DP2018-05

Combinations of Different Length
Contracts in a Multiperiod Model:
Short, Medium and Long-term
Contracts
Meg ADACHI-SATO
Kazuya KA MIYAMarch 14, 2018

Combinations of Different Length Contracts in a Multiperiod
Model: Short, Medium and Long-term Contracts
Meg Adachi-Sato ∗Royal Melbourne Institute of Technology University
Melbourne VIC 3000, Australia.
email: meg.sato at rmit.edu.au
Kazuya Kamiya
RIEB, Kobe University
2-1 Rokkodai-Machi, Nada, Kobe, Hyogo 657-8501, Japan.
email: kkamiya@rieb.kobe-u.ac.jp

Abstract
This paper develops a dynamic contracting model with verifiable and unverifiable outputs.
We prove the following properties of equilibrium wage contracts, which are new to the literature: (i) combinations of different length contracts can become equilibria, (ii) medium-term
contracts can be included in the combinations, and (iii) equilibrium wage profile differs by
the way different length contracts are combined. We also investigate a general mechanism,
which includes menu and option contracts, and show that no mechanism can perform better
than simple wage contracts in our environment. In short, above properties remain valid undergeneral mechanisms.
Keywords: Differing Length Contracts; Unverifiable Outputs; Unverifiable Investments;
Unverifiable Ability; Holdup Problems
JEL Codes: D86; J41; J31∗

This research is financially supported by the Japan Center for Economic Research.

1

1Introduction

This paper is the first to study how a multi-period wage path is designed as a combination of different length (short, medium, and long) contracts. Rey and Salanié (1990) andFudenberg, Holmstrom, and Milgrom (1990) have studied how a long-term contract could
be implemented by a combination of short-term contracts, but there has not been any
research that studies a combination of different length contract on the equilibrium. Moreover, medium-term contracts, e.g., a two-period contract when the principal and agent
lives for five periods, have never been discussed in the literature, though such contracts
are often observed in the real world. We first consider simple wage contracts and explore
why various combinations of wage profiles exist and explain the optimal length of eachwage when the agent must produce both verifiable and unverifiable outputs. We theninvestigate a general mechanism which includes menu and option contracts and show thatno mechanism can perform better than simple wage contracts in our environment. Thus
the results with simple wage contracts remain valid, even if we consider general mechanisms. Here we define a simple wage contract in which the initial wage agreement remains
unchanged for all periods as a long-term contract, whereas the agreement is only valid fora single period as a short-term contract, and for anything between them as a medium-term
contract.1 As we adopt finite horizon models, self-enforced relational contracts, such as
Levin (2003), do not work in our environment.
The overview of our model is as follows. There is a principal and an agent, and bothare risk neutral. The agent undertakes two types of investment (efforts) to accumulate thehuman capital necessary to produce the two types of output, x and y . The first type of
output x is observable and verifiable (contractible) whereas the second type of output y isobservable but unverifiable (noncontractible). Examples of x are the annual profits of thefirm or the amount of sales a salesperson makes. Examples of y are the extent to which
an employee has contributed to the work of a team, or the leadership of a high-ranking
1This is because the purpose of this analysis is to prove theoretically that an optimal wage contract can be a
combination of contracts of different lengths. As in Fudenberg, Holmstrom, and Milgrom (1990), the agent is notdismissed on the equilibrium path, even in the case of an agent repeating a number of short-term contracts.

2

employee. These investments are denoted Ic and In , and while both are observable, they
are unverifiable. When the agent makes an investment Ic , he obtains the skill to produce
x. The principal can then write a wage that depends on x. When the agent makes aninvestment In , he obtains the skill needed to produce y. Because y is unverifiable, thewage cannot reflect y. We assume that the investment in the current period becomes
effective in the next period. Both investments are made in each period after the outcomeof each period is realized.2
In this environment, suppose there are three periods, where the possible combinationsof contracts the principal can offer are as follows: a) a long-term contract in which thewages for all three periods are determined by the principal and offered to the agent at the
beginning of the first period; b) short-term contracts for all three periods in which theprincipal and the agent determine the wage at the beginning of each period; c) a shortterm contract for the wage in the first period and a medium-term contract that binds
both second and third periods; and d) a medium-term contract in which the wages for the
first and second periods are offered at the beginning of the first period and a short-term
contract in which the wage for the third period is agreed to at the beginning of the third
period.3
We show that the choice between a), b), c), and d) is made based on the endogenously
determined efficiency of investment and the relative value of the verifiable and unverifiable
outputs. That is, if the human capital useful in producing (un)verifiable outputs does
not accumulate much despite the agent’s effort, the investment in human capital can beconsidered as inefficient. If the human capital accumulates with a small amount of effort,
the investment in human capital is then said to be efficient. We show that if investment
efficiency remains fairly similar in all periods4 , the principal chooses either a) or b) (seeSato and Kamiya (2013) for details).
2

As the wage for the first period is determined prior to the investment in human capital, and as the investmentin the first period becomes effective in the second period, the first-period wage does not affect the agent’s behavior(investment decision). In other words, the first-period wage is irrelevant for the choice of the agent’s investment
and hence we focus on the wages in the second period onward.
3
We define a medium-term contract as anything between a one-period contract and the entire-period contract.
4
One example that satisfies this condition is that if it is (in)efficient in the first period, it is (in)efficient in theremaining periods.

3

If investment efficiency changes between periods, the principal chooses c) or d).5 Indeed, this is the main finding of this paper which is new to the existing literature. We
show that c) is chosen when the investment In becomes inefficient after the second period,in turn making the investment Ic relatively efficient. In reality, if a worker on probation
has invested time and effort in learning about the match between himself and his positionin the company, he does not need to spend as much time and effort in learning this same
information when he becomes a full-time employee. Even though such human capital isessential in conducting the position, the investment in In actually becomes inefficient if
he keeps doing that forever. On the other hand, d) is chosen when the investment Ic
becomes inefficient after the second period, in turn making the investment In relatively
efficient. This can explain the case of salespersons becoming middle managers. Obviously,
he will need some field experience in becoming a successful middle manager, but this field
experience for him is mostly verifiable output: sales. That is, Ic was important in thepast. However, after becoming a middle manager and receiving mainly fixed salary, heobviously needs some knowledge in leading his team or thinking about strategic plans forsales in the long run, as denoted by In .
In an n-period model, we can obtain a more complicated combination of contracts as anequilibrium, such as repeating medium-term contracts. We show that such a combination
contract is offered when the agent’s human capital depreciates. Suppose that a skill needed
to produce y depreciates at some given depreciation rate. In this case, the principal wishes
the agent to make occasional efforts In to maintain the skills needed to produce y at somecertain level. To do so, the principal repeatedly offers medium-term contracts. If theprincipal and the agent bargain over wages every two periods, the agent is given an
incentive to invest in In which compensates for any depreciation in human capital.
The structure of the remainder of this paper is as follows. Section 2 reviews the relevant
literature. Section 3 analyzes a three-period model with simple wage contracts in which
we do not impose limited liability constraints. We devote Section 4 to the five-period
5

These two cases can be considered in relation to the career concerns model. We discuss this at the beginningof Section 2.

4

case. In Section 5, we discuss limited liability constraints as an extension, and showthat we obtain nearly the same results as in the preceding section. Section 6 analyzes
a general mechanism that includes menu contracts and option contracts, and show thatany mechanism cannot perform better than simple wage contracts. Section 7 discuss theliterature, such as short-term contracts in Holmstrom and Milgrom (1990) and Rey andSalanié (1990) and the self-enforced relational contract in Levin (2003), and Section 8
presents our conclusion.

2

Three-period Model

2.1

Model

In this section, we assume that both the principal and the agent live for three periods,
and show that several interesting combinations of contracts are chosen depending on theparameters. For example, the principal chooses to contract the wage for the first period
in a short-term contract, and contract the wages for both the second and third periods ina medium-term contract at the beginning of the second period. Another example is thatthe principal may contract the wages for both the first and second periods in a mediumterm contract at the beginning of the first period, and contract the third-period wage ina short-term contract at the beginning of the third period. We do not impose limited
liability constraints in Section 3, but discuss limited liability constraints in Section 4 andsuggest that almost the same results can be obtained.
There is a principal and an agent. For simplicity, we assume both are risk neutral.6
There are two types of outputs: an observable and contractible output x and an observable
but noncontractible output y. The two contractible output levels are xH and xL , where
xH > xL > 0. The probabilities of xH and xL are denoted by P H ∈ [0, 1] and P L = 1−P H .
The two noncontractible output levels are θy H and θy L , where y H > y L > 0.7 Note thatθ ≥ 0 is a parameter introduced for later use. The probabilities of y H and y L are denoted
6

Even if we assume that the agent is risk averse, similar results can be obtained, since they are derived fromthe relative efficiency of the investments.
7
x must be stochastic, since the wage should depend on x in order to induce Ic . However, even when y is notstochastic, the same result can be obtained, since the wage cannot depend on y.

5

by QH ∈ [0, 1] and QL = 1 − QH . As will be formally stated below, for simplicity, we
assume that the random variables x and y are stochastically independent. Note thateven when x and y are correlated, our results remain almost the same. That is, they arederived from the relative efficiency of the investments on the skills needed to produce x
and y which exists even in the case that x and y are correlated.
To investigate the three-period model, we introduce human capital (the skills needed
to produce outputs), αc and αn , and investments, Ic and In . We assume that investment (or effort) accumulates the human capital. Let P H (αc ) ∈ [0, 1] be the probability
that xH occurs when a skill corresponding to a contractible output is αc ∈ [0, ∞). Let
P L (αc ) = 1 − P H (αc ). Let QH (αn ) ∈ [0, 1] be the probability that y H occurs when theskill corresponding to a noncontractible output is αn ∈ [0, ∞). Let QL (αn ) = 1 − QH (αn ).
Let fc : IR2+ → IR+ and fn : IR2+ → IR+ be the transition function of human capital.
That is, for i = c, n, αi0 = fi (Ii , αi ) means that when the skill in the current period is αiand the investment is Ii , the skill in the next period, denoted by αi0 , is fi (Ii , αi ).8 The
investments and human capital in period t are denoted It = (Ict , Int ) and αt = (αct , αnt ).
P
For a given parameter θ ≥ 0, g(αn , θ) = i=H,L Qi (αn )θy i denotes the expected value of
noncontractible output. We assume that the two types of human capital, αc and αn , andthe investments, Ic and In , are observable.
We assume the agent incurs disutility in undertaking investment, denoted by Dc (Ic )and Dn (In ). Let δ ∈ (0, 1) be the discount factor. The payment for each period is at
the end of each period following the realization of output. The wage depends on therealization of x only, as x is the only verifiable output. The wage wi , i = H, L, in period
t is denoted by wti , t = 1, 2, 3. Note that, because of the assumed risk neutrality, w2i and
w3i need not depend on the realization of an output in previous periods. (See Lemma 1 in
Subsection 2.3.)
Throughout this section, we make the following three assumptions. The assumptions
on Dc , Dn , P H , and QH are standard.
8

Another way to define the transition function is that it is a function of P (·) and Ic in the current period to
P (·) in the next period.

