Irreversible Investment under Dynamic Agency∗

Steven R. Grenadier†, Jianjun Miao‡, and Neng Wang§

December 2004

Abstract

This paper provides a recursive contracting model to analyze the irreversible investmentdecision for a decentralized firm when the manager has private information about the invest-ment cash flows. We show that compared to the full information benchmark, investment isdelayed under asymmetric information. We also show that managerial risk aversion lowersthe option value of waiting and increases information rents. The net effect of risk aversionis to delay investment. The model also predicts that according to the optimal contract, theowner punishes the manager by lowering his compensation over time when the manager doesnot invest, and rewards the manager when he makes the investment. The reward increaseswith the profits upon investing.

Keywords: irreversible investment, information asymmetry, dynamic agency, recursivecontracts, real options.

∗We thank seminar participants in Boston University for helpful comments, especially, Simon Gilchrist, LarryKotlikoff, Kevin Lang, Rasmus Lents, Bart Lipman, and Michael Manove.

†Graduate School of Business, Stanford University, Stanford, CA 94305 and National Bureau of EconomicResearch, Cambridge, MA, USA. Email: sgren@stanford.edu. Tel.: 650-725-0706.

‡Department of Economics, Boston University, 270 Bay State Road, Boston, MA 02215. Email:miaoj@bu.edu. Tel.: 617-353-6675.

§Columbia Business School, 3022 Broadway, Uris Hall 812, New York, NY 10027. Email:nw2128@columbia.edu; Tel.: 212-854-3869.

1 Introduction

One of the most important topics in corporate finance and macroeconomics is the formulation of

the optimal investment strategies of firms. The investment decision has two components: how

much to invest and when to invest. The first is the capital allocation decision, and the second is

the investment timing decision. The standard textbook prescription for the capital allocation

decision is that firms should invest in an amount such that the marginal product of capital is

equal to the user cost of capital. The standard theory for the investment timing decision is that

firms should invest in projects when their net present values (NPVs) are positive. Recently,

the alternative real options approach has been widely accepted. The real options approach

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

the investment project, and the timing of investment is economically equivalent to the optimal

exercise decision for an option. This real options approach is well summarized in Dixit and

Pindyck (1994) and Trigeorgis (1996).

Both the user cost theory and the standard real options approach fail to account for the

presence of conflicts of interests induced by information asymmetries between the owner and the

delegated manager. In most modern corporations, shareholders delegate the investment decision

to managers, using managers’ special skills and expertise. In such decentralized settings, there

are likely to be information asymmetries. In general, managers are better informed than owners

about project cash flows. A number of papers in the literature provide models of capital

budgeting under asymmetric information and agency.1 The focus of this literature is on the first

element of the investment decision: the amount of capital allocated to managers for investment.

Thus, this literature provides predictions on whether firms over- or under-invest relative to

the first-best no-agency benchmark. The focus of this paper is on the second element of the

investment decision: the timing of investment. We extend the real options framework to account

for the issues of information and agency in a decentralized firm. Analogous to the notions of

over- or under-investment, our paper provides results on hurried or delayed investment.

No agency conflicts arise in the standard real options paradigm since it is assumed that

the option’s owner makes the exercise decision.2 By contrast, in this paper, a risk-neutral

1See Stein (2003) for a summary.2While our paper focuses on the agency issues that arise from the conflict of interests between owners and

managers, similar issues exist between stockholders and bondholders.

1

owner delegates the option exercise decision to a risk averse manager. The owner’s problem

is to design an optimal contract under asymmetric information. Absent any mechanism that

induces the manager to reveal his private information voluntarily, the manager could have an

incentive to lie about the true level of the project cash flows and divert cash flows for his

private interests. For example, the manager could divert privately observed cash flows by

consuming excessive perquisites or building empires. To overcome these problems, an optimal

contract selects an investment rule and designs a compensation scheme to the manger so as to

maximize the owner’s value such that the manager follows voluntarily this investment rule and

his behavior is incentive compatible. Since the manager is risk averse, the owner will provide

partial consumption insurance to the manager in order to facilitate the manager to smooth his

intertemporal consumption. An optimal contact must trade off insurance against incentives.

Briefly summarized, unlike traditional real options models, our model incorporates at least two

important real-world features into dynamic real option models: managerial risk aversion and

informational asymmetry, using the optimal contracting approach. We show that these two

features jointly generate new and interesting implications for the investment timing decision

and the dynamics of managerial compensation.

Our model implies that investment behavior differs substantially from the one implied by the

standard real options approach with no agency problems. We show that the manager displays

greater inertia in their investment behavior, in that they invest later than implied by the first-

best solution. This is because of the agency cost induced by informational asymmetry. The

agency cost reflects the fact that the manager captures some rents from his private information

about the investment project. By waiting longer, the owner saves more information rents to

the manager. Importantly, managerial risk aversion affects the investment timing decision in

two opposite ways. First, a larger degree of risk aversion lowers the manager’s option value of

waiting due to partial insurance or market incompleteness.3 This result has been obtained by

Miao and Wang (2004) in a dynamic incomplete-markets real options model without agency

issues. Second, the information rents captured by the manager increases with risk aversion. Our

extensive numerical exercises indicate that the latter agency cost effect dominates the former.

3In the standard real options approach, capital markets are assumed to be complete. By the standard arbitrageargument, the option value and investment timing are independent of preferences (see Dixit and Pindyck (1994)).In the conracting framework, this result also holds true if the owner fully insures the manager, as shown inProposition 2.

2

Thus, we find that investment is delayed and is delayed more if the manager is more risk averse.

In addition to generating predictions on investment decisions, our model also provides pre-

dictions on the dynamic evolution of managerial compensation. We show that managerial

compensation decreases over time before investment is made and increases when the manager

invests. Moreover, the amount of increase in managerial compensation depends on the cash

flows generated by the project. This increase in managerial compensation upon investment

provides an agency based explanation to the observation that the manager is rewarded with

bonus upon completion of successful projects. Finally, our model also predicts that firm value

jumps upwards when the manager invests. This is consistent with the empirical finding that

the announcements of unexpected increases in investment lead to increases in stock prices, as

documented by McConnell and Muscarella (1985).

Since our model allows for repeated interactions between the manager and the owner in a

long-term relationship, the contracting problem becomes inherently dynamic and the optimal

contract potentially depends on the history of the manager’s reported cash flows. In order

to keep the model analytically tractable, we use the techniques developed in the recursive

contracting literature.4 The key insight of the recursive contracting methodology is to capture

the history dependence of dynamic relationship between the owner and the manager by using

a state variable so that one can formulate the history-dependent optimal contacting problem

as a recursive one. This important state variable is the promised value by the owner to the

manager. The promised value describes the manager’s lifetime utility from the consumption

(or compensation) stream delivered by the owner. It is not only a forward looking variable, but

also summarizes the histories relevant for contracting purposes.

The paper closely related to ours is Grenadier and Wang (2004). They also analyze how

informational asymmetry and agency conflicts distort investment timing decisions. Grenadier

and Wang (2004) assume that the owner pays the manager only at the time of investment,

and specify managerial compensation contingent upon the manager’s announced cash flows of

4See Abreu, Pearce and Stacchetti (1990), Green (1987), Spear and Srivastava (1987), Thomas and Worrall(1988, 1990) for seminal contributions. The recursive contract approach has been applied widely in macroeco-nomics recently, e.g., unemployment insurance (Hopenhayn and Nicolini (1997) and Shimer and Werning (2003)),taxation (Kocherlakota (2004)), and social insurance (Atkeson and Lucas (1992)). Albuquerque and Hopenhayn(2004) and Clementi and Hopenhayn (2002) study the optimal lending contracts by analyzing the conflict of in-terests between borrowers and lenders. See Ljungquist and Sargent (2004) for a textbook treatment on recursivecontracts.

3

the project at the time of investment. In our setting, the owner and the manager interact

repeatedly in a long-term relationship. The optimal contract requires the owner to pay the

manager even if the manager does not invest, since the manager is risk averse and compensating

the manager before his investment smooths his intertemporal consumption. Thus, the key

predictions of our model rely heavily on managerial preferences for intertemporal consumption

smoothing, induced by risk aversion. This leads to the next key distinction between our model

and Grenadier and Wang (2004), who assume that managers are risk neutral and are protected

by limited liability. Finally, unlike Grenadier and Wang (2004) who analyze the effects of both

informational asymmetry and moral hazard on the investment timing decision, we focus on the

role of informational asymmetry only.

Our work also relates to the capital budgeting literature, particularly to Harris and Ra-

viv (1996, 1998) and Bernardo, Cai, and Luo (2001).5 Harris and Raviv (1996) apply a static

costly state verification model to examine capital budgeting processes in a single-division firm.

