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Principal-Agent Analysis inContinuous-Time
Yuqing Zhou
First Draft, March 10, 2006Second Revision, November 3, 2006
Abstract
The principal-agent problems in continuous-time with general utilities are
analyzed. We show that, when the agents utility function is separable over
income and action, the principal-agent problems can be converted to standard
dynamic optimization ones over a space of controlled processes, which again
can be further reduced to solving static optimization problems over the space
of probability measures via the martingale approach. The optimal contract
is explicitly characterized and is shown to be a nonlinear function of some
linear aggregates when the underlying cost function of probability measure is
separable. Comparative statics analysis is performed and various applications
are given in specic situations. In terms of model tractability, the analysis of
the paper can be best understood as the nonlinear analogue of Holmstrom and
Milgrom (1987).
Keywords: Contract design, dynamic control, martingale approach, moral
hazards, principal-agent problems.
Yuqing Zhou is Associate Professor, Department of Finance, The Chinese University of HongKong (e-mail: [email protected]). I am grateful to Jim Mirrlees for his constant en-couragement in past years. I thank Tao Li and many other colleagues here at CUHK for helpfulcomments. The author alone is solely responsible for any remaining errors in the paper.
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1. Introduction
There has been a renewed interest in principal-agent analysis in economic literature
in recent years. This is particularly true when the relevant issues are addressed in
continuous-time frameworks. The principal-agent problems, formalized from a class of
nonmarket arrangements in the presence of moral hazards, have been widely studied
in earlier time (see, for example, Grossman and Hart (1983), Harris and Raviv (1979),
Holmstrom (1979), Mirrlees (1974, 1975), Shavell (1979) among others). However, the
earlier principal-agent analysis suers from a major drawback in its poor predictive
power. Specically, it fails to oer a satisfactory answer to the general properties of
the optimal solutions even in some very simple situations. The shape of the optimal
solutions is basically arbitrary; we need to place some strong restrictions on the
models in order to obtain a basic property such as monotonicity (see Milgrom (1981)or Grossman and Hart (1983)). The problems are mainly due to the fact that the
agents output level plays two conicting roles an incentive role and a signaling role.
The classical principal-agent models typically put strong restrictions on the agents
action choices, and thus tend to overemphasize the signaling role of the agents output
level. Mirrlees and Zhou (2006a,b) reformulate the existing models by considering a
richer agents action space in discrete time settings. These two papers conrm the
widely perceived idea in a rigorous way that optimal contracts can be made simple by
enriching the agents action space. Of course, enlarging the agents action space can
be best done in a continuous-time setting, where the agent can revise his/her eort
continuously according to the updated information ow and thus a rich agents action
space is obtained naturally. This motivates us to extend the work done by Mirrlees and
Zhou (2006a,b) to a continuous-time setting, and to build up a connection between
the discrete-time models and their continuous-time counterparts.
In their seminal paper, Holmstrom and Milgrom (1987) introduced a special
continuous-time principal-agent model. Assuming that the agent controls the drift
rate of a Brownian motion, that the players have an exponential utility function, and
that the agents cost of control can be expressed in terms of monetary units, theyshow that the optimal sharing rule is a linear function of some linear aggregates. The
solution is equivalent to that for the popular static model in which the agent controls
only the mean of a multivariate normal distribution and the principal is restricted
to use a linear sharing rule. They further show that, even if the principal can only
write contracts based on some coarser linear aggregates, the optimal contract is still
linear. The model was later generalized by Schattler and Sung (1993, 1997) and Sung
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(1995), among others to a more general setting by allowing the agent to control the
drift rate of more general production technology processes. However, in order to ob-
tain a closed-form solution (a linear solution indeed), this line of papers still retain the
other two key assumptions: that is, the players have an exponential utility functionand the agents cost of control can be expressed in terms of monetary units.1
There are several concerns about this line of research. First, although linear con-
tracts predicted by existing continuous-time principal-agent models provide some im-
portant insights and give useful guidelines to the design of practical incentive schemes,
there are some important nonlinear features in the real world that the linear studies
fail to capture. Second, the linear optimal incentive schemes in existing continuous-
time models are very sensitive to the two key assumptions, and relaxing any one of
them would destroy the linearity result. Third, even if the linearity result is not
destroyed when less information is used in the design of compensation rules, it is not
clear how linearity and information aggregation are related to each other and how
they altogether are related to the underlying model parameters. In other words, it
is a very important issue in practice to know how to use minimum information to
design the optimal contract, about which this current line of research was unable to
give clues.
In this paper we relax the two key assumptions given by Holmstrom and Milgrom
(1987), as mentioned above, and develop a more general and yet tractable continuous-
time principal-agent model than existing linear ones. That is, we merely assumethat both the principal and the agent have an increasing and concave utility and
that the agents utility is separable over income and action.2 However, we retain the
assumption that the agent can only control the drift rate of a Brownian motion. In
this situation, the optimal solution will for sure be a nonlinear function of output,
which is probably path-dependent. What is more important is that the rst-order
approach can now be applied to many circumstances under mild technical conditions
and that we can still develop a tractable model under a wide range of reasonable
1 There are several papers that address continuous-time principal agent problems under moregeneral frameworks (see, for instance, Cvitanic, Wan and Zhang (2005), Sannikov (2004), Williams(2005) and the references therein). Using the techniques from stochastic control theory, this line ofwork mainly focuses on necessary conditions the optimal contracts must satisfy, and, in some cases,partial characterizations of the optimal solutions. However, they are unable to deliver a solvablemodel that is exible enough to accommodate various applications.
2 Note that the assumption is standard in the classical static principal-agent models. For nota-tional convenience, we will focus on the case of additive separability, although all analysis can gothrough the case of multiplicative separability. Further note that if the agent has an exponentialutility and the cost of eort is monetary then the agents utility is multiplicatively separable, andconsequently is a special case of our model
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model parameters.
On the technical side, there is a key dierence between our approach and those
used in continuous-time literature in solving the optimal contracts. While the current
literature typically relies on the techniques from stochastic control theory, we use thestandard techniques from functional analysis, in combination with probability theory,
to deal with the dicult issues. This is done by expressing the agents expected
cost of control as a function of probability measure. Consequently, the models can
be converted to the ones considered by Mirrlees and Zhou (2006a), where explicit
solutions are obtained under a class of separable cost functions of probability measure.
The next question then becomes, what kinds of cost functionals of control in our
continuous-time setting, if any, can generate the class of separable cost functions
of probability measure. The answer is, surprisingly, yes and simple the class
of separable cost functions of probability measure can be generated by a class of
quadratic cost functionals of control.
Put another way, we show that the principal-agent model in continuous-time can
be converted to solving a standard principals optimization problem to stochastic con-
trols. Furthermore, under mild technical conditions, this standard principals dynamic
control problem can again be converted to a static optimization problem, to which
the approach developed by Mirrlees and Zhou (2006a) can be applied systematically.
The transformation of the problem follows two steps. First, for each given (indirect)
sharing rule, the agents problem can be converted to solving a static maximizationproblem over the space of probability measures, which is a concave program over a
convex set if the cost of control is convex. As a result, the rst-order approach can be
applied to transform the agents incentive constraint into a normal one. Indeed, at
optimum, the (indirect) sharing rule of the agents problem can be expressed explicitly
as a function of the agents action, and can be decomposed into three components:
the agents opportunity cost, the cost of probability measure plus the compensation
for incentive. Second, in the principals problem, by replacing the sharing rule by
the incentive constraint, we obtain a relaxed maximization problem over the space of
probability measures, which is equivalent to the static problem of Mirrlees and Zhou
(2006a) when the agents action space is the whole space of probability measures.
