Relational Contracts in a Persistent Environment
Suehyun Kwon∗
February 9, 2015
Abstract
This paper studies relational contracts with partially persistent states, where
the distribution of the state depends on the previous state. The optimal contracts have properties similar to those of stationary contracts in Levin (2003),
but stationary contracts are no longer optimal. After characterizing the optimal contracts, the paper considers two types of persistent states, and the joint
surplus in the second best increases with the state in both cases. A sufficient
condition for stationary contracts to be optimal is provided.
Keywords: Relational contracts, persistence, moral hazard.
JEL: C73, D82, D86, L14
1
Introduction
Real-world interactions don’t always take place in an i.i.d. environment. A shock to
the cost of raw material is likely to persist for some time, and if it becomes costly to
perform a task this year, a firm may not expect the cost of performing the task next
year to be distributed in the same way as it would after a good year. The production
technology this period can also depend on the past realization of the productivity.
Anticipating the persistence of the states, the employers may not expect the same
∗
Kwon: University College London, suehyun.kwon@ucl.ac.uk, Department of Economics, University College London, Gower Street, London, WC1E 6BT United Kingdom. +4402076795843 I’m
very grateful to Glenn Ellison. I thank Martin Cripps, Michael Powell, Juuso Toikka and Muhamet
Yildiz, participants at the AEA 2013, SAET 2013 and MIT Theory Lunch seminar for helpful
comments and Samsung Scholarship for financial support.
1
effectiveness of the compensation scheme every period, and the optimal compensation
scheme may in fact depend on the state.
I study a relational contract model similar to that of Levin (2003) when the
states are partially persistent and there is moral hazard. The principal and the agent
trade every period over an infinite horizon, and both parties are risk-neutral with a
common discount factor. At the beginning of each period, the payoff-relevant state
is realized and becomes observable to both the principal and the agent. Under a
relational contract, the principal offers a compensation scheme each period, and the
agent decides whether or not to accept it and how much effort to exert if he accepts
the offer. The principal doesn’t observe the agent’s effort, which leads to moral
hazard, but he observes the outcome, which is a noisy signal of the agent’s effort, and
therefore can promise contingent payments on outcomes.
There is a large literature on relational contracts, including Levin (2003) and
Baker, Gibbons, and Murphy (2002). Earlier literature on relational contracts focused
on the symmetric information case. See for example, Shapiro and Stiglitz (1984), Bull
(1987), MacLeod and Malcomson (1989), Kreps (1990). More recent papers consider
environments with asymmetric information, and most of the literature assumes that
the environment is either stationary or i.i.d. over time. My paper is most closely
related to Levin (2003), which shows that for i.i.d. states, the principal can focus
on maximizing the joint surplus and the optimal contracts can be stationary. The
necessary and sufficient condition to implement an effort schedule with stationary
contracts is that it satisfies the IC constraint and the dynamic enforcement constraint.
Levin (2003) considers both adverse selection and moral hazard while I focus on moral
hazard. In Levin (2003), the optimal contract with moral hazard either implements
the first best or is a step function. Other related literature is discussed at the end of
this section.
Section 3 considers the results that hold for any type of persistence. As was the
case with i.i.d. states, the distribution of the joint surplus between the principal and
the agent can be separated from the problem of efficient contracting, and in characterizing the Pareto-optimal contracts, it is sufficient to focus on the joint surplus
from the relationship. When the states follow a first-order Markov chain, the realization of the state this period is a sufficient statistic for the distribution of the future
states, and the principal can provide all incentives by bonus payments at the end
of this period. In particular, the principal can offer a history-independent contract.
2
However, it may not be stationary as defined in Levin (2003) in that the fixed wage
has to depend on the state. Under a relational contract, there is a temptation to
renege which leads to the dynamic enforcement constraint as in the i.i.d. case. A
necessary and sufficient condition for an effort schedule to be implementable by a
history-independent contract is that it satisfies the IC constraint and the dynamic
enforcement constraint. I also show a necessary and sufficient condition for an effort
schedule to be implementable by a stationary contract. This condition is stronger
than the condition for history-independent contracts. The optimal contract either
implements the first-best level of effort, or it takes the form of a step function.
I also consider two mechanisms through which the persistence of the states affect
relational contracts. In an optimal history-independent contract, the joint surplus
in the first best can vary with the state which will affect the dynamic enforcement
constraint, and incentive provision for a given bonus cap can also vary with the state
through the IC constraint. I consider two mechanisms separately, holding the other
constant. I find that in both cases, if the joint surplus in the first best increases with
the state, or if the implementable level of effort for a given bonus cap increases with
the state, the difference in the joint surplus between the first best and the second
best decreases with the state. The joint surplus increases with the state in the second
best. In addition, I show that in the first case when the joint surplus in the first best
increases with the state, stationary contracts are optimal.
There are a growing number of papers on games, relational contracts or an implicit
contract equilibrium with persistent states. Thomas and Worrall (2010) consider a
two-sided incentive problem where the states and the efforts are observable and the
players have limited liability. McAdams (2011) considers joint-partnership games in
which the states are persistent and both the states and efforts are observable. The
players decide whether to stay in the relationship and how much effort to exert. The
main difference from my model is that there is no asymmetric information in their
models, and there is limited liability in Thomas and Worrall (2010). In Jovanovic
(1979a), the quality of a match is fully persistent and unknown to both parties at the
beginning. In his model, the new pair gets an i.i.d. draw from a known distribution.
In Kwon (2014), I consider moral hazard with persistent states and full commitment. States are unobservable in Kwon (2014). Garrett and Pavan (2012, 2014)
have moral hazard and persistent private information. There are also papers on
dynamic adverse selection with persistent private information. Athey and Bagwell
3
(2008) study collusion with private cost shocks, and Battaglini (2005) considers consumers with Markovian types. Escobar and Toikka (2013) show folk theorem results
with Markovian types and communication.
Lastly, this paper is also related to literature on partnership games with persistent
states. Rotemberg and Saloner (1996) and Haltiwanger and Harrington (1991) study
collusion in nonstationary markets. In Rotemberg and Saloner, the potential gain
from deviating is higher in a higher state, and the future surplus is not affected by
the state. In my first model in section 4, the gain from deviating is constant across
the states, and it is the future surplus that varies with the state; my model is closer
to Haltiwanger and Harrington.
The rest of the paper is organized as follows. Section 2 describes the model,
and the general results are presented in section 3. Section 4 discusses the types of
persistence of states and their implications on the joint surplus in the second best.
Section 5 concludes.
2
Model
The principal and the agent have the opportunity to trade over an infinite horizon,
t = 0, 1, 2, · · · . Both the principal and the agent are risk-neutral, and the common
discount factor is δ < 1.
