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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Collusion
in
Hierarchical Agency
Fred Kofman and Jacques Lawarree
Working Paper
#
31
88-90-EFA
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
OCT
23
1990
Collusion
in
Hierarchical Agency
Fred Kofman and Jacques Lawarree
Working Paper
§
31
88-90-EFA
Collusion in Hierarchical
Fred
Agency
Kofman
and
Jacques Lawarree*
May
1988
Revised March 1990
Working Paper
*
#
3188-9O-EFA
Sloan School of Management, Massachusetts Institute of Technology and Department of
Economics, University of Washington
Richard Gilbert and Jean Tirole for
benefited from
Katz,
Mathew
at Seattle.
many
We
thank Patrick Bolton,
helpful discussions and
Drew Fudenberg,
comments.
We
have also
comments and suggestions by Mathias Dewatripont, Ben Hermalin, Michael
Rabin, Pierre Rdgibeau and Toshi Shibano.
'
Abstract:
We
study a model where shareholders can use auditors' reports to contract with a
privately informed manager but the
manager can
Such auditors
good information and
are useful
if
they have
In the optimal contract under collusion,
production does not reach
bribe he
is
its
bribe the auditors to manipulate their reports.
the liability
of the manager
even with unbounded punishments and costless
high.
auditing,
optimal level. Raising the punishment for the manager raises the
willing to offer the auditor raising the cost of preventing collusion.
grows without bound and
is
part of the
punishment
is
non-transferable
maximum
When
liability
deterrence will
not be optimal
To model
cross-checking mechanisms,
manager) from external
(costly but
might specify random external
the optimal contract
Finally,
we
is
we
distinguish internal (costless but
never collude)
audits.
auditors.
We
may
collude with the
prove that the optimal contract
We consider a self-interested external
auditor and find that
unchanged
present a model where allowing collusion
is
the optimal strategy for the principal.
Collusion in Hierarchical
Agency
Introduction
In recent years agency theory has begun to study the design of multi-person organizations
by a
principal
agents
who
is
interested in achieving
who posess private
performance from
self-interested, strategic
information. For instance, the board of directors of a firm must
design incentive schemes for the managers and other employees. These employess
things about themselves, their environment or their actions
which are unobservable
know
to third
parties.
The
research in this area has
by and large ignored
the possibility of collusion of various
Economic models
sorts of people within the agency.
in general, rarely consider coalitions
with side-contracting power in the design of incentive schemes. Problems of sidepayments, however, have been extensively recorded by sociologists and other behavioral
scientists; furthermore,
would
still
even
if no
evidence of collusion had been found
its
mere
possibility
constrain the set of feasible contracts.
A major exception to this kind of research is Tirole [1986], who has identified and studied
efficiency losses that can result
supervisor-agent hierarchy.
from collusion with side-payments
He notes that coalitions with (implicit)
likely to form in long-term relationships
in a principal-
side-contracts are
more
through a reputation-like mechanism. These
relationships are a mitigated blessing: repetition enhances productivity as well as
opportunities for collusion.
interactions
may be
efficiency losses.
Although job-rotation and other impediments of prolonged
effective for disrupting coalitions, they
may also
be dear in terms of
In this paper,
we
consider a different
Our
source of information.
suggestion
relationship with the first)
additional supervisor
is
way of preventing
is
to
collusion; creating an alternative
introduce a second supervisor (in a horizontal
whose purpose
is
to
discourage deviant coalitions. This
involved in a short-term contract with the hierarchy and thus less
prone to organizational-culture pressure; he
is
also less efficient because he lacks specific
experience and knowledge of the job.
An
illustration
of
this
phenomenon
Internal auditors provide
is
the coexistence
of internal and external auditing.
management and shareholders with valuable information
concerning not only the financial situation of the firm but also other aspects outside the
accounting area. Indeed, internal auditors are usually involved in evaluating the efficiency
of the firm's operating division. Labels such as "operational audits," "management audits,"
or "performance audits" are used to describe this kind of audit 1 .
Because they "live
in the firm," internal auditors
possess high quality information and,
since they also provide non-auditing services (e.g. tax assistance), the opportunity cost of
their auditing function is low.
External auditors, in contrast, have poorer information about the firm and are
Although
their use
can
still
be justified on "informational grounds," our intuition
additional information they provide does not account for the extent of their
the real world.
1
more
is
costly.
that the
employment
In addition, most external operational audits are conducted on a
in
random
While
financial audits are limited to evaluating the fairness of financial statements,
operational audits involve any activities for which operational effectiveness can be
evaluated (see, for instance, Arens and Loebbecke [19881). Although the accounting
literature links financial audits to adverse selection and operational audits to moral
we
study a model with both hidden information and hidden action which is
However, it is possible to
interpret our model in terms of a financial audit as well.
hazard,
more
easily exposited in terras of an operational audit.
basis 2 If one wants to argue that external auditors are hired only for their information, this
.
randomness seems hard to rationalize.
In this paper
explained by
we
its
present a
role in
model
in
which we show
likely to suffer pressures to give
problem
is
more
by the management, hence they are more
favorable performance treatment or, in the case of a
"window-dress" the
financial statements in favor of the
managers. This
of significant concern to regulatory agencies and the accounting profession 3
External auditors are usually hired
by
the shareholders (or the audit committee of the
of Directors), and arc therefore more independent from the manager4
The optimal
auditing policy involves a trade-off
information at low cost but subject to
poor information
at
performance. First
between
management
Board
.
internal auditors with
pressures,
.
good
and external auditors with
high cost but with unbiased reports.
In our model, the internal auditor
is still
can be better
enhancing the independence of internal auditors (See Antle [1984]).
Internal auditors are usually hired (and fired)
financial audit, to
that external auditing
we prove
and the manager can
that,
useful for the principal.
when his
falsify
information
is
evidence about the manager's
very
Then we introduce an
reliable, the internal auditor
external auditor
who
reports
2
Regarding external financial audits, most firms are legally required by the SEC to
perform them annually. One implication of this paper is that this annual requirement
might not be optimal.
3
See the various reports of the Moss Committee (U.S. Congress [1976]), the Metcalf
Committee (U.S. Congress [1977a], [1977b]), the Dingell Committee (U.S. Congress
[1985]), the Securities and Exchange Commission accounting series releases (SEC
ASR 165 [1974], ASR 247 [1978], ASR 250 [1978], ...) and the Cohen Comission's
Report on Auditors' Responsibilities [1978]. An extensive discussion of this problem
can be found in Mangold Nancy R. [1988]
4
A reputation of independence being an essential asset for an external audit
firm,
some
of these firms have also created organizations (with voluntary membership) to improve
the quality of auditing practice. For instance, the "SEC Practice Section" of the AICPA
imposes stringent auditing standards on its members (partner rotation, concurrent
partner review, proscription of certain services, mandatory peer review, etc.)
truthfully.
auditor but
For simplicity,
is
manager and
we assume
that
he has the same information as the internal
more costly 5 Therefore without the
the internal auditor, the external auditor's services
on the values of the parameters of the problem,
may
not be hired in the optimal auditing
implemented
it
possibility of collusion
.
at all,
may be optimal
they are performed
for the principal to
Another contribution of this paper
we
have no value. Depending
find that the external auditor
scheme with
on a random
between the
may
or
collusion. If external audits are
basis.
Last,
we present a model where
allow collusion.
is to
show that, under
collusion, even
when unbounded
punishments for the agent are available, the optimal contract will not specify efficient
production (inefficiencies in production are maintained
at
every level of punishment). In
addition to their deterrent effect, higher punishments induce higher bribes which are
difficult to prevent.
more
This creates a situation analogous to one where using the internal
auditor costs the principal a fixed proportion of the punishment he imposes on the agent
Literature Review
Principal-agent theory has paid considerable attention to the incentive problems arising in
two-layer hierarchies,
i.e.,
one principal and one agent (Baron-Myerson [1982], Maskin-
Riley [1984], Laffont-Tirole [1986]). With few exceptions, the standard model has been
extended naively to incorporate a third layer, usually a supervisor, as a "third arm" for the
principal (Baron- Besanko [1984]). This description of the supervisor ignores the possible
conflict of interest
problem
arises
between him and the principal. If their objectives
a
new incentive
and the self-interested supervisor must be adequately motivated to act on
behalf of the principal.
5
differ,
Demski and Sappington [1987]
studied the case of a supervisor
Assuming that the external auditor has worse information than the internal would be
more realistic but it would complicate the presentation of the results without adding to
their understanding.
subject to moral hazard in a consumer- regulator- firm hierarchy. But, in their model, they
do not consider
the possibility
of coalition-formation. Baiman, Evans and Noel [1987]
studied the gain of transferring the adverse selection
but, again, no collusion
The principal can
is
allowed
also increase his utility
[1983] formalized
[
is
of the agent's
.
utility
Stiglitz [1983]).
The
is
highly dependent on the agent's preferences; a
function can
is
make competition harmful.
similar to the
one of the tournament
literature.
A
a situation in which an agent's payment depends only on his output or rank
relative to other competitors
compatible
alternative sources of information. Hart
works harder when competition increases 6 However,
The economic intuition of these models
tournament
model.
by using
1988] showed that this result
slight alteration
the agent to the auditor
idea in a model where product market competition can alleviate the
this
incentive problem: the agent
Scharfstein
in their
problem from
But
if agents
literature
all
(Green-Stockey [1983], Lazear-Rosen [1981], Nalcbuff-
of the schemes discussed in these papers are not incentive
can collude.
on implementation has studied the problem of preventing unwanted
(collusive) equilibria.
When the direct
resort to non-revelation or indirect
revelation
game has multiple
mechanisms. This has been done
mechanism
equilibria
in three
one needs
to
ways: (1) Take
which the unwanted
the revelation
mechanism and construct an
equilibria are
knocked out by giving agents additional ("nuisance") strategies (Maskin
[1977],
Ma, Moore and Tumbull [1988]).
that his desirable equilibrium strategy
indirect
(2)
Change
in
the payoffs to one of the agents so
becomes a dominant
strategy
Sappington [1984] and Demski, Sappington and Spiller [1988].) (3)
6
(Demski and
Add
stages to the
Contestability theory (Baumol, Panzar and Willig [1982]) claims that the threat of
competition by itself can make a natural monopoly efficient. On the effects of potential
competition see also Gilbert [1989].
mechanism, and impose the requirement
that off-the-equilibrium-path strategies (threats)
must be credible (Moore and Repullo [1988]).
Ma, Moore and Tumbull [1988] study
work
in a correlated environment.
which a principal
the case in
They point out
hires
two agents
that the standard technique
to
of making
each agent's reward dependent on the other's performance has unsatisfactory features.
Namely, the agents
the principal.
(indirect)
strictly
prefer to play equilibrium strategies other than those desired
To knock-out
these undesired but incentive-compatible equilibria their
mechanism uses one agent
to police the other.
one of Demski and Sappington [1984]
incentives to the "police" agent
strategy,
by
Ma, Moore and Tumbull
in that
by making
offer
him a
Their technique differs from the
while Demski and Sappington give the right
the principal's desired choice a
set
dominant
of additional output levels from which to
choose. While Demski and Sappington strengthen the principal's constraint set -and thus
reduce his profits- Ma,
Moore and Tumbull use
the standard constraints
and the costless
technique of adding strategies which will never be played in equilibrium.
Moore and Repullo [1988] consider situations where agents
state
of the world while the designer
is
Revelation Principle has almost no cutting
are completely informed
completely ignorant of
power and they
set
it.
of the
In these cases, the
up a stage mechanism which
has a unique (the desired one) sub-game perfect equilibrium.
The
issue they
do not consider
is
that although their
coalition-proof in the sense of Bemheim, Peleg
against coalitions
when side-payments
game
set
up by the
and Whinston [1987]
it
is
unique and
need not be stable
are allowed. In other words, if side contracts are
feasible the agents might deviate together
equilibrium of the
Bayes-Nash equilibrium
even though the outcome they play was not an
principal's contract.
Grossman-Hart [1986] and Williamson [1985] discuss the difference between internal and
external auditing in their studies of integration and the limits
of
firms.
Grossman-Hart
argue that "integration in itself does not
make any new
Any audits which an employer can have done of her
variable observable to both parties.
subsidiary are also feasible
when
the
replies that interfirm auditing cannot be
company." Williamson
subsidiary
is
a separate
presumed
to
be as effective as intrafirm auditing: "[i]f a stronger mutual interest in
organizational integrity can be
would
exist
between independent trading units
receive greater cooperation
ownership boundary
is
presumed among members of an integrated organization than
is
[...]
[...]
than can be presumed
then internal auditors can expect to
when auditing across an autonomous
attempted'' Williamson, however, recognizes that internal auditing
subject to corruption
and points out
how
important
is
the question
we
address in this
paper "are the flaws of internal auditing remediable?"
