Discussion Paper No. 1009

PRODUCT DIFFERENTIATIONAND ENTRY TIMINGIN A CONTINUOUS-TIMESPATIAL COMPETITION MODELWITH VERTICAL RELATIONS
Takeshi Ebina
Noriaki Matsushima

August 2017

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 withvertical relations∗Takeshi Ebina†
Institute of Social Sciences, Shinshu University
Noriaki Matsushima‡
Institute of Social and Economic Research, Osaka University
August 17, 2017
Abstract
We study the entry timing and location decisions of two exclusive buyer–supplier
relationships in a continuous-time spatial competition model. In each relationship,
the firms determine their entry timing and location, and negotiate a wholesale pricethrough Nash bargaining. Then, the downstream firm immediately determines its retailprice. Our findings are as follows. Ordinarily, if the supplier of the first entrant (called
the leader pair) has strong bargaining power, the equilibrium location of the leader
will be closer to the center, inducing a delay in entry by the second entrant (called thefollower pair). This delay implies the stronger bargaining power of the supplier in theleader pair can also benefit the buyer of the pair. The location of the leader pair canchange non-monotonically with an increase in the supplier’s bargaining power, which
has a substantial impact on the entry timing of the follower pair. However, the greater
the bargaining power of the supplier in the follower pair, the closer the leader pair will
be to the edge. This implies that having greater bargaining power will enhance theprofitability of the supplier in the follower pair.

Keywords: Entry timing; Hotelling model; Vertical relations; Continuous-time model; Nashbargaining
JEL codes: C71, L11, L13, R32.
∗

The authors gratefully acknowledge financial support from a Grant-in-Aid for Basic Research (15H03349,
15H05728, 17H00984) and Young Scientists (15K17047) from the Japan Society for the Promotion of Scienceand the MEXT. We also thank seminar participants at Shinshu University for helpful comments. Needlessto say, we are responsible for any remaining errors.
†
Institute of Social Sciences, Shinshu University, 3-1-1, Asahi, Matsumoto, Nagano, Japan. Phone: (81)4-8021-7663. 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.jp1

1

Introduction

When firms face situations in which markets are growing, as in new product markets andnewly developed cities, the timing of entry and product positioning are important strategic
issues. For instance, mobile phone services have been growing in most countries since they
were first introduced in 1981 in Scandinavia; and for some countries, the penetration levelsof mobile phones have already reached more than 100% (Peres et al., 2010). In addition,“the massive penetration of mobile telephony is not exceptional —many commonly usedproducts and services, such as DVDs, personal computers, digital cameras, online banking,and the Internet, were unknown to consumers three decades ago” (Peres et al., 2010, p.91).
Naturally, companies in these industries consider product positions and entry decisions when
they launch new products. In addition, in the context of retailing, many leading retailers indeveloped countries have launched stores in countries that are in diﬀerent stages of economic
development (Reinartz et al., 2011). In particular, in emerging countries, including the
BRIC countries, Mexico, Poland, and South Africa, “the total population [...] considerably
exceeds the population of developed countries, making these regions particularly attractive
for mature market-based retailers and retailing innovations that are responsive to the distinctive characteristics of these markets” (Reinartz et al., 2011, p.S55). In this context, theprofitability of these retailers is significantly influenced by geographical distances and thelocation choices of rivals in each region.1
Manufacturers/franchisors usually sell their products through independent downstream
representatives/franchisees; that is, vertical relations are often observed in contexts such as
1

Several empirical studies show that geographical proximity influences the pricing decisions of retailers(Bronnenberg and Mahajan, 2001; Zhu and Singh, 2009; Cleeren et al., 2010; Orhun, 2013). In addition, many studies have empirically investigated how positioning strategies influence firm profitability (e.g.,Thomadsen, 2007; Draganska et al., 2009; Hwang et al., 2010).

2

manufacturers and retail representatives, franchisors and franchisees, and so on.2 In this context, several papers emphasize that large nationwide retailers tend to have strong bargaining
power over their upstream trading partners, enabling them to be highly competitive in retail markets (Geylani et al., 2007; Inderst and Shaﬀer, 2007, 2009). In other words, strong
bargaining power over trading partners is recognized as a source of competitiveness. In summary, when examining the growth of markets and vertical relations, we need to consider
firms’ product positioning and entry timing.
In line with the literature on positioning strategies and product diﬀerentiation (in marketing, Wernerfelt, 1986; Hauser, 1988; Moorthy, 1988; Desai, 2001; Kuksov, 2004; Sajeesh
and Raju, 2010, among others; in economics, d’Aspremont et al., 1979; Tabuchi and Thisse,
1995; Kim and Serfes, 2006; Matsumura and Matsushima, 2009; Sajeesh, 2016, among others), we investigate how market growth influences product positioning/diﬀerentiation in acontinuous-time spatial competition model with entry timing decisions and vertical relations.
In the context of economics, marketing, and related fields, with the exception of Lambertini(2002), Ebina et al. (2015), and Ebina et al. (2017),3 no studies have investigated productpositioning strategies using continuous-time Hotelling-type spatial competition models withmarket growth and entry timing. However, several related studies do use Hotelling models
to investigate the sequential entry of firms (e.g., Götz, 2005, Loertscher and Muehlheusser,2011) and buyer–supplier relations (e.g., Brekke and Straume, 2004; Matsushima, 2004,
2009; Erkal, 2007, Geylani et al., 2007; Liu and Tyagi, 2011; Matsushima and Miyaoka,
2015; Matsushima and Pan, 2016).
2

Several recent studies investigate the objectives of retailers and their increased power in channels(Dobson and Waterson, 1997; Chen, 2003; Raju and Zhang, 2005; Dukes et al., 2006). In addition, in thecontext of outsourcing by downstream firms, several studies investigate the interactions between sourcing
modes and profitability (Liu and Tyagi, 2011; Matsushima and Pan, 2016).
3

Using a non-spatial dynamic duopoly model with exclusive vertical relations, Alipranti et al. (2015)
investigate the timing of technology adoption and contract types (linear pricing and two-part tariﬀs).

3

This study substantially extends the work of Ebina et al. (2015) by incorporating two
pairs of an upstream supplier/franchisor and a downstream buyer/franchisee in their model.
Ebina et al. (2015) discuss endogenous decisions on locations and entry-timing in a Hotelling
duopoly model with market growth.4 Here, we investigate how upstream suppliers’ bargaining power over downstream buyers influences the locations of the two pairs, the timing ofentry by the follower pair, and the value of each firm.
Our findings are as follows. Ordinarily, if the supplier in a leader pair has stronger
bargaining power (denoted as β1 ), this induces the pair to locate closer to the center, and thefollower pair to enter the market later. This implies that the stronger bargaining power of the
supplier can also benefit the downstream buyer. That is, it might be better for a downstreambuyer to weaken its bargaining position over its trading supplier. In reality, there are several
examples in which buyers encourage suppliers to organize collective associations, even though
this would weaken the buyers’ bargaining power (e.g., the Carrefour case in Raynaud et al.
(2009, pp.851–2)). In addition, there can be a non-monotonic relation between β1 and theleader pair’s location. More specifically, the equilibrium location of the leader pair canjump significantly with an increase in β1 , which delays the entry timing of the follower pair
substantially. On the other hand, an increase in the bargaining power of the supplier in thefollower pair induces the leader pair to locate closer to the edge, which facilitates earlier
entry by the follower pair. This implies that when an upstream manufacturer tries to enter a
newly growing market with its downstream representative, maintaining its bargaining powerover its representative is quite important.
The remainder of the paper organized as follows. Section 2 provides the model setting,and Section 3 describes the results of the model. Section 4 provides numerical analyses of4

Ebina et al. (2017) provide another major extension to the work of Ebina et al. (2015) by incorporating
market-size uncertainty with Brownian motion in the setting of Ebina et al. (2015). However, the directionof this extension is quite diﬀerent to ours.

4the model. Then, Section 5 discusses the relation between our study and that of Ebina et al.
(2015), as well as the case in which the bargaining power in each pair is symmetric. Section
6 concludes the paper, and Section 7 is the Appendix.

