Discussion Paper No. 976

RETAILER’S PRODUCT LINE CHOICE
WITH MANUFACTURER’S
MULTICHANNEL MARKETING

Cong Pan

July 2016

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

Retailer’s Product Line Choice with
Manufacturer’s Multichannel Marketing∗
Cong Pan†
Graduate School of Economics, Osaka University
July 12, 2016

Abstract
This paper studies how a retailer decides the length of product line in a verticallyrelated industry. We study a market with two product varieties. Each retailer decides thenumber of varieties it procures from an upstream manufacturer. The manufacturer may
open an online store and encroach on the resale market. In the case of a monopoly retailer, anticipating the online store’s encroachment, the retailer may be willing to shorten its
product line, although it can choose a full-length one. In the case of duopoly retailers,on the other hand, retailers may make their product lines completely overlapped, partially overlapped, or non-overlapped. Moreover, the total surplus may decrease due tothe efficiency loss in the online channel, although the competition in the resale market
becomes more intense.

Keywords: channels of distribution; encroachment; buyer power; product line choice
JEL Classification Numbers: L14, L22, M11∗

I would like to express my sincere thanks to Noriaki Matsushima for his constructive suggestions on thispaper. Besides, I really thank Toshihiro Matsumura, Tomomichi Mizuno and Shohei Yoshida, the conference
participants at the Spring Meeting of Japanese Economic Association (2016), the seminar participants at the
University of Tokyo and Kyoto Sangyo University for their kind encouragement and suggestions during various
presentations. The author acknowledges the financial support from the Japan Society for the Promotion of
Science (Pan, 15J05223). Needless to say, all errors are mine.
†
Address:
1-7 Machikaneyama,
Toyonaka,
Osaka 560-0043,
Japan.
E-mail:pge042pc@student.econ.osaka-u.ac.jp1

1

Introduction

From the conventional wisdom, a manufacturer always decides the preferred product linethat it wishes to distribute through retailers. However, recent markets have been focusing
more on cases in which dominant retailers possess strong channel power to influence the
manufacturer to adjust its production line so that it aligns with retailers’ orders (Kadiyali etal., 2000). Faced with the loss of power in the traditional wholesale channel, more upstream
manufacturers have begun to consider the benefits of “drop-shipping” and adopting online
channels through which they gain additional profits (Randall et al., 2006). The emergenceof such manufacturer’s multichannel marketing has brought new challenges to traditional
retailers in form of competition from potential online stores (Tannenbaum, 1995; Dixon
and Quinn, 2004), which is sometimes referred to as “franchise encroachment” (Arya et al.,2007; Emerson, 2010). Compared with an online store that can simply put a product variety
on the Internet as necessary, a physical retailer is disadvantaged by limited display space andthus has to prudently decide which variety to order from the manufacturer.
Our objective is to identify how an incumbent retailer exercises the channel power of
specifying product lines when faced the potential entry of an upstream manufacturer’s online
store and how this consequences affects social welfare. Although the online store brings the
manufacturer additional channel profits, it causes intrabrand competition with the incumbent
retailer, with adverse effects on profits from the wholesale channel. Realizing this channel
tradeoff, the manufacturer strategically chooses its online store’s product line. To illustrate,
let us assume that the manufacturer carries two varieties. When there is only one variety
in the wholesale channel, it is possible that the manufacturer sells only a different variety
in the online channel (a partial encroachment). In this manner, since the retailer and online
store have a differentiated product line, direct intrabrand competition is alleviated. However,when both varieties are available in the wholesale channel, regardless of the type of variety
that the online store sells, there is always an overlap. In such a case, intrabrand competitionis always direct and intense, which cannot be alleviated by a partial encroachment. Given
that there are no benefits to be had from a partial encroachment, the manufacturer never2

sells only one variety online. Instead, it chooses either selling both varieties online (a full
encroachment) or completely shutting down the online channel. That is, the manufacturer
uses the online store as a tool to keep a balanced channel distribution of product varieties. It
does not expect either variety to be oversupplied.
Anticipating the manufacturer’s two encroaching patterns (partial or full), the retailer
strategically manages its product line to alleviate the negative effect of the online store’s
encroachment. When the online retail cost is low enough, encroachment is inevitable. Realizing this, the retailer would rather abandon a part of its product line to trigger the manufacturer’s partial encroachment. Although this means losing profits from a smaller range, theresulting partial encroachment is less harmful, which allows the retailer to stay differentiatedfrom the online store and maintain adequate profitability from a short product line.
The above result gains wider significance if we consider an oligopoly downstream market
wherein retailers’ strategic interactions play an important role. Assume the simplest duopoly
retailer case. When online retail cost is low enough, the retailers tend to overlap their productlines in only one variety, as if declaring a relatively higher profitability of the variety whose
supply is lower. In this manner, the retailers induce the online store’s partial encroachment
instead of a full one. Alternatively, when online retail cost is high enough, the retailers may
tend to make their product line non-overlapped to ensure a balanced distribution of varieties
in the wholesale channel, which then gives the manufacturer an incentive to shut down its
online channel.
From the manufacturer’s viewpoint, we show that committing not to run an online store
can sometimes be beneficial. Although this means sacrificing channel profits from the online
store, a manufacturer can benefit from retailers’ incentive to enlarge product lines, which
enhances channel efficiency when physical retailers are more adapted to resale activities.
Finally, we show that even though full encroachment results in a more intense intrabrandcompetition than a partial one, it may result in lowering social welfare. This phenomenon
occurs when the physical retailers are much more adapted to resale activities than the online store and the different varieties are close substitutes. Although a full encroachment is3

more pro-competitive than a partial one, it reallocates a greater share to the inefficient online
channel, which causes considerable social loss and may even be a dominant effect. This
result has an important policy implication–product varieties of online retailing should bewell-regulated, and the consequences of overlapping product lines between different channels should be avoided.
Our study has managerial implications for both retailers and manufacturers. First, although we often observe retailers competing with each other in product diversity to attract
more diverse consumers, our results imply that the head-to-head competition in enlarging
product lines can sometimes be inefficient even when expanding the product line does notincur additional costs. When facing a weak online store, it may be more important for retailers to better coordinate with each other so that the distribution of variety in the wholesalechannel can be well-balanced. Second, the manufacturer’s ability in online retailing may
induce its retailers’ passive behaviours and they may become less willing to carry full-length
product lines. Hence, even with relatively low online retail costs, it is important for a manufacturer to assure its retailers that they are safe in keeping a full-length product line and that
their territories will not be encroached upon.
Now, we discuss the theoretical literature. To the best of our knowledge, this study is thefirst attempt to discuss retailers’ product line choices while considering manufacturer’s direct
marketing. However, the results and methodologies of several studies are closely related tothose of our study. Dukes et al. (2009) consider a similar setting in which two retailers decide their respective product line from a multi-product manufacturer. The longer the productline, the higher is the assortment cost incurred, but this is not reckoned with in our study.
They show that one of the retailers may spontaneously cut its product line to induce the rival
retailer to carry the full-length line with higher assortment costs. Although we derive similar
results that retailers do not choose full-length product lines, the intuitive reasoning behind
this is different. In our study, which does not consider assortment costs, retailers shorten
their product lines to trigger the online store’s partial encroachment. Moner-Colonques et al.
(2011) consider a case with two single-product manufacturers and two retailers. The retail-4

ers’ expansion of their product line means a multi-sourcing pattern. The authors provide us atheoretical explanation of why some retailers choose overlapping product lines. Inderst andShaffer (2007) considers retailers’ incentives for adopting a “single-sourcing” purchasing strategy (cutting the product line) and their cross-border mergers under different bargaining
power scenarios. Gabrielsen and Sørgard (1999) allow a monopoly retailer to decide whether
producers should have exclusive dealership under a linear contract. Mills (2015) considers
a similar scenario but under a nonlinear contract setting, where contract terms are decided
by negotiations between the monopoly retailer and either or both suppliers. Gabrielsen and
Sørgard (2007) consider retailers’ incentives for carrying private labels in a setting of vertically differentiated products. Although all these studies focus on buyer power in productline choices, the upstream manufacturer’s incentive for direct marketing is not considered.
Our study complements this literature by considering the incentive for a manufacturer tostrategically recapture channel power.1
Arya et al. (2007) may be among the earliest attempts to theoretically discuss the manufacturer’s encroaching behavior. A manufacturer encroaches on a retailer’s territory if its
retail cost is low enough. The encroachment may even benefit the retailer because the manufacturer resets a lower wholesale price to maintain the retailer’s demand at an adequate level.
This benefit exists only if the retailer decides its quantity before the manufacturer does sothat the retailer’s output reaches the Stackelberg leader’s level.2 Following the main structureof Arya et al. (2007), we examine a manufacturer’s encroachment problem in which a retailer decides its preferred product line, but assume quantities to be simultaneously decided toremove the Stackelberg leader’s advantage and to focus on the effect of product line choices. Mizuno (2012) considers a case in which two manufacturers distribute their products ton retailers by competing for a wholesale market while simultaneously deciding whether to
1

For parallel researches, Inderst and Shaffer (2010) discusses how a supplier uses different contract forms(based on the share it receives of a retailer’s total purchases or on how much a retailer purchases of its products)
when retailers have out-sourcing options. Milliou and Sandonı́s (2015) examines the relation between multiproduct manufacturers’ mergers and their incentive for enlarging the product line.
2
Liao (2014) extends this model by considering a signaling game in which the manufacturer has private
information on its own retail cost and shows that in a separating equilibrium of retail competition, the manufacturer signals inefficiency in its retail behavior by setting a lower wholesale price than that under complete
information.

5

encroach upon the resale market. Li et al. (2015) considers a model with n vertical supply chains and analyzes each manufacturer’s incentive to encroach.3 Considering retailers’channel power in product line choices, we add several new insights to the literature.
The remaining paper is organized as follows. Section 2 introduces a basic model with
one manufacturer owned online store and one retailer, from which we see how the retailer
employs orders of different variety to affect the manufacturer-owned online store’s encroachment. Section 3 extends the basic model to a duopoly retailers case. We will derive retailers’
order variety in equilibrium and some related propositions. Section 4 concludes the paper.

2

The case of a monopoly retailer

We start with a simplest monopoly retailer case and demonstrate the basic result that theretailer’s product line choice affects the online store’s encroachment.

