Discussion Papers In Economics
And Business
Preemptive competition between two firms with different
discount rates

Michi NISHIHARA

Discussion Paper 20-04

February 2020

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANPreemptive competition between two firms with diﬀerent
discount rates∗
Michi NISHIHARA†,

Abstract
This paper studies a real options duopoly game between two firms with diﬀerent time
discount rates. I derive the order of investments, investment thresholds, and firm values
in equilibrium. With no cost disadvantage, the patient firm enters the market earlier andgains more value than does the impatient opponent. When the patient firm has a costdisadvantage, the order of market entry can depend on the market characteristics. Witha weaker first-mover advantage, higher market volatility, and lower market growth rate,
the impatient firm is more likely to be the first mover. Notably, the patient firm canearn more than the impatient firm, even though the patient firm enters the market later.
These results are consistent with empirical findings on market entry timing.
JEL Classifications Code: C73; G31; L13.
Keywords: real options, time discounting, market entry timing, preemption.

∗

This version was written on 12 February, 2020. The author thanks Elmar Lukas and Stefan Kupfer for

helpful comments. The author also thanks the participants at the 2019 INFORMS international conference in
Cancun and OR 2019 in Dresden for their helpful feedback. This work was supported by the JSPS KAKENHI(Grant number JP17K01254, JP17H02547).
†
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

1

1Introduction

Some firms pioneer new markets to gain the first-mover advantage, while other firms
enter the market later as followers. What are the determinants of the order of market
entry and firm value in this preemptive competition? To answer this question, several
studies combine the timing game (cf. Fudenberg and Tirole (1985)) and the real options
model (cf. Dixit and Pindyck (1994)) to develop preemption games under market demand
uncertainty (see Azevedo and Paxson (2014) for a review). For instance, Huisman (2001),
Pawlina and Kort (2006), and Kong and Kwok (2007) show that firms’ cost and payoﬀ
structures can determine the order of market entry and firm values in duopoly markets.
To the best of my knowledge, however, no study investigates the role of discount rates
in preemptive competition. This study develops the preemption game in which two firms
diﬀer in their discount rates and determines the eﬀects of the discount rates on the market
entry order and firm values in equilibrium.
Actually, firms can use diﬀerent discount rates depending on their characteristics,
even if they compete in the same market. As Harrington (1989) explains, more diversified
and less financially constrained firms tend to have lower discount rates. He examines thecollusion model among firms with diﬀerent discount rates. Harris and Siebert (2017) show
empirically that firms have diﬀerent discount rates and that merger outcomes depend on
the acquirers’ discount rates. In addition, from the behavioral viewpoint, many studies
show the dispersion of time discounting and preferences (see Frederick, Loewenstein, andO’Donoghue (2002) for a review). In the real options context, Ebert, Wei, and Zhou (2018)
investigate the investment timing of a group whose members diﬀer in their discount rates,
while Grenadier and Wang (2007) and Luo, Tian, and Yang (2020) examine the eﬀects oftime-inconsistent preferences on investment timing.
The present study develops the following preemption game to explore the roles of
discount rates in preemptive competition. Two firms with diﬀerent discount rates consider
entering a new market. Throughout the paper, I refer to the firms with the low and highdiscount rates as firms P and I, respectively, where the notations P and I represent thepatient and impatient firms, respectively. The firms can receive stochastic cash flows after
they enter the market by paying sunk costs. Cash flows decrease in the number of firms inthe market, and hence there is first-mover advantage in that the first entrant can receive
monopolistic cash flows until the other firm’s entry. The firms have complete information
about each other and optimize their market entry timing from strategic considerations.
I derive the market entry order, investment thresholds, and firm values endogenously in2

equilibrium. The results are explained below.
Compared to firm I, firm P has both higher net present value (NPV) and the option
value of waiting. Firm P may enter the market earlier because of its higher NPV, whilefirm P may enter the market later because of its higher value of waiting. When firm P has
no cost disadvantage, firm P always enters the market earlier. In other words, the NPV
eﬀect dominates the option eﬀect, and hence firm P becomes the first mover. Naturally,
firm P has a higher firm value than firm I does.
These results are similar to the standard result that the firm with an advantage ininvestment costs and/or project payoﬀs invests earlier and earns more than a disadvantageous opponent does (e.g., Chapter 8 of Huisman (2001) and Pawlina and Kort (2006)).
In fact, in the model in this paper, firm P has both the NPV and option value advantages.
Several empirical findings also support the results. For example, Robinson, Fornell,and Sullivan (1992) show that firms with more finance skills tend to enter markets earlier.
Clearly, more finance skills relax financial constraints, which leads to lower discount rates.
Schoenecker and Cooper (1998) show that larger firms tend to enter markets earlier. Notethat larger firms tend to have lower discount rates due to their more diversified project
portfolios and less financial constraints (cf. Harrington (1989)). This study can thus
potentially explain these observations from the discount rates viewpoint.
I show more notable results when firm P has a higher investment cost than firm I does.
With a small cost disadvantage, firm P enters the market earlier and earns more thanfirm I does, while with a large cost disadvantage, firm P enters the market later and earns
less than firm I does. In the intermediate cases, the order of market entry depends onthe market characteristics. Indeed, a weaker first-mover advantage, higher volatility, andlower growth rate increase the possibility that firm P becomes the second mover because
these parameters increase the option eﬀect relative to the NPV eﬀect. Moreover, firm P
can have a higher firm value even when it chooses to be the second mover.
These results contrast with the standard result that regardless of market characteristics, an advantageous firm becomes the leader and earns more than does a disadvantageous
opponent (e.g., Chapter 8 of Huisman (2001) and Pawlina and Kort (2006)). On the other
hand, Kong and Kwok (2007), Shibata (2016), and Lambrecht (2001) report similar results. For instance, Kong and Kwok (2007) show that in a duopoly market with both costand cash flow asymmetry, the level of volatility aﬀects the market entry order. Shibata(2016) shows that in a triopoly market with cost asymmetry, the firm with the second lowest cost can be the first mover, especially under higher volatility and a weaker first-mover3

