Discussion Papers In Economics
And Business

Preemption, leverage, and financing constraints
Michi NISHIHARA, Takashi SHIBATA

Discussion Paper 13-05

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANPreemption, leverage, and financing constraints
Michi NISHIHARA, Takashi SHIBATA

Discussion Paper 13-05

April 2013

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANPreemption, leverage, and financing constraints∗
Michi NISHIHARA†,

Takashi SHIBATA‡

Abstract
This paper investigates the interactions between preemptive competition and leverage.
We ﬁnd that the second mover always leaves the duopoly market before the ﬁrst mover,although the leader may exit before the follower’s entry. We also see the leverage eﬀects ofdebt ﬁnancing increasing ﬁrm values and accelerating investment, even in the presence ofpreemptive competition. In addition to the case with optimal capital structure, we analyze
a case with ﬁnancing constraints that require ﬁrms to ﬁnance investment costs by debt.
Notably, ﬁnancing constraints can delay preemptive investment and improve ﬁrm values inpreemptive equilibrium. Indeed, the leader’s high leverage due to the ﬁnancing constraints
can lower the ﬁrst-mover advantage and weaken preemptive competition. Especially withstrong ﬁrst-mover advantage, the ﬁnancing constraint eﬀects can dominate the leverage
eﬀects. These ﬁndings are almost consistent with empirical evidence that high leverageleads to competitive disadvantage and mitigates product market competition.
JEL Classifications Code: C73; G31; G33.
Keywords: Preemption; duopoly; capital structure; ﬁnancing constraints; real options.

∗†

The ﬁrst version: 15 April, 2013.
Corresponding Author. Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka,

Osaka 560-0043, Japan, E-mail: nishihara@econ.osaka-u.ac.jp, Phone: 81-6-6850-5242, Fax: 81-6-6850-5277‡
Graduate School of Social Sciences, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji, Tokyo
192-0397, Japan, E-mail: tshibata@tmu.ac.jp, Phone: 81-42-677-2310, Fax: 81-42-677-2298

1

1

Introduction

Since the seminal paper of Brander and Lewis (1986), a number of researchers haveinvestigated the interactions between ﬁnancial structure and product market competition.
Although a wide range of competition among levered ﬁrms is covered in the existing
literature (e.g., Faure-Grimaud (2000), Povel and Raith (2004)), few papers have analyzed
the eﬀects of leverage on preemptive competition. Analysis of preemptive competition
is increasingly important as ﬁrms, such as Apple, lead a new market by technologicalinnovation and gain great ﬁrst-mover advantage all over the world. This paper sheds lighton how ﬁnancial structure interacts with preemptive competition.
This paper considers a situation in which two symmetric ﬁrms compete for a newmarket. When proﬁt ﬂows in a duopoly are lower than those in a monopoly (i.e., negative
externalities), each ﬁrm has an incentive to preempt the competitor and gain monopolistic
proﬁt ﬂows before the competitor’s entry. In preemptive equilibrium, one of the ﬁrms,
denoted by the leader, enters the market earlier than the other, denoted by the follower,
but its ﬁrst-mover advantage is oﬀset by its sub-optimally early entry timing. After
Fudenberg and Tirole (1985) ﬁrst analyzed preemptive equilibrium in a case withoutmarket uncertainty, the literature including Dixit and Pindyck (1994), Grenadier (1996),
and Huisman (2001), has examined cases with market uncertainty.
We examine the interactions between preemptive competition and leverage by extending the previous analysis to a setup in which ﬁrms can access debt ﬁnancing on the market
entry. In the levered setup, we consider ﬁrms that optimize capital structure based on
the trade-oﬀ theory (e.g., Leland (1994), Goldstein, Ju, and Leland (2001)). Through themodel analysis, we ﬁnd key results as follows.
First, we ﬁnd that the last in, ﬁrst out (LIFO) scenario holds in a duopoly. This ismainly because the leader’s entry trigger is much lower than the follower’s entry trigger inpreemptive equilibrium. Because of the early entry, the leader’s debt issuance is very low
and the follower issues more debt on the later entry timing. Our result is consistent with
MacKay and Phillips (2005) who empirically show that leverage of new entrants is likely
to be higher than that of incumbents. The LIFO scenario is also consistent with empiricalﬁndings that high debt tends to lead to disadvantage in product market competition (e.g.,
Phillips (1995), Chevalier (1995a), Chevalier (1995b)).
Second, we show that the leverage eﬀects remain unchanged even if one takes accountof preemptive competition. Indeed, compared to the unlevered case, the entry triggers(ﬁrm values) become lower (higher) in preemptive competition with optimal capital structure. The leverage eﬀects are well known in corporate ﬁnance (e.g., Myers (1977)). The
leverage eﬀects are also identiﬁed in the investment timing models such as Hennessy(2004), Mauer and Sarkar (2005), and Sundaresan and Wang (2007a), but they focus ona monopoly. We ensure the robustness of the leverage eﬀects even in the presence ofpreemptive competition.

2

In addition to analysis on ﬁrms with optimal capital structure, we consider ﬁnanciallyconstrained ﬁrms in which investment costs must be ﬁnanced by debt issuance. This caseapproximates ﬁrms that have no cash reserves and cannot use external equity ﬁnancing
due to the high costs. Notably, we show that ﬁnancing constraints can delay preemptive
investment and improve ﬁrm values in equilibrium. The intuition is as follows. The
ﬁnancing constraints lead the leader to be highly leveraged, while it increases the valueof the follower’s option to wait for the leader’s exit. Thus, the constraints reduce ﬁrms’
incentive to move ﬁrst, and then it alleviates preemptive competition.
The ﬁnancing constraint eﬀects can happen with a modest level of ﬁrst-mover advantage and greatly increases as the ﬁrst-mover advantage is stronger. When the ﬁnancing
constraint eﬀects dominate the leverage eﬀects, the preemptive entry trigger can be later
than that of the unlevered case. Although the ﬁnancing constraint eﬀects in preemptivecompetition have yet to be tested rigorously, there are several ﬁndings related to thepredictions. For instance, empirical evidence indicates that higher leverage can soften
product market competition (e.g., Phillips (1995), Chevalier (1995a), Chevalier (1995b)).
In our paper, higher leverage, which is caused by the ﬁnancing constraints, delays thepreemptive entry timing and increases ﬁrm values.
Our paper is most closely related to the following papers. Lambrecht (2001) studies theentry and exit decisions of levered ﬁrms in a duopoly. The paper exogenously assumes anincumbent with debt and examines the follower’s entry and ﬁnancing decision. Our paper
complements the previous research by extending his model to a new market model in which
two ﬁrms compete for ﬁrst-mover advantage. Preemptive competition is frequently seen
in technology-oriented industries, for example, information technology (IT) industries.
Our analysis may foster better understanding of such industries. Zhdanov (2008), likeour paper, examines a preemptive competition model with leverage. His model assumes
that the leader survives as an all-equity ﬁrm after its default, while, as in Lambrecht(2001), we simply assume that the leader exits the market. Because of this simpliﬁcation,
our model is more tractable and easier to analyze. We reveal the eﬀects of ﬁrst-moveradvantage and ﬁnancing constraints that are not clariﬁed by Zhdanov (2008). Nishihara
and Shibata (2010) also study preemption with leverage, but the previous paper assumes
that the follower cannot enter the market until the leader exits it. Because of the polarassumption, the model applicability is restricted to a situation involving extremely strong
ﬁrst-mover advantage. In this paper, we relax the assumption and show how the degreeof ﬁrst-mover advantage inﬂuences the results.
The remainder of this paper is organized as follows. As a benchmark, Section 2
introduces the investment policies of unlevered ﬁrms in a duopoly. In Section 3, we
illustrate the investment and ﬁnancing policies for levered ﬁrms in a duopoly. In Section
4, we exercise numerical analysis and provide empirical implications. Section 5 brieﬂysummarizes the paper.