6

Assumption 1

2

i
1. dD
> 0, ddID2i > 0, Di (0) = 0, and
dIi
i

2.

dP H
dαc
H

> 0 andd2 P Hdα2c2d2 Di (0)= 0, i = c, n.
dIi2

< 0.

H

3. dQ
> 0 and ddαQ2 < 0.
dαnn4. The random variables x and y are stochastically independent.
Assumption 2 In the first period, the two types of human capital, αc and αn , are zero,and P H (0) = QH (0) = 0.
Assumption 3 The principal posts a take-it-or-leave-it offer for the length and wages ofa contract with the agent’s reservation utility u ≥ 0 for a contract signed at the beginningof the first period (hereafter, period 1). The principal and the agent Nash bargain over
the wages with the threat point (0, 0) for a contract signed at the beginning of the secondor third period (hereafter, periods 2 and 3, respectively).
We set Assumption 2 only to simplify the analyses. By this assumption, the principal only
has to offer w1L in the first period. Note that even if we allow for P H (0) > 0 or QH (0) > 0,
the following analyses do not change much because the investments in the first period
do not affect P H and QH in the first period but affect P H and QH in the second period
onward. That is, even if xH is realized with a positive probability and the principal offers
w1H , investment levels are unaffected.
Below, we explain Assumption 3. For the bargaining process, we suppose the market
for workers without firm-specific skills is competitive. We also assume that the agent
obtains some firm-specific skills in the first period without any particular investment (thatis, working in the firm without much effort can provide some experience, and the agent
acquires some level of firm-specific skills through this experience) and hence he obtains
bargaining power to negotiate the wage at the beginning of the second and third periods.99Alternatively, we assume that the agent who undertook investment in the first or second period obtains
bargaining power. We could assume that bargaining power is only given to the agent with αc > 0 or αn > 0.
However, we can obtain the same result even when we assume that the agent obtains bargaining power throughexperience and without undertaking any particular investment.

7

Therefore, when the principal hires an agent without firm-specific skills, she posts a take-itor-leave-it wage offer. Note that we obtain similar results even if we assume the agent hascertain bargaining power at the beginning of the first period, and hence Nash bargaining
is used for the negotiation process. We emphasize that the change in bargaining power
among different periods is not critical in obtaining our results.
After the agent has obtained skills, the principal and the agent might negotiate thewage at the beginning of the second and third periods. For simplicity, we use Nashbargaining with the threat point set at (0, 0). That is, we assume that the principal andthe agent have the same bargaining power and that they cannot find a new partner if theylose the current partner. That is, they can access the labor market only once and theirreservation utilities are zero. The case of nonzero reservation utilities is also important.
Indeed, if the agent can re-enter the job market or the principal can hire a new agent,
the threat point is nonzero. It is worthwhile noting that we obtain similar results even
if they have different bargaining power or their reservation utilities are nonzero in thesecond and third periods (see Subsection 2.9). We also note that in the discussion ofrenegotiation-proofness in the following theorems, we consider Nash bargaining games inwhich the status quo is the wage contract signed in the previous periods.
2.2

TimingAt the beginning of period 1, the parties sign a contract. The contract is either a short-,
medium-, or long-term contract. After the outputs are realized the agent makes investments, Ic and In , and the wage agreed in the contract is paid. The parties can renegotiate
the contract if they had agreed on medium or long-term contracts.
At the beginning of period 2, the parties sign either a short- or medium-term contract
if the contract in period 1 was a short-term contract. Then, the same sequence of eventsas in period 1 occurs.
At the beginning of period 3, the parties sign a contract (only a short-term contract
could be the case) when the contract in period 1 was either a medium-term contract or if
the parties have been repeating short-term contracts. At the end of period 3, the outputs
8are realized and the wage agreed in the contract is paid. Note that there is no incentive toinvest in period 3 as investment in the current period only becomes effective in the next
period.
As described in Assumption 3, at the beginning of period 1, the principal posts a takeit-or-leave-it offer for the length and wages of a contract maximizing her discounted sum of
expected utility subject to the agent’s individual rationality and incentive compatibility
constraints for investment. If the contract in period 1 is short term, at the beginningof period 2, the parties bargain over the length and wages of the contract, maximizing
the Nash product of their discounted sum of utilities in periods 2 and 3, subject to the
incentive compatibility constraint on investment. If the contract in period 2 is short term,or that in period 1 is medium term, the parties bargain over the wages of the contract atthe beginning of period 3, maximizing the Nash product of their utilities in period 3.
At the end of periods 1 and 2, the parties can renegotiate contracts and choose a newcontract if both become better off by doing so. For example, suppose the parties sign
a long-term contract at the beginning of period 1, which is before the first investment
decision is made. In this case, it might be better to change the contract at the end of
period 1, as the agent has already chosen investments Ic and In in period 1, and a newcontract for the second- and third-period wages can lead to a Pareto improvement.
2.3

Equilibria

We first prove that w2i and w3i need not depend on the realization of an output in previous
periods. If the agent has a strictly concave (risk-averse) utility, she prefers intertemporal consumption smoothing, whereas the risk-neutral principal needs not smooth theconsumption stream. In this case, the principal is better off offering wages with a small
variance, which depend on the realization of outputs in previous periods. However, in
this paper the risk-neutral agent does not require consumption smoothing, and hence,the principal does not have to offer wages that depend on the realization of an output inprevious periods.

9

Lemma 1 Even if the principal offers a contract in which w2i or w3i depend on the realiza-

tions of an output in previous periods, she cannot obtain more utility than a contract inwhich the wages do not depend on the past outputs.
Proof: See Appendix B.
We adopt the dynamic programming approach. We first present a rough sketch of theapproach. At the beginning of period one, the principal posts a take-it-or-leave-it wage
offer. She has three options. Contract only the wage in period one (a short-term contract),
contract the wages in periods one and two (a medium-term contract), or contract thewages for all three periods (a long-term contract). She chooses the one that maximizes
the discounted sum of her expected profit.
Long-term contract in this paper is quite standard. Similar to Rey and Salanie (1990)
or Fudenberg, Holmstrom, and Milgrom (1990), at the beginning of the first period, theprincipal offers wages in all periods maximizing the discounted sum of expected profit subject to the individual rationality constraint (IR) and the incentive compatibility constraint(IC) on investments, Iij , i = c, n, j = 1, 2.
In short-term contract, the principal offers a first period wage maximizing(the profit in period 1)+ (the discounted value obtained in the bargaining in period 2)
subject to the IR constraint and the IC constraint on Ii1 , i = c, n.10
Medium-term contract is what we develop in this paper and is new to the existing
literature. Suppose, the principal offers the medium-term contract at the beginning of thefirst stage. Then she offers a first and a second period wages maximizing(the discounted sum of profits in periods 1 and 2)+ (the discounted value obtained in the bargaining in period 3),
subject to the IR constraint and the IC constraint on Iij , i = c, n, j = 1, 2.
Given the above, there are two choices of contracts at the beginning of period two if
the wages for the second and third periods are not yet determined at the beginning of thesecond period. In this case, the principal and agent can either contract just second period
10Note that the (IR) constraint is included in the Nash product,i.e., the rservation utility is the threat point.

10

wage or contract the second and the third period wages at the same time. The former
is the short-term contract, and the latter is the medium-term contract. They chose thecontract that maximizes the Nash product.
The principal and the agent can bargain over the wage for period three if the wages
for the third period is not yet determined by the beginning of the third period. This is astandard Nash bargaining problem.
Below, we formally describe the principal and agent behaviors. Let the values of theprincipal and the agent (that is, the discounted sums of expected utilities when a contract
is optimally chosen) at the beginning of period t = 1, 2, 3 be Vtp (α) and Vta (α), where
α = (αc , αn ) is the human capital at the beginning of period t. That is, Vtp (α) and Vta (α)
are the principal and the agent values of α when the wages from period t onward are notyet determined at the beginning of period t. Vtp (α) and Vta (α) satisfy the following.
At the beginning of the first period, the principal posts a take-it-or-leave-it wage offer.
She has three options to chose from: one-period contract (a short-term contract), twoperiod contract (a medium-term contract), or three-period contract (a long-term contract).
Hence,
V1p (0, 0) = max{V1p3 (0, 0), V1p2 (0, 0), V1p1 (0, 0)}(1)

holds, where V1p3 (0, 0), V1p2 (0, 0), and V1p1 (0, 0) are the values in the cases of the long(three-period), medium- (two-period), and short-term (one-period) contract, respectively.
Namely, V1pk (0, 0) is the discounted sum of expected utilities when a k-period contract ischosen. Note that α = (0, 0) at the beginning of period 1. V1pk (0, 0), the principal’s value

11

of k-period contract, k = 1, 2, 3, satisfies the following:
"
#
k
X
X
p
pk
j−1
i
i
(αk+1(2))δ
P (αcj )(x − wj ) + g(αnj , θ) + δVk+1
V1 (0, 0) =max
w1 ,...,wk ,I1 ,...,Ik

s.t.

k
X

δ j−1

≥

j=1

P (αcj )wji −

X

i=H,L

i=c,n

X

X

a
Di (Iij ) + δVk+1(αk+1 ) ≥ u,

"
δ j−1

j=1
k
X

i=H,L

#
X

j=1
k
X

j=1

"

#
P (αcj )wji −

i=H,L

i=c,n

X

X

a
Di (Iij ) + δVk+1(αk+1 )

(4)

#

"
δ j−1

(3)

i=H,L

0
)wji −P (αcj0a), ∀ I10 , . . . , Ik0 ,(αk+1
Di (Iij0 ) + δVk+1

i=c,nwhere u is the reservation utility, wj = (wjH , wjL ), and Ij = (Icj , Inj ). Note that wji does
not depend on the realization of x in the previous periods as the agent is risk neutral andthere is no need for consumption smoothing (see Lemma 1). In the objective function, thefirst term of the right-hand side (RHS) is the discounted sum of the principal’s expectedutility in the k-period contract and the second term is the principal’s value of αk+1 =(αc,k+1 , αn,k+1 ), the discounted sum of expected utilities when a contract is optimallychosen in period k + 1. Note that αt (αt0 ) is derived from fc , fn and I1 , . . . , Ik (I10 , . . . , Ik0 )and that α1 = α10 = (0, 0). For example, αc2 = fc (Ic1 , 0). Expression (3) is the individual
rationality constraint and (3) is the incentive-compatibility constraint. In (3), the firstterm is the discounted sum of the agent’s expected utility in the k-period contract andthe second term is the agent’s value of αk+1 . Note that V4p (α4 ) = V4a (α4 ) = 0.
Suppose that the levels of human capital at the beginning of the second period are
α = (αc , αn ). If the wages for the second and third periods are not yet determined at the
beginning of the second period, the principal and the agent have two options: contract
one period (short-term) or contract two periods (medium-term). Note that w3 does notdepend on the realization of outputs in period 2 as the agent is risk neutral and there isno need for consumption smoothing (see Lemma 1). The Nash bargaining problem for thek-period contract, k = 1, 2, is expressed as follows:

12

k+1
X

max

w2 ,...,wk+1 ,I2 ,...,Ik+1

!
δ

j−2

j=2

×

X 

p
(αk+2 )P i (αcj )(xi − wji ) + g(αnj , θ) + δVk+2
i=H,L

k+1
X

"
δ j−2

j=2

#
X

P i (αcj )wji −Pk+1 j−2
j=2 δ

!

a
Di (Iij ) + δVk+2(αk+2 ) .

i=c,n

i=H,L

"
s.t.

