NASH EQUILIBRIA IN THE CONTINUOUS-TIME PRINCIPAL-AGENT PROBLEM
WITH MULTIPLE PRINCIPALS
By
Lening Kang

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Statistics - Doctor of Philosophy
2013

ABSTRACT
NASH EQUILIBRIA IN THE CONTINUOUS-TIME PRINCIPAL-AGENT
PROBLEM WITH MULTIPLE PRINCIPALS
By
Lening Kang
In Chapter 1, we review some basic results of backward stochastic differential equation(BSDE) to prepare for the applications in Chapter 2. BSDE has proven to be a powerful
tool in financial mathematics. It was first introduced as a tool to price contingent claims and
was later used to model utility functions. The value of a recursive utility function is essentially the solution of a BSDE. We present two versions of comparison lemma for BSDE. The
latter one allows quadratic growth in volatility term which is important for its applications
in Chapter 2.
In Chapter 2, we study the principal-agent(owner-manager) problem with moral hazard
in continuous time with a Brownian filtration, recursive preferences, and multiple principals
(one agent for each principal). In simple terms, the problem is defined in two levels, first for
the agents and then the principals. The key to the definition of the problem in both levels
is Nash equilibrium. The Nash equilibrium among the agents is straintforward and comes
through their competing(or cooperative) efforts. The Nash equilibrium among the principals
is more complicated, because the connection among them is indirect and comes only through
the agents’ Nash equilibrium in efforts. The objective is to characterize each principal’s
equilibrium control over his/her agent, taking into account the control is constrained by the
agents’ Nash equilibrium in efforts.
In technical terms, different principals’ problems are connected, because the effort of each
principal’s agent affects the common probability measure, and therefore one agent’s effort can

impact the cash-flow drifts of all the principals. This could capture, for example, the impact
of innovations by agents of one firm on the cash-flow prospects of competing firms.

The

externality of each agent’s effort results in interdependence among the principals’ optimal
contracting problems. For the class of preferences we consider, solving the equilibrium reduces
to computing a system of linked subjective cash-flow value processes, one for each principal.
We show that the system has a closed-form solution, when each principal’s cash flow is driven
by an affine-yield state process. Each principal’s optimal pay policy amounts to choosing the
component of the subjective cash-flow volatility to transfer to the agent (that is, a volatility
sharing rule). The optimal sharing rules are simple functions of each principal’s own cashflow volatility in the case when the impact of aggregate effort on drifts is additive, but are
generally functions of all the principals’ cash-flow volatilities when the impact of effort on
the drift change is diminishing in aggregate effort.
solutions to illustrate.

We provide a number of closed-form

TABLE OF CONTENTS
Chapter 1 Backwards Stochastic Differential Equation (BSDE) . . .
1.1 Introduction: What is BSDE? . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Existence and Uniqueness of a Solution for BSDEs with Linear
1.2 Comparison Lemmas for BSDEs . . . . . . . . . . . . . . . . . . . . .
1.2.1 Comparison Lemmas for BSDEs with Linear Growth . . . . .
1.2.2 Comparison Lemmas for BSDEs with Quadratic Growth . . .
1.3 Application to Economics . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Recursive Utility Functions . . . . . . . . . . . . . . . . . . .
1.3.2 European and American Contingent Claims Valuation . . . . .

. . . .
. . . .
Growth
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .

Chapter 2 Principal-Agent Problem with Multiple Principals . . .
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 The Setting and Problem . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 The Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Outline of the Problem and Solution . . . . . . . . . . . . .
2.2.3 Regularity Conditions and Feasibility . . . . . . . . . . . . .
2.3 Agent Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 A Necessary and Sufficient Condition for Agent Equilibrium
2.3.2 Some Basic Examples of Agent Equilibrium . . . . . . . . .
2.3.3 Constant Elasticity of Substitution(CES) . . . . . . . . . . .
2.4 Principal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Closed-Form Solution with Cash Flows Driven by State Processes .
2.6 Additive Measure Change and Quadratic Penalties . . . . . . . . .
2.7 Diminishing Returns to Effort (Concave Φ) . . . . . . . . . . . . . .
2.7.1 Power Measure Change . . . . . . . . . . . . . . . . . . . . .
2.7.2 Quadratic Penalty and Cobb-Douglas Measure Change . . .
2.7.3 Absolute Effort Penalty and Square-Root Measure Change .

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1
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70
86
92
105
111
118
122

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
.1 Appendix 1: Proofs Omitted from the Text . . . . . . . . . . . . . . . . . . . 128
.2 Appendix 2: Derivation of Examples . . . . . . . . . . . . . . . . . . . . . . 147
BIBLIOGRAPHY

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iv

. . . . . . . . . . . . . . . . . 155

Chapter 1
Backwards Stochastic Differential
Equation (BSDE)
Backward stochastic differential equation has proven to be a powerful tool in stochastic
analysis in the last twenty years. It has been widely applied in the problems of stochastic
control and mathematical finance. Its general form was first introduced by Paradoux and
Peng in 1990(see Pardoux and Peng (1990)). A lot work has been done since then in both
theoretical aspects and applications, such as E.Pardoux and Peng (1992), Peng (1991), Peng
(1992) and Peng (1993). In particular, Antonelli (1993) extends BSDE to a forward-backward
form(FBSDE). The collection of papers Karoui and Mazliak (1997) summarized some of the
earlier results of BSDE with linear growth. More recently, BSDE with quadratic growth
was studied in a series of papers(Kobylanski (2000), Briand and Hu (2006), Briand and Hu
(2008), and Delbaen, Hu, and Richou (2009) etc.). In this chapter, we will review some
fundamentals of BSDE theory and prove a new version of comparison lemma, which will
serve as a primary tool for the applications in chapter 2.
All uncertainty is generated by d-dimensional standard Brownian motion B over the finite
time horizon [0, T ], supported by a probability space (Ω, F, P ). {Ft : t ∈ [0, T ]} is the the
augmented filtration(satisfies the usual hypotheses) generated by B. Let B([0, T ]) denote
the Borel σ-field on [0, T ]. Let λ be the Lebesgue measure on [0, T ] and P ⊗ λ on Ω × [0, T ].

1

The qualification ”P ⊗ λ almost everywhere” is omitted throughout.
For any subset S of Euclidean space, let L (S) denote the set of S-valued FT measurable
random variables and L (S) denote the set of S-valued progressively measurable processes1
w.r.t. (Ω × [0, T ], F × B([0, T ]), P ⊗ λ).
For this chapter, we will use the following spaces:
Let Lp (S), Lp (S) and Sp (S) denote respectively:

Lp (S) = {x ∈ L (S) : E (|x|p ) < ∞} ,
T

Lp (S) =

x ∈ L (S) : E
0

Sp (S) =

x ∈ L (S) : E

|xt |p dt < ∞ ,

sup |x|p

< +∞ , 1 ≤ p < ∞.

0≤t≤T

where |·| denotes Euclidean norm.
For any real-valued random or deterministic matrix Z, we will let Z denote its transpose.

1.1

Introduction: What is BSDE?

A BSDE is an equation of the following type:
T

Yt = ξ +

T

f (s, Ys , Zs )ds −
t

t

Zs dBs ,

0 ≤ t ≤ T.

(1.1)

or equivalently
dYt = −f (t, Yt , Zt )dt + Zt dBt ,
1 In

YT = ξ.

the setting of augmented Brownian filtration, progressively measurable processes are
predictable.

2

where:
• the terminal value ξ : Ω → R is FT measurable.
• the aggregator function f : Ω × [0, T ] × R × Rd → R is progressively measurable with
respect to P × B(R) × B(Rd ) where P is the predictable σ-field on Ω × [0, T ].
The solution is a pair of progressively measurable processes (Y, Z) valued in R × Rd that
satisfies: t → Yt is continuous and Zt ∈ L2 (Rd ).
For PDEs in deterministic settings, in most situations a backward formulation can be
transformed into a forward one through reverse of time argument t. However, we can not
simply reverse time argument to transform a BSDE into SDE because of the measurability
requirement imposed on the solution. Moreover, unlike in the SDE case where the solution
has only one component, a solution of a BSDE consists of two components Y and Z.

1.1.1

Existence and Uniqueness of a Solution for BSDEs with Linear Growth

In the case of f (s, Ys , Zs ) = 0, s ∈ [0, T ], (1.1) reduces to martingale representation theorem(see the examples after Theorem 1 below). For any process Z ∈ L2 (Rd ), we know that
t
Mt = 0 Zs dBs , t ∈ [0, T ] is a martingale w.r.t. Ft and MT ∈ L2 (R). It is natural to ask

whether the converse is true. The answer is Yes and this result is the famous martingale
representation theorem.
Theorem 1 (Martingale Representation Theorem for L2 -Martingales) Suppose that
Mt is a martingale w.r.t. Ft and MT ∈ L2 (R). Then there exists a unique process Z ∈

3

L2 (Rd ) such that
t

Mt = E[MT ] +

0

Zs dBs ,

t ∈ [0, T ].

Remark 1 The theorem above can be extended to the case where Mt is a local martingale.
Before we proceed to the general case of (1.1), let us first look at two instructive examples,
which will show how to use Theorem 1 to get a solution to (1.1):
(a) In the case of f (s, Ys , Zs ) = 0, s ∈ [0, T ] and ξ ∈ L2 (R), there is a unique solution to
(1.1). We get the solution by using the theorem above. To see this, let Yt = Et (ξ), t ∈ [0, T ]
, where Et (·) ≡ E(· | Ft ). It is well-known that Yt is an L2 -martingale, so we let Mt = Yt
and the theorem gives us the existence of Zt ∈ L2 (Rd ). To see that (Yt , Zt ) is a solution
t
to (1.1), note that YT = ξ and Yt = E(ξ) + 0 Zs dBs , t ∈ [0, T ]. In this case, Y ∈ L2 (R),

because Y is L2 -martingale and T is finite.
(b) The case of f (s, Ys , Zs ) = 0, s ∈ [0, T ] can be easily extended to f (s, Ys , Zs ) =
f (s), s ∈ [0, T ], i.e. the aggregator f does not depend on the solution (Y, Z). We also
T
assume f (s) ∈ L2 (R). Let Mt = Et ( 0 f (s)ds + ξ), then Mt is an L2 -martingale.(Note
T
that f (s) ∈ L2 (R) implies 0 f (s)ds ∈ L2 (R).) We get Z ∈ L2 (Rd ) by the theorem, i.e.
t
t
Mt = E(MT ) + 0 Zs dBs . The solution Y is given by Yt = Mt − 0 f (s)ds = Et ( tT f (s)ds +
T
ξ), t ∈ [0, T ]. Note Y ∈ L2 (R), because |Yt | ≤ Et ( 0 |f (s)|ds + |ξ|), t ∈ [0, T ] and the

right-hand side of the inequality is L2 martingale.
In the general case(i.e. the aggregator f depends on the solution (Y, Z)), the theorem
below taken from Karoui and Mazliak (1997) proves the existence and uniqueness of a solution to (1.1) under certain conditions. Its proof will be based on a fixed point theorem
in addition to the martingale representation theorem. We will assume the following on the
aggregagor and terminal for Theorem 2 below.
4

Assumption 1 The aggregator and terminal satisfy: ξ ∈ L2 (R), f (·, 0, 0) ∈ L2 (R) and f is
uniformly Lipschitz in Y and Z, i.e. there exists a constant C > 0 such that ∀(y1 , z1 ), (y2 , z2 ) ∈
R × Rd
|f (t, y1 , z1 ) − f (t, y2 , z2 )| ≤ C(|y1 − y2 | + |z1 − z2 |).
Theorem 2 Assume 1. Then there exists a unique pair (Y, Z) ∈ L2 (R)×L2 (Rd ) that solves
(1.1).
Sketch of Proof. For any β > 0 and ϕ ∈ L2 (S), let ϕ β = E

T βt
2
0 e |ϕt | dt

. We will let

L2,β (S) denote the space of L2 (S) endowed with the norm · β . The proof of this theorem
is based on a fixed point theorem for a contraction mapping from L2,β (R) × L2,β (Rd ) into
itself.
For any (y, z) ∈ L2,β (R) × L2,β (Rd ), we will consider the following BSDE:
T

T

Yt = ξ +

f (s, ys , zs )ds −
t

t

Zs dBs ,

0 ≤ t ≤ T.

By the uniform Lipschitz condition, we have |f (s, ys , zs ) − f (s, 0, 0)|2 ≤ 2C(|ys |2 + |zs |2 ). By
the assumptions f (·, 0, 0) ∈ L2,β (R) and (y, z) ∈ L2,β (R) × L2,β (Rd ), we get f (s, ys , zs ) ∈
L2,β (R). By the example (b) above, there’s a unique solution (Y, Z) ∈ L2,β (R) × L2,β (Rd )
to the equation.
We will denote the mapping (y, z) → (Y, Z) by (Y, Z) = T (y, z). For any (y 1 , z 1 ), (y 2 , z 2 ) ∈
L2,β (R) × L2,β (Rd ), let (Y 1 , Z 1 ) = T (y 1 , z 1 ) and (Y 2 , Z 2 ) = T (y 2 , z 2 ). Proposition 2.2 of
Karoui and Mazliak (1997) gives the estimate:
2(2 + T )C 2
( y1 − y2 2 + z1 − z2 2 )
Y1 − Y2 2 + Z1 − Z2 2 ≤
β
β
β
β
β

5

Choosing 2(2 + T )C 2 < β, we see that T is a contraction mapping from L2,β (R) × L2,β (Rd )
into itself. Thus there exists a fixed point (Y ∗ , Z ∗ ) ∈ L2,β (R) × L2,β (Rd ), i.e. T (Y ∗ , Z ∗ ) =
(Y ∗ , Z ∗ ), which is the unique solution to the BSDE (1.1).

1.2

Comparison Lemmas for BSDEs

Since BSDEs are difficult to solve explicitly, a typical comparison lemma for BSDEs is a
useful tool in analyzing their solutions. It compares the solutions of BSDEs with different
aggregators and terminal values. Upon inspecting (1.1), it is easy to get the intuition that,
when the aggregator function and terminal value increase, the solution Y should also increase.
A comparison lemma makes this idea precise under certain conditions. In this section, we
will review a comparison lemma for BSDEs with linear growth that is a variation of Theorem
2.5 in Karoui and Mazliak (1997). We will also present a comparison lemma for BSDEs with
quadratic growth in volatility that extends the result in Briand and Hu (2008). BSDEs with
quadratic growth are heavily used in the analysis of recursive utility functions in economics,
as will be seen in section 1.3 and chapter 2. When it comes to BSDEs with quadratic growth,
uniqueness of the solution is usually proved by using a comparison lemma.

1.2.1

Comparison Lemmas for BSDEs with Linear Growth

The comparison lemma in this subsection is a direct result of the proposition below. It
deals with a linear BSDE that has an explicit solution. The solution depends on an adjoint
process, which in turn is the solution of a forward SDE.
Proposition 1 Let (β, γ) be a pair of bounded progressively measurable processes valued in

6

(R, Rd ), ϕ ∈ L2 (R) and ξ ∈ L2 (R). The linear BSDE(LBSDE)

dYt = − ϕt + βt Yt + Zt γt dt + Zt dBt , YT = ξ.

(1.2)

has a unique solution (Y, Z) ∈ L2 (R) × L2 (Rd ). Y is given explicitly by:
T

Γt Yt = Et ξΓT +

Γs ϕs ds

(1.3)

t

where the strictly positive adjoint process satisfies:

dΓt = Γt βt dt + γt dBt , Γ0 = 1.

(1.4)

Moreover, if ξ ≥ 0 and ϕ ≥ 0, then Yt ≥ 0, t ∈ [0, T ]. If in addition, Y0 = 0, then we have
ϕ = 0 and Yt = 0, 0 ≤ t ≤ T .
Proof. By Theorem 2, there exists a unique solution (Y, Z) ∈ L2 (R) × L2 (Rd ) to (1.2). By
t
t
t
Ito’s formula, we have Γt Yt + 0 ϕs Γs ds = Y0 + 0 Γs Ys γs + Γs Zs dBs , so Γt Yt + 0 ϕs Γs ds

is a local martingale. By (1.4), the adjoint process is
t

Γt = exp

0

γt dBt −

t
|γt |2
dt +
βt dt
2
0

Because β and γ are bounded processes, Γ ∈ L2 (R) and by Doob’s inequality sup0≤t≤T |Γt | ∈
L2 (R).

7

Using an equivalent form of BSDE (1.2) Yt = ξ + tT ϕs + βs Ys + Zs γs ds − tT Zs dBs ,
we get
T

sup |Yt | ≤ |ξ| +
0≤t≤T

0

T

|ϕs + βs Ys + Zs γs |ds + sup |
0≤t≤T

Zs dBs |.

t

Because ϕ, Y ∈ L2 (R), Z ∈ L2 (Rd ) and β, γ are bounded, we have by Holder’s inequality
T
0

|ϕs + βs Ys + Zs γs |ds ∈ L2 (R).

We also have

t

0≤t≤T

t

T

T

sup |

Zs dBs | ≤ |

0

Zs dBs | + sup |
0≤t≤T

0

Zs dBs |.

By Doob’s inequality, we get
t

E| sup
0≤t≤T

0

Zs dBs |2 ≤ 4E|

T
0

Zs dBs |2 = 4E

T

|Zs |2 ds < ∞,

0

so sup0≤t≤T | tT Zs dBs | ∈ L2 (R). Thus we have sup0≤t≤T |Yt | ∈ L2 (R).
By Holder’s inequality, we also have
T

sup |Ys | · sup |Γs | and
0≤s≤T

0

0≤s≤T

|ϕs Γs |ds ∈ L1 (R)

Thus
t

E sup
0≤s≤T

Γt Yt +

T

ϕs Γs ds
0

≤ E sup |Ys | · sup |Γs | + E
0≤s≤T

0≤s≤T

8

|ϕs Γs |ds < ∞
0

t
We can conclude that the local martingale Γt Yt + 0 ϕs Γs ds is a uniformly integrable mar-

tingale, so we have:
T

Γt Yt = Et ξΓT +

Γs ϕs ds
t

Because the process Γ is strictly positive, it follows directly from the above equation that,
if ξ ≥ 0 and ϕ ≥ 0, then Yt ≥ 0, t ∈ [0, T ]. If in addition Y0 = 0, then the expectation
of the non-negative random variable ξΓT + tT Γs ϕs ds is 0, so we have ξ = 0, ϕ = 0 and
Yt = 0, 0 ≤ t ≤ T .

As a direct result, we present the comparison lemma below, for which

we have the same assumption on the BSDEs as in Theorem 2. This guarantees the existence
of a unique solution for the equations in the theorem.
Theorem 3 (Comparison lemma for BSDEs with linear growth) Let (f i , ξ i ),

i =

1, 2 be the aggregator and terminal value of the BSDEs

i
i
dYti = −f i (t, Yti , Zt )dt + Zt dBt ,

i
YT = ξ i

that satisfy Assumption 1.
1
1
2
2
If ξ 1 ≥ ξ 2 and f 1 (t, Yt1 , Zt ) ≥ f 2 (t, Yt1 , Zt ) or f 1 (t, Yt2 , Zt ) ≥ f 2 (t, Yt2 , Zt ), then we

have Yt1 ≥ Yt2 , 0 ≤ t ≤ T . If in addition Y01 = Y02 we have then Yt1 = Yt2 , 0 ≤ t ≤ T .
Proof. Let δY = Y1 − Y2 and δZ = Z1 − Z2 , thus (δY, δZ) will satisfy the LBSDE:

d(δYt ) = −δft dt + δZt dBt , δYT = ξ1 − ξ2 .

9

(1.5)

where

δft

=

2
1
f 1 (t, Yt1 , Zt ) − f 2 (t, Yt2 , Zt )

1
1
=1 f 1 (t, Yt1 , Zt ) − f 2 (t, Yt1 , Zt )
1
1
f 2 (t, Yt1 , Zt ) − f 2 (t, Yt2 , Zt )
δYt
δYt
t
t
1
2
(f 2 (t, Yt2 , Zt ) − f 2 (t, Yt2 , Zt ))δZt
+IZ 1 =Z 2
δZt
|δZt |2
t
t
1
1
f 1 (t, Yt1 , Zt ) − f 1 (t, Yt2 , Zt )
=2 IY 1 =Y 2
δYt
δYt
t
t
1
2
(f 1 (t, Yt2 , Zt ) − f 1 (t, Yt2 , Zt ))δZt
δZt
+IZ 1 =Z 2
|δZt |2
t
t

+IY 1 =Y 2

2
2
+f 1 (t, Yt2 , Zt ) − f 2 (t, Yt2 , Zt ).

By proposition 1, the LBSDE (1.5) has a unique solution (δY, δZ) that satisfies δYt ≥ 0, 0 ≤
i
i
t ≤ T , if ξ 1 − ξ 2 ≥ 0 and f 1 (t, Yti , Zt ) − f 2 (t, Yti , Zt ) ≥ 0, for i = 1 or 2. (Use equality
1
1
1 for f 1 (t, Yt1 , Zt ) − f 2 (t, Yt1 , Zt ) ≥ 0 and equality 2 in the other case. Also, use the

Lipschitz assumption to bound the coefficients of δY and δZ). If in addition δY0 = 0, then
δYt = 0, 0 ≤ t ≤ T .

1.2.2

Comparison Lemmas for BSDEs with Quadratic Growth

The comparison result in Theorem 3 relies on the assumption that both aggregator functions
are uniformly Lipschitz in the corresponding arguments. In typical applications, we often
need to deal with the kind of BSDEs whose aggregator is quadratic in the volatility term,
such as recursive utility function with quadratic volatility penalty(see next subsection). Some
important properties of recursive utility are proved by using comparison lemmas. Reference
Kobylanski (2000) was the first to prove a comparison lemma for BSDEs with quadratic
10

growth in volatility and bounded terminal values. Later Briand and Hu (2006), Briand and
Hu (2008) and Delbaen, Hu, and Richou (2009) extended Kobylanski (2000) by allowing unbounded terminal values with exponential moments(the moment generating function is finite
on R). On the other hand, they added the assumption that the aggregator is concave(convex)
in the volatility term. In this subsection, we prove a comparison lemma(Theorem 4) for BSDEs with quadratic growth in volatility that extends the result of Briand and Hu (2008).
ˆ
This lemma emphasizes the symmetry between f and f (the two aggregators in the two BSDEs that we compare) with regard to assumptions (a), (b) and (c) below and also allows
ˆ
either f or f to be concave or convex in the volatility term. In chapter 2, we will apply this
theorem to solve the principal-agent problem with multiple principals. In Theorem 4 below,
we will use the following assumption on the aggregator and terminal.
Assumption 2 There exist two constants γ > 0, β > 0 and a process α(t) valued in R+
such that,
(a) ∀t ∈ [0, T ] and y ∈ R, z → f (t, y, z) is convex or concave;
(b) ∀(t, z) ∈ [0, T ] × Rd , and (y, y ) ∈ R2 ,
ˆ
(c) ∀(t, y, z) ∈ [0, T ] × R × Rd ,

|f (t, y, z) − f (t, y , z)| ≤ β|y − y |;
ˆ
ˆ

|f (t, y, z)| ≤ α(t) + β|y| + γ |z|2 ;
2

T
(d) The random variables 0 α(t)dt and |YT | have exponential moments of all orders.

ˆ ˆ
Theorem 4 Suppose (Yt , Zt ) and Yt , Zt solve

dYt = −f (t, Yt , Zt ) dt + Zt dBt ,

YT = f (T ) ,

ˆ
ˆ
ˆ ˆ
ˆ
dYt = −f t, Yt , Zt dt + Zt dBt ,

ˆ
ˆ
YT = f (T )

11

ˆ
ˆ
where f, f : Ω × [0, T ) × R × Rd → R and f (T ) , f (T ) : Ω → R. For all ω, t ∈ Ω × [0, T ), let
ˆ
either (f, YT ) satisfies Assumption 2 and sup0≤t≤T Yt has exponential moments of all orders
ˆ ˆ
or (f , YT ) satisfies Assumption 2 and sup0≤t≤T Yt has exponential moments of all orders
then




ˆ
(i) YT ≤ YT


 (ii) f (t, y, z) ≤ f (t, y, z),
ˆ

∀(t, y, z) ∈ [0, T ] × R × Rd

ˆ
implies Yt ≤ Yt , ∀t ∈ [0, T ] 2 3 .
Proof of Theorem 4.

Suppose that f satisfies Assumption 2 and f is concave in the

ˆ
ˆ
ˆ ˆ
volatility term. ∀θ ∈ (0, 1), let us define Ut = θYt − Yt , Vt = θZt − Zt and δft = f (t, Yt , Zt ) −
ˆ ˆ ˆ
f (t, Yt , Zt ) ≤ 0.
The aggregator of Ut could be written as: 4

ˆ ˆ ˆ
ˆ
ˆ
ˆ ˆ
θf (t, Yt , Zt ) − f (t, Yt , Zt ) = θf (t, Yt , Zt ) − f (t, Yt , Zt ) + f (t, Yt , Zt ) − f (t, Yt , Zt )(1.6)
ˆ ˆ ˆ
ˆ ˆ
+f (t, Yt , Zt ) − f (t, Yt , Zt )
ˆ
ˆ ˆ
f satisfies Assumption 2 (resp. f (ω, t, ·)), then it is enough to assume f (t, Yt , Zt ) ≤
ˆ(t, Yt , Zt )(resp. f (t, Yt , Zt ) ≤ f (t, Yt , Zt )). Here we assume uniform dominance over [0, T ]×
ˆ
ˆ ˆ
f
R × Rd just for presentation convenience.
3 Note that if the inequalities (i) and (ii) are reversed, then the inequality in the conclusion
is also reversed.
4 If f satisfies Assumption 2 and f is concave in the volatility term, we can define
ˆ
ˆ
ˆ
ˆ ˆ ˆ
δft = f (t, Yt , Zt ) − f (t, Yt , Zt ) ≤ 0 and write θf (t, Yt , Zt ) − f (t, Yt , Zt ) = θf (t, Yt , Zt ) −
ˆ(t, Yt , Zt ) + θf (t, Yt , Zt ) − f (t, Yt , Zt ) + f (t, Yt , Zt ) − f (t, Yt , Zt ). The rest of the steps can
ˆ
ˆ
ˆ
ˆ ˆ ˆ
ˆ
ˆ
θf
be carried out accordingly.
The case of f satisfies Assumption 2 and f is convex was covered by Briand and Hu (2008).
ˆ
ˆ
ˆ
If f satisfies Assumption 2 and f is convex in volatility term, we can define Ut = Yt −θYt , Vt =
ˆ
ˆ ˆ ˆ
ˆ
θZt − θZt and δft = f (t, Yt , Zt ) − f (t, Yt , Zt ) ≤ 0. We can write f (t, Yt , Zt ) − θf (t, Yt , Zt ) =
ˆ(t, Yt , Zt ) − f (t, Yt , Zt ) + f (t, Yt , Zt ) − θf (t, Yt , Zt ) and carry out the steps in Briand
ˆ ˆ
ˆ ˆ
ˆ ˆ ˆ
δft + f
and Hu (2008) accordingly.
2 If

12

We can rewrite

ˆ
ˆ ˆ
ˆ
ˆ
ˆ
ˆ ˆ
f (t, Yt , Zt ) − f (t, Yt , Zt ) = f (t, Yt , Zt ) − f (t, θYt , Zt ) + f (t, θYt , Zt ) − f (t, Yt , Zt )
ˆ
ˆ
= f (t, Yt , Zt ) − f (t, θYt , Zt ) + a(t)Ut

ˆ
ˆ ˆ
where a(t) = f (t, θYt , Zt ) − f (t, Yt , Zt ) /Ut , when Ut = 0 and a(t) = β, otherwise. By
(b), we have |a(t)| ≤ β and

ˆ
ˆ ˆ
f (t, Yt , Zt ) − f (t, Yt , Zt ) ≤ (1 − θ)β|Yt | + a(t)Ut

(1.7)

Since f is concave in Z, we have

ˆ
f (t, Yt , Zt ) = f

t, Yt , θZt + (1 − θ)

ˆ
Zt − θZt
1−θ

≥ θf (t, Yt , Zt ) + (1 − θ)f

t, Yt ,

ˆ
Zt − θZt
1−θ

thus by (c)

ˆ
θf (t, Yt , Zt ) − f (t, Yt , Zt ) ≤ −(1 − θ)f

t, Yt ,

ˆ
Zt − θZt
1−θ

≤ (1 − θ)(α(t) + β|Yt |) +

γ
ˆ
|Zt − θZt |2
2(1 − θ)

(1.8)

We continue with (1.6). Upon combining (1.7) and (1.8), we have

ˆ ˆ ˆ
θf (t, Yt , Zt ) − f (t, Yt , Zt ) ≤ a(t)Ut + (1 − θ)(α(t) + 2β|Yt |) +

13

γ
ˆ
|Zt − θZt |2 + δft (1.9)
2(1 − θ)

ˆ ˆ ˆ
Let Ft = θf (t, Yt , Zt ) − f (t, Yt , Zt ) − a(t)Ut . We can rewrite (1.9) as

Ft ≤ (1 − θ)(α(t) + 2β|Yt |) +

γ
ˆ
|Zt − θZt |2 + δft
2(1 − θ)

t
Let At = 0 a(s)ds. By Ito’s formula, we have

d(eAt Ut ) = −eAt Ft dt + eAt Vt dBt

(1.10)

Let c ≥ 0 and define Pt = exp{ceAt Ut }. By Ito’s formula,

dPt = −Gt dt + Qt dBt ,

where Qt = cPt eAt Vt , and

Gt = cPt eAt

Ft −

ceAt
|Vt |2
2

≤ cPt eAt {(1 − θ)(α(t) + 2β|Yt |) + δft } + cPt eAt

ceAt
γ
−
2(1 − θ)
2

|Vt |2

(1.11)

Recall that |At | ≤ βT , so if we choose c(θ) = γeβT /(1 − θ), then
γ
c(θ)eAt
−
≤0
2(1 − θ)
2

thus

Gt ≤ c(θ)Pt eAt {(1 − θ)(α(t) + 2β|Yt |) + δft } = Pt Ht
where

Ht = eAt γeβT (α(t) + 2β|Yt |) + c(θ)δft

14

(1.12)

t
0 Hs ds

˜
. By applying Ito’s formula to P

˜
˜
˜
Pt 2 − Pt 1 ≥
Qs dBs ,
t1

(1.13)

˜
˜
Let Pt = Dt Pt and Qt = Dt Qt , where Dt = exp
and (1.12), we have for any 0 ≤ t1 ≤ t2 ≤ T ,
t2

For any fixed t ∈ [0, T ], define the sequence of stopping times τn , n ≥ 1 as:
u

τn = inf u ≥ t :

˜
|Qs |2 ds ≥ n ∧ T

t

By (1.13), we have
˜
˜
Pt ≤ Pτn −

τn

˜
Qs dBs

(1.14)

t

˜
where t Upon taking conditional expectation on (1.14) using Pt = Dt Pt , we have
τn

Pt ≤ Et exp

t

eAs γeβT (α(s) + 2β|Ys |) + c(θ)δfs ds Pτn

By integrability condition (d) 5 and monotone convergence theorem, the exponential term
on the right-hand side of the above inequality converges.