advantage. Lambrecht (2001), who examines market exit timing in a duopoly market,
shows that the exit order can depend on the interest rate and volatility levels. Comparedto these studies, I focus on the diﬀerence in discount rates and explain the results by thetradeoﬀ between the NPV and option eﬀects.
The results could be also related to several observations in case studies. For instance,
Shibata (2016) demonstrates an example in which a smaller and more financially constrained firm moved earlier in the wireless telecommunication market in Japan. Shibata(2016) states that the degree of first-mover advantage in the market is weak, which isconsistent with my comparative statics result. Shackleton, Tsekrekos, and Wojakowski(2004) states that Airbus became the first mover in the very large aircrafts market, although compared to its competitor, Boeing, it is smaller and more financially constrained.
High uncertainty in the market and lower costs due to subsidies for Airbus are consistent
with my comparative statics results.
This paper proceeds as follows. Section 2 introduces the model setup. Section 3 derives
the equilibrium solution in the model. In Section 4.1, I explore the case with symmetricinvestment costs, while in Section 4.2, I explore the case with asymmetric investment
costs. Section 5 discusses the empirical implications. Finally, I conclude in Section 6 anddescribe some potential extensions.

2

Model setup

Consider firms P and I, which intend to enter a new market. Firm P (i.e., the patientfirm) has the discount rate ρP , while firm I (i.e., the impatient firm) has the discountrate ρI , where 0 < ρP < ρI . To enter the market, firm i ∈ {P, I} must pay the initialinvestment cost Ki . When only one of the firms is active in the market, the active firm
receives instantaneous cash flows X(t). When both firms are active in the market, eachfirm receives instantaneous cash flows DX(t), where D is a constant satisfying 0 ≤ D ≤ 1,
which indicates negative externalities. Following the standard literature (e.g., Dixit andPindyck (1994)), cash flows X(t) follow a geometric Brownian motiondX(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 filtered probability space(Ω, F, P, {Ft }) and µ, σ(> 0) and x(> 0) are constants. For convergence, I assume thatµ < ρP . The initial value x is suﬃciently small that each firm must wait for its entry

4condition to be met. Each firm maximizes the expected value discounted by its own
discount rate and has full information about X(t), D, ρi , and Ki (i = P, I).
Except the assumption of ρP ̸= ρI , the model follows the standard real options duopoly
model (e.g., Huisman (2001) and Pawlina and Kort (2006)). In the presence of capital
market imperfections and behavioral biases, firms do not necessarily have the same discount rate, even when operating in the same market. Firm characteristics, such as morediversified risks, less financial constraints, more patience, and optimistic biases, tend tolead to lower discount rates. The rationale behind the assumption of ρP ̸= ρI and its
implications will be discussed in full detail in Section 5.

3

Model solution

This section derives the equilibrium solution (i.e., the investment thresholds and values offirms P and I) in the real options duopoly game. Following the standard literature (e.g.,
Chapter 9 of Dixit and Pindyck (1994), Chapters 7 and 8 of Huisman (2001), and Pawlina
and Kort (2006)), I solve the game backwards. Suppose that firm i ∈ {P, I} (hereafter,
the leader) invested at time s. The remaining firm j(̸= i) (hereafter, the follower) entersthe market at the optimal time after s. The follower’s value is equal to∫ ∞F
Fj (X(s)) = sup EX(s) [e−ρj (t−s) DX(t)dt − e−ρj (Tj −s) Kj ],TjF ≥s(2)

TjF

where the follower’s investment time TjF is optimized over all stopping times later thans, and EX(s) [·] denotes the expectation conditional on X(s).
In the standard manner (for details, see Dixit and Pindyck (1994)), I can solve thefollower’s problem (2). Indeed, Fj (x) satisfies the ordinary diﬀerential equation(µ + 0.5σ 2 )xFj′ (x) + σ 2 x2 Fj′′ (x) = ρj Fj (x).

(3)

The boundary conditions are the value matching and smooth pasting conditions
DxFj− Kj ,
ρj − µD
Fj′ (xFj ) =,ρj − µFj (xFj ) =where xFj denotes the follower’s investment threshold, as well as the trivial condition
limx→0 Fj (x) = 0. By solving (3) with the three boundary conditions, I have the follower’s

5

value function:

 (
)βj (
)

DX(s)
X(s)



− Kj
(X(s) < xFj ),ρj − µxFj
Fj (X(s)) =


DX(s)



− Kj
(X(s) ≥ xFj ),ρj − µwhere βj is a positive characteristic root defined by√(µ )2 2ρjµ0.5 − 2 + 2 (> 1)
βj = 0.5 − 2 +
σ
σ
σ

(4)

(5)

and xFj is the investment threshold given byxFj =βj (ρj − µ)Kj.
βj − 1
D(6)

The follower’s investment time is equal to TjF = inf{t ≥ s | X(s) ≥ xFj }.
Suppose that the follower j enters the market at TjF . Then, the leading firm i’sinvestment at time s yields the following expected payoﬀ:
∫ TF
∫ ∞
j
X(s)
−ρi (t−s)
Li (X(s)) = E[
eX(t)dt +e−ρi (t−s) DX(t)dt] − Kis(1 − D)xFj
X(s)− Ki −=
ρi − µρi − µ

(

TjF

X(s)
max{xFj , X(s)}

)βi
.

(7)

Note that the leader i’s instantaneous cash flows are equal to X(t) before time TjF andfall to DX(t) after time TjF . In the same manner as in the derivation of the follower’s
investment threshold xFj , I can derive the leader i’s optimal investment thresholdxLi =βi (ρi − µ)Ki.
βi − 1(8)in the absence of the competitor’s preemption.
Finally, I consider the case in which neither firm entered the market. In the region Fi (X(s)) < Li (X(s)), firm i prefers to become the leader, whereas in the region
Li (X(s)) < Fi (X(s)), firm i is better oﬀ being the follower. For i = P, I, firm i’s rent
equalization threshold xE
i isxE
i = inf{x ≥ 0 | Li (x) > Fi (x)}.

(9)

Firm i can accelerate investment up to the threshold xE
i to gain the leader’s payoﬀ. For
X(s) = xE
i < ∞, firm i is indiﬀerent to whether to be the leader or the follower because
the leader’s rent is exactly equal to the follower’s rent.
∗∗ 1
Define i∗ = arg mini=P,I xE
i and j = {P, I}\i . By the standard argument in the tim-

ing game (e.g., Fudenberg and Tirole (1985), Chapter 8 of Huisman (2001), and Pawlina
1

E
∗
If xE
P = xI holds, I assume that one of the firms will randomly be firm i with probability 1/2.