X

#

a
Di (Iij ) + δVk+2(αk+2 )
i=c,n
i=H,L
#
"
P
Pk+1 j−2 P i 000a,) for all I20 , . . . , Ik+1
(αk+2
Di (Iij0 ) + δVk+2
P (αcj )wji −≥ j=2 δP

P i (αcj )wji −

i=H,L

P

i=c,n

where (αc2 , αn2 ) = (αc , αn ) and α3 (α30 ) is derived from fc , fn , and I2 (I20 ). In the objectivefunction, the term in the first (second) parentheses is the discounted sum of the principal’s(agent’s) expected utility, and the constraint is the incentive compatibility condition.
Note that the individual rationality constraint is included in the Nash bargaining withthe reservation utilities (threat point) (0, 0). Note that V4p (α4 ) = V4a (α4 ) = 0. Then,
In3 = 0 is chosen. The values of the principal and the agent (the utilities obtained fromthe bargaining) are expressed as V2pk (α) and V2ak (α). We obtain V2pk (α) = V2ak (α). It isclear that
V2p (α) = max{V2p2 (α), V2p1 (α)}.

(5)

As stated in the above, V2p1 (α) = V2a1 (α) and V2p2 (α) = V2a2 (α) hold. Thus, if V2p (α) =V2p1 (α), then V2a (α) = V2a1 (α), and otherwise V2a (α) = V2a2 (α).
Suppose that the levels of human capital at the beginning of the third period are
α = (αc , αn ). If the wage for the third period is not yet determined at the beginning of
the third period, the contracting problem (Nash bargaining) is expressed as follows:
!
!
X
X
i
i
i
i
i
(6)
max
P (αc )(x − w3 ) + g(αn , θ)P (αc )w3 ,
w3

i=H,L

i=H,L

where the terms in the first and second parentheses are the utilities of the principal andthe agent in the third period, respectively. Note that there is no incentive-compatibility
13constraint, given that the agent has no incentive to invest in the third period. With risk
neutrality, each party obtains half of the total utility available. The values, which means
the utilities obtained from the bargaining, of the principal and the agent are expressed as
V3p (α) and V3a (α), respectively.
We adopt a pure strategy subgame-perfect equilibrium as a solution concept. By thestandard argument of backward induction, there exists a subgame-perfect equilibrium:
given α in period 3, V3p (α) and V3a (α) are obtained in the Nash bargaining problem inperiod 3, then given α in period 2, V2p (α) and V2a (α) are obtained in the Nash bargaining
problem in period 2 (see (5)). Finally, V1p (0, 0) and V1a (0, 0) are obtained through theprincipal–agent problem in period 1 (see (1), (2), (3)), and (4)). There are four types of
equilibria.
Definition 1pp3

1. A long-term equilibrium contract, where V1 (0, 0) = V1 (0, 0): thewages for all periods are determined at the beginning of the first period on theequilibrium path.
p

p1

p

2. A short–short–short-term equilibrium contract, where V1 (0, 0) = V1 (0, 0) and V2 (α) =V2p1 (α): the wages for each period are determined at the beginning of each periodon the equilibrium path.
p

p1

p

3. A short–medium-term equilibrium contract, where V1 (0, 0) = V1 (0, 0) and V2 (α) =V2p2 (α): if the wage for the first period is determined at the beginning of the first period, and the remaining wages are determined in the second period on the equilibrium
path.
p

p2

4. A medium–short-term equilibrium contract, where V1 (0, 0) = V1 (0, 0): the wages

for the first and second periods are determined at the beginning of the first period,
and the remaining wages are determined at the beginning of the third period, on theequilibrium path.
We next discuss the incentives for investing Ic and In . The choice of contract depends
on the relative importance of the verifiable and unverifiable outputs and the relative
14

efficiency of the investments in human capital made for each output. As the relative
efficiency endogenously varies over time according to human capital accumulation, various
types of combinations of contracts of different lengths are obtained as equilibria. If acontract that expands for multiperiod, such as a long- or medium-term contract, is chosen
the agent is sometimes deprived of an incentive to increase αn after signing the contract, as
the wages do not depend on the realization of y. For example, if a medium-term contract
is chosen at the beginning of the first period, the agent has no incentive to increase αn in
the second period. However, the agent has an incentive to increase αn in the third period,
as he can obtain half of the gain from the investment through the Nash bargaining process
at the beginning of the third period. The benefit of contracting wages for more than two
periods is that the principal can motivate the agent to undertake a greater amount of
Ic than she could under short-term contracts, as the contract can induce the first-bestlevel of Ic in the contracting periods (see Appendix A). Conversely, under the short-term
contract, the agent can obtain half of the total utility in the following period through Nashbargaining. Therefore, the agent has an incentive to make In , which is also beneficial forthe principal.
Moreover, the relative efficiency of investment endogenously varies over time according
to human capital accumulation. For example, suppose at the beginning of period 1, In isrelatively more efficient than Ic . In this case, the principal chooses a short-term contract
to induce In1 . If αn2 = In1 is sufficiently large and Ic becomes relatively more efficientthan In at the beginning of period 2, the parties choose a medium-term contract to induce
Ic2 . Another example is Ic is relatively more efficient than In and that the differencein efficiency is not very large at the beginning of period 1. In this case, the principal
chooses a medium-term contract to induce Ic1 , predicting that in period 2 αc2 = Ic1 will
be sufficiently large and In will become relatively more efficient than Ic , leading the parties
to choose a short-term contract to induce In2 at the beginning of period 3.
In sum, to induce Ic , either a long or a medium-term contract should be chosen. To
induce In , a short-term contract should be chosen. Moreover, as the relative investment
efficiency varies over time, various combinations of contracts of different lengths are ob15

tained as equilibria. In the following subsections, we present specifications of θ, fc , fn ,Dc , Dn , P H , and QH , where we obtain various types of contracts as equilibria and also
show when a long-term contract is better than a medium-term contract or vice versa.
2.4

Renegotiation

This subsection discusses renegotiation. We consider Nash bargaining games in which thestatus quo is the wage contract signed in the previous periods. The parties can renegotiate
the contract at the end of period 1 and/or period 2. If the wage in period 3 has alreadybeen signed by the end of period 2, the Nash bargaining problem (renegotiation) at theend of period 2 is expressed as follows:
!

!
max

X

w3H0 ,w3L0

P i (αc )(xi − w3i0 ) +

i=H,L

X

X

P i (αc )(xi − w3i )

i=H,L

i=H,L

P i (αc )w3i0 −X

P i (αc )w3i

i=H,L

(7)
where (w3H , w3L ) is the status quo wages. Note that renegotiation at the end of period 2
does not improve the status quo utilities, since Ic2 and In2 are already chosen.
If the wage in period 2 has already been signed by the end of period 1, the parties canrenegotiate the contract at the end of period 1. The renegotiation is similar to the contract
in period 2. That is, they choose a short-term contract or a medium-term contract at theend of period 1: in each contract they maximize the Nash product with the threat point(V2p (α), V2a (α)) and they choose the contract with a higher utility. If both the principal
and the agent are better off, they renegotiate the contract.
The definition of renegotiation-proofness is as follows:
Definition 2 An equilibrium contract is said to be renegotiation-proof if the parties choose

not to renegotiate the contract even when they can.

2.5

Long-Term Contract

If θ is sufficiently small, it is better to induce an incentive for Ic and thus a long-term
16,

contract is chosen. Note that this holds for any fc , fn , Dc , Dn , P H , and QH satisfying
the above assumptions.
Theorem 1 There exists a θ̄ such that ∀ θ ∈ [0, θ̄), the principal chooses a long-term con-tract. In the contract, the second- and third-period wages depend on x to induce Ic . The
contract is renegotiation-proof.
Proof: See Appendix A.
2.6

Short–Short–Short-Term Contract

If θ is sufficiently large, it is always better to induce an incentive for In and offer a short–
short–short-term contract under some additional condition. The additional condition isthat the investment In is sufficiently costly and the cost function is sufficiently convex.
Then, αn is not saturated in all periods and the principal always wishes to induce an
incentive for In . For simplicity, in this subsection, we suppose QH (αn ) = αn , fn (In , αn ) =min{In + αn , 1}, and Dn (In ) = bIn2 , where b > 0 is a parameter. Note that fc , Dc , andP H can be any function satisfying the above assumptions.
Theorem 2 Suppose b > 41 θ(2δ+δ 2 )(y H −y L ). Then, there exists a θ̄ > 0 such that ∀ θ ≥ θ̄,the equilibrium contract is short–short–short term. Note that it is renegotiation-proof. The
second and third period wages can be fixed wage.
Proof: See Appendix C.
In the above theorem, as the investment Ic has already been made during the first
period, the principal does not have to offer incentive pay for the second period dependingon the realization of x, i.e., the wage can be fixed wage. The same argument applies tothe third period.
2.7

Short–Medium-Term Contract

Suppose that In is relatively efficient and that the skill (αn ) is easily saturated. In thiscase, the principal chooses a short-term contract for the first-period wage to induce In .
17

