# 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. **