Pτn = exp{ceAτn Uτn }
ˆ
ˆ
≤ exp{ceβT (θYτn − Yτn )} = exp{ceβT (θ − 1)Yτn + ceβT (Yτn − Yτn )}
ˆ
≤ exp{2ceβT (θ − 1)Yτn }/2 + exp{2ceβT (Yτn − Yτn )}/2
5 Corollary

4 in Briand and Hu (2008) shows that (d) implies that sup0≤t≤T |Yt | has
exponential moments of all orders.

15

T

Pt ≤ E t

exp
t

eAs γeβT (α(s) + 2β|Ys |) + c(θ)δfs ds PT

Because |At | ≤ βT , we have:

exp

γeβT +At
ˆ
(θYt − Yt )
1−θ

T

≤ Et
+

exp

eβT γeβT (α(s) + 2β|Ys |)

t

γe2βT
γeβT
ˆ
δfs ds +
(θYT − YT )
1−θ
1−θ

ˆ
ˆ
ˆ
ˆ
ˆ
By θYT − YT = θ(YT − YT ) + (θ − 1)YT ≤ θ(YT − YT ) + (1 − θ)|YT |, we have:

exp

γeβT +At
ˆ
(θYt − Yt )
1−θ

≤ Et

γe2βT
1−θ

exp

T

+γe2βT
t

T
t

ˆ
δfs ds + θ(YT − YT )

ˆ
(α(s) + 2β|Ys |)ds + |YT |

ˆ
Because βT + At ≥ 0, δf (s) ≤ 0 and YT − YT ≤ 0, we have:
1−θ
ˆ
θYt − Yt ≤
log Et
γ

T

exp γe2βT
t

ˆ
(α(s) + 2β|Ys |)ds + |YT |

The right hand side is finite due to (d) and the simple fact that the class of random variables
ˆ
of all exponential orders is closed under addition. Thus we can let θ go to 1 and get Yt ≤ Yt .

16

1.3
1.3.1

Application to Economics
Recursive Utility Functions

In economics, utility is a measure of relative satisfaction. Given this measure, one may
speak meaningfully of increasing or decreasing utility, and thereby explain economic behavior
in terms of attempts to increase one’s utility. Utility is often modeled to be affected by
consumption of various goods and services, possession of wealth and spending of leisure
time, etc. A utility function measures all the objects of choice on a numerical scale and a
higher measure on the scale means the consumer likes the object more.
The utility function that we will use is the continuous-time recursive utility introduced by
Duffie and Epstein in Duffie and Epstein (1992b) and Duffie and Epstein (1992a) as an extension of the popular time-additive utility. Skiadas (2008) summarized some important properties of recursive utility and its application to selection of optimal consumption-portfolio.
Let us consider an agent who can consume from 0 to T . The set of consumption plans is a
convex set C ⊆ L2 (C), where C ⊂ R (typically C = R+ ). For any c ∈ C, let ct , 0 ≤ t < T
denote the consumption rate at t. There also exists a terminal lump-sum consumption cT .
Definition 1 We will let Ut denote the agent’s utility at t, 0 ≤ t ≤ T . (U, Σ) ∈ L2 (R) ×
L2 (Rd ) solves the following BSDE:

dUt = −F (t, ct , Ut , Σt )dt + Σt dBt , UT = F (T, cT ).

(1.15)

We will assume that the terminal utility F (T, cT ) : Ω × C → R depends only on ω and cT
and the aggregator function 6 F : Ω × [0, T ] × C × R × Rd → R is increasing in c, concave
6 In

the general setup of BSDE (1.1), we allow the aggregator to depend on ω in addition
17

in (c, U, Σ) and satisfies Assumption 2.
In the next example we present the standard time-additive utility that is widely used in
asset pricing theory. It is a special example of recursive utility.
Example 1 Assume that the terminal F (T, cT ) ∈ L1 (R) and there exists some function
u : Ω × [0, T ] × C → R such that u(t, ct ) ∈ L1 (R). The aggregator function F satisfies:

F (t, ct , Ut , Σt ) = u(t, ct ) − βUt

for some constant β > 0.
By applying Ito’s formula to Ut e−βt , we have the following closed form expression for Ut .

Ut = Et F (T, cT )e−β(T −t) +

T

e−β(s−t) u(s, cs )ds .

(1.16)

t

This is the well-known standard time-additive utility.
In example 1, if in addition we allow β to vary with c, i.e. β : Ω × [0, T ] × C → R+ , we get
the following closed-form expression for Ut that extends (1.16).

T
Ut = Et F (T, cT )e− t β(s,cs )ds +

T

s
e− t β(τ,cτ )dτ u(s, cs )ds .

t

For any given consumption plan c ∈ C, we will let (U (c), Σ(c)) denote the solution of BSDE
(1.15) with consumption plan c.
In typical economical applications, we often require that the utility function satisfies
to time t and the solution (Y, Z). This allows us to include other random processes in the
aggregator such as the consumption process c in this section. In our main application in
Chapter 2, we will include more processes in aggregator functions.
18

certain properties, such as:
Monotonicity For c1 , c2 ∈ C, if c1 ≥ c2 , then Ut (c1 ) ≥ Ut (c2 ), 0 ≤ t ≤ T .
Concavity For c1 , c2 ∈ C and α ∈ (0, 1), Ut (αc1 +(1−α)c2 ) ≥ αUt (c1 )+(1−α)Ut (c2 ), 0 ≤
t ≤ T.
Dynamic consistency Let c1 , c2 ∈ C, 0 ≤ s < t ≤ T and A ∈ Fs . Assume c1 = c2
on A × [s, t] and Ut (c1 ) ≥ Ut (c2 )(or Ut (c1 ) = Ut (c2 )) on A, then U (c1 ) ≥ U (c2 )(or
U (c1 ) = U (c2 ) respectively) on A × [s, t].
Proposition 2 The recursive utility function in Definition 1 is monotonically increasing
and concave in consumption and satisfies dynamic consistency.
Proof. The proof of the proposition is based on the comparison lemma(Theorem 4). Monotonicity follows directly from Theorem 4, because the aggregator function of BSDE (1.15)
is monotonically increasing in c.
For the proof of Concavity, let cα = αc1 +(1−α)c2 , U α = αU (c1 )+(1−α)U (c2 ), and Σα =
αΣ(c1 ) + (1 − α)Σ(c2 ).
Thus (U α , Σα ) satisfies(omitting the time argument t for t ∈ [0, T )):

dU α = − αF (c1 , U (c1 ), Σ(c1 )) + (1 − α)F (c2 , U (c2 ), Σ(c2 )) dt + Σα dB
= − (F (cα , U α , Σα ) − p) dt + Σα dB,
α
UT = αF (T, c1 ) + (1 − α)F (T, c2 )
T
T

= F (T, cα ) − pT .
T

19

where

p = F (cα , U α , Σα ) − αF (c1 , U (c1 ), Σ(c1 )) − (1 − α)F (c2 , U (c2 ), Σ(c2 )),
pT = F (T, cα ) − αF (T, c1 ) − (1 − α)F (T, c2 )
T
T
T

Because F is concave in (c, U, Σ), we have pt ≥ 0, t ∈ [0, T ].
We also have that (U (cα ), Σ(cα )) satisfies:

dU (cα ) = −F (cα , U (cα ), Σ(cα ))dt + Σ (cα )dB, UT (cα ) = F (T, cα ).
T

ˆ ˆ
Applying Theorem 4 with (Y, Z) = (U α , Σα ) and (Y , Z)) = (U (cα ), Σ(cα )), we get Ut (cα ) ≥
Utα , 0 ≤ t ≤ T .
For the proof of Dynamic Consistency, we consider the BSDEs for (U (ci ), Σ(ci )),
i = 1, 2 on A×[s, t] with the aggregators F (c1 , ·) = F (c2 , ·) and the terminals Ut (c1 ) ≥ Ut (c2 ).
Thus it follows from Theorem 4 that U (c1 ) ≥ U (c2 ) on A × [s, t]. By symmetry, the claim
still holds with = replacing all the ≥.

When maximizing recursive utility with respect to

consumption plans, it is enough to maximize U0 , because the property of dynamic consistency
ensures that once an optimal consumption plan is chosen at t = 0, the agent does not have
incentive to deviate from it at any t ∈ [0, T ]. Assume U0 is maximized by the consumption
plan c ∈ C. 7 Then for any t ∈ [0, T ], there can not exist a consumption plan c that satisfies
˜
cs = cs , s ∈ [0, t) and Ut (˜) ≥ Ut (c). Otherwise, we can define c such that cs = cs , s ∈ [0, t)
˜
c
¯
¯
and cs = cs , s ∈ [t, T ]. By dynamic consistency, we have U0 (¯) ≥ U0 (c), which contradicts
¯
˜
c
7 In

typical applications such as optimal portfolio choice and optimal contracting, c is
subject to extra constraints, which is generally expressed as a forward SDE. We will leave
out this technicality for the time being and get back to it in Chapter 2.

20

the optimality of c at time 0. So if c ∈ C satisfies U0 (c) = maxc∈C U0 (˜), the agent will stick
c
˜
to it as the optimal and not deviate.
Comparing Risk Aversion: In comparison to the standard time-additive utility(Example
1), recursive utility allows us more flexibility to adjust risk-aversion through the dependence
of aggregator function on the volatility term. To see this, we will consider two recursive
utility functions U i with aggregators F i , i = 1, 2.
1
2
U 1 is more risk-averse than U 2 , if U0 (c) = U0 (c), for any deterministic plan c and
1
2
U0 (c) ≤ U0 (c) for any plan c ∈ C.

For simplicity, we assume that the two aggregators F i are deterministic functions of
the corresponding arguments (t, ci , U i , Σi ) and F i (T, ci ) is a deterministic function of ci ,
T
T
i = 1, 2. If F 1 (t, c, y, 0) = F 2 (t, c, y, 0) for any t, c, y ∈ [0, T ] × C × R, then U 1 (c) = U 2 (c) for
any deterministic plan c. While we can adjust F i such that F 1 (t, c, y, z) ≤ F 2 (t, c, y, z) for
1
2
any t, c, y, z ∈ [0, T ]×C ×R×Rd . By Theorem 4, we get U0 (c) ≤ U0 (c), for any c ∈ C. Thus

The two recursive utilities U1 and U2 have the same preference order over deterministic plans
while U1 is more risk-averse. Risk-aversion of the standard time-additive utility is totally
governed by the preference order over deterministic plans, since there is no volatility term
in the aggregator to adjust risk-aversion.

1.3.2

European and American Contingent Claims Valuation

The following section summarizes some results from Karoui, Peng, and Quenez (1997),
Karoui and Mazliak (1997) and Rogers and Talay (1997) about the application of BSDE in
pricing European and American contingent claims. In pricing European contingent claims,
these results extend the classic risk-neutral pricing results in complete markets. The application of BSDE allows us to use more flexible modeling of wealth equations including
21

consumption, nonlinear generators and incomplete markets. We also introduce a variation
of BSDE, the reflected BSDE(RBSDE) which is closely related to optimal stopping. It is used
in pricing of American contingent claims. For simplicity, we will only impose assumption 1
on the BSDEs(RBSDEs) and cover the classic pricing problems with complete markets.
We will adopt the following setting for a complete market. There are d + 1 assets. One
of them P 0 is a riskless asset. In addition, there are d risky assets P i , i = 1, . . . , d that do
not pay dividends. The n + 1 assets follow the equations below.

dPt0 = Pt0 rt dt




d
i,j

dPti = Pti bi dt +
t

j

σt dBt  , i, j = 1, . . . , d
j=1

where r is the short interest rate, b = (b1 , . . . , bd ) is a d × 1 vector representing the appreciation rates of d stocks and σ is a d × d volatility matrix.
For simplicity, r, b and σ are assumed to be uniformly bounded and predictable processes.
σ has full rank and the inverse σ −1 is uniformly bounded as well.
It is also being assumed that there exists a predictable and bounded-valued d × 1 vector
process θ that solves the following market price of risk equation.

bt − rt 1 = σt θt a.e.
where 1 is a vector with every component 1.
Under these assumptions, the market is complete.
Let π = (π 1 , . . . , π d ) denote the amount of wealth in the d risky assets, namely π is a

22

portfolio. Let Vt denote the value of the portfolio. The pair (V, π) is called self-financing if
T
0

|σt πt |2 dt < +∞ a.s.

and the following two equations hold
d
i
πt

Vt =
i=0

and

d
i
πt

dVt =
i=0

dPti
Pti
d

d
i,j

i
πt (bi dt +
t

= (Vt − πt 1)rt dt +
i=0

j

σt dBt )
j=1

= rt Vt dt + πt (bt − rt 1)dt + πt σt dBt
= (rt Vt + πt σt θt )dt + πt σt dBt

The case of European option.
First recall that an European contingent claim ξ settled at time T is an FT measurable
random variable. It is a contract that pays ξ at time T .
Without considering consumption, a hedging strategy against a short position in ξ is
defined to be a self-financing trading strategy (V, π) such that VT = ξ. We denote the class
of hedging strategies against ξ by H(ξ). The claim ξ is hedgable if H(ξ) is nonempty
The fair price X0 at time 0 of a claim ξ is defined as

23

X0 = inf{x ≥ 0; ∃(V, π) ∈ H(ξ) such that V0 = x}
With the above setup, let ξ be a nonnegative8 square-integrable contingent claim. A
hedging strategy (X, π) against ξ is a solution of the following linear BSDE

dXt = (rt Xt + πt σt θt )dt + πt σt dBt , XT = ξ

Without restrictions on the solution (Xt , πt ), the solution to such a BSDE is generally not
unique. If we require that the strategy is feasible i.e. Xt ≥ 0, t ∈ [0, T ], then the solution is
unique. As an alternative, we can require as in Theorem 2 that (X, σt πt ) ∈ L2 (R) × L2 (Rd ).
In both cases, the solution is given by a standard result on linear BSDE as appears in
Proposition 1.

t
Xt = E HT ξ|Ft
t
t
t
t
where Hs , t ≤ s ≤ T satisfies dHs = −Hs rs ds + θs dBs , Ht = 1.

The process H above is called ”state price density” or ”deflator”. The above characterization of hedging portfolio agrees with the classic risk-neutral pricing result. By Girsanov’s
t
Theorem, there exists Q a risk neutral probability measure so that Wt + 0 θs ds is a standard

d-dimensional Brownian Motion that under Q 9 .
8 For simplicity, we only consider hedging a short position of a claim that has a nonnegative

payoff at T .
we assume that θ is uniformly bounded, dQ = exp −
dP
a martingale.
9 Since

24

T
0 θs dBs

T
+ 1 0 |θs |2 ds is
2

Then the solution of the above BSDE can be written as

T
Xt = EQ e− t rs ds ξ|Ft

As we see above, the solution of the European option leads to a BSDE with generator that
is linear in the wealth and volatility. However, we can apply more general assumptions on
the generator of BSDE such as Assumption 2 to allow more flexible modeling.
The case of American option.
To motivate using RBSDE in pricing American contingent claim, let us first consider the
problem of hedging European contingent claim with consumption. Let C be an increasing,
right-continuous process representing cumulative consumption. A self-financing superhedging strategy is a collection of (V, π, C) where V is the wealth process and π the portfolio
process such that

∗
dVt = (rt Vt + πt σt θt )dt − dCt + πt σt dBt , VT = ξ
T
0

|σt πt |2 dt < +∞ a.s.

A superhedging strategy is called feasible if

Vt ≥ 0, t ∈ [0, T ] a.s.

Let H (ξ) denote the class of superhedging strategy.
The upper price at time 0 of ξ is defined as

X0 = inf{x ≥ 0; ∃(V, π, C) ∈ H (ξ) such that V0 = x}
25

By incorporating the consumption process C into the generator of the wealth equation,
a positive term dCt /dt 10 is added to the generator. Given a claim ξ, we see by using the
Comparison Lemma 3 that the upper price should be no less than the fair price. In the
setting above, the fair price agrees with the upper price i.e. X0 = X0 . This can be seen by
setting C = 0 in the above equation.
A feasible superhedging strategy is a special case of RBSDE, if we further require that
T
0

Vt dCt = 0 and V ∈ S 2 . Although these two last assumptions are not needed for pricing

European contingent claim, it turns out that for American contingent claim, due to the early
exercise feature, a consumption process is needed for the case that the option holder does not
follow an optimal exercise policy. This concludes the motivational part of the presentation.
Definition 2 A standard data for an RBSDE consists of a terminal value ξ ∈ L2 , a standard
generater f : Ω × [0, T ] × R × Rd → R that satisfy Assumption 1 and a continuous obstacle
process S ∈ L2
The solution to a RBSDE with standard data (ξ, f, S) is a tripe of F-progressively measurable process (Yt , Zt , Kt ), 0 ≤ t ≤ T taking values in R × Rn × R+ that satisfies
a. Z ∈ L2 , Y ∈ S 2 and KT ∈ L2 ;
∗
b. Yt = ξ + tT f (s, Ys , Zs )ds + KT − Kt − tT Zs dBs , 0 ≤ t ≤ T ;

c. Yt ≥ St , 0 ≤ t ≤ T ;
T
d. Kt is continuous and increasing, K0 = 0 and 0 (Yt − St )dKt = 0.

Because of the backward formulation, the above definition might look counter-intuitive
¯
at the first glance. Compare an RBSDE for Yt with a regular BSDE for Yt with the same
10 Assume

C is absolutely continuous with respect to the Lebesgue measure
26

generator and terminal value.

−dYt = f (t, Yt , Zt )dt + dKt − Zt dBt , YT = ξ
¯
¯
¯ ¯
¯
−dYt = f (t, Yt , Zt )dt − Zt dBt , YT = ξ

We see that the difference between the two equations is dKt which represents an upward
push of −dYt . Since Yt = ξ − tT dYs , increasing −dYt has the effect of pushing Yt upward.
Another way of thinking of the effect of Kt is if we let the volatility term Zt = 0 and
consider Yt to represent the price of a zero-coupon bond with a face value ξ at T . Adding Kt
to the equation has the effect of decreasing the interest rate and thus increasing the value of
the bond.
By condition (d) in the last definition, Kt is continuous and moves upward only when
Yt = St . This ensures that the minimal push is being used to make condition (c) satisfied.
Similar to the case of classic BSDE, with a set of standard data (ξ, f, S) the RBSDE has
a unique solution (Yt , Zt , Kt ). The proof of existence part is based on apriori estimates on
the solutions of two RBSDEs. The uniqueness part is based on the following comparison
theorem for RBSDEs.
Theorem 5 Let (ξ, f, S) and (ξ , f , S ) be two sets of standard data and suppose that
a. ξ ≤ ξ
b. f (t, y, z) ≤ f (t, y, z) a.e.
c. St ≤ St 0 ≤ t ≤ T a.s.
Let (Y, Z, K) and (Y , Z , K ) be the respective solutions of the RBSDEs associated with the

27

standard data above. Then
Yt ≤ Yt , 0 ≤ t ≤ T, a.s.
As in the European part, a wealth portfolio before the option is exercised is a pair of
processes (Xt , πt ) in L2 (R) × L2 (Rd ) that satisfies the following SDE

∗
dXt = −b(t, Xt , πt )dt + πt σt dWt

where b is a standard generator. The European option case corresponds to b(t, x, π) =
−rt x − π ∗ σθt . The volatility matrix σ is assumed to be invertible. Also σ and σ −1 are
uniformly bounded. We can let σ be identity matrix without loss of generality as we can
treat σ ∗ π as π.
Let ξ denote the terminal payoff of an American contingent claim in case it is not exercised
early. Let Su denote the intrinsical value of the claim which is the payoff of the claim if it is
exercised at time u for any 0 ≤ u < T . Also assume S ∈ L2 . Let

˜
Su = ξ1u=T + Su 1u<T , 0 ≤ u ≤ T

For any 0 ≤ t ≤ T , let Ψt = {τ ; τ is a stopping time and t ≤ τ ≤ T }. For any ν ∈ Ψt ,
ν ν
there exists a unique solution (Xs , πs ) to the following BSDE for the wealth process with

terminal at the stopping time ν.

ν
ν ν
ν
ν
˜
−dXs = b(s, Xs , πs )ds − (πs )∗ dWs , 0 ≤ s ≤ ν, Xν = Sν

If (X, π) are the solution to a BSDE associated with terminal time T , generator b(t, x, π)1t≤ν
˜
and terminal value Sν , then (X ν , π ν ) is just (Xs , πs )1s≤ν . This BSDE represents the wealth
28

˜
process stopped at ν and replicates the payoff Sν , if the American contingent claim is exercised at time ν.
˜
Thus the price of the American contingent claim at time t with payoff Sν at exercise time
ν ∈ Ψt is given by

ν
˜
ess sup Xt = ess sup E Sν +
ν∈Ψt

ν∈Ψt

ν
t

ν ν
b(s, Xs , πs )ds|Ft

The following proposition shows this price is exactly the solution of a RBSDE. It also
generalizes the traditional optimal stopping problem and provides a RBSDE that solves the
value of the American contingent claim.
Proposition 3 Let {Yt , Zt , Kt , 0 ≤ t ≤ T } be the solution of a RBSDE associated with a
set of standard data (ξ, f, S) then for each t ∈ [0, T ]

˜
Yt = ess sup E Sν +
ν∈Γt

ν
t

f (s, Ys , Zs )ds|Ft

˜
where Su = ξ1u=T + Su 1u<T , 0 ≤ u ≤ T
The optimal stopping time in the above equation is achieved by

Dt = inf {t ≤ u ≤ T : Yu = Su }

Note that in the above proposition, when f is an L2 process that is free of (Y, Z), the solution
Yt corresponds to the value of an optimal stopping problem.
The solution of proposition 3 is compatible with pricing by using risk-neutral probability
measure namely, with Q denoting the risk-neutral probability measure, we get that the price
˜
of the American contingent claim at time t with payoff Sν at exercise time ν ∈ Ψt is given
29

by

ν ν
˜
ess supν∈Ψt E Sν + tν b(s, Xs , πs )ds|Ft

˜
= ess supν∈Ψt EQ exp − tν rs ds Sν |Ft

30

Chapter 2
Principal-Agent Problem with
Multiple Principals

2.1

Introduction

A nontechnical overview of the problem: We study the principal-agent(owner-manager)
problem with moral hazard in continuous time with multiple principals. In simple
terms, the problem studies the interaction of a group of principals. Each has one agent
working for him/her.
To help understand the problem, we will first recall a simpler problem with just one
principal and one agent. In this problem, a utility maximizing principal offers a pay
plan to an agent in order to induce effort, which has an impact on both the principal
and agent’s utilities as well as the principal’s cash flow. In the meantime, the principal
faces two constraints. The first constraint is an incentive compatibility condition that
says the agent will make effort to maximize his/her own utility and not necessarily work
in the benefit of the principal. The second is a participation constraint namely, the
agent’s initial utility must exceed some fixed amount, because of his/her employment
opportunities elsewhere. The objective is to characterize the principal’s optimal control
over the agent, taking into account the two constraints.
31

In the principal-agent problem with multiple principals, the new element is the assumption that each principal-agent pair’s utilities and each principal’s cash-flow are
affected by the efforts of all agents. When choosing his/her controls, each principal has
to take into account not only the two constraints as in the single principal case, but
also the impact of his/her control on the other principal-agent pairs and their response.
The problem is defined in two levels, first for the agents and then the principals. The
key to the definition of the problem in both levels is Nash equilibrium. In simple language, a Nash equilibrium is a relation among a group of utility-maximizing ”players”,
in which each player’s strategy is optimal in response to the others’ strategies. In
other words, in a Nash equilibrium, each player has no incentive to change strategy,
when the others do not change theirs. In our problem, the Nash equilibrium among
the agents are through their competing(or cooperative) efforts, which have an impact
on every agent’s utility function. The Nash equilibrium among the principals is more
complicated, because the connection among them is indirect and comes only through
the agents’ Nash equilibrium in efforts. That is each principal’s incentive compatibility
constraint, from the single principal case, is replaced by a Nash equilibrium condition
among the agents. Similar to the single-principal case, each principal also faces the participation constraint of his/her agent. The objective is to characterize each principal’s
equilibrium control over his/her agent, taking into account the control is constrained
by the agents’ Nash equilibrium in efforts.
Main ideas of the paper: If agent effort is noncontractible, it is well-know from the
principal-agent literature that a principal can induce agent effort by linking pay to the
cash flows influenced by that effort.

In a single firm with several agents and effort

32

externalities (one agent’s effort affecting the output of others), the optimal contract
for each agent will generally depend on the other agents’ outputs (see Holmstrom
(1982) and, in a continuous-time model, Koo, Shim, and Sung (2008)).1 One reason
for this is the impact of an agent’s effort on the output of others (another is simple
risk reduction). But such effort externalities occur not only within firms but across
firms. For example, innovations (e.g., software, microchips, fracking technology, etc.)
by agents at one firm can affect the investment opportunities of other firms. Effort
by employees in the service sector could have spillover effects to firms in the same
industry. Alternatively, our model could represent a reduced-form model accounting
for competitive or complementary industry effects.

Within this setting there are

moral hazard and potential free-rider problems at two levels. Given the compensation
schemes offered by the principals, each agent considers only the impact of his effort
on his own compensation, ignoring the benefits to the other agents as well as the
principals. The agent may have an incentive to free ride off other agents’ efforts: in
one application we show that no more than one agent will exert effort at any time.
Each principal designs a compensation scheme to maximize own utility only, while
anticipating the impact of the promised pay on the agent equilibrium. Each principal
has an incentive to free ride off the other principals, particularly because the assumption
of binding agent participation constraints implies that the benefits of effort externalities
ultimately accrue to the principal.
We examine the multiple principals (one agent for each principal) problem in a continuoustime setting with a Brownian filtration, recursive preferences, and one agent for each
1 Even

absent the effort externality, Holmstrom (1982) shows that such relative performance evaluation can be optimal simply to reduce compensation risk when the task outputs
are correlated.
33

principal. The effort of one agent changes the drifts of the cash flows of other principals/firms as well as their own. We show that optimal contracts in this setting cannot
be designed in isolation, because the compensation policy of each principal can affect
the distributions of the other principals’s cash flows and the incentives of the other
agents. The equilibrium is simple to obtain for the class of preferences we consider
(a generalization of additive exponential utility), and reduces to computing a system
of linked subjective cash-flow value processes, one for each principal. Each principal’s
optimal pay policy amounts to choosing the component of the subjective cash-flow
diffusion, or volatility, to transfer to their agent (that is, the volatility sharing rule),
and is solved by maximizing the drift rate of this cash-flow process given the policies
of the other principals.
We first examine applications with an additive impact of effort on the Brownian motion drift (which, in turn, implies an additive impact on cash-flow drifts).

In the

additive case the marginal impact of effort is not affected by other agents’ effort. The
resulting sharing rules are simple functions of each principal’s own subjective cash-flow
volatility, but these cash-flow processes must be jointly solved because of the externality of the other agents’ effort.

When an additive measure change is combined with

quadratic effort and risk-aversion penalties, we obtain simple linear volatility sharing
rules and closed-form solutions for a class of affine type state-variable dynamics (one
example considers Ornstein-Uhlenbeck cash flows, and another example a square-root
stochastic cash-flow volatility model). In the simple case of Brownian cash flows and
no intermediate pay/consumption, the fixed component of each agent’s lump sum terminal consumption is adjusted according to the covariances among the cash-flows. For
example, in a setting with two principal-agent pairs and positive covariance between
34

the cash-flows, then each agent’s effort increases the drift of both cash flow processes.
This positive externality all accrues to the principals, who can reduce the fixed component of agent pay while still meeting the participation constraint because the fixed
fraction of the cash flow paid to each agent becomes more valuable with the larger
drift.
We next consider applications in which the impact of effort on the Brownian drift
is diminishing in total effort.
aggregate effort.