After the agent makes an investment In in the first period, they agree on a medium-term
contract for the second- and third-period wages at the beginning of the second period.
That is, following the saturation of αn , the principal wishes to induce Ic . To illustrate thispoint, in this subsection, we suppose fn (In , αn ) = In + αn , Dn (In ) = aIn , where a > 0 is
a parameter, and(bαn
Q (αn ) =1
H

if 0 ≤ αn ≤ ᾱnif ᾱn ≤ αn ,where b > 0 and ᾱn = 1b . Note that fc , Dc , and P H can be any functions satisfying theabove assumptions.
Theorem 3 Suppose a < 21 δθb(y H − y L ). Then, there exists a θ̄ > 0 such that, for allθ ≥ θ̄, the equilibrium contract is short–medium term. It is also renegotiation-proof. The
third-period wage is incentive pay depending on x, while the second-period wage can be a
fixed wage.
Proof: See Appendix D.
Low-powered wage incentives are very often observed in the real world, and this istheoretically proven by the above theorem. Indeed, in the above environment, the fixed
wage parts of w2 and w3 , i.e., w2L and w3L , include the payment for Ic1 . Thus the incentive
wage part of w3 , i.e., w2H − w2L and w3H − w3L , which induces Ic2 , is relatively small.
2.8

Medium–Short-Term Contract

Suppose that θ is sufficiently large and the agent should accumulate the first type of skill(αc ) to obtain the second type of skill (αn ). For example, the agent needs experience insales to attain a position of leadership in the sales department. Hence, the principal writes
the wages for the first and second periods in a medium-term contract to induce Ic during
the first period, and she writes the third-period wage in a short-term contract to induce
In during the second period. In this subsection, we suppose P H (αc ) = αc , fc (Ic , αc ) =min{Ic + αc , 1}, Dc (Ic ) = Ic2 , QH (αn ) = αn , fn (In , αn , αc ) = min{αc In + αn , 1}, and
Dn (In ) = In2 . Note that the transition function fn depends not only on In and αn , but
18

also on αc . More precisely, if αc is small, then the investment In is not efficient, as in
fn (In , αn , αc ) = min{αc In + αn , 1}, In is multiplied by αc .
Theorem 4 11 There exists a θ̄ such that ∀ θ ≥ θ̄, the equilibrium contract is of medium–

short term. The second-period wage depends on x, whereas the third-period wage can be a
fixed wage. Moreover, it is renegotiation-proof.
Proof: See Appendix E.
2.9

An Extension

The arguments in the previous subsections can be extended to the case with different
bargaining power and a nonzero threat point. In particular, it is worthwhile noting that
if the agent can re-enter the job market or the principal can hire a new agent, the threat
point is nonzero. In this case, the bargaining problem in period 2 is as follows:

k+1
X

max

w2 ,...,wk+1 ,I2 ,...,Ik+1

"
δ j−2

j=2

p
P i (αcj )(xi − wji ) + g(αnj , θ) + δVk+2(αk+2 ) − T2p

i=H,L

k+1
X

×

j=2

"
δ

j−2

s.t.

!1−β

#
X

i

P (αcj )wji −X

Di (Iij )a+ δVk+2(αk+2 ) − T2a

i=c,n

i=H,L

"
Pk+1 j−2
j=2 δ

!β

#
X

#

a
Di (Iij ) + δVk+2(αk+2 )
i=c,n
i=H,L
"
#
P
Pk+1 j−2 P i 0a00≥ j=2 δP (αcj )wji −Di (Iij0 ) + δVk+2(αk+2) for all I20 , . . . , Ik+1,

P

P i (αcj )wji −

i=H,L

P

i=c,n

where (T2p , T2a ) is the threat point in period 2; i.e., the utilities when the principal hires a
new agent and the agent re-enters the job market, and (β, 1−β) is the vector of bargaining
11

We can also show that a medium–short-term contract is an equilibrium if Ic is relatively efficient and easily
saturated. That is, the principal chooses a medium-term contract for the first- and second-period wages to induce
Ic , and after the investments she chooses a short-term contract on the third-period wages to induce In .

19powers. The bargaining problem in period 3 is as follows:
!1−β
!β
X
X
p
p
P i (αc )w3i − T3,max
P i (αc )(xi − w3i ) + g(αn , θ) − T3
w3

(8)

i=H,L

i=H,L

where (T3p , T3a ) is the threat point in period 3.
Even in this case, the same arguments as in the proofs of Theorems 1–4 can be applied.
In the case of Theorem 1, the proof is based on the comparison of the gains from verifiable
output in each contract; more precisely, the first-best Ic cannot be obtained in short–
short–short-, short–medium- and medium–short-term contracts but only in a long-term
contract, and thus a long-term contract is chosen when θ is close to zero. In the case of theother theorems, all proofs are based on the comparison of the gains from the verifiable andunverifiable outputs in the contracts; more precisely, comparison of the limit of differencesas θ → ∞. Even in the case with different bargaining power and a nonzero threat point,
the first-best Ic can be obtained only in a long-term contract. In addition, the comparisonsof the limit of differences as θ → ∞ are essentially the same as in the case of β = 21 andthe threat point (0, 0), though the incentive to invest is different. In the bargaining inperiod t, the agent can obtain a 1−β fraction of gain from the investments; more precisely,(1−β)((total gain)−(Ttp +Tta ))+Tta , and the investments decrease as 1−β becomes small.
Thus the gain from unverifiable output depends on 1 − β. However, as 1 − β > 0 andthe agent has an incentive to invest, the same arguments as in the proof of Theorems 1–4
can be applied, although the threshold θ̄ in the theorems becomes large as 1 − β becomes
small.

3

Five-Period ModelIn this section, we assume that the principal and the agent live for five periods. All otherthings being equal, the principal has a greater variety of combinations of contracts to offer
the agent. For example, if αn depreciates, the principal wishes to occasionally induce an
incentive for In to compensate for the depreciation. In this case, if the principal and theagent bargain over wages every two periods, then for every two periods the agent has anincentive to invest in αn , which compensates for any depreciation.
20

Under the five-period model, contracting over two, three, or four periods are all considered medium-term contracts. Therefore, to avoid confusion, instead of referring to themas “medium-term contracts” we refer to them by the length of the periods included, forexample, a two–one–two contract.
We assume that αn is a function of the investments in the previous two periods.
Namely, αnt = In,t−2 + In,t−1 . That is, the investments before period t − 2 have entirely
depreciated. Moreover, we suppose Dn (In ) = In , and(αn if 0 ≤ αn ≤ 1
QH (αn ) =1
if 1 ≤ αn .
We adopt the same environment and assumptions as in the three-period model, otherthan the length of life and the arguments for fn . Note that fc , Dc , and P H can be any
functions satisfying the above assumptions.
We can define equilibrium contracts as in Section 3. That is, Vtp (α) and Vta (α), t =1, . . . , 5, can be defined as in Section 3. Even though many types of equilibrium contracts
could exist in this model, we focus on the following contract.
One–two–two-term contract (which is a short–medium–medium-term contract):on the equilibrium path, the wage for the first period is determined at the
beginning of the first period, the wages for the second and third periods are
determined at the beginning of the second period, and the remaining wages are
determined at the beginning of the fourth period.
Note that the other combinations, such as the two–two–one-term (medium–medium–shortterm) contract, are similarly defined.
Theorem 5 Suppose xH − xL ≤ 2, y H − y L ≥ 2, and θ ≥ 5. Then, there exists a δ̄ ∈ (0, 1)

such that the equilibrium contract is a one–two–two-term contract for δ ∈ (δ̄, 1]. Moreover,
it is renegotiation-proof.
Proof: See Appendix F.
21

4Limited Liability Constraints

In this section, we discuss limited liability constraints, and show that nearly the same
results can be obtained. For simplicity, we only investigate the three-period model. It iseasy to see that almost the same arguments can be applied to the five-period model.
A standard limited liability constraint is (i) w1L ≥ 0, w1L + δw2i ≥ 0, i = H, L, andw1L + δw2i + δ 2 w3j ≥ 0, i, j = H, L: that is, the case in which the agent can save money.
In contrast, the most conservative limited liability constraint is (ii) wti ≥ 0, i = H, L, t =1, 2, 3. Below, we show that under a wide class of limited liability constraints, we canobtain the same results as in the previous section with only slight modification. Moreprecisely, our condition is as follows: there exists a real number A ≥ 0 such that wti > A
implies the limited liability constraint is not binding. In the case of (i), w1L ≥ 0, w2i ≥− 1δ w1L , and w3j ≥ − δ12 (w1L + δw2i ), i, j = H, L. Thus, the minimum wages are 0, − 1δ w1L , and− δ12 (w1L + δw2i ), and they are at most zero. Let A = 0. Then, for example, in the case of a
medium-term contract in period 2, if wti > A for all t = 2, 3, i = H, L, the limited liability
constraint in the contract is not binding.
Theorem 6 Suppose there exists a real number A ≥ 0 such that wti > A for all t, i implies

the limited liability constraint is not binding. Then, the results in Theorems 1–4 hold.
However, the thresholds are different from those in Theorems 1–4.
Proof: See Appendix G.
Below, we briefly explain the proof. Suppose θ = 0. This is synonymous with sayingthat there are no unverifiable outputs. Given that even under some limited liability
constraint any wage contract (such as a medium–short contract) can be replicated by along-term contract (see Appendix G), Ic1 and Ic2 in the contract can be induced by thelong-term contract. We can also show that the principal can make her utility strictly larger
than under any other contract. Thus, the principal chooses a long-term contract. It isthen obvious that the principal chooses a long-term contract even for a small θ. Therefore,
the same result as in Theorem 1 holds. However, the threshold is different from that in
the theorem.
22

As for the other theorems in Section 3, all proofs are based on the comparison of thegains from verifiable and unverifiable outputs in contracts; more precisely, the comparisonof the limit of differences as θ → ∞. As the gains from the verifiable output do notdepend on θ, the agent’s gain from In , which is a part of wage, goes to ∞ and exceeds
A as θ → ∞ and thus the relevant limited liability constraint is not binding. Thus theutility differences go to ∞ as θ → ∞ no matter what the limited liability constraint is.
Thus, the results in the theorems hold.

5

General Mechanism

In the previous sections, confining our attention to simple wage contracts, we found thatcombinations of contracts of different lengths arise as equilibrium contracts. Below, weinvestigate a general mechanism and show that any mechanism cannot perform better thansimple wage contracts. We assume risk neutrality of parties and renegotiation-proofness ofequilibria. Note that our mechanism is very general and includes menu contracts, changes
in ownership, such as a ‘selling option to the agent’, and some types of penalties. Notethat at the end of the next subsection, we discuss the case that a party has an option
to choose ‘no contract’; i.e., after the investments a party can choose (0, 0), the threat
point, instead of the contract outcome. Because (0, 0) is not Pareto-efficient, the parties
renegotiate (0, 0) as in Maskin and Tirole (1999a).
5.1

Mechanisms without a Change of Ownership

We consider a T ≥ 2 period model. The compensation scheme is a combination of different
length contracts.12 First, we focus on one contract starting from period τ ≥ 1. We supposethat at the beginning of period τ the parties sign an s-period contract that involves theperiod τ wage and a mechanism specified below. Let τ̄ = τ + s − 1, i.e., τ̄ is the last periodof the contract. We first focus on the case in which the principal always has ownership.
We then briefly discuss the case in which the mechanism can change the ownership. In
other words, the case in which the ownership can be moved from the principal to the12

Of course, it can consists of just one contract.

