Discussion Paper No. 1007

DEMAND UNCERTAINTY,PRODUCT DIFFERENTIATION,
AND ENTRY TIMINGUNDER SPATIAL COMPETITION
Takeshi Ebina
Noriaki Matsushima
Katsumasa Nishide

Revised July 2018
July 2017

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, JapanDemand Uncertainty, Product Diﬀerentiation, andEntry Timing under Spatial Competition∗
Takeshi Ebina†
School of Commerce, Meiji University
Noriaki Matsushima‡
Institute of Social and Economic Research, Osaka University
Katsumasa Nishide§
Graduate School of Economics, Hitotsubashi University
July 20, 2018
Abstract
We investigate the entry timing and location decisions under market size uncertainty
with Brownian motions in a continuous-time spatial duopoly competition. We show
the following results. The entry threshold of the follower non-monotonically increases
in volatility, implying that the leader’s monopoly periods get longer with volatility.
However, the leader is more likely to increase the degree of product diﬀerentiation as
the volatility rises. A larger entry cost asymmetry between the firms places the leader
closer to the edge in a preemption equilibrium although such an asymmetry places theleader closer to the center in a sequential equilibrium.

Keywords: Location; Hotelling model; Continuous-time; Entry timing; Real options.
JEL classification: C73, D81, L11, L13,∗

We gratefully acknowledge financial support from JSPS KAKENHI Grant Numbers JP25245046,
JP15H02965, JP15H03349, JP15H05728, JP15K17047, JP17H00984, JP17K03797, JP18H00847,
JP18K01627 and the Joint Research Program of KIER, Kyoto University. We also thank Takao Asanoand seminar participants at ARSC2017 (University of Tokyo), Meiji University, Shinshu University, andTohoku University for helpful comments. The usual disclaimer applies.
†
School of Commerce, Meiji University, 1-1, Kanda-Surugadai, Chiyoda-ku, Tokyo 101-8301, Japan.
Phone: (81)-3-3296-2270. Fax: (81)-3-3296-2350. E-mail: ebina@meiji.ac.jp‡
Institute of Social and Economic Research, Osaka University, Mihogaoka 6-1, Ibaraki, Osaka 567-0047,
Japan. Phone: (81)-6-6879-8571. Fax: (81)-6-6879-8583. E-mail: nmatsush@iser.osaka-u.ac.jp§
Graduate School of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo 186-8601, Japan.
Phone: (81)-42-580-8856. Fax: (81)-42-580-8856. E-mail: k.nishide@r.hit-u.ac.jp1

1

Introduction

When a firm launches new products and/or services in a market, determining the productcharacteristics is one of its important decisions. For example, a fast food chain operator
opening a new restaurant needs to decide its location considering the demand in the area
as well as the competition with other firms. Another example is a manufacturing company
that considers the kind of products it should develop while taking into account the future
demand. Several studies such as Bronnenberg and Mahajan (2001), and Cleeren et al. (2010)
show that product positions influence the pricing decisions of retailing and marketing firms
greatly.
Due to the importance of product positioning, many theoretical studies investigate thefactors determining the product positions of firms in various contexts by using the Hotelling(1929) type linear city models. These studies include d’Aspremont et al. (1979), Friedman
and Thisse (1993), Tabuchi and Thisse (1995), Tyagi (2000, 2001), Kim and Serfes (2006),
Matsushima (2009), Sajeesh and Raju (2010), Liu and Tyagi (2011), and Lai and Tabuchi(2012).1 Although the existing studies oﬀer many interesting insights, they are derived using
static or discrete-time models with consumers purchasing products at most several periods,
implying robust results over the markets where the demand is quite stable for a long time.
However, if we consider the consumers’ repeated purchases in growing or changing markets,
entry timing is an important strategic decision for firms.
In actual situations, the uncertainty of future demand for products aﬀects the decisions
on location, product positioning, entry timing, and so on. Uncertainty is especially importantfor firms facing competition because the strategies of entry timing and product positioning
with future uncertainty can have a bigger and mutual impact on the strategies of other firms
than without. In summary, a firm needs to take uncertainty and competition into account1

Several empirical studies also investigate the problems of positioning (e.g., Thomadsen, 2007; Hwanget al., 2010).

2

while deciding on product positioning and entry timing.2
The decision-making of a firm under uncertainty as well as competition is often studied
within a real options framework. McDonald and Siegel (1986) is the seminal paper in theliterature, and there are many extensions and generalizations after the paper. For example,
Décamps and Mariotti (2004) consider a duopoly market in which the irreversible cost ofeach firm at the market entry is privately observed. Thijssen (2010) studies an investment
problem under the assumption that the state variables of the two firms are player-specific inthat random terms are not perfectly correlated.
Recently, Huisman and Kort (2015) studied the problem where two asymmetric firms
choose not only their investment timing but also their production capacity at investment.
Market size is a state variable and follows a geometric Brownian motion as in the real options literature.3 Huisman and Kort (2015) is novel in that two symmetric firms optimallychoose two strategic variables. This approach can be applied to many monopoly/oligopoly
problems.4 They find that the leader firm overinvests in capacity to deter the entry of thefollower and that greater uncertainty makes entry deterrence more likely. From their article,
uncertainty is of great importance for the decision-making of firms facing competition, andthat the multiple dimension of decision-making may lead to diﬀerent results when compared
to a single-dimension case.
In addition, Ebina et al. (2015) extend Lambertini (2002) to study the entry timings offirms facing competition. They construct a continuous-time spatial competition model a lad’Aspremont et al. (1979). In the model, two firms optimally determine their locations andprices as well as the follower’s entry timing, but the leader’s entry time is exogenously fixed
as in Lambertini (2002). The main finding of the study is that the leader has an incentive to2

In the context of spatial competition with sequential entry, Neven (1987) and Bonanno (1987) are thepioneering works, which have been extended in several directions (e.g., Loertscher and Muehlheusser, 2011).
3
A good survey is presented by Chevalier-Roignant et al. (2011)4
Lobel et al. (2016) studies a monopolist’s optimization problem on product launches within a real options
framework. In addition, Nielsen (2002), Weeds (2002), Huisman and Kort (2003), and Pawlina and Kort
(2006) study monopoly or duopoly within real options frameworks, but with firms choosing only their timings
in the models. Ziedonis (2007) provides a brief overview of the real option literature.

3

locate close to the center to delay the follower’s entry, leading to a non-optimal outcome fromthe viewpoint of social welfare. The study also sheds light on the importance of multiple
dimensions for the firm’s decision-making. It is noteworthy that the demand for the productgrows at a constant rate, indicating no uncertainty in their model.
In this article, we investigate the entry decisions of firms that endogenously determine
their product positions in a market whose size is evolving stochastically. To this end, we
substantially extend Ebina et al. (2015), which does not assume uncertainty on the growthpath. Furthermore, we endogenize the leader’s entry timing; this is also a significant extensionof Ebina et al. (2015).5 We describe uncertainty as the product demand following a geometric
Brownian motion, as in Pawlina and Kort (2006) and Huisman and Kort (2015). By doing
so, we can analyze the eﬀect of uncertainty on the two firms’ entry timings and location
strategies in a Hotelling model.
The major findings of the study are as follows. First, the leader firm optimally chooses
the center as its location in case of low volatility, and the edge in case of high volatility. In
addition, in case of high volatility, the delay of the follower’s entry occurs in accordance withthe maximum diﬀerentiation by the leader, contrasting with the standard intuition in which
the leader prefers locating itself closer to the center by anticipating the delay of the follower’s
entry. The reason for our counterintuitive result is that high volatility makes the leader
firm give more importance to the future situation where two firms face price competition,
leading to the leader’s strategy locating itself as far from the center as possible. Our paper
is the first study to construct a model in which demand evolves stochastically forever, and to
examine the eﬀect of uncertainty on locations within a continuous-time setting where both
firms dynamically optimize their objectives.6
Second, the entry threshold of the follower firm is not monotonic in the volatility of the5

Ebina and Matsushima (2017) provide another major extension to the work of Ebina et al. (2015) byincorporating exclusive vertical relations. However, the direction of this extension is quite diﬀerent from ours.
6
In static frameworks, several studies investigate how uncertainty aﬀects the optimal locations in theliterature on industrial organization (e.g., Meagher and Zauner, 2004; Christou and Vettas, 2005).