6

Table 1: Baseline parameter values.
ρP

ρI

KP

KI

µ

σ

D

x

0.07

0.08

10

10

0.05

0.2

0.5 0.2

and Kort (2006)), I have the equilibrium results below. Firm i∗ , which has a stronger
incentive to move first, becomes the leader, while firm j ∗ becomes the follower. Dependingon the parameter values, either the preemption or nonstrategic equilibrium arises.
∗
Suppose that firm j ∗ ’s rent equalization threshold xEj ∗ is lower than firm i ’s nonstrate∗
gic investment threshold xLi∗ . In equilibrium, firm i enters the market as the leader at the∗Fthreshold xEj ∗ , while firm j enters the market as the follower at the threshold xj ∗ . Why
L∗does firm i∗ enter the market at xEj ∗ instead of xi∗ ? Firm i wishes to delay the market∗
Eentry up to xLi∗ , but cannot do so because firm j tries to move first for X(s) > xj ∗ . Then,∗in equilibrium, firm i∗ accelerates its market entry timing to xEj ∗ , at which firm j has no

incentive to move first. Note that I assume that the initial value x is suﬃciently smallthat x < xE
j ∗ . This equilibrium is called the preemption equilibrium. Figure 1 shows anexample of a preemption equilibrium using the parameter values set in Table 1. Firm P
E
L
E
becomes the leader because xE
P = 0.538 < xI = 0.707. By xP = 0.953 > xI = 0.707, thepreemption equilibrium arises, where firm P enters the market at xE
I = 0.707 and firm I
enters the market at xFI = 2.154.
∗
Next, suppose that firm j ∗ ’s rent equalization threshold xEj ∗ is higher than firm i ’s∗
nonstrategic investment threshold xLi∗ . In this case, firm i delays its market entry timing
∗
L
to xL
i∗ because firm j has no incentive to move first for X(s) ≤ xi∗ . Hence, in equilibrium,∗
firm i∗ enters the market as the leader at the threshold xLi∗ , while firm j enters themarket as the follower at the threshold xFj∗ . Figure 2 shows an example of a nonstrategic
equilibrium using the parameter values set in Table 1, except that ρI = 0.1. Firm P
E
L
E
becomes the leader because xE
P = 0.496 < xI = 1.059. By xP = 0.953 < xI = 1.059, thenonstrategic equilibrium arises, where firm P enters the market at xL
P = 0.953 and firm I
enters the market at xFI = 2.643.
The next proposition summarizes the investment thresholds and values in equilibrium,
denoted by xi and Vi (x) (i = P, I), respectively. Note that no joint investment occurs
in the current model because the joint investment value is not higher than the follower’s
value.
Proposition 1 Firm i∗ becomes the leader, while firm j ∗ becomes the follower.

7

L
If xE
j ∗ < xi∗ holds, then the investment thresholds and the firm’s values are given as

follows:
Preemption equilibrium:Fxi∗ = xEj ∗ , x j ∗ = xj ∗ ,()βi∗xVi∗ (x) =Li∗ (xEj ∗ ), Vj ∗ (x) = Fj ∗ (x).
xE∗jOtherwise, the investment thresholds and the firm’s values are given as follows:
Nonstrategic equilibrium:Fxi∗ = xLi ∗ , x j ∗ = xj ∗ ,( )βi∗xVi∗ (x) =Li∗ (xLi∗ ), Vj ∗ (x) = Fj ∗ (x).
xL
∗
i

4

Model analysis

4.1

Symmetric costs

In this subsection, I suppose that both firms have the same investment costs; that is,
KP = KI . For D = 0 and 1, I can show the following proposition.
Proposition 2 Assume that D is either 0 or 1. Firm P becomes the leader and firm I
becomes the follower. Firm P ’s value VP (x) is higher than firm I’s value VI (x) is.
Proof. First, consider the case when D = 0. For i = P, I, the follower’s value is Fi (x) = 0
and the leader’s value is Li (x) = x/(ρi − µ) − Ki . Then, I haveExE
P = (ρP − µ)KP < (ρI − µ)KP = xI ,L
which implies that firm P is the leader. For xE
I < xP , I have(
VP (x) =

x
xE
I

)βP (

xE
I
− KP
ρP − µ)> 0 = VI (x),Land for xE
I < xP , I have(
VP (x) =

x
xL
P

)βP (

xL
P
− KP
ρP − µ)> 0 = VI (x).

Next, consider the case when D = 1. For i = P, I, I have Li (x) ≤ Fi (x) (x ≥ 0),
which implies xE
i = ∞. Then, the nonstrategic equilibrium arises. As to the nonstrategic8

Lthresholds, I show that xL
P < xI in Appendix A. Thus, firm P is the leader. In addition,(
VP (x) =
(

x
xL
P

)βP (

xL
P
− KP
ρP − µ

)
,

)
xL
I
≥
− KP
ρP − µ( )βI ( L)xIx>
− KP = VI (x),ρI − µ
xL
I
x
xL
I

)βP (

(10)

where (10) follows the optimality of xL
P . The proof is complete. □L
For D = 0, firm P enters the market first at the threshold xP = min{xE
I , xP } =min{KI (ρI − µ), KP (ρP − µ)βP /(βP − 1)} and firm I gives up entering the market. Firm
P wins the preemption game because by virtue of the lower discount rate, firm P has ahigher NPV than firm I does; that is, x/(ρP − µ) − KP > x/(ρI − µ) − KP . In otherwords, for D = 0, the NPV eﬀect determines which firm wins the preemption game.
For D = 1, firms P and I enter the market at the nonstrategic thresholds xL
P andxL
I (see (8)), with no fear of preemption. Firm P ’s value is higher than firm I’s value
because the lower discount rate increases both the NPV and option value. Proposition 2
also shows that firm P enters the market earlier than does firm I. This result is not sosimple because firm P has the higher value of waiting than firm L does. Indeed, I have∂βi /∂ρi > 0 in (5), which leads to βP /(βP − 1) > βI /(βI − 1) in (8). This option value
eﬀect conflicts with the NPV eﬀect; that is, (ρP − µ)KP < (ρI − µ)KP . The option value
eﬀect delays firm P ’s market entry, while the NPV eﬀect speeds up firm P ’s market entry.
Proposition 2 shows that the NPV eﬀect always dominates the option value eﬀect, andhence firm P enters the market earlier than firm I does. To my knowledge, no prior workin the literature proves this result analytically. From this technical viewpoint, Appendix
A contributes to the real options literature.
Unfortunately, for intermediate levels of D ∈ (0, 1), I cannot analytically prove theresults in Proposition 2. However, I suppose that the same relationship holds true forD ∈ (0, 1) because the option eﬀect is intermediate between the two extreme cases of
D = 0 (i.e., no option eﬀect) and D = 1 (i.e., full option eﬀect). The NPV eﬀect isstronger than the option eﬀect, even for D = 1, and it will thus be stronger than theoption eﬀect for D ∈ (0, 1). Below, I numerically verify this hypothesis.
E
Figure 3 shows the nonstrategic thresholds xL
i , rent equalization thresholds xi , equi-

