Discussion Paper No. 915

PRODUCT DIFFERENTIATIONAND ENTRY TIMINGIN A CONTINUOUS TIME SPATIALCOMPETITION MODEL
Takeshi Ebina
Noriaki Matsushima
Daisuke Shimizu

October 2014

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Product diﬀerentiation and entry timing in acontinuous time spatial competition model∗
Takeshi Ebina†,
Faculty of Economics, Shinshu University,
Noriaki Matsushima‡
Institute of Social and Economic Research, Osaka University
Daisuke Shimizu§
Faculty of Economics, Gakushuin University
October 20, 2014
Abstract
We extend the well-known spatial competition model (d’Aspremont et al., 1979)
to a continuous time model in which two firms compete in each instance. Our
focus is on the entry timing decisions of firms and their optimal locations. We
demonstrate that the leader has an incentive to locate closer to the centre to delay
the follower’s entry, leading to a non-maximum diﬀerentiation outcome. We also
investigate how exogenous parameters aﬀect the leader’s location and firms’ values
and, in particular, numerically show that the profit of the leader changes nonmonotonically with an increase in the transport cost parameter.

Keywords: Location; Hotelling model; Continuous time; Entry timing
JEL Code: L13, R32, C72∗

The authors gratefully acknowledge financial support from a Grant-in-Aid for Young Scientists(24730224) and Basic Research (24530248, 24530264) from the MEXT and the Japan Society for the
Promotion of Science. The second author thanks the warm hospitality at MOVE, Universitat Autònoma
de Barcelona where part of this paper was written. Needless to say, we are responsible for any remaining
errors.
†Faculty of Economics, Shinshu University, 3-1-1, Asahi, Matsumoto, Nagano, Japan. Phone: (81)-48021-7663. Fax: (81)-4-8021-7654. E-mail: ebina@shinshu-u.ac.jp‡
Institute of Social and Economic Research, Osaka University, Mihogaoka 6-1, Ibaraki, Osaka, Japan.
Phone: (81)-6-6879-8571. Fax: (81)-6-6879-8583. E-mail: nmatsush@iser.osaka-u.ac.jp§
Faculty of Economics, Gakushuin University, 1-5-1, Mejiro, Toshima-ku, Tokyo 171-8588, Japan.
Phone: (81)-3-5992-3633. FAX: (81)-8-5992-1007. E-mail: daisuke.shimizu@gakushuin.ac.jp1

1

Introduction

Researchers in economics and marketing have emphasised the importance of (horizontal)
product diﬀerentiation in the context of firm competition (e.g. d’Aspremont et al., 1979;
Brown, 1989; Lancaster, 1990). When firms launch their new products into markets,
timing and product characteristics are some of the important factors for their profits (e.g.
Krishnan and Ulrich, 2001). Taking into account firms’ decisions regarding product differentiation, researchers theoretically and/or empirically investigate how firms determine
the timing of launching their products and those characteristics (e.g. Lambertini, 1997;
Thomadsen, 2007).
From the theoretical point of view, Lambertini (2002) presented pioneering work that
discusses the strategic interaction between the optimal locations of the inventor (themarket leader), who anticipates subsequent entry and the location choice of the follower
in a Hotelling-type spatial competition model, as in d’Aspremont et al. (1979).1 He
was the first to introduce a dynamic model in the sense that time is continuous, thefirm locations are fixed once entry is made and that firms earn their profits in each
instance. Regarding the time structure, several papers deal with sequential locational
entry in a discrete time model, which allows qualitative analyses such as how many steps
the timing of investment would change given a change in other parameters (e.g. Prescott
and Visscher, 1977). However, a more rigorous quantitative analysis, such as determining
the percentage change in the investment time attributable to a percentage change in aparameter, requires a continuous time model.2
1

Location point is interpreted as a firm’s diﬀerentiation selection because the distance between a firm’s
location point and a consumer’s address corresponds to that between a firm’s attribute and a consumer’s
ideal point. This interpretation is standard in spatial economics and marketing literature.
2
Continuous time models are often used in models such as real option game models that investigate
the timing problem of firms’ entry without the locational context (e.g. Dixit and Pindyck, 1994, Chapter
9; Azevedo and Paxson, 2014). These studies introduce one or more probabilistic fluctuations into theirmodels. Our model is deterministic and does not focus on this randomness but instead, focuses on therelation between location and entry timing. The real option game approach is useful for taking into
account the endogenous timing decision.

2This novel point is from Lambertini (2002) and diﬀers significantly from those inrelated theoretical papers discussing sequential location choices based on Hotelling-type
spatial competition models (e.g. Neven, 1987; Götz, 2005).3 Those related papers are
static Hotelling models in the sense that each firm has only one profit earning chance.4
Lambertini (2002) considered two scenarios: (i) the follower’s timing of entry is exogenousand (ii) the follower’s timing of entry is probabilistically determined. Therefore, thefollower does not endogenously determine its optimal timing of entry in either scenario.
To summarise, Lambertini (2002) considered a continuous time model, but an endogenous
entry timing model with continuous time has not been considered in locational models.
Because the entry timing of followers significantly influences market leaders as well as
followers (Kalyanaram et al., 1995; Vakratsas et al., 2003), we need to overcome theweakness in the model given by Lambertini (2002) and endogenise the follower’s entrytiming decision. Therefore, our paper substantially extends the model of Lambertini(2002).
We incorporate several aspects into the standard Hotelling duopoly model in d’Aspremont
et al. (1979). The time horizon is infinite, as in Lambertini (2002). Each firm sets a priceand earns a profit in each instance if it exists in the market, implying that a delay ofentry causes a loss of profit opportunity. In anticipation of subsequent entry by the follower, the market leader initially sets its location. Because the leader’s location decisioninfluences the profits of the follower, it also aﬀects the timing of the entry (the lengthof the monopoly period), thus representing an additional value of our paper. After thelocation choice of the market leader, the follower determines the timing of entry and its
location. When the follower enters the market, it incurs an investment cost that exponentially decreases with the standard discount rate. In contrast, consumer size increases
with a growth rate lower than the discount rate. By balancing the benefit and cost of3

Many papers discussed sequential location choices in spatial competition models. Kress and Pesch(2012) and Biscaia and Mota (2013) provided comprehensive surveys on spatial competition.
4
Lambertini (1997) and Meza and Tombak (2009) considered the endogenous timing of locations insuch static Hotelling models.

