GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.111

Only the Final Outcome Matters: Persistent
Effects of Efforts in Dynamic Moral Hazard

Ryo OgawaJanuary 2010; revised June 2010, revised May 2011

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Only the Final Outcome Matters: Persistent
Effects of Efforts in Dynamic Moral Hazard
OGAWA, Ryo∗
Institute of Social and Economic Research, Osaka University,
6-1 Mihogaoka, Ibaraki-shi, Osaka 567-0047, Japan;
r-ogawa@iser.osaka-u.ac.jp

May 12, 2011

Abstract
We analyze a dynamic principal–agent problem in which the agent’s effort in each period has strong persistent effects. We show that a simple
contract, where the reward depends only on the final outcome, is explained
as the optimal contract derived in the principal’s optimization problem. The
paper also discusses that the optimality of such a simple payment scheme
crucially depends on the first-order stochastic dominance of the final outcome under various effort sequences.
Journal of Economic Literature Classification Numbers: C61, D82, D86.
Keywords: dynamic moral hazard; history dependence; simple contract;
first-order stochastic dominance.
∗ This is a substantially revised paper of an earlier version (Ogawa (2003)). After the previous

version was prepared, I became aware of two independent works, Mukoyama and Şahin (2005) andKwon (2006), the results of which are incorporated in the paper. The current version is included as
Chapter 2 in my Ph.D. thesis (2008), and I would like to thank my advisor, Michihiro Kandori, for
his guidance and encouragement. I am grateful to the editor and anonymous referees for helpfulcomments. I also thank Eddie Dekel, Junichiro Ishida, Hideshi Itoh, Minoru Kitahara, Dan Sasaki,
and Satoru Takahashi for comments and discussions. This research was partially supported by
MEXT of the Japanese government, Grant-in-Aid for Young Scientists (B) 21730157.

1

1Introduction

In long-term principal–agent relationships with complete contracts, the principal
prepares payment schedules in advance and these schedules potentially depend on
the agent’s period-wise performance (i.e., performance related to the level of effort), in order to provide proper incentives. In the light of the celebrated sufficient
statistic theorem (Holmström (1979)), one may expect that using a very detailed
history of past performances, each of which are informative of the agent’s efforts,
is optimal for the principal in preparing payment schedules. In reality, however,
we often observe various incentive schemes that are not necessarily dependent on
the entire detailed record of performances but only on a subset of them, wheresuch subsets are sometimes much smaller than the entire set of performances.
From the viewpoint of economic studies, such simple contracts are interpreted
in several ways. One argument is that the principal incurs costs in preparing or
enforcing complex contracts, after taking such costs into consideration, some sortof simple contract is concluded as a suboptimal solution. Another possibility,
following Holmström and Milgrom (1987), is that simple payment schedules are
justified by their robustness to the change of model parameters. Understandingthat such interpretations provide insights that enable us to grasp important aspectsof contracts in reality, the present paper aims to study a third way of explaining
simple contracts.1
Consider an environment where the agent’s current efforts have persistent effects over the future performances. In such environments, a contract that provides
strong incentives in the future induces the agent to work hard in the present, andthe role of such future incentives appears to be more important in such situations
than in those where the agent’s efforts have no persistent effects. However, thisis not to say that providing incentives only in the future is sufficient: the agent’s

1 It should be noted that the current paper does not aim to provide a universal explanation forsimple contracts (as in Holmström and Milgrom (1987)), but looks for an explanation that works
in a particular class of environments.

2

current performance is still informative about present efforts,2 and we may expectthat every informative performance should be included in the optimal contract, nomatter how low their informativeness may be. The present paper shows that thisis not always the case; that is, we show that an incentive scheme that depends only
on the final performance is optimal if the agent’s effort in each period has strong
persistent effects.
We provide sufficient conditions for such simple contracts to be optimal in
dynamic moral hazard models, in which the cost of efforts is the same in all periods. The common feature of our sufficient conditions is summarized as follows:
the probability distribution of the final outcome when the agent shirks only in the
final period first-order stochastically dominates (FOS-dominates hereafter) the
distribution when the agent shirks in any other period in such a way that the expected number of shirking events is one. To grasp the idea behind this condition
intuitively, consider a two-period model in which the agent’s first-period action
also affects the probability distribution of the second-period outcome. Let (a, a′ )denote the action profile in which the first element (second element) indicates theagent’s first-period action (second-period action, respectively), and let ā (a) denote a strong effort (a shirk, respectively). Then, the sufficient condition has thefollowing two requirements (Assumption 1).
(i) The probability distribution of the second-period outcome when the agent
shirks only in the second period (ā, a) FOS-dominates the distribution when
the agent shirks only in the first period (a, ā).
(ii) The probability distribution of the second-period outcome when the agent
shirks only in the second period (ā, a) FOS-dominates the half-and-half
mixture of (a) the distribution when the agent shirks in both periods (a, a)and (b) the distribution when the agent never shirks in any period (ā, ā).
We can understand this sufficient condition in the following way. Suppose that
2 In the paper, we assume that performances are statistically independent between one period

and the next. See Section 2 for the formal model.

