Discussion Paper No. 1015

DYNAMIC PERFORMANCE EVALUATIONWITH DEADLINES:
THE ROLE OF COMMITMENT
Chia-Hui Chen
Junichiro Ishida

October 2017

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, JapanDynamic Performance Evaluation with Deadlines:
The Role of Commitment∗
Chia-Hui Chen† and Junichiro Ishida‡
October 20, 2017

Abstract
We consider an environment in which a principal hires an agent and evaluates his productivity over time in an ongoing relationship. The problem is embedded in a continuoustime model with both hidden action and hidden information, where the principal must
induce the agent to exert effort to facilitate her learning process. The value of committing to a deadline is examined in this environment, and factors which make the deadline
more profitable are identified. Our framework generates a unique recursive equilibrium
structure under no commitment which can be exploited to obtain a full characterizationof equilibrium. The analysis allows us to evaluate the exact value of commitment for any
given set of parameters and provides insight into when it is beneficial to commit to an
evaluation deadline at the outset of a relationship.

JEL Classification Number: D82, D83
Keywords: dynamic agency, deadlines, experimentation, commitment, up-or-out contracts.
∗

The first author acknowledges financial support from JSPS KAKENHI Grant-in-Aid for Scientific Research(B) 16H03615 and Young Scientists (B) 17K13699. The second author acknowledges financial support from
JSPS KAKENHI Grant-in-Aid for Scientific Research (S) 15H05728, (A) 20245031, (B) 16H03615 and (C)24530196 as well as the program of the Joint Usage/Research Center for Behavioral Economics at ISER, Osaka
University.
†
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 heis sufficiently productive. As is often the case, however, the worker’s sheer productivity is
not directly observable to the employer, who must instead make an inference from a sequenceof observed outputs. Since success cannot be achieved overnight, the employer must exercise
some patience even when things do not appear to be in good shape. Excessive tolerance for
failure, however, diminishes the worker’s incentive to take costly actions that are indispensable
to be successful. A number of issues arise in this dynamic environment. How much timeshould 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 analyzing a situation where a principal hires an agent and evaluates his productivity over time in an ongoing relationship. The
problem is embedded in a continuous-time framework with both hidden action and hidden
information. At each instant, the agent privately chooses how much effort to supply. The outcome is either a success or a failure, depending on his effort choice as well as his productivity
type. The game ends immediately when the agent achieves a success (or a “breakthrough”).
The principal’s task in this environment is to determine when to terminate the project, conditional on no success having occurred. Within this setup, we analyze two distinct cases toillustrate the role of commitment: 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 reachedwithout attaining a success; in the other, the principal makes no such commitment, thereby
retaining discretion to terminate the project at any instant, and simply terminates the project
when the continuation payoff is not high enough to justify further experimentation. By directly comparing these two cases, we evaluate the extent to which the principal benefits fromcommitting to an evaluation deadline in this dynamic environment.
The driving force of our analysis is a dynamic strategic interaction between the agent’s
effort choice and the principal’s termination strategy. On one hand, the agent’s effort choicedepends 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 choice: when the agent is less motivated and exerts low
effort, less information is revealed about his type, which makes the principal more reluctant

1

to terminate the project. This strategic interaction can generate a vicious cycle where theprincipal’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.
The main contribution of the paper is that we devise an analytical framework that is
tractable enough to admit a complete characterization of pure-strategy equilibria, both withand without commitment, while capturing this dynamic interaction. Our framework thus
allows us to evaluate the exact value of commitment to an evaluation deadline for any given
set of parameters. To this end, our main analytical focus is on the no-commitment case,which is generally far more challenging than the commitment case when dynamic interactions
are considered. In the no-commitment case, the principal’s termination strategy must be
sequentially rational along the way, and the belief off the equilibrium path plays a crucial role.
Even in this case, we can show that our framework generates a unique recursive equilibrium
structure which can be exploited to establish the existence and uniqueness of (pure-strategy)
perfect Bayesian equilibrium. We then build on this result to derive a necessary and sufficientcondition under which the principal can strictly benefit from committing to an evaluation
deadline.
As can be expected, the lack of commitment to a deadline entirely alters the dynamic
allocation of effort as well as the timing of project termination. When the principal sets a
deadline at the outset, she does so by taking into account how it affects the agent’s entire effort
sequence. In particular, since extending a deadline in general relaxes the agent’s incentive
compatibility constraint and reduces his total effort supply, the deadline must be set at apoint where the expected benefit (of achieving a success at the next instant) equals the costof a decrease in total effort. The situation changes rather drastically when she makes no suchcommitment, because her termination decision would have no influence on the agent’s past
behavior. As such, she terminates the project whenever the continuation payoff is about toturn negative while taking the agent’s effort strategy as given. This fact implies that the costof extending a deadline is smaller under no commitment and tends to give her an excessive
incentive to wait for a success.
Given this incentive structure, one may expect that the principal would always wait longer
for a success in equilibrium under no commitment. As it turns out, though, this conjecture
does not always hold true in the current setup. Among other things, an interesting, andsomewhat counterintuitive, property of our model is that the average duration of the project
may not be monotonic with respect to the initial prior belief under no commitment, whereas it
is always weakly increasing under commitment. As a direct consequence of this, the average

2duration 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
from the principal’s reluctance to terminate the project in the first place. The principal tends
to terminate the project too early when the agent’s productivity under low effort is sufficiently
small. For the sake of argument, suppose that the agent fails almost surely when he exerts
low effort, in which case the expected instantaneous payoff is negative while the principal canlearn almost nothing from failures (during the phase where the agent is supposed to exert low
effort). The principal’s belief then declines very slowly over time, forcing her to incur a large
amount of loss if she is to wait until she totally loses her confidence in the agent’s ability. If
this expected loss is prohibitively large, the principal may find it optimal to terminate theproject even when the belief is still relatively high.
Of course, in equilibrium, the agent correctly anticipates this reaction and adjusts his
effort allocation accordingly. We show that the equilibrium timing of project termination
can be pinned down by backward induction, where we start from the final critical time (tobe derived) and solve backward. This process gives rise to the aforementioned recursive
equilibrium structure which allows us to establish the uniqueness of equilibrium and also
obtain a diverse set of equilibrium dynamics under no commitment.
The implications of our analysis can be applied broadly to a range of circumstances inwhich 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, aprofessor who must evaluate graduate students, a head coach in professional sports who mustevaluate players, and so on. Among those possibilities, the most prominent example of evaluation schemes with deadlines is perhaps the “up-or-out system,” which is widely observedin academia and professional service industries such as law, accounting, and consulting. As a
specific application, our framework offers some insight for when up-or-out contracts are morevaluable by identifying several key factors – such as high ability intensity, stable job descriptions, and similar jobs across ranks – which favor the use of an evaluation deadline from apreviously unexplored channel. Each of these factors intensifies either the agent’s incentive todelay exerting high effort for a given deadline or the principal’s incentive to delay terminating
the project for a given effort sequence (or both), thereby rendering it more profitable to set
a deadline at the outset.
The paper is organized as follows. The literature review is provided in the remainder ofthis section. The model is presented in section 2 and analyzed in sections 3 and 4, where we3characterize both the commitment and no-commitment solutions. These solutions are thencompared in section 5, in order to analyze the value of commitment and derive a necessary
and sufficient condition under which it is strictly optimal to commit to a deadline. Some
extensions of the baseline model 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 through a sequenceof experiments.1 From the principal’s point of view, our model can be seen as a variant of
the canonical two-armed bandit problem with one safe arm (terminating the project) and
one risky arm (continuing the project). Our model is particularly related to the literature on
strategic experimentation which analyzes a situation where a group of individuals, rather thana single individual, face bandit problems (Bolton and Harris, 1999; Bergemann and Välimäki,
1996, 2000; Keller et al., 2005; Klein and Rady, 2011; Bonatti and Hörner, 2011). A crucial
difference from this strand of literature is that experiments in our context are “intermediated,”i.e., experiments are conducted not by the principal herself but by an informed intermediary,
the agent, who faces no uncertainty about the project.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.3 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
the agent. 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 from1

An early economic application of the bandit problem can be found in Rothschild (1974). See Bergemann
and Välimäki (2008) for a survey.
2
Several recent works analyze models of “delegated experimentation” where a principal delegates experimentation to an agent. Guo (2014) analyzes a situation where the principal can specify, with full commitment
power, how the agent should allocate the resource in all future contingencies and solves for the optimal delegation rule. Garfagnini (2011) considers a similar setting to ours but assumes that: (i) the principal and theagent are symmetrically informed about the state of nature; and (ii) the agent’s payoff is independent of thestate. Chen and Ishida (2015) consider the opposite case in which the principal, with the termination right,
may be privately informed about the project quality while the agent focuses on implementing the project
assigned to him.
3
Also, see Buisseret (2016) who considers a two-period model of this setting with a more general, convex
cost function.

4

some technical differences,4 these previous works are primarily concerned with characterizing
optimal contracts. In contrast, the aim of this paper is to compare the allocations under fullcommitment and no commitment, while considering a less complete contractual environment,in order to evaluate the extent to which the principal can gain from committing to a deadline
at the outset. Although it is hard to tell a priori which specification is more plausible, as it
depends on the details of the underlying situation, our simple and tractable framework allows
us to identify and illuminate the counterintuitive role of commitment in dynamic performance
evaluation.
Several recent works examine the role of commitment in dynamic moral hazard setting.
Mason and Välimäki (2015) consider a dynamic moral hazard problem and derive optimal
wage contracts, both with and without commitment on wage payments. Hörner and Samuelson (2016) consider a repeated-game setting in which the principal chooses the scale of the
project in addition to contingent payments and characterize the set of equilibrium payoff
vectors that can be achieved without commitment.5 A key difference from our model is thatthese two works do not consider the principal’s learning, which is our main focus, with nohidden information about the agent’s type; as a consequence, the project is never terminatedin their models.6 This stands in sharp contrast to our setting in which the principal accumulates information about the agent over time, and the project must be terminated in finite
time.7
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
in dynamic problems with multiple agents. Bonatti and Hörner (2011) analyze a dynamic
4
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 (with stochastic signals), rather
than of experimentation, from the agent’s point of view.
5
In their model, outcomes are privately observed by the agent, and the moral-hazard problem regards thetruthful disclosure of this private information.
6
Georgiadis et al. (2014) analyze the role of commitment in a dynamic contribution games where themanager has the decision right over the project size. With multiple agent types and dynamic learning, Bonattiand Hörner (2017) analyze a (symmetric-information) experimentation model in which wages are determined
competitively a la Holmström (1999) and characterize effort and wage dynamics with an exogenous terminationdate.
7
A subtle technical difference which directly arises from this fact lies in the ways in which to construct
an equilibrium. Mason and Välimäki (2016) and Hörner and Samuelson (2017) construct an equilibrium via
arguments found in the analysis of infinitely repeated games (reversion to the worst continuation equilibrium),and as such, their analyses yield a non-degenerate set of equilibria. In our analysis, an equilibrium is obtained
via backward induction which gives rise to a unique equilibrium.

5moral-hazard problem with a team in which multiple agents work on a project of unknown
quality and briefly 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 reveal that he has been successful. The focus of these works is placedon the interaction between the agents, especially the freerider problem, whereas ours is on
the dynamic interaction between the principal’s termination decisions and the agent’s effort
choices.

2

A dynamic model of performance evaluation

Environment: We employ a continuous-time model because of its greater tractability. 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) ormediocre with probability 1 − p0 . The ability type is the agent’s private information andis not directly observable to the principal who must instead evaluate it from a sequence ofobserved outcomes.
Production: The agent makes unobservable effort at ∈ {l, h} at each instant t.8 We interpretthat 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 be directly
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 as well as theagent’s type. If the good type chooses at = a over time [t, t + dt), he attains a success withprobability λa dt where λh > λl > 0. In contrast, the mediocre type can never succeed withany effort level.9 Define ∆λ := λh − λl .
Payoff: We consider an incomplete-contracting environment where contingent rewards on
the arrival of a success cannot be enforced.10 A success yields a net present value of y > 0
8

Our focus on binary effort reflects our implicit presumption that the effort cost and success probability
are linear, as usually assumed in this literature (e.g., Keller et al, 2005; Bonatti and Horner, 2011, for mostof their analysis). As long as this structure is maintained, an extension to continuous effort, say at ∈ [l, h],yields exactly the same allocation and is hence irrelevant.
9
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, 2017).
10
In reality, we rarely observe complete wage contracts in industries comprised of professionals. For instance,
few academic institutions, if any, offer rewards specifically contingent on verifiable measures of output: salaries

6to the principal and b > 0 to the agent;11 otherwise, they both receive zero. Aside from this,the principal must also pay a flow wage w > 0 to the agent as long as the project continues.12
We assume that w is exogenously given for most part; the case with an endogenous wage isbriefly discussed in the concluding section. The reservation payoff is assumed to be zero forboth players. The common discount rate is denoted by r ∈ (0, ∞).
Contract: The only contractible decision for the principal in this environment is whether
to set a deadline, and if so, at what point. If the principal commits to a deadline, sheterminates the project at the deadline (but never before) if the agent has not attained a
success up to that point. If the principal chooses not to commit to any specific deadline, on
the other hand, she retains discretion to terminate the project at any instant. As mentioned
earlier, our analytical focus is on the latter case which requires that both players’ strategies
be sequentially rational.

3

Analysis

3.1

Agent’s effort decision

The agent decides whether or not to exert high effort at each instant. 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 time τ , in whichcase the remaining time at time t is simply given by τ − 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.
are determined through bilateral negotiations, often dictated by market forces, in some countries whereas they
are subject to bureaucratic regulations in others. One possible reason for the lack of complete wage contracts
in those industries is that it is often difficult, and perhaps prohibitively costly, to measure the exact value ofa “success” in a verifiable manner. Additionally, this assumption makes our analytical framework applicable
to a wider range of circumstances. One such possibility is that the benefit of achieving a success accrues from
non-transferrable (psychological) gains such as prestige, authority, and the sense of achievement, and extrinsic
rewards are hence of secondary importance. There are also many cases where contingent monetary transfers
are neither feasible nor desirable, as in a professor-student relationship.
11
An obvious interpretation is that y and b represent the continuation payoffs of achieving a success, including
not only the intrinsic value of a successful outcome but also other benefits of identifying/signaling talent.
12
Alternatively, w can be regarded as a flow cost of production (e.g., hiring an agent). Of course, under
this interpretation, w is no longer a transfer payment to the agent and does not appear in his payoff. It is
straightforward to make the setup consistent with this interpretation without having any qualitative impact
on our results.

7

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 asU (k) = max (λa b − da + w)dt + e−rdt (1 − λa dt)U (k − dt) .
a∈{l,h}Taking the limit dt → 0, we obtain the Bellman equation:rU (k) = max λa b − da + w − λa U (k) − U̇ (k) ,a∈{l,h}(1)with limk↓0 U (k) = 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), the continuation

payoff U (k) captures the agent’s reservation payoff which he receives in case of a failure. The
cost of not succeeding today is obviously small when he has a high reservation payoff. Since
the continuation payoff is higher when the agent has more time to prove himself, the agent
has a stronger incentive to work hard as the project approaches the termination date. Thisis a manifestation of the deadline effect that lies at the core of our entire analysis.
Proposition 1 If
rb − wd>,∆λλl + r(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.

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 enough toovercome the dynamic agency cost, and the agent is willing to supply high 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.
8

Assumption 1 λhλb−d+w> b − ∆dλ > 0.
h +r
Among other things, the assumption implies that the optimal threshold k∗ is bounded fromabove and away from zero, i.e., k∗ ∈ (0, ∞).

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. Given some effort sequence {as }ts=0 , the updated belief is then given by
Rt

pt =

p0 e− 0 λas ds
Rt

1 − p0 + p0 e− 0 λas ds.

(4)

Alternatively, taking the time derivative, we obtainṗt = −λat pt (1 − pt ).

(5)

It is clear that the belief is strictly decreasing over time for any effort choice (due to the factthat λl > 0) until the agent attains a success, in which case the belief immediately jumps up
to one.
The principal’s problem is to determine when to terminate the project, conditional on no
success having occurred. Now suppose that the principal intends to terminate the project attime τ > t. Taking the effort sequence as given, the principal’s continuation payoff can also
be written as a function of the remaining time k and the current belief pt where
Z t+kRs(λas ps y − w)e− t (λau pu +r)du ds,
V (k, pt ) =tsubject to (5).
It is immediate to see that for a given effort level, there exists a threshold belief below
which the principal’s instantaneous payoff is strictly negative. We denote by qa := min{ λwa y , 1}
the “break-even” belief at which the instantaneous payoff equals zero under effort a. Combined with the fact that the belief is strictly decreasing over time for any effort sequence, qh
represents the absolute lower bound of the belief, as the principal clearly has no incentive tocontinue the project once her belief dips below this level. Since the belief must reach thislevel sooner or later (due to the fact that λl > 0 and w > 0), the presence of such a lower
bound suggests that the game must end in some finite time. This allows us to solve the gameby backward induction.
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.
9

Assumption 2 p0 > qh = λwh y .
Since the principal’s continuation payoff depends on her belief, it is often convenient
for the subsequent analysis to explicitly relate the current and termination beliefs to theremaining time. Suppose that the current belief is p and the principal intends to terminate
the project when the belief reaches q. If the agent adopts the best response, the remaining
time can be written as K(p, q) such that∗∗pe−λl max{K(p,q)−k ,0}−λh min{K(p,q),k }= q,
1 − p + pe−λl max{K(p,q)−k∗ ,0}−λh min{K(p,q),k∗}which reduces to∗pe−λl K(p,q)−∆λ k= q,
1 − p + pe−λl K(p,q)−∆λk∗
if K(p, q) > k∗ . In addition, it is also convenient to define a backward operator φ(q) suchthat∗

φ(q)e−λh k= q.
1 − φ(q) + φ(q)e−λh k∗
The backward operator suggests that if pt = φ(q) and the agent exerts high effort for t ∈[t, t + k∗ ], then pt+k∗ = q; alternatively, if the agent expects the project to be terminated at
pt = q, φ(q) indicates the belief at which he must switch to high effort. Note that these two
notions are closely related in that K(p, q) > k ∗ if and only if p > φ(q).

4Equilibrium

4.1

The cooperative solution: a benchmark

Before we move on to analyze our model, we first examine as a benchmark the case where
both the principal and the agent attempt to maximize the sum of their individual payoffs toderive 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 successminus the effort cost, i.e., λa p(y + b) − da for a given effort a.13
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.

13
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 between
participating and dropping out immediately (before time 0), as he would obtain zero payoff in either case forany given deadline. As it turns out, however, the optimal deadline is independent of the initial belief p0 andthe analysis holds irrespective of whether or not the principal can immediately screen out the mediocre type.

10

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 easily verify that the agent is generally more motivated in thecooperative case than in the noncooperative case, i.e., k̃∗ > k∗ .
Given this, it is fairly straightforward to derive the cooperative solution. Since there is
no social cost of hiring the mediocre type, the joint surplus from hiring the agent is strictly
positive for any belief level, and as such, it is efficient for the principal to continue the projectindefinitely until a success is attained. This implies that the good type will eventually succeed
at some point, although he only exerts low effort all the way if k̃∗ is finite.
Proposition 2 In the cooperative case, the principal never terminates the project, and thegame continues indefinitely. The agent always exerts low effort ifr(b + y)d>,∆λλl + r(6)

and high effort otherwise.
Proof: See Appendix.
Note the difference between (2) and (6). In the cooperative case, the joint expected payoffof exerting low effort indefinitely is r(b+y)λl +r because the value of a success is now b + y, insteadof just b, and there is no flow cost w.

