GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.132

A Theory of Multiperiod Financial Contracts that do not
Balance the Budget

Mamiko Terasaki

May 2010

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

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

A Theory of Multiperiod Financial Contracts that do not
Balance the Budget
Mamiko Terasaki∗ †Kyoto University

Abstract
This paper seeks ﬁnancial contracts that specify a repayment schedule that induces both the investor and the entrepreneur to contribute to the project when theentrepreneur privately observes the realizations. Any optimal contracts do not balance the budget, i.e., some amount of money must be transferred to or from a third
party depending on the history of the output realizations. We specify multiperiod
optimal contracts not only with budget feasibility constraint but also without it andexplain venture capital contracts as an example of the contracts that expect a third
party to break the budget without budget feasibility constraint.
JEL Classiﬁcations: D82, D86, G20, G24, G30
Keywords: ﬁnancial contract, budget-breaking, and venture capital.
∗

Email: mamiko.terasaki@gmail.com.

†

I would like to thank Hiroshi Osano, Akihisa Shibata, and Tadashi Sekiguchi for continuous guidanceand support.

1

1

Introduction

In the case in which a project’s output realizations can be observed only by an entrepreneur, a repayment must be independent of the realizations for the entrepreneur
to truthfully report them. However, an investor and an entrepreneur must obtain payoﬀs that are dependent on the realizations to induce them to contribute to the project.
Therefore any eﬃcient contracts do not balance the budget, i.e., the amount the entrepreneur repays is diﬀerent from the amount the investor obtains under some conditions. Players use a third party to diﬀerentiate between these two amounts. Withbudget feasibility constraint, i.e., the assumption that the amount the entrepreneur repays must be larger than the amount the investor obtains, the third party must receive
some amount of money from the players to diﬀerentiate between two amounts. Howeverin the real world, we do not need to assume budget feasibility; for example, venture
capitalists and venture ﬁrms expect general investors to provide money for them if theirprojects succeed by buying venture ﬁrms’ shares through initial public oﬀerings.
This paper discusses ﬁnancial contracts between an entrepreneur and an investor.
The contract speciﬁes a repayment schedule that depends on the history of the output
realizations. Here, we assume that the entrepreneur can observe the realizations but theinvestor cannot do so. We also assume that neither the investor nor the entrepreneur cancommit their contributions to the project every period. It is possible for the investor todecrease the quality his ﬁnancial services or for the entrepreneur to lower her eﬀort level,2either of which would lower the probability that the project succeeds. Furthermore, since
it is also possible for the entrepreneur to lie about the project’s realization, the investor
may be unwilling to contribute to the project.
We can ﬁnd these kinds of ﬁnancial contracts in the real world. First, one can deposit
money in money management companies and later obtain a return based on performance.
Second, under Mudarabah, one of the basic forms of Islamic banking, a bank and an
entrepreneur share a project’s proﬁts or losses according to a predetermined ratio. In
both cases, investors or depositors can choose the amount of money they contribute ineach period and their continuous contributions are important to the borrowers’ projects.
Moreover, in neither case can investors or depositors directly observe the performanceor output realization of projects, so borrowers may lie about them to increase their own
payoﬀs.
Townsend (1979), Diamond (1984), and Gale and Hellwig (1985) analyze the situation that lenders cannot observe the project’s outputs without paying a ﬁnite monitoringcost. To induce the borrower to be honest about the output realization, the lender must
audit the borrower and deprive the borrower of all of the output realizations when the
borrower reports poor performance. In contrast, we assume that the monitoring costs
are inﬁnitely high. Bolton and Scharfstein (1990) also analyze this case and propose thatﬁnancial institutions must stop lending in the second period with a positive probability
when the borrower reports poor performance at the end of the ﬁrst period to inducethem to truthfully send messages about a project’s performance. However, they fail to3give the borrower any incentive to be truthful in the last period.
This paper models the case where both entrepreneur’s contribution and an investor’s
contribution are important to the project. The model is similar to the ones that study
moral hazard in teams (Holmstrom(1982)) and double-sided moral hazard in franchisecontracts (Bhattacharyya and Lafontaine(1995)). They assume that all players canobserve project’s output realizations but we analyze the situation that an entrepreneur
can observe the realizations but an investor cannot do so. Holmstrom(1982) shows that
noncooperative behavior by players always yields an ineﬃcient result when joint output
realization is fully shared among them and that they can make an eﬃcient contract when
the sum of the amounts shared among players can be more or less than the joint output
realization, i.e., when a third party breaks the budget by seizing a part of the realizationfrom players or by paying them a bonus.
Recently, MacLeod (2003) and Kambe (2006) discussed an agency contract when aprincipal privately observes an agent’s output realizations. To induce the agent to exert
eﬀort and the principal to be truthful about the realizations, the wage the principal pays
and that the agent receives must be diﬀerent under some conditions. Since they assume
budget feasibility, i.e., the amount the principal pays must be larger than the amount theagent obtains, the diﬀerentiated amount of money must be passed to a third party. Here,a third party again plays a role in breaking the budget. Fuchs (2007) proposes a method
to reduce the amount to be passed to a third party in a repeated game setting. We apply
their ﬁndings to a ﬁnancial contract under which a principal’s eﬀort level also aﬀects the4project’s output realizations. Although they assume budget feasibility constraint, we
also seek optimal ﬁnancial contracts without this constraint because ﬁnancial contracts
that count on a third party to pay exist in the real world.
For the entrepreneur to truthfully report output realizations to the investor, theamount she repays must be independent of the output realizations. However, to induce
players to contribute to the project, the payoﬀs the investor and the entrepreneur obtainafter the projects must be dependent on the output realizations. Since repayment must
always be constant, it is the project’s property that gives the entrepreneur an incentive
to make a high level of eﬀort. In other words, the project must be likely to generatehigh enough output realization when the entrepreneur exerts a high level of eﬀort andlow enough when she exerts a low level of eﬀort. On the other hand, it is the transfer
to or from the third party that gives the investor an incentive to provide good ﬁnancial
services in each period.
The advantage of multiperiod contract in this paper is to reduce the discounted
expected amount of money to be transferred to or from the third party. When the entrepreneur hides information on the output realizations until the end of the relationship,
the contracts that give players good incentives in the ﬁrst period also can give themgood incentives in the succeeding period. When we assume budget feasibility constraint,
the optimal contract, which minimizes the discounted expected amount of the transfer
to the third party, requires no amount of the transfer if the output realization in theﬁrst period is high regardless of the realization in the succeeding period.
5Without budget feasibility constraint and with a rational third party, the optimal
contract is the one that minimizes the discounted expected amount of the transfer fromthe third party and that requires no amount of the transfer if the realization in theﬁrst period is low regardless of the realization in the succeeding period. A venture
capital contract, under which an investor provides funds and management advice to an
entrepreneur by purchasing convertible securities issued by the entrepreneur, is a good
example of this type of budget-breaking ﬁnancial contracts in that an venture capitalist
and a venture ﬁrm expect a third party to pay for them if their project succeeds.
The remainder of the paper is organized as follows. The next section presents thegeneral model framework. In section 3, we analyze one-period ﬁnancial contract withbudget feasibility constraint and the ineﬃciency of asymmetric information on output
realizations. The ﬁrst half of Section 4 considers multiperiod contract with budget feasibility constraint and illustrate the way of reducing the ineﬃciency of budget-breaking.
In the last half of the section, we seek multiperiod contract without budget feasibility
constraint and explain venture capital contracts as an example of this type of budgetbreaking contracts in the real world. Section 5 concludes. All the proofs are containedin the Appendix.