librium thresholds xi , and firm value Vi (x) for varying levels of D. The other parameterEvalues are as set in Table 1. In the top-right panel, xE
P is always lower than xI , which

implies that firm P is the leader. Note that xE
I = ∞ holds for D ≥ 0.795. In the bottom-

9right panel, VP (x) is always higher than VI (x), indicating that the results in PropositionL
2 hold true for intermediate levels of D ∈ (0, 1).2 I have xEI < xP = 0.953 for D < 0.745,

which leads to the preemption equilibrium, where xP = xE
I increases in D in the bottomleft panel. For D ≥ 0.745, the nonstrategic equilibrium arises, where xP = xL
P = 0.953
holds in the bottom-left panel. In other words, a stronger first-mover advantage (i.e.,lower D) intensifies preemptive competition. This result is consistent with the standard
result (e.g., Chapters 7 and 8 of Huisman (2001) and Pawlina and Kort (2006))).
Notably, the bottom-right panel shows the nonmonotonicity of VP (x) with respect to
D. In fact, VP (x) decreases in D until D < 0.103, after which VP (x) increases in D,
whereas VI (x) monotonically increases in D. I can explain this nonmonotonicity with thefollowing tradeoﬀ. A higher D weakens preemptive competition and increases xP (see thebottom-left panel) and cash flows in the duopoly market. This eﬀect increases VP (x). At
the same time, a higher D decreases xI (see the bottom-left panel) and the duration of
the monopoly market. This eﬀect decreases VP (x). The latter eﬀect dominates the former
eﬀect for D < 0.103, while the former eﬀect dominates the latter eﬀect for D ≥ 0.103. In
this paper, I do not discuss the sum of the two firm values VP (x) + VI (x) because theirsum under diﬀerent discount rates does not make sense.
E
Figure 4 shows the nonstrategic thresholds xL
i , rent equalization thresholds xi , equi-

librium thresholds xi , and firm value Vi (x) for varying levels of ρI (> ρP = 0.07). The
other parameter values are as set in Table 1. In the top-right panel, xE
P is always lower
than xE
I , which implies that firm P is the leader. In the bottom-right panel, VP (x) isalways higher than VI (x). These panels confirm that firm P becomes the leader and gains
Lthe higher payoﬀ. I have xEI < xP = 0.953 for ρI < 0.095, which leads to the preemption

equilibrium, where xP = xE
I increases in ρI in the bottom-left panel. For ρI ≥ 0.095, thenonstrategic equilibrium arises, where xP = xL
P = 0.953 holds in the bottom-left panel.
Thus, a higher ρI mitigates preemptive competition. Note that in the current model, ahigher ρI decreases the NPV and value of waiting. The result is similar to the standard
result (e.g., Chapter 8 of Huisman (2001) and Pawlina and Kort (2006)) that greater
asymmetries in costs and cash flows weaken preemptive competition. I also compute thecomparative statics with respect to the growth rate µ and volatility σ. I omit the figures
because I did not find a novel result.
In summary, all numerical results show that the results in Proposition 2 hold true forD ∈ (0, 1). Firm P enters the market earlier than firm I does, and firm P ’s value is higher
2

I verified the results in all numerical examples with a wide range of parameter values.

10than of firm I. Note that by virtue of the lower discount rate, firm P has advantages interms of NPV and the value of waiting. The results are similar to the standard resultthat an advantageous firm invests earlier and earns more than does a disadvantageous
firm (e.g., Chapter 8 of Huisman (2001) and Pawlina and Kort (2006)). The results arealso consistent with those of Luo, Tian, and Yang (2020), who examine the real options
duopoly game between time-consistent and time-inconsistent firms. Indeed, they showthat that the time-consistent firm enters the market earlier and earns more than its timeinconsistent competitor does. In their model, the time-consistent firm is considered morepatient because only the time-inconsistent firm’s discount rate can increase over time,
where the two firms has the same discount rate at the initial time.

4.2

Asymmetric costs

In the previous subsection, I show that with the same investment costs, firm P entersthe market earlier than firm I does because the NPV eﬀect dominates the option eﬀect.
Clearly, for KP < KI , firm P enters the market earlier and earns more than firm I because
firm P has advantages in both the discount rate and the investment cost. Now, I supposethat KI < KP and examine which firm becomes the leader and which firm gains a higher
value. For D = 0 and 1, I can show the following proposition, where I defineρP − µKP (< KP ),ρI − µβP (βI − 1) ρP − µKI∗∗ =KP (∈ (KI∗ , KP )).
βI (βP − 1) ρI − µ
KI∗ =

(11)
(12)