3

2

Unlevered ﬁrms in a duopoly

2.1

Setup

We use the same setup as the standard literature (e.g., Chapter 9 in Dixit and Pindyck(1994), Chapter 7 in Huisman (2001)). We consider two symmetric ﬁrms that have an
opportunity to enter a new market. The entry to the market requires irreversible capital
expenditure I. Throughout this paper, we assume that both ﬁrms are risk-neutral andhave full information of each other. When only one of the ﬁrms is active in the market,
the active ﬁrm receive an instantaneous cash ﬂow X(t) that is inﬂuenced by the market
demand. Following the standard real options literature, we assume that X(t) follows ageometric Brownian motion:dX(t) = µX(t)dt + σX(t)dB(t) (t > 0),

X(0) = x,(1)

where B(t) denotes the standard Brownian motion deﬁned in a probability space (Ω, F, P)
and µ, σ(> 0) and x(> 0) are constants. We assume that the initial value is suﬃciently
low to exclude a ﬁrm’ entry into a market at the initial time. For convergence, we assume
that r > µ1 , where r is a positive constant interest rate. When both ﬁrms are activein the market, the ﬁrst mover, denoted by the leader, receives an instantaneous cash
ﬂow QL X(t) while the second mover, denoted by the follower, receives QF X(t). Assume
that QL and QF are constants satisfying 0 < QF ≤ QL < 1, which means that theleader’s proﬁt in a duopoly is between the monopolistic proﬁt and the follower’s proﬁt.
We presume the negative externalities and ﬁrst-mover advantage so as to focus on theanalysis of preemptive competition.

2.2

Preemptive equilibrium

This section explains preemptive equilibrium following Dixit and Pindyck (1994), Huisman(2001), and Grenadier (1996), among others. In a duopoly game, we need consider the
problem backwards. We denote the “Unlevered” case by the subscript U . Suppose thatthe leader has invested at time s. The follower optimally enters the market by solving the
optimal stopping problem:
FU (X(s)) = sup E

X(s)TF U ≥s∫ ∞[e−r(t−s) (1 − τ )QF X(t)dt − e−r(TF U −s) I],

(2)

TF U

where TF U runs over stopping times and EX(s) [·] denotes the expectation conditional on
X(s). We denote the corporate tax rate by positive constant τ . The value FU (X(s)) corresponds to the follower’s option value at time s. Because of the strong Markov propertyof X(t), problem (2) can be reduced to
sup E

X(s)

TF U ≥s
1−r(TF U −s)

[e

(

)
(1 − τ )QF
X(TF U ) − I ]r−µFor economic rationale of the assumption, refer to Dixit and Pindyck (1994).

4and has the explicit solution as follows: ()()(1 − τ )QF x∗F U
X(s) β(X(s) < x∗F U )
−I


r−µ
x∗F U
(3)
FU (X(s)) =


(1
−
τ
)Q
F

∗

X(s) − I(X(s) ≥ xF U ),r−µ√where β := 1/2 − µ/σ 2 + (µ/σ 2 − 1/2)2 + 2r/σ 2 (> 1) is a positive characteristic root,
and x∗F U := β(r − µ)I/{(β − 1)(1 − τ )QF } is the entry trigger. The follower’s entry timeis expressed as the optimal stopping time:
TF∗ U := inf{t ≥ s | X(t) ≥ x∗F U }.

(4)

Then, the leader’s expected gain by investment at time s becomes
LU (X(s)) = E

X(s)

∫ T∗
∫ ∞
FU
−r(t−s)[e(1 − τ )X(t)dt +e−r(t−s) (1 − τ )QL X(t)dt],
TF∗ U

s

(5)

where TF∗ U is deﬁned by (4). Note that after TF∗ U the leader’s proﬁt ﬂows will decrease to
QL X(t). By straightforward calculation, we have
LU (X(s)) =where γ := 1/2 − µ/σ 2 −√(1 − τ )(1 − QL )x∗F U
1−τ
X(s) −
r−µ
r−µ

(

X(s)
x∗F U

)γ
,

(6)

(µ/σ 2 − 1/2)2 + 2r/σ 2 (< 0) is a negative characteristic root.

By comparing LU (X(s)) with FU (X(s)) we consider the situation in which neither ﬁrm
has invested. In the region FU (X(s)) < LU (X(s)), both ﬁrms are better oﬀ becoming
the ﬁrst mover, whereas in the region LU (X(s)) < FU (X(s)), both ﬁrms are better oﬀ
becoming the follower. In equilibrium, one of the ﬁrms, denoted by the leader, invests at
time
∗
TLU
:= inf{t ≥ 0 | X(t) ≥ x∗LU }.

(7)

where x∗LU is the solution to FU (x∗LU ) − LU (x∗LU ) = 0.2 This investment is called preemp∗ ). Seetive investment. The follower invests later on the best response timing TF∗ U (> TLU

the top panel of Figure 1. Note that in equilibrium the ﬁrst-mover advantage is exactly
oﬀset by ineﬃciency in the preemptive entry. Firms are indiﬀerent between the roles ofthe leader or the follower. Depending on the parameter values, there may exist joint
investment equilibrium in which both ﬁrms cooperate to invest at the same time whichis later than TLU . For example, see Huisman (2001) and Pawlina and Kort (2006). This
paper focuses only on preemptive equilibrium because the main objective is to examine
the levered case. Note that, in the levered case, no joint investment equilibrium arises.

2

Because this is a model of symmetric ﬁrms, we cannot determine which one is exactly the ﬁrst mover. Basedon most of the literature including Dixit and Pindyck (1994) and Grenadier (1996), we exclude the possibility
that both ﬁrms mistakenly invest at the same time x∗LU .

5

3

Levered ﬁrms in a duopoly

3.1

Setup

Now, we proceed to examine how leverage aﬀects preemptive competition. This section
explains how we can solve preemptive equilibrium in the levered case. In addition to thesetup in Section 2, we assume that both ﬁrms can issue console bonds on the investment
timing. This assumption is standard in the literature regarding dynamic investment andcapital structure (e.g., Sundaresan and Wang (2007a), Sundaresan and Wang (2007b),
Shibata and Nishihara (2012)). The levered ﬁrms are evaluated in the framework ofstructural models by Leland (1994) and Goldstein, Ju, and Leland (2001). In the framework, after the issuance of debt, shareholders optimize the default timing to maximize theequity value, whereas debtholders receive coupon payment until the default. The optimal
capital structure depends on the trade-oﬀ between costs associated with default and tax
beneﬁts from debt issuance. Throughout this paper, a ﬁrm determines its optimal capital
structure and investment timing to maximize its ﬁrm value.3 For simplicity, following
Lambrecht (2001) and Nishihara and Shibata (2010), we assume that a bankrupt ﬁrm
immediately leaves the market. The levered case greatly diﬀers from the unlevered casein the possibility of the market exit. The analysis in the levered case will better ﬁt preemptive competition in technological development; actually, in IT industries, the majorityof venture businesses are forced to close after a boom.

3.2

Firms without a ﬁnancing constraint

As in Section 2.2, we consider the problem backwards. We begin by deriving the follower’s
value in a duopoly. It depends on which ﬁrm exits the market ﬁrst. However, as will beshown in numerical analysis, we always have the LIFO scenario, i.e., the follower leaves
the market before the leader. From now on, we will explain the methodology to derive
preemptive equilibrium in the LIFO scenario. For the state variable X(s), the equity,debt, and ﬁrm values of the follower that has issued debt with coupon CF (> 0) are as

3

We can say that shareholders maximize the ex-ante equity value under the assumption that debtholders arecompetitive.