3the principal can (somehow) force the agent to work hard in the first period. Thenthe principal’s problem is to design the contract so that the agent finds it optimal
to work hard in the second period (given that the agent is forced to worked hard in
the first period). Such an optimal contract provides the agent with higher expected
payoff (the sum of wages and effort costs) by taking (ā, ā) than by taking (ā, a).
Note that this optimal contract depends only on the second-period outcome so that
it is a contract in which “only the final outcome matters.”
Additionally suppose that this “optimal” contract is an increasing function of
second-period outcomes.3 Then requirement (i) ensures that shirking in the first
period (a, ā) always makes the agent worse off than shirking in the second period(ā, a) does, since the cost of efforts is the same between the two action profiles(i.e., one effort) and FOSD ensures that the agent’s expected wage by taking (ā, a)is larger than by taking (a, ā).4 So given this contract, the agent never finds it
optimal to undertake (a, ā), even if there is no “forced labor” in the first period.
Requirement (ii), on the other hand, ensures that shirking in both periods (a, a)
makes the agent worse off compared with shirking in the second period (ā, a).
The argument behind this requirement is bit more complicated since the effort
costs are different between the two action profiles, (ā, a) and (a, a). As we will
see later, such a comparison is made by a FOSD between well-designed mixturesof action profiles with the same expected number of efforts. In requirement (ii),this is achieved by setting the expected number of efforts to be one on both sides(1 = 0.5 × 2 + 0.5 × 0). Thus, under requirements (i) and (ii), the agent never finds
it optimal to undertake either (a, ā) or (a, a) even if there is no forced labor in thefirst period, as long as the contract induces the agent to take (ā, ā) rather than(ā, a). This is how the contract in which only the final outcome matters provides
a sufficient incentive to work hard in both periods.
3 As is known in the literature, the monotone likelihood ratio property (MLRP) is required forthe optimal contract to be an increasing function. Subsection 2.1 discusses the case of MLRP.
In Subsection 2.2, we argue how the sufficient condition, (i) and (ii), is rewritten if MLRP is notsatisfied.
4 FOSD provides a sufficient condition for a comparison between expectations of increasing
functions.

4Strong persistent effects of efforts as characterized by the FOSD are the main
sources of our result. Historical dependence of this sort is often seen in real economic environments. For example, if an effort has a time-lag effect into the next
period, as well as the direct effect in the current period, then the probability ofsuccess in period 2 is influenced by the effort level in period 1. If the production
technology involves irreversibility, then the model becomes history dependent ina similar manner. We discuss these examples briefly in Section 2.
In Section 3, we show how the result in Section 2 is extended to a general
T-period setting. The extension is modestly straightforward and again the pointis the FOSD between mixtures of action profiles with the same expected numberof efforts. Human capital investment is discussed as an example of the sufficientcondition.

1.1

Related Literature

Much simpler models than the one in the present paper have been studied in independent works by Kwon (2006), Mukoyama and Şahin (2005), and Ogawa (2003).
In their studies, outcomes take only two values, “success” and “failure”, and simplified versions of our requirements (i) and (ii) are presented as the sufficient condition for the simple contract.5 The present paper studies a model with N possible
outcomes and the proof is given in an organized manner using the property of
FOSD, which provides clear and rigorous interpretations of the earlier works.6
Studies on the dynamic models of the moral hazard problem date back to Lam-

5 Although the distinction between MLRP and FOSD is important in the incentive theories withhidden actions (Mas-Colell, Whinston, and Green (1995, Section 14.B)), the two conditions are
given by the same inequality
Pr[“success” | ā] > Pr[“success” | ā]in the 2-outcome models.
6 The earlier works have another contributions. Mukoyama and Şahin (2005) provided somenumerical analyses when the theoretical approach is difficult. The highlight of Kwon (2006) is theempirical analysis using health insurance data. The “nonincreasing marginal returns” condition by
Kwon (2006) is a special case of our sufficient condition (see Section 3).

5bert (1983) and Rogerson (1985). Their studies are on the repeated model in thesense that there is no persistent effects as studied in the present paper. In repeatedmoral hazard, it is shown that the optimal contract depends on the entire history ofoutcomes (memory effect in Rogerson (1985)). Our paper, in contrast, shows thatthe optimal contract depends only on the final outcome under certain sufficientconditions.
Dynamic moral hazard problems then have been studied in various extensions. Problems in which the agent can access a bank are analyzed by Fudenberg,
Holmstrom, and Milgrom (1990) and Chiappori, Macho, Rey, and Salanié (1994)
among others. Renegotiation problems in dynamic moral hazard are studied by
Fudenberg and Tirole (1990), Ma (1991), Ma (1994), and Matthews (1995) among
others. In the present paper, we assume that there is no access to banks and norenegotiation.
The relationship between sufficient statistic theorem (Holmström (1979)) andour result casts an interesting light on the interpretation of “informativeness” ineconomic studies. In the view of the sufficient statistic theorem, every statistically informative signal is useful (and should be used) in the optimal contract,
whereas in our model, the principal sometimes finds it optimal to “ignore” somestatistically informative signals.
Recent theoretical studies on repeated moral hazard problems include Jarque(2010), who considers a similar problem with infinite horizon, continuum effort.
The current paper studies the model with two effort levels and we do not need toreplace incentive constraints with first-order conditions.

