Incentives and Investment Timing: Real Options in a Principal …simon.rochester.edu/fac/zhang/brown...

Incentives and Investment Timing: Real Options in a

Principal-Agent Setting∗

Steven R. Grenadier† Neng Wang‡

January 2003

Abstract

In the standard real options approach to investment, the owner of the optiondecides when to exercise the investment option. However, in many real-worldsettings, investment decisions are delegated to managers. This article providesa model of optimal contracting in a continuous-time principal-agent setting inwhich there is both moral hazard and adverse selection. We show that theunderlying option can be decomposed into two components: a manager’s optionand an owner’s option. The specification of the manager’s option is determinedby a compensation contract, and must provide an incentive for the managerto both extend effort and to exercise optimally. The residual option payoutgoes to the owner. The implied investment behavior differs significantly fromthat of the first-best no-agency solution. In particular, there will be greaterinertia in investment, as the model leads to the manager having an even greater“option to wait” than the owner. The interplay between the twin forces of hiddeninformation and hidden action leads to markedly different investment outcomesthan when only one of the two forces is at work.

∗We thank Harrison Hong, David Scharfstein, Jeff Zwiebel, and especially Erwan Morellec, forhelpful comments.

†Graduate School of Business, Stanford University and NBER. Email: [email protected].‡William E. Simon School of Business Administration, University of Rochester. Email:

[email protected].

1

1 Introduction

The real options framework has proven to be a powerful and flexible approach for

analyzing investment under uncertainty. Essentially, the real options approach posits

that the opportunity to invest in a project is analogous to an American call option on

the investment project. Once that analogy is made, the vast and rigorous machinery

of financial options theory is at the disposal of real investment analysis. Thus, the

timing of investment is economically equivalent to the optimal exercise decision for an

option. The real options approach is well summarized in Dixit and Pindyck (1994) and

Trigeorgis (1996).1

In the standard real options paradigm, there are no agency conflicts as it assumed

that the option’s owner makes the exercise decision. Thus, there is a stark assumption

that there is no delegation of investment decisions from owners to managers, and

hence no agency frictions that can impede the optimal timing of investment. A large

economics literature extensively studies principal-agent conflicts.2 In this paper, we

extend the real options framework to the principal-agent setting, where the manager is

delegated the decision as to when to invest. In this setting, managers can influence the

quality of the underlying option through their efforts, and where the information about

project quality is known only by the manager prior to investment. In the presence

of both hidden action and hidden information, we formulate an optimal contract that

provides an option value that is as close as possible to the first-best setting in which

no such agency conflicts exist.

In order to highlight the importance of including principal-agent considerations into

real options models, consider the following two examples of well-known applications.

Example 1: The US Government’s Delegation of Offshore Oil Exploration to Private

Firms

1The application of the real options approach to investment is quite broad. Brennan and Schwartz(1985) use an option pricing approach to analyze investment in natural resources. McDonald andSiegel (1986) provided the standard continuous-time framework for analysis of a firm’s investment ina single project. Majd and Pindyck (1987) enrich the analysis with a time-to-build feature. Dixit(1989) uses the real option approach to examine entry and exit from a productive activity. Triantisand Hodder (1990) analyze manufacturing flexibility as an option. Titman (1985) and Williams (1991)use the real options approach to analyze real estate development.

2See Mirrlees (1976) and Holmstrom (1979), for early formal models on principal-agent relationship.

2

A prototypical case of a real option is oil exploration. Paddock, Siegel, and Smith

(1988) model the option-like characteristics of offshore oil leases. Using an option

analogy, the right to drill for oil represents an American call option on the market value

of the oil reserves with an exercise price equal to the cost of extraction. In the case of

offshore oil exploration, there exists a clear principal-agent problem in that the U.S.

government delegates (via auctions) the production rights to private firms.3 Thus, the

principal (government) must use a contract to motivate the agent (private oil firm) to

exercise their option as efficiently as possible.4 Clearly, both moral hazard and adverse

selection problems may exist. The amount of research, effort and technology used by

the firm are not easily observed, yet can have a material impact on the ultimate costs

and efficiency of exploration. In addition, the amount and nature of the underlying oil

deposits will be known with greater certainty by the firm than the government. The

actual contract offered by the government entails that an annual royalty be paid based

on value of extracted oil, along with a fixed annual fee.

Example 2: Agency and the Management of Commercial Real Estate

Another example of a real option is the leasing of vacant space. As presented in

Grenadier (1995), the leasing of space is analogous to an American call option where the

underlying asset is the present value of lease cash flows and the exercise price equals the

lump sum costs of leasing such as tenant improvement costs. In the U.S., corporations

and institutional investors own approximately two thirds of nonresidential real estate

(Institute of Real Estate Management, 1991).5 As pointed out by Williams (2001), the

agency problems associated with owning and managing certain types of commercial

real estate can be severe. In particular, the leasing of vacant space in older, less-than-

pristine commercial buildings is prone to moral hazard and adverse selection problems.

Leasing costs such as finding tenants (or monitoring a leasing agent), negotiating with

tenants, and fixing up space to suit new tenants are often private, and not easily

verifiable or contractible. The owner of the building must provide an incentive for the

3See Hendricks, Porter and Boudreau (1987) and Hendricks and Kovenock (1989) for more detailsabout such offshore oil leases.

4While other objectives may be possible, we assume that the government’s objective is to achievethe greatest value from the potential oil deposit.

5Miles and Tolleson (1997) provide a conservative estimate of the value of U.S. commercial realestate of $4 trillion, as of 1997.

3

manager to supply the appropriate amount of effort in order to make payoff from leasing

to new tenants as close to the first-best level as possible. A compensation contract

must be constructed that is based on the observable value of the rent achieved by the

manager. For example, leasing agents (delegated the role of finding new tenants) are

typically paid a percentage of the rent obtained, with leasing bonuses and reputation

effects adding some non-linear compensation features. In addition, property managers

(delegated the role of maintaining a property) are compensated in a similar manner.

In the model, an owner delegates the option exercise decision to a manager. Thus,

the timing of investment is determined by the manager. The true quality of the project

can be high or low. However, through effort, the manager can influence the likelihood

that the quality of the project is high. This unobserved effort represents the hidden

action component of the model. In addition, while the manager learns the true qual-

ity of the project after the effort decision is made, this information is not revealed to

the owner. This represents the hidden information component of the model. A com-

pensation contract must be contingent on the observable realized value of the project

upon exercise. This contract must provide two types of incentives to the manager:

an incentive to extend effort and an incentive to exercise as close as possible to the

value-maximizing stopping time. Importantly, we show that the underlying option can

be decomposed into two components: a “manager’s option” and an “owner’s option.”

The manager’s option has a payout upon exercise that is a function of the compensation

contract. Based on this contractual payout, the manager determines the exercise time.

The owner’s option has a payout, received at the manager’s chosen exercise time, equal

to the payoff from the underlying option minus the manager’s compensation. The

model provides the solution for the optimal compensation contract that comes as close

as possible to the first-best no-agency solution.

The model implies investment behavior substantially different from that of the stan-

dard real options approach with no agency problems. In general, managers will display

greater inertia in their investment behavior, as they will invest later than implied by

the first-best solution. In essence, this is a result of the manager (even in an optimal

contract) not having a full ownership stake in the option payoff. This less than full

ownership interest implies that the manager has a greater “option to wait” than the

4

owner. Such investment delay due to incentive problems is consistent with the re-

sults in Bertrand and Mullainathan (2000), where they look at the compensation and

investment behavior of manufacturing firms following the passage of anti-takeover legis-

lation. Presumably, following such legislation, managers become more entrenched and

the divergence of interests between owners and managers becomes more pronounced.

They find that managers display more inertia in their investment policy following the

anti-takeover legislation. This is consistent with our model, and not as consistent with

the view that less disciplined managers behave like empire builders.

An important aspect of the model is the interaction of hidden action and hidden

information. In fact, we find that the nature of the optimal contract depends explicitly

on the relative importance of these two forces. While we focus on the most interesting

case in which both forces play a role in the optimal contract, it is instructive to consider

two extremes. If the cost-benefit ratio of effort (a measure of the strength of the

hidden effort component) is very low, then the hidden action component disappears

from the optimal contract terms. Thus, if the nature of the underlying option is

such that inducing effort is sufficiently inexpensive, then we are left with a simple

problem of hidden information, and the contract will simply reward the manager with

informational rent. This is the setting of Maeland (2002), which considers a real options

setting with only hidden information about the costs of investment. Conversely, if the

cost-benefit ratio of effort is very high, then the hidden action component dominates

the optimal contract. The cost of inducing effort is so high as to no longer necessitate

the payment of informational rents.

It is only when both forces are in effect that the optimal compensation contract must

trade off the relative costs and benefits of inducing effort and information revelation.