3

staying outside, the follower determines its entry timing and location.
Compared with Lambertini (2002), our contributions in this paper are threefold. The
first contribution is that we endogenise the follower’s timing. The second contribution
follows the first, as we introduce investment costs and a growth rate in consumer size tomake the model more realistic. In addition, the growth rate ensures that the entry occurs
within a finite time5 and, in turn, aﬀects the leader’s location. The third contribution
is a strategic interaction between the leader’s location and the follower’s entry timing.
In addition to the eﬀects considered by Lambertini (2002), the leader’s moving closer tothe centre increases the follower’s incentive to delay its entry, prolonging the monopoly
regime. Thus, by endogenising the follower’s timing, the leader has a stronger incentive
to move closer to the central point. Although this strategic interaction among the leader’slocation, the follower’s location and its entry timing is an important aspect of this problem,
Lambertini (2002) does not take into account this strategic interaction because of hisassumption of an exogenous entry timing by the follower.
We also show that the follower always chooses to maximise the distance between thefirms whereas the leader has an incentive to locate closer to the centre to delay the follower’s entry, possibly leading to a non-maximum diﬀerentiation outcome. Furthermore,
the location interval between the leader and the follower is negatively correlated with thelength of time for which the follower stays outside. These results are similar to those inLambertini (2002), although the mechanism behind these results definitely diﬀers between
the two papers.
Finally, we show that the entry timing becomes earlier as the growth rate of consumer
size or the parameter of consumer transport cost increases, and becomes later as thediscount rate increases. We numerically investigate how those exogenous parameters
influence their profits. A notable result is that the profit of the market leader non5

This phenomenon implies that just introducing a timing endogeneity into Lambertini (2002) without
a growth rate yields no entry and a perpetual monopoly by the leader.

4monotonically changes with an increase in the consumer transport cost parameter.6

2

The Model

Two firms, i ∈ {1, 2}, produce homogeneous goods. Consumers are uniformly distributed
over the unit segment [0, 1] as proposed by Hotelling (1929).7 Each consumer at point x ∈[0, 1] purchases at most one unit of the good and decides from which firm to purchase if hedoes make a purchase. The consumer at point x ∈ [0, 1] incurs a quadratic transportation
cost c(xi − x)2 and pays price pit at time t ∈ [0, ∞) when buying a good from firm ilocated at xi ∈ [0, 1]. To summarise, the utility of the consumer at point x ∈ [0, 1] attime t ∈ [0, ∞) is given by2ū − p1t − c(x1 − x)
ut (x; x1 , x2 , p1t , p2t ) = ū − p2t − c(x2 − x)20

if purchased from firm 1,
if purchased from firm 2,otherwise,(1)

where ū denotes the gross surplus that a consumer at point x enjoys from purchasing
the good, and c is a parameter describing the level of transportation cost or productdiﬀerentiation. Let us assume that ū is so large that each consumer prefers to purchase
one good over not buying when at least one firm is present in the market.8
Assumption 1 ū > 3c.
The game proceeds as follows: Each firm i chooses the time of entry Ti ∈ [0, ∞) andlocation xi ∈ [0, 1] at the same time, and then chooses price pit : ℜ+ → ℜ+ at each timet, which is a function from time t ∈ [0, ∞) to a real number [0, ∞) and is displayed as pit
for simplicity. In addition, we assume that firm 1 is the leader who just entered at T1 = 0,
6

The transport cost parameter can be interpreted as a parameter that describes the level of productdiﬀerentiation because the cost parameter corresponds to a consumer’s disutility between the consumer’s
ideal point and the degree of a product’s attribute. This interpretation is standard in the literature on
spatial economics and marketing.
7
This setting and the following assumptions are standard in the literature on spatial economics.
8
In other words, firm 1, located at x1 = 0, has an incentive to supply a positive amount at location 1,
after maximizing its profit.

5

whereas firm 2 is the follower who enters at time T2 , to be subsequently and endogenously
determined. In this way, firm 1 decides x1 at time T1 = 0 once and subsequently chooses
price p1t at each time t. After observing firm 1’s actions before firm 2’s entry, firm 2chooses to enter at time T2 and location x2 and thereafter chooses p2t at each time t.
Firm i can choose its location only when it makes its entry in the project, at which timeit incurs an entry cost Fi (Ti ). We also assume (without loss of generality) that x1 ≤ 1/2
holds in equilibrium.
Now, let us describe the present value of the firms at time 0 given that firm 2 would
enter at point x2 at time t = T2 . Note that firm 1 enters at point x1 at time t = T1 = 0.
The timing is exogenous9 but x1 is to be determined endogenously in the following analysis.
The total profit of firm 1, the leader, is given by∫ T2 ∫ 1
V1 (T2 , x1 , x2 , p1t , p2t ) =p1t (x; x1 )e−(r−α)t dxdt
0
∫ 0∞ ∫ x̄
+p1t (x; x1 , x2 , p2t )e−(r−α)t dxdt − F1 (0),
T2

(2)