2

The Basic Model

We study a simple dynamic moral hazard model with “history dependence.” Therelationship between a principal (she) and an agent (he) lasts for two periods (t =1, 2).
In each period, the agent chooses his action at from the action space A =6

{a, ā}. These actions are unobservable to the principal. We may find it convenient
to interpret these actions as effort levels and say that the agent works hard (shirks)
when he chooses ā (a).
In period t, after the agent has chosen his action at , the outcome xt ∈{x1 , · · · , xN } ≡ X is realized according to probabilities that depend on the historyof the agent’s actions; that is, the distribution of x1 depends on a1 , whereas thatof x2 depends on the pair (a1 , a2 ). These outcomes are immediately observed
by both parties (and assumed to be verifiable to third parties, such as a court).
We may regard these outcomes as performances and identify each of them with acorresponding revenue received by the principal.
We assume that x1 and x2 are independently distributed;7 hereafter, we write
the distributions as given below:[]
p1i (a1 ) = Pr x1 = xi | a1[]p2i (a1 , a2 ) = Pr x2 = xi | (a1 , a2 )(i = 1, · · · , N),(i = 1, · · · , N).

Throughout the paper, we assume that the distributions have full support:
p1i (a1 ) > 0
p2i (a1 , a2 ) > 0

for all (i, a1 ) ∈ {1, · · · , N} × A,
for all (i, a1 , a2 ) ∈ {1, · · · , N} × A2 .

At the beginning of the game (i.e., before t = 1), the principal and the agent
sign a contract in the manner detailed below.
First, the principal offers a long-term contract w = (w1 , w2 ), where w1 =(w1 (x1 ))x1 ∈X and w2 = (w2 (x1 , x2 ))(x1 ,x2 )∈X 2 are payment schedules for periods
1 and 2, respectively, under outcome realizations (x1 , x2 ). Such a contract stipulates N + N 2 possible payments, depending on the realizations of outcomes. Next,
the agent decides whether to accept or reject the contract offered by the principal.
7 Under this assumption, the realized value of x1 does not influence the distribution of x2 , so

the former yields no information on the current likelihood of any particular production levels inperiod 2. The “history dependence” discussed in this paper deals with the case where x2 is affectedby a1 , but not by the realization of x1 .

7

If the agent refuses the offered contract, both parties receive their reservation utilities and the game comes to an end. If the agent accepts the contract, the gameenters into the two-time moral hazard repetition discussed above.
We assume that the principal can commit to the long-term contract that she
has offered before t = 1, therefore, once the contract is accepted by the agent,
the principal cannot change the payment schedule w and must make the payment
in each period according to the history of outcome realizations up to the date ofpayment. In addition, we assume that the agent must commit to his participationin the game, therefore, once he accepts the contract, he cannot exit in the midst of
the game and must participate in it until the end of period 2.
In each period, the agent attains a payoff of u(w) − c(a), where u is strictlyincreasing and strictly concave (the agent is risk averse) and c(a) < c(ā) (harderwork involves a greater cost). We normalize this as c(a) = 0 and c(ā) = C.
Given a long-term contract w, the agent’s strategy consists of two parts: oneis the action he takes in the first period, a1 , and the other is the action schedule
for the second period a2 = (a2i )Ni=1 , each of which specifies the action he takesin period 2 under the outcome realization of x1 in period 1.8 Let Ui (a1 , a2i , w2 )denote the expected utility in period 2 for the agent when he chose a1 and theoutcome was xi in the first period:1

Ui (a , a2i , w2 ) =N∑ p2i (a1, a2i )u(w2(xi, x j )) − c(a2i ).

j=1

Using this notation, the intertemporal expected utility for the agent U(a1 ; a2 ; w)
under the agent’s strategy (a1 ; a2 ) is written asU(a ; a21 , · · · , a2N ; w) =
1

N

[

]

∑ p1i (a1) u(w1(xi)) +Ui(a1, a2i , w2) − c(a1).9i=1

8 Accordingly, we allow the agent to change his action in period 2 after he observes the outcome

realization in period 1, which is one of the standard assumptions in this literature. Once we stopmaking this assumption and assume that the agent has to commit to a pair of actions (a1 , a2 )ex ante, then the model reduces to a one-shot multitask incentive problem. We will make thesequentiality assumption to focus on the dynamics of the model, but note that the main result of
the paper also applies to the one-shot multitask model.
9 We assume that both the principal and the agent have a common discount factor of one. If

8The optimization problem for the principal when she wishes to implement an
action profile (a1 , a2 ) is written as
[
N

N

i=1

j=1

]

min ∑ p1i (a1 ) w1 (xi ) + ∑ p2j (a1 , a2i )w2 (xi , x j ) ,w(P)subject toU(a1 , a2 , w) ≥ U(a′ , a′′ , w),
Ui (a1 , a2i , w2 ) ≥ Ui (a1 , a′ , w2 ),a′ ̸= a1 ,∀a′′ ∈ AN ,a′ ̸= a2i ,U(a1 , a2 , w) ≥ 2ū,i = 1, · · · , N,

