Discussion Paper No. 745

INPUT SPECIFICITY AND PRODUCTDIFFERENTIATION

Noriaki Matsushima
andTomomichi Mizuno

June 2009

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

Input speciﬁcity and product diﬀerentiation∗
Noriaki Matsushima†
Institute of Social and Economic Research, Osaka University
Tomomichi Mizuno
Competition Policy Research Center, Japan Fair Trade Commission
June 17, 2009

Abstract
Using a simple product diﬀerentiation model with elastic demands, we investigate therelationship between diﬀerentiation strategies and vertical relations. Depending on thecompetitive structure in the upstream market, three diﬀerentiation patterns (maximum,
minimum and partial diﬀerentiation) can appear in equilibrium even though each downstream ﬁrm freely determines the degree of product diﬀerentiation. When downstreamﬁrms must incur positive investment costs to diﬀerentiate their products, they tend to doso if the upstream market is competitive.

JEL classiﬁcation numbers: L13.
Key words: oligopoly, product diﬀerentiation, upstream ﬁrm, input speciﬁcity∗
We would like to thank Masamichi Kawano, Toshihiro Matsumura, Keizo Mizuno, Dan Sasaki and theseminar participants at Kwansei Gakuin University and the Applied Economics Workshop (Tohoku University)
for their helpful comments. The ﬁrst author gratefully acknowledges ﬁnancial support from a Grant-in-Aid for
Encouragement of Young Scientists from the Japanese Ministry of Education, Science and Culture. Needlessto say, we are responsible for any remaining errors. The views expressed herein are those of the authors andshould not be interpreted as those of the Japan Fair Trade Commission.
†
Corresponding author: Noriaki Matsushima, Institute of Social and Economic Research, Osaka University, Mihogaoka 6-1, Ibaraki, Osaka 567-0047, Japan. Phone: +81-6-6877-1111 (Ex. 9170). E-mail:nmatsush@iser.osaka-u.ac.jp

1

Introduction

Manufacturers often have relationships with many input suppliers. For instance, in the aircraft
industry, the jet (turboprop) engine and aircraft industries are vertically related. To produce
a diﬀerentiated product, aircraft ﬁrms must procure suitable but costly equipment. Engine
companies also have relations with many buyers, and potentially with all of them (Bonaccorsi
and Giuri (2001)). In the automobile industry, Toyota procures electric parts from many
input suppliers, including Denso, which is one of the largest auto part manufacturers.1 Denso
sometimes supplies other automobile manufacturers, for instance Daimler Chrysler. Thus, its
electrical parts may be used in automobiles from diﬀerent manufacturers.
When manufacturers produce (horizontally) diﬀerentiated products, they sometimes faceconstraints that stem from the technological capacities of suppliers. When those manufacturers are able to negotiate with many suppliers of suitable inputs, they can procure those inputs
at reasonable prices. In contrast, when there are only a few suppliers of suitable inputs, themanufacturers must pay higher prices. That is, competitiveness in input markets aﬀects the
procurement conditions of manufacturers.
Because decisions on product positioning (diﬀerentiation), which is an important strategic
tool of ﬁrms, are inﬂuenced by such buyer–supplier relationships,2 some researchers investigate the relationship between diﬀerentiation strategies and vertical relations. Using Hotelling
location models, Brekke and Straume (2004) and Matsushima (2004, 2008) discuss how upstream units aﬀect the diﬀerentiation strategies of downstream ﬁrms.3 In Brekke and Straume(2004), ﬁrms acquire inputs through bilateral monopoly relations with suppliers. They show1

Ahmadjian and Lincoln (2001) research the vertical relationship between Toyota and Denso.

2

There are many studies on product positioning decisions. See, for instance, Hotelling (1929), d’Aspremont

et al. (1979), Hauser and Shugan (1983), Vandenbosch and Weinberg (1995), and Sayman, Hoch, and Raju
(2002).
3

Bárcena-Ruiz and Casado-Izaga (2008) incorporate the sequential move of upstream prices (wages) into

the model of Brekke and Straume (2004). Using a Hotelling location model, Erkal (2007) also investigates how
the buyer–supplier relationship aﬀects diﬀerentiation strategies.