The principal has limited commitment power and can only employ relational contracts. At the beginning of period t, the principal offers a compensation scheme to the
agent, which consists of a fixed salary wt and a contingent payment bt . Both the fixed
salary and the contingent payment can be functions of the history, which I will define
momentarily. The agent decides whether to accept the offer, and a payoff-relevant
parameter θt is realized. Both the principal and the agent observe the state. Note
that the principal offers the compensation scheme before the realization of the state;
he offers functions as the fixed salary and the bonus payment.
The state θt is drawn from the support Θ = [θ, θ̄]. The distribution of the state
θt depends only on the previous state θt−1 by P (θt |θt−1 ). It is time-invariant, and we
have P (θt |θt−1 ) = P (θ1 |θ0 ) for all t ≥ 1. In the initial period, the state θ0 is distributed
by P0 (θ0 ). The distributions P (θt |θt−1 ) and P0 (θ0 ) are common knowledge.
4
Timing in Each Period
Principal
makes
an offer.
Agent
accepts
/rejects.
θt becomes
observable.
Agent
chooses et .
Outcome yt
is realized.
Bonus
payment
is made.
Assumption 1. The distribution of state θt+1 when the previous state was θt is given
by P (θt+1 |θt ) and is identical for all t ≥ 0.
After the principal offers a compensation scheme, the agent decides whether or
not to accept, dt ∈ {0, 1}. If the agent accepts the compensation scheme, the agent
chooses how much effort to exert, et ∈ E = [0, ē]. The cost of effort, c(et , θt ), increases
with e, and c(e = 0, θ) = 0 for all θ, cee > 0. The agent’s effort generates outcome yt
with the distribution F (y|e, θ) and the support Y = [y, ȳ].1 The expected per-period
joint surplus can be written as a function of θ and e, S(e, θ) = E[y|e, θ] − c(e, θ).
Throughout the paper, when capitalized, S(e, θ) denotes the per-period joint surplus
in state θ if the agent chooses effort e.
I allow the distribution of the outcome and the cost function to depend on the
state. If neither of them depends on the state, we are back in an i.i.d. environment,
and in general, we can have one and/or the other to be state-dependent.
Each period, there are three pieces of payoff-relevant information: The costrelevant parameter θt , the agent’s effort et , and the outcome yt . The agent observes
all three parameters, but the principal observes only θt and yt . The performance
outcome is φt = {θt , yt }, and the set of all performance outcomes is denoted by Φ.
At the end of each period, the principal is obliged to pay the fixed salary wt , but
the contingent payment is only promised. The fixed salary wt : Ht × Θ → R is a
function of the state, and the contingent payment bt : Ht × Φ → R is a function of the
performance outcome, where Ht is the set of period-t histories. If bt > 0, the principal
decides whether to pay the agent, and if bt < 0, the agent has to decide whether to
make the payment. Denote the total payment to the agent by Wt ; Wt = wt + bt if the
contingent payment is made, and it is Wt = wt if not.
If the agent rejects the principal’s offer, the parties receive their outside option
for the period. The agent’s outside option is ū, and the principal’s outside option is
1
Most results of section 3 hold for any distribution of the outcome. The characterization of the
second-best contract requires an additional assumption.
5
π̄. The joint surplus from the outside option is denoted by s̄ = ū + π̄. I also assume
that the outside options ū, π̄ are independent of the state and constant over time.
Assumption 2 (Efficiency). The maximum joint surplus is strictly bigger than the
outside option for any state, but the outside option is weakly better than no effort.
For all θ ∈ Θ, maxe S(e, θ) > s̄ ≥ S(0, θ).
Given the distribution of the states, P (θt+1 |θt ), we can define the distribution of
θt+τ given θt , P (θt+τ |θt ). Let p(θt+1 |θt ) be the pdf of θt+1 , then we have
Z
p(θt+τ |θt ) =
Z
···
p(θt+τ |θt+τ −1 ) · · · p(θt+1 |θt )dθt+τ −1 dθt+1 ,
and P (θt+τ |θt ) can be constructed from p(θt+τ |θt ). The discounted payoffs to the
parties from date t given θt−1 are
∞
X
δ τ −t {dτ (Wτ − c(eτ , θτ )) + (1 − dτ )ū}|θt−1 ],
ut (θt−1 ) = (1 − δ)E[
τ =t
πt (θt−1 ) = (1 − δ)E[
∞
X
δ τ −t {dτ (yτ − Wτ ) + (1 − dτ )π̄}|θt−1 ],
τ =t
where the expectations are taken over θτ , (dτ , Wτ , eτ ), τ ≥ t, and F (·|e, θ). In period
0, the expectation is also taken over θ0 . At each period, the parties maximize their
expected payoffs. I define the expected joint surplus from period t and on as
st (θt−1 ) = ut (θt−1 ) + πt (θt−1 ).
Note that st (θt−1 ) is the expected discounted joint surplus, as it is discounted by 1−δ.
When capitalized, S(e, θ) is the expected joint surplus from the given period for e, θ.
Let ht = (w0 , d0 , φ0 , W0 , · · · , wt−1 , dt−1 , φt−1 , Wt−1 ) be the history up to period t
and Ht be the set of possible period-t histories. Given any period t and history ht ,
a relational contract specifies the compensation the principal offers, whether or not
the agent accepts it, and if the agent accepts the offer, it also specifies the effort
level. The compensations wt , bt are allowed to be functions of the history, and they
6
are functions of the following form:
wt : Ht × Θ → R,
bt : Ht × Φ → R.
A relational contract is self-enforcing if it forms a perfect public equilibrium of the
repeated game.
3
Optimal Contracts
This section characterizes the optimal relational contracts. The optimal contract
can be independent of history, but it may not be stationary as in Levin (2003), and
the fixed wage depends on the state. The self-enforcement leads to the dynamic
enforcement constraint as with i.i.d. states. The necessary and sufficient condition to
implement an effort schedule with stationary contracts is stronger than the dynamic
enforcement constraint. An optimal contract either implements the first-best level of
effort or takes the form of a step function.
A relational contract forms a perfect public equilibrium of the repeated game, and
there is multiplicity of equilibria. Instead of characterizing all relational contracts,
I focus on efficient contracting and focus on the Pareto frontier of the payoffs. The
first result is to note that the problem of efficient contracting can be separated from
the problem of distribution even if the states are persistent. The intuition is same as
in Levin (2003). The principal can always adjust the fixed salary to redistribute the
surplus.
Proposition 1. Suppose there exists a relational contract with expected joint surplus
s > s̄. Any expected payoff pair (u, π) with u ≥ ū, π ≥ π̄, u + π = s can be
implemented with a relational contract.