(IC1)
(IC2)
(PC)

where ū denotes the reservation utility for the agent.10

2.1

Simple Contract

In this subsection, we show that the optimal long-term contract is dependent only
on the second-period outcome if the probability distribution of the second-period
outcome satisfies certain conditions, as briefly discussed in the Introduction. The
result (Theorem 1) contrasts with that of the repeated moral hazard literature,
where the optimal long-term contract is always dependent on the complete historyof past outcomes.
Throughout this subsection, we assume that
p21 (ā, ā)p2N (ā, ā)
<
·
·
·
<
p21 (ā, a)p2N (ā, a)(2)

the common discount factor was less than one (but positive) and the outcome space consisted ofthree elements or more, we could not attain the plausible sufficient conditions as in Assumption 1,
which relies on the nature of (p1i (·)) and (p2i (·, ·)). Mukoyama and Şahin (2005) showed thatwhen N = 2, an extension of Assumption 1 is a sufficient condition for w1 (x1 ) to be constant, in asimilar model in which both players have a common discount factor of less than one.
10 When the model is just a repetition of two moral hazard stages (as in Lambert (1983) and
Rogerson (1985)), the action taken in period 1, a1 , does not affect the probability distribution ofoutcomes in period 2, such that Ui (a′ , a2i , w2 ) = Ui (a′′ , a2i , w2 ) for any a′ ̸= a′′ . This reduces the
incentive constraints for the first period (IC1) toU(a1 , a2 , w) ≥ U(a′ , a2 , w),(a′ ̸= a1 ),(1)under which we must only take into account the deviation strategies from a1 to the other a′ , witha2 being fixed.

9for the sake of a simple exposition. (2) is referred to as the monotone likelihood
ratio property (MLRP), and plays an important role in the monotonicity of optimal
contracts in the basic moral hazard model.11 In Subsection 2.2, we discuss how
the sufficient condition (Assumption 1 below) is described if (2) is not satisfied. At
this point we should note that the discussion in Subsection 2.2 is not restrictive,
and there we can describe an analogue of Assumption 1 for any given pairs of
p2j (ā, ā) and p2j (ā, a) which do not satisfy MLRP.
The following assumption provides a sufficient condition for such simple contracts. We may regard this assumption as relating to “strong persistent effects”
in the sense that the action chosen in period 1 has a stronger influence on theoutcome in period 2 than does the action chosen in period 2.
Assumption 1. p2j (a1 , a2 ) satisfies the following conditions:jj(i) ∑k=1 p2k (a, ā) ≥ ∑k=1 p2k (ā, a) for all j = 1, · · · , N.
()jj(ii) 12 ∑k=1 p2k (a, a) + p2k (ā, ā) ≥ ∑k=1 p2k (ā, a) for all j = 1, . . . , N.
Condition (i) states that (ā, a) (first-order) stochastically dominates (a, ā), so
it asserts that, for any increasing function γ : {1, . . . , N} → R,
N

N

j=1

j=1

∑ p2j (ā, a)γ ( j) ≥ ∑ p2j (a, ā)γ ( j).

Condition (ii) is the stochastic dominance between (ā, a) and 12 ((ā, ā) + (a, a)),and hence, again,N∑ p2j (ā, a)γ ( j) ≥j=1)
1 N ( 2p j (ā, ā) + p2j (a, a) γ ( j),∑2 j=1

for any increasing function γ .12 Given a pair of p2j (ā, ā) and p2j (ā, a), conditions(i) and (ii) are mutually independent.
11 MLRP’s implications in the principal-agent problem are discussed in Milgrom (1981).
12 For comprehensive discussions of stochastic orders, see Mas-Colell et al. (1995, Section 6.D),

Shaked and Shanthikumar (2007) for instance.

10

A different interpretation of condition (ii) can be done in the following manner.
jj

Since p j (ā, ā)/p j (ā, a) is increasing in j and therefore ∑k=1 p2k (ā, a) ≥ ∑k=1 p2k (ā, ā)
for all j, condition (ii) implies that
j (
( 2
)
)
2
p
(a,
a)
−
p
(
ā,
a)
≥
∑ k
∑ p2k (ā, a) − p2k (ā, ā) ≥ 0.
k
j

k=1

k=1

Therefore, condition (ii) implies that (ā, a) has to be “closer” to (ā, ā), comparedwith (a, a).
Here, we provide two examples of probability distributions that satisfy Assumption 1. In both examples, it is assumed that the outcome is either “success”
or “failure” (N = 2).
Example 1 (Time lag). There is a time lag between the effort and its effect.
If the agent works hard in period t, it increases the probability of success not
only in the same period by α but also in the following period by β . We assume
that 0 < α < β , and regard β as a “full effect” of the effort and α as a “partial
effect” of the effort. Let π denote the probability of success when the agent has
never taken any positive efforts. Then, we havep1success (a) = π ,p1success (ā) = π + α ,p2success (a, a) = π ,p2success (a, ā) = π + α ,p2success (ā, a) = π + β ,p2success (ā, ā) = π + α + β .

Example 2 (Irreversibility). The agent has to make a positive effort in every period to maintain the highest probability of success π̄ . If he shirks, the probabilityof success declines by γ and this degree of success is not recovered even if theagent makes a positive effort in the following period:
p1success (a) = π̄ − γ ,p1success (ā) = π̄ ,p2success (a, a) = π̄ − 2γ ,p2success (a, ā) = π̄ − γ

p2success (ā, a) = π̄ − γ ,p2success (ā, ā) = π̄ .