4.2

The commitment solution

The principal’s problem under commitment is to choose a termination date τ which maximizes
her expected payoff at time 0 while restricting attention to deterministic deadlines.14 We focus
exclusively on deterministic deadlines primarily because it is generally difficult to commit toa lottery, even when it is possible to commit to a particular termination date. To see thispoint, consider a contract where the principal terminates the project at τ 1 or τ 2 , τ 1 < τ 2 , eachwith probability strictly less than one. At time τ 1 , however, the principal’s preferences are
generically strict, i.e., it is strictly better either to stop (the continuation payoff is negative) or
to continue (positive), so that there is no incentive to randomize at that point. If it is strictly
better to stop at time τ 1 , for instance, the principal stops at time τ 1 with probability one but
14
If stochastic deadlines are feasible, the principal may screen out the low type by offering a menu ofcontracts, one with a deterministic deadline and the other with stochastic ones. We do not pursue thispossibility both because we do not think that it is realistic (given the enforcement problem) and because it isoutside the scope of our analysis (given that our focus is on comparing the commitment and no-commitment
solutions). Note that this screening cannot be done if deadlines are deterministic, because both types always
strictly prefer a longer deadline.

11

knowing that, the contract is effectively reduced to the one with a deterministic deadline. The
lack of credible enforcement is perhaps the reason why we almost never observe a contract
with stochastic deadlines.
As we have already seen, the agent exerts high effort if and only if the remaining timeis less than or equal to k∗ . We denote by V ∗ (k, pt ) the continuation payoff when the agent
adopts this best response, where k is the remaining time and pt is the principal’s belief. Thiscan be written as
Z t+ν
Rs
∗
(λl ps y − w)e− 0 (λl pu +r)du ds
V (k, pt ) =
t
Z t+k
R t+ν
Rs
+ e− t (λl pu +r)du(λh ps y − w)e− ν (λh pu +r)du dst+ν

=p t πlpt e−(λl +r)ν πh(1 − pt )w(1 − e−(λl +r)ν ) +(1 − e−(λh +r)(k−ν) ) −(1 − e−rk ),λl + rλh + rr(7)

where ν := max{k − k∗ , 0} and πa := λa y − w. The first two terms represent the expected
gain from the good type whereas the last term represents the expected loss from the mediocre
type.
Since the termination date is equivalent to the remaining time at time 0, the commitment
solution, denoted by τ C , is given byτ C (p0 ) := argmaxτ V ∗ (τ, p0 ).
The continuation payoff is continuous but kinked at τ = k∗ . For τ ∈ (0, k∗ ), the first-ordercondition is given by
p0 πh e−λh τ − (1 − p0 )w = 0.

(8)!
λl + r−(λh +r)k ∗πl −πh (1 − e) − (1 − p0 )we−rτ = 0.
λh + r(9)

For τ ∈ (k∗ , ∞), it is given by−(λl +r)(τ −k ∗ )M (τ, p0 ) := p0 e

To better understand this condition, it is instructive to rewrite this as
∗

∗
∗
∗
p0 ∆λ e−(λl +r)(τ −k )w + ry + πh e−(λh +r)k .
p0 πh e−(λl (τ −k )+λh k ) − (1 − p0 )w e−rτ =λh + r

Note that the left-hand side is the expected instantaneous payoff for the principal at the timeof termination, whereas the right-hand side is the marginal cost of extending the deadline.
Since the right-hand side is strictly positive, the principal must set the deadline at a point
where the expected instantaneous payoff is positive if τ > k ∗ .
12

Since the instantaneous payoff is negative for any effort level once the belief dips below
qh , there is clearly no incentive to set τ > K(p0 , qh ). When the initial belief is sufficiently
close to qh (φ(qh ) ≥ p0 ), it is optimal to let the agent work until the belief reaches this lower
bound. When it is far from it (p0 > φ(qh )), on the other hand, the principal must terminate
the project at a point where the belief is still above the lower bound. There are two cases,
depending on the value of M (k∗ , p0 ).15 If
λl + r∗πh (1 − e−(λh +r)k ) > πl ,λh + r(10)

M (k∗ , p0 ) < 0 for any p0 ∈ (qh , 1), and we let p̂ = 1 in this case. If (10) fails to hold, on
the other hand, there exists a unique p̂ such that M (k∗ , p̂) = 0. Then, for p0 > p̂, there
exists a unique interior solution τ̂ (p0 ) such that M (τ̂ (p0 ), p0 ) = 0. When the initial prior isin this range, it is not so costly to have a phase where the agent exerts low effort, and assuch, the principal would wait beyond time k∗ . Of course, extending the deadline beyond
time k∗ means that the agent would slack off at the beginning, thereby pushing back therealization of the high-effort phase. The principal trades off the gain of inducing low effort
for an additional instant against the cost of realizing the high-effort phase an instant laterand sets the termination date at a point where they are equalized.
For clarity, we summarize the set of parameters by Θ := (b, y, w, d, λh , λl , r). We can thenmake the following statement.
Proposition 3 For any given set of parameters (Θ, p0 ) satisfying Assumptions 1 and 2, there
exists a unique commitment solution τ C (p0 ) ∈ (0, K(p0 , qh )]. For φ(qh ) ≥ p0 , the commitment
solution is given by τ C (p0 ) = K(p0 , qh ). For p0 > φ(qh ), there exists some p̂ ∈ (ql , 1] such
that
(
k∗
τ C (p0 ) =τ̂ (p0 )if p̂ ≥ p0 > φ(qh ),if p0 > p̂,

where: (i) p̂ > ql if 1 > ql ; and (ii) p̂ = 1 if and only if (10) holds. The commitment solution
is weakly increasing in p0 .
Proof: See Appendix.
Note that K(p0 , qh ) constitutes the upper bound of τ C (p0 ). Since K(p0 , qh ) is boundedfrom above when qh > 0, the proposition suggests that the project must be terminated in
finite time, meaning that some good projects are bound to be terminated prematurely. Thisdraws clear contrast with the cooperative solution where the project is never terminated.
15

Note that M (k, p0 ) < 0 for any k > k∗ and p0 if and only if M (k∗ , p0 ) < 0.

13

4.3

The no-commitment solution

The situation becomes more complicated, and perhaps more intriguing, when the principal
makes no commitment at the outset and hence her termination decision must be sequentiallyrational. Because of this, the principal’s termination strategy now needs to be specified
in a different way, as her strategy off the equilibrium path (after the project is supposed
to be terminated) may matter. More precisely, the principal’s strategy is given by a setof termination dates T which consists of a first termination date and further dates along
off-equilibrium continuation paths. Formally, for this no-commitment case, we solve for apure-strategy perfect Bayesian equilibrium (hereafter, simply an equilibrium) in which theprincipal terminates the project with probability one at each termination date (a formal
definition is given below).16
To illustrate how we specify the principal’s strategy, consider a simple example where
the principal’s strategy consists of two distinct termination dates, i.e., T = {τ 1 , τ 2 } whereτ 1 < τ 2 . Under this strategy, the principal terminates the project with probability one ateach τ i , i = 1, 2, conditional on the continuation of the game.17 This means that the projectis surely terminated at time τ 1 on the equilibrium path. To show that this constitutes anequilibrium, however, one must assure that the principal cannot profitably deviate from thisstrategy. To see this, suppose that the principal unexpectedly continues the project at timeτ 1 . The game then enters into a new phase where the agent now expects the project tobe terminated at the next termination date τ 2 and chooses his effort accordingly (at = h ifk∗ ≥ τ 2 − t and at = l otherwise). Note also that the principal’s belief about the agent’s type
is not affected by the deviation: given T and the agent’s best response, we can still apply
Bayes’ rule to compute how the belief evolves both on and off the equilibrium path.18 This
gives the continuation payoff when the principal deviates which must be non-positive.
How long does this process continue? It generally goes on until the principal has no16

We restrict our attention to pure-strategy equilibria by assuming that the principal terminates the projectwith probability one when the continuation payoff is zero. Since the principal is actually indifferent at thispoint, however, there may exist other mixed-strategy equilibria in which the principal terminates the projectwith probability less than one at each termination date. We do not pursue this possibility for two reasons.
First, we do not believe that such an equilibrium is particularly realistic and hence appealing. Second, evenwith randomization, the game still must end with probability one by time τ 1 , and the continuation equilibrium
is exactly the same once the game enters the final interval (τ 1 , τ 2 ); solving backwards, one can see that thestructure of equilibrium is essentially the same where the agent starts exerting high effort as t approaches eachtermination date and switches back to low effort if the project is not terminated at the termination date.
17
If the game continues beyond time τ 2 , the principal terminates the project immediately as we will see
below.
18
Technically, we assume that the belief depends only on the agent’s past effort and is given by (4) even
off the equilibrium path. This is actually the “no signaling what you don’t know” condition (Fudenberg andTirole, 1991) although it is a rather obvious restriction in games with two players (it only implies that the
principal’s belief should not be affected by her own deviation).

14

incentive to continue the project regardless of the agent’s effort choice or, in other words,
until the belief reaches the lower bound qh . With slight abuse of notation, we denote by pt
the equilibrium belief at time t, conditional on the continuation of the game (including theoff-equilibrium path). Let τ n denote the time at which pt reaches qh for some n (which is
unknown at this point). We can then restrict our attention to t ∈ (0, τ n ], for it is a dominant
strategy for the principal to terminate the project once the belief reaches this lower bound.19
Lemma 1 In any equilibrium, T is a finite set.
Proof: The lemma is directly implied by the fact that any two adjacent termination dates
must satisfy τ i − τ i−1 ≥ k∗ > 0. To see this, suppose otherwise, i.e., k∗ > τ i − τ i−1 . Suppose
further that the principal deviates and continues the project at time τ i−1 . Then, it is strictly
better for the agent to exert high effort because k∗ > τ i − τ i−1 . Note, however, that sincept > qh for any t ∈ (0, τ n ) by definition, the instantaneous payoff is strictly positive until the
belief reaches the next termination belief. This is a contradiction because the principal canstrictly benefit from deviating and not stopping at time τ i−1 . Given this, it is clear that we
can only have finitely many termination dates in (0, τ n ].
Given this result, we denote each element of T by τ i where τ 1 < τ 2 < · · · < τ n for somen. Accordingly, the game is divided into n distinct segments T i , i = 1, 2, ..., n by termination
dates, where T i = (τ i−1 , τ i ) for i = 1, 2, ..., n with τ 0 = 0. The remaining time at time t is
now given by τ i − t for t ∈ T i while it is zero for all t > τ n (see footnote 19). Let q i := pτ idenote the corresponding termination belief. Only the first segment T 1 is actually played on
the equilibrium path, as the game ends with probability one by the time the game reaches
time τ 1 . For expositional clarity, we call τ 1 the no-commitment solution and denote it by
τ N C (p0 ) to indicate its dependence on p0 . The formal definition of our equilibrium is given
below.
Definition 1 A perfect Bayesian equilibrium in this game is a pair of strategies {as }∞s=0 and
T := {τ i }ni=1 and a belief system such that:• given T , the (good-type) agent chooses at at each t to maximize his continuation payoff,i.e., chooses high effort if and only if the remaining time at time t is less than or equal
to k∗ ;
19
Suppose that the principal deviates and continues beyond time τ n . In this case, the principal’s instantaneous payoff is negative even if the agent chooses at = h, so that it is a dominant strategy to terminate
immediately for all t > τ n . Given this, since the remaining time is invariably zero, the agent always chooses
at = h. This is the unique continuation equilibrium after time τ n (although, to describe this process formally,
we need to consider a discrete-time counterpart of our model where there is a minimum time unit, and takethe limit as the time unit goes to zero).

15• given {as }∞s=t and the current belief, the principal terminates the project at time t if
and only if the continuation payoff is non-positive;• The belief at time t depends only on the effort sequence {as }ts=0 and is given by (4),both on and off the equilibrium path.
To pin down the equilibrium termination dates, we need to pay closer attention to theprincipal’s continuation payoff. Since the agent’s best response is the same, the principal’s
continuation payoff can still be written as V ∗ (k, pt ). The only difference is that the principal’s
strategy must now be sequentially rational, which amount to the following two equilibrium
conditions which the principal’s termination strategy T must satisfy.
Condition T: (i) q n = qh and (ii) if n > 1, for each i = 2, ..., n, V ∗ (K(q i−1 , q i ), q i−1 ) = 0.
Condition C: For each i = 1, 2, ..., n, V ∗ (τ i − t, pt ) > 0 for all t ∈ T i .
Condition T is the usual indifference condition that requires the principal to terminate theproject when the continuation payoff (when she deviates) is non-positive at each τ i . This
indifference condition alone is in general not sufficient because the instantaneous payoff may
not be monotonically decreasing over time: in any segment T i , the agent may start off withlow 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 date τ i is reached.
The next result establishes that there exists a perfect Bayesian equilibrium that is always
unique even under no commitment.
Proposition 4 For any given set of parameters (Θ, p0 ) satisfying Assumptions 1 and 2,
there exists a (generically) unique pure-strategy equilibrium. Given Θ, the belief space (qh , 1)m(Θ)

is partitioned into m(Θ) ≥ 1 distinct intervals {(P j , P j−1 )}j=1

where P m(Θ) = qh and

P 0 = 1. For p0 ∈ (P j , P j−1 ), j = 1, 2, ..., m(Θ), the equilibrium strategies are as follows:• The principal’s equilibrium strategy is characterized by a set of n = m(Θ)−j +1 distinct
termination dates {τ i }ni=1 where q i = P i+j−1 and τ i − τ i−1 > k∗ ;20• The agent exerts high effort if the remaining time is less than or equal to k∗ and low
effort otherwise.
Moreover, if m(Θ) > 1, then ql > P 1 .
20

As discussed in footnote 19, at = 0 for all t ≥ τ n .

16

Proof: See Appendix A.
As the proposition suggests, the structure of equilibrium is thoroughly characterized by
how the belief space is partitioned into intervals. A crucial determinant of the partitionm(Θ){(P j , P j−1 )}j=1 is the profitability under low effort which is captured by πl .21 Below, wem(Θ)

first briefly illustrate how we pin down m(Θ) and {(P j , P j−1 )}j=1 from a given Θ (in what
follows, we simply denote m = m(Θ) to save notation). Once they are obtained, it is quite
straightforward to derive the equilibrium strategy T for a given p0 .
Suppose first that the profitability under low effort is sufficiently high, so that 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 where m = 1: the agent starts off with low effort and switches
to high effort when the remaining time is k ∗ ; the principal stops when the belief reaches thelower bound P 1 = qh at time τ 1 . Formally, m = 1 and hence n = 1 for all p0 ∈ (qh , 1) if andonly if V ∗ (K(ql , qh ), ql ) > 0, i.e.,ql e−(λl +r)ν πh(1 − pt )w
q l πl(1 − e−(λl +r)ν ) +(1 − e−(λh +r)(k−ν) ) >(1 − e−rk ),λl + rλh + rr(11)

where µ = max{K(ql , qh ) − k∗ , 0} (see the proof of Proposition 4, especially Lemma 2, formore detail). Note that (11) depends only on Θ. The no-commitment solution τ N C (p0 ) ismonotonic in p0 if and only if this condition holds.
[Figure 1 about here]
In contrast, the instantaneous payoff may become negative under low effort when the
success probability under low effort is relatively 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 when m = 2 while
fixing the principal’s strategy at T = {τ 1 }. 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 τ ′ (in Figure 3) such that the continuation payoff is negative for t ∈ (0, τ ′ ).
This implies that T = {τ 1 } does not constitute an equilibrium as it violates Condition C.
In this case, the game is divided into two segments, T 1 and T 2 , as illustrated in Figure 4.
The pair of strategies now satisfies the equilibrium conditions since the instantaneous payoff
is always positive in T 1 . Formally, if V ∗ (K(ql , qh ), ql ) < 0, there must exist p′ ∈ (qh , ql ) suchm(Θ)

21Note that the partition {(P j , P j−1 )}j=1
prior p0 .

is determined solely by Θ while q i depends also on the initial17

that V ∗ (K(p′ , qh ), p′ ) = 0, in which case we redefine P 1 = p′ and P 2 = qh . As above, m = 2
if and only if V ∗ (K(ql , P 1 ), ql ) > 0.
[Figures 2-4 about here]
We can continue this process until we find P 1 such that V ∗ (K(ql , P 1 ), ql ) > 0.22 Oncethe partition is pinned down, we can then easily derive the equilibrium strategies: for p0 ∈(P j , P j−1 ), the principal’s equilibrium strategy consists of n = m − i + 1 termination dates
where q i = P i+j−1 andτ N C = τ 1 = K(p0 , q 1 ), τ i =

i−1
X

′

τ i + K(q i−1 , q i ) for i = 2, 3, ..., n.

i′ =1Note also that q i−1 > φ(q i ) > q i because the continuation payoff is always strictly positive
when the belief is in (q i , φ(q i )), the range where the remaining time is less than k∗ and theagent exerts high effort. This alternatively means τ i − τ i−1 > k∗ , i.e., the length between
any two adjacent termination dates must be larger than k∗ .
Finally, we would also like to note that the constructed equilibrium is generically uniquefor any given set of parameters. This uniqueness result stems crucially from the fact that there
is a lower bound of the belief qh below which the principal would never continue the project.
As stated above, together with the fact that λl > 0, this implies that the game must end
in some finite time, which allows us to solve the game via backward induction analogously
to the “gap case” of the durable-good monopoly problem (Fudenberg et al., 1985). Moreprecisely, since the agent’s strategy depends only on the remaining time, we know exactly
how the game must end as the belief approaches the lower bound. Applying this reasoning
backward, we can identify a unique continuation equilibrium for each pt > qh and all the way
back to the initial prior p0 .

5Discussion

5.1

The value of commitment

Our framework yields unique commitment and no-commitment solutions which enable us todirectly assess the value of commitment to an evaluation deadline. Given that both τ C andτ N C can be written as functions of p0 , the expected equilibrium payoffs can also be written
as functions of p0 . Let
V C (p0 ) := V ∗ (τ C (p0 ), p0 ), V N C (p0 ) := V ∗ (τ N C (p0 ), p0 ).
22
When ql = 1, there is no such q 1 and m will go to infinity. If ql < 1, on the other hand, this process must
converge after a finite number of rounds.

18

The value of commitment is then defined as∆V (p0 ) := V C (p0 ) − V N C (p0 ).
In principle, we can compute the exact value of ∆V (p0 ) for any given p0 . The following
statement is the main result of the paper which analytically characterizes how the value ofcommitment varies with respect to p0 .
Proposition 5 (i) ∆V (p0 ) = 0 for p0 ∈ (P m , φ(P m )) where P m = qh . (ii) If m > 1,
for each j = 2, 3, ..., m, ∆V is strictly increasing in p0 for p0 ∈ (φ(P j ), P j−1 ) and strictly
decreasing for p0 ∈ (P j−1 , φ(P j−1 )) with ∆V (φ(P j )) = 0. (iii) ∆V (φ(P 1 )) = 0 if and only ifp̂ ≥ φ(P 1 ) where p̂ is the threshold defined in Proposition 3. (iv) limp0 →1 ∆V (p0 ) = 0 if andonly if 1 > p̂.
Proof: See Appendix.
The proposition suggests that the value of commitment changes in a non-monotonic way as
p0 increases, precisely due to the non-monotonic nature of V N C . Despite this, we can still findsome regularities except for the last segment (φ(P 1 ), 1).23 Take an interval (φ(P j ), φ(P j−1 ))for j = 2, 3, ..., m where φ(P j−1 ) > P j−1 > φ(P j ). In each of these intervals, (i) the valueof commitment is single-peaked and maximized at P j−1 (at which point the no-commitment
solution yields a zero payoff); (ii) the value of commitment is zero at both ends of theinterval (at which point the two solutions are identical). Important features of Proposition
5 are captured by Figure 5 which shows a typical path of the value of commitment whenm = 2.24[Figure 5 about here]
To see how ∆V changes with respect to p0 , observe that the only feasible history at any
continuation game is the one consisting only of failures up to that point, and the principal
hence has no additional information other than the fact that the agent has achieved no success.
This immediately implies that the no-commitment solution can yield no higher payoff thanthe commitment solution, i.e., ∆V (p0 ) ≥ 0 for all p0 . Clearly, the no-commitment solution
23

In the last segment, the profitability under low effort can be high enough (if p0 > ql ), and the expected
payoff can go up even during the low-effort phase. Moreover, in this range, τ C (φ(P 1 )) = τ NC (φ(P 1 )) may
not even be satisfied since τ C (φ(P 1 )) can be larger than k∗ . As a consequence, the value of commitment caneither go up or down (and can even be non-monotonic). See Appendix B for more detail.
24
For the figure, the parameters are set as follows: λh = 0.1, λl = 0.04, b = y = 1, w = 0.035, d = 0.02,
r = 0.05.