In our continuous-time model, the issue about information aggregation can be ad-
dressed as well. One may wonder that, when the optimal contracts become nonlinear,
they have to rely on ner information set to enforce the desired action taken by the
agent. Surprisingly this is not the case. For the class of separable cost functions of
probability measure, we show that, if the principals gross payo is path-independent,
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then the optimal sharing rule is, though nonlinear, path-independent as well, which
entails substantial information aggregation. It appears that linearity and aggregation
can be treated separately. This is of great economic signicance in practice as optimal
contracts are typically nonlinear in nature, but, in the meantime, we would like touse less information to economize on the costs of contract design.
Finally, we give some simple applications in various areas to illustrate the conve-
nience of our model. In particular, the benchmark case of the paper, which assumes
the quadratic cost of control but allows for general utilities, is very exible to accom-
modate various applications, as shown in the paper. It could be thought of as the
nonlinear version of the popular linear model that pairs normal distribution with ex-
ponential utility, developed by Holmstrom and Milgrom (1987). Comparative statics
analysis is performed along the way. We attempt no systematic applications of the
model in the paper, but merely point out such possibilities through some interesting
examples.
The rest of the paper is organized as follows. In section 2, we set up the model
framework, where the principal-agent model in continuous-time is formulated. In
section 3, we solve the agents problem for a given sharing rule and give the agents
optimal action a necessary and sucient condition. In section 4, we solve the prin-
cipals maximization problem, based on the result of the previous section. In section
5, we give some simple applications, using the approach developed in section 3 and 4
. Finally, in section 6, we conclude our paper and point out some potential researchdirections for the future.
2. The Model Framework
A standard principal-agent model with moral hazard in continuous-time can be vi-
sualized as follows: the owner of a rm who delegates the daily operation of the
rm to a manager for a xed period of time. The owner is referred as the principal
and the manager as the agent. The time period is normalized to be [0; 1] : The out-
come processYt2 Rn; dened on the interval [0; 1], satisesY0 = 0 and is publiclyobservable. Let (; F1; P) denote the underlying probability space generated by astandardn-dimensional Brownian motion Bt;the components of which are mutually
independent, andYt be governed by a stochastic dierential equation of the form
dYt = (t; Y)dt +dBt; (2.1)
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where the drift rate (t; Y) is the agents control vector at time t and is the dif-fusion rate matrix, which is nonsingular and therefore the inverse 1 of which is
well-dened. For notational convenience, we assume that the coecients of are
constant.3 Equation (2.1) indicates that the agent can control the drift rate butnot the diusion rate . The agents control (t; Y)is anFt-predictable process thatsatises the following technical condition
P[
Z t0
(s; Y)2ds
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agent models but a key departure point from Holmstrom and Milgrom (1987).
The principal also maximizes the expected value of a von Neumann-Morgenstern
utility functionu ()which is an increasing and concave function of the net payo over(1; +1). To do so, the principal will design an incentive scheme w (Y)so that theagent will choose an action that is in the principals best interest. Equivalently,
given his/her gross payoS(Y1), the principal solves the following dynamic controlproblem:
max2H; w(Y)
E[u(S(Y1) w(Y))] (2.4)
subject to the incentive constraint (2.3) and the participation constraint
max2H
Ev(w(Y)) Z 1
0
c(s; Y, (s; Y))ds v; (2.5)where v represents the agents outside opportunity cost.
On the whole, the principal-agent problem as described by equations (2.1)-(2.5)
would be easy to deal with if reformulated in the spirit of Grossman and Hart (1983).
LetX(Y) =v(w(Y))and w(Y) =h(X(Y));whereh = v1: X(Y)assigns each pathYan agents utility level and thus is an indirect sharing rule. With a little abuse ofnotion, we will call bothX(Y)and w(Y)a sharing rule interchangeably. Given this,the principal-agent problem becomes
max2H; X(Y)
E[u(S(Y1) h(X(Y)))] (2.6)
subject to the agents incentive constraint
max2H
E
X(Y)
Z 10
c(s; Y; (s; Y))ds
(2.7)
and the participation constraint
max2H E
X(Y) Z 1
0 c(s; Y; (s; Y))ds v: (2.8)
Note that, in this new formulation, the agents utility is not present in the incentive
constraint (2.7) and the participation constraint (2.8). However, it does appear in the
principals problem (2.6) in the inverse form ofh(): Following the same logic as thestatic models of Mirrlees and Zhou (2006a), it is easy to show that the participation
constraint (2.8) is binding. As a consequence, the participation constraint (2.8) can
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be replaced by
max2H
E
X(Y)
Z 10
c(s; Y; (s; Y))ds
= v: (2.9)
In order to apply the approach developed by Mirrlees and Zhou (2006a) to ourcontinuous-time setting, in which the probability measure is used as the agents con-
trol variable, we introduce an alternative but equivalent expression of our model. We
rst x an outcome process Yt2 Rn over (; F1; P) that is observable to both theprincipal and the agent and denote it by
dYt = dBt; Y0= 0: (2.10)
Naturally, Yt = Bt and isFt-martingale, which is independent of the agentsaction .7 However, the agents action does aect the outcome process Yt via achange of measure. To see this clearly, denote by
t = exp(
Z t0
1(s; Y)dBs 12
Z t0
jj1(s; Y)jj2ds); (2.11)
where 1(s; Y)2 Rn andjj jjrepresents the norm of a vector in the EuclideanspaceRn.
Since the agents action satises condition (2.2) and is a constant nonsingular
matrix, we have that t is an Ito process. Furthermore, by Ito lemma, it is easy to
check that t satises
dt = t1(t; Y)dBs: (2.12)
LetH2 be the space of allFt-predictable processes (t; Y)such that
E(
Z 10
2(s; Y)ds)
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utilize the L2 theory in developing our model.8 Note that H is a very large space.
For instance, it is easy to check that all bounded processes belong to H:
Given2H, we can dene a new measure P;which is equivalent to P; and thecorresponding Radon-Nikodym derivative with respect to P by
dP
dP =1: (2.15)
Therefore, by Girsanovs theorem, the process
Bt =Bt
Z t0
1(s; Y)ds (2.16)
is a Brownian motion under the new measure P: In this situation, the outcome
process Yt dened by dYt = dBt with Y0 = 0 satises the following stochastic
dierential equation
dYt = (t; Y)dt +dBt; Y0 = 0; (2.17)
under the new measureP:The new measureP can be interpreted as the likelihoods
of the sample paths Y 2 when the agents control is taken.9 Let E be theexpectation with respect toP:Given this, the agents control problem (2.7) can now
be restated as follows
max2H E[X(Y) Z 10
c(s ; Y ; (s; Y))ds]; (2.18)
or equivalently, in terms of a change of measure, it can be expressed as
max2H
E
1(X(Y)
Z 10
c(s ; Y ; (s; Y))ds)
(2.19)
under the standard measure P: The participation constraint (2.9) can be restated
accordingly.
8 This technical condition is not ncessary. For instance, we can weaken it by assuming that justsatisfy the Novikovs condition
E
1
2exp(
Z 10
2(s; Y)ds)
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Similarly, the principals problem (2.6) can be restated as
max2H; X(Y)
E[u (S(Y1) h(X(Y)))] ; (2.20)
or equivalently, in terms of a change of measure, expressed as
max2H; X(Y)
E[1u (S(Y1) h(X(Y)))] : (2.21)
Note that in equations (2.19) and (2.21), dYt = dBt; which means that Y is not
aected by control since, by assumption, is independent of control : Control
aects the expected value via 1and c(t ; Y ; ): As we will see shortly, the independence
of onsimplies the principal-agent analysis signicantly.