The main result is as follows.
11

Theorem 1. Suppose that the probability distribution of the second-period outcome satisfies Assumption 1. Then, the optimal long-term contract w that implements a1 = ā and a2 = (ā, . . . , ā) is such that(a) w1 (x1 ) is a constant for all x1 and(b) w2 (x1 , x2 ) is independent of x1 and depends only on x2 .
Proof. The proof proceeds in two steps. In the first step, we solve a “relaxed”
optimization problem as follows:
[

]

N

N

i=1

j=1

min ∑ p1i (a1 ) w1 (xi ) + ∑ p2j (a1 , a2i )w2 (xi , x j ) ,w(P’)

subject toUi (a1 , a2i , w2 ) ≥ Ui (a1 , a′ , w2 ),a′ ̸= a2i ,i = 1, · · · , N,U(a1 , a2 , w) ≥ 2ū,

(IC2)
(PC)

and show that the solution satisfies the properties (a) and (b). In the second step,
we verify that (any of the) contract satisfying properties (a) and (b) are always
compatible with the constraint (IC1). By these two steps, we conclude that thesolution to the “original” optimization problem (P) satisfies properties (a) and (b).
1. The first-order condition for w1 (xi ) in the “relaxed” problem (P′ ) is1=νu′ (w1 (xi ))for all xi ,where ν is the Lagrange multiplier with respect to (PC). Thus, w1 (xi ) is a constantfor all xi .
The first-order condition for w2 (xi , x j ) is[]
2 (ā, a)
p
1
µi
j
=
1− 2
+ ν,
′
2
u (w (xi , x j )) pi (ā)
p j (ā, ā)(3)where µi is the Lagrange multiplier with respect to (IC2) for the corresponding i. Here, w2 (xi , x j ) is independent of i (otherwise, the principal could be
12strictly better off by offering the certainty equivalence w̃′j , such that u(w̃′j ) =∑i p1i (ā)u(w2 (xi , x j )), without affecting the remaining constraints (IC2) and (PC)).
Hence, the ratio µi /pi (ā) is a constant for all i.
If µi = 0, then w2 (xi , x j ) is a constant for all j, which violates (IC2) for i.
Hence, µi > 0 should be satisfied for all i, which means that (IC2) is binding for
all i in the optimum. Therefore, w2 (xi , x j ) depends only on j. In particular, fromthe concavity of the utility function and the definition of j, w2 (xi , x j ) is increasingin j .
2. First, we check that (IC1) is satisfied for two deviation strategies, (a1 ; a2 ) =(a; ā, · · · , ā) and (a1 ; a2 ) = (a; a, · · · , a), under the optimal contract derived in step
1. Here, we write w1 (xi ) = w1 and w2 (xi , x j ) = w2j , as the contract is not dependent
on xi .
As shown in step 1, (IC2) is binding at the optimum. Therefore,NC= ∑p2j (ā, ā)u(w2j ) −j=1

N∑ p2j (ā, a)u(w2j ).

(4)

j=1

From Assumption 1 (i), we have
N

N

j=1

j=1

∑ p2j (ā, a)u(w2j ) ≥ ∑ p2j (a, ā)u(w2j ),(5)since u(w j ) is an increasing function of j. (5) together with (4) implies (IC1) witha′ = a and a′′ = ā. Similarly, from Assumption 1 (ii), we have
N

N

N

j=1

j=1

j=1

2 ∑ p2j (ā, a)u(w2j ) ≥ ∑ p2j (ā, ā)u(w2j ) + ∑ p2j (a, a)u(w2j ),which implies (IC1) with a′ = a′′ = a.
Finally, we check that (IC1) is satisfied for any deviation strategies, (a1 ; a2 ) =(a; a21 , · · · , a2N ). Suppose the agent undertakes a2i = ā if i ∈ I¯ ⊂ {1, · · · , N} and¯ The intertemporal payoff to the agent following
a2i = a if i ∈ I = {1, · · · , N} \ I.
this deviation strategy satisfies[
u(w1 ) + ∑ p1i (a)
i∈I¯

]

N

[

N

]

∑ p2j (a, ā)u(w2j ) −C + ∑ p1i (a) ∑ p2j (a, a)u(w2j )
i∈I

j=1

13

j=1

{
≤ u(w1 ) + max

N

N

j=1

j=1

}

∑ p2j (a, ā)u(w2j ) −C, ∑ p2j (a, a)u(w2j )= max {U(a; ā, · · · , ā; w), U(a; a, · · · , a; w)}≤ U(ā; ā, · · · , ā; w),
where the last inequality is derived from the previous result that (IC1) is satisfied
both for (a1 ; a2 ) = (a; ā, · · · , ā) and for (a1 ; a2 ) = (a; a, · · · , a). Hence, (IC1) issatisfied for any deviation strategy (a1 ; a2 ) = (a; a21 , · · · , a2N ).
The intuition behind the proof is as follows. For the principal who intends to
induce the agent to exert the positive effort ā in period 2, it is necessary to makethe second-period payment w2 (xi , x j ) dependent on the second-period outcome x j ,as this is the only source of incentive power available. However, such a payment
schedule induces the agent to work hard in period 1 because the distribution of
second-period outcomes is affected not only by a2 but also by a1 . Assumption 1(i) ensures that the agent always obtains a larger gross expected payoff in termsof wages by undertaking action profile (ā, a) than by undertaking (a, ā) as a resultof the FOSD. In addition, as the cost of effort, C, is the same in both periods,
the agent obtains a larger net expected payoff as well. Thus, if the contract is toinduce hard work by the agent in the second period, it automatically provides theagent with the incentive to work hard in the first period. Assumption 1 (ii), on
the other hand, ensures that the agent does not deviate to a strategy of shirking
in both periods (i.e., to (a, a)). Half-and-half mixture of the two probability distributions, (ā, ā) and (a, a), gives the agent’s gross expected payoff from taking(a, a) in accordance with the benefit of effort cost reduction normalized to C (aone-time shirk). Thus, if the contract is to induce hard work in the second period,
it automatically makes the agent worse off if he shirks in both periods, (a, a).
To summarize, if the probability distribution of the second-period outcome
when the agent shirks only in the second period (ā, a) FOS-dominates the distribution when the agent shirks in any other periods in such a way that the expected
number of shirkings is one, providing incentives to work hard in the second pe14

