GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.165

The effects of costly exploration on optimal investment timing

Michi Nishihara and Takashi Shibata

Novmber 2010

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

The eﬀects of costly exploration on optimal investment
timing∗Michi NISHIHARA†, Takashi SHIBATA‡§

Abstract
This paper investigates a principal-agent model in which an owner (principal) optimizes a contract with a manager (agent) delegated to undertake an investment project.
In the model, we explore the eﬀects of costly exploration by which the manager learns thereal value of development cost. We show that high exploration cost can lead to a poolingpolicy not contingent on project type. Further, and more notably, we show that, in thepresence of asymmetric information, higher exploration cost leads to wealth transfer fromowner to manager and can then play a positive role in preventing a greedy contract by
the owner and improving social welfare.
JEL Classifications Code: D86, G13, G31.
Keywords: Real Options; Asymmetric Information; Costly Learning; Sequential Investment; Incentive Theory∗

The ﬁrst version: 23, September, 2010. This research was supported by KAKENHI (22710146, 22710142),

a Grand-in-Aid from the Excellent Young Researcher Overseas Visit Program by JSPS (21-2171), and theTelecommunication Advancement Foundation.
†
Corresponding Author. Graduate School of Economics, Osaka University, Osaka 560-0043, Japan,
nishihara@econ.osaka-u.ac.jp‡
Graduate School of Social Sciences, Tokyo Metropolitan University, Tokyo 192-0397, Japan, tshibata@tmu.ac.jp§
Statistical Laboratory, University of Cambridge11

Introduction

The real options approach has become an increasingly standard framework for investment
timing decisions in corporate ﬁnance (e.g., (Dixit and Pindyck 1994)). Although the early
literature on real options (e.g., (Dixit 1989, McDonald and Siegel 1986)) considers investment
decisions by a monopolist, more recent studies have investigated the problem of several ﬁrms
competing in the same market from a game theoretic approach. For instance, (Grenadier
1996, Weeds 2002) derive equilibrium in real options timing games, and (Grenadier 2002)
investigates equilibrium investment strategies of ﬁrms in a Cournot–Nash framework.
While these studies have focused on strategic interactions among rival ﬁrms, (Grenadierand Wang 2005) investigated investment timing in a decentralized ﬁrm where the owner (principal) delegates the investment decision to a manager (agent) who holds private information by
combining real options theory and incentive theory. In most modern corporations, for example,
shareholders delegate investment decisions to managers, thereby taking advantage of their special skills and expertise. In their model, asymmetric information changes the investment policy
from the ﬁrst-best case because the owner designs a contract to provide a bonus-incentive forthe manager to truthfully reveal private information.1
Previous studies assume that a manager observes real project value with no costs immediately after a contract is made; however, in most real investing, a manager gathers information
about the type of project by exploratory investment which has a high cost.2 This exploration
cost is especially high in the following cases. The ﬁrst example is a resource extraction project,which has been the main application of the real options analysis (e.g., (Paddock, Siegel, andSmith 1988) among others). The exploration stage involves seismic and drilling activity to obtain information on the quantities of natural resource reserves, as well as the costs of extracting
them. It depends on the exploration results whether the manager proceeds to development
investment such as construction of platforms and drilling of production wells.
Another example is an R&D program with learning (e.g., (Childs, Ott, and Triantis 1998,
Perlitz, Peske, and Schrank 1999)). An R&D project frequently comprises multiple phases.
1(Nishihara and Shibata 2008, Shibata 2009) extend (Grenadier and Wang 2005) to a case with an audit

mechanism. (Shibata and Nishihara 2010) extends (Grenadier and Wang 2005) to a context of the dynamic
investment and capital structure. On the other hand, (Morellec and Schürhoﬀ 2009) investigates a real options
signaling model rather than the screening model. (Grenadier and Malenko 2010) investigates the real options
signaling game in a more general framework.
2
In incentive theory, (Crémer and Khalil 1992, Crémer, Khalil, and Rochet 1998a, Crémer, Khalil, andRochet 1998b, Kessler 1998, Laﬀont and Martimort 2002) investigate how cost of gathering information aﬀects
the optimal incentive mechanism. For example, (Crémer and Khalil 1992, Crémer, Khalil, and Rochet 1998a,Crémer, Khalil, and Rochet 1998b) show that high cost of information leads to a contract which provides no
incentive for an agent to accumulate information.