2.1

The basic model

Let us consider a standard vertically related market comprising one manufacturer M and one
retailer R. The manufacturer produces imperfectly substitutable products X and Y, and theretailer orders either or both varieties from the manufacturer. The products are distributed to
the retailer who sells them to consumers through a wholesale channel. In addition, the manufacturer can also open its online store and encroach upon the retailer’s territory by directly
supplying either or both product varieties to consumers. We call the competition between theretailer and the online store intrabrand competition. The retailer’s retail cost is normalized
to zero, and the online store’s cost when the encroachment occurs is c > 0.4 The manufac3

Other parallel studies discusses the manufacturer’s direct selling channel in the Salop (1979) setting. In
Balasubramanian (1998), the consumers buy products from conventional retailers incurring location-dependenttransportation cost and are charged a fixed freight per unit that is irrelevant of location, if they buy from a direct
seller. Shulman (2014) considers Balasubramanian’s setting in a vertically related market. The direct seller
can either buy wholesale products from the authorized retailers who are supplied by the manufacturer, whichis referred to as a “gray-market,” or buy wholesale products directly from the manufacturer.
4This assumption follows Arya et al. (2007). In the real world, online stores usually need to bear higher
operating costs than physical stores. Unlike the physical stores, online stores have to bear the risk of returns andredress, as consumers cannot physically inspect a product before ordering. Besides, they have to endeavor totell consumers sufficient information about products as they browse the Internet (Lieber and Syverson, 2010).

6

turer’s production cost and the fixed cost of introducing both varieties are normalized to zerofor simplicity. We assume a representative consumer’s utility to be quadratic with the form
1u(QX , QY ) = a(QX + QY ) − (Q2X + 2γQX QY + Q2Y ) + I,2
where Qn denotes the aggregate quantity of variety n with n = X or Y, γ ∈ (0, 1) denotes
the substitutability between the two varieties, and I denotes consumer income. Products ofthe same variety are perfect substitutes whether sold by the retailer or by the online store.5
Consumer demand is represented by a linear downward sloping inverse demand function
pn = a − Qn − γQ−n , where n and −n are different varieties.
Let B denote both varieties, and N denote none of the varieties. N cannot be the retailer’s
strategy because it earns profits only when it orders wholesale products and sells them toconsumers, but this is not the case for the manufacturer because it earns profits from thewholesale channel even if it does not open an online store. The game proceeds as follows:In period 1, the retailer orders variety X or both (B).6 In period 2, the manufacturer decides
whether to open the online store (N if it does not open) or which variety (or varieties) to sellthrough the online store–X, Y, or both (B). The manufacturer then sets the corresponding
wholesale price(s) wn . In period 3, the wholesale products are delivered to the retailer, and theretailer simultaneously competes with the online store in quantity if encroachment occurs;otherwise, the retailer monopolizes the resale market.7
The assumption of early ordering by retailers is observed in many industries, especiallythose in which retailers need to view the product line design as a long-term issue and forecast
potential challenges, such as an online store owned by the manufacturer, long before themarket forms (e.g., the apparel industry) (Iyer and Bergen, 1997). One explanation is that it
is costly for a traditional retailer to face the consequences of an inappropriate order variety.8
5This assumption is made for analytical simplicity. The main propositions still hold if we allow for imperfect
substitutability for the same variety sold by the retailer or the online store.
6
By symmetry, because it is the same whether the retailer chooses X or Y, we can consider either caseinstead of both.
7
For simplicity, we assume that the manufacturer cannot reject the retailer’s order. If we allow for the
possibility of rejection, we have a case of the manufacturer choosing the variety (varieties) for the retailer. If
both varieties are chosen for the retailer, it can sell either variety or both. We find our results hold irrespectiveof whether the manufacturer has the option to reject.
8
For example, the retailers are less likely to order “very-low-demand” varieties that cannot even cover the7However, given the advantages of inventory, display and product line adjustment, an online
store often regards product line design as a short-term issue. Notice that our results holdeven if the retailer and manufacturer simultaneously decide their respective product line.

2.2

Equilibrium product variety

Let K be the set of products the retailer orders in period 1, where K ⊆ {X, Y} \ ∅. Moreover,let L be the set of products the manufacturer sells through the online store, where L ⊆ {X, Y}.
Denote the retailer’s variety byr ∈ {X, B}, and that of the manufacturer by m ∈ {N, X, Y, B}.
Any pair that the retailer and the manufacturer chooses, rm, defines a product line system.
There may be seven different product line systems in period 3: rm ∈ {XN, XX, XY, XB, BN,rm
BY, BB}.9 Let qrm
nR and qnM be the retailer and the online store’s selling quantities of varietyn, where n = X or Y. In period 3, the retailer chooses qrm
nR to maximize its profit which is
given by:
πrm
R =

X

rm
[pn (Qn , Q−n ) − wrmn ]qnR .

(1)n∈K

If the manufacturer does not open the online store, its profit comes only from the wholesalechannel, which is given by
πrm
M =

X

rm
qrm
nR wn .

(2)n∈K

Otherwise, it competes with the retailer in the resale market and chooses qrm
nM (m , N) to
maximize its profits which is given by
πrm
M =

X

[pn0 (Qn0 , Q−n0 ) − c]qrm
n0 M +

n0 ∈L

X

rm
qrm
nR wn .

(3)n∈K

Solving the profit maximization problems in period 3, we obtain the equilibrium quantirmrm
rm
rm
ties, qrmnR (w) and qn0 M (w), where w ∈ wX if K = X, or w ∈ (wX , wY ) if K = X, Y.

In period 2, wn and w−n are decided by the manufacturer in anticipation of the equilibriumoutcomes in period 3.When the manufacturer does not open the online store (rm = XN or
fixed cost of storing them. However, this is not the case for the online stores because the demand comes froma larger geographic market (Lieber and Syverson, 2010).
9
By symmetry, BX is equivalent to BY.

8

r
m
wrm
X
rm
wY
I.S .

X
N

X

Y

B

N

a
2

5a−c
10

(8−4γ2 +γ3 )a−γ3 c2(8−3γ2 )

5a−c
10

\
a>0

\

\

\

c
≤ 57
a

2
c
≤ 8−2γ−γ
a
8−γ2

a
2
a
2

c
5
≤ 7+2γ
a

B
Y

a>0

5a−c
10
5a−γc
10
5γ
c
∈
(
, 5]
a
8+8γ 7

B
5a−c
10
5a−c
10
c
≤ 57
a

Table 1: Equilibrium wholesale prices
BN), it solves the following maximization problem:
max
w

X

rm
qrm
nR (w)wn .

(4)

n∈K

When the online store encroaches (rm = XX, XY, XB, BY, or BB), the manufacturer solves
the following maximization problem:
max
w

X

[pn0 (Qn0 (w), Q−n0 (w)) − c]qrm
n0 M (w) +

n0 ∈L

X

rm
qrm
nR (w)wn .

(5)

n∈K

The manufacturer decides whether to open the online store as well as the product line forthe online store based on the resulting profits. The equilibrium wholesale price, wn and w−n ,are denoted in Table 1, where “I.S.” denotes the conditions for interior solutions. For cases
in which the manufacturer encroaches, c must be low enough. Moreover, for the BY case,c must not be too low (c/a > 5γ/(8 + 8γ)), otherwise wrm
X becomes so high that the retailer
BY
gives up supplying X. Thus, the BY case with qYR= 0 becomes the XY case.

We can easily derive that given r = X or B, the wholesale prices decrease with moreproduct varieties sold online. This is summarized in the following proposition:
Proposition 1 Given the retailer’s order variety (X or B), the wholesale prices decrease
with more product varieties sold online.
Proposition 1 follows directly from the main result of Arya et al. (2007) that a manufacturer
may strategically reduce the wholesale price when it encroaches upon the retailer’s territory.
The manufacturer earns profits from both the wholesale and online channels. In our study,
the increasing number of product varieties sold online intensifies intrabrand competition,
which decreases the manufacturer’s profits in the wholesale channel. The manufacturer thus9(a) The retailer orders X(b) The retailer orders both varieties

Figure 1: The online store’s variety choice in period 2 when γ = 0.5
lowers the wholesale price to shift some business back to the retailers to keep the wholesaledemand at an adequate level. The wholesale price, conversely, reflects how intensely thepresence of an online store leads to intrabrand competition. The lower the wholesale pricecharged, the more severely is the wholesale channel affected, and the stronger is the incentive
for the manufacturer to attempt to retrieve the retailer’s demands.
For simplicity, we call the manufacturer’s encroachment with only one variety a partial
encroachment and that with both varieties a full encroachment. Considering the conditions
for interior solutions and making sure that there are no unilateral deviations in each case, wefind the equilibrium outcomes in period 2 as follows:
X

Lemma 1 (1) When the retailer orders variety X in period 1, there exist θ X (γ) and θ (γ),
X

with θ X (γ) < θ (γ), so that the online store(i) fully encroaches if c/a ≤ θ X (γ),X(ii) partially encroaches with variety Y if θ X (γ) < c/a ≤ θ (γ),X(iii) is shut down if c/a > θ (γ);(2) When the retailer orders both varieties in period 1, there exists θ B (γ), so that theonline store(i) fully encroaches if c/a ≤ θ B (γ),(ii) is shut down in period 2 if c/a > θ B (γ),
X

where θ X (γ) < θ B (γ) < θ (γ).
Notice that given r = B, m = Y is weakly dominated by m = B or N, because πBY
M ≤
10

