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'
Monitoring the Monitor:
i Structure for a Financial
An
Incentive
Intermediary
Stefan Krasa
Anne
P. Villamil
"'-
College of Commerce and Business Administration
Bureau of Economic and Business Reseai ch
University of Illinois Urbara-Champaiyn
/>'
BEBR
FACULTY WORKING PAPER NO. 90-1690
College of
Commerce and Business
University of
Illinois at
Administration
Urbana-Champaign
October 1990
Monitoring the Monitor:
An
Incentive
Structure for a Financial Intermediary
Stefan Krasa
Department of Economics
Unversity of
Illinois
University of Vienna
Anne
P. Villamil
Department of Economics
University of
Illinois
ABSTRACT
The purpose of this paper is to extend the bilateral contracting problem studied in Townsend (1979)
and Gale and Hellwig (1985) to an explicit, multilateral contracting arrangement which resembles
banking.
We
derive the structure of an optimal contract for a large but finite intermediary, establish
gains from delegated monitoring, and study the incentive problem of the monitor
in
an economy where
the intermediary has no information, risk, or cost advantage relative to individual agents.
Unlike
previous delegated monitoring studies, the law of large numbers is not sufficient to obtain our results.
Instead, we appeal to a stronger result, the large deviation principle, which establishes that convergence
in
the law of large
numbers
is
exponential.
Introduction
1
The
literatures
on both contract theory and on the nature of financial
mediaries have grown extensively in recent years. However,
literature has restricted attention to bilateral agreements
tative agents.
For example,
much
inter-
of the first
between represen-
Townsend (1979) and Gale and Hellwig (1985)
consider single representative investor-entrepreneur pairings in markets with
information asymmetries where agents must trade to facilitate production.
This restriction to bilateral representative agent contracts
the
Townsend and Gale and Hellwig
tion addressed
is
is
appropriate in
studies since the central research ques-
the allocative effect of information problems in competitive
economies. In contrast, recent contributions to the financial intermediation
literature have
extended bilateral contract models to multilateral settings
to generate intermediary-like institutions endogenously.
mond
(1984) and
For example, Dia-
Boyd and Prescott (1986) explain the existence
of financial
intermediaries as cost-minimizing institutional responses to the need for
in-
formation production. Boyd and Prescott use the theory of mechanism design
to endogenously motivate intermediation, while
may be
establishes that
it
optimal for investors to delegate monitoring tasks to an intermediary.
In this paper
mond
Diamond
we study the delegated monitoring problem posed by
in a multilateral version of
the Gale and Hellwig model.
We
an economy with a large number of investors and entrepreneurs
Dia-
consider
who must
trade to facilitate production. Entrepreneurs have private information about
their output, but a costly technology exists
which can be used to verify
re-
turns ex-post. Agents have the option to either contract directly or to elect
an intermediary to represent their interests. In the
must monitor each project that he/she has an
is
first
case, each investor
interest in independently. This
the one-sided contract problem studied by Gale and Hellwig. In the second
case, investors elect a representative to
whom
they delegate the monitoring
task (the "intermediary" or "delegated monitor").
This intermediary con-
—
.
tracts with all remaining investors
and each entrepreneur, and
is
charged with
the responsibility of monitoring the entrepreneurs. This two-sided contract
problem
The
is
the main focus of our paper.
which
crucial question
who monitors
the monitor?
arises in a delegated
Two
monitoring setting
is
previous mechanisms have been proposed
to ensure that the intermediary truthfully reports actual payoffs to the investors.
First,
Diamond
(1984, p. 369) assumes that the intermediary
is
subject to unbounded, non-pecuniary penalties such as lost time and reputation, or less plausibly, physical
and imprecise nature of
this
punishment. Unfortunately, the unbounded
enforcement mechanism
is
difficult to rationalize
on economic grounds. Further, as Gale and Hellwig (1985,
imposition of these penalties
is
649) note:
p.
The
"equivalent to perfect bond-posting," and con-
sequently the incentive problem "becomes innocuous." Second, Williamson
(1986, p. 169) assumes that the intermediary's actual monitoring costs are
fixed
and independent of
its size.
This assumption implies that the inter-
mediary's marginal monitoring costs decrease as the size of the intermediary
increases. Unfortunately, this specification confers
herent cost advantage
dominance
We
(i.e.,
in-
a natural monitoring monopoly), and hence the
of delegated monitoring over direct investment
is
not surprising. 1
study an alternative incentive mechanism, costly state verification
of both the entrepreneurs
and the intermediary,
intermediary has no information,
We
on the intermediary an
show that under
this
risk, or cost
in
an economy where the
advantage
vis-a-vis investors.
mechanism, delegated monitoring
nates direct investment and simple debt
is
an optimal contract
strictly
domi-
for all agents.
Previous delegated monitoring studies have established gains from intermedi2
ation by appealing to the law of large numbers. However, in our symmetric
l
2
Of course Williamson's main concern was credit rationing.
Diamond shows that the penalties per entrepreneur go to zero
as the
number
of entre-
preneurs (/) becomes large. Williamson shows that the expected cost of monitoring the
monitor goes to zero as / becomes large, when the costs are independent of I
cost setting an investor's cost of monitoring the
number
of entrepreneurs,
preneurs goes to
and hence becomes
from monitoring
is
infinite as the
Clearly, the law of large
infinity.
to establish gains
monitor
increasing in the
number
numbers
in this context. Instead,
of entre-
not sufficient
is
we use the
large
deviation principle, a stronger result, which shows that convergence in the
law of large numbers
is
essential in the
The paper
is
is
exponential. This alternative mathematical approach
economy that we
consider.
organized as follows. Section 2 contains a description of the
model. The main result from Section 3 establishes that there are gains from
intermediation. In Section 4
we show that
trepreneurs, two-sided simple debt
essential
mathematical
results,
if
many
there are sufficiently
en-
an optimal contract. Section 5 contains
is
and Section 6 contains concluding remarks.
The Model
2
Consider an economy with
investors
numbers
finite
of
two types of
and entrepreneurs. Each entrepreneur
i
=
1,
.
.
.
risk neutral agents,
,
I
is
endowed with
a risky investment project which transforms y units of a single input at time
into
Oi
units of ouput at time
contrast, each investor j
=
1,
.
1,
.
.
,
but has no endowment of the input.
J
is
endowed with
1
unit of the
that y
is
an integer with y
>
1.
homoge-
We
neous input, but has no direct access to a productive technology.
In
assume
Hence, the project of an entrepreneur cannot
be financed by a single investor. 3 The total available supply of investment
is
larger than the input required by all entrepreneurs, so
accomodated by the / entrepreneurs
(i.e.,
J =
yl).
J investors can be
All entrepreneurs
investors are symmetrically informed about the distribution of
t
at
and
time
but asymmetric information exists about the state of the project's actual
3
This assumption implies that there
is
0,
re-
duplicative monitoring in the absence of inter-
mediation, because more than one investor must monitor a single entrepreneur. Delegated
monitoring economizes on these costs
(c.f,
Diamond
(1984)).
alization ex-post. In particular, only entrepreneur
realization of
As
in
0,
at
time
1,
where
i
freely observes the actual
this actual realization
Townsend (1979) suppose that a technology
w
used to verify
to non-owners at time
is
denoted by w.
exists
which can be
Let this verification technology
1.
have the following characteristics:
(Tl) Use of the state verification technology
w
(T2)
is
costly;
privately revealed only to the individual
costly state verification (CSV).