2

In each phase, the manager gains information on the R&D project, such as the probability ofsuccess, the costs required to complete, and the proﬁts expected from the successfully developed
product. The manager decides whether to proceed to the next phase while taking account of thepreliminary results. In addition, several empirical works (e.g., (Aboody and Lev 2000, Barth,Kasznik, and Mcnichols 2001)) have shown a relatively large information asymmetry associatedwith R&D.
This paper extends the previous real options model with asymmetric information to a settingin which the manager learns project type by costly exploration. We consider a principal-agent
model in which the owner (principal) optimizes a contract with a manager (agent) delegatedto undertake an investment project. The investment consists of exploratory and development
stages. Development cost can take one of two possible values. Only the manager observes a
realized development cost by costly exploration. Speciﬁcally, we investigate how the ratio of
exploration cost to total cost aﬀects the investment policy. The results are summarized asfollows.
When exploration cost is relatively low, the ﬁrm, whether under symmetric or asymmetric
information, adopts a separating policy. The ﬁrm immediately proceeds to development investment for a favorable result in the exploration stage, while it delays the investment with anunfavorable result. If the project turns out to be a bad type, exploration investment leads toineﬃciency ex post. However, because of low cost, the ﬁrm attempts earlier to acquire information on project type. On the other hand, the ﬁrm takes a pooling policy when exploration
cost is relatively high. In this case, the ﬁrm invests in the development phase immediately
after the exploratory phase whether or not the exploration result is favorable, because highcost reduces the incentive to gather information by earlier investment in the exploratory stage.
Under asymmetric information, the contract becomes useless because the owner pays the manager the maximal rent for information. The pooling solution never appears in (Grenadier andWang 2005), where the manager observes project type at no cost. Our result is similar to aﬁnding by (Crémer, Khalil, and Rochet 1998a) in incentive theory. They showed that high costof information acquisition leads to a pooling contract not contingent on project type, although
their model does not treat a dynamic investment timing problem but a static problem.
A key role of asymmetric information is to provide an additional incentive for the ownerto separate investment timing of good- and bad-type projects. Consistent with the previous
studies, the owner can decrease the bonus to the manager by deferring investment in the badtype project. Thus, the ﬁrm chooses a pooling policy under symmetric information but chooses
a separating contract under asymmetric information when exploration cost is intermediate. If
this is the case, the investment timing diﬀers between symmetric and asymmetric information
cases, not only for the bad-type project but also for the good-type project. In addition, unlike in3the previous studies, the contract by which the owner cuts the information rent to the manager
can yield less social welfare than the case with no contract. This implies that wealth transferfrom manager to owner in the contract is not always eﬃcient from the social viewpoint.
Most notably, we show that unlike the case with symmetric information, a higher proportionof exploration cost can increase social welfare in the asymmetric information case. When ahigher proportion of exploration cost and asymmetric information are separately present, bothlead to ineﬃciency; however, combined with asymmetric information, costly learning can playa positive role in mitigating excessive wealth transfer from manager to owner. The intuition
behind the key result is as follows. A higher exploration cost decreases the owner’s value
and increases the manager’s value because the owner ﬁnds it diﬃcult to decrease the bonus
to the manager by a separating contract. Accordingly, high exploration cost leads to wealthtransfer from owner to manager.3 This may mitigate the social loss stemming from the owner’s
greedy contract. If information about exploration cost is also partially unknown to the owner,
the manager pretends to be ignorant of project type in order to increase his/her information
rent. This resembles the argument by (Kessler 1998), who showed that an agent’s ignorance
may generate strategic beneﬁts in a static model. The manager’s moral hazard decreases theowner’s value but does not necessarily reduce social welfare. The manager’s moral hazard
may enhance social welfare by mitigating ineﬃcient asset substitution by the owner’s greedy
contract.
The key result resembles several ﬁndings known as “two incentive problems are better thanone” in corporate ﬁnance (e.g., (Hirshleifer and Thakor 1992, Mookherjee and Png 1995, Noeand Rebello 1996)). Similar results are reported in recent papers in the real options context.
For example, (Hackbarth 2009) showed that managerial optimism and overconﬁdence, which
distort the investment and ﬁnancing policy, can increase welfare in the presence of bondholdershareholder conﬂicts. (Nishihara and Shibata 2010) showed that a ﬁnancing constraint, that
decreases ﬁrm value in a monopoly, can play a positive role in mitigating preemptive competition
and improving ﬁrm value in equilibrium.
This paper contributes to the literature of both investment timing and incentive theory as
follows. First, we complement the real options literature by pointing out the possibility that
costly learning greatly distorts the investment policy, leading especially to a pooling policy. We
also demonstrate the possibility that the owner’s greedy contract yields less social welfare thanthe case with no contract. The main contribution is our demonstration that costly learning canplay a positive role through interactions with asymmetric information. In addition, this paper
complements several works in incentive theory by extending their ﬁndings in the static models
to those of the dynamic investment timing model.
3This is also consistent with an empirical ﬁnding by (Aboody and Lev 2000).

4

The paper is organized as follows. Section 2 introduces the setup and presents the investment
policies under symmetric and asymmetric information. Section 3 investigates social welfare andsocial loss. Section 4 explores more detailed properties of the solutions in numerical examples.
Section 5 concludes the paper. All proofs appear in the appendix.

2

Model solutions

2.1

Setup

Consider an owner (principal) with an option to invest in a single project. We assume that theowner delegates the investment decision to a manager (agent). Throughout our analysis, allagents are assumed to be risk neutral and to maximize their expected payoﬀ. For simplicity,
we assume that project value follows the geometric Brownian motiondX(t) = µX(t)dt + σX(t)dB(t) (t > 0),

X(0) = x,(1)

where µ, σ > 0 and x > 0 are constants, and B(t) denotes the one-dimensional standard
Brownian motion. Throughout the paper, we assume that the initial value x is suﬃciently low
so that the ﬁrm has to wait for its investment condition to be met. For convergence, we assume
that r > µ where r is a constant interest rate.
The investment comprises two phases: exploratory and development investment.

Ex-

ploratory investment requires irreversible cost aI and reveals development cost, which cantake one of two possible values (1 − a)I (good type) or (1 − a)I + ∆I (bad type). The probabilities of drawing (1 − a)I and (1 − a)I + ∆I are equal to q and 1 − q, respectively. Assume
that I > 0, a ∈ [0, 1), and q ∈ (0, 1) are constants. We also assume that project value X(t) and
exploration cost aI is observed by both the owner and the manager while development cost
(1 − a)I or (1 − a)I + ∆I is privately observed only by the manager.4 Total investment cost
becomes I for the good-type project and I + ∆I for the bad-type project.
This model is an extension of the simpliﬁed model of (Grenadier and Wang 2005). Indeed,in the case of a = 0, the model is equal to the hidden information case of (Grenadier and Wang2005).5 As mentioned in Section 1, the setting with a positive a is suitable for real investment
4

The assumption that a portion of the project value is privately observed only by one person (here, themanager) and not observed by the other (here, the owner) is quite common in the asymmetric information
literature (e.g., (Myers and Majluf 1984)). The model is not essentially changed when the privately observable
component corresponds to part of project value, as in (Grenadier and Wang 2005, Nishihara and Shibata 2008),
rather than part of investment cost.
5
Although the manager’s one-time eﬀort, which cannot be observed by the owner, changes the likelihood q
in the original model of (Grenadier and Wang 2005), we exclude the eﬀect of this hidden action.

