Relational Incentive Contracts withHidden Action and Unequal Discounting∗
Eric S. Chou† Hideshi Itoh‡ Chien-Lung Chen§
September 3, 2024

Abstract
We analyze relational incentive contracts with hidden action when theprincipal and the agent have diﬀerent discount factors, and identify a newagency cost arising from a trade-oﬀ between rent extraction and incentive
provision, that exists only when they cannot commit themselves to longterm contracts and the principal is less patient than the agent. We characterize the condition under which the trade-oﬀ between rent extraction
and incentive provision exists, as well as the condition for the optimality of
non-stationary contracts.

1

Introduction

The purpose of this paper is to extend the theory of relational incentive contracts
with hidden action to cases where the transacting parties have diﬀerent discountfactors. We consider an infinitely repeated principal-agent model in which theagent’s binary eﬀort level is unobservable to the principal, and the principal offers an incentive contract in a take-it-or-leave-it fashion to motivate the agent toexert eﬀort. Both the principal and the agent are risk-neutral. The performancemeasure is non-verifiable, and hence the contract has to be dynamically enforceable, that is, it must satisfy self-enforcing constraints that require any bonus and∗We would like to thank the seminar participants at CTW, the 3rd Workshop on Relational
Contracts (Northwestern University), Keio University, Japanese-German Workshop on Contracts and Incentives (Osaka), Queens University, and Waseda University for helpful comments.
Itoh acknowledges the financial support from JSPS KAKENHI Grant Numbers 25245031 and
18H03640, as well as Waseda University.
†
Department of Economics, National Tsing Hua University, Hsinchu, Taiwan (e-mail: swchou@mx.nthu.edu.tw).
‡
Waseda Business School, Waseda University, Tokyo, Japan (e-mail: hideshi.itoh@waseda.jp).
§
School of Economics and Trade, Hubei University of Economics, Wuhan, China (e-mail:lungclchen@icloud.com).

1

penalty specified to be bounded above by the principal and the agent’s discounted
continuation payoﬀs (net of their reservation utilities), respectively. We derive
an optimal incentive contract in a way distinct from existing literature where theprincipal and the agent are equally patient.
The main contribution of the paper is to elucidate whether and how the standard results on relational contracts under the assumption of equal discounting,such as the optimality of stationary contracts and no rent left for the agent (Levin,2003), change with the introduction of unequal discount factors. In particular, we
identify a new agency cost that exists only under relational incentive contracting
and unequal discount factors.
While the purpose of the paper is purely theoretical, considering unequal discount factors in the ongoing principal-agent relationship has its practical motivation. The parties usually have diﬀerent time preferences such that one of themis more patient, expecting their relationship to be longer-lived, or facing a higher
cost of capital (such as internal rate of return in capital budgeting). Our analysis,hence, possibly links characteristics of the contracting parties, such as their size,
tenure, access to financial markets, and so on, to the optimal contracts between
them and the resulting division of surplus.
We allow the principal’s discount factor to be either higher or lower than thatof the agent. Since patience is one source of bargaining power, one might thinkthat the principal who makes a take-it-or-leave-it oﬀer be more patient than theagent. We however think that situations where the impatient principal can makea take-it-or-leave-it oﬀer are not unreasonable. As Lyon and Rasmusen (2004)
argue, an “advantage of economic theory over looser thinking about bargainingis that this definition of bargaining power distinguishes strong bargaining powerfrom a strong bargaining position. (p.151)” For example, the regulator and theCEO, who face elections and other turnover possibilities near future, may be moreinterested in short-term outcomes than, respectively, the regulated firms stronglyinterested in survival, and the employees who want to stay with the firm for a long
period. And some features of the optimal relational contract we derive, such as“seniority-based,” “promotion-like,” and no termination, are consistent with theemployment practices of the stylized, traditional Japanese firm where the typicalCEO has short tenure and the long-term employment is prevailing (Itoh, 1994).
Furthermore, our analysis of the less patient principal applies with minor modifications to the case in which the less patient agent makes a take-it-or-leave-it
oﬀer to the principal. For example, the contracting relationship between an entrepreneur and investors is often modeled such that the entrepreneur has scarceand valuable business ideas and unique necessary skills, but can raise money only
at a higher interest rate, while lenders behave competitively (see, for example, Ti2

role, 2006). Therefore in dynamic financial contracting literature, the entrepreneur
is often assumed to be more impatient than the financiers but has all the bargaining power at the initial date (see, for example, Biais et al., 2013, for an overview).
We first confine our attention to stationary contracts, and derive the optimal
stationary contract that maximizes the principal’s expected payoﬀ. When theprincipal’s discount factor is at least as high as a threshold, the optimal contract
simply consists of a bonus to provide eﬀort incentives and a fixed salary (or,equivalently, an up-front payment) such that the agent’s individual rationality
constraint is binding and hence no rent is left to him every period. The principal
will not renege on the bonus because she is so patient that the future payoﬀ mattersmore than the benefit from reneging. While the optimal stationary contract is thesame as the standard one under equal discount factors, the way we derive theoptimal contract is distinct from that in existing literature because adding up
the relevant constraints does not eliminate the monetary transfer parts when the
discount factors are unequal.
When the principal’s discount factor is lower than the threshold, bonuses alone
are not enough to provide eﬀort incentives because the principal’s continuation
payoﬀ is too low for her not to renege on the payment of the bonus needed for
eﬀort provision. In this case, however, if instead the agent’s discount factor is
at least as high as the threshold (and hence the agent is more patient than theprincipal), it becomes worthwhile to use some penalty to supplement incentives,
even though it requires leaving the agent some rent for him not to renege on
the payment of the penalty. The trade-oﬀ between rent extraction and incentive
provision arises even though the agent neither faces the limited-liability constraint
nor has private information at the beginning of each period. Intuitively, when theprincipal is less patient than the agent, transferring payoﬀ to the agent can increase
the joint continuation payoﬀ and making the implementation easier by relaxing
the bound on penalty more than tightening the bound on bonus. Hence, when
the principal is impatient enough (so that eﬀort cannot be implemented without
giving a rent to the agent) and the agent is patient enough, it becomes optimalfor the principal to leave some rent to the agent by using penalty to make up theeﬀort incentive that the bonus is not enough to provide by itself.
Finally, if both parties’ discount factors are lower than the threshold, the agent
reneges on the payment of the penalty even under the contract that consists ofthe penalty and a fixed salary only and hence leaves all the rent to the agent.1
Stationary contracts are in general suboptimal, however, when the principal
1

Note that our analysis clearly distinguishes between bonuses and penalties as incentive
devices, that are often pointed out as algebraically equivalent unless some psychological elements
are taken into account (see, for example, Lazear, 1995). However, in this paper we do not intend
to oﬀer any explanation concerning under what conditions these contracts are used in practice.

3and the agent have unequal discount factors. There are three reasons for this.
The first, well-known reason is that while the principal has to leave a rent to theagent under the optimal stationary contract, she can extract the rent by sellingthe whole project via the first-period contract with a low enough payment.
Second, the principal and the agent benefit from engaging in intertemporal
payoﬀ trading via non-stationary contracts. Because of this second reason, the rent
extraction by the initial contract mentioned above cannot be analyzed separately.
The optimal intertemporal trading expands the Pareto frontier, so that it is nolonger a straight line with slope minus one. Along the Pareto-optimal path, thestage payoﬀ to the more patient party is increasing and that to the less patient
party is decreasing over periods. This implies that while the stage payoﬀ to theless patient player cannot drop below his/her reservation payoﬀ, that to the morepatient player can be lower than his/her reservation payoﬀ for some periods.
If the principal is more patient than the agent, allowing non-stationary contracts does not alter the optimality of a stationary bonus contract because theprincipal must guarantee the less patient agent at least his reservation payoﬀ every period, and she can attain the maximum stage payoﬀ by a stationary bonus
contract. If the principal is less patient, however, intertemporal payoﬀ trading
via a non-stationary contract benefits her. The optimal “semi-dynamic” contract,that explores intertemporal payoﬀ trading but provides the agent with eﬀort incentives only by current payments (bonus and penalty), entails a decreasing bonus
plan, accompanied with an increasing penalty schedule, in order to ensure theagent’s eﬀort incentives. The fixed salary is an increasing, “seniority-based” planin order to compensate the diminishing payment due to the decreasing bonuses
and increasing penalties, as well as to back-load payoﬀs to the agent and guarantee
him larger stage payoﬀs in later periods.
The third reason why stationary contracts are suboptimal is that the lesspatient principal may be able to extract more rent from the agent by making
his continuation payoﬀs contingent on the current output, although they thenhave to deviate from the optimal intertemporal trading and hence their payoﬀ
vector is short of the Pareto frontier. We show that if the optimal semi-dynamic
contract has to leave some rent to the agent, the optimal “dynamic” contract is a“promotion-like” one, under which his average payoﬀ does not decrease with low
output and one-time success (high output) moves his payoﬀ to the one prescribedby the optimal semi-dynamic contract. And termination never occurs on theequilibrium path.
We also obtain several comparative statics results about the agent’s rent. The
rent is decreasing in the surplus of the relationship, and is increasing in the reservation utilities of both parties. An increase in the surplus of the relationship4makes it easier to satisfy the self-enforcing constraints, and hence shifts the optimal contract toward more use of bonuses, thereby reducing the rent to the agent.
An increase in the reservation utility of either party makes it harder to satisfy the
self-enforcing constraints, and hence increases the rent to the agent via a change ofthe optimal contract toward more use of penalties. The agent weakly prefers lessadvanced contracting technology because his rent decreases as the optimal contracting form improves from stationary to semi-dynamic, and from semi-dynamic
to dynamic contracts.
Related Literature
In the literature on repeated games, Lehrer and Pauzner (1999) study two-player
repeated game with unequal discount factors to show how intertemporal payoﬀ
trading expands the Pareto frontier and prove a folk theorem result that outcomeson the frontier can be achieved as equilibria as discount factors converge to one
with the relative patience between players fixed. Their folk theorem result is extended to n-player games (Chen and Takahashi, 2012) and to imperfect public
monitoring (Sugaya, 2015). We also follow Lehrer and Pauzner (1999) to expand
the Pareto frontier, but in our paper it is non-stationary contracts that are used
for intertemporal payoﬀ transfer. Fong and Surti (2009) confine their attention torepeated prisoners’ dilemma with unequal discounting, but instead extend Lehrer
and Pauzner (1999) by introducing the possibility of voluntary side payments atthe beginning of each period. Side payments are used to provide incentives tocooperate as well as to trade payoﬀs across periods. They show that providing
incentives for the impatient player and intertemporal payoﬀ trading may conflictwith each other, and full cooperation may be Pareto dominated by partial cooperation in which only the patient player chooses cooperation. In contrast to them,
we study the principal-agent problem and solve the optimal relational contract toimplement eﬀort provision by the agent, where the impatient principal may face a
conflict between providing incentives for the patient agent and transferring payoﬀsintertemporally.
Most literature on relational contracts assumes equal discounting: For example, a survey chapter Malcomson (2013) contains no discussion of unequal discounting. Some of the recent papers on dynamic principal-agent relationships
assume that the principal and the agent have diﬀerent discount factors (Biais etal., 2007, 2013; Opp and Zhu, 2015; Hoﬀmann et al., 2021; Krasikov et al., 2023).
However, they exclusively focus on the case where the principal is more patientthan the agent, and hence the principal benefits from front-loading rewards to theagent. In these papers, back-loading of the agent’s payoﬀs arises from the features5