BN
max{πBBM , π M }. Here, we do not take this weakly dominated strategy as an equilibrium can-didate. Figure 1 depicts the threshold values in Lemma 1. For calculations, please see
Appendix 5.2.
Notice that given r = X or B, the manufacturer’s profit curve becomes steeper when it
sells more varieties online. This is because the increasing online retail cost leads to more
losses when the online store has a wider product range.
In (a) of Figure 1, given that the retailer orders only variety X, as the online retail cost
increases, the online store first stops selling variety X. The benefits of partial encroachmentwith a different variety rather than the same as the online one come from two angles. First,
the manufacturer obtains a bigger share in the wholesale channels when its online store
XX
competes indirectly with the retailer. Second, a higher wholesale price (wXY
X > wX ) reflects

less intense intrabrand competition, implying a lower loss of profit in the wholesale channel.
BN
BY
B
In (b) of Figure 1, πBB
M , π M and π M intersects at θ (γ), implying that when the different

varieties are distributed evenly in the wholesale channel (r = B), selling only one variety online is a weakly dominated strategy for the manufacturer. This follows from the assumptionof symmetric online retail costs for both varieties. When the retailer orders both varieties andthe manufacturer starts the online channel and sells only variety Y, it lowers the wholesale
prices for both X and Y from wXBN = wYBN to wXBY and wYBY , respectively. wYBY is lower thanwXBY because the online store encroaches on the retailer’s share in Y directly but indirectlyin X. The partial encroachment leads to intense intrabrand competition in both varieties.
However, if the online store also sells X (m = B), it only further decreases the wholesale
price of X (wYBB = wYBY , wXBB < wXBY ), implying that the additional sale of X by the online
store does not further intensify the intrabrand competition in Y. The additional sale of X by
the online store reduces its sale of Y through the cannibalization effect, which prevents theretailer’s sale of Y from decreasing sharply. The manufacturer then has no further incentive
to reduce the wholesale price of Y. We see that a full encroachment does not seriously cause
more of a profit loss in the wholesale channel than a partial one. If selling one variety online
is profitable for the manufacturer than selling nothing, it always continues to sell the other11

variety. Therefore, selling one variety is a weakly dominated strategy when r = B.
If we assume that there exists a positive real number τ such that selling X incurs morecost (c + τ) than selling Y, πBB
M in (b) of Figure 1 shifts left and downward so that the range
over which the online store sells only the more efficient variety Y becomes wider. All results
here still hold true if the cost difference τ is so small that the range of selling only Y online
is negligible. We assume symmetric online retail costs for simplicity.
The case that r = X can be seen as an unbalanced order because only one variety is
distributed to the retailer; the case that r = B can be seen as a balanced order because
both varieties are distributed. The manufacturer tends to keep balance in the distribution ofvarieties–it may sell only the variety that is distributed less to the retailer (when r = X) but
never sells only one variety when both varieties are evenly distributed (when r = B).
Lemma 1 shows how the retailer’s product line choice affects the manufacturer’s incentive for opening the online store. When the online retail cost is relatively low, the manufacturer considers whether to fully encroach. The threshold values θ X (γ) < θ B (γ) show thatthe manufacturer has a stronger incentive to do so when r = B than when r = X. Reducing the number of varieties sold online affects the manufacturer’s profits in two ways: First,
it alleviates intrabrand competition and results in higher wholesale prices, from which the
manufacturer’s profit in the wholesale channel increases. Second, because of a narrower
product line, the manufacturer’s profit in the online channel decreases. We first examine theBBBB
wholesale channel. When the online store sells both varieties, wXB
X = wX = wY . If theonline store decreases its varieties, m changes from B to Y given that r = X, but changes
from B to N given that r = B. Comparing the subgame outcome under the product line system rm = XY with that under rm = BN, the manufacturer sets higher wholesale prices (wXBN
and wYBN ) when r = B than that (wXY
X ) when r = X. This is because when r = B, once the
manufacturer reduces its varieties, it directly shuts down the online channel, eliminating intrabrand competition. Thus, the manufacturer’s gain in the wholesale channel is lager whenr = B than when r = X. However, the manufacturer loses more in the online channel whenr = B than when r = X, because it loses profits from both varieties in the former case. Since

12the online retail cost is so low now that the loss dominates the gain, the manufacturer is more
likely to keep full encroachment when r = B than when r = X.
Remark 1 θ X (γ) < θ B (γ).
When the online retail cost is relatively high, the manufacturer considers whether shuttingX

down the online store. The threshold values θ B (γ) < θ (γ) show that the manufacturer has astronger incentive to do so when r = B than when r = X. Opening an online store has two
effects on the manufacturer’s profits: First, it leads to intrabrand competition, from which the
manufacturer’s profit in the wholesale channel decreases. Second, it brings the manufacturer
additional profits in the online channel. We first look at the wholesale channel. When there isBNBN
only the wholesale channel, wXN
X = wX = wY . However, when the manufacturer opens its

online store, it sets lower wholesale prices (wXBB and wYBB ) when r = B than that (wXY
X ) whenr = X. This is because when r = B, the online store competes directly with the (incumbent)
retailer in both varieties; However, when r = X, the online store can sell a different variety
so that decreases in the retailer’s share is not that large. Thus, the manufacturer’s loss inthe wholesale channel is larger when r = B than when r = X. However, the manufacturer
earns more profits in the online channel when r = B than when r = X because it profits fromboth varieties in the former case. Since online retail cost is now high enough for the loss todominate the gain, the manufacturer is more likely to shut down its online store when r = B
than when r = X.
XRemark 2 θ B (γ) < θ (γ).

2.3

The retailer’s product line choice in equilibrium

In period 1, the retailer orders either variety X or both varieties in anticipation of the manufacturer’s reaction in period 2. The following proposition indicates the equilibrium variety:
Proposition 2 In the case of a monopoly retailer, the equilibrium variety outcome is(i) the retailer orders both varieties in period 1 and the online fully encroaches in period
2 if c/a ≤ θ X (γ) (hereafter the BB variety outcome);
13

(ii) the retailer orders variety X in period 1 and the online store partially encroacheswith variety Y in period 2 if θ X (γ) < c/a ≤ θ B (γ) (hereafter the XY variety outcome);(iii) the retailer orders both varieties in period 1 and the online store is shut down inperiod 2 if c/a > θ B (γ) (hereafter the BN variety outcome).

Figure 2: The variety choices rm in equilibrium
It is important for the retailer to consider two main issues in the ordering process: First, toorder as many varieties as possible for a wider product line; Second, to weaken the negative
effect of the online store’s encroachment. When the retail cost is too low (c/a ≤ θ X (γ)),
whether the retailer orders one variety or both does not change the outcome of the online store
encroaching fully. Therefore, the retailer chooses a full-length product line. However, when
the online retail cost is not that low (θ X (γ) < c/a ≤ θ B (γ)), the retailer can order only X togive the manufacturer an incentive to encroach with the less-supplied variety Y. However, if
it orders both varieties, the manufacturer follows with a full encroachment. Since the online
store now is a relatively efficient one, alleviating the negative effect of the encroachment ismore important than enlarging the product line. Therefore, although the retailer loses fromone variety, it benefits from inducing the sale of fewer varieties by the online store. Whenthe retail cost is very high (c/a ≥ θ B (γ)), the retailer goes for a full-length product line toinduce the manufacturer to shut down the online store. Thus, the retailer counters fully thenegative effect of the encroachment and establishes a wider product line as well.

14

Figure 3: The retailer and the manufacturer’s profit function when γ = 0.3
With regard to how the change of online retail cost affects the retailer and the manufacturer’s profits (see Figure 3), we find that as the online retail cost increases, the retailer’s profit
always show an increasing tendency. This is because an increasing c makes the retailer morecompetitive and gradually removes the manufacturer’s incentive to encroach. However, themanufacturer’s profit does not always show a decreasing tendency, which is summarized in
the following proposition:
Proposition 3 As the online retail cost increases, the manufacturer’s profit drops at θ X (γ),
but jumps up at θ B (γ).
When c just surpasses θ X (γ), the retailer and the online store begins to compete in different
varieties, which alleviates intrabrand competition. However, because both variety Y in thewholesale channel and variety X in the online channel are suddenly removed, the manufacturer bears loss of profits from both channels, which overweighs the positive effect of analleviation in intrabrand competition. When c just surpasses θ B (γ), the profit from supplying
variety Y through the online channel is suddenly replaced by that through the wholesale channel. This replacement benefits the manufacturer in the following two ways: First, because
intrabrand competition is eliminated, the manufacturer earns a higher profit by specializingin the wholesale channel. Second, the channel efficiency is improved because Y is suppliedby the efficient retailer instead of the costly online store.
Proposition 3 has further implications for the manufacturer’s encroachment decision.
Assume that there is a pre-determinate period, period 0. The upward jump feature implies15

that it may be better for the manufacturer to commit to the retailer in period 0 that it will notopen its online store. This is summarized in the next corollary:
Corollary 1 When θ̂(γ) < c/a ≤ θ B (γ), it is better for the manufacturer to commit not toBNopen the online store, where θ̂(γ) is the threshold satisfying πXY
M = πM .

Given the guarantee that the resale market will not be encroached upon, the retailer chooses
to carry both varieties. Without such a commitment, at θ B (γ), the encroachment makes theretailer reduce one variety and forces the manufacturer to carry this variety inefficiently,
which implies a “lose-lose” consequence. Committing enables the manufacturer to ensurethat both varieties are supplied efficiently. Arya et al. (2007) do not consider the possibility
that the retailer specifies its product line so that the manufacturer never encroaches if doing
so causes profits to fall. Hence, such a “lose-lose” consequence never occurs in that study.
Next we see how the consumer surplus (CS ) and the total surplus (T S ) are affected by
the retailer and the online store’s choice of variety. Since the social loss only comes fromthe online retail cost, the total surplus is the representative consumer’s gross utility net of thetotal online retail cost, which is denoted by T S = U(QX , QY ) − (qXM + qY M )c. We find that asc increases, the consumer surplus always decreases, but the total surplus jumps up at θ B (γ).
This is summarized in the following proposition:
Proposition 4 As c increases, the consumer surplus always shows an increasing tendency,
and the total surplus drops at θ X (γ), but jumps up at θ B (γ).
Figure 4 depicts how c affects the consumer and the total surplus.
An increasing c gradually removes the manufacturer’s incentive for online retailing andenhances the (incumbent) retailer’s market power who finally monopolizes the resale market.
At θ X (γ), both the retailer and the online store reduce one variety and begin to compete indirectly, which results in higher prices. This harms both the consumer and total surpluses. At
θ B (γ), the online store is shut down and the retailer becomes a monopolist of both varieties.
This increases prices, which further decreases the consumer surplus. However, although the16Figure 4: The consumer and the total surplus when γ = 0.4
monopolization of the (incumbent) retailer decreases the gross utility U(QX , QY ), shutting
down the online channel wipes out the social loss, which results in a higher total surplus.

3

Extension: the case with duopoly retailers

Our basic model tells us how a monopoly retailer decides its product line when the manufacturer can strategically recapture its channel power by employing an online channel. This
raises a question of how the product line outcomes change if there are additionally strategicinteractions between retail competitors. Thus, it is natural to consider a simple extension
with duopoly retailers. Assume that there is a monopoly manufacturer, M, and two retailers,
Ri , with i = 1 or 2.
To remain consistent with the benchmark model, we assume that all players are kept
active in the market so that N (order nothing) cannot be any retailer’s choice. Besides,
we normalize both retailers’ resale costs for both varieties to zero. Let Ki be the set of
products R1 orders in period 1, where Ki ⊆ {X, Y} \ ∅. Denote each retailer’s variety choice asri ∈ {X, Y, B} and the manufacturer’s as m ∈ {N, X, Y, B}. The game proceeds as in the case of amonopoly retailer except that the retailers decide their respective product line simultaneously
and independently in period 1, and the manufacturer discriminates against the retailers byoffering different wholesale prices wrni1 r2 m in period 2. Retailers’ order combination r1 r2 inperiod 1 is simplified as follows: r1 r2 ∈ {XX, XY, XB, BB}.10
10

By symmetry, YY is equivalent to XX; Y X is equivalent to XY; Y B, BY and BY is equivalent to XB.