19

gives the principal a different, and necessarily lower, payoff when τ C 6= τ N C . Still, the nocommitment solution can replicate the same allocation as the commitment solution when
τ C = τ N C , because Proposition 1 ensures the unique equilibrium given the same horizon. It
follows from Proposition 5 that the commitment solution generically yields a strictly higher
expected payoff for the principal than the no-commitment solution if and only if p0 > φ(qh ),25
i.e.,
p0 >

1 − qhqh1−∗ ⇔qh + (1 − qh )e−λh kqh(λh + r)(b − ∆dλ )λh b − d + w

! λh

λh +r

>

1 − p0.
p0(12)Conversely, when this condition fails to hold, the absence of commitment entails no additionalcost, and the no-commitment solution becomes strictly profitable in the presence of a (possibly
very small) commitment cost.26
The proposition states that the principal benefits from committing to a deadline if theinitial prior belief p0 exceeds a certain threshold φ(qh ), i.e., if the initial prior is so highthat the belief at time k∗ is above the lower bound qh even if the agent exerts high effort
for t ∈ [0, k∗ ]. The value of a deadline depends crucially on each player’s propensity to“procrastinate.” In short, the threshold is lower and hence favors the commitment solution
for a given initial belief either when the agent delays exerting high effort for a given deadlineor when the principal delays terminating the project for a given effort sequence. Below, we
briefly summarize how each player’s propensity to procrastinate 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.
25

To be more precise, as Proposition 5 suggests, there could be some non-generic cases where the expected
payoffs are identical.
26
An obvious commitment 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 we have thus far assumed away this aspect forclarity, we extend the analysis to incorporate uncertainty into the current setup in section 6: there, we showthat the commitment solution is not always weakly optimal. The commitment cost may also arise from thecost of writing and enforcing a formal contract.

20

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.27 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 value ofa deadline is still determined in essentially the same way, although the analysis would besubstantially more complicated.

5.2

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,
τ N C (p0 ) ≥ τ C (p0 ) for all p0 . As indicated in Figures 3 and 4, however, this does not necessarily hold 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.
From Proposition 3, it is conceptually straightforward to compute the commitment solution τ C . Figures 6a and 6b illustrate τ C as a function of p0 , using the same parameters as
in Figure 5 except for λl . The qualitative nature of τ C depends largely on the productivity
under low effort as captured by λl . Figure 6a is the case where λl is relatively high. In thiscase, the principal does not lose much during the low-effort phase, and is willing to extend
beyond k∗ when the initial prior is sufficiently high. In contrast, when λl is relatively low, as
depicted in Figure 6b, the principal sets τ C ≤ k∗ to assure that the agent always exerts higheffort. In either case, for p0 > φ(qh ), τ C is set strictly below the upper bound K(p0 , qh ) sothat the project is terminated before the belief reaches qh .
[Figures 6a and b about here]
As we show in the proof of Proposition 4, we can also explicitly derive the no-commitment
solution τ N C (p0 ) although it may take a much more complicated form. Figure 7 depicts
τ N C (p0 ) along with τ C (p0 ), which confirms that the average duration of the project canbe longer with commitment than without, using the same parameters as in Figure 5. The
figure also reveals that the no-commitment may not be monotonically increasing in p0 , in27

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.

21

clear contrast to the commitment solution which is always weakly increasing. When theinitial belief is relatively high, it takes long for the belief to reach the lowerbound qh , andthe 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 agentsin some cases. This is more likely to be the case when λl is low, in which case the principal
must incur losses in the low-effort phase. Formally, we can make the following statement.
Proposition 6 τ N C (p0 ) can be larger or smaller than τ C (p0 ). Moreover, τ C (p0 ) > τ N C (p0 )for almost all p0 ∈ (φ(qh ), 1) as λl tends to zero.
Proof: It is clear that τ N C (p0 ) can be larger than τ C (p0 ) for some p0 , because τ N C (p0 ) > k ∗for p0 slightly larger than φ(qh ) while τ C (p0 ) = k∗ for p0 ∈ (φ(qh ), p̂). We can also show that
τ N C (p0 ) can be smaller if m ≥ 2, because limp0 ↓P 1 τ N C (p0 ) = 0 < k∗ ≤ τ C (P 1 ).
As for the second statement, It follows from Proposition 3 that τ C (p0 ) ≥ k∗ forp0 ∈ (φ(qh ), 1). To prove the proposition, it thus suffices to show that k∗ > τ N C (p0 ) in thisrange. Suppose otherwise, i.e., τ N C (p0 ) ≥ k∗ for some p0 ∈ (φ(qh ), 1). This happens if andonly if p0 ∈ (φ(P j ), P j−1 ) for j = 1, 2, ..., m or, in other words, P j−1 must be bounded awayfrom φ(P j ). This is a contradiction, however, as limλl →0 V ∗ (K(P j−1 , P j ), P j−1 ) < 0 for any
P j−1 > φ(P j ).

[Figure 7 about here]
Our results yield some empirical implications. An obvious one is that the initial prior
p0 is not necessarily a reliable predictor of future success under no commitment. Asidefrom this, it is also worth emphasizing that the equilibrium allocation depends cruciallyon what happens off the equilibrium path. As a consequence, two organizations that are
observationally similar may exhibit drastically different equilibrium outcomes. To put thisidea in context, consider two distinct organizations, A and B, each characterized by λil ,i = A, B, and the corresponding no-commitment solution τ N C,i (while assuming that the two
Borganizations are equivalent in every other dimension). Suppose further that λA
l > λl and
N C,B (p ) < k ∗ for almost all p , meaning
λB00
l → 0. In this case, as Proposition 6 suggests, τthat we almost never observe the low-effort phase in organization B, and hence λB
l cannot be
estimated from actual data. The two organizations are thus observationally indistinguishable
even though their respective no-commitment solutions can be totally different from eachother.
22

5.3

An application: when is up-or-out optimal?

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 professional service industries such as law, accounting, and consulting (Lazear and Gibbs, 2014;
Eriksson, 2016).28 Under a typical up-or-out contract, the employer sets a deadline by whicha promotion must occur, and the worker must leave the firm if he is not promoted by thedeadline. 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, aleading example of industries with up-or-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 acertain period of time (roughly seven to ten years). This contrasts sharply with other typicalfirm organizations which almost never specify deadlines by which workers must be promoted.
Here, we attempt to shed some light on this issue by examining under what conditions (12)is likely to be satisfied. There are of course some caveats. First and foremost, our theoreticalframework is perhaps too simplistic to capture all the important details of this diverse set ofindustries; some of our arguments may thus be applied only to a subset of those industries.29
Second, there is also a danger in comparing allocations under different incentives structures.
One particular concern is the potential endogeneity of parameter values that are taken as
exogenous here.30 With those qualifications in mind, our framework can still provide some
useful, though preliminary, insight for when it pays to commit to a promotion deadline as in
the case of up-or-out.
Ability intensity: It is often argued that knowledge intensity is one of the most fundamental
characteristics of professional service industries (von Nordenflycht, 2010). Combined with thefact that they are also less capital intensive,31 the productivity of an organization depends
28
For brevity, we refer to those industries characterized by up-or-out, including academia, broadly as professional service industries. The military is another example which is characterized by up-or-out, but we do notconsider this case because the underlying structure appears to be quite different. See Kahn and Huberman(1988), Waldman (1990), O’Flaherty and Siow (1992), and Ghosh and Waldman (2010) for formal analyses ofup-or-out contracts.
29
For instance, our exponential specification implies that success is very rare and information is coarse, asassumed in Bonatti and Hörner (2017), which might be applicable to some industries but not to some others.
Also, team production may play a bigger role in some industries than in others.
30
It is likely that y and b may vary with the contract duration when we interpret them as the net continuation
payoffs of achieving a success. In particular, we normalize the continuation payoff at zero when the projectis terminated, but this could well be a function of the contract duration (or the market belief at the time oftermination).
31
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.

23

crucially on the extent of knowledge embodied in individuals (Starbuck, 1992; Winch andSchneider, 1993). A likely consequence of this fact is that innate ability matters and createsvalue 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
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 strictly
optimal to set a deadline, thereby favoring the use of up-or-out, for λh > λ̄. To see this, werewrite (12) as
qh
w
p0
>
=
.
1 − p0(1 − qh )e−λh k∗(λh y − w)e−λh k∗(13)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, (13) is never satisfied if λh is small.

In contrast, we haveqhlim
∗ = 0,
λh →∞ (1 − qh )e−λh k
implying that (13) must hold if λh is sufficiently large.
Stable job descriptions: Another distinctive feature of professional service industries is a
professionalized workforce that builds on a particular knowledge base (Torres, 1991; von Nordenflycht, 2010). This in turn creates well-defined job boundaries and stable job descriptions.
Due to various professional requirements and accreditation processes, workers in professional
service industries are typically responsible only for a narrow and clear set of tasks, compared
to workers in other industries. This implies that the nature of 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 thejob 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,24

e.g., the arrival of a stochastic shock that makes the project completely worthless.32 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!
λ1 (b − ∆dλ )1 − p01 − qh1−>.
qh
λh b − d + w
p0
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.33 In the current setup, this
aspect may be captured by y, which can be interpreted as the expected payoff of promoting
a good-type 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.34
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.
32
Here, we allow the principal to terminate the project when the shock arrives; see section 6 for the casewhere the principal must abide by the deadline under any circumstance.
33
Ghosh and Waldman (2010) also raises job similarity as one of the distinctive characteristics of academia.
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 by Prendergast(1993). However, our argument here may be more applicable to academia (at least in humanities and social
sciences) but less so to other professional service industries such as law where the role of a partner is more
about bringing in business (or “rainmaking”).
34
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.

25

6Conclusion

This paper presents a simple stylized model of dynamic performance evaluation with particular emphasis on the role of commitment to an evaluation deadline. We consider anenvironment where the principal chooses whether to commit to a deadline and if so, at whattime. Within this framework, we obtain a complete characterization of equilibrium and derivea necessary and sufficient condition for committing to a deadline to be optimal. The simple
framework also allows us to conduct various comparative statics exercises to give insight into
when it is beneficial to set an evaluation deadline at the outset.
Since our model is deliberately stylized to obtain sharp analytical characterizations, there
are naturally several avenues to extend the current analysis. Before we conclude, we briefly
discuss two of those possible extensions (see Appendices C and D for more formal analysesof these cases).
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 cost because the only feasible history at each continuation game consists only of
failures up to that point; 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 filledwith uncertainty where there is clearly value in being flexible.
One way to cope with this possibility is to consider a setup where a permanent productivity shock may strike with some probability, which totally changes the nature of the task
and subsequently makes the agent unproductive, i.e., λh = λl = 0.35 For simplicity, supposethat (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. In this situation, the principal can retain
the flexibility to adjust to a negative shock by immediately terminating the project wheneverit strikes. We can then show that due to this benefit, there arises a situation where theprincipal strictly benefits from not committing to a deadline.
Exogenous wages: In the baseline model, we have assumed that the contractual environment is highly incomplete in that no contingent wage contracts can be written. Even insuch an environment, it may still be possible to agree on the flow wage contingent on thecontinuation of the project. Of course, since the flow wage in this context is a pure transfer
35
Once the principal chooses to commit to a deadline, the principal must abide by it and is not allowed
to terminate the project, unlike in the discussion of stable job descriptions in section 5.3 where we allow the
principal to terminate the project whenever a stochastic shock arrives. If the principal is allowed to terminate
the project at any time before the committed deadline, the commitment solution then yields a weakly higher
payoff than the no-commitment solution.

26

payment with no incentive effect on the agent’s side, there is no reason for the principal tooffer any wage beyond the minimum level if she can stop at the right time. However, as wehave seen, this is not always true when the principal makes no commitment to a deadline, inwhich case the flow wage can be used as a substitute for a deadline. This can be potentially
useful if the commitment cost is prohibitively large for some reason.
To explore whether there are circumstances in which she offers any strictly positive w tothis effect, suppose that the principal can deliberately choose any w ≥ 0 at time 0.36 The
benefit of raising w above zero is clear if we look at the case where w = 0: in this case,the instantaneous payoff is always strictly positive (as in the cooperative case), no matter
how unlikely the agent is to succeed; as a consequence, the principal can never terminate
the project, and given this, the agent never exerts high effort. The principal can alter thisstructure by raising w above zero because the flow cost of employment is now positive,
rendering the instantaneous payoff negative at some point. This can be profit-enhancing forthe principal because with the credible threat of termination, she can induce the agent toexert high effort, which is especially beneficial when the success probability under low effort
is low: formally, we can show that the optimal flow wage is strictly positive if λl is sufficiently
small. This reasoning is similar to Buisseret (2016) in that a higher flow wage is used as acommitment device to stop at the right time and reduce future rents for the agent.37

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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rUa (k) = (λa b − da + w) − λa Ua (k) − U̇a (k)(14)

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. Moreover, since Uh (k) is strictly increasing in k, the agent would never switch back30

to low effort, once U (k) reaches this critical point. Then, k∗ is obtained as a solution toUh (k∗ ) = b − ∆dλ , which can be written asd
λh b − d + w∗(1 − e−(λh +r)k ) = b −.
λh + r∆λ(15)Note that the agent always exerts high effort ifb−d
λh b − d + w
rb − wd≥⇔≥,∆λλh + rλl + r∆λwhile he never does so if ∆dλ ≥ b. For in-between cases, solving (15) yields
!
d
(λ
+
r)(b
−
)
h
1
∆
λ
,
ln 1 −k∗ = −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: The agent always exerts high effort in the noncooperative case if(2) fails to hold. An analogous condition for the cooperative case is given byddr
λh (y + b) − d
>y+b−
⇔
>
(y + b).
λh + r∆λ∆λλl + rApplying the same argument as in Proposition 1, we obtain
!
d
(λ
+
r)(y
+
b
−
)
h
1
∆λ
.
ln 1 −k̃∗ = −λh + rλh (y + b) − d
To see that the project is never terminated, we first show that if it is ever optimal to set
a deadline in some finite time, the only possibility is to set it at time k̃∗ , so that the agent
exerts high effort from the beginning. To see this, the principal extends the deadline beyond
time k̃∗ if and only if
λl (y + b) >
λl + r∗λh (y + b) − d (1 − e−(λh +r)k̃ ).
λh + r(16)

Note that this condition is independent of the deadline. As such, if this condition holds, itis then optimal to continue the project indefinitely until the project succeeds. If it fails tohold, on the other hand, it is clearly optimal to set a deadline at time k̃∗ .
Since, by definition,λh (y + b) − d
d
∗
∗
y+b−
= Ũ (k̃ ) =(1 − e−(λh +r)k̃ ).
∆λλh + r31

we can modify (16) asd
λl (y + b)
>y+b−.
λl + r∆λThe left-hand side is the expected payoff of exerting low effort indefinitely while the righthand side is that of exerting high effort from time 0 to time k̃∗ which must equal the static
gain.
This condition must hold for any finite k̃∗ when (6) is satisfied, meaning that it isstrictly better to let the project continue indefinitely. If (6) fails to hold, on the other hand,k̃∗ → ∞, and again it is better to let the project continue indefinitely. This shows that no
finite deadline can improve the social surplus in the cooperative case.
Proof of Proposition 3: First, if φ(qh ) ≥ p0 > qh (the latter inequality holds by Assumption2), there exists a unique deadline τ C ∈ (0, k∗ ] that satisfies
Cp0 e−λh τ= qh .
1 − p0 + p0 e−λh τ C
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 τ C because theinstantaneous payoff is still strictly positive. The optimal termination date is then τ C and it
is unique in this range.
Second, suppose that p0 > φ(qh ), in which case the first-order condition is given by
M (τ C , p0 ) = 0. If M (k∗ , p0 ) ≤ 0, the expected payoff is strictly decreasing in τ C for τ C > k∗ ,and the optimal termination date is τ C = k∗ . Alternatively, define p̂ ∈ (0, 1) such that
M (k∗ , p̂) = 0 if (10) holds and p̂ = 1 otherwise. Then, the optimal termination date isk∗ for p0 ≤ p̂. If p0 > p̂, on the other hand, there must exist a unique τ̂ (p0 ) such that
M (τ̂ (p0 ), p0 ) = 0. Since the instantaneous payoff is strictly positive for τ̂ (p0 ) > τ C andnegative for τ C > τ̂ (p0 ), the optimal termination date in this case is τ̂ (p0 ).
Finally, to show that p̂ > ql if 1 > ql , there are two cases. If (10) holds (but 1 > ql ), thenp̂ = 1 > ql . If (10) fails to hold, on the other hand, M (k∗ , p0 ) is increasing in p0 for any k∗ .
Given that 1 > ql , it suffices to show that M (k∗ , ql ) < 0, i.e.,−rk ∗(1 − ql )we

> ql

!
∗
λl + rπh (1 − e−(λh +r)k ) .
πl −λh + rThis can be written as
rk ∗

−e

λl + r∗ql
πh e−λh k − (1 − ql )w
λh + r
32

!

> ql

!
λl + rπl −πh .
λh + rwhich is further reduced to
rk ∗

−e

ql πl e−λhk∗

− (1 − ql )w > ql (1 − e−(λh+r)k ∗!
λl + rπh .
) πl −λh + r

Note that the left-hand side is positive by definition while the right-hand side is negative, sothat this condition holds for any k∗ . From the preceding argument, we can see that p̂ = 1 if
and only if (10) holds. It is also evident that τ C (p0 ) is weakly increasing in p0 .
Proof of Proposition 4: We show the existence and uniqueness of the equilibrium by
construction. We start with a candidate strategy T 1 = {τ 1 } and the corresponding best
response such thatat =(l for t ∈ [0, max{τ 1 − k∗ , 0}),h for t ∈ [max{τ 1 − k∗ , 0}, τ 1 ].

(17)

This pair of strategies constitutes an equilibrium if V ∗ (K(p, qh ), p) > 0 for all p ∈ (qh , p0 ].
Alternatively, we check for what interval of p0 this condition is satisfied. To this end, thefollowing result is useful.
Lemma 2 Let ql > q. Then, V ∗ (K(p0 , q), p0 ) > 0 for all p0 ∈ (q, 1) if and only if
lim V ∗ (K(p, q), p) > 0.
p↑ql

Proof: Since the necessity is evident by definition, we only prove the sufficiency part. Since
the case with ql = 1 can be proved analogously (by taking the limit), we suppose for now
that 1 > ql > q.
First, if p0 ≥ ql , V ∗ (K(ql , q), ql ) > 0 directly implies V ∗ (K(p0 , q), p0 ) > 0 for all p0 ≥ ql
because the instantaneous payoff is strictly positive for any effort choice when the belief isabove ql . Note also that V ∗ (K(p0 , q), p0 ) > 0 for all p0 ∈ (q, φ(qh )] since the high type always
exerts high effort in this case. This implies that V ∗ (K(p0 , q), p0 ) > 0 for all p0 ∈ (q, 1) if
φ(q) ≥ ql . In what follows, therefore, we assume ql > φ(q).
Given this, it suffices to show thatV ∗ (K(ql , q), ql ) > 0 ⇒ V ∗ (K(p0 , q), p0 ) > 0 for all p0 ∈ (φ(q), ql ).
Suppose on the contrary that V ∗ (K(ql , q), ql ) > 0 but V ∗ (K(p, q), p) ≤ 0 for some p ∈(φ(q), ql ). If this is the case, there must exist some z ∈ (φ(q), ql ) such that V ∗ (K(z, q), z) = 0.
Since
V ∗ (K(p, q), p) = (1 − p)wp(λl y − w)′′
1 − e−(λl +r)(K(p,q)−K(p ,q)) −1 − e−r(K(p,q)−K(p ,q))λl + rr−λl (K(p,q)−K(p′ ,q)) −r(K(p,q)−K(p′ ,q)) ∗+ 1 − p + peeV (K(p′ , q), p′ ),33

for any p > p′ > φ(q), due to the recursive structure of the expected payoff, we have
 (1 − ql )w
ql (λl y − w)
1 − e−(λl +r)∆K −1 − e−r∆K
λl + rr−λl ∆K −r∆K ∗+ (1 − ql + ql e)e
V (K(z, q), z) (1 − ql )w
ql (λl y − w)
1 − e−(λl +r)∆K −1 − e−r∆K ,=λl + rr

V ∗ (K(ql , q), ql ) =where ∆K := K(ql , q) − K(z, q). This is strictly negative by definition, and therefore acontradiction.
The lemma shows that we only need to check the continuation payoff at ql to see whether
a given strategy is sequentially rational.