Any proof that is not presented in the text is in the appendix. As long as the
expected payoff is greater than the outside option, the parties are willing to initiate
the contract. The principal can adjust the distribution of the joint surplus by the fixed
salary of the initial period, and the resulting contract is still self-enforcing because the
incentives are not affected. Given Proposition 1, we can restrict attention to optimal
relational contracts that maximize the joint surplus from the contract.
7
The next result is that despite the persistence of the states, the maximum joint surplus can be achieved with history-independent contracts. I define history-independent
contracts as follows:
Definition 1. A contract is history-independent if Wt = w(θt ) + b(φt ), et = e(θt ) at
every t on the equilibrium path for some w : Θ → R, b : Φ → R and e : Θ → E.
Compared to stationary contracts in Levin (2003), the fixed wage in a historyindependent contract may depend on the realization of the state.
Definition 2. A contract is stationary if Wt = w + b(φt ), et = e(θt ) at every t on
the equilibrium path for some w ∈ R, b : Φ → R and e : Θ → E.
Note that the contract is independent of history on the equilibrium path. Without
loss of generality, we can assume that off the equilibrium path, the parties revert to
the static equilibrium of taking the outside option every period. With a historyindependent contract, the principal offers the identical compensation scheme every
period. The compensation scheme is independent of the history if it only depends on
the performance outcome of the given period. The fixed salary may depend on the
state, but given the same state, the fixed salary is constant over time.
Proposition 2. The maximum joint surplus can be attained with a history-independent
contract.
The proof of Proposition 2 goes as follows. When the states follow a Markov
process, the current state is a sufficient statistic for the distribution of future states.
Since it is observable to both the principal and the agent, there is no information
asymmetry regarding the distribution of the states or continuation values. Together
with risk-neutrality, the principal can provide a fixed continuation value for each state
and provide all incentives by the bonus payments in the given period. The incentive
provision becomes myopic, and the principal can further isolate each state, since it is
observable at the beginning of each period. Then the principal can provide optimal
incentives in each state in each period.
It is crucial that the states are Markov, observable to both the principal and
the agent and that they are both risk-neutral. It is also important that there is
no limited liability. If the state wasn’t observable to the principal, the principal
updates his belief about the state after observing the outcome. The principal and the
8
agent can have different beliefs on the state after the agent deviates, and the agent’s
deviation payoff is different from what the principal believes he’s providing the agent
with. The agent’s IC constraint has to take into account future periods, and we can
no longer isolate the incentive provision by each period. The principal also cannot
frontload all payments if the agent wasn’t risk-neutral or has limited liability. The
key to the proof is to recognize that the principal can provide a constant continuation
value for each state, independent of the history, and all incentives can be provided by
the bonus payments. If the states are exogenous and persistent but not Markov, then
the principal can offer a constant continuation value in each state, but the payments
have to depend on more than one period, and the optimal contract is not necessarily
history-independent.
With relational contracts, neither the principal or the agent commits to the contingent payment, and there exists a temptation to renege on the promised payment.
The contract is self-enforcing if the principal and the agent have no incentives to
renege. Since we are interested in the maximum joint surplus, there is no loss of generality in assuming that a deviation leads to the static equilibrium behavior.2 The
maximum joint surplus increases when the joint surplus after a deviation decreases,
and the payoffs of the principal and the agent are bounded from below by their outside options; therefore, the maximum joint surplus is attained when the parties revert
to the static equilibrium after a deviation.
The principal makes the promised payment if and only if
δ
(πt+1 (θt ) − π̄) ≥ sup b(θt , y), ∀θt ,
1−δ
y
and for the agent to make the promised payment, we need
δ
(ut+1 (θt ) − ū) ≥ − inf b(θt , y), ∀θt .
y
1−δ
From Proposition 1, the principal can redistribute the surplus by adjusting the
fixed wage, and the above inequalities can be combined in the dynamic enforcement
2
Specifically, the strategies are as follows. The possible deviations are (i) the principal offers an
unexpected compensation scheme, (ii) the agent rejects the offer when his strategy is to accept, (iii)
the agent accepts the offer when his strategy is to reject and (iv) the parties renege on the payment.
After a deviation, the principal makes no bonus payment, the agent’s expected payoff is his outside
option and the agent exerts zero effort in all future periods. The parties have correct beliefs, and
the principal takes his outside option after a deviation.
9
constraint:
(DE)
δ
(st+1 (θt ) − s̄) ≥ sup W (θt , y) − inf W (θt , y).
y
1−δ
y
The enforceable effort schedules are characterized by the agent’s IC constraint and
the dynamic enforcement constraint. The left-hand side of the dynamic enforcement
constraint is the maximum difference between any two bonus payments for a given
state, and I call it the bonus cap.
Theorem 1 and Proposition 3 generalize the results for stationary contracts in
Levin (2003) to history-independent contracts. The main intuition is that since the
states are observable and Markov, the principal can frontload all the incentives and
provide a constant continuation value.
Theorem 1. An effort schedule e(θ) with expected joint surplus s(θ) can be implemented with a history-independent contract if and only if there exists a payment
schedule W : Φ → R such that for all θ ∈ Θ,
(IC)
(DE)
e(θ) ∈ arg max Ey [W (φ)|e, θ] − c(e, θ),
e
δ
(s(θ) − s̄) ≥ sup W (θ, y) − inf W (θ, y).
y
1−δ
y
Note that the continuation payoffs from period t + 1 matter for the dynamic
enforcement constraint, but they don’t enter the agent’s IC constraint. Since the
states are persistent, the continuation payoffs ut+1 (θt ) and πt+1 (θt ) depend on the state
θt . But the principal also observes θt , and by Proposition 2, the principal can offer a
history-independent continuation contract, and the continuation value is independent
of the outcome yt . Therefore, even though the agent’s expected payoff from period t
is W (φt ) + δut+1 (θt ), ut+1 (θt ) doesn’t matter for the agent’s IC constraint.
However, the fixed wage in the optimal history-independent contract may vary
with the state. It only depends on the current state, but since the states are partially
persistent, providing a constant fixed wage for all states may be suboptimal.
Theorem 2. An effort schedule e(θ) with expected payoffs u(θ), π(θ) can be implemented with a stationary contract if and only if there exists a payment schedule
10
W : Φ → R such that for all θ ∈ Θ,
(IC) e(θ) ∈ arg max Ey [W (φ)|e, θ] − c(e, θ),
e
δ
δ
inf [
(u(θ) − ū) + inf W (φ)] + inf [
(π(θ) − π̄) − sup W (φ)] ≥ 0,
y
θ 1−δ
θ 1−δ
y
(SE)
(IR) u(θ) ≥ ū, π(θ) ≥ π̄.