6

follows:
EF (X(s), CF )
∫ Td
F
X(s)
=E
[
e−r(t−s) (1 − τ )(QF X(t) − CF )dt]s()()
(1 − τ )QF(1 − τ )CF(1 − τ )QF d(1 − τ )CF
X(s) γ=X(s) −
−
xF −
r−µ
r
r−µ
r
xdF
DF (X(s), CF )
∫ Td
F
d
X(s)
=E
[
e−r(t−s) CF dt + e−r(TF −s) KF (xdF )]
)
(s
)(
CF
CF
X(s) γ=−− KF (xdF )
r
r
xdF

(8)

(9)

VF (X(s), CF )=EF (X(s), CF ) + DF (X(s), CF )()()(1 − τ )QF
τ CF
τ CF
(1 − τ )QF d
X(s) γd=X(s) +−+
xF − KF (xF )
,
r−µ
r
r
r−µ
xdF

(10)

where TFd stands for the exit time and KF (xdF ) denotes the liquidation value. Because of
the smooth pasting condition, we have
TFd := inf{t ≥ s | X(t) ≤ xdF },(11)

where the default trigger is deﬁned by xdF := γ(r − µ)CF /{(γ − 1)rQF }. Eq. (8)–(10)presume X(s) ≥ xdF . This argument is the same as that of the structural model byGoldstein, Ju, and Leland (2001).
For preparation, we derive a coupon that maximizes the ﬁrm value (10). Following Goldstein, Ju, and Leland (2001) and Lambrecht (2001), we assume a linear form
KF (xdF ) := (1 − αF )(1 − τ )QF xdF /(r − µ) for the liquidation value, where a constantαF ∈ (0, 1) measures costs associated with liquidation. By simple calculation, we havethe optimal coupon:CF∗ (X(s)) := arg max VF (X(s), CF ) =CFr(γ − 1)QF
X(s),(r − µ)γh(12)where h is a constant deﬁned by[(
αF )]− γ1
h = 1 − γ 1 − αF +(> 1).
τ(13)

By substituting (12) into (10), we obtain
VF (X(s), CF∗ (X(s))) =(1 − τ )QF
X(s),ψ(r − µ)where ψ is a positive constant deﬁned by[]−1
τψ = 1+(< 1).
(1 − τ )h

7

(14)

(15)

For details of derivation, refer to Sundaresan and Wang (2007a) and Nishihara and Shibata(2010).
Next, we consider a situation in which only the leader is active. We need solve theleader’s exit and follower’s entry problems simultaneously. Denote the follower’ entry
time by TF∗ := inf{t ≥ s | X(t) ≥ x∗F }. Then, we have the follower’s coupon CF∗ (x∗F ) anddefault trigger xdF = x∗F /h. We denote the leader’s default timing before and after thefollower’s entry by TLdb and TLd , respectively. Under the LIFO scenario, we have
TLd := inf{t ≥ s | X(t) ≤ xdL },(16)

where the default trigger xdL := γ(r − µ)CL /{(γ − 1)r}(< xdF ) depends on the leader’s
coupon payment CL (> 0). Denote by xdb
L the leader’s default trigger before the follower’s∗
entry. For the state variable X(s) ∈ [xdbL , xF ], the equity value of the leader that has

issued debt with coupon CL (> 0) is as follows:
EL (X(s), CL )∫ T db ∧T ∗LF
e−r(t−s) (1 − τ )(X(t) − CL )dt=EX(s) [s{∫ d
TF+ 1{T ∗ <T db }
F

=

TF∗

L

e−r(t−s) (1 − τ )(QL X(t) − CL )dt +

∫ Td

L

TFd

}
e−r(t−s) (1 − τ )(X(t) − CL )dt ](1 − τ )(1 − τ )CLX(s) −+ AL X(s)β + BL X(s)γ ,
r−µ
r

(17)

where ∧ and 1{} stand for min(, ) and the indicator function, respectively. CoeﬃcientsAL and BL in (17) are constants satisfying two value matching conditions, i.e.,(1 − τ ) db (1 − τ )CL
β
db γ
x −
+ AL xdb
L + BL xL = 0
r−µ L
r

(18)

and
(1 − τ ) ∗(1 − τ )CLxF −+ AL x∗F β + BL x∗F γ
r−µ
r
( )
(1 − τ )QL ∗τ CL (1 − τ )(1 − QL ) γ−1 ∗(1 − τ )CL x∗F γ=
xF ++h xF +
.
r−µ
r
r−µ
r−µ
xdL

(19)

(18) means that the equity value becomes zero on the default timing, while (19) denotes
the equity value just after the follower’s entry. In preparation for later, we calculate thedebt value
DL (X(s), CL )}{∫ d
∫ T db ∧T ∗TL
FLde−r(t−s) CL dt + e−r(TL −s) KL (xdL )e−r(t−s) CL dt + 1{T ∗ <T db }=EX(s) [
F

s

TF∗

L

+ 1{T ∗ ≥T db } e−r(TL −s) KLb (xdb
L )]

(20)

CL
+ A˜L X(s)β + B˜L X(s)γ ,
r

(21)

db

F

=

L

8

4
where KL (xdL ) and KLb (xdb
L ) represent the liquidation value after the follower’s exit andbefore the follower’s entry, respectively, while A˜L and B˜L are constants satisfying thevalue matching conditions:
CL
β
b
db
˜ db γ+ A˜L xdb
L + BL xL = KL (xL )
r
and
CL
CL
+ A˜L x∗F β + B˜L x∗F γ =
−
r
r

(

(22)

) ( ∗ )γxF
CL.
− KL (xdL )
r
xdL

(23)

(22) means that the debt value equals the liquidation value on bankruptcy, while theright-hand side of (23) corresponds to the debt value just after the follower’s entry. The
ﬁrm value is expressed as VL (X(s), CL ) := EL (X(s), CL ) + DL (X(s), CL ).
On the other hand, the follower’s ﬁrm value before the entry is as follows:
F (X(s), CL )∗=EX(s) [1{T ∗ <T db } e−r(TF −s) {VF (x∗F , CF∗ (x∗F )) − I} + 1{T ∗ ≥T db } e−r(TL −s) M (xdb
L )]
F

db

L

F

L

=AF X(s)β + BF X(s)γ ,(24)where CL is the leader’s coupon and M (xdb
L ) is the option value in a monopoly, i.e.,
)
(
1−τ
xdb
L [e−rTM
X(T
)
−
I
]
(25)
M (xdb
)
:=
sup
E
M
L
ψ(r − µ)TM(
) ( db )βxL(1 − τ ) ∗=xM − I.
(26)
ψ(r − µ)x∗M
Note that (25) results from the same argument as (14). We denote by x∗M := ψβ(r −∗µ)I/{(β − 1)(1 − τ )} and we presume that xdb
L ≤ xM . We also note that the follower’s

option value F (X(s), CL ) depends on the leader’s coupon CL . Coeﬃcients AF and BF in(24) are constants satisfying the following value matching conditions:() ( db )βxL(1 − τ ) ∗
db β
db γ
xM − IAF xL + BF xL =ψ(r − µ)x∗M
andAF x∗F β + BF x∗F γ =(1 − τ )QF ∗x − I.
ψ(r − µ) F

(27)

(28)

(27) means that the ﬁrm value becomes the monopolist’s option value if the leader leaves
the market before the follower’s entry, while the right-hand side of (28) corresponds tothe follower’s ﬁrm value in a duopoly minus investment costs.
∗
Now, we like to determine triggers xdb
L and xF . Recall that the leader’s exit trigger∗
xdb
L is determined so as to maximize its equity value (17), whereas the follower’s entry xF

is determined so as to maximize its option value (24). Then, by the ﬁrst order optimality
conditions, we impose the smooth pasting conditions
1−τ
β−1
γ−1
+ βAL xdb
+ γBL xdb
=0
L
L
r−µ
4

(29)

These liquidation values are not necessarily equivalent. One may assume non-linear forms for the functions,although we will assume the linear functions in Section 4.