5

such as a resource extraction and an R&D program. To preserve tractability, we assume thatinvestment takes place instantaneously, but this is not a severe restriction. The conclusions andinsights provided in this paper will be unchanged even if the period of investment is considered.

2.2

Symmetric information

As a benchmark, we investigate the case where there is no delegation of the exercise decisionand the owner observes the true value of development cost by exploratory investment. The
owner’s exploratory investment at τP = inf{t > 0 | X(t) ≥ xP } and development investment
at τi = inf{t > 0 | X(t) ≥ xi }, where i = 1 and 2 denote the investment policies for the goodand bad-type projects, yield the expected proﬁt
E[e−rτP (−aI)] + qE[e−rτ1 (X(τ1 ) − (1 − a)I)] + (1 − q)E[e−rτ2 (X(τ2 ) − ((1 − a)I + ∆I))]
( )β
( )β
( )β
x
x
x
=
(−aI) + q
(x1 − (1 − a)I) + (1 − q)(x2 − ((1 − a)I + ∆I)),
xP
x1
x2
√
where β = 1/2 − µ/σ 2 + (µ/σ 2 − 1/2)2 + 2r/σ 2 (> 1) is a positive characteristic root. Then,the owner’s optimal policy (x∗P , x∗1 , x∗2 ) solves the following problem:(πo∗ (x) :=max

xP ,x1 ,x2

s.t.

x
xP

)β

(

x
x1

)β

(−aI) + q
(x1 − (1 − a)I)( )βx+(1 − q)(x2 − ((1 − a)I + ∆I))x2
xi ≥ xP (i = 1, 2)(2)

Throughout the paper, the superscript ∗ refers to the solution under symmetric information.
Recall that the initial value x = X(0) is suﬃciently low. Clearly, in the optimum of problem(2), x∗P = min(x∗1 , x∗2 ) and x∗1 ≤ x∗2 are satisﬁed. Then, problem (2) can be reduced to
)β
( )β
x
x
(x1 − (a/q + 1 − a)I) + (1 − q)(x2 − ((1 − a)I + ∆I))
max q
x1 ,x2x1
x2(3)
s.t. x2 ≥ x1 .
(πo∗ (x) =The following proposition presents the solution to problem (3).
Proposition 1 The optimal policy (x∗P , x∗1 , x∗2 ) and the owner’s value πo∗ (x) are given as follows:
Case (i-S): a < q∆I/I (separating)x∗P = x∗1 =ββ(a/q + 1 − a)I, x∗2 =((1 − a)I + ∆I)
β−1β−1πo∗ (x) =qxβ(1 − q)xββx1

βx∗β−1
2

+
∗β−1
6

(4)

(5)

Case (ii-S): a ≥ q∆I/I (pooling)x∗P = x∗1 = x∗2 =β(I + (1 − q)∆I)β−1πo∗ (x) =xβ
βx∗β−11(6)(7)

First, we explain Case (i-S). In this case, the owner invests in the exploratory phase when
project value X(t) hits the threshold x∗P . If exploratory investment reveals the good-type
project, the owner immediately proceeds to development investment. Otherwise, the ownerdelays the investment until X(t) hits the higher level x∗2 . When costly exploration is unnecessary, i.e., a = 0 as in (Grenadier and Wang 2005), the owner can choose the best investment
timing, x∗1 = βI/(β − 1) and x∗2 = β(I + ∆I)/(β − 1) for the good- and bad-type projects,
respectively. However, when the exploratory phase requires a portion of investment cost, a > 0,
the optimal investment policy changes. The threshold x∗1 for the good-type project increases
and x∗2 for the bad-type project decreases with a. The cost of early exploration leads to ex-post
ineﬃciency when the project turns out to be the bad type. Then, a higher a and a lower q
delays exploratory investment. On the other hand, a higher a decreases remaining investment
cost. This decrease in development cost shortens the waiting time for the bad-type project.
When exploratory investment requires a relatively large portion of total cost, the ownerdoes not gather information on project type by an earlier exploratory investment. This is Case
(ii-S). The owner proceeds to the development stage immediately after exploratory investment
regardless whether the exploration result is favorable. The owner does not have to delay theinvestment because of low development cost even if the project turns out to be the bad type. As
a result of high exploration cost, information acquisition by exploratory investment is useless.
Thus, the owner makes the entire investment at the threshold β(I + (1 − q)∆I)/(β − 1) as if
investment cost were equal to the average qI + (1 − q)(I + ∆I).
For ﬁxed xi satisfying x1 < x2 , the objective function in problem (3) decreases with a. This
leads to the following corollary.
Corollary 1 In Case (i-S), ∂πo∗ (x)/∂a < 0 holds.
Corollary 1 is intuitive because a high a means diﬃculty in judging the proﬁtability of the
project. This diﬃculty distorts the owner’s investment policy (i.e., delays investment in the
good-type project and hastens investment in the bad-type project). Then, a higher a increases
loss due to the investment distortion. Note that in Case (ii-S), πo∗ (x) is a constant because theoptimal policy does not depend on a (cf. (6) and (7)).

