The Economic Theory of Agency: The Principal's Problem

Full text

The Economic Theory of Agency:
The Principal's Problem
By STEPHEN A. Ross*
The relationship of agency is one of the
oldest and commonest codified modes of
social interaction. We will say that an
agency relationship has arisen between two
(or more) parties when one, designated as
the agent, acts for, on behalf of, or as rep-
resentative for the other, designated the
principal, in a particular domain of deci-
sion problems. Examples of agency are
universal. Essentially all contractural ar-
rangements, as between employer and
employee or the state and the governed,
for example, contain important elements
of agency. In addition, without explicitly
studying the agency relationship, much of
the economic literature on problems of
moral hazard (see K. J. Arrow) is con-
cerned with problems raised by agency. In
a general equilibrium context the study of
information flows (see J. Marschak and
R. Radner) or of financial intermediaries
in monetary models is also an example of
agency theory.
The canonical agency problem can be
posed as follows. Assume that both the
agent and the principal possess state in-
dependent von Neumann-Morgenstern
utility functions, G(-) and [/(•) respec-
tively, and that they act so as to maximize
their expected utility. The problems of
agency are really most interesting when
seen as involving choice under uncertainty
and this is the view we will adopt. The
agent may choose an act, a^A, a feasible
action space, and the random payoff from
* Associate professor of economics, University of
Pennsylvania. This work was supported by grants from
the Rodney L. White Center for Financial Research at
the University of Pennsylvania and from the National
Science Foundation.
this act, wia,
6),
will depend on the random
state of nature dit9, the state space set),
unknown to the agent when a is chosen.
By assumption the agent and the prin-
cipal have agreed upon a fee schedule / to
be paid to the agent for his services. The
fee,
/, is generally a function of both the
state of the world,
6,
and the action, a, but
we will assume that the action can influ-
ence the parties and, hence, the fee only
through its impact on the
payoff.
This
permits us to write.
(1)f=fiwia,B);B).
Two points deserve mention. Obviously
the choice of a fee schedule is the outcome
of a bargaining problem or, in large games,
of a market process. Much of what we
have to say is relevant for this view but
we will not treat the bargaining problem
explicitly. Second, while it is possible to
conceive of the fee as being directly func-
tionally dependent on the act, the theory
loses much of its interest, since without
further conditions, such a fee can always
be chosen as a Dirac 8-function forcing a
particular act (see S. Ross). In some sense,
then, we are assuming that only the payoff
is operational and we will take this point
up below. Now, the agent will choose an
act, a, so as to
(2)max
where the agent takes the expectation
over his subjectively held probability dis-
tribution. The solution to the agent's
problem involves the choice of an optimal
act, a", conditional on the particular fee
schedule, i.e., a''=ai{f}), where a(-) is a
134