2

Model

There are two competing upstream firms U1 and U2 . Upstream firm Ui forms a vertical groupwith a downstream firm Di (i = 1, 2). Thus, there are four firms: two upstream firms U1 andU2 , and two downstream firms D1 and D2 . In group i, Ui and Di negotiate a wholesale price
wit , paid from Di to Ui at time t. The negotiation is based on the standard Nash bargaining(Muthoo, 1999). Here, βi ∈ [0, 1] denotes the bargaining power of Ui .
There is a mass of consumers uniformly distributed on the line segment [0, 1]. The sizeof this mass is exp(αt), where α is the growth rate of the size (market size). The utility of
the consumer at point x ∈ [0, 1] at time t ∈ [0, ∞) is given by2

if purchased from firm D1 ,ū − p1t − c(x1 − x)2
ut (x; x1 , x2 , p1t , p2t ) = ū − p2t − c(x2 − x)
if purchased from firm D2 ,0otherwise,(1)

where ū > 0 denotes the gross surplus that a consumer at point x ∈ [0, 1] enjoys frompurchasing the good, pit is the price of Di at time t, and c > 0 is a parameter describing
the level of transportation cost or product diﬀerentiation.5 From (1), the consumer who is
indiﬀerent between purchasing from D1 and D2 is x̄ = [p2t − p1t + c(x22 − x21 )]/[2c(x2 − x1 )].
We assume that ū is so large that all consumers purchase one unit of the product from oneof the firms. More specifically, we assume the following inequality:
Assumption 1 ū > m + 3c.
5

We consider a situation in which the population of consumers at time t is N exp(αt), which we normalize
to N = 1. Therefore, our interpretation is that α corresponds to the increasing rate of population and iscalled the market growth rate. This interpretation is the same as that of Ebina et al. (2015). The validityof this assumption under the Hotelling model is explained in Footnote 7 of their study.

5

The sequence of entries is as follows: Group 1 is the leader pair, who enter immediately
at T1 = 0. Group 2 is the follower pair, who enter at time T2 ∈ (0, ∞), which is determined
endogenously later. Firm Di incurs an entry cost Fi (Ti ), which is evaluated at time 0 anddecreasing in Ti . The game analyzed is an infinite-horizon game.
In group 1, U1 determines the group’s location x1 at time T1 = 0. We assume (without
loss of generality) that x1 ≤ 1/2 holds in equilibrium. After entry, w1t is determined throughNash bargaining between D1 and U1 , and then D1 sets its price p1t at t ∈ [τ, τ + dτ ). Group
1 becomes the monopoly, and can repeat the bargaining and pricing process whenever group
2 is out of the market.
In group 2, U2 oﬀers a contract (T2 , x2 ) to D2 unilaterally. This implies that U2 is in astrong position over D2 only in terms of the entry timing.6 Note that we assume that thefollower group enters if and only if entry is just acceptable for D2 , although we can derive
similar results under a diﬀerent criterion for the entry decision, which is assumed in Ebina
et al. (2015).7
After entry, competition between the groups occurs. Here, w1t and w2t are determined
simultaneously through Nash bargaining. Observing the negotiation outcomes, D1 and D2
simultaneously set their prices p1t and p2t at t ∈ [τ, τ + dτ ).
The total profits of the upstream (V1 and V2 ) and downstream (v1 and v2 ) firms of groups
1 and 2 (the leader and the follower pairs, respectively) are given by∫ T2 ∫ 1∫ ∞ ∫ x̄−(r−α)tV1 (T2 , x1 , x2 , p1t , p2t , w1t ) =[w1t − m]edxdt +[w1t − m]e−(r−α)t dxdt,
0

0

T2

0

(2)
6

This assumption is suitable to situations in which U2 is able to select one of many potential downstream
agents. However, U2 then faces a lock-in after signing the contract with D2 , owing to the necessary coordination with D2 for production, giving some bargaining power to D2 .
7

Other than incorporating a negotiation in each group and a unilateral oﬀer of the contract (T2 , x2 ) by
U2 to D2 , the basic timing structure is the same as that of Ebina et al. (2015). However, our main results
hold even under a diﬀerent criterion for the entry decision. We discuss this point in Section 5.1.

6∫ T2 ∫ 1

[p1t − w1t ]e−(r−α)t dxdt
0
∫ ∞ ∫ x̄
[p1t − w1t ]e−(r−α)t dxdt − F1 (0), (3)+v1 (T2 , x1 , x2 , p1t , p2t , w1t ) =
0

T2

0

and
∫ ∞∫ 1[w2t − m]e−(r−α)t dxdt,(4)[p2t (x; x1 , x2 , p1t ) − w2t ]e−(r−α)t dxdt − F2 (T2 ),(5)

V2 (T2 , x1 , x2 , p1t , p2t , w2t ) =T2∫ ∞∫ 1v2 (T2 , x1 , x2 , p1t , p2t , w2t ) =
T2

x̄

x̄

where m and r denote the marginal cost of each upstream firm and the interest rate, respectively. We assume that r > α to ensure that the follower enters within a finite timeframe.8
With respect to the bargaining power of group 1, we make the following assumption:
Assumption 2 β1 ∈ (0, (ū − m − 3c)/(ū − m − c)).
This assumption ensures that the monopoly outcome of group 1 at each t ∈ [0, T2 ) has aninterior solution. With respect to F2 , we also make the following assumption:
Assumption 3 (i) Fi (Ti ) = Fi exp(−rTi ). (ii) F2 > [c(2+β1 )2 (2−β2 )2 ]/[(r−α)(4−β1 β2 )2 ].
Assumption 3 (i) is made to explicitly obtain an outcome for the subgame perfect Nashequilibrium, and is the same as that in Ebina et al. (2015).9 Assumption 3 (ii) ensures that
If r ≤ α, the integral of Equation (4) could be made indefinitely larger by choosing a larger time T2 .
Thus, waiting longer would always be a better strategy, and the optimum would not exist.
89

We adopt a setting where α represents the increasing rate of population, because we consider that thisinterpretation is more natural and valid under a Hotelling location model. However, many studies consider
that α represents a cost-reducing rate when firms adopt a new technology, and assume Fi = exp(−(r − α)Ti ).
If we modify our interpretation (growing market) to that of technology improvement, our main results do
not change. Here, we can make a similar argument to that explained in the Appendix of Ebina et al. (2015,
p.912).

7

group 2 enters at a positive time T2 (> 0), and that sequential entry always occurs. This
allows us to avoid simultaneous entry at time 0 and to focus on sequential entry.10
The game proceeds as follows: At T1 = 0, U1 in group 1 determines the location of group1, x1 , in order to maximize its own total profit. Observing x1 , and to maximize its own
total profit, U2 oﬀers contract (T2 , x2 ) to D2 . At each instance t ∈ [τ, τ + dτ ), and knowing
whether group 2 is active, group i determines wit through Nash bargaining between Ui and
Di , if the group is active. Finally, Di sets its price pit .

3

Equilibrium

We derive the price outcome in the subgame perfect Nash equilibrium in Section 3.1, thetiming and the location outcomes of group 2 in Section 3.2, the location outcome of group
1 in Section 3.3, and the outcomes of the subgame perfect Nash equilibrium in Section 3.4.

3.1

Price

First, we consider the problem of prices at each t ∈ [τ, τ + dτ ), before and after the entry ofgroup 2, given x1 , x2 , and T2 . Then, we derive the local profits of U1 , U2 , D1 , and D2 at eacht ∈ [τ, τ + dτ ). Next, we provide the relations between the local profits and the bargaining
power parameters, β1 and β2 , in Remarks 1 to 3. The three remarks will be helpful tounderstand the intuition behind our main results shown in this and the next sections.
First, we consider the subgame after group 2 enters, that is, the subgame at t ∈ [T2 , ∞).
10

If simultaneous entry occurs, our model reverts to the standard location-price model with a vertical
relationship. Then, an equilibrium location pattern simply becomes the maximum diﬀerentiation. For moredetail, see Brekke and Straume (2004).

8Given w1t and w2t at t ∈ [T2 , ∞), the profits of the downstream firms are
π1t (p1t , p2t ; x1 , x2 , w1t ) = (p1t − w1t )x̄,π2t (p1t , p2t ; x1 , x2 , w1t ) = (p2t − w2t )(1 − x̄).
Solving the first-order conditions of the downstream firms, we have the following instantaneous prices:
2w1t + w2t + c (x2 − x1 ) (2 + x1 + x2 ),3
w1t + 2w2t + c (x2 − x1 ) (4 − x1 − x2 )
p2t (w1t , w2t ; x1 , x2 ) =.
3p1t (w1t , w2t ; x1 , x2 ) =

(6)
(7)

