Moral Hazard and Observability

TL;DR: In this article, the role of imperfect information in a principal-agent relationship subject to moral hazard is considered, and a necessary and sufficient condition for imperfect information to improve on contracts based on the payoff alone is derived.

Abstract: The role of imperfect information in a principal-agent relationship subject to moral hazard is considered. A necessary and sufficient condition for imperfect information to improve on contracts based on the payoff alone is derived, and a characterization of the optimal use of such information is given.

Summary

Two immediate corollaries follow:

• Under the assumption of Proposition 1, the second-best solution is strictly inferior to a first-best solution.
• Whenever f, exists, Corollary 2 indicates that a first-best solution cannot be achieved.
• The characterization in (7) has an intuitive interpretation in terms of deviating from optimal risk sharing to provide incentives for increased effort on the part of the agent.
• There are positive gains to observing the agent's action, since in that case a first-best solution can be achieved by using a forcing contract.
• Examples for which sharing rules are concave or linear or even two-peaked can be easily generated as well.

HOLMSTROM / 81

• This assumption says that the probability of an accident decreases with a so that each outcome x < 0 is less likely.
• Driving a car more carefully will presumably decrease the probability of both small and large accidents.
• This is the case if, for instance, falf is increasing in x (which holds for surprisingly many standard distributions; see Holmstrom (1977) ).
• To summarize the discussion the authors have: Proposition 2.
• Given the assumptions in (11), optimal accident insurance policies entail a deductible.

4. Optimal sharing rules based on additional information

• The interesting comparison is between s(x,0) and s(x, 1).
• Confirming their intuition, the repairman receives higher pay if it is found that the failure was outside his control than if it is found that a component that he controls failed.
• The optimal solution when y is not observed will lie initially between s(x,0) and s(x,1) and eventually go above s(x,1), since /i > /2. Notice that as k -> oo, s(x) -> s(x,0), since it becomes all the less likely that the failure will be caused by anything outside the repairman's control.

5. Value of information

• This signal is a conditional information system, where resources are invested to find out y only if the outcome is sufficiently bad (below x).
• It is readily seen that y is also informative and, depending on the costs of obtaining y, the net benefits of using y may exceed those ofy.
• The last two examples bring attention to the fact that Proposition 3 says nothing about how valuable y is, which would be important whenever costs for information acquisition and administration of more complex contracts are considered.
• An upper bound for the value is, of course, provided by the value one gets from observing a itself.
• Some indications of the value of the signal can be found by studying (13).

6. Asymmetric information

• For the sufficiency part of the proposition, an additional but insignificant qualification is needed.
• Yet, when integrating as in (25), it is conceivable that the right-hand side of (25) would become independent of y, making a function s(x) optimal and y valueless.
• This is extremely unlikely and will not happen generically; any small change in the problem data would take us out of such a situation.
• Thus, the authors can safely say that for all that matters, Proposition 3 is also valid in the asymmetric case.

7. Concluding remarks

• Of course, the analysis presented here leaves unanswered many interesting questions in contracting.
• One important aspect of the problem, which the authors have not considered, is that many contracts are based on long-term relationships.
• When the same situation repeats itself over time, the effects of uncertainty tend to be reduced and dysfunctional behavior is more accurately revealed, thus alleviating the problem of moral hazard.
• Another extension would recognize that asymmetry of information as discussed in Section 6 may warrant a renegotiation of the contract.
• One can view management by objectives and the New Soviet Incentive Scheme (Weitzman, 1976) as examples of this.