Interestingly, the interplay between hidden information and hidden action may actually

improve the efficiency in investment timing decision. This is because the compensation

that is needed to induce effort provides the manager with a greater ownership stake in

the underlying option, making the manager’s timing incentives closer to those of the

first-best. Bernardo, Cai and Luo (2001) analyze capital budgeting and compensation

in an investment context with moral hazard and adverse selection. However, their

focus in on the amount of investment rather than the timing of investment.

5

In a later section, we generalize the model to allow for managers to display greater

impatience than owners. For example, managers may have a shorter horizon (due

to job loss, death, etc.), or may appear to use a higher discount rate due to being

less diversified or liquidity constrained. This generalization leads to very different

predictions about investment timing. While the basic model predicts that investment

will occur later than the first-best case, in this generalized setting investment can

actually occur earlier than the first-best case. The intuition is that the marginal

benefit to the manager of receiving compensation at an earlier point in time is less

than the marginal cost to the owner.

The remainder of the paper is organized as follows. Section 2 describes the setup

of the model and derives the simplified optimization program. Section 3 solves for the

optimal contracts in three separate regions. In Section 4 we analyze the implications on

investment lags and the erosion of the option value due to the agency problem. Section

5 shows that both under- and over-investment problems may arise, when the manager

and the owner have different degrees of impatience. Section 6 concludes. Appendices

contain the solution details of the optimal contracts.

2 Model

In this section, we begin with a description of the model. We then, as a useful bench-

mark, provide the solution to the first-best no-agency investment problem. Finally, we

present and simplify the full principal-agent optimization problem faced by the owner.

2.1 Setup

The principal owns an option to invest in a single project. Let the value P (t) of the

underlying project evolve as a geometric Brownian motion:

dP (t) = αP (t) dt+ σP (t) dZ(t), (2.1)

where α is the instantaneous conditional expected percentage change in P (t) per unit

time, σ is the instantaneous conditional standard deviation per unit time, and dZ is

the increment of a standard Wiener process.

6

We assume that the principal (owner) delegates the exercise decision to an agent

(manager). This desire to delegate investment authority is premised on the fact that it

is more costly for the owner to exercise the option than it is for the manager, perhaps

due to the manager’s possessing some specialized skill, or to a higher opportunity cost

for the owner. Both the owner and the manager are risk neutral,6 with the risk-free

rate of interest denoted by r. For convergence, we assume that r > α. The value of

the project, P (t), is observable to both the owner and manager at all times. Let P

equal the current value of the project, P = P (0).

The exercise cost may take on two possible values: θ1 or θ2, with θ1 < θ2, determined

randomly. One may interpret a draw of θ1 as a “higher quality” project and a draw

of θ2 as a ”lower quality” project. However, the manager may affect the likelihood of

drawing θ1 by exerting a one-time effort, at time 0. If the manager exerts no effort (or

a low amount of effort with a normalized cost of zero), the probability of drawing a

low exercise cost θ1 equals qL. However, if the manager exerts effort, he incurs a cost

ξ > 0 at time 0, but increases the likelihood of drawing a low exercise cost θ1 from qL

to qH . Immediately after his exerting effort, the manager observes his realized exercise

cost, and knows if the project is of higher or lower quality.

The owner does not observe the realized exercise cost, but observes and can contract

on the trigger level at which the manager exercises the option (the value of the project

upon investment). The manager is paid immediately after he exercises the option.

The manager is always free to walk away and not exercise the option. That is, he has

limited liability, whether his exercise cost is low or high.7

In summary, the owner faces a problem with both hidden information (the owner

does not observe the true realization of θ) and hidden action (the owner cannot verify

the manager’s effort level). Before analyzing the optimal contracting, we first briefly

review the first-best no-agency solution that is used as the benchmark.

6We rule out the time-0 selling-the-firm contract between the owner and the manager. This maybe justified if the manager is liquidity constrained and cannot, for example, borrow from a bank.

7The limited-liability condition is essential in delivering the investment inefficiency result in thiscontext. Otherwise, with risk-neutrality assumptions for both the owner and the manager, and no lim-ited liability, the first-best optimal investment timing may be achieved even in the presence of hiddeninformation and hidden action [Innes (1990)]. An alternative mechanism for generating investmentinefficiency in an agency context is to assume risk aversion.

7

2.2 First-Best Benchmark (The Standard Real Options Case)

When the exercise cost θ is observed and contractible, then the first-best timing of

investment may be made. Let W (P ; θ) denote the value of the owner’s option, in a

world where θ is a known parameter. Using standard arguments [i.e. Dixit and Pindyck

(1994)], W (P ; θ) must solve the following differential equation:

0 =1

2σ2P 2WPP + αPWP − rW. (2.2)

Differential equation (2.2) must be solved subject to appropriate boundary condi-

tions. These boundary conditions serve to ensure that an optimal exercise strategy is

chosen:

W [P ∗(θ), θ] = P ∗(θ) − θ, (2.3)

WP [P ∗(θ), θ] = 1, (2.4)

limP→0

W (P, θ) = 0. (2.5)

Here, P ∗(θ) is the value of P (t) that triggers entry. The first boundary condition is the

value-matching condition. It simply states that at the moment the option is exercised,

the expected payoff is P ∗(θ)−θ. The second boundary condition is the smooth-pasting

or high-contact condition.8 This condition ensures that the exercise trigger is chosen

so as to maximize the value of the option. The third boundary condition reflects the

fact that zero is an absorbing barrier for P (t).

Closed-form solutions forW (P, θ) and the exercise trigger P ∗(θ) are easily obtained.

The value of the first-best option value and exercise trigger can be written as:

W (P ; θ) =

(

PP ∗(θ)

)β

[P ∗(θ) − θ] , for P 1. (2.8)

8See Merton (1973) for a discussion of the high-contact condition.

8

Note that P ∗′(θ) < 0, P ∗′′(θ) ≥ 0.

Since the realized value of θ can be either θ1 or θ2, we denote P ∗(θ1) = P ∗1 and

P ∗(θ2) = P ∗2 . We shall always assume the current value of the project, P , is less

than P ∗1 . The ex-ante value of the owner’s option under high effort and no agency is

qHW (P ; θ1) + (1 − qH)W (P ; θ2). We can write this first-best option value, V ∗(P ), as:

V ∗(P ) = qH

(P

P ∗1

)βθ1

β − 1+ (1 − qH)

(P

P ∗2

)βθ2

β − 1. (2.9)

It will prove useful in future calculations to define the present value of one dollar

received at the first moment that trigger P is reached, for i = 1, 2. Denote this present

value operator as D(P ; P ). This is simply the solution to differential equation (2.2)

subject to the boundary conditions that D(P ; P ) = 1, and D(0; P ) = 0. The solution

can be written as:

D(P ; P ) =

(P

P

)β

, P ≤ P . (2.10)

2.3 A Principal-Agent Setting

The owner offers the manager a contract at time zero. The contract specifies a payment

made to the manager, paid at the time of exercise. The owner is committed to

implementing the contract. The payment can be made contingent on the realized value

of the project at the time of exercise. Thus, in principle, for any value of P received

by the owner at the time of exercise, a contracted wage, w(P ), can be specified.

The principal-agent setting leads to a decomposition of the underlying option into

two options: an owner’s option and a manager’s option. The owner’s option has a payoff

function of P − w(P ), and the manager’s option has a payoff function of w(P ) − θ.

Obviously, the sum of these payoff functions equals the payoff of the underlying option.

The manager’s option is of the tradition American call option variety, since the manager

chooses the exercise time to maximize the value of his option. However, in this

optimal contracting setting, it is the owner that ultimately controls the timing of

exercise through the choice of contract parameters that induce the exercise policy that

maximizes the value of its option. In addition, the manager also possesses a compound

option, since the manager has the option to exert effort at time zero. The properties

of the manager’s option are thus contingent upon this initial effort choice.

9

Note that the problem has been framed as if the agent will pay the exercise cost of

θ. However, it is observationally equivalent to have the owner to pay the exercise cost

θ, if we redefine the net payment to the agent as s(P ) = w(P ) − θ. Then, the owner’s

payoff is P − s(P ) − θ and the manager’s payoff is s(P ). Both formulations provide

the same result. Throughout this paper, we will use the wage-value pair (w,P ) as the

contracting instruments.

Since there are only two possible values of θ, for any w(P ) schedule, there can be

at most two wage/value pairs that will be chosen by the manager.9 Thus, the contract

need only include two wage/value pairs from which the manager can choose: one that

will be chosen by a manager with an exercise cost of θ1, and one chosen by a manager

with an exercise cost of θ2. Therefore, the owner will offer a contract that promises a

wage of w1 if the option is exercised at P1 and a wage of w2 if the option is exercised

at P2. The revelation principle will ensure that a manager with an exercise cost of θ1

will exercise at the P1 trigger, and a manager with an exercise cost of θ2 will exercise

at the P2 trigger.