I named the second condition in Theorem 2 the self-enforcement constraint. The
bonus payments are self-enforcing when the self-enforcement constraint is satisfied;
it is an analogue of the dynamic enforcement constraint for history-independent contracts. Theorem 2 shows that the necessary and sufficient condition for a stationary
contract is stronger than the IC and the dynamic enforcement constraint.
Proof of Theorem 2. (⇒) Suppose e(θ) can be implemented with a stationary contract w ∈ R, b : Φ → R. We need both parties to make the bonus payment:
δ
(π(θ) − π̄) ≥ sup b(φ) = sup(W (φ) − w),
1−δ
y
y
δ
(u(θ) − ū) ≥ − inf b(φ) = − inf (W (φ) − w).
y
y
1−δ
We can combine the inequalities to
δ
δ
(u(θ) − ū) + inf W (φ) ≥ w ≥ −
(π(θ) − π̄) + sup W (φ).
y
1−δ
1−δ
y
Since the inequality holds for all θ, we get the self-enforcement constraint,
inf [
θ
δ
δ
(u(θ) − ū) + inf W (φ)] + inf [
(π(θ) − π̄) − sup W (φ)] ≥ 0.
y
θ
1−δ
1−δ
y
The agent’s IC constraint has to be satisfied, and the continuation values for both
parties are weakly greater than the outside options.
(⇐) When the conditions are satisfied, the agent chooses e(θ) in each θ, and the
parties are willing to initiate the relationship. When the self-enforcement constraint
is satisfied, we can pick w such that
inf [
θ
δ
δ
(u(θ) − ū) + inf W (φ)] ≥ w ≥ − inf [
(π(θ) − π̄) − sup W (φ)],
y
θ 1−δ
1−δ
y
11
and the parties will make the bonus payment. By construction, the contract is selfenforcing in every period.
Comparing Theorems 1 and 2, we can see that stationary contracts may be suboptimal; the self-enforcement constraint in Theorem 2 is stronger than the dynamic
enforcement constraint. Specifically, we know that providing a constant fixed wage
is no longer optimal when the states are persistent. Also note that when a contract
is stationary, the continuation value for the agent may vary with the state. The
principal cannot frontload all the payments and provide the constant continuation
value.
We also know from the dynamic enforcement constraint that the per-period joint
surplus and the expected joint surplus decrease with the outside option s̄.
Corollary 1. The per-period joint surplus and the expected joint surplus weakly decrease with the outside option s̄.
Lastly, from Theorem 1, we obtain the following characterization of optimal contracts. Assumption 3 is maintained throughout the rest of the paper.
Assumption 3. The distribution of the outcome F (y|e, θ) satisfies the MirrleesRogerson constraints: F (y|e, θ) has the monotone likelihood ratio property, (fe /f
increases with y) and F (y|e, θ) is convex in e for any θ.
Proposition 3. Suppose Assumption 3 holds. An optimal contract either (i) implements eF B (θt ) or (ii) takes the form of a step function at each θt . When e(θt ) <
eF B (θt ), there exists y(θt ) such that W (θt , y) = W̄ (θt ) for y ≥ y(θt ) and W (θt , y) =
δ
(st+1 (θt ) − s̄), and the likelihood ratio fe /f
W (θt ) for y < y(θt ). W̄ (θt ) = W (θt ) + 1−δ
changes the sign at y(θt ).
When the distribution of the outcome satisfies the Mirrlees-Rogerson constraints,
the per-period joint surplus is concave in e. Together with risk neutrality of both
parties, the principal wants to use the strongest incentives possible. If an optimal
contract cannot induce the first-best effort eF B (θt ) in state θt , the DE constraint
binds, and the strongest incentives is to provide a step function that jumps when the
likelihood raio changes the sign.
History-independent contracts generalize the stationary contracts for i.i.d. states
in Levin (2003). However, stationary contracts are suboptimal with persistent states,
and the fixed wage under the optimal contract depends on the state.
12
4
Joint Surplus in the Second Best
I consider the joint surplus in the second best for two types of persistence in this
section. From Theorem 1, we know that an effort schedule can be implemented
with a history-independent contract if and only if the IC and the DE are satisfied.
Different states can affect the IC and/or the DE differently; for the given bonus cap,
the implementable action can vary with the state, and the joint surplus can also vary
with the state which will affect the DE. I consider two channels separately.
In the first case, the joint surplus in the first best increases with the state. When
the cost function is separable and strictly decreases with the state, incentive provision
is identical in each state, and in particular, given a bonus cap, the principal can
implement the same level of effort in every state. The second type of persistence I
consider is when the incentive provision becomes easier in a higher state. The joint
surplus in the first best is identical in all states. In both cases, the joint surplus
increases with the state in the second best; the difference in joint surplus between the
first best and the second best decreases with the state. I also show that stationary
contracts are optimal in the first case, even though they are not optimal in general.
4.1
Joint Surplus Varies with the State
In this section, I consider the case in which the joint surplus varies with the state
and the incentive provision is constant across the states. Specifically, I assume the
following.
Assumption 4. The cost of effort is separable and strictly decreases with the state:
There exist c1 : E → R, c2 : Θ → R such that
c(e, θ) = c1 (e) + c2 (θ), ∀e ∈ E, θ ∈ Θ
and c02 < 0 for all θ ∈ Θ.
Assumption 5. F (·|e, θ) is independent of θ.
Assumption 6. θt > θt0 implies P (·|θt ) %FOSD P (·|θt0 ).
I also define ∆W (θ) as the minimum bonus cap to be able to induce the first-best
13
level of effort in state θ. Given a state θ, eF B (θ) can be a solution to
e(θ) ∈ arg max Ey [W (φ)|e] − c(e, θ),
e
∆W ≥ sup W (θ, y) − inf W (θ, y)
y
y
if and only if ∆W ≥ ∆W (θ).
As a benchmark, I first show the implications of Assumptions 4-6 in the first best
and in the case the principle has a within-period commitment power.
Proposition 4. Suppose Assumptions 4-6 hold. Both the per-period joint surplus
and the expected joint surplus in the first best strictly increases with the state. The
first-best level of effort is constant across all states θ ∈ Θ. The minimum bonus cap
to implement the first-best level of effort, ∆W (θ), is also constant across the states.
If the principal can credibly promise W (φ), the principal implements the same level
of effort, e∗ = eF B in all states.
Proof. The first-best effort maximizes the expected joint surplus in state θ,
Z
Ey [y|e] − c(e, θ) =
yf (y|e)dy − c1 (e) − c2 (θ).