Assumption (Tl)
in
is
is
is
4
as in
Gale and Hellwig (1985,
CSV
p.
technology
651) the veri-
comprised of both a pecuniary component and an indirect
"pecuniary equivalent" of a non-pecuniary cost.
permit negative
requests (deterministic)
similar with the specification of the
Townsend, except that
fication cost
who
and
utility
The non-pecuniary
costs
but rule out negative consumption, while pecuniary
equivalents of non-pecuniary costs ensure that the costs can be shared by
5
the contracting parties. In contrast, assumption (T2) differs fundamentally
from the specification
after
CSV
who
requests
in
Townsend.
occurs, while in our
model
CSV. Assumption (T2)
information could be
made
the monitor. However,
it
In his
w
is
is
(Al) The
5
Stochastic verification
is
publicly
announced
privately revealed only to the agent
essential for our analysis since
if all
and
most lending arrangements. 6
summarize technical aspects of the economy.
are independent, identically distributed
probability space (fi,«4, P).
4
is
also appears to accurately describe the privacy
following assumptions
6{
w
public ex-post, there would be no need to monitor
institutional features which characterize
The
model
random
variables on the
7
discussed in the concluding section.
These costs may be thought of as the money paid to an attorney to file a claim (a
pecuniary cost) and the monetary value of time lost when visiting the attorney (a pecuniary
equivalent). Note that Diamond's costs were unbounded while these costs are fixed.
6
Diamond (1984, p. 395) observes: "Financial intermediaries in the world monitor
much information about their borrowers in enforcing loan covenents, but typically do not
directly announce this information or serve an auditor's function."
7
We will always refer to this probability space without mentioning it explicitly, when
(A2) The distribution
F
has a continuously differentiable density / with
spect to the Lebesgue measure, and f(x)
(A3) The ex-post verification cost
is
>
for every
re-
x £ [0,T].
a fixed constant.
Because entrepreneurs have technologies but no input and investors have
input but no technologies,
omy
it is
to facilitate production.
clear that agents will wish to trade in this econ-
The remainder
specifying contracts which govern trade
of this section will be devoted to
among
agents and the optimization
problems from' which these contracts can be derived. Formal proofs that the
contracts are indeed optimal are deferred to Sections 3 and
The Direct Investment Problem
2.1
Let
between investors and entrepreneurs be
bilateral interactions
all direct,
regulated by a contract whose general form
Definition
is
a pair
IR +
4.
,
1.
(/?(•),
A one-sided
5),
< w
defined as follows.
contract between an investor and an entrepreneur
where R(-)
such that R(w)
is
is
an integrable, positive payment function on
every
for
w
E IR+ and S
is
an open subset of
M+
which determines the state where monitoring occurs.
As
is
standard practice
we consider
tracts that
C =
set.
under consideration
such that
restriction
S =
is
{w:
restrict the universe of con-
The
following condition ensures that
satisfy this restriction.
R(w) < R).
without
we
to the set of incentive compatible contracts.
(R(.),S) denote this
tracts
in this literature,
It is
well
loss of generality
known
There
exists
R
all
Let
con-
£ IR +
that the imposition of this
because any arbitrary contract can
be replaced by an incentive compatible contract with the same actual payoff
(c.f.,
Townsend
contracts
writing
P
is
(1979, p. 270)). Therefore, the set of
fully specified
for probability or
E
by the tuple (R(-),R).
for
expected value.
all
incentive compatible
We
will
study a particular type of contract, called a simple debt contract
(SDC), which we describe
optimal contract among
result
(R(-),R)
2.
{w < R} and R(w) =
The payment
(i)
When
(i.e.,
a simple debt contract
if
w G S c = {w >
some
(i.e.,
cutoff level),
well
will often
known
denote
Townsend
that
if:
R(w)
=w
known
wG S=
for
R}.
payment
amount
where S c
R
to the investor
SDCs
S
SDCs by
payment
to the investor
arise as
set of
state con-
is
all
bankruptcy
outcomes
states.
These contracts have been studied
R.
(1979), Gale
constant
the complement of S.
is
viewed as the
is
is
for all realizations of the
the entrepreneur pays the entire realization for
below the cutoff), where
extensively by
a well
schedules in Definition 2 resemble simple debt because:
verification does occur the
tingent
is
the
is
1.
is
the entrepreneur pays a fixed
When
We
Theorem
in
verification does not occur the
state above
(ii)
R
fact that simple debt
one-sided investment schemes
all
which we state formally
Definition
The
in Definition 2.
and Hellwig (1985), and others.
It is
optimal (cost-minimizing) responses to asym-
metric information problems in economies with costly deterministic state
In essence, agents
verification technologies.
minimize verification costs
such economies by verifying only low realizations of
payments (which do not require monitoring)
The
investor
and the entrepreneur's
in all
#,
and accepting
in
fixed
other states.
direct investment
problem can now
be stated formally:
max
f
/
T
\w
— R(w)\ dF(w)
(R(),S)ecJo
subject to
-f
J Jo
R{w)dF{w)-
This "one-sided problem'
and entrepreneurs contract
1
I
Js
cdF{w)>r.
describes the nature of trade
directly:
The expected
(1)
when
investors
utility of a representative
entrepreneur
is
maximized, subject to a constraint that the expected return
of a representative investor, net of monitoring costs (c), be at least as great as
some reservation
and I J J
problem
we
the
number
reflects the
Note that R(-)
the total payment to
is
all
investors,
of investors required to fund a single project.
The
assumption that credit markets are competitive. 8
The Delegated Monitoring Problem
2.2
We
is
level (r).
will
now
construct an alternative, multilateral contract problem that
difference
between the two problems
is
The
problem."
will refer to as the intermediary's "two-sided
as follows.
When
investors
essential
and en-
trepreneurs write direct bilateral contracts, each investor must monitor each
entrepreneur with
whom
plicative monitoring
is
he/she contracts
in certain states of nature, so du-
inherent in this setting. In contrast,
a monitor to perform the verification task,
some
of the duplicative monitoring (even
it
may be
if
investors elect
possible to eliminate
though investors must monitor the
monitor in some states of nature).
Consider now the delegated monitoring problem
intermediary).
investor
The
who wishes
lending market
is
competitive, hence
to act as an intermediary
maximize the expected
utility of
(i.e.,
must
the election of an
it
offer contracts
is
would
free entry into intermediation) with
preneurs and/or the remaining J
which
the entrepreneurs and assure each investor
of at least the reservation level of utility. Otherwise, agents
rectly or another intermediary
follows that an
—
1
offer
would trade
an alternative contract
(i.e.,
di-
there
terms that are preferable to the / entreinvestors. Let (R(-),S)
denote aspects
of the two-sided contract which pertain to the entrepreneur-intermediary re-
lationship and (R*(-), S*) denote aspects of the two-sided contract which
8
We consider an economy in which there are more agents who wish to invest than
investment opportunities. Since the supply of loans is inelastic, the level of return necessary
to attract investors
is
driven
down
to the reservation level
r.
pertain to the intermediary-investor relationship.
problem embodies optimization by
all
9
Thus, the intermediary's
agents in the economy.