1that the existence of upstream suppliers enhances the degree of product diﬀerentiation. In
Matsushima (2004, 2008), each upstream ﬁrm that locates in a linear city engages in pricecompetition for the business of downstream ﬁrms. To supply one unit of input to a downstream ﬁrm, each upstream ﬁrm incurs a transport cost quadratic in distance. These papers
show that the degree of product diﬀerentiation decreases as the coeﬃcient of the quadratictransport cost incurred by each upstream ﬁrm increases. Although the settings of Brekke andStraume (2004) and Matsushima (2004, 2008) are only slightly diﬀerent, those papers provide
quite diﬀerent results.
Although those researchers provide interesting insights into the relationship between differentiation strategies and vertical relations, they only consider markets in which downstreamﬁrms compete in price (strategic complement). Moreover, consumer demand is inelastic in
their models because they use Hotelling models. It is not obvious how vertical relations affect the diﬀerentiation strategies of downstream ﬁrms when such ﬁrms compete in quantity(strategic substitute). The aim of this paper is to investigate this problem. Moreover, afurther contribution of this paper is that our model succeeds in merging the previous threepapers into a simple duopoly model with elastic demand functions.
The model setting of the paper is as follows. There are two upstream and two downstream ﬁrms. Each downstream ﬁrm produces a diﬀerentiated ﬁnal product, and must buy
an input from the upstream ﬁrms. Each downstream ﬁrm determines the degree of product diﬀerentiation (denoted by γ). Each ﬁrm must incur investment costs to diﬀerentiate its
product (product R&D). We consider two cases concerning the competitive environment of
upstream ﬁrms. First, each upstream ﬁrm has a bilateral monopoly relation with its buyer (adownstream ﬁrm). Second, each upstream ﬁrm faces competitive forces that can potentiallydeprive it of its trade with its downstream partner. To supply one unit of input to a downstream ﬁrm, the potential suppliers must incur two additional marginal costs (k + τ di ) that
depend on the degree of product R&D that the downstream ﬁrm undertakes (di ∈ [0, 1/2],2

1 − d1 − d2 = γ), where k and τ are positive constants. We can interpret the additional costsof potential suppliers as being the conversion cost from the product of the upstream ﬁrm into
an input suitable for the downstream ﬁrm. After purchasing its input from an upstream ﬁrm,
each downstream ﬁrm sets the quantity to be supplied.
We show that when the investment cost for product R&D is zero, maximum diﬀerentiation
appears in equilibrium if k is large or τ is small. If k is small and τ is large, minimum
diﬀerentiation appears. If k has an intermediate value and τ is large, only one downstreamﬁrm engages in product R&D. We also show that both minimum and maximum diﬀerentiation
appear in equilibrium if k is small and τ has an intermediate value. Moreover, we show thatwhen the investment cost for product R&D is positive, minimum diﬀerentiation tends toappear in equilibrium if each upstream ﬁrm has a bilateral monopoly relation with its buyer(a downstream ﬁrm). A slight diﬀerence in product R&D technologies leads to quite diﬀerent
equilibrium outcomes in product diﬀerentiation. Those results are somewhat diﬀerent from
Brekke and Straume (2004) and Matsushima (2004, 2008), and provide additional insights
into the discussion of the relationship between product diﬀerentiation and vertical relations.
The setting and the results of this paper are related to the case of the aircraft industry.4
There are two major ﬁrms in this industry, Airbus and Boeing. Those ﬁrms rely heavily on
ﬁrm-speciﬁc inputs (engines, wings, horizontal stabilizers) produced by independent manufacturers. For instance, Rolls Royce and CFM International (a joint venture between GE
and the French company Snecma) make highly speciﬁc turbofan engines for airplanes. Rolls
Royce made the Trent XWB turbofan engine to meet the speciﬁc requirements of the Airbus
A350 XWB family, and developed the Trent 900 series exclusively to power the new Airbus
A380. Furthermore, CFM International designed the CFM56-3 turbofan engine exclusively
for Boeing aircraft.5 Under the procurement condition, Airbus and Boeing have considered
4This paragraph is based on the discussions in Beelaerts van Blokland et al. (2008), Nalebuﬀ (2002), Esty

and Ghemawat (2002), and Tombak (2006).
5

In reality, these engine manufacturers sometimes supply both Airbus and Boeing if possible.

3

whether their products should be diﬀerentiated. At one time, Airbus and Boeing produced
highly diﬀerentiated aircraft (A380 and Boeing 787), however at other times, they produced
less-diﬀerentiated aircraft (the A340 and Boeing 747(777)) and competed directly.6 From thisreal world example, we can say that the technological constraints of input suppliers cruciallyaﬀect the diﬀerentiation strategies of the two aircraft ﬁrms.
This paper is closely related to that of Lin and Saggi (2002). Using linear demand functions
with product diﬀerentiation, they discuss the relationship between process and product R&D.7
We add vertical relations to their model, and our motivation is quite diﬀerent from theirs. Our
main concern is how the existence of upstream ﬁrms and competition between them aﬀects
the diﬀerentiation strategies of downstream ﬁrms.8
The paper proceeds as follows. The next section outlines the basic environment. Section
3 provides an analysis of the model. Section 4 discusses the relationship between vertical
relations and diﬀerentiation strategies. Finally, Section 5 oﬀers some concluding remarks.

