The Economic Theory of Agency: The Principal's Problem
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 U(.) 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, aCA, a feasible
action space, and the random payoff from
this act, w(a, 0), will depend on the random
state of nature O(EQ 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 f to
be paid to the agent for his services. T he
fee, f, is generally a function of both the
state of the world, 0, 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. T his
permits us to write,
(1)
f = f(w(a,6);6).
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 E{G[f(w(a, 0); 0)]},
a
0
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, ao, conditional on the particular fee
schedule, i.e., ao=a((f)),
where a(.) is a
* 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.
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((f)),
he will now choose a fee so as to
max El U[wv(a((f)), 0)
(3
(f) e
(3)
- f(w(a((f)), 0); 0)]
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(X)
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 (f)
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 IG[f (w(al 0);0]
> k.
0
Since utility functions are assumed to be
independent of states, 0, one of the im-
portant reasons for a fee to depend di-
rectly on 0 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 0 and write
the fee as a function of the payoff only,
(5)
f = f(wv(a, 0)).
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 dY'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 El U[wv-f] + XG[f]},
(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] = XG'[f]
when U and G are monotone and concave.
(See H. Raiffa for a good exposition.) The
P.E. condition defines the fee schedule,
f -), as a function of the payoff' w (and the
weight, A). (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
(S)
U[w-f] = aG[f] + b
for constants a > O, b. If (f) satisfies S 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 S (for a range of A)
and show that in such situations the fee
schedule is (affine) linear, L, in the payoff.
(The class is simply that of pairs (U, G)
with linear risk tolerance,
U'
G'
= czv + d and -
=cw + e,
U"f
G"/
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' 1f(v) ]f'(7v)w,} = Oy
where a subscript indicates partial differ-
entiation. The principal's problem is now
to
max E{H} -max El U[w-f]
(8)
(f) 0
(f) 0
+ TG'f'wa + XG}
where T and X are Lagrange multipliers
associated with the constraints (7) and
(4) respectively. Changing variables to
V(0) =f(w(a, 0)) where we have suppressed
the impact of a on V and assuming, with-
out loss of generality, that 0 is uniformly
distributed on [0, t] permits us to solve
(8) by the Euler-Lagrange equation. Thus,
at an optimum
d () AH)
AH
dobVf _av
(9)dFVa
ds Wa]
dO LZVO_
or the marginal rate of substitution,
U'
d rWZaI
(10)
-=
X
I
T_ _
.
GI
~~dO -Lwo]
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
El U'[I - f']Wa + TG
'G(f'Wa)2
(1 ) 0
+ TG'f"(w0)2 + TG'f'WaaJ =
where we have made use of (7). Substitut-
ing the boundary conditions permits us to
solve for the multipliers T and X.
Like Sor 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 T is
zero, or if TX 0, we must have
VOL. 63 NO. 2
DECISION MAKING UNDER UNCERTAINTY
137
(12)
=
]
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, w(a, 0), 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 reflection 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 (a, 0), then T must be
zero and the motivational constraint (7)
must not be binding. For 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,
(f), the principal wants the act to be
chosen to maximize E0 U[w-f]
which
implies that
(13)
E { U'(1 -f')Wa}
=
0.
If (13) is to be equivalent to the motiva-
tional constraint (7) for all possible payoff
structures, then we must have
(14)
U'(1 -f') = G'f'
which, with P.E. (or (10) with T=0O)
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 T=O we
must, as before, have that the motivational
constraint is not binding for all (U, G) or
(13) must always imply (7). Ihe implica-
tion will always hold if there exists an a*
such that for all a there is some choice of
the state domain, I, for which
(15)
w(a*, 0) > w(a, 0),
6 E I.
Conversely, from P.E., we must have that
for all G(-)
(16)
E{G'[f](I -f')Wa}
= 0
0
implies (7) where f is determined by P.E.
Since (U, 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 TX 0, then to assure Pareto efficiency
we must satisfy (12). This is a partial
differential equation and its solution is
given by
(17) wv(a, 0) = H[6B(a) -C(a)],
where H(.), B(.) and C(Q) 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
1(0-a), where the object is to pick an act,
a, so as to best guess the state 0. 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
I 0-a I th(a), 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 0 yields
'aw
'07
--+
[b(a)O + c(a)]
= 0.
Along a locus of constant w,
dO
Ow/oa
- - -
______
= b(a)0 + c(a),
da
&zv/&O
is a first order Bernoulli equation that inte-
grates to
6=efb(a)
[
ef efb(a)c(a) + k
where k is a constant of integration. It fol-
lows that
w(a, 0) = H[OB(a) - C(ajj,
where
B(a) = e-Jb(a)
and
C(a) --f efb(a)c(a) + k,
and H(.) 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, The 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.