For the rest of the paper we try to develop an approach to solve the principal-agentproblem (2.6)-(2.8) or its equivalent form (2.18)-(2.21) systematically. Our analysis
will be divided into three steps. First, we solve the agents optimal control problem
(2.18) or (2.19) for a given principals sharing rule X(Y) ; and provide a necessary
and sucient condition that an agents action is implemented by a principals
sharing rule X(Y) :As a result, the agents incentive constraint can be converted to
a standard one. Second, we solve the principals control problem (2.20) or (2.21),
and show that it can be converted to a relaxed standard dynamic control problem.
Therefore, the standard optimization techniques can be applied. Third, we explicitly
solve the problem for a wide range of model parameters.
3. The Agents Problem
For a xed sharing rule X(Y) (or w(Y)); the agents optimal control problem is
equivalent to solve equation (2.18) subject to the constraint (2.17). Note that the
sharing ruleX(Y)assigns a utility level for each sample path of the outcome process
Yt;and thus depend on the entire history ofYt:Traditionally, the rst-order condition
(or the rst-order approach) is used to replace the incentive constraint (2.18) or (2.19)in solving the principals problem. However, as pointed out by Mirrlees (1974), the
rst-order approach will enlarge the constraint set in general and the resulting rst-
order condition for the principals problem is not even necessary one for the optimal
contract. Given our model setup, it is surprising that the rst-order approach will
not enlarge the constraint set, as we will see shortly. This means that the rst-order
condition is not only necessary but also sucient.
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To show the result, we rst dene the concept of implementation. A sharing rule
X(Y) is said to implement a control if control is optimal for the given sharing
rule X(Y) and if the participation constraint is binding. For simplicity, we will
call such sharing rule implementable. For the agents problem, the existence of anoptimal control should not be a major concern since the principal will not design a
sharing rule such that the agents optimal control does not exist. In other words, for
the principal-agents problem to be well-posed, we only need to focus on the set of
implementable sharing rules.
Proposition 1. A control2H is implemented byX(Y)in equation (2.18) if andonly if
X(Y) = v+ Z 1
0
c(s ; Y ; (s; Y))ds + Z 1
0
c0(s ; Y ; (s; Y))dB s (3.1)
Proof. See Appendix.
Proposition 1 transforms the agents incentive constraint into a standard one.
Even more, X(Y) can be explicitly expressed as a function of : Note that, if the
right side of equation (3.1) is well-dened for all 2 H; then all 2 Hare imple-mentable.10 As a result, the constraint (3.1) can replace both the agents incentive
constraint and participation constraint, and the resulting principals problem becomes
an unconstrained optimization problem.
The representation result in proposition 1 can be compared to that in Holmstrom
and Milgrom (1987) or in Schattler and Sung (1993). It shows that a sharing rule
X(Y) can be decomposed into three components: a). the agents opportunity cost
v; b). the actual cost occurred for the agents eort and c). the compensation for
the incentive. However, there is an additional term in the representation result of
Holmstrom and Milgrom (1987) or Schattler and Sung (1993) that is absent in our
result. That is, the risk premium due to the incentive is not present in our proposition.
This is due to the separable feature of the agents utility over income and action, for
which the agent acts like a risk-neutral person for each given sharing rule X(Y). Notethat our representation result is based on the indirect sharing rule X(Y)rather than
the direct sharing rule w(Y) on which the result of those two papers is based. It
should be noted that proposition 1 does not mean that the agents risk aversion is
irrelevant. It just transforms the issue into the principal problem via the function
h(X):
10 This requires some technical conditions on the cost of control c(t ;Y ;); which is assumed tosatisfy throughout the remaining part of the paper.
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Equation (2.19) indicates that it may be possible to develop an alternative way
of characterizing the relation between X(Y) and : In some cases, it may be more
convenient to have 1 as the agents choice variable. This is particularly true if
we can relate the principal-agent problem here to those formalized by Mirrlees andZhou (2006a), where measure as the agents decision variable is fully developed. It
then would be interesting to know how 1 and are connected to each other in our
setting. Of course, we know from the denition of1 that 1 is determined by via
equation (2.11): Dene by f() = 1 the map from H to L2(; F1; P): Clearly, f is
an one-to-one map. Let
4=f2L2++(; F1; P)jE() = 1g: (3.2)
The following lemma shows the equivalence between and 1 in a strong sense.11
Lemma 1. Letf(H)be the image off :Thenf(H) =4. That is, fis both one-to-one and onto.
Proof. See Appendix.
Lemma 1 shows that, topologically, Hand4are identical. Therefore, it providesanother way to restate the agents problem when the agent uses 1as a choice variable.
To do this, we need to write down as a function of1 explicitly. Note that, ift is
a martingale such that 12 L2++; then by martingale representation theorem, thereexists a uniqueFt-predictableRn-process b(t; Y)such that
t = 1 +
Z t0
b(s; Y)dBt: (3.3)
Dene by
(t; Y) =b(t; Y)
t: (3.4)
We thus have the inverse functionf1
:4 !H;
which is dened by equation (2.15)and (3.4). Given this, we can explicitly write down the expected cost as a function
of1;which is as follows
c(1) =E(c) =
Z
1dP
Z 10
c(s ; Y ; )ds
11 2 L2++ means > 0 a.s. We could work on the space L2+; where 2 L2+ means 0 a.s.However, since this paper does not touch the boundary issues (although it is also a very interestingissue per se), we stick to the space L2++:
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=
Z
1dP
Z 10
c(s;Y;b(s; Y)
s)ds: (3.5)
As a result, we can reformulate the agents problem in the spirit of Mirrlees and Zhou
(2006a). In other words, the agent chooses 12 4 to maximizeZ
1X(Y)dP c(1); (3.6)
wherec(1)represents expected cost of1;which is the term in equation (3.5). The
participation constraint can be similarly reformulated as well. This reformulation
allows us to use dierential analysis in L2 space to fully characterize the agents
incentive constraint. With1 being the decision variable, the concept of implemen-
tation can be dened in a similar manner. We will say that X(Y) implements 1 if
1 is an optimal choice for the given sharing rule X(Y) in equation (3.6) and if the
participation constraint is binding: The next proposition borrows from Mirrlees and
Zhou (2006a), to which we refer the proof for the interested reader.
Proposition 2. A measure choice12 4is implemented byX(Y)in equation (3.6)if and only if
X(Y) = v+c(1) +c0(1)
Z
1c0(1)dP (3.7)
Note that, similar to the case in whichis the agents decision variable, in order to
motivate the agents action1in the most ecient way;the principal needs to design
the sharing ruleX(Y)in a way that compensates the agent for: (a). the opportunity
cost; (b). the expected eort cost of taking action 1 and (c). the incentive of taking
action1:
4. The Principals Relaxed Problem
Given the fact that the agents incentive constraint and participation constraint can
be converted to the standard constraint (3.1) or (3.7), we are able to obtain a standardprincipals relaxed problem by substituting (3.1) or (3.7) into the principals problem
(2.20) or (2.21), which is as follows.
max2H
E
u
S(Y1) h(v+
Z 10
cds+
Z 10
c0dB s )
; (4.1)
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or, by Girsanovs theorem, it can be rewritten as
max124Z 1u S(Y1) h(v+ Z
1
0
(c c0)ds + Z 1
0
c0dBs) dP= max
124
Z
1u
S(Y1) h(v+c(1) +c0(1)
Z
1c0(1)dP)
dP (4.2)
Equation (4.1) would become a standard stochastic control problem with control
variable if is restricted to be Markovian: In this case, the standard techniques
developed in stochastic control theory and the HJB equations could be used to char-
acterize the optimal control and the corresponding sharing rule X(Y) :In general,
however, the control variable (t; Y) in our framework is not Markovian. Therefore
we need to develop a more general approach to tackle the principals problem (4.1) or
(4.2). Fortunately, the martingale approach, in combination with functional analysis,
provides a natural tool to solve the problem. In this section, we rst address the issue
when control is not Markovian. We then turn to the case in which control is
restricted to be Markovian.