6

The A3XX (which was rechristened the A380) program was formally launched in December 2000. Afterthat event, Boeing abandoned the Sonic Cruise development program and launched the 787 program becauseof concerns about fuel eﬃciency in the ailing airline industry in 2002. Therefore, the A380 and Boeing 787
are highly diﬀerentiated. In 1987, Airbus launched the A330 and A340 airframes, which were designed tocompete with the Boeing 747 on medium- to long-distance routes. Boeing then postponed the 7J7 project todevelop a short-range 150-seat aircraft (which has the potential to diﬀerentiate its product line from that of
the large-scale intercontinental commercial jet market) and then developed the 777 airframe with technologicalfeatures, seating capacity and range similar to those of the A340.
7

Most subsequent studies of Lin and Saggi (2002) also focus on the complementarity of process and product

R&D, which is not the focus of this paper. See, for instance, Mantovani (2006) and Braun (2008).
8

There are discussions on how vertical structure aﬀects incentives of upstream and/or downstream ﬁrms forinnovative activities. Given vertical market structures and competition environments, Ishii (2004) and Milliou(2004) discuss spillover eﬀects of R&D investments. Banerjee and Lin (2003), Brocas (2003), and Buehler
and Schmutzler (2008) focus on the relation of vertical market structure and cost-reducing investment inoligopoly. These papers do not discuss how the existence of upstream ﬁrms aﬀects the diﬀerentiation strategies
of downstream ﬁrms. Pepall and Norman (2001) also consider vertical relationships and product diﬀerentiation.
They consider vertical mergers and vertical foreclosure, but not endogenous product diﬀerentiation. In theirmodel, the degree of product diﬀerentiation of the downstream ﬁrms is exogenous.

4

2

The model

We consider a market with two downstream and two upstream ﬁrms. There is only one factor
of production, which we refer to as the input. Each ﬁrm produces a diﬀerentiated ﬁnal goodin a constant-returns-to-scale process where one unit of the input becomes one unit of the
ﬁnal good. Each downstream ﬁrm can perform this conversion at no cost.
We assume that the two downstream ﬁrms compete in quantity. Now let xi denote i’ssales. The inverse demand function of ﬁrm i is given bypi = 1 − xi − γxj , i = 1, 2, i ̸= j.

(1)where pi is the price of the ﬁnal good produced by i, and γ(∈ [0, 1]) indicates the degree of
product diﬀerentiation between the products. The lower the value of γ, the higher the degreeof product diﬀerentiation. The value of γ is determined by each ﬁrm’s investment in product
R&D, denoted by di (i = 1, 2).
We consider the downstream ﬁrm’s investments in product R&D. Following the setting in
Lin and Saggi (2002),9 we assume that the value of γ is determined by each downstream ﬁrm’s
investment in product R&D, denoted by di (i = 1, 2). The extent of product diﬀerentiationis γ = 1 − (d1 + d2 ), and 0 ≤ di ≤ 1/2. The cost of product R&D is given by F (di ), F ′ ≥ 0,
F ′′ ≥ 0.
Let zi denote the amount of input supplied by the supplier of ﬁrm i. Each input suppliersets a wholesale price wi to maximize its proﬁt function (without loss of generality, we assume
that the marginal cost is normalized to zero):πU i = wi zi , i = 1, 2.

(2)

The fact that the input suppliers unilaterally set the wholesale prices implies that they possessfull bargaining power.
9

See also Braun (2008), Cellini and Lambertini (2002), Lambertini et al. (1998), Rosenkranz (2003), andWeiss (2003).

5

In this paper, we consider two cases concerning the technologies of upstream ﬁrms:
1. Neither upstream ﬁrm faces a competitor.
2. Each upstream ﬁrm faces competitors that are potentially able to supply the downstreamﬁrms. If a competitor incurs a marginal cost, fi = τ di + k, it is able to supply input todownstream ﬁrm i, where τ and k are positive constants, and di is the level of productR&D engaged in by downstream ﬁrm i (i = 1, 2).
In the ﬁrst case, each upstream ﬁrm is in a monopoly position with respect to its trading
partner, similar to the setting in Brekke and Straume (2004). In the second case, upstreamﬁrms face potentially harmful competitive forces, which include rival upstream ﬁrms supplying
the downstream ﬁrm (that is, upstream ﬁrms i and j are potential competitors (i = 1, 2, i ̸=j)). For instance, as mentioned in the introduction, while Rolls Royce and CFM International
supply highly speciﬁc turbofan engines exclusively to Airbus and Boeing, respectively, theseengine suppliers are both potential suppliers of either Airbus or Boeing (in reality, theseengine manufacturers sometimes supply both Airbus and Boeing). Following the setting in
Matsushima (2004, 2008), we assume that the marginal cost fi is increasing in the degree of
product R&D, which is related to the degree of product diﬀerentiation. This case is a slightly
modiﬁed version of Matsushima (2004, 2008).10
We can interpret the ﬁrst case as a special case of the second one. When k is suﬃciently
high, potential competitors do not dare to supply inputs to a downstream ﬁrm because themarginal cost is prohibitively high. Because the ﬁrst case is technically simpler than thesecond, we show both cases separately.
To summarize, we model the game structure as follows.
1. Each downstream ﬁrm engages in product R&D.
10

This setting is somewhat similar to that of Aiura and Sato (2008), who discuss the diﬀerentiation strategies

of ﬁrms in a Hotelling model.