VOL.
63 NO. 2
DECISION MAKING UNDER UNCERTAINTY 135
mapping from
the
space
of fee
schedules
into
A.
If
the
principal
has
complete informa-
tion about
the fee to act
mapping,
a(
he will now choose
a fee
so
as to
(3)
max E{ U[wiai{f}),
6)
where
the
expectation
is
taken over
the
principal's subjective probability distribu-
tion over states
of
nature.
If the
principal
is
not
fully informed about a(
•),
then a( •)
will
be a
random function from
his
point
of view. Formally,
at
least,
by
appropri-
ately augmenting
the
state space
the
criterion
(3)
could still
be
made
to
apply.
In general some side constraints
on (/)
would also have
to be
imposed
to
insure
that
the
problem possesses
a
solution
(see
Ross).
A
market-imposed minimum
ex-
pected
fee or
expected utility
of
fee
by the
agent would
be one
economically sensible
constraint:
(4)
E{G[fiwia,d);e)]}>k.
e
Since utility functions are assumed
to be
independent
of
states,
6, one of the im-
portant reasons
for a fee to
depend
di-
rectly on
d
would be
if
individual subjective
probability distributions differed.
In
what
follows we will assume that both the agent
and the principal share the same subjective
beliefs about
the
occurrence
of
6
and write
the
fee as a
function
of the
payoff only.
(5)/
=
fiwia, 6)).
Notice that this interpretation would
not
in
general
be
permissible
if the
prin-
cipal lacked perfect knowledge
of a(-)-
More importantly, though, surely aside
from simple comparative advantage,
for
some questions
the
raison d'etre
for an
agency relationship
is
that
the
agent
(or
the principal) may possess different (better
or finer) information about
the
states
of
the world than the principal (agent).
If
we
abstract from this possibility
we
will have
to show that we
are not
throwing
out the
baby with
the
bath water.
Under this assumption
the
problem
is
considerably simplified
but
much
of
inter-
est does remain. Suppose, first, that we are
simply interested
in the
properties
of
Pareto-efficient arrangements that
the
agent
and the
principal will strike. Notice
that the optimal fee schedule as seen by the
principal
is
found
by
solving
(3) and is
dependent
on the
desire
to
motivate
the
agent.
In
general, then,
we
would expect
such
an
arrangement
to be
Pareto-in-
efficient,
but we
will return
to
this point
below.
The
family
of
Pareto-efficient
fee
schedules
can be
characterized
by
assum-
ing that
the
principal
and the
agent
co-
operate
to
choose
a
schedule that maxi-
mizes
a
weighted
sum of
utilities
(6) max E{U[W
- f] +
\G[f]},
where
X
is a
relative weighting factor
(and
where strategies have been randomized
to
insure convexity).
K.
Borch recognized
that
the
solution
to (6) is
obtained
by
maximizing
the
function internal
to the
expectation which requires setting
(P.E.) U'[w
- f] =
\G'[f]
when
U
and G are monotone and concave.
(See
H.
Raiffa
for a
good exposition.)
The
P.E. condition defines
the fee
schedule,
/(•),
as a
function
of
the payoff w (and
the
weight, X). (See
R.
Wilson (1968)
or
Ross
for
a
fuller discussion
of
this derivation
and
the
functional aspect
of the fee
schedule.)
An alternative approach
to
finding
op-
timal
fee
schedules
was
first proposed
by
Wilson
in the
theory
of
syndicates
and
studied
by
Wilson (1968, 1969)
and
Ross.
This
is the
similarity condition that solves
for
the fee
schedule
by
setting

136 AMERICAN ECONOMIC ASSOCIATION MAY
1973
U[w-f]
= a
(S)
for constantsa>O,b.
If (/)
satisfies5 then,
given
the fee
schedule,
it
should
be
clear
that
the
agent
and the
principal have
identical attitudes towards risky payoffs
and, consequently,
the
agent will always
choose
the act
that
the
principal most
desires. Ross
was
able
to
completely char-
acterize
the
class
of
utility functions that
satisfied both
P.E. and 5 (for a
range
of
X)
and show that
in
such situations
the fee
schedule
is
(affine) linear,
L, in the payoff.
(The class
is
simply that
of
pairs
(C/, G)
with linear risk tolerance.
=maxE{u[w-f]
if)
e
U' G'
-—
=
cw-\-d
and - — =
cw -I-
e,
where
c, d and e are
constants.)
In
fact,
it
can be
shown that
any two of S, P.E.,
or
L
imply
the
third.
A question
of
interest that naturally
arises
is
that
of the
relation that
S and
P.E. bear
to the
exact solution
to the
prin-
cipal's problem.
(A
comparable "agent's
problem"
can
also
be
posed
but we
will
not
be
concerned with that here. Some
ob-
servations
on
such
a
problem
are
contained
in Ross.)
The
solution
to the
principal's
problem
(3)
subject
to the
constraint
(4)
and
to the
constraint imposed
by the
condition that
the
agent chooses
the op-
timal
act
from
his
problem
(2) can,
under
some circumstances,
be
posed
as a
classical
variational problem.
To do so we
will
assume that
the
payoff function
is
(twice)
differentiable
and
that
the
agent chooses
an optimal
act,
given
a fee
schedule,
by the
first order condition
(7) E{G'[fiw)]f'iw)wa}
= 0,
e
where
a
subscript indicates partial differ-
entiation.
The
principal's problem
is now
to
ma.x
(8)
(/>
a
-f- \G}
where
^ and X are
Lagrange multipliers
associated with
the
constraints
(7) and
(4) respectively. Changing variables
to
ViO)
= fiwia,
0))
where we have suppressed
the impact
of a on F and
assuming, with-
out loss
of
generality, that
6 is
uniformly
distributed
on [0, 1]
permits
us to
solve
(8)
by the
Euler-Lagrange equation. Thus,
at
an
optimum
(9)
d
(dH
de
\dV
1\
dH
']
~'dV
d fWal
—
— - XG' = 0;
de
iJ
or
the
marginal rate
of
substitution,
U' d
G'
de
(10)
This is an intuitively appealing result;
the marginal rate of substitution is set
equal to a constant as in the P.E. condi-
tion plus an additional term which cap-
tures the constraint (7) imposed on the
principal by the need to motivate the
agent. To determine the optimal act, a,
we differentiate (8) with respect to a
which yields
£{f/'[l-/'K-f
^G"(/'w»)^
(11) e
-f ^G'/"(ro<.)2
+
<ifG'f'Waa}
= 0,
where
we
have made
use of (7).
Substitut-
ing
the
boundary conditions permits
us to
solve
for the
multipliers
^ and
X.
LikeSor P.E. (10) defines the
fee
schedule
as
a
function
of w.
(Notice that
we are
tacitly assuming that,
at
least
for the
optimal
act, the
payoff
is (a.e.
locally)
state invertible. This allows
the fee to
take
the
form
of (5).) It
follows that
(10)
will coincide with
P.E. if and
only
if ^ is
zero,
or
ii'^y^O,
we
must have