Since the cost of effort is separable, the maximization problems for any two states
are constant transformations of each other, and the first-best level of effort is constant across the states. The cost strictly decreases with the state, and the expected
per-period joint surplus in the first best in state θ strictly increases with the state.
By the persistence of states, the joint surplus also increases with the state. Since
the maximization problems are a constant transformation of each other, ∆W (θ) is
constant across the states.
If the principal can commit to bonus payments, the only constraint is the agent’s
IC constraint. By the efficiency assumption, it is efficient to induce the first-best level
of effort than to take the outside option in all states θ, and the principal induces the
first-best level of effort in all θ.
Now consider relational contracts under Assumptions 4-6. Define sF B (θ) as the
expected joint surplus in the first best when the previous state is θ. We know from
δ
Proposition 4 that ∆W (θ) is constant over θ. Denote ∆W ∗ = ∆W (θ). If 1−δ
(sF B (θ)−
s̄) ≥ ∆W ∗ , the principal can implement the first-best level of effort in all states with
14
relational contracts, and the problem becomes trivial. I will make the following
assumption:
Assumption 7. The principal cannot induce the first-best level of effort in the lowest
state:
δ
(sF B (θ) − s̄) < ∆W ∗ .
1−δ
Define e(θ|∆W ) to be the solution to the optimization problem
maxe Ey [y − c|e, θ] s.t. e(θ) ∈ arg max Ey [W (φ)|e] − c(e, θ),
e
∆W ≥ sup W (θ, y) − inf W (θ, y).
y
y
e(θ|∆W ) is the level of effort that maximizes the per-period joint surplus in state θ
when the bonus cap is ∆W . From the Mirrlees-Rogerson constraints, the joint surplus
is concave in e. If ∆W ≤ ∆W ∗ , the principal cannot implement the first-best level of
effort, and e(θ|∆W ) < eF B (θ). Since the principal can always mimic the payments
with ∆W 0 if ∆W ≥ ∆W 0 , the implementable level of effort weakly increases with the
bonus cap, and we have e(θ|∆W ) ≥ e(θ|∆W 0 ), ∀θ.
Lemma 1. The implementable level of effort e(θ|∆W ) weakly increases with ∆W for
all θ.
Proof. The proof follows from the fact that the principal can always mimic the compensation scheme with ∆W 0 if ∆W ≥ ∆W 0 .
Under relational contracts, the expected joint surplus from the following period
limits the principal’s ability to induce effort, and Proposition 4 states that the joint
surplus in the first best strictly increases with the state. The implementable level of
effort is lower in a worse state, and given the persistence of states, the difference in
the expected joint surplus is reinforced by the implementable effort. Under Assumptions 4-7, the joint surplus under an optimal relational contract increases with the
state, and the difference in the joint surplus between the first best and the second
best decreases with the state.
Proposition 5. Suppose Assumptions 4-7 hold. Let sSB (θ) be the expected joint
surplus under an optimal relational contract. sSB (θ) strictly increases with θ, and
FB
∂sSB
≥ ∂s∂θ > 0. The difference in the joint surplus between the first best and the
∂θ
15
second best, sF B (θ)−sSB (θ), weakly decreases with the state. The difference is strictly
positive at θ.
Proof. We know from Lemma 1 that the implementable level of effort, e(θ|∆W ),
weakly increases with ∆W . From Proposition 4, the expected joint surplus in the
first best increases with the state, and Assumption 7 says that the expected joint surplus in the state θ is not big enough to induce the first-best level of effort. Since the
distribution of the states increases with the state in the sense of first-order stochastic
dominance, the implementable level of effort under an optimal relational contract increases with the state, and the expected joint surplus in the second best also increases
with the state.
Consider the difference in per-period joint surplus between the first best and the
second best.
S(eF B , θ) − S(e(θ|∆W ), θ)
=(E[y|eF B ] − c(eF B , θ)) − (E[y|e(θ|∆W )] − c(e(θ|∆W ), θ))
=(E[y|eF B ] − c1 (eF B )) − (E[y|e(θ|∆W )] − c1 (e(θ|∆W ))).
Given ∆W , e(θ|∆W ) is constant across the states, and we also know that
E[y|e(θ|∆W )] − c1 (e(θ|∆W ))
increases with ∆W . Therefore, the difference in the per-period joint surplus,
S(eF B , θ) − S(e(θ|∆W ), θ),
decreases with the state, and by the persistence of the states, the difference in the expected joint surplus also decreases with the state. From Assumption 7, the difference
is strictly positive at θ.
When the per-period joint surplus in the first best increases with the state, the
persistence of the states enter the optimization problem through the bonus cap, and
the expected joint surplus under an optimal relational contract also increases with
the state. The dynamic enforcement constraint magnifies the impact of persistent
states, and the expected joint surplus varies more in the second best than in the first
best.
16
The next theorem shows that, however, stationary contracts are optimal in this
environment.
Theorem 3. Suppose Assumptions 4-7 hold. The joint surplus can be maximized
with stationary contracts.
Proof. I’m going to construct a stationary contract under which the joint surplus is
maximized. From Proposition 2, we know that the joint surplus can be maximized
with a history-independent contract. Let e(θ) and s(θ) be the effort and the expected
joint surplus given state θ under the history-independent contract that maximizes the
joint surplus. From Theorem 1, there exists a payment schedule W : Φ → R such
that for all θ ∈ Θ,
(IC) e(θ) ∈ arg max Ey [W (φ)|e, θ] − c(e, θ),
e
δ
(s(θ) − s̄) ≥ sup W (θ, y) − inf W (θ, y).
y
1−δ
y
(DE)
We also know from Proposition 3 that the contract either implements the first-best
action or takes a form of a step function at the given θ:
(
W (φ) =
W (θ)
W (θ) +
δ
(s(θ)
1−δ
if y < ŷ
− s̄) if y ≥ ŷ,
where fe changes the sign at ŷ. In particular, from the FOC-IC, we know that if
e(θ) 6= eF B , then e(θ), s(θ) satisfy
ȳ
Z
fe (y|e(θ))dy
ŷ
δ
(s(θ) − s̄) = ce (e(θ), θ)
1−δ
(∗)
I will now construct a payment schedule W̃ : Φ → R such that for all θ ∈ Θ,
s(θ) = u(θ) + π(θ) and
(IC) e(θ) ∈ arg max Ey [W̃ (φ)|e, θ] − c(e, θ),
e
(SE)
inf [
θ
δ
δ
(u(θ) − ū) + inf W̃ (φ)] + inf [
(π(θ) − π̄) − sup W̃ (φ)] ≥ 0,
y
θ
1−δ
1−δ
y
(IR) u(θ) ≥ ū, π(θ) ≥ π̄.