Before stating the intermediary's problem, we must
dom
variables which describe the
income from
R(w) denotes the payoff by an entrepreneur
is
and
realized
#,-
is
particular entrepreneur
from
random
the
i
Recall that
portfolio.
to the intermediary
if
output
variable which describes the output
w
of a
is
Gi(R(');u)
=
R(6i(u)).
The
variables G{ are independent for each choice of R(-) because the
Assume
are independent.
that the intermediary contracts with
neur under payment schedule R(-), denoted by
G'{R(-);»)
F
i
J
(-)
The purpose
=
WG
i
G i (R(-);uj),
1,2,...,/
(R(-);
be the distribution function of
of this paper
is
is:
U ).
G
to establish gains
(2)
!
(-).
from delegated monitor-
economy where the intermediary has no information,
ing in an
=
#,-
Thus, the average income of the intermediary per entrepre-
entrepreneurs.
Finally, let
w
Consequently, the intermediary's income
in state u.
this entrepreneur, given transfer R(-),
random
its
derive the ran-
first
risk, or cost
advantage vis-a-vis investors. Since we do not distinguish the intermediary
a priori,
neutral
it
it
has no information advantage.
has no risk advantage.
The
investors' cost structure for monitor-
ing the monitor remains to be specified.
incurred by the intermediary
when
/.
The
Let c denote the actual fixed cost
monitors entrepreneur
it
the actual cost incurred by an investor
diary with portfolio of size
Further, since investors are risk
i,
and
c\
denote
when he/she monitors the interme-
expected monitoring costs are of primary
importance to the intermediary and the investors when they make their
sions at time
9
(i.e.,
1
Js cdF(-) and Js c^dF ^) respectively). These expected
.
Recall that (R(.),S)
one-sided model.
We
is
also used to denote the entrepreneur-investor contract in the
avoid introducing additional notation at this point because the
incentive problem associated with ensuring that entrepreneurs report truthfully
same
deci-
is
the
regardless of whether they report to the investors directly or to the intermediary.
depend on three
costs
factors: the actual costs (c
and
(S and 5*), and the size of the intermediary (via
It is
may
we wish
an independent rationale
cost
F
!
(-)).
Although cost and other asymmetries
its size.
economies (providing other reasons
exist in actual
intermediaries),
and
c}
the relevant states
assume that the intermediary's actual moni-
clearly not reasonable to
toring costs are independent of
c}),
for the existence of
to evaluate the efficacy of delegated monitoring as
Hence we make the following
for intermediation.
symmetry assumption:
(CS) The costs of monitoring the monitor
Assumption
CS
for
is fulfilled,
c*j
Our
its size.
example, when state verification by
vestors involves verifying the full state
intermediary's portfolio).
are linearly increasing in
10
in-
each of the / projects in the
(i.e.,
goal in this paper
is
from
to establish gains
monitoring which stem solely from the intermediary's ability to eliminate
duplicative (but symmetric) monitoring costs.
If
investors can economize
further on these costs (e.g., by monitoring only insolvent firms or monitoring stochastically), then delegated monitoring will be even
Thus, we view
this
The two-sided
example
3.
A
case.
attractive.
11
contract between the intermediary and each entrepreneur,
and the intermediary and the
Definition
benchmark
as a
more
investors, can
two-sided contract
is
now be
defined.
a four-tuple
((/?(•), 5), (/?*(•),
5*))
having the following properties:
(i)
R(-)
is
an integrable positive payment function from an entrepreneur to
R(w) <
the intermediary such that
10
If
the intermediary's actual cost
is
w
for
every
bounded from above
its
w
6 IR+, and S
is
an
marginal cost of contracting
with additional firms is decreasing in 7 at a rate faster than 1/7, giving it an inherent cost
advantage relative to individual investors.
11
Many symmetric cost structures satisfy assumption CS, hence it is not restrictive. We
discuss this example because if we can establish gains from intermediation for a benchmark
case
(i.e.,
when
the institutional structure
costly) then clearly our
arguments
will
is
hold for
such that intermediary failures are quite
less costly cases.
Exponentially increasing
costs are also permissible, as long as they do not increase faster than the bound given by
equation (21). However, we regard such exponentially increasing costs as implausible.
10
M+
open subset of
which determines the
an
set of all realizations of
entrepreneur's project where the intermediary must monitor;
(ii)
R*{-)
is
an integrable positive payment function from the intermediary to
<
the investors such that R*(w)
For every realization
w.
the payment to an individual investor
is
given by
an open subset of 1R + which determines the
w
of
G
12
7
(-),
j^R*(w); 13 and S*
is
set of all realizations of the
intermediary's income from the entrepreneurs the investors must monitor.
We now
As
derive the set of
in the one-sided
problem, we again restrict our analysis to this set withClearly each entrepreneur will again announce an
out loss of generality.
output which minimizes
w = argmin x65 R(x)
arg
its
payment obligations
recall that
R(x).
A
portion of the contract
observed directly
in
the monitoring
R* e
B+
is
given by
similar condition holds for the intermediary-investor
(i.e., /?*(•),
lowing condition ensures that
The
is
this payoff over all non-
Consequently, the announcement by an entrepreneur
min^y, ^
exist R,
w
Let
to the intermediary.
be the output that minimizes
monitoring states, and
states S.
incentive compatible two-sided contracts.
all
such that
set of all incentive
all
S =
R*).
As
in
the one-sided problem, the
fol-
contracts are incentive compatible. There
{w:
R(w) < R] and 5* = {w: R*{w) < R*}.
compatible two-sided contracts
is
fully specified
by
the four-tuple (#(•), R), (#*(•), R*).
Finally, a two-sided simple debt contract
Definition
A
4.
contract (R(-),R),
contract (denoted by (R,R*))
(i)
R(w) =
w
lor
w
£ S
(R
m
can now be defined:
(-),R*)
is
a two-sided simple debt
if:
= {w < R}
and R(w)
= R
if
w
c
G S
= {w >
R}-
and
12
G
() is the average income per entrepreneur defined by equation (2).
It is convenient to define R"{-) as the total payment by the intermediary to investors
per entrepreneur. Therefore, in order to derive the payment to an individual investor we
Recall that
13
must multiply
this
amount with jtt-
11
(ii)
RT(w)
=w
for
w e
S'
= {w <
R*} and R*(w)
=
R*
w
if
€ S*
c
= {w >
R*}.
We
will often
The
denote two-sided simple debt contracts by (R,R*).
now be
intermediary's two-sided optimization problem can
stated:
•T
max
/
[w
— R(w)]dF(w)
,R),(R'(-),R*)Jo
subject to
^—
J —
T
l
Wo
R'(w) dF'W-), R)(w) - f
Js*
c)
dF\R(-), R)(w) >
r
(3)
r.
(4)
T
[ [w
-
R'{w)} dF'iRi-), R)(w)
Jo
-
I
Js
cdF(w)
>
This problem states that the intermediary maximizes the expected
of each ex-ante identical entrepreneur subject to
that the expected payoff to the
did not
utility.