0

where r denotes the interest rate, α denotes the growth rate parameter of the market andx̄ denotes a point at which the consumer is indiﬀerent between purchasing from firm 1 or
2. From (1), x̄ = [p2t −p1t +c(x22 −x21 )]/[2c(x2 −x1 )]. We assume that r > α to ensure thatthe follower enters in a finite time. Firm 1 earns monopoly profit flow at t ∈ [τ, τ + dτ )until firm 2 enters the market, and firm 1 earns duopoly profit flow after firm 2’s entry.
The total profit of firm 2, the follower, is given by∫ ∞∫ 1V2 (T2 , x1 , x2 , p1t , p2t ) =p2t (x; x1 , x2 , p1t )e−(r−α)t dxdt − F2 (T2 ).
T2

(3)

x̄

After entry, firm 2 earns duopoly profit flow at t ∈ [τ, τ + dτ ) to which it supplies.
Let us now make the following assumptions for the entry cost function Fi .
9

A similar interpretation is made in Chronopoulos et al. (2014) as a non-preemptive duopoly. In theirpaper, the roles of the leader and the follower are defined exogenously. Consequently, the future cash
flows of the leader are discounted to time t = 0.

6

Assumption 2 (i) Fi (Ti ) = Fi e−rTi ,∫ ∞ ∫ x̄(ii) F1 (0) = F1 < 0 0 p1t (x; x1 , x2 , p2t )e−(r−α)t dxdt for all x2 ∈ [x1 , 1],∫∞∫1(iii) F2 (0) = F2 > 0 x̄ p2t (x; x1 , x2 , p1t )e−rt dxdt for all x1 ∈ [0, x2 ].
Assumption 2 (ii) suggests that F1 is small enough that the leader can earn non-negative
total profit at time 0 whenever the follower enters and wherever the follower locates.
Similarly, Assumption 2 (iii) suggests that F2 is suﬃciently large enough to enable thefollower to enter sequentially at T2 > 0 in an equilibrium, because we would like to
avoid simultaneous entry at time 0 and focus on sequential entry.10 This assumption also
implies that the leader, but not the follower, has already learned how to enter the market
eﬃciently.

3

Equilibrium

In this section, we derive the price, location and timing outcomes in the subgame perfectequilibrium. First, given locations x1 and x2 , we consider the problem of prices at eachtime t before and after the entry of firm 2. Then, we derive the local profits of the leader
and the follower at each time t.
The following are the equilibrium prices. Notably, the maximisation of the instantaneous profit flows is equivalent to the maximisation of the total profits. In other words,
firm 1 maximises the following equation with respect to p1t before firm 2 enters t ∈ [0, T2 )when T2 is exogenously given:

∫ 1
max
p1t

p1t (x; x1 )dx.

(4)0After firm 2 enters t ∈ [T2 , ∞) when T2 is exogenously given, firm 1 maximises thefollowing with respect to p1t :

∫ x̄

max
p1t

p1t (x; x1 , x2 , p2t )dx;

(5)

0

10

If simultaneous entry occurs, this model reverts to the standard location–price model of d’Aspremont
et al. (1979).

7

and firm 2 maximises the following with respect to p2t :∫ 1max
p2t (x; x1 , x2 , p1t )dx.
p2t

(6)

x̄

Solving these maximisation problems results in the following lemma.
Lemma 1 The prices set by the leader and the follower are{
2pM
t ∈ [0, T2 )1 = ū − c(1 − x1 )p̃1t =cDt ∈ [T2 , ∞),p1 = 3 (x2 − x1 )(2 + x1 + x2 )cp̃2t = pDt ∈ [T2 , ∞).
2 = (x2 − x1 )(4 − x1 − x2 )
3

(7)
(8)

Proof. See Appendix ■For t ∈ [0, T2 ), the monopoly leader maximises its price under the constraint that all
consumers purchase its good. Assuming that x1 ≤ 1/2 without loss of generality, thefurthest consumer is located at 1. The consumer turns out to be indiﬀerent between
Dpurchasing the good at price pM
1 and not purchasing the good. pi (i = 1, 2) is derived

using the standard calculation in the context of spatial competition (e.g. d’Aspremont etal., 1979).
The instantaneous profit flows of the two firms are{∫12
π1M (x1 ) = 0 pM
1 dx = ū − c(1 − x1 )∫
π1t (x1 , x2 ) =x̄c2D
π1D (x1 , x2 ) = 0 pD
1 dx = p1 x̄ = 18 (x2 − x1 )(2 + x1 + x2 )t ∈ [0, T2 )t ∈ [T2 , ∞),
(9)

{
0
∫1
π2t (x1 , x2 ) =c2Dπ2D (x1 , x2 ) = x̄ pD
2 dx = p2 (1 − x̄) = 18 (x2 − x1 )(4 − x1 − x2 )t ∈ [0, T2 )t ∈ [T2 , ∞).
(10)Note that x̄ = (2 + x1 + x2 )/6. Substituting the outcomes of equations (7) to (10) into
equations (2) and (3), the total profits of the leader and the follower are derived as∫ ∞∫ T2−(r−α)tM(11)dt +π1D (x1 , x2 )e−(r−α)t dt − F1 ,V1 (T2 , x1 , x2 , p1t , p2t ) =π1 (x1 )eT20∫ ∞V2 (T2 , x1 , x2 , p1t , p2t ) =π2D (x1 , x2 )e−(r−α)t dt − F2 (T2 ).
(12)
T2

8

3.1

Follower

We consider the problem of the follower regarding when it enters and where it locates in
the market. With regards to the location, we have the following:
Lemma 2 The follower always locates at x2 = 1.
Proof. From equation (12), x2 only depends on π2D . From equation (10), we have
∂π2D (x1 , x2 )c=
(4 + x1 − 3x2 )(4 − x1 − x2 ),
∂x2
18