6

2

The Model

There are two risk-neutral players, an entrepreneur and an investor. The entrepreneur
has a project that stochastically generates outputs, but she does not have funds for it.
The entrepreneur has to ﬁnance the project from the investor, who can provide a variety
of ﬁnancial services in each period. The investor contributes to the project by providing
sound ﬁnancial and management assistance, but not all of it can be speciﬁed in a contract. The entrepreneur and the investor agree on a repayment schedule that depends
on the project’s output realization. The entrepreneur can observe the output realization, but the investor cannot. Therefore, the investor is worried that the entrepreneur
may lie about the output realization. In turn, the entrepreneur is concerned that theinvestor may reduce his contribution to the project. The investor is also worried thatthe entrepreneur may shirk his responsibilities while carrying out the project because
the investor cannot observe the entrepreneur’s eﬀort level.
The entrepreneur and the investor meet at the beginning of period 0, agree to cooperate on the project until the end of period T ∈ {0, 1}, and sign a contract that speciﬁes
the repayment schedule.
After signing the contract, the investor chooses his contribution to the project, It ∈{IH , IL }, at the beginning of each period. Then the entrepreneur chooses her eﬀort
level, et ∈ {eH , eL }. We assume that contributing to the project costs them It and et ,respectively. We also assume that IH > IL and eH > eL . After the project generates

7

output realization yt , the entrepreneur sends a message about it mt (yt ). The message
is observable and veriﬁable. Depending on the message, the entrepreneur repays part ofthe output realization, and the investor receives money as speciﬁed in the contract.
The output realization is also binary, yt ∈ {yH , yL }, stochastic, and depends on boththe amount of funds and the level of eﬀort. We assume that the project generates yH
with probability p(It , et ) and yL with probability 1 − p(It , et ). We deﬁne the expected
output levels as ȳsv = E(yt |It = Is , et = ev ) = psv yH + (1 − psv )yL , where psv ≡ p(It , et |It = Is , et = ev ) for s ∈ {H, L} and v ∈ {H, L}.
We assume that a riskless bond exists in the economy. When a player buys theriskless bond at the beginning of each period, he will get the same amount of the moneyat the end of the same period. We assume the interest rate of this bond is r; when player
buys x worth of this bond in period 0, he will get money worth (1 + r)x in period 1.
Moreover, we have the following assumptions.

Assumptions(a) ȳHH > ȳLH and ȳHH > ȳHL(b) ȳHH − IH − eH > ȳLH − IL − eH > 0(c) ȳHH − IH − eH > ȳHL − IH − eL(d) ȳLH − IL − eH < ȳLL − IL − eL
Assumption (a) implies that p(It , et ) is an increasing function of both It and et . In
other words, the more funds the investor lends, the more likely the project will succeed.

8Similarly, the more eﬀort the entrepreneur exerts, the more likely the project will succeed.
Assumption (b) means that, given that the entrepreneur exerts a high level of eﬀort,
more funds are more eﬃcient than fewer funds, and fewer funds are more eﬃcient thanriskless bonds. Assumption (c) implies that, given that the investor lends a large amountof money, a high level of eﬀort is more eﬃcient than a low level. On the other hand,assumption (d) denotes that, given that the investor lends a small amount, a low levelof eﬀort is more eﬃcient than a high level. It also means that the increase of the
expected output realizations by intensifying the entrepreneur’s eﬀort level is lower than
the increase of eﬀort costs by doing so when the investor lends a small amount of money.
These assumptions imply that the project’s expected output realization is maximized
if and only if both players contribute to the project. We refer to a contract under which
the project’s expected output realizations are maximized as an eﬃcient contract. In
this paper, the entrepreneur oﬀers a contract that maximizes her own expected payoﬀ
as well as maximizes the project’s expected outputs. We call such a contract an optimal
contract. An optimal contract is always eﬃcient, but an eﬃcient contract is not necessarily optimal. We also deﬁne the following expressions: ∆y ≡ yH − yL , ∆I ≡ IH − IL ,∆e ≡ eH − eL , ∆p(I) ≡ pHH − pLH , and ∆p(e) ≡ pHH − pHL . Note that all of these
expressions are nonnegative.

9

3

One-Period Contract

In this section, we discuss the case in which the project generates output only once:
when T = 0. When the output realizations are observed by both players, the contracts
can specify the repayment that depends on them. However, when the output realizations
are only observed by the entrepreneur, the repayment can no longer depend on them.

3.1

Eﬃcient and optimal contracts when output realizations are public
information

To examine the eﬀect of asymmetric information on output realizations, y0 , we ﬁrst
discuss eﬃcient contracts when both the player and the third party observe the project’s
realizations 1 . In this case, we can write contracts that depend on the output realizations
b , r b }, i.e., r b if the output realization
and assume that the entrepreneur repays rb ∈ {rHLH
b if it is y . Similarly, the investor receives r ℓ ∈ {r ℓ , r ℓ }, i.e., r ℓ if theis yH and rLL
H L
Hℓ if it is y . In this section, we assume that r b ≥ r ℓ andoutput realization is yH and rLLHHℓ . These constraints correspond to budget feasibility constraints in the context ofrLb ≥ rLb = r ℓ and r b = r ℓ , the amount the entrepreneur repays andmechanism design2 . If rHHLL

the amount the investor receives are the same, and we say that the contract balancesb > r ℓ or r b > r ℓ , we say that the contract does not balance thethe budget. If either rH
HLL
1

If players can observe the output realization but any third party cannot, the realization cannot be

veriﬁable in court. Here we assume that output realizations are observable and veriﬁable.
2

b
ℓ
b
ℓ
In section 4.2, we consider the case of rH≤ rH
and rL≤ rL, in which a third party provides money

for players.