J —
1
remaining investors
become intermediaries) must be
(4) states that the profit
two constraints.
(i.e.,
utility
(3) states
those
at least r, the reservation level of
from intermediation
net payoffs from
(i.e.,
the entrepreneurs less the payoff to the investors) must also be at least
r.
Delegated Monitoring
3
In this section
we obtain two main
simple debt
the optimal contract
This
is
orem
is
results. First,
among
all
Theorem
1
establishes that
one-sided investment schemes.
the Gale and Hellwig (1985) result for our economy.
Second, The-
2 establishes that two-sided simple debt with delegated monitoring
dominates one-sided direct investment
preneurs.
and
who
its
The proof
relies
if
there are sufficiently
relation to the law of large numbers,
we appeal
entre-
on the large deviation principle. This principle,
is
also discussed.
optimality of two-sided simple debt contracts in Section
sections
many
to certain
4.
mathematical results proved
12
We
prove the
Throughout these
in Section 5.
.
Theorem
Simple Debt
1.
among
the optimal contract
is
all
one-sided in-
vestment schemes.
Theorem
similar result
is
1
is
is
proved in Gale and Hellwig (1985) as Proposition
also
A
proved in Williamson (1986). The strategy of the proof
two optimal contracts: Let
as follows. Consider
and
4.
R be a simple debt
contract
A) be some alternative contract. Since both contracts are optimal,
{A('),
both must yield the same expected payoff to entrepreneurs. With the
contract investors request verification
w
verification occurs in all states
w < R.
that w <
if
such
first
In the alternative contract,
A> R
A. Since
(otherwise
the contracts cannot have the same return to entrepreneurs) the expected
verification costs
Theorem
strictly
must be
the simple debt contract R.
less for
Two-sided simple debt contracts with delegated monitoring
2.
dominate one-sided
investment
direct
if
there are sufficiently
many
entrepreneurs.
Proof. Our arguments depend on the continuity of the constraints, established by
Lemma
1
Section
in
5,
and can be summarized
be the simple debt contract which
described by
Theorem
1.
We
is
optimal
show that there
among
exists
all
as follows.
Let
R
one-sided schemes
an R* such that
(3) is
binding for the two-sided contract (R,R*), and that by increasing R* the
payoff to investors increases and (3) does not bind. 14
is
fulfilled
but not binding.
straints slack
Theorem
the
than
R*
14
the
number
of entrepreneurs
since by lowering
in the one-sided
We
if
if
Hence increasing R*
R we
is
We
next show that (4)
slightly
makes both con-
sufficiently large.
This proves
can make the entrepreneurs better
scheme.
begin by showing that the costs of monitoring the monitor go to zero
is less
than the intermediary's expected return from one entrepreneur,
R* because the
an increasing function of R"
In general, the investors' payoff does not increase monotonically with
probability that investors must monitor the intermediary
This
is
off
also true for one-sided
schemes
(c.f.,
is
Gale and Hellwig (1985,
13
p. 662)).
R" < E[Gi(R{-))], and
if
the intermediary
=
J
c'jdF (w)
lim
I—oo Jf
(5) follows
is
R* < E[G
for every
immediately from (21) of
sufficently large. This
Lemma 4
t
(R(-))].
in Section 5,
that the probability of a default by the delegated monitor
means:
(5)
which establishes
(i.e.,
the probability
that an investor must monitor the monitor), converges to zero exponentially.
However, by assumption (CS) the monitoring costs
The key
insight of the proof
The Theorem
(i)
The law
is
is
that, in expected terms, the costs go to zero.
proved as follows:
of large numbers implies that
lim
/— CO Jf
R'(w)dF I {R{-)){w) = R'
(6) indicates that the probability of
R' < E[G
every
for
(5)
and
(6)
that (3)
the continuity of (3) with respect to
exist a face value
contract
(R{-))].
SDC
R
R" < A" such that
(R,R m ). By
(6)
with certainty.
imply that we can find a two-sided contract (R,A
fulfilled for sufficiently large /
is
t
a default by the delegated monitor
goes to zero. Hence investors get the face value of the
(ii)
only linearly.
d] increase
but not binding.
and A" (Lemma
(3)
1),
m
)
such
Because of
there must
binding for the two-sided
is
construction, increasing R" slightly implies that the
investors' payoff increases.
(iii)
All that remains to
Because of
/.
(5),
it
show
is
that (4)
follows that fs
This and the fact that
(3)
cdF >
is
also fulfilled, but not binding.
fs .
c)
dF 1
for all sufficiently large
binding implies
is
l
if R'(w)dF <{J -l)(r+
/ cdF).
Consequently,
/
[w- R'iw^dF 1
J
> IE[Gi(R)]
-J
I
I
cdF
cdF -(J 14
l)r
> Jr
-
(J-- l)r
(7)
where the
first
inequality follows from (7) and the second inequality
R
lows from the fact that
must
constraint (1) of the one-sided
fulfill
problem by assumption. This proves Theorem
Theorem
In the proof of
principle
is
2
we use the
2.
This
large deviation principle.
numbers but
related to the law of large
is
stronger. In particular,
num-
the large deviation principle shows that convergence in the law of large
bers
is
exponential.
the proof, and
The economic
is
The
principle
is
essential in establishing equation (5) in
referenced formally in the proof of
Lemma
4 in Section
problem addressed by Theorem 2
intuition behind the
fol-
monitoring costs are linearly increasing
in the
number
is
that
of entrepreneurs
(i.e.,
as the size of the intermediary gets large, the verification costs, c*r
the bankruptcy state become large as well).
5.
However,
if
,
in the
the probability of
default goes to zero "fast enough," then the expected value of the costs of
monitoring the monitor become insignificant
ary - even though the intermediary
diversified.
The
gence result that
of large
of finite size
and hence
role of the large deviation principle
is
"fast
enough"
(i.e.,
faster
is
is
not perfectly
to provide a conver-
than that provided by the law
2 establishes that two-sided arrangements are better than one-
sided direct investment.
(4)
a well diversified intermedi-
numbers) to generate gains from intermediation.
Theorem
and
is
for
are not binding
maximization problem
is
However, the following problem remains:
it
is
not clear
who
not well defined.
gets the surplus.
The
If (3)
Thus, the
following proposition shows
that this difficulty does not arise and that optimal contracts exist.
Proposition
exist
1.
// there are sufficiently
optimal contracts
among
many
entrepreneurs, then there
the set of all two-sided simple debt contracts.
The two constraints from the intermediary's optimization problem bind for
all
optimal contracts.
Proof. The
result follows directly
from the continuity results of
15
Lemma
1
in
Section
ing to
5.
The
Lemma
existence of optimal contracts
1
(i)
for
and R*. To show that both constraints
optimal contracts, consider the following cases,
do not bind. Then
is
(ii)
R
Suppose by way of contradiction that
straints.
straightforward since accord-
both the constraints and the argument we are maximizing
over are continuous functions of
must bind
is
R
at
an optimum both constraints
can be reduced slightly without violating the con-
This, however, contradicts the optimality of the contract, so
it
not possible that both constraints are slack at an optimum,
Suppose that only
(3)
does not bind.