riod becomes sufficient to induce the agent to make strong efforts in both periods.
As we will see in Section 3, such arguments regarding FOSD and one-time shirking play central roles in T -period models as well. We also discuss the sufficientconditions for simple contracts for T -period models in a similar manner.

2.2

Non-MLRP CaseIn this subsection, we discuss how an analogue of Assumption 1 is described if
p2j (ā, a) and p2j (ā, a) do not satisfy MLRP.
For making the problem non-trivial, we suppose that
p2j (ā, ā)p2j (ā, a)̸=p2k (ā, ā)p2k (ā, a)

for all j ̸= k

is satisfied.13 Then we have a unique rearrangement (permutation) κ (1), . . . , κ (N)of 1, . . . , N so thatp2κ (1) (ā, ā)p2κ (1) (ā, a)< ··· <p2κ (N) (ā, ā)p2κ (N) (ā, a).

MLRP in the previous subsection is equivalent to (κ (1), . . . , κ (N)) = (1, . . . , N).
Then an analogue of Assumption 1 (i) is written as
j

∑

k=1

j

p2κ (k) (a, ā) ≥∑ p2κ (k)(ā, a) for all j = 1, . . . , N,

k=1

which implies, for any increasing function γ : {1, . . . , N} → R,
N

N

j=1

j=1

∑ pκ2 ( j)(ā, a)γ ( j) ≥ ∑ p2κ ( j)(a, ā)γ ( j).

The difference from the normal stochastic dominance cases is that the function γ
should be increasing in terms of the permutated numbers κ (1), . . . , κ (N).
Then (3) in the proof of Theorem 1 is rewritten as[]p2κ ( j) (ā, a)
1
µi
+ ν,
1− 2
=
u′ (w2 (xi , xκ ( j) )) pi (ā)
pκ ( j) (ā, ā)
13 Equalities can be allowed for a subset of {1, . . . , N}, but we avoid such cases for the sake of abrief exposition.

15

and, following the similar argument as in the proof, w2 (xi , xκ ( j) ) is increasing in j(and independent of i). Then (5) is rewritten as
N

N

j=1

j=1

∑ p2κ ( j)(ā, a)u(w2κ ( j)) ≥ ∑ p2κ ( j)(a, ā)u(w2κ ( j)),which is satisfied since u(w2κ ( j) ) is increasing in j and (ā, a) FOS-dominates (a, ā)in terms of κ ( j). Similar argument holds for Assumption 1 (ii) and its implicationsin the proof.
The following example depicts the role of κ ( j) in the case where MLRP is notsatisfied. Suppose N = 3 (e.g., a performance is either low, middle, or high), andthe distributions are given as in the table.
j12κ (1) = 2 κ (2) = 1 κ (3) = 3

3(ā, ā) 0.3 0.2 0.5

0.2

0.3

0.5

(ā, a) 0.3 0.4 0.3

0.4

0.3

0.3

0.4

0.4

0.2

(a, a) 0.3 0.6 0.1

0.6

0.3

0.1

LR

0.5

1.0

1.6

(a, ā) 0.4 0.4 0.2
1.0

0.5

⇒

1.6

The likelihood ratio (LR in the table) is not monotone in the original numbers
1, 2, 3, but we have a permutation κ (1), κ (2), κ (3) so that LR is monotone. It iseasy for the reader to check that (ā, a) FOS-dominates both (a, ā) and 21 ((ā, ā) +(a, a)) (in terms of κ (1), . . . , κ (N)), and hence, only the final outcome matters in
the optimal contract.

3 Extension
In this section, we extend the basic model to a T -period setup and show that asimilar result as in Theorem 1 are obtained. As in Section 2, we let at ∈ {ā, a} = A
and xt ∈ {x1 , . . . , xN } = X denote the agent’s actions and outcomes in each periodt = 1, · · · , T , respectively. We also let at = (a1 , . . . , at ) and xt = (x1 , . . . , xt ). The

16

distribution of each outcome xt is dependent on the whole past history of actions
at . We write the distributions as follows:[]pti (at ) = Pr xt = xi | at ,i = 1, · · · , N.

Throughout this section, we assume the monotone likelihood ratio property(MLRP) for the final period outcome:pT1 (ā, . . . , ā, ā)pTN (ā, . . . , ā, ā)
<
·
·
·
<
.
pT1 (ā, . . . , ā, a)pTN (ā, . . . , ā, a)This is just for a simplification as in Subsection 2.2. If the model does not satisfy
MLRP, we can rearrange the order of outcomes so that the sufficient condition forthe simple contracts as given below is written in a consistent way (see Subsection 2.3). It should be noted that MLRP is imposed only on the distributions by(ā, . . . , ā, ā) and (ā, . . . , ā, a).
We also assume that the agent decides his entire action profile aT = (a1 , · · · , aT )at the beginning of period 1 and that he never changes this profile after observing
the outcomes in each period.14 We split the agent’s action space AT = A × · · · × A
into partitions (A0 , . . . , AT ), where{{}}
Ak = (a1 , . . . , aT ) | # t | at = a = k ,k = 0, . . . , T.