other than unequal discounting, such as limited liability (Biais et al., 2007, 2013),
no hidden action (Opp and Zhu, 2015), persistent hidden action (Hoﬀmann et al.,2021), and persistent private information (Krasikov et al., 2023). And they do
not consider the dynamic enforcement issue that the principal and the agent may
renege on payments contingent on unverifiable performance measures. In contrast
to them, in our model there is no reason to either front-load or back-load the payoﬀs to the agent if his discount factor is the same as that of the principal. We thenhighlight how the optimal incentive contract is aﬀected when unequal discounting
and the dynamic enforcement condition for incentive contracts interact.
In relational contract literature, the agent may enjoy a positive information
rent when the agent’s type is his private information (Yang, 2013; Ishihara, 2016;
Malcomson, 2016) or a positive limited-liability rent when the agent is protectedby the (ex post) limited liability (Fong and Li, 2017). The agent’s rent in ourmodel is clearly diﬀerent from the information rent because there is no adverse
selection in our model.
The agent’s rent in our model is also diﬀerent from the limited-liability rent.
The latter exists because a bound is imposed on penalties exogenously, and, because there need to be enough diﬀerence between the bonus and the penalty to
induce eﬀort, the bonus is push up to the point where the agent obtains morethan his reservation payoﬀ. The agent enjoys the limited-liability rent under the
optimal one-shot contract as well as long-term contract where the principal andthe agent commit themselves to all payments (Fong and Li, 2017, Proposition0). As a result, the rent continues to exist under relational incentive contracting
when the principal and the agent are equally patient (their Proposition 1) or when
the principal is more patient than the agent (their online appendix). In contrast,
the agent receives no rent in all these cases in our paper (See Sections 3 and 6)
because we do not assume an exogenous bound for penalties.2 In our paper, theagent enjoys a positive rent only when the agent is more patient than the principal
and they cannot commit themselves to long-term contracts, because if the surplusof the relationship is too low to satisfy the non-reneging constraint for rewards
by letting the principal capture all the surplus, shifting some surplus to the agent
relaxes the non-reneging constraint for penalties more than it tighten the nonreneging constraint for rewards, making a large enough gap between rewards andpenalties possible.
Although our agency rent is distinct from theirs, there are still similarities between the optimal dynamic contract in our paper and that in Fong and Li (2017),
2

We impose a lower bound on the expected total payment to the agent in order to make theoptimization problem well-defined when we analyze non-stationary contracts. This assumption
per se does not generate a positive rent.

6because the dynamics helps the principal extract more rent in both papers. However, the diﬀerent source of the rent entails some diﬀerences in the dynamics: The
agency cost due to limited liability leads to ineﬃciency in the form of ineﬃcienttermination in Fong and Li (2017), whereas in our model, ineﬃcient termination
never happens and our new agency cost leads to ineﬃciency in the form of deviation from the eﬃcient intertemporal payoﬀ trading that expands the Pareto
frontier.
The Structure of the Paper
The rest of the paper is organized as follows. Section 2 introduces the baselinemodel. Section 3 contains the analysis of the benchmark case when the discountfactors are the same. Stationary contracts and non-stationary contracts underunequal discount factors are analyzed in Sections 4 and 5, respectively.
In Section 5, we first focus on the eﬀect of intertemporal payoﬀ transfer by
restricting attention to “semi-dynamic contracts” in which continuation payoﬀs
can vary only with time but not the revenues of the previous periods, and hence
eﬀort incentives are provided by current payments only (Subsection 5.1). We thenallow fully dynamic contracts to be designed in Subsections 5.2 and 5.3. Most of
the analysis assume that the principal is less patient than the agent. The othercase where the principal is more patient is analyzed in Subsection 5.4.
Section 6 contains the analysis of another benchmark case where the parties
can commit themselves to long-term contracts. Section 7 concludes. The appendix
provides all the proofs that are not in the main text.

2

The Model

There are one principal and one agent, both are risk neutral. In each periodt ∈ {0, 1, . . .}, the agent chooses a binary eﬀort et ∈ {0, 1}, not observable to theprincipal, by incurring cost cet where c is a positive constant. After the eﬀort ischosen, the principal observes a non-verifiable output y ∈ {h, ℓ}, where h > ℓ.
The timing of each period t is as follows.
1. The principal oﬀers a take-it-or-leave-it contract {wt , bht , bℓt }, where wt is alegally enforceable fixed salary (or equivalently, an up-front payment), byt is
a discretionary payment from the principal to the agent, contingent on y.
We say that the contract requires the principal to pay the agent a bonus if
byt is positive and the agent to pay the principal a penalty if byt is negative.
To simplify notation, we will use {Wt , bht , bℓt } where Wt is the expected total

7

payment to the agent to represent the dynamic contract in the rest of thepaper.
2. The agent accepts or rejects the contract. If he rejects the contract, theprincipal and the agent obtain, respectively, the reservation payoﬀs π ≥ 0
and u ≥ 0. Denote the total reservation payoﬀ by s ≡ u + π.
3. If the agent accepts the contract, then wt is paid, and the agent chooses an
eﬀort et .
4. The output realizes. The principal and the agent decide whether to pay thebonus or penalty, respectively.
5. Both parties observe the realization of a random variable xt ∈ [0, 1].3
We denote δP and δA as the discount factors of the principal and the agent,respectively, and let di ≡ δi /(1 − δi ) for i = P, A. To simplify notations, wewill suppress the time subscript t when there is no risk of confusion. Denote
Pr{y = h | e = 1} = P , Pr{y = h | e = 0} = p, and q ≡ P − p > 0. These areassumed to be time invariant. Let R ≡ P h + (1 − P )ℓ be the expected revenue
and s ≡ R − c be the total surplus, all conditional on e = 1. Following convention,
we assume s > s > ph + (1 − p)ℓ.
A relational contract is a complete plan for the relationship, that is, for eachperiod and every public history, it describes the fixed salary the principal shouldoﬀer the agent, whether or not the agent should accept the oﬀer, the eﬀort theagent should exert, the bonus the principal should pay, and the penalty the agent
should pay. A relational contract is self-enforcing if it describes a perfect public
equilibrium (PPE) of the repeated principal-agent game. We often refer to sucha contract as the equilibrium contract.
We solve for the optimal relational contract, which is the perfect public equilibrium that maximizes the principal’s payoﬀ at the beginning of the initial period.
For the agent to choose e = 1 in any given period, the equilibrium contract mustsatisfy the incentive compatibility constraint:
P (bh + dA uh ) + (1 − P ) (bℓ + dA uℓ ) − c ≥ p (bh + dA uh ) + (1 − p) (bℓ + dA uℓ )orc≤ bh − bℓ + dA (uh − uℓ ) ,
q
3

(IC)

In models of repeated games, it is common to assume the existence of a public randomization
device to ensure the convexity of the set of equilibrium payoﬀ. We also assume that the principal
and the agent observe the realization of a public randomization device at the beginning of period0.

8where uy is the average continuation payoﬀ to the agent when the output isy ∈ {h, ℓ}, respectively. For the payments not to be reneged, the contract needs
to satisfy the dynamic enforcement constraints for the principal and the agent,respectively, i.e.
bh ≤ dP (πh − π)
(DEP)
and
−bℓ ≤ dA (uℓ − u) ,(DEA)

where πy is the average continuation payoﬀ to the principal following output y.
Adding the three constraints yields the following incentive-compatibility-dynamicenforcement constraint:c≤ dP (πh − π) + dA (uh − u) ,q

(IC-DE)

which requires that the continuation payoﬀ vector (u, π) = (uh , πh ) to lie on or
above the line (c/q) = dP (π − π) + dA (u − u), which we refer to as the IC-DE line.
The optimal relational contract maximizes the principal’s expected average
payoﬀ, that is,(1 − δP ) (R − W ) + δP (P πh + (1 − P ) πℓ ) ,(1)
To solve for the optimal relational contract, it would be helpful if the three constraints (IC), (DEP), and (DEA) can be replaced by one constraint (IC-DE). The
following result shows that this is indeed the case.
Lemma 1 Given that (IC), (DEP), and (DEA) are the only constraints for bℓ
and bh , it is without loss of generality to replace (IC), (DEP), and (DEA) with
(IC-DE) in solving the optimal relational contract.
Proof. Denote the equilibrium contract that maximizes the principal’s payoﬀ (1) subject to (IC), (DEP), and (DEA) and some other constraints (to be
introduced later) that do not depend on bℓ or bh , called Program I, as C I ≡(W I , bIℓ , bIh , uIh , uIℓ ) and the contract that maximizes the principal’s payoﬀ (1) subject to (IC-DE) only and the same set of other constraints, called Program II, asIIIIIIC II ≡ (W II , bIIℓ , bh , uh , uℓ ). The lemma claims that the principal is indiﬀerent
between C I and C II . This can be proved by contradiction.
Suppose first that the principal is strictly better oﬀ with C I . Because C I
satisfies all the constraints in Program I, it must also satisfy all the constraints in
Program II. Given that the objective functions are the same in Programs I andII, the principal’s strict preference for C I contradicts the hypothesis that C II is
optimal for Program II.

9Suppose next that the principal is strictly better oﬀ with C II . Consider C III ≡
III
II
II
III
III
(W II , bIII
satisfy (DEP) and (DEA) as equalitiesℓ , bh , uh , uℓ ) where bh and bℓ
IIIIwith uh and uℓ plugged in, respectively. Since C II satisfies (IC-DE), C III satisfies(IC). Because C II and C III only diﬀer in bh and bℓ , and bh and bℓ do not appear in
the other constraints, C III satisfies all the constraints in Program I. In addition,
C II and C III gives the principal the same level of expected payoﬀ because (1) does
not depend on bh and bℓ . It follows that C III gives the principal a higher expected
payoﬀ than C I , contradicting the hypothesis that C I is optimal for Program I.
In each period, the contract also needs to satisfy the individual rationality
constraints (IR) of the principal and the agent:4
∞
∑
(k−t)
πt = (1 − δP )
δP(R − Wk ) ≥ π ∀ t ∈ {0, 1, . . .};
k=t
∞
∑
(k−t)
ut = (1 − δA )
δA (Wk − c) ≥ u ∀ t ∈ {0, 1, . . .},

(IR)

k=t

where πt and ut are the average payoﬀs from period t on to the principal and theagent, respectively. We say a contract implements e = 1 if it satisfies (IC-DE)and (IR).

3

Benchmark: Equal Discounting

When the principal and the agent have the same discount factor, denoted asδ ∈ (0, 1), our model collapses to a special case of Levin (2003) with binaryeﬀort moral hazard and no adverse selection. In this section, we review thisbenchmark case. Another important benchmark case, in which the principal andthe agent may have diﬀerent discount factors but can commit themselves to longterm contracts and hence (DEP) and (DEA) can be ignored, will be studied laterin Section 6.
Under equal discounting, if the principal’s objective is to implement e = 1
in every period, it is without loss of generality to restrict attention to stationary
contracts in the sense that every period the principal oﬀers an identical contract{W, bh , bℓ } regardless of the output. This implies that uh = uℓ and u ≡ uh =uℓ = W − c, and hence what are left to be decided are W , bh , and bℓ . Given thatδ = δA = δP , the first-best payoﬀ frontier is a straight line with a slope of −1.