Harris and Raviv (1998) generalize that model to a multi-division firm. Both papers assume

that all parties are risk neutral and that managerial compensation is exogenous. Bernardo,

Cai, and Luo (2001) consider the capital allocation decision for a decentralized firm under both

asymmetric information and moral hazard in a static model. They use the optimal contract-

ing approach to jointly derive the optimal investment and compensation policies. Unlike the

preceding papers, we analyze the (irreversible) investment timing decision in a dynamic frame-

work. As in Bernardo, Cai, and Luo (2001), we derive managerial compensation as part of the

optimal contract. However, our dynamic model has important implications for the dynamics

of managerial compensation.

The remainder of the paper is organized as follows. Section 2 presents and solves the optimal

contracting problem in a static model. Section 3 introduces the dynamic setting and solves for

the optimal contract using recursive methods. Section 4 analyzes the model’s implications

on investment decision and wage dynamics. Section 5 concludes. Proofs are relegated to an

appendix.

5See Harris et al. (1982), Antle and Eppen (1985), and Holmstrom and Ricart i Costa (1986) for related earlywork.

4

2 A Static Model

We first solve for the optimal contract in a static setting, which provides intuition for the nature

of dynamic contract analyzed in Section 3. Consider a decentralized firm. The risk-neutral

owner of the firm has an investment project. The project costs I and generates a stochastic

project value x, which is randomly drawn from a given distribution function F (x) over the

interval [a, b]. Assume that F has a positive continuously differentiable density f > 0 and

I ∈ (a, b) .6 The owner delegates the investment decision to the manager because the manager

has special human capital. The manager decides whether to invest in the project or not. The

manager is risk averse with an increasing, strictly concave and continuously differentiable utility

function u.

The manager and the owner sign a contract. The key assumption is that the manager has

private information about the project value, and thus has incentives to lie in order to hide

project value from the owner. An optimal contract induces the manager to truthfully reveal his

privately observed project value. Moreover, the contract also ought to provide some insurance

to the manager since the manager is risk averse and the owner is risk neutral. These two goals

often come into conflicts. For example, a fully insured manager has no incentives to reveal his

project value truthfully, when he has a high project value. We will show later that the optimal

contract trades off incentive elicitation against insurance provision. Intuitively, one should

expect that managerial compensation must be tied to the manager’s investment decision and

his reported project value, while partially help the manager smooth his consumption across

states.

The optimal contracting problem can be described formally as a mechanism design problem

or a message game. By the revelation principle, we can restrict to the direct revelation mecha-

nism in which the message space is the set of possible project values. The manager’s investment

decision is binary. The contract specifies an investment rule with a threshold value x such that

the investment is made if and only if the reported project value is above the threshold x. Sup-

pose that the manager observes the project value is x. If he reports a high value x̃ ≥ x to the

owner, the owner pays the manager a wage yA (x̃) and the manager must make the investment

6It will be clear below that the assumption on f ensures the existence of a solution under asymmetric in-formation. The assumption on investment cost I rules out the uninteresting case where the project is eitherinvested immediately or never invested.

5

and give the owner the reported project value. If the manager reports a low value x̃ < x, then

the owner pays him a wage yA (x̃) and the manager does not invest.

An optimal contract maximizes the owner’s value (or firm value) subject to meeting the

manager’s participation and truth-telling incentive constraints. Formally, we formulate the

contracting problem as

max(yA,yB ,x)

−

∫ x̄

a

yB (x) dF (x) +

∫ b

x̄

[x − yA (x) − I

]dF (x) (1)

subject to

v =

∫ x̄

a

u(yB (x)

)dF (x) +

∫ b

x̄

u(yA (x)

)dF (x) , (2)

and

u(yA (x)

)≥ u

(yA (x̃) + x − x̃

), x, x̃ ≥ x̄, (3)

u(yA (x)

)≥ u

(yB (x̃)

), x ≥ x̄ > x̃, (4)

u(yB (x)

)≥ u

(yA (x̃) + x − x̃

), x̃ ≥ x̄ > x, (5)

u(yB (x)

)≥ u

(yB (x̃)

), x̄ > x, x̃. (6)

Equation (2) is the individual rationality constraint which describes that the participating

manager has a reservation utility level v. Inequalities (3)-(6) are incentive constraints. For

example, the left side of inequality (3) states that the manager with project value x ≥ x̄, will

invest and receive compensation yA (x). The right side of (3) gives the utility for the manager if

he lies and reports x̃ ≥ x̄. Under such a scenario, the manager can hide the part of project value

(x − x̃) and receive compensation yA (x̃). Incentive constraint (3) ensures that the manager has

no incentive to lie to any x̃ ≥ x̄. The other three incentive constraints rule out the manager’s

incentives to lie in other situations. For notational convenience, let dA (x) = x − yA (x) denote

the dividend payment to the owner, when the manager invests.

The following lemma simplifies the incentive constraints.

Lemma 1 If the manager does not invest, then he receives constant compensation yB, which

is independent of the reported project value, in that yB(x) = yB(x̃) = yB, for x, x̃ < x̄. If

the manager invests, then the owner receives constant dividend dA, which is independent of the

reported project value x, in that dA(x) = dA(x̃) = dA, for x, x̃ ≥ x̄. Moreover, yB = yA (x̄) =

x̄ − dA.

6

The intuition behind the above lemma is as follows. Since the owner cannot observe the

project value, the optimal contract must be designed in such a way that the owner’s payoff may

only depend on the manager’s observable and verifiable investment/no-investment decision, and

cannot respond to the manager’s private information beyond what is conveyed by the managerial

investment decision. If the manager invests, the payoff for the owner is dA(x) = x − yA(x). If

the manager does not invest, then the owner pays the manager compensation yB (x). Lemma 1

shows that both yB and dA are constant, consistent with our intuition. It is worth noting that

the constancy of yB and dA does not hinge upon the manager’s risk aversion and solely derives

from incentive compatibility conditions. Finally, when the project value is at the threshold x̄,

the manager is indifferent between investing and not investing, and hence receives the same

compensation yB = yA (x̄) = x̄ − dA.

Using Lemma 1, we can simplify the optimal contracting problem as follows:

max(dA,x)

−(x̄ − dA

)F (x̄) +

(dA − I

)(1 − F (x̄)) (7)

subject to

v = u(x̄ − dA

)F (x̄) +

∫ b

x̄

u(x − dA

)dF (x) . (8)

Proposition 1 In the static model with asymmetric information, the optimal contract is as

follows. (i) The optimal trigger x̄∗ and the dividend payment dA satisfy the equation

x̄∗ − I =F (x̄∗)

f (x̄∗)

[λu′

(x̄∗ − dA

)− 1

], (9)

and the participation constraint (2), where

λ−1 = u′(x̄∗ − dA

)F (x̄) +

∫ b

x̄∗

u′(x − dA

)dF (x) . (10)

(ii) For x < x̄∗, the manager does not invest, and receives constant compensation yB = x̄∗−dA.

For x ≥ x̄∗, the manager invests and receives compensation yA(x) = x − dA.

Before we explain the optimal contract described in this proposition, we first outline the

efficient investment rule under symmetric information. Under symmetric information, there are

no incentive constraints (3)-(6). One can easily show that the investment trigger is equal to

the investment cost I, which is also the trigger value under the standard NPV rule. Moreover,

7

the owner fully insures the manager. That is, the owner offers a constant identical wage to the

manager no matter whether he makes the investment.

We now turn to the optimal contract under asymmetric information described in Proposition

1. Part (i) demonstrates that the investment trigger and compensation are jointly determined

by equations (9) and (2). To better understand equation (9), we rewrite it as

f (x̄∗) (x̄∗ − I) = F (x̄∗)[λu′

(x̄∗ − dA

)− 1

]. (11)

The left side of this equation represents the marginal cost from an additional unit increase in

the investment threshold. The cost arises because low “type” managers x < x̄∗ do not invest

at project values below x̄∗ and thus the NPV (x̄∗ − I) is lost. It is multiplied by f (x̄∗) which

represents the probability of this cost. The right side of the preceding equation represents the

marginal benefit. The benefit arises because the owner saves information rents λu′(x̄∗ − dA

)−1

to each “type” x below x̄∗. This happens with probability F (x̄∗) . Note that λ is the Lagrange

multiplier associated with the participation constraint (8). It represents the shadow price of

the reservation utility.

Given the above discussion, we interpret the right side of (9) as information rents or agency

costs due to asymmetric information. Managerial risk aversion (u′′(x) < 0) implies that these

agency costs are positive, and thus the investment trigger is higher than the cost of investment

(x̄∗ > I). Consequently, the investment threshold under asymmetry information is higher than

that in the full information benchmark. This means that asymmetric information leads to

underinvestment. The intuition is the following. When designing an optimal contract, the

owner faces the trade off between providing insurance and eliciting incentives. He must offer

information rents to the privately informed manager such that he will not lie. This generates

agency costs and distorts investment efficiency. Note that if the manager is risk neutral, then

investment will be efficient because the manager is equally capable of bearing risk as the owner

does.