9

for (18) andβAF x∗F β−1 + γBF x∗F γ−1 =(1 − τ )QF
ψ(r − µ)(30)

for (28). (29) corresponds to the fact that shareholders of the leader optimize the exit
timing considering the follower’s potential entry, while (30) corresponds to the fact thatshareholders maximize the option value on the entry timing considering the leader’s po∗
tential exit. For a ﬁxed CL , we can derive xdb
L , xF , AL , BL , AF , and BF by solving (18),(19), (27), (28), (29), and (30) simultaneously. Technically, we do not have to solve thesesix equations simultaneously. We easily obtain AL , BL , AF , and BF as functions of xdb
L
and x∗F from (18), (19), (27), and (28). Then, we substitute them into (29) and (30) and∗solve two equations with two variables xdb
L and xF . It is not diﬃcult to numerically solve

the problem of two equations with two variables if one uses the optimization toolbox in
the Matlab.
Finally, we consider the situation in which neither ﬁrms has invested. The leader issues
debt to maximize the ﬁrm value, and hence, the leader’s investment at time s leads to
debt with couponCL∗ (X(s)) = arg max VL (X(s), CL ).

(31)

CL ≥0

The leader’s expected gain is L(X(s)) := VL (X(s), CL∗ (X(s))) − I. In equilibrium, theleader invests at time
TL∗ := inf{t ≥ 0 | X(t) ≥ x∗L },(32)

where x∗L is the smallest solution to L(x∗L ) − F (x∗L , CL∗ (x∗L )) = 0, along with issuing debt
with coupon CL∗ (x∗L ). The follower’s optimal response is described by the investment
trigger x∗F and coupon CF∗ (x∗F ). The LIFO scenario presumes that CF∗ (x∗F ) > CL∗ (x∗L ).
Based on Lambrecht (2001) and Murto (2004), a ﬁrm with lower coupon payment wins
the exit game and survives longer. The follower can deviate from the best policy under
the LIFO scenario and choose the ﬁrst in, ﬁrst out (FIFO) scenario by decreasing coupon
CF below CL∗ (x∗L ). However, in our numerical analysis, the deviation always decreases thefollower’s value, and hence, we omit the FIFO scenario. If TF∗ < TLdb , the follower entersthe market at time TF∗ and the duopoly goes on until TFd . The center panel of Figure
1 depicts a sample pass for this case. Otherwise, the leader exits the monopoly market
ﬁrst, and the follower takes the same strategy as that of the monopolist. In that case, theduopoly is not realized. See the lower panel of Figure 1 for a sample pass in the case.
As a special case in which QF ↓ 0, we obtain preemptive equilibrium in Nishiharadand Shibata (2010). In that case, we immediately have x∗F = ∞, TF∗ = ∞, xdb
L = xL and
AL = A˜L = AF = 0. Suppose that K b (xdb ) := (1 − αF )(1 − τ )xdb /(r − µ) for the leader’sL

L

L

liquidation value. Similar to (12) and (14), we have CL∗ (X(s)) = r(γ −1)X(s)/{(r −µ)γh}
and L(X(s)) = (1 − τ )X(s)/{ψ(r − µ)} − I for the leader’s value. For the follower’s value10

we have
F (X(s), CL∗ (X(s))) =EX(s) [e−r(TL −s) M (xdL )]) ( d )β (()
xL
(1 − τ ) ∗X(s) γx −I=
ψ(r − µ) M
x∗M
xdL
(
)(
)β
(1 − τ ) ∗X(s)=xM − I
hγ−β
ψ(r − µ)
x∗M
d

(33)

where in (33) we used xdL = X(s)/h. In equilibrium, the leader invests as soon as(1 − τ )X(s)/{ψ(r − µ)} − I equals (33). Because of TF∗ = ∞, the follower never enters the market until the leader exits the market at time TLd . After the leader’s exit, thefollower takes the monopolistic strategy. This equilibrium corresponds to Proposition 2
in Nishihara and Shibata (2010).
We also note that joint investment equilibrium never appears in the levered case.
Suppose that both ﬁrms enter a new market and issue debt at the same time. Eachﬁrm always has an incentive to decrease its coupon payment just below the level of thecompetitor’s coupon because in this way it can survive and receive monopolistic proﬁt
ﬂows after the competitor’s exit. Then, there is no equilibrium in which levered ﬁrms
invest at the same time. Next, suppose that both unlevered ﬁrms enter a new market
at the same time. In this case, each ﬁrm always has an incentive to optimize its capital
structure and increase its own value. In any case, there is no possibility that both ﬁrms
invest at the same time in equilibrium when they can adjust debt ﬁnancing.

3.3

Firms with ﬁnancing constraints

In Section 3.2, we assume that ﬁrms have suﬃcient internal funds to cover investment
costs. In other words, debt issuance is merely a means for ﬁrms to optimize capital
structure. However, in some cases, debt issuance is a means for ﬁrms to fund investment projects. For instance, when one analyzes preemptive competition among venturebusinesses in IT industries, the lack of internal funds places one of critical limitations on
investment behavior. This section supplements the previous analysis of optimal capital
structure by studying ﬁnancially constrained ﬁrms. In the ﬁnancially constrained case, we
simply assume that investment costs must be ﬁnanced by debt. Povel and Raith (2004)
consider similar constraints, although they analyze a static model based on Brander andLewis (1986).
Under the ﬁnancing constraints, the follower’s optimal coupon needs to satisfy
DF (x∗F , CF∗ (x∗F )) ≥ I.

(34)

According to our numerical analysis, (34) always holds because of the follower’s high entry
trigger x∗F . Then, we explain how to compute equilibrium under the presumption of (34).
We do not need change the discussion from (12) to (30). However, the leader’s optimal

11

policy in the previous subsection does not necessarily satisfy the ﬁnancing constraint
DL (x∗L , CL∗ (x∗L )) ≥ I.

(35)

This is because the leader’s entry trigger x∗L is very low due to preemptive competition.
Note that debt value DL (x∗L , CL∗ (x∗L )) decrease as x∗L is lower. While Povel and Raith(2004) exogenously assume that one ﬁrm is ﬁnancially constrained and the other is unconstrained, our model endogenously determines which ﬁrm is ﬁnancially constrained inequilibrium. Indeed, in equilibrium, not the follower but the leader is ﬁnancially constrained whenever ﬁnancing constraints become binding.
We now turn to the case that (35) does not hold because equilibrium remains unchanged from Section 3.2 under (35). In the case, we need modify derivation after (30).
We must solve∗
CLC(X(s)) = arg

max

CL ≥0
DL (X(s),CL )≥I

VL (X(s), CL )(36)

∗ (X(s)))−and calculate the ﬁnancially constrained leader’s value LC (X(s)) := VL (X(s), CLC

I. Subscript C stands for the “Constrained” case. When the ﬁnancing constraints are∗ (X(s)) satisﬁes D (X(s), C ∗ (X(s))) = I. In equilibbinding, the optimal coupon CLC
LLC

rium, the leader invest at time∗
TLC:= inf{t ≥ 0 | X(t) ≥ x∗LC },(37)∗ (x∗ )) = 0, along with
where x∗LC is the smallest solution to LC (x∗LC ) − F (x∗LC , CLC
LC∗ (x∗ ). If T ∗ < T db , the follower enters the market at timeissuing debt with coupon CLC
LC
FLTF∗ and the duopoly goes on until TFd (cf. the center panel of Figure 1). Otherwise, theleader leaves the market prior to the follower’s entry (cf. the lower panel of Figure 1).