4

For simplicity, we present (IR) by not taking expectations with respect to public randomization.

10π(u, s − u)su + π = s + c/(qd)u+π= s
π
0

s

u

u

Figure 1: Benchmark: Equal Discounting
This means that (IC-DE) becomesc≤ d(R − c − s).
q(IC-DE-B)

where d = dA = dP and (IR) becomesR−W ≥πand W − c ≥ u.

(IR-B)The intuition for this version of (IC-DE-B) is the same as that of Levin (2003)—
the future gain from the relationship (the right-hand side) needs to be no less thanthe variation in the agent’s payoﬀ across diﬀerent outcomes required to induce theeﬀort (the left-hand side). In this case, the optimal contract will be stationary so
the current payoﬀ to the agent, u, will be the same as uℓ and uh , and henceu = (1 − δ)(W − c) + δu.
The principal’s objective can be written as max(−W ), which is independentof bh and bℓ . It is hence without loss of generality to replace (IC), (DEP), and(DEA) with (IC-DE), by Lemma 1. The optimal contract thus solves
max (−W ) subject to (IC-DE-B) and (IR-B).

{W,bh ,bℓ }We refer to the case when the eﬀort is observable and enforceable as the first11

best, and the case when the eﬀort is unobservable and contracts are relational as
the second best.
In Figure 1, we give a geometric illustration of the optimal contract. The
horizontal and vertical axes represent, respectively, the stage payoﬀs to the agent
and the principal. The line u + π = s + c/(qd) is the IC-DE line. Implementing
e = 1 results in a payoﬀ vector (u, π) on the segment of the Pareto frontier u+π = sthat gives the principal at least π and the agent at least u. Because all payoﬀ
vectors on this line segment lie above the IC-DE line, the principal can choose to
implement any one of these payoﬀs by designing an appropriate contract. Clearly,
the best for the principal is the one at the upper-left tip of this line segment.
The payoﬀ vector (u, π) = (u, s − u) characterizes the equilibrium payoﬀs to theprincipal and the agent. By setting the total expected payment to the agent as
W = u+c, the principal extracts all the rent from the agent. This implies, by (ICDE-B), that bℓ = 0, and hence there is no penalty and the agent is motivated bya bonus payment only. If (IC-DE-B) is slack, then there are multiple solutions for
bh , ranging from c/q to d(s − s), combined with diﬀerent levels of the fixed salaryw. If (IC-DE-B) is binding, then the solution is unique and bh = c/q = d(s − s).
Because the contractβ ∗ = {u + c, c/q, 0}
is an equilibrium contract no matter (IC-DE-B) is slack or not, we will use it asa benchmark in the subsequent analysis. In particular, the agent enjoys no rent
when the discount factors of the principal and the agent are the same.
One important feature of the analysis above depends crucially on the assumption of equal discount factors between the principal and the agent: When (DEP)
and (DEA) are added together to obtain (IC-DE), the expected transfer W iscanceled out, and hence disappears from (IC-DE) resulting in (IC-DE-B), i.e., thepayoﬀs to the principal and the agent are constant-sum. Because the principal’s
objective function is to minimize W , which only appears as a positive term in
the individual rationality constraint in (IR-B) for the agent, it is clear that the
principal will not leave a rent to the agent. As we will show in the next section,
this constant-sum feature disappears when discount factors are diﬀerent.

4

Optimal Stationary Contracts

When the discount factors of the principal and the agent are diﬀerent, it is conceivable that the optimal contract is non-stationary. However, the key featurethat the agent earns rents does not depend on the non-stationarity and can beseen more clearly without it. We hence restrict attention to stationary contracts

12

in this section.
Given that only stationary contracts are allowed, (IC) and (IR-B) of the benchmark remain the same. However, (DEP) and (DEA) becomebh ≤ dP (R − W − π)

(DEP-S)−bℓ ≤ dA (W − c − u).

(DEA-S)c≤ dP (R − W − π) + dA (W − c − u).
q(IC-DE-S)(IC-DE) hence becomes

The current payoﬀ to the agent, u, is the same as uℓ and uh , and henceu = (1 − δA )(W − c) + δA u.
Since the principal’s objective, max(−W ), as well as (IC-DE-S), is independentof bh and bℓ , Lemma 1 applies and we can replace (IC), (DEP-S), and (DEA-S)
with (IC-DE-S).
Note that, in contrast to the case of equal discounting, the total expected
payment W does not get canceled out when (DEP-S) and (DEA-S) are added
together to obtain (IC-DE-S). If fact, if δP < δA , while increasing W reduces thecontinuation payoﬀ to the principal, it raises that of the agent so much that it morethan makes up for the former loss so the right-hand side of (IC-DE-S)—the join
future gain of the relationship—is an increasing function of W . This means thatgiving more rents to the agent makes it easier to satisfy (IC-DE-S). Alternatively,
if δP > δA , the right-hand side of (IC-DE-S) is an decreasing function of W , so
giving more rents to the agent makes it harder to satisfy (IC-DE-S). This is a key
feature of our model.
Definec.
d∗ ≡q(s − s)
The following lemma provides the necessary and suﬃcient condition for e = 1 tobe implemented by a stationary contract.
Lemma 2 Eﬀort e = 1 can be implemented by a stationary contract if and onlyifd∗ ≤ max{dP , dA }(2)holds. If dP > d∗ , the implementation can be achieved by the bonus contractβ ∗ . If dA > d∗ , the implementation can be achieved by a penalty contract ρ∗ ={R − π, 0, −c/q}.
13

Proof. Necessity: The result follows from (IC-DE-S) and the fact thatdP (R − W − π) + dA (W − c − u) ≤ max{dP , dA }(s − s).
.
Suﬃciency: If dP > dA , (2) implies thatc
≤ dP (s − s).
qThen the bonus contract β ∗ defined in the benchmark case can implement e = 1:
(IC) and (DEA-S) bind, and then (DEP-S) becomes identical to (2).
Similarly, if dP < dA , (2) implies that andc≤ dA (s − s)q(3)holds. In this case, ρ∗ can implement e = 1 because the payoﬀs to the principal
and the agent are then π = π and u = s − π, respectively, and hence individually
rational. In addition, (IC) and (DEP-S) hold with equality, and (DEA-S) becomes
identical to (3).
The optimal stationary contract is characterized by the solution to the following program:max −W{W,bh ,bℓ }subject to (IC-DE-S) and (IR-B)

We can solve the program by analyzing three cases separately and illustrate themin Figure 2. The IC-DE line is obtained from (IC-DE-S) ascdP (π − π) + dA (u − u) = .
q(4)First suppose that dP ≥ d∗ . In this case, condition (IC-DE-S) can be satisfied
without giving any rent to the agent. The IC-DE line (4) in this case can beillustrated by the line ICDEa in the figure which must cross the line u = u at apoint below (u, s − u). According to Lemma 2, no matter δP < δA or δP ≥ δA , thepure-bonus contract β ∗ is optimal. This can be seen in the figure as follows: thepoint (u, s − u) that corresponds to β ∗ lies above line ICDEa and hence satisfies
condition (IC-DE-S) and gives the principal the highest payoﬀ. There can be otheroptimal contracts, but they are all pure-bonus contracts because all of them givethe agent no rent and hence must entail bℓ = 0.
Second, suppose that dP < d∗ and dA ≥ d∗ . Because δP < δA must hold inthis case, giving more rent makes it easier to satisfy (IC-DE-S) as we discussed
14

伊藤秀史

π
(u, s − u)

s
ICDEb

ICDEc
ICDEa

(u∗ , s − u∗ )(s − π, π)u+π= s

π
0

s

u

u

Figure 2: Optimal Stationary Contracts
above. Graphically, this means that the IC-DE line, as illustrated by line ICDEb in
Figure 2, must be steeper than the Pareto frontier u + π = s. Hence, even though(u, s − u) lies below ICDEb and hence (IC-DE-S) cannot be satisfied without
giving any rent to the agent, increasing W moves the point closer to ICDEb. The
optimal contract, called γ ∗ , can be found by the intersection of ICDEb and thePareto frontier because at this point the principal gives the agent just enoughrent (as u∗ indicated in the figure) to satisfy (IC-DE-S). The optimal contract
generally involves a combination of bonus and penalty.
Finally, if dP < d∗ and dA < d∗ , then by Lemma 2, (IC-DE-S) can never besatisfied so e = 1 can not be implemented. Graphically, this means that the whole
Pareto frontier is below the IC-DE line, as illustrated by line ICDEc in Figure 2.
We summarize the discussion above in the following proposition.
Proposition 1 The optimal stationary contract entails the following.
(a) If dP ≥ d∗ , the agent receives no rent. The pure-bonus contract β ∗ is an
optimal contract. While there may be multiple optimal contracts, they are
all pure-bonus contracts.
(b) If dP < d∗ and dA ≥ d∗ , the agent enjoys a rent. The unique optimal contract115

γ ∗ = {W ∗ , b∗h , b∗ℓ } entails that1
W =u+c+dA − dP∗[]c− dP (s − s) ;qb∗h = dP (R − W ∗ − π) ≥ 0;b∗ℓ = −dA (W ∗ − c − u) < 0.
In particular, if c/q = dA (s − s), γ ∗ = ρ∗ and hence only the penalty is used;
if c/q < dA (s − s), the optimal contract involves both bonus and penalty.
(c) If dP < d∗ and dA < d∗ , e = 1 cannot be implemented and the agent receives
no rent.
Note that if dP = dA , it must be in either (a) or (c) of Proposition 1 and hence
the agent enjoys no rent. Since we do not impose the ex post limited liability
constraint on the agent, the rent found in part (b) of Proposition 1 is diﬀerent
from the traditional limited-liability rent, and entirely due to diﬀerential discountfactors (given the restriction to stationary contracts). From Proposition 1 (b), we
can write down the formula for the agent’s rent:
1u −u=dA − dP[∗]c− dP (s − s) ,qwhich yields the following comparative statics results.
Corollary 1 Suppose that dP < d∗ and dA ≥ d∗ . The agent’s rent (agency cost)
under the optimal stationary contract is increasing in (π, u, c), and is decreasingin (R, dA , dP ), and the relationships of the principal’s payoﬀ with these parameters
are the opposite.
The rent the agent enjoys when dP < d∗ and dA ≥ d∗ ensures the agent will
not renege the penalty needed to satisfy (IC-DE-S). Whether or not changing
a parameter can increase or decrease the agent’s rent discussed in Corollary 1
depends on whether it makes it harder or easier to satisfy (IC-DE-S). When π oru increases, it makes it harder to satisfy (IC-DE-S) so the agent’s rent increases.
An increase in the total surplus s = R − c makes it easier to satisfy (IC-DE-S)
so the agent’s rent decreases. An increase in the discount factor of either the
principal or the agent has the same eﬀect.
In contrast to most existing agency models, Corollary 1 shows that principal’s
revenue and the reservation values of the principal and the agent can play important roles. Specifically, agency cost is decreasing in the revenue, exhibiting