Proposition 3
Case of D = 0: For KI < KI∗ , firm I becomes the leader and firm P becomes the follower,
where firm I’s value VI (x) is higher than firm P ’s value VP (x) is. For KI > KI∗ , firm P
becomes the leader, and firm I becomes the follower, where firm P ’s value VP (x) is higher
than firm I’s value VI (x) is.
Case of D = 1: For KI < KI∗∗ , firm I becomes the leader and firm P becomes thefollower. For KI > KI∗∗ , firm P becomes the leader and firm I becomes the follower. For
KI > KI∗ , firm P ’s value VP (x) is higher than firm I’s value VI (x) is.
This proposition shows two interesting results. First, the market entry order depends
on the degree of the first-mover advantage (i.e., D). In fact, for KI ∈ (KI∗ , KI∗∗ ), firm P
becomes the leader for D = 0 and the follower for D = 1. This result occurs because the
NPV and option eﬀects change with the level of D. For D = 0 (i.e., no option eﬀect),11

firm P becomes the leader when its NPV threshold is lower than that of firm I (i.e.,KI∗∗ < KI ). On the other hand, for D = 1 (i.e., the full option eﬀect), by virtue of thelower discount rate, firm P has a stronger incentive to delay market entry. Then, firm P
becomes the follower for KI ∈ (KI∗ , KI∗∗ ).
Second, the firm with the higher value does not always become the leader. Indeed, for
D = 1 and KI ∈ (KI∗ , KI∗∗ ), firm P becomes the follower, and its value VP (x) is higher
than VI (x). The option eﬀect explains this phenomenon. Firm P has a higher value towait than firm I does, and hence firm P prefers to be the follower and earn more value
than does firm I.
Next, I numerically examine how the market entry order depends on the key parametervalues. Figure 5 shows the market entry order for varying parameter values. The verticalaxes represent ρI , while the horizontal axes represent KI , D, µ, and σ. The regions of P
and I in each panel represent the regions in which firm P and I, respectively, become theleader. Except the top-right panel, I set firm I’s investment cost at KI = 8.8(< KP = 10)
to present the threshold between the two regions P and I. I set the other parametervalues as in Table 1.
In Figure 5, with lower ρI and KI , firm I is more likely to become the leader. This isstraightforward because lower values of ρI and KI reduce the discount rate disadvantage
and increase the cost advantage, respectively. More notably, the market entry order also
depends on the common market factors (i.e., D, µ, and σ). The top-right panel of Figure
5 shows that a higher D increases the likelihood that firm I becomes the leader. As in theargument after Proposition 3, I can explain this result by the tradeoﬀ between the NPV
and option eﬀects. Indeed, a higher D increases the option eﬀect, which delays firm P ’s
market entry compared to that of firm I. Then, with a higher D, firm I is more likely to
become the leader.
The bottom-left panel of Figure 5 shows that a lower µ increases the likelihood thatfirm I becomes the leader, whereas the bottom-right panel shows that a higher σ increases
the likelihood that firm I becomes the leader. These results also arise because the option
eﬀect changes with µ and σ. In fact, a lower µ and higher σ increase the option eﬀect, inwhich case, firm I is more likely to be the leader.
The eﬀects of the market factors on the market entry order have not been seen in thestandard real options model with cost asymmetry (e.g., Section 8 of Huisman (2001) and
Pawlina and Kort (2006)), where an advantageous firm enters the market earlier thana disadvantageous firm does, regardless of the market factors. However, my results are

12

similar to those of Kong and Kwok (2007), Shibata (2016), and Lambrecht (2001). For
instance, Kong and Kwok (2007) show that in a duopoly market model with both cost andcash flow asymmetry, the level of volatility can change the market entry order. Shibata(2016) shows that in a triopoly market model with cost asymmetry, the market entry
order depends on the levels of volatility and first-mover advantage. Lambrecht (2001)shows that the market exit order in a duopoly depends on the common economic factors,
such as the interest rate and volatility. Unlike these studies, I examine the diﬀerence indiscount rates and unveil the tradeoﬀ between the NPV and option eﬀects.
E
Figure 6 shows the nonstrategic thresholds xL
i , rent equalization thresholds xi , equi-

librium thresholds xi , and firm value Vi (x) for varying levels of D. I set the parametervalues besides those for D and KI = 8.8 as in Table 1. The top-right and bottom-leftpanels show that the leader changes from firm P to firm I at D = 0.612. As explainedwith the top-right panel of Figure 5, this occurs because with a higher D, the option eﬀect
dominates the NPV eﬀect. On the other hand, the bottom-right panel shows that firm
P ’s value VP (x) remains higher than firm I’s value VI (x). These results imply the notablephenomenon that firm P , which has a higher value, becomes the follower for D ≥ 0.612.
For D ≥ 0.612, because of the increased value of waiting, firm P prefers to be the follower
rather than the leader.
This result contrasts with the standard result that an advantageous firm enters themarket earlier and earns more (e.g., Chapter 8 of Huisman (2001) and Pawlina and Kort
(2006)). However, Shibata (2016) reports a similar result in an asymmetric real options
triopoly game. In fact, he shows that with a weaker first-mover advantage, the firm withthe least cost can be the second mover. In contrasted with his result in a triopoly market,
my result stems from the tradeoﬀ between the NPV and option eﬀects in a duopoly market.
The bottom-right panel of Figure 6 shows the nonmonotonicity of VP (x) with respect to
D. I can explain this nonmonotonicity with the same tradeoﬀ as in the bottom-right panelof Figure 3.
E
Figure 7 shows the nonstrategic thresholds xL
i , rent equalization thresholds xi , equi-

librium thresholds xi , and firm value Vi (x) for varying levels of σ, using the parametervalues (besides those for σ and KI = 8.8) in Table 1. In the top-right and bottom-leftpanels, the leader changes from firm P to firm I at σ = 0.227 for the same reason highlighted with the bottom-right panel of Figure 5 – a higher σ increases the option eﬀect.
The bottom-right panel shows that firm P ’s value VP (x) is always higher than firm I’svalue VI (x). In other words, firm P , which has a higher value, chooses to be the follower

13

for σ ≥ 0.227 due to the higher value of waiting. Shibata (2016) also shows that with ahigher volatility, the firm with the least investment cost can be the second mover. Unlikein Shibata (2016), my result holds true in a duopoly market with the tradeoﬀ between
the NPV and option eﬀects.

5

Empirical implications

5.1

Empirical predictions

In this subsection, I first explain the determinants of the discount rates, and then explain
several implications from the model. As a benchmark, I begin with the capital asset
pricing model (CAPM). Let S(t) be the price of a financial portfolio that completely
spans the risk of cash flow X(t), where S(t) follows
dS(t) = αS(t)dt + σS(t)dB(t), S(0) = s.