(13)

which is positive for x1 and x2 ∈ [0, 1]. Thus, the optimal location for firm 2 is at 1,
regardless of the other parameters. ■From Lemma 2, we show that the follower always locates as far away from the locationof the leader as possible when entering the market. This result replicates that of Lambertini (2002) and seems to be robust to the endogeneity of the follower’s entry timing.
Substituting the equilibrium profits and the location of the follower into the total profit
functions V1 and V2 , we have
V1 (T2 , x1 , 1, p̃1t , p̃2t ) =e−(r−α)T2 c
1 − e−(r−α)T2[ū − c(1 − x1 )2 ] +(1 − x1 )(3 + x1 )2 − F1 ,r−α
r − α 18(14)

e−(r−α)T2 c
V2 (T2 , x1 , 1, p̃1t , p̃2t ) =(1 − x1 )(3 − x1 )2 − F2 e−rT2 .
r − α 18

(15)

We note that if F1 ≤ F2 , the leader’s value is always greater than the follower’s value inour setting. That is, since π1D ≥ π2D , we have)πM (V1 − V2 ≥ 1
1 − e−(r−α)T2 − F1 (1 − e−rT2 ) >
r−α

(

)
(
)
π1M
− F1 1 − e−(r−α)T2 > 0.
r−α
(16)

We use Assumption 2 (ii) from the end of the last section to ensure F1 < π1M /(r − α), thepresent value of profits when the leader sustains monopoly profit forever.

9Thus, the rest of the follower’s decision problem is only its endogenously determined
entry timing, which is not present in Lambertini (2002). We have the following proposition.
Proposition 1 The entry timing of the follower is
[
[
]
]
1
rF2
1
18rF2
.
T̃2 (x1 ) = log D
= log
α
α
c(1 − x1 )(3 − x1 )2π2 (x1 , 1)

(17)

Proof. Diﬀerentiating V2 (T2 , x1 , 1, p̃1t , p̃2t ) with respect to T2 yields∂V2 (T2 , x1 , 1, p̃1t , p̃2t )= −e−(r−α)T2 π2D (x1 , 1) + rF2 e−rT2 = 0.
∂T2

(18)

Solving this equation with respect to T2 gives T˜2 (x1 ) as a solution. Note that T̃2 is positivefrom Assumption 2 (iii). Examining the second-order derivative yields the following∂ 2 V2= (r − α)e−(r−α)T2 π2D (x1 , 1) − r2 F2 e−rT2 .
∂T22

(19)

Substituting T̃2 (x1 ) into the second-order derivative shows that ∂ 2 V2 /∂T22 |T2 =T̃2 (x1 ) is negative. Because the first-order condition is uniquely satisfied and the second-order derivativeis negative at this point, the unique, positive and interior solution exists. Thus, we havethe desired result. ■The following corollaries show the change in the follower’s entry timing as the leader’slocation x1 and the exogenous parameters change.
Corollary 1 If x1 is increased, the optimal timing for the follower to enter is delayed.
x1 aﬀects the entry timing of the follower as follows. Only the denominator within thefraction inside the log in equation (17) is composed of the locations chosen by the firms.
Thus, we focus on this part of the equation, namely, π2D (x1 , 1). When firm 1 locates awayfrom firm 2 (or x1 is decreased), firm 2 can deliver the product in a broader region (x̄decreases), and firm 2 can earn a higher profit (π2D increases). Therefore, firm 2 entersearlier (later) if firm 1 locates away from (closer to) firm 2.
10

Corollary 2 If F2 or r is increased, or as c or α is decreased, the optimal timing for thefollower to enter is delayed.
As is seen in the next subsection, x1 depends on the previous parameters. Therefore, achange in these parameters aﬀects the entry timing of the follower through a direct eﬀect(Corollary 2) and an indirect eﬀect (Corollary 1).

3.2

LeaderFinally, we consider the problem of the leader. Substituting the outcomes of the follower,
Lemma 2 and Proposition 1 into the total profit function of the leader, the maximisation
problem of the leader is given bymax

x1 ∈[0,1/2]

V1 (T̃2 (x1 ), x1 , 1, p̃1t , p̃2t )=1 − e−(r−α)T̃2 (x1 )e−(r−α)T̃2 (x1 ) c[ū − c(1 − x1 )2 ] +(1 − x1 )(2 + x1 + 1)2 − F1 .
r−α
r−α
18
(20)

Because the derivation of the equilibrium is complicated, we investigate the impact on thevalue of the leader when the location of the leader changes infinitesimally, before deriving
the equilibrium outcome presented in Proposition 2. Diﬀerentiating V1 (T̃2 (x1 ), x1 , 1, p̃1t , p̃2t )with respect to x1 yields
dT̃2 (x1 ) M∂V1 (T̃2 (x1 ), x1 , 1, p̃1t , p̃2t )= e−(r−α)T̃2 (x1 )(π1 (x1 ) − π1D (x1 , 1))∂x1
dx1
1 − exp(−(r − α)T̃2 (x1 )) dπ1M (x1 )
+
r−α
dx1
exp(−(r − α)T̃2 (x1 )) dπ1D (x1 , 1)
. (21)+r−αdx1
The sign of equation (21) is the key and determines the location of the leader: whether
it is located at the centre (1/2), edge (0), or an interior point (strictly between 0 and 1/2).
The first term of (21) represents the gain from the delay of entry by firm 2 that is causedby an increase in x1 , allowing firm 1 to maintain its monopoly profit before the duopoly
11