That is, the measure-change operator is concave in

The sharing rules in this setting are more complicated, with the

optimal share of the subjective cash-flow volatility transferred to the agent (via the
promised pay) depending on all the principals’ subjective cash-flow volatility processes.
Each agent’s equilibrium optimal pay is therefore influenced by the cash-flow dynamics
of all the principals.

We obtain an explicit solution with quadratic effort penalties

and two risk-neutral principal-agents pairs in which the lump-sum terminal pay of each
agent depends on the terminal lump-sum cash-flows of both principals, increasing in
their own cash flow and decreasing in the cash flow of the other.

We also consider

the case of absolute effort penalties (with possibly risk-averse principals and agents),
in which the equilibrium results in only one agent working at any moment in time.
In the principal-agent problem with moral hazard, a utility maximizing principal pays
a compensation process to an agent in order to induce effort (which, increases expected
future cash flows). But the principal faces two constraints. First, because effort is assumed noncontractible, the contract must satisfy an incentive compatibility condition
that the agent, faced with a particular compensation process, will choose effort to
maximize his own utility. Second, because the agent has alternative employment op-

35

portunities, the agent’s initial utility must exceed some fixed amount (the participation
constraint). In the continuous-time Brownian version, first examined by Holmstrom
and Milgrom (1987), the impact of effort choice is often modeled as an equivalent
change of measure (that is, the agent’s effort changes the probabilities of the states),
which changes the drift of the driving Brownian processes. This is a convenient way
to model, for example, the impact of effort on the growth rate of a cash flow process.
In the case of multiple principals/agents, each principal chooses the pay process that
maximizes his/her own utility subject to incentive compatibility and their agent’s participation constraint. At the agent level, we find that optimal effort generally depends
on both the volatility of the agent’s utility function as well as the effort levels of the
other agents. We give necessary and sufficient conditions for a Nash equilibrium among
agent effort processes in terms of the joint agent utility-volatility processes. At the
principal level, each optimal compensation contract is specified in terms of two controls: his/her own consumption, and the volatility of agent utility. The agent volatility
control implies a unique class of pay plans. Each principal chooses the optimal controls to maximize their own utility while fully anticipating the impact of their controls
on the equilibrium efforts chosen by the agents (pay cannot be directly contingent on
effort). The resulting set of optimal contracts establishes a Nash equilibrium among
principals as well as a Nash equilibrium among agent efforts.
The dynamic contracting problem with multiple principals appears formidable, but
is simple within the class of translation-invariant (TI) recursive preferences that we
consider. The TI class of preferences is essentially as tractable as additive exponential
utility, which is a special case, because the agency problem (the fact that effort is

36

noncontractible) induces recursivity in the principal’s utility even in the time-additive
case. Furthermore, recursive preferences allow more flexible modeling of risk aversion,
as well as distinct modeling of aversion to variability in consumption across states
versus across time. We find that the important qualitative aspects of the optimal contracts, as well as equilibrium agent effort, are driven by risk aversion. Preferences for
intertemporal substitution enter only indirectly by affecting the drift of the subjective
cash-flow value processes. We show in the case of an additive measure change that
the most tractable subclass, in which the solution reduces to a set of Riccati ordinary
differential equations when uncertainty is characterized by a set of affine-type state
variables, is not necessarily time additive. It is characterized by general risk aversion,
but elasticity of intertemporal substitution restricted to be infinite (or, alternatively,
no intermediate pay/consumption at all).
Literature review: Our paper is part of a broader literature on the impact of interactions
among principals/firms or agents on optimal contracting.

Aggarwal and Samwick

(1999) show if the pay of the manager/agent is increasing in both own-firm and rivalfirm profits the incentive to compete is diminished and therefore prices and profits
are higher.2 Khanna and Schroder (2010) show that if loan non-renewal impacts the
future prospects of the rival firm, the optimal debt contract (which induces truthful
revelation of profits via the threat of loan non-renewal) is different from a standard
debt contract: If default benefits the rival, the optimal contract deters predation with
a reduced sensitivity to profits, and if default hurts the rival (say by allowing entry of
2 The

contract is assumed linear, and is not claimed to be optimal. They also consider a
principal-agent problem, but because each agent’s effort increases only own-firm profits, but
there is no interaction among efforts.

37

a more efficient firm) then the optimal contract is made more sensitive to profits to
increase prices.

There is also a literature examining the impact of product market

competition on managerial effort in a principal-agent setting (see, for example, Hart
(1983), Schrarfstein (1988), and Schmidt (1997)), but the level of competition in these
papers is assumed to be invariant to the compensation contracts.
A large literature on team contracts allows agent efforts to jointly determine the distribution of output (see Chapter 8 of Bolton and Dewatripoint (2005) for a review).
Holmstrom (1982) shows that the role of the principal (implementing pay schemes and
extracting the surplus) becomes more important in team settings because of the free
rider problem; he also examines the relationship between the information structure
(signals and noise) can be used to implement efficient pay schemes.

The incentive

of any agent to free ride can also be affected by the other agents.

For example,

Winter (2010) shows that under complementarity of the production function,3 transparency among agents increases the threat against shirking (shirking by one agent can
induce retaliatory shirking by his peers), thereby reducing the free rider problem. Edmans, Goldstein, and Zhu (2011) considers a two-period deterministic team problem
in which firm output is either zero or one (failure or success) depending on the effort
levels (each in [0, 1]) of all the agents (they focus on the cases when either output depends on the total efforts, or the minimum of the efforts). Synergy is modeled through
effort cost, which is assumed decreasing (for ”positive” symmetry), for each agent i, in
the weighted sum of other agents’ efforts. The principal announces agent wages and
then agents simultaneously choose efforts, which constitute a Nash equilibrium. In a
3 Their

production function is defined as complementarity if the marginal impact of each
agent i’s effort increases as the set of other agents who exert effort increases.

38

continuous time setting, Koo, Shim, and Sung (2008) give a general characterization
of agent equilibrium in a team setting with time-additive exponential preferences and
Brownian cash flows.
The agent-equilibrium part of our problem is similar the problem with a single principal
but multiple agents with multiple tasks (it is most closely related to the agent equilibrium in Koo, Shim, and Sung (2008)). As in the team-contract literature, there is first
an equilibrium among agents, but without a single principal to share the aggregate
output and coordinate the agents’ efforts, resulting in another layer of moral hazard
that reduces efficiency.

Furthermore the effort externality links the principals’ con-

tracting problems, because the incentives created within one firm will indirectly affect
the opportunities of other firms. Therefore the optimal contracts set by the principals
also constitute a Nash equilibrium. It seems natural in this setting to consider each
agent’s effort cost to depend on own-effort only (in contrast to the ”synergies” modeled
in Edmans, Goldstein, and Zhu (2011)), and to consider agent efforts as substitutes
rather than complements: The Brownian-motion (and therefore cash-flow) drifts are
an increasing and weakly concave function of aggregate efforts.
Our solution approach employs a number of techniques well known from the timeadditive-utility case with a single principal.

As in Schattler and Sung (1993), we

consider a general nonMarkovian Brownian setting, and use a solution technique similar to their first-order approach (see also Williams (2008) and Koo, Shim, and Sung
(2008)).

These papers contain the key ideas of letting the principal choose agent

utility diffusion (or agent effort) in order to solve for lump-sum terminal pay, which
is solved by running the agent utility forward from the participation-constraint level

39

after substituting the optimal diffusion level. This reduces the principal’s problem to
a dynamic programming problem.
Organization of the paper: The chapter is organized as follows.

In Section 2.2, we

present the setting and notation, define the translation-invariant (TI) preferences that
we use throughout, and outline the contracting problem. In Section 2.3, we develop
a necessary and sufficient condition for agent equilibrium and introduce the class of
CES(constant elasticity of substitution) measure-change operator. Section 2.4 characterizes principal equilibrium. Section 2.5 presents closed-form solutions to a set of
linked BSDEs that represent principals’ subjective PV(present value) processes. Section 2.6 and 2.7 present applications with an additive measure-change operator, and
a CES operator, respectively. Numerous examples are provided throughout. Finally,
the Appendices present proofs omitted from the text and derivations of some examples.

2.2

The Setting and Problem

All uncertainty is generated by d-dimensional standard Brownian motion B over the finite
time horizon [0, T ], supported by a probability space (Ω, F, P ). All processes appearing
in this paper are assumed to be progressively measurable with respect to the augmented
filtration {Ft : t ∈ [0, T ]} generated by B. For any subset S of Euclidean space, let L (S)
denote the set of S-valued processes, and, for any p ≥ 1,
T

L− (S) =
p

x ∈ L (S) : E

Lp (S) =

x ∈ L− (S) : E [ xT p ] < ∞ ,
p

0

40

xt p dt < ∞ ,

where xt denotes Euclidean norm.
Define the spaces

M = {x ∈ R: x is a r.v. such that E (eκx ) < ∞ for all κ > 0} ,
¯
M =

T

x ∈ L (R) : xT ,

|xs | ds ∈ M .
0

We will let B[0,T ] denote all the Borel sets on [0, T ] and λ denote the Lebesgue measure.
The qualification ”P ⊗ λ almost surely” is omitted throughout.

We will also use the

following notation: for α ∈ S N , we will let α−i ∈ S N −1 denote the collection of all but the
ith component (for example, if α ∈ RN then α−1 = α2 , . . . , αN ). Also, for any number
x ∈ R, we denote its positive part by x+ = max (0, x).

2.2.1

The Setup

Each of the N principals in our model has a single agent whom they pay to induce costly
effort. The N agents’ effort together change the probability measure, altering the drift of B
and potentially the drifts of the principals’ cash flows and cash-flow volatility. It is through
the common change of measure that the principal and agent problems are linked.
We will define the following processes that are key to our paper
Consumption The set of consumption plans is a convex set C ⊆ L2 (R). Let
cU = cU 1 , . . . cU N

∈ C N and cV = cV 1 , . . . cV N

∈ C N represent the consumption

processes of the N agents and N principals respectively. We interpret cU i , t < T ,
t
as agent i’s consumption rate, and cU i as agent i’s lump-sum terminal consumption.
T
Similar explanation applies to principal i’s consumption.

41

Efforts We define the set of effort plans as a convex set E ⊆ L− (E) for some closed set
2
E ⊂ Rd (typically E = Rd or E = Rd ). For any e = e1 , . . . eN
+

∈ E N , we interpret

ei as the time-t effort rate exerted by agent i. We assume that ei = 0 (no lump-sum
t
T
terminal effort).
Interest Rate We assume a bounded deterministic riskless short-rate process r.
¯
Pay Process We define the set of pay processes as a convex set P ⊆ M. For any p =
p1 , . . . pN

∈ P N , we interpret pi , t < T , as intermediate pay and pU i as lump-sum
t
T

terminal pay by principal i to agent i.
¯
Cash-Flow Process We define the set of cash-flow processes as a convex set X ⊆ M. For
any X = X 1 , . . . X N

i
∈ X N , we interpret Xt , t < T , as intermediate cash-flow rate

i
and XT as lump-sum terminal cash flow of principal i.

Both principal i and agent i are allowed to borrow and lend through a money-market
account. As with agent effort, we assume that the agent’s money-market account balance is
noncontractible.

We say that the pay process pi finances agent consumption cU i if there is

a wealth (money-market balance) process W U i satisfying the agent’s budget equation (2.1);
and we say that the cash-flow process X i finances pi and principal consumption cV i ∈ C if
there is a wealth process W V i satisfying the principal’s budget equation (2.2):

U
U
W0 i = w0 i ,

U
dWtU i = WtU i rt + pi − cU i dt, cU i = WT i + pi ,
t
t
T
T

V
V
W0 i = w0 i ,

i
V
i
dWtV i = WtV i rt + Xt − pi − cV i dt, cV i = WT i + XT − pi . (2.2)
t
t
T
T

(2.1)

Before the terminal date T , principal i invests the cash flow less agent i’s pay and principal i’s consumption, and agent i invests pay less the own consumption.
42

At the terminal

date, principal i’s lump-sum consumption equals the lump-sum terminal cash flow plus the
money-market balance minus the terminal lump-sum pay to agent i; the agent’s lump-sum
consumption is the sum of the agent’s money-market balance and lump-sum pay
Define the discount factor D, as well as the price process Γ of a bond paying a unit
coupon rate and unit (lump-sum) par value:

t

Dt = e− 0 rs ds ,

Γt =

1
Dt

T
t

Ds ds + DT

(2.3)

Note that

dΓt = (rt Γt − 1)dt,

ΓT = 1.

As in Koo, Shim, and Sung (2008) we model the impact of the collective agent effort
e ∈ E N as a change in probability measure to P e where
dP e
e
= ZT
dP
and the exponential supermartingale Z e is defined by

e
Zt

1 t
Φ (es ) dBs −
Φ (es ) 2 ds
2 0
0
t

= exp

for some function Φ : Rd·N → Rd which maps the agents’ effort to the change in Brownian
e
drift. We assume throughout that Z e is a martingale4 (equivalently, EZT = 1) for every
e
e ∈ E N . By Girsanov’s Theorem, dBt = dBt −Φ (et ) dt is a standard d-dimensional Brownian
4 Imposing

the well-known Novikov condition is sufficient.

43

motion under P e . The joint impact of effort on the probability measure links the agent
problems and the principal problems.
The key idea is that the time-t collection of agent effort rates et changes the measure from
e
P to P e , such dBt = dBt − Φ (et ) dt is Brownian motion under P e . For example, if the ith
i
principal’s cash flow process X i satisfies dXt = µt dt+σt dBt , then its drift under the collective
i
effort process e is augmented, under P e , by σt Φ (et ) because dXt = µt + σt Φ (et ) dt +
e
σt dBt .

Definition 3 (Translation-invariant preferences) For any (cU , cV , e) ∈ C N ×C N ×E N ,
the agents’ utility functions satisfy the following BSDEs: 5

dUti = − hU i t, xU i + k U i t, ei , ΣU i
t
t t

e
i
dt + ΣU i dBt , UT = cU i , i = 1, . . . , N, (2.4)
t
T

where
xU i = cU i − Uti , t ∈ [0, T ),
t
t
for some deterministic functions

hU i : [0, T ] × R → R and k U i : [0, T ] × E × Rd → R.

The principals’ utility functions satisfy the following BSDEs:

dVti = − hV i t, xV i + k V i t, ΣV i
t
t

e
i
dt + ΣV i dBt , VT = cV i , i = 1, . . . , N,
t
T

5 (U i , ΣU i )

(2.5)

∈ R × Rd is the solution of BSDE (2.4). U i is the utility value and ΣU i is the
utility volatility. Similar explanation applies to (2.5).

44

where
xV i = cV i − Vti , t ∈ [0, T ),
t
t
for some functions

hV i : Ω × [0, T ] × R → R and k V i : Ω × [0, T ] × Rd → R.
It is easy to verify that, for any constant v ∈ R

Uti cU i + v, e
ΣU i cU i + v, e
t

= Uti cU i , e + v,
= Σ U i cU i , e ,
t

Vti cV i + v, e = Vti cV i , e + v.

(2.6)

ΣV i cV i + v, e = ΣV i cV i , e .
t
t

We define the set of intermediate control as a convex set H ⊆ L− (R). We will use
xU = (xU 1 , . . . , xU N ) ∈ H and xV = (xV 1 , . . . , xV N ) ∈ H as part of the controls of agents
and principals respectively throughout the paper.
The following two examples give special cases of TI agent preferences.

The case of

the principal is analogous, but with no effort penalty. In both examples the effort-penalty
function is given by g : [0, T ] × Rd → R (typically assumed convex in e).
Example 2 (Risk-neutral agent) If

hU i (t, x) = βx, k U i (t, , Σ) = −g (t, ) ,

45

β > 0,

then time-t agent utility is

e
Ut = Et

T
t

ds + e−β(T −t) ci
T

e−β(s−t) βci − g s, ei
s
s

.

Example 3 (Additive exponential) If, for some γ > 0,

1
γ
1
hU i (t, x) = − exp (−γx) , k U i (t, , Σ) = − Σ Σ − g (t, ) ,
γ
2
γ

(2.7)

then the ordinally equivalent utility process ut = − exp (−γUt ) satisfies (assuming sufficient
integrability)

e
ut = −Et

T
t

exp − γci −
s

+ exp − γci −
T

T
t

s
t

g w, ei dw
w

g w, ei dw
w

ds

.

That is u is a standard time-additive exponential utility with coefficient of absolute risk
aversion parameter γ.

2.2.2

Outline of the Problem and Solution

To model moral hazard, it is assumed that the agents’ effort processes are not contractible.
However, effort can be manipulated by the principal through the pay process. At time-0,
each principal i promises a pay process to agent i and selects his/her consumption plan.(The
commitment could be enforced by some legal entity.) In response to the pay process, each
agent i chooses effort and consumption processes to maximize his/her utility.

46

Let
U
Gt = σ es , Ws , 0 ≤ s ≤ t , 0 ≤ t ≤ T.

We assume the pay process pt is not adapted to the {Gt } filtration, i.e. there exists t
such that pt ∈ Gt . The practical meaning of this assumption is that p can not be expressed
/
as a function of e and W U .
We can think of the principal and agent choices as occurring in two stages. In the first
stage, the principals simultaneously commit to a set of pay processes p and choose their own
consumption processes. In the second stage, the agents simultaneously choose efforts and
consumption processes.
In general terms, the problem is to describe a Nash equilibrium for the whole system.
This means that in both levels of the agents and principals, there is a Nash equilibrium (see
Definition 5 for agent equilibrium and Definition 8 for principal equilibrium).
The solution is obtained recursively beginning with the second-stage agent-effort equilibrium.

For any joint pay processes p, each agent i chooses effort ei and control xU i to

i
maximize utility, U0 xU i , ei , given the other agents’ strategies (xU −i , e−i ). It turns out

that the choice of agents’ controls (xU , e) is equivalent to the more natural choice (cU , e).
This is why Definition 5 is formulated in terms of (xU , e).
In Section 2.3, we define the agents’ subjective present value(PV) process (Y, ΣY )(see
equation (2.9)) and show that the Nash equilibrium joint effort e is determined by the agents’
ˆ
joint PV-diffusion processes ΣY . Technicalities aside, at time t the Nash equilibrium takes
the form et = e t, ΣY
ˆ
ˆ
t

for some deterministic function e (·) = e1 (·) , . . . , eN (·) .
ˆ
ˆ
ˆ

Theorem 6 shows that a sufficient condition for e (·) to be a Nash equilibrium is that, for
ˆ
each agent i and time t, the effort level ei maximizes the sum of the risk-effort preference
ˆt

47

function plus the effort-induced increase in utility drift:6

ei (t, ΣY ) ∈ arg max Γt k U i
ˆ
ei ∈E

ΣY i
t, ei , t
Γt

ˆ−i
+ ΣY i Φ(ei , et )
t

for all ΣY ∈ Rd×N and t ∈ [0, T ] .

This function is solved in closed form for all our applications.

Furthermore, this effort

equilibrium is dynamically consistent in the sense that the equilibrium determined by the
agents at time-0 will also constitute an equilibrium at any time t in the future.
Having obtained the agent-effort equilibrium (ˆU , e), we then solve the first-stage prinx ˆ
cipals’ problem of choosing optimal pay and principal consumption. Rather than choose pi
, from which the pay process pi
directly, we specify the principal i strategies xV i , ΣY i
t
t
t<T
is computed (as shown in Section 2.4). Choosing xV i is essentially equivalent to choosing
intermediate consumption cV i ; t < T .
t
For any xV i , we show that the joint choice of ΣY implies a unique class of payment
plans.

Given the form of the agent equilibrium,

e t, ΣY ; t ∈ [0, T ) , the control ΣY i
ˆ
t

is the natural choice to influence agent effort; it also greatly enhances the tractability of
the problem.

Taking as given the other principals’ strategies

chooses xV i , ΣY i to maximize utility V0i xV i , e t, ΣY
ˆ
t
i
constraint U0 cU i , e t, ΣY
ˆ
t

xV −i , ΣY −i , principal i

subject to agent i’s participation

≥ K i . Note that principal i’s utility depends on equilibrium

efforts by all the agents, and the principal fully anticipates the impact of ΣY on the effort
equilibrium.
In section 2.4, we define the principals’ subjective present value process (Z, ΣZ )(see
ˆ
equation (2.37)) and shows that the equilibrium strategies ΣY are interdependent and are
6 We

will assume, for simplicity, that the preference functions are deterministic (do not
depend directly on ω), and therefore ei (t, Σ) will not depend on ω.
ˆ
48

solved as a Nash equilibrium as a function of the subjective cash flow diffusion processes ΣZ .
ˆ
ˆ
A sufficient condition for ΣY = ΣY t, ΣZ
t

ˆ
ΣY i ∈ arg max Γt
t
Σ∈Rd

kV i

to be a Nash equilibrium is

ΣZi − Σ
t, t
Γt

ˆ
ˆ
+ΣZi Φ e t, Σ, ΣY −i
t
t

ˆ
+ k U i t, ei (t, Σ, ΣY −i )
ˆ
t

,

for each principal i, and all Σz ∈ Rd×N and t ∈ [0, T ] .

ˆ
Each principal i chooses the volatility ΣY i to transfer to the agent that maximizes the
sum of the principal and agent risk-effort preference functions plus the impact of effort on the
principal’s utility drift, all holding fixed the controls of the other principals (yet anticipating
the impact of the principal’s own control on the equilibrium efforts).
The resulting Nash equilibrium among principals is then consistent with a Nash equilibrium among agents’ efforts. The principal equilibrium is also time consistent, in the sense
that the equilibrium determined by the principals time-0 will also constitute an equilibrium
at any (ω, t) in the future after replacing K i with time-t value of equilibrium agent utility
processes U i .
The final step is to compute the subjective PV processes. Theorem 7 shows that these
i
are given by Zt = Γt Vti − WtV i + Yti , i = 1, . . . , N , at the optimum. Because each cash

flow X i is split between the ith principal/agent pair, it is natural to add the agent PV
process to determine the principal PV process. Maximizing principal i’s utility is equivalent
i
to maximizing Z0 because the participation constraint binds.7
7 Alternatively,

Defining the above drift

i
we can interpret Zt as a Lagrange multiplier process, incorporating the
constraint on the agent’s initial utility. It can be shown that the Lagrange multiplier,
representing the sensitivity of principal utility to a unit change in agent utility, is always one

49

functions evaluated at the optimum

i
µZi = Ht + Γt
ˆt

kV i

ˆ
ΣZi − ΣY i
t
t, t
Γt

+ kU i

ˆ
ΣY i
ˆ
t, e t, ΣY , t
ˆ
t
Γt

ˆ
ˆ
+ ΣZi Φ e t, ΣY
t
t

,

where

i
Ht = Γt hV i (t, xV i ) + hU i (t, xU i ) − xV i − xU i .
ˆt
ˆt
ˆt
ˆt
t

Then Z i , ΣZi , i = 1, . . . , N , solve the backward equation system

i
i
i
i
i
dZt = −(−rt Zt + Xt + µZi )dt + ΣZi dBt , ZT = XT , i = 1, . . . , N.
ˆt
t

Given the solution to this system, we substitute to get the principals’ controls and
ˆ
ˆ
ΣY t, ΣZ . This yields equilibrium effort e t, ΣY t, ΣZ
ˆ
t
t

and finally optimal pay, which

is obtained by running the agent subject PV equation forward from its starting value after
ˆ
substituting the equilibrium policy ΣY i .

2.2.3

Regularity Conditions and Feasibility

This section is purely technical, imposing regularity conditions on the aggregators, and
defining the class of feasible consumption and effort plans to ensure existence and uniqueness
of the utility functions.
It is assumed that the aggregators satisfy the following condition.
Condition 1 For all i = 1, . . . , N , we have
with TI preferences.
50

(a) There exists a β ∈ R+ such that

hU i (t, y) − hU i (t, y ) + hV i (ω, t, y) − hV i (ω, t, y ) ≤ β |y − y | ,
ˆ
ˆ
ˆ

for all (ω, t, y, y ) ∈ Ω × [0, T ] × R2 ;
ˆ
(b) k V i (ω, t, ·) and k U i (t, e, ·) are concave functions for all (ω, t, e) ∈ Ω × [0, T ] × E;
¯
¯
(c) Both hU i (t, ·) and hV i (ω, t, ·) are increasing functions for all (ω, t) ∈ Ω × [0, T ].

Feasibility is defined as follows. Recall xU and xV from Definition 3.
Definition 4 The set of intermediate controls and effort plans (xU , xV , e) ∈ HN ×HN ×E N
will be called feasible if there exist γ, β ∈ R+ , a process α ∈ L (R+ ) such that, for all
i = 1, . . . , N ,
(a)

˜
˜
hU i (t, y) + k U i (t, ei , z) + hV i (ω, t, y) + k V i (ω, t, z) + z Φ(et )
t
≤ αt + β |y| +

γ 2
|z| for all (ω, t, y, z) ∈ Ω × [0, T ] × R × Rd ,
2

˜
˜
where hU i (t, y) = hU i (t, cU i − y) and hV i (ω, t, y) = hV i (ω, t, cV i − y).
t
t
T
(b) 0 α(t)dt, cU i , cV i ∈ M,
T
T

where cU i and cV i are the resulting consumption plans as explained in Section 2.2.2. 8
8 By

Corollary 6 of Briand and Hu (2008), there exists a unique solution (U i , ΣU i ) to
the agent utility function in (2.4) and a unique solution (V i , ΣV i ) to the principal utility
function in (2.5).

51

N
N
We denote the set of all feasible intermediate controls and effort plans by HU f × HV f ×
N
Ef .

2.3

Agent Equilibrium

The main result in this section is Theorem 6 which gives a necessary and sufficient condition
for agent equilibrium.

Examples 4, 5 and 6 illustrate the result by obtaining the effort

equilibrium in closed form (as a function of agent utility diffusion) for some combinations of
quadratic and absolute effort penalties, and linear and concave measure-change operators Φ.
In Section 2.3.3, we show that all the examples in Section 2.3.1 with two principal-agent pairs
are special cases of the class of constant elasticity of substitution measure change operator Φ.
We will impose the feasibility condition as in Definition 4 on any collection of intermediate
controls and effort plans referred in this section and thus omit it from the text for simplicity.

2.3.1

A Necessary and Sufficient Condition for Agent Equilibrium

We first define the Nash equilibrium among agents. As explained in Section 2.2.2, the
controls are chosen to be (xU , e) which turns out to be equivalent to (cU , e).
Definition 5 (Agent Equilibrium) Holding fixed a set of pay processes p ∈ P N , the set
of agent control and effort plans (ˆU , e) where e = {ˆi , i = 1, . . . , N } and xU = {ˆU i , i =
x ˆ
ˆ
e
ˆ
x
1, . . . , N } constitute an agent equilibrium (in the sense of Nash) if, for each i = 1, . . . , N ,

i x
i
U0 (ˆU i , e) ≥ U0 (xU i , (ei , e−i )), for any ei ∈ E, xU i ∈ H
ˆ
ˆ

To motivate the characterization of agent equilibrium in Theorem 6 below, define for
any feasible policy xU , xV , e the difference between the dollar utility value and financial
52

wealth:9
Yti xU i , e = Γt Uti xU i , e − WtU i xU i ,

t ∈ [0, T ] .

(2.8)

By Ito’s lemma, and the identity ΣY i = Γt ΣU i , we get the dynamics of Yti = Yti xU i , pi , e :
t
t

dYti = − −rt Yti + pi + Γt
t
+ΣY i dBt ,
t

ΣY i
hU i t, xU i + k U i t, ei , t
t
t
Γt

i
YT = pi ,
T

− xU i + ΣY i Φ(et ) dt
t
t

i = 1, . . . , N.