Part (ii) of Proposition 1 demonstrates that the owner offers a constant wage yB to the

manager if he does not make the investment. However, the manger is rewarded if he makes

the investment. Moreover, the wage yA is positively related to the project value. Thus, wage

provides incentives to the manager such that he tells the owner the true project value and makes

the investment. In addition, the wage must also provide insurance to the risk averse manager.

8

Because of incentive problems, the manager does not obtain constant full insurance wage.

We now turn to the question as to how managerial risk aversion influences investment deci-

sion. Unfortunately, we cannot derive a general result analytically for general utility functions.

The constant absolute risk aversion (CARA) utility u (c) = −e−γc/γ is an exception, as shown

in the following Corollary. Before presenting it, we define the certainty equivalent η (x̄; γ) for

a lottery offering the payoff of max {x − x̄, 0}, where x is drawn from the distribution F .7 For

CARA utility, it is immediate to show that

η (x̄; γ) =

{− 1

γlog

[F (x̄) +

∫ b

x̄e−γ(x−x̄)dF (x)

], γ > 0,

∫ b

x̄(x − x̄) dF (x), γ = 0.

(12)

Corollary 1 Suppose u (c) = −e−γc/γ. (i) If [1 − F (x) + η′ (x; γ)] /f (x) is decreasing in x ∈

[a, b] , then the optimal trigger value x̄∗ is the unique solution to the equation

x̄∗ − I =1 − F (x̄∗) + η′ (x̄∗; γ)

f (x̄∗). (13)

Moreover, x̄∗ is increasing in γ. (ii)The optimal compensation policy is given by

yB = −1

γlog (−vγ) − η (x̄∗; γ) for x < x̄∗, (14)

yA (x) = x − x̄∗ + yB for x ≥ x̄∗ . (15)

Notice that for CARA utility, the investment trigger is determined by a single equation,

independent of compensation and reservation utility. This is because CARA utility has no

wealth effect. This feature is more important when we turn to the dynamic contract in Section

3. Thus, in most of analysis below we will adopt this utility specification.

The intuition behind Corollary 1 is the following. When risk aversion is higher, the manager

prefers to have smoother consumption. This insurance objective is in conflict with the truth-

telling incentives. In order to elicit incentives, the owner must offer more information rents to

the manager. Thus, agency costs are higher; that is, the term on the right-hand side of (13)

is higher. This implies investment is distorted more in the sense that the investment trigger is

higher.

This intuition carries over for the constant relative risk aversion utility specification u (c) =

c1−γ/ (1 − γ) , γ ≥ 0. We illustrate this point by numerical simulations. We choose a standard

7Use u(η) = E [u (max {x − x̄, 0})], where u is CARA utility.

9

uniform distribution over [0,1] and set the investment cost I = 0.5. We also set the reservation

utility v = 0.11−γ/ (1 − γ) . That is, the manager receives a consumption level of 0.1 if he does

not participate in the contract.8 Figure 1 plots the investment threshold as a function of the

risk aversion parameter. This figure indicates that the investment threshold increases with the

risk aversion parameter.

[Insert Figure 1 Here]

In the next section, we turn to the dynamic setting. When the manager and the owner

live in a dynamic environment with repeated relationship, two important distinctions from the

static setting arise. First, there is an option value of waiting even in the absence of agency

issues. Second, the owner can use an intertemporal punishment and rewarding compensation

scheme to elicit incentives. We show that the interaction between the option value of waiting

and the intertemporal incentive scheme gives interesting implications for investment timing.

3 Optimal Recursive Contract

We begin this section by first describing the dynamic setting and formulating the problem in

a recursive manner. We then provide the solution to the first-best full-information investment

problem, as a benchmark for further analysis. Finally, we derive the solution to the recursive

contracting problem under asymmetric information.

3.1 Dynamic Setting

Unlike the static setting, the manager privately observes the project’s cash flows xt in each

period t = 0, 1, . . .. The project cash flows are independently and identically drawn from a

distribution function F over the interval [a, b] .9 If the manager invests at the cash flow value

x in some period τ , the firm pays the cost I immediately, and the project generates an equal

cash flow x in period τ and each period thereafter. This assumption simplifies the contracting

8Note that we do not choose a fixed utility level, say 0, as the reservation value. Otherwise, the manager willnot participate in the contract when his risk aversion is bigger than 1 since in any contract his utility is alwaysless than 0 given u (c) = c1−γ/ (1 − γ) specification. We rule out this situation in our numerical example.

9Assume that F has a positive continuously differentiable density f > 0 and I ∈ (a/ (1 − β) , b/ (1 − β)) . Itwill be clear below that the assumption on f ensures the existence of a solution under asymmetric information.The assumption on I rules out the uninteresting case where the project is either invested immediately or neverinvested. The IID assumption is important to formulate the optimal contract recursively.

10

problem after investment. In Section 4.4, we will relax this assumption. We also assume

that investment is irreversible in the sense that if the manager invests, he does not make any

investment decision again and forgoes all future opportunities to draw high values of cash flows.

As in the static setting, the owner delegates the investment decision to the manager. The risk

averse manager has a time-additive expected utility given by E[∑∞

t=0 βtu (ct)], where β ∈ [0, 1)

is the subjective discount factor. The owner is risk neutral and discounts cash flows according

to the discount factor β.

The owner does not observe the project cash flows, which are the manager’s private infor-

mation. However, the owner observes whether or not the manager has made the investment in

the previous periods. As is standard in the contracting literature, we assume that at time zero,

the owner makes a take-it-or-leave-it offer. The terms of the dynamic contract are contingent

on histories of the manager’s reports and investment status. Let st = 0 if the investment is

not made in period t and st = 1 if the investment is made in period t. Let xt = (x0, x1, ..., xt)

and st = (s0, s1, ..., st) denote the history of cash flows and the history of investment status,

respectively. Since the investment is irreversible, st satisfies the property that sτ = 1 for all

t′ ≤ τ ≤ t if t′ is the minimum τ such that sτ = 1. By the revelation principle, we can

restrict the manager’s reporting strategy to be a function of the form x̂ ={x̂t

(xt, st

)}∞

t=0.

The dynamic contract specifies an investment rule and a managerial compensation scheme con-

tingent on the histories of reports and investment status ht = (x̂0, s0; x̂1, s1; ...; x̂t, st) .10 The

optimal contract maximizes the owner’s expected discounted payoffs subject to the manager’s

intertemporal incentive and participation constraints.

As shown by Rogerson (1985) in a related two-period moral hazard model, an optimal dy-

namic contract depends on the history of reports. This history dependence makes the analysis

complicated. In order to tackle this problem, we follow the recursive contract literature and cre-

ate a state variable that summarizes the history of reports.11 Then, we may convert the history

dependent optimization problem into a recursive (Markovian) one. The newly created state

variable is the manager’s utility promised by the owner. Intuitively, the promised continuation

utility captures what the manager cares for his future. After all, the manager’s objective is

to maximize his life-time utility. Thus, using current compensation and promised continuation

10We will offer more detailed comments on this contract form in the next subsection.11See Spear and Srivastava (1987), Thomas and Worrall (1990) and Abreu et al (1990) for early contribution

to this approach.

11

utility, the owner is able to deliver the manager’s reservation value.

3.2 Recursive Formulation

We now describe the dynamic contract in a recursive manner. Specifically, consider any date

when investment has not been made before. Suppose at that date the manager is promised an

expected lifetime utility level v. The dynamic contract is described as follows:

1. The manager observes a value of cash flows x drawn from the distribution F ( · ) and then

makes a report x̃ to the owner.

2. The owner observes the report x̃ and recommends the manager to follow a trigger in-

vestment rule. Specifically, there is a threshold value x̄, such that the investment is

undertaken when the manager reports a high value x̃ ≥ x̄; and the investment is not

undertaken, otherwise.

3. If the investment is not undertaken, the owner offers the manager compensation yB (x̃) and

a promised continuation value wB (x̃) for the next period. If the investment is undertaken,

the manager obtains the true value of cash flows x and hands over the reported cash flows

x̃ to the owner in each period after investment. Moreover, the owner offers compensation

yAn (x̃) in the nth period after investment, n ≥ 0.

Two comments on the contract form are in order. First, as in the static model described

in Section 2, the dynamic contract specifies an investment threshold such that the investment

is made if the reported cash flows are higher than this threshold and if there is no investment

before. This investment rule is intuitive and related to the trigger policy in the standard

investment model without agency issues, e.g., McDonald and Siegel (1986) or its discrete time

variant described in the proof of Proposition 2 in the appendix. Thus, we are able to compare

the investment policy in our model with that in the standard model without agency issues.

Note that the investment trigger x̄ may depend on the state variable – promised utility v. In

Section 3.4, we will show that it is a constant independent of v under the CARA specification.

