V
ot
i
omm
HD28
.M414
no.
lAUG 15 1991
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
On Optimality of Allowing Collusion
Fred Kofman
Massachusetts Institute of Technology
Jacques Lawarree
University of Washington, Seattle
WP#3315-EFA
July 1991
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
On Optimality of Allowing Collusion
Fred Kofman
Massachusetts Institute of Technology
Jacques Lawarree
University of Washington, Seattle
WP#3315-EFA
July 1991
TXl'T. LIBRARIES
AUG 1 5 1991
^..wMi
On the
Optimality of Allowing Collusion
Fred Kofman*
and
Jacques Lawarree**
April 1990
Revised July 1991
*
Sloan School of Management, M.I.T.
**Department of Economics, University of Washington,
We thank Yoram Baizel,
Seattle
Richard Gilbert Michael Katz and Jim RatUff for helpful comments.
,
The Sloan Foundation's fmancial support of Fred Kofman
is
gratefully acknowledged.
Abstract
In this paper,
we examine
a
model wherein a
privately informed manager. There are
auditors satisfy their
may
principal can use an auditor's report to contract with a
two kinds of auditors:
own self-interest, which may differ from
of information in order
falsify their report
to
maximize
that
strategic
and non-strategic. Strategic
of the principal.
As a consequence
their payoff. Non-strategic auditors
they
always
report their information truthfully.
"We allow the manager
principal.
Given
to
collude with strategic auditors in order to manipulate the information sent to the
the impossibility of a screening equilibrium
— our model
be distinguished from one emother
—
strategic
and non-strategic auditors cannot
explicitly considers the principal's decision to allow or to
deter collusion.
Deterring collusion
is
costly for the principal because he has to reward both strategic and non-strategic
auditors for not accepting a potential bribe from the manager. Collusion
of the bribe he has
enough
to
pay
costly for the
to the (strategic) auditor. If the principal believes that there is
that his auditor is non-strategic,
manager and a strategic
is
he will offer a contract
that
manager because
a high probability
does not prevent collusion between the
auditor.
Keywords: Adverse Selection, Agency, Auditing, Bribery, Collusion, Hierarchies, Monitoring.
1.
Introduction
Pick any newspaper, in any
city, at
any
date;
it
highly probable that there will be a report of a corruption
is
or collusion scandal. Examples abound: from customs officials bought with drug
money
to regulators
lured with promises of high paying salaries at the end of their tenures; from fire inspectors
safety standards to procurement officials
which
who award
principal to pursue his
own economic
pattern of a controller
benefit
is
equilibrium outcome of principal-agent models.
show up
in our
"relax"
contracts to highest-cost bidders; from auditing firms
certify imprudently aggressive accounting practices to "old
who never check on management. The
who
boy network" corporate boardrooms
who
uses the
power vested
in
a very familiar one. Collusion, however,
him by
is
the
never an
Why is it that such a widespread phenomenon does not
models of the world?
Optimal contract models do not involve collusion because of two underlying assumptions: the screening
condition and the revelation principle.
The screening condition allows
by which
different types of agents distinguish themselves
actions.
The revelation
the principal to devise
mechanisms
from one another by taking different observable
principle asserts that any incentive-compatible individually- rational optimal
mechanism can be mapped onto a scheme wherein every type of agent
truthfully reveals his privately-
known characteristics.
The
fact that
optimal contracts do not involve collusion does not
optimality of allowing
it.
where collusion persists
The
mean
that they
logic of the revelation principle implies that if there
in equilibrium, then
it
is
is
deny
the possible
an optimal contract
possible to replicate the payoff vector of this contract
by
using a truth-telling mechanism. Since direct-revelation problems are tractable, whereas general problems
may
1
be non-tractable, researchers focus on the formeri. So, although
we know
Another problem with the Revelation Principle is the potential multiplicity of equilibria
direct one. For a complete treatment see Dasgupta, Ilammond and Maskin ( 1979).
that are
that collusion
can be
being mapped onto the
prevented by designing cross-checking mechanisms, the question remains:
is it
always optimal
to deter
collusion?
In answer to this question, our institutions seem to say "no"
mentioned above.
scheme
in
It
might be
an organization
costs of putting
some
— as suggested by
that collusion is too costly to deter.
may demand complete
"bite" into his
Although
the
newspaper
articles
the designer
of an incentive
conformity from his agents, he will need
to evaluate the
demand. Some members of the organization might have strong
incentives to collude in order to increase their utility. If they are to be discouraged from doing so, a
carefully crafted (and costly!) structure of rewards and punishments
If agents are
homogeneous, and
if it is
own weight (in terms of profit ),
with the agent without attempting
the limit case of
mixed
to
risk averse,
illicit
he will have
to
make
know which auditors are
sure that the incentives for
compUance
are strong
may fmd
enough
is
of cheating by a few
his beliefs about the likelihood
it
strategic
are
more
and which are
Depending on
costly,
he
it
to
aUgn
worthwhile
all his
the lowest
of
not.
auditors
common
to accept the possibiUty
of collusion, the principal
optimal to either allow or deter collusion.
In this paper we explore this problem using a model similar to that of Kofman-Lawarree
latter paper,
some
to see opportunities for collusion or to foresee the benefits
denominator. If providing for these incentives
might fmd
than others,
non-strategic auditors. If he decides to prevent collusion for
outliers.
it.
A may accept, under identical
B would not. Some people are more honest
practices. Unfortunately, the principal does not
would only hire
the principal will choose
equal.
and some are simply unable
If he did, he
the principal
always deter collusion or always allow
Indeed, different types of agents are present in any organization. Auditor
circumstances, a bribe that auditor
why would
reduce the information gap between them. Except for
strategies, the principal will either
However, not all people are created
into place.
too costly to deter auditors from colluding,
ever hire an auditor? If an auditor caimot pull his
to deal directly
must be put
we
studied a principal-auditor-manager hierarchy and
in his interest to prevent collusion
between the
(strategic) auditor
showed
[
199
that the principal
la].
fmds
In the
it
to
be
and the manager whenever the auditor is
hired, and, that if he is not hired, incentives are given to the agent to cause
Therefore, in the optimal contract,
To
him
behave
to
truthfully.
nobody colludes.
explore the more plausible case wherein
self-enforcing collusive arrangements are
some deviant behavior persists
more
easily achieved in
For example, take the case where there are two types of auditors:
some
in equilibrium,
states
strategic
assume
of the world than
in others.
and non-strategic. The
strategic
auditors accept bribes with positive expected value, and the non-strategic never take bribes.
we
introducing a binary random variable,
supporting collusion
may be attained and
Suppose, for example,
firm
—
ties that
make
that
that
that,
this case, the court will not
with probability (1-c) side-contracts are not feasible.
— and
between them self-enforcing
enforce any contract that
and Nagarajan [1989] claim
of the auditor are unenforceable;
i.e.
,
by using a screening
the
with the manager of the
perse illegal. In
that these ties are
tie.
Baiman, Evans
an inherent characteristic of an
contract. Contracts based
owner cannot write a
we concur with them
ties
contingent on a report of this
is
that since the option to collude is not
individual, collusion cannot be avoided
Next,
with probability "c", the state of the world
with probability "c" the auditor has pre-contractual
side-contracts
reveal his type. Although
could say
that
contract
which
on a "type" report
will induce the auditor to
that a screening contract is not feasible
we
don't agree
with their argument^.
Although
it
may be
clause contingent
on
impossible to sign a contract based on a report of type, the court can enforce any
the
outcome of a publicly-observed
agree upon. For example, the principal and the auditor
red
tie
he
is strategic
what is it
2
by a
tie
he
is
which
the principal
(in secret) that
non-strategic.
when
and
on type?
for suggesting this line of thought.