10budget and interpret that the diﬀerence between the amount the entrepreneur repays
and the amount the investor receives is passed to a third party. We postulate that bothplayers are reluctant to transfer any hard-won value to a third party. We denote theb − r b and the diﬀerence
diﬀerence of the amount the entrepreneur repays as ∆rb ≡ rHL
ℓ − r ℓ . Next, we discuss the constraintsof the amount the investor receives as ∆rℓ ≡ rH
L

that eﬃcient contracts must satisfy and the conditions under which eﬃcient contracts
exist.
First of all, any eﬃcient contracts must guarantee expected payoﬀs that exceed theplayer’s burdens, or guarantee nonnegative net expected payoﬀs. The entrepreneur
writes and oﬀers the contracts if and only if the expected payoﬀ that she obtains byexerting eﬀort eH for the project is higher than or identical to her eﬀort costs. Similarly, the investor signs the contract if and only if the expected payoﬀ that he receives
by investing IH is higher than or identical to the payoﬀ that he receives by investing
the same amount of money into riskless bonds. Therefore, any eﬃcient contracts mustsatisfy the following constraints:
E(rℓ |I0 = IH , e0 = eH ) ≥ IH ,(1)

E(y0 − rb |I0 = IH , e0 = eH ) ≥ eH .

(2)

We refer to constraints (1) and (2) as individual rationality constraints for the investor
and for the entrepreneur respectively. These constraints can be rewritten in the following

11

way:
ℓ
ℓ
pHH rH+ (1 − pHH )rL≥ IH ,(3)bb
pHH (yH − rH
) + (1 − pHH )(yL − rL
) ≥ eH .

(4)

In addition to individual rationality constraints, any eﬃcient contract must satisfy
constraints that induce players to greatly contribute to the project:
E(rℓ |I0 = IH , e0 = eH ) − IH ≥ E(rℓ |I0 = IL , e0 = eH ) − IL ,(5)

E(y0 − rb |I0 = IH , e0 = eH ) − eH ≥ E(y0 − rb |I0 = IH , e0 = eL ) − eL .

(6)

The left-hand side of each inequity is each player’s net expected payoﬀ when he greatly
contributes to the project, and the right-hand side is that when he do not so, given thatthe other player greatly contributes to the project. We refer to constraints (5) and (6) as
incentive compatibility constraints for the investor and for the entrepreneur respectively.
When both players can observe the project’s output realizations, these constraints canbe rewritten with diﬀerence expressions in the following way:
∆rℓ ≥

∆I
∆p(I)

∆rb ≤ ∆y −

∆e
.
∆p(e)

(7)
(8)

Thus, we have the following lemma about the existence of eﬃcient contracts whenboth players can observe the project’s output realizations.

Lemma 1(i) If ∆y ≥ ∆I/∆p(I) + ∆e/∆p(e), there exists an eﬃcient contract thatb =balances the budget. An eﬃcient contract speciﬁes a repayment schedule such that rH

12

ℓ and r b = r ℓ ; thus, no amount of money is transferred to the third party.
rH
LL(ii) If ∆y < ∆I/∆p(I)+∆e/∆p(e) and yH ≥ IH +eH +(1−pHH ) (∆I/∆p(I) + ∆e/∆p(e)),
there are no eﬃcient contracts that balance the budget, but there is an eﬃcient contract that does not balance the budget. The eﬃcient contract that minimizes the transb = r ℓ and r b =fer to the third party speciﬁes a repayment schedule such that rH
H
L

rLℓ +(∆I/∆p(I) + ∆e/∆p(e) − ∆y). In the case of y0 = yL , players transfer ∆I/∆p(I)+∆e/∆p(e) − ∆y to the third party, i.e., the ex ante expected transfer to the third partyis (1 − pHH )(∆I/∆p(I) + ∆e/∆p(e) − ∆y).
(iii) If ∆y < ∆I/∆p(I)+∆e/∆p(e) and yH < eH +IH +(1−pHH ) (∆I/∆p(I) + ∆e/∆p(e)),
there are no eﬃcient contracts.

The case of (i) implies that, when the diﬀerence of output realizations between highand low is large enough, players can make eﬃcient contracts under which they do notneed to transfer any amount of money to a third party. When this diﬀerence is small,an eﬃcient contract has to make a large enough diﬀerence by transferring some money
to a third party. However, this transfer reduces the players’ expected payoﬀs. Thus,
when yH is small, players cannot make contracts that satisfy the individual rationality
constraints for them.
The eﬃcient contract that minimizes the transfer to the third party requires thefollowing rules. If the output realization is high, the amount of money paid by theentrepreneur is directly passed to the investor. However, if it is low, the amount of13

money paid by the entrepreneur must be divided into two parts; one is passed to theinvestor and the other to the third party.
It is impossible for these players to negotiate and share the amount to be transferred
to a third party because this renegotiation increases the investor’s payoﬀ when the output
realization is low and violates the incentive compatibility constraint for the investor.
The entrepreneur writes and oﬀers contracts that not only satisfy all these constraints
but also maximize her own expected payoﬀ. We obtain an optimal contract without
asymmetric information on output realizations as follows.

Proposition 1 Suppose that both players and a third party can observe project’s output
b , rb , rℓ , rℓ )realization y0 ∈ {yH , yL } and the contract speciﬁes repayment schedule (rHL H L

that depends on project’s output realization y0 .
(i) If ∆y > ∆I/∆p(I) + ∆e/∆p(e), optimal contracts exist inﬁnitely. Under any
optimal contracts, the investor’s expected payoﬀ is IH , and the entrepreneur’s is ȳHH −IH . The optimal contract that binds the incentive compatibility constraint for the investor
speciﬁes a repayment schedule such that
b
ℓ
rH
= rH
= IH + (1 − pHH )
b
ℓ
rL
= rL
= IH − pHH

∆I
,
∆p(I)

∆I
.
∆p(I)

(ii) If ∆y ≤ ∆I/∆p(I)+∆e/∆p(e) and yH ≥ IH +eH +(1−pHH ) (∆I/∆p(I) + ∆e/∆p(e)),
an optimal contract exists and it is unique. The investor’s expected payoﬀ is IH , and14

the entrepreneur’s is ȳHH − IH − (1 − pHH ) (∆I/∆p(I) + ∆e/∆p(e) − ∆y). The optimal
contract speciﬁes a repayment schedule such that
b
ℓ
rH
= rH
= IH + (1 − pHH )brL= IH + (1 − pHH )ℓrL= IH − pHH

∆I
,
∆p(I)
∆I
∆e
+
− ∆y,
∆p(I) ∆p(e)

∆I
.
∆p(I)

For y0 = yL , the amount of ∆I/∆p(I) + ∆e/∆p(e) − ∆y is transferred to a third party,i.e., the ex ante expected transfer to the third party is (1−pHH )(∆I/∆p(I)+∆e/∆p(e)−∆y).