4

state variable. In the real options literature, the threshold is always an increasing functionof volatility, irrespective of whether the firm is a leader or a follower if there is a negative
externality.7 To the authors’ best knowledge, this article is the first to show that greater
uncertainty can lead to early entry of the follower firm in some cases. This finding indicatesthe importance of multiple dimensions for decision-making. We also present some intuitive
remarks on why non-monotonicity occurs.
Third, the eﬀect of entry cost asymmetry between the two firms on the leader’s location
diﬀers, depending on the type of equilibrium. More concretely, in a preemptive equilibrium,a higher entry cost asymmetry makes the leader locate itself closer to the edge, while it
makes the leader’s location closer to the center in a sequential equilibrium. As we will see
later, there are positive and negative eﬀects of the leader’s location on its profits, and themagnitude of the two eﬀects is dependent on the type of equilibrium that is realized.
In summary, the analysis shows that multiple dimensions of strategies complicate theproblem and may give diﬀerent results from those in single-dimension cases which are studied
by many papers in the real options literature.
The remaining part of the article is organized as follows. Section 2 sets up our model andformulates our problem. Section 3 derives the optimal timing and location of the follower
firm. In Section 4, we first categorize the equilibrium type and then derive the optimal timing
and location of the follower firm in a sequential equilibrium. We implement numerical calculations to examine how the leader’s entry threshold and location are aﬀected by exogenous
parameters in Section 5. Finally, some concluding remarks are presented in Section 6.

2

The Model

In this section, we construct our model based on Ebina et al. (2015) and introduce uncertainty
into it as in Pawlina and Kort (2006).
7

Mason and Weeds (2010) show that the threshold of a leader is non-monotonic if there is a positive
externality. However, under the assumption of a negative externality, the thresholds of the leader and thefollower are increasing in the volatility.

5

Two firms, indexed by i ∈ {1, 2}, produce homogeneous goods. Consumers are uniformly
distributed over the unit segment [0, 1], as proposed by Hotelling (1929).8 The density of
consumer distribution at time t is Yt , which stochastically changes as explained later. Eachconsumer is indexed by x ∈ [0, 1] and repeatedly purchases at each instance [t, t + dt) at
most one unit of the good. He chooses the firm to purchase from when he actually decides
to purchase.9 The consumption of a unit of the good entails a positive utility. On the other
hand, the consumer at point x ∈ [0, 1] incurs a quadratic transportation cost c(xi − x)2 andthe price pit at time t ∈ [0, ∞) while buying a good from firm i located at xi ∈ [0, 1].10 To
summarize, the utility of the consumer at point x ∈ [0, 1] and time t ∈ [0, ∞) is given by2

if purchased from firm 1,
ū − p1t − c(x1 − x)2
ut (x; x1 , x2 , p1t , p2t ) = ū − p2t − c(x2 − x)(1)
if purchased from firm 2,0otherwise,
where ū denotes the gross surplus each consumer enjoys from purchasing the good and c > 0
is a parameter describing the level of transportation cost or product diﬀerentiation.11 Alternatively, we can regard c as the parameter representing the width of the market given the
fixed length of the Hotelling line segment.12
The following assumption can make the equilibrium meaningful.
Assumption 1. ū > 3c.
Assumption 1 guarantees that at least one of the two firms has an incentive to supply apositive amount of goods after maximizing its profit wherever it is located. It also guarantees
that a monopolist prefers serving all consumers regardless of its location.
8

This setting and the following assumptions are standard in the literature on spatial economics.
In equilibrium, all consumers purchase a unit of the product at all times, according to Assumption 1
presented below.
10
The problem in which both firms can locate themselves outside the interval [0, 1] is studied in a diﬀerent
paper.
11
If c = 0, the follower can earn no profit after its entry, implying that our setting corresponds to amonopolistic situation because the follower never enters the market. To avoid this simple case, we assumec > 0.
12
See the discussion in Section 5.
96

Now, we introduce uncertainty into our market model. The density of consumer distribution, or the market size Yt , is dynamically stochastic. We impose the following assumption
on Y .
Assumption 2. The process Y follows a geometric Brownian motion as
dYt = αYt dt + σYt dWt ,where α is the expected growth rate, σ is the market volatility, and {Wt }t≥0 is a standard
Brownian motion. The initial value of the state process, Y0 ≡ y0 , is suﬃciently low.
Assumption 2 states that the future profit flow of each firm is uncertain and follows ageometric Brownian motion.13 The assumption on the initial value y0 is standard in the real
options literature, and means that the market is too small and that neither firm has madean entry into the market at the initial time.
The game in this article proceeds as follows. Firm i chooses the time of entry Ti ∈ [0, ∞)and the location xi ∈ [0, 1] simultaneously. The entry incurs an irreversible cost Fi at Ti .
Existing firm(s) simultaneously choose the price pit at each time t, observing all available
information such as the realization of Y and their location(s) xi . Although firm i can vary theprice pit at any time, its location is fixed forever after the determination of xi . We consider
the Nash equilibrium in which firm i maximizes its present value of cash flows with respect
to (Ti , xi , {pit }t≥Ti ) given the other firm’s strategy. Here is a remark on our assumption. We
assume that firms locate within the line segment [0, 1], which contrasts with the assumption inseveral previous papers that firms are allowed to locate outside the line segment (e.g., Tyagi,
2000; Sajeesh and Raju, 2010; Liu and Tyagi, 2011). If we follow the latter assumption, we
will obtain an outcome in which the leader firm, say firm 1, locates at x1 = 1/2 irrespectiveof the exogenous parameters.
We present the following assumptions for the entry cost Fi .
13

Exponential market growth is seen to be valid in many industries; for example, see Lages and Fernandes(2005) on telecommunication services, Victor and Ausubel (2002) on DRAM, and Vakratsas and Kolsarici(2008) on pharmaceuticals.

7

Assumption 3. F1 = F < κF = F2 , where κ > 1.
Assumption 3 states that firm 1 has an advantage in the entry cost over firm 2. Since
asymmetry occurs only in the entry cost, we speculate that firm 1 is always the leader formarket entry in our model.
Now, let us describe the present value of the firms at time t ∈ [0, ∞) given that firm jenters the market at point xj at time t = Tj . Here, Tiℓ denotes the entry time of firm i when
it is the leader, and Tjf denotes that of firm j when it is the follower. Superscripts ℓ and f
represent the leader and the follower, respectively. As the model is time-homogeneous, theoptimal entry time is expressed as the first hitting time; that is,
Tiℓ = Ti (yiℓ ) ≡ inf{t ≥ 0; Yt ≥ yiℓ } and Tjf = Tj (yjf ) ≡ inf{t ≥ 0; Yt ≥ yjf },where yiℓ and yjf are the thresholds of the leader and the follower, respectively.
Let x̄ ∈ [0, 1] be a point at which the consumer is indiﬀerent between purchasing fromfirm 1 and purchasing from 2. We can easily verify from (1) thatx̄ =p2t − p1t + c(x22 − x21 ),2c(x2 − x1 )(2)which indicates that the optimal price of each firm depends on the locations of both firms(x1 , x2 ). Therefore, the value function of firm j at time t when it is the follower is writtenwith discount rate r as
Ey[∫∞
Tj (yjf )e−r(s−t) Ys∫ 1]
pfjs dxds − e−r(

Tj (yjf )−t

)F

(3)

,

j

x̄

where Ey denotes the expectation operator conditional on Yt = y. We assume that r > α toensure finiteness of the value function.14
On the other hand, the value function of firm i as the leader, denoted by Viℓ , is expressed
as

[∫
Ey

Tj (yjf )Ti (yiℓ )e−r(s−t) Ys∫ 1
pℓis dxds +0∫ ∞Tj (yjf )e−r(s−t) Ys∫ x̄]−r(Ti (yiℓ )−t)pℓis dxds − e

Fi .

(4)

0

If r ≤ α, the integral of equation (3) diverges to positive infinity by choosing a larger time T2 , meaningthat waiting for a longer time would always be a better strategy, and an optimal entry timing would notexist.
14

8

The first term in the right-hand side of (4) describes the discounted cash flow whereas firm
i is the monopolist, and the second term is the discounted cash flow after the other firm’s
entry. More concretely, if firm i is the leader, it earns a monopoly profit flow for t ∈ [Ti , Tj )and a duopoly profit flow for t ≥ Tj .