regime begins. This term is not present in Lambertini (2002) and captures the essence
behind our results. Therefore, if this term turns out to be large, our result may become
significantly diﬀerent from that of Lambertini (2002). The second term of (21) signifies
the increase in the monopoly profit attributable to moving closer to the centre. The third
term of (21) shows how the duopoly profit decreases as firm 1 moves closer to firm 2, thus
intensifying competition.11 The first two terms are positive and the last term is negative.
Thus, if the eﬀect of the last term is relatively small, the optimal location of firm 1 is 1/2.
We now investigate the eﬀect of each parameter on the equilibrium location of firm 1,
E
which we denote as xE
1 . First, consider the eﬀects of parameters α and r on x1 . As αapproaches r, the weight on the periods after which firm 2 enters the market gets larger.
Therefore, firm 1 needs to take into account profitability under the duopoly situation.
The significance of the last term dominates that of the other two terms, which implies
that the optimal location of firm 1 is x1 = 0. In contrast, if α becomes smaller, theconverse holds. Thus, the eﬀect of parameter r is essentially the opposite of that of α.
Parameter ū appears only in the first term, which is increasing in ū. As the locationof firm 1 becomes closer to the centre, the leader obtains the monopoly profit for a longer
time because the follower’s timing of entry is delayed from Corollary 1. An increase in ū
yields a stronger incentive to obtain this profit. Thus, as ū increases, the leader is more
likely to locate at xE
1 = 1/2.
Regarding F2 and c, they aﬀect equation (21) through T̃2 in that an increase in F2 ora decrease in c increases T̃2 . Thus, the eﬀects of these two parameters counter each other.
Note that the term dT̃2 (x1 )/dx1 is independent of c or F2 . From Corollary 2, increasing
F2 delays the entry of the follower (increasing T̃2 ), and F2 does not aﬀect equation (21) inother ways. The discounted present value of the increase in profit from a longer monopoly
regime attributable to firm 1 locating closer to the centre is decreased. Thus, the firstterm is decreasing in F2 . The second term is increasing in F2 because the increase in the
11

Although T2 is not endogenised, the second and third terms describe how Lambertini (2002) determined the leader’s location.

12monopoly profit for firm 1 from its moving closer to the centre is sustained longer due tothe entry delay. Similarly, the third term is also increasing in F2 , as the decrease in the
duopoly profit for firm 1 from its moving closer to the centre is devalued from the entry
delay. These three eﬀects are complicated, and none of them are analytically dominant.
We numerically examine this issue in the next section.
The eﬀect of parameter c counters that of F2 with respect to how they aﬀect T̃2 . In
addition, c enters in all the profit levels. Namely, c has two contrasting eﬀects on the profit
of firm 1. First, the monopoly profit of firm 1 is decreasing in c. When the firm supplies
to all consumers, it needs to compensate consumers for transport costs by lowering its
monopoly price. The compensation is higher as the consumer transport cost parameterincreases. Second, the parameter c is positively related to the duopoly profit of firm 1
as in the standard Hotelling model with price competition. The eﬀect of the first term
is lower as the parameter c increases, whereas those of the second and third terms arehigher. Therefore, the relative importance of the three terms influences the eﬀect of con the location choice of firm 1. Unlike the previous argument, this eﬀect depends on
the parameters α and r. We apply the previous argument regarding α to the eﬀect ofparameter c on the optimal location of firm 1. If α approaches r, the significance of thelast term dominates that of the second term, which implies that the optimal location offirm 1 is more likely to become xE
1 = 0 as c increases. In contrast, if α becomes smaller,
the significance of the second term dominates the last, which implies that the optimal
location of firm 1 is more likely to become xE
1 = 1/2 as c increases. If r is suﬃciently
large, for example r > 1 + α, the optimal location of firm 1 is more likely to becomexE
1 = 0 as c increases because the first term is relatively large.
Solving (20), we have the following proposition.
Proposition 2 (a) If (21) is positive for any x1 ∈ [0, 1/2], the outcome of the subgame

13perfect equilibrium is[]
1
144rF21
7∗
T2 = log, x∗1 = , x∗2 = 1, x̄∗ = ,
α
25c
2
12
{
c
∗
p̃M
t ∈ [0, T2∗ )5c
1 (x1 ) = ū − 4∗∗p∗1t =
p∗2t = pD
2 (x1 , x2 ) =
7c
D ∗
∗
∗
12
p1 (x1 , x2 ) = 12t ∈ [T2 , ∞),(22)t ∈ [T2∗ , ∞).

(23)(b) If (21) is negative for any x1 ∈ [0, 1/2], the outcome of the subgame perfect equilibrium
is1
T2∗∗ = log

[

]
1
2rF2
∗∗
∗∗
= ,
, x∗∗
1 = 0, x2 = 1, x̄
c
2

α
{
∗∗
pM
1 (x1 ) = ū − c
∗∗
p1t =
∗∗
∗∗
pD
1 (x1 , x2 ) = ct ∈ [0, T2∗∗ )
D ∗∗
∗∗
∗∗
p∗∗
2t = p2 (x1 , x2 ) = c t ∈ [T2 , ∞).
∗∗t ∈ [T2 , ∞),

(24)
(25)

(c) If the equilibrium location of firm 1, x∗∗∗
1 , is strictly between 0 and 1/2, then theoutcome of subgame perfect equilibrium is
[
]
18rF2
1
3 + x∗∗∗
1
∗∗∗
∗∗∗
∗∗∗
T2 = log
,
x
=
1,
x̄
=
,
2
∗∗∗ 2
α
c(1 − x∗∗∗
)(3
−
x
)
6
1
1
{
∗∗∗
∗∗∗ 2
pM
t ∈ [0, T2∗∗∗ )1 (x1 ) = ū − c(1 − x1 )
∗∗∗
p1t =
c
∗∗∗
∗∗∗
∗∗∗
∗∗∗
t ∈ [T2∗∗∗ , ∞),pD
1 (x1 , x2 ) = 3 (1 − x1 )(3 + x1 )
c
∗∗∗
∗∗∗
D ∗∗∗
∗∗∗
t ∈ [T2∗∗∗ , ∞).
p∗∗∗2t = p2 (x1 , x2 ) = (1 − x1 )(3 − x1 )
3