Proposition 1 provides two types of optimal contracts and the conditions under which
they exist. When ∆y is large, that is, when the diﬀerence between a large output
realization and a low one is large, players can make optimal contracts that provide
good incentives for them without transferring any money to a third party. Since players
are risk-neutral, they are indiﬀerent to contracts that give constant expected payoﬀs.
Therefore, optimal contracts exist inﬁnitely. Among these optimal contracts, the onethat binds the incentive compatibility constraint for the investor is unique. When ∆y issmall but yH is large, players can make a unique optimal contract; however, the budget
won’t balance.
The investor’s expected payoﬀ is always IH . However, the entrepreneur’s expected
payoﬀ varies depending on the project’s property. When players must transfer money15

to a third party, the contract charges the entrepreneur this amount since the investor
always receives, at most, the same amount of burden. When both ∆y and yH are small,players cannot make any eﬃcient contracts by Lemma 1, nor can they make any optimal
contracts.
The optimal contracts expressed in Proposition 1 depend on the assumption that boththe players and the third party can observe the project’s output realization. If neither
the investor nor the third party can observe the output realizations, the entrepreneur
may lie about the project’s output realization to increase her own payoﬀ. In the next
section, we discuss eﬃcient and optimal contracts when there is asymmetric informationon the project’s output realizations.

3.2

The eﬃcient and optimal contracts when output realizations are
private information

Suppose that neither an investor nor a third party can observe the output realizations,but the messages sent by the entrepreneur are observable and veriﬁable. In this case,the contract speciﬁes a repayment schedule that depends on those messages. First web > r b , i.e.,
consider a case in which the contracts balance the budget. Then, if rHL
b regardless∆rb > 0, the entrepreneur always sends message m0 (y0 ) = yL and repays rL

whether yH or yL is achieved. This is because the entrepreneur can increase her payoﬀ
by minimizing the repayment. However, if the investor anticipates the entrepreneur’sb −I if he lends
behavior, he always lends IL . The reason is that the investor’s payoﬀ is rLH16

b − I if he lends I ; thus, the latter is always larger than the former. Given
IH and rL
L
L

that the entrepreneur always repays rLb , then the contract cannot satisfy the incentive
compatibility constraint for the entrepreneur, since we havebbȳHL − rL− eH < ȳLL − rL
− eL

by assumption (d).
In contrast, if ∆rb < 0, the entrepreneur always sends message m0 (y0 ) = yH , andthe investor always lends IL . As a result, the necessary and suﬃcient condition for theentrepreneur to be honest about the project’s output realization is ∆rb = 0. This constraint is equivalent to ∆rℓ = 0 because we are now considering a contract that balances
the budget. However, by constraint (7), ∆rℓ = 0 violates the incentive compatibility
constraint for the investor. Therefore, no contract can balance the budget when there isasymmetric information on output realization. Consider the following proposition.

Proposition 2

Suppose that neither the players nor the third party can observe project’s

b , rb , rℓ , rℓ )output realization y0 ∈ {yH , yL } and the contract speciﬁes repayment schedule (rHL H L

that depends on message m0 ∈ {yH , yL }.
(i)No eﬃcient contract balances the budget.
(ii)If ȳHH ≥ eH + IH + (1 − pHH )∆I/∆p(I) and yL ≥ IH + (1 − pHH )∆I/∆p(I),there is eﬃcient contracts that do not balance the budget. The unique optimal contract17

speciﬁes a repayment schedule such that
b
b
ℓ
rH
= rL
= rH
=IH + (1 − pHH )ℓrL=IH − pHH

∆I
,
∆p(I)

∆I
.
∆p(I)

For m0 = yL , players transfer ∆I/∆p(I) to the third party, i.e., the ex ante expectedamount of the transfer is (1 − pHH )∆I/∆p(I). The investor’s expected payoﬀ is IH , andthe entrepreneur’s is ȳHH − IH − (1 − pHH )∆I/∆p(I).
(iii)If either ȳHH < eH +IH +(1−pHH )∆I/∆p(I) or yL ≥ IH +(1−pHH )∆I/∆p(I),there are no eﬃcient contracts.

Compared to Proposition 1, when output realizations are unobservable, players cannot make any eﬃcient contract that balances the budget. The reason is that the entrepreneur will not truthfully send a message without requiring the amount she repays
to be constant, and the investor will not lend a large amount of money without differentiating the amount he receives depending on the output realizations. The ex ante
expected amount of money transferred to the third party is always (1 − pHH )∆I/∆p(I),which is larger than that in the case of Proposition 1 (ii) if ∆I/∆p(I) ≥ ∆I/∆p(I) +∆e/∆p(e) − ∆y, i.e., ∆y ≥ ∆e/∆p(e), which is always true by assumption (c).
For an eﬃcient contract to exist, ȳHH and yL must be larger than the case of symmetric information. Unlike the case of Proposition 1 (ii), there is no requirement on ∆y
except for assumption (c).
18The investor’s expected payoﬀ is the same as that in the case of symmetric information. In contrast, the entrepreneur’s expected payoﬀ is smaller than that in the case ofsymmetric information. The reason for this diﬀerence is that the expected amount theinvestor receives equals his burden regardless whether he can observe the output realizations, but the entrepreneur must repay a larger amount for asymmetric information.

4

Multiperiod Contract

In this section, we illustrate the gain of a multiperiod relationship between an investor
and an entrepreneur. The previous section showed that no eﬃcient contract that balances
the budget exists in the case of asymmetric information on a project’s output realizations.
Any eﬃcient contract has to specify the amount of money to be passed to a third party.
Players minimize this amount so as to maximize their expected payoﬀs. In this section,
we apply Fuchs’s idea (2007) and seek an optimal contract when the business relationshipcontinues for two periods.
Fuchs (2007)’s T -period principal-agent model reveals the condition that minimizes
the amount of transfer to the third party when the principal privately observes the signalof an agent’s eﬀort3 . He shows that, when the relationship continues for ﬁnite periods,
players can minimize the expected amount of the reluctant transfer to a third party
by forcing the principal to repay only once at the end of the last period instead of at
the end of each period. He does not specify the bargaining power between the agent
3

Fuchs (2007) also seeks eﬃcient contracts in inﬁnite horizon model.

