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Federal Reserve Bank of Chicago
Equilibrium Lending Mechanism and
Aggregate Activity
By: Cheng Wang and Ruilin Zhou
WP 2000-30
Equilibrium Lending Mechanism and Aggregate Activity
Cheng Wang
Graduate School of Industrial Administration
Carnegie Mellon University
Pittsburgh, PA 15213
chewang@wang.gsia.cmu.edu
and
Ruilin Zhou
Research Department
Federal Reserve Bank of Chicago
P.O. Box 834 Chicago, IL 60690-0834
rzhou@frbchi.org
December, 2000
Abstract
This paper develops a model of the credit market where the equilibrium lending mechanism,
as well as the economy's aggregate investment and output, are endogenously determined.
It predicts that the optimal contract is one of two kinds: either with intensive monitoring
by investors to overcome entrepreneurs' incentive problems, such as most of intermediated
nancing, or with heavy reliance on entrepreneurs, such as market nancing. We show that the
observation that bank lending falls relative to corporate bond issuance during recessions can be
explained by movements in the economy's real factors, such as a decline in average investment
returns, a contraction of credit supply, and paradoxically, maybe even an increase of investment
demand (which worsens credit market condition and intensi es incentive problems).
J.E.L Classi cation: E44, G32.
We thank Ed Green, Rick Green, Je Lacker, Rafael Repullo for helpful discussions. The views expressed herein
are those of the author and not necessarily those of the Federal Reserve Bank of Chicago or the Federal Reserve
System.
1. Introduction
What determines a rm's choice of its mechanism of investment nancing? How is the choice of
a rm's nancing mechanism at the micro level related to investment and output at the aggregate
level? These questions are at the center of recent discussions with respect to the nature and role
of the so-called \credit crunch" which occurred during the most recent 1990-91 U.S. recession.
During this recession, the economy's total outstanding loans fell dramatically and more important,
the fraction of intermediated loans fell dramatically relative to unintermediated loans including
public bonds and commercial paper (Friedman and Kuttner 1993).
There are very few existing theories of the relationship between the mechanism of nancing and
aggregate economic activity. Kashyap, Stein and Wilcox (1993) nd that following a tightening
of monetary policy, while there usually is a sharp increase in the amount of commercial paper
outstanding, bank loans fall. They argue that monetary contraction tightens the supply of bank
credit and hence forces borrowers to switch to commercial paper, and they further view this as
evidence on the existence of a loan supply channel of monetary policy transmission. On the
other hand, Bernanke, Gertler and Gilchrist (1996) advocate a ight to quality view on the same
subject. They postulate both that demand for short-term credit is countercyclical, and that rms
di er in their degree of access to credit markets. Thus, during a recession, high-grade rms borrow
relatively easily by way of issuing commercial paper while low-grade rms, rms which can only
borrow from banks, are constrained.
This paper develops a model of the credit market where the equilibrium lending mechanism,
as well as the economy's aggregate investment and output, are endogenously determined. We then
use the model to examine how the relationship between the equilibrium nancing mechanism and
aggregate output varies in response to disturbances to the model's exogenous variables. Suppose
the economy receives a negative \real" shock. Speci cally, suppose there is a decrease in the
potential returns of an average investment project. Then the credit market may respond by
switching from intermediated bank loans to unintermediated market lending. Meanwhile, fewer
projects will be implemented, the success rates of the implemented projects will be higher ( ight
to quality), and total investment and output will both fall. Thus we provide a \real" explanation
for the observation that economic downturns are often accompanied not only by contractions in
total lending, but also by declines in the ratio of bank loans to non-bank lending.
Our theoretical ndings are based on a lender (investor)-borrower (entrepreneur) relationship
that features adverse selection, moral hazard and costly monitoring. In the model, adverse selection arises in that after the project is funded, the entrepreneur observes a random signal 2 [0; 1]
which indicates the project's success rate. This signal is private to the entrepreneur unless the
investor pays a xed cost to monitor. The project can then be liquidated or fully undertaken. In
the latter case, the entrepreneur must make an unobservable e ort to carry out the rest of the
1
investment process.
We show that the optimal contract has the following characteristics. First, it is always optimal
to fully undertake projects with suÆciently high success rates. Let denote the cut-o level of
the realization of the random signal below which the project is liquidated and above which
the project is fully implemented. Second, either monitoring is never optimal, in which case any
project with continues to be funded without being monitored; or monitoring is optimal, in
which case there exists a second cut-o level of the success rate n 2 (; 1] such that a project
with a that falls between and n is monitored. Third, the optimal compensation scheme is
a debt contract if the contract prescribes no monitoring, and a combination of debt and equity
contract otherwise.
If the optimal policy involves a positive probability of monitoring the entrepreneur, we brand
the optimal contract as a form of bank lending; and, if the optimal contact involves no monitoring at all, we classify the optimal contract as market nancing. This interpretation of the
model is essential for our purpose. In practice, some business enterprises seek nancing from
nancial intermediaries while others borrow directly from the credit market (e.g., commercial
paper, corporate bond). A key distinction between the two nancing mechanisms is that nancial intermediaries often engage in extensive monitoring during the process of nancing, whereas
typical individual lenders do not monitor, or do so much less. A theoretical explanation for this
distinction is that monitoring of private information is more e ective when it is delegated to a
nancial intermediary rather than when done repetitively by individual lenders (Diamond 1984).
The idea that banks are delegated monitors is central to the models of nancial intermediation
based on costly state veri cation (e.g., Williamson 1986, 1987). Recent studies on the choice of
the optimal nancing mechanism by Diamond (1991) and Holmstrom and Tirole (1997) have also
taken seriously the notion that bank nancing is closely related to monitoring. In both papers,
nancial intermediaries are modeled as monitors who can detect bad projects.
We now explain why a negative productivity shock can cause both the aggregate output and
the ratio of intermediated loans to unintermediated loans to fall. In our model, it holds that in the
absence of monitoring, the agency costs that must be incurred by the lender are higher if projects
with lower success rates are undertaken. Now suppose the economy receives a productivity shock
that lowers the return of a successful project. Then fewer projects should be fully funded (that is,
should be higher). But this implies monitoring would be less eÆcient relative to no-monitoring,
which in turn implies the ratio of intermediated loans to unintermediated loans would decrease.
Meanwhile, because fewer projects are fully funded, total outstanding loans and aggregate output
would be lower.
Our model is also rich enough to permit studies of other interactions between the credit
market and the aggregate variables. In particular, in our model, it can be the case that the
economy's total output is higher when it has less investment opportunities than when it has more
2
investment opportunities. This seemingly counterintuitive result can be explained as follows.
When the economy is endowed with more investment opportunities, competition for loans will
lower the equilibrium expected utility of the borrowers. This, given limited liability, makes the
agency problem more severe, thereby causing more liquidation and less output. On the other
hand, if the economy is endowed with less investment opportunities, competition for projects will
shift the bargaining power from the lender to the borrower, thus raising the equilibrium expected
utility of the borrower and lowering agency costs, resulting in less liquidation and more output.
Another lesson we learn from the model is that when the economy is experiencing a decline
in bank lending, the economy's total output may rise or fall, depending on the source of the
decline. Put di erently, \credit crunch" is not necessarily bad news. It depends on what causes
the crunch. We show that if the decline in bank lending is caused, say, by an increase in the cost
of monitoring or by a decrease in the potential returns of the project, then total output falls as
bank lending declines. If the decline in bank lending is caused by a decrease in the economy's
endowment of investment projects, then under some conditions the economy's total output could
increase while total bank loans fall.
An important feature of our model is that whenever there is a shortage of funds, in equilibrium
there is always credit rationing of the type discussed by Stiglitz and Weiss (1981) and Williamson
(1986, 1987), where among a group of identical borrowers, those who receive loans are strictly
better o than those who do not. Credit rationing in our model is motivated sometimes by costly
monitoring (as in Williamson 1987) and sometimes by costly over-liquidation. A lower reservation
utility of the borrower may imply that his project must be liquidated with an excessively higher
probability, which lowers the lender's expected returns on a loan. The notion that credit rationing
is a mechanism to avoid excessive liquidation has not been discussed in the literature.
This paper builds on the large literature in contract theory that follows Townsend (1979)
in modeling the role of costly monitoring in optimal nancial arrangements, including Gale and
Hellwig (1985), Williamson (1986, 1987), and Boyd and Smith (1997). At the heart of our model
is the interaction between the investor's optimal monitoring policy and optimal nancing strategy.
In which states of the project should the investor monitor, and what happens subsequently? Could
it be optimal that in some states the project is not monitored but fully nanced, whereas in other
states the project is monitored but subsequently abandoned? These questions, though obviously
important for the study of investment nancing, have not been addressed explicitly by the existing
literature. Holmstrom and Tirole (1998) also model the optimal liquidation decision conditional
on the realization of a random signal (a liquidity shock in their environment). But they abstract
from the problem of costly monitoring by assuming the random shock is observable (or when
it is not, it still does not a ect the structure of the optimal contract). Admati and P eiderer
(1994) have a model which is somewhat similar to ours and they assume monitoring is not costly.
Modeling explicitly the process of costly monitoring allows us to study the interaction between
3
costly information acquisition and the investment nancing decision. But as we will show, solving
the optimization problem is by no means a trivial task.
Section 2 presents the model. In Section 3, we study the two-agent optimal contract assuming
a certain credit market outcome. Section 4 embeds the optimal contract in a perfect competitive
credit market, and analyses the market equilibrium. It then considers the implications of the
model's comparative statics. Section 5 concludes the paper.
2. The Model
There are three periods, = 0; 1; 2: There are two types of agents, investors and entrepreneurs,
and there is a continuum of each type such that the measure of the investors is , and that of the
entrepreneurs is Æ. All agents are risk neutral. Investors maximize their expected consumption
in period 2, entrepreneurs maximize the expected value of u(c; e) = c e, where c is consumption
in period 2 and e is e ort exerted in period 1.
In period 0, each investor has one indivisible unit of investment good, which can either be
invested in the credit market which matches worthy projects (entrepreneurs) with investment
goods (investors), or earn a certain gross return of one unit of consumption in period 2 through
storage. Each investor also has access to (> 0) units of the consumption good in period 2. We
will assume that is large enough to ful ll all payments speci ed by nancial contract. Each
entrepreneur owns a risky investment project, which requires an investment of one unit of the
investment good in period 0, or it simply perishes. No entrepreneur has any initial wealth, and
hence he must rely on external nancing in order to undertake his project. All investors are ex
ante identical, so are all entrepreneurs (hence, their projects).