An immediate corollary of the lemma is thatV ∗ (K(p0 , qh ), p0 ) > 0 for all p0 if φ(qh ) ≥ ql . In this case, m(Θ) = 1, P 1 = qh and T = {τ 1 }constitutes an equilibrium for any p0 .
We now suppose that ql > φ(qh ) and check if the candidate strategy can satisfy theequilibrium conditions. From Lemma 2, this is the case if and only if V ∗ (K(ql , qh ), ql ) > 0,i.e.,
(1 − ql )w
ql πh −(λl +r)(K−k∗ )q l πl∗∗ e
1 − e−(λh +r)k >1 − e−(λl +r)(K−k ) +1 − e−rK .
λl + rλh + rr(18)
where K = K(ql , qh ). If (18) holds, the expected payoff is always positive for any p0 , so
we can apply the same argument as in Case 2 to show the existence and uniqueness of theequilibrium.
The situation becomes more complicated when (18) fails to hold, in which case the principal has an incentive to deviate and stop at some point, and the candidate strategy no longer
constitutes an equilibrium. In this case, there instead exists a critical belief that satisfies (18)with equality, because V ∗ (K(p, qh ), p) is continuous in p with V ∗ (K(φ(qh ), qh ), φ(qh )) > 0 >V ∗ (K(ql , qh ), ql ). We now redefine P 1 = min{q : V ∗ (K(q, qh ), q) = 0} and P 2 = qh . Given
the critical belief, we can then consider another candidate strategy T 2 = {τ 1 , τ 2 } for p0 > P 1
where pτ 1 = q 1 , and the best response is given byl for t ∈ [0, max{τ 1 − k∗ , 0}),
h for t ∈ [max{τ 1 − k∗ , 0}, τ 1 ),at =l for t ∈ (τ 1 , τ 2 − k∗ ),h for t ∈ [τ 2 − k∗ , τ 2 ].
Note that q 1 > φ(q 2 ) > q 2 = qh or τ 2 > τ 1 + k∗ .

The game is now divided into two segments. First, we can show that the equilibrium is
unique after time τ 1 (as done in Case 2). Before time τ 1 , we can apply the same procedure to34

the truncated interval [0, τ 1 ] to see whether T 2 can be an equilibrium strategy. More precisely,
we need to check if V ∗ (K(p, P 1 ), P 1 ) > 0 for all p ∈ [p0 , φ(P 1 )). Applying Lemma 2, T 2
is an equilibrium strategy for all p0 ∈ (P 1 , 1) if V (K(ql , P 1 ), ql ) > 0. For p0 ∈ (P 1 , 1), theprincipal’s equilibrium strategy consists of two termination dates. Given that the continuation
equilibrium after time τ 1 is unique, we can establish the uniqueness by applying the same
argument.
If V (K(ql , P 1 ), ql ) ≤ 0, on the other hand, we can find yet another critical belief in(φ(P 1 ), ql ]. We again redefine P 1 analogously and repeat the same process until we findP 1 such that V (K(ql , P 1 ), ql ) > 0. In the end, this process would give us m(Θ) intervals{(P j , P j−1 )}m
j=1 where the principal’s equilibrium strategy consists of n = m(Θ) − j + 1
termination dates if p0 falls into (P j , P j−1 ). Note also that since V ∗ (K(P 1 , P 2 ), P 1 ) = 0 by
definition, it must be that ql > P 1 if m(Θ) > 1.
Proof of Proposition 5: (i) For p0 ∈ (qh , φ(qh )), τ C (p0 ) = τ N C (p0 ) = K(p0 , qh ). Since theagent’s strategy depends only on the remaining time, the two cases yield the same expected
payoff for the principal.
(ii) To show this, we first examine how V C and V N C vary with respect to p0 for p0 ∈ (φ(qh ), 1).
It is evident that V C must monotonically increase with the initial prior. To see this, note
that
dV C
∂V ∗ ∂V ∗ dτ C
=
+ C
.
dp0
∂p0
∂τ dp0The last term is the indirect effect though an increase in τ C which is invariably zero because
∗

C

∗

∂V
= 0 or dτeither ∂V
dp0 = 0 must hold by the optimality condition. Since the direct effect ∂p0∂τ C

is unambiguously positive, V C is strictly increasing in p0 . More precisely, we obtain
dV C
∗
w
∗
πh
(1 − e−(λh +r)k ) + (1 − e−rk )=dp0
λh + rr
∗∗= p0 πh − (1 − p0 )w − (p0 λh + r)V C (p0 ) − p0 e−λh k πh − (1 − p0 )w e−rk ,(19)for p0 ∈ (φ(qh ), p̂), and

C

∗

dV C
w
πl
e−(λl +r)(τ −k ) πh
C
∗
∗
C
=
(1 − e−(λl +r)(τ −k ) ) +(1 − e−(λh +r)k ) + (1 − e−rτ )dp0
λl + rλh + rrC∗C∗λh + r(1 − e−(λl +r)(τ −k ) ) + p0 πh e−(λl +r)(τ −k ) − (1 − p0 )w= p 0 πlλl + r

C
C
∗
∗
(20)
− (p0 λh + r)V C (p0 ) − p0 πh e−λl (τ −k )−λh k − (1 − p0 )w e−rτ ,for p0 ∈ (p̂, 1).

35In contrast, V N C changes with p0 in a non-monotonic way if m > 1. Consider p0 ∈(φ(P j ), φ(P j−1 )) for j = 2, 3, ..., m. Since V N C (p0 ) = V ∗ (K(p0 , q), p0 ) where q is the closest
termination belief, it follows that∂V ∗∂V ∗ ∂V ∗ ∂K
∂V ∗1
dV N C
=
+
=
+
.
dp0
∂p0
∂k ∂p0∂p0
λl p0 (1 − p0 ) ∂k
First, for p0 ∈ (φ(P j ), P j−1 ), we obtain∗dV N Cπl∗πh e−(λl +r)(K−k )∗w=(1 − e−(λl +r)(K−k ) ) +(1 − e−(λh +r)k ) + (1 − e−rK )dp0
λl + rλh + r
r
!
∗
πl
πh
∗
we−rK
e−(λl +r)(K−k )−(1 − e−(λl +r)k ) −+λl (1 − p0 )λl + r λh + r
λl p 0= p0 πl − (1 − p0 )w − (p0 λl + r)V N C (p0 ),(21)which is negative because ql > P 1 > p0 . For p0 ∈ (P j−1 , φ(P j−1 )), we obtain

w
e−rK
πh
dV N C(1 − e−(λh +r)K ) + (1 − e−rK ) +=p0 πh e−λh K − (1 − p0 )w
dp0
λh + rr
λh p0 (1 − p0 )= p0 πh − (1 − p0 )w − (p0 λh + r)V N C (p0 ),(22)which is positive for any p0 > qh .
We are now ready to see how ∆V changes with respect to p0 . First, it is clear that ∆V
is strictly increasing in p0 for p0 ∈ (φ(P j ), P j−1 ) because V C is increasing while V N C isdecreasing. In contrast, for p0 ∈ (P j−1 , φ(P j−1 )), both V C and V N C are increasing in p0 .
For p̂ > p0 , it follows from (19) and (22) that


d∆V
∗
< 0 ⇔ (p0 λh + r) V C (p0 ) − V N C (p0 ) > − p0 e−λk − (1 − p0 )w e−rk∗ .
dp0∗This condition always holds because p0 e−λh k > (1 − p0 )w and V C (p0 ) ≥ V N C (p0 ) by definition. Similarly, for p0 > p̂, comparing (20) and (21) yields!
λh + rC∗
p 0 πh −πl (1 − e−(λl +r)(τ −k ) ) + (p0 λh + r) V C (p0 ) − V N C (p0 )λl + rCC∗∗> − p0 e−λl (τ −k )−λk − (1 − p0 )w e−rτ ,which again always holds.

(iii) If p̂ ≥ φ(P 1 ), then τ C (φ(P 1 )) = k∗ . Since τ N C (φ(P 1 )) = k∗ by definition, ∆V (φ(P 1 )) =0. To see the necessity, note that ∆V (p0 ) = 0 implies τ C (p0 ) = τ N C (p0 ). Since τ N C (φ(P 1 )) =k∗ , we must have τ C (φ(P 1 )) = k∗ which holds if and only if p̂ ≥ φ(P 1 ).

36

(iv) Suppose first that 1 > p̂, in which case τ C and τ N C solve−λτ C (p0 )e1 − p0=p0−λl τ NC (p0 )e∗(λh + r)we−(λ+r)k∗(λh + r)πl − (λl + r)πh (1 − e−(λh +r)k )1 − p0=p0

q1(1 − q 1 )e−∆λ k∗

!

=:

!

=:

1 − p0 CΦ ,p0

1 − p0 N CΦ ,p0respectively, for p0 > p̂. Note that both ΦC and ΦN C are independent of p0 . From this, since
limp0 →1 τ C (p0 ) = ∞ and limp0 →1 τ N C (p0 ) = ∞, we obtainlim V C (p0 ) = lim V N C (p0 ) =

p0 →1

p0 →1

(1 − p0 )w
p 0 πl−.
λl + rr

If p̂ = 1, on the other hand, τ C (p0 ) = k∗ for all p0 , but as we have seen, limp0 →1 τ N C (p0 ) = ∞.

Appendix B: the value of commitment in (φ(P 1 ), 1)
As we have seen in Proposition 5, the value of commitment moves with some regularities forp0 ∈ (qh , φ(P 1 )). The exception is the last segment (φ(P 1 ), 1). The reason is that for this(and only this) segment, we could have φ(P 1 ) > p̂ > ql (see Propositions 3 and 4), so thatthe instantaneous payoff under low effort can be positive and τ C (p0 ) > k∗ for p0 ∈ (φ(P 1 ), 1).
There are three cases we need to consider, depending on the value of p̂.
Case 1: φ(P 1 ) > p̂In this case, τ C (φ(P 1 )) > τ N C (φ(P 1 )) = k∗ and hence ∆V (φ(P 1 )) > 0 (as we show in(iii) of Proposition 4). For p0 > φ(P 1 ), τ C (p0 ) and τ N C (p0 ) increase proportionally (see theproof of Proposition 4), with both diverging to infinity as p0 → 1. As we show in (iv) of
Proposition 4, ∆V (p0 ) converges to zero as p0 → 1.
Case 2: p̂ > φ(P 1 )In this case, τ C (φ(P 1 )) = τ N C (φ(P 1 )) = k∗ and hence ∆V (φ(P 1 )) = 0. For p0 ∈(φ(P 1 ), p̂), τ N C (p0 ) continues to go up while τ C (p0 ) stays at k∗ ; as a consequence, ∆V (p0 )gradually increases in this range. Once p0 goes above p̂, τ C (p0 ) also gradually increase, but
again proportionally to τ N C (p0 ). Since τ N C (p0 ) > τ C (p0 ) for p0 ∈ (p̂, 1) in this case, ∆V (p0 )gradually decreases and converges to zero as p0 → 1.38
38
If p̂ = φ(P 1 ), then τ C (p0 ) = τ NC (p0 ) and hence ∆V (p0 ) = 0 for p0 ∈ (p̂, 1). We ignore this non-generic
case.

37

Case 3: p̂ = 1
This is the most straightforward case. Given that p̂ = 1, τ C (p0 ) = k∗ for p0 ∈ (p̂, 1) andhence τ C (p0 ) and τ N C (p0 ) diverge away from each other in this range. As a consequence,∆V (p0 ) increases as in all the preceding intervals.

Appendix C: The case with uncertainty
We analyze an extended version of the baseline model where a permanent productivity shock
may strike with some probability. For simplicity, we assume that a shock arrives with aPoisson arrival rate β; once it arrives, the agent becomes totally unproductive thereafter,i.e., λh = λl = 0. We also assume that the arrival of a shock is publicly observable. Other
more complicated cases, e.g., those with transitory shocks, can in principle be analyzed in an
analogous manner.
The agent: The agent’s problem requires a slight modification. The augmented Bellman
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 7 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: We first consider the case where the principal commits to a deadline. Note that the
principal cannot terminate the project even if the shock arrives before the deadline. Since theagent has no incentive to exert high effort, we have Ũ C (k) = w(1−e
r

−rk )

. Let UaC (k) denotethe value function under commitment when the effort level is fixed at a. We then obtain!
−(λh +β+r)k−rk (1 − e−(λh +β)k )
1
−
e
βw
e
bλ
−
d
+
w
h
.
(1 − e−(λh +β+r)k ) +−UhC (k) =λh + β + rr
λh + β + r
λh + β38

The 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 + rd
N C given by
h −d+wIf bλ
λh +β+r > b − ∆λ > 0, there exists an interior threshold k

!
d
(λ
+
β
+
r)(b
−
)
h
1
∆λ
kN C = −.
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 obtainbλh − dλh w
UhC (k) =(1 − e−(λh +β+r)k ) −λh + β + rr

1 − e−(λh +β+r)ke−rk (1 − e−(λh +β)k )−λh + β + r
λh + βw(1 − e−rk )r−rk )bλh − d − λh w(1−eλh we−rkr(1 − e−(λh +β+r)k ) +=λh + β + r
r

!

+

+

w(1 − e−rk ).
r
39

1 − e−(λh +β)k1 − e−(λh +β+r)k−λh + βλh + β + r

!

−αk

First, 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−eCrUh (k) >λh + β + r
+

−rk )

)

−(λh +β+r)k(1 − eλh we−rk)+r

1 − e−(λh +β)k1 − e−(λh +β+r)k−λh + βλh + β + r!

w(1 − e−rk ),rfor k ≤ kC . This is further reduced to
!
!
−rk )
λh (1 − e−(λh +β+r)k )w(1−e
λh we−rk
1−
UhC (k) −>λh + β + rrr
which gives UhC (k) > w(1−e
r

−rk )

!
1 − e−(λh +β+r)k1 − e−(λh +β)k−,λh + βλh + β + r.

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, eitherwith or without commitment (kN C > k C > k∗ ). 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 thelowest possible payoff as long as the project survives. The situation is even worse withoutcommitment, however, because the principal can immediately terminate the project, thereby
leaving no rent for the agent. The extent of the loss is therefore larger when the principal
makes no commitment, which amounts to the fact that kN C > kC and 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 (k, p0 ) denote the principal’s expected payoff at time 0 under commitment,
which can be written as
Z νCRs(λl ps y − w)e− 0 (λl pu +β+r)du ds
V C (k, p0 ) =
0
Z k
R C
Rs
− 0ν (λl pu +β+r)du+e(λh ps y − w)e− ν C (λh pu +β+r)du ds
νC

Z νC

Rs
β
w(1 − e−r(k−s) )e− 0 (λl pu +β+r)du dsr 0ZRs
β − R ν C (λl pu +β+r)du k− e 0w(1 − e−r(k−s) )e− ν C (λh pu +β+r)du ds.
r
νC

−

40

(23)

where ν C := max{τ C − kC , 0}. The difference from the previous case is that a negative shock
may now strike with some probability, in which case the principal earns zero from that point
on. The commitment solution is obtained asτ C = argmaxτ V0C (τ ).
As can be seen from (23), 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) :=pNC .
p + (1 − p)e−λh kGiven 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(k, p) =Z ν NCRs(λl ps y − w)e− 0 (λl pu +β+r)du ds0
Z k
R ν NCRs+ e− 0 (λl pu +β+r)du(λh ps y − w)e− ν NC (λh pu +β+r)du ds,(24)

ν NC (q)where ν N C is defined analogously. Note that for the same time horizon, V N C (k, p0 ) >V0C (k, p0 ) for any p0 .
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. We can thenmake the following statement.
Proposition 8 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 to41

set a deadline before the belief reaches qh . The profit is clearly lower than V N C (τ N C , p0 ). 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 commitment through two channels. First,
by making no commitment, the principal can retain the flexibility to adjust to the negative
shock by 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 ofproject termination, and thus, he starts exerting high effort earlier. Since φN C (qh ) > φ(qh )for any β > 0 by Proposition 7, we could have φN C (qh ) ≥ p0 > φ(qh ), in which case the valueof commitment is strictly negative with uncertainty whereas it is strictly positive without.

7

Appendix D: The case with endogenous wages

We consider a setup where the principal can deliberate choose w ≥ 0 at time 0. In general,
an increase in w above zero yields two opposing effects. On one hand, it raises the criticalvalue qh , which may give the principal the incentive to stop earlier than otherwise. On theother hand, it also raises the expected future payoff which in turn lowers k∗ and gives theagent the incentive to procrastinate even more. A necessary condition for w > 0 is that thefirst effect dominates the second, or alternatively that τ N C − k∗ decreases with w.39 We canthen make the following statement.
Proposition 9 It is optimal to set w > 0 if λl is sufficiently small.
Proof: As w → 0, we have qh → 0 and ql → 0, which implies τ N C (p0 ) → ∞ for any p0 . The
expected payoff then converges tolim V ∗ (τ N C , p0 ) =w→0

p 0 λl y,
λl + r(25)

for any p0 . Define µ(w) such that φ( λwh y ) = µ(w), i.e.,∗µ(w)−λh kw.
∗ = qh =−λk
1 − µ(w) + µ(w)λh y39Note that τ NC − k∗ is the interval during which the agent exerts low effort. If it is longer with a smallerk , the principal’s expected payoff is necessarily lower.
∗

42

Note thatd(λh + r)(b − ∆λ )1 − qh −λh k∗
λh y − w
1 − µ(w)
1−
=
e
=
µ(w)
qh
w
λh b − d + w

! λh

λh +r

.

Since limw→0 µ(w) = 0 and µ(λh y) = 1, there must exist at least one w̃ such that µ(w̃) = p0 .
If we set w = w̃, then τ N C = k∗ > 0 by definition, and the expected payoff is∗

V (k , p0 ) =

Z k∗

Rs

(λh yps y − w̃)e− 0 (λh pu +r)du ds,0which is always strictly positive and larger than (25) if λl is sufficiently small.

43

pt

ql

P 1 = qhk∗
τ 1 − k∗τ1t

Figure 1: Evolution of the belief on the equilibrium path (m(Θ) = 1)pt

V ∗ (τ 1 − t, pt ) < 0

qlq′

P 1 = qhk∗
τ 1 − k∗τ′

τ1Figure 2: A violation of Condition C: the belief (m(Θ) = 2)

44

t

V∗

V ∗ (τ 1 − t, pt ) < 00k∗τ 1 − k∗

τ′

τ1

t

Figure 3: A violation of Condition C: the continuation payoff
pt

ql

P1

P 2 = qhk∗
τ NC − k∗k∗τ 2 − k∗

τ 1 = τ NC

τ2

Figure 4: Evolution of the belief, on and off the equilibrium path

45

t

V∗
V C (p0 )V NC (p0 )φ(P 2 )

P 2 = qhP1

φ(P 1 )

p0

Figure 5: The expected payoffs with and without commitment
τK(p0 , qh )

τ C (p0 )qh

φ(qh )

p0

Figure 6a: The commitment solution when λl is relatively high (λl = 0.07)46τK(p0 , qh )

τ C (p0 )qh

φ(qh )

p0

Figure 6b: The commitment solution when λl is relatively low (λl = 0.04)
ττ NC (p0 )τ C (p0 )φ(P 2 )

P 2 = qh

P1

Figure 7: The no-commitment solution

47

φ(P 1 )p0

Dynamic Performance Evaluation with Deadlines:
The Role of Commitment∗
Chia-Hui Chen† and Junichiro Ishida‡
October 20, 2017

Abstract
We consider an environment in which a principal hires an agent and evaluates his productivity over time in an ongoing relationship. The problem is embedded in a continuoustime model with both hidden action and hidden information, where the principal must
induce the agent to exert effort to facilitate her learning process. The value of committing to a deadline is examined in this environment, and factors which make the deadline
more profitable are identified. Our framework generates a unique recursive equilibrium
structure under no commitment which can be exploited to obtain a full characterizationof equilibrium. The analysis allows us to evaluate the exact value of commitment for any
given set of parameters and provides insight into when it is beneficial to commit to an
evaluation deadline at the outset of a relationship.