In Section 4.3 we will also consider the case where the investment policy is not written in the

contract and follows a simple NPV rule often recommended in practice. We will contrast the

resulting solution with the optimal contract. Second, after investment the owner knows that

12

the project cash flows are locked in at the reported value upon investment, even though he does

not know the true value.12 Thus, any non-flat reports after investment are irrelevant to the

owner. Consequently, the owner will offer wages in every period after investment, contingent

on the reported value x̃ at the time of investment only.

We solve the contracting problem by dynamic programming. Let P (v) denote the value

function of the owner or firm value. The optimal contract solves the following problem:

P (v) = max(yB ,wB ,yA

n ,x̄)

∫ x̄

a

[−yB (x) + βP

(wB (x)

)]dF (x) (16)

+

∫ b

x̄

[∞∑

n=0

βn(x − yA

n (x))− I

]dF (x)

subject to

v =

∫ x̄

a

[u

(yB (x)

)+ βwB (x)

]dF (x) +

∫ b

x̄

∞∑

n=0

βnu(yA

n (x))dF (x) (17)

and

∞∑

n=0

βnu(yA

n (x))

≥∞∑

n=0

βnu(x − x̃ + yA

n (x̃)), x, x̃ ≥ x̄, (18)

∞∑

n=0

βnu(yA

n (x))

≥ u(yB (x̃)

)+ βwB (x̃) , x ≥ x̄ > x̃, (19)

u(yB (x)

)+ βwB (x) ≥

∞∑

n=0

βnu(x − x̃ + yA

n (x̃)), x̃ ≥ x̄ > x, (20)

u(yB (x)

)+ βwB (x) ≥ u

(yB (x̃)

)+ βwB (x̃) , x̄ > x, x̃. (21)

The interpretation of the participation and incentive constraints (17)-(21) is similar to that

given in the previous section. When x < x̄, the manager does not invest. The owner offers

the manager compensation yB (x) in the current period and continuation utility wB (x). The

12As an example, suppose that the contracted trigger value is 5. The manager’s draw of cash flow value hasbeen below the trigger level 5 from the initial period to period 9. Thus, there is no investment before (andincluding) period 9. Suppose that the manager draws a high value 7 of cash flows in period 10. However, themanager reports the cash flow value to be 6 in period 10. According to the contract, the manager invests inperiod 10, and hands over the reported cash flow value 6 in period 10 and each period thereafter. But he obtainsthe true value 7 in period 10 and each period thereafter. The owner does not observe the true value, which is 7.If the manager decides to change his reported cash flow to another false level, say he reports 5 in period 12, thisnew report in period 12 is payoff irrelevant to the owner. The wage in any period after investment is contingenton the report 6.

13

owner does not receive any cash flow, but obtains continuation firm value P(wB (x)

). When

x ≥ x̄, the manager invests. The owner offers the manager compensation yAn (x) in the nth

period after investment. The owner pays the investment cost I at the time of investment and

obtains truthfully reported cash flows x in each period thereafter. By the principle of optimality,

summing up the owner’s value under both investment and no-investment regions and integrating

with respect to the distribution F give the owner’s current value function P (v) .

3.3 Full Information Benchmark

As a benchmark, we analyze the case in which the manager and the owner have symmetric infor-

mation about the cash flows. The optimal contract is the solution to the optimization problem

(16) subject to the promise-keeping constraint (17). The following proposition summarizes the

optimal contract.

Proposition 2 The optimal contract under full information is characterized as follows: (i)

The investment threshold xFB is the unique solution to the following equation:

xFB − (1 − β) I =β

1 − β

∫ b

xFB

(x − xFB

)dF (x) . (22)

(ii) The managerial compensations, yB (x) and yAn (x), before and after investment are equal

and given by yFB = u−1 ((1 − β) v) for all x and n. (iii) The continuation value if investment

is not made is given by wB (x) = v. Firm value is given by

P (v) =−u−1 (v (1 − β))

1 − β+

∫ b

xFB (x − (1 − β) I) dF (x)

(1 − β) [1 − βF (xFB)]. (23)

The interpretation of (22) is as follows. The left side of equation (22) represents the marginal

cost of waiting when increasing the threshold value xFB by one unit. This cost is the forgone

NPV of the project, measured in the flow sense. The right side of (22) represents the option

value of waiting for one more period to receive better draws x ≥ xFB. Equation (22) prescribes

the firm to invest at the threshold value xFB such that the marginal benefit from waiting for

one more period equals the marginal cost of waiting. Note that under full information, the

manager is fully insured and obtains constant wages. Thus, he behaves in a risk neutral way

and the investment threshold is independent of preferences. In fact, in the appendix we show

that xFB is equal to the investment trigger under risk neutrality.

14

For much of the remainder, we use the CARA utility. With such a utility specification, the

managerial compensation under first-best benchmark is given by yFB = − log (−vγ (1 − β)) /γ.

3.4 Model Solution

We now turn to the optimal contract under asymmetric information. We first simplify the

incentive constraints and derive some general properties of the contract.

Lemma 2 (i) The incentive constraints (19)-(21) can be replaced with the condition

u(yB (x)

)+ βwB (x) =

∞∑

n=0

βnu(yA

n (x̄))

for all x < x̄. (24)

(ii) Let dAn (x) = x− yA

n (x) . Suppose u (c) = −e−γc/γ. If P (·) is strictly concave and differen-

tiable, then at the optimum yB (x) and wB (x) are independent of x for all x < x, and dAn (x)

is independent of n and x for all x ≥ x, and n ≥ 0.

The intuition behind this lemma is similar to that behind Lemma 1. Before investment,

the manager does not receive any cash flows. Thus, offering a constant wage and a constant

continuation value to the manager can induce his truth-telling incentives. The resulting lifetime

utility of the manager is the same as that upon investment at the threshold value x̄, as shown

in (24). Moreover, under CARA specification the incentive constraint (18) is redundant. This

is because for the CARA utility the manager “type” variable x can be cancelled out in the

incentive constraint (18). Consequently, an optimal contract cannot distinguish the manager

upon investment and hence requires all types of manager to hand over identical dividends to

the owner. By this lemma, without risk of confusion we may simply use dA, yB, and wB to

denote dAn (x) , yB (x) , and wB (x) , respectively. Let wA (x) =

∑∞n=0 βnu

(yA

n (x))

denote the

continuation value after investment. We are now ready to present our main result.

Proposition 3 Suppose the assumptions in Corollary 1 hold. Then the optimal contract under

asymmetric information is characterized as follows:

(i) The trigger value x̄∗ is the unique solution to the equation

x̄∗ − (1 − β) I =β

1 − βη (x̄∗; γ) +

[1 − F (x̄∗) + η′ (x̄∗; γ)] [1 − βF (x̄∗)]

(1 − β) f (x̄∗). (25)

15

(ii) The optimal compensation policy is given by

yB = −1

γlog (−vγ (1 − β)) − η (x̄∗; γ) for x < x̄∗, (26)

yA (x) = x − x̄∗ + yB for x ≥ x̄∗ . (27)

(iii) The continuation values are given by

wB = v exp (γη (x̄∗; γ)) for x < x̄∗, (28)

wA (x) = v exp (γη (x̄∗; γ) − γ (x − x)) for x ≥ x̄∗. (29)

(iv) The owner’s value function is given by

P (v) =log (−vγ (1 − β))

γ (1 − β)+

(x̄∗ − I (1 − β)) (1 − F (x̄∗)) + η (x̄∗; γ)

(1 − βF (x̄∗)) (1 − β). (30)

In general, an optimal solution to the contracting problem (16)-(21) is of the following

form: x̄∗ = x̄∗ (v) , wB (x) = wB (x, v) , yB (x) = yB (x, v) , yAn (x) = yA

n (x, v) , n ≥ 0. Note that

the promised utility is a state variable, which summarizes the history of reports. The contract

dynamics can be described in terms of the evolution of the promised utility. Specifically, suppose

there is no investment before date t and the promised utility level is vt at the beginning of

date t. Then the investment trigger at date t is given by x̄∗ (vt) . If the manager truthfully

reports a high value of cash flows xt ≥ x̄∗ (vt) , then the manager invests and obtains cash

flows yAn (xt, vt) , at the nth period after date t. If the manager truthfully reports a low value of

cash flows xt < x̄∗ (vt) , then the manager does not invest, and obtains compensation yB (xt, vt)

and continuation value wB (xt, vt) . In date t + 1, the starting promised utility level is given by

vt+1 = wB (xt, vt) . The contract then has the same form as that described earlier for date t.

Proposition 3 demonstrates under the CARA utility specification, the above contract form

can be simplified significantly. First, the investment trigger is a constant independent of the

state variable – promised utility. This is similar to Corollary 1 due to the lack of wealth effect.

Second, the compensation yB and continuation utility wB before investment are independent

of cash flows x. Moreover, the dividend dA = x − yA after investment is also independent of x.

We next turn to the discussion of model implications.