4

Numerical analysis and implications

4.1

Base caseThis section numerically examines the properties of equilibrium in the precious section.
To do this, we need specify the leader’s liquidation values, i.e., KL (xdL ) after the follower’s
exit and KLb (xdb
L ) before the follower’s entry. For simplicity, similar to Goldstein, Ju,
and Leland (2001) and Lambrecht (2001), we assume the linear functions KL (xdL ) :=bdb
(1−αL )(1−τ )xdL /(r −µ) and KLb (xdb
L ) := (1−αL )(1−τ )xL /(r −µ). Although both values

are the liquidation values in a monopoly, we should reduce the latter by taking account ofb = α /Q . The
the potential entry of the other ﬁrm. For simplicity, we assume that αL
LLbase parameter values are set as follows:r = 0.08, µ = 0.06, σ = 0.2, τ = 0.15, αL = αF = 0.3, QL = QF = 0.5.

12

(38)

The parameter values, except for QL and QF , are similar to Leland (2004) and Sarkar(2008) based on empirical evidence. The investment costs I and initial value x = X(0) arenot substantial because they can be normalized. For expositional purpose, we set I = 10
and x = 0.2.
[Insert Figure 2 here]∗ (X(s)) in the levered
The upper panel of Figure 2 presents LC (X(s)) and F (X(s), CLC

case, whereas the lower panel presents LU (X(s)) and FU (X(s)) in the unlevered case. To
check whether ﬁnancing constraints are binding, the center panel shows the leader’s debt
∗ (X(s)). D (X(s), C ∗ (X(s)) is equal to I(= 10) when X(s) is lower thanDL (X(s), CLC
L
LC

0.54. In this region, the ﬁnancing constraints bind the leader, and then the leader needs
adjust the debt issuance to meet the investment costs. Especially for X(s) <= 0.4,
the leader under the ﬁnancing constraints cannot ﬁnance the project because we have
maxCL DL (X(s), CL ) < I. Thus, in the upper panel, we show the leader’s and follower’s
values only for the region X(s) > 0.4. In preemptive equilibrium, the leader’s trigger x∗LC∗ (x∗ )) = 0. The trigger is equal to 0.65,
is the smallest solution to LC (x∗LC ) − F (x∗LC , CLC
LC

for which the ﬁnancing constraints are not binding. Accordingly, we have x∗LC = x∗L =0.65, i.e., ﬁnancing constraints are not binding in the base case.
As noted in Section 3, we need check that the follower does not prefer the strategic
choice of the FIFO scenario. When the leader’s strategy is ﬁxed, we compute the optimal∗ (x∗ ) = 1.03. In
policy of the follower issuing debt with a coupon that is lower than CLC
LC

the FIFO scenario, the follower invests at the trigger 1.97 with issuing debt with coupon
just below 1.03. The follower also invests when X(t) decreases below 0.4, which is theleader’s exit trigger in duopoly under the assumption of the FIFO scenario. However,
the optimal policy in the FIFO scenario leads to the value 9.14, which is lower than
∗ (x∗ )) = 9.31 in the LIFO scenario, and then the follower has no incentive to
F (x∗LC , CLC
LC

deviate from the LIFO scenario. Throughout our numerical analysis, the follower never∗ (x∗ ) is low
likes the FIFO scenario mainly because the leader’s coupon payment CLC
LC

due to the the early entry, whether ﬁnancial constraints bind the leader or not. Then, we
have the ﬁrst observation as follows.
Observation 1 In equilibrium, the LIFO scenario usually happens.
This result is consistent with ﬁndings by Lambrecht (2001) and Zhdanov (2008). Lambrecht (2001) shows that the LIFO scenario prevails, except for the case in which theleader has an extraordinary debt to repay. Zhdanov (2008) shows that the LIFO scenario
happens because the leader’s preemptive entry makes its coupon payment lower than thefollower’s. Observation 1 is also in line with the standard ﬁndings that more establishedﬁrms with lower debt repayment are more likely to survive.
[Insert Table 1 here.]13

In equilibrium, we have Table 1. “Value” denotes the option value, which is evaluatedfor an initial value x = 0.2. Note that the option value is the same for both ﬁrms because
in equilibrium the leader invests on the preemptive entry timing in which the ﬁrst-moveradvantage is oﬀset by the follower’s option value. The ﬁrm value in the levered case ishigher than that of the unlevered case. This is because both ﬁrms can optimize the capital
structure. We can see from “Entry” of Table 1 that the entry triggers for both levered
leader and follower are lower than those of the unlevered ﬁrms.
Observation 2 Compared to the unlevered case, levered ﬁrms’ entry takes place earlier
and levered ﬁrms’ values become higher.
Access to debt ﬁnancing increases ﬁrm value, and thus levered ﬁrms can invest earlier thanunlevered ﬁrms. The leverage eﬀects are similar to that of the seminal work by Myers(1977). The same results are seen in the real options studies, including Hennessy (2004),
Mauer and Sarkar (2005), and Sundaresan and Wang (2007a), based on the monopoly
setup. Observation 2 complements the existing literature by showing the robustness of
the leverage eﬀects in the presence of preemptive competition.
In “Exit” for the leader of Table 1, we present two exit triggers xdL and xdb
L . We always
have xdL < xdb
L . This can be naturally explained by the leader who faces the follower’s
potential entry as being more likely to exit compared to the case with no fear. If X(t)hits 0.28 ﬁrst, then the leader leaves the market prior to the follower’s entry. In this case,the follower chooses the monopolist’s entry trigger 1.13 and exit trigger 0.56. In the basecase, the probability that the leader exits before the follower’s entry is approximately 30%
according the simulations.
“Leverage,” “Credit spread,” and “Coupon” of Table 1 show the values on the entry
timing. In other words, we present DL (x∗L , CL∗ (x∗L ))/VL (x∗L , CL∗ (x∗L )), CL∗ (x∗L )/DL (x∗L , CL∗ (x∗L ))−r, and CL∗ (x∗L ) for the leader and CF∗ (x∗F )/DF (x∗F , CF∗ (x∗F ))−r, DF (x∗F , CF∗ (x∗F ))/VF (x∗F , CF∗ (x∗F ))and CF∗ (x∗F ) for the follower. As ﬁnancing constraints are not binding for either ﬁrm, theleverage and credit spreads are optimal. The leader’s values are slightly lower than thefollower’s values, which are equal to the optimal levels in a monopoly. This is probably because the leader decreases the level of coupon payment taking into account thefollower’s potential entry. This ﬁnding is consistent with that of Zhdanov (2008). It isalso consistent with MacKay and Phillips (2005) who empirically showed that the average leverage ratio of incumbents is lower than that of new entrants. Although Table 1
shows the leader’s leverage on the leader’s entry trigger 0.65, the leader’s leverage on thefollower’s entry trigger 2.27 is equal to 0.20, which is much smaller than the follower’s∗ (x∗ ) = 1.03 is muchleverage 0.66. This is because the leader’s coupon payment CLC
LC

smaller than the follower’s coupon payment 2.94. There are many empirical ﬁndings thathighly levered ﬁrms suﬀer from competitive disadvantage (e.g., Phillips (1995), Chevalier(1995a), Chevalier (1995b)). Observation 1 is also in line with the ﬁndings.