JEL Classification Number: D82, D83
Keywords: dynamic agency, deadlines, experimentation, commitment, up-or-out contracts.
∗

The first author acknowledges financial support from JSPS KAKENHI Grant-in-Aid for Scientific Research(B) 16H03615 and Young Scientists (B) 17K13699. The second author acknowledges financial support from
JSPS KAKENHI Grant-in-Aid for Scientific Research (S) 15H05728, (A) 20245031, (B) 16H03615 and (C)24530196 as well as the program of the Joint Usage/Research Center for Behavioral Economics at ISER, Osaka
University.
†
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 heis sufficiently productive. As is often the case, however, the worker’s sheer productivity is
not directly observable to the employer, who must instead make an inference from a sequenceof observed outputs. Since success cannot be achieved overnight, the employer must exercise
some patience even when things do not appear to be in good shape. Excessive tolerance for
failure, however, diminishes the worker’s incentive to take costly actions that are indispensable
to be successful. A number of issues arise in this dynamic environment. How much timeshould 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 analyzing a situation where a principal hires an agent and evaluates his productivity over time in an ongoing relationship. The
problem is embedded in a continuous-time framework with both hidden action and hidden
information. At each instant, the agent privately chooses how much effort to supply. The outcome is either a success or a failure, depending on his effort choice as well as his productivity
type. The game ends immediately when the agent achieves a success (or a “breakthrough”).
The principal’s task in this environment is to determine when to terminate the project, conditional on no success having occurred. Within this setup, we analyze two distinct cases toillustrate the role of commitment: 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 reachedwithout attaining a success; in the other, the principal makes no such commitment, thereby
retaining discretion to terminate the project at any instant, and simply terminates the project
when the continuation payoff is not high enough to justify further experimentation. By directly comparing these two cases, we evaluate the extent to which the principal benefits fromcommitting to an evaluation deadline in this dynamic environment.
The driving force of our analysis is a dynamic strategic interaction between the agent’s
effort choice and the principal’s termination strategy. On one hand, the agent’s effort choicedepends 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 choice: when the agent is less motivated and exerts low
effort, less information is revealed about his type, which makes the principal more reluctant

1

to terminate the project. This strategic interaction can generate a vicious cycle where theprincipal’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.
The main contribution of the paper is that we devise an analytical framework that is
tractable enough to admit a complete characterization of pure-strategy equilibria, both withand without commitment, while capturing this dynamic interaction. Our framework thus
allows us to evaluate the exact value of commitment to an evaluation deadline for any given
set of parameters. To this end, our main analytical focus is on the no-commitment case,which is generally far more challenging than the commitment case when dynamic interactions
are considered. In the no-commitment case, the principal’s termination strategy must be
sequentially rational along the way, and the belief off the equilibrium path plays a crucial role.
Even in this case, we can show that our framework generates a unique recursive equilibrium
structure which can be exploited to establish the existence and uniqueness of (pure-strategy)
perfect Bayesian equilibrium. We then build on this result to derive a necessary and sufficientcondition under which the principal can strictly benefit from committing to an evaluation
deadline.
As can be expected, the lack of commitment to a deadline entirely alters the dynamic
allocation of effort as well as the timing of project termination. When the principal sets a
deadline at the outset, she does so by taking into account how it affects the agent’s entire effort
sequence. In particular, since extending a deadline in general relaxes the agent’s incentive
compatibility constraint and reduces his total effort supply, the deadline must be set at apoint where the expected benefit (of achieving a success at the next instant) equals the costof a decrease in total effort. The situation changes rather drastically when she makes no suchcommitment, because her termination decision would have no influence on the agent’s past
behavior. As such, she terminates the project whenever the continuation payoff is about toturn negative while taking the agent’s effort strategy as given. This fact implies that the costof extending a deadline is smaller under no commitment and tends to give her an excessive
incentive to wait for a success.
Given this incentive structure, one may expect that the principal would always wait longer
for a success in equilibrium under no commitment. As it turns out, though, this conjecture
does not always hold true in the current setup. Among other things, an interesting, andsomewhat counterintuitive, property of our model is that the average duration of the project
may not be monotonic with respect to the initial prior belief under no commitment, whereas it
is always weakly increasing under commitment. As a direct consequence of this, the average

2duration 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
from the principal’s reluctance to terminate the project in the first place. The principal tends
to terminate the project too early when the agent’s productivity under low effort is sufficiently
small. For the sake of argument, suppose that the agent fails almost surely when he exerts
low effort, in which case the expected instantaneous payoff is negative while the principal canlearn almost nothing from failures (during the phase where the agent is supposed to exert low
effort). The principal’s belief then declines very slowly over time, forcing her to incur a large
amount of loss if she is to wait until she totally loses her confidence in the agent’s ability. If
this expected loss is prohibitively large, the principal may find it optimal to terminate theproject even when the belief is still relatively high.
Of course, in equilibrium, the agent correctly anticipates this reaction and adjusts his
effort allocation accordingly. We show that the equilibrium timing of project termination
can be pinned down by backward induction, where we start from the final critical time (tobe derived) and solve backward. This process gives rise to the aforementioned recursive
equilibrium structure which allows us to establish the uniqueness of equilibrium and also
obtain a diverse set of equilibrium dynamics under no commitment.
The implications of our analysis can be applied broadly to a range of circumstances inwhich 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, aprofessor who must evaluate graduate students, a head coach in professional sports who mustevaluate players, and so on. Among those possibilities, the most prominent example of evaluation schemes with deadlines is perhaps the “up-or-out system,” which is widely observedin academia and professional service industries such as law, accounting, and consulting. As a
specific application, our framework offers some insight for when up-or-out contracts are morevaluable by identifying several key factors – such as high ability intensity, stable job descriptions, and similar jobs across ranks – which favor the use of an evaluation deadline from apreviously unexplored channel. Each of these factors intensifies either the agent’s incentive todelay exerting high effort for a given deadline or the principal’s incentive to delay terminating
the project for a given effort sequence (or both), thereby rendering it more profitable to set
a deadline at the outset.
The paper is organized as follows. The literature review is provided in the remainder ofthis section. The model is presented in section 2 and analyzed in sections 3 and 4, where we3characterize both the commitment and no-commitment solutions. These solutions are thencompared in section 5, in order to analyze the value of commitment and derive a necessary
and sufficient condition under which it is strictly optimal to commit to a deadline. Some
extensions of the baseline model 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 through a sequenceof experiments.1 From the principal’s point of view, our model can be seen as a variant of
the canonical two-armed bandit problem with one safe arm (terminating the project) and
one risky arm (continuing the project). Our model is particularly related to the literature on
strategic experimentation which analyzes a situation where a group of individuals, rather thana single individual, face bandit problems (Bolton and Harris, 1999; Bergemann and Välimäki,
1996, 2000; Keller et al., 2005; Klein and Rady, 2011; Bonatti and Hörner, 2011). A crucial
difference from this strand of literature is that experiments in our context are “intermediated,”i.e., experiments are conducted not by the principal herself but by an informed intermediary,
the agent, who faces no uncertainty about the project.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.3 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
the agent. 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 from1

An early economic application of the bandit problem can be found in Rothschild (1974). See Bergemann
and Välimäki (2008) for a survey.
2
Several recent works analyze models of “delegated experimentation” where a principal delegates experimentation to an agent. Guo (2014) analyzes a situation where the principal can specify, with full commitment
power, how the agent should allocate the resource in all future contingencies and solves for the optimal delegation rule. Garfagnini (2011) considers a similar setting to ours but assumes that: (i) the principal and theagent are symmetrically informed about the state of nature; and (ii) the agent’s payoff is independent of thestate. Chen and Ishida (2015) consider the opposite case in which the principal, with the termination right,
may be privately informed about the project quality while the agent focuses on implementing the project
assigned to him.
3
Also, see Buisseret (2016) who considers a two-period model of this setting with a more general, convex
cost function.

4

some technical differences,4 these previous works are primarily concerned with characterizing
optimal contracts. In contrast, the aim of this paper is to compare the allocations under fullcommitment and no commitment, while considering a less complete contractual environment,in order to evaluate the extent to which the principal can gain from committing to a deadline
at the outset. Although it is hard to tell a priori which specification is more plausible, as it
depends on the details of the underlying situation, our simple and tractable framework allows
us to identify and illuminate the counterintuitive role of commitment in dynamic performance
evaluation.
Several recent works examine the role of commitment in dynamic moral hazard setting.
Mason and Välimäki (2015) consider a dynamic moral hazard problem and derive optimal
wage contracts, both with and without commitment on wage payments. Hörner and Samuelson (2016) consider a repeated-game setting in which the principal chooses the scale of the
project in addition to contingent payments and characterize the set of equilibrium payoff
vectors that can be achieved without commitment.5 A key difference from our model is thatthese two works do not consider the principal’s learning, which is our main focus, with nohidden information about the agent’s type; as a consequence, the project is never terminatedin their models.6 This stands in sharp contrast to our setting in which the principal accumulates information about the agent over time, and the project must be terminated in finite
time.7
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
in dynamic problems with multiple agents. Bonatti and Hörner (2011) analyze a dynamic
4
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 (with stochastic signals), rather
than of experimentation, from the agent’s point of view.
5
In their model, outcomes are privately observed by the agent, and the moral-hazard problem regards thetruthful disclosure of this private information.
6
Georgiadis et al. (2014) analyze the role of commitment in a dynamic contribution games where themanager has the decision right over the project size. With multiple agent types and dynamic learning, Bonattiand Hörner (2017) analyze a (symmetric-information) experimentation model in which wages are determined
competitively a la Holmström (1999) and characterize effort and wage dynamics with an exogenous terminationdate.
7
A subtle technical difference which directly arises from this fact lies in the ways in which to construct
an equilibrium. Mason and Välimäki (2016) and Hörner and Samuelson (2017) construct an equilibrium via
arguments found in the analysis of infinitely repeated games (reversion to the worst continuation equilibrium),and as such, their analyses yield a non-degenerate set of equilibria. In our analysis, an equilibrium is obtained
via backward induction which gives rise to a unique equilibrium.

5moral-hazard problem with a team in which multiple agents work on a project of unknown
quality and briefly 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 reveal that he has been successful. The focus of these works is placedon the interaction between the agents, especially the freerider problem, whereas ours is on
the dynamic interaction between the principal’s termination decisions and the agent’s effort
choices.

2

A dynamic model of performance evaluation

Environment: We employ a continuous-time model because of its greater tractability. 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) ormediocre with probability 1 − p0 . The ability type is the agent’s private information andis not directly observable to the principal who must instead evaluate it from a sequence ofobserved outcomes.
Production: The agent makes unobservable effort at ∈ {l, h} at each instant t.8 We interpretthat 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 be directly
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 as well as theagent’s type. If the good type chooses at = a over time [t, t + dt), he attains a success withprobability λa dt where λh > λl > 0. In contrast, the mediocre type can never succeed withany effort level.9 Define ∆λ := λh − λl .
Payoff: We consider an incomplete-contracting environment where contingent rewards on
the arrival of a success cannot be enforced.10 A success yields a net present value of y > 0
8

Our focus on binary effort reflects our implicit presumption that the effort cost and success probability
are linear, as usually assumed in this literature (e.g., Keller et al, 2005; Bonatti and Horner, 2011, for mostof their analysis). As long as this structure is maintained, an extension to continuous effort, say at ∈ [l, h],yields exactly the same allocation and is hence irrelevant.
9
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, 2017).
10
In reality, we rarely observe complete wage contracts in industries comprised of professionals. For instance,
few academic institutions, if any, offer rewards specifically contingent on verifiable measures of output: salaries

6to the principal and b > 0 to the agent;11 otherwise, they both receive zero. Aside from this,the principal must also pay a flow wage w > 0 to the agent as long as the project continues.12
We assume that w is exogenously given for most part; the case with an endogenous wage isbriefly discussed in the concluding section. The reservation payoff is assumed to be zero forboth players. The common discount rate is denoted by r ∈ (0, ∞).
Contract: The only contractible decision for the principal in this environment is whether
to set a deadline, and if so, at what point. If the principal commits to a deadline, sheterminates the project at the deadline (but never before) if the agent has not attained a
success up to that point. If the principal chooses not to commit to any specific deadline, on
the other hand, she retains discretion to terminate the project at any instant. As mentioned
earlier, our analytical focus is on the latter case which requires that both players’ strategies
be sequentially rational.

3

Analysis

3.1

Agent’s effort decision

The agent decides whether or not to exert high effort at each instant. 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 time τ , in whichcase the remaining time at time t is simply given by τ − 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.
are determined through bilateral negotiations, often dictated by market forces, in some countries whereas they
are subject to bureaucratic regulations in others. One possible reason for the lack of complete wage contracts
in those industries is that it is often difficult, and perhaps prohibitively costly, to measure the exact value ofa “success” in a verifiable manner. Additionally, this assumption makes our analytical framework applicable
to a wider range of circumstances. One such possibility is that the benefit of achieving a success accrues from
non-transferrable (psychological) gains such as prestige, authority, and the sense of achievement, and extrinsic
rewards are hence of secondary importance. There are also many cases where contingent monetary transfers
are neither feasible nor desirable, as in a professor-student relationship.
11
An obvious interpretation is that y and b represent the continuation payoffs of achieving a success, including
not only the intrinsic value of a successful outcome but also other benefits of identifying/signaling talent.
12
Alternatively, w can be regarded as a flow cost of production (e.g., hiring an agent). Of course, under
this interpretation, w is no longer a transfer payment to the agent and does not appear in his payoff. It is
straightforward to make the setup consistent with this interpretation without having any qualitative impact
on our results.

7

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 asU (k) = max (λa b − da + w)dt + e−rdt (1 − λa dt)U (k − dt) .
a∈{l,h}Taking the limit dt → 0, we obtain the Bellman equation:rU (k) = max λa b − da + w − λa U (k) − U̇ (k) ,a∈{l,h}(1)with limk↓0 U (k) = 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), the continuation

payoff U (k) captures the agent’s reservation payoff which he receives in case of a failure. The
cost of not succeeding today is obviously small when he has a high reservation payoff. Since
the continuation payoff is higher when the agent has more time to prove himself, the agent
has a stronger incentive to work hard as the project approaches the termination date. Thisis a manifestation of the deadline effect that lies at the core of our entire analysis.
Proposition 1 If
rb − wd>,∆λλl + r(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.

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 enough toovercome the dynamic agency cost, and the agent is willing to supply high 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.
8

Assumption 1 λhλb−d+w> b − ∆dλ > 0.
h +r
Among other things, the assumption implies that the optimal threshold k∗ is bounded fromabove and away from zero, i.e., k∗ ∈ (0, ∞).

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. Given some effort sequence {as }ts=0 , the updated belief is then given by
Rt

pt =

p0 e− 0 λas ds
Rt

1 − p0 + p0 e− 0 λas ds.

(4)

Alternatively, taking the time derivative, we obtainṗt = −λat pt (1 − pt ).

(5)

It is clear that the belief is strictly decreasing over time for any effort choice (due to the factthat λl > 0) until the agent attains a success, in which case the belief immediately jumps up
to one.
The principal’s problem is to determine when to terminate the project, conditional on no
success having occurred. Now suppose that the principal intends to terminate the project attime τ > t. Taking the effort sequence as given, the principal’s continuation payoff can also
be written as a function of the remaining time k and the current belief pt where
Z t+kRs(λas ps y − w)e− t (λau pu +r)du ds,
V (k, pt ) =tsubject to (5).
It is immediate to see that for a given effort level, there exists a threshold belief below
which the principal’s instantaneous payoff is strictly negative. We denote by qa := min{ λwa y , 1}
the “break-even” belief at which the instantaneous payoff equals zero under effort a. Combined with the fact that the belief is strictly decreasing over time for any effort sequence, qh
represents the absolute lower bound of the belief, as the principal clearly has no incentive tocontinue the project once her belief dips below this level. Since the belief must reach thislevel sooner or later (due to the fact that λl > 0 and w > 0), the presence of such a lower
bound suggests that the game must end in some finite time. This allows us to solve the gameby backward induction.
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.
9

Assumption 2 p0 > qh = λwh y .
Since the principal’s continuation payoff depends on her belief, it is often convenient
for the subsequent analysis to explicitly relate the current and termination beliefs to theremaining time. Suppose that the current belief is p and the principal intends to terminate
the project when the belief reaches q. If the agent adopts the best response, the remaining
time can be written as K(p, q) such that∗∗pe−λl max{K(p,q)−k ,0}−λh min{K(p,q),k }= q,
1 − p + pe−λl max{K(p,q)−k∗ ,0}−λh min{K(p,q),k∗}which reduces to∗pe−λl K(p,q)−∆λ k= q,
1 − p + pe−λl K(p,q)−∆λk∗
if K(p, q) > k∗ . In addition, it is also convenient to define a backward operator φ(q) suchthat∗

φ(q)e−λh k= q.
1 − φ(q) + φ(q)e−λh k∗
The backward operator suggests that if pt = φ(q) and the agent exerts high effort for t ∈[t, t + k∗ ], then pt+k∗ = q; alternatively, if the agent expects the project to be terminated at
pt = q, φ(q) indicates the belief at which he must switch to high effort. Note that these two
notions are closely related in that K(p, q) > k ∗ if and only if p > φ(q).

4Equilibrium

4.1

The cooperative solution: a benchmark

Before we move on to analyze our model, we first examine as a benchmark the case where
both the principal and the agent attempt to maximize the sum of their individual payoffs toderive 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 successminus the effort cost, i.e., λa p(y + b) − da for a given effort a.13
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.

13
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 between
participating and dropping out immediately (before time 0), as he would obtain zero payoff in either case forany given deadline. As it turns out, however, the optimal deadline is independent of the initial belief p0 andthe analysis holds irrespective of whether or not the principal can immediately screen out the mediocre type.

10

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 easily verify that the agent is generally more motivated in thecooperative case than in the noncooperative case, i.e., k̃∗ > k∗ .
Given this, it is fairly straightforward to derive the cooperative solution. Since there is
no social cost of hiring the mediocre type, the joint surplus from hiring the agent is strictly
positive for any belief level, and as such, it is efficient for the principal to continue the projectindefinitely until a success is attained. This implies that the good type will eventually succeed
at some point, although he only exerts low effort all the way if k̃∗ is finite.
Proposition 2 In the cooperative case, the principal never terminates the project, and thegame continues indefinitely. The agent always exerts low effort ifr(b + y)d>,∆λλl + r(6)

and high effort otherwise.
Proof: See Appendix.
Note the difference between (2) and (6). In the cooperative case, the joint expected payoffof exerting low effort indefinitely is r(b+y)λl +r because the value of a success is now b + y, insteadof just b, and there is no flow cost w.