(2.9)

Theorem 6 (Agent Equilibrium) Assume Y i , ΣY i solves the the set of BSDEs (2.9).
Then a necessary and sufficient condition for (ˆU , e) to constitute a Nash equilibrium among
x ˆ
the agents is for each i = 1, . . . , N ,

Yi

Σ
Γt hU i t, xU i + k U i t, ei , Γ
ˆt
ˆt t
t
Yi

Σt
≥ Γt hU i t, xU i + k U i t, ei , Γ
t
t
t

− xU i + ΣY i Φ(ˆt )
ˆt
e
t

(2.10)

− xU i + ΣY i Φ(ei , e−i )
t
t
t ˆt

t ∈ [0, T ], for any ei ∈ E, xU i ∈ H.
Proof.
Sufficiency Holding fixed a set of and principal controls xV and pay processes p, consider
a set of agent control and effort plans xU , e such that (2.10) is satisfied. Let ht denote
ˆ ˆ
the nonnegative process that represents the following difference
9 The

wealth process W U i xU i

satisfies (2.1) after substituting consumption cU i =
t

xU i + Uti xU i , e .
t

53

ht =

hU i t, xU i + k U i
ˆt

Γt
− Γt

hU i t, xU i + k U i
t

ΣY i
t, ei , t
ˆt
Γt
ΣY i
t, ei , t
t
Γt

e
− xU i + ΣY i Φ(ˆt )
ˆt
t
− xU i + ΣY i Φ(ei , e−i ) .
t ˆt
t
t

ˆ
With xU i , ei , the corresponding discounted process Dt Y i solves the following BSDE
ˆ ˆ

ˆ
dDt Yti = −Dt

ht + pi + Γt
t

ˆ
+Dt ΣY i dBt ,
t

On the other hand, for any

hU i t, xU i + k U i
t

ΣY i
t, ei , t
t
Γt

− xU i + ΣY i Φ(ei , e−i ) dt
t
t
t ˆt

ˆi
YT = pi .
T

(xU i , xU −i ), (ei , e−i )
ˆ
ˆ

the corresponding discounted process

Dt Y i solves the BSDE

dDt Yti = −Dt

pi + Γ t
t

+Dt ΣY i dBt ,
t

ΣY i
hU i t, xU i + k U i t, ei , t
t
t
Γt

− xU i + ΣY i Φ(ei , e−i ) dt
t
t
t ˆt

i
YT = pi .
T

Theorem 4 implies
Y0i (ˆU i , e) ≥ Y0i (xU i , (ei , e−i )).
x ˆ
ˆ

(2.11)

i x
By the definition of Y process in (2.8) and the identical initial wealth, we have U0 (ˆU i , e) ≥
ˆ
i
U0 (xU i , (ei , e−i ))
ˆ

Necessity Suppose (ˆU i , ei ) is optimal for agent i given (ˆU −i , e−i ) and (2.10) is violated
x ˆ
x
ˆ
by some (xU i , ei ); that is,

54

Γt
Γt

hU i t, xU i + k U i
ˆt
hU i t, xU i + k U i
t

ΣY i
t, ei , t
ˆt
Γt
ΣY i
t, ei , t
t
Γt

e
− xU i + ΣY i Φ(ˆt ) <
ˆt
t

(2.12)

− xU i + ΣY i Φ(ei , e−i )
t ˆt
t
t

on some subset of Ω × [0, T ] that belongs to F × B[0,T ] with a strictly positive P ⊗ λ
measure.
Let

(¯U i , ei ) =
xt ¯t


 Ui i
 (ˆ , e ) if (2.10) is true
x ˆ
 Ui i
 (x , e ) if (2.12) is true

Then comparison Theorem implies

Y0i (ˆU i , e) < Y0i (¯U i , (¯i , e−i ))
x ˆ
x
e ˆ

i x
i x
and U0 (ˆU i , e) < U0 (¯U i , (¯i , e−i )), which contradicts the assumption.
ˆ
e ˆ

From Theorem 6, at any equilibrium corresponding to p, for each i = 1, . . . , N agent i’s
time-t optimal control maximizes the (negative) time-t instantaneous drift of Y under the
original measure that is

ei (t, ΣY ) ∈ arg max Γt k U i
ˆ
for all

ΣY i
t, ei , t
Γt

ei ∈E
ΣY = (ΣY 1 , . . . , ΣY N )

55

+ ΣY i Φ(ei , et )
ˆ−i
t

∈ Rd×N and t ∈ [0, T ] .

and

xU i ∈ arg max Γt hU i t, xi − xi
ˆt

(2.13)

xi ∈R

ˆ
The effort equilibrium at t is determined by only t, ΣY . The optimal xU i depends only
ˆ
t
on agent i’s own preferences and not the other agents. Thus in our setting, agent equilibrium
is totally characterized by the equilibrium effort processes. At time t, the Nash equilibrium
et = e t, ΣY
ˆ
ˆ
t

is some deterministic function of t and ΣY .
t

We will let µY i (t, ΣY ) denote
ˆ
t

µY i
ˆ

t, ΣY
t

= Γt

hU i

t, xU i
ˆt

+ kU i

t, ei ,
ˆt

ΣY i
t
Γt

− xU i + ΣY i Φ(ˆ(t, ΣY )),
ˆt
e
t
t

(2.14)

so the Y process defined in (2.8) at the equilibrium controls (ˆU , e) follows the following
x ˆ
set of BSDEs.

dYti = − −rt Yti + pi + µY i t, ΣY
ˆ
t
t

dt + ΣY i dBt ,
t

i
YT = pi , i = 1, . . . , N
T

(2.15)

With the µY of (2.14) and corresponding equilibrium policy functions e, xU of (2.13),
ˆ
ˆ ˆ
we can proceed directly to the principals’ problem in Section 2.4, but for completeness, the
solution of the equilibrium agent-policy sample paths
xU t, ΣY , e t, ΣY ; t ∈ [0, T ]
ˆ
ˆ
t
t

given p requires the solution Y, ΣY

to the set of

BSDEs (2.15), which then yields the ΣY to substitute into the policy functions.

56

We interpret Y as the subjective present value (PV) of pay process.10

Note that

Γt Uti xU i , et represents the dollar cost of financing the optimal excess consumption stream
ˆt ˆ
(relative to the zero-utility optimal consumption stream), and therefore Yti = Γt Uti xU i , et −
ˆt ˆ
WtU i represents part of that cost financed by promised pay. The uncertainty driving Y, ΣY
is entirely due to the pay process, because the agents’ aggregators are deterministic functions. With xU i of (2.13) and the solution Y i of (2.15), agent i’s wealth process process
ˆ
ˆ
W U i is obtained from (2.1) after substituting optimal intermediate consumption cU i =
ˆt
ˆ
xU i + Yti + WtU i /Γt :
ˆt

U
ˆU
W0 i = w0 i ,

ˆ
dWtU i =

1
ˆ
WtU i rt −
Γt

+ pi
t

− xU i
ˆt

Yti
−
Γt

dt.

(2.16)

ˆU
Optimal lump-sum terminal consumption is cU i = WT i + pi , and optimal agent utility
ˆT
T
ˆ ˆ
is Uti cU i , ei

=

ˆ
Yti + WtU i /Γt . Equation (2.16) also shows that (xU , e) are effective

controls, because the process Y i of (2.8) depends only on (xU i , pi , e).
The solution of (2.15) is not needed to solve the principal’s problem, although we show
in Section 2.4 that the principals must solve analogous BSDEs representing their subjective
cash-flow PV process. We will assume that the equilibrium controls in (2.13) are well defined,
as is the case in all our applications. For sufficient conditions for the existence of a solution,
see Fudenberg and Tirole (1992).
A Nash equilibrium among the agents is obtained by defining the correspondence Γ :
[0, T ] × RN ·d → B E N

11

10 More

precisely, if we let Y i (p) denote the solution corresponding to the pay process p,
then we can interpret Yti (p) − Yti (0) as the agent’s time-t subjective value of pay.
11 In our Section 2.7 application there will be cases in which multiple time-t equilibria exist
for a given ΣY (that is, Γ t, ΣY is indeed set-valued). However, we also show that such a
t
t

57

Γ (t, Σ) =

∈ E N : i = I i t, −i , Σi ,

i = 1, . . . , N .

(2.17)

where I i : [0, T ] × E N −1 × Rd → E represents agent i’s optimal effort, given the other
agents’ efforts e−i ∈ E N −1 , i.e.12

I i t, −i , Σi = arg max

i ∈E

2.3.2

Γt k U i

i

Σ
t, i ,
Γt

+ Σi Φ( i , −i ) .

Some Basic Examples of Agent Equilibrium

We now give some examples of agent equilibria assuming, for simplicity, additive separability
of the risk-aversion and effort penalty terms:

k U i (t, , Σ) = f i (t, Σ) + g i (t, ) ,

t ∈ [0, T ] ,

i = 1, . . . , N,

(2.18)

for some concave f i , g i : Ω × [0, T ] × Rd → R. Separability implies that the effort equilibria
depend directly only on the effort-penalty functions g i . The problem of finding equilibrium
efforts is reduced to

e t, ΣY
ˆ
t

= arg max

ei ∈E
t

i

Γt g i t, ei + ΣY Φ(et ) ,
t
t

t ∈ [0, T ] .

(2.19)

Examples 4 assume a linear measure-change operator, which implies that equilibrium
agent-i effort depends only on agent-i utility diffusion, and not the diffusions of the other
ΣY would never be part of an equilibrium strategy among the principals. That is why we
t
define the function e t, ΣY which is some selection from Γ t, ΣY . In our applications, e
ˆ
ˆ
t
t
is always well defined.
12 In all our applications below I is well-defined: the arg max in (2.3.1) exists and is unique.

58

agents.

Examples 5 and 6 assume a concave measure-change operator, which generally

implies that agent-i effort will depend on all the agent diffusion processes. The examples
also illustrate the different implications of linear versus quadratic effort penalties. Unlike the
case of a quadratic penalty, a linear penalty results in a threshold effect such that agent i’s
time-t effort is zero for sufficiently small ΣY i compared to the other agents’ time-t diffusion
t
(in the concave case).
Example 4 (Quadratic effort penalties & linear Φ) Suppose (2.18) with a quadratic
effort penalty and a linear measure-change operator:

1
g i t, ei = − ei Qei ei ,
t
2 t t t

N

i = 1, . . . , N,

i
βt ei ,
t

Φ (et ) =
i=1

where β i , Qei ∈ L Rd×d , and Qei is assumed symmetric positive definite. Each agent’s
optimal effort in this case does not depend on the other agents’ efforts, and there is a unique
equilibrium ei = ei t, ΣU , i = 1, . . . , N . Applying (2.19), we have
ˆt ˆ
t
If E = Rd (no constraints on effort) then ei = (1/Γt ) Qei
ˆt
t

−1 i
βt

If E = Rd (nonnegative effort) then ei = (1/Γt ) max 0, Qei
ˆt
+
t

ΣY i .
t

−1 i
βt

ΣY i (the maximum
t

operator is applied to each dimension).
If E = v ∈ Rd : v1 = · · · = vd

(effort is restricted to be the same in every dimension)

i
then13 ei = 1d 1d βt ΣU i /Γt 1d Qei 1d where 1d is a length-d vector of 1s.
ˆt
t
t
i
If, in Example 4, βt is invertible for every i and t ∈ [0, T ] we can redefine effort as
i
ei = βt ei (preserving the quadratic penalty form) giving Φ (˜t ) =
˜t
e
t

N
i=1

ei . We henceforth use
˜t

this normalization.
13 The

same result is obtained by letting E = Rd and replacing obtained Qei with
t
i
i
1d Qei 1d I and βt with βt 1d 1d ), where I denotes an identity matrix.
t
59

In the next two examples the marginal impact of any agent’s effort is diminishing in
aggregate effort, which we model with a measure-change operator Φ (·) = (Φ1 (·) , . . . , Φd (·))
that is concave in aggregate effort along each dimension:.
N
i=1

Φk (et ) =

1−δ

ei (t)
k
,

1−δ

δ ∈ [0, 1),

k = 1, . . . , d.

(2.20)

Unlike the additive case, equilibrium effort of agent i depends on the utility-diffusion processes of all agents.
Example 5 (Quadratic effort penalties & concave Φ) Suppose (2.18); E = Rd (non+
negative effort); a quadratic effort penalty

gi

t, ei
t

1
=−
2

d

Qei ei (t)2 ,
k k

i = 1, . . . , N,

k=1

with Qei > 0 for all i, k; and the power measure-change operator (2.20).
k
Equilibrium agent-i’s effort in the kth dimension is uniquely given by

ei
ˆk

ΣY

ΣY i
k

+



N


=
ei 
Γt Qk
j=1

 −δ
Y j + 1+δ
Σk

ej 
Γt Qk

,

k = 1, . . . , d.

(2.21)

Agent i’s effort is increasing in his/her own utility diffusion value but diminishing in the
diffusion values of the others.

The final example combines a linear effort penalty with a concave measure-change operator, resulting in equilibria with only a single agent working in each dimension when

60

penalty-scaled diffusions are different, and multiple equilibria with more than one agent
working in each dimension when penalty-scaled diffusions match.
Example 6 (Linear effort penalties & concave Φ) Suppose preferences satisfy (2.18);
E = Rd (nonnegative effort); a linear effort penalty
+
d

gi

t, ei
t

i
qk ei (t) ,
k

=−

i = 1, . . . , N,

k=1
ei
with qk > 0 for all i, k; and the power measure-change operator (2.20). With two principal-

agents pairs, i ∈ {a, b}, we obtain for each dimension k and time t (henceforth omitting time
in efforts:14

arguments) the following possible Nash equilibria ek ∈ Γk t, ΣY

ea
k
eb
k

= 0,

eb
k

= 0,

ea
k

ea , eb
k k

> 0,

ea
k

=

ΣY b
k

=

ΣY a
k

+ eb
k

=

+
+

ΣY a
k

1/δ
a
b
if ΣY a /qk ≤ ΣY b /qk
k
k

b
/Γt qk
1/δ
a
/Γt qk
+

b
a
if ΣY b /qk ≤ ΣY a /qk
k
k

(2.22)

1/δ
a
b
if ΣY a /qk = ΣY b /qk
k
k

a
/Γt qk

The agent with the smaller scaled diffusion value will not work, whereas the agent with
the larger value will work if that value is positive.

Multiple time-t equilibria exist only if

a
b
ΣY a /qk = ΣY b /qk for some dimension k.15
k
k
14 Given

the other agents’ efforts, optimal agent-i effort in dimension k is

+
i
Ik t, e−i , ΣY i = 
k
k

ΣY i
k

+

i
/Γt qk

2

j

−

ek  .
j=i

15 For

N > 2 the results are analogous: Total effort in dimension k is
d

ei (t)
k
k=1

=

max

i=1,...,N

61

ΣY i
k
i
Γt q k

+ 1/δ

,

Derivation. See Section .2 in the Appendix.
The marginal cost of each agent i’s effort is constant, and the marginal benefit is the
product of the agent’s diffusion and the derivative of the common measure-change operator.
Each working agent therefore equates the ratio of diffusion and penalty term to the same
quantity: the inverse of the common derivative. Each agent whose ratio falls short of the
maximum will find the fixed marginal cost of effort too high at any effort level, and will
therefore shirk, free riding off the other agents’ effort.

2.3.3

Constant Elasticity of Substitution(CES)

In this section, we define a new class of measure change operator: CES production function. The CES class deals with the case of two principal-agent pairs. It covers the linear
and concave Φ in Section 2.3.1 as special cases and many other production functions in
Economics. Lemma 1 below presents closed-form solutions for agent equilibrium with CES
measure-change operator and quadratic effort penalty. We achieved additional relaxation
of the restrictions on parameters by working with quasiconcavity and FOC in the proof of
Lemma 1.
We assume two principal-agents pairs, i ∈ {a, b} throughout this section.
Definition 6 (Constant Elasticity of Substitution) For the measure change operator
Φ(·) = (Φ1 (·), . . . , Φd (·)), we define

Φk (e) = κ

α(ea )γ
k

+ (1 − α) (eb )γ
k

v
γ

,

κ > 0, α ∈ (0, 1), 0 = γ ≤ 2,

0 < v < 2, 1 ≤ k ≤ d.
and positive effort is exerted only by those agents whose utility to penalty ratio equals the
maximum ratio.
62

Elasticity of substitution is 1/ (1 − γ), and v is the elasticity of scale.
Some special cases follow:
Cobb Douglas production function:

Φk (e) = κ

ea
k

α

eb
k

1−α v

. This is achieved by letting γ −→ 0.

Leontief production function (or perfect complements):

Φk (e) = κ min ea , eb . This is achieved by letting v = 1 and γ −→ −∞.
k k

Infinite elasticity (linear production if v = 1; diminishing returns to scale if v < 1):

Φk (e) = ea + eb
k
k

v

1
. This is achieved by letting γ = 1, α = , and κ = 2v .
2

Lemma 1 Assume preferences satisfy (2.18); E = Rd (nonnegative effort); a quadratic
+
effort penalty
1
g i t, ei = −
t
2

d

Qei ei (t)2 ,
k k
k=1

with Qei > 0 for all i, k; Φ (e) satisfies (2.23).
k
Suppose ΣY a , ΣY b > 0 and define

Sa =

(1 − α) vΣY b
αvΣY a
, Sb =
.
Γt Qea
Γt Qeb

63

a. 0 = γ < 2. The unique Nash equilibrium among the agents’ efforts is

ea = κ1/(2−v) (S a )1/(2−γ) α (S a )γ/(2−γ) + (1 − α) S b
1/(2−γ)

eb = κ1/(2−v) S b

γ/(2−γ) (v−γ)/{γ(2−v)}

α (S a )γ/(2−γ) + (1 − α) S b

,

γ/(2−γ) (v−γ)/{γ(2−v)}

.

b. γ = 2. The Nash equilibria are

ea

=

eb =

v
2(2−v)
α

vΣY a κ
Γt Qa

v
(1 − α) 2(2−v)

1
2−v

,

eb

αΣY a
(1 − α) ΣY b
if
>
,
Qa
Qb

= 0,

1

vΣY b κ 2−v
,
Γt Qb

ea = 0,
2

α(ea )2 + (1 − α) (eb )2 =

αvΣY a κ 2−v
, if
Γt Qa

if

αΣY a
(1 − α) ΣY b
<
,
Qa
Qb

(1 − α) ΣY b
αΣY a
=
.
Qa
Qb

c.(Cobb Douglas) γ → 0. The Nash equilibria are

ea
eb

=

2−(1−α)v
1/(2−v) (S a ) 2(2−v)
κ

=

αv
1/(2−v) (S a ) 2(2−v)
κ

(1−α)v
b 2(2−v)
S

2−αv
b 2(2−v)
S

,

,

and the additional equilibrium ea = eb = 0.
d.(Leontief ) γ → −∞. The Nash equilibria are

ea = eb = K, for any K ∈ 0,

64

κ
min
Γt

(2.23)

ΣY a ΣY b
,
Qa Qb

.

Proof.
By Theorem 6, each agent solves, for each dimension k,
v
Γt ei i 2
Y κ α(ea )γ + (1 − α) (eb )γ γ ,
max − Qk (ek ) + Σk
k
k
2
ei ∈E

(2.24)

k

holding fixed the other’s effort. Henceforth omit the k subscript. Consider agent a’s problem
(2.24). The first derivative of the RHS of (2.24) w.r.t effort is
v
∂
a ea + αvΣY a κ α(ea )γ + (1 − α) (eb )γ γ −1 (ea )γ−1
= −Γt Q
∂ea
Φ (e)
.
= −Γt Qa ea + αvΣY a (ea )γ−1
α(ea )γ + (1 − α) (eb )γ

∂
First suppose ΣY a ≤ 0 then ∂ea ≤ 0. The maximum is attained by ea = 0, if γ ≥ 0. The

maximum is unattainable, if γ < 0. In this case, the Nash equilibrium does not exist and
instead the -Nash equilibrium exists. However, in all our applications, we have ΣY i > 0,
i ∈ {a, b}.
Now suppose ΣY a > 0, ΣY b > 0.
The stationary point is positive, because v > 0. The second derivative is
∂2
(∂ea )2

= −Γt

Qa

+ αvΣY a

∂
∂ea

α(ea )γ

+ (1 − α) (eb )γ

v
γ −1

(ea )γ−1

where

∂
∂ea
=

α(ea )γ

+ (1 − α) (eb )γ

Φ (e) (ea )γ−2
2
α(ea )γ + (1 − α) (eb )γ

v
γ −1

(ea )γ−1

(γ − 1) α(ea )γ + (1 − α) (eb )γ + α (v − γ) (ea )γ

65

A sufficient condition for concavity is that the square bracketed term is nonpositive. That
is,
α (v − 1) (ea )γ + (γ − 1) (1 − α) (eb )γ ≤ 0,
which is satisfied if v, γ ≤ 1 and because of our parameter restrictions. So the FOCs are
sufficient for optimality.
More generally, from

∂
= ea
∂ea

−Γt Qa + αvΣY a (ea )γ−2

Φ (e)
α(ea )γ + (1 − α) (eb )γ

.

we get quasiconcavity(A function f : Rn → R is called quasiconcave, if the upper-level
set {x ∈ Rn : f (x) ≥ r} is convex, for any r ∈ R.) if the term in the brackets is decreasing
in ea , because then the objective function is either always decreasing or first increasing and
then decreasing as ea increases. Assuming ΣY a > 0 the term in parenthesis is decreasing if
and only if

h (ea )

=

(ea )γ−2

Φ (e)
a )γ + (1 − α) (eb )γ
α(e

=

κ(ea )γ−2

α(ea )γ

+ (1 − α) (eb )γ

v
γ −1

decreasing in ea . From

h

(ea )

=

k(ea )γ−3

α(ea )γ

+ (1 − α) (eb )γ

v
γ −2

(v − 2) α(ea )γ + (γ − 2) (1 − α) (eb )γ

we get
h (ea ) ≤ 0

⇐⇒

(v − 2) α(ea )γ + (γ − 2) (1 − α) (eb )γ ≤ 0

66

(2.25)

This holds if
γ, v ≤ 2 (strictly for ea > 0 if v < 2, γ ≤ 2)
We now deal with each case separately.
a. γ < 2. The FOCs for the two agents can be written (if γ < 2)

(ea )2−γ
Sa

eb
=

2−γ

=

Sb

v
Φ (e)
γ −1
= κ α(ea )γ + (1 − α) (eb )γ
,
α(ea )γ + (1 − α) (eb )γ

Substituting out eb yields

(ea )2−γ
Sa

= κ (ea )v−γ




α + (1 − α)



Sb
Sa

v
γ/(2−γ)  γ −1


so the FOC implies

ea
ˆ

=

κ1/(2−v) (S a )1/(2−γ)

Use the equality

eb
ˆ

=

κ1/(2−v)

eb

2−γ

Sb

=

α (S a )γ/(2−γ)

(ea )2−γ
Sa

1/(2−γ)
Sb

+ (1 − α)

(v−γ)/{γ(2−v)}
b γ/(2−γ)
S
.

to get the solution

α (S a )γ/(2−γ)

+ (1 − α)

(v−γ)/{γ(2−v)}
b γ/(2−γ)
S

The next step is to verify that ea is indeed a unique maximum point given eb and vice
ˆ
ˆ
versa.
∂
This is achieved by noting that strict monotonicity of h(see (2.25)) implies ∂ea > 0 for

67

∂
e
any ea ∈ (0, ea ) and ∂ea < 0 for any ea ∈ (ˆa , ∞).
ˆ

b. γ = 2 The first derivative of the RHS of agent a’s problem (2.24) w.r.t effort is
v −1
∂
= ea −Γt Qa + αvΣY a κ α(ea )2 + (1 − α) (eb )2 2 .
∂ea

(it is easy to confirm quasiconcavity because

α(ea )2
k

v −1
b )2 2
+ (1 − α) (ek

is mono-

tonically decreasing in ea ). The FOC (also sufficient) for a and b are, respectively,
2

αvΣY a κ 2−v
Γt Qa
(1 − α) vΣY b κ
Γt

If

α(ea )2

>

(1−α)vΣY b κ
Γt Qb

2
2−v

Qb

2
2−v

= α(ea )2 + (1 − α) (eb )2 ,

(2.26)

= α(ea )2 + (1 − α) (eb )2 .

(2.27)

, then from (2.27) the optimal eb = 0, (2.26) implies

the first possible equilibrium below.
Similarly, the second possible equilibrium holds if

(1 − α) (eb )2

The last possible equilibrium follows if α(ea )2 , (1 − α) (eb )2 ≤
αvΣY a κ
Γt Qa

2
2−v

.

Thus all the possible equilibria are (2.23).

68

>

αvΣY a κ
Γt Qa
(1−α)vΣY b κ
Γt Qb

2
2−v

.

2
2−v

=

c.(Cobb Douglas) γ → 0.
γ/(2−γ) (v−γ)/{γ(2−v)}

α (S a )γ/(2−γ) + (1 − α) S b
γ→0



 ln α (S a )γ/(2−γ) + (1 − α) S b γ/(2−γ) 




(v − γ) 

= exp  lim
γ→0 (2 − v) 

γ




lim

v
α ln S a + (1 − α) ln S b
2 (2 − v)

= exp
=

vα
a ) 2(2−v)
(S

v(1−α)
b 2(2−v)
S

d.(Leontief ) γ → −∞. If Φ (e) = κ min ea , eb , then

ea = min

κΣY a b
,e
ΓQa

and eb = min

κΣY b a
,e .
ΓQb

Therefore any equal nonnegative effort level less than or equal to min

κΣY a , κΣY b
ΓQa
ΓQb

is an equilibrium.

The CES measure change operator with quadratic effort penalty includes power measure
change operator with linear effort penalty as a special case. The notation change defined
below transforms CES with quadratic penalty into Example 6.
First define ei =
˜k

2ei , i ∈ {a, b}.
k

For any δ ∈ [0, 1), let

γ = 2, v = 1 − δ, α = 1/2 and κ = 1/(1 − δ).

69

With the new notations, the measure change operator (2.23) agrees with (2.20).
The agent equilibrium solution in (2.23) agrees with (2.22).

2.4

Principal Equilibrium

Having solved the agent equilibrium efforts as function of the agent subjective pay PV diffusion, we now solve the equilibrium in the principal level. The principal equilibrium is defined
based on agent equilibrium and subject to participation constraints. The participation constraint
i
U0 ≥ K i , i = 1, . . . , N.

(2.28)

U
is equivalent to Y0i ≥ Γ0 K i − w0 i , i = 1, . . . , N .

To simplify notations, for any set of ΣY , we will let Π(ΣY ) denote the set of agent
equilibrium efforts under ΣY , i.e.

Π(ΣY ) = e ∈ E N : for any t ∈ [0, T ), et ∈ Γ(t, ΣY )
t

(2.29)

,
where Γ(·) is defined in (2.17).
Recall our definition, xV i = cV i − Vti , t < T .
t
t
We denote by
xV = (xV 1 , . . . , xV N ) and p = (p1 , . . . , pN )
the collection of principals’ strategies. As we explained in Section 2.2.2, choosing xV is
essentially equivalent to choosing cV .
70

Because of the availability of money-market trading, the class of pay processes p that
generates the same agents’ and principals’ utilities is not unique. This is well known, but
shown in Lemma 2 below for completeness.
The following lemma shows that there is no unique optimal pay process because with
unrestricted trading in the money-market security, both principal and agent are indifferent
between shifting some intermediate pay to the money-market account, and modifying terminal pay. The next lemma applies to any principal-agent pair i, i = 1, . . . , N , so for simplicity
we will omit the superscript i.
Lemma 2 Suppose X finances

p, cV , and p finances cU .

Let pt , t < T , be some
˜

intermediate pay process. If
T

pT = pT +
˜

then X finances

0

T
˜
e t rs ds (pt − pt ) dt,

p, cV , and p finances cU .
˜
˜

(2.30)

That is, the same agent and consumption

streams are attained by investing the difference in intermediate pay in the money-market
account and adding the terminal money-market balance to lump-sum terminal pay.
˜
Proof. Fix the consumption processes cU and cV , and let W k and W k , k ∈ {U, V }, denote
the money-market balances corresponding to pay processes p and p, respectively.
˜
˜
∆i = Wti − Wti , i ∈ {U, V }, satisfy
t

∆U = 0,
0

d∆U = ∆U rt + pt − pt dt,
˜
t
t

0 = ∆ U + pT − pT ,
˜
T

∆V
0

d∆V = ∆V rt − pt + pt dt,
˜
t
t

˜
0 = ∆V − pT + pT ,
T

= 0,

71

Then

which have the solutions

∆U =
t

t

t

e s ru du (ps − ps ) ds,
˜

0

∆V = −∆U ,
t
t

t ∈ [0, T ] .

If (2.30) holds then (X, p) and (X, p) finance the same principal and agent consumption
˜
processes.