VOL.
63 NO.
2
DECISION MAKING UNDER UNCERTAINTY 137
(12)
d
r
dd Lwe
\=
J
bia).
a function
of a
alone.
In particular, using these conditions we
can
ask
what class
of
(pairs
of)
utility
functions
(U, G) has the
property that,
for any payoff structure, wia, 6),
the
solu-
tion
to the
principal's problem
is
Pareto-
efficient. Conversely, we can ask what class
of payoff structures has the property that
the principal's problem yields
a
Pareto-
efficient solution
for any
pair
of
utility
functions {U,
G).
A little reffection reveals that
the
only
pairs
of
{U, G) that could possibly belong
to
the
first class must
be
those which
satisfy
S
and P.E.
for
a range
of
schedules
(indexed
by
the
X
weight
in
P.E.). Clearly
if
(10) is to be
equivalent
to
P.E.
for all
payoff functions,
w
ia, d), then
^
must
be
zero
and the
motivational constraint
(7)
must
not be
binding.
Eor
this
to be the
case,
for
an interval
of
values
of k (in
(4)),
the satisfaction
of
P.E. must imply that
the agent chooses
the
principal's most
desired
act by (7). For any fee
schedule,
(/),
the
principal wants
the act to be
chosen
to
maximize Ee{U[w—f]] which
implies that
(13)
E{U'i\ -/>„}
= 0.
If (13) is to be equivalent to the motiva-
tional constraint (7)
for
all possible payoff
structures, then we must have
(14)
- /')
=
G'f
which, with
P.E. (or (10)
with
^
= 0)
yields
a
linear
fee
schedule
in the
payoff.
But,
as
shown
in
Ross, linearity
of the
fee schedule
and
P.E. imply
the
satisfac-
tion
of S
and
the
{U, G) pair must belong
to
the
linear risk-tolerance class
of
utility
functions described above.
Since
the
linear risk-tolerance class,
while important,
is
very limited,
we
turn
now
to
the converse question
of
what pay-
off structures permit
a
Pareto-efficient
solution
for all
{U,
G)
pairs.
If ^
= 0
we
must, as before, have that the motivational
constraint
is not
binding
for all
(U, G)
or
(13) must always imply (7). The implica-
tion will always hold
if
there exists
an
a*
such that
for all a
there
is
some choice
of
the state domain,
/, for
which
(15)
wia*,e)>wia,e),
9
6/.
Conversely, from P.E., we must have that
foraUG(-)
(16) E{G'[f]il
-
f')w.}
=0
implies
(7)
where/
is
determined
by P.E.
Since
(f/,
G) can always be chosen so as
to
attain any desired weightings
of
Wa
in (7)
and (16) the special case
of
(15) is the only
one
for
which motivation
is
irrelevant.
Given (15) all individuals have
a
uniquely
optimal
act
irrespective
of
their attitudes
towards risk.
If ^?^0, then
to
assure Pareto efficiency
we must satisfy
(12).
This
is a
partial
differential equation
and its
solution
is
given
by
(17)
wia,
e)
=
H[eBia)
-
Cia)],
where
Hi-), Bi-) and C(-) are
arbitrary
functions. (The detailed computations are
carried
out in an
appendix.) This
is a
rich
and
interesting class
of
payoff func-
tions.
In particular, (17) is
a
generalization
of
the
class
of
functions
of the
form
lid—a),
where the object is
to
pick
an
act,
a, so
as to
best guess
the
state d.
It
there-
fore includes,
for
example, traditional
estimation problems, problems with
a
quadratic payoff function,
and all
prob-
lems with payoff functions
of the
form
d—a\^hia),
and many asymmetric ones as
well.
It is not,
however, difficult
to
find
plausible payoff functions which
do not
take
the
form
of
(17).
(The
class
of the
form (15) will generate such functions.)