4 Model Implications

We first analyze the model’s implication for investment timing. We next turn to the implication

for the dynamics of managerial compensation. After that, we contrast our model with one in

16

which investment is assumed to follow the NPV rule. We finally outline some extensions.

4.1 Investment Timing

Investment timing is determined by equation (25), which characterizes the investment threshold

x̄∗. Its interpretation is similar to that for (9). The left side of (25) is the marginal cost from

waiting (measured in the current period flow payoff). The right side of (25) contains two types

of marginal benefit from waiting. The first term reflects the option value of waiting as in (22).13

Unlike the full information model, risk aversion impacts the option value. From the classical

utility theory, we know that the certainty equivalent η (x̄∗; γ) decreases in risk aversion γ (Mas-

Colell et al. (1995)). Thus, risk aversion lowers the first term on the right side of (25), the

option value of waiting, ceteris paribus. The second term on the right side of (25) is analogous

to the term on the right side of (13). It reflects information rents saved by the owner from

waiting. One can show that this information rents term is positive. These information rents

represent agency costs due to asymmetric information. The owner rationally incorporates both

the option value of waiting and managerial information rents in designing the optimal contract.

Consequently, the optimal investment rule trades off the cost from waiting against the option

value of waiting plus saved agency costs. Importantly, risk aversion affects both components of

the benefit.

As a reference for comparison, we first consider the special case in which the manager is risk

neutral. When γ = 0, the owner does need to take risk sharing motive into account when he

designs the contract. Therefore, even in the presence of asymmetric information, the first-best

efficient investment decision is achieved14. When the manager is risk averse, investment timing

decisions reflect both the owner’s motives to meet the manager’s incentive constraint and to

offer consumption smoothing insurance. We may show that

Proposition 4 (i) Holding investment trigger x̄∗ fixed, the option value β1−β

η (x̄∗; γ) is de-

creasing in γ and the agency cost term

[1 − F (x̄∗) + η′ (x̄∗; γ)] [1 − βF (x̄∗)]

(1 − β) f (x̄∗)(31)

13We call this term the option value component because it vanishes in the static model in (13).14Mathematically, the certainty equivalent η is simply given by η(x̄∗; 0) =

∫ b

x̄(x − x̄) dF (x) and the derivative

of η with respect to the investment trigger is given by η′(x̄∗; 0) = − (1 − F (x̄∗)). Thus, the second term in(25) vanishes and the investment trigger under risk neutrality is the same as the value maximizing trigger underperfect information.

17

is increasing in γ. (ii) If E [x − y|x ≥ y] < 1−βF (y)βf(y) for all y ∈ [a, b] , then there is underinvest-

ment under asymmetric information when risk aversion is high enough; that is, x̄∗ > xFB for

γ large enough.

Part (i) shows that risk aversion has two opposing effects. It lowers the option value of

waiting, thereby speeding up investment. However, it also increases agency costs, thereby

delaying investment. The overall impact of risk aversion and asymmetric information depends

on which effect dominates. Part (ii) demonstrates that when risk aversion is high enough, the

agency cost effect may dominate. This happens when a condition on distribution F is satisfied.

One can verify that this condition is satisfied for exponential and uniform distributions.

For general distributions and risk aversion parameter values, we cannot analytically charac-

terize the effect of risk aversion and asymmetric information on investment timing. We thus turn

to numerical analysis. We choose three specifications for the distribution F : normal distribu-

tion with mean µ and variance σ2, uniform distribution over [0, b], and exponential distribution

F (x) = 1− e−αx over [0,∞). Throughout numerical calculations in this section, we choose the

subjective discount factor β = 0.98 and fix the investment cost at unity (I = 1).

[Insert Figures 2-4 here.]

Figure 2 graphs both the left side and the right side of (25), for a standard normal distri-

bution. The left side (x − (1 − β)I) measures the net benefit from investment. The right side

captures both the option value of waiting and agency costs. We plot the right side of (25) for

various levels of risk aversion coefficient γ. The intersection in Figure 2 gives the investment

trigger. Figure 2 shows that the investment trigger increases with risk aversion coefficient γ.

This is because the agency cost effect dominates the option value of waiting. Similarly, Figures

3 and 4 graph the left side and the right side of (25) for exponential distribution and uniform

distribution, respectively. Both figures also show that investment trigger increases with risk

aversion.

While there are two opposing effects of risk aversion on investment timing, our extensive

numerical exercises show that the agency cost effect dominates the option effect. As a result,

the investment trigger increases in the degree of risk aversion. We next turn to the effects of

volatility on the investment timing decision. Consider normal distribution with mean zero and

18

variance σ2. Figure 5 plots the investment trigger as a function of the volatility parameter σ

and the risk aversion coefficient γ. It reveals that a higher volatility leads to a larger investment

trigger for any value of risk aversion parameter. Moreover, the impact of volatility on investment

is larger for more risk averse managers. This is because our numerical simulations indicate

that both the agency cost component and the option value component increase with volatility,

reinforcing each other and leading to greater incentives to wait. This effect is stronger for more

risk averse managers.

[Insert Figure 5 here.]

4.2 Wage Dynamics

Unlike the static model described in Section 2, in a dynamic setting the owner can choose

an intertemporal incentive scheme to punish or reward the manger such that the managerial

incentives are aligned with the owner. We now describe the wage dynamics implied by the

optimal contract.

Let the initial promised utility to the manager be v0 < 0 in period 0 (the utility function

is negative exponential). According to the optimal contract described in Proposition 3, when

the manager reports truthfully the initial value of cash flows x0 ≥ x̄∗, the manager makes

the investment and hands over x0 to the owner. It follows from (27) that the owner pays the

manager compensation yA0 (x0) = x0 − x̄∗ + yB

0 , which is higher than yB0 , the compensation if

the manager does not invest. Notice that managerial compensation yA0 depends on the project

cash flow x0. The manager who invests in a higher cash-flow project captures more information

rents, ceteris paribus. The owner also promises the manager with a continuation utility value

wA0 (x0) = v0 exp (−γ (x0 − x̄∗ − η (x̄∗; γ))) = wB

0 e−γ(x0−x̄∗) , x0 ≥ x . (32)

It is immediate to note that wA0 is higher than wB

0 . Thus, the owner rewards the manager for

his investment both in terms of current compensation and his future value (via continuation

utility). In period 1, the contract starts with an initial utility level v1 = wA0 . In all periods t ≥ 1,

the owner offers a completely flat consumption profile to the manager in that yAt (x0) = yA

0 (x0) .

When the manager reports truthfully a low value of cash flows in period 0, x0 < x̄∗, the

manger will rationally choose not to make investment according to the optimal contract. The

owner pays the manager a constant compensation in the amount of yB0 = yFB(v0) − η (x̄∗; γ) ,

19

which is lower than yFB(v0), the compensation under full information to yield the manager

with utility level v0. Moreover, to induce the manager to stay in the long-term contract, the

owner also promises the manager a continuation utility given by wB0 = v0 exp (γη (x̄∗; γ)) . Since

the initial utility v0 < 0, it is immediate to see that wB0 < v0. That is, upon a no-investment

decision, the owner punishes the manager by lowering not only his current compensation but

also his continuation utility. In period 1, the contract repeats with the manager’s initial utility

level v1 = wB0 , which is promised by the owner to the manager in period 0. This reflects the

recursive nature of the optimal contract: The promised utility in period 0 is the manager’s

starting utility level at the beginning of period 1.

We next turn to period 1 when there is no investment in period 0. If the manager reports

truthfully a value of cash flows x1 < x∗ in period 1, the optimal contract instructs the manager

not to make investment. By (26), the owner pays the manager

yB1 =

− log (−v1γ (1 − β))

γ−η (x̄∗; γ) =

− log(−wB

0 γ (1 − β))

γ−η (x̄∗; γ) = yB

0 −η (x̄∗; γ) , (33)

which is lower than yB0 . Moreover, the owner promises a continuation value to the manager

wB1 = v1 exp (γη (x̄∗; γ)) = wB

0 exp (γη (x̄∗; γ)) , (34)

which is lower than wB0 . Thus, the owner punishes the manager by lowering both managerial

compensation and continuation utility over time during the period when the manger does not

make investment.

If there is no investment in period 0 and the manager reports truthfully a value of cash

flows x1 ≥ x∗ in period 1, then the optimal contract instructs the manager to make investment.

The owner pays the manager yA1 (x1) = x1 − x̄∗ + yB

1 , and promises him utility

wA1 (x1) = v1 exp (−γ (x1 − x − η (x̄∗; γ))) = wB

0 exp (−γ (x1 − x − η (x̄∗; γ))) . (35)

Thus, the owner gives a current period reward x1 − x̄∗ and promises a higher continuation

utility to the manager if he makes the investment. In all periods t ≥ 2, the owner offers a fat

compensation yAt (x1) = yA

1 (x1) .