3

Follower’s Value Functions

In this section, we derive the optimal price, location, and timing outcomes of the follower firm.
As in the literature, we implement backward induction for the derivation of the solutions.
First, given the locations x1 and x2 , we consider the problem of optimal prices at eachtime t before and after the entry of firm 2. From (2), we can conclude that the uncertainty
of Y does not aﬀect the equilibrium prices. Therefore, we obtain the following lemma, which
describes the equilibrium prices of both firms.
Lemma 1. We assume the leader firm i’s location xi is xi ∈ [0, 1/2] without loss of generality.
The prices set by the leader firm i and the follower firm j(̸= i) are respectively,{
2pMt ∈ [Ti , Tj ),i (xi ) = ū − c(1 − xi ) ,ℓp̂it =cDℓpi (xi , xj ) = 3 (xj − xi )(2 + xi + xj ), t ∈ [Tj , ∞),cp̂fjt = pDft ∈ [Tj , ∞).
j (xi , xj ) = (xj − xi )(4 − xi − xj ),
3

(5)
(6)

Proof. See the proof of Lemma 1 in Ebina et al. (2015).
Firm i has monopolistic power over the price at t ∈ [Ti , Tj ) such that all consumers
purchase its goods. Then, the optimal price of firm i before the entry of firm j is the price atwhich the consumer at location 1 is indiﬀerent between purchasing and not purchasing thegood. The optimal monopolistic price pM
i in Lemma 1 satisfies this condition. The optimal
Dfprices for the duopoly pDℓi (xi , xj ) and pj (xi , xj ) are based on the standard calculation in

the context of spatial competition (e.g., d’Aspremont et al., 1979).

9

With the prices p̂ℓit and p̂fjt , the instantaneous profit flows of the two firms are expressedfas Yt πitℓ and Yt πjt, respectively, where{πiM ℓ (xi ) = ū − c(1 − xi )2 ,ℓπit (xi , xj ) =c(xj − xi )(2 + xi + xj )2 ,πiDℓ (xi , xj ) = 18
{
0,
f
πjt
(xi , xj ) =c(xj − xi )(4 − xi − xj )2 ,πjDf (xi , xj ) = 18t ∈ [Ti , Tj ),t ∈ [Tj , ∞),(7)t ∈ [0, Tj ),t ∈ [Tj , ∞),(8)

as x̄ = (2 + x1 + x2 )/6 in equilibrium.

3.1

Optimal location and entry threshold

First, we consider the problem of the follower relating to when it enters and where it locatesin the market. We assume that firm i is the leader and has already invested and located atxi ∈ [0, 1/2]. We write the value function of the follower as
Vjf ∗ (y; xi ) =

where

[∫
Vjf (y; yjf , xi , xj ) =Eymaxf(y ,xj )jf
yj ≥0,xj ∈[0,1]Vjf (y; yjf , xi , xj ),]∞Tj (yjf )−t

Tj (yjf )e−r(s−t) πjDf (xi , xj )Ys ds − e−r(

( Df
) ( )βπj (xi , xj )yjf
y
=
− Fj
r−α
yjf

)F

j

.

(9)

and
1
α
β= − 2+
2 σ

√(

1
α
− 2
2 σ

)2
+

2r
.
σ2

Note that the function Vjf depends on the other firm’s location as the instantaneous profit
flow πjDf depends on xi . This implies that the pair of optimal strategies (yjf , xj ) also depends
on xi . However, with regard to the location of the follower firm j, we have the following
lemma.
10

Lemma 2. In equilibrium, the follower firm j always locates at xfj = 1.
Proof. Since Vjf is given by (9), the optimal location of the follower maximizes its instantaneous profit πjDf in (8) given xi . The lemma immediately follows.
Once the optimal location is obtained, the optimal timing can easily be derived as in thefollowing lemma. We omit the proof because it is an easy exercise.
Lemma 3. The value function of firm j as the follower is given by
(
)(
)β
f
Df

y
 ȳj (xi )πj (xi ,1) − F, if y < ȳjf (xi ),jr−αȳjf (xi )Vjf ∗ (y; xi ) = yπjDf (xi ,1)− Fj ,
if y ≥ ȳjf (xi ),r−αwhere the investment threshold of the follower isȳjf (xi ) =(r − α)βFj .
β − 1 πjDf (xi , 1)

(10)

From Lemma 2, the follower firm always locates as far away from the location of theleader as possible while entering the market. This replicates the results of Lambertini (2002)
and Ebina et al. (2015) and seems to be robust to the endogeneity of the follower’s entry
timing under uncertainty.
From (10), we easily obtain the following corollary:
Corollary 1. If x1 is increased, the optimal threshold for the follower to enter ȳ2f (x1 ) is
increased.
The corollary shows that the leader can obstruct the follower’s entry by choosing its
location close to 1/2. On the other hand, setting xi close to 1/2 can induce a tougher pricecompetition after the follower’s entry. The leader firm faces the trade-oﬀ between the two
with regard to location.

11

4

Leader’s Strategy and Equilibria

In this section, we derive the outcome of the subgame perfect Nash equilibrium (SPNE).
We face two diﬃculties while deriving the equilibria, when compared to the process used in
Pawlina and Kort (2006). The first diﬃculty arises because the value functions of our modelare determined by the firms’ entry timings as well as location choices, but in the previous
study, these functions were determined by only their entry timings. Thus, the location choicein our model entails complexity, making it diﬃcult to derive an analytical solution.
The second diﬃculty arises from the asymmetry of the firms’ profits. Since the firms’
location choices may not be symmetric on the Hotelling interval [0, 1], the profits πiDℓ (xi , 1)
and πjDf (xi , 1) may have diﬀerent values, leading to a situation where an equilibrium becomes
more complicated.
To derive equilibrium, we need to consider the leader’s optimization problem given thefollower’s strategy. Suppose that firm i is a leader and enters the market with location xi at
Ti (yiℓ ). Given the other firm’s strategy (yj , xj ) = (ȳjf (xi ), 1), the present value of the leader
firm i is written asV̂iℓ (y; yiℓ , xi ) = Ey
whereViℓ (y; ȳjf (xi ), xi ) =Ey[∫
Tj (ȳjf (xi ))[−r(Ti (yiℓ )−t)ee−r(s−t) πiM ℓ (xi )Ys ds +

t

=

]

Viℓ (yiℓ ; ȳjf (xi ), xi )Tj (ȳjf (xi ))( yπiM ℓ (xi ) − F − ȳjf (xi )(πiM ℓ (xi )−πiDℓ (xi ,1))
r−α

i


 yπiDℓ (xi ,1) − F
r−α

r−α

]

∫ ∞

y
ȳjf (xi )

e−r(s−t) πiDℓ (xi )Ys ds − Fi)βfor y ≤ ȳjf (xi ),for y > ȳjf (xi ).

i

If firm i is a leader and can choose its location and the entry threshold irrespective of theother’s strategy, the value function is expressed asViℓ∗ (y) =max(y ℓ ,xi )iyiℓ ≥0,xi ∈[0,1/2]

12V̂iℓ (y; yiℓ , xi ).

(11)

We define (ȳiSℓ , xSℓi ) to be the entry threshold and the location that attain (11). It is
worth noting that Viℓ∗ does not take into account the preemptive action by the other firmj. In other words, Viℓ∗ may not be the value function of firm i in equilibrium even if it isactually a leader.
Now, we investigate the outcome of the SPNE. Following Pawlina and Kort (2006), three
types of equilibria can occur: sequential, preemptive, and simultaneous. The leader’s location
choice and threshold are key to investigate the equilibrium, because the follower’s strategy
for location and entry timing is dominant in that the follower chooses its location xj = 1 andthreshold ȳjf after the leader enters the market.