Baiman, Evans and Nagarajan [1989] also study a model of auditing with collusion. Their
approach
is
quite different
the realization of a
from ours. The manager does not take any
random output
to the principal.
who observes output perfectly. The manager and
not
model the incentives for collusion; a
he transfers
principal can use a costly auditor
the auditor can collude but the authors
move by
enforcing collusive arrangements are feasible.
internal
The
effort;
do
"nature" determines whether self-
Finally, they
do not
distinguish
between
and external auditors.
The Model
We study a three-layer hierarchy consisting of a principal,
principal
owns
the vertical structure; the
information about
its
an auditor and a manager. The
manager runs a productive
unit with private
efficiency; the auditor collects information for the principal.
Following Tirole (1986]
required to supervise the
we assume that the principal lacks either the time or the knowledge
manager and
required to run the vertical structure.
that the auditor lacks either the time
or the resources
Tirole relies on a third
among
risk
axiom which
rules out the possibility
several auditors. This allows the auditor
of dividing the auditing job
and the manager
of being discovered. The point of this paper
is
to collude without
any
to explore the consequences of
introducing a second auditor together with a system of rewards
and punishments
to prevent
deviant coalitions.
The
first
auditor
hired anyway;
an
—
internal
we assume
— performs several other tasks inside the firm and has
that the opportunity cost
audit, the internal obtains information
about the privately
manager. "Living in the firm", the internal
second auditor
—
— serves only
external
of him auditing
is likely to
to
is
zero.
known
to
be
By conducting
productivity of the
collude with the manager. The
perform his audit;
we assume
that
he
is
"collusion-free" (he always reports truthfully), that his services are costly and that his
information
is
perfectly correlated with the internal's information.
Players
Four people
interact in this
agency relationship: the principal, the
external auditor and the manager. All are
The manager
assumed
exerts effort e which, together
to
be
internal auditor, the
risk neutral.
with a productivity parameter
6, determines
output x
x=6 +
We
assume
function.
e
that the cost for the
is
an increasing, convex
For simplicity we take
g(e) = c2/2
7
manager of exerting effor g(e)
'
We have obtained
(available
the same qualitative conclusions in an earlier version of this paper
upon request) considering a more general specification of g(e).
The output belongs
to the principal
manager maximizes
his
who compensates
the
manager with a
transfer
t.
The
payoff
Manager's payoff =
t -
g(e)
Since the manager signs the contract after learning the realization of 6, he must receive his
(We assume
expected reservation utility in any state of the world.
principal to
The
that is optimal for the
employ the manager even when the lowest 6 occurs.)
internal auditor
observes a signal s imperfectly correlated with 6.
He
receives a
wage
w from the principal, and his objective is to maximize his expected monetary income. The
same signal
external receives the
s
about
and reports
it
truthfully to the principal. His
services are provided at a cost z.
The
principal receives the residual profits of the vertical structure
when
is called for)
it
the other
members of the
and pays (or punishes
hierarchy. His objective is to
maximize
expected profits
E(;r)
= E(x
- 1 -
w
-
z + punishments)
Uncertainty and information structure
We assume that 8 can take only two values,
with probability
suppose
The
(1
-
q).
that the output
internal
6i
< 82
.
8i obtains with probability
While 8 and e are private information of the manager, we
x (- 8 +
will
e) is publicly observable (and verifiable).
and the external obtain an identical signal
signal can be si or S2
q and 6 2
which occur with the conditional
s
imperfectly correlated with 8
probabilities described in Table
The
1:
10
si
r
S2
11
Note
that
we
rule out the possibility of
blackmail by the internal. If he gets a signal
favorable to the manager, he can not threaten the
manager with
revelation of a less
favorable signal since he needs his help to falsify evidence. In that sense, information
neither completely "soft" nor "hard".
convey information
productivity but
is
By
this
we mean
to the principal in a credible
verifiable
when
it
is
that the internal's report can not
way when
it
announces a bad
of
state
announces high productivity.
In Tirole's model the supervisor can only hide information (reporting he observed
"nothing") but not forge evidence.
An auditor's report is usually a point estimate of the cost
of the firm or a acceptance/rejection of the firm's financial report, therefore "nothing"
a
realistic alternative for his report.
(See Shibano
lead to a posterior distribution that
is
( 1
988]).
It
may be
that
is
not
some observations
These uninformative
identical to the prior.
observations are formally equivalent to observing "nothing," so that our information
structure is not inconsistent
with Tirole's.
We will only study collusion between the manager and the auditor since they constitute the
single nexus with ability to manipulate the information received
by
the rest of the vertical
structure.
To avoid the distracting problem of bargaining, we will assume
Pareto efficient outcome for
party at least
further
the
its
members, and that the division of payoffs guarantees each
what they would get without collusion. For the
assume
that the
that the coalition chooses a
last part
of the paper
outcome of the negotiation between the manager and
we will
the auditor is
Nash bargaining solution.
Strategy spaces
The
principal designs the contract
manager t(x) and uses the
and offers
it
to the
manager and
internal with probability y(x).
the auditors.
He pays
the
For each 6 the contract determines
12
when 8
an optimal level of effort (ei and e 2 ) yielding Xi
pays the internal w(x,s') and uses the external
the internal's report and
If an auditor's
it
may be
Si, s 2
or
at cost
when 6
= 82
9
probability d(x,s>).
was not
He
).
s' is
used.
announced signal differs from the manager's (implicit) report, the principal
from the one of the
internal, the principal
P™
.
If the external reports a signal different
can impose a penalty on the internal of up to P1
feature of limited liability enables us to solve the
many of
obtain
—with
z
if the internal
can impose a penalty on the manager of up to
The
—
= 9i and x 2
the features
model with
risk-neutral players
.
and
of a model with risk-averse players and unbounded
punishments (see Sappington [1983] and Baron-Besanko [1984]).
The manager chooses effort
of a side-payment
He
e.
More
t(s>).
However the manager has an
why
signal obtains). This is
can collude with the internal to manipulate
explicitly, t(«) should
receives the signal s and reports
it
in
exchange
be written as a function of s' given
interest in colluding only
t(«) is written as
s'
when s' =
a function of
s»
s'
s 2 (the high-productivity
only. Finally, the external
truthfully (se * s.) All players
must be awarded
their
reservation utility level to sign the contract
Timing
(1)
Nature chooses the state of productivity (0) and the signal for the auditors
according to Table
(s)
2.
(2)
The manager learns
(3)
The
8.
(He doesn't observe
s yet.)
principal offers a contract specifying the transfer for the
manager
(t(x,s',s e ))
as a
e
function of output and the auditors' reports, the transfer for the internal (w(x,s',s )) as
a function of output, his report and the external's report, the (fixed) transfer for the
9
Ifx
x2
.
£
{x h x 2 }, the agent gets punished, so
we
can
restrict attention to
outputs X|,
13
external (z), the probabilities
d(x,s'))
all
these functions will
probability of a
punishment
low type
for the
and external audits
contract
is
signed.
(y(x)
P').
and
At the
be depend on the parameters of the problem: the
(q), the
manager (P™
and the cost of the external
The
internal
and the punishments for the manager and the internal (P"3 and
optimum
(4)
of conducting
informativeness of the signal
),
the
maximum punishment
(r),
the
maximum
for the internal (P**)
(z).
The manager chooses his
effort (e).
Output (x)
is
realized
and
observed by all parties.
(5)
The
internal is sent
and the
(6)
(8)
observed by the manager
internal can sign a side-contract specifying
a transfer dependent
internal's report ("Ks*)).
The internal
The
is
auditors.
The manager and the
on the
(7)
with probability y(x). The signal (s)
reports
s'.
Side-transfers are realized.
principal asks for the external's report with probability d(x,s').
reports s e -s.
(9) Transfers are realized.
si
e,
6:
qr
The
external
14
internal ignores the real state
of productivity 6
is
irrelevant since the punishment can only
depend on observed variables.
Note
that
we do
not define the transfer as being net of punishments.
collusion, the
manager gets the transfer corresponding to the
announced by
the external
minus
the
punishment and the
When
state
convicted of
of productivity
internal gets the transfer
corresponding to his report minus the punishment. This simplifies the presentation of the
results
without further consequences
10
.
A remark must be made about the side-contract between the internal and the manager. Since
this contract is usually illegal,
mechanism
'
'
to the validity
making
must be self-enforceable.
We assume that there exists some
the side contract self-enforceable. Finally,
of the contracts
are not renegotiation-proof.
and proposals
it
we will
an objection can be raised
characterize since the resulting equilibrium strategies
We will consider the renegotiation problem in our conclusions
for further research-
Preview of the Results
As a benchmark, we derive
necessary condition for
enough
to
make him
the first best contract
its feasibility.
and show
In the first best
that
symmetric information
scheme
sign the contract and the marginal cost
the
manager
is
is
a
paid just
of compensating the manager
10 With a limited liability interpretation, we should model the maximum punishment as a
function of the transfer received by the manager. In our two- type model, however, the
punishment is only applied to the low type. Therefore,
level of maximum punishment.
1
it
is
correct to consider a unique
For example, the firm could pay the auditor with "half dollar
his audit report.
Once
bills" before
he submits
the report specified in the side-contract has been filed, the firm
has no interest to keep the other halves of the dollar bills. Making a deposit on an
account with 2 signatures (auditor and manager) would be a similar device. Reputation
could be another alternative to make the side-contract enforceable.
15
for increasing his effort is
manager's
equated
to the
marginal benefit for the principal of increasing the
effort.
When the principal
cannot observe the manager's effort the
because the manager's incentive compatibility constraint
first
is
best contract
is
not feasible
not satisfied: if the principal
A standard
pays the manager a fixed amount, the high productivity manager will shirk.
result in contract
theory
is
that the second best
scheme
will induce an inefficient level of
from the low type as a means of achieving incentive compatibility with the
effort
a small reduction in the
relaxation
low type's
effort involves
a second-order loss in profits while the
of the incentive compatibility constraint generates a first-order gain (see Baron-
Myerson [1982] and Maskin-Riley [1984]).
best level
of effort
Finally, the
(this result is
known
low type does not get any
On
the other hand, the high type exerts the
in the literature as
we
introduce a faithful and costless auditor.
correlated with the state
to the relation
"no inefficiency
of the world and reports
it
He
is slack).
to the principal.
principal reduces the informational rent of the high type. In the
been reduced to zero and the effort of the low type
punishment, the
Next
we
let
first
is
We show that according
maximum liability of the
best (third region)
is
first
region, the
second region, the
adjusted towards the
sufficiently accurate
the faithful auditor be costly.
new regions appear.
is
I.)
receives a signal imperfectly
between the accuracy of the auditor's signal and the
information of the auditor
at the top").
(See proposition
manager, the effect of the auditor can be separated in three regions. In the
if the
first
rent (his individual rationality constraint is binding)
while the high type does (his individual rationality constraint
Next
least cost;
rent has
first best.
Finally
with respect to the available
reached. (See proposition 2.)
Compared
to the
scheme with
free auditing,
In the first one, the principal does not use the auditor because he
expensive relative to his information and the
maximum punishment available.
one, the principal controls the probability of sending the auditor.
is
two
too
In the second
We show that
it
is
best to
16
decrease the probability of audit while the effort of the low type
best can
still
be reached, but only at the limit
(See proposition
Next
we
allow the manager
best
i.e., r
<
is
and the auditor
The reason
punishment capacity,
that
The
first
grows without bound.
model
that, at
is
manager chose
We show that the
(jointly) observe the signal.
(as long as the auditor's information
some
(expected)
i.e.,
to sign a side-contract after the
and the auditor
never reached in this
1).
the punishment
suboptimal.
3.)
his level of effort and both he
first
when
is still
want to use
point, the principal does not
maximum
deterrence
not optimal.
is
not perfect;
is
The
all his
intuition is
high punishments induce high bribes which are costly to deter. (See proposition 4.)
The next
step
is to
introduce a second auditor.
To model
achievement of coalition incentive compatibility,
reports truthfully but
the external
is
is
costly.
regions depending on the values of r
external audit
enough
we
best
is
of
r
and P™
called to audit on a
random
basis (the probability
third region,
assuming the
internal to
to
is
r
and
P™
When r =
1
of
are high
and
P™ is
achieved. (See proposition 5.)
be so
far)
and completely
be so far) types. In this case,
parameter space, the principal will find
it
This non-screening mechanism produces a
We let the internal
truthful (as
we have been
self-interested (as
we have been
drawn from a distribution which has
assuming the external
main
are such that the internal auditor is
extend the model to consider auditors of different "types."
auditor be randomly
auditor always
contract specifies four
for the optimal coalition-free contract to use only the internal.
first
new
inexpensive and can be bribed,
The optimal
between zero and one.) In the
is strictly
high enough the
Finally,
is
is
that this
and P™. For low values of r and P™ no auditor
used. In the second region, the values
always used, but the external
we assume
While the inside auditor
expensive but cannot be bribed.
the trade-offs involved in the
we
claim that for some range of the
optimal to allow collusion (See proposition 6)
more
realistic
account of empirical observations:
some people do not collude and do not get caught, some people do not collude and get
17
punished by mistake, some people collude and do not get caught, some people collude and
get punished. 12
First Best Allocation
Consider the case
principal
I3
.
in
which output and
effort exerted
by
the
manager are observable by the
Auditors have no role and a contract which specifies the manager's transfer as
a function of effort and the state of productivity
maximizes the
{81
+
The optimal
minus the cost of inducing the manager to take
profit
The principal's problem
Max q
is feasible.
e,
is to
- 1,}
+
choose
(1
-
level
of effort
that effort.
ej, e^, ti, t2 to
q) {6 2 + e 2
-
2}
subject to:
ti*
(MIRl)
t2
(MIR2)
The
*
7
solution of the above problem has C\ - e 2
e™
1
and
tx
t2
= t re
1/2.