Then R* can be reduced
slightly
without violating the constraint. This reduces the total payment of the
intermediary to the investors, but
(4)
will
no longer bind and we can
apply the above argument to get a contradiction,
(iii)
suppose that only
Finally,
longer binds
if
R*
is
(4)
does not bind.
We
must show that
increased by a small amount. This
is
(3)
not straightfor-
ward since by increasing R* we automatically increase the expected
of monitoring the monitor.
However,
no
(6) establishes that this
cost
expected
monitoring cost goes to zero as the size of the intermediary increases.
Further,
it
as / increases.
15
from
if
Thus, the expected cost of monitoring the monitor
re-
(4) that
(i.e.,
changes very
(3)
little) if
we
increase
R*
slightly.
only (4) does not bind the gain from a higher payment
to investors exceeds the loss
and
EG
(R)
mains close to zero
Consequently,
R" remains bounded away from
l
follows
from an increase
does not bind, a contradiction.
in
monitoring expenditures
Hence, both constraints must
bind at an optimum.
Optimality of Two-Sided Simple Debt
4
The proof
that two-sided simple debt
is
optimal requires a slightly stronger
result than that used in the previous section.
15
In particular, in the Proof of
Divide both sides of (4) by / and take the limit for /
16
—
oo.
Theorem
we used Lemma
2
bilities of default
from Section
From
zero.
5,
4
from Section 5 to establish that the proba-
We now
converged exponentially to zero.
show that the
to
we
this result
use
Lemma
5
densities also converge exponentially to
get the following Corollary which establishes that
the difference in probability between the realization being below x\ and x 2
respectively,
is
bounded by the absolute value of the
difference
and X2 times a term which converges exponentially to
Corollary
Let
1.
R>
such that |P(G 7 (£)
x2
<
z,
<
>
for every I
This Corollary
is
xx )
I
z
R>
and for every
,
of the proof
zero.
>
there exist a
e~
aI
between Xi
\x 1
-
and I >
x 2 for every x x
\
,
R.
essential for establishing the
The main problem
tion.
< EGi(R). Then
- P{G r (R) < x 2 )\ <
and
,
main Theorem
of this Sec-
that two-sided simple debt contracts
is
do not necessarily minimize the expected costs of monitoring the monitor, as
the following example shows:
Example
In order to simplify the computations,
1.
distribution. Using simple approximation
example to the case
entrepreneurs
and
1
with
i
=
of continuous distributions.
1,2,
and that the
and 2 each with probability
R =
that A(0)
1
and R*
=
the contract
A(l)
is
the
arguments
=
=
0.7;
16
0.3.
discrete
it is
easy to extend the
Assume
that there are two
t
is
with probability
0.4;
Let (P, R*) be a simple debt contract
and (A(-),R*) be an alternative contract such
and A(2)
same
realization of
we consider a
=
2.
The investor-intermediary
part of
both cases, and both contracts yield the same
in
expected return to the entrepreneurs. However, the probability of a default
by the intermediary
is
lower with the second contract. Specifically,
for the alternative contract
As we have seen
in
the one-sided problem.
16
Recall that
Ft*
is
and 0.64
Theorem
Any
the total
1,
for the
it is
0.49
simple debt contract.
simple debt contracts are optimal for
other contracts generate higher expected costs
payment by the intermediary
17
to the investors per firm.
monitoring the entrepreneurs. In the two-sided case we essentially must
for
minimize the sum of the expected costs of monitoring the entrepreneurs and
not be minimal for two-sided simple debt as Example
of the proof of the following
Theorem
sided problems are essentially the
in the
summand need
Unfortunately, the second
of monitoring the intermediary.
is
same
to
1
shows.
The main
idea
show that the one-sided and two-
for large intermediaries (i.e.,
changes
intermediary-entrepreneur part of the contract have a very small effect
on investors' payoffs because of Corollary
Consequently minimizing the
1.
expected costs of monitoring the entrepreneurs
Theorem
If there are sufficiently
3.
many
Proof.
dominate
We
all
the main concern).
entrepreneurs then the optimal
Two-sided simple debt con-
contracts are two-sided simple debt contracts.
tracts strictly
is
other types of contracts.
proceed by contradiction. Assume without
some
that there exists
loss of generality,
alternative two-sided contract (A/(-), /!}(•)), for every
which improves upon the optimal two-sided simple debt contract of The-
/.
orem
We
2,
we can
restrict
our analysis to two-sided contracts.
show:
The
(i)
By Theorem
2.
investor's part of contract A}(-)
must be a simple debt contract,
Next we choose a two-sided simple debt contract (Rf,
R*j)
A'j.
such that entre-
preneurs have the same expected return and the expected payments from the
intermediary to the investors remain constant. 1
in Section 5.
Corollary
1
A) >
R"j.
18
The
<
z
Lemma
1.
contracts (R[, R]) must
< EG^R) < Ri
a
SDC,
EG
fulfill
R>
Lemma 3
the conditions of
and
for all sufficiently large I).
19
z
<
We
EG
!
{R)
show:
/
{Ri) = £'G (.4/()). Because of the continuity results
R], can be chosen such that / R'I {w)dF{R{)) = f A'I {w)dF{A{)).
^First choose Rj such that
18
Furthermore, from
for all sufficiently large / (i.e., there exists
such that R]
of
'
!
This holds since by Lemma 3, f A'j(w) dF(R{)) > J A}(w)dF{A(-)). Thus, to get
equality we must choose R] < A).
1
By the (indirect) assumption of the proof, (Aj(-),A mj(-)) dominates the optimal twosided SDCs of Theorem 2. This is only possible if the probability that investors must mon-
18
(ii)
Using (R^R}) instead of (A/(-), AJ), the
by at most
least
c*I e~
aI
—
\A I
Ri\.
20
m=
Icm\Aj — R[\, where
The
hand
left
min x€
,T]
[
left
f( x ),
hand
side of (3) decreases
side of (4) increases by at
21
because with simple debt
contracts the intermediary must monitor entrepreneurs in fewer states of
nature. This
(iii)
is
essentially the two-sided analogue of
Since for large / the surplus
that
possible to distribute
is
it
much
is
Theorem
greater than the loss,
some
1.
we can show
of the intermediary's gain to the
investors by increasing the face value of
R} such that both constraints are
and not binding. Hence, the face value of Rj can be lowered such
satisfied
that both constraints
hold.
still
The entrepreneurs
are better off with a
contract with a lower face value. Consequently, the two-sided simple debt
contract (Rj, R]) dominates
Therefore
We now
(/!/(•),
A\), which provides the contradiction.
optimal contracts must be simple debt contracts.
all
prove claim
cause the intermediary
is
by the random variable
Next we prove claim
This follows immediately from Theorem
(i).
an entrepreneur whose production
like
G
(because the
(ii).
In order to
compute the change
intermediary goes to zero as
would go to
infinity.