16

increasing return. In addition, Corollary 1 shows an increase in the agent’s reservation value makes the agent less likely to quit, as opposed to the case in many
existing models, and an increase in the principal’s reservation value has the sameeﬀect. Since the limited-liability rent is independent of the principal’s revenue andthe reservation payoﬀs, the comparative statics result also shows that the rent westudy here is very diﬀerent from limited-liability rent.
In Proposition 1, both the cases with dP < dA and dP ≥ dA are presented
and the changing parameter is the discount factor. In the analysis of dynamic
contracts, it is easier to characterize the diﬀerent cases using the expected revenue
R under e = 1, and the presentation is less messy by considering the cases withdP < dA and dP ≥ dA separately (as in the main body of Section 5). For a better
connection between the results here and that under dynamic contracts, we restate
Proposition 1 in terms of R.
To this purpose define RA as the solution to dA (R − c − s) = c/q, that is,RA ≡ s + c +

c
.
qdA

(5)

RA is the expected revenue under e = 1 such that (s − π, π) is on the IC-DE line.
Similarly, define RP as the expected revenue such that (u, s − u) is on the IC-DE
line:c
RP ≡ s + c +
.
(6)
qdP
Hence, R < RA if and only if dA < d∗ and R < RP if and only if dP < d∗ . Given
the assumption that dP < dA , the agent’s rent exists under the optimal stationary
contract if RA ≤ R < RP according to Proposition 1 (b). As RP will be used as
an important threshold that can be compared with the case of dynamic contract,
denote R∗ = RP .
We can now restate Proposition 1, focusing on the case with dP < dA , as thefollowing corollary.5
Corollary 2 Suppose the principal is less patient than the agent (dP < dA ). The
optimal stationary contract entails the following.
(a) If R ≥ R∗ , the agent receives no rent. The pure-bonus contract β ∗ is an
optimal contract.
5

When the principal is more patient than the agent (dP > dA ), cases (a) and (c) of Corollary
2 still hold with the definition of R∗ = RP . However, case (b) disappears so that the morepatient principal either extracts all the rent from the agent by the pure-bonus contract or finds
no stationary contract implementing e = 1. In Subsection 5.4, we will show that this resultcontinues to hold even if the more patient principal can design non-stationary contracts. In
particular, we will show that threshold R∗ = RP does not change by allowing non-stationary
contracts.

17(b) If RA ≤ R < R∗ , the agent enjoys a rent. The unique optimal contract isγ ∗.
(c) If R < RA , e = 1 cannot be implemented and the agent receives no rent.

5

Optimal Dynamic Contracts

As discussed in Section 1, dynamics brings three beneficial eﬀects. The first, wellknown eﬀect is that the principal, if she is less patient and has to leave a rent to theagent under the optimal stationary contract, can be better oﬀ by modifying only
the initial contract so as to extract the rent via a suﬃciently low fixed payment.
The second, more fundamental change is that, given the discount factors of theprincipal and the agent being diﬀerent, they can increase the total surplus byengaging in intertemporal utility transfer (i.e., front-loading the payoﬀ to theimpatient one and back-loading that to the patient one), as discussed in Lehrer
and Pauzner (1999). Importantly, the rent extraction via the initial contract andintertemporal payoﬀ transfer cannot be separated and must be jointly analyzed.
In this section, we first focus on the case where the principal is less patientthan the agent (dP < dA ), and examine these two eﬀects of intertemporal utility
transfer by allowing continuation payoﬀs to vary with time but not history (“semidynamic contracts”) in Subsection 5.1.
The third eﬀect of dynamics is that allowing continuation payoﬀs to be contingent on the outputs of the previous periods helps providing incentive to the agent.
This general case where the continuation payoﬀ can vary with history (“dynamic
contracts”) is analyzed in Subsection 5.2 under the assumption that the principal
induces the agent to work in every period. The possibility that the principal asks
the agent to work in one period and shirk in another period, or shut down production in certain periods is considered in Subsection 5.3. Finally in Subsection
5.4, we consider the case in which the principal is more patient than the agent.

5.1

Semi-Dynamic Contracts

In this subsection, we study the eﬀect of intertemporal utility transfer by restricting attention to “semi-dynamic contracts” in which continuation payoﬀs can vary
only with time but not the revenues of the previous periods, and hence u = uℓ = uhmust hold for each period. The objective of the principal ismax (1 − δp )(R − W ) + δp f (u),{W,bh ,bℓ ,u}18

where f (u) is the principal’s highest average continuation payoﬀ in PPE when
the agent’s payoﬀ is u. Note that the objective function is independent of bh andbℓ . Hence by Lemma 1, it is without loss of generality to replace (IC), (DEP),
and (DEA) with (IC-DE). In the following, we first find the first-best Pareto
frontier, denoted by F (u), that gives the highest average payoﬀ to the principal
when the eﬀort is observable and enforceable and the agent’s average payoﬀ is u,
and then find its intersection with the IC-DE line. For u that satisfies (IC-DE),f (u) = F (u) holds.
When the impatient principal and the patient agent engage in intertemporal
payoﬀ trading, the total expected payment Wt must be increasing in t, implyingthat along the Pareto-optimal path, the stage payoﬀ Wt − c to the agent is increasing and that to the principal R − Wt is decreasing. Thus the principal’s stage
payoﬀ cannot drop below π in any period because if it does in some period t′ , (IR)will be violated for all t ≥ t′ . In contrast, the agent’s stage payoﬀ can drop below
u in some period because as long as (IR) is satisfied for t = 0, it holds for all t.
In order to make the Pareto frontier well-defined, we impose a lower bound on
Wt , the expected total compensation to the agent, for all t.
Assumption 1 Wt ≥ W for all t ∈ {0, 1, . . .}, where W ≤ u + c.
Assumption 1 ensures that the stage payoﬀ to the agent is no lower than somefinite level. While the standard (ex post) limited-liability constraint is a suﬃcientcondition for Assumption 1 to hold, we emphasize that Assumption 1 does notdirectly give him a positive rent. In fact, since W ≤ u + c, the agent receives
no rent in the stage game and hence in the dynamic game as well if the discountfactors are the same. Furthermore, we will show later in Section 6 that under
the same condition, the agent earns no rent if the principal and the agent, whose
discount factors may be diﬀerent, can commit themselves to long-term contracts.
In the following analysis, we will show that if they cannot make a commitment
and the principal is less patient than the agent, he may enjoy rents even thoughW ≤ u + c holds.
The first step of deriving the Pareto frontier F (u) is a direct application of
Lehrer and Pauzner (1999). Given Assumption 1 and dP < dA , the set of stage
payoﬀs that can be used along the Pareto-optimal path of the repeated game is theconvex hull of (u, π), (s − π, π), and (W − c, R − W ). The first-best Pareto frontier, which contains an infinite numbers of vertices, can be constructed as follows.
The agent’s most preferred vertex, apart from (s − π, π), is supported by playing(W − c, R − W ) in the first period and playing (s − π, π) forever afterwards; theagent’s second preferred vertex is supported by playing (W − c, R − W ) in thefirst two periods and afterwards playing (s − π, π) forever, and so on. Thus the
19

0

u

s

u
π
(v (u) , F (v (u)))(u, F(u))

W − c, R − WIC-DE line

(ud , F(ud ))s

π = F(u)(s − π, π)u+π= s
π
0

s

u

u

Figure 3: The Pareto Frontier
agent’s n’th preferred vertex is given by(ûn , π̂n ) = ((1 − δAn ) (W − c) + δAn1(s − π) , (1 − δPn ) (R − W ) + δPn π)
for n ∈ {0, 1, 2, . . .}.
The access to public randomization device ensures that all the line segments
connecting (ûn , π̂n ) and (ûn+1 , π̂n+1 ) for n ∈ {0, 1, . . .} are also on the frontier.
The slope of the line segment connecting (ûn , π̂n ) and (ûn+1 , π̂n+1 ), denoted by λn ,
is
( )n
δP
1 − δPλn = −.
(7)
δA
1 − δA
Note that λn is increasing in n, and limn→∞ λn = 0. The Pareto frontier F (u)that satisfies (IR) is therefore characterized by the following functional form: Foru ≤ u ≤ s − π,
π = F (u) = π̂n+1 + λn (u − ûn+1 ) if ûn+1 < u ≤ ûn .
This frontier is depicted in Figure 3 as a green kinked curve π = F (u). Startingfrom any initial average payoﬀ pair (u, F (u)), where u ≤ u < s − π, the averagecontinuation payoﬀ pairs gradually move towards the bottom-right until (s − π, π)
is reached.
In the second step we go beyond Lehrer and Pauzner (1999) and analyze how20the intertemporal payoﬀ transfer aﬀects the incentive and enforcement conditions
for relational contracts, by imposing the IC-DE constraints on F (u). Finding the
optimal semi-dynamic contract requires finding a sequence of payoﬀs for the agent
and a corresponding sequence of payoﬀs for the principal (on the Pareto frontier)for t = 0, 1, ..., that satisfy the IC-DE constraints for t = 1, 2, ... (the IC-DE
constraints are for continuation payoﬀ, not initial payoﬀ).
According to the way the first-best Pareto frontier is derived, the principal
and the agent obtain (W − c, R − W ) in every period before certain period T ≥ 1
and then switch to (s − π, π) forever afterward. It then follows that, given a levelof average payoﬀ to the agent u in period t = 0, the subsequent average payoﬀ,
denoted as v t (u) for t = 1, . . . , T − 1 must satisfy the following equations:u = (1 − δA ) (W − c) + δA v 1 (u)v τ (u) = (1 − δA ) (W − c) + δA v τ +1 (u)for all τ = 1, . . . , T − 1. Similarly,()
F (u) = (1 − δP ) (R − W ) + δP F v 1 (u)()
F (v τ (u)) = (1 − δP ) (R − W ) + δP F v τ +1 (u)for all τ = 1, . . . , T − 1.
The sequence of the agent’s payoﬀ described above suggests that it can becharacterized by a transition functionv : [u, s − π] → [u, s − π](8)that determines the transition from any current average payoﬀ u to the agent to
his average continuation payoﬀ in the next period, such that v (v (u)) = v 2 (u),v (v (v (u))) = v 3 (u), and so on. It is clear that v is an increasing function so thatv τ is increasing in τ .
For a point (u, F (u)) in a given period to be attained with a semi-dynamic contract, Lemma 1 implies that the transition function associated with the contract
must satisfyc≤ dP (F (v (u)) − π) + dA (v (u) − u)
(IC-DED)
q
and
( (
)
)
(
)
c
≤ dP F v τ +1 (u) − π + dA v τ +1 (u) − uq21

for τ = 1, . . . , ∞.