In period 0, there is a competitive credit market in which investors o er lending contracts to
entrepreneurs who exchange investment opportunities for credit and compensation. Given that
all agents are risk neutral, and all projects are identical ex ante, without loss of generality, we
assume that each entrepreneur obtains funds from at most one investor, and each investor invests
in at most one project. That is, any contract is formed exclusively between one entrepreneur and
one investor. At the end of period 0, there may be projects unfunded or investment goods unused,
depending on the size of each side of the market Æ and . All contracts traded are identical at
a credit market equilibrium, each promises an expected utility equal to u0 to the entrepreneur
party. This equilibrium expected utility of the entrepreneur will be determined endogenously.
Without loss of generality, we discuss the contracting and investment problem between two
generic agents of each type: an investor I and an entrepreneur E. Assume that E's project is
worth funding and is funded by I. The timing of the events unfold as indicated in Figure 1.
At the beginning of period 1 the entrepreneur E observes a signal . Here 2 [0; 1] is a
random variable that represents the potential success rate of the project. We assume that is a
4
continuous random variable on with a distribution function G() and a density g(), and for
all 2 , g() > 0. The signal is directly observable only to the entrepreneur. The investor I
can observe the realization of through a costly monitoring process, which requires 0 units
of I's e ort. Here, may be interpreted as e ort required to discover the project's technical
feasibility and market pro tability. If the investor monitors, then she learns the true realization
of . Otherwise, she knows only the entrepreneur's report.
Figure 1. The Timing of a Project Development
E observes
reports
=0
6
=1 ?
contracting
investment
6
I
I
8
>
>
<
>
>
:
liquidate ) Stop
return
continue
??
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
8
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
:
6
monitor ) truth
report
cost
not
monitor ) report
no cost
report
E
project return realized:
if e = t
return = H
0
if e = 0
return = 0
=2
(
wp
wp 1
?
8
>
<
e ort e = t
>
:
no e ort e = 0
(unobservable)
After the observation of the signal by at least one party, a decision must be made as for
whether to continue the project or liquidate it. The liquidation value of the project is 0; measured in units of period 2 consumption. If the investor I monitors E's report of , then of course
the liquidation/continuation decision can be based on the true realization of . Otherwise, the
liquidation/continuation decision may take into account only E's report. Suppose the investment
is continued, then the entrepreneur E makes an unobserved e ort of either t > 0 or zero. In other
words, there is moral hazard.
If the project is continued, at = 2, the return is realized. If E makes the required e ort t
to his continued project, then with probability the project succeeds with return H > 0, and
with probability 1 it fails and yields nothing. If E does not make the required e ort, then the
project fails with probability one. We call H the potential return of the project, and the project's
success rate. At the end of period 2, the contract ends with transferring the speci ed payment,
5
depending on the commonly observed events occurred during the project development process
(monitoring, continuation/liquidation, and project return), from the investor to the entrepreneur.
The investor receives the residual return of the project.
We make some further assumptions. First, all payments to the entrepreneur must be nonnegative (limited liability). Second, renegotiation is not allowed. In other words, we assume that
once the contract is signed, both parties can fully commit, at all stages of the investment process,
to the terms of the initial contract. Third, a necessary condition that a project is worthy of
investment ex ante is that at = 1, with the best possible realization of (= 1), the project
return H is higher than all the potential costs from then on. That is,
Assumption (1) t + + H:
Finally, we make a technical assumption to guarantee the uniqueness of solution to the optimal
contracting problem in the sections to follow.
Assumption (2)
H
H
(t + )
gg(()) t + H +
0
:
Clearly, there is a wide range of distribution functions with support [0; 1], including the uniform
distribution, which satisfy the above condition.
In what follows, we rst investigate the optimal contract between an investor and an entrepreneur, assuming a certain outcome of credit market competition. We then use the obtained
optimal contract to study credit market equilibrium.
3. Two-Agent Optimal Contract
In this section, we determine the form of the optimal contract between a representative investor
I and entrepreneur E pair. Suppose that the period-0 credit market competition yields that an
entrepreneur's compensation from his funded project is at least u0 0 in terms of expected utility.
The optimal contract, then, maximizes the investor's expected utility subject to the constraint
that the entrepreneur's expect payo is no less than u0. As u0 varies, the contract moves along
the Pareto frontier between the two agents.
3.1. The First-Best Contract
Consider the case where both the realization of and entrepreneur E's e ort t are publicly
observable. Given that there is no private information and moral hazard problem, entrepreneur
E can be compensated with a xed payment, denote it x. In addition, a contract must specify
a liquidation/continuation policy , a subset of : if 2 , then the project is continued;
6
otherwise, it is liquidated. Given the environment, the optimal contract must implement t as the
entrepreneur's e ort.
Let 0 denote the complement of the set . The problem of optimal contracting can then be
formulated as follows.
Z
(P 0)
max
HdG( ) +
x;
subject to
Z
x
Z
0
dG( )
x
tdG( ) u0 :
(1)
(2)
The objective function (1) represents investor I's expected payo .1 Condition (2) is entrepreneur
E's participation constraint: his expected return is no less than the expected credit market payo
u0 . Clearly, constraint (2) must be binding, since otherwise reducing the value of x can improve
I's expected payo while holding the participation constraint satis ed. By substituting constraint
(2) into the objective function, we can rewrite the optimal contracting problem as
max
Z
t dG( ) +
H
Z
0
dG( )
u0 :
(3)
Obviously, the optimal must be an upper interval of .2 Let this interval be [fb; 1], fb =
argmaxx2[0;1] F (x) where
1
F (x)
H t dG( ) + G(x):
x
It can be shown that function F (x) is strictly concave under assumption (2),3 and therefore the
maximization problem (3) has a unique solution:
= [ ; 1]; where = t + :
(4)
Z
fb
fb
fb
Given fb, E's compensation is determined by x = u0 + (1
(expected net returns on an investment) is given by
Vfb
=H
1
Z
fb
(
fb )dG( ) +
H
G(fb ))t,
u0
1:
and I's expected payo
(5)
For convenience, we omit the constant unit cost of date-0 investment in all of the objective functions.
If a project0 with a lower success rate is continued,0 then a project
with a higher should also be continued.
3
We have F (x) = [xH (t + )]g(x): Obviously, F (0) > 0, F 0 (1) < 0. Also, F 00 (x) = [xH (t + )]g0 (x)
Hg (x): So for the function F (x) to be concave, we need
h
i
g 0 (x)
xH (t + ) < H:
g (x)
Now if xH (t + ) 0, then the above inequality certainly holds. If xH (t + ) < 0, then the concavity condition
becomes
g 0 ( x)
H
<
g (x)
(t + ) xH ;
which holds, by the second inequality of assumption (2).
1
2
7
Note that given both parties are risk neutral, it is straightforward to show the following holds.
The rst-best outcome is achievable when there is only moral hazard concerning the entrepreneur's
e ort, but no information asymmetry with respect to the project's success rate . The rst-best
outcome is also achievable if there is only asymmetric information concerning the success rate ,
but there is no moral hazard.
3.2. Contract with Costly Monitoring and Moral Hazard
Now, we consider our original model where the realization of the project's success rate
is directly observable only to entrepreneur E. Investor I can observe the value of at a cost
> 0. Moreover, there is moral hazard: entrepreneur E's e ort is not observable. In this
standard nite-horizon, two-person game with adverse-selection and moral-hazard problems, the
revelation principle applies. That is, every equilibrium allocation of any arbitrary mechanism
can be implemented as an equilibrium of a revelation mechanism. Therefore, we will focus on
incentive compatible mechanisms to characterize the optimal contract.
3.2.1.
The De nition of contract
With costly monitoring and moral hazard, there are now three components to a loan contract: (i) a monitoring policy M for verifying the state of the success rate , (ii) a liquidation/continuation policy which determines whether or not the project is liquidated after the
realization of the state , and (iii) a scheme for state contingent compensations to the entrepreneur.
Formally, a contract takes the following form:
= M ; ; x; y (~); ~ 62 ; R0 (~); R(~); ~ 2 :
n
o
We abstract throughout from stochastic monitoring; thus, the monitoring policy M is a subset of
in which veri cation of the reported state will occur. That is, let ^ denote the entrepreneur's
report of , then monitoring takes places if and only if ^ 2 M .
The liquidation/continuation policy is also a subset of . Unlike in the case of complete
information, here as well as compensation schedule must take into account the fact that there
is information asymmetry between investor I and entrepreneur E concerning the realization of .
Let ~ denote investor I's knowledge of the realization of on which the liquidation/continuation
decision must be conditioned:
;
if ^ 2 M;
~(^; ) =
^;
otherwise.
8
>
<
>
:
Then the project is continued if ~ 2 , and it is liquidated if ~ 62 .
8
In the state of liquidation, investor I seizes the scrap value of the project , and entrepreneur
E receives a payment y(~). Conditional on the project being continued, entrepreneur E is paid
R0 (~) 0 if the project eventually fails; and he is paid R0 (~) + R(~) 0 if the project succeeds
with realized return H . Finally, the contract speci es a xed payment x 0 that entrepreneur E
receives in period 2. This payment is not contingent on the state of nor the realization of the
project's random return.4
Given that monitoring is costly, it may not be eÆcient to always monitor. To simplify matters,
we assume that if entrepreneur E is indi erent between reporting truthfully and lying, he reports
truthfully. Therefore, if both the realization and the report ^ are in the monitoring region
M , there is no point to lie. Furthermore, for any realization not in the monitoring region M ,
entrepreneur E gains nothing by giving a false report in M to induce monitoring. Since in either
case, E's payo will depend on the truth only. This implies that entrepreneur E will not submit
a false report of for monitoring. In other words, we have:
Lemma 1. If ^ 2 M; then ^( ) = :
As mentioned in the introduction, a focus of this paper is the joint determination of the
optimal monitoring policy M and liquidation/continuation policy . It is thus useful to de ne
the following subsets of :
A \ M;
B
0 \ M;
C
\ M 0;
D 0 \ M 0 ;
where 0 and M 0 are the complements of and M , respectively. By Lemma 1, if E's report of
the state ^ is in A[B ], then monitoring will occur and the project will [will not] continue. On the
other hand, if ^ 2 C [D] , then the project is not monitored and it will [will not] continue.