19and the principal; however, in this paper, we seek optimal contracts that maximize theentrepreneur’s payoﬀ, which corresponds to the principal’s payoﬀ in his model.
We assume that the business relationship continues for two periods. Let the discountfactors of the players be identical and denote them by β. Moreover, we assume that thediscounted factor is identical to the interest rate of the riskless bond, i.e., β = r. In
Section 4.1, we seek optimal contracts with budget feasibility constraint, i.e., rsb ≥ rsℓ , fors ∈ {H, L}, as we have done so far, and the conditions under which they exist. When the
optimal contract speciﬁes a repayment schedule such that rsb > rsℓ , the diﬀerence between
the two values must be transferred to the third party. Section 4.2 relaxes the budget
feasibility constraint, i.e., analyzes the case of rsb ≤ rsℓ . In this case, when the optimal
contract speciﬁes a repayment schedule such that rsb < rsℓ , the diﬀerence between thetwo values must be transferred by the third party. An example of this type of contract
is a venture capital contract since a venture capitalist expects to get money not only
from a venture ﬁrm but also from a third party, such as general investors, through an
initial public oﬀering if a project succeeds.

4.1

The eﬃcient and optimal contracts when the entrepreneur defers
reporting output realizations

In the previous section, we analyzed a case in which an entrepreneur sends messagem0 (y0 ) = ys after the project ends in period 0 for s ∈ {H, L}. Fuchs (2007) allows theplayer who observes the output realizations to defer sending message m0 until the end of20

the relationship. For s ∈ {H, L}, the entrepreneur is indiﬀerent whether he should send
message m0 (y0 ) = ys and repay rsb at the end of period 0 or send message m0 (y0 ) = ys
and repay β −1 rsb at the end of period 1. Similarly, for the investor to receive m0 = ys
and rsℓ at the end of period 0 is equivalent to receiving m0 = ys and β −1 rsℓ at the end of
period 1. Therefore, there exists a contract under which the entrepreneur defers sendinga message until the end of a relationship; this contract is still eﬃcient because the players
are indiﬀerent to an optimal contract under which the entrepreneur sends a message atthe end of each period.
After repeating the project twice, four types of histories emerged: (yH , yH ), (yH , yL ),(yL , yH ), and (yL , yL ). The entrepreneur sends one of them as message m1 (y0 , y1 ) at theend of period 1. The amounts the investor receives are denoted by z ℓ ∈ {zHH , zHL , zLH , zLL }depending on each message. We relate the amount the investor receives to each history
rather than to each period-output realization. As in the single-period model, to give theentrepreneur an incentive to send an honest message, the amount to be repaid by theentrepreneur must be constant regardless of the message. We denote this amount by
z b . The entrepreneur oﬀers the investor an optimal contract that minimizes z b subject
to the incentive compatibility constraints and the individual rationality constraints forboth players.
First, we consider the constraints that an optimal contract must satisfy to preventthe investor from lending a small amount of money. Given that the investor lends a large
amount in period 1, the constraint that induces him to lend a large amount in period 0
21

is
{
}
β E(z ℓ |I0 = I1 = IH ) − E(z ℓ |I0 = IL , I1 = IH ) ≥ ∆I.

(IC0)

Similarly, given that he lends a large amount in period 0, the constraint that induces
him to lend a large amount in period 1 is{}β E(z ℓ |I0 = I1 = IH ) − E(z ℓ |I0 = IH , I1 = IL ) ≥ β∆I.

(IC1)Finally, the constraint that prevents him from lending a small amount in both periodsis{}
β E(z ℓ |I0 = I1 = IH ) − E(z ℓ |I0 = I1 = IL ) ≥ (1 + β)∆I.

(IC2)Next, we consider the constraint for the entrepreneur to exert an eﬀort both times.
Given that the amount to be repaid is constant, the constraints that prevent any possibility of shirking responsibilities only in period 0, only in period 1, or in both periods
are as follows:E(y0 + βy1 |e0 = e1 = eH ) − E(y0 + βy1 |e0 = eL , e1 = eH ) ≥ ∆e,
E(y0 + βy1 |e0 = e1 = eH ) − E(y0 + βy1 |e0 = eH , e1 = eL ) ≥ β∆e,E(y0 + βy1 |e0 = e1 = eH ) − E(y0 + βy1 |e0 = e1 = eL ) ≥ (1 + β)∆e.
We also consider the individual rationality constraints for each player. Given thatboth players greatly contribute to the project, these constraints are
βE(z ℓ |I0 = I1 = IH , e0 = e1 = eH ) ≥ (1 + β)IH ,E(y0 + βy1 − z b |I0 = I1 = IH , e0 = e1 = eH ) ≥ (1 + β)eH .
22

They lead to another proposition that deals with optimal contracts in a case involving
a relationship that continues for two periods.

Proposition 3 The project continues for two periods, 0 and 1, and generates output
realizations at the end of each period. The entrepreneur privately observes output realizations y0 and y1 . Assume that she sends a message about the project output only onceand repays at the end of period 1. Denote the amounts of the transfer to the third party
depending on each history by a ≡ zHH − zHL , b ≡ zHH − zLH , and c ≡ zHH − zLL .
(i) If ȳHH ≥ IH + eH + (1 − pHH )∆I/(1 + β)∆p(I) and yL ≥ IH + (1 − pHH )∆I/(1 +β)∆p(I), there is a set of optimal contracts that does not balance the budget. Under
optimal contracts, the amounts that must be transferred to the third party are as follows:a = 0,[]
1 − β ∆Ib ∈ 0,,β ∆p(I)
c=

1
∆I
pHH
−
b.
β(1 − pHH ) ∆p(I) 1 − pHH

Therefore, an optimal repayment schedule (z b , zHH , zHL , zLH , zLL ) isz b = zHH = zHL =1 − pHH ∆I
1+β
IH +
,
β
β
∆p(I)

zLH = z b − b,zLL = z b − c.
The discounted expected amount of the transfer from a third party is (1−pHH )∆I/∆p(I).
23The investor’s discounted expected payoﬀ is (1 + β)IH and the entrepreneur’s is (1 +β) {ȳHH − IH − (1 − pHH )∆I/(1 + β)∆p(I)}.
(ii) If either ȳHH < IH + eH + (1 − pHH )∆I/(1 + β)∆p(I) or yL < IH + (1 −pHH )∆I/(1 + β)∆p(I), there are no eﬃcient contracts.