17

r1 r2
XX

XY

XB

BB

m
N
X
Y
B
N
Y
B
N
X
Y
B
N
Y
B

1 r2 m
wrX1

a
2
3a−c
6
(6−4γ2 +γ3 )a−γ3 c6(2−γ2 )
3a−c
6
a
2
2
(10−5γ +γ3 )a−2γc4(5−2γ2 )(5+γ)a−(1+γ)c2(5+γ)a2
3a−c
6(15−4γ2 +γ3 )a−(3γ+γ3 )c6(5−γ2 )
3a−c
6
a
2
3a−γc
6
3a−c
6

r1 r2 m
wY1
\
\
\
\
\
\
\
\
\

\
\
a
2
3a−c
6
3a−c
6

1 r2 m
wrX2

wrY21 r2 m
\
\
\
\

\
\
\

a
2
(10−γ−3γ2 )a−2c4(5−2γ2 )(5+γ)a−(1+γ)c2(5+γ)a23a−γc6(15−γ−2γ2 )a−(3+γ2 )c6(5−γ2 )15a−(3+2γ)c
30
a
2
3a−c
6
3a−c
6

a
2
3a−c
6
(6−4γ2 +γ3 )a−γ3 c6(2−γ2 )
3a−c
6

a
2
3a−c
6
(15−4γ2 +γ3 )a−(3γ+γ3 )c6(5−γ2 )
3a−c
6
a
2
3a−γc
6
3a−c
6

I.S .
a>0
c
≤ 35
a
2
c
≤ 6−2γ−γa6−γ23c≤ 5+2γaa>0γ−γ2 5−γ−γ2c∈ ( 8−4γ2 , 7−2γ2 ]a5+γ
c≤ 7+3γaa>03γc∈ ( 4(1+γ), 53 ]a2γc∈ ( 2(1+γ), 15−2γ−γ
]
a
21−γ2
c
15
≤ 25+4γaa>0γc∈ ( 2(1+γ), 53 ]ac≤ 35
a

Table 2: Equilibrium wholesale prices

3.1

Equilibrium Product Variety

In period 3, the Cournot competition proceeds based on the product line system, r1 r2 m, which
are of 14 kinds: XXN, XXX, XXY, XXB, XY N, XYY, XY B, XBN, XBX, XBY, XBB, BBN,
BBY, and BBB.11 The optimization system is the same as in the basic model.
In period 2, the manufacturer decides whether to open its online store and which variety(varieties) to sell based on the resulting profits. The equilibrium wholesale price, wrni1 r2 m and1 r2 mwr−ni, are denoted in Table 2, where “I.S.” denotes the conditions for interior solutions.

The online store encroaches only if its retail cost is low enough. Moreover, for the interior solutions in the XBY case, the online retail cost must not be too low. When the online
store is very efficient in selling variety Y, even if R2 orders both varieties, it will be charged
XBY
an unacceptable wY2
so that R2 does not really sell Y. The XBY case with a corner solution
XBY
(qYR2
= 0) becomes the XXY case. Thus, we need the online retail cost to be higher thanθ̂ XB (γ) ≡ γ/(2 + 2γ) in the XBY case. After several calculations, considering the corner solu11

By symmetry, XY X is equal to XYY, and BBX is equal to BBY.

18

tions and checking the manufacturer’s incentive to deviate, we find the equilibrium outcomesin period 2 as follows:XX

Lemma 2 (1) When both retailers order variety X in period 1, there exist θ XX (γ) and θ (γ),
XXwith θ (γ) > θ XX (γ), such that the online store(i) fully encroaches in period 2 if c/a ≤ θ XX (γ),XX(ii) partially encroaches with variety Y in period 2 if θ XX (γ) < c/a ≤ θ (γ),XX(iii) is shut down in period 2 if c/a > θ (γ);(2) When one retailer orders variety X, and the other one orders both varieties in period
XB

XB

1, there exist θ1XB (γ), θ2XB (γ), θ1 (γ) and θ2 (γ), such that the online store(i) fully encroaches in period 2 when γ ≤ 0.690 if c/a ≤ θ1XB (γ), as well as when γ >0.690 if c/a ≤ θ2XB (γ),(ii) partially encroaches with variety Y in period 2 when γ ≤ 0.690 if θ1XB (γ) < c/a ≤XB

XB

θ1 (γ), as well as when 0.69 ≤ γ ≤ 0.897 if θ̂ XB (γ) < c/a ≤ θ1 (γ).
XB(iii) is shut down in period 2 when γ ≤ 0.897 if c/a > θ1 (γ), as well as when γ > 0.897
XB

if c/a > θ2 (γ);(3) When one retailer orders variety X and the other one orders variety Y in period 1,
there exist θ XY (γ), such that the online store(i) fully encroaches in period 2 if c/a ≤ θ XY (γ),(ii) is shut down in period 2 if c/a > θ XY (γ);(4) When both retailers order both varieties in period 1, there exist θ BB (γ), such that theonline store(i) fully encroaches in period 2 if c/a ≤ θ BB (γ),(ii) is shut down in period 2 if c/a > θ BB (γ).
Please see Appendix 5.4 for the calculations. The ranges for equilibrium outcomes in period
2 are summarized in Figure 5. For simplicity in notation, we denote the upper bound and
XB

lower bound values of the XB case as θ (γ) and θ XB (γ) respectively.12
XB
XB
12 XB
θ1 and θ2XB (γ) are connected at γ = 0.690; θ1 and θ2 (γ) are connected at γ = 0.897.

19

(a) r1 r2 = XX in period 1(b) r1 r2 = XY in period 1(c) r1 r2 = XB in period 1(d) r1 r2 = BB in period 1

Figure 5: The online store’s variety choice in period 2 when γ = 0.5
Following the logic of the monopoly retail case, the case that r1 r2 = XX or XB corresponds to r = X in Section 2 (see (a) and (b) in Figure 5). These two cases can be seen as
the unbalanced orders because variety X is distributed to both retailers, but Y is distributed to at most one retailer. Knowing that retailers will supply less of Y compared with X,
XX

the online store sells only Y when c is an intermediate value (θ XX (γ) < c/a ≤ θ (γ) or
XB

XB

XX

θ XB (γ) < c/a ≤ θ (γ)). The threshold values, θ XX (γ) < θ XB (γ) < θ (γ) < θ (γ), imply
that partial encroachment with Y is more likely to happen in the XX case than in the XB case.
This is because the distribution imbalance is more serious in the former case. In the XB case,
R2 ’s share of Y makes selling Y online less profitable for the manufacturer than in the XX
case.
The case that r1 r2 = XY or BB corresponds to r = B in Section 2 (see (c) and (d) in Figure5). These two cases can be seen as the balanced orders, because both varieties are evenly
distributed to the retailers. To maintain this balance, the manufacturer chooses either to fully
encroach or completely shut down the online channel. The threshold values, θ BB (γ) < θ XY (γ),

20

imply that a full encroachment is more likely to happen in the XY case than in the BB case.
In the XY case, the retailers have differentiated product lines so that the resale market before
the presence of encroachment is less competitive for online retailing compared with the BB
case where retailers have overlapping product lines and thus compete intensely.
Notice that in the case r1 r2 = XY, a partial encroachment is strictly dominated by a full
encroachment or a shutdown of the online channel,13 which differs from the case r1 r2 = BBin the case of duopoly retailers and r = B in the basic model. Partially encroaching withN
XYY
Y forces the manufacturer to reduce the wholesale prices of X and Y from wXY
X1 to wX1
XY N
XYYand from xY2
to wY2, respectively. If the manufacturer additionally sells X online, then it
B
XYY
would sell at a lower wholesale price of X (wXY
X1 < wX1 ) but at a higher wholesale price of Y
XY B
XYY
(wY2
> wY2
), implying that selling X additionally alleviates intrabrand competition in Y.

Since now each variety is ordered by only one retailer, the demand in the wholesale channel
is small, which stimulates the manufacturer’s strong incentive for online retailing. When the
online store sells only Y, R2 ’s share is severely invaded so the manufacturer has to greatly
lower the wholesale price of Y. However, if the manufacturer additionally sells X online, thecannibalization effect diminishes the online sale of Y, thus alleviating the negative impact
on R2 ’s share in Y. Selling X additionally not only helps the manufacturer to enlarge theonline store’s product line, but also enhances the wholesale channel’s profit in Y. If fully
encroaching is more profitable than shutting down the online store, a manufacturer would
never partially encroach.

3.2

Retailers’ order varieties at equilibrium

In period 1, the retailers decide which variety (varieties) to order, anticipating the potential
online store’s encroachment. The equilibrium variety outcome in period 1 is summarized asfollows:
Proposition 5 (i) The retailers order both varieties, and the online store fully encroaches(hereafter the BBB outcome) if c/a ≤ θ XB (γ);
13 XY BXXY
π M and πXY
M intersects at a point that is larger than θ (γ), implying that partial encroachment with either

variety is a strictly dominated strategy for the manufacturer.