The owner’s option has a payout function of P1 − w1, if θ = θ1, and P2 − w2, if

θ = θ2. Thus, using function D( · ; · ) derived in (2.10), conditional on the manager

exerting effort, the value of the owner’s option, πo(P ;w1, w2, P1, P2), can be written as:

πo(P ;w1, w2, P1, P2) = qHD(P ;P1)(P1 − w1) + (1 − qH)D(P ;P2)(P2 − w2). (2.11)

The manager’s option has a payout function of w1−θ1, if θ = θ1, and w2−θ2, if θ = θ2.

Similarly, conditional on the manager exerting effort, the value of the manager’s option,

πm(P ;w1, w2, P1, P2), can be written as:

πm(P ;w1, w2, P1, P2) = qH D(P ;P1)(w1 − θ1) + (1 − qH)D(P ;P2)(w2 − θ2). (2.12)

For notational simplicity, we will sometimes drop the parameter arguments and simply

write the owner’s and manager’s option values as πo(P ), and πm(P ), respectively.

The owner’s objective is to maximize its option value through its choice of the

contract terms w1, w2, P1, and P2. Thus, the owner solves the following optimization

9We allow for the possibility of a pooling equilibrium in which only one wage/value pair is offered.However, this pooling equilibrium will always be dominated by a separating equilibrium with twowage/value pairs.

10

problem:

maxw1, w2, P1, P2

qH

(P

P1

)β

(P1 − w1) + (1 − qH)

(P

P2

)β

(P2 − w2) . (2.13)

This optimization is subject to a variety of constraints induced by the hidden infor-

mation and hidden action of the manager. The contract must induce the manager to

accept the contract, exert effort, exercise at the trigger P1 if θ = θ1, and exercise at

the trigger P2 if θ = θ2. It is the specification of these constraints to which we now

turn.

There are both ex-ante and ex-post constraints involving the manager’s option

value, πm( · ), derived in (2.12). The ex-ante constraints ensure that the manager

exerts effort and that the contract is accepted. These are the traditional constraints in

a static moral hazard setting.

• ex-ante incentive constraint:

qH

(P

P1

)β

(w1 − θ1) + (1 − qH)

(P

P2

)β

(w2 − θ2) − ξ ≥

qL

(P

P1

)β

(w1 − θ1) + (1 − qL)

(P

P2

)β

(w2 − θ2). (2.14)

The left side of this inequality is the value of the manager’s option if effort is exerted

minus the cost of effort. The right side is the value of the manager’s option if no effort

is exerted. This constraint ensures that the manager will exert effort.

• ex-ante participation constraint:

qH

(P

P1

)β

(w1 − θ1) + (1 − qH)

(P

P2

)β

(w2 − θ2) − ξ ≥ 0. (2.15)

This constraint ensures that the total value to the manager of accepting the contract

is non-negative.

The ex-post constraints ensure that the manager, in forming its optimal exercise

policy, acts in accordance with the owner’s beliefs. These are in keeping with the

revelation principle, in that the θ1-manager will exercise at P1 and the θ2-manager will

exercise at P2.

11

• ex-post incentive constraints (IC):

w1 − θ1 ≥(P1

P2

)β

(w2 − θ1) , (2.16)

w1 − θ2 ≤(P1

P2

)β

(w2 − θ2) . (2.17)

The first inequality ensures that a manager with exercise cost θ1 does indeed choose

to exercise at the trigger P1. This is because the value of the payout at trigger P1 is

seen to be at least as great as the present value of the payout obtained by waiting until

trigger P2 is reached.10 The second inequality ensures that a manager with exercise

cost θ2 does indeed choose to exercise at the trigger P2. This is because the value of

the payout at trigger P1 is seen to be less than or equal to the present value of the

payout obtained by waiting until trigger P2 is reached. Thus, these ex-post constraints

ensure that the revelation principle holds.

• ex-post limited-liability constraints:

wi − θi ≥ 0, i = 1, 2. (2.18)

The manager is thus never forced to lose any money by participating in the deal, ex-ante

or ex-post.

Therefore, the owner’s problem has a total of six inequality constraints: the ex-

ante incentive and participation constraints, and each of the two ex-post incentive and

limited-liability constraints. Fortunately, the following three Lemmas simplify the

problem in that we can reduce the number of constraints to three.

Lemma 1. The limited-liability condition for type-θ1 manager does not bind. That is,

w1 > θ1.

Proof 1. (P

P1

)β

(w1 − θ1) ≥(P

P2

)β

(w2 − θ1) ≥(P

P2

)β

∆θ > 0,

where ∆θ = θ2 − θ1 > 0. The first and second inequality follow from (2.16) and (2.18),

respectively.

10This argument is contingent upon P1 ≤ P2. We show below that this is always the case.

12

Re-arranging the IC constraint (2.14) gives

(P

P1

)β

(w1 − θ1) −(P

P2

)β

(w2 − θ2) ≥ ξ

∆q, (2.19)

where ∆q = qH − qL. The left-hand side of (2.19) is the expected excess surplus for a

type-θ1 manager over the surplus for a type-θ2 manager. The right-hand side measures

cost of exerting high effort, normalized by ∆q, the incremental change of the likelihood

of drawing low exercising cost θ1. Inequality (2.19) states that the manager will exert

high effort, if the incremental benefit from doing so is high enough.

In order to motivate the manager to exert high effort, we need to reward the manager

with a surplus larger than zero, which is the manager’s reservation value. This leads

to the following result.

Lemma 2. The ex-ante participation constraint (2.15) does not bind.

Proof 2.(P

P1

)β

(w1 − θ1) +1 − qH

qH

(P

P2

)β

(w2 − θ2) − ξ

qH

≥ ξ

∆q− ξ

qH

> 0,

where the first inequality follows from ex-ante IC constraint (2.19) and the limited

liability condition for type-θ2 manager.

The following result states that there is no rent (not counting time-0 sunk effort

cost) for the type-θ2 manager.

Lemma 3. The limited liability for type-θ2 manager binds, in that w2 = θ2.

The intuition is straightforward. Giving type-θ2 manager any positive rent ex-

post implies a higher rent for type-θ1 manager in order to meet the type-θ1 manager’s

IC constraint. In order to minimize the rents subject to the manager’s IR and IC

constraints, the owner shall give type-θ2 manager zero ex-post rent: w2 = θ2. A

formal proof of this Lemma appears in Appendix ??.

Together, Lemmas 1, 2, and 3 simplify the owner’s optimization problem as follows:

maxw1, P1, P2

qH

(P

P1

)β

(P1 − w1) + (1 − qH)

(P

P2

)β

(P2 − θ2), (2.20)

13

subject to

(P

P1

)β

(w1 − θ1) ≥(P

P2

)β

(θ2 − θ1) (2.21)

0 ≥(P

P1

)β

(w1 − θ2) (2.22)

(P

P1

)β

(w1 − θ1) ≥ ξ

∆q. (2.23)

At least one of (2.21) and (2.23) binds. Otherwise, reducing w1 will increase the

owner’s value strictly, without violating constraint (2.22). Based on which of these two

constraints binds, the resulting contract will differ.

In the next section we explicitly solve this (now simplified) constrained optimization.

We will find that there exist three equilibrium solutions to the principal-agent problem,

the particular solution depending on which of three disjoint regions a function of the

initial parameters resides.

3 Model Solution: Optimal Contracts

In this section, we provide the solution to the optimal contracting problem described

in the previous section: maximizing (2.20) subject to inequality constraints (2.21),

(2.22), and (2.23). We find that the nature of the solution depends on the initial

parameter values. In particular, the solution depends explicitly on the magnitude

of the cost-benefit ratio of the manager’s effort. Depending on this magnitude, the

optimal contract can take on three possible types: a “joint hidden information/ hidden

action” type, a “pure hidden information” type, and a “pure hidden action” type.

We first define the three regions that determine the nature of the optimal contract.

The key to the contract is the magnitude of the ratio of costs to benefits of the manager’s

effort, defined by ξ/∆q. The numerator is the direct cost of extending effort, and the

denominator is the change in the likelihood of achieving a low exercise cost due to

effort. The regions are then defined by where this cost-benefit ratio falls relative to the

present value of receiving a cash flow at three particular trigger values: P ∗1 = P ∗(θ1),

P ∗2 = P ∗(θ2), and KH = P ∗(ψH), where ψH = θ2 + qH∆θ/(1 − qH) > θ2. Using the

present value operator D( · ; · ), these present values can be expressed as D(P ; P ∗1 )∆θ,

14

D(P ; P ∗2 )∆θ, and D(P ; KH)∆θ, respectively. These present values are ordered by

D(P ; KH)∆θ < D(P ; P ∗2 )∆θ < D(P ; KH)∆θ. We now define the three critical

regions by whether the cost-benefit ratio of extending effort is medium, low or high.