3 Unless the court discovers that red = collude.
the auditor can
the auditor
As long as the
third party, the court will enforce a contract contingent
that fails, if not the contractibility
We thank Eric Maskin
may agree
and when the auditor wears a blue
the auditor is observable
variable,
on
its
tie
wears a
worn by
color^.
So
In this setup, what
fails to
The auditors have no
separate types
is
the single-crossing, screening or Spence-Mirrlees condition.
which enables
feature
the principal to discriminate
between them by means of
providing different incentives. If the principal were simply to ask for type reports, promising a high
reward for strategic auditors, every auditor would claim
for the strategic auditors, every auditor
that the
two types of auditor have
the
would claim
same
to
to
be strategic;
if he
were
punishment
to threaten
be non-strategic. The problem as modeled here
utility function, but different strategy spaces; the existence
different strategy spaces is not sufficient to allow the principal to screen the different types
is
of
by means of
profitable contractual arrangements.
We present a model wherein the principal decides to hire an auditor, who is non-strategic (i.e., never
accepts a bribe) with probability (1-c) and strategic
with probability "c". Either type of auditor seeks
refuses to take bribes, but he does take any
strategic but
he
is
not stupid!).
To keep
assumption does not affect the main
principal could get a signal that
would
the
(i.e.,
to
would
maximize
payment offered
to
take bribes if they enhance his welfare)
his welfare.
him by
model simple we assume
results.
The
non-strategic auditor
the principal.
that his reservation
(He
may be
wage
is
non-
zero; this
We also discuss how our model would be affected if the
reveal the occurrence of collusion.
For some parameter values, the optimal contract will not specify a complete avoidance of collusion. If
there is a large
enough probability
that non-collusion situations will occur, the principal will
allow the few
cases of collusion to occur rather than paying every type of auditor the reward necessary to deter the few
colluding ones.
This argument
occurs
at
is
a point
similar in flavor to the theory of single-price monopoly.
at
which price exceeds marginal cost. This
marginal revenue and price. In order
to sell
result is driven
The monopolist's optimum
by
the
wedge
existing
an additional unit of output, the monopolist needs
price not only of the marginal unit, but of cill the units
which would have sold
at the
to
between
lower the
higher price, as well.
Similarly, our principal, in order to deter the strategic auditors, needs to reward the non-strategic as well.
As
in
Kofman-Lawarree [ 199 la], when
the principal deters collusion,
comply.
the contract provides all agents with incentives to
shirked, everyone involved
knows
punishment must be applied
to the agent.
shirks or colludes, since
When an auditor indicates that the manager has
that the reported signal is
wrong.
Still, the
ex-post irrational
This commitment to potential ex-post irrational actions
required in order to provide, ex-ante, the right incentives'*.
in the first place,
nobody
Of course,
if the
agent
is to
is
accept the contract
he must be compensated in advance for the potential mistaken punishments he will
suffer.
The above suggests
that the principal
exempts himself from the need
to
may
expect some benefits in allowing collusion. In doing so he
compensate the agent for the
entire punishment.
value of the bribe will have to be provided to the agent in advance. This lends
intuition that since
We will explore the soundness of this intuition and show that
collusion.
will
punishments are only applied by mistake, the principal
examine
in addition the
assumption
that strategic agents
it
Only the expected
some support
to the
may actually gain by allowmg
does not always hold
true.
We
produce a higher payoff for the principal than
non-strategic agents do.
2.
Can
Before
Collusion Enhance Welfare?
we
study the optimality of allowing collusion,
we will examine
the possibility that collusion
welfare-enhancing. In his insightful book Controlling Corruption Robert Klitgaard notes that
.
is
many
scholars have argued that corruption can play a useful role. This statement goes beyond saying that the
optimal level of corruption
is
not zero.
be worthwhile, but that corruption
It is
itself
not simply that fighting corruption
may
may be
so costly as not to
create economic, political or managerial benefits.
distinguishes three categories of arguments that
may
suggest that corruption
is
socially beneficial: the
"economist's reminder", the "political scientist's reminder" and the "manager's reminder". The
third are relevant
''
ones
to
our present concerns.
We ignore the possibility of renegotiation at the interim
stages.
He
first
and
The economist's reminder. Corrupt payments introduce a mechanism similar
goods and services are allocated by queue,
allocate
in the
goods according
hands of people
to willingness
who value them
sense, as a result of corrupt actions,
back the market through
the
and
the
politics,
random
ability to pay.
Corruption
efficiently. In the
services
economic
Corruption brings
efficiently.
form of a bidding or bargaining process.
The manager's reminder. Corruption may have uses within an organization.
may sometimes
constraining, the organization
may instead
may thereby put goods and
most and who can use them most
in the
market system. While
selection, or "merit," corruption
goods and services are allocated more
back door
to the
by
benefit
the
If bureaucratic rules are
employees' corrupt circumvention of the
rules.
These reminders, claims Klitgaard, have some
common
many
or most
Second, the putative benefits of the corrupt actions depend upon the assumption
that the
gained from specific corrupt actions as opposed
decisions.
features. First, they refer to the benefits to be
corruption transgresses a
wrong
organizational rules. In short,
if
to
systematic corruption that pervades
or inefficient economic policy or gets around imperfections in
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
is
one with a
rigid,
overcentralized, honest bureaucracy."
The upshot of this discussion
there are
where
some
inefficiencies
the principal designs
is that
which
literature contains
The apparent
to
to
information asymmetries, collusion will never
a mechanism where the "controlling" agent behaves non-strategically.
numerous examples of papers
contradiction disappears
these papers use.
when
are impossible to remove. In the realm of optimal contract theory,
mechanisms subject only
enhance welfare when compared
The
corruption can only be helpful as a second-best mechanism
The usual argument
in
which collusion appears
upon consideration of the
different
for allowing sub-coaUtions
is that,
to
be welfare enhancing.
benchmarks
that
our paper and
since agents can observe things
that are not accessible to the principal, then,
by
forcing the agents to play as a team, the principal can
use of the private information they hold. Examples are
[1990],
Macho
Che
[1990], Itoh [1989],
make
Hohnstrom and Milgrom
Stadler and Perez-Catrillo [1989], Ramakrishnan and Thakor [1989], Varian [1989] and
Villadsen [1991]. In these papers, the agents obtain information not accessible to the principal. Either
these papers exclude
communication and
alone. Consequently
Nash
revelation
transfers
it
is
were able
higher payoff.
It is
for this information, or they
between principal and agents are
assume
less efficient than those
We do not
to get accurate information
the fact that
it
is
take issue with this argument.
Our
that the
between agents
only by letting the agents play as a team that the principal can make
of the agents' private information.
principal
mechanisms
point
efficient use
is that if the
from one of the agents, he would then be able
to
achieve a
impossible or inefficient to get that information that makes collusion a
lesser evil for the principal.
Tirole [1990] also mentions a case
a signal S2
signal S3
where collusion
constraints are "linked": inducing truthful reporting of
when a bribe would reward a signal si makes it more
when a bribe would reward
$2-
In such a case
it
might be optimal
of S2) in order
minor cases
(e.g.,
crimes
preventing S2 from being reported in place of S3)
(e.g.,
Our present paper
having
fits
si reported instead
into a third category,
screen the potential colluders.
for the principal to bear the cost
3.
expensive to induce truthful reporting of a
to
to tolerate
some
collusion in
lower the cost of fighting more serious
wherein the principal allows collusion because he cannot
When the proportion of potential coUuders is low,
it
becomes suboptimal
of deterring collusion 100 percent of the time.