14

4.2

Eﬀects of ﬁrst-mover advantage

In this section, we examine the eﬀects of ﬁrst-mover advantage from two aspects. First,
following Pawlina and Kort (2006), we consider the symmetric case, i.e., QL = QF , andchange the level of QL = QF . A decrease in the level of QL = QF denotes the proﬁt
decrease in a duopoly, increasing the incentive for ﬁrms to move ﬁrst and monopolize
proﬁt ﬂows until the follower’s entry. To examine preemptive equilibrium, we focus only
on the case of negative externalities, i.e., QL = QF < 1.
[Insert Figures 3 and 4 ]Figures 3 and 4 plot ﬁrm values, entry and exit triggers, leverage, and credit spreadfor varying levels of QL = QF . The other parameter values are set at the base case (38).
Throughout the paper, except for ﬁrm values, the left panels represent the leader’s values,
whereas the right panels represent the follower’s values. In all ﬁgures, the triangle, cross,and circle marks in the panels denote the levered case with ﬁnancing constraints, thelevered case without a constraint, and the unlevered case, respectively. When QL = QF
is lower than 0.4, ﬁnancing constraints become binding for the ﬁrst mover. For any
QL = QF , as noted in Section 3, the ﬁnancing constraints are not binding for the follower.
A proﬁt decrease in a duopoly –in other words, an increase in competition in a duopoly–
forces the leader to suﬀer from the constraints.
We can see from the top panel that a decrease in QL = QF decreases ﬁrm values
in any cases. This result is natural because the proﬁt decrease in a duopoly damages
ﬁrm values. More interestingly, we ﬁnd that the decrease in ﬁrm values is mitigated byﬁnancing constraints. Indeed, for QF = QL = 0.2 and 0.3 in the top panel of Figure 3 ,ﬁrm values in the constrained case are highest among three cases. If ﬁrms face no ﬁnancing
constraints, a decrease in QL = QF straightforwardly intensiﬁes preemptive competition
and greatly reduces ﬁrm values in equilibrium. However, the ﬁnancing constraints, which
increase the ﬁrst mover’s leverage beyond the optimal level, decrease the leader’s value
while they enhance the follower’s value of the option to invest after the leader’s exit.
This eﬀect of high leverage of the leader on the potential entrant is the same as that ofLambrecht (2001), although he exogenously assumes an incumbent with debt repayment.
Thus, the ﬁnancing constraints decrease ﬁrst-mover advantage and moderate preemptivecompetition. Firm values with ﬁnancing constraints become higher than that with noﬁnancing constraints. A similar mechanism is explained in Nishihara and Shibata (2010),
although the previous model focuses only on the polar case (QF = 0). Especially for a low
level of QF = QL , the gap between ﬁrm values in the constrained and unconstrained cases(ﬁnancing constraint eﬀect) is larger than the gap between ﬁrm values in the unconstrained
and unlevered cases (leverage eﬀect).
We turn to entry and exit triggers (see center and lower panels in Figure 3). A decrease
in QL = QF decreases the leader’s entry trigger and increases the follower’s entry trigger.

15

The former is due to the intensiﬁed preemptive competition, while the latter is due to thedecreased proﬁt of the follower. In the center panels of Figure 3, the entry triggers in thelevered case without a ﬁnancing constraint are always lower than those in the unlevered
case. As mentioned in Section 4.1, this is due to the leverage eﬀect. However, withﬁnancing constraints, which weaken preemptive competition, the leader’s entry trigger
can increase beyond the level of the unlevered case (see QL = QF = 0.2 in the center-leftpanel). In this case, the ﬁnancing constraint eﬀects overwhelm the leverage eﬀects. For
QL = QF = 0.2 and 0.3 in the lower-left panel of Figure 3, we see that the default triggers
in the case with ﬁnancing constraints are higher than those without a ﬁnancing constraint.
This is because the ﬁnancing constraints impose higher leverage on the leader. On theother hand, whether the leader suﬀers from the ﬁnancing constraints little inﬂuences thefollower’s entry and exit triggers. Note that the follower’s exit trigger always satisﬁes
xdF = x∗F /h because of its optimal capital structure.
We now examine leverage and credit spreads (see Figure 4). We see from the rightpanels that the follower’s leverage and credit spreads do not depend on the level of QL =QF . Because ﬁnancing constraints never bind the follower, the follower maintains the
optimal capital structure in which leverage is 0.66 and credit spread is 0.0047. On theother hand, the leader’s leverage and credit spread depend on the level of QL = QF .
Without a ﬁnancing constraint, a decrease in QL = QF slightly decreases the leverageand credit spread. This is because the leader lowers the leverage and credit spread byconsidering the potential decrease in proﬁt ﬂows in a duopoly. With ﬁnancing constraints
binding, a decrease in QL = QF adversely increases the leader’s leverage and credit spread.
This means that a decrease in QL = QF forces the leader to take riskier capital structure.
The leader’s leverage being higher than the follower’s is not necessarily inconsistent with
MacKay and Phillips (2005). Indeed, if one sees the leader’s leverage on the follower’s
entry trigger, it remains around 0.2, which is much lower than the follower’s leverage.
Next, we examine the other aspect of the ﬁrst-mover advantage. We ﬁx QL + QF = 1
and change the leader’s share QL /(QL + QF ) in a duopoly. This measures the degreeof barriers to the follower’s entry. For instance, customers accustomed to the leader’s
product might prefer the leader’s brand to the follower that has the same quality. This
sort of asymmetric case is also treated in Lambrecht (2001) and Kong and Kwok (2007).
[Insert Figures 5 and 6]
Figures 5 and 6 plot ﬁrm values, entry and exit triggers, leverage, and credit spreads
for varying levels of QL . The other parameter values are set at the base case (38).
Overall, we ﬁnd that an increase in QL in Figures 5 and 6 leads to the similar eﬀects to
a decrease in QL = QF in Figures 3 and 4. The eﬀects are much stronger because thelevel of QL /(QL + QF ) changes the ﬁrst-mover advantage more directly than the level of
QL = QF . Indeed, ﬁnancing constraints become binding when QL is larger than 0.55. As
mentioned, the ﬁnancing constraints can play a positive role in moderating preemptive

16

competition and increasing the ﬁrm value. See the top panel of Figure 5. Especiallyfor QL = 0.7, 0.8, and 0.9, the value enhanced by the ﬁnancing constraints are greaterthan the value enhanced by the leverage eﬀects. An increase in QL , which increases theﬁrst-mover advantage, decreases the leader’s entry trigger, but the decrease is mitigatedby the ﬁnancing constraints. The leader’s entry trigger can be either higher or lower thanthat of the unlevered case depending on the trade-oﬀ between the ﬁnancing constraint
and leverage eﬀects. Although the ﬁnancing constraints increase ﬁrm values, they makethe leader’s investment riskier in terms of the leverage and credit spread.
Finally, we summarize interesting ﬁndings as follows:
Observation 3 Financing constraints can delay the leader’s preemptive entry and improve ﬁrm values. When the ﬁnancing constraint eﬀects dominate the leverage eﬀects, theleader’s entry trigger can be higher than that of the unlevered case.
The eﬀect of ﬁnancing constraints on the investment trigger is similar to the standard
result that ﬁnancially constrained ﬁrms invest less than unconstrained ﬁrms (e.g., Fazzari,
Hubbard, and Petersen (1988), Hubbard (1998)). Although Nishihara and Shibata (2010)
point out the possibility that ﬁnancing constraints can soften preemptive competition andimprove ﬁrm values, we extend the previous setup (QF = 0) into a more general setup(QF > 0) and show that the positive eﬀect can arise in such a general case. Indeed,
we ﬁnd that, with slight ﬁrst-mover advantage, ﬁnancing constraints can be binding andplay a positive role. Thus, the empirical implications from this paper are not limited to
extremely intensiﬁed competition but also apply to a wide range of market competition.
Observation 4 Stronger ﬁrst-mover advantage speeds up the leader’s preemptive entry
and reduces ﬁrm values. This ﬁrst-mover advantage eﬀect can be greatly mitigated by theﬁnancing constraint eﬀects that increase with stronger ﬁrst-mover advantage.
The ﬁrst sentence is in line with the standard result in preemptive competition (e.g., Huisman (2001), Pawlina and Kort (2006)). The interactions between ﬁnancing constraints
and ﬁrst-mover advantage generate another empirical prediction. There are a number of
papers that study the relation between leverage and product market competition, though
few papers consider preemptive competition. The majority of research in this area shows
empirical evidence that more leverage can lead to weaker product market competition(e.g., Phillips (1995), Chevalier (1995a), Chevalier (1995b)). These ﬁndings support our
results of the ﬁnancing constraint eﬀects. Indeed, in our paper, ﬁnancing constraints
increase the leader’s leverage and soften preemptive competition, Lambrecht (2001) also
shows that high leverage of an incumbent moderates competition, but the paper does notanalyze preemptive equilibrium. We complement the previous research by showing thatsimilar results hold in preemptive equilibrium.