We will say two feasible pay processes p and p are equivalent if (2.30) is satisfied.
˜
Based on the above lemma, the set of pay P is a union of mutually exclusive equivalent
classes. We will treat the pay processes that belong to the same equivalent class as the same
throughout the paper. Any pay process can be replaced by its peers in the same equivalent
class.
Denote by V0i xV i , p, e the initial principal i utility, given by the solution of the BSDE
(2.5). We now give our initial formulation of principal equilibrium. The final formulation
will appear later, see Definition 8.
An initial formulation of principal equilibrium: The set of principal strategies p =
ˆ
{ˆi , i = 1, . . . , N } ∈ P N and xV = {ˆV i , i = 1, . . . , N } ∈ HN constitute a principal
p
ˆ
x
equilibrium (in the sense of Nash), if for all (xV , p) ∈ HN × P N and each i = 1, . . . , N ,

V0i (ˆV i , p, e) ≥ V0i (xV i , (pi , p−i ), e),
x
ˆˆ
ˆ
given that p and (pi , p−i ) induce agent equilibrium efforts e and e, respectively. Also the
ˆ
ˆ
ˆ
participation constraint (2.28) is satisfied. As we saw in Section 2.3.1, agent equilibrium

72

and participation constraint can be expressed by the following:

e ∈ Π(ΣY (ˆ)),
ˆ
p
e ∈ Π(ΣY (pi , p−i )) and
ˆ

U
p
Y0i (˜) ≥ Γ0 K i − w0 i ,

p ∈ p, (pi , p−i )
˜
ˆ
ˆ

where (Y i = Y i (˜), ΣY i = ΣY i (˜)), i = 1, . . . , N satisfy
p
p

dYti = − −rt Yti + pi + µY i t, ΣY
˜t ˆ
t

dt + ΣY i dBt ,
t

i
YT = pi , i = 1, . . . , N,
˜T

and µY i (·) is defined in (2.14).
ˆ
In the formulation above, (Y i , ΣY i ) represents the agent i’s subjective PV process with
pay plan p and evaluated at the agent equilibrium solution. As we discussed in Section
˜
2.3.1, the uncertainty driving Y, ΣY

is entirely due to the set of pay processes p, in
˜

addition to the driving Brownian motion B.
We follow the solution approach as in the single principal/agent case examined in Schroder
(2013). The paper shows that letting principals choose xV , p is essentially equivalent to
letting principals choose xV , ΣY . By using the strategy xV i , ΣY i , the principal’s probˆ ˆ
lem is amenable to a simple dynamic programming solution. Furthermore, with this choice
of principal strategy, principal i need not consider whether his/her strategies will impact the

73

participation constraints of other principal-agent pairs16 ; all participation constraints are
shown to be binding under the TI preferences. The added complexity of the multiple principal problem is the interdependence of the problems and the determination of the equilibrium
strategies among principals and agents.
The first step in applying the dynamic programming approach is to confirm that the
participation constraints will bind at principal equilibrium.
Lemma 3 The participation constraints all bind in any principal equilibrium.
Proof. Suppose (ˆV , p) ∈ HN × P N is a principal equilibrium, and there exists i, such
x ˆ
i x
that under the resulting agent equilibrium (ˆU , e), U0 (ˆU i , e) > K i . Let (ˆU , cV ) be the
x ˆ
ˆ
c ˆ
i x
implied sets of agents’ and principals’ consumptions and = U0 (ˆU i , e) − K i . By the wealth
ˆ

equations (2.1) and (2.2), the same principal cash flow process would finance a principal
consumption rate cV i + and a pay plan rate of pi − . This pay plan would finance an agent
ˆ
ˆ
consumption rate cU i − and , while ΣY , and therefore e t, ΣY
ˆ
ˆ
t

remain unchanged17 . By

i c
quasilinearity (2.6), the resulting agent i’s utility is U0 (ˆU i , e) = K i , a binding participation
ˆ

constraint. The resulting principal i’s utility is V0i (ˆV i , e) + > V0i (ˆV i , e). Therefore (ˆV , p)
c ˆ
c ˆ
x ˆ
can not be a principal equilibrium.
i
Upon substituting the binding participation constraints U0 = K i , which implies Y0i =
16 If

a strategy choice by principal i were to cause a violation of agent j’s participation
constraint, then principal i would have to account for the effect of agent j’s rejection of the
contract.
17 This can be seen from (2.15). Because Y i is in the TI form, a constant change in pay
process results a deterministic change in Y i and ΣY i remains unchanged.

74

U
Γ0 K i − w0 i , into (2.15), we have the following set of forward equations

dt + ΣY i dBt ,
t

ˆ
dYti = − −rt Yti + pi + µY i t, ΣY
t
t

U
Y0i = Γ0 K i − w0 i , i = 1, . . . , N

(2.31)
where µY i (t, ΣY ) is given by (2.14). Observe that the agent equilibrium solution (ˆU , e) is
ˆ
x ˆ
t
part of (2.14), so it is already built into (2.31).
With a binding participation constraint, for principal i, choosing pi is essentially equivalent to choosing ΣY i . Once the set of ΣY has been chosen, principal i can choose any feasible
T
intermediate pay process(pi , s < T is feasible, if 0 pi ds ∈ M.) with terminal pay implied
s
s

by (2.31) i.e. for any ΣY and feasible intermediate pay process pi , t < T , the terminal pay is
t

pi
T

=

T
Y0i e 0 rs ds

T

−

e

T
t rs ds

0

pi
t

+ µY i (t, ΣY )
ˆ
t

T

dt +
0

T
e t rs ds ΣY i dBt
t

(2.32)

Each set of ΣY corresponds to a class of equivalent pay processes for each principal i. This
is because for any other feasible intermediate pay processes pt , t < T , let the terminal pays
˜
U
pi be implied by the forward equation (2.31) with the same initial value Y0i = Γ0 K i − w0 i
˜T

and ΣY , t < T chosen by the N principals as in (2.32), i.e.
t
T
pi = Y0i e 0 rs ds −
˜T

T
0

T
e t rs ds pi + µY i (t, ΣY ) dt +
˜t ˆ
t

T
0

T
e t rs ds ΣY i dBt (2.33)
t

Upon subtracting (2.32) from (2.33), we see that (2.30) is satisfied. Thus p and p are
˜
equivalent.
Based on the above argument, we will define ΣY as principals’ strategies instead of p.

75

From now on, we denote by

xV = (xV 1 , . . . , xV N ) and ΣY = (ΣY 1 , . . . , ΣY N )

the collection of principals’ strategies.
The following definition specifies the set of implementable principal strategies such that
the resulting equilibrium effort and control plans are feasible.
Definition 7 For a set of principals’ strategies (xV , ΣY ) ∈ HN × L2 (Rd )N , let (ˆU , e) be
x ˆ
the resulting agent equilibrium control and effort plans in (2.13).
N
N
N
If (ˆU , xV , e) ∈ HU f × HV f × Ef , we will call (xV , ΣY ) implementable and denote the
x
ˆ

set of the principals’ implementable strategies by I N .
We will impose the above implementability condition on any collection of principals’
strategies and thus omit it from the text. Each principal i chooses the optimal xV i , ΣY i
to maximize utility holding fixed the strategies of the other principals, xV −i , ΣY −i , while
anticipating the impact of their strategy on the equilibrium efforts of all the agents.
Based on the explanation above, for any set of principals’ strategies (xV , ΣY ), we will let
Ψi (ΣY ) denote the class of equivalent pay plans of principal i induced by ΣY , i.e.

Ψi (ΣY ) = pi ∈ P : pi satsifies (2.32) for any feasible pi , t < T
t
T

We will also let Λ(ΣY ) denote the pay plans of all principals induced by ΣY , i.e.

76

(2.34)

Λ(ΣY ) = p ∈ P N : pi ∈ Ψi (ΣY ), i = 1, . . . , N

(2.35)

Since we have used (xV , ΣY ) as principals’ strategies, we give the following appropriate
Definition for principal equilibrium.
Definition 8 (Principal Equilibrium) A set of strategies (ˆV , ΣY ) ∈ HN × L2 (Rd )N
x ˆ
constitutes a principal equilibrium, if for all (xV i , ΣY i ) ∈ H × L2 (Rd ) and each i = 1, . . . , N ,

ˆ
V0i xV i , ΣY
ˆ

ˆ
≥ V0i xV i , ΣY i , ΣY −i ,

subject to

ˆ
V0i xV i , ΣY
ˆ

= V0i xV i , p, e ,
ˆ
ˆˆ

where

ˆ
e ∈ Π ΣY
ˆ

ˆ
and p ∈ Λ(ΣY ),
ˆ

and

ˆ
V0i xV i , ΣY i , ΣY −i = V0i xV i , p, e ,

where

ˆ
e ∈ Π ΣY i , ΣY −i

ˆ
and p ∈ Λ(ΣY i , ΣY −i ).

The main result of the section is Theorem 7 and its corollary below, which show that
the determination of the equilibrium pay contracts reduces to the computation of a set of
subjective present-value (PV) processes and associated diffusion processes for the principals,
Z i , ΣZi , i = 1, . . . , N (see (2.36) for the definition). The principals’ equilibrium controls
ˆ
ˆ
ˆ
ΣY take the form ΣY = ΣY ω, t, ΣZ .
t

The equilibrium time-t diffusion that principal
77

i chooses for his/her agent is a function of the time-t subjective PV diffusion of all the
principals.

In the case of a linear measure-change operator Φ, the equilibrium simplifies

ˆ
because ΣY i depends only ith PV diffusion value ΣZi .
t
t

But even in the linear case, the

principals’ optimal contracts are linked because the subjective PV processes ΣZ must be
solved jointly as the drift of each depends on the diffusion processes of the others.

This

follows because even though agent i’s effort under the equilibrium compensation contract
will depend only on ΣY i , this effort affects the cash-flow drifts and therefore the subjective
t
PV processes of all the principals.
To motivate the solution of principal equilibrium, for any implementalbe principal policy
xV , ΣY

i
define for principal i, Zt xV i , ΣY

as his/her subjective PV process, which is

dollar utility value plus the subjective pay liability minus the principal’s financial wealth:

i
Zt xV i , ΣY

= Γt Vti xV i , ΣY

− WtV i xV i + Yti ΣY .

(2.36)

i
Applying Ito’s lemma, with (2.5), (2.31), and (2.2), we get the dynamics of Zt =
i
Zt xV i , ΣY :

i
i
i
dZt = − −rt Zt + Xt + µZi xV i , ΣY
t

i
i
dt + ΣZi dBt , ZT = XT , i = 1, . . . , N. (2.37)
t

where

µZi
t
+Γt

xV i , Σ Y

= Γt

hV i

hU i (t, xU i ) + k U i
ˆt

t, xV i
t

t, e
ˆ

+ kV i

t, ΣY
t

78

ΣY i
, t
Γt

ΣZi − ΣY i
t
t, t
Γt

− xV i
t

− xU i + ΣZi Φ e t, ΣY
ˆt
ˆ
t
t

,(2.38)

ΣZi = Γt ΣV i + ΣY i ,
t
t
t
and xU i and e t, ΣY
ˆt
ˆ
t

(2.39)

are the agent equilibrium given by Theorem 6 for any set of ΣY .

Theorem 7 below shows that each principal’s optimality is essentially equivalent to maximizing the drift term of his/her cash-flow PV processes.
Theorem 7 (Principal optimality) Suppose

Zi, ΣZi , i = 1, . . . , N solves the BSDE

system (2.37) with a set of principals’ strategies

xV , ΣY

∈ HN × L2 (Rd )N .

Then

xV i , ΣY i ∈ H × L2 (Rd ) is optimal for principal i holding fixed other principals’ strategies,
˜
˜
˜
i.e. V0i (xV i , ΣY ) ≥ V0i xV i , ΣY i , ΣY −i , if and only if for any other strategy xV i , ΣY i ∈
˜
H × L2 (Rd ),

µZi xV i , ΣY
t

˜
≥ µZi xV i , ΣY i , ΣY −i , t ∈ [0, T ].
˜
t

(2.40)

where µZi (·) is defined in (2.38).
t
Proof.
Sufficiency Consider a set of principals’ strategies (xV , ΣY ) and let the process Z i be
defined as in (2.36) with its dynamics specified by (2.37) and (2.38). If principal i’s strategy
˜
(xV i , ΣY i ) is switched to (˜V i , ΣY i ), analogously define
x

˜i ˜
˜
˜
˜
Zt xV i , ΣY i , ΣY −i = Γt Vti xV i , ΣY i , ΣY −i − WtV i xV i + Yti ΣY i , ΣY −i .
˜
˜

and it has the following dynamics

i
˜i
˜i
˜
dZt = − −rt Zt + Xt + µZi xV i , ΣY i , ΣY −i
˜
t

79

i
˜
˜i
dt + ΣZi dBt , ZT = XT .
t

Using (2.40), define the nonnegative process

˜
− µZi xV i , ΣY i , ΣY −i , t < T.
˜
t

ht = µZi xV i , ΣY
t

i
˜i
The discounted processes Dt Zt and Dt Zt follow the dynamics below.

i
i
i
i
˜t
dDt Zt = −Dt (ht + Xt + µZi )dt + Dt ΣZi dBt , ZT = XT ,
t

(2.41)

i
i
˜i
˜
˜i
dDt Zt = −Dt (Xt + µZi )dt + Dt ΣZi dBt , ZT = XT .
˜t
t

The comparison theorem (Theorem 5 of Briand and Hu (2008)) implies

i
˜i
Z0 ≥ Z0 .

˜
By the definition of Z and Z, the identical initial financial wealth and a binding participation
˜
constraint for both, we have V0i (xV i , ΣY ) ≥ V0i xV i , ΣY i , ΣY −i .
˜
Necessity Suppose that

xV i , ΣY i

is the optimal strategy for principal i and there

˜
exists some other strategy xV i , ΣY i such that the process h < 0 on some subset of Ω×[0, T ]
˜
that belongs to F × B[0,T ] with a strictly positive P ⊗ λ measure. Here h is defined the same
as in the sufficiency part. We define

=

xV i , ΣY i




¯
xV i , Σ Y i
¯





˜
xV i , ΣY i
˜

if h ≥ 0,
(2.42)
otherwise.

Analogously, define

¯
¯i ¯
¯
¯
¯
Zt xV i , ΣY i , ΣY −i = Γt Vti xV i , ΣY i , ΣY −i − WtV i xV i + Yti ΣY i , ΣY −i .
¯
80

¯
then the discounted process Dt Zt solves

i
¯
¯i
¯
dDt Zt = −Dt Xt + µZi xV i , ΣY i , ΣY −i
t

¯
By (2.42), µZi xV i , ΣY i , ΣY −i
¯
t

i
¯i
¯
dt + Dt ΣZi dBt , ZT = XT .
t

> µZi xV i , ΣY
t

on some subset of Ω × [0, T ] with

i
a strictly positive measure. Upon comparing the above BSDE with the BSDE for Dt Zt in
i
¯
¯i
¯
(2.41), the comparison Theorem implies Z0 < Z0 and thus V0i (xV i , ΣY ) < V0i xV i , ΣY i , ΣY −i ,

which contradicts the optimality of xV i , ΣY i .

The corollary below characterizes principal equilibrium based on Theorem 7.
Corollary 1 (Principal equilibrium) Suppose xV , ΣY
ˆ ˆ

∈ HN × L2 (Rd )N satisfies

xV i ∈ arg max Γt hV i (t, x) − x,
ˆt

(2.43)

x∈R

ˆ
ΣY i ∈ arg max Γt
t
Σ∈Rd

kV i

ΣZi − Σ
t, t
Γt

ˆ
+ΣZi Φ e t, Σ, ΣY −i
ˆ
t
t

where Z, ΣZ

ˆ
+ k U i t, ei (t, Σ, ΣY −i ),
ˆ
t

i = 1, . . . , N,

Σ
Γt

all (ω, t) ∈ Ω × [0, T ),

solves the BSDE system

i
i
i
i
i
dZt = −(−rt Zt + Xt + µZi (t, ΣZ ))dt + ΣZi dBt , ZT = XT , i = 1, . . . , N.
ˆ
t
t

where we have defined

81

(2.44)

i
µZi = Ht + Γt
ˆt

kV i

ˆ
ΣZi − ΣY i
t
t, t
Γt

ˆ
+ΣZi Φ e t, ΣY
ˆ
t
t

+ kU i

ˆ
ΣY i
ˆ
t, e t, ΣY , t
ˆ
t
Γt

,

(2.45)

i
Ht = Γt hV i (t, xV i ) + hU i (t, xU i ) − xV i − xU i ,
ˆt
ˆt
ˆt
ˆt
t

ˆ
and xU i , e t, ΣY
ˆt ˆ
t
Then xV , ΣY
ˆ ˆ

ˆ
are the agent equilibrium given by Theorem 6 for the set ΣY .

is a set of equilibrium strategies among the principals.

ˆ
The equilibrium xV i only depends on principal i’s own preferences, while ΣY i depends on
t
t
the diffusion of cash-flow PV of all the principals, ΣZ . The equilibrium among the principals
ˆ
is constructed by first solving (2.43) for xV i and ΣY i ; the equilibrium thus takes the form
ˆt
t

ˆ
ˆ
xV i (ω, t) and ΣY i = ΣY i ω, t, ΣZ ,
ˆ
t
t

t ∈ [0, T ),

i = 1, . . . , N,

ˆ
for some functions xV i : Ω × [0, T ] → R and ΣY i : Ω × [0, T ] × Rd×N → R.
ˆ

(2.46)

These are

substituted into (2.45) to obtain the BSDE system (2.44). In Section 2.5, we obtain closedform solutions for (2.44), when the cash-flow dynamics for X i are driven by affine-yield state
ˆ
variables. The functions xV (·) and ΣY (·) evaluated at the BSDE solution Z, ΣZ
ˆ

yields

a principal equilibrium.
i
Note that the final ”payoff” of ZT is principal i’s lump-sum terminal cash flow, and

the intermediate cash flow enters the drift of Z i .

If the agent and principal preference

functions do not depend explicitly on ω as in all our applications, then the only source of
82

uncertainty driving the Z i s is the cash-flow uncertainty. This justifies the interpretation of
Z i as principal i’s subjective PV of the cash-flow process X i .
The subjective PV process defined by (2.44) is within a multidimensional version of
the TI class of the utility functions and inherits their quasilinearity property: each unit
increase in the cash-flow process X i (including the terminal lump-sum component) results
in a deterministic increase in Z i .
Recalling the identity ΣZi = Γt ΣV i + ΣY i in (2.39), the equilibrium diffusion stratt
t
t
egy can be interpreted as a diffusion ”sharing rule” of the subjective PV process Z i , with
ˆ
ΣY i ω, t, ΣZ
t

ˆ
of the time-t risk allocated to agent i and ΣZi − ΣY i ω, t, ΣZ
t
t

allocated

to principal i. Unlike the single principal-agent case, principal i’s optimal sharing rule can
depend on all the subjective diffusion processes, ΣZ = ΣZ1 , . . . ΣZN .
As seen from the corollary, it is simple to compute the equilibrium controls xV because
ˆ
the problems are unlinked across the principals. We will henceforth assume that the optimal
xV i is well defined for each i. Also, the equilibrium controls xV have no direct impact on
ˆ
t
either equilibrium agent effort or on the equilibrium rule for sharing the subjective cash-flow
diffusion processes ΣZ among principals and agents. As a simple example, if for some γ > 0,
1
1
hV i (t, x) = − γ exp(−γx), then xV i = Γ ln(Γt ).
ˆt

ˆ
The more interesting problem is the computation of the equilibrium strategies ΣY ,
ˆ
which must be jointly solved, and each component ΣY i may depend on joint diffusion
processes ΣZ .

ˆ
Furthermore it is the set of ΣY that determine equilibrium agent effort,

e t, ΣY ; t ∈ [0, T ) . Our applications will focus on this control, for which we find exˆ
t
plicit solutions.
Having solved for Z, ΣZ and the equilibrium principal strategies xV , ΣY , substitute
ˆ ˆ

83

ˆ
ΣY into (2.31) to get
t

t

ˆ
Yti = Y0i e 0 rs ds −

0

t

t

ˆ
ˆ
e s ru du pi + µY i (s, ΣY ) ds +
s
s

0

t

e s ru du ΣY i dBs .
s

for any feasible intermediate consumption process. Then the terminal pay is

ˆi
pi = YT ,
ˆT

(2.47)

i
ˆ
ˆ
ˆ
equilibrium principal i’s utility is Vti = Zt − Yti + WtV i /Γt , and equilibrium principal i’s

ˆ
wealth satisfies (2.2) after substituting cV i = xV i + Vti :
ˆt
ˆt

V
ˆV
W0 i = w0 i ,

ˆ
dWtV =

1
ˆ
WtV i rt −
Γt

ˆ
Z i − Yti
i
+ X t − p i − xV i − t
ˆt
t
Γt

dt,

i
ˆV
and equilibrium lump-sum terminal consumption is cV i = WT i + XT − pi .
ˆT
ˆT

Remark 2 (Terminal consumption only) The following modifications are required if there
is no intermediate consumption (by either principal or agent).

Let cU i = cV i = 0 in the

wealth equations (2.1) and (2.2), omit the excess consumption arguments in the aggregators,
T
and replace the bond price Γt by DT /Dt = e− t rs ds in the following equations.

The corresponding equations (2.9) for the Y process are changed to

D
dYti = − −rt Yti + pi + T k U i
t
Dt

t, ei ,
t

ΣY i
t
DT /Dt

i
+ ΣY i Φ(et ) dt + ΣY i dBt , YT = pi ,
t
t
T

i = 1, . . . , N.

The agent equilibrium condition (2.13) is the same except that the maximization over
84

intermediate consumption is dropped.

DT U i
k
ei ∈E Dt

ei (ω, t, ΣY ) ∈ arg max
ˆ

t, ei ,

ΣY i
t
DT /Dt

ˆt
+ ΣY i Φ(ei , e−i )
t

The corresponding equations (2.37) for the Z process are changed to

i
i
i
i
i
dZt = −(−rt Zt + Xt + µZi )dt + ΣZi dBt , ZT = XT , i = 1, . . . , N.
t
t

where

µZi
t

D
= T
Dt

kV i

ΣZi − ΣY i
t
t, t
DT /Dt

+ kU i

t, e
ˆ

t, ΣY
t

ΣY i
t
,
DT /Dt

+ ΣZi Φ e t, ΣY
ˆ
t
t

The principal equilibrium condition (2.43) is the same except that the maximization over
intermediate consumption is dropped.

D
ˆ
ΣY i ∈ arg max T
t
Σ∈Rd Dt

kV i

ΣZi − Σ
t, t
DT /Dt

ˆ
ˆ
+ΣZi Φ e t, Σ, ΣY −i
t
t

85

ˆ
ˆ
+ k U i t, ei (t, Σ, ΣY −i ),
t

Σ
DT /Dt

2.5

Closed-Form Solution with Cash Flows Driven by
State Processes

In the previous sections, we showed that the key step to solving principal equilibrium is
solving a subjective PV of cash flow process denoted by Z and given by (2.44). The equilibrium strategies ΣY are then functions of the corresponding subjective diffusion process ΣZ .
In general, this diffusion process is itself a stochastic process, with dynamics dependent on
the preferences of the principal and agent. We show in this section that if the principals’
aggregators in (2.5) are deterministic functions, and uncertainty is driven by state processes
in (2.48) with affine dynamics, then the subjective cash-flow diffusion ΣZ is deterministic,
solvable in closed form, and invariant to preferences (i.e., invariant to the form of the aggregators). Of course the subjective cash-flow process will depend on preferences, but it will
always be affine in the state process for this class of state dynamics. The cash-flow-diffusion
solution below can be used in the subsequent sections to obtain explicit solutions for the
equilibrium policies under various specifications for the aggregators.

We follow the main

result of this section with two examples, in which current Brownian shock have short-run
and long-run impacts on future cash-flows.
We introduce N state processes ζ i ∈ L (Rn ) with dynamics

i
i
dζt = µi + β i ζt dt + Σζi dBt , i = 1, . . . , N

(2.48)

where18 µi ∈ Rn , β i ∈ Rn×n , and Σζi ∈ Rd×n . Assume that the cash-flow process X
18 All

the results, except the closed-form expression for Θ in Proposition 4, apply with
time-varying deterministic parameters replacing the constant parameters throughout.

86

satisfies
i
i
Xt = Mti ζt , i = 1, . . . , N

where M i ∈ L (Rn ) is deterministic.

(2.49)

Our examples will all assume constant M i , but by

allowing time-dependency we can also model the case of no intermediate cash flow by letting
Mti = 0 for t < T . The following proposition gives the closed-form solution Z i , ΣZi for
the subjective cash-flow PV processes. Z i is affine in the state process ζ and ΣZi is affine
in the state process diffusion Σζ .
Proposition 4 (Closed-form solution with affine state process) Suppose the principals aggregators hV i , k V i , i = 1, . . . , N are deterministic functions, the state process ζ
satisfies (2.48), and the cash flow X i satisfies (2.49).

Furthermore, let the deterministic

vector process Θi ∈ L(Rn ), i = 1, . . . , N solve the linear ODE system:

˙
Θi + β i − rt I Θi = −Mti ,
t
t

t < T,

i
Θi = MT , i = 1, . . . , N.
T

(2.50)

The system has the closed-form solution

T
i
Θi = exp((T − t)β i )e− t rs ds MT +
t

T
t

s
i
exp((s − t)β i )e− t ru du Ms ds.

Finally, let

i
θt =

T
t

Ds
Θi µi + µZi t, Σζ1 Θ1 , . . . , ΣζN ΘN
ˆ
s
s
s
Dt

where µZi (·) is defined in (2.45).
ˆ

87

ds

(2.51)

Then the solution (Z, ΣZ ) to the BSDE system (2.44) satisfy

i
i
i
Zt = θt + Θi ζt , ΣZi = Σζi Θi , t ∈ [0, T ] , i = 1, . . . , N.
t
t
t

(2.52)

The equilibrium strategies ΣY are therefore deterministic and only the lump-sum components
of terminal pay pT are stochastic.
Because ΣZi is independent of preferences, it matches the diffusion obtained using riskneutral discounting of cash flows (that is, solving (2.44) with µZi = 0). Risk aversion enters
ˆ
only the θi of Z i .
The first example considers Ornstein-Uhlenbeck cash flows. As mean reversion increases,
the impact of current Brownian shocks (and therefore current effort) diminishes more quickly,
and has only a transient impact on future cash flows. Higher mean reversion therefore implies
a smaller ΣZi .
Example 7 (Short-run effort impact) Suppose r is constant (for simplicity), and the
cash-flow dynamics are

i
i
dXt = η i − κi Xt dt + σ i dBt , i = 1, . . . , N

(2.53)

for some σ i ∈ Rd , η i ∈ R, and κi ∈ R+ . This is just a special case of Proposition 4 by
letting M i = (1, . . . , 1) , µi = η i /n(1, . . . , 1) , β i = −κi I and σ i = Σζi M i .
Then
i
i
Zt = Φi (t) + Φi (t) Xt ,
0
1

ΣZi = Φi (t) σ i , i = 1, . . . , N
t
1

88

where

Φi (t) =
1
Φi (t) =
0

T
t

1
− r+κi (T −t)
−e
r + κi

1
−1 ,
r + κi

ˆ
e−r(s−t) η i Φi (s) + µZi s, ΣZ1 , . . . , ΣZN
s
s
1

(2.54)
ds.

Note that Φ1 (t) = Γt r + κi , where Γ r + κi denotes the bond price with interest rate
r+κi replacing r. If κi = 0 then Φi (t) = Γt , which follows because a unit time-t shock in the
1
kth dimension of Bt increases the present value of future cash flows by Γt σ i [k], and therefore
i
i
increases Zt by this amount (by the quasilinearily property of Zt ). As κi increases, the

impact of effort diminishes through additional discounting, and as κi → ∞, the contribution
to present value vanishes.
The second example considers long-run Brownian shock impact, modeled along the lines
of Bansal and Yaron (2004).

Current Brownian shocks (and therefore current effort) is

allowed to affect not only the current cash-flow shock, but also the drift of the cash-flow
drift.
Example 8 (long-run effort impact) Suppose a constant r (for simplicity) and the cash
flow dynamics

i
i
dXt = αt dt + σ i dBt ,

i
i
dαt = η i − κi αt dt + Σi dBt ,

for some σ i , Σi ∈ Rd , µi ∈ R, and κi ∈ R+ .

89

(2.55)

This is a special case of Proposition 4 by letting

i
i
i
ζt = αt , Xt

,

µi = η i , 0

,


i
βt = 

−κi
1


0 
,
0

ζi

Σt = Σi , σ i ,

Mti = (0, 1) .

Then
i
i
i
Zt = Φi (t) + Φi (t) αt + Γt Xt ,
1
0

ΣZi = Φi (t) Σi + Γt σ i ,
t
1

where 19

Φi (t) =
1
Φi (t) =
0

T − r+κi (s−t)

e

Γs ds,

(2.56)

t
T
t

e−r(s−t) η i Φi (s) + µZi s, ΣZ1 , . . . , ΣZN
ˆ
s
s
1

ds.

A time-t unit shock in the time-t Brownian motion in dimension k has two impacts: 1)
it increases the future cash-flow path by σ i [k], which increases its present value by Γt σ i [k];
i
and 2) it increases the time-s cash flow drift, for each s ≥ t, by e−κ (s−t) Σi [k], resulting
− r+κi (s−t) i
Σ [k] Γs .

in a time-s present-value increment of e

This second effect reflects a

higher-order persistence effect.
Assuming r < 1 (less than 100%), then both Γt and Φi (t) are increasing T − t, time
1
remaining to the terminal date. Letting T → ∞, then Γt is the time-t present value of a
19 The

assumption of constant r implies

1
Γt = −e−r(T −t)
r

1
−1 ,
r

Φ1 (t) =

1
e−r(T −t) −κi (T −t)
+
e
κi
r r + κi

90

1
−1 −
r + κi

1
−1
r

.

unit perpetuity and Φi (t) is the time-t present value of a growing perpetuity
1
i
1
1 − e−κ (s−t) /κi ; s ≥ t : Γt → 1 , Φi (t) →
, and therefore
1
r
r r+κi

ΣZi =
t

1
r

1
Σi + σ i ,
i
r+κ

as T → ∞.

Examples 7 and 8 show that increased persistence of current Brownian shocks on future
cash-flow increments results in higher subjective cash-flow diffusion (i.e., higher sensitivity of
the cash-flow PV process to Brownian shocks). The diffusion is sensitive to mean reversion
and the interest rate.

Lower interest rates imply a larger impact of current shocks on

the subjective cash-flow PV, and therefore a larger diffusion, particularly with the long-run
dynamics. The sensitivity of the diffusion to the interest rate also increases with a lower
interest rate.
Because ΣZ is determined by the impact of current Brownian shocks on the present value
of future cash flows, the terminal-date effects are driven by the principal’s lifespan, not the
agent’s. The same diffusion process would be obtained in a model with a short-lived agent
employed by a long-lived principal.

91

2.6

Additive Measure Change and Quadratic Penalties

We assume throughout this section unrestricted effort20 , E = Rd , and the following quadratic
penalties and linear measure change specification:

k U i t, ei , ΣU i
t t
k V i ΣV i
t

1
1
= − ΣU i QU i ΣU i − ei Qei ei ,
t
t
t
2
2 t t t
1
= − ΣV i QV i ΣV i , i = 1, . . . , N,
t
t
2 t

(2.57)

N

ei ,
t

Φ (et ) =
i=1

where the deterministic Qei , QU i , QV i ∈ L Rd×d are assumed symmetric positive definite.
From Example 4, the set of equilibrium effort plans as functions of agent utility diffusion is
uniquely given by

ei = (1/Γt )(Qei )−1 ΣY i ,
ˆt
t
t

t ∈ [0, T ] ,

i = 1, . . . , N.