7

2.3

Asymmetric information

The owner delegates the investment decision to the manager who will acquire private information with exploratory investment. Before contracting, both the owner and the manager knowthat the probability of drawing (1 − a)I (good type) equals q. Immediately after exploratory
investment, only the manager knows the true development cost. As in (Grenadier and Wang
2005, Nishihara and Shibata 2008, Shibata and Nishihara 2010), we focus on the owner’s optimal contract at the initial time. At time 0, the owner oﬀers the manager a contract thatcommits the owner to pay the manager at the time of exercise. We assume that no opportunity
for renegotiation exists. Although the commitment may lead to ex post ineﬃciency in investment timing, it increases ex-ante value of the project. In fact, if the owner makes no contract
with the manager, the owner’s value becomes πo∗ (x) with q = 0 (see (7)). This is because themanager requires the owner the higher cost (1 − a)I + ∆I and makes ∆I his/her own even if
the true development cost is (1 − a)I.
As in (Grenadier and Wang 2005), the optimal contract is included in a mechanism represented by (xP , x1 , x2 , w1 , w2 ). In the contract, the owner forces the manager to make exploratory
investment at the threshold xP and report project type. For a report of the good-type (badtype) project, the owner speciﬁes the investment threshold x1 (x2 ) and pays the manager w1(w2 ) at the development investment time. Since the revelation principle (cf. (Laﬀont andMartimort 2002, Bolton and Dewatripont 2005)) ensures that the manager reveals the true
∗∗
∗∗
∗∗
∗∗
development cost in the optimum, the owner’s optimal contract (x∗∗P , x1 , x2 , w1 , w2 ) solves

the following problem:(πo∗∗ (x) :=max

xP ,x1 ,x2 ,w1 ,w2

s.t.

x
xP

)β

(

x
x1

)β

(−aI) + q
(x1 − (1 − a)I − w1 )( )βx+(1 − q)(x2 − ((1 − a)I + ∆I) − w2 )x2
xi ≥ xP (i = 1, 2)
wi ≥ 0 (i = 1, 2)
( )β
( )β
x
x
w1 −
(w2 + ∆I) ≥ 0
x1
x2
( )β
( )β
x
x
w2 −
(w1 − ∆I) ≥ 0,
x2
x1

(8)

Throughout the paper, the superscript ∗∗ refers to the solution under asymmetric information.
In the constraints of problem (8), the second inequalities correspond to the ex-post limitedliability constraints, while the last two inequalities are the ex-post incentive-compatibility constraints. The incentive-compatibility constraint means that with a truthful report, the manager
who observes (1−a)I ((1−a)I +∆I) obtains the expected payoﬀ (x/x1 )β w1 ((x/x2 )β w2 ), whichis larger than the expected payoﬀ with a false report, (x/x2 )β (w2 + ∆I) ((x/x1 )β (w1 − ∆I)).
8

∗∗
∗∗
∗∗
Clearly, in the optimum of problem (2) x∗∗
P = x1 ≤ x2 and w2 = 0 are satisﬁed. Then,problem (8) can be reduced to
( )β
( )β
x
x
∗∗
πo (x) = max q(x1 − (a/q + 1 − a)I − w1 ) + (1 − q)(x2 − ((1 − a)I + ∆I))
x1 ,x2 ,w1
x1
x2
s.t.
x2 ≥ x1
w1 ≥ 0
( )β
( )β
x
x
w1 −
∆I = 0.
x1
x2(9)
The following proposition shows the solution to problem (9).
∗∗
∗∗
∗∗
∗∗
∗∗
Proposition 2 The optimal contract (x∗∗P , x1 , x2 , w1 , w2 ), the owner’s value πo (x), and the∗∗
manager’s value πm(x) are given as follows:

Case (i-A): a < q∆I/(1 − q)I (separating)

∗∗
∗
x∗∗
P = x1 = x1 ,

x∗∗
2 =

β
((1 − a)I + ∆I/(1 − q)), w1∗∗ =β−1πo∗∗ (x) =

qxβ

+
∗β−1

(

x∗1
x∗∗
2

)β
∆I, w2∗∗ = 0(1 − q)xββx∗∗β−1
2
(
)β
x
∗∗
πm
(x) = q∆I
x∗∗
2
βx1

(10)
(11)
(12)

Case (ii-A): a ≥ q∆I/(1 − q)I (pooling)
∗∗
∗∗
x∗∗
P = x1 = x2 =β
(I + ∆I), w1∗∗ = ∆I, w2∗∗ = 0β−1πo∗∗ (x) =xβ

βx∗∗β−11()β
x∗∗πm (x) = q∆I
x∗∗
1

(13)
(14)
(15)

Proposition 2 can be interpreted similarly to Proposition 1. Indeed, exploration cost plays
a role in deferring the good-type project and accelerating the bad-type project for the same
reason as Proposition 1. However, under asymmetric information, the owner widens the gap
between the investment timing for the good- and bad-type projects to decrease the bonus to the∗∗∗∗
manager. Actually, x∗∗
2 /x1 ≥ x2 /x1 ≥ 1 is satisﬁed for any a. Note that, by (10), the bonus∗∗
to the manager, w1∗∗ , decreases as x∗∗
2 /x1 increases. This spread increases the critical value