(26)
(27)
(28)

Proposition 2 shows that three types of equilibrium location for the leader can emerge.
Similar to Lambertini (2002), we show that firm 1 can locate at 0, 0.5 or an interior location depending on the parameter values. However, a case exists in which the introductionof endogenous entry timing leads to diﬀerent equilibrium location outcomes comparedwith Lambertini (2002) using the same parameter values. To grasp the intuition behind
this result, we proceed with a numerical analysis in the next section.

4

Numerical Analysis

In this section, we investigate in detail the underlying properties of our model using
numerical analysis. First, we investigate the eﬀects of the key parameters, α, c and F2 ,
14

∗
∗∗
∗∗∗
on firm 1’s equilibrium location, which we denote as xE
1 ∈ {x1 , x1 , x1 }. Then, we show

the importance of the endogeneity of T2 , namely the first term of equation (21). Finally,
we examine the eﬀects of the parameters on the total values of the firms in equilibrium.

4.1

The leader’s location

This subsection examines the eﬀects of parameters (α, c, F2 ) on the equilibrium locationof the leader.
First, let us consider the eﬀect of α on xE
1 . Let us set the parameters as r = 0.1, ū = 4
and c = 1. We use three tables to illustrate the relationship between α and the equilibrium
location for the leader. F2 = 20, 50 and 100 for Tables 1 to 3, respectively. The valuesof α are incremented by one ten-thousandth. As explained in the previous section, we
pointed out that the location xE
1 moves from 1/2 to 0 as α approaches r. We confirm thisresult from all three tables.
Additionally, we investigate the eﬀects of F2 on xE
1 when α approaches r. α denotes
the lowest value for which the equilibrium xE
1 is 0, and α denotes the highest value forwhich the equilibrium xE
1 is 1/2. Tables 1 to 3 imply that both α and α are increasingin F2 . This implication is consistent with the intuition oﬀered in the previous section. In
addition, the range of the interior location, α − α, is increasing in F2 . Therefore, as F2
increases, the equilibrium location is likely to become interior or 1/2 and not 0.
α
xE
1

···
0.5

0.09610.5

0.0962
0.432

···
···

0.0969
0.171

···
···

0.0976 0.0977 · · ·
0.014
0
0

Table 1: Location of firm 1, x1 , when r = 0.1, ū = 4, F2 = 20, and c = 1. We have
α = 0.0961, α = 0.0977.

Second, we investigate the eﬀect of c on the equilibrium location of the leader. Figure
1 illustrates the equilibrium location of firm 1 for diﬀerent values of c when F2 = 20 and
F2 = 13. Consider the case in which an interior solution can exist. An interior solution
exists when α approaches r. We consider the same values as in the previous numerical
15

α
xE
1

···
0.5

0.09640.5

0.0965
0.453

···
···

0.0973
0.192

···
···

0.0981 0.0982 · · ·
0.011
0
0

Table 2: Location of firm 1, x1 , when r = 0.1, ū = 4, F2 = 50, and c = 1. We have
α = 0.0964, α = 0.0982.
α
xE
1

···
0.5

0.0966 0.0967 · · ·0.5
0.46 · · ·

0.975
0.212

···
···

0.09830.024

0.09840···
0

Table 3: Location of firm 1, xE
1 , when r = 0.1, ū = 4, F2 = 100, and c = 1. We have
α = 0.0966, α = 0.0984.

analysis; r = 0.1, α = 0.096 and ū = 4. As ū > 3c, c ∈ (0, 4/3). When F2 = 20, thevalue of xE
1 declines from 1/2 to approximately 1/10 as c increases. In this case, firm 1
never locates at point 0. When F2 = 13, xE
1 moves from 1/2 to 0 as c increases. Figure 1
demonstrates the following concept. If the interior solution exists in the equilibrium, xE
1
is decreasing in c. As F2 increases, xE
1 is more likely to take a higher value for a given c.
Third, we show the eﬀects of both c and F2 on the equilibrium location of the leader.
Tables 4 and 5 illustrate the equilibrium location of firm 1 for diﬀerent values of c andF2 when r = 0.1, ū = 4 and α = 0.098 for Table 4 and α = 0.097 for Table 5. For thesevalues of r and ū, these values of α allow interior solutions of x1 . Thus, for α very close
to r, at approximately 0.099, we primarily have x1 = 0; however, for most values of α, wetend to have the other corner solution at x1 = 1/2.
By examining the interior solutions, we can better understand the eﬀects of c and F2 .
Values of c increase by 0.1 up to 1.3, as ū > 3c, c ∈ (0, 4/3). F2 is increased from 10
to 200 in increments of 10. Both tables show two trends. As c increases, if an interiorEsolution exists in the equilibrium, xE
1 decreases. As F2 increases, x1 increases if theinterior solution exists in the equilibrium. As noted in the previous section, the eﬀects ofthese two parameters tend to counter each other.

16

0.5

0.4

x1E

0.3

0.2

0.1

0.0
0.9

1.0

1.1

1.2

1.3

c

Figure 1: Plotted values of optimal location for firm 1 with respect to c ∈ (9/10, 4/3)
when F2 = 20 (dotted) and F2 = 13 (solid).