Substituting these into the profits of the firms and x̄, we have
(w2t − w1t + c(x2 − x1 )(2 + x1 + x2 ))2,18c(x2 − x1 )(w1t − w2t + c(x2 − x1 )(4 − x1 − x2 ))2π2t (w1t , w2t , x1 , x2 ) =,18c(x2 − x1 )(w1t − m)(w2t − w1t + c(x2 − x1 )(2 + x1 + x2 ))Π1t (w1t , w2t , x1 , x2 ) =,6c(x2 − x1 )(w2t − m)(w1t − w2t + c(x2 − x1 )(4 − x1 − x2 )),Π2t (w1t , w2t , x1 , x2 ) =6c(x2 − x1 )2 + x1 + x2
w2t − w1tx̄(w1t , w2t , x1 , x2 ) =+.
66c(x2 − x1 )
π1t (w1t , w2t , x1 , x2 ) =Next, we consider the bargaining problems in the vertical groups. Substituting the profits
into the objective functions of the bargaining problems in the vertical chains, Ω1t and Ω2t ,we haveΩ1t (w1t , w2t ; x1 , x2 ) = [Π1t (w1t , w2t , x1 , x2 )]β1 [π1t (w1t , w2t , x1 , x2 )]1−β1 ,Ω2t (w1t , w2t ; x1 , x2 ) = [Π2t (w1t , w2t , x1 , x2 )]β2 [π2t (w1t , w2t , x1 , x2 )]1−β2 .
Each vertical group i determines wi to maximize Ωit . Solving the maximization problems of9the groups, we havecβ1 (x2 − x1 ) [2(2 + x1 + x2 ) + (4 − x1 − x2 )β2 ](≡ w1D (x1 , x2 )),4 − β1 β2cβ2 (x2 − x1 ) [2(4 − x1 − x2 ) + (2 + x1 + x2 )β1 ]
w2t (x1 , x2 ) = m +(≡ w2D (x1 , x2 )).
4 − β1 β2

w1t (x1 , x2 ) = m +Here, wit (x1 , x2 ) monotonically increases with βi for any βi (∈ (0, 1)).
Substituting these wholesale prices into Equations (6) and (7), we have2c(1 + β1 ) (x2 − x1 ) [2(2 + x1 + x2 ) + (4 − x1 − x2 )β2 ](≡ pD
1 (x1 , x2 )),3(4 − β1 β2 )2c(1 + β2 ) (x2 − x1 ) [2(4 − x1 − x2 ) + (2 + x1 + x2 )β1 ]
p2t (x1 , x2 ) = m +(≡ pD
2 (x1 , x2 )).
3(4 − β1 β2 )

p1t (x1 , x2 ) = m +In this case, pit (x1 , x2 ) monotonically increases with βi for any βi (∈ (0, 1)).
The indiﬀerent consumer isx̄(x1 , x2 ) =(2 − β1 )[2(2 + x1 + x2 ) + (4 − x1 − x2 )β2 ],6(4 − β1 β2 )where x̄(x1 , x2 ) monotonically decreases (resp. increases) with β1 (resp. β2 ) for any β1 (∈(0, (ū − m − 3c)/(ū − m − c))) (resp. β2 (∈ (0, 1))).
Next, we consider the subgame before group 2 enters, that is, the subgame at t ∈ [0, T2 ).
Since D1 is the monopolist in the market, it chooses p1t = ū − c(1 − x1 )2 if w1t is not
suﬃciently large.11 The objective function of the bargaining problem in the vertical chain ofgroup 1 is
1−β1
Πβ1t1 × π1t= (w1t − m)β (ū − c(1 − x1 )2 − w1t )1−β .

The wholesale price is
w1M (x1 ) = [ū − c(1 − x1 )2 ]β1 + m(1 − β1 ).
Thus, we have the following lemma.
11

In the Appendix, we discuss the possibility that the market demand is partially covered by the monopolist.

10Lemma 1 The prices set by the upstream and the downstream firms are[ū − c(1 − x1 )2 ]β1 + m(1 − β1 )t ∈ [0, T2 ),w̃1t (x1 , x2 ) =(8)cβ (x − x1 ) [2(2 + x1 + x2 ) + (4 − x1 − x2 )β2 ]m + 1 2t ∈ [T2 , ∞),4 − β1 β22ū − c(1 − x1 )t ∈ [0, T2 ),p̃1t (x1 , x2 ) =2c(1 + β1 ) (x2 − x1 ) [2(2 + x1 + x2 ) + (4 − x1 − x2 )β2 ]m +t ∈ [T2 , ∞),3(4 − β1 β2 )
(9)cβ2 (x2 − x1 ) [2(4 − x1 − x2 ) + (2 + x1 + x2 )β1 ]w̃2t (x1 , x2 ) = m +t ∈ [T2 , ∞),(10)
4 − β1 β22c(1 + β2 ) (x2 − x1 ) [2(4 − x1 − x2 ) + (2 + x1 + x2 )β1 ]p̃2t (x1 , x2 ) = m +t ∈ [T2 , ∞). (11)3(4 − β1 β2 )The instantaneous profit flows of the two groups are

∫1 M
2
ΠM
t ∈ [0, T2 ),
1 (x1 ) ≡ 0 [w1 (x1 ) − m]dx = [ū − c(1 − x1 ) − m]β1∫ x̄ D
 D
Π1t (x1 , x2 ) = Π1 (x1 , x2 ) ≡ 0 [w1 (x1 , x2 ) − m]dx
cβ1 (2 − β1 ) (x2 − x1 ) [2(2 + x1 + x2 ) + (4 − x1 − x2 )β2 ]2t ∈ [T2 , ∞),=6(4 − β1 β2 )2∫1M2
π1M (x1 ) ≡ 0 [pM
1 (x1 ) − w1 (x1 )]dx = [ū − c(1 − x1 ) − m](1 − β1 ) t ∈ [0, T2 ),∫ x̄ D D
D
π1t (x1 , x2 ) = π1 (x1 , x2 ) ≡ 0 [p1 (x1 , x2 ) − w1 (x1 , x2 )]dxc(2 − β1 )2 (x2 − x1 ) [2(2 + x1 + x2 ) + (4 − x1 − x2 )β2 ]2t ∈ [T2 , ∞),=18(4 − β1 β2 )20t ∈ [0, T2 ),



∫1 D
 D
Π2t (x1 , x2 ) = Π2 (x1 , x2 ) = x̄ [w2 (x1 , x2 ) − m]dx
c(2 − β2 )β2 (x2 − x1 ) [2(4 − x1 − x2 ) + (2 + x1 + x2 )β1 ]2 =t ∈ [T2 , ∞),6(4 − β1 β2 )20t ∈ [0, T2 ),



∫1 D
 D
D
π2t (x1 , x2 ) = π2 (x1 , x2 ) = x̄ [p2 (x1 , x2 ) − w2 (x1 , x2 )]dxc(2 − β2 )2 (x2 − x1 ) [2(4 − x1 − x2 ) + (2 + x1 + x2 )β1 ]2 =t ∈ [T2 , ∞).
18(4 − β1 β2 )2Second, before presenting the nature of the locations and the bargaining power parameters, we briefly summarize the eﬀects of bargaining power on the ratio of the monopoly andDthe duopoly profits for U1 , namely ΠM
1 (x1 )/Π1 (x1 ), which directly influences the location

decision of U1 .
11

D
Lemma 2 (i) If β1 increases, the ratio of the monopoly and the duopoly profits, ΠM
1 (x1 )/Π1 (x1 ),increases.
D(ii) If β2 increases, the ratio of the monopoly and the duopoly profits, ΠM
1 (x1 )/Π1 (x1 ), de-creases.
As β1 increases, the competitiveness of D1 weakens owing to an increase in w1t in the duopoly,although this does not matter in the monopoly. This implies that the relative importance ofthe monopoly profit increases as β1 increases. In addition, as β2 increases, the competitiveness of D1 strengthens owing to the increase in w2t in the duopoly, although this does notmatter in the monopoly. This implies that the relative importance of the duopoly profit forgroup 1 increases as β2 increases.
Before considering the problem of each group’s entry timing, we summarize three properties with respect to instantaneous profits. These properties are discussed and utilized in
Subsection 4.2.
Remark 1 The instantaneous profit of a firm in a vertical chain is completely correlatedwith that of another firm in the sense that the locations, x1 and x2 , influence the profits inthe vertical chain almost equally (compare Πit (x1 , x2 ) with πit (x1 , x2 ) right after Lemma 1).
Remark 2 Here, we examine the eﬀect of an increase in βi (i = 1, 2). In the duopoly situation, the larger β1 is, the higher w1t is, reducing the competitiveness of D1 in the downstream
market. Here, U1 in group 1 is able to overcome the weakness by choosing a location that iscloser to the center, although this location accelerates the downstream competition. Because
group i anticipates this strong downstream competition, w1t is determined at a lower level
than when D1 locates far from the center. Under the assumption that firms’ locations are
restricted within the line segment, the competition-accelerating eﬀect dominates the demand
expansion.12
12

If firms’ locations are not restricted within the line segment, a higher β1 works as a commitment to12Remark 3 The larger β2 is, the higher w2t is, enhancing the competitiveness of D1 in thedownstream market. However, this competitive advantage is diminished if U1 chooses alocation closer to the center, because this location accelerates the downstream competition,inducing group 2 to set a lower w2t . The larger β2 is, the greater the wholesale price reductionis, which implies that x1 decreases.
Substituting the results of Equations (8) to (11) into Equations (2) to (5), the total profits
of the upstream and the downstream firms in both groups are derived as follows:
V1 (T2 , x1 , x2 ) ≡ V1 (T2 , x1 , x2 , p̃1t (x1 , x2 ), p̃2t (x1 , x2 ), w̃1t (x1 , x2 ), w̃2t (x1 , x2 ))
∫ T2
∫ ∞
M
−(r−α)t−(r−α)t=Π1 (x1 )e
dt +
ΠD
dt,
1 (x1 , x2 )e
0

