, for a given p ∈ Θh , we may find some κ(p) ∈ [K ∗ +1, T −1]such thatV̂ (Bκ(p) (p); κ(p)) ≤ 0 and V̂ (Dl (Bκ(p) (p)); κ(p) − 1) > 0.
If there is no such κ(p), there is no incentive for the principal to terminate the project until
period T and we let κ(p) = T in this case. Moreover, define p̄ := max{p ∈ Θh : κ(p) < T },κ̄ := κ(p̄), and κ := κ(Dh (qh )). Note that κ ≥ K ∗ + 1 because the continuation payoff must
be positive if the number of remaining periods is less than or equal to K ∗ .
Let p̄K := max{p ∈ Θh : κ(p) = K} for K ∈ [κ, κ̄] and p̄κ−1 = Dh (qh ). We constructa continuation equilibrium for the interval νκ̄ in which the principal may need to terminate
the project before period T .35 To this end, note first that if qh > p̄, there exists a nonempty
interval (Bκ̄ (p̄), Bκ̄ (qh )] in which the principal’s continuation payoff is invariably positive
until period T under Â: in this case, there exists a well-behaved continuation equilibrium inwhich the principal never terminates the project until the belief reaches Θh . It thus suffices
to consider a (possibly truncated) interval ν̄κ̄ := (Bκ̄ (Dh (qh )), Bκ̄ (p̄)].
For p ∈ ν̄κ̄ , the principal needs to terminate the project at some point. More precisely,
we go through the following procedure, starting from step 0, until we reach a point where theprincipal has an incentive to stop.
Step jConsider a continuation game with κ̄ − j remaining periods. Any p ∈ νκ̄−j is classified into
one of the following four cases.
Case 1: p ∈ (Bκ̄−j (p̄κ̄−1−j ), Bκ̄−j (p̄κ̄−1 )]There always exists a nonempty interval (Bκ̄−j (p̄κ−1−j ), Bκ̄−j (p̄κ̄−j )] in which the continuation payoff is weakly negative but is positive from the next period. When the current belief
falls in this interval, there exists a well-behaved continuation equilibrium in which the principal continues the project until period T and the players play for κ̄ periods. The principal
thus has an incentive to stop when the belief is in this interval.
Case 2: p ∈ (Bκ̄−1−j (qh ), Bκ̄−j (p̄κ−1−j )]35
Note that each νK is by definition sufficiently long, so that any belief path starting from above must
eventually fall in this interval under an arbitrary strategy.

42

For p ∈ (Bκ̄−1−j (qh ), Bκ̄−j (p̄κ̄−1−j )], which may or may not be empty, the continuation payoffin the next period is also negative under Â. The principal must then continue the project forone more period, and the agent exerts high effort. Since Dh (p) ∈ νκ̄−1−j by definition, we
proceed to step j + 1 with the updated belief Dh (p).
Case 3: p ∈ (max{Bκ̄−1−j (p̄κ−1−j ), Bκ̄−j (Dh (qh ))}, min{Bκ̄−1−j (qh ), Bκ̄−j (p̄κ−1−j )}]Note that p ∈ νκ̄−j ∩ νκ̄−1−j in this case, meaning that there are (at least) two ways to reachΘh . The problem is that neither constitutes an equilibrium by itself: if the players play forκ̄ − j periods,V̂ (p; κ̄ − j) < 0 and V̂ (Dl (p); κ̄ − 1 − j) ≤ 0,
whereas if they play for κ̄ − 1 − j periods,V̂ (p; κ̄ − 1 − j) > 0 and V̂ (Dl (p; κ̄ − 2 − j) > 0.
Our equilibrium construction in this interval exploits the idea of public randomization, muchin the spirit of repeated games. SinceV̂ (Dl (p); κ̄ − 2 − j) > V̂ (p; κ̄ − 1 − j) > 0 > V (Dl (p); κ̄ − 1 − j) > V̂ (p; κ̄ − j),there exists a range of γ such thatγ V̂ (p; κ̄ − j) + (1 − γ)V̂ (Dl (p); κ̄ − 1 − j) ≤ 0,γ V̂ (p; κ̄ − 1 − j) + (1 − γ)V (Dl (p); κ̄ − 2 − j) > 0.
That is, possibly with the aid of some public randomization device, we can construct a wellbehaved continuation equilibrium where the players play for κ̄ − j periods with probability
γ and for κ̄ − 1 − j periods with probability 1 − γ. Given this, the principal has an incentive
to stop when the belief falls in this interval.
Case 4: p ∈ (Bκ̄−j (Dh (qh )), Bκ̄−1−j (p̄κ−1−j )]This is an interval, if any, where V̂ (p; κ̄ − 1 − j) ≤ 0, meaning that κ(p) < κ̄ − j. We directly
proceed to step j + 1 with the current belief.
In cases 2 and 4, the process goes to the next step. This process must stop eventually as
it cannot go beyond K ∗ + 1 > κ̄ − j. To see this, suppose that there exists a well-defined
κ = K ∗ + 1 and the belief falls in (BK ∗ +1 (Dh (qh )), BK ∗ +1 (p̄K ∗ +1 )] in step κ̄ − K ∗ − 1. Sincep̄K ∗ = Dh (qh ), however, this must correspond to Case 1.
43

Once we identify when to stop, we can again go back in time from that point and see if
the continuation payoff stays positive on the equilibrium path. If there is a point at which thecontinuation payoff becomes negative, we go through the same procedure again to constructa continuation equilibrium.
Finally, it is clear that as ∆ → 0, Θh degenerates to qh . As such, all the terminationintervals eventually converge to q 2 , q 3 ,..., q n .

44

pt

ql

q 1 = qhk∗tFigure 1: Evolution of the belief on the equilibrium pathptV ∗ (p, qh ) < 0
qlq2

q 1 = qhk∗tFigure 2: A violation of Condition C: the belief

45

V∗

V ∗ (p, qh ) < 00k∗t
Figure 3: A violation of Condition C: the continuation payoff
pt
P2

P1

ql
q2 = qN C

q 1 = qhk∗k∗tFigure 4: Evolution of the belief, on and off the equilibrium path46