We now use the martingale approach to provide a necessary condition for the prin-
cipals relaxed problem (4.2). According to the martingale representation theorem,
letrJt be the uniqueFt-predictableRn-vector process such that
dudXjY =u0h0(X(Y)) =u0h0 +Z 10
rJsdBs ; (4.3)
whereu0h0 =E[u0h0(X(Y))]:
Proposition 3. Suppose that is an optimal control of the principals relaxed prob-
lem (4.2). Then must satisfy
u [S(Y1) h(X)] =Z 10
rJsTc00dB s +0; (4.4)
where is a constant. In particular, if the principal is risk neutral, we have
S(Y1) h(X) =Z 10
rJsTc00dB s +0: (4.5)
Proof. See Appendix.
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Note thath(X) =w(Y)is the principals compensation to the agents action, and
u [S(Y1) h(X)] =u [S(Y1) w(Y)] (4.6)
is the principals utility level of net payo. Equation (4.4) (or (4.5)) shows that, at
optimum, the principals utility level of net payo (or net payo) is an Ito integral
plus a constant. Proposition 3, together with proposition 1, shows a striking result of
the general principal-agent problem in continuous-time. At optimum, the expected
net utility processes of both the principal and the agent are martingales.
So far we have used the martingale approach to formulate the principal-agent
problem in continuous-time. The advantage of the approach is that it is very general,
and the solution in some cases can be expressed very neatly. For instance, when the
principal is risk neutral and Tc00= kIis the scaled identity matrix, we immediatelyhave
S(Y1) w= kZ 10
rJsdBs +0
=k(h0(X) h0) +0 = kv0(w)
+ : (4.7)
For other applications using the martingale approach, see the next section. In
particular, for certain model parameters, the martingale approach leads to a class of
static models that has already been fully developed by Mirrlees and Zhou (2006a,b).
The major weakness is that, due to the martingale representation theorem,rJsis not constructive. That is, in general, it is dicult to relaterJs to u0h0(X) inan explicit manner. However, if we restrict to the class of controls to be Markov
processes, then we can use the standard dynamic approach to solve the principals
relaxed problem.
Here we outline the standard dynamic approach. Suppose that (t; Yt)is a Markov
process. Let
dY =(t; Yt)dt +dBt withY0 = 0 (4.8)
and
dX=c( (t; Yt))dt +c0( (t; Yt))dB
t (4.9)
Given an initial value(t;y;x)wherexandy are a realization ofXandY respectively;
let the principals expected value J(t;y;x)be
J(t;y;x) =E(t;y;x) [u (S(Y1) h(X))] (4.10)
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when the agents control is : Then the Hamilton-Jacobi-Bellman condition implies
that, at optimum,J(t;y;x)must satisfy a system of second-order partial dierential
equations
@J
@t +
nXi=1
i @J
@yi+c @J
@x +1
2
Xij
aij @2
J
@yi@yj
+nX
i=1
(c0T)i@2J
@yi@x+
1
2
nXi=1
(c0Tc0)@2J
@x2 = 0 (4.11)
whereaij is the component of matrix T; and
@J
@yi+c0i
@J
@x + (
nXj=1
aij )c00i
@2J
@yi@x+ c0Tc00
@2J
@x2 = 0; 8i (4.12)
with the boundary condition
J(T ; y ; x) =u (S(Y1) h(X)) : (4.13)
We do not attempt to develop a full-edged approach to solve the resulting HJB
equations (4.11)-(4.13) as they are standard in the stochastic control literature. As a
result, the mathematical techniques and numerical methods developed in stochastic
control theory can be applied. In general, the HJB equation (4.11)-(4.13) admit no
closed-form solution. However, as we will see in the next section, the optimal controland the optimal sharing rule can be solved explicitly for a class of cost functions of
probability measure.
5. Applications
Because of their simplicity and computational ease, linear optimal contracts gradu-
ally become popular in the principal-agent literature. The paper of Holmstrom and
Milgrom (1987) is the rst one that provides a theoretical foundation for the use of
them, followed by Schutter and Sung (1993) and many others. The linear contractscan be optimal in a continuous time model if the players utilities are exponential and
the cost of control can be expressed as monetary units. However, linear contracts are
not in accordance with observations in the real world. Our model keeps the basic
assumption about the technology process, but allows for more general utilities. Yet
it is as tractable as the linear ones. Here we provide some simple applications in
several areas to give the reader some avor of our models exibility. No systematic
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applications are attempted, although it is a fruitful area to explore.
5.1. Quadratic Costs of Control
The rst application relates our model to the one developed by Mirrlees and Zhou
(2006a), where the probability measure is used as the agents choice variable and
static models are studied in details. The underlying cost functions of probability
measure are shown to be critical in determining the shape of the optimal contracts.
In particular, the optimal contracts are solved explicitly when the underlying cost
functions of probability measure are separable. The remaining subsection shows how
the cost functions of probability measure studied in Mirrlees and Zhou (2006a) can
be generated by a class of cost functionals of control in continuous-time. These cost
functionals of control have their economic contents and their underlying parameterscan be rich enough to accommodate various applications.
Example 1. Suppose that n = 1andc() = 12k2;wherek >0 is a constant:Then,
from equation (2.11), we have
Z 10
(t; Y)
dBs 1
2
Z 10
(t; Y)
2ds= ln 1:
Taking the expectation of two sides with respect toP;we have
c (1) =k2
Z
1ln 1dP: (5.1)
In the n-dimensional case, ifc() = 12
kjjjj2 and = I(whereIis then nidentitymatrix and is a real number), then the cost function of probability measure is the
same as equation (5.1), except for the fact that is replaced by.
Given the cost function of probability measure (5.1), equation (3.7) implies that
a sharing rule X(Y) implements 1 if and only if
X(Y) = v+k2 ln 1: (5.2)
As a result, the principals relaxed problem (4.2) becomes
max124
E
1u
S(Y1) h(v+k2 ln 1)
; (5.3)
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which is a concave program over the convex set 4:Therefore, there is a unique optimal1 (a unique sharing rule X(Y)accordingly) in the problem (5.3).
This example, in its slightly dierent form, has been studied in details in Mirrlees
and Zhou (2006a,b), from which we know that, in our context, the optimal sharingrulew(Y)must satisfy
u0 (S(Y1) w)v0(w)
= +u(S(Y1) w)
k2 ; (5.4)
where is a constant that is determined by the participation constraint:In particular,
if the principal is risk neutral, we must have
S(Y1)
w= k2
1
v0
(w)
+ : (5.5)
Equation (5.5) plays an important role in the risk-neutral case when the model
in example 1 is further enriched. To simplify the analysis in later applications, we
introduce a simple version of equation (5.5) that ignores the constant, which is as
follows
s= w + a
v0(w); (5.6)
wherea >0 is a parameter. It is easy to check that s is an increasing function ofw
anda:Depending on the agents utilityv; scan be either a concave or convex function
of w: For each a; dene the inverse function of s in equation (5.6) by w = f(s; a):
Clearly, @f@s
> 0 and @f@a
< 0: Given this, the optimal sharing rule in equation (5.5)
can be written as
w(Y) =w(Y1) =f(S(Y1) ; k2); (5.7)
where is determined by solving
E[1] =E
expf 1
k2v(f(S(Y1) ; k2)) 1
k2vg
= 1 (5.8)
(see Mirrlees and Zhou (2006a)), and the agents optimal action 1 is
1= expf 1k2
v(f(S(Y1) ; k2)) 1k2
vg: (5.9)
Finally, the agents optimal control that implements the optimal action 1 is
=1
kS0v0f0(S(Yt) ; k2): (5.10)
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Note that the optimal sharing rule depends only on the nal outcome if the prin-
cipals gross payo does the same. Equation (5.10) implies that, in contrast to the
linear case of Holmstrom and Milgrom (1987), the optimal control is in general no
longer constant even ifS0 is constant (or Sis linear). Also note that depends on v;kand :A simple calculation shows that @
@v
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It is easy to show that, when the principal is risk neutral, the rst-best solution
is independent of the outcome volatility : Lemma 3 gives a surprising and yet an-
ticipated result. It explicitly addresses the risk eect on the principals welfare in a
nonlinear context, and points out an important ineciency source of the second-bestcontract; the volatility of outcome process negatively aects the principals welfare.