4.2

The commitment solution

The principal’s problem under commitment is to choose a termination date τ which maximizes
her expected payoff at time 0 while restricting attention to deterministic deadlines.14 We focus
exclusively on deterministic deadlines primarily because it is generally difficult to commit toa lottery, even when it is possible to commit to a particular termination date. To see thispoint, consider a contract where the principal terminates the project at τ 1 or τ 2 , τ 1 < τ 2 , eachwith probability strictly less than one. At time τ 1 , however, the principal’s preferences are
generically strict, i.e., it is strictly better either to stop (the continuation payoff is negative) or
to continue (positive), so that there is no incentive to randomize at that point. If it is strictly
better to stop at time τ 1 , for instance, the principal stops at time τ 1 with probability one but
14
If stochastic deadlines are feasible, the principal may screen out the low type by offering a menu ofcontracts, one with a deterministic deadline and the other with stochastic ones. We do not pursue thispossibility both because we do not think that it is realistic (given the enforcement problem) and because it isoutside the scope of our analysis (given that our focus is on comparing the commitment and no-commitment
solutions). Note that this screening cannot be done if deadlines are deterministic, because both types always
strictly prefer a longer deadline.

11

knowing that, the contract is effectively reduced to the one with a deterministic deadline. The
lack of credible enforcement is perhaps the reason why we almost never observe a contract
with stochastic deadlines.
As we have already seen, the agent exerts high effort if and only if the remaining timeis less than or equal to k∗ . We denote by V ∗ (k, pt ) the continuation payoff when the agent
adopts this best response, where k is the remaining time and pt is the principal’s belief. Thiscan be written as
Z t+ν
Rs
∗
(λl ps y − w)e− 0 (λl pu +r)du ds
V (k, pt ) =
t
Z t+k
R t+ν
Rs
+ e− t (λl pu +r)du(λh ps y − w)e− ν (λh pu +r)du dst+ν

=p t πlpt e−(λl +r)ν πh(1 − pt )w(1 − e−(λl +r)ν ) +(1 − e−(λh +r)(k−ν) ) −(1 − e−rk ),λl + rλh + rr(7)

where ν := max{k − k∗ , 0} and πa := λa y − w. The first two terms represent the expected
gain from the good type whereas the last term represents the expected loss from the mediocre
type.
Since the termination date is equivalent to the remaining time at time 0, the commitment
solution, denoted by τ C , is given byτ C (p0 ) := argmaxτ V ∗ (τ, p0 ).
The continuation payoff is continuous but kinked at τ = k∗ . For τ ∈ (0, k∗ ), the first-ordercondition is given by
p0 πh e−λh τ − (1 − p0 )w = 0.

(8)!
λl + r−(λh +r)k ∗πl −πh (1 − e) − (1 − p0 )we−rτ = 0.
λh + r(9)

For τ ∈ (k∗ , ∞), it is given by−(λl +r)(τ −k ∗ )M (τ, p0 ) := p0 e

To better understand this condition, it is instructive to rewrite this as
∗

∗
∗
∗
p0 ∆λ e−(λl +r)(τ −k )w + ry + πh e−(λh +r)k .
p0 πh e−(λl (τ −k )+λh k ) − (1 − p0 )w e−rτ =λh + r

Note that the left-hand side is the expected instantaneous payoff for the principal at the timeof termination, whereas the right-hand side is the marginal cost of extending the deadline.
Since the right-hand side is strictly positive, the principal must set the deadline at a point
where the expected instantaneous payoff is positive if τ > k ∗ .
12

Since the instantaneous payoff is negative for any effort level once the belief dips below
qh , there is clearly no incentive to set τ > K(p0 , qh ). When the initial belief is sufficiently
close to qh (φ(qh ) ≥ p0 ), it is optimal to let the agent work until the belief reaches this lower
bound. When it is far from it (p0 > φ(qh )), on the other hand, the principal must terminate
the project at a point where the belief is still above the lower bound. There are two cases,
depending on the value of M (k∗ , p0 ).15 If
λl + r∗πh (1 − e−(λh +r)k ) > πl ,λh + r(10)

M (k∗ , p0 ) < 0 for any p0 ∈ (qh , 1), and we let p̂ = 1 in this case. If (10) fails to hold, on
the other hand, there exists a unique p̂ such that M (k∗ , p̂) = 0. Then, for p0 > p̂, there
exists a unique interior solution τ̂ (p0 ) such that M (τ̂ (p0 ), p0 ) = 0. When the initial prior isin this range, it is not so costly to have a phase where the agent exerts low effort, and assuch, the principal would wait beyond time k∗ . Of course, extending the deadline beyond
time k∗ means that the agent would slack off at the beginning, thereby pushing back therealization of the high-effort phase. The principal trades off the gain of inducing low effort
for an additional instant against the cost of realizing the high-effort phase an instant laterand sets the termination date at a point where they are equalized.
For clarity, we summarize the set of parameters by Θ := (b, y, w, d, λh , λl , r). We can thenmake the following statement.
Proposition 3 For any given set of parameters (Θ, p0 ) satisfying Assumptions 1 and 2, there
exists a unique commitment solution τ C (p0 ) ∈ (0, K(p0 , qh )]. For φ(qh ) ≥ p0 , the commitment
solution is given by τ C (p0 ) = K(p0 , qh ). For p0 > φ(qh ), there exists some p̂ ∈ (ql , 1] such
that
(
k∗
τ C (p0 ) =τ̂ (p0 )if p̂ ≥ p0 > φ(qh ),if p0 > p̂,

where: (i) p̂ > ql if 1 > ql ; and (ii) p̂ = 1 if and only if (10) holds. The commitment solution
is weakly increasing in p0 .
Proof: See Appendix.
Note that K(p0 , qh ) constitutes the upper bound of τ C (p0 ). Since K(p0 , qh ) is boundedfrom above when qh > 0, the proposition suggests that the project must be terminated in
finite time, meaning that some good projects are bound to be terminated prematurely. Thisdraws clear contrast with the cooperative solution where the project is never terminated.
15

Note that M (k, p0 ) < 0 for any k > k∗ and p0 if and only if M (k∗ , p0 ) < 0.

13

4.3

The no-commitment solution

The situation becomes more complicated, and perhaps more intriguing, when the principal
makes no commitment at the outset and hence her termination decision must be sequentiallyrational. Because of this, the principal’s termination strategy now needs to be specified
in a different way, as her strategy off the equilibrium path (after the project is supposed
to be terminated) may matter. More precisely, the principal’s strategy is given by a setof termination dates T which consists of a first termination date and further dates along
off-equilibrium continuation paths. Formally, for this no-commitment case, we solve for apure-strategy perfect Bayesian equilibrium (hereafter, simply an equilibrium) in which theprincipal terminates the project with probability one at each termination date (a formal
definition is given below).16
To illustrate how we specify the principal’s strategy, consider a simple example where
the principal’s strategy consists of two distinct termination dates, i.e., T = {τ 1 , τ 2 } whereτ 1 < τ 2 . Under this strategy, the principal terminates the project with probability one ateach τ i , i = 1, 2, conditional on the continuation of the game.17 This means that the projectis surely terminated at time τ 1 on the equilibrium path. To show that this constitutes anequilibrium, however, one must assure that the principal cannot profitably deviate from thisstrategy. To see this, suppose that the principal unexpectedly continues the project at timeτ 1 . The game then enters into a new phase where the agent now expects the project tobe terminated at the next termination date τ 2 and chooses his effort accordingly (at = h ifk∗ ≥ τ 2 − t and at = l otherwise). Note also that the principal’s belief about the agent’s type
is not affected by the deviation: given T and the agent’s best response, we can still apply
Bayes’ rule to compute how the belief evolves both on and off the equilibrium path.18 This
gives the continuation payoff when the principal deviates which must be non-positive.
How long does this process continue? It generally goes on until the principal has no16

We restrict our attention to pure-strategy equilibria by assuming that the principal terminates the projectwith probability one when the continuation payoff is zero. Since the principal is actually indifferent at thispoint, however, there may exist other mixed-strategy equilibria in which the principal terminates the projectwith probability less than one at each termination date. We do not pursue this possibility for two reasons.
First, we do not believe that such an equilibrium is particularly realistic and hence appealing. Second, evenwith randomization, the game still must end with probability one by time τ 1 , and the continuation equilibrium
is exactly the same once the game enters the final interval (τ 1 , τ 2 ); solving backwards, one can see that thestructure of equilibrium is essentially the same where the agent starts exerting high effort as t approaches eachtermination date and switches back to low effort if the project is not terminated at the termination date.
17
If the game continues beyond time τ 2 , the principal terminates the project immediately as we will see
below.
18
Technically, we assume that the belief depends only on the agent’s past effort and is given by (4) even
off the equilibrium path. This is actually the “no signaling what you don’t know” condition (Fudenberg andTirole, 1991) although it is a rather obvious restriction in games with two players (it only implies that the
principal’s belief should not be affected by her own deviation).

14

incentive to continue the project regardless of the agent’s effort choice or, in other words,
until the belief reaches the lower bound qh . With slight abuse of notation, we denote by pt
the equilibrium belief at time t, conditional on the continuation of the game (including theoff-equilibrium path). Let τ n denote the time at which pt reaches qh for some n (which is
unknown at this point). We can then restrict our attention to t ∈ (0, τ n ], for it is a dominant
strategy for the principal to terminate the project once the belief reaches this lower bound.19
Lemma 1 In any equilibrium, T is a finite set.
Proof: The lemma is directly implied by the fact that any two adjacent termination dates
must satisfy τ i − τ i−1 ≥ k∗ > 0. To see this, suppose otherwise, i.e., k∗ > τ i − τ i−1 . Suppose
further that the principal deviates and continues the project at time τ i−1 . Then, it is strictly
better for the agent to exert high effort because k∗ > τ i − τ i−1 . Note, however, that sincept > qh for any t ∈ (0, τ n ) by definition, the instantaneous payoff is strictly positive until the
belief reaches the next termination belief. This is a contradiction because the principal canstrictly benefit from deviating and not stopping at time τ i−1 . Given this, it is clear that we
can only have finitely many termination dates in (0, τ n ].
Given this result, we denote each element of T by τ i where τ 1 < τ 2 < · · · < τ n for somen. Accordingly, the game is divided into n distinct segments T i , i = 1, 2, ..., n by termination
dates, where T i = (τ i−1 , τ i ) for i = 1, 2, ..., n with τ 0 = 0. The remaining time at time t is
now given by τ i − t for t ∈ T i while it is zero for all t > τ n (see footnote 19). Let q i := pτ idenote the corresponding termination belief. Only the first segment T 1 is actually played on
the equilibrium path, as the game ends with probability one by the time the game reaches
time τ 1 . For expositional clarity, we call τ 1 the no-commitment solution and denote it by
τ N C (p0 ) to indicate its dependence on p0 . The formal definition of our equilibrium is given
below.
Definition 1 A perfect Bayesian equilibrium in this game is a pair of strategies {as }∞s=0 and
T := {τ i }ni=1 and a belief system such that:• given T , the (good-type) agent chooses at at each t to maximize his continuation payoff,i.e., chooses high effort if and only if the remaining time at time t is less than or equal
to k∗ ;
19
Suppose that the principal deviates and continues beyond time τ n . In this case, the principal’s instantaneous payoff is negative even if the agent chooses at = h, so that it is a dominant strategy to terminate
immediately for all t > τ n . Given this, since the remaining time is invariably zero, the agent always chooses
at = h. This is the unique continuation equilibrium after time τ n (although, to describe this process formally,
we need to consider a discrete-time counterpart of our model where there is a minimum time unit, and takethe limit as the time unit goes to zero).

15• given {as }∞s=t and the current belief, the principal terminates the project at time t if
and only if the continuation payoff is non-positive;• The belief at time t depends only on the effort sequence {as }ts=0 and is given by (4),both on and off the equilibrium path.
To pin down the equilibrium termination dates, we need to pay closer attention to theprincipal’s continuation payoff. Since the agent’s best response is the same, the principal’s
continuation payoff can still be written as V ∗ (k, pt ). The only difference is that the principal’s
strategy must now be sequentially rational, which amount to the following two equilibrium
conditions which the principal’s termination strategy T must satisfy.
Condition T: (i) q n = qh and (ii) if n > 1, for each i = 2, ..., n, V ∗ (K(q i−1 , q i ), q i−1 ) = 0.
Condition C: For each i = 1, 2, ..., n, V ∗ (τ i − t, pt ) > 0 for all t ∈ T i .
Condition T is the usual indifference condition that requires the principal to terminate theproject when the continuation payoff (when she deviates) is non-positive at each τ i . This
indifference condition alone is in general not sufficient because the instantaneous payoff may
not be monotonically decreasing over time: in any segment T i , the agent may start off withlow 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 date τ i is reached.
The next result establishes that there exists a perfect Bayesian equilibrium that is always
unique even under no commitment.
Proposition 4 For any given set of parameters (Θ, p0 ) satisfying Assumptions 1 and 2,
there exists a (generically) unique pure-strategy equilibrium. Given Θ, the belief space (qh , 1)m(Θ)

is partitioned into m(Θ) ≥ 1 distinct intervals {(P j , P j−1 )}j=1

where P m(Θ) = qh and

P 0 = 1. For p0 ∈ (P j , P j−1 ), j = 1, 2, ..., m(Θ), the equilibrium strategies are as follows:• The principal’s equilibrium strategy is characterized by a set of n = m(Θ)−j +1 distinct
termination dates {τ i }ni=1 where q i = P i+j−1 and τ i − τ i−1 > k∗ ;20• The agent exerts high effort if the remaining time is less than or equal to k∗ and low
effort otherwise.
Moreover, if m(Θ) > 1, then ql > P 1 .
20

As discussed in footnote 19, at = 0 for all t ≥ τ n .

16

Proof: See Appendix A.
As the proposition suggests, the structure of equilibrium is thoroughly characterized by
how the belief space is partitioned into intervals. A crucial determinant of the partitionm(Θ){(P j , P j−1 )}j=1 is the profitability under low effort which is captured by πl .21 Below, wem(Θ)

first briefly illustrate how we pin down m(Θ) and {(P j , P j−1 )}j=1 from a given Θ (in what
follows, we simply denote m = m(Θ) to save notation). Once they are obtained, it is quite
straightforward to derive the equilibrium strategy T for a given p0 .
Suppose first that the profitability under low effort is sufficiently high, so that 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 where m = 1: the agent starts off with low effort and switches
to high effort when the remaining time is k ∗ ; the principal stops when the belief reaches thelower bound P 1 = qh at time τ 1 . Formally, m = 1 and hence n = 1 for all p0 ∈ (qh , 1) if andonly if V ∗ (K(ql , qh ), ql ) > 0, i.e.,ql e−(λl +r)ν πh(1 − pt )w
q l πl(1 − e−(λl +r)ν ) +(1 − e−(λh +r)(k−ν) ) >(1 − e−rk ),λl + rλh + rr(11)

where µ = max{K(ql , qh ) − k∗ , 0} (see the proof of Proposition 4, especially Lemma 2, formore detail). Note that (11) depends only on Θ. The no-commitment solution τ N C (p0 ) ismonotonic in p0 if and only if this condition holds.
[Figure 1 about here]
In contrast, the instantaneous payoff may become negative under low effort when the
success probability under low effort is relatively 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 when m = 2 while
fixing the principal’s strategy at T = {τ 1 }. 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 τ ′ (in Figure 3) such that the continuation payoff is negative for t ∈ (0, τ ′ ).
This implies that T = {τ 1 } does not constitute an equilibrium as it violates Condition C.
In this case, the game is divided into two segments, T 1 and T 2 , as illustrated in Figure 4.
The pair of strategies now satisfies the equilibrium conditions since the instantaneous payoff
is always positive in T 1 . Formally, if V ∗ (K(ql , qh ), ql ) < 0, there must exist p′ ∈ (qh , ql ) suchm(Θ)

21Note that the partition {(P j , P j−1 )}j=1
prior p0 .

is determined solely by Θ while q i depends also on the initial17

that V ∗ (K(p′ , qh ), p′ ) = 0, in which case we redefine P 1 = p′ and P 2 = qh . As above, m = 2
if and only if V ∗ (K(ql , P 1 ), ql ) > 0.
[Figures 2-4 about here]
We can continue this process until we find P 1 such that V ∗ (K(ql , P 1 ), ql ) > 0.22 Oncethe partition is pinned down, we can then easily derive the equilibrium strategies: for p0 ∈(P j , P j−1 ), the principal’s equilibrium strategy consists of n = m − i + 1 termination dates
where q i = P i+j−1 andτ N C = τ 1 = K(p0 , q 1 ), τ i =

i−1
X

′

τ i + K(q i−1 , q i ) for i = 2, 3, ..., n.

i′ =1Note also that q i−1 > φ(q i ) > q i because the continuation payoff is always strictly positive
when the belief is in (q i , φ(q i )), the range where the remaining time is less than k∗ and theagent exerts high effort. This alternatively means τ i − τ i−1 > k∗ , i.e., the length between
any two adjacent termination dates must be larger than k∗ .
Finally, we would also like to note that the constructed equilibrium is generically uniquefor any given set of parameters. This uniqueness result stems crucially from the fact that there
is a lower bound of the belief qh below which the principal would never continue the project.
As stated above, together with the fact that λl > 0, this implies that the game must end
in some finite time, which allows us to solve the game via backward induction analogously
to the “gap case” of the durable-good monopoly problem (Fudenberg et al., 1985). Moreprecisely, since the agent’s strategy depends only on the remaining time, we know exactly
how the game must end as the belief approaches the lower bound. Applying this reasoning
backward, we can identify a unique continuation equilibrium for each pt > qh and all the way
back to the initial prior p0 .

5Discussion

5.1

The value of commitment

Our framework yields unique commitment and no-commitment solutions which enable us todirectly assess the value of commitment to an evaluation deadline. Given that both τ C andτ N C can be written as functions of p0 , the expected equilibrium payoffs can also be written
as functions of p0 . Let
V C (p0 ) := V ∗ (τ C (p0 ), p0 ), V N C (p0 ) := V ∗ (τ N C (p0 ), p0 ).
22
When ql = 1, there is no such q 1 and m will go to infinity. If ql < 1, on the other hand, this process must
converge after a finite number of rounds.

18

The value of commitment is then defined as∆V (p0 ) := V C (p0 ) − V N C (p0 ).
In principle, we can compute the exact value of ∆V (p0 ) for any given p0 . The following
statement is the main result of the paper which analytically characterizes how the value ofcommitment varies with respect to p0 .
Proposition 5 (i) ∆V (p0 ) = 0 for p0 ∈ (P m , φ(P m )) where P m = qh . (ii) If m > 1,
for each j = 2, 3, ..., m, ∆V is strictly increasing in p0 for p0 ∈ (φ(P j ), P j−1 ) and strictly
decreasing for p0 ∈ (P j−1 , φ(P j−1 )) with ∆V (φ(P j )) = 0. (iii) ∆V (φ(P 1 )) = 0 if and only ifp̂ ≥ φ(P 1 ) where p̂ is the threshold defined in Proposition 3. (iv) limp0 →1 ∆V (p0 ) = 0 if andonly if 1 > p̂.
Proof: See Appendix.
The proposition suggests that the value of commitment changes in a non-monotonic way as
p0 increases, precisely due to the non-monotonic nature of V N C . Despite this, we can still findsome regularities except for the last segment (φ(P 1 ), 1).23 Take an interval (φ(P j ), φ(P j−1 ))for j = 2, 3, ..., m where φ(P j−1 ) > P j−1 > φ(P j ). In each of these intervals, (i) the valueof commitment is single-peaked and maximized at P j−1 (at which point the no-commitment
solution yields a zero payoff); (ii) the value of commitment is zero at both ends of theinterval (at which point the two solutions are identical). Important features of Proposition
5 are captured by Figure 5 which shows a typical path of the value of commitment whenm = 2.24[Figure 5 about here]
To see how ∆V changes with respect to p0 , observe that the only feasible history at any
continuation game is the one consisting only of failures up to that point, and the principal
hence has no additional information other than the fact that the agent has achieved no success.
This immediately implies that the no-commitment solution can yield no higher payoff thanthe commitment solution, i.e., ∆V (p0 ) ≥ 0 for all p0 . Clearly, the no-commitment solution
23

In the last segment, the profitability under low effort can be high enough (if p0 > ql ), and the expected
payoff can go up even during the low-effort phase. Moreover, in this range, τ C (φ(P 1 )) = τ NC (φ(P 1 )) may
not even be satisfied since τ C (φ(P 1 )) can be larger than k∗ . As a consequence, the value of commitment caneither go up or down (and can even be non-monotonic). See Appendix B for more detail.
24
For the figure, the parameters are set as follows: λh = 0.1, λl = 0.04, b = y = 1, w = 0.035, d = 0.02,
r = 0.05.