The Model
We examine a three-layer hierarchy consisting of a principal, an auditor and a manager.
owns
the vertical structure; the
manager runs
principal
a productive unit with private information about
efficiency; the auditor collects information for the principal. Following Tirole [1986],
principal lacks either the time or the
The
knowledge required
to
we assume
its
that the
supervise the manager, and that the auditor
lacks either the time or the resources required to run the vertical structure.
utility
of the manager and the auditor
The manager exerts
effort e,
to
be zero and further assume
We normalize the reservation
that all agents are risk neutral.
which, together with a productivity parameter 6, determines output x =
We assume that the cost for the manager of exerting effort, g(e), is an increasing,
simplicity
transfer,
t.
we
take g(e) = e2/2.
The output belongs
The manager maximizes
to the principal,
his payoff equal to
e.
convex function. For
who compensates
the
The manager signs
g(e).
t -
+
manager with
a
the contract after
learning the realization of 0. Therefore he must receive his expected reservation utility in any state of the
world (we assume
that is
optimal for the principal to employ the manager even
The auditor observes a signal
and
his objective is to
maximize
his expected
when
is to
maximize expected
0.
He
receives a
wage
the lowest
w from
occurs).
the principal,
monetary income. The principal receives the residual
of the vertical structure and pays (or punishes when
His objective
with
s imperfectly correlated
when
profits E(;r)
it
called for) the other
is
= E(x
w
- 1 -
+
?•").
members of the
The manager
is
profits
hierarchy.
punished with
?^
the auditor reports that he has shirked.
Uncertainty and information structure:
obtains with probability
manager,
q and 02 with
we will suppose
assume
probability
that the output
The auditor obtains a signal
We
x (=
+
( 1
(si
|
0i) = Prob (s2
short, the signal is right with probability r
Equating the probabilities across
states
with
I
assume
productivity. Auditors
While
and e are private information of the
0.
02) =
The
r
;
and
verifiable.
signal can be si or S2 with the following
Prob
(si
02) = Prob (s2
|
and wrong with probability
I
0i) =
(1-r) regardless
1
- r.
In
of the true
0.
of the world has no major consequences and greatly simplifies the
presentation of our results. Without loss of generality
It is realistic to
q).
e) is publicly observable
s imperfectly correlated
conditional probabilities: Prob
•
can take only two values, 0i < 02- 0i
that
we assume
that
r>
1/2.
that the auditor receives a signal imperfectly correlated
do not have the material
possibilities to
examine
and examine only a sample of them and make inferences regarding the
all the
with the true state of
firm's records; they select
situation of the firm,
which they
then report to the principal. This inference
is
We also assume that the manager can
subject to mistakes.
observe this signal: the auditee usually knows what records were examined by the auditor and can deduce
the inferences they support.
Finally,
we assume
that
with probability "c" the auditor
and with probabiUty (1-c) he
is strategic,
is
non-
strategic.
Collusion
and Audit Technology:
rule out the possibility
If the auditor
and the manager collude, they can forge evidence. To
of the auditor framing the manager,
two must cooperate. If the auditor gets a signal favorable
we assume
to the
with revealing a less favorable signal, since he needs his help
For
the bargaining process
between
the
manager and
coalition are distributed according to the
Nash
that in order to forge evidence the
manager, he cannot direaten the manager
to falsify evidence.
the strategic auditor
solution.
we assume
that the gains
of the
We will develop this idea further in the next
sections of the paper after laying out the basic structure of the model.
Strategy spaces:
the
The
principal designs the contract and offers
manager t(x) and uses
level of effort (ei for 0i
auditor w(x,s). s
If the auditor's
is
02), yielding Xi
the auditor's report
announced signal
and
differs
it
may be
to the
manager and
when
si, S2
= 0i and X2
or
the auditor.
the contract determines
For each
the auditor with probability y{x).
and 62 for
it
if the
from the manager's (implicit)
when
= 02
an optimal
He
^.
He pays
pays the
auditor was not used.
report, the principal
can impose a
penalty on the manager of up to P^"* (P'" ^ P""*). Limited liability enables us to solve the model with
risk-neutral players
and
to obtain
many of the
features of a
model with
risk-averse players and
unbounded
punishments (see Sappington [1983] and Baron-Besanko [1984]).
The manager chooses
payment
^ If X
C'
t(s).
More
effort e.
He can collude
explicitly, t(«) should
{xi, X2}, the agent gets punished, so
with the auditor
to
manipulate
be a function of s given
we can
restrict attention to
s,
s in
exchange for a
side-
but the manager has an interest in
outputs Xj, X2.
10
colluding only
when
s
= S2 (the high-productivity signal obtains). This
why we
is
take t(«) to be a
function of s only. All players must be awarded their reservation utiUty level to sign the contract.
Timing :
(1) Nature
chooses the
state
of productivity (0) and the signal for the auditor (s) according
also determines whether or not the auditor
is
to
Table
of the colluding type with probabilities c and
(
1
.
It
1-c)
respectively.
(2)
The manager learns
(he has not yet observed either the type of the auditor or his signal).
The
auditor learns his type.
(3)
The
principal offers a contract specifying the transfer for the
and the auditor's
The
report.
for the
report,
and the transfer for
manager (P""). At
strategic auditor (c),
The
the auditor (w(x,s)) as a function
of output and his
contract also determines the probability of conducting an audit (y(x)) and the punishment
the
optimum each of these
problem: the probability of a low type
(4)
manager (t(x,s)) as a function of output
and
the
(q), the
on
informativeness of the signal
maximum punishment
The manager chooses his
contract is signed.
functions will depend
for the
effort (e).
manager (P™
Output (x)
is
the parameters of the
(r),
the probability of a
).
realized
emd observed by
all
parties.
(5)
The
principal sends the auditor with probabiUty y(x).
The manager and
the auditor obseive the signal
(s).
(6)
The manager and a
strategic auditor
the auditor) dependent
(7)
The auditor reports
s.
(8) Transfers take place.
on
can sign a side-contract specifying a transfer (from the manager
the auditor's report (t(s)).
Side- transfers take place.
to
11
Sl
e,
qr
S2
12
We then study the optimal contract, when the auditor is strategic with probabiUty
the optimal
perfect.
mechanism
The reason
is that
is that, at
the first best
some
is
we assume
point, the principal will not
that the auditor is strategic
feature of
never reached as long as the auditor's information
Higher punishments induce higher bribes, which are more costly
Next,
The saUent
1.
want
to use all his
when he
not
punishment capacity.
to deter.
with probability less than
attempt to deter collusion by rewarding the auditor
is
and
that the principal
does not
reveals incriminating information.
While a
1
non-strategic auditor will truthfully convey this information to the principal, a strategic auditor will collude
with the manager. This extreme case (c=l; no deterrence) will help us
between allowing and deterring collusion, and
parameter space in which allowing collusion
to
defme
the indifference locus
the indifference locus will separate the region of the
is strictly
prefered from the region in which deterrence
is
strictly prefered.
The important decision
the principal faces
whenever he decides
to hire
an auditor is whether he
allowing or deterring collusion between the manager and the auditor. If he decides
will have to
pay
the auditor a reward at least as large as the bribe proposed
by
the
why would
better off
to deter collusion,
he
manager. If he allows
collusion he will not pay any reward, but a strategic auditor will never report that the
punished. If the strategic auditor never reports a non-compliance
is
manager should be
the principal
fmd
his
services useful?