While we have extensively discussed the contract dynamics for periods 0 and 1, the pre-

ceding analysis applies for any period. We now summarize the dynamics of compensation and

continuation utility. Let T be the first time at which the manager invests. Then, for all t < T ,

20

no investment is made. By (26) and (28), the compensation and continuation value evolve

according to

yBt = yB

t−1 − η (x̄∗; γ) , wBt = wB

t−1 exp (γη (x̄∗; γ)) , (36)

for all t ≤ T. At time T , xT ≥ x̄∗, the investment is made. By (27) and (29), the compensation

and continuation utility are given by

yAT (xT ) = xT − x̄∗ + yB

T , wAT (xT ) = wB

T exp (−γ (xT − x̄∗ − η (x̄∗; γ))) . (37)

After period T , the manager receives the same level of compensation yAT (xT ) in every period

thereafter. The owner makes a constant stream of profits xT in each period thereafter.

[Insert Figure 6 here]

Now we graph the implied contract dynamics. Figure 6 plots a sample path of dynamic

evolution of managerial compensation and managerial continuation utility, when the underlying

distribution F (x) is uniform on [0,1]. Suppose that xt < x̄∗, for 0 ≤ t ≤ 4 and x5 > x̄∗. Then,

the manager finds optimal to invest in period 5. We have shown that before investment,

compensation decreases by an amount of certainty equivalent η each period and the promised

utility wBt decreases by a factor of (eγη − 1) each period. The left side panel plots the scenario

under which the manager draws a very high cash flow project x5 in period 5. His compensation

yAt after investment (for t ≥ 5) is higher than the full-information compensation level yFB(v0).

The right side panel plots the scenario under which the manager draws a sufficiently high cash

flow project in period 5 to justify his investment decision. However, the cash flow in period

5 is not high enough to give the manager a compensation higher than the full-information

compensation level yFB(v0).

The certainty equivalent η describes how the owner uses the risk averse manager’s con-

sumption smoothing motive to induce managerial truth-telling in an intertemporal setting.

The standard consumer theory tells us that the certainty equivalent for any lottery decreases in

managerial risk aversion.15 Thus, the owner shall offer a flatter consumption path to the more

risk averse manager by decreasing managerial compensation at a slower pace in each period.

Our model also has implications for the dynamics of firm value. Since the owner does not

earn any profits and pays decreasing wages before investment, firm value is increasing over time

15In our model, the lottery offers the payoff max {x − x̄, 0}, for given x̄.

21

before investment. Upon investing, the owner starts to obtain a flat stream of project cash

flows. Thus, firm value jumps to a higher level and remains at the highest level forever.

In our model, commitments in period 0 from both the owner and the manager’s sides are

crucial. Neither side can walk away from the contract after signing it at time 0. The optimal

contract implies that the owner rewards the manager with larger compensation when he invests

in high cash flow projects, and punishes the manager who does not invest by lowering his

compensation over time. One implication of this contract is that a manager who is sufficiently

unlucky for many periods and thus has not invested after many periods will have a decreasing

wage over time. This result to a large extent derives from the manager’s inability to walk away

from the contract, ex post. It is worth generalizing our model to a setting in which the manager

is protected by the limited liability condition. For example, the legal institution may require

that compensation to be higher than certain minimum wage. We leave this extension for future

research.

4.3 Comparison with the NPV rule

We have derived the investment timing decision as part of the optimal contract. One implication

is that information asymmetry delays investment. However, in reality, it is sometimes difficult

to contract on the manager’s investment decisions. To quantify the loss of firm value when

the owner cannot contract with the manager on his investment decision, we need a comparison

model with investment rules specified. The most widely taught investment rule is the NPV rule.

Obviously, the NPV rule in our setting ignores both the option value of waiting and conflicts of

interests between the owner and the manager. However, it seems natural as a benchmark for

our comparison. We now quantify the utility cost of adopting the NPV rule.

According to the NPV rule, investment is made if the cash flow x is larger than the threshold

xm ≡ (1 − β) I. It is straightforward to show that other results of Proposition 3 on compensation

and continuation utility still hold, by replacing the optimal investment threshold x̄∗ with the

trigger rule xm. When the manager chooses the NPV rule over the optimal investment rule,

firm value is lowered since it not optimal.

[Insert Figure 7 here.]

Figure 7 plots the firm value reduction when the NPV rule is adopted. We choose the

22

underlying distribution to be normal with mean 0. The upper and lower graphs plot the value

reduction as a function of coefficient of risk aversion γ, when standard deviation σ = .5 and

σ = 1, respectively. We see that firm value reduction increases with γ. This is because we have

shown earlier that the more risk averse the manager is, the later he invests according to the

optimal contract. Consequently, the simple NPV rule deviates farther away from the optimal

investment rule. Thus, following the NPV rule is more costly to the owner.

4.4 Extensions

We now consider two extension of our model.

Allowing the Manager to Borrow and Save. As in other dynamic principal-agent models,

we have assumed that the owner directly controls the consumption of the manager. That is, the

manager is not allowed to save and borrow at the interest rate R = β−1. However, restricting

the manager from borrowing and lending seems restrictive. We now show that the optimal

contract derived in our model remains optimal after allowing the manager to borrow and lend.

We only need to verify that the manager’s consumption Euler equation is satisfied, in that

u′ (ct) = βREt [u′ (ct+1)] , where Et [·] is the conditional expectation given information available

at period t. This Euler equation after investment is obviously satisfied, because managerial

compensation is constant over time. To show that the preceding Euler equation is also satisfied

before investment, one can use Proposition 3 to verify that16

u′(yB

t

)=

∫ x∗

a

u′(yB

t+1

)dF (xt+1) +

∫ b

x∗

u′(yA

t+1

)dF (xt+1) . (38)

The consumption Euler equation confirms that the dynamic optimal contract described in

Proposition 3 remains optimal, even when the manager can borrow and save.17

16We only need to show that

e−γyB

=

∫ x∗

a

e−γ(yB−η)dF +

∫ b

x∗

e−γ(x−x̄∗+yB−η)dF (x) .

Tis is equivalent to

e−γη = F (x̄∗) +

∫ b

x∗

e−γ(x−x̄∗)dF (x) .

This equation is true by the definition of the certainty equivalent η.17However, in other related settings, the agent may be savings constrained (Rogerson (1985)). That is, if the

agent could save part of the compensation without knowledge of the principal, he would choose to do so. Thatis often reflected in a violation of the Euler equation.

23

Fluctuating Cash flows After Investment. So far, we have assumed that once the man-

ager makes the investment, the value of cash flows is locked in thereafter even though the owner

does not know this value. We now extend our model by considering the scenario under which

the value of cash flows after investment can fluctuate over time. That is, the cash flow in each

period after investment is stochastic and drawn identically and independently from the distri-

bution F . The manager observes the cash flows, but the owner does not. We briefly sketch the

setup. A complete analysis is beyond the scope of the present paper. The key difference between

the problem here and the one analyzed earlier is the contracting problem after investment.

We solve the problem backward. After investment, the optimal contract solves the following

problem18

Π (v) = maxz,w

∫ b

a

[x − z (x) + βΠ (w (x))] dF (x), (39)

subject to

v =

∫ b

a

[u (z (x)) + βw (x)] dF (x), (40)

u (z (x)) + βw (x) ≥ u (z (x̃) + x − x̃) + βw (x̃) , ∀ x̃ . (41)

where (40) is the promise-keeping constraint and (41) is the incentive constraint. We now

consider the optimal contracting problem before investment. It is described by the following

program:

P (v) = max(yB ,wB ,yA,wA,x̄)

∫ x̄

a

[−yB (x) + βP

(wB (x)

)]dF (x) (42)

+

∫ b

x̄

[x − yA (x) − I + βΠ

(wA (x)

)]dF (x)

subject to

v =

∫ x̄

a

[u

(yB (x)

)+ βwB (x)

]dF (x) +

∫ b

x̄

[u

(yA (x)

)+ βwA (x)

]dF (x) (43)

and

u(yA (x)

)+ βwA (x) ≥ u

(yA (x̃) + x − x̃

)+ βwA (x̃) , x, x̃ ≥ x̄, (44)

u(yA (x)

)+ βwA (x) ≥ u

(yB (x̃)

)+ βwB (x̃) , x ≥ x̄ > x̃, (45)

u(yB (x)

)+ βwB (x) ≥ u

(yA (x̃) + x − x̃

)+ βwA (x̃) , x̃ ≥ x̄ > x, (46)

u(yB (x)

)+ βwB (x) ≥ u

(yB (x̃)

)+ βwB (x̃) , x̄ > x, x̃. (47)

18This problem is similar to Thomas and Worrall (1990).

24

Here wA (x) is the continuation value promised by the owner after the manager makes the

investment at reported value x.

5 Conclusion

This paper extends the real options framework to account for informational asymmetry that is

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

agers, contracts must be designed to provide incentives for the manager to reveal his private

information. The owner designs the optimal dynamic contract by both awarding information

rents to the manager and at the same time providing consumption insurance to the manager.