In the optimal contract under symmetric information, the principal pays the agent just
enough
to
make him
agent's effort to
is
its
sign the contract;
i.e.,
eV2 -
1/2
and equates the marginal
cost
of the
marginal value product Also note that the transfer received by the agent
independent of the state of productivity; t(xi)
Kx^ -
1/2.
12 For future research and preliminary speculate about a mode where two auditors of
unknown type are available to the principal. Only with a second auditor collusion may
be uncovered and punished.
1
And therefore,
the state of productivity can be inferred with certainty.
18
Graphically,
and find
we
can represent the agent's indifference curves in the product-transfer space
the solution
where
the slope of these curves
is 1.
t
4
Figure
1
.
First Best
Scheme
No Auditor
From now
on,
we will assume
that the principal
can neither observe the
state
of nature nor
the manager's effort. Thus, the contract can only specify transfers as a function
output (x). If the principal tried to apply the
first
best
of the
scheme described above, the manager
would always produce x i
Under these circumstances, the problem (which we
ei, C2,
ti,andt2to
Max
q {8i +
ei
-
1!
}
subject to
(MIR1)
(MIR2)
-!
h*
+
(1
-
q) {8 2
+ e2
-
2)
label
BO)
for the principal is to choose
19
2
(MIC1)
2
2
'
2
2
l
(MIC2)
2
2
where A9 = 62-61.
The manager's individual
compensated for the cost of his effort in each
are that the
(MIR)
rationality constraints
manager signs the contract
state
state that he
must be
at least
of the world. Our implicit assumptions
after learning
6 and
that his reservation utility is
normalized to be zero. The manager's incentive compatibility constraints (MIC) appear
because the principal can not differentiate between states of nature one and two, therefore
some
incentives
must be provided
to the
manager
if he is to exert the effort desired
by
the
principal.
As
usual, only
MIR1 and MIC2
post that the other
are binding at the
two constraints are
optimization program.
satisfied at the
optimum.
It
can indeed be verified ex
optimum when
they are ignored in the
20
Graphically, the solution
is:
x
Figure
Proposition
1
2.
l
x
l
Optimal Contract with
(BO Scheme): In the absence of auditors, the optimal contract between the
manager specifies:
principal and the
ei- l-
—3-A8<
^-A9<
1
1
q
C2-1
tl
-4-i.(!isW±(^l
q
q
2
2
i+
1
Proof:
(e,
ei
2
*
•
A6)
Ae^
2 1
j
i
2
No Auditor
2
2
The proof is standard and,
therefore, omitted.
21
When the principal
because
it
cannot observe the manager's effort the
MIC
does not satisfy the
's
scheme
shirk.
is
not feasible
pays the manager a fixed
A standard result in contract theory
will induce an inefficient level
of effort from
the
low type
Q\
means of achieving incentive compatibility with
the least cost: a small reduction in type
effort involves a second-order loss in profits
while the relaxation of the incentive
as a
1
the second best
best contract
constraint. If the principal
amount, a highly productive manager will always
is that
first
compatibility constraint generates a first-order gain (see Baron-Myerson [1982] and
Maskin-Riley
[
1984]).
effort (this result
is
On
the other hand, the high type 62 exerts the first best level of
known in
the literature as
"no inefficiency
at the top"). Finally, the
type does not obtain any rent (his individual rationality constraint
type does (his individual rationality constraint
One Free
The
is
low
binding) while the high
is slack.)
Faithful Auditor
principal can
now use an auditor who
observes a signal s (imperfectly) correlated with
the true state of productivity 8, and reports
reservation salary to zero and
assume
it
that his
truthfully (s
-
s).
We normalize the auditor's
wage can be conditioned
contractually on the
manager's output and his report.
If the principal receives a report indicating that the
s^Xi-ej
manager shirked
(i- 1,2)
he can impose a penalty on the manager of P"1 bounded above by P111
[1984] have shown that the principle of
hence,
we will
consider from
14 For the general result of
[1987].
now on
that
maximum
maximum
P™
- P" *.
1
deterrence
It is
.
Baron and Besanko
holds in this model
also easy to check that there
M
,
is
deterrence in the presence of mistakes see Bolton
22
no need to use the auditor when the high output x2
will not exceed his reservation value
The problem faced by
Maxq
{6,
+
e,
the principal
- tj
+
(1
-
r)
is
realized and that the auditor's salary
of zero.
now
15
P»J +
is to
(1
-
choose
c\, ci, t\
q) {8 2 + e 2
-
and
t2
to
2}
subject to:
(MIR1)
V
;
2
'
4
(MIR2)
(MIC1)
2
2
2
2
15 Notice that he risk neutrality
his expected transfer.
l
2
of the agent allows us to
state
MIR1
as a lower
bound
for
23
Graphically, the solution
is:
Figure
3.
Optimal Contract with a Free Faithful Auditor
24
Proposition
The optimal contract with a
2:
regions according to the values of
AG
2r -
(Bl) IfPm <
the
scheme (BO)
(B2)
If
AG
2r -
r
free faithful auditor
can be divided
in three
and P™.
i-*, a
- q
,
(MIR2)
is
slack and the difference with
1
is that
t2
1
=
h 30
-
(2r
- AG
(
-
1)
P™
(the rent is reduced)
< pm <
2 - q
2q
1
AG
-
1
2r - 1
—
~ AG
.
,
(MIR2)
L
binds and the differences with the scheme (Bl) are that
e,
= 2r -
1
pm
+
AG
t2
-
AG
(the effort
is
adjusted)
2
11 = 1
(the rent is zero)
AG
(FB) IfF11 *
1
- AG
2r - 1
the
first
best
is
achieved.
Proof: The proof is standard and, therefore, omitted.
The mechanism can be described sequentially
as follows: the principal observes the output
produced by the manager and gives him the corresponding transfer. If output
is
no
further action. If output is
low
auditor reports a 'high' signal the
In the
first
region, with
the principal requires the report
manager
is
is
high, there
of the auditor. If the
punished with P™.
low accuracy and small punishments, only informational
rents are
decreased while effort levels for each type of manager remain unchanged. This
is
25
analogous
made
to the
Baron-Besanko's separation result which stated
that the
output decision
independently of the auditing decision (although the auditing decision
is
not
is
made
independently of the output decision). In other words, the effort exerted by each type of
manager
in the optimal contract
As
contract without auditing.
punishment
increases, the rent
with auditing
is
the
same one he exerted
in the optimal
the accuracy of the auditor's signal or the
of the high type manager decreases but
the effort
maximum
of the low
type manager is not adjusted.
In the second region, the rent of the high type has been reduced to zero because the
information of the auditor
is
sufficiently accurate relative to the available
individual rationality constraint of the high type
is
now binding. The principal
reduce the inefficiency imposed on the low type's effort and
compatibility. If the information improves or the punishment
distortion disappears and the first best is reached.
this result is that infinite
is
Another (more
punishments can support the
arbitrarily close to non-informative (r 5 1/2 in
punishment The
our case).
still
can therefore
achieve incentive
increased even more, the
intuitive)
first best
consequence of
with a signal that
is
26
1/2
1
r
Figure 4. Regions of the Optimal Contract with a Free Faithful Auditor
The following graphs are
effort
and the high type's
useful to see the
rent:
change in the principal's
profits, the
low type's
27
l
2
- «<«2?
1
1
oT'CP"
Figure
)
jf (P
m
)
and Rent in the Optimal Contract
with a Free Faithful Auditor
5. Profits, Efibrt
28
One Costly
Faithful Auditor
Normalizing the auditor's reservation wage to zero was not without loss of generality.
new
come
effects
when
into play
the auditor is costly. First, the principal might decide not
to audit at all because the auditor is too expensive.
punishment
is
high enough,
the effort of the
low type
z.
Second, if the
maximum
available
will be optimal to reduce the probability of auditing before
restored to
is
Suppose the auditor costs
it
Two
The
its first
best level.
principal's problem
is to
choose
ej, ti, X\, ti,
and y
(the
probability of auditing) to
Maxq {8,
+
e, - 1,
+Y
[(1
-
r)
Pm -
z]}
+
(1
-
v
Y
r
q) {82
+ c2
-
2)
subject to:
s 5- + vYU
( 1
t,
(M1R1)
'
2
(M1R2)
r)
;
Pm
* 2
b2
(MIC)
-
2
—2 ^
(e,-A6)
—
—
2
e?
-
til
-
2
-
Pm
Define:
x(r,z), the
minimum
having information
(1
value of
r
-q)(2r-
1)
P m such
that
it is
profitable to use an auditor costing z
and
29
value of
co(r,z), the
P™
at
which the benefit of adjusting
reducing the probability of sending the auditor (yz),
2P m2
which
(2r-l) 2 + P m A8(2r-
1)
(A8
is
2) +
-
effort is equal to the benefit
implicitly given
of
by
2zA6 2
implies:
A8'(2-A6) +V(2-A6) 2
16
-
u(r,z)
4(2r-l)
Notice that
w
<q
Proposition 3: The optimal contract can be divided in four regions according to the values
P^andz.
ofr,
(BO) If (2r
If(2r-
-
1)
—'-^- P™ s z, the auditor is not used (y = 0) and scheme BO is applied
1)—^-pnjsz, the auditor is uscd.(y>0)
AG
IfP™*
(Bl)
is
2r -
-it'*--
1
;
y
1,
(MIR2)
is
slack and
scheme Bl
applied.
(B2) If
-^2r 1
scheme B2
(E3)
If
is
1 -
**
< pm s
I
y -
1,
(MIR2) binds and
applied.
1^ >
io(r,z); y
=
z
= 1 >m
Pm (2r - l)
I
m
P
(2r - l)
A6 [(2 - A6)
"
2
ei
co(r,z);
2q
I
A9
(2r - l)
2
- 2 z A9
<
1
and
30
This
we
(FB)
When P™ -
call the
scheme E3 (where E
refers to "external").
the first best is approximated arbitrarily closely.
°°,
.m
Tiff!'
"
»
!
1
!',!
1
',
!
!',
i
!
!
!
!',!
I
!
'
!
ES^
mZvWmmgmm^
^s^
'
:-xx-:-:<-:x-:x::-:-::o:x::-:::v:
1/2
Figure 6. Regios of the Optimal contract with a Costly Faithful Auditor
We can observe again how
change with n
the principal's profits,
low type's
effort
and audit probability
31
TT
J
FB
TT
-1
-1 /^rris
1/2 *
e
,_m
(P
)
«
(P
(P
)
«
(P
)
^(P™)
)
uTW
s
iA
,FB
,BC
K
1/2
6
k
"
1/2
Figure
(P
)
«
(P
)
w~
l
(P
m
Profit, Low Type's Effort and Audit Probability
Optimal Contract with a Costly Faithful Auditor
7.
)
in the
32
Compared
scheme of proposition
to the
2,
two new regions appear.
First, in
region
BO
the
principal does not use the auditor because he is too expensive relative to his information
and the maximum punishment available. If the cost of the auditor increases, the region BO
increases as well.
If the principal does use the auditor, the optimal
in proposition 2. In region
Bl
,
scheme
is
the principal extracts rent
very similar
of the high
to the
one presented
type. If the
maximum
punishment and the informativeness of the auditor become sufficiently high,
informational rent of the high type can be extracted
adjusted towards the
proposition 2,
type
it
is
first
However, and
this is the
effort of the
low type can be
second major difference with
optimal to decrease the probability of audit (y) while the effort of the low
suboptimal.
is still
best level.
and the
all the
The
first
best
is
only reached
grows without bound. The trade-off driving
at the limit
this result has,
on
when
the
punishment
the one hand, the possible
of the low type and, on the other hand, the
increase in profits due to adjusting the effort
possible increase in profits due to decreasing the probability of a costly audit.
Consider, for instance, the case in which the
Bl when P
111
,
is
increasing, the principal can
and extracting more
decrease
rent
The reason
v.
is
rent extraction, the cost
use the auditor at
So
if y
any
r,
becomes
first
that in order to
punishment increases. In region
choose between decreasing the probability y
Lemma
make the
3.