138 AMERICAN ECONOMIC ASSOCIATION MAY 1973
We may conclude, then, that the class of
payoff structures that simultaneously solve
the principal's problem and lead to Pareto
efficiency for all {U, G) pairs is quite im-
portant and quite likely to arise in practice.
In general, though, it is clear that the
solution to the principal's problem will not
be Pareto-efficient. This is, however, a
somewhat naive view to take. Pareto effi-
ciency as defined above assumes that per-
fect information
is
held by the participants.
In fact, the optimal solution to the prin-
cipal's problem implied that the fee-to-act
mapping induced by the agent was com-
pletely known to the principal. In such a
case it might be thought that the principal
could simply tell the agent to perform a
particular act. The difficulty arises in
monitoring the act that the agent chooses.
Michael Spence and Richard Zeckhauser
have examined this problem in detail in
the case of insurance. In addition, if agents
are numerous the fee may be the only com-
munication mechanism. While it might in
principle be feasible to monitor the agent's
actions, it would not be economically
viable to do so.
The format of this paper has been such
as to allow us to only touch on what is
surely the most challenging aspect of
agency theory; embedding it in a general
equilibrium market context. Much is to
be learned from such attempts. One would
naturally expect a market to arise in the
services of agents. Furthermore, in some
sense, such a market serves as a surrogate
for a market in the information possessed
by agents. To the extent to which this
occurs, the study of agency in market
contexts should shed some light on the
economics of information. To mention one
more path of interest—in a world of true
uncertainty where adequate contingent
markets do not exist, the manager of the
firm is essentially an agent of the share-
holders. It can, therefore, be expected that
an understanding of the agency relation-
ship will aid our understanding of this
difficult question.
The results obtained here provide some
of the micro foundations for such studies.
We have shown that, for an interesting
class of utility functions and for a very
broad and relevant class of payoff struc-
tures,
the need to motivate agents does not
conflict with the attainment of Pareto
efficiency. At the least, a callous observer
might view these results as providing some
solace to those engaged in econometric
activity.
APPENDIX
This appendix solves the partial differen-
tial equation (12) in the text.
Integrating (12) over e yields
dw . ^ dw
+
[hia)e
-I- cia)] = 0.
da de
Along a locus of constant w,
de dw/da
da dw/de = bia)e + cia).
is a first order Bernoulli equation that inte-
grates to
4.
J
where ;fe is a constant of integration. It fol-
lows that
where
and
wia,e) = H[eBia) - Cia)],
Bia) =
Cia)^Je-J ia)
-\-
k.
and Hi-) is an arbitrary function.

VOL.
63 NO. 2 DECISION MAKING UNDER UNCERTAINTY 139
REFERENCES
K. J. Arrow, Essays in the Theory of
Risk-
Bearing, Chicago 1970.
K. Borch, "Equilibrium in a Reinsurance Mar-
ket," Econometrica, July 1962, 30, 424-444.
J. Marschak and R. Radner, Tke Economic
Theory of Teams, New Haven and London
1972.
H. Raiffa, Decision Analysis; Introductory
Lectures on Choices Under Uncertainty,
Reading, Mass. 1968.
S. Ross, "On the Economic Theory of Agency:
The Principle of Similarity," Proceedings of
the NBER-NSF Conference on Decision
Making and Uncertainty, forthcoming.
M. Spence and R. Zeckhauser, "Insurance, In-
formation and Individual Action," Amer.
Econ.
Rev. Proc, May 1971, 61, 380-387.
R. Wilson, "On the Theory of Syndicates,"
Econometrica, Jan. 1968, 36, 119-132.
, "The Structure of Incentives for De-
centralization Under Uncertainty," La De-
cision, Editions Du Centre National De Le
Recherche Scientifique, Paris 1969.