The eﬃcient and optimal contracts inﬁnitely exist if and only if ȳHH and yL are
suﬃciently large. The optimal contract speciﬁes a repayment schedule such that a = 0,i.e., zHH = zHL . This means that no money has to be transferred to a third party
to induce players to contribute signiﬁcantly to the project in period 1 if the output
realization is large in period 0. Note that under such a contract, the investor will nothave any incentive to lend a large amount of money in period 1 as long as he knowsthat y0 = yH has been realized at the end of period 0. The optimal contract provides
good incentive to the investor by hiding information on y0 . Thus, an optimal contract
minimizes the discounted expected amount of money that must be transferred to a third
party; the minimum achieved value is (1 − pHH )∆I/∆p(I), which is the amount that,at most, gives the investor a good incentive in period 0.
Since the discounted expected amount of the transfer is (1 + β)(1 − pHH )∆I/∆p(I)when the entrepreneur sends a message in each period by the result of Proposition 2(ii), the amount of transfer to a third party is reduced by not sending a message atthe end of period 0. Since players can reduce a reluctant transfer to a third party andkeep the investor’s expected discounted payoﬀ identical to his discounted burden, the24entrepreneur’s expected discounted payoﬀ increases. In addition, players can agree to
optimal contracts under smaller ȳHH and yL here than when the entrepreneur sends a
message in each period4 .

4.2

The mutiperiod optimal ﬁnancial contract without budget feasibility constraint

The characteristics of venture capital contracts are similar to those of optimal contracts
presented in Proposition 3, except that in the former contracts a third party breaks
the budget by providing money for players but in the latter contracts he does so byreceiving money from players. Venture capitalists often invest in new ﬁrms by purchasing
convertible securities (Sahlman (1990)). If ﬁrms are proﬁtable, venture capitalists often
recoup their investments through initial public oﬀerings (Black and Gilson (1998) and
Gompers and Lerner (2004)). Venture capitalists contribute to projects not only byﬁnancial assistance but also by management assistance, as do ﬁrms that contribute toprojects through their eﬀorts. However, not all of their contributions can be speciﬁed ina contract. Venture capitalists contribute to each project several times (stage ﬁnancing)but obtain a return only once: at the end of the relationship. If a venture ﬁrm is
proﬁtable, the venture capitalist exercises a conversion option from debt to equity and4

When the entrepreneur sends a message and repays in each period, the eﬃcient and optimal contacts

exist if and only if ȳHH ≥ eH + IH + (1 − pHH )∆I/∆p(I) and yL ≥ IH + (1 − pHH )∆I/∆p(I) as I soughtin Proposition 2.

25

obtains the right to take the ﬁrm’s surplus. If the venture ﬁrm is not proﬁtable, theventure capitalist requires the ﬁrm to pay back the debt or liquidate the ﬁrm.
Suppose that a venture capital contract requires that a ﬁrm pay z b as dividends when
he sends a good message and z b as liquidated values when he sends a bad message. The
ﬁrm is then indiﬀerent to which messages he sends. Thus, ﬁrms always honestly report
the history of the output realizations. If the history is good, venture capitalists canobtain capital gains through initial public oﬀerings in addition to dividends. Therefore,
in this case, the amounts obtained by venture capitalists diﬀer depending on the history,
but the amounts repaid by ﬁrms are constant. A third party or general outside investors
contribute funds to the players if the history is good, which is in contrast to the resultsof Proposition 3, where the third party receives money from these players if the history
is bad. Here we assume z b ≤ z ℓ for z ℓ ∈ {zHH , zHL , zLH , zLL } and denote the amountof the transfer from the third party depending on each history as a′ ≡ zLH − zLL ,b′ ≡ zHL − zLL , and c′ ≡ zHH − zLL .
If third party or general outside investors are too generous and buy a venture ﬁrm’s
share through initial public oﬀerings for enough money to satisfy the individual rationality constraint for the venture capitalist, the venture ﬁrm, which minimizes therepayment, never repays anything to the venture capitalist. In Proposition 4, we seek an
optimal venture contract when outside investors are rational and are not so generous tominimize the amount they pay when buying the venture ﬁrm’s share through an initialpublic oﬀering.
26

Proposition 4

The project continues for two periods and generates output realization

at the end of each period. The entrepreneur privately observes output realizations y0 andy1 . She sends a message about the history of the output realizations and repays only onceat the end of the last period. We assume that markets for an initial public oﬀering aredeveloped and a third party is willing to contribute a minimum fund that ensures highcontributions by both the entrepreneur and the investor.
(i) If yL ≥ IH −pHH ∆I/(1+β)∆p(I), there is a set of optimal contracts that does notbalance the budget.5 Under optimal contracts, minimum values that must be transferredfrom a third party are as follows:a′ = 0,[]1 − β ∆I′b ∈ 0,,β ∆p(I)c′ =∆I
1 − pHH ′1−b.
βpHH ∆p(I)
pHH

Therefore, an optimal repayment schedule (z b , zHH , zHL , zLH , zLL ) isz b = zLL = zLH =pHH ∆I
1+βIH −,ββ ∆p(I)zHL =z b + b′ ,zHH =z b + c′ .
The discounted expected amount of the transfer from a third party is pHH ∆I/∆p(I).
5

In fact, there is a set of optimal contracts if and only if ȳHH ≥ eH + IH − pHH ∆I/(1 + β)∆p(I) andyL ≥ IH − pHH ∆I/(1 + β)∆p(I). The condition on ȳHH is always true because of assumption (b).

27The investor’s discounted expected payoﬀ is (1 + β)IH and the entrepreneur’s is (1 +β) {ȳHH − IH + pHH ∆I/(1 + β)∆p(I)}.
(ii) If yL < IH − pHH ∆I/(1 + β)∆p(I), there are no eﬃcient contracts.

The optimal contract presented in Proposition 4 provides the suﬃcient conditions
under which such contracts exist, the minimum contribution by outside investors, andthe amounts the venture ﬁrm repays and the venture capitalist obtains. The optimal
contract speciﬁes a repayment schedule such that a′ = 0, i.e., zLH = zLL . This meansthat no money must be transferred by a third party if the output realization is small inperiod 0. As in the case of Proposition 3, players minimize the amount of the transferfrom the third party by hiding information on the output realization in period 0. If thistype of outside investors and a market for initial public oﬀering exist, an entrepreneur
can oﬀer venture capital contracts to an investor. There is no requirement on ȳHH underassumption (b) and the expected payoﬀ for the entrepreneur is larger than that in thecase of Proposition 3; the entrepreneurs are more likely to oﬀer eﬃcient contracts toinvestors than in the case of Proposition 3.
Even if players do not expect outside investors to pay suﬃciently large amounts
for them or the markets for initial public oﬀerings are underdeveloped or sluggish, theentrepreneur can oﬀer a contract such as that presented in Proposition 3 and pursue
eﬃcient results. They are free to pay money to a third party, such as a charity.