(9)

Because this requires evaluating dP (F (u) − π)+dA (u − u) for any point (u, F (u)),
the following result is useful.
Lemma 3 Suppose that the principal is less patient than the agent and Assumption 1 holds. Then (i) dP (F (u) − π) + dA (u − u) is increasing in u. (ii) If constraint (IC-DED) is satisfied in a given period, then constraints (9) hold for all
subsequent periods.
Proof. (i) The conclusion follows from the fact that the slope of π = F (·) is larger
than (−dA /dP ) by (7). (ii) The conclusion follows from (i) since v τ is increasingin τ .
We now characterize the optimal semi-dynamic contract that implements e = 1
every period, for the diﬀerent ranges of the expected revenue R. We first argue
that if R < RA , no contract, whether stationary or non-stationary, can implement
e = 1. Remember that if R < RA , e = 1 cannot be implemented by any stationary
contract (Lemma 2 (c)), where RA is defined by (5). This threshold level does
not change even though we allow contracts to be non-stationary: e = 1 cannot be
implemented by any contract if R < RA . To see this, note that in both Figures
2 and 3, point (s − π, π) is on the Pareto frontier. And it is on the IC-DE line
(4) as well if R = RA . Furthermore, the slope of the IC-DE line, − (dA /dP ), isalways steeper than that of the Pareto frontier, λn , defined by (7). Therefore, for
R < RA , the whole first-best Pareto frontier is below the IC-DE line, and hencee = 1 is not implementable even by non-stationary contracts.
Second, note that the optimal stationary contract leaves no rent to the agent
if R ≥ R∗ (Lemma 2 (a)), where R∗ = RP , defined by (6), is the expected
revenue under which the IC-DE line intersects with u + π = s at (u, s − u). Sinces − u ≤ F (u), the optimal semi-dynamic contract leaves no rent to the agent forR ≥ R∗ as well.
From now on we focus on the intermediate case R ∈ [RA , R∗ ). Referring to
Figure 3, denote the intersection of the Pareto frontier with the IC-DE line as(ud , F (ud )), so thatcdP (F (ud ) − π) + dA (ud − u) = .
q(10)While (ud , F (ud )) satisfies constraint (IC-DED), it is better for the principal to defer playing it to period-1 and play his favorite stage game payoﬀ pair (W − c, R − W )in period 0, because constraint (IC-DED) only needs to be satisfied for continuation payoﬀs (in period 1). Doing so, the principal can lower the agent’s payoﬀ
in period 0 further to ue such that the agent’s continuation payoﬀ is ud , i.e.
v(ue ) = ud or(11)
ue = (1 − δA ) (W − c) + δA ud ,22and give herself
F (ue ) = (1 − δP ) (R − W ) + δP F (ud )provided that ue > u.
When ue > u holds, the optimal semi-dynamic relational contract gives theprincipal F (ue ) in the initial period because (a) f (ue ) = F (ue ) > F (u) for allu > ue and (b) f (ue ) = F (ue ) > F (u) for all u < ue . To see (a), note that F (u) is
decreasing in u ∈ (u, s − π). To see (b), note that for u < ue , constraint (IC-DED)
is violated and e = 1 cannot be implemented. Furthermore, by r < u + π, f is a
upward-sloping straight line connecting (ue , f (ue )) and (u, π).
On the other hand, if ue ≤ u, the payoﬀs to the agent and the principal are u
and F (u), respectively, under the optimal semi-dynamic relational contract sincef (u) = F (u) is decreasing in u ≥ u. In this case, the agent receives no rent.
Whether ue > u or ue ≤ u depends on R in a clear way because the relation
between ue and ud in (11) and the relation between ud and R imply that ue isdecreasing in R. And ue < ud = u for R = R∗ by (11). Hence if ue > u is satisfied
for R = RA , then there exists a threshold R∗∗ ∈ (RA , R∗ ) such that ue > u if andonly if R < R∗∗ , so the agent can obtain a positive rent only when R ∈ [RA , R∗∗ ).
A condition for ue > u to hold for R = RA is derived as follows. When R = RA ,(s − π, π) intersects with the IC-DE line, and hence by the definition of ud givenby (10), ud = u + c/(qdA ). (11) then implies that ue > u for R = RA and thusR∗∗ exists if and only ifc
W >u+c− .
(12)q
The dynamics of the optimal semi-dynamic contract goes as follows. First
suppose R ≥ R∗∗ . Then in the initial period, (u, F (u)) can be attained by asemi-dynamic contract and the agent receives no rent. In this case, an optimal
semi-dynamic contract entails))( ( t+1b∗∗(u) − πht = dP F v
and

( t+1
)
b∗∗
=
−d
v
(u)
−
u
.
A
ℓt

∗∗
Both b∗∗ht and bℓt are decreasing in t < T , where T is the period when (s − π, π)
is reached, since F (v t (u)) is decreasing and v t (u) is increasing in t. The fixedsalary wt∗∗ is increasing in t for t < T by∗∗
wt∗∗ + P b∗∗
ht + (1 − P ) bℓt = W .

Next suppose RA ≤ R < R∗∗ . This interval is non-degenerate if (12) holds,
23and the agent receives a rent ue − u > 0 in the initial period. The principal’s
maximum average payoﬀ in period 0 is
F (ue ) = (1 − δP ) (R − W ) + δP F (ud ) < F (u) .
An optimal semi-dynamic contract entails( ( t+1))(ue ) − πb∗∗ht = dP F v
and

( t+1
)
(ue ) − u .
b∗∗ℓt = −dA v∗∗∗∗
Both b∗∗ht and bℓt are decreasing in t and the fixed salary wt is “seniority-based,”in the sense that it is increasing in t for t < T to compensate for the decreasing
bonuses and increasing penalties.
We summarize the preceding analysis as follows:

Proposition 2 Suppose that the principal is less patient than the agent (dP < dA )and Assumption 1 holds.
∗∗(a) There is an optimal semi-dynamic contract {wt∗∗ , b∗∗ht , bℓt } implementing e =∗∗∗∗1 every period such that b∗∗
ht and bℓt are decreasing in t, and wt is increasing
in t.

(b) If (12) holds, then there exists R∗∗ ∈ (RA , R∗ ) such that under the optimal
semi-dynamic contract the agent receives a rent in the initial period if andonly if R, the expected revenue under e = 1, satisfies R ∈ [RA , R∗∗ ).
Figure 4 illustrates Proposition 2 (b). The highest first-best Pareto frontier
π = Fa (u) corresponds to the case Ra = R∗∗ and hence the IC-DE line intersects
with the Pareto frontier at (va (u), Fa (va (u))) where va is the transition function
defined in (8) under R = Ra . In this case, ue = u holds. The principal attains
Fa (u) and the agent enjoys no rent in the initial period for all R ≥ R∗∗ .
The middle first-best Pareto frontier π = Fb (u) depicts the case R < Rb < R∗∗ ,and hence ud > vb (u), where vb is the transition function defined in (8) under
R = Rb . In this case, the agent enjoys a positive rent ue − u in the initial period.
Finally, the lowest first-best Pareto frontier in the figure depicts the case Rc < RA
so that the principal cannot implement e = 1.
Note that the rent the agent enjoy is not directly caused by the lower bound on
the payments to the agent imposed by Assumption 1 because the rent only exist
when the revenue is at an intermediate level and it disappears when the revenue
is high enough.
24𝜋𝑢, 𝐹( 𝑢𝑊 − 𝑐, 𝑅( − 𝑊𝑣( 𝑢 , 𝐹( 𝑣( 𝑢Figure 3.

𝑢, 𝐹) 𝑢𝑣) 𝑢 , 𝐹) 𝑣) 𝑢

𝑊 − 𝑐, 𝑅) − 𝑊(𝑢. , 𝜋. )(𝑢0 , 𝜋0 )𝑊 − 𝑐, 𝑅* − 𝑊

IC−DE
𝜋

𝑢

𝑢
Figure 4: Optimal Semi-dynamic Contracts

The intuition behind Proposition 2 (a) is as follows. Because the principal
is less patient than the agent, the join surplus can be increased if the principal“borrows” payoﬀs from the agent and then pays back in later periods. This kindof intertemporal payoﬀ trade implies that the principal’s continuation payoﬀ will
decrease and the agent’s will increase over time. However, the principal’s continuation payoﬀ imposes an upper bound on the bonus he is willing to pay. Hence
if the bonus keeps constant as in stationary contracts, the principal would renege on paying it in some point of time. To cope with this problem, the optimal
semi-dynamic contract must entail a decreasing bonus plan, accompanied with anincreasing penalty schedule, ensuring the agent’s incentives to exert eﬀort.
The optimal semi-dynamic contract clearly dominates the optimal stationary
contract, as can be see by comparing (ue , πe ) for the former and (u∗ , s − u∗ ) forthe latter in Figure 4. The improvement of the principal’s payoﬀ comes from twosources. First, intertemporal payoﬀ trading expands the first-best Pareto frontier.
Second, the expansion of the first-best Pareto frontier makes it easier to satisfy(IC-DED), thereby increasing the principal’s power to extract rents from the agent
in the initial period. Note, however, that because of non-stationarity, the agent
continues to earn rents beyond the initial period.

25

5.2

Dynamic Contracts

In the previous subsection 5.1, we show that if incentives are provided by current
payments only, it is possible that the agent still enjoys a positive rent ue − u in theinitial period. This happens when RA < R∗∗ (equivalently W > u + c − (c/q)) andRA ≤ R < R∗∗ . In this subsection, we focus on this case, that is, ud ∈ (v(u), s − π]and ue ∈ (u, ud ), and examine whether the principal can extract more rent if fully
dynamic contracts are allowed.
Under either stationary or semi-dynamic contracts, the equilibrium eﬀort is notallowed to vary across period. When fully dynamic contracts are allowed, however,
it is in general possible that in equilibrium the principal asks the agent to workin one period and shirk in another period, or shut down production in certain
periods. We say that the average payoﬀ pair (u, f (u)) on the Pareto frontier is
supported with eﬀort, shirking, or termination if the agent puts in eﬀort, shirks,
or quit, respectively, in the initial period under the corresponding PPE.
For u ≥ u, the payoﬀ pair (u, f (u)) is supported with eﬀort if it satisfies thefollowing Bellman equation:f (u) = fe (u) ≡max(W,bℓ ,bh ,uh )(1 − δP ) (R − W ) + δP [P f (uh (u)) + (1 − P ) f (uℓ (u))] ,(PPe)subject to (IC), (DEP), (DEA), W ≥ W , uh (u) ≥ u, uℓ (u) ≥ u, andu = (1 − δA ) (W − c) + δA (P uh (u) + (1 − P ) uℓ (u)) ,(UUe)where uy (u) is the agent’s average continuation payoﬀ following the realizationof y ∈ {h, ℓ}. Because bh and bℓ do not aﬀect the objective function nor theconstraints other than (IC), (DEP), and (DEA), Lemma 1 applies once again, and
it is without loss of generality to replace (IC), (DEP), and (DEA) with (IC-DE),i.e.,c≤ dP (f (uh (u)) − π) + dA (uh (u) − u) ,(IC-DE)q
Even though fully dynamic contracts are allowed, f (u), the highest average
payoﬀ to the principal, coincides with the first-best Pareto frontier F (u) for u ≥ue , with W = W , uh (u) = uℓ (u), and fe (u) = f (u) = F (u). In other words,(u, f (u)) is supported with eﬀort and f (u) = fe (u) for u ≥ ue . Furthermore,uh (ue ) = uℓ (ue ) = ud and f (ue ) > π hold by ue < ud ≤ s − π.
For u < ue , however, f (u) must be lower than the first-best Pareto frontier, and(u, f (u)) in this range may be supported not only with eﬀort but also with shirking
or termination. In this subsection, we focus on the analysis of problem (PPe),and relegate the analysis of payoﬀ pairs to be supported with eﬀort, shirking, or26

termination to Subsection 5.3.
The first lemma derives three properties that the solution to the maximization
problem in (PPe) must satisfy.6
Lemma 4 The solution to the maximization problem in (PPe) for u < ue satisfies
the following three properties: (a) W = W ; (b) uℓ (u) < uh (u); and (c) (IC-DE)
binds.
Lemma 4 (c) implies that (IC), (DEP), and (DEA) must bind at the solution.
We can then use these binding constraints to determine bℓ , bh , uℓ (u), uh (u), andw = W − P bh − (1 − P )bℓ . The binding (DEP) and (DEA) yieldbh − bℓ = dP (f (uh (u)) − π) + dA (uℓ (u) − u) .
By plugging this equation into the binding (IC) we obtaincdP (f (uh (u)) − π) + dA (uh (u) − u) = ,q
and hence (uh (u), f (uh (u))) = (ud , F (ud )) by the definition of ud given by (10),Note that ud is independent of u and hence so is uh = uh (u). This implies
that no matter how many times low output (failure) has happened, once the firsthigh output realizes, the agent’s average payoﬀ increases to ud , and then follows
the path of ud , v 1 (ud ), v 2 (ud ), and so on, entailed by the optimal semi-dynamic
contract. In this sense, the contract only exhibit fully dynamic features until thefirst high output appears. After that, it is semi-dynamic. Since R−W > f (ud ) ≥ π
and f (uℓ (u)) ≥ π by definition, fe (u) > π holds for u < ue by (PPe).
On the other hand, uℓ (u) depends on u via (UUe), where the current payoﬀ isreplaced with W − c. Hence, there exists a unique ug such that that uℓ (ug ) = ug ,that is,ug =(1 − δA ) (W − c) + δA P ud.
1 − δA (1 − P )(13)

If ug > u, and (ug , f (ug )) is supported with eﬀort, the principal’s payoﬀ corresponding to ug is f (ug ) given by
f (ug ) = (1 − δP ) (R − W ) + δP P f (ud ) + δP (1 − P ) f (ug ) ,orf (ug ) =(1 − δP ) (R − W ) + δP P f (ud ).
1 − δP (1 − P )

6

(14)

Note that this lemma does not depend on whether or not the initial payoﬀ pair (u, f (u)) is
supported with eﬀort. That is, whether or not f (u) = fe (u) holds.