Then it follows from Theorem 2 that e(θ), u(θ), π(θ) can be implemented with a
17
stationary contract.
From Proposition 5, there exists θ̂ such that the DE binds if and only if θ ≤ θ̂.
The principal implements eF B for θ > θ̂. Since the contract implements the first-best
action or takes a form of a step function, we know that on [θ, θ̂]
(
W (φ) =
W (θ)
W (θ) +
if y < ŷ
δ
(s(θ) − s̄) if y ≥ ŷ,
1−δ
If a stationary contract maximizes the joint surplus, the second term in the SE becomes
δ
δ
δ
(π(θ) − π̄) − sup W̃ (φ) =
(π(θ) − π̄) − (inf W̃ (φ) +
(s(θ) − s̄))
y
1−δ
1−δ
1−δ
y
δ
=−
(u(θ) − ū) − inf W̃ (φ),
y
1−δ
and the SE simplifies to
δ
(u(θ) − ū) + inf W̃ (φ) = w
y
1−δ
for some w ∈ R and all θ ≤ θ̂.
Denote inf y W̃ (φ) = Ŵ (θ). Construct W̃ such that
all θ, and define


 Ŵ (θ)
W̃ (φ) =
Ŵ (θ) +


Ŵ (θ) +
δ
(s(θ)
1−δ
δ
(s(θ̂)
1−δ
δ
(u(θ) − ū) + Ŵ (θ)
1−δ
= w on
if y < ŷ,
− s̄) if y ≥ ŷ, θ ≤ θ̂
− s̄) if y ≥ ŷ, θ > θ̂.
By construction, the IC and the SE are satisfied, and it remains to verify that the IR
is also satisfied. We can express the expected payoff of the agent and the principal
18
as below:
Z
(1 − δ)(Ey [W̃ (φ)] − c(e, θ0 )) + δu(θ0 )dP (θ0 |θ)
u(θ) =
θ̂
Z
(1 − δ)w + δū + (1 − F (ŷ))δ(s(θ0 ) − s̄) − (1 − δ)c(e, θ0 )dP (θ0 |θ)
=
θ
Z
+
θ̄
(1 − δ)w + δū + (1 − F (ŷ))δ(s(θ̂) − s̄) − (1 − δ)c(eF B , θ0 )dP (θ0 |θ),
θ̂
Z
π(θ) =
Z
=
(1 − δ)S(θ0 ) + δs(θ0 )dP (θ0 |θ) − u(θ)
θ̂
(1 − δ)(E[y|e(θ0 )] − w) + δπ̄ + F (ŷ)δ(s(θ0 ) − s̄)dP (θ0 |θ)
θ
Z
+
θ̄
(1 − δ)(E[y|eF B ] − w) + δπ̄ + F (ŷ)δ(s(θ̂) − s̄) + δ(s(θ0 ) − s(θ̂))dP (θ0 |θ).
θ̂
On [θ, θ̂], we have
d
δ
[(1 − F (ŷ|e(θ0 )))
(s(θ0 ) − s̄) − c(e(θ0 ), θ0 )]
0
dθ
1−δ
∂
δ
= 0 [(1 − F (ŷ|e(θ0 )))
(s(θ0 ) − s̄) − c(e(θ0 ), θ0 )]
∂θ
1−δ
de
d
δ
+ 0 (θ0 ) [(1 − F (ŷ|e(θ0 )))
(s(θ0 ) − s̄) − c(e(θ0 ), θ0 )]
dθ
de
1−δ
∂
δ
= 0 [(1 − F (ŷ|e(θ0 )))
(s(θ0 ) − s̄) − c(e(θ0 ), θ0 )]
∂θ
1−δ
by the IC. We know from Proposition 5 that s(θ0 ) increases with θ0 , and c02 < 0
from Assumption 4. Therefore, the integrand in u(θ) increases with θ0 , and from
Assumption 6, u(θ) also increases with θ. It is sufficient to verify u(θ) ≥ ū.
19
For the principal, on [θ, θ̂], we have
δ
d
δ
s(θ0 ) − ((1 − F (ŷ|e(θ0 )))
(s(θ0 ) − s̄) − c(e(θ0 ), θ0 ))]
[S(θ0 ) +
0
dθ
1−δ
1−δ
∂
δ
δ
= 0 [S(θ0 ) +
s(θ0 ) − ((1 − F (ŷ|e(θ0 )))
(s(θ0 ) − s̄) − c(e(θ0 ), θ0 ))]
∂θ
1−δ
1−δ
δ
δ
de
d
s(θ0 ) − ((1 − F (ŷ|e(θ0 )))
(s(θ0 ) − s̄) − c(e(θ0 ), θ0 ))]
+ 0 (θ0 ) [S(θ0 ) +
dθ
de
1−δ
1−δ
∂
δ
δ
= 0 [S(θ0 ) +
s(θ0 ) − ((1 − F (ŷ|e(θ0 )))
(s(θ0 ) − s̄) − c(e(θ0 ), θ0 ))]
∂θ
1−δ
1−δ
de
d
+ 0 (θ0 ) S(θ0 )
dθ
de
by the IC. From the Mirrlees-Rogerson constraints, the per-period joint surplus is concave in e. Since s(θ0 ) increases on [θ, θ̂] from Proposition 5, it follows from Lemma 1
that e(θ0 ) also weakly increases on this interval. The second term in the above expression is weakly positive. By rearranging the terms, we get
∂
δ
δ
[S(θ0 ) +
s(θ0 ) − ((1 − F (ŷ|e(θ0 )))
(s(θ0 ) − s̄) − c(e(θ0 ), θ0 ))]
0
∂θ
1−δ
1−δ
∂
∂
δ
δ
s(θ0 ) − ((1 − F (ŷ|e(θ0 )))
(s(θ0 ) − s̄)]
= 0 [S(θ0 ) + c(e(θ0 ), θ0 )] + 0 [
∂θ
∂θ 1 − δ
1−δ
It follows that the integrand in π(θ) increases with θ0 , and from Assumption 6, π(θ)
increases with θ. It is sufficient to verify π(θ) ≥ π̄.
As long as the expected joint surplus at θ is weakly greater than s̄, we can always find w such that both IR constraints are satisfied. Fix w such that the IR is
satisfied, then u(θ), π(θ) are pinned down for all θ. From our construction of W̃ (φ)
δ
(u(θ) − ū) + Ŵ (θ) = w, b(φ) is pinned down for all φ ∈ Φ. The IC and the
and 1−δ
SE are satisfied. Therefore, the joint surplus can be maximized with a stationary
contract.