21

(ii) the retailers order X, and the online store partially encroaches with Y in period 2
XB(hereafter the XXY outcome) if θ XX (γ) < c/a ≤ θ XB (γ), or if θ XB (γ) < c/a ≤ min{θ (γ), θ̂ XB (γ)}.
(iii) one retailer orders X and the other retailer orders both varieties, and the online
store partially encroaches with Y (hereafter the XBY outcome) if max{θ XB (γ), θ̂ XB (γ)} <c/a ≤ θ BB (γ);(iv) each retailer orders a different variety, and the online store is shut down (hereafter
XB

the XY N outcome) if θ XY (γ) < c/a ≤ θ (γ);(v) the retailers order both varieties, and the online store is shut down (hereafter the
BBN outcome) if c/a > θ BB (γ).
Here, because quantities are decided simultaneously, encroachment always harms retailers’ profits. Although charged lower wholesale prices, retailers’ losses in market share cannever be compensated.14 The main issue for each retailer is to find a balance between enlarging the product line and weakening the negative impact of the online store’s encroachment.
When the online retail cost is too low (c/a ≤ θ XB (γ)), whether r1 r2 = BB or XB, theonline store always fully encroaches. Hence, each retailer chooses a full-length product line(the BBB outcome).
As the online retail cost gradually increases, retailers’ unbalanced order variety may induce the manufacturer’s partial encroachment instead of a full one. This is because these
unbalanced orders signal to the manufacturer that variety Y would be supplied less. Observing this, the manufacturer sells only Y so that the resulting intrabrand competition is not that
XB

intense. In the range that θ XX (γ) < c/a ≤ θ XB (γ) or θ XB (γ) < c/a ≤ min{θ (γ), θ̂ XB (γ)}, only
when r1 r2 = XX does the online store partially encroach (the XXY outcome).15 Althoughboth retailers give up variety Y, they benefit from weakening the negative impact from theonline store. In the range that max{θ XB (γ), θ̂ XB (γ)} < c/a ≤ θ BB (γ), either when r1 r2 = XX
14

If the online store moves after the retailers, it is possible that retailers’ share losses caused by the encroachment are compensated for by being charged lower wholesale prices (see Arya et al., 2007).
15
If θ XX (γ) < c/a ≤ θ XB (γ), when r1 r2 = XY, XB or BB, the online store sells both varieties. If θ XB (γ) <XB

c/a ≤ min{θ (γ), θ̂ XB (γ)}, when r1 r2 = XY or BB, the online store sells both varieties. But when r1 r2 = XB,
XBY
the online store sells only Y. This because when c/a ≤ θ̂ XB (γ), R2 is offered an unacceptable wY2
so that its
XB
quantity of Y becomes zero. The r1 r2 m = XBY case with qYR2 = 0 becomes the XXY case.

22

or XB, the online store’s partial encroachment can be induced. In equilibrium, one retailer iswilling to give up one variety and let its retail rival have a full-length product line (the XBY
outcome). Although the retailer who cuts product line earns lower profit than its retail rival,
it would lose more if it insists on choosing a full-length line because the online store would
then fully encroach.
When the online retail cost is relatively high, retailers’ balanced order variety may induce
the manufacturer’s incentive to shut down its online store because running it means a full encroachment, which causes a serious loss of profit in the wholesale channel. In the range
XB

that θ XY (γ) < c/a ≤ θ (γ), the manufacturer partially encroaches in the face of imbalanced
orders but shuts down the online channel when r1 r2 = XY (the XY N outcome). When one of
the retailers orders one variety, the other one would spontaneously order a different variety.
Although both of them give up the full-length product line, their non-overlapping productline choices eliminate the online store’s partial encroachment and make the competition between them indirect and mitigating. In the range that c/a > θ BB (γ), the manufacturer shuts
down its online store when r1 r2 = BB (the BBN outcome). No retailer would unilaterally
reduce the number of varieties because this not only gives the retail rival the advantage of
XB

diversity but also results in the online store’s partial encroachment (when c/a ≤ θ (γ)).
Comparing the equilibrium ranges, we find that the following three sets of variety outcomes can coexist: the XXY and BBB outcomes, XXY and BBN outcomes, XY N and BBN
outcomes. This implies the possibility of a coordination failure in some ranges, which issummarized in the following corollary:
Corollary 2 Retailers’ coordination failure may occur in the following ranges, in which
each retailer’s profit is higher in the former outcome than in the latter one:(i) If θ XX (γ) < c ≤ θ XB (γ), the XXY and BBB variety outcomes coexist;
XB(ii) If θ BB (γ) < c ≤ min{θ̂ XB (γ), θ (γ)}, the BBN and XXY variety outcomes coexist;
XB(iii) If θ XY (γ) < c ≤ θ (γ), the XY N and BBN variety outcomes coexist.
Figure 6 depicts the variety outcomes in equilibrium. The shady areas denote the ranges inwhich coordination failure may exist.
23

Figure 6: The variety outcomes r1 r2 m in equilibrium
In (i) of Corollary 2, although the BBB variety outcome earns each retailer profits from afull-length product line, the resulting online store’s full encroachment makes the competition
too intense. In (ii) of Corollary 2, comparing with the BBN variety outcome, the XXY
outcome not only removes each retailer’ profit from selling Y but also triggers the online
store’s partial encroachment. In (iii) of Corollary 2, although both cases, r1 r2 = BB andr1 r2 = XY, make the manufacturer shut down its online store, the former case is clearly lessbeneficial than the latter one because the competition in the latter case is more mitigating.
Despite this, neither retailer is willing to deviate from the less profitable variety outcome.
Finally, we examine the consumer surplus and total surplus in the duopoly retailer case.
The consumer surplus here is similar to that in the case of a monopoly retailer. It always
shows a decreasing tendency as the online retail cost increases within the ranges in which
each variety outcome exists. Moreover, it always jumps down at some threshold points where
the number of varieties in the resale market decreases or where the online store is shut downby the manufacturer. However, the total surplus shows a distortion, which is summarized in
the following proposition.
Proposition 6 When γ > 0.751, the total surplus jumps up at θ XX (γ), wherein the variety
outcome changes from BBB to XXY.

24

Figure 7: The total surplus when γ = 0.9
This proposition is counterintuitive at first glance because now less intense intrabrandcompetition may even benefit society (see Figure 7).16 Recall that the social loss only comesfrom the online store’s sale. We first examine the XXY outcome. Assume that γ is almost
zero so that the different varieties are almost heterogeneous. Since the online store almost
independently monopolizes the market for Y, the social loss is very high. As γ grows larger,
the markets for X and Y become more interconnected. Since competition between the retailers and online store intensifies, the online retail cost shifts more business from the online
store to the retailers, which decreases the social loss. However, in the BBB outcome, because
the retailers and the online store always compete in the same variety, the impact of a growing
γ on the online store’s share is quite limited. When γ is large enough, the online store’s
share is lower in the XXY outcome than in the BBB outcome, implying less social loss in theformer case. When the variety outcome changes from BBB to XXY, although the less intense
intrabrand competition harms the consumer’s gross utility, U(QX , QY ), the total surplus may
still improve by saving on social costs. Notice that in the duopoly retailer case, the shutdown
XBof the online store (when c/a = θ (γ)) does not improve the total surplus as in the case ofa monopoly retailer (Proposition 4). This is because partial encroachment causes less social
loss here than in the case of a monopoly retailer. Shutting down the online store (removing
partial encroachment) is anti-competitive, which cannot be compensated for by the benefit
16

Each variety is carried by both channels in the BBB case but by only one channel in the XXY case. Thus,
intrabrand competition changes from a direct one to an indirect one.

25

of eliminating the social loss.

4

Concluding remarks

Our study discusses a market in which a retailer decides its product line from a manufacturer who may encroach upon the resale market through an online store. We show that the
manufacturer uses the online channel as a tool to keep the variety in distribution balanced.
Specifically, the manufacturer tends to use an online store to sell the variety that is lesssupplied by the (incumbent) retailers but will not start the online channel if the variety distribution in the wholesale channel is already balanced. As discussed in the case of a monopoly
retailer, if the online retail cost is moderately low, a retailer chooses a shorter product line
instead of a full-length one to induce the manufacturer’s incentive to sell the less-suppliedvariety through the online store. The retailer benefits from doing so by creating for itself amore mitigating market condition.
For analytical simplicity, we consider the case in which there are two varieties and at
most two retailers. It is mathematically difficult to consider a more general case because wealready have 27 types of subgames to consider under the two-variety, and two-retailer case.
The situation becomes more complicated if we generalize either or both of the two parameters to n. Examining a continuous product variety offers one way to solve this difficulty.
However, our results already provide strong implications of a more general case. Besides
the assumption of horizontally differentiated products, it is natural to extend this setting tothe one with vertically differentiated products. If consumers show their heterogeneity in evaluating product quality and the (incumbent) retailer(s) can decide products’ quality fromthe manufacturer who also decides the quality of products sold online, the question arises as
to what would be the product quality outcomes and how such outcomes would affect social
welfare.
Moreover, as noted earlier, the result that the online store never sells one variety when
receiving retailers’ orders for both varieties follows from the assumption of symmetric retailcosts. This is just for analytical simplicity. Our results continues to hold even when there is26

a small retail cost gap between the two varieties. How large this retail cost gap can be andstill sustain our main results can make for an interesting issue to examine future studies.

5

Appendix

5.1

The case of a monopoly retailer

We first consider the equilibrium outcomes in period 3. By solving the optimization system
of Eq. (1) and (3), the equilibrium quantities denoted by wholesale prices are as follows:(i) when rm = XN:
qXN
XR (w) =a − wXN
X;
2(ii) when rm = XX:qXX
XR (w) =a + c − 2wXX
a − 2c + wXX
X
XXX, qXM (w) =;33(iii) when rm = XY:qXY
XR (w) =(2 − γ)a + γc − 2wXY
(2 − γ)a − 2c + γwXY
X
X
XY
(w)
=
,
q
;
YM
4 − γ24 − γ2(iv) when rm = XB:qXB
XR (w) =(2 − γ)a − (4 + γ)c − (2 + 2γ)γwXB
a + c − 2wXB
a−c
X
X
, qXB
(w)
=
, qYXBM (w) =;XM
36(1 + γ)2(1 + γ)(v) when rm = BN:
BN
qXR
(w) =

(1 − γ)a + γwXBN − wYBN
(1 − γ)a − wXBN + γwYBN BN
,
q
(w)
=
;
YR
2(1 − γ2 )2(1 − γ2 )(vi) when rm = BY:
BY
qXR
(w) =

(2 − 3γ + γ2 )a + (2 − 2γ2 )c − (4 − γ2 )wXBY + 3γwYBY,
6(1 − γ2 )

BY
qYR
(w) =

(1 − γ)a + γwXBY − wYBY BY
a − 2c + wXBY
,
q
(w)
=
;
XM
2(1 − γ2 )3(vii) when rm = BB:
BB
qXR
(w) =

(1 − γ)a + (1 − γ)c − 2wXBB + 2γwYBB,
3 − 3γ2
27

BB
qYR
(w) =

(1 − γ)a + (1 − γ)c + 2γwXBB − 2wYBB,
3 − 3γ2

BB
qXM
(w) =

(1 − γ)a − (2 − 2γ)c + wXBB − γwYBB,
3 − 3γ2qYBBM (w) =(1 − γ)a − (2 − 2γ)c − γwXBB + wYBB.
3 − 3γ2