Figures

FIGURE 2

FIGURE 1

Full text

The RAND Corporation
Moral Hazard and Observability
Author(s): Bengt Holmstrom
Source:
The Bell Journal of Economics,
Vol. 10, No. 1 (Spring, 1979), pp. 74-91
Published by: The RAND Corporation
Stable URL: http://www.jstor.org/stable/3003320
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Moral
hazard and
observability
Bengt
Holmstrom
Swedish School
of Economics and Business Administration
The role
of imperfect
information
in a
principal-agent relationship subject
to
moral
hazard
is considered.
A
necessary
and
sufficient
condition
for imperfect
information
to
improve
on
contracts based
on the
payoff
alone is
derived,
and
a
characterization
of
the
optimal
use
of
such
information
is
given.
1.
Introduction
U
It
has
long
been
recognized
that a
problem
of moral hazard
may
arise when
individuals
engage
in
risk
sharing
under conditions
such that
their
privately
taken
actions
affect the
probability
distribution
of
the
outcome.1 This situation
is common
in
insurance,
labor
contracting,
and
the
delegation
of
decisionmaking
responsibility,
to
give
a
few
examples.
In
these instances
Pareto-optimal
risk
sharing
is
generally
precluded,
because
it will
not
induce
proper
incentives for
taking
correct actions.
Instead,
only
a second-best
solution,
which trades
off
some
of
the
risk-sharing
benefits for
provision
of
incentives,
can
be achieved.
The source of this moral
hazard
or incentive
problem
is
an
asymmetry
of
information
among
individuals
that
results because individual
actions cannot
be observed
and
hence contracted
upon.
A
natural
remedy
to
the
problem
is
to invest resources
into
monitoring
of actions and use
this information
in
the
contract.
In
simple
situations
complete
monitoring
may
be
possible,
in
which
case
a
first-best solution
(entailing
optimal
risk
sharing)
can be achieved
by
employing
a
forcing
contract that
penalizes
dysfunctional
behavior.
Generally,
however,
full
observation of actions
is
either
impossible
or
prohibitively
costly.
In
such situations interest centers around the
use of
imperfect
estimators
of actions
in
contracting.
Casual observation indicates that
imperfect
informa-
tion
is
extensively
used
in
practice
to alleviate
moral
hazard,
for instance
in
the
supervision
of
employees
or
in
various forms
of
managerial accounting.
A
natural
question
then arises:
when
can
imperfect
information about
actions
be
used
to
improve
on
a
contract which
initially
is based
on the
payoff
alone?
Secondly,
how should
such
additional
information be
used
optimally?
This
paper
is
partly
based on
Chapter
4
of the author's
unpublished
dissertation,
"On
Incentives
and Control in
Organizations,"
submitted
to Stanford
University,
December 1977. It was
written
while the
author was
visiting
the
Center for
Operations
Research and
Econometrics,
Universit6
Catholique
de
Louvain,
Belgium.
An earlier version was
presented
at
the
European
Meeting
of the
Econometric
Society
in
Geneva,
1978. I am
much
indebted to Joel
Demski,
Fr0ystein Gjesdal,
Charles
Holloway,
David
Kreps,
and
Robert
Wilson
for
many helpful
discussions
and
to David
Baron
and Gerald Kramer
for detailed
comments
on
an
earlier
manuscript.
1
See
for
instance Arrow
(1970),
Zeckhauser
(1970),
Pauly
(1974),
and
Spence
and
Zeckhauser
(1971).
74

HOLMSTROM
/
75
A
recent
interesting
paper by
Harris
and
Raviv
(1976)
addresses
these
questions
in
the context
of
a
principal-agent
relationship
in
which the
agent provides
a
productive input
(e.g.,
effort)
that cannot
be observed
by
the
principal
directly.2
Their results relate to
a
very
specific
kind
of
imperfect monitoring
of
the
agent's
action.
They
study
monitors
which
provide
information that
is
independent
of
the state
of
nature and allows
the
principal
to
detect
any
shirking by
the
agent
with
positive probability.
Such monitors are of
limited
interest, however,
since
they
are
essentially equivalent
to
observing
the
agent's
action
directly,
because
a
first-best
solution can
be
approximated arbitrarily
closely
in
this case.3
Clearly,
one cannot
expect
imperfect monitoring
to
possess
such
strong
characteristics
in
general.
Employing
a
different
problem
formulation from Harris
and
Raviv's,
we
are able to
simplify
their
analysis
and
generalize
their
results
substantially.
Both
questions posed
above are
given
complete
answers
(in
our
particular
model).
It
is
shown that
any
additional information about the
agent's
action,
however
imperfect,
can be
used to
improve
the
welfare
of both the
principal
and
the
agent.
This
result,
which formalizes earlier references to
the
value of
monitoring
in
agency
relationships
(Stiglitz,
1975;
Williamson,
1975),
serves
to
explain
the extensive use of
imperfect
information
in
contracting.
Further-
more,
we characterize
optimal
contracts based on such
imperfect
information
in
a
way
which
yields
considerable
insight
into the
complex
structure of actual
contracts.
The formulation
we
use is an extension of that introduced
by
Mirrlees
(1974, 1976).
We start
by
presenting
a
slightly
modified
version
of Mirrlees'
model
(Section
2),
along
with
some
improved
statements about the
nature of
optimal
contracts when the
payoff
alone
is
observed.
In Section 3 a
detour is
made to
show how these
results
can be
applied
to
prove
the
optimality
of
deductibles
in
accident
insurance
when moral hazard is
present.
Section
4
gives
the
characterization
of the
optimal
use of
imperfect
information and
Section
5
presents
the result
when
imperfect
information is valuable.
Up
to this
point
homogeneous
beliefs are
assumed,
but
in
Section
6
this
assumption
is
relaxed
to
the
extent
that
we allow the
agent
to be more informed
at
the time he chooses his
action.
The
analysis
is
brief,
but indicates
that
qualitatively
the same results
obtain as for the case
with
homogeneous
beliefs.
Section
7
contains
a
summary
and
points
out some directions for
further
research.
2.
Optimal
sharing
rules
when the
payoff
alone
is
observed
*
We
study
a
principal-agent
relationship,
where
the
agent
privately
takes
an
action
a
E A
C
R,
A
being
the set of
all
possible
actions,
and a
together
with
a
random
state
of
nature
0,
determines
a
monetary
outcome
or
payoffx
=
x(a, 0).
The
problem
is
to determine
how
this
payoff
should
be shared
optimally
between
the
principal
and the
agent.
The
principal's
utility
function
is
G(w),
defined over
wealth
alone,
and
the
agent's
utility
function
is
H(w,a),
defined
over
wealth
2
The
main
results of Harris
and Raviv
(1976)
are
reported
in
their
1978
paper.
For
earlier
work
on
principal-agent
models,
see
Wilson
(1969),
Ross
(1973),
and
Mirrlees
(1976).
3
This
fact,
which is
not
observed
by
Harris and
Raviv
(1976),
can be
verified
by
using
an
argument
similar to the
one
given by
Mirrlees
(1974,
p.
249),
or
by Gjesdal (1976) (cf.
example
in
footnote
7).
Obviously,
it
implies
that
monitoring,
which
satisfies
Harris and
Raviv's
conditions,
is
valuable.
This is
their
partial
answer to
the first
question
raised
above.