In Mirrlees and Zhou (2006a), the case in which the cost function of probability
measure c(1) is separable has been extensively explored. However, the economic
environment under which the cost function of probability measure is separable is not
elaborated. In what follows we show that, surprisingly, the separable cost functions
of probability measure can be generated by a class of state-dependent quadratic cost
functionals of control.
Proposition 4. Let g (z) be a smooth, convex function over(0; 1). Suppose thatc(t;;) = 1
2ktg
00(t)jjjj2 and= I : Then we have
c (1) =k 2
Z
g(1)dP: (5.14)
More general, let t be martingale and1 = (Y) : Then if
c(t;;) =1
2ktg
00(t)tjjjj2; (5.15)
we have
c (1) =k 2
Z
(Y) g(1)dP: (5.16)
Proof. See Appendix.
Equation (5.16) shows that any value-weighted separable cost function of proba-
bility measure can be constructed from a class of simple state-dependent quadratic
cost functionals of control. Note that g is only required to be convex, and thus
is very exible to accommodate various applications.14 When g(z) = zln z and t
is constant; we are back to the canonical case in example 1. Another particularly
interesting case would be
g(z) =
1 z1 if6= 1 and g(z) = ln z if= 1: (5.17)
14 Note that g dened in proposition 4 implies that the cost functional of control depends on thepast history of eorts, which is not incoporated into the model in the previous section. However,if we add the process t to the outcome process Yt; then we are back to the normal case. In thissituation, the diusion rate will no longer be free of control :
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In this case,
c(t;;) =1
2k2t
t jjjj2: (5.18)
Note that t
captures the eect of the past eorts on the marginal cost of eort at
time t. 0 means the opposite. Given the cost function of probability measure
c (1)dened by g (z)in equation (5.17), we are back to the static models developed
by Mirrlees and Zhou (2006a). As a result, the optimal sharing rule can be written
as a nonlinear form of some linear aggregates if the principals gross payo does the
same, as has been shown for the case of equation (5.7). See Mirrlees and Zhou (2006a)
for details.
5.2. The Role of Signals
Signals are important in the design of contracts. In general, additional information
increases the value of contract (see Holmstrom (1979)). It would be nice to see how
the optimal sharing rule in our continuous-time model responds to signals. In this
subsection we demonstrate the role of signals by solving two cases explicitly. For
simplicity, in the remaining part of this section we will assume that the principal is
risk neutral and the cost function of probability measure is canonical, as specied in
example 1.
5.2.1. The Eort-Free Signal
One case is related to a signal that is independent of the agents eort. Consider a
two-dimensional process. Let Yt be a one-dimensional outcome process. Now lets
further assume that the principal can observe an additional signal Zt; and thus can
write contracts on the signal, in addition to the outcome process Yt. The signal could
be thought of as a market index, an observed competitors performance or anything
that might be relevant to the principals payo. For simplicity, we assume that
dYt = 11dB1t +12dB
2t (5.19)
and
dZt = 21dB1t +22dB
2t ; (5.20)
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whereB1t andB2t are two standard Brownian motions and the component of covari-
ance matrix is ij :Dene a new measure P such that dP =1dP and
dt = t(
22
jjjjdB1
t 21
jjjjdB2
t ) = t
Z
jjjjdB
t ; (5.21)
whereZ=p
222+221;and B
t =
22Z
B1t 21Z B2t is a Brownian motion. Under thisnew measure P;we have
dYt= dt +11dB1t +12dB
2t (5.22)
and
dZt = 21dB1t +22dB
2t : (5.23)
Note that, by assumption, the agents action only aects the outcome process Yt;
and the signal process Zt is driftless and not aected by the agents action.
Let the cost functional of control be c () = 12k2 and the principals gross payo
be a linear function ofY1 andZ1 or
S(Y1; Z1) =k1Y1+k2Z1; (5.24)
wherek1andk2are some constants. A direct calculation shows that the cost function
of probability measurec
(1)is given by
c(1) =kjjjj2
2Z
Z
1ln 1dP =k2Y(1 2)
Z
1ln 1dP; (5.25)
where 2Z = 221+
222 and
2Y =
211+
212 are the variances of outcome process Y1
and signal Zt respectively;and = 1121+1222
YZis the correlation coecient.
Equation (5.21) indicates that the agents action space is not full in4:Thus theresult in the previous section can not be applied to this situation directly. In order
to obtain an explicit solution, we rst need a lemma
Lemma 3. Suppose that X implements the agents action 1. LetfFt gt0 be theltration generated byBt andX
be the uniqueF1 -measurable random variable thatimplements the action E(1jF1 ) :Then we have
E[1S] =E[1E(SjF1 )] =E[E(1jF1 ) E(SjB1)] (5.26)
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and
E[1h(X)]E[E(1jF1 ) h(X)] (5.27)
Proof. See Appendix.Lemma 3 implies that, ifX implements 1;then we have
E[1(S h(X))]E[E(1jF1 ) (E(SjB1) h(X))] : (5.28)
Note that, in equation (5.28), the left side is the principals expected net payo,
and the right side is the principals expected net payo when the coarser information
ltration F1 generated byBt is used in the design of contract. Equation (5.28) showsthat the optimal sharing rule must be
F
1
-measurable.
It is easy to show that
E[SjB1 ] =k1jjjjZ
B1 (5.29)
and thatB1 can be expressed as a linear combination ofY1andZ1;which is as follows
B1 = Z
jjjjY1 Cov(Y1; Z1)
jjjjZ Z1: (5.30)
ReplacingB1 in equation (5.29) by (5.30), we have
E[SjB1 ] =k1
Y1 Cov(Y1; Z1)2Z
Z1
: (5.31)
As a result, equation (5.7) implies that the optimal sharing rule X can be written as
w(Y) =f(s; a) =f(k1
Y1 Cov(Y1; Z1)
2ZZ1
; k2Y(1 2)) (5.32)
and, similar to the case in example 1, the principals optimal expected payo
u(v ;k; ;Y; Z) =k2Y(1 2) Z
1
v0(f)dP+: (5.33)
Note that the term Y1 Cov(Y1;Z1)2Z
Z1 can be interpreted as the additional infor-
mation that Y1 can provide over the signal Z1 about the agents eort. It conrms
the popular idea in practice that the principal should not base the agents compensa-
tion on the factors that are out of the agents control. Two extreme cases are worth
mentioning. First, whenZ1 is uncorrelated to the nal outcome Y1;expression (5.32)
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becomes
w(Y) =f(s; a) =f(k1Y1 ; k2Y): (5.34)
Thus, in this situation, we are back to the no signal case. Second, whenZ1is perfectly
correlated to the nal outcome Y1; we are back to the rst-best situation the
principal can perfectly infer the agents action by observing the signal Z1:In general,
expression (5.32) clearly shows the eects of all relevant parameters on the optimal
sharing rulew(Y). It is a nonlinear function of the index Y1 Cov(Y1;Z1)2Z
Z1;the linear
combination of the two linear aggregates Y1 andZ1:
To see the role of the signal Z1 clearly, we rewrite expression (5.32) as follows
w(B1) =f(k1Yp1 2B1 ; k2Y(1 2)): (5.35)
Note that the right side of expression (5.35) has exactly the same form as that in
example 1, with being replaced by Yp
1 2: Therefore, the signals function in theprincipals contract design is equivalent to reduce the volatility of the outcome process
in the no signal case and its value from the principals perspective is completely
determined by its correlationjj with Y1:There are some additional comparative statics results beyond those of the no
signal case in example 1. The parameters and Yaect the sharing rule via both s
anda: For instance, a high valuejj makes the sharing rule relatively more sensitiveto the signal Z1 and less sensitive to the output Y1; while in the same time reducesthe marginal cost of eort. In addition, they also have a complicated eect on :
Thus, the overall eect of on the shape of the optimal sharing rule is mixed. The
parameterZaects the sharing rule via s: A high Zwill make the principal rely
relatively less on the signal and more on the output, and vice versa.