27

Note that f (ug ) > π holds since R − W > π and f (ud ) ≥ π. The solutions to w,
bℓ , and bh are obtained as follows:bℓ = −dA (uℓ (u) − u) ;bh = dP (f (ud ) − π) ;w = W − P bh − (1 − P ) bℓ .
The next lemma determines the order of the magnitude between ug and ue , aswell as that between f (ug ) and f (ue ).7
Lemma 5 If ug > u and (ug , f (ug )), defined by (13) and (14), is supported witheﬀort, then ug < ue and f (ug ) > f (ue ).
Suppose (u, f (u)) is supported with eﬀort. The dynamic of uℓ (u) can now be
described as follows. At u = ug , low output keeps the agent’s average continuation
payoﬀ unchanged. If u ∈ (ug , ue ), the agent’s payoﬀ increases to uℓ (u) > u after
low output, and if u < ug , low output decreases his payoﬀ to uℓ (u) < u. These
patterns arise because given that high output will raise the agent’s average payoﬀ
from u to ud , his payoﬀ after low output must be adjusted to satisfy equation(UUe).
For u ∈ (ug , ue ), every time output is low, the agent’s continuation payoﬀ, as
well as the penalty and base wage, increases. Although the resulting payoﬀ pairis below the first-best Pareto frontier, the agent’s average payoﬀ is still increasingover time, which feature shares with that of the Pareto-optimal intertemporal
payoﬀ transfer.
However, for u < ug , every time output is low, the agent’s continuation payoﬀ
decreases and hence both the penalty and base wage do, too. The agent’s average
payoﬀ is now too low to sustain with an increasing payoﬀ stream. It now has
to decrease in contrast to the Pareto-optimal intertemporal payoﬀ transfer. Notethat since uℓ (u) ≥ u must hold, (u, f (u)) cannot be supported with eﬀort if u issuch that (UUe) holds with uℓ (u) < u. This is the case for u = u < ug . Since uℓ (u)is decreasing, (u, f (u)) cannot be supported with eﬀort for u suﬃciently small if
u < ug .
Define the payoﬀ pair in the initial period under the optimal relational contractas (ũ, f (ũ)) such that ũ ≡ min {u′ |u′ ∈ arg maxu≥u f (u)}. We call (ũ, f (ũ)) the
optimal payoﬀ pair. Because the optimal eﬀort pair cannot be in the middle ofa straight line but a vertex of the Pareto frontier, it must be supported by apure-strategy equilibrium (eﬀort, shirking, or termination). In this subsection,
we analyze the optimal relational contract under the assumption that (ũ, f (ũ))
7Remember ue < ud and f (ue ) > f (ud ) as we have shown after the definition of ue (11).

28is supported with eﬀort. Later in Subsection 5.3 we allow payoﬀ pairs to besupported with shirking and termination as well, and obtain a suﬃcient condition
for the payoﬀ pair in the initial period under the optimal relational contract to besupported with eﬀort only.
Given that the optimal payoﬀ pair is supported with eﬀort, the next lemmashows that the optimal relational contract gives the principal f (ug ) if ug ≥ u, orf (u), otherwise.
Lemma 6 Suppose (ũ, f (ũ)) is supported with eﬀort. Then ũ = max{u, ug }.
Proof. We prove the lemma by ruling out the other three cases: ũ = u < ug ,ũ ∈ (u, ug ), and ũ > max{u, ug }.
First, ũ = u < ug is not possible since, as we argue above, (ũ, f (ũ)) thencannot be supported with eﬀort.
Second, if ũ ∈ (u, ug ), then f+′ (u) > 0 for u in the left neighborhood of ũ, andhence by the concavity of f , f+′ (u) > 0 for all u < ũ. In addition, ũ < ug implies
that uℓ (ũ) < ũ, and hence f+′ (uℓ (ũ)) > 0. Then by the definition of ũ, (PPe), and(UUe),
δP ′′0 > f+′ (ũ) ≥ fe+(ũ) =f (uℓ (ũ)) > 0.
δA +A contradiction.
Finally, suppose ũ > max{u, ug }. Then f−′ (u) < 0 for u in the right neighborhood of ũ, and hence by the concavity of f , f−′ (u) < 0 for all u > ũ. Sinceuℓ (ũ) > ũ, f−′ (uℓ (ũ)) < 0 must hold. Then by the definition of ũ, (PPe), and(UUe),
δP ′′0 < f−′ (ũ) ≤ fe−(ũ) =f (uℓ (ũ)) < 0.
δA −A contradiction.
When the optimal relational contract can be supported with eﬀort, the questionof our focus is whether the agent will receive a positive rent or no rent, which canbe the case when ũ = ug > u or ũ = u ≥ ug , respectively. The relation between
ug and ud in (13) and the relation between ud and R imply that ug is decreasingin R. And ug < ue = u for R = R∗∗ by (11) and (13). Hence if ug > u holds for
R = RA , there exists R∗∗∗ ∈ (RA , R∗∗ ) such that ug > u if and only if R < R∗∗∗ .
To obtain a condition for ug > u to be satisfied at R = RA , note that when
R = RA , ud = u + c/(qdA ), and hence (13) implies that ug > u holds and R∗∗∗
exists if and only ifP
W > u + c − c.
(15)qNote that (15) implies (12) so that R∗∗∗ > RA implies R∗∗ > RA .

29

If R ≥ R∗∗∗ , and hence ug ≤ u, ũ = u holds, and the principal can extract
all the rent from the agent in the initial period by designing a dynamic contract
with the following, “promotion-like” feature: In period 0, the average payoﬀ pairis (u, f (u)). The first time high output realizes, the average payoﬀ pair becomes(ud , F (ud )), where ud is defined by (10) as the intersection of the first-best Pareto
frontier and the IC-DE line. After that, the pair follows the path of the optimal
semi-dynamic contract in Proposition 2 and the agent continues to earn rents
beyond the initial contract. When output is low, the average payoﬀ to the agent
either stays at ug if ug = u, or increases with every failure if ug < u. In the latter
case, the increment of the average payoﬀ is increasing in the number of previous
failures, until the average payoﬀ goes beyond ud . Once the average payoﬀ goesbeyond ud , the payoﬀ pair again follows the path of the optimal semi-dynamic
contract in Proposition 2.
If R < R∗∗∗ , and hence ug > u, the principal cannot extract all the rent in
the initial period even though she designs the dynamic contract optimally (ũ =ug > u): In period 0, the average payoﬀ pair is (ug , f (ug )). Until the first highoutput realizes, the average payoﬀ pair remains at (ug , f (ug )). The average payoﬀ
pair becomes (ud , F (ud )) the first time high output appears. After that, the pairfollows the path of the optimal semi-dynamic contract in Proposition 2. Note,however, that by exploiting the fully dynamic nature of the contract, the principal
can reduce the rent in the initial period from ue − u, the rent under the optimal
semi-dynamic contract, to ug − u.
We can summarize the preceding analysis as follows:
Proposition 3 Suppose that the principal is less patient than the agent and the
optimal payoﬀ pair is supported with eﬀort, and Assumption 1 holds.
(a) There is an optimal, “promotion-like” dynamic contract implementing e = 1
every period such that (i) the agent’s continuation payoﬀ will move to thepath of the optimal semi-dynamic contract after high output realizes for the
first time; (ii) his continuation payoﬀ after every low output is constant orincreasing; and (iii) his continuation payoﬀ does not decrease as time goeson, regardless of the output, and there will never be termination.
(b) If (15) holds, then there exists R∗∗∗ ∈ (RA , R∗∗ ) such that under the optimal
dynamic contract the agent receives a rent in the initial period if and only
if R, the expected revenue under e = 1, satisfies R ∈ [RA , R∗∗∗ ). This rent,ug − u, is smaller than the rent under the optimal semi-dynamic contract,ue − u.
Our result in Proposition 3 (a) that there will never be termination is in sharp
contrast with the result of Fong and Li (2017), where there is a positive probability30𝜋𝑢.( , 𝜋.(𝑊 − 𝑐, 𝑅( − 𝑊

Figure 4.

𝑢+( , 𝜋+(𝑢0( , 𝜋0((𝑢0) , 𝜋0) )𝑊 − 𝑐, 𝑅) − 𝑊(𝑢.) , 𝜋.) )(𝑢+) , 𝜋+) )𝑊 − 𝑐, 𝑅* − 𝑊

IC−DE
𝜋

𝑢

𝑢
Figure 5: Optimal Dynamic Contracts

of ineﬃcient termination in equilibrium. Under our framework, the ineﬃciency
takes the form of deviation from the eﬃcient intertemporal utility transfer. This
is why the payoﬀ pair (ug , f (ug )) does not lie on the first-best Pareto frontier, as
illustrated in Figure 5.
Figure 5 illustrates Proposition 3 (b). The highest first-best Pareto frontier inthe figure corresponds to the case Ra = R∗∗∗ , and ug is denoted by uas , which isequal to u. The optimal payoﬀ for the principal in period 0 is πsa = f (uas ) = f (u),and the agent obtains no rent. The expected revenue Rb satisfies RA < Rb < R∗∗∗for the middle first-best Pareto frontier. In this case, ug is denoted by ubs , which isgreater than u. Hence, the agent enjoy a positive rent ubs − u, and the principal’s
payoﬀ is πsb = f (ubs ).
Comparative Statics
We present how the agent’s rent changes with some of the parameters in the model
under the optimal dynamic contract.
Corollary 3 Suppose that the principal is less patient than the agent and the
optimal payoﬀ pair is supported with eﬀort, and Assumption 1 holds. Furthermore,
suppose RA < R < R∗∗ is satisfied so that the optimal semi-dynamic contract
leaves a rent ue − u to the agent. The agent’s rent under the optimal dynamic
contract is increasing in (π, u, W ), and is decreasing in R. The opposite happens
for the principal’s payoﬀ.
31

The eﬀects of the other parameters are diﬃcult to pin down. For example,
increasing the cost of eﬀort c raises ud by shifting F inward as well as by increasing
the right-hand side of (10). However, ug is decreasing in c by (13), and hence wecannot determine the net eﬀect of c on the agent’s rent. Similarly, the eﬀects ofthe discount factors δA and δB are diﬃcult to pin down.
Finally, we compare the agent’s rent under optimal stationary, semi-dynamic,
and dynamic contracts. The following result is immediate from the preceding
analysis.
Corollary 4 Suppose that the principal is less patient than the agent and the optimal payoﬀ pair is supported with eﬀort, and Assumption 1 holds. The agent’s rent
decreases as contracting technology advances from stationary contracts to semidynamic contracts if R < R∗ (so that he enjoys a rent under the former technology), and from semi-dynamic contracts to dynamic contracts if R < R∗∗ .