4.1

Sequential equilibrium

Here, we consider the first type of equilibrium, sequential equilibrium.
In the beginning, we discuss suﬃcient conditions for a sequential equilibrium to occur.
One simple and apparent suﬃcient condition is that
ξ2 (y; x1 , x2 ) ≤ 0
for any y ∈ (0, ȳ2f (x1 )), x1 ∈ [0, 1/2], and x2 ∈ [0, 1/2], whereξ2 (y; x1 , x2 ) ≡ V2ℓ (y; ȳ1f (x2 ), x2 , 1) − V2f ∗ (y; x1 ).
The inequality means that firm 2 has no incentive to be a leader before it decides to enter
the market as a follower in any situation.
Another suﬃcient condition is as follows. Define the lowest level of the state variable yat which firm 2 becomes willing to enter the market as a leadery2ξ (x1 , x2 ) = inf{y ≥ 0; ξ2 (y; x1 , x2 ) > 0}
for fixed (x1 , x2 ).15 Suppose on the contrary to the above trivial condition that ξ2 (y; x1 , x2 ) >0 for some (y, x1 , x2 ) ∈ (0, ∞) × [0, 1/2] × [0, 1/2]. If ȳ1Sℓ ≤ y2ξ (x1 , x2 ) for any x1 ∈ [0, 1/2]
15

We set y2ξ = ∞ if ξ2 ≤ 0 for any (x1 , x2 ).

13and x2 ∈ [0, 1/2] satisfying ξ2 (y; x1 , x2 ) > 0, firm 1 actually enters the market before firm 2
becomes willing to do so as a leader. Then, a sequential equilibrium occurs.
Intuitively, if the cost asymmetry between the two firms is substantial, there is only asequential equilibrium. In other words, a sequential equilibrium occurs if κ = F2 /F1 , themagnitude of cost asymmetry, is suﬃciently high.16 Pawlina and Kort (2006) show thenecessary and suﬃcient condition for a sequential equilibrium to occur in their model where
two firms only consider the optimal entry timing, not choosing their locations.
Now, we suppose that a sequential equilibrium occurs. Then, (11) is actually the valuefunction of the leader before the time of its entry, and the optimal strategy is given bySℓSℓ(ȳ1Sℓ , xSℓ1 ). To derive (ȳ1 , x1 ), we calculate
(
)( )
[ Mℓ
] f
(
)β
β
Dℓ
Mℓ
ℓ
ℓ
π
(x
)
−
π
(x
,
1)
ȳ
(x
)
π
y
y
(x
)y
1
1
1
1
2
1
1
1
1
1
ℓ
ℓ
V̂1 (y; y1 , x1 ) =
−
− F1
.
r−α
r−α
y1ℓ
ȳ2f (x1 )(12)

From (12) with (10), the maximization problem of the leader is formulated as
maxℓ ,x )(y1
1
y ℓ ≥0,x1 ∈[0,1/2]v̂1ℓ (y1ℓ , x1 ; β),where(v̂1ℓ (y1ℓ , x1 ; β) = [π1M ℓ (x1 )y1ℓ − (r − α)F1 ](y1ℓ )−β − [π1M ℓ (x1 ) − π1Dℓ (x1 , 1)]r−απ2Df (x1 )

)1−β
F2

.

Thus, we have the first-order condition with respect to y1ℓ to obtainȳ1Sℓ (x1 ) =

r−α
β
F1 .
β − 1 π1M ℓ (x1 )(13)Now, we are ready to show the following lemma regarding xSℓ
1 .
Lemma 4. There are constants β and β such that(i) xSℓ
1 = 1/2 for β > β,16

We numerically show in Section 5.3 that this intuition is correct, in the sense that a type of equilibrium
switches from a preemptive equilibrium to a sequential one as κ increases.

14

(ii) xSℓ
1 = 0 for β < β.
Proof. After diﬀerentiating v̂1ℓ with respect to x1 , we substitute (13) to obtain
(
)1−β
∂v̂1ℓ
r−α
M ℓ′
ℓ 1−β
M ℓ′
Dℓ′
=π1 (x1 )(y1 )− [π1 (x1 ) − π1 (x1 , 1)]
F2∂x1
π2Df (x1 )
(
)−β
r−α
(r − α)π2Df ′ (x1 )MℓDℓ− (1 − β)[π1 (x1 ) − π1 (x1 , 1)]
F2
F2
π2Df (x1 )π2Df (x1 )2)1−β [(
)1−β
(
)1−β
(
β
r−α
r−α
M ℓ′
F2
=
F1
π1 (x1 ) −β−1
π1M ℓ (x1 )π2Df (x1 , 1)
(
)]
Df ′
π
(x
,
1)
1
× [π1M ℓ′ (x1 ) − π1Dℓ′ (x1 , 1)] + (β − 1)[π1M ℓ (x1 ) − π1Dℓ (x1 , 1)] 2Df.
π2 (x1 , 1)(14)

If β → 1, the right-hand side of (14) converges to π1Dℓ′ (x1 , 1), which is always negative. On
the other hand, consider the ratio of the first and the second terms in the square bracket of
(14):
(
(

=

r−α
F2
π2Df (x1 ,1))1−β (r−αF
π1M ℓ (x1 ) 1)1−β

π1M ℓ′ (x1 )π Df ′ (x ,1)[π1M ℓ′ (x1 ) − π1Dℓ′ (x1 , 1)] + (β − 1)[π1M ℓ (x1 ) − π1Dℓ (x1 , 1)] π2Df (x 1,1)
1
2
(
)β−1
Df
F2 /π2 (x1 ,1)F1 /π1M ℓ (x1 )π M ℓ (x1 )−π1Dℓ (x1 ,1) π2Df ′ (x1 ,1)
π Dℓ′ (x ,1)+ 1 − π1M ℓ′ (x11 )π1M ℓ′ (x1 )π2Df (x1 ,1)1(β − 1) 1).

It follows from L’Hôpital’s Theorem that the above equation diverges if β → ∞ since
F1 /π1M ℓ (x1 ) < F2 /π2Df (x1 , 1). Therefore, we confirm that (14) is positive for any x1 if β
is suﬃciently large. Now the lemma follows from noting that all functions to be examined
are continuous.
Note that the volatility σ aﬀects only through the parameter β and that ∂β/∂σ < 0.
The above proposition implies that in a sequential equilibrium, the location of the leader isthe center, if the volatility of the market size is suﬃciently small. On the other hand, if theuncertainty over the future size of the market is suﬃciently large, the optimal location of the15

leader in a sequential equilibrium moves closer to the edge. The observation is presented in
the following proposition.
Proposition 1. There exist constants σ and σ such that(a) xSℓ
1 = 1/2 for σ < σ,(b) xSℓ
1 = 0 for σ > σ.
As we already showed in Lemma 2, the follower chooses its location at xf2 = 1 to minimize
the price competition between the two firms. Given this fact, there are three eﬀects concerninga decision on the location by the leader. First, when the cash flows before the follower enters
are considered, it is optimal for the leader to be located at 1/2. This maximizes the leader’smonopolistic profit by charging a higher price while still attracting all consumers. Second,
being located at 1/2 deters the future entry by the follower, since the price competition
becomes severe and causes a delay of the follower’s entry. Third, when the cash flows after
the follower enters are considered, being located at 1/2 is less attractive for the leader firm
because it maximizes price competition in a duopolistic situation.
It is a well known result in the real options literature that when uncertainty is large, avalue of waiting with entry seems to be high, leading to the follower’s late entry. Then, theleader’s monopoly period being likely to last long, the first eﬀect should dominate the third
eﬀect and the leader should choose xℓ1 = 1/2. However, Proposition 1 is inconsistent withthe above intuition. How can this be understood?
We learn the above interpretation through mathematical analysis. First, we rewrite (12)at y = y1ℓ as[(()β−1 ]
)β−1
ℓ
Mℓ
ℓ
(x
)y
π
y
y1ℓ
π1Dℓ (x1 )y1ℓ
1 1
1
1
ℓ ℓ
ℓ
V̂1 (y1 ; y1 , x1 ) =1−− F1 .
+r−αr−αȳ2f (x1 )ȳ2f (x1 )(15)Equation (15) shows that the value function of the leader is the weighted average of themonopolistic and duopolistic cash flows with weight (y1ℓ /ȳ2f (x1 ))β−1 . It should also be noted
16

that
( ℓ )β
y1
ȳ2f

[
]
f f
ℓ ℓ
= Ey e−r(T2 (ȳ2 )−T1 (y1 )) .