(12)

T2

v1 (T2 , x1 , x2 ) ≡ v1 (T2 , x1 , x2 , p̃1t (x1 , x2 ), p̃2t (x1 , x2 ), w̃1t (x1 , x2 ), w̃2t (x1 , x2 ))∫ ∞=
π1D (x1 , x2 )e−(r−α)t dt − F1 ,T2

V2 (T2 , x1 , x2 ) ≡ V2 (T2 , x1 , x2 , p̃1t (x1 , x2 ), p̃2t (x1 , x2 ), w̃1t (x1 , x2 ), w̃2t (x1 , x2 ))∫ ∞−(r−α)t=ΠDdt,
2 (x1 , x2 )eT2

v2 (T2 , x1 , x2 ) ≡ v2 (T2 , x1 , x2 , p̃1t (x1 , x2 ), p̃2t (x1 , x2 ), w̃1t (x1 , x2 ), w̃2t (x1 , x2 ))∫ ∞=
π2D (x1 , x2 )e−(r−α)t dt − F2 e−rT2 .
T2

3.2

Timing and location of group 2Here, we derive the timing and the location outcomes. First, we derive the equilibrium
location of group 2. Then, we derive the timing of group 2. In Section 3.3, we derive theequilibrium location of group 1.
First, we derive the equilibrium location of group 2.
locate closer to the center, inducing group 2 to locate outside the line segment. However, the commitment
device does not work in the current setting.

13

Lemma 3 The upstream firm U2 oﬀers location xE
2 = 1 in a contract with downstream firm
D2 .
Proof. Diﬀerentiating V2 with respect to x2 , we have∂V2
e−(r−α)T2 cβ[2(4 + β1 ) − (3x2 − x1 )(2 − β1 )][2(4 + β1 ) − (2 − β1 )(x1 + x2 )]
=
> 0,
∂x2
6(r − α)(4 − β1 β2 )2■implying that xE
2 = 1.

This result shows that xE
2 = 1 is a dominant strategy for the follower group. The result
is the same as that in Ebina et al. (2015), who do not consider the vertical relationship. We
show that this result holds for the case with a vertical relationship and for any bargaining
power βi ∈ [0, 1] with regard to the firms.
Hereafter, to avoid notational clutter, we substitute xE
2 = 1 into the equations andEDomit x2 in the third term, where it is not necessary (e.g., ΠD
1 (x1 , x2 = 1) ≡ Π1 (x1 ) andV1 (T2 , x1 , xE
2 = 1) ≡ V1 (T2 , x1 )).
Proposition 1 The equilibrium timing oﬀered by U2 to D2 is as follows:[][](r − α)F218(4 − β1 β2 )2 (r − α)F211T̃2 (x1 ) = log= log. (13)ααπ2D (x1 )c(2 − β2 )2 (1 − x1 ) [2(3 − x1 ) + (3 + x1 )β1 ]2Proof. First, we show that V2 is decreasing in T2 , as follows:∂V2−(r−α)T2= −ΠD< 0,
2 (x1 )e∂T2
which implies that the U2 wants D2 to enter as soon as possible. Since U2 can unilaterally oﬀer
T2 to D2 , U2 can induce entry by D2 by setting T2 = T̃2 (x1 ), which satisfies v2 (T2 , x1 ) = 0.
Therefore, we have the following equation:[](r − α)F2π2D (x1 ) −(r−α)T21−rT2ev2 (T2 , x1 ) = 0 ⇔= F2 e(≡ T̃2 (x1 )).
⇔ T2 = logr−ααπ2D (x1 )Because v2 is negative if and only if T2 < T̃2 (x1 ), this solution is unique.
14■

Although the entry timing of group 2 presented in Proposition 1 looks similar to thatof Proposition 1 in Ebina et al. (2015), the results are diﬀerent because of the vertical
relationships and the form of competition. We discuss this point in more detail and how thisdiﬀerence aﬀects the equilibrium behavior of the firms in Section 5.
The following corollaries show the change in group 2’s entry timing as the leader’s location
x1 and the exogenous parameters change.
Corollary 1 Given x1 , the optimal timing of U2 , T̃2 (x1 ), is decreasing in the bargaining
power of U1 , β1 , and increasing in that of U2 , β2 .
Proof. Diﬀerentiating T̃2 (x1 ) with respect to β1 and β2 , we have
∂ T̃2 (x1 )
4[2 (3 + x1 ) + β2 (3 − x1 )]
=−
< 0.
∂β1
α (4 − β1 β2 ) [2 (3 − x1 ) + β1 (3 + x1 )]∂ T̃2 (x1 )
4 (2 − β1 )
=
> 0.
∂β2
α (2 − β2 ) (4 − β1 β2 )■This proves the corollary.

We discuss the intuition behind Corollary 1. The larger β1 is, the higher w1t is, reducing
the competitiveness of D1 in the downstream market when group 2 stays. This implies thata larger β1 makes the entry of group 2 easier. The larger β2 is, the higher w2t is, enhancing
the competitiveness of D1 in the downstream market when group 2 exists. This implies thata larger β2 makes it more diﬃcult for D2 to cover its entry cost through its total discounted
profit.
Corollary 2 If x1 , F2 , or r increase, or if c or α decrease, the optimal timing of group 2 toenter is delayed.

15

Proof. Diﬀerentiating T̃2 (x1 ) with respect to each parameter, we have∂ T̃2
10 + β1 (1 + 3x1 ) − 6x1
=
> 0,
∂x1
α (1 − x1 ) [6 + β1 (3 + x1 ) − 2x1 ]
∂ T̃2
1
=
> 0,
∂F2
αF2
∂ T̃2
1
=
> 0,
∂r
α(r − α)
∂ T̃2
1
=−
< 0,
∂c
cα
{
}
(r−α)F2
α
+
log
D
r−α
∂ T̃2
π2 (x1 )
=−
< 0.
2
∂α
α
■

This proves the corollary.

These properties in Corollary 2, which describe whether each parameter aﬀects the timingof group 2 positively or negatively, are the same as those in Ebina et al. (2015).
Finally, we derive the cross-partial derivatives of T̃2 (x1 ) with respect to β1 or β2 .
Lemma 4 The cross-partial derivatives of T̃2 with respect to x1 and βi are given as follows:∂ 2 T̃224=−< 0,∂β1 ∂x1
α [2(3 − x1 ) + β1 (3 + x1 )] 2
∂ 2 T̃2= 0.
∂β2 ∂x1
We utilize the above lemma to explain how each parameter aﬀects the outcomes in thesubgame perfect Nash equilibrium in Subsection 4.2.

3.3

Location of group 1

Here, we consider the problem of the leader. Substituting the results of the lemmas and the
proposition into Equation (12) of V1 , we have the following maximization problem for U1 :max V1 (x1 ) ≡ V1 (T̃2 (x1 ), x1 )x1 ∈[0,1/2]∫ T̃2 (x1 )=−(r−α)tdt +ΠM1 (x1 )e∫ ∞T̃2 (x1 )016−(r−α)tdt.
ΠD
1 (x1 )eIn order to obtain the optimal location of U1 , diﬀerentiating V1 with respect to x1 , we havedT̃2 (x1 ) M
dV1 (x1 )= exp(−(r − α)T̃2 (x1 ))(Π1 (x1 ) − ΠD
1 (x1 ))
dx1
dx1
1 − exp(−(r − α)T̃2 (x1 )) dΠM
1 (x1 )+r−α
dx1
exp(−(r − α)T̃2 (x1 )) dΠD
1 (x1 )+. (14)r−αdx1
The sign of Equation (14) is significant and determines the location of group 1 at the center(1/2), the edge (0), or an interior point (strictly between 0 and 1/2). If the sign of Equation(14) is positive, U1 should locate at the center, whereas if it is negative, U1 locates at theedge. If Equation (14) is equal to 0 for x1 ∈ (0, 1/2), U1 should locate at the interior point
x1 .
Now, we examine the signs of the three terms in order to obtain the equilibrium location ofgroup 1, and to give economic interpretations of the three terms. The first term of Equation(14) (call it (I), the entry-delay eﬀect) signifies the gain from the delay of entry by group 2that is caused by an increase in x1 , allowing group 1 to maintain its monopoly profit beforeDthe duopoly regime begins. The sign of the first term is positive, because ΠM
1 (x1 ) > Π1 (x1 )and, thus, the following inequality holds:
10 + β1 (1 + 3x1 ) − 6x1
dT̃2 (x1 )
=
> 0.
dx1
α(1 − x1 )[6 + β1 (3 + x1 ) − 2x1 ](15)The second term of Equation (14) (call it (II), the monopoly-gain eﬀect) signifies theincrease in the monopoly profit, which is increased by moving closer to the center. As shown
below, this term becomes positive:dΠM
1 (x1 )= 2c(1 − x1 )β1 > 0.
dx1