76
/ THE BELL JOURNAL
OF
ECONOMICS
and
action.
The model is
further
restricted
by assuming
that
H(w,a)
=
U(w)
-
V(a),
with V'
>
0
and Xa
>
0.4 The
interpretation
is that
a
is
a
productive
input
with direct
disutility
for the
agent
and
this
creates
an inherent difference
in
objectives
between the
principal
and the
agent.
It is
convenient to
think
of
a
as
effort
and this
term
will be used
interchangeably
with action. Since the
problem
of moral
hazard
can
be
avoided when the
agent
is risk-neutral
(Harris
and
Raviv,
1976),
we
shall assume U"
<
0. The
principal
may
or
may
not be risk-
neutral,
i.e.,
G"
<
0.
In this
section,
we consider
the
case
where
the
principal
observes
only
the outcome
x.
Thus,
sharing
rules have
to be functions
of
x
alone.
Let
s(x)
denote the
share of
x
that
goes
to the
agent
and
r(x)
=
x
-
s(x)
denote
the
share
that
goes
to the
principal.
It
is assumed
that
both
parties
agree
on the
probability
distribution of 0
and
that
the
agent
chooses
a
before
0
is
known.5
In
this
case
(constrained)
Pareto-optimal
sharing
rules
s(x)
are
generated
by
the
program:
max
E{
G(x
-
s(x))}
(1)
s(x),a
subject
to
E{
H(s(x),a)}
-
Hf, (2)
a
E
argmax E{H(s(x),a')},
(3)
a'eA
where the notation
"argmax"
denotes
the set of
arguments
that maximize the
objective
function
that
follows.6
Constraint
(2) guarantees
the
agent
a
minimum
expected
utility
(attained
via
a market
or
negotiation
process).
Constraint
(3)
reflects
the restriction
that
the
principal
can observe
x but
not
a.
If
he also
could
observe
a,
a
forcing
contract
could
be
used to
guarantee
that the
agent
selects
a
proper
action even
when
s(x)
is
chosen
to solve
(1)-(2)
ignoring
(3).
The latter we
will refer to
as
thefirst-
best
solution,
which
entails
optimal
risk
sharing.
It differs
in
general
from
the
solution
of
(1)
subject
to
(2)
and
(3),
which
we call a second-best
solution.
Two
approaches
can be used
to solve
the
program
above.
The
earlier
one,
used
by
Spence
and Zeckhauser
(1971),
Ross
(1973),
and Harris
and
Raviv
(1976),
recognizes explicitly
the
dependence
of
x on a and
0,
so that
the
expectations
in
(1)-(3)
are
taken
with
respect
to
the distribution
of
0.
They
proceed
to characterize
an
optimal
solution
by
replacing
(3)
with
the
first-order
constraint
E{H
s'
sXa
+
H2}
=
0,
and
then
apply
the calculus
of
variations.
To
validate these
steps
one has
to assume
that
an
optimum
exists
and
is
differentiable.
However,
as
an
example
by
Mirrlees
(1974)
shows,
there
may
commonly
exist
no
optimal
solution
among
the class
of unbounded
sharing
rules,
and
for
this
reason
s(x)
has
to be restricted
to
a
finite
interval in
general.
As a
result,
the solution
will
become
nondifferentiable
and the above-mentioned
approach
can
no
longer
be
applied.7
4
Subscripts
denote
partial
derivatives
with
respect
to
corresponding
variables.
5
This
assumption
corresponds
to
model
1
in Harris
and
Raviv
(1976),
which is
the model
they
use for
studying
imperfect
information.
We
shall
relax
it
in
Section
6.
6
As
usual,
E
denotes the
expectation
operator.
Since
E{H(s(x),a)}
need
not be
concave
in
a,
there
may
exist
multiple
solutions,
hence the
inclusion
symbol.
7
Even
when
an
optimal
solution
exists
among
unbounded
sharing
rules,
it
may
be
nondiffer-
entiable. This has
been
observed
by
Gijesdal
(1976).
To
illustrate his
ideas one can
look
at
the
follow-