1 The Joint Hidden-Information/Hidden-Action Region (medium cost-benefit

ratio): (P

KH

)β

∆θ ≤ ξ

∆q≤(P

P ∗2

)β

∆θ. (3.24)

2 The Hidden-Information Region (low cost-benefit ratio):

ξ

∆q<

(P

KH

)β

∆θ. (3.25)

3 The Hidden-Action Region (high cost-benefit ratio):

(P

P ∗2

)β

∆θ <ξ

∆q<

(P

P ∗1

)β

∆θ. (3.26)

Note that another potential region in which ξ/∆q > D(P ; P ∗1 )∆θ = ∆θ exists,

however in this range the costs of effort are so high as to no longer justify the exertion

of effort in equilibrium. Thus, we do not consider this region.11

We now describe the optimal contract in each of the three regions. Each contract is

specified by two wage/trigger-value pairs: (w1, P1), and (w2, P2). The proofs detailing

the solution are provided in Appendix A.

3.1 Joint Hidden Information/Hidden Action Region

In the parameter range defined by D(P ;KH)∆θ ≤ ξ/∆q ≤ D(P ; P ∗2 )∆θ, the optimal

contract can be written as:

P1 = P ∗1 , (3.27)

P2 = K2 = P

(∆q∆θ

ξ

)1/β

, (3.28)

w1 = θ1 +

(P ∗

1

K2

)β

∆θ, (3.29)

w2 = θ2. (3.30)

11A proof of this result is available from the authors by request.

15

Note that in this range, P2 > P ∗2 , and thus a θ2-type manager delays investment

beyond the first-best full information trigger. In addition, while the θ2-type manager

receives no net compensation (i.e., w2 = θ2), the θ1-type manager receives positive

information rents (i.e., w1 > θ1).

Since the manager must be induced into providing effort, w1 must be high enough

to provide enough compensation for the ex-ante IC constraint (2.23) to bind. This

reflects the “hidden action” component of the contract. In addition, given the surplus

wage paid to the θ1-type manager, the exercise trigger P2 must be high enough to

dissuade the θ1-type manager from deviating from the equilibrium first-best trigger

P ∗1 . Thus, in this region P2 is set so that the ex-post IC constraint (2.21) binds,

ensuring that the revelation principle holds. This requires that P2 be above the full-

information trigger P ∗2 . This deviation from the full-information trigger reflects the

“hidden information” component of the contract.

We can use these contract terms to place a value on the owner’s and manager’s op-

tion values. The owner’s and manager’s option values, πo(P ) and πm(P ), respectively,

can be written as:

πo(P ) = qH

(P

P ∗1

)βθ1

β − 1+ (1 − qH)

(P

K2

)β

(K2 − θ2) − qHξ

∆q, (3.31)

πm(P ) = qHξ

∆q. (3.32)

The owner’s option deviates from the first-best value, V ∗(P ) in (2.9), for two reasons.

First, the exercise trigger in the high exercise cost state is equal to K2 rather than P ∗2 ;

this is seen in the second term of the expression for πo(P ). Second, the manager must

be compensated for its effort through the wage paid in the low exercise cost state; this

is evidenced in the third term of the expression for πo(P ). The manager’s option value

is simply equal to the present value of the surplus he receives should he be a θ1-type

manager.

16

3.2 Hidden Information Only Region

In the parameter range, defined by ξ/∆q < D(P ; KH)∆θ, the optimal contract can

be written as:

P1 = P ∗1 , (3.33)

P2 = KH = P ∗(ψH), (3.34)

w1 = θ1 +

(P ∗

1

KH

)β

∆θ, (3.35)

w2 = θ2, (3.36)

where we recall that ψH = θ2 + qH∆θ/(1 − qH) > θ2.

In this region, the net costs of effort are low enough so that there is no need for

the firm to have to compensate the manager for extending effort. In this range, the

ex-ante IC constraint does not bind, and therefore the cost of effort does not find

its way into the optimal contract.12 The compensation the θ1-type manager receives

is purely an information rent that induces the manager to exercise at the first-best

trigger P ∗1 , in accordance with the revelation principle. Since w1 no longer needs to

be increased in order to induce effort (as in the joint region previously discussed), the

P2 trigger needs to be higher (relative to the first-best trigger P ∗2 ) in order to dissuade

the θ1-type manager from deviating from the equilibrium first-best trigger P ∗1 . That

is, in this region, P2 is higher than it was in the joint region. Therefore, perhaps

surprisingly, moral hazard serves to increase investment efficiency since the increased

share of the firm that must go to compensate the manager leads the manager to more

fully internalize the benefits of efficient investment timing.



can be written as:

πo(P ) = qH

(P

P ∗1

)βθ1

β − 1+ (1 − qH)

(P

KH

)βψH

β − 1, (3.37)

πm(P ) = qH

(P

KH

)β

∆θ. (3.38)

12In this region, where there is no moral hazard component, the contract takes essentially the sameform as that in Maeland (2002).

17

It is interesting to note that the owner’s option value is observationally equivalent to

the first-best solution V ∗(P ) characterized in (2.7), when one substitutes the higher

exercise cost ψH for θ2. Thus, the impact of the costs of hidden information is fully

embodied by an increase in the cost of exercise in the high exercise cost state. The

manager’s option value is simply equal to the surplus the θ1-type manager receives

from the necessity of being induced into exercising at the first-best trigger P ∗1 .

3.3 Hidden Action Only Region

In the parameter range, defined by D(P ; P ∗2 )∆θ < ξ/∆q < D(P ;P ∗

1 )∆θ, the optimal


P1 = P ∗1 , (3.39)

P2 = P ∗2 , (3.40)

w1 = θ1 +ξ

∆q

(P ∗

1

P

)β

, (3.41)

w2 = θ2. (3.42)

In this region, the contract triggers equal those of the first-best outcome. The

moral hazard costs are so high that the rent needed for motivating high effort (via the

ex-ante IC constraint) is sufficiently high so that the ex-post incentive constraints do

not demand addition rents. That is, by motivating the manager to extend effort, w1

ends up being high enough so that the θ1-type manager no longer needs P2 to exceed

P ∗2 in order to dissuade him from deviating from the low-state trigger P ∗

1 . Thus, the

contract is entirely driven by the need to motivate effort, as the ex-post IC constraints

that reflect hidden information fail to bind.



can be written as:

πo(P ) = V ∗(P ) − qHξ

∆q, (3.43)

πm(P ) = qHξ

∆q. (3.44)

The owner’s option value is equal to the first-best solution V ∗(P ) characterized in

(2.7), minus the present value of the rent paid to the manager in order to induce

18

effort. The manager’s option value is equal to the present value of this effort-inducing

compensation.

3.4 An Observationally Equivalent Setting

In the first two regions discussed, investment in the high-cost state is delayed beyond

the first-best trigger of P ∗2 . It is only in the pure hidden-action region that the first-

best trigger is achieved. In order to enhance the reader’s intuition, we provide an

observationally equivalent model that gives the same delayed investment trigger (in

the high cost state). The key point will be that increasing the costs of moral hazard

mitigates the inefficiency induced by informational asymmetry, and therefore increases

the efficiency of the investment timing decision.

Consider an alternative setting, without agency issues. However, suppose the owner

only gets a fraction δ ≤ 1 of the total project P (t) at the time of exercise. For example,

the owner may need to pay a fee proportional to the project’s value. Therefore, his

net payoff is δP (t) − θ2. Thus, the optimal trigger is simply P ∗2 /δ. The owner delays

his exercising timing, relative to the first-best (δ = 1), because the private marginal

benefit is lower than the social marginal benefit.

The exercise in each of the three regions can be replicated through an appropriately

chosen δ. In the joint hidden information/ hidden action region,

δj =P ∗

2

K2

=

(ξ

∆q∆θ

)1/βP ∗

2

P< 1. (3.45)

In the hidden information region,

δi =P ∗

2

KH

=

(1 +

qH

1 − qH

∆θ

θ2

)−1

=1 − qH

1 − qHθ1/θ2

< 1.

In the hidden action region, there is no inefficiency in exercising the option, in that

δa = 1. (3.46)

It can be seen that increasing the costs of moral hazard lead to increased investment

efficiency (although at a price of increased ex-ante effort compensation paid to the

manager). As the costs of moral hazard move us from the hidden information region

19

to the joint region, the percentage of the project retained by the firm increase, since

δj > δi in this intermediate range. As the costs of moral hazard then move us into the

hidden action region, the informational rents arising from the informational asymmetry

is smaller than the limited-liability rents induced by the hidden action. Therefore,

the owner may recommend the first-best investment timing decision to the manager

and compensate him with enough limited-liability rents. The manager will no longer

have an incentive to make costly delayed investment decision. This increase in the

“ownership stake” in the high exercise cost state represents an equivalent representation

of the timing efficiency induced by moral hazard costs.13

4 Model Implications

In this section, we analyze some of the more important implications of the optimal

contracting model. First, a clear prediction of our model is that the principal-agent

problem will introduce inertia in a firm’s investment behavior. Investment projects

are expected to be undertaken later than in the first-best setting. We thus consider

the factors that influence the expected lag in investment due to the principal-agent

problem. Second, specifically because the timing of investment differs from that of the

first-best outcome, the principal-agent problem results in a welfare loss. Obviously,

factors that make such welfare losses more pronounced will introduce an incentive for

firms and industries to alter their structures to more closely align the interests of owners

and managers.