The eects ofv,k and Y onuare the same as those in example 1, and the eects
of and Z on uare@u
@ =
Y1 2
@u
@ Yand
@u
@ Z= 0: (5.36)
From lemma 2, we have @u@Y 0 if
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where is the principals rst-best expected payo, which is independent of ; Y
and Z. As a result, the principals welfare uincreases with the absolute valuejj:Itis equivalent to the no signal case when = 0 and reaches to the rst-best outcome
whenjj= 1:
5.2.2. The Common Eort Signal
An alternative application is in the area of managerial compensation, where the rms
prot and the stock price, among others, are typically used as information to design
contracts. Our model is highly stylized as we do not attempt to target any specic
situation. Rather, we try to use the model to show the power of our approach and
to highlight some key economic insights. LetYt andZt be the same process as that
in equation (5.19) and (5.20), where now Yt is interpreted as the rms accumulatedprot process and Zt as the rms price process.
15 Again, we dene a new measure
P such that dP =1dP; and the density process t by
dt = t
22 12
jjjj dB1t +
21+11jjjj dB
2t
: (5.39)
Let
dBt = 22 12
jjjj dB1t +
21+11jjjj dB
2t ; (5.40)
whereB
t is a new Brownian motion under P: Then, under the new measure P
; wehave
dYt= dt +11dB1t +12dB
2t (5.41)
and
dZt = dt +21dB1t +22dB
2t : (5.42)
These specications imply that both the rms prot process Yt and price process
Pt are aected by the managers eort : Thus, by specifying the new measure P;
we implicitly assume that the eect of eort on both the accmulated prot process
and price process is identical.
While possibly highly correlated, the accumulated prot process and the price
process are in general dierent, as price process reects more information than prot
15 In what follows Yt and Zt may be considered to be the log of the prot process and the priceprocess respectively, and the principal has a log utility over the ratio of gross payo and incentivecost. As a result, the forthcoming sensitivity analysis becomes elasticity analysis, which is more inline with empirical literature. There are several ways of modelling the price process according todierent capital market conditions. However, the basic insights remain the same as captured in ourmodel.
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process. For instance, the stock price may aggregate information that professional
investors actively acquire in their seek of prot, such as information about the man-
agers eorts and project selections, among others (see Holmstrom and Tirole (1993)
and Bolton and Dewatripont (2005) for details).16 Let the shareholders gross payobe
S(Y1; Z1) = Y1+ (1 )Z1; (5.43)
where01: The parameter can be thought of as the percentage of the rmsaccumulated prot at the end period that will be distributed to the owner, while
1 can be thought of as the rms market value of the remaining asset as a goingconcern. = 1 means that the shareholder intends to liquidate the rms asset at
the end period, and thus does not care about the rms market value. = 0 means
that the shareholder will not receive any dividend and will sell the rm as a goingconcern at market value at the end period. In this case, the shareholder only cares
about the rms market value. Alternatively, could also be thought of as the rms
dividend payout ratio.
The cost functional of control is, as before, c = 12k:Given this, a direct calculation
shows that the cost function of probability measure c(1)is
c(1) = kjjjj2
jj21 11jj2 + jj22 12jj2Z
1ln 1dP
=k 2Y
2Z(1 2)
2Y +2Z 2YZ
Z
1ln 1dP =k
Z
1ln 1dP: (5.44)
Note that the new Brownian motionB t can be expressed as a linear combination
ofYt andZt:
dBt =kYdYt+kZdZt; (5.45)
where
kY = 2Z Cov(Y; Z)
jjjj2 andkZ= 2Y Cov(Y; Z)
jjjj2 : (5.46)
Next, we calculate E[SjB1 ] ;which is as follows
E[SjB1 ] = 1
kY
2Y + (1 )kZ2Z+ (kZ+ (1 )kY)YZ
B1 : (5.47)
Similar to the case of the eort-free signal, it is straightforward to show that the
16 Indeed, the model in this subsection when = 1 is a nonlinear analogue of the linear contractcase given by Bolton and Dewatripont (2005) in section 4.6.
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agents optimal action1 can be restricted to be F1 -measurable. As a result, we have
w(Y1; Z1) =w(kYY1+kZZ1) =f
Cov(S; B1)
(kYY1+kZZ1) ; k : (5.48)
Again, expression (5.48) indicates that the optimal sharing rule can be written
as a nonlinear function of an aggregate index that is a linear combination of a linear
aggregate and the signal value at the end period. We rst look at the eect of
on the contract; it aects the sharing rule only through Cov(S; B1): A dierent
places dierent weights on the parameters related to Yt andZt: In a particular case
Y = Z; becomes irrelevant in the design of contract; that is, the optimal sharing
rule is independent of the principals intention of selling or holding the asset in the
end-period.
Compared to the case of the eort-free signal, there are some subtle dierences
for the use of the signal here. Since the agents action aects the drift rate ofYt and
Zt in the same magnitude, the role ofY1 andZ1 in the optimal sharing rule (5.48) is
symmetric. Moreover, unlike the case of eort-free signal, the volatility Zof signal
Zt plays a symmetric role as that of prot process Yt and the correlation coecient
has a signicant dierent eect on the incentive scheme and the principals welfare.
Lets rst look at the eects of the volatilities. For simplicity, we set = 0:In this
case, we have Cov(S; B1) = 1; kY = 12Y
; kZ= 12Z
; k = kkY+kZ
and =p
kY +kZ:
Therefore,w(Y1; Z1) =w
(B1
) =f
B1
; kkY +kZ
=
f
2Z
2Y +2Z
Y1+ 2Y
2Y +2Z
Z1 ; kkY +kZ
= f
B1pkY +kZ
; kkY +kZ
(5.49)
The left side of expression (5.49) clearly shows the symmetry ofY1 andZ1; it implies
that as the stock price Z1 (or the accumulated prot Y1) becomes more volatile, the
optimal sharing rule will weigh less on the price signal (or the accumulated prot),
and vice versa. Similar to equation (5.35), the right side of expression (5.49) showsthat the eect of bothY and Zon the principals welfare is negative; the principals
optimal expected payo decreases with Y andZ:17
17 As in the canonical case of example 1, the eect of the volatilities Y and Z on the shape ofthe optimal sharing rule, though symmetric, is mixed.