Now consider the set D, the non-monitoring/liquidation region. Suppose that 1; 2 2 D and
y (1 ) > y (2 ). Then, whenever 2 is realized, E could lie and report 1 to get the higher payo
y (1 ), given that both 1 and 2 are not monitored. This implies that y ( ) must be constant on
D in order for the contract to be incentive compatible.
Lemma 2. An incentive compatible contract satis es y ( ) = YD for all 2 D .
Given Lemmas 1 and 2, an incentive compatible contract has to satisfy the following three
sets of incentive constraints. First, there should be no incentives for entrepreneur E to report
untruthfully a ^ 2 C in order to continue the investment process without monitoring (conditions
(6) and (8)). Second, there should be no incentives for entrepreneur E to report untruthfully
a ^ 2 D so that a good project is abandoned to avoid making e ort or to receive a better
compensation YD (conditions (7) and (9)). Third, there should be no incentives for entrepreneur
Clearly, x is a mathematically redundant component of the contract, and we have introduced x only for
analytical convenience.
4
9
E to shirk whenever the project is continued (condition (10)). Formally,
Truth-telling constraints:
8 2 A [ C; 8 ^ 2 C R() + R0() t max R(^) + R0(^) t; R0(^)
(6)
R( ) + R0 ( ) t YD
(7)
8 2 B [ D; 8 ^ 2 C y() max R(^) + R0 (^) t; R0 (^)
(8)
8 2 B y ( ) Y D
(9)
E ort constraint:
8 2 A [ C R() + R0() t R0 ()
(10)
The entrepreneur's participation constraint is as follows,
x+
R( ) + R0 ( ) t dG( ) +
y ( )dG( ) u0 :
(11)
A[C
B [D
We are now in a position to de ne optimality. We call a contract optimal if it maximizes
investor I's expected payo , subject to the incentive constraints, the participation constraint,
and the limited-participation constraint for entrepreneur E. That is, an optimal contract solves
the following problem,
(P 1) max
[H R() R0 ()] dG() +
[ y()]dG()
(12)
A[C
B [D
(A [ B )
x
subject to (6)|(11)
x 0;
8 2 B [ D y() 0;
8 2 A [ C R0() 0; R0 () + R() 0
(13)
where denotes the probability measure on : for any set Z , (Z ) = Z dG().
n
n
Z
Z
o
o
Z
Z
R
3.2.2.
The optimal contract
We now set out to analyze the properties of the optimal contract. Our rst task is to simplify
the incentive constraints. The approach we take is to consider a class of optimal contracts, all of
which deliver the same expected utilities to both entrepreneur E and investor I, and then show
that each contract in that class is equivalent to a contract whose compensation scheme resembles
that of a debt or an equity contract. In the following, any two contracts are said to be equivalent
if they satisfy the same set of constraints and promise the same expected payo s to both the
investor and the entrepreneur.
Proposition 1.
For any contract that solves (P 1), there exists a contract ^ which is
equivalent to , and ^ has the following properties: for all 2 A [ C , R0 () = 0, and for all
2 C , R( ) = RC , where RC 0 is a constant.
10
Proposition 1 implies that we can focus on the set of contracts which have a relatively simple
compensation structure: conditional on the project being continued, the entrepreneur's compensation is zero if the project fails. Moreover, if the project succeeds, and if there is no monitoring,
entrepreneur E's compensation is independent of his report of . The intuition for this result
is simple. The debt structure is eÆcient here partly because it imposes the largest possible
punishment for a bad outcome. The constant compensation on C is required by truth-telling
constraint. The technical proof of this proposition, however, is somewhat involved because of the
tangled truth-telling and e ort-making incentive constraints. The proof of Proposition 1 is in the
appendix.
Proposition 1 allows us to focus on a set of simpler contracts where the compensation schemes
are debt-looking in the continuation regions A and C . Note that by constraint (10), we have
R( ) t= > 0 for any in A or C , which implies the non-negativity of R( ) on A and RC .
Hence, the optimal contracting problem can be simpli ed as follows:
(P 2)
max
Z
Z
Z
[H R()] dG() + H RC dG() + ( y())dG()
(14)
C
B
+( YD )(D) (A [ B ) x
subject to
x 0; 8 2 B y ( ) 0; YD 0;
(15)
x + (R( ) t)dG( ) + (RC t)dG( ) +
y ( )dG( ) + YD (D ) u0 (16)
A
Z
Z
A
Z
C
B
and the following set of incentive constraints,
8 2 A
R( )
8 2 B
y ( ) RC
8 2 C
8 2 D
8 2 A
8 2 C
R( )
t RC
t YD ;
y ( ) YD ;
RC
t YD ;
RC
R( ) t
RC t:
YD
t;
t;
t;
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
The next proposition shows that the optimal liquidation/continuation policy is monotonic.
That is, if a project with success rate is continued, then any project with a higher rate of
success is also continued.
Proposition 2. For any optimal contract that solves problem (P 2), there exists a contract
^ satisfying: for all 1 2 and 2 2 0 , 1 > 2 and ^ is equivalent to .
11
Suppose a project with success rate 1 is continued, but a project with success rate 2 (2 > 1)
is liquidated. Then by switching the positions of 1 and 2, and by re-arranging the compensation
schemes properly, one can achieve a Pareto improvement. The proof of Proposition 2 is in the
appendix.
Given Proposition 2, we can then focus, without loss of generality, on contracts with monotonic
liquidation/continuation policies; that is, contracts in which the set is an upper interval of .
Our next lemma shows that for optimality, this upper interval must not be empty.
Lemma 3. If is an optimal contract, then 0 6= ;.
The proof of Lemma 3 is in the appendix. The next contract speci es the main structure of
optimal contract.
Proposition 3.
(i)
(ii)
B
An optimal contract has the following characteristics:
= ; and 0 = D.
There are constants m and n , 0 < m n 1 such that
A = [m ; n ); C
(iii)
= [n; 1];
D = [0; m ]:
Moreover, the following compensation scheme is optimal:
8 2 A = [m; n) R() = t=;
8 2 C = [n ; 1]
RC
= t=n;
YD
= 0:
By Proposition 3, it is never optimal to have the project monitored and then abandoned.
Moreover, the optimal monitoring strategy is to monitor those success rate which are neither too
low, nor too high. Put di erently, it is optimal not to monitor when the \news" from entrepreneur
E is suÆciently good or suÆciently bad. The entrepreneur's compensation is zero when the project
is abandoned and when the project is continued but fails. When the project is continued and
succeeds with return H , entrepreneur E's compensation is nonlinear in the realization of : it is
relatively high but decreasing in the success rate in the region where monitoring occurs, and it
is low (but positive) and constant across the region where monitoring does not occur.
Note that the optimal monitoring strategy is not monotonic over the whole state space ,
although it is monotonic conditional on the investment being continued. Also, given that the
investment is continued, entrepreneur E's expected net compensation is monotonic and piecewise
linear in : it is zero for all 2 A and (t=n ) > 0 for all 2 C .
The intuitions for Proposition 3 are as follows. There is no need to monitor a project that
is to be abandoned (B = ;), since it is a waste of resources without any return. Conditional
on continuation, it is optimal to monitor lower rather than higher reports of because doing so
minimizes the cost of monitoring. To see this, suppose there is a 0 in A that is greater than the
lowest in C . Then continuing a project with report 0 without monitoring (that is, move 0 into
12
C)
can result a net gain for investor I without violating any other constraints. Entrepreneur E's
expected payo can be maintained by increasing the xed payment x by the amount of reduction at
0 from R( 0 ) to RC . The change is also incentive compatible. First, the e ort-making constraint
is satis ed at 0 since it is satis ed at the lowest in C . Second, there are no incentives for the
entrepreneur to misreport a realization 0 as some other 00 in D, since his payo in the liquidation
stage YD is lower than 0RC t, his expected payo if the project continues. Third, there are no
incentives for him to misreport a di erent 00 in C other than 0, since the payo is the same with
both reports on C . However, this change strictly improves investor I's expected payo because
it reduces monitoring cost in state 0.
To explain part (iii), notice that it is optimal to set the entrepreneur's compensation, YD in
the liquidation region D and R() and RC in the continuation regions A and C , just high enough
that proper incentives are given for truthful reporting and e ort making. Holding the levels of
YD , R( ) and RC too high is potentially costly: it may cause the entrepreneur's overall expected
compensation to exceed his reservation utility, since x 0 must hold. Now given (i) and (ii), it
is easy to check that the compensation scheme speci ed in the proposition is the lowest possible
that still satis es all the incentive constraints.
Given Proposition 3, the optimal contract is characterized fully by variables x, m, and n,
with x 0 and m n. Hence, the optimal contracting problem (P 2) can be rewritten as follows:
(P 3)
Z
n
max
x; ;
subject to
m
n
m
H
t
dG( ) +
x + T (n ) u0 ;
x 0;
where
Z
m ; n
n
2 [0; 1];
Z
1
1 t
H
m n ;
t
dG( ) + G(m )
n
x
(25)
(26)
(27)
(28)
The term T (n) is the lowest reservation utility of entrepreneur E in order to induce his full e ort
for all continued projects in non-monitoring region [n; 1]. We have
T (n )
T 0 (n ) =
n
t
t dG( ):
n
Z
1
(n)2 dG() < 0:
Thus, in the absence of monitoring, more must be promised to entrepreneur E if more projects
are continued.
Obviously, the problem (P 3) has a solution, denote it fx ; m ; ng. To solve problem (P 3),
we consider the cases where the entrepreneur's participation constraint (26) is binding and slack
separately.
n
13
First, suppose that constraint (26) is binding at the optimum. Then, substituting x = u0
T (n ) into the objective function (25) and maximize it without constraint (27), we have
1
= = = (t + )
(29)
m
and
fb
n
H
= u0 T (fb):
(30)
That is, no monitoring is necessary, and the optimal cuto level for project continuation coincides
with the rst-best continuation level fb. Of course, for fx ; m ; ng, given by (29) and (30), to be
the solution, it has to satisfy x > 0, given that fb 2 [0; 1] by assumption (1). That is,
x
T (fb ) u0 :
(31)
When condition (31) holds, fx ; m ; ng given by (29) and (30) is the solution to (P 3). In fact,
condition (31) is the necessary and suÆcient condition for this to be the solution and for the
rst-best continuation/liquidation to be achievable.