Therefore, the weight is closely related to the discount factor for the period T2f − T1ℓ with
rate r. Thus, the change in the volatility σ has an impact on(i) the monopolistic profits represented by

π1M ℓ (x1 )y1ℓ,r−α((ii) the follower’s entry timing represented by(iii) the duopolistic profits represented by

y1ℓ
ȳ2f (x1 ))β−1,π1Dℓ (x1 )y1ℓ.
r−α

We can present some intuitive explanation on the negative relationship between σ andfℓxSℓ
1 in the following manner. Suppose that σ becomes large. For fixed y1 and y2 , the weight(y1ℓ /y2f )β−1 in (15) becomes large and close to 1, since y1ℓ /y2f < 1 and β is decreasing in σ.
This implies that the leader should take this into more account for the future situation where
the two firms face fiercer price competition. To put it diﬀerently, if the eﬀect of x1 on themonopolistic cash flow is rather small, then the leader firm should give more importance tothe future cash flow, causing its location to be close to 0.
From our result, we can say that a firm should consider uncertain future cash flows
rather than the current ones in the case of higher uncertainty. Our result in Proposition
1 is consistent with the results of the extant papers in this sense, and contributes to theliterature by presenting the impact of volatility on the firms’ positioning strategies in aneﬀective manner.
In the literature on standard real options, most studies consider the case of the optimal
entry timing alone. Then, σ positively aﬀects the value function of the leader alone throughthe second eﬀect. However, under our setting, where each firm chooses the timing as well as
location, the problem is not simple, with results diﬀering from those in the standard model.
In fact, Section 5 presents other novel results due to the multi-dimensionality of strategies.
17

Finally, by substituting xSℓ
1 for each case in Proposition 1, we have the following proposition stating the outcomes of SPNE.
Ef
Eℓ Ef
Proposition 2. Let (y1Eℓ , y2Ef , xEℓ
1 , x2 , p1 , p2 ) be the equilibrium solution of the locations,thresholds, and prices. Then, we have the following three cases.
(a) If Equation (14) is positive for any x1 ∈ [0, 1/2], the outcome of the subgame perfect
Ef
Eℓ Ef
∗ ∗
∗
∗ ∗ ∗
equilibrium is expressed as (y1Eℓ , y2Ef , xEℓ
1 , x2 , p1 , p2 ) = (y1 , y2 , x1 , x2 , p1 , p2 ), where

β 4(r − α)F
β 144(r − α)κF
, y2∗ ≡,
β − 1 4ū − c
β−1
25c
1
x∗1 ≡ , x∗2 ≡ 1,
2{
M∗
p (x ) = ū − 4c for t ∈ [T1 (y1∗ ), T2 (y2∗ ))p∗1t ≡ 1D ∗1 ∗7c
for t ∈ [T2 (y2∗ ), ∞),p1 (x1 , x2 ) = 125c∗∗p∗2t ≡pD
for t ∈ [T2 (y2∗ ), ∞).
2 (x1 , x2 ) =
12

y1∗ ≡

7
The consumer at x̄∗ ≡ 12
is indiﬀerent between purchasing the good from firm 1 andpurchasing it from firm 2.
(b) If Equation (14) is negative for any x1 ∈ [0, 1/2], the outcome of the subgame perfect
Ef
Eℓ Ef
∗∗ ∗∗
∗∗
∗∗ ∗∗ ∗∗
equilibrium is expressed as (y1Eℓ , y2Ef , xEℓ
1 , x2 , p1 , p2 ) = (y1 , y2 , x1 , x2 , p1 , p2 ),where
β 2(r − α)κF
β (r − α)F, y2∗∗ ≡,β − 1 ū − c
β−1
c
∗∗
x∗∗
1 ≡ 0, x2 ≡ 1,
{
∗∗
∗∗
∗∗
pM
1 (x1 ) = ū − c for t ∈ [T1 (x1 ), T2 (x2 ))
≡
p∗∗
1t
∗∗
∗∗
for t ∈ [T2 (x∗∗pD2 ), ∞),
1 (x1 , x2 ) = cy1∗∗ ≡

∗∗
∗∗
D ∗∗
p∗∗
2t ≡p2 (x1 , x2 ) = c for t ∈ [T2 (x2 ), ∞).

The consumer at x̄∗∗ = 12 is indiﬀerent between purchasing the good from firm 1 andpurchasing it from firm 2.

18

Remark 1. We cannot analytically prove the presence of an inner solution
Ef
Eℓ Ef∗∗∗ ∗∗∗∗∗∗∗∗∗ ∗∗∗ ∗∗∗(y1Eℓ , y2Ef , xEℓ
1 , x2 , p1 , p2 ) = (y1 , y2 , x1 , x2 , p1 , p2 )with x∗∗∗∈ (0, 1/2) when β < β < β. Fortunately, from our numerical analysis, there is an1
inner solution under a wide range of parameter settings if β is intermediate and ū is large
enough.
Proposition 2 with Remark 1 shows three cases for the leader’s location in our sequential
equilibrium. Which case occurs depends on the parameters describing the dynamics of themarket size α, σ, and so on. To grasp the intuition behind the two propositions, we proceedwith a numerical analysis in Section 5.

4.2

Preemptive equilibrium

Second, we consider another type of equilibrium, namely, preemptive equilibrium. Pawlina
and Kort (2006) define a preemptive equilibrium as the situation in which firm 2, which isdisadvantaged in an investment cost, has an incentive to become the leader. In this case,firm 1 needs to note that firm 2 would enter the market before the state variable Y reaches
the optimal threshold for firm 1.
Sℓ
Sℓ
Assume that y2ξ (xSℓ
1 , x2 ) < ȳ1 for some x2 ∈ [0, 1/2]. In this case, ξ2 (y; x1 , x2 ) > 0 forSℓsome y ∈ (y2ξ (xSℓ
1 , x2 ), y1 ), meaning that firm 2 has an incentive to become a leader in theinterval. Hence, firm 1 should take the following two actions: (i) lowering the entry thresholdy1ℓ to deter the other firm’s entry, and (ii) changing its location x1 to fit with the adjusted
entry threshold y1 . If firm 1 chooses y1ℓ for the entry threshold as a leader, the location shouldbex̃1 (y1ℓ ) = argmax V1ℓ (y1ℓ ; ȳ2f (x1 ), x1 ).

(16)

x1 ∈[0,1/2]

Let ȳ1P ℓ and xP1 ℓ be the entry threshold and the location of firm 1 in a preemptive equilibrium.
With ȳ1P ℓ and xP1 ℓ , the other firm has no incentive to be a leader for y < ȳ1P ℓ . In other words,
19

ȳ1P ℓ and xP1 ℓ satisfyȳ1P ℓ = y2ξ (xP1 ℓ , xξ2 (xP1 ℓ )) and xP1 ℓ = x̃1 (ȳ1P ℓ ),where xξ2 (x1 ) is defined by the equation
ξ2 (y2ξ (x1 , xξ2 ); x1 , xξ2 ) = 0.
From the above observation, a preemptive equilibrium is rather complicated and seems
hard to solve, analytically. More concretely, xP1 ℓ is the solution of the non-linear equations
x = x̃1 (y2ξ (x, xξ2 (x))) and the function x̃1 (·) includes the maximization problem (16). The
diﬃculty in our case comes from the fact that the payoﬀ of a Hotelling-type model is a cubic
function with respect to the state variable.17
In the next section, we numerically calculate the optimal strategies, if an equilibrium
is of a preemptive-type, and conduct comparative statics to examine how the volatility orother parameters aﬀect both firms’ entry thresholds and locations. Note for the numericalcalculation that the leader’s equilibrium location xP1 ℓ is derived from the fact thatxP1 ℓ = sup X1P ℓ ,where
X1P ℓ = {0}

4.3

∪{

}
∂ ℓ ξ
ξ
f
x1 ∈ (0, 1/2];
V (y (x1 , x2 ); ȳ2 (x1 ), x1 ) > 0 .
∂x1 1 2

(17)

Simultaneous equilibrium

Finally, let us consider a simultaneous equilibrium and show that it cannot occur.
Lemma 5. A simultaneous equilibrium cannot occur as an outcome of SPNE.
17

On the contrary, Huisman and Kort (2015) assume a quadratic payoﬀ function, which allows them toexplicitly solve a preemptive equilibrium.