between the separating and pooling solutions from a = q∆I/I in the symmetric informationcase to a = q∆I/(1 − q)I in the asymmetric information case. Accordingly, Case (i-S) is
included in Case (i-A).
9∗∗In the region a < q∆I/I, i.e., Case (i-S), the thresholds x∗∗
P and x1 remain unchanged from∗x∗P and x∗1 , while x∗∗
2 becomes larger than x2 . The manager’s private information distorts theinvestment timing for the bad-type project rather than the good-type project. This result isthe same as that of previous studies such as (Grenadier and Wang 2005, Nishihara and Shibata
2008, Shibata and Nishihara 2010). Because of low exploration cost, the owner designs thecontract contingent on information obtained with exploratory investment.
In the region q∆I/I ≤ a < q∆I/(1 − q)I, i.e., Case (ii-S) and Case (i-A), the investment
policy is quite diﬀerent from that of the symmetric information case. To decrease the information rent to the manager, the owner chooses a separating contract instead of a pooling contract.
Although exploration cost is not very low, the owner forces the manager to gather information
on project type earlier for the purpose of saving the bonus to the manager. Unlike the previous
studies, the investment timing for both bad- and good-type projects is delayed from that of thesymmetric information case. This complements the previous works by enforcing the results ofunderinvestment caused by agency costs.
When a increases above q∆I/(1 − q)I, i.e., Case (ii-A), the owner makes a pooling contract
independent of the exploration result. Because exploration cost is very high, the owner gives
up utilizing information regarding project type. Naturally, the owner must pay the entire
rent ∆I of the manager’s private information. This means that the owner cannot oﬀer any
eﬀective contract to the manager. Indeed, the owner’s value πo∗∗ (x) is equal to πo∗ (x) withq = 0 (cf. (7) and (14)). This type of pooling contract never appears in the previous studies
with no exploration cost. However, a similar result is found in a static model by (Crémer,
Khalil, and Rochet 1998a). They showed that a high cost of information acquisition leads to apooling contract independent of project type, although their model cannot account for dynamic
investment timing.
For ﬁxed w1 and xi satisfying x1 < x2 , the objective function in problem (9) decreases witha. In Case (i-A), we have ∂x∗∗
2 /∂a < 0. These immediately lead to the following corollary.
∗∗
Corollary 2 In Case (i-A), ∂πo∗∗ (x)/∂a < 0 and ∂πm(x)/∂a > 0 hold.
∗∗(x) depends on a. Corollary 2 shows that an
Note that in Case (ii-A) neither πo∗∗ (x) nor πm

increase in exploration cost transfers a portion of the owner’s value to the manager. A higher∗∗
a decreases the investment threshold x∗∗
2 and, hence, increases the manager’s value πm (x). In

addition to the increased rent to the manager, distortion due to exploration cost decreases
the owner’s value πo∗∗ (x) with a. The wealth transfer is consistent with empirical evidence by(Aboody and Lev 2000). They found insider gains in R&D-intensive ﬁrms substantially larger
than insider gains in ﬁrms without R&D, and then identiﬁed R&D as a major contributor
to information asymmetry. Our result provides a complementary explanation for their ﬁnding.
10Large insider gains in R&D-intensive ﬁrms originate not only from large information asymmetry
but also from high exploration cost for R&D activities.
Now, consider a case in which the proportion of exploration cost is also privately observedby the manager. For example, assume that only the manager observes the realized value ofa ∈ [aL , aR ] at time 0 while the owner knows only a range [aL , aR ]. In this case, the manager
always reports aR with no additional mechanism, because a higher a increases the manager’s∗∗
value πm(x). In other words, the manager pretends to be ignorant of project type. This result

is similar to that of (Kessler 1998) in incentive theory. He showed that, in a static model anagent may be better oﬀ if he/she can commit to remain ignorant with some probability.

3

Social welfare

In the previous subsection, we have focused on the owner’s optimal contract with the manager
who will acquire information on project type with exploratory investment. In this subsection,
we explore how the agency problem aﬀects social welfare. We deﬁne the social welfare and∗∗the social loss by V ∗∗ (x) := πo∗∗ (x) + πm(x) and L∗∗ (x) := πo∗ (x) − V ∗∗ (x), respectively. As

documented by (Grenadier and Wang 2005), this social loss stemming from agency costs canbe indicative of a corporate structure. With potentially large social loss, a ﬁrm will be forced
to be privately held or to be organized in a manner that provides the closest alignment between
the owner and the manager. The following proposition shows how the ratio of exploration cost
to total investment cost, a, aﬀects the social welfare and loss.
Proposition 3 Assume that q∆I/(1 − q)I ≤ 1. There exists a unique ã ∈ (q∆I/I, q∆I/(1 −q)I) such that
 < 0 (0 ≤ a < ã)∂V ∗∗ (x) = 0 (a = ã)∂a  > 0 (ã < a < q∆I/(1 − q)I),(16)

which means that the social welfare V ∗∗ (x) is minimized at a = ã. The social loss L∗∗ (x)satisﬁes
 > 0 (0 ≤ a < ã)∗∗∂L (x) = 0 (a = ã)∂a  < 0 (ã < a < q∆I/(1 − q)I),(17)

which means that the social loss L∗∗ (x) is maximized at a = ã.
Note that in the region a ≥ q∆I/(1 − q)I neither V ∗∗ (x) nor L∗∗ (x) depends on a. Proposition 3 contrasts asymmetric and symmetric information cases. In the absence of asymmetric information, a higher a leads to less eﬃciency because exploration cost distorts the in11

vestment policy; however, the outcome changes with asymmetric information. In the regionq∆I/I ≤ a < q∆I/(1 − q)I, the owner, unlike in the symmetric information case, chooses a
separating contract to decrease the information rent to the manager. For a ≈ ã, the owner’s
contract maximizing his/her interest becomes far from eﬃcient from the social viewpoint. The
owner’s contract may lead to less social welfare than the case with no contract. Indeed, thesocial welfare with no contract agrees with (14) + (15), which is larger than V ∗∗ (x)|a=ã by (16).
This can also be veriﬁed from the opposite side, i.e., the social loss L∗∗ (x).
In the region a ∈ (ã, q∆I/(1 − q)I), V ∗∗ (x) increases with a. When costly learning andasymmetric information are separately present, both lead to ineﬃciency. However, higher exploration cost can lead to less ineﬃciency due to asymmetric information by preventing theowner’s greedy contract. This complements the literature by revealing interactions between
costly learning and asymmetric information. Similar to this result, several corporate ﬁnancestudies have shown that “two incentive problems are better than one” (e.g., (Hirshleifer andThakor 1992, Mookherjee and Png 1995, Noe and Rebello 1996)). Related results are seen in
the context of real options. In (Hackbarth 2009), managerial optimism and overconﬁdence,
which distort the investment and ﬁnancing policy, play a potentially positive role in ameliorating bondholder-shareholder conﬂicts such as debt overhang, asset stripping, and risk-shifting.
(Nishihara and Shibata 2010) developed a model that reveals complex interactions between
preemptive competition and a ﬁnancing constraint and showed that a ﬁnancing constraint can
mitigate preemptive competition.
As discussed after Corollary 2, let us look at the situation where the manager has private
information about a ∈ [aL , aR ]. The manager always reports the maximal value a = aR because
a higher a increases the manager’s value (cf. Corollary 2). This moral hazard decreases theowner’s value but does not necessarily reduce the social welfare. For example, consider a caseof aL = ã. By Proposition 3, we have V ∗∗ (x)|a=ã < V ∗∗ (x)|a=aR and L∗∗ (x)|a=ã > L∗∗ (x)|a=aR .
This means that the manager’s moral hazard may enhance the social welfare by mitigating
ineﬃcient asset substitution by the owner’s greedy contract.