(2.58)

That is, each agent’s effort in equilibrium is linear in the the PV process diffusion. The folˆ
lowing proposition gives the equilibrium principal controls ΣY and the resulting equilibrium
effort and subjective cash-flow PV processes.
Proposition 5 Under the linear measure change and quadratic penalties (2.57), equilibrium
agent utility diffusion and agent effort are

ˆ
ΣY i = Wti ΣZi ,
t
t
20 Except

ei = (1/Γt ) Qei
ˆt
t

−1

Wti ΣZi ,
t

i = 1, . . . , N,

in Example 12 where effort is allowed only in one dimension.
92

where
Wti =
and Z i , ΣZi ; i = 1, . . . , N
t

−1

Qei
t

+ QV i + QU i
t
t

−1

Qei
t

−1

+ QV i ,
t

solve the system of linked BSDEs

1 Zi Zi Zi
Σ Qt Σt
2Γt t


−1
ej
j Zj
Qt
Wt Σt
dt + ΣZi dBt ,
t


i
i
i
i
dZt = − −rt Zt + Xt + Ht −

+

1 Zi
Σ
Γt t

j=i

(2.59)
i
i
ZT = XT , i = 1, . . . , N,

where
QZi = Wti QU i − Qei
t
t
t

−1

,

(2.60)

i
Ht = Γt hV i (t, xV i ) + hU i (t, xU i ) − xV i − xU i ,
ˆt
ˆt
ˆt
ˆt
t

and xV i is part of principal i’s equilibrium strategy defined in (2.43) and xU i is part of
ˆ
ˆ
agent i’s equilibrium strategy defined in (2.13).
For any intermediate pay pi , t < T chosen by the principal,
t
the terminal pay

T
U
pi = (Γ0 K i − w0 i )e 0 rs ds −
T

−

T

T
e t rs ds pi + Γt hU i t, xU i − xU i
ˆt
ˆt
t
0


1
1 Zi pi Zi
Σt Qt Σt + ΣZi Wti
2Γt
Γt t

ej
j Zj
(Qt )−1 Wt Σt  dt +
j=i

where
pi

Qt = Wti

QU i − Qei
t
t

Proof of Proposition 5.

93

−1

Wti .

T

0

T
e t rs ds ΣZi Wti dBt
t

With the preferences and Φ in (2.57), after substituting the equilibrium agent i effort
ei = (1/Γt ) Qei
ˆt
t

−1

ΣY i , we have
t

i
ˆ
µZi = Ht + k Zi (t, ΣY i , ΣY −i , ΣZi )
ˆt
t
t

where

1
ˆ
k Zi (t, ΣY i , ΣY −i , ΣZi ) = −
ΣZi − ΣY i QV i ΣZi − ΣY i
t
t
t
t
t
2Γt
−1
1 Yi
1
Σ
−
QU i + Qei
ΣY i + ΣZi
t
t
2Γt
Γt t
1
ej −1 ˆ Y j
Qt
Σt .
+ ΣZi
t
Γt

Qei
t

−1

ΣY i
(2.61)

j=i

Maximizing over ΣY i , the FOC for principal i (which is necessary and sufficient) is

ˆ
QV i ΣZi − ΣY i − QU i + Qei
t
t
t
t
t

−1

ˆ
ΣY i + Qei
t
t

−1

ΣZi = 0,
t

(2.62)

ˆ
which has the solution ΣY i = Wti ΣZi .
t
t
1ˆ
Multiplying (2.62) by 2 ΣY i , we get
t

1 ˆY
1 ˆY i V i
ˆ
Σt Qt ΣZi − ΣY i − Σt i
t
t
2
2

QU i + Qei
t
t

94

−1

1ˆ
ˆ
ΣY i + ΣY i
t
2 t

Qei
t

−1

ΣZi = 0.
t

Upon substituting this into (2.61) we have

1 Zi
1 Zi V i
ˆ
ˆ
ˆ
Σt Qt ΣZi − ΣY i +
Σ
k Zi (t, ΣY i , ΣY −i , ΣZi ) = −
t
t
t
t
2Γt
2Γt t
1
ej −1 ˆ Y j
+ ΣZi
Qt
Σt
t
Γt

Qei
t

−1

ˆ
ΣY i
t

j=i

1 Zi
Σ
2Γt t
1
+ ΣZi
Γt t

= −

Wti QU i − Qei
t
t
ej −1

Qt

j

−1

ΣZi
t

Zj

Wt Σt ,

j=i

where we used the identity QV i I − Wti − Qei
t
t

−1

Wti = Wti QU i − Qei
t
t

−1

for the last

equality.
For any intermediate pay pi , t < T chosen by the principal, recall the expression of (2.32)
t
and the definition of µY i in (2.14). We get the terminal pay (2.61).
ˆ

The optimal diffusion sharing rule is simple: Principal i lays off the (matrix-valued)
proportion Wti of the appreciated subjective-cash-flow diffusion ΣZi /Γt to the agent, and
t
ˆ
bears the rest (from ΣV i = I − Wti ΣZi /Γt ). The weight process Wti depends only on the
t
t
risk aversion and effort penalties, and is invariant to the cash-flow dynamics. In the extreme
case of agent-i risk neutrality (QU i = 0) then W i = I and all the risk of Z is transferred to
the agent. At the other extreme, as the agent becomes infinitely risk averse, then W i → 0,
the principal bears all the risk of Z, and the agent exerts no effort.
Each agent i’s equilibrium time-t effort depends only on his/her own subjective cash-flow
diffusion ΣY i , but the principal problems are nonetheless linked because of the common
t
impact of each agent’s effort on the measure (and therefore the distribution of cash flows).
This linkage appears in the summation term in the drift of Z, which adds the distorted
95

covariances of the subjective cash-flow processes.

Optimal agent terminal pay above has

the following components: a) a fixed component to satisfy the participation constraint; b)
an adjustment for utility from intermediate consumption and pay (through the hU i term); c)
compensation for agent risk aversion; d) an adjustment depending equilibrium effort of the
other agents; e) a martingale part typically driven by innovations in the cash-flow process.
The following example obtains an expression for equilibrium lump-sum terminal pay in
terms of the lump-sum component of terminal cash flow in the case of constant diagonal
preference parameters.
Example 9 Assume constant diagonal preference parameters: Qei = q ei I, QU i = q U i I, and
t
t
QV i = q V i I, for some q ei , q U i , q V i ∈ R++ , i = 1, . . . , N . Then Wti = wi I, QZi = q Y i I,
t
t
pi

and Qt = q pi I where

wi =

1 + q ei q V i
,
1 + q ei q V i + q ei q U i

1
q Zi = q U i wi − ei ,
q

q pi = wi

2

1
q U i − ei
q

.

(2.63)

Principal i’s equilibrium control and agent ´’s equilibrium effort are
ı

ˆ
ΣY i = wi ΣZi ,
t
t

ei =
ˆt

wi Zi
Σ ,
Γt q ei t

and equilibrium terminal pay for agent i is

i
pi = w i XT +
T

−wi e
T

−
0

T
0

T
T
i
U
e t rs ds Xt dt + (1 − wi )e 0 rs ds (Γ0 K i − w0 i )

T
i
0 rs ds (Γ0 V0

V
− w0 i ) +

T

e
0

T
i
t rs ds ξt dt +

T
e t rs ds pi dt
t

T
e t rs ds wi (1 − wi ) Zi Zi
Σt Σt dt
2Γt q ei
0
T

(2.64)

96

where

i
ξt = wi Γt hV i (t, xV i ) − xV i − (1 − wi ) Γt hU i (t, xU i ) − xU i .
ˆt
ˆt
ˆt
ˆt

The first component of terminal pay is a proportion wi of the terminal cash flow and the
cumulative intermediate cash flow. The second and third terms adjust for the participation
constraint and intermediate consumption. The fourth term compensates the agent for the
cumulative risk of the cash-flow process. The last term subtracts the cumulative intermediate
pay . The effort of other agents affects the solution Z i , ΣZi , which then impacts terminal
i
V
U
pay through Z0 = Γ0 V0i − w0 i + Γ0 K i − w0 i and the quadratic variation term in (2.64).

Derivation. See Section .2 in the Appendix.
In the single principal/agent case (N = 1), the BSDE (2.59) is of the same form as the
BSDE (21) in Schroder and Skiadas (2005) (which applies to the optimal portfolio problem).
They provide sufficient conditions on the BSDE parameters and Markovian state-variable
processes such that the Z will be an affine function of the state variables, the coefficients
of which satisfy a set of Riccati ordinary differential equations (ODEs). In Section 2.5,
we showed that an analogous result holds in the multiple principal/agent case, in which
the solution reduces to a tractable linked system of ODEs. The examples below follow the
Examples 7 and 8 in Section 2.5. We will assume the the same diagonal preferences as in
Example 9 throughout.
The next example follows Example 7 and focuses on the role of mean reversion and its
impact on equilibrium controls and optimal terminal pay. We obtain a simple sharing rule
for the terminal lump-sum cash flow, proportional to the volatility sharing rule.
sharing rule depends on the distribution of cash flows.
97

Neither

Therefore an increase in cash-

flow mean reversion, which causes the impact of effort to become more transient and reduces
equilibrium effort rates, has no impact on the sharing rules. In fact, additional compensation
may be necessary in order to satisfy the participation constraint.
Example 10 (Ornstein-Uhlenbeck cash flow) This example follows Example 7 in Section 2.5 where the cash-flow dynamics are

i
i
dXt = η i − κi Xt dt + σ i dBt , i = 1, . . . , N

for some σ i ∈ Rd , η i ∈ R, and κi ∈ R+ .
Assume the same constant diagonal preference parameters as in Example 9 and that r is
constant (for simplicity).
In Example 7, we showed that

i
i
Zt = Φi (t) + Φi (t) Xt ,
0
1

ΣZi = Φi (t) σ i , i = 1, . . . , N
t
1

The equilibrium principal i’s control and agent effort are

ˆ
ΣY i = wi Φi (t) σ i ,
t
1

ei =
ˆt

wi i
Φ (t) σ i
Γt q ei 1

where

Φi (t) =
1
Φi (t) =
0

1
− r+κi (T −t)
−e
r + κi

T

t

1
−1
r + κi

i
e−r(s−t) η i Φi (s) + Hs −
1

1 i 2 Zi i i
1
j
Φ1 (s) q σ σ + Φi (s)Φ1 (s)σ i
2Γs
Γs 1

(2.65)


j=i

wj j 
σ ds
q ej

In the absence of mean reversion, with an interest rate r ∈ (0, 1) the Φi (t) increases as
1
98

time to the terminal date increases so are the absolute agent diffusions and efforts.

1
Φi (t) ↑ ,
1
r

wi i
ˆ
ΣY i →
σ,
t
r

ei →
ˆt

wi
σi
Γt q ei r

as

T → ∞.

With a positive mean reversion (κi > 0) that is large enough such that r + κi > 1, Φi (t)
1
decreases as time to the terminal date increases, and therefore absolute agent diffusion and
absolute effort also decrease.

If r + κi > 1, then Φi (t) ↓
1

1
wi i i
wi
ˆ
, ΣY i →
σ , et →
ˆ
σ i as T → ∞.
t
r + κi
r + κi
Γt q ei (r + κi )

This is because the drift impact of effort is more likely to be reversed by mean reversion
(in fact both converge to zero as κi → ∞).
The terminal pay is



wj

pi = wi Li −
T
j=i

+

(wi )2
2

q ej


U
Cov(X i , X j ) + erT (Γ0 K i − w0 i )

1
q U i − ei
q

T

V ar(X i ) −
0

where

99

(2.66)

er(T −t) pi + Γt hU i (t, xU i ) − xU i dt
ˆt
ˆt
t

T

Li =
0

=

i
XT

er(T −t) Φi (t) σ i dBt
1
T

+
0

i
i
er(T −t) Xt dt − erT Φi (0)X0
1

T

=

1
i
er(T −t) dXt −
r 0
T

−η i
0

(2.67)
T

− ηi
0

er(T −t) Φi (t)dt
1

1
1
−
r r + κi

1
i
− 1 XT +
r

erT +

i
1
i
− 1 e−κ T X0
i
r+κ

er(T −t) Φi (t)dt,
1

Cov(X i , X j )

j

er(T −t) Φi (t)Φ1 (t) i j
1
dXt dXt ,
Γt
0
T er(T −t) Φi (t)2
i
i
1
dXt dXt .
Γt
0
T

=

Var(X i ) =

The mean reversion term κi affects the deterministic function Φi (t).
1
j

The impact of other agents’ effort is through the deterministic the terms Φ1 (t), j = i.
In the case of terminal consumption only(cU i = cV i = 0) and zero cash flow drift (η i =
κi = 0), the solution is 21

1
Φi (t) = Γt = − e−r(T −t)
1
r

q Zi σ i σ i
Φi (t) = −
+ σi
0
2

1
−1
r
wj
j=i

q



σj 
ej

T
t

er(T −2s+t) Φi (s)2 ds
1

and the terminal pay is
21 As discussed in Remark 2, the Γ in (2.65) needs
s
i (s) equals the old Γ , it should not be replaced.
Φ1
s

100

to be replaced by e−r(T −s) . Although



wj

pi = wi Li −
T
j=i

+

(wi )2
2

q ej


U
Cov(X i , X j ) + erT (Γ0 K i − w0 i )

1
q U i − ei
q

T

V ar(X i ) −
0

er(T −t) pi dt
t

(2.68)

where

T

Li =
0

i
er(T −t) Φi (t)dXt =
1

Cov(X i , X j )

1 T r(T −t) i
e
dXt −
r 0

T

er(T −t) Φi (t)
1

0
T
0

2

2

(2.69)

i
dXt dXt = σ i σ j S(2)

er(T −t) Φi (t)
1

=

Var(X i ) =

1
i
i
− 1 (XT − X0 )
r

i
i
dXt dXt = σ i σ i S(2)

j

with 22

T

S(k) =
0

er(T −t) Φi (t)
1

k

dt.

If the cash-flow covariance is positive, the jth agent’s effort increases the drift of X i
i
thereby increasing the value of agent i’s share of the terminal value of the cash flow, wi XT .

This allows principal i to reduce the fixed component of pay, while still satisfying the partic22 Here

S(2) = e2rT − 1 − 4(1 − r)(erT − 1) / 2r3 + 1 − 1
r

101

2

T

ipation constraint.
Example 11 (long-run effort impact) This example follows Example 8 in Section 2.5
where the cash flow dynamics are

i
i
dXt = αt dt + σ i dBt ,

i
i
dαt = η i − κi αt dt + Σi dBt ,

for some σ i , Σi ∈ Rd , µi ∈ R, and κi ∈ R+ .
Assume the same constant diagonal preference parameters as in Example 9. and that r
is constant (for simplicity),
In Example 8, we showed that

i
i
i
Zt = Φi (t) + Φi (t) αt + Γt Xt ,
0
1

ΣZi = Φi (t) Σi + Γt σ i ,
t
1

The equilibrium principal i’s control and agent effort are

ˆ
ΣY i = wi (Φi (t) Σi + Γt σ i ),
t
1

ei
ˆt

wi
(Φi (t) Σi + Γt σ i )
=
ei 1
Γt q

where

Φi (t) =
1
Φi (t) =
0

T − r+κi (s−t)

e

Γs ds

t
T
t

+

i
e−r(s−t) η i Φi (s) + Hs −
1

1
Γs

j=i

wj
Φi (s)Σi + Γs σ i
1
ej
q

q Zi
Φi (s)Σi + Γs σ i
1
2Γs


102

j
Φ1 (s)Σj + Γs σ j  ds

Φi (s)Σi + Γs σ i
1

The terminal pay is

U
pi = wi Li − erT Φi (0)) + erT (Γ0 K i − w0 i )
0
T

+

(2.70)

wi (1 − wi ) T er(T −t)
Φi (t)Σi + Γt σ i
1
Γt
2q ei
0
T

+
0

Φi (t)Σi + Γt σ i dt
1

i
er(T −t) (ξt − pi )dt
t

where Li is given by (2.69).
As the time to the terminal date T increases, both Φi (t) and Γt increases.
1

Φi (t) ↑
1

1
,
r(r + κi )

Γt ↑

1
r

as

T → ∞.

The equilibrium principal and efforts approach the following limits.

wi
ˆ
ΣY i →
t
r

1
Σi + σ i ,
r + κi

wi
ei → ei
ˆt
q

1
Σi + σ i
r + κi

as

T → ∞.

Example 12 (Square-root state-variable dynamics) Assume the same constant diagonal preference parameters as in Example 9. We introduce a state process ζ ∈ L (Rn ) with
dynamics23
dζt = µζ + β ζ ζt dt + Σ diag

υ + V ζt dBt .

(2.71)

where24 µζ ∈ Rn , β ζ ∈ Rn×n , υ ∈ Rd and V, Σ ∈ Rd×n . Let mi ∈ R and M i ∈ Rn and
√
√
δ1 , . . . , δN and let
any length-N vector δ = (δ1 , . . . , δN ) , we define δ =
diag(δ) denote the N × N matrix with ith diagonal element δi .
24 All the results apply with time-varying deterministic parameters replacing the constant
parameters throughout.
23 For

103

assume that the cash-flow process X i satisfies:

i
Xt = mi + Mti ζt .
t

We obtain a solution for the subjective cash-flow value process Z i in equation (2.59) which
is affine in the state variable:

i
i
Zt = θt + Θi ζt ,
t

ΣZi = diag
t

υ + V ζt ΣΘi
t

i = 1, . . . , N,

i
for deterministic θt ∈ L (R) and Θi ∈ L (Rn ) satisfying the Riccati ODE system
t

i
i
˙i
0 = θt + µZ Θi − rt θt + mi + Ht +
t
t
j=i

−

q Zi
2Γt

j=i

−

2Γt

(2.72)

Θi Σ diag(υ)ΣΘi ,
t
t

˙
0 = Θi + β Z Θi − rt Θi + Mti +
t
t
t
q Zi

wj i
j
Θ Σ diag(υ)ΣΘt
q ej Γt t

Θi Σ diag Θi Σ
t
t

wj
j
Θi Σ diag Θt Σ
t
ej Γ
q t

V

V

i
i
with the terminal conditions θT = mi and Θi = MT , i = 1, . . . , N .
T
T

From Example 9, the equilibrium principal controls and agent efforts are

ˆ
ΣY i = wi diag
t

υ + V ζt ΣΘi ,
t

ei =
ˆt

wi
diag
Γt q ei

and terminal pay/consumption is

104

υ + V ζt ΣΘi ,
t

i = 1, . . . , N,

i
pi = w i XT +
T

T
0

T
T
U
i
i
e t rs ds Xt dt − e 0 rs ds (θ0 + Θi ζ0 ) + erT (Γ0 K i − w0 i )
0
T

wi (1 − wi ) T e t rs ds
+
Θi Σ diag (υ) ΣΘi + V diag Θi Σ
t
t
t
Γt
2q ei
0
T

+
0

Σ Θi ζt dt
t

i
er(T −t) (ξt − pi )dt.
t

The impact of the other principal-agents pairs enters both the fixed component of pay (via
i
θ0 and Θi ) and the path dependent state-variable term(the ζt part of the third term). Note
0

that Θ = Θ1 , . . . , ΘN

are jointly solved from (2.72), and each Θi generally depend on the

preference parameters of all agents and principals.

Once the system Θ is solved, it is a

simple matter to solve the first-order linear ODE for each θi .
Derivation. See Section .2 in the Appendix.

2.7

Diminishing Returns to Effort (Concave Φ)

Section 2.6 showed that with a linear Φ, the sharing rule allocating the time-t subjective
PV volatility, ΣY i , between the ith principal-agent pair depends on ΣY i , but not on the
t
t
other principals’ subjective PV volatility terms.

In this section, we build on Lemma 1

and Examples 5 and 6. We solve the principal equilibria under a CES measure-change
operator Φ. A CES measure change operator implies that the marginal impact of an agent’s
effort is diminishing in the aggregate effort of all agents. As a result, each agent’s optimal
effort generally depends on the volatility-controls of all the principals, and therefore each
principal’s optimal (own-agent) volatility control generally depends on the subjective PV

105

volatility processes of all the principals.
In this Section, we will assume two pairs of principals/agents (N = 2), labeled {a, b} and
the CES measure-change operator as defined in Definition 6 throughout.

Φk (e) = κ

α(ea )γ
k

+ (1 − α) (eb )γ
k

v
γ

,

κ > 0,

α ∈ (0, 1),

0 = γ ≤ 2,

0 < v < 2.
(2.73)

We will also assume the diagonal quadratic penalties below, except in Section 2.7.3.

kU i

t, ei , ΣU i
t t

1
= −
2

k V i (t, ΣV i ) = −
t

1
2

d

Qei (ei )2
k k
k=1
d

1
−
2

d

QU i (ΣU i )2 ,
k
k

(2.74)

k=1

QV i (ΣV i )2 ,
k
k
k=1

with Qei , QU i , QV i > 0 for all i, k.
k
k
k
Before proceeding to each special case, we first present a general solution of principal
equilibrium for risk-neutral principal-agent pairs in the following Lemma. A relaxation of
the restrictions on parameters is achieved by working with quasiconcavity and FOC in the
Proof of the Lemma.
Lemma 4 Assume the aggregator functions satisfy (2.74) and Φ(·) satisfies (2.73). Suppose
that in addition to our usual parameter restrictions on Φ(κ > 0, α ∈ (0, 1), 0 = γ ≤ 2, 0 <
v < 2) we assume γ ≤ v or

2 > γ > v and γ(2 − γ)(2 − v) + v − γ > 0.
106

(2.75)

Note that (2.75) is implied by 0 < v < γ ≤ 1.
Henceforth, omit the dimensional argument. Let

Sa =

αvΣY a
(1 − α) vΣY b
, Sb =
.
Γt Qea
Γt Qeb

If the agents and principals are risk-neutral i.e. QU i = QV i = 0, i ∈ {a, b}, then principal
equilibrium satisfies

α (S a )γ/(2−γ)
=

γ/(2−γ)
2−γ
+ (1 − α) S b
Sa
2−v

α (S a )γ/(2−γ) + (1 − α) S b

α (S a )γ/(2−γ) + (1 − α) S b
=

γ/(2−γ)

γ/(2−γ)

α (S a )γ/(2−γ) + (1 − α) S b

αΣZa v (2 − γ)
t
Γt Qea (2 − v)

2−γ
2−v

γ/(2−γ)

(2.76)

Sb

(1 − α)ΣZb v (2 − γ)
t
(2 − v)
Γt Qeb

Proof.
We showed in Lemma 1 that if γ < 2, agent equilibrium is

ea = κ1/(2−v) (S a )1/(2−γ) α (S a )γ/(2−γ) + (1 − α) S b
eb = κ1/(2−v) S b

1/(2−γ)

α (S a )γ/(2−γ) + (1 − α) S b

107

γ/(2−γ) (v−γ)/{γ(2−v)}
γ/(2−γ) (v−γ)/{γ(2−v)}

and therefore

Φ (e) = κ2/(2−v) α (S a )γ/(2−γ) + (1 − α)

γ/(2−γ)
Sb

v(2−γ)
γ(2−v)

The principal a’s problem is

max
Sa

γ/(2−γ) 2(v−γ)/{γ(2−v)}
1
− Γt Qea κ2/(2−v) (S a )2/(2−γ) α (S a )γ/(2−γ) + (1 − α) S b
2

+ΣZa κ2(2−v)
t

α (S a )γ/(2−γ)

+ (1 − α)

γ/(2−γ)
Sb

v(2−γ)
γ(2−v)

Let
z = (S a )γ/(2−γ) , K = (1 − α) S b

γ/(2−γ)

and then the above problem is equivalent to
2(v−γ)

v(2−γ)

1
max − Γt Qea κ2/(2−v) z 2/γ {αz + K} γ(2−v) + ΣZa κ2/(2−v) {αz + K} γ(2−v)
t
z
2

The derivative of the RHS is (using the abbreviation {} = {αz + K})

∂
=
∂z

1
− Γt Qea κ2/(2−v)
2

2(v−γ)

2(v−γ)

2 2/γ−1 γ(2−v)
2 (v − γ) γ(2−v) −1
z
{}
+ αz 2/γ
{}
γ
γ (2 − v)

v (2 − γ)
+αΣZa κ2/(2−v)
t
γ (2 − v)

2(v−γ)
{} γ(2−v)

108

This is equivalent to

∂
=
∂z

{−Γt

Qea }

(v − γ)
1 2/γ−1
1 + αz
z
{αz + K}−1
γ
(2 − v)

v (2 − γ)
+ αΣZa
t
γ (2 − v)

2(v−γ)
{} γ(2−v)

κ2/(2−v)

To prove quasiconcavity, it is sufficient the term in the large parentheses above is monotonically decreasing in z. This is equivalent to showing h (z) is increasing in z where

h (z) =

1 + αz

(v − γ)
{αz + K}−1
(2 − v)

1 2/γ−1
z
=
γ

1+

(v − γ) αz
(2 − v) αz + K

1 2/γ−1
z
γ

The first derivative
2 −1

2
2 − γ γ −2 v − γ αz γ [(2 − γ)αz + 2K]
h (z) =
z
+
2−v
γ2
(αz + K)2 γ 2

It is obvious that h (z) ≥ 0 if 0 = γ ≤ v < 2.
Now suppose 2 > γ > v > 0.
αz
Let ε = αz+K . From (2.77), we have

2 −2

zγ
h (z) =
γ2

2−γ+

v−γ
(2 − γ)ε2 + 2ε(1 − )
2−v

2 −2

z γ (2 − γ)(2 − v) + 2(v − γ)ε + γ(γ − v)ε2
=
2−γ
γ2

And h (z) ≥ 0 is equivalent to

g(ε) = (2 − γ)(2 − v) + 2(v − γ)ε + γ(γ − v)ε2 ≥ 0
109

(2.77)

for any 0 ≤ ε ≤ 1.
1
g (ε) = 0 yields ε = γ .

If 1 ≥ γ > v > 0, then the minimal of g(ε) is attained by ε = 1 and

g(1) = (2 − γ)2 > 0 when 1 ≥ γ > v > 0.

1
If γ > 1 and γ > v > 0, then the minimal of of g(ε) is attained by ε = γ and

1
γ

g

=

γ(2 − γ)(2 − v) + v − γ
γ

g(ε) > 0 is equivalent to

γ(2 − γ)(2 − v) + v − γ > 0.

So quasiconcavity holds for any 0 = γ ≤ v < 2 or 0 < v < γ ≤ 1 or
1 < γ and v < γ such that

γ(2 − γ)(2 − v) + v − γ > 0.

Note that the above inequality is implied by 0 < v < γ ≤ 1.
Under these conditions, the optimality for principal a, holding fixed principal b, is equiv110

alent to a positive solution z (there can be only one, because of the monotonicity of h (z))
to the principal’s FOC, which is equivalent to a positive solution z to

αz

γ/(2−γ)
(2 − γ)
+ (1 − α) S b
z 2/γ−1 =
(2 − v)

αz + (1 − α) S b

γ/(2−γ)

which is a quadratic polynomial in z when γ = 1.
If γ = 1, α = 1/2 and κ = 2v , then
the measure change function is

Φk (e) = ea + eb
k
k

v

and the FOC is

Sa
Sb
+
2(2 − v)
2

2.7.1

Sa

Za v
1
a + S b αΣt
=
S
2
Γt Qea 2 − v

Power Measure Change

For any δ ∈ [0, 1), let

γ = 1, v = 1 − δ, α = 1/2 and κ = 21−δ /(1 − δ).
The measure change operator is (2.20).

111

αΣZa v (2 − γ)
t
Γt Qea (2 − v)

In Example 5, we show the agent equilibrium solution is (2.21) in the case of two principalagent pairs i ∈ {a, b}.
Proposition 6 (Risk neutrality) Suppose Φ satisfies the above specification, and preferences satisfy the risk-neutral specification (2.74) with QU i = QV i = 0. There are two pairs
of principal and agent i ∈ {a, b}. Assume ΣZa , ΣZb > 0 all t and define the processes
t
t
ΣZa ΣZb
k − k ,
Qea
Qeb
k
k

∆k =

ΣZa ΣZb
k + k .
Qea
Qeb
k
k

Sk =

The unique principal equilibrium is at each dimension k 25

ˆ
S +
ΣY a
k = k
Qea
k

2
Sk + (2 + δ) δ∆2
k

2 (2 + δ)

ˆ
ΣY b
k

∆
+ k,
2

Qeb
k

=

Sk +

2
Sk + (2 + δ) δ∆2
k

2 (2 + δ)

−

∆k
(2.78)
2

and the corresponding equilibrium agent-effort processes are

ˆ
ei t, ΣY
ˆk
t

=

ˆ
ΣY i
k
Γt Qei
k

ˆ
ˆ
ΣY b
ΣY a
k
+ k eb
Γt Qea Γt Q
k
k

−δ
1+δ

,

i ∈ {a, b}

k = 1, . . . , d.

(2.79)

Total equilibrium agent effort is


ea
ˆk

Proof.

+ eb
ˆk

=

Sk +

2
Sk + (2 + δ) δ∆2
k

(2 + δ)Γt

 1

1+δ



.