23

agent.
First, we define histories from period one to t. Let I t = (I1 , . . . , It ), xt = (x1 , . . . , xt ),y t = (y1 , . . . , yt ), ht = (I t , xt , y t ), and htn = (I t , y t ), where It = (Ict , Int ) is an investment
vector in period t, xt and yt are realizations of x and y in period t. Note that htn is ahistory of unverifiable variables. The parties play a game (a mechanism) at the beginningof t = τ , . . . , τ̄ . The mechanism is a pair of a function f = (fτ , . . . , fτ̄ ) and a message
space M = Πτ̄t=τ Mt , where Mt = Mtp × Mta and ft is a function from the message space
to the space of outcomes specified below. Note that Mtp , the principal’s message space,and Mta , the agent’s message space, can be any sets. Each element of the message spaces
is observable and verifiable. Let htc = (xt , mt ), i.e., a history of observable and verifiablevariables, where mt = ((ma1 , mp1 ) . . . , (mat , mpt )). The space of outcomes Ω is IR×IR2 , where
the first IR is the set of transfers from the agent to the principal, denoted by qt , paid before
the realization of xt and yt , and IR2 is the set of the payments to the agent in period t,
denoted by (vtH , vtL ), where vti is the payment in the case of xit , i = H, L. For a given ht−1c ,ft assigns an element of Ω for each mt ∈ Mt , i.e., it can be written as ft (mt ; ht−1c ). It is
straightforward that the simple wage contract is the contract that the message space M
is a singleton and qt = 0.
In principle, the mechanism includes all possible outcomes when the principal always
has ownership. First, the mechanism can force the agent to pay a ‘penalty’ q depending on
the message. However, given renegotiation-proofness, a penalty cannot be paid to a third
party (see Maskin and Tirole 1999a). Thus, the principal must obtain the penalty. In
general, any monetary transfer between parties is included in the mechanism. Note that
if at least one of the parties were strictly risk averse, a penalty using stochastic payment
is useful. Suppose that only the agent is risk averse and that the mechanism forces theagent to pay q with probability 12 to the principal and to obtain q (pay −q) from theprincipal with probability 12 . This mechanism is renegotiation-proof because q is not paid
to the third party, and the agent’s utility is less than the case without the payment due tothe risk aversion. Maskin and Tirole (1999) investigate mechanisms with such penalties.
However, there is no point to use it in our case because both parties are risk neutral.
24

Second, the mechanism includes the case that the principal (the agent) chooses acontract from a menu of several simple wage contracts. That is, the case that ft assigns
a simple wage contract depending on the message.
Remark 1 Moore and Repullo (1988) use a sequential-type mechanism and investigate

implementation by subgame-perfect equilibria. Considering our message space as the setof bundles of messages in all nodes of the game tree, sequential-type mechanisms arecovered by our model. In other words, we show below that any Nash equilibria, including
subgame-perfect equilibria, cannot be better than the simple wage contract equilibria.
Remark 2 Even if the parties are risk averse, Maskin and Tirole’s mechanism does notwork in our environment. That is, the welfare neutrality which is the necessary condition
for their theorem is violated in standard principal–agent models as in this paper (seeSections 2 and 8 in Maskin and Tirole 1999a).
In period t, depending on ht−1 , the principal (the agent) chooses mpt ∈ Mtp (mat ∈ Mta )and a strategy of the principal (the agent) is a function of ht−1 , denoted by spt (ht−1 ) ∈ Mtp(sat (ht−1 ) ∈ Mta ). Let st = (spt , sat ). Moreover, in period t the agent chooses It after
observing the realization of xt and yt , and the strategy is denoted by sIt (ht−1 , xt , yt ).
t−1
L
t−1
H
t−1
Let mt = (mpt , mat ) and ft (mt ; ht−1c ) = (qt (mt ; hc ), vt (mt ; hc ), vt (mt ; hc )). Then

we recursively define the agent’s (discounted sum of) utilities before and after investmentsin period t, denoted by uat and Uta , and the principal’s utility in period t, denoted by upt ,as follows. First, let Ra (hτ̄ ) and Rp (hτ̄ ) be the value of the agent and the principal which
are determined by the contracts from period τ̄ + 1 on. Note that Ra (hτ̄ ) and Rp (hτ̄ ) are
zero when τ̄ = T and they are determined by backward induction when τ̄ < T . Then we
define the utilities:

25

Uτ̄a (Iτ̄ ; hτ̄ −1 , xτ̄ , yτ̄ ) = −Dc (Icτ̄ ) − Dn (Inτ̄ ) + δRa (hτ̄ ),X
X
Qiτ̄ (Inτ̄ −1 )y i
P i (Icτ̄ −1 )(xi − v i (mτ̄ ; hτ̄c −1 )) +upτ̄ (mτ̄ ; hτ̄ −1 ) = qτ̄ (mτ̄ ; hτ̄c −1 ) +
i=H,L

i=H,L

+δE(R (sIτ̄ (hτ̄ −1 , xτ̄ , yτ̄ ), hτ̄ −1 , xτ̄ , yτ̄ ))X
P i (Ict−1 )vti (mt ; ht−1
−qt (mt ; ht̄−1
c )
c )+
i=H,L
τ̄ −1
+E(Wt (h , xτ̄ , yτ̄ )), t = τ , . . . , τ̄
−Dc (Ict ) − Dn (Int ) + δuat+1 (st+1 (ht ); ht ),puat (mt ; ht−1 ) =Uta (It ; ht−1 , xt , yt ) =upt (mτ̄ ; hτ̄ −1 ) = qt (mt ; hct−1 ) +X

P i (Ict−1 )(xi − vti (mt ; ht−1c )) +

i=H,Lp+δE(ut+1 (st+1 (ht ); ht )), t = τ , . . . , τ̄ − 1,X

Qit (Int−1 )y i

i=H,Lwhere E is the expectation operator and Wt (ht−1 , xt , yt ) = Uta (sIt (ht−1 , xt , yt ); ht−1 , xt , yt )),For simplicity, P i (αct ) and Qi (αnt ) are denoted by P i (Ict−1 ) and Qi (Int−1 ).
Definition 3 For a given (wτ , f, M ), (sp , sI , sa )) is said to be an equilibrium strategy if

Uta (sIt (ht−1 , xt , yt ); ht−1 , xt , yt ) ≥ Uta (It , ; ht−1 , xt , yt ) for all It ,(9)and, for any given I t−1 ,upt ((spt (ht−1 ), sat (ht−1 ); ht−1 ) ≥ up2 ((mpt , sat (ht−1 ); ht−1 ) for all mp ∈ Mtp ,(10)

uat ((spt (ht−1 ), sat (ht−1 )); ht−1 ) ≥ uat ((spt (ht−1 ), ma ); ht−1 ) for all mat ∈ Mta .

(11)

The first inequality is the condition that the agent’s investment choice in period t is
optimal, and the second and third inequalities imply that the principal’s and the agent’s
choices of messages are optimal in period t.
We first show the revelation principle.
Theorem 7 Given (w1L , f, M ), suppose (sp , sI , sa ) is an equilibrium strategy. Then there
2(t−1)

exists a direct revelation mechanism (game) {(f˜t , IR+2(t+1)(second) IR+2(t−1)× IR+)}τ̄t=τ where the firstis the set of principal’s (agent’s) messages of ht−1
n , which can be different

from the true one, such that26

p

1. s̃at (ht−1 ) = s̃t (ht−1 ) = htn , the truth telling, is an equilibrium strategy of the game,
t−1
t−1
p t−1
2. f˜t (ht−1), sa (ht−1 )); ht−1
n , hn ; hc ) = ft ((s (hc ).

Proof: See Appendix H.
We are now ready to present the main theorem in this section. In the following theorem,
we consider all contracts in periods t = 1, . . . , T .
Theorem 8 Any mechanism cannot improve welfare in the simple wage contract. That is,

the sum of the utilities of the principal and the agent cannot be better off.
Proof: See Appendix I.
5.2

Change of Ownership

In the above arguments, the principal is the residual claimer. The other possible outcome
is to sell ownership to the agent and the agent becomes the residual claimer. We exclude
this case by limited liability constraints. That is, the agent cannot borrow money and thus
the agent cannot buy the ownership and the payment to the agent cannot be negative.
In this case, the principal cannot sell the ownership in any mechanisms, and the proof of
Theorem 8 clearly remains valid under the condition that the payment cannot be negative.
5.3

An Option to Choose ‘No Contract’

Next we suppose that in addition to the mechanism above one (or both) party has an
option to choose ‘no contract’. That is, a party can choose (0, 0), the threat point, insteadof the contract outcome. In this case, the parties renegotiate (0, 0), i.e., Nash bargain
with threat point (0, 0), since it is not Pareto efficient.
Suppose in equilibria at least one party chooses the option. Then the outcome is thatof the Nash bargaining with threat point (0, 0). Predicting this, the agent chooses theinvestments that of the short-term wage contract because the Nash bargaining outcome
is the same as that of the contract. Thus the mechanism cannot perform better thanshort-term wage contract. In sum, the general mechanism with the option can performbetter neither simple wage long-term contracts nor simple wage short-term contracts.
27Remark 3 Finally, note that Edlin and Reichelstein (1996) show that in a certain envi-

ronment, where the threat point of a renegotiation is a function of investment, an appropriately chosen initial contract can provide the correct incentive for investment andlead to the first-best output. As shown in the proof of the renegotiation-proofness of thelong-term contract in the three-period model (Theorem 1), the investment do not affectthe threat point of renegotiation in the second period. Although the investment affect thethreat point of renegotiation in the third period, the risk-neutral parties do not renegotiate because they share only the outputs in the third period. Thus, the initial contracting
has no value, as verified in Che and Hausch (1999).

6

Literature

Labor contracts tend to be depicted as either short- or long-term contracts (for example,
Fudenberg, Holmstrom and Milgrom 1990, Rey and Salanié 1990, Dutta and Reichelstein
2003, and Sato and Kamiya 2013)13 . Moreover, although many outputs in practice areobservable but unverifiable, most models tend to incorporate only verifiable outputs (forexample, Mirrlees 1976, Harris and Raviv 1979, Holmstrom 1979, and Grossman andHart 1983). The main contribution of our model is that it is the first to combine contracts
of different lengths in an incomplete contracting environment where both verifiable andunverifiable outputs are incorporated.
This paper is related to earlier work by Fudenberg, Holmstrom, and Milgrom (1990),
Rey and Salanié (1990), and Salanié (2005, Chapter 6) in which they discuss the environment where an efficient long-term contract can be implemented as a sequence of one-period
short-term contracts. However, they do not investigate the situation when a multiperiod
optimal contract is implemented as a combination of contracts of different lengths. Therefore, as far as we are aware, this paper is the first to show that the optimal contract in themultiperiod principal–agent relationship comprises several contracts of different lengths.
13
In Dutta and Reichelstein (2003), the principal sometimes chooses a short-term contract even when outputs
are verifiable. This is because in their model, the optimal short-term contract requires dismissing (or firing) the
incumbent agent and hiring a new agent in the second period. It is clear that this scenario is ruled out underlong-term contracting with the same incumbent agent.