4

Numerical examples

We investigate more detailed properties of the solutions in numerical examples. Section 4.1
focuses on the eﬀects of the ratio of exploration cost, a. Section 4.2 explores the comparative
statics with respect to the volatility of project value, σ.

12

4.1

The eﬀects of exploration cost

For comparison, we use the same base parameter values as (Shibata 2009, Shibata and Nishihara2010):6
q = 0.5, σ = 0.2, r = 0.07, µ = 0.03, I = 50, ∆I = 30, X(0) = x = 100(18)

For the base parameter values (18) the critical value between Case (i-S) (separating) and Case
(ii-S) (pooling) in the symmetric information case becomes q∆I/I = 0.3 (cf. Proposition1), while the critical value between Case (i-A) (separating) and Case (ii-A) (pooling) in theasymmetric information case becomes q∆I/(1 − q)I = 0.6 (cf. Proposition 2).
The upper-left and upper-right panels in Figure 1 depict the investment thresholds in thesymmetric and asymmetric information cases, respectively, with varying levels of a. In theabsence of exploration cost, i.e., a = 0, the solution is equal to that of the benchmark caseby (Grenadier and Wang 2005). Indeed, as in (Shibata 2009), the investment thresholds are∗∗x∗1 = 128.43, x∗2 = 205.49, x∗∗
1 = 128.43, x2 = 282.56. In the previous studies, the agency

problem does not distort the investment threshold for the good-type project, because the owner
can eﬀectively decrease the bonus to the manager by changing the investment threshold forthe bad-type project. However, as discussed after Proposition 2, together with exploration
cost a > 0.3, the investment is delayed whether the project is the good- or bad-type. The
upper panels also show that distortion due to high exploration cost increases with asymmetric
information. For example, we compare the cases of a = 0 (separating) and a = 0.6 (pooling).
∗∗∗∗
For a = 0.6 the investment thresholds are x∗P = x∗1 = x∗2 = 166.96, x∗∗
P = x1 = x2 = 205.49.

The distortion due to a = 0.6 is 166.96 − 128.43 = 38.53 under symmetric information and
205.49 − 128.43 = 77.06 under asymmetric information.
Next, let us turn to the owner’s and manager’s values. The lower-left and lower-right panels
in Figure 1 plot the owner’s value under symmetric information and the owner’s and manager’s
values under asymmetric information, respectively, with varying levels of a. As shown in
Corollaries 1 and 2, the owner suﬀers from ineﬃciency due to exploration cost while the manager
beneﬁts from increased information rent. Comparing πo∗ (x) in the lower-left panel and πo∗∗ (x)in the lower-right panel, we ﬁnd that the decrease in the owner’s value is more serious in theasymmetric information case than in the symmetric information case. Asymmetric information
reinforces ineﬃciency caused by costly exploration from the owner’s viewpoint.
Finally, we explore the eﬀects of a on the social welfare and loss. The left and right panels in∗∗(x) and L∗∗ (x) = πo∗ (x) − V ∗∗ (x), respectively, with varying
Figure 2 plot V ∗∗ (x) = πo∗∗ (x) + πm

levels of a. As shown in Proposition 3, V ∗∗ (x) is U-shaped and L∗∗ (x) is unimodal. The worst
6

We carried out a lot of computations with varying parameter values and distilled robust results into thissection.

13

a is ã = 0.3894. We see from the right panel that the agency cost is very high for a ≈ 0.3894.
As mentioned above (cf. the lower-right panel in Figure 2), an increase in a plays a role intransferring a portion of the value from the owner to the manager. In the region a < 0.3894,
the decrease in the owner’s value dominates the increase in the manager’s value. Then, V ∗∗ (x)decreases with a. This is straightforward and the same as the symmetric information case. On
the other hand, in the region a ∈ (0.3894, 0.6), the increase in the manager’s value dominates
the decrease in the owner’s value. This leads to a counter-intuitive result that V ∗∗ (x) increases
with a. A higher a moderates ineﬃcient asset substitution from owner to manager underasymmetric information.

4.2

The eﬀects of uncertainty

The eﬀect of the volatility σ on the investment thresholds are straightforward; hence, we omit
a ﬁgure illustrating it. Actually, an increase in σ increases the investment thresholds, as well∗∗
as most real options models. Figure 3 illustrates πo∗∗ (x), πm(x), V ∗∗ (x), and L∗∗ (x) for σ =0.1, 0.2, 0.3, and 0.4. Note that the critical value between Case (i-A) (separating) and Case(ii-A) (pooling) in the asymmetric information case, q∆I/(1 − q)I = 0.6, does not depend onσ.
The upper panels show that a higher volatility increases the owner’s value and decreases
the manager’s value. This asset substitution is consistent with previous ﬁndings by (Shibata
2009, Shibata and Nishihara 2010). In the lower panels, we verify that asset substitution iseﬃcient from the social viewpoint. The lower panels demonstrate that V ∗∗ (x) increases with σ
and L∗∗ (x) decreases with σ. The impact of σ is especially great for L∗∗ (x). Indeed, an increasein σ from 0.2 to 0.4 reduces L∗∗ (x) by half. We also see from the ﬁgure that the eﬀects of σ
are relatively robust with respect to a.