The self-enforcement constraint for stationary contracts is stronger than the dynamic enforcement constraint for history-independent contracts in general. However,
when the cost function is separable and the joint surplus increases with the state,
then stationary contracts are optimal, and the principal can maximize the joint surplus with stationary contracts. In general, if we don’t know that u(θ) and π(θ) are
monotone, it is difficult to verify whether a fixed salary w such that the IR is satisfied
at all θ exists. However, we know from Proposition 5 that the joint surplus increases
20
with the state, and one can show that stationary contracts are in fact optimal in this
setting.
4.2
Incentive Provision Varies with the State
This section considers the alternative case in which the joint surplus in the first best
is constant across the states but the incentive provision varies with the state.
I assume that the first-best level of effort is constant across the states. This is
without loss of generality for any interior solution eF B . I also assume that for a given
bonus cap, the maximum per-period joint surplus strictly increases with the state,
and the principal cannot implement the first-best level of effort in the worst state,
even if the bonus cap was given by the first best.
Assumption 8. The first-best level of effort is constant in all states. The per-period
joint surplus in the first best is constant across the states: S(eF B , θ) = S ∗ for all θ.
Assumption 9. For a given bonus cap ∆W , if the principal cannot induce the firstbest level of effort, the maximum per-period joint surplus strictly increases with the
state. i.e., S(e(θ|∆W ), θ) strictly increases with θ for all e(θ|∆W ) < eF B .
Assumption 10. The principal cannot implement the first-best level of effort in the
lowest state, that is e(θ|sF B ) < eF B .
Under Assumptions 6, 8-10, the expected joint surplus in the second best strictly
increases with the state, and the difference in the expected joint surplus between
the first best and the second best decreases with the state. We have the following
proposition which is an analogue of Proposition 5.
Proposition 6. Suppose Assumptions 6, 8, 9 and 10 hold. There exists θ∗ ∈ Θ such
that sSB (θ) strictly increases with θ for θ ≤ θ∗ , and sSB (θ) = S ∗ for θ > θ∗ . The
difference in the joint surplus between the first best and the second best, S ∗ − sSB (θ),
weakly decreases with the state. The difference is strictly positive at θ.
Proof. By Assumptions 9, 10 and the persistence of the states, the per-period joint
surplus in the second best weakly increases with θ, and it increases strictly for all θ
such that e(θ|sSB (θ)) < eF B . Therefore, the expected joint surplus in the second best
also increases with the state. Since the first-best joint surplus is constant across the
states, the difference between the first best and the second best decreases with the
state.
21
I have considered two types of persistent states. In both environments, the difference in the expected joint surplus between the first best and the second best decreases
with the state. If the two factors, the level of joint surplus in the first best and the
difficulty of incentive provision, move in the same direction, the effect will be magnified. If they move in the opposite directions, the difference in the joint surplus will
be determined by which effect dominates.
5
Conclusion
I study relational contracts in a persistent environment. I show that when there is
no asymmetric information about the state, history-independent contracts are optimal, and I derive necessary and sufficient conditions to implement an effort schedule
with history-independent contracts. These properties show that history-independent
contracts are appropriate generalization of stationary contracts. Stationary contracts
are no longer optimal when the states are partially persistent. The necessary and sufficient condition to implement an effort schedule with stationary contracts is stronger
than the dynamic enforcement constraint.
Suboptimality of stationary contracts means that the persistence of the underlying
environment changes the optimal contract qualitatively. If the environment is persistent, providing bonus payments may not be sufficient, and the fixed wage may also
have to vary with the state. Given that many compensation schemes have a constant
fixed wage in every state, the firms will be strictly better off with state-dependent
wages.
When the states are observable and follow a first-order Markov chain, the state
in any given period is a sufficient statistic for the distribution of future states. In
particular, the outcome doesn’t carry any information about the distribution of future
states, and the principal can provide incentives by the bonus payments in the given
period. It is optimal to provide the same expected per-period payoff in every state.
If in some period the continuation contract for a given state provides the maximum
joint surplus for the given state, the principal can provide the same continuation
contract in every period for the given state. Since the agent gets the same expected
payoff in all states, the agent’s IC constraints are still satisfied when the principal
replaces the continuation contract, and the optimal contract can be independent of
history. The principal can also redistribute the surplus through the fixed wage, and
22
we get the dynamic enforcement constraint as with i.i.d. states. An effort schedule
can be implemented with history-independent contracts if and only if it satisfies the
IC constraint and the dynamic enforcement constraint. I also show that an effort
schedule can be implemented with stationary contracts if and only if it satisfies the
IC, the IR and the self-enforcement constraint which is stronger than the dynamic
enforcement constraint.
Persistent states can affect the relational contracts through two mechanisms. The
persistence of the states implies that if the joint surplus depends on the state, the
bonus cap in the DE also varies with the state, and the implementable level of effort
depends on the state, even if the incentive provision for the given bonus cap is identical
in each state. On the other hand, in the IC, the incentive provision for the given
bonus cap can also change with the state. Analyzing those channels separately, I
have shown that if the joint surplus in the first best increases with the state, or if
the implementable level of effort for a given bonus cap increases with the state, the
joint surplus in the first best and the second best increase with the state, and the
gap between the two decreases with the state. I also show that with separable cost
functions, stationary contracts are optimal. When the cost function is separable and
decreasing with the state, the joint surplus in the second best increases with the state,
and one can construct a stationary contract that maximizes the joint surplus.
I have considered partially persistent environments where the states are observable
and the persistence is of first-order. If the states are observable, the optimal contract
can be independent of history. However, if the states were unobservable, there could
be information asymmetry between the principal and the agent about the future
states. The belief about the agent’s effort matters for the future, and the relational
contract will likely have to take into account the private information. It will be
interesting to study relational contracts and their implications for the market when
the information about the future states is no longer symmetric.
A
Proofs
Proof of Proposition 1. Consider the relational contract that provides s. In the initial
period, the principal offers w(θ0 ), b(φ0 ), and if the agent accepts, he exerts effort e(θ0 ).
The continuation payoffs under the contract are denoted by u(φ0 ) and π(φ0 ), and the
expected payoffs from the contract are u0 and π0 . Without loss of generality, we can
23
assume that off the equilibrium path, the parties revert to the static equilibrium of
(ū, π̄). The first period payment W is a function of φ0 .
The contract is self-enforcing if and only if the following conditions hold:
(i) u0 ≥ ū, π0 ≥ π̄,
(ii) e(θ0 ) ∈ arg max Ey [(1 − δ)W (φ0 ) + δu(φ0 )|e, θ0 ] − c(e, θ0 ),
e
δ
δ
u(φ0 ) ≥
ū,
1−δ
1−δ
δ
δ
π(φ0 ) ≥
π̄,
− b(φ0 ) +
1−δ
1−δ
(iii) b(φ0 ) +
and (iv) each continuation contract is self-enforcing.