(16)

Finally, the third term of Equation (14) (call it (III), the duopoly-gain eﬀect) signifies
how the duopoly profit decreases as group 1 moves closer to group 2, thus intensifying the17competition. The sign of this term becomes negative:dΠDc(2 − β1 )β1 [2 + 6x1 + (5 − 3x1 )β2 ][6 + 2x1 − (3 − x1 )β2 ]1 (x1 )=−< 0.
dx1
6(4 − β1 β2 )2(17)Thus, the first and second terms are positive, and the last term is negative. If the eﬀects of(I) and (II) are relatively large, the optimal location of group 1 is 1/2.
We now investigate the eﬀect of each parameter on the equilibrium location of group 1,xE
1 . If T̃2 (x1 ) converges to zero (α is close to r), group 2 enters the market immediately,yielding the duopoly competition forever. This implies that xE
1 = 0 holds. With regard to
the possibility that xE
1 = 1/2, we have the following proposition.
E
Proposition 2 (i) If ū is suﬃciently large, xE
1 = 1/2. (ii) If α is suﬃciently small, x1 =1/2.
Proof. If ū is suﬃciently large, since ū only exists in the first term being positive, Equation(14) becomes positive.
Related to Equation (14), (15), (16), and (17) and the following inequalities hold:e−(r−α)T̃2 (x1 ) ≥ 0,DMDΠM
1 (x1 ) − Π1 (x1 ) > Π1 (0) − Π1 (0)[()]
3 (2 − β1 ) (β2 + 2) 2
= β1 ū − m + c+1> 0,
2 (4 − β1 β2 ) 2([])(r−α)F2
1
exp
−(r
−
α)
log
D
α
exp(−(r − α)T̃2 (x1 ))
π2 (x1 )
lim
= lim
= 0.
α→0
α→0
r−α
r−α

Thus, as α approaches 0, the last terms converge to 0, whereas the first term becomes positiveand infinitely large, and the second term becomes positive and finite. Therefore, when αapproaches 0, Equation (14) is positive, implying that the equilibrium location of group 1 is
■

1/2.

18

When α approaches r, the timing of group 2’s entry becomes earlier, which induces group
1 to weigh the importance of its profitability under the duopoly more highly.13 The last termin Equation (14) dominates the other two terms,14 which implies that the optimal location
Eof group 1 is xE
1 = 0. The relationships between the other parameters, βi , c, r, F2 , and x1

are discussed in Section 4 using a numerical analysis.

3.4

Outcomes in the subgame perfect Nash equilibrium

Before we proceed with our numerical analysis, we explicitly show analytical results under
two polar cases, where α is suﬃciently small, and then suﬃciently large. We have thefollowing proposition.
Proposition 3 (i) If Equation (14) is positive, the outcome in the subgame perfect Nashequilibrium is]
144 (4 − β1 β2 ) 2 F2 (r − α),α(10 + 7β1 ) 2 (2 − β2 ) 2 c
1(2 − β1 ) (14 + 5β2 )x∗1 = , x∗2 = 1, x̄∗ =,2
12 (4 − β1 β2 ){)(
w1M (x∗1 ) = m + β1 ū − m − 4ct ∈ [0, T2∗ ),∗w1t =1 (14+5β2 )ct ∈ [T2∗ , ∞)w1D (x∗1 ) = m + β4(4−β1 β2 ){c∗t ∈ [0, T2∗ ),pM
1 (x1 ) = ū − 4p∗1t =(β1 +1)(14+5β2 )c∗∗pDt ∈ [T2∗ , ∞)1 (x1 , x2 ) = m +6(4−β1 β2 )1
T2∗ = log[

(10 + 7β1 ) β2 ct ∈ [T2∗ , ∞)4 (4 − β1 β2 )(10 + 7β1 ) (1 + β2 ) c∗∗
p∗2t = pDt ∈ [T2∗ , ∞).
2 (x1 , x2 ) = m +6 (4 − β1 β2 )∗w2t
= w2D (x∗1 , x∗2 ) = m +13

In contrast to Proposition 2(ii), for higher α, we cannot rigorously specify the sign of Equation (14)owing to the upper bound of α, which is imposed in Assumption 3(ii). If we could ignore the upper boundof α and could make α approach r, we could show that the value in (14) becomes negative.
14

When T̃2 approaches 0, the second term of Equation (14) becomes zero.

19

(ii) If Equation (14) is negative, the outcome in the subgame perfect Nash equilibrium is[]1
2 (4 − β1 β2 ) 2 F2 (r − α)∗∗T2 = log,
α(2 + β1 ) 2 (2 − β2 ) 2 c(2 − β1 ) (2 + β2 )
∗∗
∗∗
,
x∗∗
=
1 = 0, x2 = 1, x̄
2 (4 − β1 β2 ){
w1M (x∗∗t ∈ [0, T2∗∗ ),1 ) = m + (ū − m − c)β1∗∗
w1t=
3β1 (2+β2 )cw1D (x∗∗t ∈ [T2∗∗ , ∞)1 ) = m + 4−β1 β2{∗∗pMt ∈ [0, T2∗∗ ),1 (x1 ) = ū − c=p∗∗1t2(1+β1 )(2+β2 )c∗∗∗∗t ∈ [T2∗∗ , ∞)pD
1 (x1 , x2 ) = m +4−β1 β23 (2 + β1 ) β2 ct ∈ [T2∗∗ , ∞)4 − β1 β2
2 (2 + β1 ) (1 + β2 ) c
D ∗∗
∗∗
p∗∗
t ∈ [T2∗∗ , ∞).
2t = p2 (x1 , x2 ) = m +4 − β1 β2

∗∗
∗∗
w2t
= w2D (x∗∗1 , x2 ) = m +Proof. Substituting the equilibrium locations, xEi , into Equations (8)–(11) of Lemma 1 andEquation (13) of Proposition 1, we have the desired result.

■

Remark 4 We also obtain an implicit-form solution, as follows. If there exists α̂, such thatx∗∗∗is between 0 and 1/2, the outcome in the subgame perfect Nash equilibrium is1[]
18(4 − β1 β2 )2 (r − α)F2
1
∗∗∗
∗∗∗
T2 = T̃2 (x1 ) = log
2 ,
∗∗∗
∗∗∗
α
c(2 − β2 )2 (1 − x∗∗∗1 ) [2(3 − x1 ) + (3 + x1 )β1 ](2 − β1 ) [3(2 + β2 ) + (2 − β2 ) x∗∗∗
1 ]
∗∗∗
∗∗∗
,
x∗∗∗
∈
(0,
1/2),
x
=
1,
x̄
=
1
2
6 (4 − β1 β2 ){ M ∗∗∗2
w1 (x1 ) = [ū − c(1 − x∗∗∗t ∈ [0, T2∗∗∗ ),1 ) ]β1 + m(1 − β1 )
∗∗
w1t
=
∗∗∗
∗∗∗
∗∗∗
cβ1 (1−x1 )[2(3+x1 )+(3−x1 )β2 ]t ∈ [T2∗∗∗ , ∞),w1D (x∗∗∗1 ) = m+4−β1 β2{ M ∗∗∗2
p1 (x1 ) = ū − c(1 − x∗∗∗t ∈ [0, T2∗∗∗ ),
1 )
∗∗
∗∗∗
∗∗∗
p1t =
2c(1+β1 )(1−x∗∗∗
1 )[2(3+x1 )+(3−x1 )β2 ]∗∗∗∗∗∗t ∈ [T2∗∗∗ , ∞),pD
1 (x1 , x2 ) = m +3(4−β1 β2 )∗∗∗∗∗∗
cβ2 (1 − x∗∗∗1 ) [2(3 − x1 ) + (3 + x1 )β1 ]t ∈ [T2∗∗∗ , ∞),4 − β1 β2∗∗∗∗∗∗2c(1 + β2 ) (1 − x∗∗∗1 ) [2(3 − x1 ) + (3 + x1 )β1 ]
∗∗∗
D ∗∗∗
)
=
m
+
,
x
(x
=
p
p∗∗∗
t ∈ [T2∗∗∗ , ∞).
2
1
2
2t3(4 − β1 β2 )∗∗∗∗∗∗= w2D (x∗∗∗w2t
1 , x2 ) = m +As shown above, the equilibrium prices, locations, and entry timing depend on the parameters, c, r, ū, F2 , α, and βi . In addition, the above-mentioned outcomes are complicated. We
20

confirm how these outcomes change using a numerical analysis, shown in Table 4 in Section
4.