5

Conclusion

This paper has investigated a principal-agent model in which the owner (principal) optimizes thecontract with the manager (agent) delegated to undertake the investment project. The model
assumes that the manager learns the real value of development cost by exploratory investment.
For low cost in the exploration phase, the ﬁrm invests in the exploratory stage early andseparates the investment timing for good- and bad-type projects. On the other hand, for highcost in the exploration phase, the ﬁrm proceeds to the development stage immediately after
exploratory investment whether or not the exploration result is favorable. High exploration costleads to a pooling solution not contingent on the exploration result. Asymmetric information
increases the owner’s incentive to take a separating contract rather than a pooling contract
14

because the owner can decrease the information rent to the manager with a separating contract.
However, the owner’s greedy contract may seriously reduce the social welfare especially when
exploration cost is intermediate. Most notably, costly learning, or equivalently, the manager’s
pretense of ignorance, could improve social welfare by preventing ineﬃcient wealth transfer by
the owner’s greedy contract.

A

Proof of Proposition 1

We can ignore xβ . Deﬁne the LagrangianL(x1 , x2 , λ) := qx1 −β (x1 − (a/q + 1 − a)I) + (1 − q)x2 −β (x2 − ((1 − a)I + ∆I)) + λ(x2 − x1 ),where λ denotes the Lagrangian multiplier. The Karush-Kuhn-Tucker conditions areqx∗−β−1((−β + 1)x∗1 + β(a/q + 1 − a)I) − λ∗ = 01(19)((−β + 1)x∗2 + β((1 − a)I + ∆I)) + λ∗ = 0(1 − q)x∗−β−12(20)x∗2 ≥ x∗1 , λ∗ ≥ 0, λ∗ (x∗2 − x∗1 ) = 0(21)

First, consider the case of x∗1 < x∗2 . We have λ∗ = 0 by (21). Substituting λ∗ = 0 in (19) and(20), we have (4). Taking account ofββ(a/q + 1 − a)I <((1 − a)I + ∆I)
β−1
β−1⇔ a < q∆I/I,x∗1 < x∗2 ⇔we have the optimal policy in Case (i-S). The maximum value can be easily calculated as (5).
Next, consider the case of x∗1 = x∗2 . By (19) + (20), we have (6). By (19) we have((−β + 1)x∗1 + β(a/q + 1 − a)I) ≥ 0
λ∗ ≥ 0 ⇔ qx∗−β1⇔ −(I + (1 − q)∆I) + (a/q + 1 − a)I ≥ 0⇔ a ≥ q∆I/I,
which leads to the optimal policy in Case (ii-S). The maximum value becomes (7).

B

Proof of Proposition 2

We can ignore xβ . Deﬁne the LagrangianL(x1 , x2 , w1 , λ1 , λ2 , λ3 ) := qx1 −β (x1 − (a/q + 1 − a)I − w1 ) + (1 − q)x2 −β (x2 − ((1 − a)I + ∆I))−β
+λ1 (x2 − x1 ) + λ2 w1 + λ3 (x−β1 w1 − x2 ∆I),

15

where λi denotes the Lagrangian multiplier. The Karush-Kuhn-Tucker conditions are∗∗∗∗∗∗ ∗∗−β−1 ∗∗qx∗∗−β−1((−β + 1)x∗∗w1 = 01
1 + β((a/q + 1 − a)I + w1 )) − λ1 − βλ3 x1

(22)

∗∗
∗∗ ∗∗−β−1(1 − q)x∗∗−β−1((−β + 1)x∗∗∆I = 022 + β((1 − a)I + ∆I)) + λ1 + βλ3 x2(23)∗∗ ∗∗−β−qx∗∗−β+ λ∗∗=012 + λ3 x 1

(24)

∗∗
∗∗
∗∗ ∗∗
∗∗
x∗∗
2 ≥ x1 , λ1 ≥ 0, λ1 (x2 − x1 ) = 0

(25)

∗∗ ∗∗
w1∗∗ ≥ 0, λ∗∗2 ≥ 0, λ2 w1 = 0

(26)

x∗∗−β
w1∗∗ − x∗∗−β∆I = 0
1
2

(27)

∗∗ β
∗∗
∗∗
By (27) we have w1∗∗ = (x∗∗1 /x2 ) ∆I > 0. Then, by (26) and (24) we have λ2 = 0 and λ3 = q.

We can rewrite (22) and (23) as follows:∗∗qx∗∗−β−1((−β + 1)x∗∗1
1 + β(a/q + 1 − a)I) − λ1 = 0

(28)

∗∗
(1 − q)x∗∗−β−1((−β + 1)x∗∗2
2 + β((1 − a)I + ∆I/(1 − q))) + λ1 = 0

(29)

∗∗
∗∗
First, suppose x∗∗
1 < x2 in (25). Substituting λ1 = 0 in (28) and (29), we have (10) in Case(i-A). Note thatββ(a/q + 1 − a)I <((1 − a)I + ∆I/(1 − q))β−1
β−1⇔ a < q∆I/(1 − q)I.

∗∗
x∗∗
⇔
1 < x2

We can easily show (14) and (15).
∗∗Next, suppose x∗∗
1 = x2 in (25). By (28) + (29), we have (13). We have∗∗−β−1λ∗∗((−β + 1)x∗∗
1 ≥ 0 ⇔ qx1
1 + β(a/q + 1 − a)I) ≥ 0⇔ −(I + ∆I) + (a/q + 1 − a)I) ≥ 0⇔ a ≥ q∆I/(1 − q)I,

which leads to the solution in Case (ii-A).