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Next we consider the eect of:For simplicity, let 2Y =2Z:In this case,
E[SjB1 ] = B1
=
1
2(Y1+Z1) ; k= k
1 +
2 2Y (5.50)
and
kY =kZ= 1
1 +
1
2Y: (5.51)
Therefore, the optimal sharing rule is
w(Y1; Z1) =w(Y1+Z1) =f
Y1+Z1
2 ; k
=fYr1 + 2
B1
; k1 +
2 2Y! : (5.52)
Following the same logic of equation (5.35) in the case of eort-free signal, expression
(5.52) indicates that the principals optimal expected payo decreases with :18 This
is a little surprising but makes intuitive sense as a high gives a high variance of the
index Y1+Z12 : From equation (5.52), we see that, as goes to 1, signal Z1 becomes
less informative and thus the optimal sharing rule is close to the no signal case. In
contrast, when is close to1, the optimal sharing rule converges to the rst-bestsituation. Therefore, an alternative explanation for the eect of on the principals
optimal payo is that a high reduces the overall quality of information generatedby both Y1 andZ1 about the agents eort, and thus makes the principal worse-o,
and vice versa.
Finally, we consider an interesting case in which 11 =21 and12 = 0: That is,
Z1 is a more noisy signal than Y1: In this case, E[SjB1 ] =Y1; k=k2Y; kZ= 0 andkY =
12Y
:As a result,
w(Y1; Z1) =w(Y1) =f
Y1 ;k2Y
: (5.53)
In other words, we are back to the no signal situation. Thus, if Z1 is a pure noisysignal, then it becomes irrelevant andY1 will be a sucient statistics.
18 Since aects the shape of the optimal sharing rule only through the volatility of the indexY1+Z1
2 ; its eect, just llike that of volatilities, is mixed.
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5.3. Multi-Task Analysis
Another potential application area is multi-task analysis. Consider a simple case
in which = I ; where is a real number and I identity matrix, and c () =12
kjjjj2: This means that the agent can allocate his/her eort among n tasks Yit(1 i n), and that the n tasks are stochastically independent. S(Y1) representsthe principals gross payo, which is a function of the nal outcomeY i1 :It summarizes
the contribution of each taski to the principals payo. Proposition 2 shows that the
optimal sharing rule
w(Y) =w(S(Y1)) = f(S(Y1) ; k2): (5.54)
Equation (5.54) indicates that the optimal sharing rule is a nonlinear function of
then linear aggregates. Note that
rw= f0rSor w0i
w0j=
S0iS0j
; 8i;j: (5.55)
The optimal action can be calculated as follows.
k2
2
2
Z 10
jjjj2ds + 1Z 10
dBs
=v+X(S(Y1))
=12
2Z 1
0
((X(S))00 dt + Z 1
0
v0f0rSdBs (5.56)
As a result,
= 1
kv0f0rSor i
j=
S0iS0j
: (5.57)
IfS=P
kiYi1 ; we then have S
0i =ki: It means that the contribution of all tasks
to the principals gross payo is independent, and that the marginal payo of task i
is ki: As a result, w0
i
w0j
= ij
= kikj
: That is, the principals marginal compensation rate
of substitution is equal to the agents relative eort on dierent tasks, which again is
equal to the tasks relative contribution to the principals gross payo. Note that
aects the absolute value of the marginal compensation of tasks but not the relative
one.
For a general non-singular matrix ; we can only solve the optimal sharing rule
numerically. However, ifS=Pn
i=1 kiYi1 and the principal can only design contracts
based on information generated by the total payo S; then, following the similar
procedure as that in the previous subsection, we can calculate the optimal sharing
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rule explicitly, which is as follows
w(Y) =w(S) =f(S ; k0); (5.58)
where
k0 =kPn
j=1(Pn
i=1 ij)2Pn
i=1 k2i
: (5.59)
Of course, our example only touches the very surface. It remains an interesting
research topic to further conduct a systematic nonlinear multi-task analysis and to see
what new insights one can draw as compared to the current popular linear multi-task
analysis (see Holmstrom and Milgrom (1991)).
6. Conclusion
This paper conducts a nonlinear principal-agent analysis in continuous-time. By
relaxing two of the three key assumptions of the model developed by Holmstrom and
Milgrom (1987), we are still able to obtain a closed-form solution for a wide range of
model parameters. The optimal contracts are in general nonlinear, which are more
in accordance with reality. Furthermore, the property of information aggregation can
be retained under a class of separable cost functions of probability measure. We do
not need to use all available information to implement an optimal contract. In some
cases some linear aggregates are sucient to achieve the objective.
The agents action space can be naturally enriched in continuous-time frameworks.
Indeed, in our model, each probability measure P can be implemented by a unique
control process : As a result, the agents action space has a full dimension in the
spirit of Mirrlees and Zhou (2006a), and the techniques developed there can be applied
exactly to our setting. In the meantime, the problems caused by the rst-order
approach in the classical principal-agent models disappear completely due to the
richness of the agents action space. One of the striking facts is that, although the
technical conditions as compared to those in Holmstrom and Milgrom (1987) are mild,yet the model is tractable and exible enough to accommodate various applications.
In this paper we mainly focus on extracting economic insights at the expenses
of sacricing technical generality. A few lines of future research arise. For instance,
it would be interesting to see how the model can be generalized to the production
processes in which the agent can control both the drift rate and the diusion rate. In
this situation, there may be many control s that give the same probability measure
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P; and one needs to choose the least cost control to implement the probability
measure: As a result, a cost function of probability measure is still well-dened, and
it is quite interesting to see how specic situations lead to explicit solutions. Another
line of research would be to incorporate intertemporal consumption and compensationinto analysis.19 This may involve new conceptual and technical issues, such as the
well known time-inconsistency issue that arises from the fact that the principal may
not be able to fully commit himself/herself at the middle points in time. Mirrlees
and Zhou (2006b) give some insights along this line. A continuous-time version of
the work has yet to be seen. Finally, in recent nance literature, there is a growing
interest in formulating eort and portfolio choice into a unied framework, where the
drift rate and diusion rate cannot be controlled independently. Our analysis may
be able to shed some lights on portfolio management in continuous-time (see Dybvig,
Fansworth and Carpenter (2004) and Li and Zhou (2006) among others).
19 It is not an easy problem, even for the rst-best risk sharing case (see Cadenillas, Cvitanic andZapatero (2005) and the references therein).
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Appendix A
Proof of Proposition 1:
Proof. Because of the separability of the agents utility over income and action, with
respect to the (indirect) sharing rule X(Y)the agent acts like a risk-neutral person.
The proof follows a similar logic as that of theorem 3.1 in Schattler and Sung (1993).
That is, we use the martingale approach to our general stochastic control problems.