Thus,
in
/
let
of the left
hand
expected monitoring costs
on the left-hand side of
construction of (R, R*)). In the following,
itor the
described
(-).
integral
first
be-
!
we need only compute the change
side of (3)
is
1
(3)
does not change by
Aj be the simple debt contract
gets large. Otherwise the expected monitoring costs
in the limit investors receive the face
value
with certainty,
A*j
lim/^oo f A'j(w) dF(A()) = lim/_ 00 A'j. Clearly, the costs of monitoring an individual entrepreneur J cdF(w) remain bounded away from zero as / —* oo. Dividing both
s
i.e.,
and taking the limit we conclude that lim/_ 00 A*t < lim/^oo EG I (A( •)),
so there exist z,z' such that A] < z < z' < EG I (A()) for all sufficiently large /. Since
R) < A) by Lemma 3 (c.f., previous footnote) and EG '(Ri(-)) = EG^Aji)) by definisides of (4) by /
1
tion,
it
follows that
all sufficiently
20
The
two-sided
intermediary.
21
m
>
large
ft}
/.
<
This
SDC
Thus the
z
<
EG
I
{R I {)). Now choose
R
<
z'
is
exactly the condition of Corollary
such that
z
< R < Rj
for
1.
does not necessarily minimize the probability of default by the
left
hand
side of (3)
may be
by assumption (A2).
19
smaller under contract (Ai(-),A*f).
with the same face value as Ai(-). Observe that:
c}dF(M-))<
cldFiRri-f
J
/
Js k
%
}
I
$dF(Ri)-'[.
Js
Js
h
%dF{Ai).
(8)
h
This inequality follows from two factors: First, the income of the intermediary from contract Aj
is
higher than from contract Aj(-) in
monitoring occurs; and second A}
less
Al -
£,.
22
R}.
from an individual entrepreneur to the
Clearly, the difference in payoff
SDCs
intermediary between the two
>
hence
all states,
with face values Aj and Rj
is
Therefore
I
G\Ri) = ji^Gii&t) > ji2lGi(Ai)-(A t -Ri)] = G (A)-(A I -R I
Hence
7
(8), (10)
/
The
P{G'(Ri) < R] - (A/ - Ri)}.
%} >
and Corollary
1
now
it
/
(-))<cJe-
(A / - R / ).
a/
the decrease of the
(i.e.,
can be computed using the main idea of Theorem
must monitor
A[(-) instead of Ri, the intermediary
hand
left
If
1:
EG^R) <
total loss
we prove
Icm\Aj —
at least
is
(iii).
Let e
>
0.
side of (4))
agents use contract
w 6
cdF > cm(.4; —
in additional states
[Ri,Aj). Hence, expected monitoring costs increase by f^
Finally
(11)
J
Js*,.
intermediary's loss
and the
(10)
follows that
c^F(A
c'jdFiRj)- I
Js*
jR/),
(9)
).
(9) implies,
P{G* (A/) <
From
most
at
1
Rj\. This proves
From footnote
19
(ii).
we have
R[. Hence, by the law of large numbers there exist h
R"j
>
<
z
and
<
/
such that
P({G ; (^ 7 >
)
"A{ — Rj >
(hence
it
R'j
+
h})
>
1
since both contracts have the
has the lowest face value
among
all
-
£,
for all I
I.
same expected return but Ri
same expected
contracts with the
20
>
(12)
is
a
SDC
return).
.
and Corollary
(12)
1
imply that by increasing the face value of
the payoff to investors
—e) — hc*je~ aI
j^-rh(l
by jh(l
hand
—
If
/
is
hand
left
choosing h
is
sufficiently large, this
decreased by at most Ih(\
= j^~c*I e~
aI
—
(Ai
amount
—
is
bounded below
profit
Ri) constraint
(3) is fulfilled
which
is
Icm,
chosen independently of Ai, such that the gain
means that constraint
is
(4)
is
left
it is
and not
Given
the profit of the intermediary decreases by at most I j Yz^c*je~
this to the total gain,
the
(i.e.,
e).
binding (by the computations in the previous paragraph).
Comparing
/i,
side of (3)) increases by at least
Again because of (12) the intermediary's
2e).
side of (4))
By
.
the
(i.e.,
by h <
R*j
aI
clear that
this h,
—
(Aj
Ri).
an / can be
greater than the
loss.
23
This
not binding as well, which proves the Theorem.
Proofs of Mathematical Results
5
The
following notation will be useful in the analysis that follows. Given
two-sided contract,
denote the aggregate payoff of the J —
let Ti(-)
from the intermediary who interacts with
/ entrepreneurs,
and
7!
1
some
investors
denote the
expected payoff of a single investor from the intermediary. For the two-sided
SDC
with delegated monitoring, {R{-), R), {R*(-), R') these payoffs are:
m
=
T I ({R(-),R),(R (-),R'))
I
R'(w)dF I (R(-),R)(w)-
(13)
./o
7 /((tf(0, A), (*•().#))=
The
following
Lemma establishes
tinuous in the face values
constraint (3).
We also
R
j^ryM-)]-
J/ dF\-).
(u)
that the above payoff functions are con-
and R"
for
SDCs.
This proves continuity of
prove results which are necessary to get continuity of
constraint (4) and of the argument of the two-sided optimization problem.
23
and
Clearly this
is
is
the case
if
jYz^ c ) e
aI
< cm which
independent of Aj and Rj
21
holds for
all sufficiently
large /
Lemma
R(w)
Let
1.
R
the functions
f R(w) dF(w) and
t—>
Furthermore, (R,R*)
is
»->
Ti(R,R*)
continuous at every (R,R*) £
The proof
Proof.
R.
be one-sided simple debt with face value
is
M
2
,
R
Jw< RcdF(w) are continuous.
i—>
continuous; and (R,R*)
such that R*
two functions
for the first
Then
<
i—
~fi(R,R*)
>•
R.
We now
straightforward.
is
prove continuity of Yj. Note that,
R
PiiG'iR) >
h)
>
h
for
< P({G*(R + h)> R m }) < P({G ; (#) > R* -
m
})
because of
dF(R)(x),
for
— ^(Oloo —
\A
—
\T I (R
>
for every h, h"
Tj.
R\, for all simple debt contracts
+ h,R* + h*)-T I (R,Rm <
)\
for every
bounded by a
is
implies
|
f*^
c
m
Kn
dF T (R) - f*^
holds for every R"
<
£ IR
< R and
c'
h,h* £
(R,R m
to prove that
every (R, R*) 6 IR such that R"
which
h)(x)
< f f(x +
/ by a standard approximation argument. Further,
and hence
now remains
It
f{x)dF I {R +
(15)
every increasing step-function /, and hence for every arbi-
trary increasing function
|A(-)
Therefore /
(10).
h}),
R.
in
The
\h\+
h)\
R" h+ f_ ao
<
dF(R)
c"
G
Lemma
Kn
(16)
This proves continuity of
f_ 00 c~
(c.f.
and R. Thus,
\h*\,
distribution of
(— oo,i?)
dF\R +
since
i—*
)
IR.
A
\h\.
dF ! (R)
l
is
continuous at
(R) has a density
4).
Therefore (15)
Since this inequality
is
clearly continuous,
this proves continuity of 7/.
To prove optimality
nical
Lemma
total
payments
2.
we need an additional
tech-
which shows that two-sided simple debt contracts maximize
to investors (not including
formally stated in
Lemma
of two-sided simple debt
Let
f.t
Lemma
3.
It
monitoring costs). The result
follows immediately
be a probability
is
from the next Lemma.
measure on [0,A/]. Let R(-) and A(-) be
two contracts with the same expected value.