Next, suppose that condition (31) does not hold. Then it must hold that x = 0, for otherwise,
given T 0(n ) < 0, it is possible to reduce the values of x and n simultaneously and to increase
the investor's expected payo without violating the participation constraint (26). With x = 0,
constraint (26) becomes T (n ) u0 . Let n be the level of at which the constraint binds, that
is,
T (n ) = u0 :
(32)
Since u0 = T (n) < T (fb), and T is decreasing in n , we have fb < n 1. Also because T (n) is
decreasing, the participation constraint T (n ) u0 is equivalent to n n. Now we can rewrite
the optimal contracting problem (P 3) as follows:
(P 4)
Z
1
Z
n
max
O (m ; n )
H t dG( ) + G(m ) T (n )
dG( )
(33)
;
subject to m; n 2 [0; 1]; n m; n n:
(34)
For future reference, let (^m; ^n) be the solution for the unconstraint problem (33). Then,
t++
^m =
(35)
H
m
n
m
m
1
dG( )
(^n )2 ^n
t
Z
g (^n ) = 0:
Furthermore, let ~n be the solution to the following equation,
1
1
t
dG( ):
~n = (t + ) +
H
(~n )2 g(~n )H ~
Z
n
14
(36)
(37)
It is easy to show that fb < ^m 1, 0 ^n < 1, and fb < ~n < 1.
Problem (P 4) is well de ned: the objective function O(m; n ) is strictly concave, and the
constraint set de ned by (34) is convex. So it has a unique solution. The detailed solution is
given in the appendix. The following proposition summarizes the solution for problem (P 3).
Proposition 4.
(i)
The optimal contract takes one of the following three forms:
If u0 T (fb), then
= fb; M = ;; = [fb; 1]; and x = u0 T (fb):
(ii) If u0 < T (fb), and ^m < minf^n ; ng, then m = ^m, n = minf^n; ng,
sb :
< ; M = [ ; ); = [ ; 1], and x = 0,
m
fb < m
n
m n
m
(iii) If u0 < T (fb) and ^m minf^n ; ng, then m = n = minf~n ; ng,
nsb : fb < n ; M = ;; = [n ; 1], and x = 0.
nfb
:
n
Proposition 4 states the precise conditions under which the optimal contract takes what particular form. One critical element is u0, the reservation utility of entrepreneur E. If u0 is high
enough, the optimal contract nfb achieves the rst-best outcome without monitoring. When u0 is
below a certain threshold; u0 < T (fb), only second-best outcome can be obtained. In such a case,
the optimal contract may be one with some monitoring (reports in region M = [m ; n), contract
sb ), or without monitoring at all (contract sb ). In either cases, the cut-o level for continuation
m
n
(m under msb, n under nsb ) is the higher than that of the rst-best solution. That is,
Corollary 4.1.
the optimum.
Whenever the rst-best is not attainable, there is always over-liquidation at
Proposition 4 also shows that unless the rst-best is attainable, the non-state-contingent
component of the compensation x must be zero. Thus, when the rst-best is not attainable,
the optimal contract takes the form of debt (nsb ) or combined debt and equity (msb ) contract.
Moreover, at the optimum, the entrepreneur earns positive compensation only in states where
the project is carried out without being monitored; that is, only when the project is suÆciently
good and when the project is ultimately successful. Given (m ; n), the optimal payo schedule
for entrepreneur E is given by Proposition 3. That is,
8 2 [0; m ) y() = 0;
8 2 [m ; n)
R ( ) = t=;
8 2 [n; 1]
R ( ) = t=n :
(38)
Let V (u0 ) denote investor I's net expected payo from the project as a function of the reservation utility u0 of entrepreneur E. Then,
V (u0 ) = H
Z
1
m
fb dG( )
Z
n
m
15
dG( ) +
(x + T (n)) 1:
(39)
Not surprisingly, when the rst-best is attainable, V (u0 ) coincides with Vfb as de ned in (5). We
call V the investor's value function. For the optimal contract to be valid, the project has to yield
positive expected net return for the investor, that is, V (u0 ) 0.
By proposition 4, given the parameters of the model, H; ; ; and t, there are two possible
cases: (1) ^m < ^n at which for some values of the entrepreneur's reservation utility u0 , the optimal
contract involves monitoring, and (2) ^m ^n at which there is never monitoring at optimum
regardless of u0. De ne
= ^n ^m:
(40)
Consider case (1) rst ( > 0). By proposition 4, the value function (39) can be divided into
ve segments, as depicted in Figure 2. When, u0 u^0, where
u^0
= T (fb)
(41)
the optimal contract is the non-monitoring rst-best nfb. The threshold between contract msb or
nsb being optimal, call it u0 , is determined by n = ^m , that is,
u0 = T (^m ):
(42)
When the optimal contract is msb, i.e. when u0 2 [0; u0), the cuto level above which continued
projects are not monitored, n, can be either ^n or n. The value function V is di erent depending
on which value n takes. In particular, when n = n, entrepreneur E is paid exactly u0 , and
V is strictly decreasing as u0 rises. But when n = ^n , entrepreneur E is paid T (^n ), which is
independent of u0, and as a result V is constant in u0. Let u10 be the boundary between the two
regions, which is determined by ^n = n, that is,
u10 = T (^n ):
(43)
Similarly, when the optimal contract is nsb, i.e. when u0 2 [u0 ; u^0 ), the cuto level below which
projects are liquidated, n, can be either ~n or n. When n = n , more projects is liquidated as
u0 decreases, entrepreneur E is paid exactly u0 , and function V is strictly decreasing in u0 . But
when n = ~n, the liquidation cuto level is xed at ~n, entrepreneur E is paid T (~n ), which is
independent of u0 , and as a result V is constant in u0 . Let u20 be the boundary between these
two regions, which is determined by ~n = n , that is,
u20 = T (~n ):
(44)
It is easy to verify that 0 u10 u0 u20 u^0 . In case (2) ( 0), the value function
V corresponds to only a portion of the function under case (1): segments III, IV and V. To
summaries, the value function V may consist of either ve or three segments.
16
If > 0, V (u0 ) may be divided into ve segments, as depicted in Figure 2:
I: 8 u0 2 [0; u10 ), msb is optimal, (m ; n) = (^m; ^n), V is constant.
II: 8 u0 2 [u10 ; u0 ), msb is optimal, (m ; n) = (^m; n), V is strictly decreasing.
III: 8 u0 2 [u0 ; u20 ), nsb is optimal, m = n = ~n, V is constant.
IV: 8 u0 2 [u20; u^0), nsb is optimal, m = n = n, V is strictly decreasing.
V: 8 u0 u^0 , nfb is optimal, m = n = fb, V is linear and strictly decreasing.
If 0, then u10 = u0 = 0, and V is divided into III, IV and V three segments as above.
Figure 2. The Investor's Value Function
V
6
JJ
SS
@l
Q
HH
= ^
m
m
n = ^n
HH PPX
X
II
u10
-
nsb
sb
m
I
3.2.3.
=
^m
m
n = n
III
u0
n
= ~n
``hh
@@
@
@
V @
IV
u20
n
= n
nfb
u^0
- u0
n = fb
Analyzing the optimal contract
Suppose that in addition to assumption (1) and (2), V (u0 ) 0, that is, investment in the
project is bene cial for both parties. Two questions are of particular interest. First, under what
conditions does T (fb) u0 hold and hence the rst-best is attainable? Second, suppose that the
rst-best is not attainable, then under what conditions is monitoring optimal? Both questions
can be studied with the criterion given in Proposition 4.
Consider rst the condition T (fb) u0 , which is necessary and suÆcient for attaining the rstbest outcome. From the analysis above, T (fb) is the minimum amount of expected compensation
17
needed, in absence of monitoring, to induce truth-telling and e ort-making if all projects with
potential success rate above fb are to be continued. For this reason, we call T (fb) the incentive
cost to rst-best nancing. Condition (31) requires that this incentive cost to be lower than
entrepreneur E's reservation utility u0 . The following proposition summarizes the conditions
under which the rst-best is attainable. For convenience, de ne
1
(H; t; ; u0 ) ut0 fb
( fb)dG() 0:
(45)
Then, condition T (fb) u0 is equivalent to 0. Note that is independent of the monitoring
cost .
Proposition 5. Suppose that assumptions (1) and (2) hold, and V (u0 ) 0. Then holding
Z
fb
other parameters constant,
(i)
(ii)
(iii)
0 if and only if u0 u^0.
There exists H^ t + such that 0 if and only if H H^ ;
If 0 holds for some ^, then it holds for all ^.
Part (i) of the proposition is obvious. Parts (ii) and (iii) are given by the facts that function
is increasing in fb and that fb is decreasing in H but increasing in . The proof is omitted.
When the rst-best is not attainable, the optimal contract is either one at which some projects are
monitored or one at which no project is monitored. The following proposition characterizes the
parameter space for each of these two forms of second-best nancing mechanism to be optimal.
Proposition 6. Suppose that assumptions (1) and (2) hold, V (u0 ) 0, and that
< 0.
^
Let H and u^0 be de ned as in Proposition 5. Holding other parameters constant,
(i) For all u0 2 [0; u0 ), msb is optimal, and for all u0 2 [u0 ; u^0 ), nsb is optimal.
^ H ), nsb is optimal, and for all H > H , msb
(ii) There exists H > H^ suc that for all H 2 (H;
(iii)
(iv)
is optimal.
There exists 2 [0; H t ) such that for all , msb is optimal; and for all
nsb is optimal.
Suppose that msb is optimal for some > 0, then msb is optimal for all .
,
The proof for Proposition 6 is given in the appendix. In both lemmas, we omit the discussion
of the e ect of moral hazard cost t on the form of optimal contract because of its complexity.5
5
We know only when the e ort cost t is very large or very small, the rst-best nancing is achievable. This is
because in both cases, the incentive cost T (fb ) is small: when t is very small, the direct e ect of a small e ort cost
implies that the incentive cost is small; when t is large, the indirect e ect of inducing more liquidation (fb is high)
implies that the incentive cost required for the small amount of continued project is also low. However, in general,
there is no monotone relationship between t and the incentive cost. As t increases, the compensation required to
overcome the moral hazard problem for the continued project increases, but the optimal level of total amount of
the project continued decreases. These two opposite e ects are what create the potential non-monotonicity of the
relationship.