5.3

Possibility of Supporting with Termination and Shirking

In this subsection, we present the analysis on the possibility of supporting the
optimal payoﬀ pair with shirking or termination. If (u, f (u)) is supported withshirking, it must satisfy the following Bellman equation:f (u) = fs (u) ≡ max (1 − δP ) (r − W ) + δP f (us (u)) ,
W

(PPs)

subject to W ≥ W , us (u) ≥ u, andu = (1 − δA ) W + δA us (u),(UUs)

where r = ph + (1 − p)ℓ is the expected revenue under shirking, which we haveassumed smaller than s = π + u, and us (u) is the average continuation payoﬀ tothe agent under shirking.
If (u, f (u)) is supported with termination, it must satisfy the following equations:(PPx)f (u) = fx (u) ≡ (1 − δP ) π + δP f (ux (u)) ,subject to ux (u) ≥ u and
u = (1 − δA ) u + δA ux (u),(UUx)

where ux (u) is the expected continuation payoﬀ to the agent under termination.

32

(u, f (u)) can also be supported with randomization, meaning that it is a convex combination of vertexes that are supported with either eﬀort, shirking, or
termination. The principal’s highest payoﬀ f (u) is the concave upper envelope offe (u), fs (u), and fx (u) for u ≥ u.
As in our analysis in Subsection 5.2, we focus on the optimal relational contract
and the optimal payoﬀ pair in the initial period, (ũ, f (ũ)). The first lemma showsthat the optimal payoﬀ pair cannot be supported with termination.
Lemma 7 Given R ≥ RA , (ũ, f (ũ)) cannot be supported with termination.
The next lemma shows that if the optimal payoﬀ pair is supported with shirking, the optimal payoﬀ to the principal must be lower than r − W .
Lemma 8 If (ũ, f (ũ)) is supported with shirking, the agent must receive no rent(ũ = u), and fs (ũ) < r − W must hold.
These characterizations has strong implications. In Fong and Li (2017), they
implies that f must be linear between u and us (u) if (u, f (u)) is supported withshirking, or between u and ux (u) if (u, f (u)) is supported with termination (seethe proof of their Proposition 1). In the current paper, δP ̸= δA , and in particularδP < δA in this section, and they rule out the possibility that the equilibrium
supported with termination, and the possibility that the equilibrium is supportedwith shirking under which the agent enjoys a positive rent.
We now obtain a suﬃcient condition for the optimal payoﬀ pair (ũ, f (ũ)) tobe supported with eﬀort. By Lemma 7, it cannot be supported with termination.
If it is supported with shirking, f (ũ) = fs (u) must be lower than r − W . Hencethe optimal payoﬀ pair must be supported with eﬀort if
r − W ≤ fe (ug )holds. The left-hand side of the inequality is highest when W = u + c − (P c)/q
(as required by Proposition 2 (b)). In that case,r−W =r−u−c+Ppc = r − u + c.
q
q

(16)

The next lemma shows that fe (ug ), the right-hand side of the inequality, isincreasing in R and decreasing in W .
Lemma 9dfe (ug ) > 0 anddRdfe (ug ) < 0 hold.
dW

33

Hence the right-hand side of the inequality is lowest when R = RA (in whichcase ug > u) and W = u,8 which leads tofe (ug ) = π +1 − δP
1 + dA qc.
1 − δP (1 − P ) dA q

(17)

Combining (17) with (16) yields a suﬃcient condition for r − W ≤ fe (ug ), which
we present as the following assumption that the expected revenue under shirking
r is suﬃciently small.
Assumption 2{
r ≤u+π+1 − δP
1 + dA q−p
1 − δP (1 − P ) dA

}

c
.
q

The result of this subsection is summarized as the following proposition.
Proposition 4 If Assumption 2 is satisfied, the optimal payoﬀ pair (ũ, f (ũ)) mustbe supported with eﬀort.
Equilibrium Supported with Shirking
We consider the parameter range in which equilibrium supported with shirking,
i.e.
(18)
fs (u) = (1 − δP )(r − W ) + δP f (us (u)) > fe (ug ),arises, by considering some extreme cases. First, we assume that both R and W
are at their lowest possible value and r is at its highest possible value, i.e. R = RA ,W = u + c − (P/q)c, and r = u + π. In this case, (14) yields
f (ug ) =(1 − δP ) (RA − W ) + δP P π.
1 − δP (1 − P )In addition, us (u) = u + ((1 − δA )/δA )(p/q)c. To find a lower bound for f (us (u)),
first note that ug = u < us (u) < ud . Second, the Pareto frontier is bounded
below by a straight line between (ug , f (ug )) = (u, f (u)) and (ue , f (ue )), definedas (u, L(u)), andL(us (u)) = pπ + f (ug (u))(u + c/(qdA )) − us (u)c/(qdA )= pπ + f (ug (u))(1 − p),

While W ≤ u + c in Assumption 1, we show in the proof of Lemma 8 in Appendix that if(u, f (u)) is supported with shirking, W ≤ u must hold.
834Third,(1 − δP )(r − W ) + δP L(u) > fe (ug ),holds if and only ifp>

1 − δP (1 − p) 1 + P dA,
1 − δP (1 − P ) qdA

(19)

(20)

It follows that (20) is a suﬃcient condition for (18). In (20), the first ratio on theright-hand side is smaller than 1 and the second ratio on the right-hand side islarger than 1. However, there can clearly be cases that (20) holds. The observationthat (20) cannot hold if Assumption 2 holds and r = u+π serves as a confirmationfor (20) to be a suﬃcient condition for the equilibrium to be supported withshirking. Given this and the fact that fs (u) is continuous and increasing in r, andfe (ug ) is increasing in R the following result hold.
Lemma 10 Suppose that (20) holds for R = RA , r = u + π, and W = u + c −(P/q)c.
(i) Given that R = RA and W = u + c − (P/q)c, there is a cutoﬀ r1 such thatthe equilibrium is supported with shirking if and only if r > r1 .
(ii) Given that r > r1 and W = u + c − (P/q)c, there exists a cutoﬀ R1 suchthat the equilibrium is supported with shirking if and only if R < R1 .
The second extreme point worth noting is when R and r remain the same as in
the previous case and W is increased to its highest possible value, i.e. u. In thiscase, it is clear that the equilibrium can not be supported with shirking because
us (u) = u, so that supporting the equilibrium means the agent will shirk in every
period. Note that in Fong and Li (2017), the possibility of shirking is ruled out
under the assumption that the surplus from eﬀort is high enough.

5.4

When The Principal Is More Patient

So far in this section we have focused on the case where the principal is less patientthan the agent (dP < dA ). In this subsection, we consider the other case in which
the principal is more patient (dP > dA ).
If dP > dA , the first-best Pareto frontier has to be modified from that underdP < dA . The procedure of finding it is similar to the preceding analysis exceptthat the directions of movement for the payments are reversed: Wt must be decreasing in t. This means that the agent’s stage payoﬀ cannot drop below u inany period since otherwise (IR) will be violated for some t. Hence, the principal’s
maximum stage payoﬀ is s−u, which implies that F (u) = s−u. When (IC-DE) istaken into consideration, the result is the same whether contracts are restricted to35

π(u, sa − u)IC-DE line

π = Fa (u)
π


0
u

u
W − c, Ra − W

π = Fb (u)W − c, Rb − W

Figure 6: When The Principal Is More Patient
stationary ones, or semi-dynamic or dynamic contracts are feasible: It all depends
on whether the payoﬀ pair most preferred by the principal (u, s − u) satisfied theIC-DE constraint. The following result is immediate, where RP is the expected
revenue under e = 1, defined by (6), such that the IC-DE line intersects with thefirst-best Pareto frontier at (u, s − u).9
Proposition 5 Suppose that the principal is more patient than the agent (dP >dA ).
(a) If R ≥ RP , then the stationary, pure-bonus contract β ∗ is optimal, and theagent receives no rent: The principal’s stage payoﬀ is s − u for all periods.
(b) If R < RP , e = 1 cannot be implemented and the agent receives no rent.
Figure 6 illustrates cases (a) and (b) of Proposition 5, where W is an upperbound on the expected total payment to the agent. The first-best Pareto frontiers
are represented by green kinked curves for two diﬀerent levels of R. Case (a)corresponds to the higher frontier π = Fa (u) under Ra = RP , and hence thestationary pure-bonus contract shown as (u, sa − u) is optimal. If R increases
from Ra , the Pareto frontier moves upward. Then the point corresponding to
the pure-bonus contract remains above the IC-DE line, and hence optimal. It
9We denote R∗ = RP in the previous sections.

36

is important to observe that the slope of the Pareto frontier is anywhere steeper
than that of the IC-DE line, and thus if there exists a point on the frontier above
the IC-DE line, the one corresponding to the pure bonus contract is also above it.
Case (b) of the proposition is illustrated by the Pareto frontier π = Fb (u)under Rb < RP . It is everywhere below the IC-DE line, and hence e = 1 cannot
be implemented.

6Commitment

In Section 3, we present a benchmark in which the principal and the agent are
equally patient and engage in relational contracting. Another important benchmark is when they cam commit themselves to long-term contracts, that is, when
payments specified by any contract signed between the principal and the agent
cannot be reneged, so (DEP) and (DEA) can be ignored.
As long as W is allowed to be set below u + c in every period, the agent enjoys
no rent in our model if the principal and the agent could make the long-term commitment. This result holds not only when the principal’s discount factor is equal
to that of the agent, based on the analysis of the benchmark in Section 3, butalso whether her discount factor is higher or lower than that of the agent. Given
that the set of stage payoﬀs that can be used along the Pareto-optimal path of
the repeated game is a right triangle with a hypotenuse that has a negative one
slope, the first-best Pareto frontier always has a negative slope (see, for example,(7) for the case of the impatient principal). Without the restriction of the enforcement constraints, the whole first-best Pareto frontier can be implemented with acurrent bonus and penalty satisfying (IC). The principal will clearly choose thetip (u, F (u)) and leave no rent to the agent. In sum, there will be no rent for theagent no matter how the the discount factors of the principal and the agent are
ranked.
As we explain in Section 1, this benchmark distinguishes the current paper
from the closely related paper on relational contracts by Fong and Li (2017).
In their model, the agent enjoys the standard limited-liability rent under the
optimal one-shot contract, and it remains so after the principal extracts someof the limited-liability rent back by using the optimal long-term contract or the
optimal relational contract.
Based on Section 3 and the current section, we can argue that the rent we
identify is not due to limited liability but due to both 1) limited revenue and 2)
diﬀerent discount factors between the principal and the agent. Our rent exists
even though the expected total payment to the agent can be decreased below
u + c, unless it can be low enough to satisfy W ≤ u + c − (P/q)c. Limited liability
37

rent will not disappear when the revenue increases or when the discount factors
are the same.