In this section, we focus our analysis on the contract that prevails in the joint hidden

information/ hidden action region. It is in this region that the incentive problems are

the most rich and meaningful.14 Recall from Section 3 that the optimal contract in

13We are only considering increases in the costs of hidden action up to a point, equal to the rightendpoint of the hidden action region. Increases in such costs beyond this point are not supported inequilibrium, since effort will no longer be extended.

14In the hidden information region, exerting effort is worthwhile for the manager even withoutadditional payment; effectively there is really no hidden action. In the hidden action region, thereis no distortion in investment timing; hidden action and hidden information together do not lead tosocial welfare losses.

20

this joint region, D(P ;KH)∆θ ≤ ξ/∆q ≤ D(P ;P ∗2 )∆θ, is:

P1 = P ∗1 , (4.47)

P2 = K2 = P

(∆q∆θ

ξ

)1/β

, (4.48)

w1 = θ1 +

(P ∗

1

K2

)β

∆θ, (4.49)

w2 = θ2, (4.50)

where the owner’s and manager’s option values, πo(P ) and πm(P ), respectively, can be

written as:

πo(P ) = qH

(P

P ∗1

)βθ1

β − 1+ (1 − qH)

(P

K2

)β

(K2 − θ2) − qHξ

∆q, (4.51)

πm(P ) = qHξ

∆q. (4.52)

4.1 Agency Problems and Investment Lags

In the principal-agent setting, investment will occur later than it does in the standard

real options setting in which no principal-agent conflict is assumed to exist. Such in-

vestment delay due to incentive problems is consistent with the results in Bertrand

and Mullainathan (2000). In that paper, they look at the compensation and invest-

ment behavior of manufacturing firms following the passage of anti-takeover legislation.

Presumably, following such legislation, managers become more entrenched and the di-

vergence of interests between owners and managers becomes more pronounced. They

find that managers display more inertia in their investment policy following the anti-

takeover legislation. This is consistent with our model, and not as consistent with

the view that less disciplined managers behave like empire builders. They also find

that wages (including those for white collar workers) rise following the passage of the

legislation, which is also consistent with the information rents paid in our model.

In the standard real options setting, investment is triggered at the value maximizing

triggers, P ∗1 and P ∗

2 , for the low and high exercise cost outcomes, respectively. However,

in our setting, while the trigger for investment in the low exercise cost state remains at

P ∗1 , investment in the high exercise cost state may be triggered at K2, which is higher

than the first-best benchmark level P ∗2 .

21

Let T and T ∗ be the stopping times at which the option is exercised, in our model

and the first-best setting, respectively. We denote Γ = E (T − T ∗) as the expected

time lag due to the principal-agent problem. A solution for such an expectation can

be derived using Harrison (1985, Chapter 3). The expected lag is given by

Γ =

(1 − qH

α− σ2/2

)log

(K2

P ∗2

)(4.53)

=

(1 − qH

α− σ2/2

)[log

(P

θ2

)+

1

βlog

(∆q∆θ

ξ

)− log β + log (β − 1)

], (4.54)

where we assume that α > σ2/2 in order for this expectation to exist.

An important insight from Section 3 is that increases in the cost-benefit ratio of

effort leads to less distortion in investment timing. That is, as the ratio ξ/(∆q∆θ)

increases, the equilibrium trigger P2 becomes closer to the first-best trigger P ∗2 . The

intuition for this result is that as the incentive problems become more pronounced, the

owner must compensate the manager with a greater fraction of the underlying project’s

cash flows. This greater ownership stake implies that the manager’s exercise decision

will be more closely aligned with that of the owner’s. This intuition is confirmed by

the following comparative statics:

∂Γ

∂ξ= −

(1 − qH

α− σ2/2

)1

βξ< 0, (4.55)

∂Γ

∂(∆q)=

(1 − qH

α− σ2/2

)1

β∆q> 0, (4.56)

∂Γ

∂(∆θ)=

(1 − qH

α− σ2/2

)1

β∆θ> 0. (4.57)

Thus, increases in the cost of effort lead to a shorter expected investment lag, while

increases in the benefits of effort lead to a longer expected investment lag.

An increase in the volatility of the underlying project, σ2, has an ambiguous effect

on the expected time lag Γ. This can be seen from the following comparative static:

∂Γ

∂σ2= −

(1 − qH

α− σ2/2

)1

β2

[log

(∆q∆θ

ξ

)− β

β − 1

]∂β

∂σ2+

1 − qH

2 (α− σ2/2)2 log

(K2

P ∗2

),

(4.58)

where ∂β/∂σ2 < 0. An increase in σ2 raises the option value and makes waiting more

worthwhile, implying that both P ∗2 and K2 are larger, ceteris paribus. However, if the

22

marginal cost-benefit ratio for exerting effort is relatively high, in that

log

(ξ

∆q∆θ

)>β − 1

β, (4.59)

then the change of K2 relative to the change in P ∗2 is larger. Therefore, under such

conditions the expected time lag increases in σ2.

An increase in the expected growth rate of the project, α, also has an ambiguous

effect on the expected time lag Γ. This can be seen from the following comparative

static:

∂Γ

∂α= − 1 − qH

(α− σ2/2)2

log

(K2

P ∗2

)− 1

β

(log

(∆q∆θ

ξ

)− β

β − 1

)α− σ2/2√

(α− σ2/2)2 + 2rσ2

.

(4.60)

However, if (4.59) holds, then expected time lag decreases with drift α.

4.2 Welfare Loss and Option Values

Although the owner chooses the value-maximizing contract to provide an incentive for

the manager to extend effort, the agency problem ultimately still proves costly. In an

owner-managed firm, the manager will extend effort and will exercise the option at the

first-best stopping times. However, in firms with delegated management, there will be

a welfare loss due to the firm’s suboptimal exercise strategy.

By a welfare loss, we are referring to the difference between the values of the first-

best option value, V ∗(P ) in (2.9), and the sum of the owner and manager options,

πo(P ) and πm(P ) in (4.51) and (4.52). Thus, define the welfare loss due to agency

issues as L, where L = V ∗(P ) − (πo(P ) + πm(P )). Simplifying, we have:

L = (1 − qH)

[(P

P ∗2

)β

(P ∗2 − θ2) −

(P

K2

)β

(K2 − θ2)

]. (4.61)

This welfare loss is likely to have economic ramifications on the structure of firms.

For firms in industries with potentially large welfare losses due to agency, there will

be powerful forces that will push them to be privately held, or to be organized in a

manner that provides the closest alignment between owners and managers.

The size of the welfare loss is driven by the distance of the equilibrium trigger

K2 from the first-best trigger P ∗2 . As previously discussed, the firm’s exercise timing

23

becomes less distorted as the net cost-benefit ratio of effort increases. That is, as the

ratio ξ/( ∆q∆θ) increases, the equilibrium triggerK2 gets closer to the first best trigger

P ∗2 , and thus:

∂L

∂ξ< 0, (4.62)

∂L

∂∆q> 0, (4.63)

∂L

∂∆θ> 0. (4.64)

In terms of the owner’s option value, the incentive problem represents a trade-off

between timing efficiency and the surplus that must be paid to the manager to extend

effort. One can obtain better intuition on the forces at work in the agency problem

through the following decomposition. In the first-best solution, the owner pays the cost

of effort ξ and obtains the first-best option value V ∗(P ). In the agency equilibrium,

the owner delegates the cost of effort to the manager, but then holds the sub-optimal

option value πo(P ). The loss in the owner’s option value due to the incentive problem

is therefore given by:

∆πo(P ) ≡ V ∗(P ) − ξ − πo(P ) = L+ V m, (4.65)

where L is the total welfare loss given in (4.61), and V m is the ex-ante expected surplus

paid to the manager to exert effort, and is given by:

V m = πm(P ) − ξ = qH

(P

P ∗1

)β

(w1 − θ1) − ξ = qHξ

∆q− ξ =

qL

∆qξ. (4.66)

The decomposition of the welfare cost to the owner from (4.65) into the sum of

the timing component (L) and the compensation component (V m), is very useful for

providing intuition. The impact of a higher effort cost ξ represents a trade-off in

terms of the timing and compensation components. As shown in (4.62), a higher effort

cost results in more efficient investment timing. This must be traded-off against the

increased surplus that must be paid to provide appropriate incentives to the manager,

as seen in (4.66). Therefore, the total effect on the loss of owner’s option value due

24

to an increase in ξ depends on whether the “timing effect” or “compensation effect” is

larger, in that

∂

∂ξ∆πo(P ) = − (1 − qH) (β − 1)

(P

K2

)β (1 − P ∗

2

K2

)K2

βξ+qL

∆q(4.67)

=β − 1

β

1

∆q∆θ[− (1 − qH) (K2 − P ∗

2 ) + qL (P ∗2 − P ∗

1 )] . (4.68)

If the investment trigger K2 is significantly larger than P ∗2 , in that

(1 − qH) (K2 − P ∗2 ) > qL (P ∗

2 − P ∗1 ) , (4.69)

then an increase in ξ leads to a smaller loss in the owner’s option value, as the gain in

timing efficiency overcomes the loss due to the manager’s incentive surplus.15

5 Impatient Managers and Earlier-than-Optimal In-

vestment

In the standard model, both owners and managers value payoffs identically. However,

it may be the case that owners and managers have different discount rates. A manager

very well may have a higher discount rate for the project’s cash flows than the owner.