Substituting the equilibrium outcomes in period 3 back to Eq. (4) or (5) and solving the
manufacturer’s optimization problem, we derive the equilibrium wholesale prices in period
2 as in Table 1. Substituting the equilibrium wholesale prices, we derive the retailer andmanufacturer’s equilibrium quantities and profits in period 2 as follows:(i) when rm = XN:

a XN a2 XN a2, π = ;qXN
XR = , πR =
4
16 M
8

(ii) when rm = XX:
qXX
XR =

5a + 3c XX 4c2 XX 5a2 − 10ac + 9c2
5a − 7c XX, qXM =, πR =
, π =
;
10
10
25 M
20

(iii) when rm = XY:qXY
XR =πRXY =

2(1 − γ)a + 2γc XY(8 − 2γ − γ2 )a − (8 − γ2 )c
,
,
q
=
YM
8 − 3γ22(8 − 3γ2 )4[(1 − γ)a + γc]2 XY (12 − 8γ + γ2 )a2 − (16 − 8γ + 2γ2 )ac + (8 + γ2 )c2;, πM =(8 − 3γ2 )24(8 − 3γ2 )(iv) when rm = XB:
qXB
XR =

2c XB
5a − (7 + 2γ)c XB
a−c, qXM =, qY M =,510(1 + γ)
2(1 + γ)πRXB =

4c2 XB 5a2 − 10ac + (7 + 2γ)c2
, π =
;
25 M
10(1 + γ2 )(v) when rm = BN:
BN
BN
qXR
= qYR
=

a
a2
a2
, πRBN =, πBN=;
4(1 + γ)8(1 + γ) M
4(1 + γ)(vi) when rm = BY:
BY
qXR
=

−5γa + 8(1 + γ2 )c BYa5a − 7c
BY
,
q
=
,
q
=
,
YR
XM
20(1 + γ2 )4(1 + γ)1028

πRBY =

25(1 − γ)a2 + 64(1 + γ)c2 BY 5(3 + γ)a2 − 20(1 + γ)ac + 18(1 + γ)c2
, πM =;
400(1 + γ)40(1 + γ)(vii) when rm = BB:
BB
BB
qXR
= qYR
=

2c
5a − 7c8c2
5a2 − 10ac + 9c2
BB, qXM= qYBBM =, πRBB =, πBB=.
5(1 + γ)10(1 + γ)25(1 + γ) M
10(1 + γ)

Making all quantities positive, we obtain the I.S . in Table 1.

5.2

Proof of Lemma 1

In period 2, we solve for the manufacturer’s equilibrium outcome, taking the corner solutions
into consideration and checking the incentive for deviation. Simply comparing the threshold2

5< 57 < 8−2γ−γ
values in Table 1 shows that 7+2γ
. We show that the online store never
8−γ2
5
5
c
XY
< ac ≤ 57 , we find that πXX
encroaches with variety X in this case. If 7+2γ
M < π M . If a ≤ 7+2γ ,XYXB
we find that πXX
M < max{π M , π M }. Thus, we have proved that encroaching with variety X
5
is a strictly dominated strategy in this case. The manufacturer chooses B if ac ≤ 7+2γ
and
XN
XX
XY
πXB
M ≥ max{π M , π M , π M }, from which we obtain
p
2
5(8
+
4γ
+
γ
)
−
2
40 + 80γ + 25γ2 − 30γ3 − 15γ4c
≤ θ X (γ) ≡.
a72 + 64γ + 17γ2

The outcome that the manufacturer chooses Y must satisfy the following conditions: if 57 <
2
c
c
5
5
5
c
XN
XN
XY
XX
≤ 8−2γ−γand πXYM ≥ π M , or if 7+2γ < a ≤ 7 and π M ≥ max{π M , π M }, or if a ≤ 7+2γ and
a
8−γ2

XN
XX
XB
πXY
M ≥ max{π M , π M , π M }, from which we obtainp2(8 − 4γ + γ2 ) − γ 16 − 6γ2cXθ (γ) ≤ ≤ θ (γ) ≡.
a2(8 + γ2 )X

The manufacturer does not open its online store ifcX≥ θ (γ).
a
We next see the case in which the retailer orders both varieties in period 1. Notice that for
5γc
BY≤ 8(1+γ), qXR
become zero so that the outcome is the same with that in the Y X case. Simplya5γ
comparing the threshold values in Table 1 shows that 8(1+γ)< 75 . We show that the online
5γ
store never encroaches with either variety except for a single point. If 8(1+γ)< ac ≤ 57 , we find

29

√

5γ
10− 10BNBB
BBthat πBY. If ac ≤ 8(1+γ), we find that πYMX < max{πBNM ≥ max{π M , π M } for c/a =M , π M }.
18

Thus, we have proved that encroaching with either variety is a weakly dominated strategy
5γ
BNBY
in this case. The manufacturer chooses B if 8(1+γ)< ac < 75 and πBB
M ≥ max{π M , π M }, or if
5γ
c
BN
YX
≤ 8(1+γ)and πBBM ≥ max{π M , π M }, from which we obtain
a

√
c
10
−
10
≤ θ B (γ) ≡.
a
18
The manufacturer does not open its online store ifc≥ θ B (γ).
a


5.3

The case of duopoly retailers

r1 r2 mr1 r2 mLet qnR(ŵ) and qnM(ŵ) be the retailer i and the online store’s equilibrium quantities in
i
XXm
XBm
XBm
XBm
period 3, where ŵ ∈ (wXXm
X1 , wX2 ) if r1 r2 = XX, ŵ ∈ (wX1 , wX2 , wY2 ) if r1 r2 = XB, ŵ ∈
XYm
BBm
BBm
BBm
BBm
(wXYm
X1 , wY2 ) if r1 r2 = XY, or ŵ ∈ (wX1 , wX2 , wY1 , wY2 ) if r1 r2 = BB. The equilibriumoutcomes in period 3 are as follows:(i) when r1 r2 m = XXN:qXXN
XR1 (ŵ) =XXN
XXN
a − 2wXXN
a + wXXN
X1 − 2wX2
X1 + wX2, qXXN
(
ŵ)
=
;
XR2
3
3

(ii) when r1 r2 m = XXX:qXXX
XR1 (ŵ) =XXX
XXX
a + c − 3wXXX
a + c + wXXX
X1 + wX2
X1 − 3wX2, qXXX
(
ŵ)
=
,
XR2
4
4

qXXX
XM (ŵ) =XXX
a − 3c + wXXX
X1 + wX2;
4(iii) when r1 r2 m = XXY:qXXY
XR1 (ŵ) =2XXY(2 − γ)a + γc − (4 − γ2 )wXXY
X1 + (2 − γ )wX2,
2(3 − γ2 )qXXY
XR1 (ŵ) =2XXY(2 − γ)a + γc + (2 − γ2 )wXXY
X1 − (4 − γ )wX2,
2(3 − γ2 )30

qYXXYM (ŵ) =XXY(3 − 2γ)a − 3c + γwXXY
X1 + γwX2;2(3 − γ2 )(iv) when r1 r2 m = XXB:qXXBXR1 (ŵ) =XXB
XXB
a + c − 3wXXB
a + c + wXXB
X1 + wX2
X1 − 3wX2, qXXB
(
ŵ)
=
,
XR2
4
4

qXXB
XM (ŵ) =XXB(1 − γ)a − (3 + γ)c + (1 + γ)wXXB
X1 + (1 + γ)wX2,4(1 + γ)
a−c
qYXXB
;
M (ŵ) =2(1 + γ)(v) when r1 r2 m = XY N:
NqXY
XR1 (ŵ) =N
XY N(2 − γ)a − 2wXY
X1 + γwY2,
4 − γ2XY N
qYR2(ŵ) =N
XY N(2 − γ)a + γwXY
X1 − 2γwY2;4 − γ2(vi) when r1 r2 m = XY X:
X
qXY
XR1 (ŵ) =X
XY X(2 − γ)a + (2 − γ2 )c − (4 − γ2 )wXY
X1 + γwY2,2(3 − γ2 )

XY X
qYR2
(ŵ) =XqXY
XM (ŵ) =X
XY X(3 − 2γ)a + γc + γwXY
X1 − 3wY2,
2(3 − γ2 )X
XY X
(2 − γ)a − (4 − γ2 )c + (2 − γ2 )wXY
X1 + γwY2;2(3 − γ2 )(vii) when r1 r2 m = XY B:
B
qXY
XR1 (ŵ) =B
XY B(3 − γ)a + (3 − γ)c − 6wXY
X1 + 2γwY2,9 − γ2XY B
qYR2(ŵ) =B
XY B(3 − γ)a + (3 − γ)c + 2γwXY
X1 − 6wY2,
9 − γ2BqXY
XM (ŵ) =B
2XY B(3 − γ)a + (6 + γ − γ2 )c + 3(1 + γ)wXY
X1 − (γ + γ )wY2,
9 + 9γ − γ2 − γ3qYXYMB (ŵ) =B
XY B(3 − γ)a + (6 + γ − γ2 )c − (γ + γ2 )wXY
X1 + 3(1 + γ)wY2;
9 + 9γ − γ2 − γ3(viii) when r1 r2 m = XBN:qXBN
XR1 (ŵ) =XBN
a − 2wXBN
X1 + wX2
,
3

31

qXBN
XR2 (ŵ) =

XBN
XBN
2
(2 − 3γ + γ2 )a + 2(1 − γ2 )wXBN
X1 − (4 − γ )wX2 + 3γwY2,6(1 − γ2 )qXBN
XR2 (ŵ) =XBN(1 − γ)a + γwXBN
X2 − wY2;
2(1 − γ2 )(ix) when r1 r2 m = XBY:qXBY
XR1 (ŵ) =qXBY
XR2 (ŵ) =XBY
a + c − 3wXBY
X1 + wX2
,
4

2
XBY
XBY
(2 − 2γ + γ2 )a + (1 − γ2 )c + (1 − γ2 )wXBY
X1 − (3 − γ )wX2 + 2γwY2,4(1 − γ2 )XBYqYR2(ŵ) =XBY(1 − γ)a + γwXBY
X2 − wY2,
2(1 − γ2 )qXBY
XM (ŵ) =XBY
a − 3c + wXBY
X2 + wY2;4(x) when r1 r2 m = XBY:qXBY
XR1 (ŵ) =XBY
XBY(3 − γ)a + 2γc − 6wXBY
X1 + 3wX2 − γwY2,
9 − γ2qXBY
XR2 (ŵ) =

2
XBY
3
XBY
(3 − 4γ + γ2 )a − (1 − γ2 )c + 3(1 − γ2 )wXBY
X1 − 2(3 − γ )wX2 + (5γ − γ )wY2,
9 − 10γ2 + γ4

XBY
qYR2
(ŵ) =

3XBY
2XBY(3 − 4γ + γ2 )a + 3(1 − γ2 )c − (γ − γ3 )wXBY
X1 + (5γ − γ )wX2 − 2(3 − γ )wY2,
9 − 10γ2 + γ4qXBY
XM (ŵ) =XBY
a − 3c + wXBY
X2 + wY2;4

qXBB
XR1 (ŵ) =XBB
a + c − 3wXBB
X1 + wX2,
4(xi) when r1 r2 m = XBB:qXBB
XR2 (ŵ) =