18
We now explain the intuition behind Propositions 5 and 6.
There are four cost factors at work. First, monitoring cost is paid for project whose success
rate is in the monitoring range M . Second, entrepreneur E has to be paid incentive cost T (n ),
which is a result of giving the entrepreneur enough incentive to report truthfully and to make
the required e ort t for all continued but not-monitored projects as if they are the ones with
the lowest signal in the region. Third, as Corollary 4.1 indicated, there will be over-liquidation
when the rst-best is not attainable. And fourth, entrepreneur E's expected net payo from the
project has to meet his reservation utility u0. The form of optimal contract is usually a result of
balancing two of the relevant factors out of these four. We discuss the in uence of each of the
parameters of the model on the choice of optimal nancing through their e ects on these factors,
assuming assumptions (1) and (2) are satis ed, and V (u0 ) > 0.6
E 's reservation utility u0 :
sb
m
nsb
u0
u^0
nfb
All four factors may be at play here, although not concurrently. Figure 3 depicts the relationship between the incentive cost function T () and the reservation utility u0 , and their e ects
on the optimal contract, when monitoring contract is optimal for some u0 (corresponds to Figure
2, the ve segmented value function V ). When the credit market awards a high expected utility
u0 (> u^0 ) to entrepreneur E, the rst-best outcome is achievable. In such a case, the incentive
cost T (fb) is part of the u0 payment. No monitoring is necessary, since entrepreneur E's stake in
the project is high enough to avoid any of his incentive problems.
When u0 is in the intermediate range (u0 2 (u0 ; u^0), note that it is possible u0 = 0), it is
not worthwhile for investor I to pay the incentive cost to rst-best nancing T (fb), which is
higher than u0 . It is also too costly to pay the monitoring cost and liquidating more projects
than necessary. Continuity argument suggests that raising the liquidation threshold n, and hence
reducing the incentive cost T (n) to u0 preserve the best interests of both parties. This leads to
the optimal contract to be the one with non-monitoring, nsb. However, when u0 2 (u0 ; u20 ], setting
the liquidation threshold n by equating T (n ) to u0 (n = n ) may resulting too much liquidation.
In such a case, entrepreneur E is paid more than his market share of the project u0 since the gain
to continue the pro table project (or the cost of liquidating it) outweighs the incentive cost.
When u0 is low (u0 < u0 , in the case of u0 > 0, which may be true when is low), it
is optimal to monitoring some projects, and the chosen contract is msb . While the trade-o
between monitoring cost and liquidation cost determines the boundary between liquidation and
continuation m , the boundary between monitoring and non-monitoring among the continued
projects is more complicated. Its determination partitioned this set of u0 into two subsets. If
u0 2 [
u0 ; u10 ), the non-monitoring region is chosen so that the incentive cost to induce e ort T (n )
6
Note that the cut-o points, such as u0 , H^ , etc. may be zero.
19
is exactly u0 , and the rest of continuation region are monitored. If u0 u10, the trade-o between
paying the monitoring cost and the incentive cost due to non-monitoring tip the balance in
favor of the incentive cost: it is bene cial for the investor to pay a higher incentive cost than
u0 in exchange for monitoring less projects. Therefore, even though the credit market may have
assigned zero or a very small share of project surplus to the entrepreneur, his expect payo will
not go down that low since it is optimal for the investor to not monitor him and hence pay him
the incentive cost than paying the high monitoring cost.
Figure 3. Reservation Utility u0, Incentive Cost T (), and Optimal Contract
u0
u^0
6 AA
JJ
J
T (n )
9
>
=
nfb
>
;
JS
SS
: x + T (fb) = u0
9
>
>
>
>
>
=
@@
@ll
l
>
>
>
>
>
;
u20
u0
u10
nsb
:
n
= n ; T (n) = u0
QQQ
HHHP
P
)
= ~n; T (n) > u0
PXXX``` g msb : n = n; T (n) = u0
```hhh
sb : = ^ ; T ( ) > u
m
- n n n 0
^m
^n
nsb
:
n
o
fb
Project return H :
~n
nfb
^
H
nsb
H
sb
m
With very low project return H (< H^ ), a project's potential success rate has to be very high
in order for it to be continued, that is, the rst-best liquidation level fb is very high. With few
high-success-rate projects being continued, the incentive cost of not monitoring these projects are
also small. In particular, it is lower than the entrepreneur's reservation utility u0 . Therefore,
the non-monitoring contract achieves the rst-best nancing. With very high return H (> H ),
on the other hand, even projects with low potential success rate is worth to be continued. Because many projects are continued, paying the incentive cost of non-monitoring according to the
lowest-success-rate continued project for the entire set of continued projects is very expensive.
Hence, monitor the lower success rate projects and not monitoring the higher ones (contract msb )
20
is optimal. There is a lower bound for H , H , below which paying incentive cost of non-monitoring
is cheaper than monitoring. In such a case, the non-monitoring contract nsb is optimal.
Monitoring cost
sb
m
:
nsb
This parameter is only relevant when the rst-best nancing is not attainable. The key tradeo here is paying the monitoring cost or the incentive cost due to non-monitoring. It is intuitive
when the monitoring cost is low, the optimal contract should be the one with monitoring msb .
And when it is high, not monitoring but paying the incentive cost to induce truth-telling and
e ort-making is less costly, and hence, nsb is optimal.
Liquidation value :
sb
m
^
nsb
nfb
The liquidation value works exactly the opposite way as project return H , through its e ect
on the level of project continuation. With very high (> ), the opportunity cost of continue
a project is high. Hence, few projects are continued. But with very low (< ^), the close to
nothing scrape value of a project will not have much e ect on its continuation decision, and hence
relatively, more projects are continued. The e ect of on the amount of projects continued is
monotonic.
This concludes our discussion of the two-agent optimal contract.
4. Equilibrium
In this section we rst describe what a credit market equilibrium is, and then compare allocations across equilibria under di erent parameter values of the model. We focus on the following
questions: What determines the equilibrium total number of projects which are fully implemented?
(What determines the economy's equilibrium output?) How are the economy's total investment
and output related to the equilibrium lending mechanism?
We have assumed up to now that the parameters of the model, H; t; ; ; and the density
function g(), satisfy assumptions (1) and (2). We need to make a third assumption: at least for
some credit-market solution u0 , it is worthwhile for investors to invest in projects rather than the
storage technology. That is,
Assumption (3) There exists u0 0 such that V (u0 ) 0.
Since investors' value function V (u0 ) is a weakly decreasing function of u0 by Lemma 4,
assumption (3) implies the following lemma.
21
Suppose that assumptions (1)|(3) hold. Then for the optimal contract , there
satisfying V (u0 ) = 0, such that for all u0 u0 , V (u0 ) 0, and for all u0 > u0 ,
Lemma 6.
exists u0 > 0
V (u0 ) < 0.
We need to assume that the parameters of the model satisfy assumptions (1)|(3) in order to
have any investment made.
The equilibrium notion we use is competitive: the short side of the market extract all the surplus from trades. There are two possibilities. The rst is the case when the economy's total supply
of loanable funds exceeds the total demand for funds, that is, Æ < . In this case, competition for
projects among lenders will work to maximize the expected payo of entrepreneurs. Speci cally,
it will drive the expected payo of each entrepreneur u0 down to u0 at which the expected net
payo s for investors are zero. Thus in equilibrium entrepreneurs will extract all of the surplus
associated with the invested projects. Depending on where u0 is located, the optimal contract
can be the monitoring msb (u0 2 [0; u0]), the non-monitoring second-best nsb (u0 2 [u0 ; u^0]), or
the non-monitoring rst-best nfb (u0 u^0).
The second case occurs when the economy's total supply of funds is less than the total demand
for funds, that is, Æ > . In this case, not all projects will be funded. Competition for funds will
drive u0 to zero, although the expected payo of each entrepreneur may be above zero (due to
incentive problems) at a level where the investor's expected returns are maximized. The optimal
contract is the the monitoring msb if regions I and II in Figure 2 are not empty (u0 > 0). Otherwise,
it is the non-monitoring second-best nsb .
In the special case where Æ = , the two parties can divide the surplus from invested projects
in any arbitrary way. For simplicity, we assume that in such a case, entrepreneurs get all the
surplus, since this is the most likely case to achieve the rst-best. Formally,
Suppose that the parameters of the model satis es assumptions (1)|(3).
Then, there are two possible equilibria.
Proposition 7.
(i)
(ii)
If Æ , then in equilibrium all projects are funded, and an equilibrium is a pair (u0 ; ),
where u0 = u0 (that is, V (u0 ) = 0), and is the optimal contract which gives expected
utility u0 to the entrepreneur.
If Æ > , then in equilibrium the measure of the projects funded is , and an equilibrium is
a pair (u0 ; ), where u = 0 and is the optimal contract which presumes zero expected
payo to the entrepreneur, but its actual prescription is T (n ).
Obviously we have imposed a very simple market structure in the de nition of the creditmarket equilibrium. This simple competition mechanism may be interpreted as a special case of
a bargaining process which in principle can take a very general form.7
7
For example, this bargaining process may dictate that the equilibrium fraction of the surplus associated with
22
4.1. Credit Rationing
Notice that whenever Æ > , in equilibrium there is always credit rationing of the type discussed
by Stiglitz and Weiss (1981) and Williamson (1987), where among a group of identical borrowers
some receive loans and some don't, and those who do are strictly better o than those don't.
Why is credit rationed? Sometimes it is because of costly monitoring, as in Williamson (1986,
1987).8 Speci cally, when msb is optimal, lowering the entrepreneur's reservation utility u0 implies
a higher n, which in turn implies that the expected monitoring cost is higher. Sometimes credit
is rationed because of costly liquidation. In particular, when nsb is optimal, as u0 decreases,
more projects must be liquidated in order to make the contract incentive compatible. The notion
that credit rationing is a mechanism to avoid excessive liquidation has not been discussed in the
literature. Stiglitz and Weiss (1981) model credit rationing as a mechanism to reduce costly ex
post default on loans.