That is, Ak is the set of the action profile aT in which there are k weak effortsa (and hence, T − k strong efforts ā).15 For instance, if T = 3, then we have
A0 = {(ā, ā, ā)}, A1 = {(ā, ā, a), (ā, a, ā), (a, ā, ā)}, etc.
Let V (a, w) denote the agent’s gross expected payoff from payment schedule
w when he takes action profile a. Then, the agent’s net expected payoff is written
14 In the basic model presented in Section 2, it is assumed that the agent makes his second-

period action a2 after observing the first-period outcome x1 ; therefore, the action profile consistsof N + 1 components (a1 ; a21 , . . . , a2N ), where a2i denotes the second-period action when the firstperiod outcome is xi . For the T -period model in this section, we may also consider the possibility
that the agent’s actions depend on past outcomes (the action profile in such a model consists of(N T − 1)/(N − 1) components), but such a consideration does not change the result in Theorem 2.
See the Appendix for more on this point.
∪
15 It is obvious that (A , . . . , A ) satisfies A ∩A = 0/ for any k, l (k ̸= l), and k Ak = A. Hence,T0kl(A0 , · · · , AT ) is a partition of A.

17as
V (a, w) −C · m(a),where C is the cost of a strong effort ā (as in Section 2) and m(a) is the numberof strong efforts in action profile a. Then, the incentive compatibility constraint in
the T -period model is simplified as follows: for k = 1, . . . , T ,V (ā, . . . , ā, w) −C · k ≥ V (a′ , w),

for all a′ ∈ Ak .

(ICk )

Now, we have an extension of Assumption 1.
Assumption 2. For all k = 1, . . . , T and all a′ ∈ Ak ,]j [j∑i=1 (k − 1)pTi (ā, . . . , ā, ā) + pTi (a′ )≥ ∑ pTi (ā, . . . , ā, a)
k
i=1

(6)

holds for all j = 1, . . . , N.
The following is an important example of Assumption 2.
Example 3 (Human capital investment). Suppose that the distributions of outcomes in each period are dependent not on the detail of past actions, but on thenumber of strong efforts that the agent has taken to date. To be specific, we letqi (k) denote the probability distribution when the agent has undertaken k strong
efforts, and provide pti (·) aspti (a1 , . . . , at ) = qi (#{τ ≤ t | aτ = ā}).
Such distributions depict the agent’s human capital investment or the learning-bydoing effect of the agent’s effort.
In this example, the requirement of Assumption 2 is the following: For all
k = 2, · · · , T , the inequalityjj∑i=1 [(k − 1)qi (T ) + qi (T − k)]≥ ∑ qi (T − 1)
k
i=1

18

holds for all j = 1, . . . , N. This condition seems rather artificial, but it includes
an important economic environment in which the investment has nonincreasing
marginal returns:jj∑ [qi(k + 1) − qi(k)] ≥ ∑ [qi(k) − qi(k − 1)] .

i=1

(7)

i=1

This inequality states that the marginal “benefit” in the probability distribution of
one additional effort is decreasing in k. It is easy to show that (7) is a special caseof condition (ii).16
Theorem 2. Suppose that pTi (·) satisfies Assumption 2. Then, payments in the
optimal long-term contract are dependent only on the final outcome xT .
Remark. Assumption 2 is imposed only on the distribution of outcomes in the
final period (t = T ), and how the efforts of the agent affect the outcomes in otherperiods (t = 1, . . . , T − 1) is irrelevant to the result of Theorem 2. We should
discuss the reason briefly.
As long as the principal wants to make the agent work hard in the final period(t = T ), the optimal wage should depend (at least) on the final outcome, xT . If theeconomic environment satisfies Assumption 2, then the agent works hard in otherperiods (t = 1, . . . , T − 1) under the wage scheme that gives enough incentives towork hard in the final period. As long as the principal finds it optimal to implement
high efforts (as usually assumed in studies on moral hazard, including the present
16 Kwon (2006) investigated a similar model with binary outcomes (N = 2) and showed thatthe optimal long-term contract is dependent only on the final outcome under the assumption of“nonincreasing marginal returns.” Although nonincreasing marginal returns is an easy assumption
to interpret economically, it is a rather strong assumption if T is a substantially large number, as it
requires that the inequality (7) be satisfied for all possible numbers of high efforts, k = 1, . . . , T −1.
On the other hand, consider the following distributions when T = 3 and N = 2:
q1 (0) = 0.9,

q1 (1) = 0.8,

q1 (2) = 0.2,

q1 (3) = 0.1,

(q1 (·) represents the probability of “failure” when N = 2). This is not “nonincreasing returns,” butit satisfies our FOSD condition. Our result suggests that what is central to the incentives in simple
contracts is the FOSD relationship, and nonincreasing marginal returns is just one example of thecondition. In addition, note that the two conditions coincide if (and only if) T = 2.