7

Conclusion

We solved for the optimal dynamic relational incentive contract when the principal
and the agent have diﬀerent discount factors, and identified a new agency costarising from a trade-oﬀ between rent extraction and incentive provision that exists
only when the parties cannot commit themselves to long-term contracts and theprincipal is less patient than the agent.
The analysis is restricted to the case the agent’s eﬀort choice is binary. The
optimality of “one-step” payment schemes under the assumption of the validityof the first-order approach (Levin, 2003) suggests that we be able to extend ouranalysis to the continuous-eﬀort case as long as the eﬀort implemented is fixed.
However, the optimal eﬀort itself may change over time. Such an extended analysis
is left for future research.
Another topic to be left for future research is an application to the relationshipbetween an entrepreneur and investors, as we mentioned in Section 1. Most existing literature in financial contracting with diﬀerent discount factors between themalso assumes limited liability. Our model and analysis may be able to contributeto the literature by distinguishing eﬀects of limited liability from those of diﬀerent
discount factors.

References
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Appendix
Proof of Lemma 4b(a) Suppose that W > W at the solution. Consider an alternative contract w
with bh and bℓ decreased by ϵ > 0, that is small enough that W ≥ W still holds,
and with uh (u) and uℓ (u) increased by ϵ/dA . This alternative contract satisfies(UUe) and, by Lemma 3, increases the right hand side of (IC-DE) and hence alsosatisfies it. The principal is better oﬀ with the alternative contract if the changeb given byin the principal’s payoﬀ from switching to w,(1 − δP ) ϵ + δPϵ(P f ′ (uh (u)) + (1 − P ) f ′ (uℓ (u))) ,dA

is positive. We show it is in fact positive, which is a contradiction.
To show this, note first that (7) implies F ′ (z) > −dA /dP for all z. Because
f (z) is concave and coincides with F (z) for the part of F (z) that has the steepest
slopes and f (z) ≤ F (z) for all z, we havef ′ (z) ≥ F ′ (z) > −

dA
,
dP

(A1)

which then implies−dA< P f ′ (uh (u)) + (1 − P ) f ′ (uℓ (u)) ,dPand henceϵ(P f ′ (uh (u)) + (1 − P ) f ′ (uℓ (u))) ,
dA
(
)
ϵ
dA
> (1 − δP ) ϵ + δP
−
= 0.
dA
dP

(1 − δP ) ϵ + δP(b) First note uh (ue ) = uℓ (ue ) = ud . Furthermore, (IC-DE) and (A1) imply
uh (u) ≥ ud . Then (UUe), along with W = W , implies that uℓ (u) < ud , that is,uℓ (u) < uh (u) must hold for u < ue .
(c) Note that after plugging W = W into the objective function, the decisionvariables W , bℓ , and bh disappear from the program (PPe), and hence can beignored. The program is then reduced to finding uh = uh (u) and uℓ = uℓ (u) so40

that P uh +(1−P )uℓ is equal to u (to satisfy (UUe)), uh satisfies (IC-DE) (which nolonger consists of uℓ ), and P f (uh ) + (1 − P )f (uℓ ) is maximized. Because uℓ < uhas shown in part (b) and f (z) is a concave function, reducing uh to the pointuh = ud , the lowest uh satisfying (IC-DE), and then increasing uℓ to satisfy (UUe)
increases the objective function. Hence, (IC-DE) must be binding.

Proof of Lemma 5(13) implies thatue − ug =δA (1 − δA ) (1 − P ) (ud − (W − c))> 0.
1 − δA (1 − P )and hence ug < ue .
Using f (ue ) = (1 − δP )(R − W ) + δP f (ud ) and (14) yields
f (ue ) − f (ug ) =δP (1 − δP ) (1 − P ) [f (ud ) − (R − W )]< 0,
1 − δP (1 − P )and hence f (ug ) > f (ue ) holds.

Proof of Corollary 3
First, note that R and W aﬀect the payoﬀ frontier F : An increase in R shifts
F outward, while W shifts F inward. Hence by (10), increasing R decreases ud ,which in turn increases f (ud ). The conclusion then follows from (13) and (14).
And by (10), increasing W raises ud , which in turn reduces f (ud ). Furthermore,
ug is increasing in W by (13) and f (ug ) is decreasing in W by (14). The conclusion
follows.
Next, increasing u does not aﬀect F , but via (10) raises ud , which in turn
reduces f (ud ). The conclusion then follows from (13) and (14).
Finally, an increase in π shifts F inward, and by (10), resulting in an increasein ud and a decrease in f (ud ). The conclusion then follows from (13) and (14).

Proof of Lemma 7
First suppose ũ = u. Then if (u, f (u)) is supported by termination, ux (u) = u,and hence the agent’s payoﬀ is equal to u every period, implying that termination
happens forever. Then fx (ũ) = π. This is impossible because the principal canobtain at least f (ue ) > π with eﬀort. We then show below that (ũ, f (ũ)) cannot
be supported with termination in the case of ũ > u, either, and this completes theproof.

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Suppose that ũ > u and (ũ, f (ũ)) can be supported with termination so f (ũ) =fx (ũ). We prove the lemma by showing′
f−′ (ũ) > fx−(ũ) > 0,(A2)which, along with fx (ũ) = f (ũ) implies that fx is higher than f in the left neighborhood of ũ, contradicting the definition of f as the Pareto frontier.
Equation (UUx) implies u′x (ũ) = 1/δA and hence′fx−(ũ) = δP f−′ (ux (ũ))u′x (ũ) =δP ′f (ux (ũ))
δA −

(A3)

Because fx (ũ) = f (ũ) > π, (PPx) implies
f (ũ) = fx (ũ) < f (ux (ũ)).

(A4)

In addition, (UUx) implies ũ < ux (ũ), and hencef−′ (ũ) ≥ f−′ (ux (ũ)),(A5)f−′ (ux (ũ)) > 0,

(A6)

by the concavity of f , and′(ũ) > 0, andby (A4) and the concavity of f . (A3) and (A6) then lead to fx−
hence, given δP < δA ,′f−′ (ux (ũ)) > fx−
(ũ).
(A7)

(A2) follows from (A5), (A6), and (A7).

Proof of Lemma 8
Before proving the lemma, we first prove the following two lemmas.
Lemma A1 At the solution to the maximization problem (PPs), W = W holds.
Proof. Suppose that W ≥ W is not binding at the optimum. Consider anb with W decreased by ϵ > 0, that is small enough thatalternative contract w
W ≥ W still holds, and us (u) increased by ϵ/dA . This alternative contract satisfies(UUs). The principal is better oﬀ with the alternative contract if the change inb
her payoﬀ from switching to w,(1 − δP ) ϵ + δP42

ϵ ′f (us (u))
dA

is positive. However, using (A1) in the proof of Lemma 4 yields
ϵ ′f (us (u))
dA
(
)
ϵ
dA
> (1 − δP ) ϵ + δP
−
dA
dP
= 0.
(1 − δP ) ϵ + δP

A contradiction.
Lemma A2 If (u, f (u)) is supported with shirking, then (a) W ≤ u ≤ us (u); (b)f−′ (u) ≤ 0 and f+′ (u) ≤ 0; and (c) f (z) ≤ r − W for all z ≥ u.
Proof. (a) Given (UUs), the definition of us (u), the relationship between W , u,
and us (u) can be either (i) W ≤ u ≤ us (u) or (ii) us (u) < u < W . Suppose that(ii) is the case and hence W > u and r − W < π, the latter of which is due to theassumption r < s. Given that f (us (u)) > π by definition, this implies thatf (u) = fs (u) = (1 − δP )(r − W ) + δP f (us (u)) < f (us (u)).
Given us (u) < u, this implies f+′ (us (u)) < 0, which, combined with the factthat f is concave, implies that f+′ (z) < 0 holds for all z > us (u). Hence usingu′y (z) = 1/δA yields′
fs+(z) = δP f+′ (us (z))u′y (z) =δP ′f (us (z)) > f+′ (us (z)) ≥ f+′ (z)δA +for all z ∈ [u, W ]. This implies that∫ W
f (W ) ≥ fs (W ) = fs (u) +′fy+(z)dz > f (u) +u∫ W

f+′ (z)dz = f (W ),uwhich is a contradiction.
(b) It suﬃces to show that f−′ (u) ≤ 0 because it implies f+′ (u) ≤ 0 by the
concavity of f . Suppose, on the contrary, that f−′ (u) > 0. Because f (u) = fs (u)′(u) > 0. This implies f−′ (us (u)) > 0
and f (z) ≥ fs (z) for all z, it follows that fy−
since
δP ′
′
fy−
(z) =
f (us (z)) for all z.
(A8)
δA −
It then follows from the convexity of f that f−′ (z) > 0 for all z ≤ us (u) and hence,by (A8) again,′fy−(z) > 0 for all z ≤ u.
(A9)

43

This implies that
′
fy−
(z) =

δP ′
f (us (z)) < f−′ (us (z)) ≤ f−′ (z) for all z ∈ [W , u],δA −where the first equality is given by (A8), the second inequality is implied by (A9)and δP /δA < 1, and the third inequality is implied by the concavity of f . It follows
that
∫ u
∫ u
′
fy− (z)dz > f (u) −f−′ (z)dz = f (W ),f (W ) ≥ fs (W ) = fs (u) −WWwhich is a contradiction.
(c) Because f is concave, (b) implies that f+′ (z) ≤ 0 for all z ≥ u. Hence,fs (u) = f (u) is no lower than any point on the second-best Pareto frontier tothe right of u, and in particular, fs (u) ≥ f (us (u)). By definition (PPs), fs (u) is aweighted average between r − W and f (us (u)). This implies that r − W ≥ fs (u) =f (u) ≥ f (us (u)) and hence f (z) ≤ r − W for all z ≥ u.
We now prove Lemma 8. By Lemma 7, (ũ, f (ũ)) cannot be supported withtermination. And if ũ > u, (ũ, f (ũ)) cannot be supported with shirking becausef (ũ) = fs (ũ) implies that f−′ (ũ) > 0, violating Lemma A2 (b). Hence, the agent
cannot enjoy a rent if it is supported with shirking.
Finally, if ũ = u and (u, f (u)) is supported with shirking,f (ũ) = fs (ũ) = (1 − δP ) (r − W ) + δP f (us (ũ))
andf (ũ) = fs (ũ) > f (us (u)) .
It follows that fs (ũ) < r − W .

Proof of Lemma 9
Proof.
dddfe (ug ) = (1 − δP ) + δP P
fe (ud ) + δP (1 − P ) fe (ug ) ;
dR
dR
dR
d
d
d
fe (ug ) = − (1 − δP ) + δP P
fe (ud ) + δP (1 − P )fe (ug ) .
dW
dW
dW

44

Hence,d1 − δPδP P
dfe (ug ) =+fe (ud ) > 0;dR
1 − δP (1 − P ) 1 − δP (1 − P ) dRd1 − δPδP P
dfe (ug ) = −+fe (ud ) < 0,dW
1 − δP (1 − P ) 1 − δP (1 − P ) dW
because F (u) increases (decreases) as R (W , respectively) increases.

45