HOLMSTROM
/
77
A
better
approach
to
solving
(1)-(3),
which
also
gives
a more
intuitive
characterization of
an
optimum,
has been introduced
by
Mirrlees
(1974,
1976).
He
suppresses
0
and
views
x as
a
random
variable
with a
distribution
F(x,a),
parameterized by
the
agent's
action. Given
a
distribution of
0,
F(x, a)
is
simply
the distribution induced on
x via the
relationship
x
=
x(a, 0).8
It is
easy
to see
that
xa
>
0
implies
Fa(x, a)
<
O.
It will be assumed
that for
every
a,
Fa(x, a)
<
0
for some
x-values,
so
that a
change
in
a has
a
nontrivial effect
on the distribu-
tion of
x.
In
particular,
it
will
shift the distribution of x to the
right
in
the sense
of first-order stochastic dominance.
For
the
moment,
assume
F has
a
density
function
f(x,a)
with
fa
and
fa
well
defined
for
all
(x,a).9 Replacing
(3)
with a first-order
constraint
yields
the
program:
s(x)E[c,d+x],a
G(x
-
s(x))f(x,a)dx (4)
subject
to
I
[U(s(x))
-
V(a)]f(x,a)dx
H, (5)
U(s(x))fa(x,a)dx
=
V'(a).
(6)
Note
that
s(x)
is
restricted
to lie
in the interval
[c,
d +
x]
to avoid nonexistence
of
a
solution.10
This restriction
is
natural from
a
pragmatic point
of
view as
well,
since
the
agent's
wealth
puts
a
lower
bound,
and the
principal's
wealth
(augmented
with
x)
an
upper
bound on
s(x).
Let
X
be
the
multiplier
for
(5)
and
tx
the
multiplier
for
(6).
Pointwise
optimization
of
the
Lagrangian yields
the
following
characterization
of
an
optimal
sharing
rule:
G'(x
-
s(x)) fa(x,a)
(
=
h
+
ux
,
(7)
U'(s(x))
f(x,a)
ing insightful example.
Let
x(a,z)
=
a
+
z
and
z
-
Unif(0,1),
so that
x
-
Unif(a,a
+
1).
If
(a*,s*(x))
is a
first-best
solution
it
is
easy
to see that a contract of the form
s(x)
=
s*(x)
when
x
>
a*,
s(x)
=
w
otherwise,
will make the
agent
choose
a
=
a*
for
w
sufficiently
low. But in that
case
x 2 a*
for
all
outcomes of
2,
and the
first-best solution
s(x)
=
s*(x)
is
effectively
realized.
In other
words,
a
nondifferentiable
sharing
rule,
which
penalizes
the
agent
for outcomes
x
<
a*,
will
give
both the
principal
and the
agent
the same
expected utility
as
a
first-best
solution. In this
example
no
optimal
differentiable
sharing
rule exists
for
(1)-(3).
Gjesdal's
analysis
shows that both
Spence
and
Zeckhauser
(1971,
p.
383,
footnote
5)
and
Harris
and Raviv
(1976,
pp.
36-37)
err in
giving
incorrect
characterizations
(based
on the Euler
equation)
for
examples
similar to this.
We
will
avoid situations like these
by essentially
assuming
that the
support
of the distribution
of
x
will
not
change
with
a,
as
explained
below. For a more detailed
comparison
of the
state-space
approach
with
Mirrlees'
approach,
see Holmstr6m
(1977).
8
Thus,
it
is
always possible
to
go
from the state
space
approach
to Mirrlees'
approach,
while
the reverse is not
always
true.
9
In
Section
3
we
shall allow discrete distributions as well. The crucial
assumption
is
thatfa
exists. Note that
this
assumption
is not satisfied
by
the
example
in footnote
7.
10
More
precisely,
existence
of a solution to
(1)-(3)
can
be
proved
for the class of functions:
SK
{s(x)
E
[c,d
+
x]
Vb\'(s)
<
K
(b'
-b)},
where
Vb'
(s)
is
the total variation of
s
in
the
interval
[b,b']
(Kolmogorov
and
Fomin,
1970),
under some technical
assumptions
about
integrability
and the
behavioral
assumption
that the
agent,