19one), other distributions are irrelevant in the principal’s cost minimization problem
and only the final distribution is central to the contract in which only the final
outcome matters.
We should also discuss that Assumption 2 is close to a “necessary” conditionin the following informal manner. Suppose that k = 1 and only a′ = (a, ā, . . . , ā) ∈A1 violates (6). If the contract gives enough incentive to work hard in the final
period (and the IC in the final period is binding), the agent finds it optimal to work
hard in periods t = 2, . . . , T , but may not in the first period.17 If the utility functionof the agent is such that the incentive to work hard in the first period is not provided
by the simple contract, the principal needs to rewrite the contract. However, theprecise form of the optimal contract is complicated in dynamic models, so wecannot argue whether or not only the final outcome matters in the optimal contract
in the way using the FOSD condition as shown in the present paper.
Sketch of the proof. As in the proof of Theorem 1, we can show that the optimal
long-term contract is independent of outcome history xT −1 up to the period T − 1,
if all incentive constraints are not binding with the exception ofV (ā, . . . , ā, ā, w) −C ≥ V (ā, . . . , ā, a, w)(8)

In the following, we show that the derived contract, which is dependent only on
the final outcome xT , satisfies all incentive compatibility constraints.
As the derived contract w∗ satisfies (8) with equality, we have
C = V (ā, . . . , ā, ā, w∗ ) −V (ā, . . . , ā, a, w∗ ).

(9)

For each k = 1, . . . , T , substituting (9) into (ICk ) yields:k ·V (ā, . . . , ā, a, w∗ ) ≥ (k − 1) ·V (ā, . . . , ā, ā, w∗ ) +V (a′ , w∗ ).
Condition (6) ensures that this inequality holds.
17 FOSD between random variables X and Y is a sufficient condition for that E[u(X)] ≥ E[u(Y )]holds for an exogenously given increasing function u(·), but not a necessary one (Shaked andShanthikumar (2007)).

20

4Concluding Remarks

This paper explores the role of history dependence in a dynamic moral hazard
model. It is shown that, under certain conditions on the probability distributionsof outcomes, the optimal long-term contract is such that the payment schedules
are not contingent upon the realization of past outcomes. This finding contrasts
strikingly with the results in repeated moral hazard models, where the optimal
long-term contracts are generally dependent on the complete history of past outcomes.
An important point of the results is the relationship between statistical informativeness of the signals (outcomes) and its effects on incentive problems in the
optimal contracts. In the light of statistical inference of agent’s past efforts, thehistory of all outcomes must be valuable to the principal (i.e., informative), because the distributions of outcomes are assumed to be independent over time in
the present analysis (Apart from the information provided by x2 about a1 , x1 provides a statistically independent information about a1 ). The principal, however,sometimes finds it optimal to ignore such x1 in the present paper. We believe theresult that the principal finds it optimal to “ignore” informative signals is found inother (non-trivial) contract theory models, but we leave this for future research.
This paper demonstrated a particular class of dynamic moral hazard models inwhich the optimal contract is written in a simple manner. This extra conclusion
brings some new insights to the incentive provision in moral hazard problems,
but does not extend to general models on which we pose no conditions on theinformation structure. Perhaps more important question is to explore the general
structure of dynamic incentives in which efforts have persistent effects over time.
The general theoretical examination of these problems remains to be done.

21Appendix
In this Appendix, we provide some mathematical arguments for footnote 14 inSection 3.
Suppose that the agent is to undertake actions dependent on past outcomes.
We think of such an agent’s strategy as a sequence of “behavior strategy”; forexample,α = (α 1 , α 2 (x1 ), α 3 (x1 , x2 ), . . . , α T (x1 , . . . , xT −1 )),where each α t : {1, . . . , N}t−1 → A is a mapping from the history of past outcomes(up to period t − 1) to the action in period t.
The problem in footnote 14 is whether the agent improves his payoff by selecting such a history-dependent strategy α (rather than a history-independent
strategy a = (a1 , a2 , . . . , aT )).
Theorem 3. If the contract is simple, then the agent cannot improve his payoff by
selecting a history-dependent strategy α .
Proof. The expected payoff to the agent undertaking such a strategy α is written
as




T −1
Pr[xt | α t (x1 , . . . , xt−1 )]
∑T  1 T∑∏a∈A
(x ,...,x −1 )∈I(a,α ) t=1
(
)}
×

T −1

N

t=1

j=1

∑ u(wt ) + ∑ pTj (a)u(wTj ) −C · m(a)

(10)

where
{
}
I(a, α ) = (x1 , . . . , xT −1 ) | α t (x1 , . . . , xt−1 ) = at for t = 1, . . . , T ;that is, I(a, α ) is the set of historical outcomes with the positive probability that
action profile a is played under strategy α .
Because every history (x1 , . . . , xT −1 ) generates exactly one action profile, agiven α , the first parentheses in (10) is seen as a probability distribution of a over22

AT ; that is,∑T  1a∈A(x ,...,x

T −1∏ Pr[xt | α t (x1, . . . , xt−1)] = 1 for any α .

∑T −1)∈I(a,α ) t=1

As the expected value of random variables does not exceed the maximum of thevariables, we establish that((10) ≤ max
a∈AT

T −1

N

t=1

j=1

)

∑ u(wt ) + ∑ pTj (a)u(wTj ) −C · m(a)= max (V (a, w) −C · m(a)) .
a∈AT

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