4.2. Comparative Statics
In the section of two-agent optimal contract, we discuss the e ect of the parameters of the
model on the optimal contract, for a given credit market outcome u0. In this section we study the
general-equilibrium e ects of shocks to project return H , monitoring cost , the investment fund
supply (the measure of investors ), and the demand for funds (the measure of entrepreneurs Æ)
on the equilibrium nancing mechanism and output. The results obviously depend on the initial
condition, in particular, which party gets bigger share of the investment return, that is, u0 = 0 or
u0 = u0 . We rst summarize some properties the value function V which are needed to analyze
the response of u0 to changes in the parameters of the model.
The value function V are implicit function of the exogenous parameters H; ; ;, and explicit
function of u0.
Holding other parameters of the model constant, the sign of the derivative of
V with respect to parameter x, @V =@x, for x being H; ; ; as well as u0 on the ve (or three)
Lemma 4.
a project that goes to the investor is a function of Æ and in the form of say ( Æ ), with 2 [0; 1] and 0 < 0. The
special case we use in this paper simply sets
Æ
1 if Æ > 1
(46)
( ) =
0 if Æ < 1:
Adopting
the more general form of the competition mechanism does not a ect our results qualitatively.
8
Williamson considers a standard costly state veri cation model.
23
segments of V are as follows.
H
I
+
II
+
u0
+
0
+
III
+
0
0
0
IV
+
0
+
V
+
0
+
overall
increasing
decreasing
increasing
decreasing
This lemma can be veri ed directly against the solution of the optimal contract given in
Proposition 4. As parameters H; and changes, in addition to the level of V , the validity
of condition > 0 (the case where for some u0 , the optimal contract involves monitoring), as
well as the boundaries u^0 ; u0 ; u10; u20 that divide the value function V into ve or three segments
may change correspondingly. Using the de nition of ; u^0 ; u0 ; u10 ; u20 given in equations (40)|
(44) as well as that of n; ^n and ~n given in equations (32), (36) and (37), the following lemma
summarizes these e ects.
Holding other parameters of the model constant, the sign of the derivative of
y with respect to parameter x, @y=@x, for y being ; u10 ; u0 ; u20 ; u^0 ; and x being H; ; are as
follows.
u10
u0
u20
u^0
Lemma 5.
H
+
0
+
0
+
+
0
+
0
The derivations of both lemmas are omitted here.
4.2.1.
Disturbances to project return H
Consider rst the e ect of a change in the level of H . Fix the model's other parameters
at levels such that assumptions (1)|(3) are satis ed. Suppose the economy is experiencing a
negative technology shock which lowers the level of H from H o to H n. Assume that with H n,
assumptions (1)|(3) remain valid.
Suppose rst that regardless the initial environment, after the shock, n 0 (at which only
non-monitoring contract is optimal). Then, equilibrium lending mechanism either shifts from
bank lending to market nancing or continue to be market nancing. With lower H , it is less
pro table to continue some projects that would have been fully funded under H o. Lower expected
return of each continued project and increased liquidation implies that the total output of the
economy will fall, more than the drop in H .
Next, suppose that in the initial environment, o > 0 (so it is possible to have intermediated
lending to be optimal), and after the shock, by Lemma 5, decreases but n > 0. Suppose that
24
the economy has an over-supply of projects, that is, u0 = 0. Then the nancing mechanism was
and continues to be intermediated lending (msb ). If the economy has an over-supply of funds,
that is, u = u0 , the e ect of the shock is more complicated. On the one hand, by Lemma 4,
the investors' value function V is an increasing function of H and a decreasing function of u0.
Hence, as H decreases from H o to H n, u0 is also reduced, say from uo0 to un0 . On the other hand,
by Lemma 5, three of the four boundaries between segments of V , u0 , u20 and u^0 also decrease.
The end result is likely that the lending mechanism does not change in response to the shock.
However, it is possible, although rare, that the initial funding mechanism is market lending (nsb
or nfb), and the new market clearing u0 (= un0 ) drops enough to a level that is less than the new
boundary un0 (see Figure 2). In this case, the lending activities moves to borrowing from banks.
Whether or not the lending mechanism is a ected, the amount of projects liquidated increases,
and hence total output falls.
Thus, our model predicts that, as H decreases, the economy will see more projects being
liquidated. There is a ight for quality in the sense that projects which are fully executed have
higher probabilities to succeed. Furthermore, if rms borrow from banks to nance investment,
the drop in H is likely to trigger a switch to bond nancing (although not necessarily). This is
consistent with what happened during the 1990-91 recession. Note that as H falls, the economy's
total output falls more than proportionally. This is because, as H decreases, not only each rm
produces less, but also there are fewer rms producing. In other words, an earnings shock is
ampli ed through the credit market.
Conversely, as H rises, which is often associated with economic boom, more projects will be
fully implemented, total output of the economy rising, and the optimal lending mechanism is
likely to be intermediated nancing.
4.2.2.
Disturbances to monitoring cost
Consider now the e ect of a change in the values of , while other parameters of the model
hold constant. Suppose the economy is experiencing an improvement in monitoring technology
such that monitoring cost drops from o to n. Assume that both before and after the change,
assumptions (1)|(3) are satis ed.
First, suppose that in the initial environment, o 0, that is, the initial optimal contract is
the non-monitoring nsb or nfb. The drop of monitoring cost raises the value of to n > 0 (by
Lemma 5). If the economy has an over-supply of projects, that is, u0 = 0, then the equilibrium
nancing mechanism shifts to bank lending (msb ). If the economy has an over-supply of funds,
that is, u = u0, bank-lending is optimal if u0 is now in the msb region, otherwise, bond nancing
remains optimal. Next, suppose that in the initial environment, o > 0. Then by Lemma 5,
n > 0. In this case, regardless who has the upper hand on the credit market, the equilibrium
nancing form does not change. (When u = u0, a drop in induces a rise in u0 since V is
25
decreasing in in regions I and II, but it also leads to a right shift of the boundary between msb
and nsb , u0 . Given that the rest of function V does not change, the new u0 remains in region II.)
In all these scenarios, lending mechanism either remains unchanged or moves into intermediated
nancing. Either way, the drop in will leads to less liquidated projects, hence, higher total
output.
If there is an increase in the monitoring cost , the e ect will be complete opposite, with
increased liquidation, lower output, and more likely market lending.
4.2.3.
Shocks to demand for funds Æ or supply of funds
Suppose the economy resides initially at an equilibrium where there is an over-supply of funds,
i.e., Æ < . So all projects are funded, and the entrepreneurs have the upper hand in the credit
market, earning the maximum payo possible u0 . The equilibrium lending mechanism can be
either borrowing from banks or issuing corporate bond.
Imagine now the economy receives a \real" shock which increases the number of investment
opportunities from Æo to Æn , and Æn > . That is, there is an increase in the demand for funds
while the supply of funds remain unchanged. This reversal of power on the credit market implies
that now the entrepreneurs' share of the investment return is reduced to zero, i.e, u0 = 0. The
new optimal contract can be either msb or nsb . The only possible change of lending mechanism
occurs when > 0, the initial optimal contract prescribes non-monitoring (nsb or nfb), and the
new one requires monitoring (msb ). In such a case, lending activities shift from bond market to
bank loans. Regardless of the lending mechanism, the induced drop in entrepreneurs' reservation
utility is likely to trigger an increase in the amount of projects liquidated, unless u0 is in the at
portion of the value function V (region I or III). However, the increase of investment opportunities
implies that the amount of projects funded rises from Æo to , to fully utilize the available funds.
The combined e ect on total output is unclear, depending on whether the downward push of
the increased liquidation can be overturned by the upward lift of the increased investment. It is
possible that the e ect of more liquidation dominates that of more funding, hence total output
falls despite the increased investment. This seemingly counterfactual result is rooted in the logic
that the tightening of credit market for the entrepreneurs may produce the adverse e ect of
intensifying incentive problems.
Next, consider the implications of a \monetary" shock that changes the supply of loanable
funds, starting from the same initial credit market equilibrium (excess of funds, u0 = u0). Specifically, suppose that a sequence of monetary policy innovations manage to bring to below Æ.9
The reduction in available funds shifts the market power to the side of the investors, and leads
to a similar reduction of the entrepreneurs' entitlement of investment return u0 from u0 to zero.
We do not provide explanations for why a tight monetary policy induces a contraction in the total supply of
credit.
9
26
The response to this decline in u0 is the same as those induced by the increased demand for funds
discussed above, with increased liquidation and a potential shift of lending mechanism from issuing bond to borrowing from banks. The only di erence here is that when available loans shrink,
total investment also shrink. Therefore, the economy will see a de nite fall in total output.
In our model, a fall in the supply of credit can create two e ects: an interest rate e ect and
a credit e ect. The interest rate e ect occurs right when crosses Æ, where there is a discrete
downward jump in both total investment and total output caused by a sudden increase in the
investor's expected return on a loan (rate of interest). As the expected return on loans increases,
the utility of the entrepreneurs fall, agency costs increase, more projects are liquidated after the
observation of the random signal, and aggregate output is lower.
After has crossed Æ, the credit e ect takes over. As keeps falling, total investment and
total output fall continuously while the rate of interest remains at. The same credit e ect is
discussed in Stiglitz and Weiss (1981). But a somewhat interesting point this paper o ers is: a
decrease in the total supply of loans may cause aggregate output to fall more than proportionally.
A fall in loan supply causes less projects to be funded initially (a pure credit e ect), and among
those receive initial funding, more are to face liquidation subsequently (an agency e ect).
5. Concluding Remarks
We have constructed and studied a model of the credit market in which both the economy's
total output and the equilibrium source of nancing are endogenously determined. In contrast
to the literature, we focus on two important elements of external nancing. One is that the
equilibrium contract is optimal with respect to the environment rather than exogenously imposed.
The other is the e ect of credit market condition on the equilibrium lending mechanism. Among
other things, we show that the observation that bank lending falls relative to corporate bond
issuance during recessions can be explained by movements in the economy's real factors, including
the availability of investment opportunities and the potential returns of an average investment
project.
A major simplifying assumption of the model is that the economy's total demand for and
supply of funds are exogenously xed. This can be relaxed. For instance, one could imagine that
the availability of funds is an increasing function of the expected return on a loan to the investor,
or one could also assume that a higher expected return on a project to the entrepreneur brings a
supply of more projects. But as long as these relationships are not suÆciently elastic, it is clear
that the comparative statics properties of the model will remain valid.
27
Appendix
Proof of Proposition 1.