1
proves that
never optimal to
reduction of v profitable compared to the
of auditing would have to be so large that
less than one,
P™
for
it
has to be in region B2. Indeed,
which y <
1
it
would not be optimal
lemma
3.2
shows
to
that for
even though the effort level of the low type
is
best This means that at some point the principal becomes indifferent
(at the
E3
starts.
margin) between decreasing v and adjusting the effort. This
The
it is
all.
there exists a
below the
from the high type.
maximum
line separating
B2 and E3
is
called to(r,z).
Along
co(r,z)
is
where
and
the region
in region
E3, the marginal
33
of decreasing the probability of auditing equates the marginal
profit
profit
of restoring the
effort.
At
this point,
makes
it is
useful to highlight
two
facts: (1)
P™ via
the profit function concave in
The convexity of the
adjustment becomes less and less profitable while approaching
best, the benefit
auditing
is
of adjusting
effort is zero.) (2)
The
profit
To
manager's incentives to
see this, notice that
tell
the truth.
So
it
is
its
in region
maximum punishment
which
first
(an
affects the
of punishment increases, the expected
punishment can be kept constant by decreasing the probability of audit
proportional
effort
optimal level (at the
the expected punishment
if the level
B2
from lowering the probability of
a concave (increasing) function of the available
inverted hyperbola).
Hence,
effort adjustment.
effort function
at
an inversely
rate.
Therefore, above line co(r,z)
the probability of auditing
(i.e., in
region E3) the principal will simultaneously decrease
and increase the
effort for the
low type so
marginal profit of those two actions. Finally, if the punishment
bound, the optimal contract goes to the
fist
is
that
he equates the
allowed to grow without
best level of effort with an arbitrarily low
probability of audit
Notice that the punishment capacity will be fully exercised in the optimal contract;
strongly optimal to use
all).
The
decision of the principal
adjust effort and
One
maximum
when
deterrence (of course, whenever the auditor
is
when
to use this
to reduce the probability
punishment
is
to extract rents,
used
at
when
to
of auditing.
Free Unfaithful Auditor
Next we allow the manager and the auditor to collude signing a side-contract
manager chose his
It is
it is
level
of effort and both him and the auditor
readily checked that the only case for collusion
is
when
(jointly)
after the
observed the signal.
the productivity of the
manager
34
is
low and the
signal
of the auditor
is
high. If the
manager has high productivity (and
produces high output) the principal does not use the report of the auditor. If the manager
has low productivity and the auditor receives a low signal there
transfer since the auditor
According
to the audit
the signal is right or
world he knows
signal obtains.
shirks.
He
is
punishment
that
technology
wrong
is
his effort
And, since the optimal contract
is
P™ the manager
is
would not
above
is
if a
up
knows
it
true state
of the
high productivity
incentive compatible, the
willing to pay the auditor
game, whether
At the moment of the
and regardless of the
only punished by mistake and everybody
I6 .
manager never
To avoid
the
to P™. Thus, in the presence
of
report payoff-relevant information and the
no longer sustainable. The auditor would be
between releasing or not information leading to the punishment of the manager,
but the manager
is
rise to
ready to "buy the auditor's silence".
a new constraint, the coalition incentive compatibility (CIC) constraint
To induce truth-telling when
must pay him
at least as
the auditor has evidence to punish the manager, the principal
much
as the manager
auditor must be rewarded with at least the
punished
the timing of the
even with truthful reporting he will be punished
allocation described in section
This gives
we have specified and
irrelevant for studying the coalition.
a covert contract, the auditor
indifferent
for a side-
cannot forge evidence by himself.
manager has already chosen
negotiation, the
no reason
is
if the auditor
Also, as stated above,
would pay him
to change his report;
amount by which
the
i.e.,
the
manager would be
revealed his incriminating information.
we assume
that is Pareto efficient
that the
manager and
the auditor choose a side-contract
rational with respect to the
no side-contract
theory. For a similar result see
Green and Porter
and individually
outcome.
16 This
is
[1984].
a standard result from
game
35
The problem
for the principal
Maxq{6i+e,
subject to:
-
1[
now
+ y(l
-
is to
choose e i e2,
rHP™- w)} +
,
(1
-
1 1
,
t2
q) {e 2
and
w to:
+ e2-t 2 }
36
2 q A6 + q A8 2
a
-
2
A6 2
2q(2r-l)
as the value of P
m at which MIR2 becomes binding and
A6 (4r
p _
A9
-
2q(2r-l)
as the value of
P
111
at
2)
-
2
which the benefit of adjusting
effort is equal to the cost
of increasing
the expected punishment (yP™).
Proposition 4: The optimal contract can be divided in five regions according to the values of
randPm
(BO)
If r
>
(Bl)
(B2)
If r
^
-z
i
,
^
,
-
the auditor is not used (y
the auditor is used (v
Ifl>r>=
i
If
1
>
r
>
and
-
If
(FB)
If r
1
> r>
1
q
F
11
*
>
a(r),
scheme Bl
is
applied.
and a(r) s pm < p( r) scheme
B2
y^—
and P(r) < P™ scheme B3
applied.
and
q
(5(r)
< P™ the
first
is
best is attained.
Definition of scheme B3:
KrJ^-BW
is
0).
tA—
i
-
(B3)
and scheme BO
0)
q
.
Proof: see appendix
w=
pm
.
eB3 (r)=l
.^iLll)
is
applied.
applied.
37
1/(2-q)
Figure
8.
Optimal Contract with a Free Unfaithful Auditor
38
We
can plot the evolution of the principal's profits and the probability of an audit in the
1
case where
2-q
<r<
1:
77
TT^
a
1/2
Figure
If the signal
It is
time
is
9.
Probability of Audit and Profit
too noisy
—
i.e., r is
,m
3
when
the Auditor
small (r s (1/2-q)) the auditor
not optimal to use the auditor even though his reservation
it
is
manager
costly to induce truthful revelation.
for mistaken punishments.
The
cost
To understand why
is
is
Used
not used (region BO).
wage
is
zero because this
comes from compensating
the critical value of
r
the
depends on
39
q,
remember
that
our optimal mechanism
is
collusion free and incentive compatible.
Therefore, the principal has to pay the auditor's reward
mistake (which happens with probability
(which happens with probability
q).
To
1-r) after the
is,
the incentive scheme
similar to the one in proposition
If the auditor's signal
(r>
is
the better the information
good enough with
manager and the contract
obtains in regions
is restored).
is
1 )
even
if the
intuition for this result is that increasing the
effects: first,
it
lower cost but,
at the
same time,
when
the fulfillment
p\r), the
colluding with the
of the low type
effort
however lower because
1
-r)
to prevent collusion.
is that the first best is
liability
the
never
grows without bound.
manager's expected punishment has two
allows the principal to achieve incentive compatibility from the manager at a
auditor. Indeed,
making
maximum
low productivity
A new separation result
and B2 (where the
Another major difference with the scheme of proposition 2,
The
him from
w to the auditor with probability q(
reached for an imperfect signal (r <
the higher the
When the auditor not is used,
respect to the probability of
rent is extracted)
makes a
1
to proposition 2, the profits are
pay a reward
principal has to
must be.
similar to the one of proposition 2.
Bl (where the
Compared
of an auditor,
rewards the auditor to discourage
l/(2-q)), the principal
the auditor
low productivity type obtained
justify the use
proportion of low type
is
w only when
it
increases the cost of getting truthful revelation from the
P"11 is high, the potential
bribe to the auditor will be high as well,
of the CIC constraint expensive. At some point, determined by
second effect outweighs the
first
line
and the principal does not want to increase the
expected punishment anymore while production
is
bounded away from
the
first best.
A major difference with the scheme E3 in proposition 3 is that the effort is never adjusted in
region B3.
The reasoning
is
as follows. In this model, the product
yP m must be kept
constant in order to induce incentive compatibility. Recall that the cost of the auditor for the
principal
is
yP m (weighted by
1-r).
Suppose
principal finds profitable to decrease y instead
that at
some point when P™
of adjusting the
effort.
Once
increases, the
y
and P™ have
40
been adjusted, the product (yP™)
yP™
not to increase
Compared with
the
before,
the
same
as before (by
has to be the case that
scheme E3
in proposition 3, the
it
proposition 3 (the cost
if the
zero.
not optimal. In this model,
However this
indifferent
profitable
do so now.
P™
in proposition
maximum
4 but not in
result
it is
its first
-
Alternatively
is
why
P™ could be
maximum
best level, expected
comes from
the fart that the auditor's reservation
was
costly,
it
it
is
by
multiplied
would be optimal
in proposition
replaced
by
0(r)
4 the B3 scheme
< P™ and
y
best
is
with
is
earning
unbounded punishments even though
depend crucially on the
right to audit, or to
is
fact that
we do
P" - P
111
1
would be always equal
puzzling features appear in this model: the auditor
results
wage
y, the
to decrease
deterrence above the line P(r). In our model, the principal
Two
not attainable with
In
not optimal to use them.
between decreasing y or not increasing P™. What really matters here
product yP01 This
fact
we can also claim that maximum deterrence is not
probability of sending the auditor. If the auditor
These
not profitable to
Each time P111 appears in our maximization problem,
Y and always use
was
maximum punishment available increases.
So, even though production does not reach
is
it
major difference comes from the
other words, even if higher punishments are available
optimal either.
if
manager, the effort of the low productivity type and the profit
of the principal are kept constant even
is
MIC). So
is z).
the punishment for the
deterrence
is still
of the auditor varies proportionately with
that the cost
Then
it
is still
all
some
*
rent
the
is
and y <
to
is
1.
1
and the
first
players are risk neutral.
not allow the principal to "sell" the
impose on the auditor negative transfers when he reports that the
manager complied. Either possibility would take us back
to the case
of a faithful auditor
(see footnote 17).
The following graph
such
that
it
all
is
an illustration of the profit-loss caused by collusion
regions are crossed
when
r
goes from
1/2 to
1. i.e.,
when P "
1
is
41
^(Ae-^W,Ae- A-f
».
ttcfb)
n(BO)
1/2
l
i/(2-q)
<x"
(p
_1
m
)
B
m
(P )
1
Figure 10. Profits as a function of r
Self-interested Internal, Truthful External
Usefulness of a second auditor
Suppose
that 2 signals are available
informative (n >
r2
>
and have the same cost
Suppose the principal gets the
1/2).
manager shirked and should be punished.
n
2-a.
*=*
)
(A8
A
y-)
is
the
20 but the first
first
one
a(r)
more
signal saying that the
If the second signal confirms the
P m where
is
and
first,
the
cross.
P(r)
Afl9
(A8
-
—j- )
is
the
minimum P ra necessary to achieve
the
first
best
when r =
1
20 If the cost and informativeness are different, the principal should consider the
informativeness per dollar.
42
principal
first,
he
"more certain"
is
is 'less
he shirked
is
that the
manager shirked. If the second
would
certain" but he
still
punish the manager because the probability that
higher than the probability that he did not.
The upshot of the argument
signal is altered
by
is
that in
no
state
of the world the decision prior
Therefore, the second signal
it.
is
correlated. If the
two
second
one of them even
if
they are not
signals are equally informative, the least costly will be used, if they
are equally costly, the most informative will be chosen
is
to the
worthless. If two signals (auditors)
are available, the optimal contract will only consider
This result
signal contradicts the
a very particular feature of our model.
will later enable us to highlight the specific role
police.' Since the external
21
We do not think
pay
it
is realistic,
of the external auditor
adds no value for the principal
the only reason for the principal to
.
when
but
it
as 'policing the
the internal does not collude
for his services is to audit the internal.
Optimal Contract
Suppose now
at
a cost z >
the
that the principal
0.
The two
has the additional opportunity of hiring an external auditor
auditors get the
manager (6) according
to table
same
signal imperfectly correlated with the type of
1
We are interested in the tradeoff involved in the achievement of incentive compatibility: the
more expensive, the
internal auditor requires a
payment of
external auditor
is
P
manager's bribe, and they both impose an extra burden on the individual
111
to refuse the
truthful, but
rationality constraint
of the manager in the low state of productivity (the manager could be
punished by mistake).
21 Contrast this with Tirole's [1986] model where there is a chance of the auditor
(supervisor) getting no signal at all. In that case, a second signal would be valuable.
43
A new
parameter which comes into play
auditor, P'*, this
bound has the same
is
the
limited-liability interpretation
Let y be the probability of sending the internal
sending the external
when
now to choose
Max q
{8,
+
e,
- 1,
+y
U-q){8 2
[(1
23
-
when
ei
r)
,
e2
(Pm
,
-
when
tj
,
w)
t2,
-
the internal
w,
r
output
y,
d and
6 z] +
(1
-
is
of
is
for the internal
P™ .—
low, d the probability of
low productivity
the internal reported a
probability of sending the external
principal is
maximum punishment
not sent.
signal
and
The problem
<p
the
for the
cp to:
y) [<p((l
-
r)
P™ - z)]} +
+ e 2 -t 2 }
subject to:
•».4 +p,(l
<Mnu)
2
(MIR2)
b
'-* if+ « (| -' JI
2
.|, .(^?)l. r p. lY+rt! - y,i
tl
w-pm ^-sd^+p*)
(cic)
Definitions:
Let
a and
(3
be the same as in proposition 4.