Suppose that (t; Y)is implemented by a sharing rule X(Y). Let
dYt= (t; Y)dt +dB
t (6.1)
and
Vt = E[fZ 1
t
c(s ; Y ; (s; Y))ds +X(Y)gjFt]: (6.2)
Given equation (6.2), we immediately have V0 = v: Let (t; Y)2 H be anotheragents control and dene a new process
Mt =Vt
Z t0
c(s ; Y ; (s; Y))ds: (6.3)
Note that Mt = M
t dened by equation (6.3) represents the agents optimal ex-pected value at time t if the optimal control is used on [0; t]. It is easy to show
that Mt is a supermartingale on [0; 1] for all control 2 H: In particular, Mt is amartingale if and only if= : Since Mt is a martingale, the martingale represen-
tation theorem implies that there exists a uniqueFt-predictable Rn-vector processb(t; Y)such that
Mt = v+
Z t0
b(s; Y)dB
s : (6.4)
From the denition ofMt ; we have
Mt =M
t +
Z t0
c(s ; Y ; (s; Y))ds Z t0
c(s ; Y ; (s; Y))ds: (6.5)
Therefore,Mt can be written as
dMt =btdB
t + [c(t ; Y ; (t; Y)) c(t ; Y ; (t; Y))]dt: (6.6)
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Since, by the Girsanov theorem, a change of measure leads to
dB
t =dBt +
1( )dt; (6.7)
we have
Mt = v+
Z t0
b(s; Y)dBs +
Z t0
b(s; Y)1( )ds
+
Z t0
c(s ; Y ; (s; Y))ds Z t0
c(s ; Y ; (s; Y))ds: (6.8)
Since Mt is a supermartingale, it has a unique Doob-Meyer decomposition as a
martingale minus an increasing process. In other words, the term
Z t0
b(s; Y)
1
(
) c(s ; Y ;
(s; Y)) + c(s ;Y;(s; Y))
ds (6.9)
must be an increasing process. This implies that, at each (t; Y);
b(t; Y)1( ) c(t ; Y ; ) +c(t ; Y ; ) (6.10)
must be maximized a.e. at :Since c(; ; )is an increasing and convex function of;we have that the necessary and sucient condition for to be maximized at is
that
b(t; Y) =c0(t ; Y ; )=rc(t ; Y ; ): (6.11)Putting b(t; Y)in equation (6.11) into equation (6.4) and setting t= 1we immediately
have
X(Y) = v+
Z 10
c(s ; Y ; )ds +
Z 10
c0(t ; Y ; )dB
s : (6.12)
Conversely, if equation (6.12) is satised, then
Mt = v+E[
Z 10
c0(s ; Y ; (s; Y))dB
s jFt]: (6.13)
Since
M1 = v+
Z 10
c0(s ; Y ; (s; Y))dB
s ;
we immediately have
b(t; Y) =c0(s ; Y ; (s; Y)): (6.14)
Putting equation (6.14) into equation (6.10) we obtain that must be maximized at
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uniquely. This nishes the proof of proposition 1.
Proof of Lemma 1:
Proof. First note that if2H; then12L2++:This is true by denition ofH .Next, we show that if 12 L2++; then there exists a unique 2 H such that
f() =1:
Let tbe the martingale generated by 1. From martingale representation theorem
there exists a uniqueFt-predictable process vector b (t; Y) such that b (t; Y)2H2anddt = b
(t; Y)dBt:Let zt = ln t: From Ito lemma, we have
dzt =12jjb
(t; Y)
tjj2dt + b
(t; Y)
tdBt:
Note that
1= exp z1 = exp(12
Z 10
jjb (t; Y)
tjj2dt +
Z 10
b (t; Y)
tdBt)
Dene (t; Y) = b(t;Y)
t: Then 12 L2++ implies that zt is an Ito process. As a
result, (t; Y) satises condition (2.2). Combining with the factb (t; Y)2 H2; wenish the proof.
Proof of Proposition 3:
Proof. We use1as the decision variable to maximize (4.2). Note that the direction
derivative dd
alongt will be
d
dj =1(
Z 10
1ds +
Z 10
dBs) =1
Z 10
1T
dBs : (6.15)
Let
1 = Z 1
0
1T dBs : (6.16)As a result, the rst-order necessary condition implies that
Z
1u [S(Y1) h(X(Y))] dP =Z
1u0h0Z 10
c00ds +
Z 10
c00dBs
dP
=
Z
Z 10
c00dB s
u0h0dP =
Z
Z 10
c00dB s
Z 10
rJsdBs
dP
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=
Z
Z 10
c00rJsds
dP =
Z
1
Z 10
rJsTc00dB s
dP: (6.17)
Hence we have the required result
u [S(Y1) h(X(Y))] =Z 10
rJsTc00dB s +0: (6.18)
Proof of Lemma 2
Proof. Without loss of generality, let two outcome processes Y1 and Z1 be such that
Y1 = Z1+ B01 where Z1 andB
01 are two independent, standard Brownian motions.
Thus, the variance ofY1 is1 + 2: Let u(Y1) and u(Z1) be the principals optimal
payo when Y1 andZ1 are the underlying outcome process respectively. We need to
show that u(Y1) < u(Z1): This result directly comes from lemma 3, which impliesthat when both Z1 andB
01 are observed and can be used in the design of contract,
noise information B01 is not used. As a result, u(Y1)< u(Z1):
As for the result of the second part (without loss of generality, we set m= 1 and
b= 0); we rst note that, from equation (5.5), the optimal sharing rule converges to
w= Y1 when goes to zero, where = u(v;k; 0) is the principal optimal payoat= 0:Note that
Y1 =
Z 10
(s; Y) ds when= 0:
Given this, we have that, from the agents maximization problem, the optimal action
=(t; Y) = 1
kv0(Y1 ) (6.19)
is a constant. ThereforeY1 = ; w = . To determine = u(v;k; 0); we note
that the participation constraint gives
v ( ) = v+12
k2; (6.20)
which implies that= h(v+1
2k2): (6.21)
As a result, equation (6.19) exactly becomes the principals rst-best condition, which
is dened by
max
Z 10
dt h
v+1
2k
Z 10
2dt
(6.22)
Proof of Proposition 4:
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Proof. This is because, by Ito lemma,
g(t)
t= Z
t
0
(1
2g00s s g0s+
gs
s)jj
jj2ds + Z
t
0
(g0s gs
s)
dBs: (6.23)
Under the new measure P; we have
g(t)
t=
Z t0
1
2g00s sjj
jj2ds +
Z t0
(g0s gs
s)
dBs : (6.24)
As a result,
k2Z
g(1)dP =k 2
Z
g(1)
1dP =k 2
Z
dPZ 10
1
2g00s sjj
jj2ds
=
Z
dPZ 10
c(t;;Y)ds= c(1): (6.25)
Using the fact t is a martingale, the case of value-weighted separable costs of
probability measure can be proved similarly.
Proof of Lemma 3:
Proof. Equations (5.29)-(5.30) implies that
S=E[SjB1 ] +
k2+k1Cov(Y1; Z1)
2Z
Z1: (6.26)
Since Z1 is orthogonal to B1 ;we have E[1Z1] = 0: Therefore
E[1S] =E[1E[SjB1 ]] ; (6.27)
and, by assumption on S; we must have
E[S
jF1 ] =E[S
jB1 ] : (6.28)
Combining equation (6.27) with (6.28), we immediately have
E[1S] =E[1E(SjF1 )] =E[E(1jF1 ) E(SjB1)] : (6.29)
To show the second result, we rst note that, following the approach developed
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by Mirrlees and Zhou (2006a), X(!) implements 1 if and only if
X(!) = v+kjjjj2
2Zln 1+; (6.30)
where is a random variable that is orthogonal to all 1:Therefore, there are an in-
nite number of incentive schemes that implement an action 1;which are represented
by equation (6.30): Clearly,
E
1h(v+k
jjjj22Z
ln 1+)
E
1h(v+k
jjjj22Z
ln 1)
(6.31)
for all : In other words, the incentive scheme X(!) in equation (6.30) with = 0
is the one that implements 1 with the minimum expected cost. Without loss of
generality, we will set = 0 in equation (6.30) in solving the optimal sharing rule.
Let
h(1) =1h(v+kjjjj2
2Zln 1): (6.32)
It is easy to show that h(1)is a convex function of1: Given this, we have
E[1h(X)] =Eh
h(1)i
= Eh
Eh
h(1)jF1ii
Eh^h(E(1jF1 ))i= E[E(1jF1 ) h(E(XjF1 ))] ; (6.33)which nishes the proof .
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