The manager’s higher discount rate will be result in greater impatience; the manager

will be more willing than the owner to take a reduced future payoff in exchange for

receiving it earlier. The most interesting result of this generalization is that the

manager may now invest earlier than the first-best optimum. Thus, while the standard

model results in a form of under-investment, in the generalized model we can have both

over and under-investment.

Consider the following potential motivations for a manager possessing a higher

discount rate than the owner. For example, even if shocks to the underlying project’s

returns are solely idiosyncratic and uncorrelated with the market, an undiversified

15Note that the above condition is non-empty. This can be seen as follows. Condition 4.69 isequivalent to

K2 >1

1 − qH[(1 − qH)P ∗

2 + qL(P ∗2 − P ∗

1 )] = KH − ∆q

1 − qH(P ∗

2 − P ∗1 ) .

The joint moral hazard/hidden information region is characterized by P ∗2 ≤ K2 ≤ KH . Therefore,

the above condition is met for some K2.

25

manager may discount project cash flows at a rate greater than the riskless rate. As

another example, the manager may use a higher discount rate if his effective horizon

is shorter. More specifically, suppose the manager faces an exogenous termination

driven by a Poisson process with intensity λ, where termination may occur at any

time prior to the realization of P1. This termination could be due to the manager’s

death, finding another job, or forced termination, but is not explicitly modeled.16 The

addition of stochastic termination transforms the manager’s option to one in which his

discount rate r is elevated to r + λ, a higher discount rate, to reflect the stochastic

termination.17

In this section, we consider a setting where the manager and the owner have different

discount rates. We suppose that the manager is more impatient than the owner.

Mathematically, this implies separate discount functions for the owner and manager,

Do(P ; P ) and Dm(P ; P ), where P ≤ P :

Do(P ; P ) =

(P

P

)β

, (5.70)

Dm(P ; P ) =

(P

P

)γ

. (5.71)

Since ∂β/∂r > 0, the manager’s higher discount rate is embodied by the condition

γ > β.

The problem is basically the same as that of Section 2.3, with the exception that

the constraints all use the higher discount factor γ rather than β. In addition, much

of the solution is similar. For example, Lemmas 1 and 2 apply as before, using the

same proof. In addition, Lemma 3 also holds, and is demonstrated in the appendix.

Thus, the optimal contracting problem in the generalized setting can be written as:

maxw1, w2, P1, P2

(P

P1

)β

(P1 − w1) +1 − qH

qH

(P

P2

)β

(P2 − w2) (5.72)

16We assume that the owner can costlessly replace the manager in the event of separation.17See Yaari (1965), Merton (1971) and Richard (1975) for this result.

26

subject to (P

P1

)γ

(w1 − θ1) ≥(P

P2

)γ

(θ2 − θ1) (5.73)

0 ≥(P

P1

)γ

(w1 − θ2) (5.74)(P

P1

)γ

(w1 − θ1) ≥ ξ

∆q. (5.75)

Just as in Section 3, there are three contracting regions: a hidden information

region, a joint hidden information-hidden action region, and a hidden action region.

In this section, we focus on the joint hidden information/hidden action region, as it

provides the richest setting in which to analyze the model.18

The joint hidden information-hidden action region is defined by the following region:(P

KH

)γ

∆θ < ξ∆q

<(

PP ∗

2

)γ

∆θ, where KH > P ∗2 is defined in (B.25). In this region the

optimal contract can be written as:

P1 = K1, (5.76)

P2 = K2 = P

(∆q∆θ

ξ

)1/γ

, (5.77)

w1 = θ1 +

(P1

K2

)γ

∆θ < θ2, (5.78)

w2 = θ2, (5.79)

where K1 is the root of H(x) = 0, where:

H(x) =β

β − 1

[θ1 +

(1 − γ

β

)( xP

)γ ξ

∆q

]− x. (5.80)

Importantly, unlike the basic model, we now have the possibility of investment

occurring before the first-best trigger, in that P1 < P ∗1 . To see this, note that H(0) =

P ∗1 and

H(P ∗1 ) =

β

β − 1

(1 − γ

β

)(P ∗

1

P

)γξ

∆q< 0. (5.81)

The derivative of H( · ) is

H ′(x) =β

β − 1γ

(1 − γ

β

)( xP

)γ−1 1

P

ξ

∆q− 1 < 0. (5.82)

18The derivations for the optimal contracts in the other regions are shown in the appendix.

27

Therefore, there exists a unique solution P1 = K1 P ∗

2 in the region

Dm(P ; KH)∆θ < ξ/∆q < Dm(P ;P ∗2 )∆θ. However, for γ > β, the trigger is closer to

the first-best trigger than for the standard case in which γ = β. This is true, since for

γ > β,

K2 = P

(∆q∆θ

ξ

)1/γ

< P

(∆q∆θ

ξ

)1/β

= K2. (5.83)

Thus, we have the result that in a model in which the manager is more impatient

than the owner, equilibrium investment occurs sooner than it does in the standard

model, with investment actually occurring prior to the first-best trigger in the low

exercise cost state. The greater impatience on the part of the manager implies that it

is in the owner’s interest to offer a contract that motivates earlier exercise. This results

in both costs and benefits to the owner. By motivating earlier investment in the high-

cost state, investment timing moves closer to first-best. Since the manager receives no

surplus in this state, the owner is the sole beneficiary of this timing efficiency. However,

earlier investment in the low-cost state implies investment that occurs sooner than the

first-best outcome. In this state, the owner is worse off for two reasons: investment

occurs too early, and the wage paid to the manager in this state must be higher (than

in the standard model) in order to motivate early investment. The net effect of these

costs and benefits is ambiguous and is driven by the relative parameter values.

6 Concluding Remarks

This paper extends the real options framework to account for the agency issues that are

prevalent in many real-world applications. When investment decisions are delegated

to managers, contracts must be designed that provide an incentive for the manager to

both extend effort and to exercise optimally. This article provides a model of optimal

contracting in a continuous-time principal-agent setting in which there is both moral

hazard and adverse selection. The implied investment behavior differs significantly from

that of the first-best no-agency solution. In particular, there will be greater inertia in

investment, as the model leads to the manager having an even greater “option to wait”

28

than the owner. The interplay between the twin forces of hidden information and

hidden action leads to markedly different investment outcomes than when only one of

the two forces is at work. Allowing the manager to have an effort choice that affects the

likelihood of getting a high quality project mitigates the investment inefficiency due to

informational asymmetry. When the model is generalized to include differing degrees

of impatience between owners and managers, we find that investment may occur either

earlier or later than optimal.

Some extensions of the model would prove interesting. First, the model could allow

for repeated investment decisions. This richer setting would permit owners to update

their beliefs over time, and for managers to establish reputations. Second, the model

could also be generalized to include competition in both the labor and product markets.

As shown by Grenadier (2002), the forces of competition greatly alter the investment

behavior implied by real options models.

29

Appendices

A Solution to the Optimal Contracting Problem

This appendix provides a derivation of the optimal contracts detailed in Section 3.

We proceed by first proving Lemma 3. Then, we present the simplified optimization

problem and derive the optimal contract in the three critical regions.

The owner’s objective is to maximize the value of his option, as expressed in (2.13).