We apply the recursive contract approach to solve this dynamic contracting problem.

We show that compared to the full information benchmark, investment is delayed or there

is underinvestment under asymmetric information. We also show that the optimal contract

punishes the manager over time if he does not make the investment, and rewards him when

he makes the investment. We also show that the option value of waiting decreases with risk

aversion and information rents increases with risk aversion. Moreover, our extensive numerical

simulations indicate that the incentive effect (via information rents) dominates the option effect,

and thus investment is delayed longer when the manager is more risk averse. We also show that

agency costs influence the investment-uncertainty relationship, which has not been emphasized

in the real options literature. In particular, the impact of uncertainty on investment is larger

for more risk averse managers. Finally, we show that the firm value reduction from following

the simple NPV rule increases with managerial risk aversion.

Since our model provides the first recursive analysis of dynamic incentive problems in a

real options model, we aim to highlight the mechanisms by using the simplest possible setting.

This leads us to make some simplifying assumptions. For example, we assume CARA utility.

We also assume that the manager cannot walk away from the contract after signing it at time

0. However, in most employment relationships, the manager has the option to quit his job by

simply forgoing his compensation. Incorporating such limited commitment or limited liability

conditions for the manager makes the owner’s ability to punish the manager for not investing

rather limited.

25

Appendix

A Proofs

Proof of Lemma 1: Since u is increasing, (6) implies that yB(x) ≥ yB(x̃) and yB(x̃) ≥

yB(x) for x, x̃ < x̄. The joint restriction of the two equalities immediately gives a constant

compensation, denoted by yB. Similarly, incentive constraint (3) implies that yA(x) ≥ yA(x̃) +

x − x̃ and yA(x̃) ≥ yA(x) + x̃ − x for x, x̃ ≥ x̄. The joint restriction of the two equalities

immediately gives a constant dividend dA(x̃) = dA(x) = dA, using the definition of dividend

dA(x) = x−yA(x). The incentive constraint (5) implies that yB ≥ yA (x)+x−x. The incentive

constraint (4) implies that yA (x) ≥ yB. Combining these two inequalities and letting x → x,

we obtain yB = yA (x) = x − dA. Q.E.D.

Proof of Proposition 1: We form a Lagrangian as follows:

L = −(x̄ − dA

)F (x̄) +

(dA − I

)(1 − F (x̄)) (A.1)

−λ

[v −

(u

(x̄ − dA

)F (x̄) +

∫ b

x̄

u(x − dA

)dF (x)

)]

= −x̄F (x̄) + dA − I (1 − F (x̄)) − λ

[v −

(u

(x̄ − dA

)F (x̄) +

∫ b

x̄

u(x − dA

)dF (x)

)].

The first-order conditions are

∂L

∂dA= 1 − λ

[u′

(x̄ − dA

)F (x̄) +

∫ b

x̄

u′(x − dA

)dF (x)

]= 0 , (A.2)

∂L

∂x̄= −F (x̄) − (x̄ − I) f (x̄) + λu′

(x̄ − dA

)F (x̄) = 0 . (A.3)

Eliminating the Lagrangian multiplier gives the formula reported in Proposition 1. Q.E.D.

Proof of Corollary 1: It is straightforward to derive it from Proposition 1. It is also a

special case of Proposition 3 when β = 0. Q.E.D.

Proof of Proposition 2: We conjecture that the value function P (v) takes the following

form

P (v) = −u−1 ((1 − β) v)

1 − β+ H, (A.4)

26

where H is a constant to be determined. Substituting it into (16) and using the Lagrange

method, one can derive the desired result and

H =

∫ b

xFB (x − (1 − β) I) dF (x)

(1 − β) [1 − βF (xFB)]. (A.5)

We next show that xFB is the same as the investment trigger for a risk neutral investor. To

this end, let V (x) be the value function of the risk neutral investor. Then V (x) satisfies the

Bellman equation

V (x) = max

{x

1 − β− I, β

∫V

(x′

)dF

(x′

)}. (A.6)

Since V (x) is increasing, there is a trigger value x∗ such that when x < x∗, there is no invest-

ment, and when x ≥ x∗, investment is undertaken. That is

V (x) =

{x

1−β− I if x ≥ x∗,

β∫ b

aV (x′) dF (x′) if x < x∗.

(A.7)

By continuity of V (x) at the threshold value x∗, we have

x∗

1 − β− I = β

∫ b

a

V(x′

)dF

(x′

)= β

∫ x∗

a

V(x′

)dF

(x′

)+ β

∫ b

x∗

V(x′

)dF

(x′

)(A.8)

= β

∫ x∗

a

(x∗

1 − β− I

)dF

(x′

)+ β

∫ b

x∗

(x′

1 − β− I

)dF

(x′

).

Subtracting β∫ b

x∗

(x∗

1−β− I

)dF (x′) on each side of the above equation and simplifying yield

(22). To prove the existence of uniqueness, we use the fact that the left-side of (22) is an

increasing function of x∗ and the right-side of (22) is a decreasing function of x∗. Using the

assumption I ∈ (a/ (1 − β) , b/ (1 − β)) , we apply the intermediate value theorem to obtain the

desired result. Q.E.D.

Proof of Lemma 2: (i) The incentive constraint (21) implies that u(yB (x)

)+ βwB (x) is

constant for all x < x̄. The incentive constraint (20) implies that

u(yB (x)

)+ βwB (x) ≥

∞∑

n=0

βnu(yA

n (x̄) + x − x̄). (A.9)

But (19) implies that∞∑

n=0

βnu(yA

n (x̄))≥ u

(yB (x)

)+ βwB (x) . (A.10)

27

Combining the above two inequalities and letting x → x, it follows from the continuity of u

that u(yB (x)

)+ βwB (x) is continuous at x̄ and equal to the value at x̄ for all x < x̄. Thus,

we obtain (24).

We now show that given (18), condition (24) implies (19)-(21). First, (21) is trivially true.

Use (18) and (24) to deduce

u(yB (x)

)+ βwB (x) =

∞∑

n=0

βnu(yA

n (x̄))≥

∞∑

n=0

βnu(yA

n (x̃) + x − x̃)

(A.11)

≥∞∑

n=0

βnu(yA

n (x̃) + x − x̃)

for all x̃ ≥ x > x,

which gives (20). Finally, use (18) and (24) to deduce

∞∑

n=0

βnu(yA

n (x))

≥∞∑

n=0

βnu(yA

n (x̄) + x − x̄)≥

∞∑

n=0

βnu(yA

n (x̄))

(A.12)

= u(yB (x)

)+ βwB (x) for all x ≥ x > x̃,

which gives (19).

(ii) Substitute dAn (x) = x − yA

n (x) into (18). By the CARA utility specification, x can be

cancelled out from (18). We can then deduce that∑∞

n=0 βn −eγdn(x)

γis independent of x. Denote

this constant value by W. We can rewrite the contract problem (16)-(21) as

P (v) = maxy,z,w,x̄

∫ x̄

a

[−yB (x) + βP

(wB (x)

)]dF (x) +

∫ b

x̄

[∞∑

n=0

dAn (x) − I

]dF (x) (A.13)

subject to

v =

∫ x̄

a

[u

(yB (x)

)+ βwB (x)

]dF (x) + W

∫ b

x̄

e−γxdF (x) , (A.14)

u(yB (x)

)+ βwB (x) = We−γx̄, x < x̄, (A.15)

W =∞∑

n=0

βn−eγdAn (x)

γfor all x ≥ x̄. (A.16)

If P (·) is differentiable and strictly concave, one can use the Lagrange method to show that

dAn (x) is independent of x and n, and that yB (x) and wB (x) are independent of x for x < x.

Q.E.D.

28

Proof of Proposition 3: (i) We conjecture that the value function P takes the following

form

P (v) =log (−v)

γ (1 − β)+ K, (A.17)

where K is a constant to be determined.19 We use Lemma 2 to simplify the optimal contracting

problem as

P (v) = maxyB ,dA,wB ,x̄

[−yB + βP

(wB

)]F (x̄) +

(dA

1 − β− I

)(1 − F (x̄)) (A.18)

subject to

v =

[−e−γyB

γ+ βwB

]F (x̄) +

∫ b

x̄

−e−γ(x−dA)

γ (1 − β)dF (x) , (A.19)

−e−γyB

γ+ βwB =

−e−γ(x̄−dA)

γ (1 − β). (A.20)

Substitute (A.17) into (A.18) and use the first-order conditions for yB and wB to obtain

wB =−e−γyB

γ (1 − β). (A.21)

Substituting this expression into (A.20) to solve for dA, we obtain

dA = x̄ − yB. (A.22)

Substituting (A.17), (A.21) and (A.22) into (A.18) and (A.19), we can simplify the problem to

P (v) = maxyB ,x̄

[−

yB

1 − β−

β log [γ (1 − β)]

γ (1 − β)+ βK

]F (x̄) +

(x̄ − yB

1 − β− I

)(1 − F (x̄)) (A.23)

subject to

v =−e−γyB

γ (1 − β)

[F (x̄) +

∫ b

x̄

e−γ(x−x̄)dF (x)

]. (A.24)

We use (A.24) to solve for yB to obtain

yB =− log (−v)

γ+

− log γ (1 − β)

γ+

1

γlog

{F (x̄) +

∫ b

x̄

e−γ(x−x̄)dF (x)

}. (A.25)

Substitute this expression into the right-side of (A.23) and use the definition (12) to obtain

P (v) = maxx̄

log (−v)

γ (1 − β)+

log (γ (1 − β))

γ (1 − β)+

η (x̄; γ)

1 − β(A.26)

+

[−

β log [γ (1 − β)]

γ (1 − β)+ βK

]F (x̄) +

(x̄

1 − β− I

)(1 − F (x̄)) .