17

4.3

Eﬀects of volatility σ

In this section, we analyze comparative statics with respect to the other key parameterσ. As our model includes strategic interactions, leverage, and ﬁnancing constraints, welike to focus on how the market volatility inﬂuences those factors.
[Insert Figures 7 and 8]
Figures 7 and 8 plot ﬁrm values, entry and exit triggers, leverage, and credit spreads forvarying levels of σ. The other parameter values are set at the base case (38). In all panels,
we see that the dot marks overlap the cross marks. Actually, ﬁnancing constraints are
not binding for any σ.
We see from the top panel of Figure 7 that the ﬁrm value increases with σ in bothlevered and unlevered cases. In the center (lower) panels, we also see that the entry (exit)
triggers increase (decrease) with σ in all cases. The standard real option theory argues
that higher uncertainty increases a ﬁrm’s option value and delays the exercise of its option.
Our results regarding entry and exit triggers are in line with the standard theory. Most of
the papers including Pawlina and Kort (2006) and Nishihara and Shibata (2010) show thesame result in preemptive equilibrium. We conclude that the standard results are robust
even if preemptive competition and leverage are taken into account.
In the center panels of Figure 7, we ﬁnd another interesting property. The diﬀerence
between the follower’s entry triggers in the levered and unlevered cases clearly decreases
with σ, although the diﬀerence in the leader’s entry triggers scarcely changes. Sincea higher σ increases the option value of waiting for the leader’s exit, the follower has
more incentive to delay the entry. For σ = 0.4, this eﬀect becomes almost as strong as
the leverage eﬀect, and hence the diﬀerence in the levered and unlevered cases becomes
nearly zero.
Figure 8 depicts the leverage and credit spreads on the entry timing. In all panels, we
ﬁnd that the leverage monotonically decreases with σ while the credit spread monotonically increases with σ. Because debt becomes riskier under high uncertainty, ﬁrms reduce
debt issuance. These results are consistent with the standard results in the absence ofstrategic interactions (e.g., Leland (1994), Sundaresan and Wang (2007a)) as well as being
consistent with empirical ﬁndings (e.g., Bradley, Jarrell, and Kim (1983)).
Now, we explore the eﬀects of σ on ﬁnancing constraints. To do this, we replace
QL = QF = 0.5 with QL = QF = 0.4 in (38).
[Insert Figures 9 and 10]
Figures 9 and 10 show ﬁrm values, entry and exit triggers, leverage, and credit spreads forvarying levels of σ. We see that ﬁnancing constraints become binding when σ increases
beyond 0.2. A higher σ intensiﬁes the ﬁnancing constraint eﬀects. This result is similar
to that of Nishihara and Shibata (2010). The intuition is as follows. For a higher σ, the18

optimal leverage (see the upper-left panel of Figure 10) becomes lower and so does thedebt issuance. When σ exceeds a threshold level (in this example, 0.2), the debt value
under the optimal capital structure falls short of the investment costs. Then, the ﬁnancing
constraints become binding under higher uncertainty about the market demand.
Observation 5 A higher volatility delays ﬁrms’ entry and improve ﬁrm values. This
volatility eﬀect can be slightly magniﬁed by the ﬁnancing constraint eﬀects that increase
with a higher volatility.
Although a higher σ increases ﬁrm values and entry triggers in all cases, the ﬁnancing
constraint eﬀects amplify the volatility eﬀect for levered ﬁrms with ﬁnancing constraints.
Recall that, as explained in Section 4.2, ﬁnancing constraints alleviate preemptive competition, thus increasing the leader’s entry trigger and ﬁrm values. In Figures 9 and 10,
the ﬁnancing constraint eﬀects are not very large compared to Figures 3–6. Indeed, forσ = 0.25, 0.3, 0.35, and 0.4, the ﬁrm value enhanced by the ﬁnancing constraints are muchsmaller than the value enhanced by the leverage eﬀects (see the top panel of Figure 9).
We conclude that the ﬁnancing constraint eﬀects can be caused by a higher σ but they
are relatively small.

5

Conclusion

In this paper, we shed light on how ﬁrms’ ﬁnancial structure inﬂuences preemptive competition. We found the LIFO scenario, i.e., the follower exits the duopoly market prior to
the leader. This is primarily because, due to the entry lag, the follower issues much moredebt than the leader. We also showed that the well-known leverage eﬀects remain true inpreemptive equilibrium. Actually, access to debt ﬁnancing increases ﬁrm values and accelerates investment. This paper examined not only the case with optimal capital structure
but also the case with ﬁnancing constraints that require ﬁrms to ﬁnance investment costs
by debt. Notably, we showed that the ﬁnancing constraints can delay preemptive investment and improve ﬁrm values in equilibrium. Indeed, the leader’s leverage increased by
the ﬁnancing constraints could lower ﬁrst-mover advantage and mitigate preemptive competition. The ﬁnancing constraint eﬀects increase in a market with stronger ﬁrst-moveradvantage and higher volatility. Especially in the presence of relatively strong ﬁrst-moveradvantage, the ﬁnancing constraint eﬀects can potentially dominate the leverage eﬀects
and then preemptive investment occurs later than that in the unlevered case. Our ﬁndings
are in line with empirical evidence that high leverage leads to competitive disadvantage
and mitigates product market competition.

19

Acknowledgment
This work was supported by KAKENHI (23310103, 21241040).

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21

Table 1: Base case.
Value Entry
Exit
Leverage Credit spread
Levered caseCoupon2.18

Leader

0.65

0.20/0.28

0.63

0.00391.03

Follower

2.27

1.12

0.66

0.0047

2.94

Unlevered case

1.90

Leader0.77

Follower

2.46

22

3
X(s)
2.5
x∗

FU

X(s)

2
1.5
1
x∗LU
0.5
0

0

T∗

LU

2

4

∗

6

8 TFU

10

∗
6 TF

8

10

8

10

Time s

3
X(s)

X(s)

2.5
x∗F
2
1.5
1
x∗L
0.5
xdb
L
0

0

T∗L

2

4
Time s

3
X(s)

X(s)

2.5
x∗F
2
1.5
1
x∗L
0.5
xdb
L
0

0

2

∗

db

4 TL
6 TL
Time sFigure 1: Simulation. We simulated X(s) with the base parameter values (38) in Section 4. The
top panel shows the leader and follower’s entry triggers in the unlevered case. The center panel
corresponds to the case in which the duopoly is realized, while the lower panel corresponds tothe case in which the leader exits the market before the follower’s entry.

23

25
LC(X(s))∗FC(X(s),CLC(X(s)))

Firm value

20
15
10
5
0
0.2

0.4

0.6

0.8

1

1.2

0.8

1

1.2

0.8

1

1.2

X(s)
25
∗

DL(X(s),CL(X(s)))

Debt value

20
15
10
5
0
0.2

0.4

0.6
X(s)

25
LU(X(s))FU(X(s))Firm value

20
15
10
5
0
0.2

0.4

0.6
X(s)

Figure 2: The base case. The top and bottom panels represent the leader and follower’s values
in the levered case with financing constraints and the unlevered case, respectively. The center∗
panel show the leader’s debt value DL (X(s), CLC(X(s)).