99
We assume
that R(-)
is
a simple
debt contract R.
Then
the simple debt contract
sense of Rothschild and Stiglitz;
for
all
If
f u(A(w)) dfi(w)
i.e.,
Lemma
interpret u as a utility function, then
riskiness of
Theorem
is
one of the
two distribution of Rothschild and
The
straightforward.
the proof of
Proof of
Lemma
Lemma
T(y)
=
3
2.
is
due to Martin Hellwig.
T(M) =
(17)
Ha and Hr
Let
and
criteria for
value of
t
We now
< R, and
more
more
is
be
3.
Lemma
give the proof.
= H^t) — Hpt(t),
risky than R(-)
if
the following
fulfilled:
and
0,
T(y)
(18)
that G(t)
Theorem
<
for
every
>
for
2 of Rothschild
R < t < M.
and
risky than R{-). This proves the
Lemma
using
be the cumulative density functions
every y e
T(M) =
— Jq xg(x)dx = 0, since the integral over xg(x)
A(-) — R(-). This proves (17). Now note that
proves (18).
is
By
the proof of the
ff,
Let g be the density of G. Partial integration yields
xG(x)\q I
comparing the
Stiglitz (1970).
R{-), respectively. Let G(t)
y
/ G(x) dx. Then A(-)
two conditions are
<
2 essentially says that
idea to use the Rothschild and Stiglitz criterion for
of the distributions of A(-)
let
f u(R(w)) dfi(w),
2 of their paper which proves the equivalence of three different
concepts, and by using their argument on p. 230
and
<
averse consumers prefer the simple debt contract to any other contract
with the same expected value. This
is
risky than A(-) in the
concave functions u.
we
all risk
is less
is
EG
I
23
G(x) dx
=
G(t)
>
for every
This together with (17)
Stiglitz (1970) implies that A(-)
Lemma.
(A(-))
ri(A('),tir).
M}.
just the expected
Let (R, R") be a two-sided simple debt contract
an alternative contract where
A/
/
[0,
= EG
!
(R).
and (A(-),R m )
Then r[(R,R") >
Proof. Note that
r,(£(.).
&()) =
Since R*(-)
is
JJ-"J
R ^jE R wi)) dF(w
(
Next we show that convergence
The proof
Lemma
random
is
—>
Let
4.
ff.)
variables with values in
~Y^=
\
Then
X{.
—
df.i(x)
implies that
[0,
/^ n
M] and
distribution
<
X
€
IR.
exponential.
decreasing on
\x n
be the
{
Then Cramers theorem (Stroock
=
Since H(0)
for
< -
x \
[
the large
satisfies
J(x),
inf
(Stroock
every b
>
X(x)
,6]
<
0, it is sufficient
(20)
for
Lemma
EX
to
.
t
every
(3.3)).
For fixed
6
<
EX
:
t
Therefore we
b let
show that H'{0)
^
H(£)
0.
=
This,
verified:
H'(f)
Uj
there exists an a
fi n
.
— oo,EX
£b- logM(£).
=
has a compact
f.i
X Consequently
prove that inf x6
>
consequently H'(0)
Let
x€[0,6]
need only prove that X(b)
t
is
\l.
JR, since
a "rate function", and that
is
We now
however, can be easily
E
oo for every £
n£N
EX
number
converges to zero exponentially as
([0, b\)
limsupl/nlog/£ n ([0,6])
is
2 and inductively
independent identically distributed
supe € jj(£x— log M(£)).
deviation principle with rate
X(x)
(19)
< EX{.
M(£) = f e^ x
for every 6
.
Lemma.
in the law of large
be a sequence of
(X,-),- e jv
support. Let X(x)
p. 30
dF( Wl ).
.
follows immediately from the large deviation principle.
oo for every b
Proof.
.
used in Section 3 to establish gains from delegated monitoring.
distribution of
n
)dF(w 2 )
we can apply Lemma
a concave function
substitute R(-) by A(-) in (19). This proves the
This result
l
b
>
- EX, ^
b
0.
f xe ^f( x ) dx
Jet* f{x) dx
This and (20) imply that
and an integer n such that \/n
24
log
for
/i n
every
([0, b])
b
<
< —a
for every
n
>
n.
Consequently
M[0, b]) <
e~
^ every n>n.
an
(21)
For the optimality of two-sided simple debt contracts proved in Section 4
we need a stronger
the densities of
where
random
=
jj,
Let
5.
(X
t
\8r where supp
M\. Let
ji
—*
on every interval
[0, 6]
The
M] and
fi
=
n
-\-
jl n
all
distribution
Y%=i Xis an d
\ n 8R where supp/i n
idea of the proof of
Lemma
t
5
).
is
nomial term
if
ercise 14.24.
is
[0,
as follows.
G
differentiable with
straighforward
compact support
R].
ji.
has a density
n
!
(R(-))
is
We
is
to zero as
exponential.
show that
first
bounded by a poly-
the densities are continuously
in IR, see for
example Floret (1981) Exand
In our case the densities are discontinuous at
R
which
The
requires us to deal with the derivatives at these two points separately.
idea of the proof, however,
is
essentially the same.
If
the
Lemma
hold then this immediately implies that the measure of an interval
only converge to zero with polynomial speed (25). However, by
probability of every such interval with b
exponential speed as /
Proof:
We first
problem
is
that
—
>
< G
l
is
oo (21). This contradiction proves
discontinous at
and
25
R
does not
[0, b]
Lemma
can
4 the
[R{-)) converges to zero with
Lemma
derive an upper boundary for the derivative of
/
that
Dirac point
be the
This convergence
the derivative of the density functions of
This
C
^ $R
measure which converges uniformly
compact subsets of[0,EX
(24).
We assume
/i.
C [0, R] and \i has a density f with respect to the
We assume that f is continuously differentiable on
respect to the Lebesgue
oo on
[0,
be the distribution of -
ft n
Then
measure.
n
that
be a sequence of independent, identically distributed
)j 6 /v
variables with values in
+
fn with
we must show
In particular,
4.
also converge to zero exponentially
fi n
Lebesgue measure on JR.
[0,
Lemma
< EX{.
b
Lemma
H
than
result
5.
fn The main
.
as a function defined
on
IR.
Claim
Let
1:
v be functions on IR with support in [0,T].
it,
Let u be
continuous from the right and bounded. Let v be continuously differentiate
Then the convolution u
in [0,T].
< IMLbll
u)'I
Let z(t,x,h)
x
-
<
t
=
+2|«|
B2 h =
let
,
=
z(t,x,h)dt
-
-
h
+
th,x,h)dt
z(x
<
t
<
x
T}.
=
from the
z(t,x,h)dt
A
right.
/
Blfh =
let
{t:
-h <
Then
u(x
+
th)v(h
—
th) dt.
Jo
Jo
Consequently lim^ois
*
|(it
.