4

Numerical analysis

To investigate the optimal location of group 1 and the total discounted values of the firms,
we conduct numerical analyses in Sections 4.1 and 4.2. First, Section 4.1 investigates theeﬀects of the parameters ū (each consumer’s gross surplus for the product), c (consumer’stransport cost), α (the growth rate of the market size), r (the interest rate), and F2 (group∗∗∗∗∗∗2’s entry cost) on group 1’s equilibrium location, denoted as xE
1 ∈ {x1 , x1 , x1 }. Then,

Section 4.2 investigates the eﬀects of the bargaining power, βi , on xE
1 . In particular, weshow that a non-monotonic relationship exists between β1 and xE1.

4.1

The Eﬀects of c, m, r, ū, F2 , and α on xE
1

Before discussing the eﬀects of the suppliers’ bargaining power βi , which are the most important factors in our model, we summarize the eﬀects of c, m, r, ū, F2 , and α on xE
1 byconsidering Equation (14). The fundamental properties behind the eﬀects of c, m, r, ū, F2 ,and α on x1 are the same as those demonstrated by Ebina et al. (2015).
With regard to ū, r, and F2 , if one of these parameters increases, group 1 weighs theimportance of the monopoly phase more highly, yielding higher xE
1 . Intuitively, an increasein ū increases only the profit of group 1 in the monopoly phase. Both an increase in r andin F2 delay entry by group 2 by increasing its real entry cost, implying a longer duration of
the monopoly phase.
With regard to c and α, if one of these parameters increases, group 1 weighs the duopoly
phase more highly, because such an increase makes entry by group 2 easier as a result of thegreater product diﬀerentiation between D1 and D2 and the faster market growth.

21Changing these parameters aﬀects xE
1 . The fundamental direction of these eﬀects issummarized in Table 1
xE
1
xE
2

c↗
↘
1

r↗
↗
1

ū ↗
↗
1

F2 ↗
↗
1

α↗
↘
1

Table 1: Locations of firms 1 and 2, when c, r, ū, F2 , or α increases. xE
1 ↗ means thelocation of firm 1 approaches 0.5 (the center), whereas xE
↘
means
the
locationof firm 1
1
approaches 0 (the edge).

4.2

The eﬀects of β1 and β2 on xE
1

We illustrate the relationship between the bargaining powers, β1 and β2 , and the equilibrium
location of group 1, xE
1 , in the case where ū = 25 (see Table 2).
β1 \β2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9

0.05
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5

0.1
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5

0.15
0.259
0.283
0.336
0.5
0.5
0.5
0.5
0.5
0.5

0.2
0.140
0.149
0.167
0.197
0.251
0.5
0.5
0.5
0.5

0.25
0.3 0.35 · · ·
0.035
0
0
···
0.039
0
0
···
0.048
0
0
···
0.063
0
0
···
0.088
0
0
···
0.128
0
0
···
0.198 0.019
0···
0.5 0.074
0
···
0.5
0.5
0
···

1
0
0
0
0
0
0
0
0
0

Table 2: The equilibrium location xE
1 for F2 = 1000, c = 1, m = 1, r = 0.1, ū = 25,
α = 0.0988, when β1 ∈ (0, 21/23) and β2 ∈ [0, 1].

First, we consider how β2 aﬀects xE
1 . Table 2 shows that as the bargaining power of
U2 , β2 , increases, the optimal location of group 1, xE
1 , monotonically decreases for given
β1 . The reason has already mentioned in Remark 3; that is, accelerating the downstream
competition through an increase in x1 substantially decreases w2t , which is higher owing to22

larger β2 . Thus, x1 becomes lower when β2 is larger. The inverse relation between β2 andxE
1 holds on most of the parameter space (c, m, r, ū, F2 , α).
Second, we consider how β1 aﬀects xE
1 . As in the decomposition in (14), the location
decision of group 1 depends on three eﬀects: (I) the entry-delay eﬀect, (II) the monopoly-gaineﬀect; and (III) the duopoly-gain eﬀect. From (I), the center (xE
1 = 1/2) is the best option
and the edge (xE
1 = 0) is the worst option. This is because the ex post stronger competition,achieved by a central location of D1 , reduces the profitability of group 2, inducing group
2 to delay its entry. This eﬀect is multiplied by the diﬀerence between the profits underDthe monopoly and the duopoly, ΠM
1 (x1 ) − Π1 (x1 ). The higher β1 is, the stronger the factor

is, because an increase in β1 increases ΠM
1 (x1 ) directly, owing to the stronger bargaining
15
power, and diminishes ΠD
From (II),1 (x1 ) owing to the weaker competitiveness of group 1.

the center is the best option and the edge is the worst option, because the monopoly priceis constrained by the transportation cost of the furthest consumer from D1 , and x1 = 1/2
minimizes this cost. As in (I), the higher β1 is, the stronger the factor is.16 From (III),the edge is the best option and the center is the worst option, as in the standard spatialcompetition. The higher β1 is, the weaker the competitiveness of group 1 is, inducing group
1 to locate closer to the edge.17 If the eﬀects of (I) and (II) dominate that of (III), an increasein β1 induces group 1 to locate closer to the center, as in Table 2.
Note that the monotonic relation between β1 and xE
1 does not always hold. To understand15

Note that, from Lemma 4, the higher β1 is, the lower dT̃2 (x1 )/dx1 is, because a higher β1 leads to ahigher w1 , which eases the barrier to entry for group 2. This eﬀect is not that strong, because an increasein β1 indirectly influences group 2’s timing decision through the strategic interaction in the downstream
market.
16

Note that an increase in β1 diminishes the first fraction of the second term in (14) through a decreasein T̃2 (x1 ), which diminishes the second term. In the subsequent numerical example, this decrease is oﬀset
by the enhancement mentioned in the main text.
17

Note that an increase in β1 increases the first fraction of the third term in (14) through a decrease inT̃2 (x1 ), which increases the absolute value of the third term.

23the complexity of the relation between β1 and xE
1 , we investigate the relative sizes of thethree terms in Equation (14). Using the parameters and group 1’s location xE
1 , we derive
concrete values of Equation (14) for β1 ∈ (0, 0.9]. The common parameter values are c = 1,m = 1, r = 0.1, ū = 25, F2 = 1000, α = 0.0982, and β2 = 0.5. The result is summarized inTable 3. The “Sum” column in Table 3 indicates how U1 decides its location. If the value inthe column is negative, then U1 locates at 0. However, if the value is positive, then U1 locates
at 1/2. If the value is 0, an interior location between 0 and 1/2 arises. As discussed earlier,each term in Table 3 monotonically changes with an increase in β1 , except for the change
between β1 = 0.45 and β1 = 0.5 in the second term (from 12.1 to 10.9). The exception comesfrom the substantial change of xE
1 , from 0.051 to 0.5. This table shows an example in which
the relation between β1 and xE
1 is non-monotonic.
Third, we consider how α and β2 influence the relationship between β1 and xE
1 . Supposethat c = 1, m = 1, r = 0.1, ū = 25, and F2 = 1000. The three figures in Figure 1 depict how
α influences the relation between β1 and xE
1 under β2 = 0.25, β2 = 0.5, and β2 = 0.75. The
top inequality in each figure (e.g., α ≤ 0.0986 in Figure 1(a)) and the bottom inequality inEeach figure (e.g., α ≥ 0.0991 in Figure 1(a)) show that xE
1 = 0.5 and x1 = 0 become therespective equilibrium locations under these ranges of α.
Figure 1(a) indicates that as α increases (decreases, respectively), xE
1 becomes closer tothe edge (center, respectively). This result, which is explained in the previous section, isthe same as shown in Ebina et al. (2015). As α increases, the duopoly phase becomes
longer, inducing group 1 to locate closer to the edge. On the other hand, as α decreases,
the monopoly phase becomes longer, inducing group 1 to locate closer to the center. Whenβ2 = 0.25 in Figure 1(a), xE
1 is increasing with β1 . In contrast, when β2 = 0.5 and β2 = 0.75
in Figures 1(b) and 1(c), respectively, xE
1 is not always increasing with β1 . Figures 1(b)and 1(c) show an interesting property that the equilibrium location becomes U-shaped with24

β10.0001
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9

xE
1
0.5
0.5
0.090
0.072
0.059
0.050
0.044
0.043
0.044
0.051
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5

(I) First term
0.0623
30.7
37.4
54.3
70.4
85.9
101
116
131
146
277
302
326
351
375
399
422
446
469

(II) Second term
0.00289
1.41
3.43
4.99
6.45
7.79
9.02
10.1
11.2
12.1
10.9
11.6
12.3
13.0
13.5
14.1
14.5
15.0
15.3

(III) Third term
−0.0611
−30.2
−40.8
−59.3
−76.8
−93.7
−110
−126
−142
−158
−264
−285
−305
−323
−340
−355
−369
−382
−393