28

In this paper, if the noncontractible investment is important in every period, a sequenceof one-period short-term contracts is optimal. By contrast, Joskow (1987) suggests thatthe contracting parties prefer to rely on the long-term contract as the specific investment
becomes more important. The difference arises from the differences of the frameworkof the two models: Joskow (1987) considers the investment that benefits the investing
party, whereas this paper discusses the investment that benefits the counter party of theinvesting party.
Guriev and Kvasov (2005) also develop a dynamic buyer-seller model in which timeis one of the most important variables in a contract either as the duration of contractual
obligations or as the advance notice time for certain unilateral actions. They show that theholdup problem can be resolved either through a sequence of certainly renegotiated fixedterm contracts during the contract duration or through a renegotiation proof perpetual
contract that allows unilateral termination with advance notice by the buyer. However,as they consider only observable outputs and not verifiable outputs, their analysis cannot
clarify how the contract length depends on the relative value of the verifiable and unverifiable outputs and the endogenously determined efficiency of investment. In our model,
we consider a combination of different length contract on the same equilibrium path.
Hellmann and Thiele (2011) consider a compensation scheme where the agent is confronted with a multitasking choice between a standard task (verifiable) and the development of innovation (unverifiable). However, their model is quite different from ours: itis essentially a one-period model, i.e., the agent makes each effort just once, and thus
the length of contract cannot be discussed, and in their model the wage contract can beseparable for each task.
Bernheim and Whinston (1998) demonstrate that if there are some unverifiable actions and if agents’ actions are sequential, there are cases in which an efficient outcome isobtained only by the incomplete contracting of verifiable actions. More precisely, incompletely restricting the second mover’s (verifiable) action space in the contract, the shapeof the second mover’s best-response function can be modified such that the first moverchooses an (unverifiable) action that leads to an efficient outcome. That is, in the incom29

plete contract, the second mover can adjust its action to punish the deviation by the firstmover. This is in some sense similar to our logic. In this paper, without specifying thefuture wage (incomplete contract), the bargaining process can adjust in response to theinvestments.
One may think that self-enforced relational contracts, such as Levin (2003), may induce
the first best In . However, such contracts work effectively only in infinite horizon models.
That is, similar arguments as in repeated games are used and it works only when themodel does not have the final period. As our model has the final period, the arguments
can not be applied.
Edlin and Reichelstein (1996), Maskin and Tirole (1999a, 1999b), Moore and Repullo(1988), Kahn and Huberman (1988), and Noldeke and Schmidt (1995) also examine relatively complicated contracts. In Section 6, we investigate a general mechanism, andshow that no mechanism can perform better than simple wage contracts. Our mechanism
is very general and therefore includes most complicated contracts, including menu andoption contracts.
Finally, seminal work by Fama (1980) suggests that there is no need to resolve incentive
problems using explicit output-contingent contracts, because the agent is concerned about
his reputation in the labor market. However, Holmstrom (1999) provides a formal modelin which Fama’s conclusion is only correct under some narrow assumptions: namely, careerconcerns induce the efficient action of the agent. That is, in most cases, explicit contracts
play an important role. In our model, investment and human capital are to some extent
firm specific. One way to relate our model to Holmstrom (1999) is to interpret firm
specificity as an observable signal of the agent’s investment. Firm specificity allows thefirm to observe the investment perfectly, but the market can only receive a noisy signal
about the agent’s investment. The agent knows that the market can learn the agent’s
investment through this noisy signal over time; hence, he might make some efforts to
influence this learning process of the market. However, the learning process of the market
is imperfect and slow, so an explicit contract is the only way to induce a large amount ofagent investment in the environment concerned within this model.
30

7Conclusion

In this paper, we investigated multiperiod contracts of different lengths in an incomplete
contracting framework. Confining our attention to simple wage contracts, we found thatcombinations of contracts of different lengths arise as equilibrium contracts when the principal’s output is determined by both verifiable and unverifiable outputs and the investment
efficiency endogenously changes over time. We also showed that any sophisticated contract(mechanism) could not do better than simple wage contracts.

AppendicesA

The Proof of Theorem 1

We first show that the first-best level investments of Ic1 and Ic2 are obtained in a longterm contract. Given the risk neutrality of the principal and the agent, the first-best
investments are the maximizer of the following problem:
"
#
"
#
X
X
max xL − Dc (Ic1 ) + δ
P i (αc2 )xi − Dc (Ic2 ) + δ 2
P i (αc3 )xi ,
i=H,L

i=H,L

where αc2 = fc (Ic1 , 0) and αc3 = fc (Ic2 , αc2 ). Then, setting w2j = xj − r2 , j = H, L andw3j = xj − r3 , j = H, L, where r2 and r3 are the principal’s utilities in periods 2 and 3, theincentive-compatibility constraint in the long-term contract indeed yields the maximizerof the above problem. That is, the verifiable part of the constraint is reduced to theabove problem. On the other hand, in the cases of the short–short–short-term contract,
the medium–short-term contract, and the short–medium-term contract, the total utilities
obtained from the contractible output, denoted by Sc , M Sc , and SMc , are smaller thanthat of the long-term contract, denoted by Lc . Indeed, in the cases of the short–short–
short-term and short–medium-term contracts, the agent chooses Ic1 to maximize half ofthe second- and third-period utilities obtained from the contractible output, and in thecase of the medium–short-term contract, the agent chooses Ic2 to maximize half of the31third- period utility obtained from the contractible output. Thus, in these cases, Ic1
and/or Ic2 are different from their first-best levels. Thus,
Sc < Lc , SMc < Lc , M Sc < Lcholds. Accordingly, if θ = 0, the long-term contract shown above is chosen. Given thatthe first-best value of the gain from the unverifiable output is a continuous function of θ,
there exists a θ̄ such that the principal chooses the long-term contract for θ ∈ [0, θ̄).
Next, we show that the above long-term equilibrium contract is renegotiation-proof.
There is no need to discuss renegotiation at the beginning of the third period, as theparties do not invest. Moreover, any renegotiation on the wages does not induce a Pareto
improvement because both the principal and the agent are risk neutral. Below, we investigate a renegotiation at the beginning of the second period, where the threat point is thediscounted sums of the utilities in periods 2 and 3 obtained from the long-term contract.
Suppose their utilities resulting from this renegotiation are Pareto superior to those of thethreat point at the beginning of the second period. Given this, further suppose the agent
chooses Iˆc1 and Iˆn1 by maximizing the discounted sum of his expected utility. (Note thatthe investments do not affect the threat point of the renegotiation in period 2.) Then,
their utilities are even Pareto superior to those of the long-term contract, even at the beginning of the first period, as the agent can choose Ic1 and In1 in the long-term contract,
and the outputs in the first period are assumed to be always xL and y L . If a medium-term
contract is chosen in the renegotiation, Iˆc1 and Iˆn1 can be considered as the investmentsin the case of the short–medium-term contract, and if a short-term contract is chosen in
the renegotiation, then Iˆc1 and Iˆn1 can be considered as the investments in the case of
the short–short–short-term contract. However, it has been shown that the total utility islarger under the long-term contract than under the short–medium-term contract ∀ θ ≤ θ.
This is a contradiction. Thus, the long-term contract is renegotiation-proof if θ ≤ θ.

B

The Proof of Lemma 1

As in the proof of Theorem 1, (i) even if the wages do not depend on the realization of32

x in previous periods, Ic1 and Ic2 can be the first best level, and (ii) even if the wages
depend on the realization of x in previous periods, the principal cannot induce In1 andIn2 . Therefore, the Lemma follows.

C

The Proof of Theorem 2

In (i)–(vi) below, we focus on each contract and obtain its total equilibrium utility from thenoncontractible output. We use backward induction (if necessary). Then, in (v), we showthat if θ is sufficiently large, the principal chooses a short–short–short-term contract.
(i) We first focus our attention on a long-term contract. That is, we derive the maximum total utility from the noncontractible output produced under the long-term contract. The agent chooses In1 = In2 = 0, and thus g(αn , θ) = θy L holds throughout all
periods. The total utility obtained from the noncontractible output, denoted by Ln , is
(1 + δ + δ 2 )θy L .
(ii) Next, we focus on a short–short–short-term contract. The wages for the second
and third periods are determined by Nash bargaining at the beginning of each period.
Below, we consider only the utilities obtained from the noncontractible outputs. In thePthird period, the agent obtains half of the total utility, i.e., 21 i=H,L Qi (αn3 )θy i . Thus, inthe second period, the agent chooses In2 , satisfying the incentive compatibility constraint:
1 X i
2
Q (αn3 )θy i − bIn2
,
max δ
In2 2
i=H,L
where αn3 = min{αn2 + In2 , 1}. Below, suppose that the optimal αn2 and αn3 are less thanone, i.e., the optimal In1 and In2 are determined by the first-order condition. (Later, weshow that the optimal αn2 and αn3 are indeed less than one.) Then,
∗
In2
=

1
δθ(y H − y L ).
4b∗
Note that In2
does not depend on αn2 . In the second period, the agent obtains half of thetotal utility:
1
2

!
X

i

Q (αn2 )θyi∗ 2− b(In2
)

i=H,L

33

1 X i
∗
+ δ
Q (αn2 + In2)θy i ,
2 i=H,Lwhere αn2 = In1 . Thus, in the first period, the agent chooses In1 , satisfying the incentivecompatibility constraint:
1
max
In1 2

!
X

∗ 2
Qi (In1 )θy i − b(In2
)

i=H,L

1 X i
∗
2
+ δ
Q (In1 + In2)θy i − bIn1.
2 i=H,L

Then, the optimal In1 is obtained as follows:
∗
=
In1

1
θ(δ + δ 2 )(y H − y L ).
4b

∗
∗
= In1
and
By the premise of the theorem, b > 41 θ(2δ + δ 2 )(y H − y L ) holds; thus, αn2
1
∗
∗
∗
∗
∗
= 4b
+ In2
are indeed less than one because In1
+ In2
= In1
αn3
θ(2δ + δ 2 )(y H − y L ).