Applying Lemma 4 with γ = 1, v = 1 − δ, α = 1/2(note that the restriction on

the parameters is obviously satisfied), the FOC (2.76) implies
ˆ
ˆ
shown in the proof, when ΣZa ≤ 0 the optimum is ΣY a = 0 and ΣY b = ΣZb , and
k
k
k
k
vice versa.
25 As

112

Za
a
a b
a
b Σ (1 − δ)
(Sk )2 + (1 + δ)Sk Sk = (Sk + Sk ) k
2Γt Qea
k

(2.80)

Zb
b
a b
a
b Σ (1 − δ)
(Sk )2 + (1 + δ)Sk Sk = (Sk + Sk ) k
2Γt Qeb
k

Subtracting the second equation above from the first yields

a
b
Sk − Sk =

1−δ
2Γt

ΣZa ΣZb
k − k
Qea
Qeb
k
k
Ya

Yb

Σ
Σ
a
b
Using the fact that Sk = A 1−δ and Sk = B 1−δ , where A = Qk and B = k , we get
ea
2Γt
2Γt
Qeb

A − B = ∆k .

(2.81)

Plugging (2.81) into the first equation in (2.80), we get an equation for A below

(2 + δ)A2 − [(2 + δ)∆k + Sk ] A +

∆k (Sk + ∆k )
= 0.
2

This yields the solution (2.78).
For verification, in Section .1 in the Appendix, we provide another proof by directly
applying Corollary 1.

Lemma 5 (Comparative statics) Under the assumptions of Proposition 6, the equilibrium optimal controls

ˆ
ˆ
ΣY a , ΣY b , ea , eb
t
t ˆ ˆt

as functions of

t ∈ [0, T ] and i, j ∈ {a, b}, j = i, the following:

113

ΣZa , ΣZb , δ , satisfy, for all
t
t

a)
ˆ
dΣY i
1
k ≤ 1,
≤
1+δ
dΣZi
k

−

ˆY i
δ Qei
k ≤ dΣk ≤ 0,
Zj
1 + δ Qej
dΣk
k

b)
ˆY b
ˆ
1 dΣY a
k = 1 dΣk ≤ 0,
Qea dδ
Qeb dδ
k
k

(2.82)

c)
dˆi
ek
≤ 0,
dδ
d)
dˆi
ek
dΣZi
k

dˆi
ek

≥ 0,

Zj

≤ 0.

dΣk

The inequalities in b)-d) are all strict if ΣZa , ΣZb > 0.
k
k
Proof. See Section .1 in the Appendix.
In the kth dimension holding fixed ΣZb , an increase in ΣZa implies an increase in both
k
k
ˆ
principal a’s control, ΣY a , and agent a’s effort, ea ; but a decrease in principal b’s control,
ˆk
k
ˆ
ΣY b , and agent b’s effort (though total time-t effort exerted increases). (Analogous results
k
hold for an increase in ΣY b .) The special case of δ = 0 corresponds to an additive measure
k
change and the optimally bearing all the cash-flow risk under the principal’s optimal policy:
ˆ
ˆ
ΣY i = ΣZi ; the corresponding optimal agent effort is ei = ΣZi /(Γt Qei ), which is the firstˆk
k
k
k
k
ˆ
best effort level for principal i. We also obtain ΣY a = ΣZa if ΣY b = 0, and the corresponding
k
k
k
ˆ
optimal agent effort is ea = ΣY a /(Γt Qea )
ˆk
k
k

1/(1+δ)

, which is again first-best for principal a

(given zero effort by agent b). Finally, an increase in the concavity parameter δ implies a
decrease in both principals’ controls and both agents’ efforts.
The negative dependence of principal a’s sharing rule on principal b’s cash-flow volatility
is different from seemingly related results in some team contract settings. Here the interde114

pendence is a free rider problem. With diminishing marginal productivity to effort, higher
sensitivity of firm b’s cash flows in any dimension will cause b to offer it’s agent a larger cashflow-volatility share in that dimension, inducing more effort by that agent. Higher effort by
agent b causes agent a to work less. Principal a responds to the diminished marginal value
of its own agent’s effort by reducing the cash-flow-volatility share is offers.
The following example illustrates the interdependence of the contracts in a two-dimension
setting with constant-volatility cash flows.
Example 13 (Terminal consumption only) Suppose risk-neutral preferences, the Brownian dimension d = 2 and terminal consumption only. The cash flows satisfy

i
dXt = σ i dBt ,

σ i > 0,

i ∈ {a, b} ,

t ∈ [0, T ]

, Qea = Qeb = Qk , k = 1, 2 and a constant interest rate r. By Example 7(letting η i =
k
k
κi = 0), the subjective cash-flow volatilities match the discounted actual cash-flow volatilities,
ΣZi = Γt σ i , i ∈ {a, b}, where Γt = 1/r − (1/r − 1)e−r(T −t) and the optimal principal
volatility controls are

 a
 σ + σb +
 k
k

ˆ
ΣY a = Γt
k




a
b
+ (2 + δ) δ σk − σk
2 (2 + δ)

a
b
σk + σk

a
b
ˆ
ˆ
ΣY b = Γt ΣY a − σk − σk
k
k

2

,

2




σa − σb 

+ k

k

2

,





k = 1, 2.

i
i
ˆ
There exist constants vi ∈ (0, 1) and vj ∈ (−1, 0), i = j, such that the control ΣY i is a linear

115

combination of the cash-flow diffusions:26

i
i
ˆ
ΣY i = Γt (va σ a + vb σ b ),

i ∈ {a, b} .

That is, each principal’s volatility sharing rule depends on the volatility of both principals’
cash flows. With any feasible intermediate pay pi , t < T , the lump-sum terminal pay for
t
agent i is affine in the terminal cash flows of both firms27 :

pi = Y0i erT −
T

T
0

i
i
er(T −t) pi dt − Πi + va La + vb Lb
t

i ∈ {a, b} ,

where the constant Πi is defined as 28

Πi

=S

2
1+δ

− 1−δ
2
Qk 1+δ
k=1

1−δ






1+δ
2

2

j a
i b
i b
i a
i a
va σk + vb σk + (va σk + vb σk )(va σk
2δ
a + v b )σ a + (v a + v b )σ b 1+δ
(va
a k
b
b k



j b
+ vb σk ) 




with
ˆ
d > 2 we cannot generally replicate ΣY i with a linear combination of cash-flow
diffusion vectors because of the nonlinear relationship given in (2.78). With d = 2, let
i i
the 2 × 2 matrix σ = [σ a , σ b ], then v i = (va , vb ) is the solution of the linear equation
σv i = ΣY i , i ∈ {a, b}.
27 If δ = 0 then v i = 1, v i = 0, i = j and solution matches (2.68).
i
j
28 The Γ in equation (2.79) needs to be replaced by D /D = e−r(T −t) throughout the
t
t
T
calculation. Apply (2.32) with
26 With

ˆ
µY i (t, ΣY ) = −
ˆ
t

e−r(T −t)
2

2

2

ˆ
Qk ei (t, ΣY )2 +
k
k
k=1

ˆ
ˆ
ΣY i Φk ek (t, ΣY )
k
k
k=1

116

T

S(k) =
0

er(T −t) Γt

k

dt

and

T

Li =
0

i
er(T −t) Γt dXt =

1 T r(T −t) i
e
dXt −
r 0

1
i
i
− 1 (XT − X0 ).
r

Agent a’s terminal pay is increasing in La and decreasing in Lb (and analogously for
agent b). An increase in either cash-flow volatility (in either dimension) results a reduction
in the fixed component of both agent’s pay because the increase in aggregate effort increases
the cash-flow drifts.

Relaxing the N = 2 assumption, in the special case of identical cash-flow processes
Xt = σ Bt and identical Qei s (letting Qe denote the common effort penalty and ΣZ the
common subjective PV diffusion process), the optimal controls and effort are

ˆ
ΣY i =
t

1
1+ 1−
N

−1

δ

Γt σ,

ei
ˆt

=

−δ
N 1+δ

 1



1+δ

σ


1
Qe 1 + 1 − N δ

When δ = 0 (additive Φ), the effect of N vanishes.



, i = 1, . . . , N.

As N → ∞ the optimal control

ˆ
ΣY i converges to (1 + δ)−1 Γt σ (and ΣU i converges to (1 + δ)−1 σ and ΣV i converges to
t
t
t
[1 − (1 + δ)−1 ]σ) and individual agent effort converges to zero (though aggregate effort is of
order N 1/(1+δ) ).
The next proposition shows that in the case of a strictly concave measure-change operator,
117

as N goes to infinity the optimal principal policies are motivated solely by risk-sharing, and
individual agent effort goes to zero.

The sharing rules converge, as N gets large, to the

sharing rule of Proposition 5 with linear Φ and infinite effort cost.
Proposition 7 Assume, for simplicity, terminal consumption only, deterministic principal
i
aggregator functions hV i , k V i for each i, and Ornstein-Uhlenbeck cash flows: Xt = (η i −
i
κi Xt ) + v i Bt , for each i and t ∈ [0, T ], where v i ∈ Rd and the power measure-change
+

specification (2.20) with δ > 0. Also assume

QV i ≥

1
, Qei , QU i ≤ κ,
κ

for some constant κ > 0.

N

i = 1, 2, . . . ,

i
vk = ∞,

lim

N →∞

k = 1, . . . , d,

i=1

Then the limiting equilibrium principal and agent controls as

N → ∞ are given by
QV i Φi (t) i
ˆ
lim ΣY i = V i 1 U i vk and lim ei → 0,
ˆ
N →∞ k
N →∞ k
Q +Q

k = 1, . . . , d.

i
1 −1 .
where Φi (t) = 1 i − e−(r+κ )(T −t)
1
r+κ
r+κi

Derivation. See Section .1 in the Appendix.

2.7.2

(2.83)

Quadratic Penalty and Cobb-Douglas Measure Change

For any δ a , δ b ∈ [0, 1), δ a + δ b > 0, let κ = 1, αv = 1 − δ a and (1 − α)v = 1 − δ b .
As γ → 0, the measure change operator converges to

a

b

Φk (e) = (ea )1−δ (eb )1−δ ,
k
k

118

k = 1, . . . , d

In Section 2.3.3, we obtain for each dimension k and time t the following possible Nash
equilibria

ea =
k

eb =
k

ΣY b+ (1 − δ b )
k
Γt Qeb
k

1+δ b
2(δ a +δ b )

ΣY a+ (1 − δ a )
k
Γt Qea
k

1+δ a
2(δ a +δ b )

ΣY b+ (1 − δ b )
k
Γt Qeb
k
ΣY a+ (1 − δ a )
k
Γt Qea
k

1−δ b
2(δ a +δ b )
1−δ a
2(δ a +δ b )

or

ea = 0,
k

eb = 0.
k

Proposition 8 (Risk neutrality) Suppose Φ satisfies the above specification, and preferences satisfy the risk-neutral specification (2.74) with QU i = QV i = 0. The principal
equilibrium solution is

ΣY a =

2ΣZa+
,
1 + δb

ΣY b =

2ΣZb+
1 + δa

Proof. From the parameter specification αv = 1 − δ a and (1 − α)v = 1 − δ b , we get
v = 2 − (δ a + δ b ) and α =

1−δ a .
2−(δ a +δ b )
Ya

Applying Lemma 4 with the above values and the identity S a = αvΣ ea and letting
Γt Q
γ → 0, (2.76) implies the following equation for ΣY a .

119

2
1 − δa
2
1 − δb
ΣY a = ΣZa+ a
+
a + δb) δa + δb
a + δb)
2 − (δ
2 − (δ
δ + δb
Using the identity 2(1 − δ a ) + (1 − δ b )(δ a + δ b ) = [2 − (δ a + δ b )](1 + δ b ), we get the
Za
solution ΣY a = 2Σ b . Similarly, we can get the solution for ΣY b .
1+δ

For verification, in Section .1 in the Appendix, we provide another proof by directly
applying Corollary 1.
In the case of identical principal-agent pairs and constant return to scale i.e.

1
QU a = QU b = QU , QV a = QV b = QV , Qea = Qeb = Qe , δ a = δ b =
2

The solution is

ΣY =

16Qe QV + 4
ΣZ+
e (QV + QU ) + 3
16Q

Example 14 (Terminal consumption only) Assume the cash-flow dynamics are

i
dXt = σ i dBt , i ∈ {a, b}

for some σ i ∈ (R+ )d , QV i = 0, QU i = 0, i ∈ {a, b} and terminal consumption only.
The equilibrium principal i’s control and agent effort are

120

ΣY i =

σi

2Γt
,
1 + δj

i
2σk (1 − δ i )
(1 + δ j )Qei
k

ei =
ˆk

1+δ j
2(δ i +δ j )

j
2σk (1 − δ j )
ej
(1 + δ i )Qk

1−δ j
2(δ i +δ j )

1

er(T −t) Γt δ i +δ j

where

Γt =

1
− e−r(T −t)
r

1
−1 .
r

The terminal pay is

U
pi = (Γ0 K i − w0 i )erT −
T

−S

2
i + δj
δ

d

T
0

er(T −t) pi dt
t

(2.84)

1−δ i
1−δ j
i ) δ i +δ j (C j ) δ i +δ j
(C

k=1

j

2
+
Li .
1 + δj

where

T

S(k) =
0

er(T −t) Γt

and

121

k

j

1−δ
1+δ
1 + δ i i δ i +δ j j δ i +δ j
(σ )
(σk )
1 + δj k

dt

T

Li =
0

i
er(T −t) Γt dXt =

1 T r(T −t) i
e
dXt −
r 0

1
i
i
− 1 (XT − X0 ).
r

Derivation. See Section .2 in the Appendix.

2.7.3

Absolute Effort Penalty and Square-Root Measure Change

In this section we build on Example 6 and solve for the principal equilibrium with an absolute
effort penalty and, to obtain a closed form solution, a square-root measure change (i.e., the
power measure-change operator (2.20) with δ = 1/2). Unlike the quadratic-effort-penalty
solution in the previous section, any equilibrium with absolute effort penalty has at most
one agent working in any given dimension at any moment. Furthermore, there are regions
with more than one possible principal equilibrium.
For simplicity, we assume throughout this section one-dimensional Brownian motion29
(d = 1), nonnegative effort (E = R+ ), and two principals/agents (N = 2), which we label a
and b. We also assume the preferences and measure-change operator

1
k V i t, ΣV i = − QV i (ΣV i )2 ,
t
t
2 t

1
i
k U i t, ei , ΣU i = − QU i (ΣU i )2 − qt ei ,
t
t
t
t t
2
Φ (e) = 2

i ∈ {a, b},

ea + eb ,
t
t

where q i , QU i ∈ L (R++ ) and QV i ∈ L (R+ ). When N = 1 this specification is equivalent
to the quadratic/linear specification 30 (2.57) with E = R+ .
29 The

extension to d > 1 with diagonal preference parameters is simple: as in the agent
equilibrium in Example 6, the principal equilbrium below applies to each dimension.
30 This is seen by redefining effort as e = 2√e .
˜t
t

122

Define, for i ∈ {a, b},

wi =
t

QV i
t
QV i
t

,

+ QU i
t

2
i
qt

+ QV i
t

1 i
1
wt = qt QV i wi − wi + wi .
˜i
t
t
t
4
2 t

wi = 2
,
t
+ QV i + QU i
t
t
qi
t

We assume that QV a , QV b ≥ 0 are sufficiently small that

1 Vi
Q +
2 t

1
i
qt

wi >
t

1 Vi
Q +
2 t

2
i
qt

wi ,
t

t ∈ [0, T ] ,

i ∈ {a, b} .

(2.85)

Condition (2.85) implies wt ∈ wi , wi .
˜i
t t
The agent equilibrium was given by (2.22) in Example 6. The following proposition
ˆ
ˆ
characterizes the principal equilibrium controls (ΣY a , ΣY b ) and agent equilibrium efforts
(ˆa , eb ).
e ˆ
ˆ
Proposition 9 If ΣZa , ΣZb ≤ 0 then ΣY i = wi ΣZi for each i ∈ {a, b}. If ΣZi > 0 and
t
t
t
t t
t

ΣZi
t

>

Zj
Σt

i
qt
j

qt

j

max

j

wt w t
˜
, i
w i wt
t ˜

,

i, j ∈ {a, b} ,

i = j,

j Zj
ˆ
ˆY j
then there is an equilibrium with ΣY i = wi ΣZi and Σt = wt Σt .
t
t t

The corresponding

agent-effort equilibria are

ˆ
ΣY i = wi ΣZi
t
t t
ˆ
ΣY i
t

=

wi ΣZi
t t

=⇒

ei = 0,
ˆt

=⇒

ei
ˆt

123

=

2
ˆ
ΣY i
t
,
i
Γt qt

i ∈ {a, b},

and the BSDE system for Z i , ΣZi satisfies (2.44) with

µZi t, ΣY
ˆ
t

 1
+ −
2

=

2
1
1
− QV i 1 − wi −
t
t
2
2

2
QV i 1 − wi
t
t

2
QU i + i
t
qt
ΣZi

2
+ QU i wi
t
t

Γt

2

2
wi
wi + 2 it
t
qt
Yj

j Yi
wt Σt Σt
+2 j
Γt
q
t

ΣZi
Γt

+

2

1 i
et >0
ˆ


(2.86)
1 i
et =0 .
ˆ

Derivation. See Section .1 in the Appendix.
An equilibrium with principal-i sharing the proportion wi of ΣZi corresponds to zero
t
t
equilibrium agent-i exert, and the higher proportion wi corresponds to strictly positive equit
librium agent-i effort. The lower proportion is motivated purely by risk sharing, and the
higher proportion by both risk sharing and agent-effort incentive. If ΣZi is not positive, then
t
ˆ
the only principal equilibrium is with ΣY i = wi ΣZi and zero agent-i effort. If ΣZi > 0 then
t
t t
t
Zj

there is at least one principal equilibrium depending on the magnitude of ΣZi /Σt . Defint
ing λt =

b
qt
a
qt

max

wt wa
˜a
, t
b
wb wt
t ˜

¯
and λt =

b
qt
a
qt

min

wt w a
˜a
, t
b
w b wt
t ˜

, there is a time-t equilibrium

with only agent b working if ΣZb /ΣZa > λt and an equilibrium with only agent a working if
t
t
¯
¯
¯
ΣZb /ΣZa < λt . Because λt > λt we have both equilibria possible if ΣZb /ΣZa ∈ λt , λt .
t
t
t
t
For example, if QV a = QV b = 0 (risk-neutral principals) and the preference parameters
a
b
¯
are identical (QU a = QU b and qt = qt ), then λt = 1/2 and λt = 2.
t
t

Risk-neutrality of

principal i implies that agent-i diffusion is zero in a zero-effort-i equilibrium because there
is no risk-sharing motive on the part of the principal to share cash-flow risk with the agent.
For comparison, in a team setting with a single principal, identical agents and absolute
effort penalties, it can be shown that the optimal principal policy is to choose identical agent

124

diffusions, resulting in a continuum of agent equilibria.
Example 15 (Equilibrium with only agent a working) We assume constant preference
parameters and interest rate r, terminal consumption only and constant cash-flow volatility:

i
dXt = σ i dBt ,

σ i > 0,

i ∈ {a, b}.

(2.87)

Note that by Example 7, ΣZi = Γt σ i , where Γt = 1 − e−r(T −t) ( 1 − 1).
r
r
Assume condition (2.85) as well as
σa
>
σb

qa
qb

max

wb wb
˜
,
wa wa
˜

.

Then Proposition 9 implies that equilibrium principal and agent controls are

ˆ
ΣY a = wa Γt σ a ,
ea
ˆt

=

ˆ
ΣY b = wb Γt σ b ,

2
wa σ a
r(T −t)
Γt e
,
qa

eb = 0.
ˆt

Equilibrium terminal pay are

a
P T = w a La +
¯
T

−
0

1
2

2
QU a − a
q

U
(wa σ a )2 S(2) + erT (Γ0 K a − w0 a )

(2.88)

er(T −t) pa dt,
t

2
wa
1
b
U
PT = wb Lb − 2 a σ b σ a S(2) + QU b wb σ b S(2) + erT (Γ0 K b − w0 b ) (2.89)
q
2
T

−
0

er(T −t) pb dt.
t

where
125

T

S(k) =
0

er(T −t) Γt

k

dt

and

T

Li =
0

i
er(T −t) Γt dXt =

1 T r(T −t) i
e
dXt −
r 0

1
i
i
− 1 (XT − X0 )
r

Derivation. See Section .2 in the Appendix.
If, for simplicity, agent preferences are identical, and principal preferences are identical,
then σ a = σ b implies that principal a has a higher utility in an equilibrium with agent b
working rather than an equilibrium with only agent a working.

This follows because the

same effort is exerted in either equilibrium, but principal b pays for the disutility of agent
effort if agent b works. However, if σ b is sufficiently close to zero, then principal a can be
better off with his/her own agent working because under the agent-b-working equilibrium
insufficient effort is exerted.
The example is easily extended to multiple dimension (d > 1), in which case the same
results holds dimension by dimension, and equilibria are characterized by only one agent
exerting effort in each dimension at any point in time.

126

APPENDICES

127

.1

Appendix 1: Proofs Omitted from the Text

Proof of Proposition 4.
Matching the diffusion term of (2.44) and (2.52), we get

ζi

ΣZi = Σt Θi
t
t

Matching the drifts, we get

iζ
i
i
i
˙i
˙
θt + Θi µ + (Θi + β i Θi ) ζt = rt θt − µZi (t, Σt Θi ) + (rt Θi − Mti ) ζt
t
t
t
t
t

i
Matching coefficients of ζt yields the ODE (2.50), and matching the other terms yields the

ODE
i
˙i
θt − rt θt = −µZi t, Σζ1 Θ1 , . . . , ΣζN ΘN − Θi µi ,
t
t
t

i
θT = 0,

which has the solution (2.51). It is easy to then confirm that (2.52) solve the BSDE system
(2.44).
Below is a derivation of the closed form solution for Θi .
To solve

˙
Θt = −(β − rt I)Θt − Mt ,

t < T,

, ΘT = MT

where Θt , Mt are n × 1 vectors, β is an n × n matrix, rt is a function, I is an identity
matrix.
To avoid confusion, for the exponential functions we will use exp(·) with matrix arguments
128

and e· with scalar arguments.
First note

t
t
t
de− 0 rs ds Θt = −β e− 0 rs ds Θt − e− 0 rs ds Mt dt

t

t

Substituting ξt = e− 0 rs ds Θt and Nt = e− 0 rs ds Mt , we get the equation

T
ξT = e− 0 rs ds MT

dξt = −β ξt − Nt dt,

Let yt = exp(tβ )ξt .

dξt
dyt
= β exp(tβ )ξt + exp(tβ )
dt
dt
= β exp(tβ )ξt + exp(tβ ) −β ξt − Nt dt
= − exp(tβ )Nt dt

where we have used β exp(tβ ) = exp(tβ )β .
T
Integrating the above equation from t to T and use yT = exp(T β )e− 0 rs ds MT to get

T
yt = exp(T β )e− 0 rs ds MT +

T

exp(sβ )Ns ds
t

s
t
Substitute in yt = exp(tβ )e− 0 rs ds Θt and Ns = e− 0 ru du Ms to get

129

t
T
exp(tβ )e− 0 rs ds Θt = exp(T β )e− 0 rs ds MT +

T

s
exp(sβ )e− 0 ru du Ms ds

t

t

Left multiply both sides by exp(−tβ )e 0 rs ds and use the fact that t1 β and t2 β are
commutable where t1 and t2 are any arbitrary real numbers to get

T
Θt = exp((T − t)β )e− t rs ds MT +

T

s
exp((s − t)β )e− t ru du Ms ds

t

Proof of Proposition 6.
Let
A=

ΣY a
k

+

Γt Qea
k

, B=

ΣY b
k

+

Γt Qeb
k

For notational simplicity, omit the dimension subscripts k. From equation (2.43) and
(2.21), at principal equilibrium given B

A ∈ arg max f (A)
A∈R+

where
f (A) = −

−2δ
1−δ
Γt Qea 2
ΣZa
A (A + B) 1+δ +
(A + B) 1+δ
2
1−δ

Zb
If ΣZa <= 0, then it is easy to check the optimal A = 0, B = Σ eb and vice versa. We
Γt Q

will assume ΣZa > 0, ΣZb > 0 and calculate the following derivatives.

130

−2δ

f (A) = −Γt Qea A(A + B) 1+δ +

−3δ−1
−2δ
Γt Qea δA2
ΣZa
(A + B) 1+δ +
(A + B) 1+δ
1+δ
1+δ

−4δ−2

(A + B) 1+δ
f (A) =
(1 + δ)2

Γt Qea g(a) − 2δΣZa (A + B)

where

g(A) = 4δ(1 + δ)A(A + B) − (1 + δ)2 (A + B)2 − δ(3δ + 1)A2

It is easy to verify that g(A) ≤ 0 and thus f (A) ≤ 0.
Zb

Za

Let σ a = Σ e a and σ b = Σ e b . Finding the equilibrium is equivalent to solving the
Q
Q
following FOC equations

−Γt (1 + δ)A(A + B) + Γt δA2 + σ a (A + B) = 0
−Γt (1 + δ)B(A + B) + Γt δB 2 + σ b (A + B) = 0

a −σ b
Upon subtracting the second equation from the first, we get A−B = σ Γ . Substituting
t

this equality back to get

(2 + δ)(Γt A)2 − [(2 + δ)∆ + S]Γt A +
131

(S + ∆)∆
=0
2

where S = σ a + σ b and ∆ = σ a − σ b .
This yields the solution (2.78).

Proof of Lemma 5.
Each proof is for i = a and j = b (the other case is proved by reversing the roles of the
two labels).
Proof of a) From the expression (2.78) we have
ˆ
dΣY a
t

1
1
{1 + g (∆t , St )} +
=
2 (2 + δ)
2
dΣZa
t

where
St + (2 + δ) δ∆t

g (∆t , St ) =

.

(90)

2
St + (2 + δ) δ∆2
t

Holding fixed St we have ∂g (∆t , St ) /∂∆t ≥ 0 and therefore g (−St , St ) ≤ g (∆t , St ) ≤
g (St , St ).

Substituting g (−St , St ) =

(2+δ)(1−δ)
1+δ

− 1 and g (St , St ) = 1 + δ yields the

ˆ
first inequality. Note that dΣY a /dΣZa is minimized at ∆t = −St (i.e.,at ΣY a = 0) and
t
t
t
maximized at ∆t = St (i.e., ΣY b = 0), holding fixed S.
t
Again from (2.78) we have

d

ˆ
ΣY a
t
Qea
t

/d

ΣZb
t
Qeb
t

=

1
1
{1 + g (−∆t , St )} −
2 (2 + δ)
2

(91)

Because g is increasing in ∆t it follows that the left side in decreasing in ∆t , and therefore
maximized at ∆t = −St (i.e., at ΣY a = 0) and minimized at ∆t = St (i.e., at ΣY b = 0).
t
t
Substituting the above expressions for g (−St , St ) and g (St , St ) give the bounds.

132

Proof of b) From the solution (2.78) we have

S +
ˆ
2 dΣU a
t = d t
Qea dδ
dδ
t
=

1
2

2
St + (2 + δ) δ∆2
t

2+δ

(2 + δ) (2 + 2δ) ∆2
t

(2 + δ)2

−

St +

2
St + (2 + δ) δ∆2
t

(2 + δ)2

2
St + (2 + δ) δ∆2
t

Derivative is negative if and only if

(2 + δ) ∆2 ≤ St
t

2
2
St + (2 + δ) δ∆2 + St
t

which follows easily. (The derivative equals zero when St = ∆t .)
Proof of c) Differentiating (2.79) and using the equality in (2.82)
ˆ
∂ ΣY a /Qea
t
t
ea
1/(1+δ) ∂ˆ
(Γt )
=
∂δ
∂δ
ˆ
∂ ΣY a /Qea
t
t
=
∂δ

ˆ
ˆ
ΣY a ΣY b
t + t
Qea
Qeb
t
t

ˆ
ˆ
ΣY a ΣY b
t + t
Qea
Qeb
t
t
−δ −1 
 St +
1+δ


−δ −1
1+δ

ˆ
ˆ
ˆY a
ΣY a ΣY b
t + t − 2 δ Σt
Qea
1 + δ Qea
Qeb
t
t
t

2

St + (2 + δ) δ∆2
δ
t
−
∆t
(2 + δ)(1 + δ)
1+δ 

The braced term is positive iff

St +

2
St + (2 + δ) δ∆2 > (2 + δ) δ∆t .
t

This is obviously true for ∆t ≤ 0. Now consider ∆t > 0. Because the left side is increasing
in S and St ≥ ∆t the proof is competed by showing the inequality holds at St = ∆t , which
follows because δ < 1.