19By a standard dynamic programming argument (e.g., Thomas and Worrall (1990)), one can show that thereis a unique value function P (·) solving the contract problem.

29

This verifies our conjectured functional form of P (v) in (A.17). Moreover, matching equations

(A.26) and (A.17) gives an equation for K,

K = maxx̄

log (γ (1 − β))

γ (1 − β)+

η (x̄; γ)

1 − β(A.27)

+

[−β log [γ (1 − β)]

γ (1 − β)+ βK

]F (x̄) +

x̄ − (1 − β) I

1 − β(1 − F (x̄)) .

By the first-order condition for (A.26), we have

η′ (x̄; γ)

1 − β+

[−β log [γ (1 − β)]

γ (1 − β)+ βK

]f (x̄) +

1 − F (x̄)

1 − β−

x̄ − (1 − β) I

1 − βf (x̄) = 0 (A.28)

Evaluated at the optimum, equation (A.27) implies that

K =log [γ (1 − β)]

γ (1 − β)+

(x̄ − I (1 − β)) (1 − F (x̄)) + η (x̄; γ)

(1 − βF (x̄)) (1 − β). (A.29)

Substitute this expression into the preceding first-order condition to obtain that the optimal

trigger satisfies

x̄ = (1 − β) I +1 − F (x̄) + η′ (x̄; γ)

f (x̄)+

β (x̄ − I (1 − β)) (1 − F (x̄)) + βη (x̄; γ)

1 − βF (x̄). (A.30)

Multiplying 1 − βF (x̄) on each side and simplifying yield (25).

We next check the second-order condition for an optimum is satisfied. Taking a second-order

derivative in (A.26) gives

η′′ (x̄; γ)

1 − β+

[−β log [γ (1 − β)]

γ (1 − β)+ βK

]f ′ (x̄) −

2f (x̄)

1 − β−

x̄ − (1 − β) I

1 − βf ′ (x̄) . (A.31)

We will show that this expression is negative at x̄∗. We first substitute the preceding expression

for K into it and multiply (1 − β) to obtain

η′′ (x̄∗; γ) − 2f (x̄∗) +βf ′ (x̄∗) η (x̄∗; γ)

1 − βF (x∗)− [x̄∗ − (1 − β) I]

f ′ (x̄∗) (1 − β)

1 − βF (x∗). (A.32)

We now use equation (25) to substitute [x̄∗ − (1 − β) I] . Simplifying the resulting expression

yields

η′′ (x̄∗; γ) − 2f (x̄∗) −f ′ (x̄∗) (1 − F (x∗) + η′ (x̄∗; γ))

f (x̄∗). (A.33)

It suffice to show that this expression is negative. In fact, by the assumption that

1 − F (x) + η′ (x; γ)

f (x)(A.34)

30

is decreasing in x, we differentiate this expression to derive

f ′ (x̄∗) (1 − F (x∗) + η′ (x̄∗; γ))

f (x̄∗)> η′′ (x̄∗; γ) − f (x̄∗) . (A.35)

This inequality delivers our desired result.

We finally show that the existence and uniqueness of the solution to (25). Since η (x; γ) is

decreasing in x, it follows from the assumption that the right side of (25) is decreasing in x̄∗.

One can also check the right side is equal to zero when x̄∗ = b and is positive when x̄∗ = a. The

desired result follows from the intermediate value theorem.

(ii) Equation (26) follows from (A.25) and the definition of η. Since yB = x−dA, (27) follows

from (A.22).

(iii) Equation (28) follows from (A.21) and part (ii). Equation (29) follows from Lemma 3

and (A.22).

(iv) It follows from (A.17) and (A.27). Q.E.D.

Proof of Proposition 4: (i) First, η (x̄; γ) is decreasing in γ by Mas-Colell et al. (1995).

Moreover, one can check that

η′ (x̄, γ) =−

∫ b

x̄e−γ(x−x̄)dF (x)

F (x̄) +∫ b

x̄e−γ(x−x̄)dF (x)

(A.36)

is increasing in γ.

(ii) When γ → ∞, η (x̄; γ) → 0 and η′ (x̄, γ) → 0, and thus the investment trigger satisfies

x̄∗ − (1 − β) I =[1 − F (x̄∗)] [1 − βF (x̄∗)]

(1 − β) f (x̄∗). (A.37)

When γ = 0, the investment trigger xFB satisfies (22). If the condition in the proposition is

satisfied, then the expression on the right side of the preceding equation is larger than that

on the right side of equation (22), when x̄∗ and xFB are replaced with y. This implies that

x̄∗ > xFB for γ = ∞. By continuity, this is also true for γ large enough. Q.E.D.

31

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34

00.5

0.55

0.6

0.65

0.7

0.75

1 7 92 843 5 6 10

risk aversion

Inves

tmen

tth

resh

old

Figure 1: Investment threshold and risk aversion in the static setting. Thisfigure plots the investment threshold at varying levels of risk aversion for CRRA utility(u(c) = c1−γ/ (1 − γ)

)and uniform distribution over [0, 1]. Set parameter value I = 0.5 and

reservation utility v = 0.11−γ/(1 − γ).

35

2

20

4

6

the value of x

x − (1 − β)I

γ = 0

γ = ∞

Figure 2: Investment threshold and risk aversion in the optimal dynamic contract.Utility is CARA (u(c) = −e−γc/γ) and F is the standard normal. Set β = 0.98 and I = 1.From the lowest to the highest graphs, the manager’s coefficient γ of risk aversion is 0 (riskneutral), 1, 5, and ∞, respectively.

36

1

10.75 0.85 0.950.8 0.9

0.5

1.5

2.5

3.5

2

0

3

the value of x

Figure 3: Investment threshold and risk aversion in the optimal dynamic contract.Utility is CARA (u(c) = −e−γc/γ) and F is the uniform distribution over [0, 1]. Set β = 0.98and I = 1. From the lowest to the highest graphs, the risk aversion parameter γ is 0 (riskneutral), 1, 5, and ∞, respectively.

37

4.5 5.5 6.552

4

4

6

6

8

10

12

14

16

7

the value of x

Figure 4: Investment threshold and risk aversion in the optimal dynamic contract.Utility is CARA (u(c) = −e−γc/γ) and F is an exponential distribution over [0,∞). Set β = 0.98and I = 1. From the lowest to the highest graphs, the risk aversion parameter γ is 0 (riskneutral), 1, 5, and ∞, respectively.

38

1

1

1

3

1.5

1.5

1.5

2.5

0.5

0.50.5

3.5

Inves

tmen

tth

resh

old

γσ

2

2

2

0

0 0

Figure 5: Investment Threshold, volatility, and risk aversion. This figure plots invest-ment threshold as a function of volatility and risk aversion parameters for normal distributionswith mean zero and variance σ2. Utility is CARA (u(c) = −e−γc/γ). Set β = 0.98 and I = 1.

39

-3-3

-4-4

-2.5-2.5

-3.5-3.5

0.50.5

0.520.52

0.540.54

0.560.56

0.580.58

0.60.6

22

22

44

44

66

66

88

88

1010

1010

00

00

manager

ialutility

com

pen

sation

Figure 6: The dynamics of managerial compensation, and continuation utility. Thetwo graphs on the left (right) panel plot the dynamics for a high (low) value of cash flows uponinvestment. Utility is CARA (u(c) = −e−γc/γ) and F is the uniform distribution over [0, 1].Set β = 0.98 and I = 1.

40

1

1

35

45

55

50

3

3

2

2

40

17

18

19

20

21

22

23

0

0

σ = .5

σ = 1

firm

valu

ere

duct

ion

∆P

firm

valu

ere

duct

ion

∆P

risk aversion

risk aversion

Figure 7: Firm value reduction from using the NPV rule. Utility is CARA(u(c) = −e−γc/γ) and F is a normal distribution with mean zero. Set β = 0.98 and I = 1.

41