Then, the total utility obtained from the noncontractible output, denoted by Sn (θ), is
obtained as follows:

∗ 2
∗
∗
∗ 2
Sn (θ) = θy L − b(In1) + δ In1
θy H + (1 − In1)θy L − b(In2
)

∗
∗
∗
∗
+δ 2 (In1+ In2)θy H + (1 − In1
− In2
)θy L
3 H
(y − y L )2 δ 2 θ2 (δ 2 + 3δ + 1).
= (1 + δ + δ 2 )θy L +16b(iii) Next, we focus on a medium–short-term contract. The agent’s utility in the thirdP
period is 21 i=H,L Qi (αn3 )θy i . The wage for the second period is determined by a take-itor-leave-it offer at the beginning of the first period. Thus, the agent is interested only in
αn3 , as the wage for the second period does not depend on αn2 . That is, the agent solves
the following problem with respect to In1 and In2 in the first period:
1 2 X i22
− δbIn2,δQ (αn3 )θy i − bIn1
In1 ,In2 2
i=H,L
max

∗
∗
where αn1 = min{In1 , 1} and αn2 = min{αn1 +In2 , 1}. Suppose the optimal αn3 = In1+In2∗∗
is less than one. Then, In1
and In2
are obtained as follows:
∗
In1
=

1 2
δ θ(y H − y L ),
4b

∗
In2
=

1
δθ(y H − y L ).
4b

34

∗
∗
By the premise of the theorem, b > 41 θ(2δ + δ 2 )(y H − y L ) holds; thus, In1+ In2
is indeed

less than one. Then, M Sn (θ), the total utility obtained from the noncontractible output,
is obtained as follows:
M Sn (θ) = (1 + δ + δ 2 )θy L +1 H(y − y L )2 δ 3 θ2 (3δ + 7).
16b(iv) Finally, we consider a short–medium-term contract. By definition, the wage forthe third period is determined at the beginning of the second period. Thus, the agent
chooses In2 = 0 in the second period. Therefore, the agent solves the following problem
with respect to In1 in the first period:
1 X i
1 X i
2
max δ
Q (αn2 )θy i + δ 2
Q (αn3 )θy i − bIn1
,
In1 2
2
i=H,L
i=H,L
∗
is less than one.
where αn2 = min{In1 , 1} and αn3 = αn2 . Suppose the optimal αn3 = In1

Then, it is obtained as follows:
∗
=
In1

1
θ(δ + δ 2 )(y H − y L ).
4b∗
By the premise of the theorem, b > 41 θ(2δ + δ 2 )(y H − y L ) holds; thus, In1
is indeed lessthan one. Then, the total utility obtained from the noncontractible output, denoted bySMn (θ), is obtained as follows:
SMn (θ) = (1 + δ + δ 2 )θy L +3 H(y − y L )2 δ 2 θ2 (δ + 1)2 .
16bBelow in (v), we compare the total utilities obtained from both the contractible andnoncontractible outputs.
(v) As shown in the proof of Theorem 1, the total utilities obtained from the contractible output, denoted by Lc , are first-best in the case of a long-term contract. In
the cases of the short–short–short-term contract, the medium–short-term contract, andthe short–medium-term contract, the total utilities obtained from the contractible output,
denoted by Sc , M Sc , and SMc , are smaller than Lc . That is,
Sc < Lc , SMc < Lc , M Sc < Lc .
35

On the other hand,
3 H(y − y L )2 δ 2 θ2 (δ 2 + 3δ + 1) > 0,
16b
1 H
Sn (θ) − M Sn (θ) =(y − y L )2 δ 2 θ2 (2δ + 3) > 0,
16b
3 H
Sn (θ) − SMn (θ) =(y − y L )2 δ 3 θ2 > 0
16b
Sn (θ) − Ln =hold. As Sn (θ)−Ln , Sn (θ)−M Sn (θ), and Sn (θ)−SMn (θ) are strictly increasing functionsof θ and go to +∞ as θ goes to +∞, there exists a θ̄ > 0 such that ∀ θ ≥ θ̄,
Lc + Ln < Sc + Sn (θ), SMc + SMn (θ) < Sc + Sn (θ), M Sc + M Sn (θ) < Sc + Sn (θ).
That is, Sc + Sn (θ) is the largest. Thus, the short–short–short-term contract is chosenfor ∀ θ ≥ θ̄. Suppose the contrary. Then, the equilibrium contract derived from backward
induction is either a long, a medium–short, or a short–medium term, and the equilibrium
total utility is larger than Sc + Sn (θ). This contradicts the above inequalities.
It is straightforward that there is no need to discuss the renegotiation-proofness of theshort–short–short-term contract.

D

The Proof of Theorem 3

As in the proof of Theorem 2, Lc , Ln , Sc , Sn (θ), M Sc , M Sn (θ), SMc , and SMn (θ) are
obtained.
Suppose a short–medium-term contract is chosen. By a < 21 δθb(y H − y L ), the marginal
cost of In is strictly smaller than the marginal utility, so that the agent chooses In1 = ᾱnin the first period. Thus,
SMn (θ) = θy L + δθy H + δ 2 θy H −abholds. Note that Sn (θ) = SMn (θ) holds.
Then, suppose a medium-term contract on the first- and second-period wages is signed
at the beginning of the first period. Given that the marginal cost of In is strictly smaller36

than the marginal utility, then the agent chooses In1 = 0 and In2 = ᾱn because of the
discount factor. Thus,
M Sn (θ) = θy L + δθy L + δ 2 θy H −δa.
b

For a sufficiently large θ,SMn (θ) − M Sn (θ) = δθ(y H − y L ) −holds.

a δa
+
>0
b
b

Given SMn (θ) − M Sn (θ) and SMn (θ) are strictly increasing, and SMn (θ) −M Sn (θ) → +∞ and SMn (θ) → +∞ as θ → +∞,Lc + Ln < SMc + SMn (θ), M Sc + M Sn (θ) < SMc + SMn (θ)
holds for a sufficiently large θ. Moreover, as shown in the previous section, Sc < SMcholds and thus
Sc + Sn (θ) < SMc + SMn (θ).
Using the same argument as in the proof of Theorem 2, the short–medium-term contract
is an equilibrium contract.
Finally, as shown in the proof of Theorem 1, the above equilibrium is renegotiationproof, as we should consider only the renegotiation in the third period.

E

The Proof of Theorem 4

By fn (In , αn , αc ) = min{αc In + αn , 1}, αn in the second period is zero even if In > 0 in thefirst period, because αc = 0 holds at the beginning of the first period. Thus, the principal
does not choose a short-term contract at the beginning of the first period; she insteadchooses either a long- or a medium–short-term contract. If a medium–short-term contract
is chosen, QH in the third period is αn3 = min{αc2 In2 , 1}. Note that αc2 is positive andan increasing function of θ in these contracts. Accordingly, the agent maximizes2δ(αn3 θy H + (1 − αn3 )θy L ) − In237with respect to In2 . Suppose the optimal αn3 is less than one, then the optimal In2 is equal
to 21 δθαc2 (y H − y L ). Therefore, the following total utility from noncontractible output isobtained:
1 1(1 + δ + δ 2 )θy L + δ 2 θ2 αc2 (y H − y L )2 ( − αc2 ).
2 4

(12)

∗
Suppose the optimal αn3 is equal to one, then the agent chooses In2= α1c2 . Therefore, thefollowing total utility from noncontractible output is obtained:(1 + δ + δ 2 )θy L + δ 2 θy H −

1
.
2
αc2

(13)

∗
∗
If the principal chooses a long-term contract, In1= In2= 0 holds and the total utility

from the noncontractible output becomes (1 + δ + δ 2 )θy L . Given that αc2 is an increasing
function of θ, (12) and (13) go to +∞ as θ → +∞. Using the same arguments as in the
proof of Theorem 2, there exists a θ̄ such that ∀ θ ≥ θ̄, the medium–short-term contract is
an equilibrium contract.
Finally, as shown in the proof of Theorem 1, the above equilibria are renegotiationproof.

F

The Proof of Theorem 5

Suppose δ = 1. Under the one–two–two-term contract (short–medium–medium-term
contract), the agent chooses In1 = In3 = 1 and In2 = In4 = 0; as in periods 2 and 4, theprincipal and the agent bargain over the wages; and in periods 1 and 3, the marginal costof In is one, and the marginal utility is 12 (δ + δ 2 )θ(y H − y L ) = θ(y H − y L ) ≥ 10 for In < 1.
Thus, αn = 1 holds in periods 2, 3, 4, and 5, and the total utility obtained from In is(δ + δ 2 + δ 3 + δ 4 )θy H − In1 − δ 2 In3 = 4θy H − In1 − In3 ≥ 38.
Other than a one–two–two-term contract, logically there are three other combinations ofcontracts under a five-period principal–agent relationship: (i) combinations that involvea contract that covers at least three periods, e.g., a one–three–one-term contract (short–medium–short-term contract); (ii) combinations that involve at most one contract that38

covers two periods, e.g., a one–one–two–one-term contract (short–short–medium–shortterm contract); and (iii) combinations that include two contracts that cover two periods,
e.g., a two–two–one-term contract (medium–medium–short-term contract). In (i), as there
exists a period in which αn = 0, they lose at least θ(y H − y L ) − 1 ≥ 9 in the total
noncontractible utility and obtain at most 4(xH − xL ) ≤ 8 in the total contractible utility,
where the number of periods in which αc could be increased is four. In (ii), although thetotal utility obtained from In is the maximum amount, the total utility obtained from Ic
is smaller than the case of the one–two–two-term contract. This is because combinations
that fall in the category of (ii) involve more bargaining periods than the one–two–two-term
contract. In the case of (iii), a two–two–one-term contract (medium–medium–short-term
contract) and a two–one–two-term contract (medium–short–medium-term contract) are
the only possibilities. In both cases, given In1 = 0 in the first period, they lose at leastθ(y H −y L )−1 ≥ 9 in the total noncontractible utility and obtain at most 4(xH −xL ) ≤ 8 in
the total contractible utility, where the number of periods in which αc could be increased
is four. Therefore, using the same arguments as in the proof of Theorem 2, the one–two–
two-term contract is an equilibrium contract. The above arguments apply to any δ > 0
close to one. The maximal length of the contract is two, and the parties might renegotiate
on wages in the last period of each contract. However, given risk neutrality, a Pareto
improvement is impossible. Thus, the equilibrium is renegotiation-proof.