19

gives the principal a different, and necessarily lower, payoff when τ C 6= τ N C . Still, the nocommitment solution can replicate the same allocation as the commitment solution when
τ C = τ N C , because Proposition 1 ensures the unique equilibrium given the same horizon. It
follows from Proposition 5 that the commitment solution generically yields a strictly higher
expected payoff for the principal than the no-commitment solution if and only if p0 > φ(qh ),25
i.e.,
p0 >

1 − qhqh1−∗ ⇔qh + (1 − qh )e−λh kqh(λh + r)(b − ∆dλ )λh b − d + w

! λh

λh +r

>

1 − p0.
p0(12)Conversely, when this condition fails to hold, the absence of commitment entails no additionalcost, and the no-commitment solution becomes strictly profitable in the presence of a (possibly
very small) commitment cost.26
The proposition states that the principal benefits from committing to a deadline if theinitial prior belief p0 exceeds a certain threshold φ(qh ), i.e., if the initial prior is so highthat the belief at time k∗ is above the lower bound qh even if the agent exerts high effort
for t ∈ [0, k∗ ]. The value of a deadline depends crucially on each player’s propensity to“procrastinate.” In short, the threshold is lower and hence favors the commitment solution
for a given initial belief either when the agent delays exerting high effort for a given deadlineor when the principal delays terminating the project for a given effort sequence. Below, we
briefly summarize how each player’s propensity to procrastinate 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.
25

To be more precise, as Proposition 5 suggests, there could be some non-generic cases where the expected
payoffs are identical.
26
An obvious commitment 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 we have thus far assumed away this aspect forclarity, we extend the analysis to incorporate uncertainty into the current setup in section 6: there, we showthat the commitment solution is not always weakly optimal. The commitment cost may also arise from thecost of writing and enforcing a formal contract.

20

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.27 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 value ofa deadline is still determined in essentially the same way, although the analysis would besubstantially more complicated.

5.2

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,
τ N C (p0 ) ≥ τ C (p0 ) for all p0 . As indicated in Figures 3 and 4, however, this does not necessarily hold 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.
From Proposition 3, it is conceptually straightforward to compute the commitment solution τ C . Figures 6a and 6b illustrate τ C as a function of p0 , using the same parameters as
in Figure 5 except for λl . The qualitative nature of τ C depends largely on the productivity
under low effort as captured by λl . Figure 6a is the case where λl is relatively high. In thiscase, the principal does not lose much during the low-effort phase, and is willing to extend
beyond k∗ when the initial prior is sufficiently high. In contrast, when λl is relatively low, as
depicted in Figure 6b, the principal sets τ C ≤ k∗ to assure that the agent always exerts higheffort. In either case, for p0 > φ(qh ), τ C is set strictly below the upper bound K(p0 , qh ) sothat the project is terminated before the belief reaches qh .
[Figures 6a and b about here]
As we show in the proof of Proposition 4, we can also explicitly derive the no-commitment
solution τ N C (p0 ) although it may take a much more complicated form. Figure 7 depicts
τ N C (p0 ) along with τ C (p0 ), which confirms that the average duration of the project canbe longer with commitment than without, using the same parameters as in Figure 5. The
figure also reveals that the no-commitment may not be monotonically increasing in p0 , in27

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.

21

clear contrast to the commitment solution which is always weakly increasing. When theinitial belief is relatively high, it takes long for the belief to reach the lowerbound qh , andthe 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 agentsin some cases. This is more likely to be the case when λl is low, in which case the principal
must incur losses in the low-effort phase. Formally, we can make the following statement.
Proposition 6 τ N C (p0 ) can be larger or smaller than τ C (p0 ). Moreover, τ C (p0 ) > τ N C (p0 )for almost all p0 ∈ (φ(qh ), 1) as λl tends to zero.
Proof: It is clear that τ N C (p0 ) can be larger than τ C (p0 ) for some p0 , because τ N C (p0 ) > k ∗for p0 slightly larger than φ(qh ) while τ C (p0 ) = k∗ for p0 ∈ (φ(qh ), p̂). We can also show that
τ N C (p0 ) can be smaller if m ≥ 2, because limp0 ↓P 1 τ N C (p0 ) = 0 < k∗ ≤ τ C (P 1 ).
As for the second statement, It follows from Proposition 3 that τ C (p0 ) ≥ k∗ forp0 ∈ (φ(qh ), 1). To prove the proposition, it thus suffices to show that k∗ > τ N C (p0 ) in thisrange. Suppose otherwise, i.e., τ N C (p0 ) ≥ k∗ for some p0 ∈ (φ(qh ), 1). This happens if andonly if p0 ∈ (φ(P j ), P j−1 ) for j = 1, 2, ..., m or, in other words, P j−1 must be bounded awayfrom φ(P j ). This is a contradiction, however, as limλl →0 V ∗ (K(P j−1 , P j ), P j−1 ) < 0 for any
P j−1 > φ(P j ).

[Figure 7 about here]
Our results yield some empirical implications. An obvious one is that the initial prior
p0 is not necessarily a reliable predictor of future success under no commitment. Asidefrom this, it is also worth emphasizing that the equilibrium allocation depends cruciallyon what happens off the equilibrium path. As a consequence, two organizations that are
observationally similar may exhibit drastically different equilibrium outcomes. To put thisidea in context, consider two distinct organizations, A and B, each characterized by λil ,i = A, B, and the corresponding no-commitment solution τ N C,i (while assuming that the two
Borganizations are equivalent in every other dimension). Suppose further that λA
l > λl and
N C,B (p ) < k ∗ for almost all p , meaning
λB00
l → 0. In this case, as Proposition 6 suggests, τthat we almost never observe the low-effort phase in organization B, and hence λB
l cannot be
estimated from actual data. The two organizations are thus observationally indistinguishable
even though their respective no-commitment solutions can be totally different from eachother.
22

5.3

An application: when is up-or-out optimal?

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 professional service industries such as law, accounting, and consulting (Lazear and Gibbs, 2014;
Eriksson, 2016).28 Under a typical up-or-out contract, the employer sets a deadline by whicha promotion must occur, and the worker must leave the firm if he is not promoted by thedeadline. 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, aleading example of industries with up-or-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 acertain period of time (roughly seven to ten years). This contrasts sharply with other typicalfirm organizations which almost never specify deadlines by which workers must be promoted.
Here, we attempt to shed some light on this issue by examining under what conditions (12)is likely to be satisfied. There are of course some caveats. First and foremost, our theoreticalframework is perhaps too simplistic to capture all the important details of this diverse set ofindustries; some of our arguments may thus be applied only to a subset of those industries.29
Second, there is also a danger in comparing allocations under different incentives structures.
One particular concern is the potential endogeneity of parameter values that are taken as
exogenous here.30 With those qualifications in mind, our framework can still provide some
useful, though preliminary, insight for when it pays to commit to a promotion deadline as in
the case of up-or-out.
Ability intensity: It is often argued that knowledge intensity is one of the most fundamental
characteristics of professional service industries (von Nordenflycht, 2010). Combined with thefact that they are also less capital intensive,31 the productivity of an organization depends
28
For brevity, we refer to those industries characterized by up-or-out, including academia, broadly as professional service industries. The military is another example which is characterized by up-or-out, but we do notconsider this case because the underlying structure appears to be quite different. See Kahn and Huberman(1988), Waldman (1990), O’Flaherty and Siow (1992), and Ghosh and Waldman (2010) for formal analyses ofup-or-out contracts.
29
For instance, our exponential specification implies that success is very rare and information is coarse, asassumed in Bonatti and Hörner (2017), which might be applicable to some industries but not to some others.
Also, team production may play a bigger role in some industries than in others.
30
It is likely that y and b may vary with the contract duration when we interpret them as the net continuation
payoffs of achieving a success. In particular, we normalize the continuation payoff at zero when the projectis terminated, but this could well be a function of the contract duration (or the market belief at the time oftermination).
31
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.

23

crucially on the extent of knowledge embodied in individuals (Starbuck, 1992; Winch andSchneider, 1993). A likely consequence of this fact is that innate ability matters and createsvalue 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
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 strictly
optimal to set a deadline, thereby favoring the use of up-or-out, for λh > λ̄. To see this, werewrite (12) as
qh
w
p0
>
=
.
1 − p0(1 − qh )e−λh k∗(λh y − w)e−λh k∗(13)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, (13) is never satisfied if λh is small.

In contrast, we haveqhlim
∗ = 0,
λh →∞ (1 − qh )e−λh k
implying that (13) must hold if λh is sufficiently large.
Stable job descriptions: Another distinctive feature of professional service industries is a
professionalized workforce that builds on a particular knowledge base (Torres, 1991; von Nordenflycht, 2010). This in turn creates well-defined job boundaries and stable job descriptions.
Due to various professional requirements and accreditation processes, workers in professional
service industries are typically responsible only for a narrow and clear set of tasks, compared
to workers in other industries. This implies that the nature of 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 thejob 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,24

e.g., the arrival of a stochastic shock that makes the project completely worthless.32 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!
λ1 (b − ∆dλ )1 − p01 − qh1−>.
qh
λh b − d + w
p0
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.33 In the current setup, this
aspect may be captured by y, which can be interpreted as the expected payoff of promoting
a good-type 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.34
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.
32
Here, we allow the principal to terminate the project when the shock arrives; see section 6 for the casewhere the principal must abide by the deadline under any circumstance.
33
Ghosh and Waldman (2010) also raises job similarity as one of the distinctive characteristics of academia.
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 by Prendergast(1993). However, our argument here may be more applicable to academia (at least in humanities and social
sciences) but less so to other professional service industries such as law where the role of a partner is more
about bringing in business (or “rainmaking”).
34
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.

25

6Conclusion

This paper presents a simple stylized model of dynamic performance evaluation with particular emphasis on the role of commitment to an evaluation deadline. We consider anenvironment where the principal chooses whether to commit to a deadline and if so, at whattime. Within this framework, we obtain a complete characterization of equilibrium and derivea necessary and sufficient condition for committing to a deadline to be optimal. The simple
framework also allows us to conduct various comparative statics exercises to give insight into
when it is beneficial to set an evaluation deadline at the outset.
Since our model is deliberately stylized to obtain sharp analytical characterizations, there
are naturally several avenues to extend the current analysis. Before we conclude, we briefly
discuss two of those possible extensions (see Appendices C and D for more formal analysesof these cases).
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 cost because the only feasible history at each continuation game consists only of
failures up to that point; 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 filledwith uncertainty where there is clearly value in being flexible.
One way to cope with this possibility is to consider a setup where a permanent productivity shock may strike with some probability, which totally changes the nature of the task
and subsequently makes the agent unproductive, i.e., λh = λl = 0.35 For simplicity, supposethat (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. In this situation, the principal can retain
the flexibility to adjust to a negative shock by immediately terminating the project wheneverit strikes. We can then show that due to this benefit, there arises a situation where theprincipal strictly benefits from not committing to a deadline.
Exogenous wages: In the baseline model, we have assumed that the contractual environment is highly incomplete in that no contingent wage contracts can be written. Even insuch an environment, it may still be possible to agree on the flow wage contingent on thecontinuation of the project. Of course, since the flow wage in this context is a pure transfer
35
Once the principal chooses to commit to a deadline, the principal must abide by it and is not allowed
to terminate the project, unlike in the discussion of stable job descriptions in section 5.3 where we allow the
principal to terminate the project whenever a stochastic shock arrives. If the principal is allowed to terminate
the project at any time before the committed deadline, the commitment solution then yields a weakly higher
payoff than the no-commitment solution.

26

payment with no incentive effect on the agent’s side, there is no reason for the principal tooffer any wage beyond the minimum level if she can stop at the right time. However, as wehave seen, this is not always true when the principal makes no commitment to a deadline, inwhich case the flow wage can be used as a substitute for a deadline. This can be potentially
useful if the commitment cost is prohibitively large for some reason.
To explore whether there are circumstances in which she offers any strictly positive w tothis effect, suppose that the principal can deliberately choose any w ≥ 0 at time 0.36 The
benefit of raising w above zero is clear if we look at the case where w = 0: in this case,the instantaneous payoff is always strictly positive (as in the cooperative case), no matter
how unlikely the agent is to succeed; as a consequence, the principal can never terminate
the project, and given this, the agent never exerts high effort. The principal can alter thisstructure by raising w above zero because the flow cost of employment is now positive,
rendering the instantaneous payoff negative at some point. This can be profit-enhancing forthe principal because with the credible threat of termination, she can induce the agent toexert high effort, which is especially beneficial when the success probability under low effort
is low: formally, we can show that the optimal flow wage is strictly positive if λl is sufficiently
small. This reasoning is similar to Buisseret (2016) in that a higher flow wage is used as acommitment device to stop at the right time and reduce future rents for the agent.37

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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rUa (k) = (λa b − da + w) − λa Ua (k) − U̇a (k)(14)

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. Moreover, since Uh (k) is strictly increasing in k, the agent would never switch back30

to low effort, once U (k) reaches this critical point. Then, k∗ is obtained as a solution toUh (k∗ ) = b − ∆dλ , which can be written asd
λh b − d + w∗(1 − e−(λh +r)k ) = b −.
λh + r∆λ(15)Note that the agent always exerts high effort ifb−d
λh b − d + w
rb − wd≥⇔≥,∆λλh + rλl + r∆λwhile he never does so if ∆dλ ≥ b. For in-between cases, solving (15) yields
!
d
(λ
+
r)(b
−
)
h
1
∆
λ
,
ln 1 −k∗ = −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: The agent always exerts high effort in the noncooperative case if(2) fails to hold. An analogous condition for the cooperative case is given byddr
λh (y + b) − d
>y+b−
⇔
>
(y + b).
λh + r∆λ∆λλl + rApplying the same argument as in Proposition 1, we obtain
!
d
(λ
+
r)(y
+
b
−
)
h
1
∆λ
.
ln 1 −k̃∗ = −λh + rλh (y + b) − d
To see that the project is never terminated, we first show that if it is ever optimal to set
a deadline in some finite time, the only possibility is to set it at time k̃∗ , so that the agent
exerts high effort from the beginning. To see this, the principal extends the deadline beyond
time k̃∗ if and only if
λl (y + b) >
λl + r∗λh (y + b) − d (1 − e−(λh +r)k̃ ).
λh + r(16)

Note that this condition is independent of the deadline. As such, if this condition holds, itis then optimal to continue the project indefinitely until the project succeeds. If it fails tohold, on the other hand, it is clearly optimal to set a deadline at time k̃∗ .
Since, by definition,λh (y + b) − d
d
∗
∗
y+b−
= Ũ (k̃ ) =(1 − e−(λh +r)k̃ ).
∆λλh + r31

we can modify (16) asd
λl (y + b)
>y+b−.
λl + r∆λThe left-hand side is the expected payoff of exerting low effort indefinitely while the righthand side is that of exerting high effort from time 0 to time k̃∗ which must equal the static
gain.
This condition must hold for any finite k̃∗ when (6) is satisfied, meaning that it isstrictly better to let the project continue indefinitely. If (6) fails to hold, on the other hand,k̃∗ → ∞, and again it is better to let the project continue indefinitely. This shows that no
finite deadline can improve the social surplus in the cooperative case.
Proof of Proposition 3: First, if φ(qh ) ≥ p0 > qh (the latter inequality holds by Assumption2), there exists a unique deadline τ C ∈ (0, k∗ ] that satisfies
Cp0 e−λh τ= qh .
1 − p0 + p0 e−λh τ C
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 τ C because theinstantaneous payoff is still strictly positive. The optimal termination date is then τ C and it
is unique in this range.
Second, suppose that p0 > φ(qh ), in which case the first-order condition is given by
M (τ C , p0 ) = 0. If M (k∗ , p0 ) ≤ 0, the expected payoff is strictly decreasing in τ C for τ C > k∗ ,and the optimal termination date is τ C = k∗ . Alternatively, define p̂ ∈ (0, 1) such that
M (k∗ , p̂) = 0 if (10) holds and p̂ = 1 otherwise. Then, the optimal termination date isk∗ for p0 ≤ p̂. If p0 > p̂, on the other hand, there must exist a unique τ̂ (p0 ) such that
M (τ̂ (p0 ), p0 ) = 0. Since the instantaneous payoff is strictly positive for τ̂ (p0 ) > τ C andnegative for τ C > τ̂ (p0 ), the optimal termination date in this case is τ̂ (p0 ).
Finally, to show that p̂ > ql if 1 > ql , there are two cases. If (10) holds (but 1 > ql ), thenp̂ = 1 > ql . If (10) fails to hold, on the other hand, M (k∗ , p0 ) is increasing in p0 for any k∗ .
Given that 1 > ql , it suffices to show that M (k∗ , ql ) < 0, i.e.,−rk ∗(1 − ql )we

> ql

!
∗
λl + rπh (1 − e−(λh +r)k ) .
πl −λh + rThis can be written as
rk ∗

−e

λl + r∗ql
πh e−λh k − (1 − ql )w
λh + r
32

!

> ql

!
λl + rπl −πh .
λh + rwhich is further reduced to
rk ∗

−e

ql πl e−λhk∗

− (1 − ql )w > ql (1 − e−(λh+r)k ∗!
λl + rπh .
) πl −λh + r

Note that the left-hand side is positive by definition while the right-hand side is negative, sothat this condition holds for any k∗ . From the preceding argument, we can see that p̂ = 1 if
and only if (10) holds. It is also evident that τ C (p0 ) is weakly increasing in p0 .
Proof of Proposition 4: We show the existence and uniqueness of the equilibrium by
construction. We start with a candidate strategy T 1 = {τ 1 } and the corresponding best
response such thatat =(l for t ∈ [0, max{τ 1 − k∗ , 0}),h for t ∈ [max{τ 1 − k∗ , 0}, τ 1 ].

(17)

This pair of strategies constitutes an equilibrium if V ∗ (K(p, qh ), p) > 0 for all p ∈ (qh , p0 ].
Alternatively, we check for what interval of p0 this condition is satisfied. To this end, thefollowing result is useful.
Lemma 2 Let ql > q. Then, V ∗ (K(p0 , q), p0 ) > 0 for all p0 ∈ (q, 1) if and only if
lim V ∗ (K(p, q), p) > 0.
p↑ql

Proof: Since the necessity is evident by definition, we only prove the sufficiency part. Since
the case with ql = 1 can be proved analogously (by taking the limit), we suppose for now
that 1 > ql > q.
First, if p0 ≥ ql , V ∗ (K(ql , q), ql ) > 0 directly implies V ∗ (K(p0 , q), p0 ) > 0 for all p0 ≥ ql
because the instantaneous payoff is strictly positive for any effort choice when the belief isabove ql . Note also that V ∗ (K(p0 , q), p0 ) > 0 for all p0 ∈ (q, φ(qh )] since the high type always
exerts high effort in this case. This implies that V ∗ (K(p0 , q), p0 ) > 0 for all p0 ∈ (q, 1) if
φ(q) ≥ ql . In what follows, therefore, we assume ql > φ(q).
Given this, it suffices to show thatV ∗ (K(ql , q), ql ) > 0 ⇒ V ∗ (K(p0 , q), p0 ) > 0 for all p0 ∈ (φ(q), ql ).
Suppose on the contrary that V ∗ (K(ql , q), ql ) > 0 but V ∗ (K(p, q), p) ≤ 0 for some p ∈(φ(q), ql ). If this is the case, there must exist some z ∈ (φ(q), ql ) such that V ∗ (K(z, q), z) = 0.
Since
V ∗ (K(p, q), p) = (1 − p)wp(λl y − w)′′
1 − e−(λl +r)(K(p,q)−K(p ,q)) −1 − e−r(K(p,q)−K(p ,q))λl + rr−λl (K(p,q)−K(p′ ,q)) −r(K(p,q)−K(p′ ,q)) ∗+ 1 − p + peeV (K(p′ , q), p′ ),33

for any p > p′ > φ(q), due to the recursive structure of the expected payoff, we have
 (1 − ql )w
ql (λl y − w)
1 − e−(λl +r)∆K −1 − e−r∆K
λl + rr−λl ∆K −r∆K ∗+ (1 − ql + ql e)e
V (K(z, q), z) (1 − ql )w
ql (λl y − w)
1 − e−(λl +r)∆K −1 − e−r∆K ,=λl + rr

V ∗ (K(ql , q), ql ) =where ∆K := K(ql , q) − K(z, q). This is strictly negative by definition, and therefore acontradiction.
The lemma shows that we only need to check the continuation payoff at ql to see whether
a given strategy is sequentially rational.