C

Proof of Proposition 3

In Case (i-A), by (11) and (12), we have
(
)
(
)β (
( ∗∗ )β )∗∗2∂V (x)
x
x2
β q∆I
=I
(1 − q) 1 −
+
.
∗∗
∗
∂a
x2
x1
(β − 1)x∗∗
2

16

(30)

2 ∗∗
2
Because of ∂x∗1 /∂a > 0 and ∂x∗∗
2 /∂a < 0, we have ∂ V (x)/∂ a > 0. Further, we have∂V ∗∗ (x)∂aa=q∆I/I
(
)
(
)β (
(
)β )
x
q∆I
βq∆I
= I
(1 − q) 1 − 1 ++x∗∗(1 − q)(I + (1 − q)∆I)
I − q∆I + ∆I/(1 − q)
2
(
)
)β (
x
βq∆I
βq∆I
≤ I
+
(31)
−
∗∗
x2
I + (1 − q)∆I I − q∆I + ∆I/(1 − q)
< 0,where (31) follows from the inequality (1 + y)β ≥ 1 + βy (y ≥ 0), and we also have()β
x∂V ∗∗ (x)β 2 q∆I
=I
> 0.
∂a
x∗∗
(β − 1)x∗∗2
2
a=q∆I/(1−q)I
Then, there exists a unique ã ∈ (q∆I/I, q∆I/(1 − q)I) satisfying (16).
Considering ∂πo∗ (x)/∂a = 0 for a ≥ q∆I/I, we have only to show that ∂L∗∗ (x)/∂a > 0 fora ∈ (0, q∆I/I). For a ∈ (0, q∆I/I), by (5), (11), and (12) we have

=
=
≥
>

∂L∗∗ (x)
∂a
((
)
)
(
)β (
)β
2
x
x∗∗
β
q∆I
2
I
(1 − q)
−1 −
∗
x∗∗
x
(β − 1)x∗∗
2
2
2
((
)
)
)β (
)β
(
x
q∆I
βq∆I
(1 − q)
1+
−1 −
I
x∗∗
(1 − q)((1 − a)I + ∆I)(1 − a)I + ∆I/(1 − q)
2
(
)β (
)
x
βq∆I
βq∆I
I
−
(32)
x∗∗
(1 − a)I + ∆I
(1 − a)I + ∆I/(1 − q)20,where (32) follows from the inequality (1 + y)β ≥ 1 + βy (y ≥ 0). This leads to (17).

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19

300

300

Case (i−S) Case (ii−S)

x∗∗
1
x∗∗
2

250
Trigger

Trgger

250

x∗1
x∗2

200

150

200

150
0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

a

0.8

1

50

40
Case (i−S) Case (ii−S)3040π∗o(x)
Value

Value

0.6
a

50

20
10
0

Case (ii−A)

Case (i−A)

π∗∗
(x)
o

30

π∗∗
(x)
m

20

Case (i−A)

Case (ii−A)

10

0

0.2

0.4

0.6

0.8

0

1

a

0

0.2

0.4

0.6

0.8

1

a

Figure 1: The investment thresholds and values under symmetric and asymmetric information.
The upper-left panel plots the investment thresholds for good- and bad-type projects, x∗1 and x∗2in the symmetric information case. The critical value between Case (i-S) (separating) and Case
(ii-S) (pooling) is a = 0.3. The upper-right panel plots the investment thresholds for good- and∗∗bad-type projects, x∗∗
1 and x2 in the asymmetric information case. The critical value between

Case (i-A) (separating) and Case (ii-A) (pooling) is a = 0.6. The lower-left panel plots theowner’s value under symmetric information, πo∗ (x), while the lower-right panel plots the owner’s∗∗(x). The parameter values
and manager’s values under asymmetric information, πo∗∗ (x) and πmare set at the base case (18).

20

45V∗∗(x)

44.5

43.5

1.2

Case (ii−A)

Case (ii−A)

1

43
42.5
42

Case (i−A)

Loss

Value

44
Case (i−A)

L∗∗(x)

1.4

0.8
0

0.2

0.4

0.6

0.8

1

0

0.2

a

0.4

0.6

0.8

1

a

Figure 2: The social welfare and loss. The left panel plots the social welfare V ∗∗ (x), while theright panel plots the social loss L∗∗ (x). At a = ã = 0.3894, the social welfare (loss) is minimized(maximized). The parameter values are set at the base case (18).

21

55

σ=0.4

50

σ=0.3

45

Case (i−A)
σ=0.2

40

0

0.2

0.4

0.6

5
σ=0.2
Case (ii−A)

4

σ=0.3
σ=0.4

3

Case (i−A)

2
0.8

1

0

0.2

0.4

a

0.8

1

2.5

50
45

σ=0.1

∗∗

σ=0.4

V (x)

Case (i−A)σ=0.3

Case (ii−A)

σ=0.2

40

σ=0.1

35
0

0.2

0.4

0.6

L∗∗(x)

2
Loss

55

Value

0.6

Case (ii−A)

a

60

30

π∗∗
(x)
m

σ=0.1

σ=0.1

35
30

6

π∗∗
(x)
o

Value

Value

60

1.5

σ=0.2

1

σ=0.3

0.5

σ=0.4
Case (i−A) Case (ii−A)

0.8

0

1

a

0

0.2

0.4

0.6

0.8

1

a

Figure 3: The comparative statics with respect to the volatility σ. The upper-left and upperright panels plot the owner’s and manager’s values under asymmetric information, πo∗∗ (x) and∗∗(x), respectively. The lower-left and lower-right panels plot the social welfare and loss,πm

V ∗∗ (x) and L∗∗ (x), respectively. The parameter values other than σ are set at the base case(18).

22