62. Each upstream input supplier sets its wholesale price.
3. Given these prices, each ﬁrm simultaneously decides on a quantity to supply.

3

Analysis

3.1

No potential supplierIn this case, there is no potential competitor to the upstream ﬁrms. Upstream ﬁrm i unilaterally oﬀers its wholesale price wi (i = 1, 2) to downstream ﬁrm i.
The proﬁts of downstream ﬁrms 1 and 2 are:πD1 ≡ (1 − x1 − γx2 − w1 )x1 − F (d1 ), πD2 ≡ (1 − x2 − γx1 − w2 )x2 − F (d2 ).

(3)

The ﬁrst-order conditions lead to:x1 =2 − γ − 2w1 + γw2
2 − γ − 2w2 + γw1, x2 =.
24−γ
4 − γ2(4)

The proﬁts of the input suppliers are:πU 1 =

w1 (2 − γ − 2w1 + γw2 )w2 (2 − γ − 2w2 + γw1 ), πU 2 =.
4 − γ24 − γ2(5)

The ﬁrst-order conditions lead to:
w1 = w2 =

2−γ
.
4−γ

(6)

The equilibrium proﬁts of the downstream and the upstream ﬁrms are:
πDi =

3.2

4
(4 − γ)2 (2 + γ)2− F (di ) (i = 1, 2), πU 1 = πU 2 =2(2 − γ).
(4 − γ)2 (2 + γ)

(7)

Potential suppliers

In this case, potential suppliers can supply inputs to both downstream ﬁrms. If fi = τ di + k is
insuﬃciently large, upstream ﬁrm i faces the constraint that its wholesale price for downstreamﬁrm i cannot exceed fi because other potential suppliers can undercut its wholesale price to7

downstream ﬁrm i. Therefore, when we consider the determination of wholesale prices, wemust take into account the constraints.
The proﬁts of downstream ﬁrms 1 and 2 are:πD1 = (1 − x1 − γx2 − w1 )x1 − F (d1 ), πD2 = (1 − x2 − γx1 − w2 )x2 − F (d2 ).

(8)

The ﬁrst-order conditions lead to:x1 =2 − γ − 2w2 + γw1
2 − γ − 2w1 + γw2, x2 =.
24−γ
4 − γ2(9)The proﬁts of the input suppliers are:πU 1 =
πU 2 =

w1 (2 − γ − 2w1 + γw2 ), s.t. w1 ≤ f1 ,4 − γ2w2 (2 − γ − 2w2 + γw2 ), s.t. w2 ≤ f2 .
4 − γ2(10)(11)

Without loss of generality, we assume that f1 ≥ f2 , that is, d1 ≥ d2 . In other words, the eﬀort
of ﬁrm 1 in product R&D is more signiﬁcant than that of ﬁrm 2. The ﬁrst-order conditions
lead to:
w1 = k + τ d1 , w2 = k + τ d2 ,if k ≤ k1 ,






2 − γ + γk + γτ d2
w1 =, w2 = k + τ d2 , if k1 < k ≤ k2 ,
4



2−γ


,
if k2 < k,
 w1 = w2 =4−γ
2 − γ − τ (4d1 − γd2 )2 − γ − τ (4 − γ)d2
where k1 ≡, k2 ≡
.
4−γ
4−γ

8

(12)

The equilibrium proﬁts of the downstream and the upstream ﬁrms are:
((2 − γ)(1 − k) − 2τ d1 + γτ d2 )2− F (d1 ), πD1 =
(2 − γ)2 (2 + γ)2(1)
if k ≤ k1 ,((2 − γ)(1 − k) − 2τ d2 + γτ d1 )2 πD2 =− F (d2 ),(2 − γ)2 (2 + γ)2(2 − γ + γk + γτ d2 )2− F (d1 ), πD1 =
4(2 − γ)2 (2 + γ)2(2)
if k1 < k ≤ k2 , (13)2 )(1 − k) − 2γ − (8 − γ 2 )τ d )2
((8
−
γ