20

Proof. Let V̂iT be the value function of firm i in the case of a simultaneous equilibrium. In
a simultaneous equilibrium, the optimal locations are apparently given by (x1 , x2 ) = (0, 1).
Therefore, if y T is the entry threshold of both firms, we have
) ( )β
( T T


π
y
y


− Fi
, y < yT ,

T
r
−
α
y
T
T
V̂i (y; y ) =


πT y



− Fi ,y ≥ yT ,
r−α
where
c
πT = .
2
Defineζ1 (y, x) =V̂1ℓ (y; y T , x) − V̂1T (y; y T ),
ζ2 (y) =V2f ∗ (y) − V̂2T (y; y T ).
A necessary condition that a simultaneous equilibrium happens is(i) ζ1 ≤ 0 for all y ∈ [ȳ1Sℓ , ȳ2f (0)],and(ii) ζ2 ≤ 0 for all y ∈ [y T ∧ ȳ2f (0), y T ∨ ȳ2f (0)].
ffSℓFirst, we suppose that xSℓ
1 > 0. Since ȳ2 (0) < ȳ2 (x1 ), we have

V2T ∗ (y; y T ) ≤ V̂2T (y; ȳ2f (0)) = V2f (y; ȳ2f (0), 0, 1) < V2f ∗ (y) = V2f (y; ȳ2f (xSℓ ), xSℓ
1 , 1)for y < y2f (0), implying that a sequential equilibrium dominates a simultaneous equilibrium
for firm 2.
Suppose next that xSℓ
1 = 0. Since (x1 , x2 ) = (0, 1) for both sequential and simultaneous
equilibria, firm 2 always enters the market at y = ȳ2f (0) irrespective of the type of equilibrium.
However,

)(
( Mℓ
)β
f
Dℓ
Mℓ
(0,
1)
ȳ
(0)
−
π
(0)
π
(0)
yπ
y
2
1
1
1
Sℓ
ζ1 (y; x1 ) =− F1 −r−αr−αȳ2f (0)
21

(

)(
)β
π1Dℓ (0, 1)ȳ2f (0)
y
−
− F1
r−α
ȳ2f (0)()()β
yπ1M ℓ (0)ȳ2f (0)π1M ℓ (0)y=− F1 −
− F1
.
r−α
r−α
ȳ2f (0)fSℓ
We now show that ζ1 (y; xSℓ1 ) > 0 for some y < ȳ2 (0). Diﬀerentiating ζ1 (y; x1 ) with respect

to y, we have
π1M ℓ (0)
ζ1′ (y; xSℓ
)
=
−β
1
r−α

(

ȳ2f (0)π1M ℓ (0)
− F1
r−α

)

y β−1
ȳ2f (0)β.

Therefore,Mℓζ1′ (0; xSℓ1 ) =π1 (0)/(r − α) > 0andβ−1
ζ1′ (ȳ2f (0); xSℓ
1 ) =
r−α

{

F1
−π1M ℓ (0) + π2Df (0, 1)
F2

}
< 0.

f
f
Sℓ
Since ζ1 (0; xSℓ1 ) = −F1 < 0 and ζ1 (ȳ2 (0); x1 ) = 0, we have ζ1 > 0 for y close to ȳ2 (0). Thisimplies that a sequential equilibrium dominates a simultaneous equilibrium for firm 1 in thecase xSℓ
1 = 0.
We can argue for the case of xP1 ℓ , in a manner similar to xSℓ
1 , which completes the proof
Intuitively, when there is no cash flow in advance of the entry, the two firms have no
incentive to coordinate under cost asymmetry, and hence, a simultaneous equilibrium cannot
occur.

5

Numerical Analysis

In this section, we investigate the underlying properties of our model using numerical analysis
in depth.

22

5.1

The eﬀect of volatility

This subsection numerically examines how the volatility σ aﬀects the leader’s location xEℓ
1 .
As we have already seen, the type of equilibrium is mainly determined by the cost asymmetry.
That is, a sequential equilibrium occurs if the cost asymmetry is significant, and a preemptive
equilibrium occurs otherwise. Therefore, we analyze with two cases, in one of which case thevalue of κ is large, and in the other of which it is small. Other parameter values are given in
Table 1.
ū c
30 1

r
α
0.1 0.099

F
100

Table 1: Base case parameter values.

Case 1 (κ = 10): We consider the situation where the cost asymmetry is large. It can bechecked using numerical calculations that a sequential equilibrium in fact occurs irrespectiveof the volatility σ.
The outcomes of SPNE (the locations, entry thresholds, and prices of both firms) are presented in Table 2 for σ ∈ {0.0001, 0.1, 0.2 . . . , 1}, confirming the analytical result in ProposiEℓEℓtion 2. That is, xEℓ
1 = 1/2 for a small σ and x1 = 0 for a large σ. Further, x1 is decreasingin σ and is between 0 and 1/2 when σ takes an intermediate value, again confirming theexistence of an inner solution.18
Several important observations are obtained from the numerical results in Table 2. First,
Ef
we briefly discuss how the volatility σ influences the thresholds xEℓ
1 and x2 . Table 2 shows
Efthat the locations of the two firms xEℓ= 1 always holds.
1 is decreasing in σ, whereas x2

Thus, σ aﬀects only the equilibrium location of firm 1. Numerical calculations confirm thevalidity of Lemma 2 and Proposition 1.
18
We conclude with numerical calculations for a wide variety of parameter settings that xEℓ
is non1
increasing in σ, even for an inner solution.

23

σ0.0001
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

xEℓ
1
0.5
0.5
0.5
0.5
0.5
0.241
0.108
0.018
0
0
0

xEf
2
1
1
1
1
1
1
1
1
1
1
1

x̄E
0.583
0.583
0.583
0.583
0.583
0.540
0.518
0.503
0.5
0.5
0.5

y1Eℓ
y2Ef
0.336 576
0.353 605
0.404 692
0.488 837
0.607 1039
0.767 703
0.961 677
1.191 712
1.451 842
1.744 1012
2.072 1202

ℓ
pM
1
29.75
29.75
29.75
29.75
29.75
29.42
29.20
29.04
29
29
29

pDℓ
1
0.583
0.583
0.583
0.583
0.583
0.820
0.924
0.988
1
1
1

pDf
2
0.417
0.417
0.417
0.417
0.417
0.698
0.860
0.976
1
1
1

Table 2: The outcome of the subgame perfect equilibrium in Case 1 with κ = 10. In thiscase, a sequential equilibrium occurs.

Second, we discuss the relationship between y1Eℓ or y2Ef and σ. To investigate the eﬀectof σ, the following decomposition is useful:dy Ek (β(σ), xEℓ∂y Ek ∂β ∂yiEk ∂xEℓ
dyiEk
1
1 (σ))
= i
= i
+
,
dσ
dσ
∂β ∂σ
∂x1 ∂σ(i, k) = (1, ℓ), (2, f ).

(18)

We call the first term of (18) the option eﬀect; this is induced through the parameter β.
The option eﬀect always appears in a standard real options model, and is monotonicallyincreasing in the volatility σ. The second term of (18), called the location eﬀect, is the eﬀect
induced through the leader’s location xEℓ
1 . The location eﬀect is new in the literature andshould be considered in the model where the strategy of each firm is multi-dimensional.
The eﬀect of σ on the threshold of the leader firm, y1Eℓ can be explained as follows. Noteagain that the first term of (18) is strictly positive. For the second term, we easily verify that∂y1Eℓ /∂x1 and ∂xEℓ
1 /∂σ are both non-positive. In other words, the option and location eﬀects
are both positive and the leader’s threshold is always increasing with respect to volatility.
However, the story is diﬀerent for the eﬀect of σ on y2Ef . Although the option eﬀect
is still positive for y2Ef , the location eﬀect, or the eﬀect of σ via the leader’s location, canEℓbe negative, because ∂y2Ef /∂x1 > 0 and ∂xEℓ
1 /∂σ < 0 when x1 is on (0, 1/2). In other24

words, the option eﬀect is positive but the location eﬀect can be negative for the follower’s
threshold if the equilibrium has an inner solution. Further, the total eﬀect (18) is negativeif the location eﬀect dominates the option eﬀect. Our numerical result shows that this canhappen with our parameter setting (see y2Ef when σ = 0.4, 0.5, and 0.6 in Table 2).
Now, let us explain why σ has a negative eﬀect on y2Ef if it takes an intermediate value.
Eℓ
Eℓ
If xEℓ
1 = 0 or x1 = 1/2, the follower knows that the leader would not change its location x1