(13)

The diﬀerence between the rates of return, δ = α − µ is positive and is called the convenience yield from the real investment project (see Dixit and Pindyck (1994)). Let ρ be
the correlation coeﬃcient between S(t) and the market portfolio price. By the CAPM
formula, we have α = r + ϕσρ, where ϕ denotes the market price of risk. Then, thefollower’s value (4) is∫ ∞F
Fj (X(s)) = sup EX(s) [e−(r+ϕσ)(t−s) DX(t)dt − e−r(Tj −s) Kj ],

(14)

TjF

TjF ≥s= sup Ẽ

X(s)

∫ ∞
F
[
e−r(t−s) DX(t)dt − e−r(Tj −s) Kj ],

(15)

TjF

TjF ≥swhere Ẽ[·] represents the expectation under the risk-neutral measure. Note that under
the risk-neutral measure, X(t) follows
dX(t) = (r − δ)X(t)dt + σX(t)dB̃(t),rather than (1), where B̃(t) denotes the standard Brownian motion under the risk-neutral
measure. Similarly, the leader’s value (7) is
Li (X(s)) = Ẽ

X(s)

∫ TF
j

−r(t−s)e[∫ ∞X(t)dt +TjF

se−r(t−s) DX(t)dt] − Ki .

(16)

Thus, the discount rate (or the risk-neutral measure) is determined only by the projectcharacteristics; namely, the correlation coeﬃcient ρ and the convenience yield δ. In other14

words, the two firms must have a uniform discount rate for the same project because any
risk besides the market risk is diversifiable in CAPM.
In the real world, however, two firms may have diﬀerent discount rates for the same
project because the risk of X(t) may neither be spanned by any combination of financial
assets nor be diversifiable. For instance, Harris and Siebert (2017) show empirical evidence
that firms diﬀer in their discount factors. Firms with a wide range of businesses may
be able to diversify the risk of X(t), which decreases their discount rate. In practice,firms may substitute the weighted average cost of capital (WACC) for the discount rate
for simplicity. Then, firms with diﬀerent WACC have diﬀerent discount rates. Moregenerally, firms suﬀer from capital market imperfections such as financing constraints.
As Harrington (1989) discusses, more financially constrained firms (e.g., low cash, tightliquidity constraints, etc.) tend to have higher discount rates.
Furthermore, firms (especially entrepreneurs) are not always rational and have behavioral biases (e.g., Frederick, Loewenstein, and O’Donoghue (2002), Grenadier and Wang(2007), and Hackbarth (2009)). Clearly, more patient firms have lower discount rates.
In addition, the following model shows that more optimistic firms have lower discountrates. In addition to the baseline model setup in Section 2, I assume that cash flows X(t)disappear by an external shock, which arrives according to an exponential distribution.
Suppose that the two firms have the same discount rate ρ, but diﬀerent estimations of theintensity of the exponential distribution (say λH > λL ). Then, the more optimistic firm,(i.e., the firm with λL ) discounts X(t) by the rate ρ + λL , which is lower than the otherfirm’s rate ρ + λH .
To summarize, in the presence of capital market imperfections or behavioral biases, discount rates depend on firm characteristics. More diversified, less financially constrained,
more patient, and more optimistic firms tend to have lower discount rates, and I thus
refer to them as firm P in this model. Larger, older, and more successful firms can be
regarded as firm P because they have these characteristics. Section 4.1 shows that withno cost disadvantage, firm P enters the market earlier and earns more. Section 4.2 showsthat with a cost disadvantage, firm I can be the first mover, especially in a market witha weak first-mover advantage, high volatility, and low growth rate. It also predicts thatfirm P can earn more than firm I can, even if firm P enters the market later.

15

5.2

Empirical findings

Although no study investigates the eﬀects of discount rates on market entry timing empirically, I find several results related to my predictions in Section 5.1. Schoenecker andCooper (1998) show that larger firms tend to enter markets earlier. This finding is consistent with my result. Although Schoenecker and Cooper (1998) explain their finding by
the conventional theory of Schumpeter (1950) – larger firms have capability advantages(e.g., lower investment cost), my study complements this argument. Indeed, even in theabsence of capability advantages, larger firms can enter markets earlier by virtue of lower
discount rates.
Robinson, Fornell, and Sullivan (1992) show that firms with more finance skills tend toenter markets earlier, which is also consistent with my result. More finance skills decreasefinancing costs and relax financial constraints, which leads to lower discount rates. Thus,
my result can potentially explain their finding – with no cost disadvantage, firm P entersthe market first.
Although the above observations show that larger and less financially constrained
firms tend to enter markets earlier, many case studies show that smaller and younger
firms (especially IT venture businesses) can enter new markets earlier than larger andolder competitors. Many researchers argue that these first movers beat their opponents
by virtue of lower cost and faster decision-making. Section 4.2 shows that with a costadvantage, firm I can enter the market earlier; nevertheless, firm P can gain a higher
value than firm I can. This result points to the possibility that larger and older firms
may not be losers in competition, but be more successful, even if they enter markets later.
Shibata (2016) illustrates an example in which a smaller and more financially constrained firm invested earlier in the wireless telecommunication market in Japan. In
fact, in 1999, KDDI Corporation started providing wireless telecommunication services,which were called “2.5G” technology services, earlier than its larger competitor, NTT DOCOMO. According to Shibata (2016), the first-mover advantage in the market is weak.
This example is therefore consistent with the result that firm I is more likely to be theleader in a market with a weaker first-mover advantage.
Shackleton, Tsekrekos, and Wojakowski (2004) studies the competition between Boeing and Airbus. Boeing is a larger and less financially constrained firm than Airbus is,
but Airbus became the leader by launching the A380 in the very large aircrafts market.
Airbus receives large amounts of subsidies, which decrease investment costs compared to
Boeing. The very large aircraft market has high uncertainty. Thus, this example is also16

consistent with my result that firm I, which has a larger cost advantage, is more likely tobe the leader in a market with higher volatility.