Sum
0.00407
1.88
0
0
0
0
0
0
0
0
23.5
28.3
34.1
41.0
48.5
57.3
67.4
78.8
91.7

Table 3: Values of the three terms in Equation (14) when c = 1, m = 1, r = 0.1, ū = 25,
F2 = 1000, α = 0.0982, and β2 = 0.5.

respect to β1 . In particular, with regard to Figure 1(b), xE
1 is in the range (0, 1/2) if β1
is neither small nor large; otherwise, xE
1 is 0 or 1/2. Because there is no clear-cut relation
between β1 and the three significant eﬀects (I), (II), and (III), a non-monotonic relationshipbetween β1 and xE
1 can emerge, as in Table 3.
From the three figures in Figure 1, we determine how β2 influences the curve of x1 foreach α. When β2 = 0.25, xE
1 is (almost) monotonically increasing with β1 ; when β2 = 0.5,
xE
1 forms a U-shape with respect to β1 and reaches the bottom around β1 ∈ [0.25, 0.3]; whenβ2 = 0.75, xE
1 suddenly changes from one corner solution to the other, and reaches thebottom around β1 ∈ [0.45, 0.65].
It may seem that the discussion in this subsection implies that a manufacturer that needs
25

Figure 1: The relationship between β1 ∈ (0, 21/23) and xE
1 , when (a) β2 = 0.25 (upper left),(b) β2 = 0.5 (upper right), and (c) β2 = 0.75 (bottom).

a downstream agent to launch a new product faces a diﬃculty in determining its optimal
product position if it anticipates the subsequent entry of another supplier, and the market
growth rate α is close to the interest rate r (if these do not hold, the optimal location is thecenter of the Hotelling line). However, we do not think that the non-monotonic relationshipbetween β1 and xE
1 is a serious problem for U1 . In other words, we do not think that thesignificant jump of xE
1 as a result of an increase in β1 makes U1 diﬃcult to choose its optimal
location. In fact, under the parameter set in Figure 2, which is explained later, V1 has two
locally maximum points with respect to x1 , and this multiplicity of locally optimal points
causes the substantial jump of xE
1 from one of the two to the other. Thus, as shown in
Figure 2 later, V1 smoothly changes with β1 , even when x1 jumps substantially from one of26

the locally optimal points to another as a result of an increase in β1 . Therefore, the complex
relation between β1 and xE
1 is not that serious for U1 , but would be serious for U2 , as shownin Figure 2, later.

4.3

The eﬀect of β1 on T̃2 (xE
1)

Table 4 shows a non-monotonic relationship between β1 and T̃2 (xE
1 ) through a monotonic
increase in xE
1 . An increase in β1 has two contrasting eﬀects. First, increasing β1 itself
decreases T̃2 (x1 ) for a fixed x1 , as shown in Corollary 1. Second, increasing β1 makes xE
1closer to the center, yielding an increase in T̃2 (xE
1 ) through stronger duopoly competition. We
explain the two contrasting eﬀects here using Table 4. When β1 ∈ (0, 0.5], the former eﬀect
Edominates the latter eﬀect; T̃2 (xE
1 ) decreases, because x1 moves toward the center slowly inaccordance with an increase in β1 . In contrast, when β1 ∈ [0.5, 0.7], the movement of xE
1accelerates, so that the latter eﬀect dominates the former eﬀect; T̃2 (xE
1 ) increases. Moreover,when β1 ∈ [0.7, 21/23), this trend suddenly changes, because xE
1 reaches 1/2 (the center),and the latter eﬀects vanish. This is why T̃2 (xE
1 ) decreases again when β1 ∈ [0.7, 21/23).
β2
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1

β1
0.001
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
21/23

xE
xE
1
2
0.120 10.126 10.136 10.153 10.178 10.215 10.276 1
0.5
1
0.5
1
0.5
1
0.5
1

T2E
10.1
9.14
8.26
7.52
6.94
6.60
6.72
10.3
9.33
8.39
8.27

pM
1
24.2
24.2
24.3
24.3
24.3
24.4
24.5
24.8
24.8
24.8
24.8

w1M
1.02
3.32
5.65
7.98
10.3
12.7
15.1
17.6
20
22.4
22.7

pD
1
1.96
2.05
2.14
2.22
2.29
2.33
2.34
2.05
2.11
2.17
2.18

w1D
1.00
1.14
1.28
1.42
1.55
1.67
1.75
1.65
1.74
1.83
1.85

pD
2
1.93
1.97
2.01
2.04
2.05
2.05
2.00
1.70
1.73
1.76
1.77

w2D
1.13
1.13
1.14
1.14
1.14
1.14
1.14
1.09
1.10
1.10
1.10

Table 4: The outcome of the subgame perfect Nash equilibrium depending on β1 and β2
when c = 1, m = 1, r = 0.1, ū = 25, F2 = 1000, and α = 0.099.

27

4.4

Eﬀects of bargaining power on the total values

In this subsection, we investigate how the bargaining power of group 1, β1 , aﬀects thetotal values of the firms (U1 , D1 , U2 , and D2 ), V1 , v1 , V2 , and v2 . These relationships are
summarized in Figure 2. Suppose that c = 1, m = 1, r = 0.1, ū = 25, F1 = 0, and F2 = 1000.

Figure 2: The relationship between β1 ∈ (0, 21/23) and the total values V1 and v1 in equilibrium when (a) α = 0.0982 (upper left), and (b) α = 0.09825 (upper right); and therelationship between β1 ∈ (0, 21/23) and V2 in equilibrium when (c) α = 0.0982 (lower
right), and (d) when α = 0.09825 (lower left).

Figures (a) and (b) in Figure 2 depict the relationship between β1 and V1 and thatbetween β1 and v1 under α = 0.0982 and α = 0.09825, respectively. These include the caseof β2 = 0.5, which is the same as that shown in Figure 1(b). We can easily confirm that the
total value of U1 strictly increases with β1 , although that of D1 strictly decreases with β1 .

28Note that v1 drops at some β1 when xE
1 changes substantially with an increase in β1 . For
example, in Figure 2(a), v1 |β2 =0.5 drops at β1 = 0.1. Furthermore, Figures 2(a) and (b) showa non-monotonic relationship between v1 and β2 , although the relationship between V1 andβ2 is monotonic. With regard to Figure 2(a), v1 |β2 =0.4 ≥ v1 |β2 =0.5 holds for β1 ∈ [0.1, 0.2],
though v1 |β2 =0.4 < v1 |β2 =0.5 holds for β1 ∈ (0, 0.1) and for β1 ∈ (0.2, 21/23). With regard to
Figure 2(b), we see a similar relationship, where v1 |β2 =0.4 ≥ v1 |β2 =0.5 holds for β1 ∈ (0, 0.2],
though v1 |β2 =0.4 < v1 |β2 =0.5 holds for β1 ∈ (0.2, 21/23).
As mentioned in the last paragraph in Section 4.2, the non-monotonic relationship between xE
1 and β1 has a significant impact on U2 . To understand the relation between the
non-monotonic relationship and V2 , we focus on the case of α = 0.0982 in Figure 1(b) andV2 |β2 =0.5 in Figure 2(c). When α = 0.0982 in Figure 1(b), xE
1 drops substantially from 1/2
to about 0.1 when β1 moves from 0.05 to 0.1; xE
1 moves around 0.075 when β1 is in the range[0.1, 0.45]; and xE
1 jumps up substantially from about 0.05 to 1/2 when β1 moves from 0.45 to
0.5. Related to the substantial drop and the substantial jump around β1 = 0.1 and β1 = 0.5,
respectively, V2 |β2 =0.5 in Figure 2(c) jumps significantly at β1 = 0.1, and drops significantlyat β1 = 0.5. Furthermore, for other values of β2 , in Figures 2(c) and (d), we observe thatV2 drops or jumps substantially, which is related to the non-monotonic relationship betweenExE
1 and β1 . Therefore, we conclude that the complex relation between x1 and β1 has asignificant impact on U2 .

5

Discussion

We discuss how changing the assumption of U2 ’s strong position over D2 in the entry timing
aﬀects our results in Section 5.1. Then, we discuss whether changing from asymmetric tosymmetric bargaining power aﬀects our result in Section 5.2.