4.2

On the importance of endogenous timing by the follower

One of the main points of this paper in contrast to previous papers, including Lambertini(2002), is that the follower’s entry timing is endogenised and the leader determines its
location considering this move. To determine its importance, we examine the relative
sizes of the three terms in equation (21). Equation (21) is the first-order derivative of
V1 with respect to x1 . The first term represents how the marginal change in the leader’slocation delays the follower’s entry, allowing the leader to prolong its monopoly regime.
Using the parameters and the leader’s location x1 in Table 2, we derive concrete valuesof equation (21) in four cases. The common parameter values are r = 0.1, ū = 4, c = 1
and F2 = 50. The result is summarised in Table 6.
The sum column in Table 6 indicates how firm 1 decides its location. If the value inthe column is negative (positive), then firm 1 locates at 0 (0.5). If it is 0, an interior
17

F2 /c
200
190
180
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10

0.4
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5

0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5

0.6
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5

0.7
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5

0.8
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.467
0.400
0.266

0.9
0.5
0.5
0.5
0.5
0.5
0.497
0.489
0.480
0.470
0.460
0.448
0.436
0.422
0.406
0.387
0.365
0.334
0.298
0.240
0.123

1.0
0.418
0.414
0.408
0.403
0.397
0.391
0.384
0.377
0.369
0.360
0.350
0.339
0.326
0.311
0.294
0.272
0.244
0.206
0.15
0.191

1.1
0.353
0.348
0.343
0.338
0.332
0.323
0.319
0.312
0.303
0.295
0.285
0.273
0.261
0.246
0.228
0.205
0.176
0.136
0.072
0

1.2
0.304
0.299
0.294
0.288
0.283
0.276
0.269
0.262
0.254
0.245
0.234
0.223
0.210
0.194
0.175
0.152
0.122
0.079
0.011
0

1.3
0.265
0.260
0.255
0.249
0.243
0.236
0.229
0.222
0.213
0.204
0.193
0.181
0.167
0.151
0.132
0.107
0.075
0.030
0
0

Table 4: Location of firm 1, xE
1 , depending on the values of F2 and c, when r = 0.1, α =0.097, ū = 4.

location between 0 and 0.5 may arise.
The last three cases are interesting. If the first term is not present, the sum will be
negative, and firm 1 has an incentive to move closer to 0. Therefore, compared with
Lambertini (2002), firm 1 is more likely to locate closer to the centre, forcing firm 2 todelay its entry. This eﬀect of delaying entry does not exist in Lambertini (2002).

4.3

On total values of the firms

Finally, from the firms’ point of view, their total values are more or at least as importantas their entry timing and locations. We examine how the parameters ū and c, commonlyused in the Hotelling settings, aﬀect firm values. The result for ū is straightforward, but
18

F2 /c
200
190
180
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10

0.4
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5

0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.399

0.6
0.420
0.417
0.413
0.409
0.404
0.400
0.395
0.389
0.383
0.377
0.370
0.362
0.353
0.341
0.331
0.317
0.299
0.275
0.239
0.173

0.7
0.305
0.302
0.298
0.294
0.290
0.286
0.281
0.276
0.271
0.265
0.258
0.250
0.242
0.232
0.220
0.206
0.188
0.163
0.127
0.056

0.8
0.235
0.231
0.228
0.224
0.220
0.215
0.210
0.205
0.200
0.193
0.186
0.178
0.170
0.159
0.147
0.132
0.113
0.087
0.048
0

0.9
0.184
0.180
0.177
0.173
0.168
0.164
0.159
0.153
0.147
0.141
0.133
0.125
0.116
0.105
0.092
0.077
0.057
0.030
0
0

1.0
0.144
0.140
0.136
0.132
0.128
0.123
0.118
0.112
0.106
0.099
0.092
0.083
0.074
0.062
0.049
0.033
0.012
0
0
0

1.1
0.112
0.108
0.104
0.099
0.095
0.090
0.085
0.079
0.072
0.065
0.058
0.049
0.039
0.027
0.013
0
0
0
0
0

1.2
0.084
0.080
0.076
0.072
0.067
0.062
0.056
0.050
0.044
0.037
0.029
0.019
0.009
0
0
0
0
0
0
0

1.3
0.061
0.057
0.053
0.048
0.043
0.038
0.032
0.026
0.019
0.012
0.004
0
0
0
0
0
0
0
0
0

Table 5: Location of firm 1, xE
1 , depending on the values of F2 and c when r = 0.1, α =0.098, ū = 4.

α = 0.0985
α = 0.0973
α = 0.0965
α = 0.092

xE
1
0
0.192
0.453
0.5

First term
40.84453.82680.42077.479Second term
45.94342.39834.58431.673

Third term
−107.283−96.224−115.004−45.367Sum−20.4960063.785

Table 6: Values of the three terms in equation (21) when r = 0.1, ū = 4, F2 = 50, andc = 1.

that for c is more complicated.
An increase in ū improves the leader’s value and possibly worsens the follower’s value.
ū only appears in the monopoly profit phase of the leader. As it increases, firm 1 has