An immediate corollary of the lemma is thatV ∗ (K(p0 , qh ), p0 ) > 0 for all p0 if φ(qh ) ≥ ql . In this case, m(Θ) = 1, P 1 = qh and T = {τ 1 }constitutes an equilibrium for any p0 .
We now suppose that ql > φ(qh ) and check if the candidate strategy can satisfy theequilibrium conditions. From Lemma 2, this is the case if and only if V ∗ (K(ql , qh ), ql ) > 0,i.e.,
(1 − ql )w
ql πh −(λl +r)(K−k∗ )q l πl∗∗ e
1 − e−(λh +r)k >1 − e−(λl +r)(K−k ) +1 − e−rK .
λl + rλh + rr(18)
where K = K(ql , qh ). If (18) holds, the expected payoff is always positive for any p0 , so
we can apply the same argument as in Case 2 to show the existence and uniqueness of theequilibrium.
The situation becomes more complicated when (18) fails to hold, in which case the principal has an incentive to deviate and stop at some point, and the candidate strategy no longer
constitutes an equilibrium. In this case, there instead exists a critical belief that satisfies (18)with equality, because V ∗ (K(p, qh ), p) is continuous in p with V ∗ (K(φ(qh ), qh ), φ(qh )) > 0 >V ∗ (K(ql , qh ), ql ). We now redefine P 1 = min{q : V ∗ (K(q, qh ), q) = 0} and P 2 = qh . Given
the critical belief, we can then consider another candidate strategy T 2 = {τ 1 , τ 2 } for p0 > P 1
where pτ 1 = q 1 , and the best response is given byl for t ∈ [0, max{τ 1 − k∗ , 0}),
h for t ∈ [max{τ 1 − k∗ , 0}, τ 1 ),at =l for t ∈ (τ 1 , τ 2 − k∗ ),h for t ∈ [τ 2 − k∗ , τ 2 ].
Note that q 1 > φ(q 2 ) > q 2 = qh or τ 2 > τ 1 + k∗ .

The game is now divided into two segments. First, we can show that the equilibrium is
unique after time τ 1 (as done in Case 2). Before time τ 1 , we can apply the same procedure to34

the truncated interval [0, τ 1 ] to see whether T 2 can be an equilibrium strategy. More precisely,
we need to check if V ∗ (K(p, P 1 ), P 1 ) > 0 for all p ∈ [p0 , φ(P 1 )). Applying Lemma 2, T 2
is an equilibrium strategy for all p0 ∈ (P 1 , 1) if V (K(ql , P 1 ), ql ) > 0. For p0 ∈ (P 1 , 1), theprincipal’s equilibrium strategy consists of two termination dates. Given that the continuation
equilibrium after time τ 1 is unique, we can establish the uniqueness by applying the same
argument.
If V (K(ql , P 1 ), ql ) ≤ 0, on the other hand, we can find yet another critical belief in(φ(P 1 ), ql ]. We again redefine P 1 analogously and repeat the same process until we findP 1 such that V (K(ql , P 1 ), ql ) > 0. In the end, this process would give us m(Θ) intervals{(P j , P j−1 )}m
j=1 where the principal’s equilibrium strategy consists of n = m(Θ) − j + 1
termination dates if p0 falls into (P j , P j−1 ). Note also that since V ∗ (K(P 1 , P 2 ), P 1 ) = 0 by
definition, it must be that ql > P 1 if m(Θ) > 1.
Proof of Proposition 5: (i) For p0 ∈ (qh , φ(qh )), τ C (p0 ) = τ N C (p0 ) = K(p0 , qh ). Since theagent’s strategy depends only on the remaining time, the two cases yield the same expected
payoff for the principal.
(ii) To show this, we first examine how V C and V N C vary with respect to p0 for p0 ∈ (φ(qh ), 1).
It is evident that V C must monotonically increase with the initial prior. To see this, note
that
dV C
∂V ∗ ∂V ∗ dτ C
=
+ C
.
dp0
∂p0
∂τ dp0The last term is the indirect effect though an increase in τ C which is invariably zero because
∗

C

∗

∂V
= 0 or dτeither ∂V
dp0 = 0 must hold by the optimality condition. Since the direct effect ∂p0∂τ C

is unambiguously positive, V C is strictly increasing in p0 . More precisely, we obtain
dV C
∗
w
∗
πh
(1 − e−(λh +r)k ) + (1 − e−rk )=dp0
λh + rr
∗∗= p0 πh − (1 − p0 )w − (p0 λh + r)V C (p0 ) − p0 e−λh k πh − (1 − p0 )w e−rk ,(19)for p0 ∈ (φ(qh ), p̂), and

C

∗

dV C
w
πl
e−(λl +r)(τ −k ) πh
C
∗
∗
C
=
(1 − e−(λl +r)(τ −k ) ) +(1 − e−(λh +r)k ) + (1 − e−rτ )dp0
λl + rλh + rrC∗C∗λh + r(1 − e−(λl +r)(τ −k ) ) + p0 πh e−(λl +r)(τ −k ) − (1 − p0 )w= p 0 πlλl + r

C
C
∗
∗
(20)
− (p0 λh + r)V C (p0 ) − p0 πh e−λl (τ −k )−λh k − (1 − p0 )w e−rτ ,for p0 ∈ (p̂, 1).

35In contrast, V N C changes with p0 in a non-monotonic way if m > 1. Consider p0 ∈(φ(P j ), φ(P j−1 )) for j = 2, 3, ..., m. Since V N C (p0 ) = V ∗ (K(p0 , q), p0 ) where q is the closest
termination belief, it follows that∂V ∗∂V ∗ ∂V ∗ ∂K
∂V ∗1
dV N C
=
+
=
+
.
dp0
∂p0
∂k ∂p0∂p0
λl p0 (1 − p0 ) ∂k
First, for p0 ∈ (φ(P j ), P j−1 ), we obtain∗dV N Cπl∗πh e−(λl +r)(K−k )∗w=(1 − e−(λl +r)(K−k ) ) +(1 − e−(λh +r)k ) + (1 − e−rK )dp0
λl + rλh + r
r
!
∗
πl
πh
∗
we−rK
e−(λl +r)(K−k )−(1 − e−(λl +r)k ) −+λl (1 − p0 )λl + r λh + r
λl p 0= p0 πl − (1 − p0 )w − (p0 λl + r)V N C (p0 ),(21)which is negative because ql > P 1 > p0 . For p0 ∈ (P j−1 , φ(P j−1 )), we obtain

w
e−rK
πh
dV N C(1 − e−(λh +r)K ) + (1 − e−rK ) +=p0 πh e−λh K − (1 − p0 )w
dp0
λh + rr
λh p0 (1 − p0 )= p0 πh − (1 − p0 )w − (p0 λh + r)V N C (p0 ),(22)which is positive for any p0 > qh .
We are now ready to see how ∆V changes with respect to p0 . First, it is clear that ∆V
is strictly increasing in p0 for p0 ∈ (φ(P j ), P j−1 ) because V C is increasing while V N C isdecreasing. In contrast, for p0 ∈ (P j−1 , φ(P j−1 )), both V C and V N C are increasing in p0 .
For p̂ > p0 , it follows from (19) and (22) that


d∆V
∗
< 0 ⇔ (p0 λh + r) V C (p0 ) − V N C (p0 ) > − p0 e−λk − (1 − p0 )w e−rk∗ .
dp0∗This condition always holds because p0 e−λh k > (1 − p0 )w and V C (p0 ) ≥ V N C (p0 ) by definition. Similarly, for p0 > p̂, comparing (20) and (21) yields!
λh + rC∗
p 0 πh −πl (1 − e−(λl +r)(τ −k ) ) + (p0 λh + r) V C (p0 ) − V N C (p0 )λl + rCC∗∗> − p0 e−λl (τ −k )−λk − (1 − p0 )w e−rτ ,which again always holds.

(iii) If p̂ ≥ φ(P 1 ), then τ C (φ(P 1 )) = k∗ . Since τ N C (φ(P 1 )) = k∗ by definition, ∆V (φ(P 1 )) =0. To see the necessity, note that ∆V (p0 ) = 0 implies τ C (p0 ) = τ N C (p0 ). Since τ N C (φ(P 1 )) =k∗ , we must have τ C (φ(P 1 )) = k∗ which holds if and only if p̂ ≥ φ(P 1 ).

36

(iv) Suppose first that 1 > p̂, in which case τ C and τ N C solve−λτ C (p0 )e1 − p0=p0−λl τ NC (p0 )e∗(λh + r)we−(λ+r)k∗(λh + r)πl − (λl + r)πh (1 − e−(λh +r)k )1 − p0=p0

q1(1 − q 1 )e−∆λ k∗

!

=:

!

=:

1 − p0 CΦ ,p0

1 − p0 N CΦ ,p0respectively, for p0 > p̂. Note that both ΦC and ΦN C are independent of p0 . From this, since
limp0 →1 τ C (p0 ) = ∞ and limp0 →1 τ N C (p0 ) = ∞, we obtainlim V C (p0 ) = lim V N C (p0 ) =

p0 →1

p0 →1

(1 − p0 )w
p 0 πl−.
λl + rr

If p̂ = 1, on the other hand, τ C (p0 ) = k∗ for all p0 , but as we have seen, limp0 →1 τ N C (p0 ) = ∞.

Appendix B: the value of commitment in (φ(P 1 ), 1)
As we have seen in Proposition 5, the value of commitment moves with some regularities forp0 ∈ (qh , φ(P 1 )). The exception is the last segment (φ(P 1 ), 1). The reason is that for this(and only this) segment, we could have φ(P 1 ) > p̂ > ql (see Propositions 3 and 4), so thatthe instantaneous payoff under low effort can be positive and τ C (p0 ) > k∗ for p0 ∈ (φ(P 1 ), 1).
There are three cases we need to consider, depending on the value of p̂.
Case 1: φ(P 1 ) > p̂In this case, τ C (φ(P 1 )) > τ N C (φ(P 1 )) = k∗ and hence ∆V (φ(P 1 )) > 0 (as we show in(iii) of Proposition 4). For p0 > φ(P 1 ), τ C (p0 ) and τ N C (p0 ) increase proportionally (see theproof of Proposition 4), with both diverging to infinity as p0 → 1. As we show in (iv) of
Proposition 4, ∆V (p0 ) converges to zero as p0 → 1.
Case 2: p̂ > φ(P 1 )In this case, τ C (φ(P 1 )) = τ N C (φ(P 1 )) = k∗ and hence ∆V (φ(P 1 )) = 0. For p0 ∈(φ(P 1 ), p̂), τ N C (p0 ) continues to go up while τ C (p0 ) stays at k∗ ; as a consequence, ∆V (p0 )gradually increases in this range. Once p0 goes above p̂, τ C (p0 ) also gradually increase, but
again proportionally to τ N C (p0 ). Since τ N C (p0 ) > τ C (p0 ) for p0 ∈ (p̂, 1) in this case, ∆V (p0 )gradually decreases and converges to zero as p0 → 1.38
38
If p̂ = φ(P 1 ), then τ C (p0 ) = τ NC (p0 ) and hence ∆V (p0 ) = 0 for p0 ∈ (p̂, 1). We ignore this non-generic
case.

37

Case 3: p̂ = 1
This is the most straightforward case. Given that p̂ = 1, τ C (p0 ) = k∗ for p0 ∈ (p̂, 1) andhence τ C (p0 ) and τ N C (p0 ) diverge away from each other in this range. As a consequence,∆V (p0 ) increases as in all the preceding intervals.

Appendix C: The case with uncertainty
We analyze an extended version of the baseline model where a permanent productivity shock
may strike with some probability. For simplicity, we assume that a shock arrives with aPoisson arrival rate β; once it arrives, the agent becomes totally unproductive thereafter,i.e., λh = λl = 0. We also assume that the arrival of a shock is publicly observable. Other
more complicated cases, e.g., those with transitory shocks, can in principle be analyzed in an
analogous manner.
The agent: The agent’s problem requires a slight modification. The augmented Bellman
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 7 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: We first consider the case where the principal commits to a deadline. Note that the
principal cannot terminate the project even if the shock arrives before the deadline. Since theagent has no incentive to exert high effort, we have Ũ C (k) = w(1−e
r

−rk )

. Let UaC (k) denotethe value function under commitment when the effort level is fixed at a. We then obtain!
−(λh +β+r)k−rk (1 − e−(λh +β)k )
1
−
e
βw
e
bλ
−
d
+
w
h
.
(1 − e−(λh +β+r)k ) +−UhC (k) =λh + β + rr
λh + β + r
λh + β38

The 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 + rd
N C given by
h −d+wIf bλ
λh +β+r > b − ∆λ > 0, there exists an interior threshold k

!
d
(λ
+
β
+
r)(b
−
)
h
1
∆λ
kN C = −.
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 obtainbλh − dλh w
UhC (k) =(1 − e−(λh +β+r)k ) −λh + β + rr

1 − e−(λh +β+r)ke−rk (1 − e−(λh +β)k )−λh + β + r
λh + βw(1 − e−rk )r−rk )bλh − d − λh w(1−eλh we−rkr(1 − e−(λh +β+r)k ) +=λh + β + r
r

!

+

+

w(1 − e−rk ).
r
39

1 − e−(λh +β)k1 − e−(λh +β+r)k−λh + βλh + β + r

!

−αk

First, 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−eCrUh (k) >λh + β + r
+

−rk )

)

−(λh +β+r)k(1 − eλh we−rk)+r

1 − e−(λh +β)k1 − e−(λh +β+r)k−λh + βλh + β + r!

w(1 − e−rk ),rfor k ≤ kC . This is further reduced to
!
!
−rk )
λh (1 − e−(λh +β+r)k )w(1−e
λh we−rk
1−
UhC (k) −>λh + β + rrr
which gives UhC (k) > w(1−e
r

−rk )

!
1 − e−(λh +β+r)k1 − e−(λh +β)k−,λh + βλh + β + r.

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, eitherwith or without commitment (kN C > k C > k∗ ). 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 thelowest possible payoff as long as the project survives. The situation is even worse withoutcommitment, however, because the principal can immediately terminate the project, thereby
leaving no rent for the agent. The extent of the loss is therefore larger when the principal
makes no commitment, which amounts to the fact that kN C > kC and 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 (k, p0 ) denote the principal’s expected payoff at time 0 under commitment,
which can be written as
Z νCRs(λl ps y − w)e− 0 (λl pu +β+r)du ds
V C (k, p0 ) =
0
Z k
R C
Rs
− 0ν (λl pu +β+r)du+e(λh ps y − w)e− ν C (λh pu +β+r)du ds
νC

Z νC

Rs
β
w(1 − e−r(k−s) )e− 0 (λl pu +β+r)du dsr 0ZRs
β − R ν C (λl pu +β+r)du k− e 0w(1 − e−r(k−s) )e− ν C (λh pu +β+r)du ds.
r
νC

−

40

(23)

where ν C := max{τ C − kC , 0}. The difference from the previous case is that a negative shock
may now strike with some probability, in which case the principal earns zero from that point
on. The commitment solution is obtained asτ C = argmaxτ V0C (τ ).
As can be seen from (23), 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) :=pNC .
p + (1 − p)e−λh kGiven 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(k, p) =Z ν NCRs(λl ps y − w)e− 0 (λl pu +β+r)du ds0
Z k
R ν NCRs+ e− 0 (λl pu +β+r)du(λh ps y − w)e− ν NC (λh pu +β+r)du ds,(24)

ν NC (q)where ν N C is defined analogously. Note that for the same time horizon, V N C (k, p0 ) >V0C (k, p0 ) for any p0 .
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. We can thenmake the following statement.
Proposition 8 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 to41

set a deadline before the belief reaches qh . The profit is clearly lower than V N C (τ N C , p0 ). 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 commitment through two channels. First,
by making no commitment, the principal can retain the flexibility to adjust to the negative
shock by 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 ofproject termination, and thus, he starts exerting high effort earlier. Since φN C (qh ) > φ(qh )for any β > 0 by Proposition 7, we could have φN C (qh ) ≥ p0 > φ(qh ), in which case the valueof commitment is strictly negative with uncertainty whereas it is strictly positive without.

7

Appendix D: The case with endogenous wages

We consider a setup where the principal can deliberate choose w ≥ 0 at time 0. In general,
an increase in w above zero yields two opposing effects. On one hand, it raises the criticalvalue qh , which may give the principal the incentive to stop earlier than otherwise. On theother hand, it also raises the expected future payoff which in turn lowers k∗ and gives theagent the incentive to procrastinate even more. A necessary condition for w > 0 is that thefirst effect dominates the second, or alternatively that τ N C − k∗ decreases with w.39 We canthen make the following statement.
Proposition 9 It is optimal to set w > 0 if λl is sufficiently small.
Proof: As w → 0, we have qh → 0 and ql → 0, which implies τ N C (p0 ) → ∞ for any p0 . The
expected payoff then converges tolim V ∗ (τ N C , p0 ) =w→0

p 0 λl y,
λl + r(25)

for any p0 . Define µ(w) such that φ( λwh y ) = µ(w), i.e.,∗µ(w)−λh kw.
∗ = qh =−λk
1 − µ(w) + µ(w)λh y39Note that τ NC − k∗ is the interval during which the agent exerts low effort. If it is longer with a smallerk , the principal’s expected payoff is necessarily lower.
∗

42

Note thatd(λh + r)(b − ∆λ )1 − qh −λh k∗
λh y − w
1 − µ(w)
1−
=
e
=
µ(w)
qh
w
λh b − d + w

! λh

λh +r

.

Since limw→0 µ(w) = 0 and µ(λh y) = 1, there must exist at least one w̃ such that µ(w̃) = p0 .
If we set w = w̃, then τ N C = k∗ > 0 by definition, and the expected payoff is∗

V (k , p0 ) =

Z k∗

Rs

(λh yps y − w̃)e− 0 (λh pu +r)du ds,0which is always strictly positive and larger than (25) if λl is sufficiently small.

43

pt

ql

P 1 = qhk∗
τ 1 − k∗τ1t

Figure 1: Evolution of the belief on the equilibrium path (m(Θ) = 1)pt

V ∗ (τ 1 − t, pt ) < 0

qlq′

P 1 = qhk∗
τ 1 − k∗τ′

τ1Figure 2: A violation of Condition C: the belief (m(Θ) = 2)

44

t

V∗

V ∗ (τ 1 − t, pt ) < 00k∗τ 1 − k∗

τ′

τ1

t

Figure 3: A violation of Condition C: the continuation payoff
pt

ql

P1

P 2 = qhk∗
τ NC − k∗k∗τ 2 − k∗

τ 1 = τ NC

τ2

Figure 4: Evolution of the belief, on and off the equilibrium path

45

t

V∗
V C (p0 )V NC (p0 )φ(P 2 )

P 2 = qhP1

φ(P 1 )

p0

Figure 5: The expected payoffs with and without commitment
τK(p0 , qh )

τ C (p0 )qh

φ(qh )

p0

Figure 6a: The commitment solution when λl is relatively high (λl = 0.07)46τK(p0 , qh )

τ C (p0 )qh

φ(qh )

p0

Figure 6b: The commitment solution when λl is relatively low (λl = 0.04)
ττ NC (p0 )τ C (p0 )φ(P 2 )

P 2 = qh

P1

Figure 7: The no-commitment solution

47

φ(P 1 )

p0