G

The Proof of Theorem 6

Suppose θ = 0. In other words, this is the case in which there are only verifiable outputs(no unverifiable outputs). Below, we show that any contract can be replicated by a longterm contract. Consider a short–short–short-term contract as an example. Then, setting
wtH = 12 xH > 0 and wtL = 12 xL > 0, t = 2, 3, Ic1 and Ic2 in the short–short–short-term
contract can be induced by the long-term contract. On the other hand, in the long-term

39

contract,
w1L = u − δX

P i (αc2 )w2i − δ 2

i=H,L

X

P i (αc3 )w3i + Dc (Ic1 ) + δDc (Ic2 )

i=H,L

1
1 X i
= u− δ
P (αc3 )xi + Dc (Ic1 ) + δDc (Ic2 ).
P i (αc2 )xi − δ 2
2 i=H,L2 i=H,LX

The last line is equal to the first-period wage in the short–short–short-term contract. Notethat the discounted sum of wages is the same in the contracts. Moreover, the principal
can choose wage differences (w2H − w2L ) larger than 21 (xH − xL ) and can also keep the
expected wages constant. Therefore, she can obtain a larger gain. Thus, the principal
strictly prefers a long-term contract. The same argument applies to medium–short andshort–medium contracts. It is clear that she chooses a long-term contract even for a small
θ. Thus, the same results as in Theorem 1 hold. However, the threshold is different fromthat of Theorem 1.
Next, we prove the results in Theorems 2–4. If θ is small, the limited liability constraint
might be binding and the agent might not invest the same In as in the case without theconstraint. However, if θ is sufficiently large, half of the gain from In , which is a part ofwages, is larger than A, and thus it is better to invest the same In as in the case without
the constraint. In the proofs of these theorems, the gains from the unverifiable output arecompared, and we take the limit of differences as θ → ∞. For example, in Theorem 2,
Sn (θ) − Ln , Sn (θ) − M Sn (θ), and Sn (θ) − SMn (θ) are strictly increasing functions of θ andgo to +∞ as θ goes to ∞. Given that the least upper bound of gains from the verifiable
output does not depend on θ, the utility differences of the principal go to ∞ as θ → ∞,no matter what the limited liability constraint. Thus, the results in the theorems hold.
However, the thresholds differ from those in the theorems.

40

H

The Proof of Theorem 7

a t−1
t−1
t−1
), sp (ĥt−1 )); ht−1
Setting f˜t ((ht−1c ), the truth telling is an equin , ĥn ); hc ) = ft ((s (hlibrium strategy. Indeed, since
t−1
a t−1
q̃t (hnt−1 , ĥt−1), sp (ĥt−1 ); ht−1
n ; hc ) = qt (s (hc ),t−1
H a t−1ṽtH (hnt−1 , ĥt−1), sp (ĥt−1 ); ht−1
n ; hc ) = vt (s (h
c ),

and
t−1
L a t−1ṽtL (hnt−1 , ĥt−1), sp (ĥt−1 ); ht−1
n ; hc ) = vt (s (hc ),then (9), (10), and (11) follow from the definition of equilibrium. Finally, it is clear thats̃at (ht−1 ) = s̃pt (ht−1 ) = htn holds.

I

The Proof of Theorem 8(i) We first show that, for any given hτ̄ −2 , xτ̄ −1 , yτ̄ −1 , and Ic,τ̄ −1 , uaτ̄ − E(Wτ̄ ) does notdepend on In,τ̄ −1 in equilibria. That is, the agent’s equilibrium temporal utility in period0τ̄ does not depend on In,τ̄ −1 . Suppose the contrary. Then there exist In,τ̄ −1 6= In,τ̄−1 suchthatuaτ̄ (hτ̄n−1 , hnτ̄ −1 ; hτ̄ −1 ) − E(Wτ̄ (hτ̄ −1 , xτ̄ , yτ̄ )) > uaτ̄ (hnτ̄ −10 , hnτ̄ −10 ; hτ̄ −10 ) − E(Wτ̄ (hτ̄ −10 , xτ̄ , yτ̄ ))(14)holds in an equilibrium, where the arguments in hτ̄ −10 and hτ̄ −1 are the same except00In,τ̄ −1 6= In,τ̄−1 . Of course, at least one of Iτ̄ −1 and Iτ̄ −1 must be an off-equilibrium choice.

From the definition of an equilibrium,uaτ̄ (hτ̄n−10 , hτ̄n−10 ; hτ̄ −10 ) − E(Wτ̄ (hτ̄ −10 , xτ̄ , yτ̄ )) ≥ uaτ̄ (hnτ̄ −10 , hτ̄n−1 ; hτ̄ −10 ) − E(Wτ̄ (hτ̄ −10 , xτ̄ , yτ̄ ))(15)
0holds. That is, even if the agent announces Inτ̄ −1 instead of the true investment Inτ̄−1 ,he cannot be better off. Moreover, since the Ict are the same in hτ̄ −1 and hτ̄ −10 for41

t = τ , . . . , τ̄ − 1 and the probability P does not depend on In , then from the definition ofuaτ̄uaτ̄ (hτ̄n−10 , hτ̄n−1 ; hτ̄ −10 ) − E(Wτ̄ (hτ̄ −10 , xτ̄ , yτ̄ )) = uaτ̄ (hnτ̄ −10 , hτ̄n−1 ; hτ̄ −1 ) − E(Wτ̄ (hτ̄ −1 , xτ̄ , yτ̄ )).
(16)
Thus, from (14), (15), and (16),uaτ̄ (hτ̄n−1 , hτ̄n−1 ; hτ̄ −1 ) − E(Wτ̄ (hτ̄ −1 , xτ̄ , yτ̄ )) > uaτ̄ (hnτ̄ −10 , hτ̄n−1 ; hτ̄ −1 ) − E(Wτ̄ (hτ̄ −1 , xτ̄ , yτ̄ ))(17)
holds. Sinceupτ̄ (hτ̄n−1 , hτ̄n−1 ; hτ̄ −1 ) =XX

P i (Icτ̄ −1 )xi +i=H,LQi (Inτ̄ −1 )y i +E(Gτ̄ (hτ̄ −1 , xτ̄ , yτ̄ ))−uaτ̄ (hτ̄n−1 , hτ̄n−1 ; hτ̄ −1 )

i=H,L

and
upτ̄ (hτ̄n−10 , hnτ̄ −1 ; hτ̄ −1 ) =X

P i (Icτ̄ −1 )xi +

i=H,L

X

Qi (Inτ̄ −1 )y i +E(Gτ̄ (hτ̄ −1 , xτ̄ , yτ̄ ))−uaτ̄ (hτ̄n−10 , hτ̄n−1 ; hτ̄ −1 )

i=H,L

(18)
hold, where Gτ̄ (hτ̄ −1 , xτ̄ , yτ̄ ) = Wτ̄ (hτ̄ −1 , xτ̄ , yτ̄ ) + δRp (hτ̄ −1 , xτ̄ , yτ̄ , sIτ̄ (hτ̄ −1 , xτ̄ , yτ̄ )), thenfrom (17), (I), and (18),upτ̄ (hτ̄n−10 , hτ̄n−1 ; hτ̄ −1 ) − δE(Rp (hτ̄ −1 , xτ̄ , yτ̄ , sIτ̄ (hτ̄ −1 , xτ̄ , yτ̄ )))> upτ̄ (hτ̄n−1 , hτ̄n−1 ; hτ̄ −1 ) − δE(Rp (hτ̄ −1 , xτ̄ , yτ̄ , sIτ̄ (hτ̄ −1 , xτ̄ , yτ̄ ))).
That is,upτ̄ (hτ̄n−10 , hτ̄n−1 ; hτ̄ −1 ) > upτ̄ (hτ̄n−1 , hτ̄n−1 ; hτ̄ −1 )0holds and the principal chooses In,τ̄−1 instead of the true investment In,τ̄ −1 . This contra-

dicts the definition of equilibrium.
(ii) First, we consider the case τ̄ = T . Since T is the last period, then there does
not exist any contract from T and Ra (hT ) = 0. Thus the agent’s period T investment
in an equilibrium, denoted by Iˆn,T , is zero. Next, we show that the agent’s period T − 1
investment in an equilibrium, denoted by Iˆn,T −1 , is zero. Suppose the contrary. Then42Iˆn,T −1 > 0 holds. If the agent chooses In,T −1 = 0, then from (i), uaT remains the same andDn (Iˆn,T −1 ) > Dn (0). Thus the agent prefers Inτ̄ −1 = 0. and Iˆn,T −1 = 0 holds in equilibria.
That is, the mechanism cannot induce In,T −1 at all. Next, consider the choice of In,T −2
which might affect Ic,T −1 and/or uaT −1 − δE(WT ). First, Ic,T −1 is a maximizer of UTa −1 andit does not depend on In,T −2 . Second, by the same argument as in (i), uaT −1 − δE(WT )does not depend on In,T −2 . Thus Iˆn,T −2 = 0 holds in equilibria. The same argument canbe applied to all periods and In,t = 0 for t = τ , . . . , T − 3 hold. Thus the mechanism
can induce at most the first-best Ict , t = τ , . . . , T − 1. Recall that a simple wage contract
can induce the first-best Ict , but cannot induce Int . Thus even if the best mechanism ischosen, it can induce the same investments as in simple wage contracts.
(iii) Suppose the contract in (ii) is not the last one, i.e., τ > 1. Suppose the second
last contract is from period τ 2 to τ̄2 . First, in the equilibrium Iˆn,τ̄2 is a maximizer of−Dn (Inτ̄2 ) + δRa (hτ̄2 ). Then from the same arguments as in (i), in the equilibrium Iˆn,τ̄2 −1
is a maximizer of −Dn (Inτ̄2 −1 ) + δE(Ra (hτ̄2 )). That is, Inτ̄2 −1 has no effect on equilibrium
temporal utility in period τ2 and thus the agent takes the gain from period τ2 + 1 into
account. The same argument can be applied to all periods and In,t for t = τ 2 , . . . , τ̄2 − 2,
are maximizers of −Dn (In,t ) + δ t−τ 2 +2 E(Ra (hτ̄2 )). That is, the In,t is induced only by thecontract from period τ̄2 + 1, and it is the one induced by the simple wage contract fromperiod τ̄2 + 1. On the other hand, Ict , t = τ 2 , . . . , τ̄2 , are induced by the mechanism in thesecond last contract and the contract from period τ̄2 +1. Both contract do not work better
than simple wage contracts as in the arguments in (ii). Thus even if the best mechanism ischosen, it can induce the same investments as in simple wage contracts, since the contract
outcome is determined by Nash bargaining and it depends only on the total utility. If
τ 2 > 1, the same argument applies. This process ends in finite times, since T is finite.
Acknowledgement: The authors would like to thank Bruce Chapman, Benjamin Hermalin, Hideshi Itoh, Mary Milne, and Hiroshi Osano for helpful comments on an earlier
version of this paper. This research is financially supported by the global Center of Excellence (COE) program, “The Research and Training Center for New Development in
43

Mathematics.”

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