19more incentive to move closer to the centre, giving firm 2 an incentive to delay its entry
if the leader has not already located at the centre. Because there is only one eﬀect to beconsidered, the result is simple.
Examining the impact of the change in c on the values of the leader and follower
requires greater care. We set the parameters as r = 0.1 and ū = 4. Figure 2 demonstrates
that the value of the follower is monotonically increasing in c whereas that of the leader
does not move monotonically. In a Hotelling setting, transport cost c can be interpreted
as the degree of product diﬀerentiation. A larger c indicates a greater degree of productdiﬀerentiation.
In our model, if c increases, the follower always enjoys a positive eﬀect on its total
profit, which is not always true for the leader because the eﬀects of an increase in c
are classified into three categories: (i) a decrease in T2 , (ii) a decrease in π1M and (iii)
an increase in π1D . In other words, as product space becomes more diﬀerentiated, (i) the
follower enters earlier, (ii) the monopoly profit decreases and (iii) the duopoly profits after
entry increase. The follower only faces (i) and (iii), which are both positive eﬀects forthe follower, yielding that V2 is increasing in c. However, eﬀects (i) and (ii) are negative
for the leader, whereas eﬀect (iii) is positive. As c increases, (ii) and (iii), presented
by π1M and π1D respectively, are linearly changed whereas (i), presented by T̃2 , decreases
proportionally to − log c. Thus, when c is close to zero, the eﬀect (i) dominates (iii), andV1 is decreasing in c. When c is relatively large, eﬀect (i) is small, eﬀect (iii) dominates(i) and (ii) and V1 is increasing in c.
In the usual static Hotelling game, only eﬀect (iii) exists, yielding that greater diﬀerentiation on a product space always enlarges the values of duopoly firms. This result is wellknown in the related literature. However, in our dynamic Hotelling game, greater diﬀerentiation in a product space may not benefit both firms, thus indicating the importanceof introducing timing into the consideration.

20

300
250

V1 ,V2

200

V1H

150
V1L
100
V2H

50

V2L

0
0.0

0.2

0.4

0.6

0.8

1.0

1.2

c

Figure 2: Plotted values of the values of the leader and the follower with respect toc ∈ {0.001, 0.1, 0.2, 0.3, · · · , 1.3}, when r = 0.1, ū = 4. V1H and V2H are the values of V1
and V2 when α = 0.098 and F1 = F2 = 60. V1L and V2L are the values of V1 and V2 when
α = 0.097 and F1 = F2 = 20.

5Concluding Remarks

In this paper, we develop a duopoly model that determines the follower’s entry timing,
firms’ locations and their prices. Examining the timing of investments is important when
considering firms’ entry strategies. Hence, we extend Lambertini (2002), which in turnextends the location-price competition model (d’Aspremont et al., 1979) by using a continuous time model in which firms earn profits in each instance and the follower’s entry
timing is given exogenously. Our model endogenises the follower’s entry timing. In doing
so, we introduce parameters such as investment costs and a market growth rate to makethe model and the outcome more realistic.
We find that these changes create a strategic interaction between the leader’s locationand the follower’s entry timing. As a result, the leader has greater incentive to locate
closer to the centre to delay the follower’s entry. We also find that the follower always
locates as far away as possible from the leader, which is a robust result also seen in Lam21

bertini (2002). Numerical analyses are also presented that investigate how the exogenous
parameters aﬀect the leader’s location and the firms’ values. In particular, the profit of
the leader changes non-monotonically with an increase in the transport cost parameter.
A potential future study should address the last result, for which the relationshipbetween the firm values and the transport cost parameter c is nontrivial. c can be interpreted as a parameter that describes the degree of product diﬀerentiation. Therefore, c
is important for firms and their marketing strategies, necessitating its careful estimation.
Another issue that needs to be addressed is making the entry timing fully endogenous. In
this paper, we allow the follower to enter endogenously; however, the game starts when
the leader enters at T = 0, thus exogenously. This setting is intentional in order to compare our result to Lambertini (2002). However, we may allow the leader to endogenously
enter and, more interestingly, allow a preemption by the firms to enable full analysis of
the endogenous entry timing model, as indicated by Fudenberg and Tirole (1985).

Appendix
Proof of Lemma 1:
The duopoly result is a straightforward maximisation of profit by the two firms, as
previously stated. See, for example, d’Aspremont et al., (1979).
As for a monopoly, firm 1 is located at or to the left of 1/2, which we denote as x1 .
Firm 1 sets up its price so that the consumer at 1 or at a point to the left of 1 is indiﬀerent
between purchasing and not purchasing. That is, the consumer at 1 has no surplus, asotherwise firm 1 has an incentive to increase its price without losing any of its consumers.
Let this indiﬀerent consumer be denoted x̂.
Firm 1’s profit is given by
π1 = p1 x̂,s.t.

ū − p1 − c(x̂ − x1 )2 = 0.

22

Substituting p1 of the constraint into the profit and maximising with respect to x̂ gives∂π1 p1∂ x̂

= ū − c(3x̂ − x1 )(x̂ − x1 ) > 0,from Assumption 1. Thus, the firm sets x̂ to be at 1 and we have the monopoly result.
Finally, we must show that the monopoly and duopoly results are sensitive to time
only with respect to the entry timing. We show in three steps that this point holds.
First, after the follower enters the market (at a given T2 ), the corresponding profits
are independent of time t. That is, firm 1 maximises the following with respect to p1t :∫ x̄
p1t (x; x1 , x2 , p2t )dx
max
p1t

0

and firm 2 maximises the following with respect to p2t :∫ 1max
p2t (x; x1 , x2 , p1t )dx.
p2tx̄The result is the aforementioned simple duopoly.
Second, given this result, firm 2 determines T2 . Firm 2 maximises equation (3) withrespect to T2 , and notably, this process does not depend on how firm 1 sets its prices
before firm 2 enters the market.
Third, because firm 1’s price before firm 2’s entry, p1t , t ∈ [0, T2 ), does not aﬀect postentry competition (the first step) or entry timing (the second step), firm 1 maximises thefollowing with respect to p1t :
∫ 1
max
p1t

p1t (x; x1 )dx.
0

Therefore, the profit maximisation problem corresponds to the maximisation of the integrand. Therefore, firm 1 achieves the monopoly instantaneous profit flow as previously
given.
Hence, we show that the maximisation of the instantaneous profit flows is equivalent
to the maximisation of total profits.
23

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25