6

Conclusion

This study investigates preemptive competition between firms P and I, which have different discount rates. I derive the market entry order, investment thresholds, and firm
values in equilibrium. The results are summarized below. With no cost disadvantage,
firm P enters the market earlier and gains more value than does firm I. With a costdisadvantage, firm P can be the follower in the market, especially with a weak first-moveradvantage, high volatility, and low growth rate. Notably, firm P can earn more than firm
I can, even when firm P becomes the follower. These results can potentially account forseveral empirical findings on market entry.
Lastly, I explain some potential directions for future research. Similar results to those
I report here hold true in a model of two firms with diﬀerent growth rate parameters (sayµH > µL ). Indeed, the firm with µH has both higher NPV and option value, and hencecorresponds to firm P in the model in this study. With the same investment cost, the firmwith µH enters the market first because the NPV eﬀect dominates the option eﬀect. Whenthe firm has a higher investment cost, the market entry order can change with market
characteristics, and the firm with the higher value can become the follower. However,
the assumptions of diﬀerent growth rate parameters may not be plausible because firms
can update their parameters by observing X(t). One interesting issue for future researchwould be to investigate the dynamic learning eﬀects on the preemption game (cf. Décamps,
Mariotti, and Villeneuve (2005)).
Another limitation is the assumption that firms have complete information about eachother (especially, discount rates). It could actually be diﬃcult to precisely know anotherfirm’s discount rate. However, as I explained in Section 5.1, observable factors (e.g.,business diversification, financial constraints, size, age, etc.) are related to discount rates,
so firms can estimate each other’s discount rates to a certain degree. One important
future direction is to study such incomplete information rigorously (cf. Lambrecht andPerraudin (2003)).
An increasing number of studies examine the interactions between investment and debt
financing (e.g., Sundaresan, Wang, and Yang (2015) and Shibata and Nishihara (2018)).
Although these studies use the risk-free interest rate under the risk-neutral measure, it

17would be interesting to examine the roles of the discount rate in the preemption gamewith debt financing (cf. Nishihara and Shibata (2010) and Nishihara and Shibata (2014)).

References
Azevedo, A., and D. Paxson, 2014, Developing real option game models, European Journalof Operational Research 237, 909–920.
Décamps, J., T. Mariotti, and S. Villeneuve, 2005, Investment timing under incomplete
information, Mathematics of Operations Research 30, 472–500.
Dixit, A., and R. Pindyck, 1994, Investment Under Uncertainty (Princeton University
Press: Princeton).
Ebert, S., W. Wei, and X. Zhou, 2018, Discounting, diversity, and investment, Available
at SSRN: https://ssrn.com/abstract=2840240.
Frederick, S., G. Loewenstein, and T. O’Donoghue, 2002, Time discounting and time
preference: A critical review, Journal of Economic Literature 40, 351–401.
Fudenberg, D., and J. Tirole, 1985, Preemption and rent equalization in the adoption of
new technology, Review of Economic Studies 52, 383–401.
Grenadier, S., and N. Wang, 2007, Investment under uncertainty and time-inconsistentpreferences, Journal of Financial Economics 84, 2–39.
Hackbarth, D., 2009, Determinants of corporate borrowing: A behavioral perspective,
Journal of Corporate Finance 15, 389–411.
Harrington, J., 1989, Collusion among asymmetric firms: The case of diﬀerent discountfactors, International Journal of Industrial Organization 7, 289–307.
Harris, J., and R. Siebert, 2017, Firm-specific time preferences and postmerger firm performance, International Journal of Industrial Organization 53, 32–62.
Huisman, K., 2001, Technology Investment: A Game Theoretic Real Options Approach(Kluwer Academic Publishers: Boston).
Kong, J., and Y. Kwok, 2007, Real options in strategic investment games between two
asymmetric firms, European Journal of Operational Research 29, 1405–1428.
Lambrecht, B., 2001, The impact of debt financing on entry and exit in a duopoly, Review
of Financial Studies 14, 765–804.

18

Lambrecht, B., and W. Perraudin, 2003, Real options and preemption under incomplete
information, Journal of Economic Dynamics and Control 27, 619–643.
Luo, P., Y. Tian, and Z. Yang, 2020, Real option duopolies with quasi-hyperbolic discounting, Journal of Economic Dynamics and Control 111, 103829.
Nishihara, M., and T. Shibata, 2010, Interactions between preemptive competition and afinancing constraint, Journal of Economics and Management Strategy 19, 1013–1042.
Nishihara, M., and T. Shibata, 2014, Preemption, leverage, and financing constraints,
Review of Financial Economics 23, 75–89.
Pawlina, G., and P. Kort, 2006, Real options in an asymmetric duopoly: Who benefits
from your competitive disadvantage?, Journal of Economics and Management Strategy
15, 1–35.
Robinson, W., C. Fornell, and M. Sullivan, 1992, Are market pioneers intrinsically stronger
than later entrants?, Strategic Management Journal 13, 609–624.
Schoenecker, T., and A. Cooper, 1998, The role of firm resources and organizational
attributes in determining entry timing: A cross-industry study, Strategic Management
Journal 19, 1127–1143.
Schumpeter, J., 1950, Capitalism, Socialism and Democracy (Harpar: New York).
Shackleton, M., A. Tsekrekos, and R. Wojakowski, 2004, Strategic entry and market
leadership in a two-player real options game, Journal of Banking and Finance 28, 179–
201.
Shibata, T., 2016, Strategic entry in a triopoly market of firms with asymmetric cost
structures, European Journal of Operational Research 249, 728–739.
Shibata, T., and M. Nishihara, 2018, Investment timing, reversibility, and financing constraints, Journal of Corporate Finance 48, 771–796.
Sundaresan, S., N. Wang, and J. Yang, 2015, Dynamic investment, capital structure, anddebt overhang, Review of Corporate Finance Studies 4, 1–42.