Step 1. We show that there exists a contract ~ = fM ; ; x~; y(); 2 B; YD ; R(); R0 (); 2
~ (); R~0 (); 2 C g such that ~ is equivalent to , and that R~ () and R~0() are constants on
A; R
~ (), R~ 0(), and x~. Without loss of generality, assume
C . Note that ~ is identical to except for R
that C has a minimum point, and let 1 min2C . Let R~ () = R(1), R~0 () = R0 (1) for all
2 C , and allow x~ to be determined later.
(i) We rst show ~ is incentive compatible. We need to show only that the revision on C
satis es conditions (6)|(8), and (10). Note that conditions (6) and (8) are obviously satis ed on
~ () = R(1), R~ 0() = R0 (1), for all 2 C . Since condition (7) holds
C , given that 1 2 C , and R
for = 1, we have for any 2 C , 1,
~ () + R~0 () 1R(1) + R0(1 ) YD :
R
Thus ~ also satis es (7) for all 2 C . Similarly, condition (10) holds for = 1, which implies
~ () 1R(1) t. That is, (10) is satis ed with
1 R(1 ) t. Then, for any 2 C , we have R
any 2 C .
(ii) With ~ instead of , the entrepreneur's expected utility is di erent only in C . Let x~ be
de ned as follows:
x
~ =x+
Z
C
R( ) + R0 ( ) dG( )
Z
C
R(1 ) + R0 (1 ) dG( ):
We need to show x~ 0. But by (6), 8 2 C ,
~ () + R~0():
R( ) + R0 ( ) R(1 ) + R0 (1 ) = R
That is, with ~ , for all 2 C , entrepreneur E's expected payo is less than or equal to that of the
original contract. We therefore have: x~ x 0.
By (ii), investor I's payo is the same with contract ~ as with contract . So, we have shown
that ~ is equivalent to the original contract .
Step 2. We further demonstrate that the contract ~ is equivalent to a third contract ^ =
fM ; ; x^; y(); 2 B; Y^D ; R^ (); R^0 (); 2 A [ C g; which is otherwise identical to ~ except
8 2 A R^ () = 0; R^ () = R() + 1 R ()
(47)
8 2 C
Y^D = YD
0
^0() = 0;
R
R0 (1 );
0
^ () = RC R~ () = R(1)
R
x^ = x~ + (C )R0 (1 ) + (D )(YD
28
R0 (1 )):
(48)
(49)
(i) We show that this new contract ^ promises the same expected utilities as does ~ to both
entrepreneur E and investor I. The entrepreneur's expected payo on A under ^ is the same
pointwise as under since for each 2 A,
1
^ () + R^0 ():
R( ) + R0 ( ) = (R( ) + R0 ( )) + 0 = R
By (48), under ^ , if the project with in C succeeds, entrepreneur E receives the expected payo
RC that he would receive under
~ . His total expected payment on C when the project fails,
(C )R0 (1 ), and part of the payment on D , (D )(YD R0 (1 )) (which is positive by (8)), are
moved from C and D, respectively, into the constant payment x^ (an increase from x~). Therefore,
the two contracts give the same expected payo s to both agents.
(ii) We show that the new contract ^ is incentive compatible. First, since the changes on
A do not a ect the entrepreneur's expected payo pointwise, the left-hand side of the relevant
constraints (6) and (7) are the same as those under ~ . Seconde, note that for any 2 , and
any 0 2 C , R~ 0(0) = RC 0 = R^0 (0), and R~ (0) + R~0 (0) t RC t = R^ (0) + R^0(0) t.
Furthermore, YD Y^D . That is, the right-hand sides of conditions (6)|(9) under ^ are all
smaller than that under ~ . Therefore, for any 2 A [ B , conditions (6)|(9) are satis ed under
^ . Next, given that the de nition of Y^D by (49), and that conditions (6)|(8) are satis ed for
any 2 C [ D under ~ , they are also satis ed under contract ^ . Last, since constraint (10) is
satis ed under ~ , for any 2 A, R() t, and 1RC t. By (47), R^ () R(), so for any
^ () R() t. For any 2 C , 1, so R^ () = RC 1RC t. Therefore,
2 A, R
condition (10) holds under ^ .
We have shown that incentive constraints (6)|(10) hold for ^ , and that both agents receive
the same expected payo under contract ^ as under ~ . Therefore, the two contracts are equivalent.
Proof of Proposition 2.
We rst introduce some notation. De ne
X > Y () 8 x 2 X; 8 y 2 Y; x > y:
Let P = be the product measure on , and let X >Y denote \X > Y almost surely",
X >Y () P (x; y ) j x 2 X; y 2 Y and y > x = 0:
The proposition states that for any given optimal contract , there is contract which is equivalent
to and which satis es ^ > ^ 0:
We rst show that satis es >0; which is equivalent to showing A >B; A >D, C >B , and
C >D . Before proceeding, assume each of the four sets A; B; C and D has positive measure. (If
one of the sets has measure 0, the corresponding assertion holds automatically.)
e
n
o
e
e
e
e
29
e
e
(i) We show A >B: Suppose not, then there exist A A and B B such that B > A:
Without loss of generality, suppose that A and B satisfy (A) = (B ) 6= 0,10 and that R()
has a minimum on A. Now consider an alternative contract ~ which is identical to except
(a) B~ = B [ A n B , and 8 A 2 A,
1
y~(A ) =
y ( )dG( ):
(B )
e
Z
B
(b) A~ = A [ B n A, and 8 B 2 B ,
~ (B ) = Rmin (A) min R();
R
2
A
(c) If
R
B B Rmin (A )dG(B ) < A AR(A )dG(A ), then
R
x~ = x +
Z
A
A R(A )dG(A )
Z
B
B Rmin (A )dG(B )
otherwise, x~ = x.
We need only verify that the incentive constraints (17)|(20) and (23) hold for ~ . Since
~ (B ) = Rmin(A) RC , for all B 2 B ; or R~ (B )
R(A ) RC for all A 2 A , it holds that R
satis es (17). Now de ne A argmin2 R(A): By (18) and (23), AR(A) YD t t. Thus
for all B 2 B ;
~ (B ) AR~ (B ) = AR(A) YD t t:
B R
That is, R~ (B ) satis es (18) and (23). Since satis es constraints (19) and (20), we have for all
2 B , y ( ) maxfRC t; YD g. Therefore, for all A 2 A ,
1
RC
y~(A ) =
y ( )dG( ) max
dG( ) t; YD max RC A t; YD ;
(B )
(B )
or y~(A) satis es (19) and (20). Thus we have shown that ~ is incentive compatible.
Next, we show that ~ Pareto dominates . By construction, y~(A)dG(A) = y(B )dG(B ).
Suppose B Rmin (A)dG(B ) AR(A)dG(A). Then moving from to ~ the entrepreneur's expected payo is changed by
A
Z
Z
B
n
o
B
R
R
A
R
B
R
B
A
Z
B
Z
B Rmin (A )dG(B )
A
A R(A )dG(A ) > 0;
and the investor's expected payo is changed by
Z
10
B
B (H
Z
Rmin (A ))dG(B )
A
A (H
R(A ))dG(A ) > 0;
Given is continuously distributed, the sets A and B can be cut arbitrarily small to satisfy this property.
30
since B > A. Thus both parties are better o under ~ than under .
Suppose B Rmin(A)dG(B ) < AR(A)dG(A). Then under ~ the entrepreneur's
expected payo decreases on B compared to what she receives under on A, but the decrease
is made up exactly by the increase of x to x~, so her total expected payo remains the same.
Now the investor's expected payment to the entrepreneur is the same, but the investor's expected
payo is increased by
H
B dG(B )
A dG(A ) > 0:
This is because projects with higher success rates are continued. Again, ~ Pareto dominates .
(ii) We show A >D: Suppose not, then there exist A A and D D such that D > A:
Without loss of generality, suppose A and D satisfy (A) = (D ) 6= 0, and R() has a
minimum on A. Now consider an alternative contract ~ which is identical to except
(a) D~ = D [ A n D , and 8 A 2 A,
R
R
B
A
Z
Z
B
A
e
y~(A ) = YD :
(b) A~ = A [ D n A, and 8 D 2 D ,
~ (D ) = Rmin(A) min R(); 8 D 2 D :
R
2 A
(c) If
R
D D Rmin (D )dG(D ) < A A R(A )dG(A ),
R
x~ = x +
Z
A
Z
A R(A )dG(A )
D
D Rmin (A )dG(D );
otherwise, x~ = x.
Now since D > A and (22) holds for , we have YD D RC t ARC t holds for all
A and D , and hence constraint (22) is satis ed by contract
~.
~
As in the proof for A >B , we can show that R(D ) satis es constraints (17), (18), and (23), and
thus ~ is incentive compatible. As in the proof for A >B , we can show that ~ Pareto dominates
, a contradiction.
(iii) We show C >B: Suppose not. Without loss of generality, assume that there exists B B
and C C such that B > C ; and (B ) = (C ) 6= 0: Now consider an alternative contract
~ which is identical to except
(a) B~ = B [ B n C , and 8 C 2 C ,
1
y~(C ) =
y ( )dG( ):
(B )
e
e
e
Z
B
31
(b) C~ = C [ C n B ; and 8 B 2 B ,
~ (B ) = RC :
R
Since every B 2 B satis es (19) and (20), y(B ) maxfB RC t; YD g. Then, for any
C 2 C ,
1
RC
y ( )dG( ) max
dG( ) t; YD max RC C t; YD :
y~(C ) =
(B )
(B )
That is, y~(C ) satis es (19) and (20). Also, take an arbitrary C 2 C , C RC t YD and
C RC t. Since for any B 2 B , B > C , we have B RC t YD and B RC t, or,
constraints (21) and (24) are satis ed on B . So the modi ed contract satis es all the relevant
incentive constraints.
By construction, y~(C )dG(C ) = y(B )dG(B ). But since B > C , the entrepreneur's expect payo is increased by
Z
Z
B
n
o
B
R
R
C
B
Z
B
B RC dG(B )
Z
C
C RC dG(C ) > 0;
and the investor's expected payo is increased by
Z
B
B (H
RC )dG(B )
Z
C
C (H
RC )dG(C ) > 0:
That is, both agents' expected payo s are strictly higher under ~ than under .