{t:T
h
JB lh
tiable
continuous from the right and
is
u(t)±[v(x + h-t)-v(x-t)]. Furthermore
0} and
/
||u|
* v
=
u(x)u(0), since u and v are differen-
argument yields lim^o fB2
similar
h
z(t,x,h) dt
=
u(x)v(T). Therefore
=
{u * v)'(x)
=
lim [ z{t,x,h)dt
I u(t)v'(x
'
h[0 J
which proves claim
Claim
point
The
2:
^
x
<
<G
and
+
^I/ll
for
2
n
The right-hand
.
every x €
suc ^
+
u(x)v(0)
+
u(x)v(T),
^ at
[0, rci?).
^
ItfuWI
n2
The
derivative g'n (x) exists for
Further, there exists a constant
K
f° r
W and
ever Y n €
for
every
[0,ni2).
For the proof of claim 2
We now
to prove.
formula
9n(x)
we proceed by
=
sum
for the "point
n -i(t)f(x
R
mass" A at
-t)dt + X n
n
=
for
1
n
is
— .1.
Using the
~l
and A n_1
f(x
-
(n
-
at (n
l)R)
+
— \)R we
Xgn ^(x
get
-
every x £ [0,ni?]. (22) implies
l^nlL
nothing
there
two independent random variables and
of
1
Jg
induction. For
assume that the inequality holds
for the density of the
accounting
for
dt
1.
probability A
W
-ll/'loo
t)
distribution of ££=1 X{ has a density g n on [0,nR).
{nR} has
every n E
-
J
< bn-l|J/L + |/L + bn-lL < 21/^ +
26
WOn-ll^-
H),
(22)
.
Consequently
lfl»L
Using
and claim
(22), (23)
1
< 2n|/|„„.
substituting the induction hypothesis for
the proof of claim
= ng n (nx)
We get
Lebesgue measure.
<
4
n K,
is
EX
<
K=
2
/
2||/
|
oo
+4|/||
,
we conclude
the density of
jl
with respect to the
n
every n £ iV, and for every x E [0,R).
for
In the second part of the proof
exists a 6
A-M/'L + Wn-iL-
2.
Note that fn (x)
|/n( x )l
(23)
yields
\S»L $ ISn-iU/'l. + 4n|/£ +
By
'
we proceed
indirectly.
Assume
(24)
that there
such that fn does not converge uniformly to zero on
t
[0, 6]
with exponential speed. That means that there exist a sequence e n that does
not converge exponentially to zero and a sequence (y n ) n ^pj in
fn{Vn)
>
implies that
dicts
//
Lemma
We
n E IN.
£ f° r every
n ([0,6])
[0, b]
such that
prove that under these circumstances, (24)
cannot converge to zero exponentially which contra-
4.
Let
l
=
\
/
h n (x)
(
for
b'
6 [b,EX
every n
>
x
n.
).
4
K{x
-y n
if
),
yn
<
x
<
+ -^;
yn
<
.
{
Let
e-n
otherwise.
0,
Then
there exists an integer n such that y n
Consequently
<
h n (x)
< fn {x)
for
every n
>
+
rc,
-frr
<
b'
because
of (24). Therefore
6
/*n([0,6'])
By
(21)
and
(25)
This proves the
we
=
/
Jo
get ^477
fn (x)dx >
<
I* h n (x)dx
Jo
e~ an for every n
Lemma.
27
=
-i—
2n 4 A
> max(n,
(25)
n), a contradiction.
We
are
now ready
Proof of Corollary
/ which
is
to prove Corollary
The
1.
i
—
+
/|[o,ffl
from Section
distribution of
continuously different able on
then given by Jr
1
<$r,
[0,
0,
4.
has by assumption a density
M]. The densities of
so the constant
K
F
1
(R) are
of (24) can be chosen
independently of R. Thus inequality (25) holds uniformly for every R. Further,
P(G n (R) <
x)
< P{G n (R) <
uniformly for every distribution
R>
holds uniformly for every
6
\i
n of
e M. Thus, (21) holds
x) for every x
G n (R),
where
R>
R, so
Lemma 5
also
R. This proves the Corollary.
Concluding Remarks
In this paper
we extend the Gale and
Hell wig (1985) contract
the delegated monitoring setting proposed by
mond, a
crucial
problem
truthfully to investors.
in the analysis
We
is
Diamond
(1984).
to ensure that the
problem
As
to
in Dia-
monitor reports
use a two-sided version of Townsend's (1979)
fixed cost, deterministic state verification technology to solve the monitor's
incentive problem.
We
show that delegated monitoring dominates
investment and that two-sided simple debt
is
optimal.
direct
Our economic
envi-
ronment requires us to introduce new mathematical arguments based on the
This mathematical technique
large deviation principle.
model because when the
cost of monitoring the monitor
the law of large numbers
large deviation
argument
is
essential in our
depends on
its size,
not sufficient to establish gains from interme-
is
In addition, even
diation.
is
if
monitoring costs are independent of
size,
the
necessary to derive the optimality of simple debt
(since uniform convergence of the densities does not follow
from the law of
large numbers).
We
obtain our results
herent information,
ministic.
in
an economy where the intermediary has no
risk, or cost
advantage, and where monitoring
Recently, costly state verification studies
28
(c.f.,
is
in-
deter-
Townsend (1988)
and Mookherjee and Png (1989)) have shown that the form of the optimal
may be
contract
altered under stochastic monitoring. In contrast, our dele-
gated monitoring result
from the
is
not affected by stochastic monitoring. This follows
fact that as in the deterministic case, the probability that a state
occurs in the stochastic case which triggers monitoring goes to zero exponentially.
as well,
Our
Hence the expected
cost of monitoring the monitor goes to zero
and delegated monitoring continues
dominate direct investment.
to
rationale for studying deterministic monitoring
for the cost
symmetry assumption
in Section 2.
from delegated monitoring which stem
solely
which further reduce monitoring costs
intermediation will be even more attractive.
29
similar to that given
wish to establish gains
from the intermediary's
to eliminate duplicative verification costs for a
tors exist
We
is
benchmark
(e.g.,
ability
case. If other fac-
stochastic monitoring),
References:
J.
Boyd and E.C. Prescott
nal of
(1986), "Financial Intermediary Coalitions," Jour-
Economic Theory
D.W. DIAMOND
38,
211-232.
(1984), "Financial Intermediation and Delegated Monitoring,"
Review of Economic Studies
51,
393-414.
K. FLORET (1981), "Mafi und Integrationstheorie, eine Einfiihrung," Teubner:
Stuttgart.
D.
Gale and M. Hellwig
(1985), "Incentive-Compatible Debt Contracts:
One Period Problem," Review of Economic Studies
D.
MoOKHERJEE and
I.
Png
J.
Stroock
Economics 104, 399-415.
Stiglitz (1970), "Increasing Risk:
Journal of Economic Theory
J.
647-663.
52,
(1989), "Optimal Auditing, Insurance and Redis-
tribution," Quarterly Journal of
M. Rothschild and
The
2,
A
I.
Definition,"
225-243.
(1984), "An Introduction to the Theory of Large Deviations," Uni-
versitext, Springer.
R.M. Townsend (1979), "Optimal Contracts and Competitive Markets with
Costly State Verification," Journal of Economic Theory 21,
R.M. Townsend (1988), "Information Constrained
Principle Extended," Journal of
S.D. Williamson
Insurance:
Monetary Economics
1-29.
The Revelation
21,
411-450.
(1986), "Costly Monitoring, Financial Intermediation, and
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30