As discussed in Section 2.3, this optimization is subject to the ex-ante and ex-post

incentive constraints (2.14), (2.15), (2.16), (2.17) and (2.18). Lemmas 1 and 2 allow

us to simplify the problem as follows:

maxw1, w2, P1, P2

(P

P1

)β

(P1 − w1) +1 − qH

qH

(P

P2

)β

(P2 − w2), (A.1)

subject to (P

P1

)β

(w1 − θ1) ≥(P

P2

)β

(w2 − θ1) (A.2)

(P

P2

)β

(w2 − θ2) ≥(P

P1

)β

(w1 − θ2) (A.3)

(P

P1

)β

(w1 − θ1) −(P

P2

)β

(w2 − θ2) ≥ ξ

∆q(A.4)

w2 ≥ θ2. (A.5)

Using the method of Kuhn-Tucker, we form the Lagrangian as follows:

L =

(P

P1

)β

(P1 − w1) +1 − qH

qH

(P

P2

)β

(P2 − w2)

+ λ1

[(P

P1

)β

(w1 − θ1) −(P

P2

)β

(w2 − θ1)

]

+ λ2

[(P

P2

)β

(w2 − θ2) −(P

P1

)β

(w1 − θ2)

]

+ λ3

[(P

P1

)β

(w1 − θ1) −(P

P2

)β

(w2 − θ2) − ξ

∆q

]+ λ4 (w2 − θ2) , (A.6)

with corresponding complementary slackness conditions for the four constraints. The

first-order condition with respect to w1 implies:

λ1 − λ2 + λ3 = 1. (A.7)

30

The first-order condition with respect to w2 implies:(−λ1 + λ2 − λ3 − 1 − qH

qH

)(P

P2

)β

+ λ4 = 0. (A.8)

Using (A.7) to simplify (A.8),

λ4 =

(P

P2

)β1

qH> 0. (A.9)

Therefore, the complementary slackness condition λ4(w2−θ2) = 0 implies that w2 = θ2.

This proves Lemma 3.

Using Lemmas 1, 2 and 3, we can now write the simplified problem as:

maxw1, P1, P2

(P

P1

)β

(P1 − w1) +(1 − qH)

qH

(P

P2

)β

(P2 − θ2) (A.10)

subject to (P

P1

)β

(w1 − θ1) ≥(P

P2

)β

(θ2 − θ1) (A.11)

0 ≥(P

P1

)β

(w1 − θ2) (A.12)

(P

P1

)β

(w1 − θ1) ≥ ξ

∆q. (A.13)

The first-order conditions with respect to P1 and P2 imply:

P1 =β

β − 1(θ1 − λ2∆θ) , (A.14)

P2 =β

β − 1

(θ2 +

qH

1 − qH

λ1 ∆θ

). (A.15)

We conjecture that the ex-post IC constraint for the type-θ2 manager, (A.12), does

not bind, in that λ2 = 0, for all three regions that we analyze. We will verify this

conjecture for each region. By combining the conjecture that λ2 = 0 with Lemma 3,

we have that P1 = P ∗1 and w2 = θ2 in each of the relevant contract regions.

With λ2 = 0, then (A.7) may be written as λ1 + λ3 = 1. Therefore, it must be the

case that at least one of (A.11) and (A.13) binds. This is consistent with our intuition.

Otherwise, lowering w1 increases the expected surplus to the owner, without violating

any other constraints.

31

A.1 The Joint Hidden Information-Hidden Action Region:(P

KH

)β

∆θ ≤ ξ∆q ≤

(PP ∗

2

)β

∆θ

We derive the optimal contract in this region by conjecturing that both (A.11) and

(A.13) bind. Solving these two equality constraints gives us:

P2 = K2 = P

(∆q∆θ

ξ

)1/β

, (A.16)

w1 = θ1 +

(P ∗

1

K2

)β

∆θ.

The solution for P2 implies that λ1 can be written as:

λ1 =β − 1

β(K2 − P ∗

2 )1 − qH

qH∆θ. (A.17)

The only possible regions under which both constraints may bind19 is characterized by(P

KH

)β

∆θ <ξ

∆q<

(P

P ∗2

)β

∆θ. (A.18)

The above argument implies that the only possible region under which both constraints

hold must lie within (A.18). But it does not necessarily imply that all the region will

require that both constraints bind. However, we now show that both (A.11) and

(A.13) bind throughout this entire region. The region characterized by (A.18) can be

equivalently expressed as P ∗2 < K2 < KH . Because (A.17) implies that λ1 is monotonic

increasing in K2, therefore, the Lagrangian multiplier is characterized by 0 < λ1 < 1,

in this region. Since λ3 = 1 − λ1, we also have 0 < λ3 < 1. Therefore, by the

complementary slackness conditions, both (A.11) and (A.13) bind in this joint region,

confirming the result that (A.18 is the whole region, with both constraints binding.

A.2 The Hidden Information Region: ξ∆q <

(P

KH

)β

∆θ

Suppose that the ex-ante IC constraint (A.13) does not bind. Since (A.11) must hold

as an equality, and since λ1 = 1, we therefore have

P2 = KH =β

β − 1

(θ2 +

qH

1 − qH

∆θ

), (A.19)

19If(

PP ∗

1

)β

∆θ > ξ∆q >

(PP ∗

2

)β

∆θ, then only the third constraint binds. If(

PKH

)β

∆θ > ξ∆q , then

only the first constraint binds. If ξ∆q >

(PP ∗

1

)β

∆θ, then supporting high effort is no longer in theowner’s interest.

32

and

w1 = θ1 +

(P ∗

1

KH

)β

∆θ < θ2. (A.20)

The inequality in (A.20) confirms that (A.12) does not bind, as conjectured. Finally,

in order to be consistent with the assumption that (A.13) doesn’t bind, we require that

ξ

∆q<

(P

KH

)β

∆θ. (A.21)

This is indeed confirmed by the definition of this contract region.

A.3 The Hidden Action Region:(

PP ∗

2

)β

∆θ < ξ∆q <

(PP ∗

1

)β

∆θ

Suppose that (A.11) does not bind and (A.13) binds, then λ1 = 0 by complementary

slackness, and λ3 = 1. Therefore,

P2 = P ∗2 , (A.22)

w1 = θ1 +

(P ∗

1

K2

)β

∆θ.

We need to verify that (A.11) and (A.12) do not bind. The constraint (A.12) is

non-binding if and only if P ∗1 < K2. The constraint (A.11) is non-binding if and only

if K2 < P ∗2 . Thus, together this implies that P ∗

1 < K2 (

PP ∗

1

)β

∆θ,

then it can be shown that the owner will not choose to motivate the manager to exert

effort. The cost of effort is so high as to overwhelm any potential benefits of motivating

effort. A proof of this result is available from the authors upon request.

B Solution to the Generalized Optimal Contracting

Problem in Section 5

This appendix provides a derivation of the optimal contract detailed in Section 5. Since

Lemma 1 and Lemma 2 apply as in the standard model, the owner solves the following

optimization problem:

maxw1, w2, P1, P2

(P

P1

)β

(P1 − w1) +1 − qH

qH

(P

P2

)β

(P2 − w2) (B.1)

33

subject to (P

P1

)γ

(w1 − θ1) ≥(P

P2

)γ

(w2 − θ1) (B.2)(P

P2

)γ

(w2 − θ2) ≥(P

P1

)γ

(w1 − θ2) (B.3)(P

P1

)γ

(w1 − θ1) −(P

P2

)γ

(w2 − θ2) ≥ ξ

∆q(B.4)

w2 ≥ θ2. (B.5)

Using the method of Kuhn-Tucker, we form the Lagrangian as follows:

L =

(P

P1

)β

(P1 − w1) +1 − qH

qH

(P

P2

)β

(P2 − w2)

+ λ1

[(P

P1

)γ

(w1 − θ1) −(P

P2

)γ

(w2 − θ1)

]

+ λ2

[(P

P2

)γ

(w2 − θ2) −(P

P1

)γ

(w1 − θ2)

]

+ λ3

[(P

P1

)γ

(w1 − θ1) −(P

P2

)γ

(w2 − θ2) − ξ

∆q

]+ λ4 (w2 − θ2) . (B.6)

with corresponding complementary slackness conditions for the four constraints.The

first-order condition with respect to w1 implies:

(λ1 − λ2 + λ3)

(P

P1

)γ

=

(P

P1

)β

. (B.7)

The first-order condition with respect to w2 implies:

(−λ1 + λ2 − λ3)

(P

P2

)γ

− 1 − qH

qH

(P

P2

)β

+ λ4 = 0. (B.8)

Using (B.7) to simplify (B.8),

λ4 =

(P

P2

)β[(

P2

P1

)β−γ

+1 − qH

qH

]> 0. (B.9)

The complementary slackness condition implies that w2 = θ2. This proves Lemma 3

for the generalized model.