28

5

conclusion

This paper seeks ﬁnancial contracts that specify a repayment schedule that induces
both an investor and an entrepreneur to contribute to a project in every period and thatmaximizes the entrepreneur’s expected payoﬀ when the entrepreneur privately observes
project’s output realizations. To induce the entrepreneur to truthfully report output
realizations, repayments must be identical regardless of the history of the output realizations. However, to induce both players to greatly contribute to the project, their
payoﬀs must reﬂect the history. Therefore, one type of optimal contract requires players
to directly transfer repayment from the entrepreneur to the investor if the history is good
and to pass some part of the money repaid by the entrepreneur to a third party and givethe rest to the investor if it is bad. Another type of optimal contract requires players to
directly transfer repayment from the entrepreneur to the investor if the history is badand to give money to the investor, money that the entrepreneur repays and that thethird party provides, if it is good.
When the project continues for two periods, contracts that require the entrepreneur
to deter reporting on output realizations and repaying the loan until the end of the lastperiod can reduce the amount of money that must be passed to or be provided by thethird party. When the entrepreneur hides information on the output realizations until
the end of the relationship, the contracts that give players good incentives in the ﬁrst
period also can give them good incentives in the succeeding period. An example of the29budget-breaking ﬁnancial contracts that expect a third party to pay money for players
depending on project’s performance is a venture capital contract, which expects generalinvestors to purchase venture ﬁrm’s shares if the project succeeds.

References
Bhattacharyya, Sugato and Francine Lafontaine (1995), “Double-Sided Moral Hazard
and the Nature of Share Contracts,” RAND Journal of Economics, Vol.26, No.4, pp.761781.
Black, Bernard S. and Ronald J. Gilson (1998),“Venture capital and the structure ofcapital markets: banks versus stock markets ,” Journal of Financial Economics, Vol.47,
No.3, pp.243-277.
Bolton, Patrick and David S. Scharfstein (1990), “A Theory of Predation Based on
Agency Problems in Financial Contracting,” American Economic Review, Vol.80, No.1,
pp.93-106.
Diamond, Douglas W. (1984), “Financial Intermediation and Delegated Monitoring,”
Review of Economic Studies, Vol.51, No.3, pp.393-414.
Fuchs, William (2007), “Contracting with Repeated Moral Hazard and Private Evaluations,” American Economic Review, Vol.97, No.4, pp.1432-1448.
Gale, Douglas and Martin Hellwig (1985), “Incentive-Compatible Debt Contracts: The
One-Period Problem,” Review of Economic Studies, Vol.52, No.4, pp.647-63.

30

Gompers, Paul A. and Lerner Josh (2006), The Venture Capital Cycle, 2nd Edition, The
MIT Press.
Holmstrom, Bengt (1982), “Moral Hazard in Teams,” Bell Journal of Economics, Vol.13,
No.2, pp.324-340
Kambe, Shinsuke (2006), “Subjective Evaluation In Agency Contracts,” Japanese Economic Review,Vol. 57, No.1, pp.121-140.
MacLeod, W. Bentley (2003), “Optimal Contracting with Subjective Evaluation,” American Economic Review, Vol.93, No.1, pp.216-240.
Sahlman, William A. (1990), “The structure and governance of venture-capital organizations,” Journal of Financial Economics, Vol.27, No.2, pp.473-521.
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Appendix
Proof of Lemma 1(i) Suppose ∆y ≥ ∆I/∆p(I) + ∆e/∆p(e). Then, there exists ∆rb such that ∆y −∆e/∆p(e) ≥ ∆rb ≥ ∆I/∆p(I). When the budget balances , the schedule satisﬁes
constraints (7) and (8) simultaneously. Constraints (3) and (4) implybℓℓȳHH ≥ IH + eH + pHH (rH− rH) + (1 − pHH )(rLb − rL
).

31

(9)

When the budget balances, this constraint is ȳHH ≥ IH + eH and always satisﬁed byassumption (b).
(ii) Suppose ∆y < ∆I/∆p(I) + ∆e/∆p(e) and ∆rb satisﬁes (8). Then, there is no∆rℓ which satisﬁes (7) when the budget balances. If ∆rℓ satisﬁes (7) and ∆rb satisﬁesℓ ≤ r ℓ − ∆I/∆p(I) and r b ≥ r b − (∆y − ∆e/∆p(e)), we have(8), i.e., rLHLHℓrLb − rL
≥

{
(
)} {
(
)}
∆e
∆I
b
ℓ
rH
− ∆y −
− rH
−
∆p(e)
∆p(I)

b
ℓ
= (rH
− rH
)+

∆I
∆e
+
− ∆y.
∆p(I) ∆p(e)To minimize the reluctant transfer to the third party, the repayment schedule satisﬁes
b = r ℓ and r b = r ℓ +∆I/∆p(I)+∆e/∆p(e)−∆y. Considering this transfer, constraint
rH
H
L
L

(9) can be written as ȳHH ≥ IH + eH + (1 − pHH ) (∆I/∆p(I) + ∆e/∆p(e) − ∆y), i.e.,yH ≥ IH + eH + (1 − pHH ) (∆I/∆p(I) + ∆e/∆p(e)).
(iii) Suppose ∆y < ∆I/∆p(I) + ∆e/∆p(e). By the proof of Lemma 1(ii), if yH <IH + eH + (1 − pHH ) (∆I/∆p(I) + ∆e/∆p(e)) holds, players cannot make any eﬃcient
contracts.

Proof of Proposition 1(i) Suppose ∆y ≥ ∆I/∆p(I)+∆e/∆p(e). Then, eﬃcient contracts are those that Lemma
1(i) speciﬁes. The optimal contracts maximize the entrepreneur’s expected payoﬀ; thus,ℓ + (1 − pℓthe individual rationality constraint for investor (3) binds, i.e., pHH rH
HH )rL =ℓ , r ℓ ) that satisﬁes this constraint and constraint (7) exists inﬁnitely.
IH . The pair (rHL32Therefore, the optimal contracts exist inﬁnitely.
ℓ = r ℓ + ∆I/∆p(I), constraint (3) is
If we assume that constraint (7) binds, i.e., rHL( ℓ)ℓ = I . Then, we have r ℓ = I − p
pHH rL− ∆I/∆p(I) + (1 − pHH )rL
HH
HH ∆I/∆p(I),Lℓ = I +(1−pand therefore, rHH
HH )∆I/∆p(I). Since the repayment balances the budget,

we have the results.
(ii) Suppose ∆y < ∆I/∆p(I) + ∆e/∆p(e). Then, eﬃcient contracts are those that
Lemma 1(ii) speciﬁes. Since the optimal contract binds (3) and (7), the repaymentℓℓ = I −pschedule is rLH
HH ∆I/∆p(I) and rH = IH + (1 − pHH )∆I/∆p(I). By the result
b = r ℓ andof Lemma 1(ii), we have rHHℓrLb = rL
+

(

∆I
∆e
+
− ∆y
∆p(I) ∆p(e)= IH + (1 − pHH )

)

∆I
∆e
+
− ∆y.
∆p(I) ∆p(e)Then, the entrepreneur’s expected payoﬀ isbb
E(y0 − rb |I0 = IH , e0 = eH ) = ȳHH − pHH rH− (1 − pHH )rL()∆I= ȳHH − pHH IH + (1 − pHH )
∆p(I)
)
(
∆e
∆I
+
− ∆y
− (1 − pHH ) IH + (1 − pHH )∆p(I) ∆p(e)
(
)
∆I
∆e
= ȳHH − IH − (1 − pHH )+− ∆y .
∆p(I) ∆p(e)Considering the transfer to the third party, constraint (9) can be written as yH ≥IH + eH + (1 − pHH ) (∆I/∆p(I) + ∆e/∆p(e)).