The
strategic auditor is useful because,
assume
the
Nash Bargaining
depending on the bargaining power of auditor and manager (we
Solution), the
manager will
still
suffer a loss
when
the auditor receives a
high-productivity signal after low-productivity production has been obtained. Namely, the manager will
have
to
pay
the auditor a bribe equal to half the punishment,
payoff, but the deterrent effect of the punishment
that the principal
must compensate
the
manager
still
i.e.,
?W2. The
operates on the manager.
for the expected mistaken
complies, so punishments occur only by mistake.) Thus, the principal
manager and a
principal does not get this
strategic auditor to play cooperatively, since
On the
other hand, recall
punishment (the agent always
may be better off if he
no compensation
will
be necessary.
allows the
13
Our conclusion is
that the
lower the proportion of colluders, the higher the punishment, and the worse the
auditor's information, the
more
profitable
probability of having a strategic auditor
prevent collusion. This
is
so because
it
is
is
it
becomes
for the principal to allow collusion.
When
the
low, the principal will be reluctant to give a high reward to
highly probable that the auditor
is
non-strategic and that he will
not collude anyway.
When the punishment is high our assumption about the Nash bargaining solution becomes very important.
Even
the strategic auditor will inflict a loss of P'"/2
enough, half of it can suffice
to
on
the
manager;
if the
punishment becomes high
achieve the most profitable second-best contract given the informativeness
of the auditor's signal.
It
also appears that a
low correlation between
the auditor's signal
manager will be punished by mistake. In order
needs
to
compensate hun
in
to get the
manager
when the
signal
is
advance for these expected mistakes. This advance compensation increases
less noisy.
When collusion is allowed,
is that it
becomes
less effective to use the
screening of types. However, if the punishment
is
First Best Allocation : Consider the case in
the
manager and can,
which
when
is less
to the
expensive to deter
manager. The
punishment as a
really high, half of the
deterrent and allowing collusion is always optimal, even
it
the strategic auditor only takes half the
make a lower advance payment
punishment. Therefore the principal has to
of this beneficial effect
of the world makes
to sign the contract, the principal
with the noise level of the auditor's signal. So, everything else being equal,
collusion
state
When the manager is low-type and the signal is high
allowing collusion more profitable for the principal.
the
and the true
threat to support the
punishment
the auditor's signal has
the principal can observe output
is still
little
and
principal's problem
feasible.
The
maximize
his expected profit, Err =
is to
choose efforts and transfers
q (Oi+ei-ti) +
to
state
an effective
noise.
effort exerted
therefore, infer the state of productivity with certainty. Auditors have
and a contract which specifies the manager's transfer as a function of effort and the
flip-side
no
by
role here,
of productivity
each type of agent in order
is
to
(1-q) (02+e2-t2), subject to the participation (individual
14
of the two types,
rationality) constraints
constraints
(MIR)
of the world.
and
state that the
tj
s eiV2 and
^
t2
t^}-/!.
manager must be compensated
Our assumptions are
that either the
The manager's
at least for the cost
manager signs
individual rationality
of his
each
effort in
state
the contract, or he quits after learning
that his reservation utility is zero.
In the optimal contract under symmetric information, the principal equates the marginal cost of the
manager's effon
make him
sign the contract
=
(t
of the state of productivity, t(xi)
From now on, we will assume
The
effort (e).
first
6^/2
=
=
1/2).
t(x2)
=
best contract
When
the strategic type,
i.e.,
Note
is
the Auditor
is
=
62
1)
and pays the manager just enough
that the transfer received
by the agent
no longer optimal. Auditors now have a role
we
first
assume
that the auditor is
Strategic with Probability 1 :
always
Suppose
in evaluating the
strategic.
that the auditor is
signal of the auditor
P''^.
is
high.
for collusion is
To avoid
the
when
P^
punishment
the productivity of the
the
always of
may collude after the manager has
chosen his level of effort and both he and the auditor have (jomtly) observed the signal.
made
independent
can observe neither the productivity parameter (0), nor the
c=l. Then, the meuiager and a strategic auditor
verify that the only case to be
is
to
1/2.
that the principal
productivity parameter. In what follows,
Optimal Scheme
=
to the marginal value of his product (ei
manager
is
One can
manager
is
readily
low and
the
willing to pay the auditor up to
Thus, in the presence of a covert contract, the auditor will not report payoff- relevant information.
The auditor would be
indifferent
between revealing or not revealing
punishment of the manager, but the manager
This gives
is
ready
to
rise to the coalition incentive compatibility
"buy
the auditor's silence".
(CIC) constraint. To induce
auditor has evidence to punish the manager, the principal must pay
would pay him
the cimount
information.
to
change his
by which
the
Of course,
report;
i.e.,
the auditor
him
at least as
truth-telling
when
much
manager
must be rewarded with an amount
manager would be punished
this constraint is
the information leading to the
only relevant
if the
when
as the
the
at least as great as
auditor were to reveal his incriminating
the auditor is used,
i.e.,
when y >
0.
15
We
also
assume
that the
manager and
individually rational with respect to the
Finally, the
the auditor choose a side-contract that is Pareto efficient and
no side-contract outcome.
manager incentive compatibility (MIC)
specified for his type.
The problem
As usual,
for the principal
Max q
{01 + ei
subject
to:
- ti
+Y
(1
-
r)
only one
now is
(P""
-
to
constraint induces the
manager
MIC constraint is binding at the optimum.
choose ei e2,
w)} +
,
(1
-
q) {62
ti
,
t2
+ e2
-
and
12}
w to:
to take the action
16
condition can be justified
auditor has neither wealth nor the capacity to borrow. This
if the
we examined
assumption, which
in
Kofman-Lawarr6e [1991a,
is
a strong
b].
We now define two functions before stating our first proposition.
a as the value of P"" at which (MIR2) becomes binding: a
Define the function
Define the function P as the value of P'"
at
which
increasing the expected punishment (y P'"): P =
The optimal
1:
—
.
2q(2r-l)
the benefit of adjusting effort is equal to the cost of
A0(4r-A0-2)
2q(2r-
Proposition
=
1)2
contract can be divided into five regions according to the values of r and P'"-
with P"" = pm*.
(NA)
If r
>
If r
-r-^
2
is
-
—
q
,
^
—
-*
,
the auditor is not used (y = 0)
the auditor is used (y
^'
(RE)
if P"^
(EA.)
if
(CP)
if P""
kept constant.
(FB)
The
—
^
>P
The
if r
=
1
•
scheme
P, the effort-adjustment
of the low type
and Pi" >
applied.
'
the constant-punishment
efibrt
is
> 0) and
a, the rent-extraction
a ^ P"^ <
and the no-auditor scheme
is
is
applied.
scheme
scheme
is
is
applied.
applied.
kept constant below
The expected punishment
its first-best level.
P, the first best is attained.
definition of the different regions
and a graphical example are presented below.
(y P^")
17
RE:
Identical to
NA except t2 =
12"^^^
-
(2r
— '^67=1 '^ti=^
FA- e,=^^^^P'" +
Ae
2
'
CP:
2
-
1) P'", i.e., the rent
t7
'
^
=
of the high type
is
reduced.
J-
2
e7=l.^i^; e.M;.,-^;
y-i; w-P
..-i;
Proof: see Appendix.
1/(2-q)
Figure
If the signal
is
too noisy —i.e., if r
optimal to use the auditor even
revelation.
1.
The
cost
Optimal Contract with a Strategic Auditor
is
smaller than
when his
27-—
reservation
comes from compensating
the
the auditor
wage
is
manager
is
not used (region NA).
zero because
it
is
It is
not
costly to induce truthful
for mistaken punishments.