for a marginal change in σ and thus considers only the option eﬀect, leading to σ having a
positive eﬀect on y2Ef . On the other hand, if xEℓ
1 is between 0 and 1/2, the follower actuallytakes the marginal change of xEℓ
1 that aﬀects the present value of its own future profit flows.
Since the leader chooses a location closer to 0, the present value of the follower becomes
higher and the follower can enter the market more easily. Hence, the follower optimallychooses a lower y2Ef .
Mason and Weeds (2010) show that the threshold of a leader is non-monotonic if there isa positive externality. However, to the best of the authors’ knowledge, the negative eﬀect ofthe volatility on a threshold for some parameter settings is new in the real options literature
under the assumption of a negative externality. Our model setup in which each firm optimizes
the threshold and location, confirms the importance of the analysis of multi-dimensional
strategies.
Case 2 (κ = 1.1): We consider the situation where the cost asymmetry is small. It can bechecked using numerical calculations that a preemptive equilibrium in fact occurs irrespectiveof the volatility σ.
The outcomes of SPNE (the locations, entry thresholds, and prices of both firms) in apreemptive equilibrium are presented in Table 2 for σ ∈ {0.0001, 0.1, 0.2 . . . , 1}.
A similar observation to Case 1 is obtained from the above results. First, the locationof the leader xEℓ
1 is monotonically decreasing in the volatility σ, meaning that the productdiﬀerentiation widens as the demand uncertainty becomes larger. Second, the eﬀect of σ on25

σ0.0001
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

xEℓ
1
0.5
0.5
0.5
0.5
0.5
0.236
0.102
0.011
0
0
0

xEf
2
1
1
1
1
1
1
1
1
1
1
1

x̄E
0.583
0.583
0.583
0.583
0.583
0.539
0.517
0.502
0.5
0.5
0.5

y1Eℓ y2Ef
0.050 63
0.052 67
0.059 76
0.069 92
0.083 114
0.117 770.161 740.225 770.280 93
0.336 111
0.399 132

ℓ
pM
1
29.75
29.75
29.75
29.75
29.75
29.42
29.19
29.02
29
29
29

pDℓ
1
0.583
0.583
0.583
0.583
0.583
0.824
0.929
0.993
1
1
1

pDf
2
0.417
0.417
0.417
0.417
0.417
0.704
0.867
0.985
1
1
1

Table 3: The outcome of the subgame perfect equilibrium in Case 2 with κ = 1.1. In thiscase, a preemptive equilibrium occurs.

the follower’s entry threshold y2Ef is not monotonic (See y2Ef when σ = 0.4, 0.5, 0.6 in Table3). The eﬀect is negative if σ takes an immediate value, or in a situation where an equilibrium
has an inner solution xEℓ
1 ∈ (0, 1/2).
As we have already explained, the leader firm should give more importance to the future
situation, if uncertainty of the cash flow is higher. Thus, the leader takes the duopoly market
into more account with a higher σ, and xEℓ
1 is non-increasing as a result. We confirm theanalytical result in Proposition 1 with numerical examples.

5.2

The eﬀect of growth rate

In this subsection, we investigate the eﬀect of α on the leader’s location, by conducting acomparative statics with numerical calculations. Here, we study Case 1 (κ = 10), meaningthat a sequential equilibrium occurs.19
Figure 1 illustrates how the parameters σ and α aﬀect the equilibrium location of theleader. We observe from the figure that the leader’s location is decreasing in α. In otherwords, the leader optimally locates itself at the center if the market grows slowly in average.
19

We omit the analysis in Case 2, because similar observations are obtained.

26

Figure 1: The relationship between xEℓ
1 and α with diﬀerent values of σ.

Especially, xEℓ
1 = 1/2 ways holds when α = 0.096; this is depicted at the upper left-hand side.
When α = 0.099 the location of the leader depends on the volatility σ; xEℓ
1 = 0.5 if σ ≤ 0.4
and xEℓ1 = 0 if σ ≥ 0.75. Ebina et al. (2015) obtain a similar result with their numericalanalysis on pages 908–910.
An intuitive explanation on the above observation can be presented as follows. A large
value of α means the rapid market size growth on average, inducing the follower to enter themarket earlier. Then, the leader should avoid price competition with the follower. Thus, theleader firm optimally chooses xEℓ
1 = 0.

5.3

The eﬀect of cost asymmetry

Next, we consider how the degree of asymmetry κ aﬀects the equilibrium outcome. Table 4shows the eﬀect of κ on the outcome of SPNE when σ = 0.5.
First, we can see from Table 4 that a preemptive equilibrium occurs for a smaller κ, anda sequential equilibrium for a larger κ. The reason is almost self-evident. If the asymmetry
27

κ
1.1
2
3
4
5
6
7
8
9

xEℓ
xEf
2
1
0.23591
0.234110.232910.231910.231110.232610.23531
0.2376
1
0.23961y1Eℓ
y2Ef
0.1166 77
0.2338 99
0.3779 149
0.5336 198
0.6994 248
0.7669 415
0.7668 487
0.7667 558
0.7667 631

Type of Equilibrium
Preemptive
Preemptive
Preemptive
Preemptive
Preemptive
Sequential
Sequential
Sequential
Sequential

Table 4: The outcome of the subgame perfect equilibrium with diﬀerent values of κ. The
type of equilibrium changes at κ between 5 and 6.

in costs becomes large, the entry of the follower firm is significantly late in comparison to
the leader’s entry. Thus, the leader does not need to take the follower’s entry timing into
consideration and can optimally choose when to enter.
Second, we observe that the leader firm locates itself farther from the center as the cost
asymmetry becomes larger in the case of a preemptive equilibrium. On the other hand, if asequential equilibrium occurs, the leader’s location is closer to the center. The above observations imply that the eﬀect of cost asymmetry diﬀers, depending on the type of equilibrium.
The intuition for the eﬀect of κ is as follows. It should be noted that the leader’s locating itself closer to the center has two eﬀects on the follower’s strategy. The first one ispositive for the leader in that it enjoys higher monopolistic profits and deters the follower’s
entry. The second one is negative in that it lowers its profits in a duopolistic market. In apreemptive equilibrium where a disadvantaged firm has an incentive to become a leader, theleader should deter the other firm’s entry by entering the market earlier. The leader’s earlier
entry to preempt signifies the importance of the discounted sum of instantaneous monopoly
profits. To increase the discounted sum, the leader needs to locate closer to the center whenpreemption occurs. If κ increases from a lower level (e.g., κ = 1.1 in Table 4), the leader’sentry timing gets delayed (see y1Eℓ in Table 4), implying that the discounted sum of instanta28

neous monopoly profits becomes less important and xEℓ
1 decreases. On the other hand, in asequential equilibrium where κ is large, the leader firm is not forced to enter earlier and sets
its threshold satisfactorily. In fact, κ hardly aﬀects the leader’s threshold in each sequential
equilibrium in Table 4. On the contrary, a higher value of κ means a higher entry cost F2 inthis analysis, leading to a higher entry threshold of the follower. Thus, if the cost asymmetryis large, the leader firm can enjoy a longer monopoly cash flow, and sets its location to becloser to the center.

5.4

The eﬀect of transportation cost

Finally, we investigate the eﬀects of the transportation cost c on the leader’s location. In
our setting, an agent who is located far from the firm should pay high transportation costs.
Therefore, given the fixed length of the Hotelling line, we can interpret that widening themarket area corresponds to the increase in the transportation cost parameter c.
Figure 2 illustrates how the parameter c aﬀects the equilibrium location of the leader.
Again, we study Case 1 with κ = 10 and σ = 0.3, where a sequential equilibrium occurs.
One can confirm that the equilibrium location of the leader is decreasing in c, implying thatthe distance between the two firms is increasing as market areas are widened.
Our theoretical result is closely related to the empirical finding presented by Thomadsen(2007), who computationally investigates the product positioning (location) strategies of two
asymmetric firms in the fast food industry, McDonald’s and Burger King, in the UnitedStates. In his study, McDonald’s is advantaged in cost and quality and can firstly choose its
positioning (leader in our study), whereas Burger King is disadvantaged and secondly chooses
it (follower in our study). The major findings by Thomadsen (2007) are that (1) McDonald’sand Burger King should pursue diﬀerent location strategies, which means that the minimum
diﬀerentiation never occurs, and that (2) under some conditions, McDonald’s prefers locating
at or near the center, whereas Burger King tries to locate far away from McDonald’s to attract
diﬀerent types of customers. In addition, the prices set by McDonald’s and Burger King are29

Figure 2: The relationship between the transportation cost c and the equilibrium location of
the leader xEℓ
1 .

increasing as the two firms are located farther apart in most cases. These results seemconsistent with ours, as shown in Proposition 2 and Table 2.
Note that Ebina et al. (2015) also obtain similar results as the above with a model where
no uncertainty exists. However, our findings strengthen Thomadsen (2007) in the following
sense. If the demand evolves stochastically, the volatility signifies the tendency that a leader
locates farther away from the center. This means that demand uncertainty is also one of thecauses for Thomadsen (2007)’s findings. In summary, our theoretical results give one possible
explanation on the empirical results.