(iv) Last, we show C >D: Constraints (21) and (22) directly imply that C > D; which
further implies C >D:
To summarize, we have shown that >0: Given that contract satis es >0; it is trivial
to show that there is an equivalent contract ^ that satis es ^ > ^ 0: Since > 0 can only be
violated on a measure zero set, we can rearrange monitoring and continuation/liquidation policies
on this measure-zero set to eliminate the violations without a ecting the payo s.
e
e
e
e
Proof of Lemma 3.
Consider an optimal contract . By Proposition 2, we assume = [1; 1] and 0 = [0; 1 ).
Suppose 0 = ;. Then consider contract ^ which is otherwise identical to except
(a) D^ = [0; t=H ); A^ = A \ [t=H; 1]; C^ = C \ [t=H; 1]:
(b) Y^D = 0.
(c) If RC > H , then R^C = H ; if RC H (including the case C = ;), then R^ C = RC .
(d) x^ = x + A\D^ (R() t)dG() + C \D^ (RC t)dG() + C^ (RC R^ C )dG().
R
R
R
32
Notice that since t > 0, (D^ ) 6= 0.
By construction, for all 2 D^ , R^ C t Ht R^ C t 0 = Y^D , and hence constraint (22)
is satis ed on D^ . The contract ^ satis es constraint (17) since RC R^C . ^ also satis es (18)
since (23) holds under . If R^ C = RC , then clearly constraints (21) and (24) are both satis ed.
If H = R^ C < RC , then for all 2 C^ , t=H , R^ C = H t, hence constraints (21) and (24)
are also satis ed. Therefore, ^ is incentive compatible. Finally, the expected payment to the
entrepreneur under contract ^ is the same as under contract , but the investor's expected payo
is increased by (D^ ) D^ (H t)dG() > 0. This contradicts the fact that the is optimal.
R
Proof of Proposition 3.
We rst show that given the optimal continuation/liquidation policy , the optimal monitoring
region A is a lower interval of and the non-monitoring region C is the complement upper interval
of . Suppose this is not true. That is, suppose there is an optimal contract such that a subset
A of A is embedded in C , that is, for all A 2 A, A > inf C . Without loss of generality
assume (A) 6= 0.
Consider contract ^ which is otherwise identical to except
(a) A^ = A n A; C^ = C [ A, and 8 A 2 A,R^ (A) = RC :
(b) x^ = x +
A ( R ( )
R
RC )dG( ):
To show that ^ is incentive compatible, we need only check that constraints (21) and (24) are
satis ed for all A 2 A. Since for all 2 C , RC t YD , we have (inf C )RC t YD , which
in turn implies for all A 2 A, ARC t YD given that A > inf C . That is, (21) is satis ed.
Constraint (24) is implied by (21) since YD 0.
By construction of x^, the entrepreneur's expected payo remains the same under ^ . But the
investor gains by the savings of the monitoring cost (A) > 0. This contradicts the fact that
the contract is optimal.
Next, we show that B = ;. Let be an optimal contract which has B 6= ;. By Proposition
2, > 0, and from the above proof, C > A. Hence we can let A = [m; n) and C = [n ; 1],
where m n . Consequently, = A [ C = [m; 1] and 0 = B [ D = [0; m). Notice Lemma 3
implies m > 0.
Consider an alternative contract ^ which is otherwise identical to except
(a) R^ C = t=n .
(b) D^ = D [ B , and Y^D = 0.
33
(c) x^ = x + C (RC R^C )dG() + B y()dG() + D YD dG().
Using the equations RC t=n = R^C and Y^D = 0, it is easy to check that the contract
^ satis es all the incentive constraints including (17), (18), (21), (22), (24), as well as the nonnegative constraints (13). Moreover, the construction of x^ implies that the entrepreneur's expected
compensation under ^ is the same as under . However, under ^ the investor's expected payo is
increased by the savings of the monitoring cost (A [ B ) > 0. This contradicts the assumption
that is optimal.
Finally, we show (iii) holds. Suppose is optimal and it has a compensation scheme that
di ers from what is given by the proposition. We need only show that is equivalent to a
contract ^ whose compensation scheme takes the form that is given by the proposition. Let
the compensation scheme of ^ be given by R^ () = t= for all 2 A, R^C = t=n, Y^D = 0, and
^C )dG()+ (D)YD . It is easy to check that ^ satis es
x^ = x + A (R( ) t= )dG( )+ C (RC R
incentive constraints (17){(24). Since the compensation schedule of the contract also satis es
these constraints, in particular, for all 2 A, R() t= = R^ (), for all 2 C , RC t=n = R^C ,
and YD 0 = Y^D , we have x^ x. Clearly, the compensation scheme of ^ conforms with the
proposition, and ^ is equivalent to .
R
R
R
R
R
( 4)
We rst show that by assumptions (1) and (2), the objective function O(m; n ) is strictly
concave in both m and n.
The function O(m; n ) is strictly concave if its Hessian matrix is negative de nite. By the
de nition of function O(m; n ) in equation (33), @O(m; n)2 =@n @m = 0. So, we only need to
show that the second derivatives with respect to m and n are strictly negative.
@O (m ; n )2
= [ H (t + + )]g0 ( ) Hg( ):
Solution to Problem P
2
@m
m
m
m
If mH (t + + ) 0, then @O(m; n )2 =@m2 < 0 since by the rst inequality of assumption
(2),
t
g 0 (m )
H
<
:
H (t + + )
H (t + )
g ( )
m
If mH (t + + ) < 0, then @O(m; n)2 =@m2 < 0 is equivalent to
g 0 (m )
H
<
g (m )
(t + + ) mH
which is implied by the second inequality of assumption (2). With respect to n,
1
@O (m ; n )2
= t g( ) 2t dG() g0 ( ) < t g( ) g0 ( ) J ( ):
Z
@n2
n
n
n3
n
n
34
n
n
n
n
By assumptions (1) and (2),
0
H (I +t t + ) gg((m))
1
m
hence, J (n ) < 0, or equivalently, @O(m; n)2 =@n2 < 0. So, O(m; n ) is strictly concave in both
m and n .
Given that the objective function O(m; n ) is strictly concave, and that the constraint set
de ned by (34) is convex, problem (P 4) has a unique solution. We can solve (P 4) with Lagrange's
method. Let 1 be the multiplier for constraint m n, and 2 be the multiplier for constraint
n n . Then the Lagrange is given by
t
<
n
L(m ; n ; 1 ; 2 ) =
Z
1
m
t
t dG( ) + G(m )
H
T (n )
Z
n
m
dG( ) + 1 (n
m) + 2 (n
n ):
The rst-order conditions are
@L
@m
@L
@n
@L
@1
@L
@2
=
Hm + t + +
Z
1
g (m )
1
=0
(50)
= (t )2 dG() g(n ) + 1 2 = 0
n
= n m 0; 1 0 with complementary slackness
(51)
= n
(53)
n
n
0;
2
0 with complementary slackness.
(52)
Depending on which of the constraints binds, there are four possible solutions for fm ; ng.
(a) 1 = 0, 2 = 0. Then neither constraint binds. By (50) and (51), m = ^m and n = ^n.
= 0, 2 > 0. Then by (50), m = ^m, and n is given by the binding constraint: n = n.
(c) 1 > 0, 2 = 0. Then m = n and n < n . Substituting 2 = 0 and m by n , we get
equation (37) from (50) and (51). Given that ~n is its solution, n = m = ~n:
(b)
1
(d) 1 > 0, 2 > 0. Then both constraints binds: m = n = n.
To summarize, the solution to (P 4) can be one of the two classes, depending on whether m = n:
(1) When ^m < minf^n ; ng, which includes cases (a) and (b), m < n. Then the monitoring
region M is not empty, M = [m ; n), and the project-continuation region is = [m ; 1],
where m = ^m, n = minf^n ; ng, and m > fb.
(2) When ^m minf^n ; ng, which includes cases (c) and (d), m = n. Hence, the monitoring
region is empty, M = ;, and the project-continuation region is given by = [n; 1], where
n = minf~n ; n g, and n > fb .
35
Proof of Proposition 6.
(i) This is given by the discussion of value function V .
(ii) Both ^n and n do not depend on H . When H = H^ , n = fb (t + + )=H = ^m,
since u0 = T (fb) = T (n). Then, regardless of ^n , ^m > n minf^n; n g. When H ! 1,
^m ! 0 < minf^n ; n g. Since ^m is a continuous function of H , and minf^n ; n g does not depend
^ H ], ^m minf^n; n g, which by
on H , there exists an H > H^ , such that for all H 2 (H;
Proposition 4, implies that the optimal contract is the one without monitoring nsb , and for all
, ^m < minf^n ; n g, which, by the same proposition, implies that the optimal contract is
H>H
the one with monitoring msb.
(iii) The following facts are relevant to the proof of this statement.
(a) Given condition (31) does not hold, fb < n, and both fb and n are not functions of .
(b) When ! 0, ^m ! fb < n and ^n = 1, hence, ^m < minf^n; n g = n. When = H t
(where H t is the maximum that is allowed by assumption (1)), ^m = 1 > minf^n ; n g.
(c) It is obvious that ^m is an increasing function in . Also, ^n as a solution to equation (36)
is a decreasing function of , since totally di erentiate (36) with respect to n and at ^n , we
have
@O (m ; n )2
d ^n =d = g (^n )
<0
=^
@n2
given that function O is strictly concave. Since n does not depend on , minf^n ; n g is also a
decreasing function of .
Both ^m and minf^n; n g are continuous functions of . By (c), ^m is increasing in and
minf^n ; ng is decreasing in . By (b), as ! 0, ^m < minf^n; n g, but at = H t ,
^m > minf^n ; n g. Therefore, there exists a 2 (0; H t ) such that ^m = minf^n ; n g, for all
< , ^m < minf^n ; n g, and for 2 [ ; H t ], ^m minf^n ; n g. Hence by Proposition 4, the
optimal contract is the one with monitoring msb for < , and it is the one without monitoring
nsb for 2 [ ; H t ].
(iv) Both ^n and n do not depend on . By assumption, there exists an > 0 satisfying
assumptions (1) and (2) such that the optimal contract is the one with monitoring msb . Hence,
by Proposition 4, ^m = (t + + )=H < minf^n ; ng. Therefore, for any ,
t++
t + + < minf^n; n g:
^m =
.
n
H
n
H
That is, the condition for case (ii) of Proposition 4 is satis ed. It then follows the optimal contract
is the one with monitoring msb .
36
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@FedFRASER