29

5.1

Diﬀerence between this model and the model of Ebina et al.
(2015)In this subsection, we examine whether changing the assumption of U2 ’s strong position overD2 in the entry timing to reflect that in Ebina et al. (2015) aﬀects our main results.
Before proceeding with the analysis, we briefly check the entry decisions of group 2 (firm2) in this study and those in Ebina et al. (2015). The optimal entry timing of firm 2(follower) under the setting of Ebina et al. (2015) is
[
]
[
]
1
rF2
1
18rF2
= log
,
T̄2 (x1 ) = log Dααc(1 − x1 )(3 − x1 )2π̄2 (x1 )(18)

where π̄2D (x1 ) = c(1 − x1 )(3 − x1 )2 /18 denotes the duopoly profit of firm 2, whereas the
optimal timing of group 2 in our setting is given in Equation (13). There are two diﬀerences
between the two studies. The first stems from the diﬀerence in the duopoly profits, which
is based on whether vertical relations exist. This diﬀerence in the duopoly profits changes
the diﬀerence from π̄2 (x1 ) to π̃2 (x1 ). The second stems from the diﬀerence in the criteria
for market entry. In Ebina et al. (2015), the follower has the freedom to choose when it
enters the market. In contrast, U2 maximizes its profit by choosing T2 , which satisfies thezero-profit condition for D2 . This diﬀerence makes group 2 enter sooner, with the timingchanging from r to r − α, which corresponds to the numerator inside the logarithm in (13).
Thus, changing the second assumption on the criteria for market entry made in this study
to that made in Ebina et al. (2015), we have
∂v2
=0
∂T2

[
[
]
]
1
rF2
18(4 − β1 β2 )2 rF2
1⇒T̂2 (x1 ) = log D
= log
2 .
∗∗∗
∗∗∗
α
α
π̃2 (x1 )c(2 − β2 )2 (1 − x∗∗∗
)
[2(3
−
x
)
+
(3
+
x
)β
]
1
1
1
1
Next, we conduct a numerical analysis using the same parameter values in Section 4.
Suppose that r = 0.1, F2 = 1000, c = 1, and m = 25, which are the same as in Figures 1(b)30and (c). When β2 = 0.5 and 0.75, the equilibrium location pattern of group 1 is as shown in
Figures 3(a) and (b), respectively.

Figure 3: The relationship between the equilibrium location of group 1 and β1 ∈ (0, 21/23)
under the setting of Ebina et al. (2015) when (a) β2 = 0.5 (left) and (b) β2 = 0.75 (right).

Comparing Figures 1(b) and 3(a) (Figures 1(c) and 3(b), respectively), a slight increase18
in β2 has a similar impact on xEThis is1 in both this study and that of Ebina et al. (2015).

because the sign of the first-order derivatives of T̃2 and T̂2 with respect to the parameters are
the same. Therefore, we verify that the U-shaped nature of xE
1 on β2 , which is an interestingfinding of this study, depends on the vertical relationship, but does not depend on the market
entry criteria for the follower.

5.2

Symmetric bargaining power

Many studies assume symmetric bargaining power, in the sense that the bargaining powers of all firms are equal (β1 = β2 (≡ β)), and investigate firms’ behavior under vertical
relationships.19 In this subsection, we briefly discuss how employing the assumption of sym18

We also conduct a numerical analysis under the setting of Ebina et al. (2015) when β2 = 0.9, and obtaina figure that shows a sudden change of xE
1 from one corner solution to the other corner solution, similarly
to Figure 1(c). We omit the figure owing to space limitations.
19

See, for instance, the non-spatial oligopoly model in Horn and Wolinsky (1988).

31

metric bargaining power aﬀects our main results.
Figure 4(a) depicts the relationship between β and xE
1 when α changes. Suppose thatc = 1, m = 1, r = 0.1, ū = 25, and F2 = 1000. This setting is the same as that in
Figure 1, and the result is summarized in Table 3. In contrast to the asymmetric case shown
Ein Figure 1, where xE
1 can be increasing or U-shaped, we confirm that x1 is decreasingin β. This relationship is strong, because the rival’s wholesale price reduction owing to thenarrower distance between D1 and D2 is strong. Thus, although many studies consider a Nashbargaining problem under symmetric bargaining in a discrete time model, it is important toconsider asymmetric bargaining power when examining a vertical relationship through Nashbargaining by employing a continuous-time model.

Figure 4: The relationship between (a) the equilibrium location of group 1 and symmetricbargaining power β (left) and (b) xE
1 and the bargaining power of group 2 β2 when β1 = 0.5(right).

Finally, we briefly discuss the relationship between xE
1 and β2 . Figure 4(b) depicts therelationship between β2 ∈ [0, 1) and xE
1 when β1 = 0.5. The settings of the other parameters
are the same as in the symmetric case. The fundamental shapes of the graphs, which are
downward sloping, are similar to the symmetric case, and for the same reason. Thus, we
focus only on the relationship between β1 and xE
1 , and on the values of the firms in the32

previous section.

6

Conclusion

This study investigates a dynamic Hotelling duopoly model with endogenous entry timing
and location choices under vertical contracting. We focus on how upstream suppliers’ bargaining power over their downstream buyers (β1 and β2 ) influence the locations (x1 and x2 )and the entry timing of the follower pair (T2 ). A larger β1 usually induces a larger x1 and,thus, a larger T2 . This implies that the stronger bargaining power of the supplier can also
benefit the downstream buyer; that is, it might be better for a downstream buyer to weakenits bargaining position over its trading supplier. In addition, under some sets of exogenous
parameters, there is a non-monotonic relation between β1 and x1 . More specifically, theequilibrium location of the leader pair can jump substantially with an increase in β1 , whichsignificantly delays T2 and diminishes the profitability of the follower pair. Moreover, alarger β2 induces a smaller x1 , which facilitates the earlier entry of the follower pair. Thisimplies that when an upstream manufacturer tries to enter a newly growing market withits downstream representative, it needs to maintain its strong bargaining power over the
representative.
This study shows that the bargaining power of an upstream manufacturer/franchiser
over a downstream representative/franchisee aﬀects the entry timings and location points.
As mentioned in the introduction, several studies emphasize that large national retailers tend
to have strong bargaining power over their upstream firms (Geylani et al., 2007; Inderst andShaﬀer, 2007, 2009). Our results indicate that location points and entry timings are strongly
aﬀected by such bargaining power. Our theoretical results suggest that as the bargaining
powers strengthen, the location of group 1 will move closer to an interior, or center (not theedge), and that group 2 will wait longer before entering a new market.
33

We assume that the leader immediately enters the market (i.e., T1 = 0). As in Ebina etal. (2017), considering the endogenous entry timing of the leader pair is a topic for future
research, although it substantially complicates the analysis. In addition, future research can
explore a more sophisticated contract term, such as a two-part tariﬀ contract, as in Alpranti
et al. (2015).

7

Appendix

We explain the condition in which the market is fully covered by D1 in the monopoly case.
The profit of D1 is given as(p1t − w1t )if p1t ≤ u − c(1 − x1 )2 ,



[√
]



u − p1t
(p1t − w1t )+ x1 if u − c(1 − x1 )2 < p1t ≤ u − cx21 ,Mπ1t(x1 ) =
c


√




 2(p1t − w1t ) u − p1t
if u − cx21 < p1t ≤ u.
c
The first-order condition is given as1
if p1t ≤ u − c(1 − x1 )2 ,
2u − 3p1t + w1t
M
if u − c(1 − x1 )2 < p1t ≤ u − cx21 ,∂π1t (x1 )  x1 + √=2 c(u − p1t )

∂p1t


2u − 3p1t + w1t
if u − cx21 < p1t ≤ u.

 √2 c(u − p1t )We show the condition that the optimal price of D1 at each t ∈ [0, T2 ) is p1t = u − c(1 −x1 )2 , which is the maximum price among those that lead to full coverage of the market.
When u − c(1 − x1 )2 < p1t ≤ u − cx21 , the second-order condition isM−4u + 3p1t + w1t∂ 2 π1t(x1 )√= √(< 0).
2∂p1t
4 c(u − p1t ) u − p1tM(x1 ) monotonically decreases with p1t if the following inequality
Under this range of p1t , π1t

holds:
Mu − c(1 − x1 )2 − w1t
(x1 )
∂π1t
=1−
< 0;
∂p1t ↓p1 =u−c(1−x1 )22c(1 − x1 )34

that is,
w1t < u − c(1 − x1 )(3 − x1 ).

(19)

If w1t does not satisfy the inequality, this w1t does not maximize the joint profit in group 1
because the market demand is not fully covered.
When the firms in group 1 anticipate that D1 sets p1t (x1 ) = u − c(1 − x1 )2 in the pricing
stage, the bargaining outcome in the negotiation stage is
β1 : 1−β1 = (w1t −m) : u−c(1−x1 )2 −w1t → w1t (x1 ) = β1 (u−c(1−x1 )2 )+(1−β1 )m. (20)
If this w1t (x1 ) satisfies the inequality in (19), D1 actually sets p1t (x1 ) = u − c(1 − x1 )2 , given
w1t (x1 ) = β1 (u − c(1 − x1 )2 ) + (1 − β1 )m. Substituting w1t in (20) into (19), we obtainc(1 − x1 )(3 − x1 − β1 (1 − x1 )) < (1 − β1 )(u − m).
This becomes the most stringent condition when the left-hand side is maximized. For x1 ∈[0, 1/2], it is maximized at x1 = 0, and then the condition becomesc(3 − β1 ) < (1 − β1 )(u − m) → β1 <u − m − 3c.
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