Let
22 Since fraud
is
punished more harshly by the courts than breach of contract one could
argue that P'* > P 1"*. This will increase the value of the external auditor for the
principal. (See below.)
23 The principle of maximum deterrence holds for P 1
.
44
A
r
1
be the
2r
-
1
maximum
1
ni
-
value of P™ as a function of r, P' and z such that
if r
£
if r
a
.
optimal
it
is
lt
^ optimal
not to use any auditor.
Let
1
v
be the
2-P
1-r
maximum
value of P™ as a function of r, P' and z such that
y
to use the internal alone.
Proposition
5:
The optimal contract can be divided
of the informativeness of the auditors' signal
(P111 ), the
auditor
maximum punishment
(z).
[1]
(BO)
[2]
If
Two of these
probability
1,
the
maximum
for the internal auditor (P')
punishment for the manager
and
the cost
of the external
regions have three subregions each.
Ifr<min{ j-
r)*HPm ,P',z) s
r,
in four regions according to the values
r
;
Tr 1 (P m ,P i ,z)
}
no auditor is used. See
Prop.
I.
s p-l(P™,pi,z) the two auditors are used, the internal with
the external with probability
d <
region are analogous to the ones of Prop. 3
1.
The
where
incentive schemes applied in this
the cost
of the auditor would be dz
instead of z.
[3]
[4]
Ifr&max{=—
z-q
(FB) If r =
1
and
;
p-l(Pm ,P i ,z)
p(r)
< P"1
the
}
only the internal
first
is
best is achieved.
used. See Prop. 4.
45
Proof: Sec appendix.
1/(2-q)
1/2
Figure 11. Regions in the Optimal Contract with
Notice that
^>0
dz
;
*1<0
^->0
0pi
dz
d?
1
1
Two Auditors
r
46
.m
T
T7
r
1
'
'
'
'
'
'
'
'
1/2
1/(2-q)
Figure 12. Regions and Subregions in the Optimal Contract with
We
find that
it is
optimal not to use the auditors
that "too noisy" has a different
internal. If auditing
is
to
when
meaning now than
Two Auditors
their signal is too noisy, but notice
in the case without external
be used, the principal must decide whether to use the
or both the internal and the external. In the previous model (proposition 4)
with
L/(2-q).
This time, however,
of an external
auditor.
This
we
r
1
and costly
internal alone
we compared
have an extra condition due to the possible presence
is reflected
by
the comparison of
r
with
nKP^P^z).
In other
words, the principal compares the accuracy of the auditor's signal with a function that
increasing in the
internal
maximum punishment
to the
manager, the
and decreasing in the wage of the external.
by looking at two extreme
auditing
is
cases: z
always useful (recall
r
=
>
r
Some intuition on
(the external is free)
1/2).
maximum punishment
and P 1 =
The case where z =
is
rj'K*)
°°.
is
to the
can be obtained
In these situations,
obvious.
When P
1
is
47
unbounded, the principal
may use
induce truthful revelation from the
the costly external with a very small probability and
manager and
still
the internal auditor.
If the principal uses the internal alone, the incentive
scheme
is
one of
identical to the
proposition 4.
If the principal uses the
two auditors
internal auditor is truthful but costs
we
are in a situation similar to the one where the
dz (the probability of sending the external auditor times
his reservation wage). Indeed, the internal will report truthfully, even though he is not paid
anything.
The reason
the principal will set the probability of sending the external (d)
is that
sufficiently high to discourage
him from colluding with
the
manager when threatened with
punishment F*. The optimal incentive scheme has the three familiar subregions (El, E2
and E3). The
intuition is identical to the
the appendix) that
it
is
one of proposition
3.
We also show (lemma 3
never optimal to send the external alone. If he
be used as a "second opinion" required with probability less than
the external auditor
result
is
used
it
must be on a random
depends crucially on the fact that the internal
On
the Opt imality of Allowing Collusion
So
far,
we have shown
the auditor and the
that the principal finds
it
basis.
1
is
24
.
used
at all,
he will
In other words, if
However, we notice
that this
is free.
in his interest to prevent collusion
manager whenever the auditor
in
is
hired. In the optimal contract,
between
nobody
colludes.
To
explore the more plausible case
where some deviant behavior persists
assume
that self-enforcing collusive
24 6
always
is
less than
1
in equilibrium 25 ,
arrangements are more easily achieved
(whenever P' >
in
some
states
0).
25 Without changing the model one could claim that the Revelation Principle is just a
solution technique that facilitates the characterization of the optimal contract but does
48
of the world
For example, take the case where there are two types of
that in others.
and non-collusive.
auditors: collusive
positive expected value
The collusive auditors
and the non-collusive will never take bribes. Then, introducing a
binary random variable
we
could say that with probability c the state of the world
supporting collusion obtains and with probability
Suppose
for
example
manager of the firm
(1
-c) side-contracts are not feasible26 .
with probability c the auditor has pre-contractual
that
—
will accept bribes with
make side-contracts between them
ties that
these ties are "per se" illegal. Then, the court will not enforce
"type" report of the auditor are unenforceable, the
induce the auditor to reveal his type;
i.e.,
the
owner cannot
owner cannot
when
with the
— and
self-enforcing
any contract that
of a report of this variable. Baiman-Evans-Nagarajan say that
ties
is
that
contingent
contracts based on a
write a contract which will
write screening contracts.
Although we concur with them on the point that a screening contract
is
not feasible
we
don't agree with their argument 27 .
Even though
a report of type
clause contingent
may be
non-contractible upon, the court could enforce any
on the outcome of a publicly observed variable which the principal and
the auditor can agree to determine. For example, the principal
(although nobody
colluding type and
may know
when the
this) that
when
and the auditor
the auditor wears a red
auditor wears a blue tie he
is
"truth-telling
he
is
of the
a non colluder. As long as the
not claim the uniqueness of the characterized optimal contract.
mechanism may be a
tie
may agree
tie
In fact, the described
map" of a very complex mechanism where
bribes,
punishments and rewards happen in equilibrium.
26 Baiman, Evans and Nagarajan claim that since this approach implies that whether or not
an individual will collude is not an inherent characteristic of that individual, collusion
cannot be avoided by using a screening contract. We will state our disagreement with
this approach later.
27
We thank Eric Maskin
for suggesting this line
of thought.
49
worn by
the auditor
contingent on
What
its
is
So what
color.
fails in this
observable by third parties, the court will enforce a contract
is it that fails if
setup to separate types
is
not the contractibility on type?
the single-crossing, screening or Spence-
no feature of the auditors which enables the principal
Mirrlees condition.
There
discriminate between
them by means of providing different incentives. If he simply asked
is
for type reports promising a high
colluder; if he promised a
reward for the colluder, every auditor would claim to be a
punishment for the colluder, every auditor would claim to be a
non colluder. The problem of the two types of auditor as modeled here
same
utility function
to
is that
they have the
although different strategy spaces, and different strategy spaces are
not enough to screen by
means of a menu of contractual arrangements.
There are two models that are interesting to study depending on whether an honest external
auditor
is
available to
explore the
first,
check the report of the internal auditor. (In
take bribes with probability
who
is
we will
honest with probability ( 1 -c) and
In order to illustrate our model,
c.
Internal auditors are usually
manager.
paper
leaving the second as a topic for further research.) (1) In the
the principal decides to hire an auditor
internal.
this
we will
first
only
model
who would
label this auditor the
employees of the firm hired and
Therefore, there might be likely to suffer pressures from
fired
by the
management
to
window-dress the statements of the firm. In our model, with probability (1-c) an internal
auditor never takes a bribe and always reports honestly. (2)
second auditor (the police of the police). That auditor
is
The second model
assumed
to
introduces a
be always honest but
expensive.
For some parameter values,
it
will be the case that in either
contract will not specify a complete avoidance of collusion.
of these models the optimal
The
intuition is that if there is
large probability of non-collusion situations, the principal will prefer to let the
a
few out layer
50
cases get
away with
collusion instead of paying every type of auditor the reward that
would
deter the few colluding ones.
In the model without the external there
be discovered. Everybody would
would be
know
the bribing. In the
some
that in
bribe the auditor but, since the principal has
collusion in equilibrium but
cases the
enough probability
previous
management) but not often enough
ties to
collusion will again occur.
to deter the "soft" types
The difference
by
to
he cannot stop
Indeed, the external
of auditor (the ones without
to deter the
"tough" types. Thus,
now sometimes it will be
is that
external auditor will find that the signal reported
audit,
this is not the case.
will be sent with
would never
manager would be able
no way of verifying the
model with an honest external,
it
discovered: the
the internal auditor is not correct
there will be not only collusion in equilibrium, but collusion that
is
Then
discovered (and
punished) observed in the real world.
When the principal
deters collusion
nobody
shirks or colludes
and when an auditor receives
a signal implying that the manager shirked, everybody knows
punishment has
to
allowed. Since punishments are only applied
principal to allow collusion
we
because
will explore the
the loss
examine
collusive agents
collusive agents.
it
is
Still,
bad for the principal when collusion
by mistake
it
may be
will prevent the application
soundness of
this intuition
actually
good
is
for the
of these punishments. In
and show
that
it
of control that collusion brings about. Another claim
compared with
is that
was mistaken.
be applied to the manager or the contract would not be incentive
compatible. This suggests that not everything
this section
it
must be
we
will
might produce a higher payoff for the principal than non-
51
Can
Before
we
Collusion
Enhance Welfare?
study the convenience or not of allowing collusion
Klitgaard notes that
many scholars have argued
that fighting corruption
that corruption
may be
can play a useful
is
not zero.
economic, political or managerial benefits.
arguments
that
may make corruption
He distinguishes
not just
three categories of
reminder and the manager's reminder.
We will describe the first and
our present concerns.
The economist's reminder. Corrupt payments introduce a mechanism similar
to the
market
goods and services are allocated by queue, politics, random selection, or
"merit," corruption
Corruption
It is
This
socially beneficial: the economist's reminder, the
third since they are the relevant ones for
When
role.
so costly as not to be worthwhile, but that corruption itself
may create
system.
examine the
book Controlling Corruption.
statement goes beyond saying that the optimal level of corruption
political scientist's
will
We will base our considerations on Robert
possibility that collusion is welfare-enhancing.
Klitgaard's insightful
we
may instead allocate goods according
may thereby put goods and services
most and who can use them most
efficiently.
in the
In
to willingness
hands of people
some
and ability to pay.
who value them the
sense, then, after the corrupt act
those goods and services are more "efficiently" allocated in the economics sense.
The manager's reminder.
Corruption
may have
uses within an organization.
bureaucratic rules are constraining, the organization
employees' corrupt circumvention of the
may sometimes
benefit by the
rules.
These reminders, claims Klitgaard, have some
common
features.
First,
they refer to the
benefits from specific corrupt acts, not from systematic corruption pervading
decisions. Second, the putative benefits
transgresses a
wrong or
inefficient
If
depend upon the assumption
many or most
that the corruption
economic policy or gets around imperfections
in
52
organizational rules.
In short, if the prevailing system
is
bad, then corruption
may be
good.
Klitgaard quotes political scientist Samuel Huntington: "In terms of economic growth, the
only thing worse than a society with a rigid, overcentralized, dishonest bureaucracy
with a
rigid, overcentralized,
The upshot of
this
is
one
honest bureaucracy."
discussion
is that
mechanism when there are some
corruption can only be helpful as a second-best
inefficiencies impossible to remove.
In the realm of
optimal contract theory, where the principal designs mechanisms only subject to
information asymmetries collusion will never enhance welfare.
Results
The important decision
he
is
the principal faces
to hire an auditor at all is if
whenever he decides
better off allowing or deterring collusion
between the manager and the auditor. If he
decides to deter collusion, he will have to pay the auditor a reward at least as large as the
punishment imposed on the manager. If he allows collusion he will not pay any reward,
but a collusive auditor will never report that the
collusive auditor never reports a non-compliance
manager should be punished.
why would the principal
If the
find his services
useful?
The auditor is
(2)
useful for the principal
on two accounts
( 1 )
Not
auditors are collusive
all
and
depending on the bargaining power of auditor and manager (we assume the Nash
Bargaining Solution) the manager will
still
suffer a loss
when
the auditor receives a high-
productivity signal after low-productivity production obtained.
have to pay the auditor a bribe equal
to half the
punishment;
Namely,
i.e.,
On
the other hand, recall that the principal
manager will
P"V2. The principal does
not get this money, but the deterrence effect of the punishment will
manager.
the
must compensate
still
the
operate on the
manager
for the
53
expected mistaken punishment (the agent always complies so punishments occur only by
mistake.)
So
the principal
may be
better
ofTby
letting the
manager and a collusive auditor
play cooperatively.
Our
result is that the
lower the proportion of colluders, the higher the punishment and the
worse the auditor's information, allowing collusion becomes more profitable for the
principal.