33

Proof of Proposition 2(i) To induce the entrepreneur to be honest about project output, players must set∆rb = 0; in addition, to induce the investor to lend a large amount of money, they mustset ∆rℓ > 0. These constraints do not hold simultaneously if the repayment schedulebalances the budget.
(ii) There do not exist any eﬃcient contracts which balance the budget. Then, theℓ and r ℓ are thus the as those in Proposition
optimal contract binds (3) and (7), and rH
L

1(ii). Since ∆rb = 0, constraint (8) can be rewritten by ∆y ≥ ∆e/∆p(e). ∆rb = 0 alsomeans that the transfer to the third party is ∆I/∆p(I) when the message is m0 = yL .
ℓ , i.e., r b = r b = I + (1 − p
The entrepreneur always repays rH
H
HH )∆I/∆p(I). Then,H
LyL ≥ IH +(1−pHH )∆I/∆p(I) must hold and the entrepreneur’s expected payoﬀ is ȳHH −IH − (1 − pHH )∆I/∆p(I). The individual rationality constraint for the entrepreneur
requires ȳHH ≥ IH + eH + (1 − pHH )∆I/∆p(I).
(iii) By the proof of Proposition 2(ii), if either ȳHH < IH + eH + (1 − pHH )∆I/∆p(I)or yL < IH + (1 − pHH )∆I/∆p(I), players cannot make any eﬃcient contract.

Proof of Proposition 3
First, we seek optimal contracts which prevent the investor from deviating to lend asmall amount of money in either period 0 or period 1. Later, we show that this contract
is also the solution to the problem with the possibility to deviate twice by the investor.

34

Constraints (IC0) and (IC1) are rewritten as follows:β{pHH (zHH − zLH ) + (1 − pHH )(zHL − zLL )} ≥

∆I
,
∆p(I)

(IC0′ )

β{pHH (zHH − zHL ) + (1 − pHH )(zLH − zLL )} ≥ β

∆I
.
∆p(I)

(IC1′ )

These constraints mean the discounted expected amount that must be transferred to thethird party. Using the deﬁnitions of a ≡ zHH − zHL , b ≡ zHH − zLH , and c ≡ zHH − zLL ,those constraints can be
pHH (a + b) + (1 − pHH )c ≥∆I+ a,β∆p(I)(IC0′′ )

∆I
+ b.
∆p(I)

(IC1′′ )pHH (a + b) + (1 − pHH )c ≥The individual rationality constraint for the investor isβ{(pHH )2 zHH + pHH (1 − pHH )(zHH − a + zHH − b) + (1 − pHH )2 (zHH − c)}≥ (1 + β)IH .
This constraint can be
zHH ≥

1+β
IH + (1 − pHH ){pHH (a + b) + (1 − pHH )c}.
β(10)The entrepreneur who wants to maximize her discounted expected payoﬀs minimizes
zHH subject to the constraints (IC0′′ ), (IC1′′ ), and (10); therefore, he minimizes the
expected amount of the transfer to the third party subject to these constraints.
Note that ∆I/β∆p(I) ≥ ∆I/∆p(I). To set a = 0 and b ∈ [0, (1 − β)∆I/β∆p(I)]minimizes the amount of pHH (a + b) + (1 − pHH )c required by constraints (IC0′′ ) and35

(IC1′′ ). Constraint (IC0′′ ), which prevents period 0 deviation, must bind, but constraint(IC1′′ ), which prevents period 1 deviation, does not have to bind. c will be determinedby c = ∆I/β(1 − pHH )∆p(I) − pHH b/(1 − pHH ).
The minimum amount of pHH (a + b) + (1 − pHH )c equals ∆I/β∆p(I); thus, theoptimal zHH is given as
zHH =

1+β
∆I
1+β
1 − pHH ∆I
IH + (1 − pHH ) ·
=
IH +
.
β
β∆p(I)ββ∆p(I)Next, we show that the optimal contracts we have sought are optimal when we
add the possibility of deviating twice by the investor. Consider the constraint which
prevents the investor from deviating in period 1 given that he has deviated in period 0.
This constraint is
β {E(z|I0 = IL , I1 = IH ) − E(z|I0 = I1 = IL )} ≥ β∆I.

(IC3)Like other constraints, it can be transformed aspLH (zHH − zHL ) + (1 − pLH )(zLH − zLL ) ≥

∆I
,
∆p(I)

(IC3′ )

i.e.,
pLH a + (1 − pLH )(c − b) ≥

∆I
.
∆p(I)

(IC3′′ )Constraint (IC1), which prevents period 1 deviation given that the investor has notdeviated in period 0, can be rewritten similarly as
pHH a + (1 − pHH )(c − b) ≥
36

∆I
.
∆p(I)

(IC1′′′ )Note that pHH > pLH and a = 0, and any pair (b,c) which satisﬁes (IC1′′′ ) satisﬁes(IC3′′ ). It follows that any repayment schedule (z b , zHH , zHL , zLH , zLL ) which satisﬁes(IC1) satisﬁes (IC3). Combining (IC0) and (IC3), we have constraint (IC2). Therefore,
any repayment schedule (z b , zHH , zHL , zLH , zLL ) which satisﬁes (IC0) and (IC1) satisﬁes(IC2), and the optimal repayment schedule under constraints (IC0) and (IC1) is thenalso optimal under constraints (IC0), (IC1), and (IC2).
Since the entrepreneur always repays zHH , the project’s property must satisfy (1 +β)yL /β ≥ zHH , i.e., yL ≥ IH +(1−pHH )∆I/(1+β)∆p(I). The entrepreneur’s discounted
expected payoﬀ is (1+β)(ȳHH −zHH ) = (1+β) {ȳHH − IH − (1 − pHH )∆I/(1 + β)∆p(I)}and therefore the individual rationality constraint for her implies ȳHH ≥ IH + eH + (1 −pHH )∆I/(1 + β)∆p(I).

Proof of Proposition 4
We will obtain the results in the same way as the proof of Proposition 3.

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