A

Proof of xLP < xLI

Definef (ρ) =

β(ρ)(ρ − µ)β(ρ) − 1

19(ρ > µ),whereµβ(ρ) = 0.5 − 2 +
σ

√(

0.5 −

µ )2 2ρ+ 2 (> 1).
σ2σI can prove f ′ (ρ) > 0 as follows.
)
(
1
ρ−µ ′
′
f (ρ) =β (ρ)β(ρ) −β(ρ) − 1β(ρ) − 1(√(1µµ )2 2ρ
=0.5 − 2 +0.5 − 2 + 2β(ρ) − 1
σ
σ
σ



ρ−µ
1

√(
√
2ρ 2 (2ρµ )2µ )20.5 − σ2 + σ2 σ
0.5 − σ2 + σ2(√(
1µµ )2 2ρ
=0.5 − 2 +0.5 − 2 + 2β(ρ) − 1
σ
σ
σ
(
)

√(
)2
σ 2 0.5 + σµ2 +0.5 − σµ2 + σ2ρ2

1

√(
−
)
2
2ρ 
µ 2
2
σ
0.5 − σ2 + σ2


√(
)2 2ρ
0.5 + σµ2
1
µ
µ
− +

=
0.5 − 2 + 2 − √(β(ρ) − 1
σ2
σ
σ
µ )2
2ρ
2 0.5 − σ2 + σ2


√
µ
(
)
2
0.5 + σ22(ρ − µ)1µ− µ +=− √(0.5 + 2 +β(ρ) − 1σ2σσ22(ρ−µ)µ )2
2 0.5 + σ2 + σ2
)
(
√
(
µ
µ )2 2(ρ − µ)
1
(17)
>
− 0.5
− 2+
0.5 + 2 +β(ρ) − 1σσσ2−−0.5 − σµ2 +

> 0,

(18)

where (17) and (18) follow from ρ > µ. The proof is complete. □

20

80

60

40

LP (X(s))
F P (X(s))

20

0
x EP
-20
0

0.5

1

1.5

X(s)

x FP 2

x FI

2.5

3

40

30
LI (X(s))

20

F I (X(s))

10

0

-10
0

0.5

x EI

1

1.5

x FP 2

x FI

2.5

3

X(s)

Figure 1: Li (X(s)) and Fi (X(s)). The parameter values are those in Table 1. The preemption
equilibrium arises because xLP = 0.953 is higher than xE
I = 0.707.

21

80
LP (X(s))
F P (X(s))

60

40

20

0
x EP

-20
0

0.5

1

1.5

X(s)

x FP 2

2.5 x F
I

3

20
15
10
LI (X(s))

5

F I (X(s))

0
-5
-10
0

0.5

1 xE
I

1.5

X(s)

x FP 2

2.5

x FI

3

Figure 2: Li (X(s)) and Fi (X(s)). The parameter values are those in Table 1, except ρI = 0.1.
The nonstrategic equilibrium arises because xLP = 0.953 is lower than xE
I = 1.059.

22

1.2

1.1

x EP

1
x LP
x LI

1.05

x EI

0.8
0.6

1
0.4
0.95

0.2
0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

D

0.6

0.8

1

D

10

6
V P(x)xP

V I (x)

5

xI

8

4
6
3
4
2
2

1

0

0
0

0.2

0.4

0.6

0.8

1

0

D

0.2

0.4

0.6

0.8

1

D

Figure 3: Comparative statics with respect to D. This figure shows the nonstrategic thresholds
xLi , rent equalization thresholds xE
i , equilibrium thresholds xi , and firm value Vi (x). The otherparameter values are those in Table 1. The preemption equilibrium arises for D < 0.745, while
the nonstrategic equilibrium arises for D ≥ 0.745.

23

1.4
1

x LP

1.3

x LI

0.9

x EP
x EI

0.8

1.2

0.7
1.1
0.6
1
0.9
0.07

0.5

0.08

0.09

0.4
0.07

0.1

0.08

I

0.09

0.1

I

3

3
xP

2.5

xI

2.5
V P(x)V I (x)

2
2
1.5
1.5
1
1
0.5
0.07

0.5

0.08

0.09

0
0.07

0.1

I

0.08

0.09

0.1

I

Figure 4: Comparative statics with respect to ρI . This figure shows the nonstrategic thresholds
xLi , rent equalization thresholds xE
i , equilibrium thresholds xi , and firm value Vi (x). The otherparameter values are those in Table 1. The preemption equilibrium arises for ρI < 0.095, while
the nonstrategic equilibrium arises for ρI ≥ 0.095.

24

0.082

0.1

0.081

0.095
0.09

0.079
I

I

P

0.08

P
0.085

I

0.077

I

0.08

0.078

0.076
0.075

0.075

0.07

0.074
7

8

9

10

0

0.2

0.4

KI

0.6

0.8

1

D

0.08040.083
0.08250.0802P

0.082
0.0815II

0.08

0.0798P

0.081
0.0805II

0.08

0.07960.07950.0794
0.03

0.04

0.05

0.079
0.15

0.06

0.2

0.25

0.3

0.35

Figure 5: The market entry order in equilibrium. The notations P and I in the figure represent
the regions in which firms P and I become the leader, respectively. Except the top-right panel,
firm I’s investment cost is set at KI = 8.8. The other parameter values are those in Table 1.

25

0.954

1

0.953

0.9

x EP
x EI

0.952

x LP

0.8

0.951

x LI

0.7

0.95

0.6

0.949

0.5

0.948

0.4

0.947

0.3

0.946

0.2
0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

0.6

0.8

1

0.6

0.8

1

D

D

10

6
V P(x)

xP

5

xI

8

V I (x)

4
6
3
4
2
2

1

0

0
0

0.2

0.4

0.6

0.8

1

0

D

0.2

0.4

D

Figure 6: Comparative statics with respect to D. This figure shows the nonstrategic thresholds
xLi , rent equalization thresholds xE
i , equilibrium thresholds xi , and firm value Vi (x). Firm I’s
investment cost is set at KI = 8.8. The other parameter values are those in Table 1. Firm
P becomes the leader for D < 0.612, while firm I becomes the leader for D ≥ 0.612. The
preemption equilibrium arises for any D < 1.

26

1.4
1.3

0.9
x LP

0.85

x LI

0.8

1.2

0.75

1.1

0.7

x EP
x EI

0.65

1

0.6
0.9
0.8
0.15

0.55
0.2

0.25

0.3

0.5
0.15

0.35

3

0.3

0.35

0.25

0.3

0.35

V P(x)

xI

2.5

2

2

1.5

1.5

1

1

0.5
0.15

0.25

3
xP

2.5

0.2

0.2

0.25

0.3

0.5
0.15

0.35

V I (x)

0.2

Figure 7: Comparative statics with respect to σ. This figure shows the nonstrategic thresholds
xLi , rent equalization thresholds xE
i , equilibrium thresholds xi , and firm value Vi (x). Firm I’s
investment cost is set at KI = 8.8. The other parameter values are those in Table 1. Firm
P becomes the leader for σ < 0.227, while firm I becomes the leader for σ ≥ 0.227. The
preemption equilibrium arises for the full range.

27