24

3.5

Firm value

3
2.5
2
1.5
1
0.5

0.2

0.3

0.4
0.5
QL=QF

0.6

0.7

0.2

0.3

0.4
0.5
QL=QF

0.6

0.7

0.2

0.3

0.4
0.5
QL=QF

0.6

0.7

1
6
5

0.8

Entry trigger

Entry trigger

0.9

0.7
0.6

4
3

0.5
2
0.4

0.2

0.3

0.4
0.5
QL=QF

0.6

0.7

0.45

3

0.4
2.5
Exit trigger

Exit trigger

0.35
0.3
0.25

2
1.5

0.2
1
0.15
0.1

0.2

0.3

0.4
0.5
QL=QF

0.6

0.5

0.7

Figure 3: Option values, entry triggers, and exit triggers with varying levels of QL = QF . The
other parameter values are set at the base case (38). The left panels represent the leader’svalues, whereas the right panels represent the follower’s values. The triangle, cross, and circle
marks in the panels denote the levered case with financing constraints, the levered case without
a constraint, and the unlevered case, respectively. In the lower-left panel, triangle (downpointing triangle) and cross (square) marks represent the leader’s exit triggers after (before)
the follower’s entry in the constrained and unconstrained cases, respectively.

25

0.8

0.75

0.75

0.7

0.7

Leverage

Leverage

0.8

0.65
0.6

0.65
0.6

0.55

0.2

0.3

0.4
0.5
QL=QF

0.6

0.55

0.7

0.01

0.01

0.3

0.4
0.5
QL=QF

0.6

0.7

0.2

0.3

0.4
0.5
QL=QF

0.6

0.7

Spread0.015

Spread

0.015

0.2

0.005

0

0.005

0.2

0.3

0.4
0.5
QL=QF

0.6

0

0.7

Figure 4: Leverage and credit spreads with varying levels of QL = QF . The other parametervalues are set at the base case (38). The left panels represent the leader’s values, whereas theright panels represent the follower’s values. The triangle, cross, and circle marks in the panels
denote the levered case with financing constraints, the levered case without a constraint, andthe unlevered case, respectively.

26

Firm value

2

1.5

1

0.5
0.5

0.6

0.7
QL

0.8

0.9

14

0.8

12
Entry trigger

Entry trigger

0.7
0.6
0.5
0.4

0.5

0.6

0.7
QL

0.8

8
6

2

0.9

0.35

6

0.3

5
Exit trigger

Exit trigger

10

4

0.3
0.2

0.25
0.2
0.15
0.1

0.9

0.5

0.6

0.7
QL

0.8

0.9

0.5

0.6

0.7
QL

0.8

0.9

4
3
2

0.5

0.6

0.7
QL

0.8

1

0.9

Figure 5: Option values, entry triggers, and exit triggers with varying levels of QL . The
other parameter values are set at the base case (38). The left panels represent the leader’svalues, whereas the right panels represent the follower’s values. The triangle, cross, and circle
marks in the panels denote the levered case with financing constraints, the levered case without
a constraint, and the unlevered case, respectively. In the lower-left panel, triangle (downpointing triangle) and cross (square) marks represent the leader’s exit triggers after (before)
the follower’s entry in the constrained and unconstrained cases, respectively.

27

0.8

0.8
Leverage

0.85

0.75

0.7

0.65

0.65

0.6

Spread

0.75

0.7

0.5

0.6

0.7
QL

0.8

0.6

0.9

0.02

0.02

0.015

0.015
SpreadLeverage

0.85

0.01

0.005

0

0.5

0.6

0.7
QL

0.8

0.9

0.5

0.6

0.7
QL

0.8

0.9

0.01

0.005

0.5

0.6

0.7
QL

0.8

0

0.9

Figure 6: Leverage and credit spreads with varying levels of QL . The other parameter values are
set at the base case (38). The left panels represent the leader’s values, whereas the right panels
represent the follower’s values. The triangle, cross, and circle marks in the panels denote thelevered case with financing constraints, the levered case without a constraint, and the unlevered
case, respectively.

28

Firm value

3

2.5

2

1.5

0.1

0.2

0.3

0.4

σ
1.3

4.5

1.2

4
Entry trigger

Entry trigger

1.1
1
0.9
0.8

3.5
3
2.5

0.7
2

0.6
0.5

0.1

0.2

0.3

1.5

0.4

0.1

0.2

σ
0.45

1.4

0.4

1.35

Exit trigger

Exit trigger

0.4

0.3

0.4

1.3

0.35
0.3
0.25
0.2

1.25
1.2
1.15
1.1

0.15
0.1

0.3
σ

1.05
0.1

0.2

0.3

1

0.4

0.1

σ

0.2
σ

Figure 7: Option values, entry triggers, and exit triggers with varying levels of σ. The otherparameter values are set at the base case (38). The left panels represent the leader’s values,
whereas the right panels represent the follower’s values. The triangle, cross, and circle marksin the panels denote the levered case with financing constraints, the levered case without aconstraint, and the unlevered case, respectively. In the lower-left panel, triangle (down-pointing
triangle) and cross (square) marks represent the leader’s exit triggers after (before) the follower’s
entry in the constrained and unconstrained cases, respectively.

29

0.9

0.8

0.8

0.7

0.7

Leverage

Leverage

0.9

0.6
0.5

0.6
0.5

0.4

0.1

0.2

0.3

0.4

0.4

0.1

0.2

0.025

0.025

0.02

0.02

0.015

0.015

0.01
0.005
0

0.3

0.4

0.3

0.4

σ

SpreadSpread

σ

0.01
0.005

0.1

0.2

0.3

0

0.4

σ

0.1

0.2
σ

Figure 8: Leverage and credit spreads with varying levels of σ. The other parameter values are
set at the base case (38). The left panels represent the leader’s values, whereas the right panels
represent the follower’s values. The triangle, cross, and circle marks in the panels denote thelevered case with financing constraints, the levered case without a constraint, and the unlevered
case, respectively.

30

Firm value

2.5

2

1.5

1

0.1

0.2

0.3

0.4

1.1

5.5

1

5

0.9

4.5
Entry trigger

Entry trigger

σ

0.8
0.7

4
3.5

0.6

3

0.5

2.5

0.4

0.1

0.2

0.3

2

0.4

0.1

0.2

σ

0.3

0.4

0.3

0.4

σ

0.4

1.7

0.35
1.6
Exit trigger

Exit trigger

0.3
0.25
0.2

1.5
1.4

0.15
1.3

0.1
0.05

0.1

0.2

0.3

0.4

0.1

σ

0.2
σ

Figure 9: Option values, entry triggers, and exit triggers with varying levels of σ. We set
QL = QF = 0.4. The other parameter values are set at the base case (38). The left panels
represent the leader’s values, whereas the right panels represent the follower’s values. The
triangle, cross, and circle marks in the panels denote the levered case with financing constraints,
the levered case without a constraint, and the unlevered case, respectively. In the lowerleft panel, triangle (down-pointing triangle) and cross (square) marks represent the leader’sexit triggers after (before) the follower’s entry in the constrained and unconstrained cases,
respectively.

31

0.9

0.8

0.8

0.7

0.7

Leverage

Leverage

0.9

0.6
0.5

0.6
0.5

0.4

0.1

0.2

0.3

0.4

0.4

0.1

0.2

0.025

0.025

0.02

0.02

0.015

0.015

0.01
0.005
0

0.3

0.4

0.3

0.4

σ

SpreadSpread

σ

0.01
0.005

0.1

0.2

0.3

0

0.4

σ

0.1

0.2
σ

Figure 10: Leverage and credit spreads with varying levels of σ. We set QL = QF = 0.4. The
other parameter values are set at the base case (38). The left panels represent the leader’svalues, whereas the right panels represent the follower’s values. The triangle, cross, and circle
marks in the panels denote the levered case with financing constraints, the levered case without
a constraint, and the unlevered case, respectively.

32