Therefore, we may simplify the optimization problem as follows:

maxw1, P1, P2

(P

P1

)β

(P1 − w1) +1 − qH

qH

(P

P2

)β

(P2 − θ2) (B.10)

34

subject to (P

P1

)γ

(w1 − θ1) ≥(P

P2

)γ

(θ2 − θ1) (B.11)

0 ≥(P

P1

)γ

(w1 − θ2) (B.12)(P

P1

)γ

(w1 − θ1) ≥ ξ

∆q. (B.13)

The first-order condition with respect to P1 is given by

P1 =β

β − 1

[w1 − γ

β[(λ1 + λ3) (w1 − θ1) − λ2 (w1 − θ2)]

(P

P1

)γ−β]. (B.14)

The first-order condition with respect to P2 is given by

P2 =β

β − 1

[θ2 + λ1

qH

1 − qH

γ

β

(P

P2

)γ−β

∆θ

]. (B.15)

Also note that

λ1 + λ3 = λ2 +

(P

P1

)β−γ

. (B.16)

We conjecture that the ex-post IC constraint for the type-θ2 manager, (B.12), does not

bind, in that λ2 = 0, for all three regions that we analyze. We will verify this conjecture

for each region. With λ2 = 0, then (B.16) may be written as λ1 + λ3 =(

PP1

)β−γ

> 1.

Therefore, it must be the case that at least one of (B.11) and (B.13) binds. This is

consistent with our intuition. Otherwise, lowering w1 increases the expected surplus

to the manager, without violating any other constraints.

B.1 The Hidden Information Region: ξ∆q <

(P

KH

)γ

∆θ

Suppose that the third constraint (B.13) does not bind and thus λ3 = 0. Then,

λ1 =

(P

P1

)β−γ

> 1. (B.17)

A binding ex-ante IC constraint (B.11) implies that the wage payment is

w1 = θ1 +

(P1

P2

)γ

∆θ. (B.18)

35

Using the first-order conditions (B.14) and (B.15), the optimal trigger levels are

P1 =β

β − 1

[θ1 +

(1 − γ

β

)(P1

P2

)γ

∆θ

](B.19)

P2 =β

β − 1

[θ2 +

qH

1 − qH

γ

β

(P1

P2

)γ−β

∆θ

]. (B.20)

Note that with γ > β, we immediately have P1 P ∗

2 . Therefore, w1 < θ2,

as conjectured, confirming that (B.12) does not bind.

Note that the system of equations (B.19) and (B.20) can be solved as follows. The

ratio x∗ = P1/P2 solves

G(x) = 0, (B.21)

where

G(x) = x

[θ2 +

γ

β

(qH

1 − qH

)xγ−β∆θ

]−[θ1 +

(1 − γ

β

)xγ∆θ

]= 0. (B.22)

First, note that G(0) = −θ1 < 0 and G(1) = γ/(β(1 − qH)) > 0. Second,

G′(x) = θ2 +γ + γ(γ − β)

β

qH

1 − qH

xγ−β∆θ + γxγ−1

(γ

β− 1

)∆θ > 0, (B.23)

for γ > β. Therefore, there exist a unique x∗ ∈ (0, 1) solving (B.21).

Therefore, for the region defined by ξ∆q

<(

PKH

)γ

∆θ, the optimal contract can be

written as:

P1 =β

β − 1

[θ1 +

(1 − γ

β

)(x∗)γ ∆θ

], (B.24)

P2 = KH =β

β − 1

[θ2 +

qH

1 − qH

γ

β(x∗)γ−β∆θ

], (B.25)

w1 = θ1 +

(P1

P2

)γ

∆θ, (B.26)

w2 = θ2. (B.27)

Finally, since in this region(

PKH

)γ

∆θ > ξ∆q

, constraint (B.13) is indeed not binding.

B.2 The Hidden Action Region:(

PP ∗

2

)γ

∆θ < ξ∆q <

(PK1

)γ

∆θ

Suppose that (B.13) binds, while (B.11) does not. Thus, λ1 = 0 and

λ3 =

(P

P1

)β−γ

> 1. (B.28)

36

Since λ1 = 0, (B.14) implies that

P2 = P ∗2 . (B.29)

Substituting λ3 =(

PP1

)β−γ

into (B.15) results in:

P1 = K1 =β

β − 1

[θ1 +

(1 − γ

β

)(P1

P

)γξ

∆q

]=

β

β − 1

[θ1 +

(1 − γ

β

)(P1

K2

)γ

∆θ

].

(B.30)

In Section 5 we prove that a unique K1 exists, where K1 ∈ (0, P ∗1 ).

A binding (B.13) implies that the wage payment is

w1 = θ1 +

(K1

P

)γξ

∆q= θ1 +

(K1

K2

)γ

∆θ, (B.31)

where K2 = P(

∆q∆θξ

)1/γ

. To ensure are conjecture that (B.12) doesn’t bind, (B.31)

implies that we need K1 < K2. This inequality can be written as ξ∆q

<(

P

K1

)γ

∆θ,

which is thus assured to hold in this region.

In order to be consistent with the fact that (B.11) does not bind, we need(

PP ∗

2

)γ

∆θ >ξ

∆q, which holds in this region.

Therefore, for the region defined by(

PP ∗

2

)γ

∆θ < ξ∆q

<(

PK1

)γ

∆θ, the optimal


P1 = K1, (B.32)

P2 = P ∗2 , (B.33)

w1 = θ1 +

(K1

K2

)γ

∆θ, (B.34)

w2 = θ2. (B.35)

B.3 The Joint Hidden Information/Hidden Action Region:(P

KH

)γ

∆θ ≤ ξ∆q ≤

(PP ∗

2

)γ

∆θ

We derive the optimal contract in this region by conjecturing that both (B.11) and

(B.13) bind. Solving these two equality constraints gives us:

P2 = K2 = P

(∆q∆θ

ξ

)1/γ

, (B.36)

w1 = θ1 +

(P1

K2

)γ

∆θ. (B.37)

37

Plugging the above two equations into the first-order condition (B.14) gives

P1 = K1 =β

β − 1

[θ1 +

(1 − γ

β

)(P1

K2

)γ

∆θ

], (B.38)

the same solution for P1 as in the hidden action region. In Section 5 we prove that a

unique K1 exists, where K1 ∈ (0, P ∗1 ). In addition, w1 in (B.37) is the same solution

as in the hidden action region. Thus, as before, we have verified that (B.12) does not

bind in this region.

We know that the only possible regions in which both (B.11) and (B.13) bind is(P

KH

)γ

∆θ < ξ∆q

<(

PP ∗

2

)γ

∆θ, since we have already shown that in the other regions

only one of these constraints binds.20 Equivalently stated in terms of K2, this region

is characterized by P ∗2 < K2 < KH . We now verify that the above solutions are indeed

optimal for this entire region.

In order to confirm that both the first and third constraints bind, we require that

λ1, λ3 �= 0. Since λ2 = 0 in this entire region, λ1 + λ3 =(

PP1

)β−γ

. Thus, we require

that

0 < λ1 <

(P

K1

)β−γ

. (B.39)

The first-order condition with respect to P2 implies that

λ1 =

[β

β − 1

qH

1 − qH

γ

β

(P

K2

)γ−β

∆θ

]−1 (K2 − P ∗

2

)(B.40)

Since K2 > P ∗2 , we have confirmed that λ1 > 0. We now prove that λ1 <

(PK1

)β−γ

.

We now consider λ1 as a function of K2, and rewrite (B.40) as:

λ1(K2) =

(P

K1(K2)

)β−γ (K1(K2)

K2

)β−γ [qH

1 − qH

γ

β(P ∗

2 − P ∗1 )

]−1 (K2 − P ∗

2

).

(B.41)

Note that from (B.38), K1 is a function of K2; we make this functional dependence

explicit in the above equation.

20Note that in the region ξ∆q >

(PK1

)γ

∆θ, it can be shown that effort cannot be induced. Thisresult is available from the authors by request.

38

We need to show that λ1(K2) <(

P

K1(K2)

)β−γ

over the region K2 ∈ (P ∗2 , KH). This

is equivalent to showing that

K2 0 for x ∈ (P ∗2 , KH).

Using implicit differentiation in (B.38), we can write:

dK1(x)

dx=K1(x)

x

γ(K1(x) − P ∗1 )

γ(K1(x) − P ∗1 ) − K1(x)

> 0, (B.44)

because K1(x) < P ∗1 in this region. Therefore,

dL(x)

dx= (γ − β)

qH

1 − qH

γ

β(P ∗

2 − P ∗1 )

(K1(x)

x

)γ−β−1(xdK1(x)

dx− K1(x)

x2

)− 1. (B.45)

From (B.44),

xdK1(x)

dx− K1(x) =

[K1(x)

]2γ(K1(x) − P ∗

1 ) − K1(x)< 0, (B.46)

because K1(x) 0 for x ∈ (P ∗2 , KH). This confirms that λ1, λ3 > 0

in this entire region, and therefore both (B.11) and (B.13) bind.

Therefore, for the region defined by(

P

KH

)γ

∆θ ≤ ξ∆q

≤(

PP ∗

2

)γ

∆θ, the optimal


P1 = K1, (B.47)

P2 = K2, (B.48)

w1 = θ1 +

(K1

K2

)γ

∆θ, (B.49)

w2 = θ2. (B.50)

39

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42

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