2

 πD2 =− F (d2 ),16(2 − γ)2 (2 + γ)24
− F (d1 ), πD1 =
2(4 − γ) (2 + γ)2(3)
if k2 < k.
4
 πD2 =−
F(d),
2
(4 − γ)2 (2 + γ)2(k + τ d1 )((2 − γ)(1 − k) − 2τ d1 + γτ d2 ), πU 1 =(2 − γ)(2 + γ)
if k ≤ k1 ,(1)
(k + τ d2 )((2 − γ)(1 − k) − 2τ d2 + γτ d1 ), πU 2 =(2 − γ)(2 + γ)
(2 − γ + γk + γτ d2 ), πU 1 =8(2 − γ)(2 + γ)(2)
if k1 < k ≤ k2 , (14)(k + τ d2 )((8 − γ 2 )(1 − k) − 2γ − (8 − γ 2 )τ d2 ), πU 2 =4(2 − γ)(2 + γ)2(2 − γ), πU 1 =(4 − γ)2 (2 + γ)(3)
if k2 < k.
2(2 − γ), πU 2 =(4 − γ)2 (2 + γ)

4Endogenous product diﬀerentiation

We now discuss how each downstream ﬁrm determines its products positions.

4.1

No investment cost (F = 0)

We now suppose that F = 0 for any di (i = 1, 2). Each downstream ﬁrm freely sets γ =1 − d1 − d2 to maximize its own proﬁt. We consider two cases: that with no competing
potential supplier and that with potential suppliers.

94.1.1

No potential supplierDiﬀerentiating πDi with respect to di , we have:
∂πD1
∂πD2
16(d1 + d2 )
=
=
≥ 0.
∂d1
∂d2
(3 − d1 − d2 )3 (3 + d1 + d2 )3(15)

Therefore, each downstream ﬁrm chooses di = 1/2 (i = 1, 2).
Proposition 1 When there is no potential competitor for either upstream ﬁrm, if F = 0 for
any di (i = 1, 2), d1 = d2 = 1/2 in equilibrium.
4.1.2

Potential suppliers

To clarify the analysis, we now restrict the choices of the downstream ﬁrms. We tentatively
assume that each downstream ﬁrm has only two options, di = 0 and di = 1/2 (i = 1, 2).
Because of the simpliﬁcation, the game is reduced to a simple 2×2 matrix. In the main text,
we only solve the 2 × 2 matrix. Fortunately, we can show that the equilibrium outcomes
derived by the 2 × 2 matrix really appear as equilibrium outcomes in the game in which
each downstream ﬁrm freely sets di ∈ [0, 1/2] (i = 1, 2).11 Roughly speaking, in this setting,the proﬁt functions of the downstream ﬁrms are convex with respect to di . Therefore, theboundaries, di = 0 and di = 1/2, may be the best candidates.
There are three cases, depending on the endogenous and exogenous variables: (1) k ≤ k1 ,(2) k1 < k ≤ k2 , and (3) k2 < k (see equation (12)). When each ﬁrm chooses di = 0(i = 1, 2), that is, γ = 1, the ﬁrst case appears if k ≤ 1/3 and the third case appears if
1/3 < k. When one downstream ﬁrm chooses di = 0 and another chooses dj = 1/2 (i = 1, 2,i ̸= j), that is, γ = 1/2, the ﬁrst case appears if k ≤ (3 − 4τ )/7, the second case appears if(3 − 4τ )/7 < k ≤ 3/7, and the third case appears if 3/7 < k. When each ﬁrm chooses di = 1/2(i = 1, 2), that is, γ = 0, the ﬁrst case appears if k ≤ (1 − τ )/2 and the third case appears if(1 − τ )/2 < k. We summarize the discussion in the following ﬁgure.
11

The calculus is highly complex but does not provide any economic information. The calculus is available

from the authors on request.

10

*************************
Figure 1 here*************************In Figure 1, (1), (2), and (3) refer to the equation number in equation (13) that applies to eachregion. For instance, suppose that the values of k and τ are in region (e). When d1 = d2 = 0,
the third proﬁt function (3) in equation (13) applies; when di = 0 and dj = 1/2, the second
proﬁt function (2) in equation (13) applies; when d1 = d2 = 1/2, the ﬁrst proﬁt function (1) inequation (13) applies. The payoﬀ matrix is described in Table 1 (note that, γ = 1 − d1 − d2 ).
*************************
Table 1 here*************************In region (e), both ﬁrms choose di = 1/2 (i = 1, 2) in equilibrium if k > 15τ − 3, otherwise
a ﬁrm chooses di = 0 and another ﬁrm chooses dj = 1/2 (i = 1, 2, i ̸= j) in equilibrium. We
can apply this argument to all eight regions. Solving the payoﬀ matrices for the eight regions,
we have Proposition 2.
Proposition 2 When there are potential competitors for the upstream ﬁrms, if F = 0 for
any di (i = 1, 2), depending on the exogenous values (k and τ ), the following outcome appears
in equilibrium:

d1 = d2 = 1/2 k ≤ 33/109 and k ≤ (6 − 19τ )/6 or
33/109 ≤ k ≤ 12/31 and k ≥ 15τ − 3 orif
12/31 ≤ k,

d1 = 0 d2 = 1/2 if 1/3 ≤ k ≤ 12/31 and k ≤ 15τ − 3,d1 = d2 = 0

if k ≤ 1/3 and k ≥ 1 − 4τ.