When the probability of having a colluding auditor is low
the principal will be
reluctant to promise a high reward to the auditor in order to prevent collusion. This is so
because with high probability collusion will not happen anyway since the auditor
When the punishment is high
most
non-maximum expected
To
see
why
a loss of Y^^O. to the manager.
becomes high enough, half of it can be enough
collusion
low-type and the signal
manager to sign
expected mistakes.
principal
to achieve the
of the auditor's signal
(recall
deterrence result in proposition 4).
a low correlation between the auditor's signal and the true state of the world
makes allowing
the
inflict
profitable second-best contract given the informativeness
the
honest.
our assumption about the Nash bargaining solution becomes
very important. Even the collusive auditor will
Therefore, if the punishment
is
is
more
profitable for the principal recall that
the contract the principal
everybody
his expected
payment back.
If the auditor
neutral, the
always colludes the
punishment imposed on the manager, therefore loosing
advanced payment to the manager. But in
this case, with
colludes taking only half of the punishment and this
make a lower advanced payment
beneficial effect is that the
screening of types.
is risk
by rewarding the auditor for an
principal can only achieve coalition incentive compatibility
principal has to
manager is
needs to compensate him in advance for these
If the auditor never colluded, since
to the
the
high the manager will be punished by mistake. In order to get
would simply get
amount equal
when
to the
is
some
all his
probability, the auditor
common
knowledge.
So, the
manager. The counter-face of this
punishment threat becomes
less effective to support the
54
To
we
solve this problem,
considering
now a
will derive a modification of the
scheme
andassuming that the principal does not offer any
free unfaithful auditor
We will
compare the
reward and,
thus,
scheme with
the profits of proposition 4 to study the optimality
does not deter collusion
No Collusion
We will now allow
in proposition 2 but
profits
of
of allowing
this
modified
collusion.
Deterrence bv Assumption
for a probability c<
that the auditor is unfaithful, that the principal
1
does not offer any reward to the auditor and, thus, does not deter collusion, and the Nash
Bargaining solution for the coalition between the unfaithful auditor and the manager. The
problem for the principal
Max q
{0!
+
e!
is
- 1,
now to choose
+y
(1
-
c) (1
-
t\, e2, t\,
r)
P
111
}
+
(1
t2,
-
v to
q) {6 2
+ e2
-
2}
subject to:
(MIR1)
ti*|
(MTR2)
t2
-^
+ Y (l-r)Pm(
*|-
(2-c)
Notice that this
is
the
same problem
solution to this problem
is
as in proposition 2 replacing
P™ by P™ -y-
.
The
given by
Define:
=
PM
U- C)
as the
2-c-2q(l-c)
(2-q)(2-c)-2q(l-c)
minimum
value of r such that
him being a colluder is
c.
it is
profitable to use
an auditor when the probability of
55
Notice that
^ >0 and that when c = 0,
as the vahie of P^at which
A6
ll "
(2-c)(2r-l)
as the value of P™
at
1
,
5 = l/(2-q).
MIR2 becomes binding.
2A9
a/a^
P(A)=
c =
when
£ = 1/2;
A9(l-r)
*(2r-l)(2-c)
2
which the benefit of adjusting
effort is equal to the cost
of increasing
the expected punishment (y P™).
Lemma 6.1: The optimal
contract
when
of r, P"1 and
in four regions according to the values
IBO]
If
1
<
If r
>
r
>
5(c), the
the principal never deters collusion can be divided
^(c), the auditor is not
auditor
used (y >
is
used
IfP^sctfAhv- 1,(MIR2)
[B2(A)]
If
[B3(A)]
Ifp™> P(A) scheme B3(A)
[FB]
If r
<Pm s
ft
(A)
,
y =
,
1
and P m >
= 0) and scheme BO
is
applied.
and scheme B1(A)
is
applied.
0).
[B1(A)]
a(A)
(y
c
1
,
is
slack
(MIR2) binds and scheme B2(A)
is
applied.
(5(A) then the first best
Definition of Schemes:
Scheme B1(A)
is
identical to
scheme Bl.
Scheme B2(A)
is
identical to
scheme B2, except fore
t;
is
achieved.
is
applied.
56
e ,B2(A )
= 4e + P(2-c)(2r.l)
>
2A6
2
Scheme B3(A)
P(A)
Y(rP m ); = pm
YU,r
B3(A)
C
e,
.
j
.
ABO^r)
2r-
1
Proof: See appendix
Remember
that
we assume
here that since the principal does not deter collusion, he only
has two options: he can either not use the auditor's report and give a second best incentive
scheme (BO) or ask
latter contract
knowing
for the auditor's report
may be
optimal because the agent
has to bribe him to avoid the punishment.
who
that this report
may
be biased. This
slacks and gets caught
The amount of the
by
the auditor
bribe might deter the agent
from slacking.
The
line £ determines
which incentive schemes the principal chooses:
r
^
on q and
c.
does not use the auditor's report (BO scheme). If
collusion will occur. Notice that £ depends
will only occur
when
the agent
is
of type
(mistaken) signal received by the auditor
0, i.e., all auditors are faithful
is
1
(this
the
1, i.e., all
<
£,
the principal
£, the auditor's report is
It
used and
depends on q because collusion
happens with probability q) and the
one of type
and never collude, £ =
always uses the auditor report. If c =
if r
1/2.
2.
It
also depends
on
c.
If c
=
This means that the principal
auditors accept bribes,
Notice that in this case the region where the auditor's report
is
used
is
the
same as
in
proposition 4.
In proposition 4, line l/(2-q)
where
it
became more
was
the level of informativeness of the auditor's signal r
profitable for the principal to punish the agent (by mistake!) than to
not use the auditor's report. Recall that l/(2-q)
is
independent of the punishment level. So
57
if instead
of imposing a loss of
P™ on
manager
the
(as
it
happened
in proposition 4) the
pm
principal imposes a loss
ofV-
(as in
lemma
1
when
c
1)
the condition for using the
auditor remains invariant.
When
the 5
<
£
r
1
the auditor's report
applied depending on the value of
Bl
is
high,
the rent of the high type
scheme B2(A)
non-maximum
of the low type
does not use
is
restoring the effort.
relaxation of the
extracted.
When
contract of
and P™ are sufficiently
if r
The
its first
best level
becomes too high
and P m are even higher,
however
the
MIR1
P
111
large
that
is.
The
The
one developed
constraint outweighs
principal
P™ imposes on the
its
benefit through the
lemma
6.1.
was derived assuming
is
Unknown Type
that the principal
would never
not necessarily true. If we abandon
can obtain our main proposition.
Define:
2-c-2q(l-c)
as the as the
The
in proposition 4.
MIC constraint.
optimal to deter collusion. This assumption
*
effort
relative to the potential extra profit gained
intuition is similar to the
P™ through
principal can use
deterrence result obtains with scheme B3(A).
Optimal Collusion Deterrence with a Free Auditor of
The
r
punishment capacity he has because the burden
constraint of the agent
cost of increasing
and low P™, the standard scheme
of the low type. Finally,
kept fixed below
all the
is
r
manager has been extracted and the
to restore the effort
another expected
by
and P™. For low
applied and the rent of the high type
all
MIR1
r
used and three different incentive schemes are
is
(2-q)(2-c)-2q(l-c)
minimum
value of r such that
probability of him being a colluder
is c.
it
is
profitable to use an auditor
when
the
find
it,
we
58
g
£
2q(l-c) + c
q(2-3c) + 2c
as the
a
minimum
more
value of r such that
it is
profitable to deter collusion. (See the
appendix
for
precise definition of £.)
P(D) =
p"
in proposition 4.
Proposition 6: The optimal contract with a free auditor of unknown type can be divided in
four regions according to the values of the informativeness of the auditor's signal
maximum punishment
manager P™ and
for the
Two of these regions have
c.
[1]
If r< £(c),
[2]
If ^(c)
<
r
no auditor
If r
>T
,
is
used and a scheme BO
< "£ collusion
,
collusion
is
is
is
If r
-
1
and
applied.
in this region
Lemma 6.1.
deterred.
P™ a 0(D)
Proof: See appendix.
collusive auditor
allowed The incentive schemes applied
The
incentive
schemes applied
ones of proposition 4.
[4]
of having a
the
three sub-regions each.
are the ones defined in
[3]
the probability
r,
,
the
first
best
is
achieved.
in this region are the
59
Figure 13. Regions of the Optimal Contract with a Free Auditor of Unknown
When the informativeness
find
it
of the signal
optimal to not use the auditor.
is
sufficiently
Even though
bad
(i.e., r
the auditor
Type
below §) the principal will
is free,
by submitting the
agent to an audit he places him under the risk of being punished by mistake.
participation or individual rationality constraint the principal needs to
To
satisfy the
pay the agent
the
60
expected value of this mistaken punishment. This would not influence the contract in the
event that the principal got the punishment back from the manager since both are risk
neutral.
However,
in proposition 4
if the principal deters
(where
we
say that
if r
collusion the punishment will go to the auditor as
was
less than 1/2-q the auditor
would not be used)
or if the principal does not deter collusion there will be a transfer to the auditor since the
when
agent will have to pay him 7^/2 as a bribe. Hence,
there are
many mistakes
(r is
low) no auditor will be used.
Given the probability of a colluding
auditor, the principal will be better off
collusion in the diagonally striped region of the
deter collusion he will have to pay
compliance and with probability 1-c
all
above graph-
If the principal decides to
the auditors the reward
this
by allowing
when they
report a non-
reward was not needed; on the other hand,
if
he
does not deter collusion some auditors will be bribed, reducing the deterrence effect of the
punishment. The indifference locus of this tradeoff is reflected in
Depending on
will
the
scheme chosen,
be the ones derived
the subregions
in proposition 4
and
—
lemma
line~£.
as depicted in the following graph
6.
1
61
P(A)
o<D)
:!
p;Hii!iJiii
" ,:i! '- !li -'" !,
1i{iji;
B1(D)
1/2
$
'"
$
Figure 14. Regions and Subregions of the Optimal Contract
with a Free Auditor of Unknown Type
This proposition "merges" the results of proposition 4 and
no auditor
is
used
identical to the
is
the
same as
the one derived in
one given for £ in lemma 6.1. If § <
lemma
lemma
r
s
6.1.
6.1.
The
The region where
definition of £
~£, collusion is
is
allowed and the
62
scheme of lemma
6.1 is applied.
collusion applying the
facts
(1) (Expected)
non-maximum
>~£, the principal finds more profitable to deter
4.
should be noticed:
both schemes don't use
is
r
scheme of proposition
Two important
best
If
deterrence always obtains
maximum deterrence.
when
the auditor
is
used. Indeed
Therefore, in presence of collusion, the
first
never reached whether the principal deters collusion or not
(2) If the punishment grows,
obtains because the effort
collusion
is
when
deterred (e B3 ( D >).
always becomes optimal to allow collusion. This
it
collusion
The
is
allowed (eB3 ( A
auditor).
)
higher than the effort
principal can afford to restore the effort
allowing collusion because the cost of increasing
would expect
deterred collusion he
is
1
'
to loose
P
111
the expected punishment he will collect
scaled
^(l-Oy
If he allowed collusion, his expected cost
payment required by the low-type manager
is
down by c/2.
(1
-
when
more when
If the principal
(the expected reward to the
would be the difference between
to sign the contract (see (2-c)/2 in
( 1 - r)
result
c) y
P™ when an
the
MIR1) and
honest auditor convicts
the manager.
Finally, each region has the 3 familiar subregions
and expected punishment
The graphs presented
6, = l/2,8 2
is
where
rent is extracted, effort is restored
kept constant.
in figures 13
=l,q=l/2,c=
and 14 are an
illustration
of the optimal contract when
1/2.
The following graphs show how
the regions of the optimal contract change with
c.
63
FB
C
-0
C
-0
Figure 15. Regions of the Optimal Contract with a Free Auditor of Unknown Type
when c -»
64
FB
c
1
C
1
6.
1
Regions of the Optimal Contract with a Free Auditor of Unknown Type
when
When
-
1/2-q
1/2
Figure
-
c -*
the proportion of collusive auditors goes
1
down
(c-*0), the contract resembles the
standard second best contract with a faithful auditor.(see proposition 2). At the limit, if
c=0, no collusion ever occurs and the optimal contract always uses the free honest auditor
(see figure 15).
At
the other limit, if c=
and the optimal contract
is
1
,
identical to the
the principal
always has
to
worry about collusion
one of proposition 4 (see figure
16).
Conclusions and Directions for Further Research
We have analyzed the role of auditing in a simplified hierarchy (principal-auditor-manager)
in
which the auditor and the manager can collude;
we have
distinguished between internal
65
(costless, self-interested)
technology which
is
and external
(costly, truthful) auditors
and developed an audit
imperfect and allows forging of evidence under collusion.
We have shown that the internal auditor alone may be useful
collude with the manager.
The
possibility
to the principal
is
if he
can
of collusion imposes an additional cost on the use
it.