To understand
18
why
the critical value
of r depends on
q,
remember
that
pay the auditor's reward
incentive- compatible. Therefore, the principal has to
makes a mistake (which happens with
(which happens with probability
is,
the better the information
If the auditor's signal
the auditor's report
As
in
is
q).
our optimal mechanism
probability 1-r) after the
To justify
the use
is
collusion- free and
w only when the auditor
low productivity type has been obtained
of an auditor, the higher the proportion of low type
must be.
good enough with respect
to the probability
and rewards the auditor in order
to
of low productivity, the principal uses
discourage him from colluding with the manager.
Baron and Besanko [1984], a separation between
occurs in region RE, where rent of the high-type manager
and the production policy
the the audit policy
is
extracted.
When the principal uses the auditor
with punishment capacity below a, the prescribed productions xj and X2 are the same as they would be
were he not
result
to
use the auditor. In region
EA, where
the effort
of the low type
is
adjusted, the separation
breaks down.
The
first
The
intuition
first, it
best
is
never reached for an imperfect signal even
behind
this result lies that increasing the
allows the principal
effort distortion
to
fulfillment of the
when P'"
CIC
is
at the
same
time,
liability
grows without bound.
manager's expected punishment has two
it
effects:
increases the cost of getting truthful revelation from
high, the potential bribe to the auditor will be high as well,
constraint expensive.
Above curve
principal keeps the expected punishment (region
from the
maximum
achieve incentive compatibility from the manager at a lower cost in terms of
of the low type;
the auditor. Indeed,
if the
CP)
at
P, the
second effect outweighs the
a constant level while production
is
making
first,
the
and the
bounded away
first best.
No Collusion Deterrence by Assumption We now consider a situation where
:
probability less than one (c
<
1)
and where the principal offers no reward
the auditor Is strategic with
to the auditor and, thus,
does
not deter collusion.
We assume the Nash Bargaining solution for the coalition between the strategic auditor and the manager.
By convicting
the manager, the auditor can get a reward
w, while the manager suffers a punishment
P'".
19
We
call the pair (w, -Pi") a "threat point."
By
colluding with the auditor the manager avoids the
punishment while the auditor foregoes the reward. The net gains of the coalition are then P"i
-
w,
i.e.,
the
gain for the manager minus the loss for the auditor. The Nash bargaining solution assumes that the gains
of the coalition will be
split equally:
each
member receivmg
point. Then, the strategic auditor receives a transfer of
(^y
+
?"
•
W
- ?»" +
W
for the principal is
Max q
{0,
ei
subject
to:
- ti
+ Y
'
(1
-
now
c) (1
-
r)
to
choose
?"} +
(1
'
ej, e2, t], t2, y to:
-
^
above
w + ^'" ^
)
The problem
+
^"^
q) {02 + e2
-
12}
= ^'"
their coordinate
"*"
^
and
of the threat
the agent receives
-
20
probability y the principal will send an auditor,
auditor
non-strategic, the
is
manager wiU
manager will
who with probability ( 1
-
r)
will get a
wrong
suffer a loss (punishment) of P'". If the auditor
signal. If the
is strategic,
the
suffer a loss (bribe) of P'"/2. Similar reasoning clarifies the expression of the manager's
incentive compatibility constraint.
Define ^ as the
minimum
being a coUuder
and when
c
=
1,
is c: ?(c)
value of r such that
2
- c —-- 2q
is
it
( 1 -
profitable to use an auditor
.
^ = l/(2-q);
,
—
-L^ I s
2
2
i.e.,
the probability of his
c)
r
\
(2-q)(2-c)-2q(l-c)
=
when
.
Notice that ? > 0; and that
when
c = 0, ^ = 1/2
1
-
q
7 Afl
Define a(A)
^
as the value of Pi" at which
Define P(A) as the value of P°i
at
which
expected punishment (y P™): P(A) =
Lemma 2.1:
The optimal
contract
according to the values of r, P^" and
(NA)
If
1
>
If r
r
>
<
^(c), the auditor is not
^(c), the auditor is
RE(A)
if Pt"
EA(A)
if
used
s a(A)
a(A) <
appUed.
*
A stands
for
Allowing collusion.
,
Pni
(y
MIR2 becomes binding: a(A)=
the benefit
of adjusting
2A0
J
(2-c)(2r-l)
when
c,
(2-c)(2r-l)
effort is equal to the cost
of increasing the
Ae ( 1 - r)
A0
'2q"(2-c)(2r-I)
the principal allows collusion
can be divided in four regions
with P"i = P"^*
used (y = 0) and the no-auditor scheme
>
.-f(2-,)
is
appUed.
0) and:
the rent-extraction-when-allowing-collusion
^ P(A)
,
scheme
is
applied.
the effort adjustment-when-allowing-collusion
scheme
is
21
The
CP(A)
if Pi"
(FB)
if r
=
> P(A)
1
,
the constant-punishment-when-allowing-collusion
and Pi" > P(A) then the
first
best
scheme
is
applied.
achieved.
is
different regions are defined as follow:
RE(A)
is identical to
EA(A)
is
identical to
scheme RE.
scheme EA, except
^
for ei;
ei^2(A)
=
—
+
—
CP(A):Y(r.P-) =
^
,cp(A) .
1 .
A
i
2A8
2
AGil^
Proof: See appendix.
Remember that we assume
here that since the principal never deters collusion, he has only two options:
either he can choose not to use the auditor's report
no-auditor or
This
and give instead a second best incentive scheme (the
NA contract) or he can ask for the auditor's report knowing that this report may be biased.
may be optimal because, even disregarding that with some probability the auditor is a
latter contract
manager who
non-strategic, the
shirks and gets caught has to bribe the strategic auditor to avoid
punishment. The amount of this bribe might deter the manager from shirking in the
The
line ^
determines which incentive schemes the principal chooses:
the auditor's report
that ^
(this
It
(NA scheme).
depends on q and
c.
It
If
the principal
on
c.
If c = 0,
the auditor receives a
i.e., if all
always uses the auditor's
that in this case, the region
5, the auditor's report is
<
place.
^, the principal
does not use
used and collusion will occur. Notice
depends on q because collusion will occur only when the agent
happens with probability q) and
also depends
r^
if r
first
where
wrong
signal (this
=
the auditor's report
1
,
is
i.e., if all
used
is
of type
happens with probability
auditors are faithful and never collude, ^ = 1/2. This
report. If c
is
means
1
r).
that
auditors accept bribes, ^ = 2^-. Notice
the
same as
in proposition
1
22
In proposition
l/(2-q)
1,
was
the level
of informativeness of the auditor's signal where
it
became more
profitable for the principal to punish the agent (by mistake!) than not to use the auditor's report. Recall
that l/(2-q) is
independent of the punishment level. Thus
manager (as happened
in proposition 1), the principal
if,
instead of imposing a loss of P'" on the
imposes a loss o(^z— (as in
lemma
2.1
when c =
1),
then the condition for using the auditor remains invariant.
When
? < r
2
the auditor's report
1,
depending on the value of r and
P"".
the rent of the high type is extracted.
manager has been
extracted,
type. Finally, if r
and
and
P™ are
is
For a low
The
P™
and a low
P"i, the standard
MIRl
the principal can then use
through the
MIRl
The
RE
scheme EA(A)
Ls
is
applied, and
to restore the effort
even higher, another expected non-maximum deterrence
constraint of the agent
restoring the effort.
scheme
be applied
P'" are sufficiently high, all the rent of the high type
kept fixed at a level below
principal does not use all the punishment capacity available to
imposes on the
by
r
When r and
with scheme CP(A). The effort of the low type
P'".
may
used, and three separate incentive schemes
becomes
its first
result is obtained
best,
him because
of the low
however large
the
burden
is
that P'"
too high relative to the potential extra profit gained
intuition is similar to that developed in proposition
I
.