Proof: See Appendix.
The result is summarized in Figure 2.

11

(16)

*************************
Figure 2 here*************************
We now mention the intuition behind Proposition 2. In this setting, when a downstreamﬁrm enhances its degree of product diﬀerentiation (increases in di ), it has a beneﬁt and acost. The beneﬁt stems from the mitigation of competition between the downstream ﬁrms.
The cost stems from the increase in its procurement cost if the competitive force of potential
suppliers (their marginal costs are k + τ di ) is eﬀective.
When τ is small, because the increase in di is less eﬀective in increasing the procurement
cost (k + τ di ), both downstream ﬁrms enhance the degree of product diﬀerentiation (see theleft side in Figure 2).
When k is small (k ≤ 1/3), there can be two equilibria if τ is in the medium range((1 − k)/4 ≤ τ ≤ 6(1 − k)/19 if k < 33/109, (1 − k)/4 ≤ τ ≤ (k + 3)/15 if 33/109 ≤ k < 1/3).
That is, strategic complementarity concerning the strategies of product diﬀerentiation exists(see the lower central area in Figure 2). When a downstream ﬁrm engages in product R&D,
this investment mitigates the competition between the downstream ﬁrms and enhances the
proﬁts of both. The increase in the proﬁt of the downstream ﬁrm that does not engage inproduct R&D provides greater incentive to do so. On the other hand, when a downstreamﬁrm does not engage in product R&D, the proﬁt levels of both downstream ﬁrms are notlarge. Because of the limited proﬁtability, product R&D undertaken by a downstream ﬁrm
does not compensate for the increase in its procurement cost k + τ di . Therefore, if τ is in themedium range, two equilibrium outcomes can appear.
If k is small and τ is suﬃciently large, no downstream ﬁrm engages in product R&D
because the increase in the procurement cost (k + τ di ) is signiﬁcant (see the lower right sideof Figure 2).
When k is large, another interesting property can appear. Before we mention this property,
12we explain the relationship between wholesale prices and the value of k. In this model, k = 1/3
is a critical value concerning the strategies of product diﬀerentiation. If k is larger than 1/3,
potential suppliers do not function as a constraint on upstream ﬁrms when downstream ﬁrms
do not engage in product R&D (that is, d1 = d2 = 0 and γ = 1 − d1 − d2 = 1). From
the discussion of Section 3.1, when each upstream ﬁrm monopolistically supplies input, thewholesale price is wi = (2 − γ)/(4 − γ) = 1/3, which is smaller than k if no downstream ﬁrm
engages in product R&D (d1 = d2 = 0).
When k is larger than 1/3 but is insuﬃciently large, if a downstream ﬁrm engages in
product R&D, the price set by the upstream ﬁrm that supplies to another downstream ﬁrm
becomes binding because the increase in the degree of product diﬀerentiation enhances the
incentive for the upstream ﬁrms to set their prices at higher levels. This is because wi in(6) is decreasing in γ. Because the price of the upstream ﬁrm that supplies the nonengaging
downstream ﬁrm (call it ﬁrm i) is binding, the wholesale price for ﬁrm i is positively correlatedwith the value of di . This feature of the wholesale price diminishes the incentive of ﬁrm i toengage in product R&D. Therefore, if k is in the medium range (1/3 < k < 12/31) and thevalue of τ is large (τ > (k + 3)/15), only one downstream ﬁrm engages in product R&D (seethe upper right side of Figure 2). In this case, the proﬁt of the engaging downstream ﬁrm issmaller than that of the nonengaging downstream ﬁrm.

4.2

Positive investment cost

From the discussion in the previous subsection, one may conclude that the degree of productdiﬀerentiation is large even when k is large. We now show that this conclusion is not always
true.
Following the discussion of Lin and Saggi (2002), we now suppose that each downstreamﬁrm has to incur an investment cost for its product R&D. We now specify the investment cost
as F (di ) = cd2i /2. The marginal investment cost is F ′ (di ) = cdi .

13

4.2.1

No potential supplier

Substituting γ = 1 − (d1 + d2 ) into πD1 and πD2 , we have:πDi =4(3 + d1 + d2 )2 (3 − d1 − d2 )2− cd2i /2, (i = 1, 2).