2
XBB
XBB
(3 − 4γ + γ2 )a + (3 − 4γ + γ2 )c + 3(1 − γ2 )wXBB
X1 − (9 − γ )wX2 + 8γwY2,12(1 − γ2 )XBB
qYR2(ŵ) =qXBB
XM (ŵ) =XBB(1 − γ)a + (1 − γ)c + 2γwXBB
X2 − 2wY2,3(1 − γ2 )

2
XBB
XBB
(3 − 4γ + γ2 )a − (9 − 8γ − γ2 )c + 3(1 − γ2 )wXBB
X1 + (3 + γ )wX2 − 4γwY2,12(1 − γ2 )qYXBB
M (ŵ) =XBB(1 − γ)a − 2(1 − γ)c − γwXBB
X2 + wY2;
3(1 − γ2 )32

(xii) when r1 r2 m = BBN:
BBN
qXR1(ŵ) =

BBN
BBN
BBN
BBN
− γwY2+ 2γwY1+ wX2(1 − γ)a − 2wX1,3(1 − γ2 )

BBN
qYR1
(ŵ) =

BBN
BBN
BBN
BBN
+ wY2
− 2wY1− γwX2(1 − γ)a + 2γwX1,3(1 − γ2 )

BBN
qXR2
(ŵ) =

BBN
BBN
BBN
BBN
(1 − γ)a + wX1− 2wX2− γwY1+ 2γwY2,
3(1 − γ2 )

BBN
qYR2
(ŵ) =

BBN
BBN
BBN
BBN
− 2wY2+ wY1+ 2γwX2(1 − γ)a − γwX1;3(1 − γ2 )(xiii) when r1 r2 m = BBY:
BBY
BBY
BBY
BBY
− (3 + γ2 )wX2
− 8γwY1+ 4γwY2(3 − 4γ + γ2 )a + 3(1 − γ2 )c (9 − γ2 )wX1
BBY−,
qXR1 (ŵ) =2
2

12(1 − γ )

BBY
qYR1
(ŵ) =BBY
qXR2(ŵ) =

12(1 − γ )

BBY
BBY
BBY
BBY
(1 − γ)a + 2γwX1− γwX2− 2wY1+ wY2,
3(1 − γ2 )

BBY
BBY
BBY
BBY
− (9 − γ2 )wX2
− 4γwY1+ 8γwY2(3 − 4γ + γ2 )a + 3(1 − γ2 )c (3 + γ2 )wX1+,
12(1 − γ2 )12(1 − γ2 )BBYqYR2(ŵ) =

BBY
BBY
BBY
BBY
(1 − γ)a − γwX1
+ 2γwX2+ wY1− 2wY2,3(1 − γ2 )BBYqXM(ŵ) =BBY
BBY
a − 3c + wX1
+ wX2
;
4

(xiv) when r1 r2 m = BBB:BBB
qXR1(ŵ) =

BBB
BBB
BBB
BBB
(1 − γ)a + (1 − γ)c − 3wX1
+ wX2
+ 3γwY1− γwY2,4(1 − γ2 )

BBB
qYR1
(ŵ) =

BBB
BBB
BBB
BBB
(1 − γ)a + (1 − γ)c + 3γwX1− γwX2− 3wY1+ wY2,
4(1 − γ2 )

BBB
qXR2
(ŵ) =

BBB
BBB
BBB
BBB
(1 − γ)a + (1 − γ)c + wX1− 3wX2− γwY1+ 3γwY2,
4(1 − γ2 )BBB(ŵ) =
qYR2

BBB
BBB
BBB
BBB
(1 − γ)a + (1 − γ)c − γwX1
+ 3γwX2+ wY1− 3wY2,4(1 − γ2 )BBBqXM(ŵ) =

BBB
BBB
BBB
BBB
(1 − γ)a − 3(1 − γ)c + wX1+ wX2− γwY1− γwY2,4(1 − γ2 )qYBBB
M (ŵ) =

BBB
BBB
BBB
BBB
(1 − γ)a − 3(1 − γ)c − γwX1
− γwX2
+ wY1
+ wY2
.
4(1 − γ2 )33

By substituting the equilibrium outcomes in period 3 and solving the manufacturer’s
optimization problem, we derive the equilibrium wholesale prices in period 2 as in Table
2. Substituting the equilibrium wholesale prices, we derive the retailer and manufacturer’s
equilibrium quantities and profits in period 2 as follows:(i) when r1 r2 m = XXN:
a XXN
a2 XXN a2
XXN
XXN
qXXN
=
q
=
,
π
=
π
=
, π
= ;
XR1
XR2
R2
6 R1
36 M
6
(ii) when r1 r2 m = XXX:
c XXX 3a − c XXX
c2 XXX 3a2 − 6ac + 7c2
XXX
XXX
qXXX
=
q
=
,
q
=
,
π
=
π
=
, π
=
;
XR1
XR2
R1
R2
3 XM
6
9 M
12
(iii) when r1 r2 m = XXY:XXYqXXY
XR1 = qXR2 =

XXY
XXY
πR1
= πR2
=

(1 − γ)a + γc XXY (6 − 2γ − γ2 )a − (6 − γ2 )c, qY M =,3(2 − γ2 )6(2 − γ2 )[(1 − γ)a + γc]2 XXY (10 − 8γ + γ2 )a2 − (12 − 8γ + 2γ2 )ac + (6 + γ2 )c2;, πM =
9(2 − γ2 )212(2 − γ2 )(iv) when r1 r2 m = XXB:c XXB 3a − (5 + 2γ)c XXB
a−c
XXB
qXXB
, qY M =,XR1 = qXR2 = , qXM =36(1 + γ)2(1 + γ)
XXB
XXB
πR1
= πR2
=

c2 XXB 3a2 − 6ac + (5 + 2γ)c2
, π
=
;
9 M
6(1 + γ)(v) when r1 r2 m = XY N:
XY N
N
qXY
XR1 = qYR2 =

a
a2
a2
XY N
XY N
XY N
=
, πR1
= πR2
,
π
=
;
2(2 + γ)4(2 + γ)2 M
2(2 + γ)(vi) when r1 r2 m = XY X:
X
qXY
XR1 =

4(2 − γ2 )c − (γ − γ2 )a XY X 5(1 − γ)a + 4γc XY X (5 − γ − γ2 )a − (7 − 2γ2 )c, qYR2 =, qXM =,
4(5 − 2γ2 )4(5 − 2γ2 )2(5 − 2γ2 )
XY X
πR1
=

[(γ − γ2 )a − (8 − 4γ2 )c]2 XY X [5(1 − γ)a + 4γc]2, πR2 =,
16(5 − 2γ2 )216(5 − 2γ2 )2

X
πXY
=
M

(15 − 10γ + γ2 )a − (20 − 8γ)ac + (18 − 4γ2 )c2;
8(5 − 2γ2 )34

(vii) when r1 r2 m = XY B:
B
XY B
qXY
XR1 = qYR2 =

XY B
XY B
πR1
= πR2
=

2c
(5 + γ)a − (7 + 3γ)c
B
XY B
, qXY
,
XM = qY M =5+γ2(5 + 6γ + γ2 )4c2
(5 + γ)a2 − 2(5 + γ)ac + (9 + 5γ)c2
XY B
,
π
=
;
(5 + γ)2 M
2(5 + 6γ + γ2 )(viii) when r1 r2 m = XBN:
(2 − γ)a
a
a XBN
XBN
, qYR2=,qXBN
XR1 = , qXR2 =6
12(1 + γ)
4(1 + γ)XBN
πR1=

a2 XBN (13 − 5γ)a2 XBN (7 + γ)a2
, π=
, πM =;
36 R2
144(1 + γ)24(1 + γ)(ix) when r1 r2 m = XBY:
c XBY 4(1 + γ)c − 3γa XBY
a3a − γcqXBY, qYR2 =, qXBY,XM =XR1 = , qXR2 =3
12(1 + γ)
4(1 + γ)
6
XBY
πR1
=

c2 XBY 9(1 − γ)a2 + 16(1 + γ)c2 XBY 3(3 + γ)a2 − 12(1 + γ)ac + 14(1 + γ)c2,π=
, πM =;
9 R2
144(1 + γ)24(1 + γ)(x) when r1 r2 m = XBY:
qXBY
XR1 =

5(1 − γ)a + 8γc XBY (5 + γ2 )a − 4(γ + γ2 )c XBY (15 − 2γ − γ2 )a − (21 − γ2 )c, qXR2 =q=,
6(5 − γ2 )6(5 + 5γ − γ2 − γ3 ) YR2
6(5 − γ2 )XBYa − 3c + wXBY
X2 + wY2,
4[5(1 − γ)a + 8γc]2 XBY (25 − 25γ + 11γ2 − 11γ3 )a2 − 64(γ − γ3 )ac + 16(9 + 9γ − 5γ2 − 5γ3 )c2
XBY, πR2 =πR1=,
36(5 − γ2 )236(25 + 25γ − 10γ2 − 10γ3 + γ4 + γ5 )

qXBY
XM =

πXBY
=
M

(25 + 5γ − 7γ2 )a2 − 2(15 + 11γ − 3γ2 + γ3 )ac + (27 + 27γ + γ2 + γ3 )c2;
5 + 5γ − γ2 − γ3(xi) when r1 r2 m = XBB:
(5 − γ)c2c15a − (25 + 4γ)c XBB
5a − 7cc XBB
XBB, qYR2=, qXBB, qY M =
,
qXBB
XM =
XR1 = , qXR2 =3
15(1 + γ)
5(1 + γ)30(1 + γ)10(1 + γ)
XBB
πR1
=

c2 XBB (61 − 11γ)c2 XBB 15a2 − 30ac + (31 + 4γ)c2, π=, πM =;
9 R2
225(1 + γ)30(1 + γ)(xii) when r1 r2 m = BBN:
BBN
BBN
BBN
BBN
qXR1
= qYR1= qXR2= qYR2
=

a
a2
a2
BBN
BBN
, πR1
= πR2
=
, πBBN=;6(1 + γ)18(1 + γ) M
3(1 + γ)35

(xiii) when r1 r2 m = BBY:
BBY
BBY
qXR1
= qXR2
=

BBY
BBY
πR1
= πR2
=

a
3a − 5c2(1 + γ)c − γa BBY
BBY
BBY, qYR1 = qYR2
=
, qXM
=
,
6(1 + γ)6(1 + γ)6

(1 − γ)a2 + 4(1 + γ)c2 BBY (5 + γ)a2 − 6(1 + γ)ac + 7(1 + γ)c2
, πM =;
36(1 + γ)12(1 + γ)(xiv) when r1 r2 m = BBB:
BBB
BBB
BBB
BBB
qXR1
= qXR2= qYR1= qYR2
=

BBB
BBB
πR1
= πR2
=

c
3a − 5c
BBB, qXM= qYBBB,M =3(1 + γ)6(1 + γ)2c2
3a2 − 6ac + 7c2, πBBB=.
9(1 + γ) M
6(1 + γ)Making all quantities positive, we obtain the I.S . in Table 2. We will prove that encroachingwith variety X through the online store when receiving the retailer’s XX or XB type order,and encroaching with either variety through the online store when receiving the retailer’s XY
or BB are dominated strategies later.