The
cost of increasing
constraint outweighs the benefits incurred through the relaxation of the
MIC
constraint (each time collusion occurs, the principal loses P'"/^).
Optimal Contract
2.
1
When
the Auditor
was derived assuming
is
Strategic with Probability Less
that the principal
Than
1:
The
contract of lemma
would never deter collusion.
We abandon now to address
minimum
it
it
our main proposition.
We first define the following two functions.
Define
when
E'
=
—
—
—-
7-^;
-r
^
(2-q)(2-c)-2q(l-c)
the probability
as the
of his being a colluder is
c.
value of r such that
is
profitable to use
an auditor
23
Ph^ 2q(l -c) + c
q(2-3c) + 2c
Define*^
^ the minimum value of r such that
it
is
profitable to deter collusion. See
appendix for a more precise defmition of ^^.
Defme P(D) = P
Proposition
in proposition
2:
1.
The optimal contract can be divided
informativeness of the auditor's signal
and
(AC)
If r
<
r
maximum
Two of these regions have three sub-regions each.
no auditor is used and a scheme
< ^Kc), collusion
is
punishment for the manager P^n (with P^" = pm*),
NA is applied.
allowed. The incentive schemes applied in this region are the ones
Lemma 2. 1.
If r > ^^(c), collusion
proposition
(FB)
^'(c),
If ^'(c)
defined in
(DC)
<
the
of having a strategic auditor c.
the probability
(NA)
r,
into four regions according to the values of the
is
deterred.
The
incentive schemes applied in this region are the ones of
1
If r =
1
and
Pm s
(3(D)
,
the first best is achieved.
Proof: See appendix.
The obvious corollary
collusion.
is
that there exists a region
of the parameter space where
it is
optimal
to
allow
24
FB
1
'
r
'
1
'
1
1
r
'
'
'
1
1
'
'
1
25
reducing the deterrence effect (expected value) of the punishment. The indifference locus of this tradeoff
is
reflected in line ^^.
Notice that imposing
offer a positive
w=
w, then
it is
in the region
where collusion
optimal to offer
w = P"i
is
allowed
is
optimal. Indeed, if it
and deter collusion. This indicates
will never offer a reward simply to raise the ante of collusion. If he offers a reward at
enough
is
optimal
to
that the principal
all, it
will be high
to deter collusion altogether.
Depending on
the
scheme chosen,
derived in proposition
1
the subregions
and lemma
2.1:
—
—
as depicted in the following graph
will be the ones
26
Figure
3.
Regions and Subregions with an Auditor of Unknown Type
27
This proposition "merges" the results of proposhion
used
the
is
lemma
same as
2.1.
If^'<r2
principal finds
Two
more
it
^''i
lemma
in
collusion
is
2.
1
.
The
and lemma
2.1.
definition of ^'
is
The region where no auditor
is
one given for ^
in
identical to the
allowed and the scheme of lemma
profitable to deter collusion, applying the
2.
applied.
1 is
scheme of proposition
Ifr>^^,
the
1.
important facts should be noticed:
non-maximum
(1) (Expected)
uses
one derived
the
1
maximum
deterrence always occurs
when
the auditor is used^. Indeed neither
deterrence. Therefore, in presence of collusion, the
first
best
is
scheme
never reached whether the
principal deters collusion or not.
punishment increases in amount,
(2) If the
true
because the effort
deterred (e^^^)).
The
when
collusion
is
it
always becomes optimal
allowed (e^P(^))
is
allow collusion. This result holds
higher than the effort
principal can better afford to restore the effort
down by cjl.
to
were
to deter collusion,
lose pni (1-r) Y (the expected reward to the auditor). If he
were
to
would be
is
the difference
(2-c)/2 Pn™ (1-r) Y in
scaled
between the payment required by
MIRl) and
the
is
the
he would expect
to
allow collusion, his expected cost
the low-type
punishment he expects
collusion
when allowing collusion because
If the principal
cost of increasing P^"
when
manager
to receive (1
-
to sign the contract (see
r) (1 - c)
y
P™ when an honest
auditor convicts the manager.
Finally, each region has the 3 familiar subregions
punishment
is
1/2,
c=
full
and expected
in figures 2
and 3 are
illustrations
of the optimal contract
when
0i = 1/2, 02 =
1/2.
Sensitivity Analysis I:
' For a
rent is extracted, effort is restored,
kept constant.
The graphs presented
q=
where
Changes
in the prohabilitv
of drawing a strategic auditor
discussion of the non-optimality of maximum deterrence under collusion see Kofman-Lawarree (1991a).
1,
:
The following graphs show how
the regions
of the optimal contract change with
c.
FB
No Auditor
Allow Collusion
Deter CoUusion
1/2
5h
1/2-q
Figure 4. Regions of the Optimal Contract
When the proportion of strategic auditors goes down (c->0),
best contract with a faithful auditor (see proposition
1
in
1
,
the principal always has to
proposition
spans [0,1 ]
the contract resembles the standard second
Kofman-Lawarree [1991a]). At
no collusion ever occurs, and the optimal contract always uses
if c=
when c
worry about collusion and
the free, honest auditor.
the optimal contract
is
the limit, if c=0,
At
the other limit,
identical to the
one of
1
Sensitivity Analysis II:
Changes
in the
bargaining power of the manager ond auditor
We have cissumed that the bargaining powers of the manager and the auditor were identical and, thus, that
the
Nash bargaining
solution should occur within the coaUtion these two agents form.
More
generally,
we
28
29
could model the bargaining power of the manager and the auditor by a variable p € (0,1)"^ and a transfer
(bribe) of
w + (Pi"
-
w)
p=
p.
e represents a situation
where
the
manager has
all the
bargaining power
and can bribe the auditor by infinitesimally exceeding the reward offered by the principal, p =
represents a situation where the auditor has
manager almost equal
to the potential
all the
punishment
1
-
£
bargaining power and will extract a payment from the
manager would
that the
suffer
were
the auditor to report
his signal to the principal.
An increase in the auditor's bargaining power (p going to
1) will
have an
effect similar to that caused
by an
increase in the probability of hiring a non-strategic auditor (c going to zero): the auditor will always be
used
(^'
« 1/2) and collusion will never be deterred (^^ ~
auditor, he will
still
have
to
bear (almost) the
full
1).
Even
if the
manager succeeds
in bribing the
weight of the punishment. This helps the principal;
although he recaptures only the punishment from the non-strategic auditors, the strategic auditors with
full
bargaining power are equally useful in that they align the incentives of the manager with the interests of the
principal.
The
effect of an increase in the bargaining
principal.
Whereas
influence on the
in the
power of the manager (p going
Nash framework (p =
to 0), is not as
1/2) the strategic auditor exercises
manager even when colluding, now he
is
good
some
for the
beneficial
basically useless unless the principal deters
collusion. This will reduce the principal's willingness to use the auditor.
Adding an External Signal
What
if there
were an external
auditor and the
signal that
would give
manager had colluded? Say,
announcement of an event correlated with
the principal information about the likelihood that the
for example, that after the audit
is
the true state of the world. After the
level of effort there is a chance that the principal will find out through a
performed there
manager makes
random
is
the
or p "
1
,
there
would be compactness problems.
wrong
event. If the building
bums down without any of its alarms going off, one may deduce with a fair amount of confidence
10 If p
a public
diat the
30
fire
inspector did not do his job; if a drug dealer
might
testify that
recent history
is
caught by the police and
he was bribing some law-enforcement agents;
comes under
close scrutiny,
if a
is
promised immunity, he
company goes into bankruptcy and
somebody might make a strong case on
its
the failure of the audit
controls.