(17)

Diﬀerentiating πDi with respect to di , we have:∂πDi
16(d1 + d2 )=− cdi .
∂di
(3 + d1 + d2 )3 (3 − d1 − d2 )3(18)

We now provide three equilibrium outcomes that include minimum diﬀerentiation. Given
that d2 = 0, the marginal beneﬁt from the investment (the partial derivative of ΠD1 ≡
πD1 + F (d1 )) is zero when d1 = 0.
*************************
Figure 3 here*************************
If c is large enough, ﬁrm 1 also chooses d1 = 0. We have the following proposition:
Proposition 3 Suppose that F (di ) = cd2i /2 for any di ∈ [0, 1/2] (i = 1, 2). d1 = d2 = 0 is
an equilibrium outcome if c > 2272/99225 ≃ 0.02230. The proﬁts of the downstream and the
upstream ﬁrms are:
πDi =

4
2
, πU i = .
81
27

(19)

Proof: Given that d2 = 0, diﬀerentiating πD1 with respect to d1 , we have,¯
d1 (16 − c(3 − d1 )3 (3 + d1 )3 )∂πD1 ¯¯=.
∂d1 ¯d2 =0(3 − d1 )3 (3 + d1 )3(20)

Because (3 − d1 )3 (3 + d1 )3 is decreasing in d1 ∈ [0, 1/2], the partial derivative is (i) always
positive or (ii) negative (resp. positive) if d1 is small (resp. large) or (iii) always negative.
Therefore, the net proﬁt of ﬁrm 1 is maximized when d1 = 0 or d1 = 1/2. If d1 = 0 maximizes
14the net proﬁt, d1 = d2 = 0 is an equilibrium outcome. When d1 = 0, the net proﬁt is 4/81.
When d1 = 1/2, it is (512 − 1225c)/9800. A simple calculation (4/81 − (512 − 1225c)/9800)
leads to this proposition.

Q.E.D.

Given that d2 = 1/2, the marginal beneﬁt from ﬁrm 1’s investment is always positive (seeFigure 4).
*************************
Figure 4 here*************************
If c is small enough, ﬁrm 1 also chooses d1 = 1/2. We have the following proposition:
Proposition 4 Suppose that F (di ) = cd2i /2 for any di ∈ [0, 1/2] (i = 1, 2). d1 = d2 = 1/2 isan equilibrium outcome if c < 1/16 ≃ 0.0625. The proﬁts of the downstream and the upstreamﬁrms are:
πDi =

c
1
1
− , πU i = .
16 8
8

(21)

Proof: Given that d2 = 1/2, the partial derivative of ΠD1 is¯∂ΠD1 ¯¯
512(2d1 + 1)=.
¯
∂d1 d2 =1/2 (7 + 2d1 )3 (5 − 2d1 )3(22)

We now consider two functions.
y1 =

512(2x + 1)
, y2 = cx.
(7 + 2x)3 (5 − 2x)3

On the domain [0, 1/2], there is only one intersection of y1 and y2 if and only if c ≥ 1/16
otherwise there is no intersection. In other words, for any d1 ≤ 1/2, the partial derivative of
ΠD1 is larger than the marginal investment cost if and only if c < 1/16. Therefore, d1 = 1/2
is the optimal choice of ﬁrm 1 if c ≤ 1/16.

Q.E.D.

The downstream ﬁrm’s proﬁt in which d1 = d2 = 1/2 is larger than that in which d1 =d2 = 0 if and only if c < 17/162. When 1/16 < c < 17/162, maximum diﬀerentiation does
not appear in equilibrium even though it is more proﬁtable for the downstream ﬁrms.
15

For intermediate values of c, there can be another symmetric equilibrium. We now supposethat d1 = d2 = x. The ﬁrst-order condition of each ﬁrm issµ ¶1/332x
1
4e− cx = 0 → x ≡
9−2
.
3
3
(3 + 2x) (3 − 2x)
2
c

(23)

We can show that given dj = xe , the optimal choice of ﬁrm i is di = dj = xe . In this case,the proﬁts of the downstream and upstream ﬁrms are (see Figure 5):πDi =4c(xe )22(1 + 2xe )
−
,
π
=
.
U
i
(3 + 2xe )2 (3 − 2xe )22(3 + 2xe )2 (3 − 2xe )(24)*************************
Figure 5 here*************************
Figure 6 provides an example in which multiple equilibria appear when c is an intermediate
value. In Figure 6, given that dj = 1/10, the optimal choice of ﬁrm i is also di = 1/10.
Moreover, given that dj = 1/2, the optimal choice of ﬁrm i is also di = 1/2. Both cases canappear as equilibrium outcomes. xe has an interesting property. As the coeﬃcient of themarginal investment cost increases, xe increases.
*************************
Figure 6 here*************************
Summary

The above discussion is summarized in Figure 7.
*************************
Figure 7 here*************************