6

Conclusion

In this article, we construct a Hotelling-type spatial model with uncertainty of market demandin a real options framework. Contrary to the standard real options literature, the strategy
of each firm is multi-dimensional in that each firm maximizes its value function with respect30

to multiple variables, entry timing, and location. This multi-dimensional approach leads tomany interesting results. One important finding is that the threshold of the follower is not
monotonically increasing in volatility, which is new in the literature, and can be obtained
only with multi-dimensional strategies. Further, although the follower’s entry timing tends
to be late as the volatility increases, the leader is more likely to locate at the edge as thevolatility gets higher.
Finally, we mention three possible directions for future studies. The first is to consider acircular city model, which is another representative model of spatial competition. The second
is to study an oligopoly model where more than two firms consider market entry. The third
is to generalize the stochastic process Y . Considering these extensions are useful to clarify
relationships between firms’ locations, their thresholds, and a level of the volatility. However,as a final remark, we argue that the results obtained in this study are not restrictive and canbe applied to many actual competitive situations.

References
Bonanno, G. (1987), “Location Choice, Product Proliferation and Entry Deterrence,” Review
of Economic Studies, 54(1), 37–45.
Bronnenberg, B. J. and V. Mahajan (2001), “Unobserved Retailer Behavior in Multimarket
Data: Joint Spatial Dependence in Market Shares and Promotion Variables,” Marketing
Science, 20(3), 284–299.
Chevalier-Roignant, B., C. M. Flath, A. Huchzermeier, and L. Trigeorgis (2011), “Strategic
Investment under Uncertainty: A Synthesis,” European Journal of Operational Research,
215(3), 639–650.
Christou, C. and N. Vettas (2005), “Location Choices under Quality Uncertainty,” Mathematical Social Sciences, 50(3), 268–278.

31

Cleeren, K., F. Verboven, M. G. Dekimpe, and K. Gielens (2010), “Intra- and Interformat
Competition among Discounters and Supermarkets,” Marketing Science, 29(3), 456–473.
d’Aspremont, C., J. J. Gabszewicz, and J.-F. Thisse (1979), “On Hotelling’s ‘Stability inCompetition’,” Econometrica, 47(5), 1145–1150.
Décamps, J.-P. and T. Mariotti (2004), “Investment Timing and Learning Externalities,”
Journal of Economic Theory, 118(1), 80–102.
Ebina, T. and N. Matsushima (2017), “Product Diﬀerentiation and Entry Timing in aContinuous-Time Spatial Competition Model with Vertical Relations,” working paper.
Ebina, T., N. Matsushima, and D. Shimizu (2015), “Product Diﬀerentiation and Entry Timing in a Continuous Time Spacial Competition Model,” European Journal of Operational
Research, 247(3), 904–913.
Friedman, J. W. and J.-F. Thisse (1993), “Partial Collusion Fosters Minimum Product Differentiation,” RAND Journal of Economics, 24(4), 631–645.
Hotelling, H. (1929), “Stability in Competition,” Economic Journal, 39(153), 41–57.
Huisman, K. J. M. and P. M. Kort (2003), “Strategic Investment in Technological Innovations,” European Journal of Operational Research, 144(1), 209–223.
Huisman, K. J. M. and P. M. Kort (2015), “Strategic Capacity Investment under Uncertainty,” RAND Journal of Economics, 46(2), 376–408.
Hwang, M., B. J. Bronnenberg, and R. Thomadsen (2010), “An Empirical Analysis of Assortment Similarities across U.S. Supermarkets,” Marketing Science, 29(5), 858–879.
Kim, H. and K. Serfes (2006), “A Location Model with Preference for Variety,” Journal ofIndustrial Economics, 54(4), 569–595.

32

Lages, L. F. and J. C. Fernandes (2005), “The SERPVAL Scale: a Multi-Item Instrumentfor Measuring Service Personal Values,” Journal of Business Research, 58(11), 1562–1572.
Lai, F.-C. and T. Tabuchi (2012), “Hotelling Meets Weber,” Regional Science and Urban
Economics, 42(6), 1017–1022.
Lambertini, L. (2002), “Equilibrium Locations in a Spatial Model with Sequential Entry inReal Time,” Regional Science and Urban Economics, 32(1), 47–58.
Liu, Y. and R. K. Tyagi (2011), “The Benefits of Competitive Upward Channel Decentralization,” Management Science, 57(4), 741–751.
Lobel, I., J. Patel, G. Vulcano, and J. Zhang (2016), “Optimizing Product Launches in the
Presence of Strategic Consumers,” Management Science, 62(6), 1778–1799.
Loertscher, S. and G. Muehlheusser (2011), “Sequential Location Games,” RAND Journalof Economics, 42(4), 639–663.
Mason, R. and H. Weeds (2010), “Investment, Uncertainty and Pre-emption,” International
Journal of Industrial Organization, 28(3), 278–287.
Matsushima, N. (2009), “Vertical Mergers and Product Diﬀerentiation,” Journal of Industrial
Economics, 57(4), 812–834.
McDonald, R. and D. Siegel (1986), “The Value of Waiting to Invest,” Quarterly Journal of
Economics, 101(4), 707–727.
Meagher, K. J. and K. G. Zauner (2004), “Product Diﬀerentiation and Location Decisions
under Demand Uncertainty,” Journal of Economic Theory, 117(2), 201–216.
Neven, D. J. (1987), “Endogenous Sequential Entry in a Spatial Model,” International Journal of Industrial Organization, 5(4), 419–434.

33

Nielsen, M. J. (2002), “Competition and Irreversible Investments,” International Journal ofIndustrial Organization, 20(5), 731–743.
Pawlina, G. and P. M. Kort (2006), “Real Options in an Asymmetric Duopoly: Who Benefits
from Your Competitive Disadvantage?” Journal of Economics and Management Strategy,
15(1), 1–35.
Sajeesh, S. and J. S. Raju (2010), “Positioning and Pricing in a Variety Seeking Market,”Management Science, 56(6), 949–961.
Tabuchi, T. and J.-F. Thisse (1995), “Asymmetric Equilibria in Spatial Competition,” International Journal of Industrial Organization, 13(2), 213–227.
Thijssen, J. J. (2010), “Preemption in a Real Option Game with a First Mover Advantage
and Player-Specific Uncertainty,” Journal of Economic Theory, 145(6), 2448–2462.
Thomadsen, R. (2007), “Product Positioning and Competition: The Role of Location in theFast Food Industry,” Marketing Science, 26(6), 792–804.
Tyagi, R. K. (2000), “Sequential Product Positioning under Diﬀerential Costs,” Management
Science, 46(7), 928–940.
Tyagi, R. K. (2001), “Cost Leadership and Pricing,” Economics Letters, 72(2), 189–193.
Vakratsas, D. and C. Kolsarici (2008), “A Dual-Market Diﬀusion Model for a New Prescription Pharmaceutical,” International Journal of Research in Marketing, 25(4), 282–293.
Victor, N. M. and J. H. Ausubel (2002), “DRAMs as Model Organisms for Study of Technological Evolution,” Technological Forecasting and Social Change, 69(3), 243–262.
Weeds, H. (2002), “Strategic Delay in a Real Options Model of R&D Competition,” Review
of Economic Studies, 69(3), 729–747.

34

Ziedonis, A. A. (2007), “Real Options in Technology Licensing,” Management Science,
53(10), 1618–1633.

35