Suppose, in the
spirit
of the previous paragraph,
positives condition. That
is, it
that the
non-manipulable signal
when it
will indicate that collusion took place only
non- false-
satisfies the
actually did take place.
In a more sophisticated model, collusion would be a necessary but not sufficient condition for the signal.
That
is,
the signal does not indicate directly if collusion has or has not occured, but instead indicates the
true state
To keep
of the world. In that sense, even a non-strategic auditor
the
by mistake:
model simple, we
will
assume
The question now becomes: how much
on
the
is this
pay more for the signal since
we would still be unable
If the punishment
is
to justify
worth
to the principal?
would not deter
is that
The answer
Ls
that
it
will enable
him
to stop collusion altogether.
this
company has gone
signal
is that it
is
is
big, he
In this case
much. However, suppose
free as in the case
from colluding. In
the colluders get caught (and
reasoning
depends
that the
extremely low. In that case, the principal would use the
is
fire, the
the strategic auditors
sometimes
The problem with
it
small, the principal will not be willing to spend as
signal (for example, in the case of the
the
false positives.
our observation of collusion in the news.
signal is extremely cheap and the punishment
game
signal
and avoid punishing auditors
punishment capacity he can exercise on the manager and the auditor. If the punishment
will be willing to
signal
caught "colluding" by mistake.
that the signal points to collusion
and there are no
the signal points to collusion
may be
assumes
make
this case,
of the drug bribes) but
this
an equilibrium outcome of the
the headlines).
that collusion is discovered
under, for instance), and that generating the signal
is
only after
it
is
too late (after
not a possible strategy for the
principal.
In the spirit of Kofman-Lawarree [1991a]
external auditing
body
to
we
can consider the possibility
that the principal set
check the collusive tendencies of strategic auditors and managers
up an
(e.g., as in the
31
department of Internal Affairs for police forces, the General Accounting Office for government related
entities, etc.).
This body
information before
collusion
is
it is
may
act as the external signal of the previous paragraph but releases
too late. If the cost of external auditing
is fairly
not too high", collusion will persist in equilibrium, and
it
will
low and
the
its
punishment for
sometimes be uncovered by
cross-checking agents.
5.
Conclusion
This paper presents a model where the principal of a hierarchy must make the decision of whether
or to deter collusion
among his two employees: an auditor and a manager.
the principal because he
must reward
his auditor in order to prevent
from the manager. If the principal believes
Deterring collusion
him from accepting a
that his auditor is non-strategic
is
to
allow
costly for
potential bribe
with a sufficiently high
probability, he will not try to deter collusion, thereby allowing strategic auditors to collude with the
manager. Even in such a case, hiring an auditor
whenever
it is
discovered by the auditor,
therefore refuses to collude)
1 1
Low punishments
and a bribe
for collusion presumption
from administrative guidelines.
Congress (1990)).
is
useful because the
may cost him
a punishment if the auditor
if the auditor is strategic
is
that shirking,
non-strategic (and
(and agrees to collude).
the non-verifiability of the external auditors* evidence or
investigation, for example, can stop the process by resigning (U.S.
may derive from
An IRS agent under
manager knows
32
References
Baiman
Evans, and N, Nagarajan, "Collusion in Auditing", Mimeo, Camegie-Mellon University
S., J.
[1989].
Baron D. and D. Besanko, "Regulation, Asymmetric Information and Auditing", Rand Journal of
Economics, vol. 15 [1984], pp. 447-470.
Che Y.-K, "Revolving Doors and Optimal Tolerance
for Agency Collusion",
Mimeo, Stanford
University,
[1990].
Dasgupta
P., P.
Hammond and E.
Maskin, "The Implementation of Social Choice Rules: some General
Results on Incentive Compatibility",
Felli L. "Collusion in Contracts
Review of Economic Studies vol 46
,
(2),
[1979].
with Lock-in: The optimal Timing of Commitment", Muneo, M.I.T.,
[1990a].
Felli L., "Collusion in Incentive Contracts:
Holmstrom
B.,
Does Delegation Help?", Mimeo, M.I.T., [1990b].
and P. Milgrom, "Regulating Trade
Among
Agents", Journal of Institutional and
Theoretical Economics. 46, [1990].
Itoh H., "Collusion,
Incentives,
and Risk Sharing", Mimeo,
UCSD,
[1989].
KUtgaard, Robert. Controlling Corruption University of California Press, Berkeley, California. [1988].
.
Kofman F., and J. Lawarree,
"Collusion
m Hierarchical Agency", University of Washington Discussion
paper 9 1/01, [1991a].
Kofman
F.,
and
J.
Lawarree, "Auditor Bonding, Collusion, and
Maximum
Deterrence",
[1991b].
Laffont
J.-J.,
and J.Tirole, "Cost Paddmg Auditing, and Collusion", Mimeo, M.I.T., [1990].
Mimeo,
33
Macho-Stadler I. and D.Perez-Catrillo, "Moral Hazard with Several Agents: the Gains from Cooperation",
Mimeo, Universitad
del Pais Vasco, Bilbao, [1989].
Ramakrishnan R. and A.Thakor, "Cooperation
vs.
Competition in Agency", Mimeo, Indiana University,
[1989].
Sappington D., "Limited Liability Contracts Between Principal and Agent", Journal of Economic Theorv
.
vol. 29 [1983], pp. 1-21.
Tirole
J.,
On the Role of Collusion in Organization",
"Hierarchies and Bureaucracies:
Economics and Organization, Vol.2, No.
Tirole
J.,
in Journal
of Law,
2, [1986].
"Collusion and the Theory of Organizations",
Mimeo, M.LT.,
[1990].
U.S. Congress, Committee on Govenmient Operations, "Misconduct by Senior Managers in The Internal
Revenue Service", House Report 101-800, October, [1990]
Varian H., "Monitoring Agents with other Agents", Mimeo, University of Michigan, [1989].
Villadsen B. "Communication and Delegation in Collusive Agencies",
,
Mimeo, Yale
University, [1991].
34
Appendix
Proof of Proposition
The Lagrangian
L=q
{6i + ei
-
(Kofman-Lawarree [1991a])
1
for this
1,
+ Y
problem
- r)
(1
is:
(P"i
-
w)} +
(1
-
q) {02 + e2
-
12)
+
Xl(ti-^-Y(l-r)P'"}+X2{t2-^}+X3{t2-^-ti
^^^
^^'
+ Y
'
r
Pm) + X4 {w
-
pm} + X5
{ 1
-
+
y)
with the additional constraints:
ej^O
62^0
ti^O
t2^0
w^O
O^Y^l
Xi^O
X2^0
Xb^O
X4SO
given that y and P"^ always appear together,
The
alternative choice
the reservation
—
to fix y
=
1
and
we
X5SO
fix P"^ at P""*
let P"i
and use y as the argument for maximization.
— does not
vary
alter the results
because
we assume
wage of the auditor is zero.
The Kuhn-Tucker conditions
for maximization are:
|^=q-X,e,+X3(ei-A0)sO
|^=(l-q)-(X2
+ X3)e2^0
;
;
062
ei|^=0
e2^=0
(1)
(2)
062
+ X,-X3^
f^=-q
Oti
f^=-(l-q)
;
t,f^=0
Oti
+ X2 + X3 ^
;
ow
(4)
Ot2
Ot2
^=-q(l-r)Y + X4
t2|^=0
(3)
^
;
wf^=0
ow
(5)
that
35
§^=q(l-r)(P'»-w)-Xi(l-r)P'" + X3rP'"-X5^
;
ay
Y$^=0
oy
(6)
Plus the constraints and their complementary slackness conditions.