5.4

Proof of Lemma 2

In period 2, we solve for the manufacturer’s equilibrium outcome, taking the corner solutions
into consideration and checking the incentive for deviation.
First we see the case in which the manufacturer receives the XX type order in peri2

3
od 1. Simply comparing the threshold values in Table 2 shows that 5+2γ
< 53 < 6−2γ−γ.
6−γ2We show that the online store never encroaches with variety X in this case. If ac ≤ 35 ,we find that πXXX
< πXXBM
M . Thus, we have proved that encroaching with variety X is astrictly dominated strategy in this case. The manufacturer chooses B if ac ≤

3
5+2γ

and

XXX
XXY
πXXB
≥ max{πXXN
M
M , π M , π M }, from which we obtainp2
2 + 4γ + γ2 − 2γ3 − γ4
(6
+
4γ
+
γ
)
−
2
c
≤ θ XX (γ) ≡.
a
14 + 16γ + 5γ2

The outcome that the manufacturer chooses Y must satisfy the following conditions: if 35 <
2
c
3
c
3
c
3
XXY
XXX
≤ 6−2γ−γand πXXY≥ πXXN≥ max{πXXN
M
M , if 5+2γ < a ≤ 5 and π M
M , π M }, if a ≤ 5+2γ and
a
6−γ2

XXX
XXB
πXXY
≥ max{πXXN
M
M , π M , π M }, from which we obtainp(6 − 4γ + γ2 ) − γ 2(2 − γ2 )cXX
θ (γ) ≤ ≤ θ (γ) ≡.
a
6 + γ2
XX

36

The manufacturer does not open its online store ifcXX≥ θ (γ).
aSecond, we see the case in which the manufacturer receives the XY type order in period
γ−γ
XY X
1. Notice that for ac ≤ 4(2−γ2 ) , q XR1 becomes zero so that the outcome is the same with that2γ−γ
in the NY X case. Simply comparing the threshold values in Table 2 shows that 4(2−γ
2) <
2

5+γ
7+3γ

2

< 5−γ−γ. We show that the online store never encroaches with variety X in this case.
7−2γ2

γ−γ
5+γ
5+γ
c
N
X
XY X
< πXY
<
< ac ≤ 5−γ−γ, we find that πXY
If 7+3γ
M
M . If 4(2−γ2 ) < a ≤ 7+3γ , we find that π M
7−2γ2
2

2

γ−γ
c
N
NY X
N
XY B
XY B
max{πXY< max{πXY
M , π M }. If a ≤ 4(2−γ2 ) , we find that π M
M , π M }. Thus, we have proved
2

that encroaching with variety X is a strictly dominated strategy in this case. The outcome
2

5+γ
1−γ
c
that the manufacturer chooses B must satisfy the following conditions: if 4(2−γ2 ) < a < 7+3γ
2

1−γ
c
NY X
N
N
B
XY X
XY B
and πXY≥ max{πXY≥ max{πXY
M
M , π M }, or if a ≤ 4(2−γ2 ) and π M
M , π M }, from which we

obtain

p
2
(10
+
7γ
+
γ
)
−
10 + 27γ + 25γ2 + 9γ3 + γ4c≤ θ XY (γ) ≡.
a
18 + 19γ + 5γ2

The manufacturer does not open its online store ifc≥ θ XY (γ).
aThird, we see the case in which the manufacturer receives the XB type order in period 1.
3γ
Notice that for ac ≤ 4(1+γ), qXBY
XR2 becomes zero so that the outcome is the same with that inγ
XBY
the XY X case; for ac ≤ 2(1+γ), qYR2
becomes zero so that the outcome is the same with thatγ−γX
in the XXY case; for ac ≤ 4(2−γ), qXY
XR1 becomes zero so that the outcome is the same with
2γ−γthat in the NY X case. Simply comparing the threshold values in Table 2 shows that 4(2−γ
2) <
2

γ
2(1+γ)23γ
15< 4(1+γ)< 25+4γ< 53 < 15−2γ−γ. We show that the online store never encroaches
21−γ2

15
with variety X in this case. If 25+4γ< ac ≤ 35 in which the XBB equilibrium outcome does
3γ
c
15
XBY
XBY
not exist, we find that πXBY
< max{πXBN<
M
M , π M }. If 4(1+γ) < a ≤ 25+4γ , we find that π M
3γ
γ
c
XY X
XBY
XBB
XBY
XBB
max{πXBN< max{πXBN
M , π M , π M }. If 2(1+γ) < a ≤ 4(1+γ) , we find that π M
M , π M , π M }. If
γ−γ2
γ
γ−γ2
c
X
XXY
XBB
< ac ≤ 2(1+γ), we find that πXY< max{πXBN
M
M , π M , π M }. If a ≤ 4(2−γ2 ) , we find that4(2−γ2 )
X
XXY
XBB
πNY
< max{πXBN
M
M , π M , π M }. Thus, we have proved that encroaching with variety X is astrictly dominated strategy in this case. The outcome that the manufacturer chooses B must
37

3γ
15
XBY
XBY
satisfy the following conditions: if 4(1+γ)≥ max{πXBN< ac ≤ 25+4γ
and πXBB
M
M , π M , π M },
γ
γ−γ
3γ
γ
c
XBY
XY X
or if 2(1+γ)≥ max{πXBN< ac ≤ 4(1+γ)
and πXBB
M
M , π M , π M }, or 4(2−γ) < a ≤ 2(1+γ) and
2

γ−γ
c
NY X
XXY
XXY
XY X
XBB
≥ max{πXBN≥ max{πXBN
πXBBM
M , π M , π M }, from which
M , π M , π M }, or a ≤ 4(2−γ) and π M
2

we obtainp5(15 + 4γ + γ2 ) − (5 + 2γ) 10(5 − γ2 )c
XBwhen γ ≤ 0.690, ≤ θ1 (γ) ≡,a
175 + 80γ + 13γ2p5(6 + 4γ + γ2 ) − 2 5(−2 + 20γ + 5γ2 − 10γ3 − 2γ4 )cXBas well as when γ > 0.690, ≤ θ2 (γ) ≡.
a
94 + 80γ + 13γ2
The outcome that the manufacturer chooses Y must satisfy the following conditions: if 35 <
2
3γ
c
c
3
c
15
XBY
XBY
≤ 15−2γ−γ≥ {πXBNand πXBY≥ πXBNM
M , π M }, or if 4(1+γ) < a ≤M , or if 25+4γ < a ≤ 5 and π M
a
21−γ2

γ
3γ
15
c
XBY
XBB
XBY
XY X
XBB
and πXBY≥ {πXBN≥ {πXBNM
M , π M , π M }, or if 2(1+γ) < a ≤ 4(1+γ) and π M
M , π M , π M },25+4γfrom which we obtain
2(15 − 4γ + γ2 ) − (3 + γ)c
XB
XB
when γ ≤ 0.690, θ1 (γ) ≤ ≤ θ1 (γ) ≡2p2(5 − γ2 )2(27 + γ )aas well as when 0.690 < γ ≤ 0.897, θ̂ XB (γ) ≡

,

γ
c
XB
≤ ≤ θ1 (γ).
2(1 + γ) a

The outcome that the manufacturer does not open its online store must satisfy the following
2

2

15
c
3
conditions: if ac > 15−2γ−γ, or if 35 < ac ≤ 15−2γ−γ
and πXBN≥ πXBYM , or 25+4γ ≤ a < 5
M
21−γ2
21−γ2
3γ
c
15
XBN
XBY
XBY
XBB
and πXBN≥ max{πXBY≥ max{πXBY
M , π M }, or 4(1+γ) < a ≤ 25+4γ and π M
M , π M , π M }, orM
γ2(1+γ)

3γ
γ−γ
γ
c
XBN
X
XBY
XBB
< ac ≤ 4(1+γ)
and πXBN≥ max{πXY≥
M , π M , π M }, or 4(2−γ2 ) < a ≤ 2(1+γ) and π M
M
2

γ−γ
c
XBN
X
X
XXY
XBB
XXY
XBB
max{πXY≥ max{πNY
M , π M , π M }, or a ≤ 4(2−γ2 ) and π M
M , π M , π M }, from which we
2

obtainc
XB> θ1 (γ),
acXBas well as when γ > 0.897, > θ2 (γ)
ap(1 + γ)(12 − 8γ + 2γ2 ) − 2(1 + γ)(36 − 36γ − 4γ2 + 20γ3 − 7γ4 − γ5 )≡.
2(1 + γ)(6 + γ2 )when γ ≤ 0.897,

Fourth, we see the case in which the manufacturer receives the BB type order in period
γ
BBY
BBY
1. Notice that for ac ≤ 2(1+γ), qXR1and qXR2
becomes zero so that the outcome is the samewith that in the YY X case. Simply comparing the threshold values in Table 2 shows that38γ
2(1+γ)< 53 . We show that the online store never encroaches with either variety except forγ
BBB≥ max{πBBN
a single point in this case. If 2(1+γ)< ac ≤ 35 , we find that πBBY
M
M , π M } whenγBBBX
c/a = θ BB (γ). If ac ≤ 2(1+γ)< max{πBBN, we find that πYYM
M , π M }. Thus, we have proved thatencroaching with either variety is a weakly dominated strategy in this case. The outcomeγthat the manufacturer chooses B must satisfy the following conditions: if 2(1+γ)< ac ≤ 35 and
γ
c
BBY
YY X
BBB
≥ max{πBBN≥ max{πXBN
πBBB
M
M , π M }, or if a ≤ 2(1+γ) and π M
M , π M }, from which we obtain

√
c
3− 2
BB
≤ θ (γ) ≡.
a
7The outcome that the manufacturer does not open its online store must satisfy the following conditions:c≥ θ BB (γ).
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