133

Proof of d) We first have

d
Za /Qea
d Σt
t

ˆ
ˆ
ΣY a ΣY b
t + t
Qea
Qeb
t
t

=

1
{1 + g (∆t , St )}
2+δ

where g is defined in (90). Differentiating (2.79) yields

dˆa
e
(Γt )1/(1+δ)
d ΣZa /Qea
t
t

=

1
1
{1 + g (∆t , St )} +
2 (2 + δ)
2
ˆ
δ ΣY a
t
−
1 + δ Qea
t

ˆ
ˆ
ΣY a ΣY b
t + t
Qea
Qeb
t
t

−δ −1
1+δ

ˆ
ˆ
ΣY a ΣY b
t + t
Qea
Qeb
t
t

−δ
1+δ

1
{1 + g (∆t , St )}
2+δ

and therefore

 ˆY a
 Σ
ˆY b
1/(1+δ)
t + Σt
(Γt )
 Qea
Qeb

t
t
=

1
2

ˆ
ˆ
ΣY a ΣY b
t + t
Qea
Qeb
t
t

+



δ +1 
1+δ 

dˆa
e
 d ΣZa /Qea

t
t

1 + g (∆t , St )
2+δ

ˆ
ˆ
ΣY a ΣY b
t + t
Qea
Qeb
t
t

1
2

−

ˆ
δ ΣY a
t
1 + δ Qea
t

.

δ
Positivity of dˆa /dΣY a follows because 1+δ < 1/2.
e
t

Now use
d
d ΣZb /Qeb
t
t

ˆ
ˆ
ΣY a ΣY b
t + t
ea
Qt
Qeb
t

=

1
{1 + g (−∆t , St )}
2+δ

and (91) to get

 ˆY a
 Σ
ˆY b
1/(1+δ)
t + Σt
(Γt )
 Qea
Qeb

t
t
1
=−
2

ˆ
ˆ
ΣY a ΣY b
t + t
Qea
Qeb
t
t



δ +1 
1+δ 

dˆa
e

 d ΣZb /Qeb

t
t

1 + g (−∆t , St )
1−
2+δ

134

−

1 + g (−∆t , St )
2+δ

ˆ
δ ΣY a
t
1 + δ Qea
t

The inequality g (−∆t , St ) ≤ 1 + δ implies that ea is decreasing in ΣY b .
ˆ
t

Proof of Proposition 7.
The maximization problem can be handled dimension by dimension. In each dimension
Yi
k, omitting the dimension argument k let y i = Σ ei , i = 1, . . . , N , y −i =
Γt Q

j=i y

j

and

Zi
σ i = Σ ei . The following Lemma implies the following bounds on principal i’s optimal policy:
Q

yi ∈
ˆ

QV i
1
σi,
QV i +QU i (1+δ)Γt

σi
Γt

.

By Corollary 1, the principal i’s problem is

max f (y i )
yi

where

 QV i (Qei )2
i) = Γ
f (y
t −

2

σi
Γt

2

− yi

−

QU i
2

(y i Qei )2 −

Qei
2



2
yi

−2δ 
i + y −i 1+δ
y



1−δ

y i + y −i 1+δ
+σ i Qei
1−δ

(92)

Lemma 6 Assume σ i , QV i , δ > 0. Then the function f in (92) satisfies

f

y i > 0 for y i ∈ 0,

QV i
σi
Γt (1 + δ) QV i + QU i

Proof.
The first derivative is
135

and

f

σi
y i < 0 for y i ≥ .
Γt

f (y i ) = (Qei )2 QV i σ i − Γt (QV i + QU i )y i
−2δ

(y i + y −i ) 1+δ ei i
yi
Q σ − Γt y i − Γt y −i δ i
+
1+δ
y + y −i

(93)

Vi
σi
If y i ∈ 0, Γ (1+δ) VQ U i then both braced term are positive, the second because
Q i +Q
t

yi
σ i − Γt y i − Γt y −i δ i
≥ σ i − Γt (1 + δ) y i > 0.
y + y −i
If y i ≥ σ i , then both braced terms are negative.
The second derivative of f (y i ) is

− 2δ −2 ei
Q

(y i + y −i ) 1+δ
f (y i ) = −Γt (Qei )2 (QV i + QU i ) −
1+δ
+Γt

1−δ i 2
(y ) + 2(1 − δ)y i y −i + (1 + δ)(y −i )2
1+δ

2δσ i (y i + y −i )
1+δ
≤0

The concavity of f together with differentiability imply that i’s optimal policy y i is uniquely
ˆ
given by the FOC f

y i = 0. The next Lemma bounds principal i’s optimal policy as a
ˆ

function of the other principals’ policies,
Lemma 7 The optimal y i satisfies

yi =
ˆ

QV i σ i
+ ε y −i
(QV i + QU i )Γt

136

(94)

−2δ

where ε y −i is of order y −i 1+δ given by
−2δ
δ
i y −i 1+δ ≤ ε y −i
− ei V i
σ
Q Q + QU i (1 + δ)Γt

1

≤

−2δ
1
σ i y −i 1+δ
Qei QV i + QU i (1 + δ)Γt

1

Proof. Substituting (94) into (93) and equating to zero yields

Qei QV i + QU i Γt ε y −i =

1
1+δ

The inequality σ i − Γt y − Γt δy −i

ε y −i ≤

yi
ˆ
σ i − Γt y i − Γt δy −i i
ˆ
y + y −i
ˆ

y
y+y −i

−2δ

y i + y −i 1+δ .
ˆ

(95)

< σ i implies

−2δ
1
σ i y −i 1+δ .
Qei QV i + QU i (1 + δ)Γt

1

Substitute
yi
ˆ
σ i − Γt y i − Γt δy −i i
ˆ
= σi −
y + y −i
ˆ

y i + (1 + δ) y −i
ˆ
y i + y −i
ˆ

Γt y i ≥ σ i − (1 + δ) Γt y i > −δσ i
ˆ
ˆ

into (95) (the last inequality because Γt y i < σ i ) to get
ˆ

Qei QV i + QU i Γt ε y −i ≥ −

Because ε y −i

−2δ
−2δ
δ
δ
ˆ
σ i y i + y −i 1+δ ≥ −
σ i y −i 1+δ .
1+δ
1+δ

→ 0 as y −i → ∞, and

ˆj
j=i y

→ ∞ as N → ∞ (by the lower

bound in Lemma 6 and the assumptions of the proposition), we get ε

137

ˆj
j=i y

→ 0 and

therefore y i →
ˆ

QV i σ i
(QV i +QU i )Γt

V i Zi
as N → ∞, which is equivalent to ΣY i → Q i Σ U i Finally,
QV +Q

the assumptions of the proposition and Example 7 imply ΣZi (t) = Φi (t)v i , i = 1, . . . , N ,
1
i
1 −1 .
where Φi (t) = 1 i − e−(r+κ )(T −t)
1
r+κ
r+κi

Proof of Proposition 9.
We omit the arguments (ω, t) throughout the proof. From the expression of equilibrium
efforts above, we obtain

ea ΣU + eb ΣU = max 0,
ˆ
ˆ

ΣY a ΣY b
, b
qa
q

/Γt .

Fixing some ΣY b ∈ R, and suppressing the dependence on ΣY b in the notation, by Corollary
1, principal a’s problem is maxΣY a ∈R J a ΣY a where

J a ΣY a = −

1
2

QV a ΣZa − ΣY a

2

+ QU a ΣY a

2

+ 2ΣZa max 0,




+ 2






ΣY a 
a 
2 eb 1
−q
− (Γt )



 ΣY a /qa =ΣY b /qb
qa






+ 2
ΣY a 

−q a 
 1 Ya a Yb b
Σ /q >Σ /q
qa

(note that eb ∈ 0,

(ΣY a )+
q a Γt

ΣY a ΣY b
, b
qa
q

(96)

2

on ΣY a /q a = ΣY b /q b ). Switching a and b gives principal

b’s problem.
We first consider, in the following lemma, the case of nonpositive ΣY a .

138

Lemma 8 For any fixed ΣZb ,

ΣZa ≤ 0

=⇒

ˆ
ΣY a = wa ΣZa .

ˆ
Proof. We first conclude that ΣY a ≤ 0, because it is easy to be seen from (96) that
J a (x) > J a (−x) if x < 0. Principal a’s problem therefore simplifies to

2
1
max − QV a ΣZa − ΣY a + 2ΣZa
ΣY a ≤0 2

ΣY b
qb

+

2
1
− QU a ΣY a .
2

From the FOC
∂J a
= QV a ΣZa − QU a + QV a ΣY a = 0
Ya
∂Σ
(Note that J a (x) is concave on x ≤ 0 and the right-hand derivative at 0 is negative) we get
the result.
ˆ
From Lemma 8 we also get that ΣZb ≤ 0 and ΣZa > 0 imply ΣY a = wa ΣZa .
¯

This

ˆ
follows because ΣY b = wb ΣZb ≤ 0 and (from (96)) J a (x) < J a (0) for any x < 0; therefore
principal a’s problem simplifies to

2
1
max − QV a ΣZa − ΣY a + 2ΣZa
ΣY a ≥0 2

ΣY a
qa

−

1
2

2
QU a + a
q

ΣY a

2

.

(Note that J a (x) is concave on x ≥ 0 and the right-hand derivative at zero is positive). The
result is obtained from the FOC
∂J a
=
∂ΣY a

2
QV a + a
q

2
ΣZa − QV a + QU a + a
q

139

ΣY a = 0.

The next step, in the following Lemma, is to solve principal a’s problem given any
ΣY b ≥ 0 chosen by principal b.
ˆ
Lemma 9 Let ΣY a denote principal a’s optimal control given ΣY b ≥ 0. For any ΣZa > 0
then ΣY b > wa
˜

qb
qa

ˆ
ΣZa implies ΣY a = wa ΣZa (which corresponds to zero agent-a effort)

and 0 ≤ ΣY b < wa
˜

qb
qa

ˆ
ΣZa implies that ΣY a = wa ΣZa (which corresponds to positive

agent-a effort).
Proof. Fixing some ΣZa > 0 throughout and define principal a’s objective function corresponding to zero and positive agent-a effort, respectively:
2
1
ΣY b
1
= − QV a ΣZa − Σ − QU a Σ2 + 2ΣZa b , ΣY b ≥ 0,
2
2
q
2
1
1
2
Σ
J p (Σ) = − QV a ΣZa − Σ −
QU a + a Σ2 + 2ΣZa a .
2
2
q
q

J n Σ; ΣY b

Principal a solves (note that ΣY a /q a = ΣY b /q b cannot be optimal unless principal a knows
with certainty that agent b will exert all the joint effort at that point)

sup J n (Σ) 1
Σ

Σ/q a <ΣY b /q b

+ J p (Σ) 1

Σ/q a >ΣY b /q b .

ˆ
The maximum of J n Σ; ΣU b occurs at Σn = wa ΣZa and

1
ˆ
J n Σn ; ΣU b = − QV a (1 − wa ) ΣZa
2

140

2

+ 2ΣZa

ΣY b
.
qb

(97)

where we have used the equality

QV a (1 − wa )2 + QU a (wa )2 = QV a (1 − wa )

ˆ
The maximum of J p (Σ) occurs at Σp = wa ΣZa and

ˆ
J p Σp =

wa
1
− QV a (1 − wa ) + a
2
q

ΣZa

2

.

where we have used the equality

2
2wa
QV a (1 − wa )2 + (QU a + a )(wa )2 − a = QV a (1 − wa )
q
q
Define

ˆ
ˆ
f (Σ) = J n Σn ; Σ − J p Σp
=

1 Va a
wa
Q (w − wa ) − a
2
q

Σ
(ΣZa )2 + 2ΣZa b
q

which is continuous and strictly increasing in Σ. It is easily confirmed that
f

q b a Za
˜
qa w Σ

q b a Za
˜
qa w Σ

ˆ
ˆ
= 0 and therefore principal a’s optimal control is ΣY a = Σp if ΣY b <
b

q
ˆ
ˆ
and ΣY a = Σn if ΣY b > qa wa ΣZa .
˜

We now apply Lemma 9 to obtain the Nash equilibria among principals when ΣZa , ΣZb >
0.
ˆ
ˆ
Case (i) (on agent b works): Lemma 9 implies that ΣY a = wa ΣZa , ΣY b = wb ΣZb holds

141

qb
qa

if ΣY b > wa
˜

qa
qb

ΣZa and ΣY a < wb
˜

ΣZb
>
ΣZa

qb
qa

ΣZb ; that is

max

wa wa
˜
,
˜
wb wb

.

ˆ
ˆ
Case (ii) (on agent a works): Lemma 9 implies that ΣY a = wa ΣZa , ΣY b = wb ΣZb holds if
ΣY b < wa
˜

qb
qa

ΣZa and ΣY a > wb
˜

ΣZb
<
ΣZa

qa
qb

qb
qa

ΣZb ; that is

min

wa wa
˜
,
wb wb
˜

.

The same Lemma easily rules out equilibria with both agents working.

Proof of Proposition 8.
Let

ΣY a
,
Γt

B=

ΣY b
Γt

1 − δa
,
Qea

fb =

1 − δb
Qeb

A=

and

fa =

By (2.43), at each dimension k principal a is solving holding fixed B(omitting dimension
arguments)

142

max f (A)

A∈R+

where

Γt QV a
f (A) = −
2

ΣZa
−A
Γt

2

b

b

1−δ
1+δ
Γt QU a 2 Γt Qea
a ) δ a +δ b (Bf b ) δ a +δ b
A −
(Af
−
2
2

1−δ a
1−δ b
+ΣZa (Af a ) δ a +δ b (Bf b ) δ a +δ b

If ΣZa <= 0, the optimal A = 0. We will assume ΣZa > 0 and calculate the following
k
derivatives.

f (A) = Γt

QV a

ΣZa
−A
Γt

b

− Γt

QU a A −

1−δ b
1−δ a
Za (Bf b ) δ a +δ b (f a ) δ a +δ b
+Σ

a

1 − δ a 1−2δ −δ
A δ a +δ b
a + δb
δ

b

f (A) = −Γt

QV a

− Γt

QU a

b

a

1−δ
1+δ
1−δ
b
Γt Qea
b ) δ a +δ b (f a ) δ a +δ b 1 + δ A δ a +δ b
(Bf
2
δa + δb
b

b

a

1−δ
1+δ
1−2δ −δ
b
a
Γt Qea
b ) δ a +δ b (f a ) δ a +δ b (1 + δ )(1 − δ ) A δ a +δ b
−
(Bf
2
(δ a + δ b )2

1−δ b
1−δ a
Za (Bf b ) δ a +δ b (f a ) δ a +δ b
+Σ

a

−2δ
(1 − δ a )(1 − 2δ a − δ b ) 1−3δ+δ b
δa
A
(δ a + δ b )2

f (A) < 0 if

143

b

b

−

(1 + δ b )(1 − δ a )ΣY a
+ ΣZa (1 − 2δ a − δ b ) < 0
2

(98)

Similarly, for principal b the solution has to satisfy

−

(1 + δ a )(1 − δ b )ΣY b
+ ΣZb (1 − 2δ b − δ a ) < 0
2

(99)

Assuming ΣZi > 0, i ∈ {a, b}, (98) and (99) are satisfied if

2δ a + δ b > 1

(100)

2δ b + δ a > 1

(100) holds if for example δ a + δ b > 1.
We will solve the following FOC equations.

Γt

QV a

Γt QV b

1−δ b
1+δ b
1−δ a
ea
U a A − Γt Q (Bf b ) δ a +δ b (f a ) δ a +δ b 1+δ b A δ a +δ b
− A − Γt Q
2
δ a +δ b
1−δ b
1−δ a
1−2δ a −δ b
Za (Bf b ) δ a +δ b (f a ) δ a +δ b 1−δ a A δ a +δ b = 0
+Σ
δ a +δ b
1−δ a
1+δ a
1−δ b
eb
ΣZb − B − Γ QU b B − Γt Q (Af a ) δ a +δ b (f b ) δ a +δ b 1+δ a B δ a +δ b
t
2
Γt
δ a +δ b

ΣZa
Γt

1−δ a
1−δ b
1−2δ b −δ a
Zb (Af a ) δ a +δ b (f b ) δ a +δ b 1−δ b B δ a +δ b
+Σ
δ a +δ b

144

=0

(101)

Case 1 Both pairs of principal and agent are risk-neutral.
By letting QV i = 0, QU i = 0, i ∈ {a, b}, (101) yields solution

ΣY a =

2ΣZa+
,
1 + δb

ΣY b =

2ΣZb+
1 + δa

Case 2 Only pair a of principal and agent are risk-neutral.
Let QV a = 0, QU a = 0 and assume QV b > 0, QU b > 0. (101) yields solution

ΣY a =

2ΣZa+
1 + δb

If ΣZa <= 0, then

ΣY b =

QV b ΣZb+
QV b + QU b

If ΣZa > 0, B is the solution to the nonlinear equation

Γt QV b

ΣZb
Γt

1−δ a
1+δ a
1−δ b
Γt Qeb
a +δ b b δ a +δ b 1+δ a δ a +δ b
− B − Γt QU b B − 2 (Af a ) δ
(f )
B
δ a +δ b
1−2δ b −δ a
1−δ a
1−δ b
a +δ b b δ a +δ b 1−δ b
+ΣZb (Af a ) δ
(f )
B δ a +δ b = 0
δ a +δ b

145

where Af a =

2ΣZa+ (1−δ a )
Γt Qea (1+δ b )

Case 3 Both pairs of principal and agent are risk-averse
The nonlinear equation system (101) has to be solved to obtain principal equilibrium.
In the case of identical principal-agent pairs and constant return to scale i.e.

1
QU a = QU b = QU , QV a = QV b = QV , Qea = Qeb = Qe , δ a = δ b =
2

(101) simplifies to

QV ΣZ

− Γt

(QV

+ QU )A −

fa Z
3Γt e a 2
Q (f ) A + Σ = 0
4
2

which yields solution

ΣY =

16Qe QV + 4
ΣZ
16Qe (QV + QU ) + 3

146

.2

Appendix 2: Derivation of Examples

Derivation of Example 6.
j

By Theorem 6, at each dimension k agent i holds fixed ek ≥ 0 and seeks ei to maximize
k

max f (ei )
k

ei ≥0
k

where

f (ei )
k

=

i
−Γt qk ei
k

i
Y i (ek
+ Σk

j

+ ek )1−δ
, i, j ∈ {a, b} , i = j.
1−δ

The first derivative is

j

i
f (ei ) = −Γt qk + ΣY i (ei + ek )−δ
k
k
k

Yj

If ΣY i ≤ 0, the maximum is achieved by ei = 0. So we assume ΣY i , Σk > 0.
k
k
k
The second derivative is

j

f (ei ) = −δΣY i (ei + ek )−δ−1
k
k
k

If

j
ek

= 0, the FOC implies the maximum is achieved by

j

If ek > 0, the FOC implies

147

ei
k

=

ΣY i
k
Γt q i
k

1/δ

and vice versa.

j

(ei + ek )δ =
k

If

j
ek

ΣY i
k
Γt q i
k

>

ΣY i
k
i
Γt qk

1/δ

, the above FOC can not be satisfied. It is easy to see that f (ei ) < 0,
k

so the maximum is achieved by ei = 0. Thus the agent equilibrium is
k
1/δ

Yj

ei
k

= 0,

j
ek

=

Σk

Yj

, if

j

ΣY i
> k
i
j
qk
qk

Σk

Γ t qk
Yj

Yi

Σ
Σ
j
If ei , ek > 0, symmetry implies k = k . So the agent equilibrium is
j
i
k
q
q
k
k

ei
k

j
+ ek

ΣY i
k

=

1/δ

, if

i
Γt qk

ΣY i
k
i
qk

Yj

=

Σk

j

qk

Derivation of Example 9.
i
Applying Ito’s formula on Dt Zt , using (2.44) and integrating from 0 to T , we get

T

T

−
0

e

T
T
i
V
e t rs ds ΣZi dBt = XT − e 0 rs ds (Γ0 V0i − w0 i + Y0i ) +
t

0
T r ds
t s
q Zi ΣZi
t
2Γt

ΣZi dt
t

T

+
0

e

T
t rs ds

Γt

Substituting (102) into (2.61) to get

148

ΣZi
t
j=i

wj Zj
Σ dt
q ej t

T
0

T
i
i
e t rs ds (Xt + Ht )dt

(102)

i
p i = w i XT +
T

T
0

T
i
e t rs ds Xt dt + wi

T
0

T
ˆt
e t rs ds Γt hV i (t, xV i ) − xV i dt
ˆt

T
T
V
−e 0 rs ds (Γ0 V0i − w0 i ) + (1 − wi ) e 0 rs ds Y0i −
T

+

e

0

T
t rs ds w i (1 − w i ) Zi
Σt
2Γt q ei

ΣZi dt −
t

T
0

T
0

T
ˆt
e t rs ds Γt hU i (t, xU i ) − xU i dt
ˆt

T
e t rs ds pi dt
t

Derivation of Example 12.
Apply Ito’s formula to

i
i
Zt = θt + Θi ζt ,
t

and match diffusion to get

ΣZi = diag
t

υ + V ζt ΣΘi
t

Match drift to get (2.72).
Substitute the above into (2.64) to get (2.70)
Derivation of Example 15.
Apply (2.32)

149

T
pi = Y0i e 0 rs ds −
T

T
0

T
ˆ
e t rs ds pi + µY i (t, ΣY ) dt +
t
t

T
0

T
e t rs ds ΣY i dBt , i = a, b.
t

with

ΣY a = wa Γt σ a ,

ΣY b = wb Γt σ b

and



Y a (t, ΣY ) = e−r(T −t) − 1 QU a ΣY a er(T −t) 2 − q a
µ
ˆ
t
 2
+ΣY a

2

qa



2ΣY a er(T −t)
qa

2
1
µY b (t, ΣY ) = e−r(T −t) − QU b ΣY b er(T −t)
ˆ
t
2

to get (2.88).

Derivation of Example 14.
The change of drift function is

150



ΣY a er(T −t)

+ ΣY b

2ΣY a er(T −t)
qa

Φk (ˆk ) =
e

a
2σk (1 − δ a )
(1 + δ b )Qea
k

1−δ a
δ a +δ b

b
2σk (1 − δ b )
(1 + δ a )Qeb
k

1−δ b
δ a +δ b

2−(δ a +δ b )
δ a +δ b
er(T −t) Γt

In Example 7, we showed that

i
i
Zt = Φi (t) + Φi (t) Xt ,
1
0

ΣZi = Φi (t) σ i , i = 1, . . . , N
t
1

where

Φi (t) = Γt =
1

1
− e−r(T −t)
r

The Φi is given by
0

151

1
−1
r

Φi (t) =
0

T

e−r(s−t)

t






−

2





2
er(T −s) Γs δ i +δ j

er(T −s) Γs

d

e−r(T −s)

Qei
k
k=1

d
i
σk

+Γs

k=1

i +δ j )
2−(δ

δ i +δ j

1+δ j
δ i +δ j

i
2σk (1 − δ i )
(1 + δ j )Qei
k

1−δ i
δ i +δ j

i
2σk (1 − δ i )
(1 + δ j )Qei
k

j
2σk (1 − δ j )
ej
(1 + δ i )Qk

j
2σk (1 − δ j )
ej
(1 + δ i )Qk

1−δ j
δ i +δ j

ds.





 1 d
−
=
Qei
k
 2


k=1
d
i
σk

+
k=1

d

= F (t)

i
2σk (1 − δ i )
(1 + δ j )Qei
k

i
2σk (1 − δ i )

1+δ j
δ i +δ j

1−δ i
δ i +δ j

(1 + δ j )Qei
k

1−δ j
δ i +δ j

j
2σk (1 − δ j )
ej
(1 + δ i )Qk

1−δ j 

δ i +δ j 



j

2σk (1 − δ j )
ej

(1 + δ i )Qk

1−δ i
1−δ j i
i ) δ i +δ j (C j ) δ i +δ j δ
(C

F (t)




j

j

1−δ
1+δ
+ δ j i δ i +δ j j δ i +δ j
(σk )
(σk )
1 + δj

k=1

where

T

F (t) =

2

e−r(s−t) er(T −s) δ i +δ j

−1

2

(Γs ) δ i +δ j ds

t

Ci

2(1 − δ i )
=
,
(1 + δ j )Qei
k

Cj

=

2(1 − δ j )
ej

(1 + δ i )Qk

and the terminal pay is

pi = Y0i erT −
T

T
0

ˆ
er(T −t) pi + µY i (t, ΣY ) dt +
ˆ
t
t

152

T
0

1−δ j
δ i +δ j

T
ˆ
e t rs ds ΣY i dBt
t

where


1+δ j
2
1−δ j
 e−r(T −t) Qei
Y i (t, ΣY ) =
k C i σ i δ i +δ j C j σ j δ i +δ j er(T −t) Γ δ i +δ j
ˆ
µ
ˆ
−
t
t
k
k

2
k=1

2−(δ i +δ j ) 
1−δ i
1−δ j
i
2Γt σk
j
i
δ i +δ j
+
C i σk δ i +δ j C j σk δ i +δ j er(T −t) Γt
j

1+δ
d

Thus we get the expression (2.84).

153

BIBLIOGRAPHY

154

BIBLIOGRAPHY
Aggarwal, R., and A. Samwick (1999): “Executive Compensation, Strategic Competitiion, and Relative Performance Evaluation: Theory and Evidence,” Journal of Finance,
pp. 1999–2043.
Antonelli, F. (1993): “Backward-forward stochastic differential equations,” The Annals
of Applied Probability, 3(3), 777–793.
Bansal, R., and A. Yaron (2004): “Risks for the Long Run: A Potential Resolution of
Asset Pricing Puzzles,” Journal of Finance, 59, 1481–1509.
Bolton, P., and M. Dewatripoint (2005): Contract Theory. MIT Press, Cambridge.
Briand, P., and Y. Hu (2006): “BSDE with quadratic growth and unbounded terminal
value,” Probability Theory and Related Fields, pp. 604–618.
(2008): “Quadratic BSDEs with convex generators and unbounded terminal conditions,” Probability Theory and Related Fields, 141, 543–567.
Delbaen, F., Y. Hu, and A. Richou (2009): “On the uniqueness of solutions to quadratic
bsdes with convex generators and unbounded terminal conditions,” Ann. Inst. Henri
Poincar Probab. Stat.
Duffie, D., and L. G. Epstein (1992a): “Asset Pricing with Stochastic Differential
Utility,” Review of Financial Studies, 5, 411–436.
(1992b): “Stochastic Differential Utility,” Econometrica, 60, 353–394.
Edmans, A., I. Goldstein, and J. Y. Zhu (2011): “Contracting with Synergies,” NBER
Working Paper 17606.
E.Pardoux, and S. Peng (1992): “Backward stochastic differential equations and quasilinear parabolic partial differential equations,” Lecture Notes in CIS, Springer, 176, 200–217.
Fudenberg, D., and J. Tirole (1992): Game Theory. MIT Press, Cambridge.
Hart, O. (1983): “The market mechanism as an incentive scheme,” Bell Journal of Economics, 14, 366–82.
155

Holmstrom, B. (1982): “Moral Hazard in Teams,” Bell Journal of Economics, 13, 324–
340.
Holmstrom, B., and P. Milgrom (1987): “Aggregation and Linearity in the Provision
of Intertemporal Incentives,” Econometrica, 55, 303–328.
Karoui, N. E., and L. Mazliak (eds.) (1997): Backward Stochastic Differential Equation.
Addison Wesley Longman Inc.
Karoui, N. E., S. Peng, and M.-C. Quenez (1997): “Backward stochastic differential
equations in finance,” Mathematical Finance, 7, 1–71.
Khanna, N., and M. Schroder (2010): “Optimal debt contracts and product market
competition with replacement,” Journal of Economic Theory, 145, 156188.
Kobylanski, M. (2000): “Backward Stochastic Differential Equations and Partial Differential Equations with Quadratic Growth,” The Annals of Probability, 28, 558–602.
Koo, H. K., G. Shim, and J. Sung (2008): “Optimal Multi-Agent Performance Measures
For Team Contracts,” Mathematical Finance, 18:4, 649–667.
Pardoux, E., and S. Peng (1990): “Adapted Solution of a Backward Stochastic Differential Equation,” Systems and Control Letters, 14, 55–61.
Peng, S. (1991): “A generalized dynamic programming principle and hamiltonjacobibellman equation,” Stochastics, 37, 61–74.
(1992): A nonlinear feynman-kac formula and applications, World Scientific, Singapore. Proceedings of Symposium of System Sciences and Control Theory.
(1993): “Backward stochastic differential equation and it’s application in optimal
control,” Appl. Math. Optim., pp. 125–144.
Rogers, L. C. G., and D. Talay (eds.) (1997): Numerical Methods in Finance. Cambridge University Press.
Schattler, H., and J. Sung (1993): “The First-Order Approach to the ContinuousTime Principal-Agent Problem with Exponential Utility,” Journal of Economic Theory,
61, 331–371.

156

Schmidt, K. (1997): “Managerial incentives and product market competition,” Review of
Economic Studies, 64, 191–213.
Schrarfstein, D. (1988): “Product market competition and managerial slack,” Rand
Journal of Economics, 19, 147–155.
Schroder, M. (2013): “The Dynamic Continuous-Time Principal-Agent Problem with
Moral Hazard and Translation-Invariant Recursive Preferences,” .
Schroder, M., and C. Skiadas (2005): “Lifetime Consumption-Portfolio Choice under
Trading Constraints and Nontradeable Income,” Stochastic Processes and their Applications, 115, 1–30.
Skiadas, C. (2008): “Dynamic Portfolio Choice and Risk Aversion,” in Handbooks in OR
& MS, Vol. 15, ed. by J. R. Birge, and V. Linetsky, chap. 19, pp. 789–843. Elsevier.
Williams, N. (2008): “On Dynamic Principal-Agent Problems in Continuous-Time,” working paper.
Winter, E. (2010): “Transparency and incentives among peers,” RAND Journal of Economics, 41, 504–523.

157