Suppose Y =
0, then
X5 =
by the complementary slackness condition and, using
q(l-r)(P'"-w)-Xi(l-r)pm + X3rPmsO
(7)
If Y = 0, the auditor is not used. Therefore the optimal
In
NA, X 1 =
1
Suppose Y =
and X3 =
1»
then,
1 -
q equation
,
(7) then
scheme
becomes
r
<
NA and we can let w = P""
is
:r^
w.Lo.g.
—
2-q
EA and the line a are derived from
schemes RE,
(6)
equations (1) to (4). Also, by the
coaUtion incentive compatibility constraint P"" = w.
<Y<
Suppose
1
(i-c,
we are
in region CP).
Using equations
( 1 )
to (4),
q(l-ei)
Xi = X3 + q and q- Xi ei + X3 (ei
-
A0) =
0;
which imply
By the coalition incentive compatibility constraint, P"^
P"" Xi (1
Note
-
r)
+ P"™
r
that ei in region
X3 =
which impUes
that ei
=
=
yTT
and
The function
-4r2 +
(4
+
that
P
when yyT" <
p(r) is increasing if
-
2A0)r-(Ae+ 1)<0
4
r2
<
r
+
^q
since y <
+ 2 AO)
,
X5 =
0.
A0(l-r)
2r-
EA and
1
in region
RA.
r
-
(A0 +
1)
>
—
EA when P"" =
It is
easy to check that
1
(4
1
w and using (6) we find that
RA is independent of P"^ and equal to ei
a<
X3 =
1
This defines the line P, the border between regions
r
=
that
and decreasing
if
a
=
(3
.
when
36
This means that P(r)
increasing in the interval
.2
bound of the
interval. Therefore,
(1+A6y2 could be
1+A0
1
is
'
and decreasing when
2
r
depending on the value of AO satisfying the no shut
less than l/(2-q),
between
l/(2-q)
and
1,
or greater than
exceeds the upper
down condition^^,
1.
Proof of Lemma 2.1
The Lagrangian
L
= q {6, + ei
+
for this
- ti
problem
+ Y (1-c) (1
-
is:
r) Pn>}
+
(1
-
q) {62 + e2
-
12}
X,{t,-^.Y(l-r)Pni(i^) -HX2{t2-^)
.X3{t2-^-t..(^i:^.YrP-^}.X4{l-Y}
with the additional constraints:
ei^O
e2S:0
tis:0
t2
Y^O
Xi^O
X2^0
X3 2:0
The Kuhn-Tucker conditions
^
X4SO
for maximization are:
^=q-X,ei+X3(e,-A0)^O
Ce
;
ei^ =
(3e
1
(1)
1
|^=(l-q)-(X2-HX3)e2.0;e2|^=0
(2)
|=-q-X,-A3.0:t,f. =
(3)
|=-(l-q).X2.X3.0;t2|=0
'2
The maximum A6
for
no shut down
is q/(
1
-q).
Therefore the function 3(r)
starts
(4)
decreasing
at
most when
r
= l/(2-2q).
37
q(l-c)(l-r)-(^^j[A,(l-r) +
X4^0;y|^=0
A3r]
ay
(5)
dy
Plus the constraints and their complementary slackness conditions
If no auditor
is
used, y = 0; therefore X4 =
Plugging those values
and using
(1) to (4)
we
obtain Xi =
1
and X3 =
1
-
q.
in (5) yields
2-c-2q(l-c)
'^(2-q)(2-c)-2q(l-c)
Note
c
=
that
it is
when
c
=
this
always optimal
condition boils
to
use an auditor.
to r
^
When c =
1
down
we fmd
that
the effort is constant with respect to r
and
Also,
when y =
0,
using
( 1 )
to (4)
1/2
,
which
is
never
the condition
In other words,
true.
becomes
r
< -J—.
eNA=i.L:qA0
q
'
i.e.,
When Y =
1
and
P"".
MIR2 is not binding, we are in region B 1
and using
( 1)
to (4),
eRE=i.i-l9.A0
q
'
and the informational rents are extracted from
If r or P*" are high
enough
to
make MIR2
by MIC. Therefore, using MIRl we can
Equating
a
=
ei^ and ei^A gives us the line
l-2^(2-q)
^(M't
the high type while the effort level remains constant.
binding,
obtain,
a, separating the
two regions
EA and RE:
when
38
At some point above a,
it
may be
profitable to decrease y
with equality. Also, using (1) to (4)
1
<
If
.
y
<
1
,
then X4 =
and
(5) holds
obtain:
Ae(l.r)c
cp_.
'
(2r-l)(2-c)
The region where
<y<
1
is
called
obtained by solving ei^'^ = ej^^
(3
we
below
=
2r-l
A6
AG
^Q
'
A2-C
CP
(constant punishment) and the border between
.
( 1 - r)
c
J
2
(2r-l)(2-c)
Therefore, above P, the effort of the low productivity type (ei^^)
the
EA and CP is
kept constant below
is
1
however big
punishment becomes.
Proof of P roposition 2
For
this
proof we will use the
results derived in
precisely the choices of the principal
or deter (proposition
1)
when
and
2.1
2-q
-^
^ of lemma 2.
the incentive
1
1
and proposition
memager and a strategic
Comparing
these profits
Derivation of $': The optimal contract of proposition
The value of r =
2.
These two contracts are
.
was found
scheme of lemma
to
1
auditor.
2.1
For each alternative
we will characterize
the four
main
'j
z
-
r
.
q
So
in the region
We call the line
dominates the scheme NA.
2.1)
we wiU
regions.
NA (no auditor) when r <
uses the scheme
be always smaller than
(lemma
Z,
±
2-q
between
^
of lemma
V.
Remark: At
this point,
we have also proved
the existence
of a region of the parameter space where
optimal for the principal to allow collusion. Namely, whenever
Pi",
1
facing an auditor of unknown type: he can allow
collusion between the
derive the profits for the principal.
lemma
r
£
(^'
,
2
.
it
is
g) regardless of the value of
allowing collusion dominates both other schemes (no auditor and deterring collusion.)
39
Derivadon of 5^:
than the border between regions
^
a(D) =
border between regions
First note that the
a(A) and c €
RE
and
EA when
RE and EA when deterring collusion is lower
allowing collusion;
i.e.,
(0,1).
We can then compare the profits of schemes B 1(A) and B 1(D) below a(D) and
equates
B 1(A) and B1(D). When r <
when r >
When
r
^^
<
it
will
be optimal
which implies
that
/T>w^
t
and
^1j
?^
-
2r +
2q (
1 -
c)
+c
.J^
.
/
^
q(2
=
will be optimal to allow collusion (using
to deter collusion (using
{c[q(3r- 2)
when r^
7r(A)^7r(D)
r
jt
scheme
B
= a(D),
-
3c) + 2c
we saw
l]
+ 2q(l
scheme B 1(A)) and
(D)).
1
that 7r(A)
is
"deterring"
scheme has already extracted
better since
it
-
r)}
and vice versa.
"allowing"
= 7r(D). If a(D) <
pm < a(A)
r
makes
and
r
= ^^ the scheme
continues to extract rent at the same constant rate as before while the
all the rent
and
is
restoring effort at a decreasing rate.
border between "allowing collusion" and "deterring collusion" i^^) has
increasing
which
2-q
^
For
^h
find the locus ^^
—L and pm < a(D),
7r(A)-7r(D)=
/ * N
a(D) ^ a(A) since
deterring relatively
more
profitable because
it
to
So
the
head north-east. Indeed,
uses the whole punishment P^".
55143
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