Given any (u, π) such that u ≥ ū, π ≥ π̄, u + π = s, the principal can offer the
same b(φ0 ) and continuation contracts and adjust w(θ0 ) to
ŵ(θ0 ) ≡ w(θ0 ) −
π − π0
.
1−δ
The conditions are satisfied with the new contract, and it provides (u, π) as the
expected payoffs.
Proof of Proposition 2. Suppose a contract that maximizes the joint surplus provides
wt , bt and the agent chooses et . The first step is to construct an alternative contract
ŵt , b̂t under which the agent chooses the same level of effort et and his expected
payoff is equal to ū in every state.
The distribution of the states from period t + 1 only depends on θt , and the
outcome yt doesn’t carry any information about the future states. The principal can
adjust the contingent payment bt and keep the expected payoff in each state constant.
Specifically, consider the following contract. Let ut (ht , φt ) be the continuation value
of the agent under the given contract, and define ŵt , b̂t as follows:
δ
(ut (ht , φt ) − ū),
1−δ
ŵt (ht , θt ) ≡ ū − Eyt [b̂t (ht , φt ) − c(et (ht , θt ), θt )|et (ht , θt )].
b̂t (ht , φt ) ≡ bt (ht , φt ) +
From
b̂t (ht , φt ) +
δ
δ
ū = bt (ht , φt ) +
ut (ht , φt ),
1−δ
1−δ
24
the agent chooses the same level of effort et under the new contract. The agent’s
expected payoff is ū for all t, ht , θt .
The next step is to show that we can choose w̃ : Θ → R, b̃ : Φ → R such that
the principal offers w̃, b̃ in every period. Consider ŵt and b̂t . The agent’s expected
payoff is constant over all t, ht , and θt , which implies that the agent’s IC constraint
is determined by the within-period compensation scheme. Specifically, the agent
chooses e such that
et (ht , θt ) ∈ arg max Eyt [b̂t (ht , φt )|e, θt ] − c(e, θt ).
e
When the agent’s IC constraints are myopic, the principal can replace a compensation scheme for any given period with another compensation scheme without affecting
the incentives. The principal can also treat each θ separately, because the state is
observable before the agent chooses the effort. Specifically, let b̃ be the compensation
scheme that maximizes the expected per-period joint surplus for state θ:
b̃(θ, ·) ≡ arg maxb̂t (ht ,θ,·),ht Ey [y|et (ht , θ), θ] − c(et (ht , θ), θ)
s.t. et (ht , θ) ∈ arg max Ey [b̃t (ht , θ, ·)|e, θ] − c(e, θ).
e
If there’s a multiplicity of the compensation schemes, we can pick one without loss of
generality.
Given b̃ : Φ → R, the agent chooses e : Θ → E such that
e(θ) ∈ arg max Ey [b̃(φ)|e, θ] − c(e, θ).
e
Define w̃ as
w̃(θ) ≡ ū − Ey [b̃(φ) − c(e(θ), θ)|e(θ), θ],
and we have a history-independent contract that maximizes the expected joint surplus. By construction, it is self-enforcing, and it provides the same expected payoff
to the agent in all t, ht , θt . Let s∗ be the minimum expected per-period joint surplus
over the states under b̃, w̃:
s∗ ≡ min{Ey [y|e(θ), θ] − c(e(θ), θ)}.
θ
The principal can adjust the fixed salary and can provide any u such that ū ≤ u ≤
25
s∗ − π̄ to the agent as the constant expected payoff.
Proof of Theorem 1. (⇒) Suppose e(θ) is implementable. Let u(θ) and π(θ) be the
continuation value for the agent and the principal when the previous state was θ. The
IC constraint has to be satisfied, and we also know that
δ
(π(θ) − π̄) ≥ sup b(θ, y), ∀θ,
1−δ
y
δ
(u(θ) − ū) ≥ − inf b(θ, y), ∀θ
y
1−δ
(1)
(2)
have to hold. Adding the two inequalities, we have the dynamic enforcement constraint.
(⇐) Suppose W (φ) and e(θ) satisfy the IC constraint and the dynamic enforcement constraint. Define
b(φ) = W (φ) − inf W (φ̂),
φ̂
w(θ) = ū − Ey [b(φ) − c(e(θ), θ)|e(θ), θ],
and consider the history-independent contract with w(θ), b(φ) and e(θ). The parties
revert to the static equilibrium if a deviation occurs. The agent receives ū as the
expected payoff in each state, and the principal receives π(θ) = s(θ)− ū if the previous
state was θ. By the dynamic enforcement constraint, s(θ) ≥ s̄ and π(θ) ≥ π̄ for all
θ. From the IC constraint, the agent chooses e(θ) in each state θ, and it can be
verified that Inequalities (1) and (2) are satisfied. By construction, the contract is
self-enforcing in every period.
Proof of Corollary 1. From the dynamic enforcement constraint, the bonus cap decreases with the outside option s̄. The maximum per-period joint surplus weakly
decreases with s̄, which further suppresses the bonus cap through the expected joint
surplus. Therefore, both the per-period joint surplus and the expected joint surplus
decrease with s̄.
Proof of Proposition 3. We know from Proposition 1 that we can focus on maximizing the joint surplus, and Proposition 2 implies that we can focus on historyindependent contracts. By the Mirrlees-Rogerson constraints, we can replace the
26
agent’s IC constraint with the first-order condition. The optimal history-independent
contract solves
max Eθ,y [y − c|e(θ), θ]
e(·),W (·,·)
subject to
d
{Ey [W (θ, y) − c(e, θ)|e = e(θ), θ]} = 0, ∀θ,
de
δ
(s(θ) − s̄) ≥ sup W (θ, y) − inf W (θ, y),
θ,y
1−δ
θ,y
X
s(θ0 ) = (1 − δ)E[
δ t {dt (yt − c(et , θt )) + (1 − dt )s̄}|θ0 ].
t=0
From the Mirrlees-Rogerson constraints, the principal wants to maximize e when
e(θt ) < eF B (θt ). We get
(
W (θ, y) =
W̄ (θ)
W (θ)
if y ≥ y(θ)
,
if y < y(θ).
and fe changes the sign at y(θ), and W̄ (θ) = W (θ) +
δ
(s(θ)
1−δ
− s̄).
References
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Econometrica 76 (2008): 493-540.
[2] Baker, George, Robert Gibbons, and Kevin J. Murphy. “Relational Contracts
and the Theory of the Firm.” Quarterly Journal of Economics 117 (2002): 3984.
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