Zhou 2006 (Principal Agent Analysis in Continuous Time)

8/13/2019 Zhou 2006 (Principal Agent Analysis in Continuous Time)

1/39

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.


2/39

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

1


3/39

(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

2


4/39

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,

3


5/39

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)

4


6/39

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


7/39

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

6


8/39

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)


9/39

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)


10/39

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.

9


11/39

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.

10


12/39

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.


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++:

11


13/39

=

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)

12


14/39

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)


13


15/39

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)

14


16/39

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

15


17/39

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)

16


18/39

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)

17


19/39

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


20/39

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)


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 :

19


21/39

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)

20


22/39

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)

21


23/39

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)

22


24/39

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


25/39

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.

24


26/39

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.

25


27/39

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.

26


28/39

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.

27


29/39

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

28


30/39

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

29


31/39

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).

30


32/39

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)

31


33/39

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

32


34/39

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. 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

33


35/39

=

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)


34


36/39

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

35


37/39

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 .

36


38/39

References

Bolton, Patrick and Mathias Dewatripont, 2005, Contract Theory, The MIT Press,

Cambridge, Massachusetts

Cadenillas, A., J. Cvitanic and F. Zapatero, 2005 Dynamic Principal-Agent Prob-

lems with Perfect Information, working paper, USC

Cvitanic, Jaksa, Xuhu Wan and Jianfeng Zhang, 2005, Continuous-Time Principal-

Agent Problems with Hidden Action: The Weak Formulation, working paper,

USC

Diamond, Peter, 1998, On the Near Linearity of Optimal Compensation, Journal of

Political Economy, 106, 931-957

Dybvig, Philip, Heber Farnsworth and Jennifer Carpenter, 2004, Portfolio Perfor-

mance and Agency, Working Paper

Grossman, Sanford and Oliver Hart, 1983, An Analysis of the Principal-Agent Prob-

lem, Econometrica, 51, 7-45

Holmstrom, Bengt, 1979, Moral Hazard and Observability, Bell Journal of Eco-

nomics, 10, 74-91

Holmstrom, Bengt and Paul Milgrom, 1987, Aggregation and Linearity in the Pro-

vision of Intertemporal Incentives, Econometrica, 55, 303-328

Holmstrom, Bengt and Paul Milgrom, 1991, Multitask Principal-Agent Analyses:

Incentive Contracts, Asset Ownership, and Job Design, Journal of Law, Eco-

nomics, and Organization, 7, 24-52

Holmstrom, Bengt and Jean Tirole, 1993, Market Liquidity and Performance Mon-

itoring, Journal of Political Economy, 101, 678-709

Jewitt, Ian, 1988, Justifying the First-Order Approach to Principal-Agent Problems,

Econometrica, 5, 1177-1190

Li, Tao and Yuqing Zhou, 2006, Optimal Contracts in Portfolio Delegation: The

Case of Complete Markets, Working Paper, The Chinese University of Hong

Kong

37


39/39

Milgrom, Paul, 1981, Good News and Bad News: Representation Theorem and

Applications, Bell Journal of Economics 12, 380-391

Mirrlees, James, 1974, Notes on Welfare Economics, Information, and Uncertainty

in Balch, McFadden and Wu (ed.), Essays on Economic Behavior Under Uncer-

tainty, North Holland, Amsterdam.

Mirrlees, James, 1975, The Theory of Moral Hazard and Unobservable Behavior

Part I, Nueld College, Oxford, Mimeo

Mirrlees, James, 1979, The Implications of Moral Hazard for Optimal Insurance,

Seminar given at Conference hel in honour of Karl Borch, Bergen, Norway,

Mimeo

Mirrlees, James and Yuqing Zhou, 2006a, Principal-Agent Problems Revisited, Part

I: The Static Models, working Paper, The Chinese University of Hong Kong

Mirrlees, James and Yuqing Zhou, 2006b, Principal-Agent Problems Revisited, Part

II: The Dynamic Models, working Paper, The Chinese University of Hong Kong

Palomino, Frederic and Andreu Prat, 2003, Risk Taking and Optimal Contracts for

Money Manager, Rand Journal of Economics, 34, 113-137

Rogerson, William, 1985, The First-Order Approach to Principal-Agent Problems,Econometrica, 53, 1357-1367

Ross, Stephen, 1973, The Economic Theory of Agency: The Principals Problem,

American Economic Review, 63, 134-139

Sannikov, Yuliy, 2004, A Continuous-Time Version of the Principal-Agent Problem,

Working paper, UC Berleley

Shavell, Steven, 1979, Risk Sharing and Incentive in the Principal and Agent Rela-

tionship, Bell Journal of Economics, 10, 55-73

Sinclair-Desgagne, Bernard, 1994, The First-Order Approach to Multi-signal Principal-

agent Problems, Econometrica, 62, 459-465

Williams, Noah, 2005, On Dynamic Principal-Agent Problems in Continuous-Time,

ki P i t U i it

Zhou 2006 (Principal Agent Analysis in Continuous Time)

Documents

Transcript of Zhou 2006 (Principal Agent Analysis in Continuous Time)