We
always hired on a random basis because his
role
of the auditor but under certain parameter configurations the principal prefers
have also found
even
that the external auditor is
to
bear
mainly to police the internal auditor.
We have also shown that expected maximum
presence of collusion.
his effort but also
The
intuition
is that
makes collusion more
deterrence
is
not necessarily optimal in the
increasing the punishment for the manager raises
profitable: the bribe tends to be higher
and thus
harder to deter.
Finally,
we have
presented a model where the principal has to decide whether to allow or
deter collusion. Deterring collusion
is
costly for the principal because he has to reward his
auditor for not accepting a potential bribe from the manager. If the principal believes that
his auditor
is
and therefore
honest with a sufficiently high probability, he will not try to deter collusion
let
the collusive auditor and the
manager collude. Even
an auditor is useful because the manager knows that shirking
in
such a case, hiring
may cost him
and gets a
the punishment
signal of shirking.
if the auditor is
honest and the bribe
In this analysis
we have assumed that the bargaining power of the manager and
was
identical and, thus, the
if the
auditor
Nash bargaining
is
collusive
the auditor
solution obtained within the coalition formed
by these two agents. More generally, we could model the bargaining power of the manager
and the auditor by a variable p € [0,1] and a transfer (bribe) of
setup,
by
p=
just
means
that the
manager has all
the bargaining
matching the principal's offered reward, p =
1
W+(P m-W)p.
power and can
means
In this
bribe the auditor
that the auditor has all the
66
bargaining power and will extract a
punishment
that the
payment from
manager would sufier
We have also assumed that the principal
the
manager equal
to the potential
if the auditor reported his signal to the principal.
only offers a reward when his purpose
collusion. This need not be optimal since the
bargaining process within the coalition.
reward offered by the principal
In a future version
we
is
to deter
will alter the
plan to investigate this
issue.
One
objection to the contracts characterized in
strategies are not renegotiation-proof.
and
the
manager may prefer
penalty).
The
incentive to
original contract
tell
the principal
Once a low
to renegotiate
principal does not audit and the
our paper
is that
output has been revealed, the principal
and reach a Pareto-superior allocation
manager gives the
To avoid
(e.g., the
principal one half of the expected
would be renegotiated and
the truth in the first place.
the resulting equilibrium
this
would
alter the
the renegotiation issue,
commitment power over the probability of
audit.
manager's
we have given
Although we plan to
characterize the optimal renegotiation-proof contract in a future extension of this paper, for
the
moment we
will accept this simolifying
assumption.
An alternative to the ad-hoc power of commitment with which we endow the principal is to
invoke a reputation argument: in an infinite supergame where the principal and the manager
could only sign one-period contracts the original contract would be renegotiation-proof.
Introducing reputation to
make endogenous assumptions about
auditors opens the issue of dynamics.
two-period model
in
the principal
and the
We are exploring the consequences of considering a
which the "faithfulness" of the external and the
"self-interest"
of the
external are a consequence of the utility-maximizing behavior.
Another alternative
for the
a
is to
solve the renegotiation-proof contract relying
two types of manager.
game where a
principal
It is
who
a well
known
result
on mixed
of the crime-deterrence
strategies
literature that
cannot commit to a (possibly contingent) investigation
67
strategy by a costly enforcer polices an agent
who may
or
may
not
comply with
provisions of a contract or law does not have a pure-strategy equilibrium.
If the agent
complies with probability one, the enforcer should not be used. If the enforcer
the agent should not comply. If the enforcer
is
the
is
not used,
used with probability one, the agent should
comply. If the agent complies the enforcer should not be used.
Our model can
be extended in a variety of directions. If we consider a costly self-interested
internal auditor, for example, the probability
any more. If all
is
auditors
behave
of sending the external alone need not be zero
strategically the interesting question arises as to
whether it
possible to police the police without falling in an infinite regress.
Another
interesting direction
correlated.
would be
to
allow the auditors' information to be imperfectly
A new trade-off would appear,
his individual rationality constraint
the internal could be punished
by mistake and
would gain importance (he won't work unless he
is
compensated for the possible mistakes of the external).
There
is
an important reason to study this extension. In our model,
we assume
that the
external auditor never colludes. But even if he could collude, the optimal contract
not be altered since the external auditor has exactly the
auditor.
To
prevent the external from colluding
same information
we need only
to
as the internal
add a new coalition
incentive compatibility constraint promising a reward for truthful revelation.
model of Demski and Sappington, whatever that reward
is
As
in the
the principal never has to pay
because the external can only confirm the internal's report 28
28 The implicit assumption here
would
it
.
is that any side contract that the external could sign with
the agent or the internal is non-enforceable. Denying him this commitment capacity is
crucial since if he could bind himself to take, say, half of the bribe; the coalition would
be in a Pareto superior situation.
68
If the information of the
external
two auditors
is
imperfectly correlated, the study of a self-interested
becomes more interesting (and more complex).
coalitions
In this case several other types
of
between the manager and the external, the external and
the internal and the three
of them must be considered.
69
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vol. 19 [1988], pp. 147-155.
"An Equilibrium Model of Auditing
for Attestation and Compliance", Miraeo,
Stanford University [1988].
Tirole
J.,
"Hierarchies and Bureaucreacies:
On
the Role of Collusion in Organizations",
Journal of Law, Economics and Organi7arinns vol. 2 [1986], pp. 181-214.
,
Williamson O., The Economic Institutions of Capitalism
,
New York:
Free Press [1985].
73
Appendix
Proof of proposition 3
For proving proposition 3
Lemma 3.1:
we
will need the following
In region Bl, y
lemmas:
1.
Proof: In region Bl, the slope of the profit function with respect to
extracted from the high type (with
=
^Tn
dp
at*
TT
at^
~ln
m
ap
The
1
is
"
" q)
(1
slope of the profit function with respect to
extracted)
dn
is
1
dn
1
atf
m
aP
a*
Decreasing y below
-%
(2r - 1)
P™ when y is
reduced (and no rent
1
q
M
P
would only be
(1 - q)
>
z
=
m
profitable if
(2r - 1)
*»
z
>
^-2
29 To achieve incentive compatibility, y P" must equal a constant
1
k=
tf°
-
g(e?0)
^
r
But
at the limit
dy _
ap m
l
*
is
29
m
dP
q
1 )
rent is
1
.i
,i
y
P™ when more
pm
f
'
where y=
1
,
i
pm
L_
^.._
(pm)
^pm
k = F" and then
2
k,
-
(2r - l)
P
m
74
But
when
this condition is true
profitable not to use the auditor at
it is
Lemma 3.2: If the cost of the auditor is sufficiently low
there exists a punishment
For z
this
P™ such that
occurs before the
s-^3 (2r
-
1)
minimum punishment
Proof: In
lemma
1
P™
for
best level of effort
first
there exists
which
ei
we have shown
for the principal to use
the probability y of sending the auditor
1
is
On the
first
that this
besL (The profit function
other hand, the slope
of sending the auditor
is
of the
adjusted
<
that y
can only happen
(q
1,
is
in region
P™ when
less than
where P^fb
is
the
first
B2.
effort is adjusted is positive
best and smoothly
becomes
increasing and concave in P™)
profit function with respect to
is
is
at all,
in the optimal contract.
punishments below the level necessary to achieve the
zero at the
him
reached. Formally,
P m < P^fb such
In B2, the slope of the profit function with respect to
for
(See definition of
n
k below.)
one and
all.
zVP™ which is positive
,
P™ when
fo all P™.
the probability
n
Proof of Proposition 3 (continued)
The Lagrangjan
L»q{6
1
for this
problem
is:
+ e 1 -t 1 + Y[(l-r)Pn»-z]} +
+ X2
C
{t 2 -
-f}
+ X3
C
{t 2 -
(1
-
q) {9 2 + e 2
4-t,}+ ( £i-L
-
with the additional constraints:
£0 Osysl
e,sOe 2 &0
ti*0
t2
X,sO X 2 &0
Xj^O
X4
^0
-
}i- +
^
2
}
+Xj
e ,2
{t, -
-j-- y(l
YrP^}+X4{l-Y}
-
r)Pn>}
75
The Kuhn-Tucker conditions
J*age
ae
for
= q - A e + A
3
x
1
=
(1
v"
maximization
(e,
are:
- AG)
- q)
e,
v> 2 + Xj
M/ - (X,
"3' ^2
<
2
age
e
'
2
"
ae
2
JjL , _ q +
dt
Jfj-
<
2
aje_
=
e,
;
^
- x
<
3
= -(l - q) + X, + X, <
;
f*
tl
;
(l)
76
u
By
t?\
eq. (3)
j/<\
q
Ml P
and
(5), Xi •>
X3 =
In region B2, ci =
'
=
Xi-q
pm(2 r
_az_
P m (2r-
—— + A8
.
i)
.-=-
rrr
A0
*
equating the above two expressions
AG
•6
find y
=
find
(AG - 2)
J
- 16 z
4 (2r - 1)
we use the
AG
IR1 and IC constraints:
- AG) P
(2
P
2
Since at
(See proposition 2)
=
0)(r,z)
To
we
- AG) +
(2
1)
zA8
m
P (2r-1)
d-1
Byeq.(l)
m (2r- l) + z
'
Pm (2r- 1)
to,
y =
1
and
m
m2
in region
- 2 z AG
(2r - 1)
2
(2r - l)
E3 :p- <
=> y
<
1
in region
E3.
+
+
n
Proof of proposition 4
The Lagrangian
L=q
{8,
+
e!
problem
for this
- 1,
+y
(1
-
r)
is:
(Pm - w)} +
(1
-
q) {0 2
Mt,-^-Y(l-r)Pro}+X 2 {t 2 (Cl
Q)2
'
2
+Y
r
P"1 } + X 4 {w
-
e2
-
2}
X }+X3{t2
Pm} + X 5
{ 1
-
y}
C
-
"2"-
t,)
+
77
with the additional constraints:
2
e,2
e2
Osysl
^1^0
given that y and
ti^O
because
*0
P™ always appear
for maximization.
results
X2
The
tj^O
w^O
X32O
X42O
together,
alternative choice
we assume
—
we
to fix v
that the reservation
The Kuhn-Tucker conditions
fix
=
*0
P™ at P™* and
1
and
wage of the
for maximization are:
X5
let
use v as the argument
P"1 vary
— does not
internal auditor is zero.
alter the
78
If y = 0, the auditor is not used.
Therefore the optimal scheme
is
BO and we can
let
w = P™
w.l.o.g.
In BO, Xj =
Suppose y
1
**
and X3 =
li
then,
since the relevant
pm =
q
-
1
,
equation (7) then becomes
schemes Bl, B2 and the
line a(r) are the
in proposition 2
FOC are identical. Also, by the coalition incentive compatibility constraint
w-
<
Suppose
y
<
(i.e.,
1
we
are in region B3).
Using equations (1)
X, = X 3 + q
which imply
since y
we
<
1
to (4),
(3)
q-X, C!+X 3 (e!-Ae) =
(6)
same as
,
that
X3 =
X5
0.
(1)
q(1
ei)
^Q
By the
coalition incentive compatibility constraint,
P" =
1
w and using
find
-PmX, (l-r) + PmrX 3
=
which implies that
A9
V
Note
m
1
"
(1
-
2r-
1
that c\ in region
B3
_
r)
is
independent of P™ and equal to c\ in region
A9 (4r - A 6 2
2)
(2r - 1)
This defines the line P(r), the border between regions
B2 and B3.
B2 when
79
It is
a(r)
easy to check that
<
when
p(r)
function
-4r2 +
(4
(5(r) is
(4
1
2
r
-
and
j
i
that
- Cj
1.
- cj
increasing if
if
that (3(r) is increasing in the interval
1
+ A6
I
J
2
'
<
when
+ 2A8)r-(A6 + 1)<0
This means
f
L
r
= P(r)
2A6)r-(A6+l)>0
+
and decreasing
-4r2 +
<
=
i
The
cc(r)
and decreasing when
r
exceeds the upper bound of the interval.
Therefore, depending on the value of A8 satisfying the no shut
(1+A8V2 could be less
rise to the
30 The
l/( 2 -q),
than
between l/(2-q)and
1 ,
down condition^,
or greater than
1.
This gives
following 3 graphs:
maximum A0
decreasing at most
for
no shut down
when
r
= l/(2-2q).
is
q/(l-q). Therefore the function P(r) starts
80
.m
Mr)
oc(r)
1
2 - q
r
2
Figure
A 1. a.
First Possible
Shape of P(r)
.m
MO
«(r)
2 - q
Figure
A
1
.b.
Second Possible Shape of P(r)
1
r
81
Figure
Note
2
Third Possible Shape of P(r)
graphs imply a rate of increase in the effort function different
that such
and B3. Indeed,
a
A I.e.
it
is
easy to check that
2
e? 2
a
=
ef
and
3
<
2
a(r)
a(r)'
Proof of Proposition 5
The Lagrangian
for this
L-q
1,
{B, + e,
-
+y
(l-q){e 2 +
problem
[(1
e 2 -t 2
- r)
is:
(P™
}+X
-
w)
e
1
{t 1 -
-
r6
z]
+
(1
-
y)
^--(l-r)Pm
I
[Y
«P ((1 - r)
P" - z)]}
+ cP (l.Y)]} +
+
in
region
B2
82
e
X2{t 2
X4
-
4}+Mt
P
{w + 6
-
(1
5)
-
^'-^
e
2
-
-f
2
+ rP"[Y + <P(l-Y)]} +
-t,-
P™} + X 5
{ 1
-
y)
+ X6
{ 1
-
5} + X 7
{ 1
-
<p}
with the additional constraints:
e^O
e2
^0
ti^O
6
*0
cp
£0
w^O
Xi^0X 2 ^0
X3
^0
X^O
*0 X 6 *0
X7
*0
Y
*0
X5
where y
is
t2
^0
the probability of sending the internal auditor
probability of sending the external auditor
productivity and
cp
= q -
|f
=
f*-
%-
(1
\
low
.,
for maximization are:
+ X
3
(,,
- A9)
- q) - (X + X,)
2
= -q +
the internal auditor reported
the
the probability of sending the external alone.
The Kuhn-Tucker conditions
g-
when
when output is low, 6
\
c
- A, S
<
<
e
,
2
,
= -CI - q) + X + A <
2
3
&--qft-r)*-\*0
t,
.,
;
2
gj- =
(1)
=
(2)
|f
=
f^
(3)
M-
,
t
i
w
2
=
^-0
(4)
(5)
83
= q {[(1 -
||
- X
(1
1
*
H
2g- = -q
||-
= q
-
r)
m
(1
- w) - r6z] - [9((l - 9) P
m
+ A3 P
m
r)
P
m
r(l - 9) -
- z)]}
\
=
r z
m
<
(6)
5 + X
- *)
(1
+ X3 P
(P
r)
r
1
+ P
(P
4
[(1 - r)
-
(1
tJ)
P
m
m
- X
)
- X
<
-
z]
- X
1
?
6
;
6
<
-
(1
;
r)
^
(1
=
-
9^=0
(7)
<5)
P
m
(8)
Plus the constraints and their complementary slackness conditions
For proving the proposition
Lemma 5.1:
If the outsider
the outsider
is
8=
pm +
Proof: If the outsider
1
>6
=»
<8
=»
X6 =
is
the following three lemmas:
sent after the insider, the conditional probability
—
pi
sent after the insider,
8>
0.
Suppose
by the complementary slackness condition
9J
Then,A4=^ pm-^->0
+ pi
X4
^>
that
by the complementary slackness condition
38. = o
d6
is
we will use
implies
that 8 =
y
pm . w
— by' the CIC
pm + pi
constraint.
1
>6>
0.
of sending
84
Next,
we
If 6 =
1
,
will
show
X4 =
that
because
but then, from
(7),
Lemma 5.2: The
X6 =
6
1
leads to a contradiction.
w + P* >
-q
rz
y
<
(from
0.
CIC
constraint),
Contradiction since
X6 a
n
0.
reward for the internal auditor (w) can only be
or P™.
Proof: Substituting our above result for 6 in the objective function, the principal's
involves the following maximization:
Max m
(1
-
r)
(P
m
-
w) -
r
0<w<P
whose
(p
solution can be always found at a corner.
pm +
Ifr<—
pm
+
-2
pi
pi
+z
m
1-2^
+ P )
1
z
problem
85
(Notice that this result depends crucially on the fart that the internal
is free.)
Proof of proposition 5 (continued):
We
must consider 2 alternative ways by which the principal could achieve incentive
compatibility:
(P111) or
(ii)
(i)
paying the internal auditor a reward no lower than the
enough (with
sending an external with probability (6) high
maximum
respect to
bribe
P and
111
pi) to deter collusion.
pnj
As we have proved
r
lemmas
in
5.
1
8
and 5.2 the optimal
r
optimal to use a "mix" of the two methods;
w is either
If
(i)
i.e.,
_
is
^y
r
pra + pi
and
it
is
never
strictly
w and 6 strictly positive. This implies that
or P™.
w = P™
then 6 = 0. Then, equation (6) implies that
1
2 - q
P <
pm
(ii)
If
w - 0, then 8 - pm pi—
+
Pm
*
2r-1
The condition
< min
r
*
for not
^
Then, equation (6) implies that
» - P* - nfr.P'.ri
using any auditor
-A—
2
.
q
L_
•
- q)
(P
m
the principal hires only the internal auditor, y =
w = P™
3
Recall that when the internal
without loss of generality.
= tT 1
3-1
2
and
1
then,
'
(1
When
is
is
costless
we can
1
+ P
1
)
31,6 =
adjust
^, p\z)
(this implies that
P™ setting
X6 =
0)
y always equal to one
86
From
equation (5),
Substituting in (7),
And
At
this takes
this point
=
Tl(r)
X4 = q
r
( 1 - r)
pm + pi
pm + pi + z
*
-.
.
= p" l (Pm )
us back to the schemes Bl
we can show
r
that
n,
q
2r - 1
1
- q
,
B2 and B3 derived
in proposition 4.
and p cross when
z -
r
_i
P =
1
-
z -
l
P = p(r)
r
1
which occurs
%
at r
2-q
m
P
1
FB
/(2-q)
Figure A2. Regions for the use of internal and external auditors
pm
When
=
the principal hires both the internal
and ?m > max
{
n(r),p(r)}
and the external auditors, y - 1,6
pm +
pi
,w
87
We find a situation analogous to the one described in proposition 3. The only difference is
that the cost
of auditing
is
now bz
instead of z.
Recall that the line a(r) gives the locus of pairs (r.P™) such that
the high productivity type of agent
between
n,(r)
Bl derived
by a reduction
to(r)
ae
.
If
P™ is
a straightforward extension of the scheme
in y is
equivalent to the increase in benefits via effort restortion of the
(see proposition 3).
2 - AG +
/
Therefore
(2 - AG)
go is
implicitly defined
2
- 16 z
=
in proposition 3, the regions
be below
and P1
gives the locus of pairs (r,P m) such that the decrease in auditing
4
As
is
extracted from
in proposition 3.
low productivity type
m
P
that a(r) is independent of z
and a(r) the scheme applied (El)
Recall that the line
costs
and
all rent is
n(r).
p
by
m
P^P
1
(2r - 1)
El and E2
may not exist
Indeed, line a(r) and
co(r)
could
88
pm
1
\ \ X X \
f * f * /
X \ \ \ X
y / • / /
x x \ \ x
'
• • •
•
XXXXXX**
xxxxxxxxxxx
xxxxxxxxxxx
***********
xxxxxxxxxxx
***********
xxxxxxxxxxx
**************
xxxxxxxxxxxxxx
**************
xxxxxxxxxxxxxx
***************
xxxxxxxxxxxxxxxx
ft************
xxxxxxxxxxxxx
/
'
Figure A3. Regions
Proof of
The Lagrangian
L=
q {6, +
e,
+ X,{t
^
1
Lemma
for this
- 1!
E 1 and E2 Do Not
6.1.
problem
+ y (1-c)
-
(1
is:
r)
F™} +
(1
-
q) {6 2
+ e2
-
2)
-Y-Y(l-r)Pm (^L-)} +X 2 {t 2 -^-}
C
3 {t 2
(ei
4--t^
M
2
-f
2
)2
+Y rP"(^)}^X4{l-Y}
with the additional constraints:
d^O
e 2 *0
y *
Xi
=
Exist
^0
t|^0
t2
X 2 s=0
X3 *
The Kuhn-Tucker conditions
for
X4 *
maximization are:
89
age
= q - X
de.
+ X
e
3
- A9)
(e
Pae
<
1
=
(1)
1
age
- q) - (A + X ) e
2
3
2
(1
ae.
aie
= -q + X
- X
at.
age
3
<
age
q (1 - c)
as
(1
r)
(3)
at.
i
'
- c^
(2
+ X
(2)
1^- =
<
-
=
^
2
-(1 - q) + X + X <
2
3
at.
age
e
;
d£
t
;
2
jn
4
p " - X
t
(1
=
(4)
dt
-
(2 - c)
2
r)
<m
.m
=
r
(5)
3
Plus the constraints and their complementary slackness conditions
If no auditor is used, y
1
- q.
0; therefore
- c - 2q (1 (2 - q) (2 - c) - 2q
<
Note
that
when
c
words, when c -
becomes
Also,
c
r
£
we
obtain X\ -
1
and \3
=
-
(1
this condition boils
it
always optimal
is
c)
down
to use
to r
*
1/2
which
an auditor.
never true. In other
is
When
c =
1
,
the condition
—
2-q
when
y
the effort
When y
=
c)
~
0,
bo =
1 _
i.e.,
to (4)
Plugging those values in (5) yields
2
r
and using (1)
X*=
1
using
( 1 )
to (4)
we
find that
i__q. Ae
is
and
constant with respect to
MIR2
is
not binding,
we
r
and P™.
are in region
B and using
1
(
1 )
to (4),
90
e
Bi = 1 .
i^_i
and the informational
If
r
or
P
171
^
rents are extracted
from the high
type.
MIR2,
are high enough to bind
2
(e«
- A9)
=
l
+
r
P
m
(2 - cl
l
by MIC. Therefore, using MIR1 we can
e B2
= AG
2
1
Equating
ex
p
,
ei
f2 - c)
AG
B1 and e] B2 gives us the line a, separating the
= (2
c^
2
j
I
(2r - 1)
obtain,
two regions Bl and B2:
91
Proof of Proposition 6
For
proof we will use the
this
results
lemma
derived in
6.
1
contracts are precisely the possible choices of the principal
unknown
type: he can allow
alternative
we
(lemma
main
when
r
s
2
q
< ^
So
.
-
.
dominates the scheme BO.
Remark: At
this point,
space where
it
2
)
is
For each
and be comparing these
profits
we
regions.
The value of
in the region
6
These two
facing a free auditor of
Derivation of §: The optimal contract of proposition 4 uses the scheme
when
4.
or deter (proposition 4) collusion.
1)
will derive the profits for the principal
will characterize the four
and proposition
= £ of
r
lemma
1.
was found
between I and ~
2
We call the line
we have
also
1;
to
auditor)
be always smaller than
the incentive
-
BO (no
scheme of lemma
1
q
of lemma
§
1
.
proved the existence of a region of the paramenter
optimal for the principal to allow collusion. Namely, whenever
r
€ (J
,
regardless of the value of P"1 allowing collusion dominates both other schemes
,
(no auditor and deterring collusion.)
Derivation ofT[i: First note that the border between regions Bl and
collusion
i.e.,
is
below than
cc(D) s cc(A) since
deterring
between regions Bl and B2 when allowing collusion;
-c
^— a(A) and c € (0,1).
the border
a(D) =
We can then compare the
2
—
profits
of schemes B1(A) and B1(D) below a(D) and find the
border^ which equates B1(A) and B1(D). When
i
collusion (using scheme
B 1(A)) and when
scheme B 1(D)).
When r > =——
B2 when
and Pm < a(D),
r
>!;
i
it
will
r
<~^i
it
will
be optimal
be optimal to allow
to deter collusion (using
92
,m
IT
which implies
n
/
IT
^-
(A) - 17(D) =
]
+ 2q
-
(1
r)
}
that
,
^
when
- tt(D) >
(A)
[q (3r - 2) - 2r + 1
{c
2q (1 -
<
r
q
c)
- 3c)
(2
+
c
+ 2c
and viceversa.
For
r
"1
1
and P™ = a(D),
scheme "allowing"
is
we saw
MA)
that
clearly better since
If
7r(D).
a(D) < P™ < a(A) and
So
the border
am
=
^ (A)
I
e
2
-
B3(A)
1
,
(5(A)
>
2 -
2^7
>
we
1
.
,
X
l
(3(A), the
makes
and
is
restoring
deterring relatively
Also
_c
2 -
note that e B3 < A > > e
i
A6
(1
A6 (4r - A6 2 (2r - l)
.
-
\
more
r)
scheme B3(A)
_
is
optimal.
_
"
A8
MD)
(l - r)
e
2r - 1
2r - 1
c
2)
^^>\
Therefore, below (3(A) and above (3(D) and for
above
all the rent
(3(D). Indeed,
A6 >
^2~
2r^~T
_A9_
2
c
l
uses the whole punishment P™.
we note that when r =
Finally,
since =
it
-~£ the
between "allowing collusion" and "deterring
collusion" (£) has to head north-east. Indeed, increasing r
profitable because
r
continues to extract rent at the same constant rate
scheme has already extracted
as before while the "deterring"
effort at a decreasing rate.
it
=
r
>T
the
scheme B3(D)
B 3(D)
l
is
optimal.
And
-
-.
8.
-
Date Due
:\
Lib-26-67
MIT LIBRARIES
DUPL
3
TOflO
OObTTlZb
b