This result is in contrast with that in Brekke and Straume (2004). Using a Hotelling location
model with upstream units, they show that the existence of those upstream units enhances
16the degree of product diﬀerentiation.12 In our setting, however, upstream units may diminish
the degree of product diﬀerentiation. To prove that the statement is true, we reconsider thesetting of Lin and Saggi (2002) (there is no upstream unit). In their model, the proﬁt ofeach (downstream) ﬁrm and the marginal gain of product innovation are (we assume that themarginal cost is h and the demand functions are the same ones discussed here):πi =(1 − h)2∂πi2(1 − h)2,=
, i ̸= j.
(3 − di − dj )2 ∂di(3 − di − dj )3(25)

Because the marginal gain from the product innovation is positive for any dj ∈ [0, 1/2], theequilibrium investment levels must be positive for any c (note that the marginal cost of theinvestment is cdi ). Therefore, we ﬁnd that if c is large, the existence of upstream units
diminishes the degree of product diﬀerentiation.
4.2.2

Potential suppliers

In this part, our main focus is the comparison between the case in which each upstream ﬁrmmonopolistically supplies its input, and that in which it faces potential input suppliers. We
only discuss region (a) in the previous subsection. That is, for any di ∈ [0, 1] the ﬁrst equationin equation (13) applies. Substituting γ = 1 − (d1 + d2 ) into πD1 in equation (13), we have:πD1 =((1 + d1 + d2 )(1 − k) − (2d1 − d2 + d1 d2 + d22 )τ )2− cd21 /2.
(3 − d1 − d2 )2 (1 + d1 + d2 )2(26)

Diﬀerentiating πDi with respect to di , we have:∂πD12((1 + d1 + d2 )(1 − k) − (2d1 − d2 + d1 d2 + d22 )τ )=∂d1
(3 − d1 − d2 )3 (1 + d1 + d2 )3(27)×((1 + d1 + d2 )2 (1 − k) − (6 + 2d21 + (9 − 2d1 + d21 )d2 − 2(2 − d1 )d22 + d32 )τ ) − cd1 .

12Note that, in their model, there is no investment cost for product innovation. Even though we introduce

the product innovation costs discussed here into their model, the main result of their paper holds. That is,
upstream units enhance the degree of product diﬀerentiation in Brekke and Straume (2004).

17

Given that d2 = 0, if the partial derivative is strictly positive, then the optimal value of d1 is
positive. Substituting d2 = 0 into the partial derivative, we have:¯∂πD1 ¯¯2((1 + d1 )(1 − k) − 2d1 τ )=¯∂d1 d2 =027
× ((1 − k + 2τ )d21 + 2(1 − k)d1 + (1 − k − 6τ )) − cd1 . (28)
If k + 6τ < 1, when d1 = 0, this is positive for any c(> 0). From the discussion, we have thefollowing proposition:
Proposition 5 If k + 6τ < 1, the value of γ is lower than 1 for any c.
The small values of k and τ mean that upstream market competition is intense. In other words,
if the procurement conditions for the downstream ﬁrms are preferable, the downstream ﬁrms
tend to diﬀerentiate their products.

5

Concluding remarks

Using a simple product diﬀerentiation model with elastic demands, we investigate the relationship between diﬀerentiation strategies and vertical relations. Depending on the competitive
structure in the upstream market, three diﬀerentiation patterns (maximum, minimum, andpartial) can appear in equilibrium even though each downstream ﬁrm freely determines thedegree of product diﬀerentiation. These results are somewhat diﬀerent from those of previous
studies (Brekke and Straume (2004) and Matsushima (2004, 2008)) that discuss how upstream
units aﬀect the diﬀerentiation strategies of downstream ﬁrms. Our model provides additional
insights into the relationship between product diﬀerentiation and vertical relations.
We only consider the quantity-setting model. If we employed a price-setting model, thecompetition structure in the downstream market would be quite diﬀerent. Although a pricesetting model is more complex than the quantity-setting model employed in this paper, investigating a price-setting model may be considered for future research.

18

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22

Table 1: An example of a payoﬀ matrix
d2 = 0d1 = 04/81
4/81(27 − 31k)2 /3600

d1 = 1/2(3 + k)2 /225

23

d2 = 1/2(3 + k)2 /225
(27 − 31k)2 /3600(2 − 2k − τ )2 /16(2 − 2k − τ )2 /16

k
1/2
3/7
1/3

(g)

(h)

(d)

(e)

(f)
(c)
(b)

(a)

0

τ

Region
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)

di = 0
(1)
(1)
(1)
(3)
(3)
(3)
(3)
(3)

di = 0, dj = 1/2
(1)
(2)
(2)
(1)
(2)
(2)
(3)
(3)

di = 1/2
(1)
(1)
(3)
(1)
(1)
(3)
(1)
(3)

1/2

Figure 1: The relationship between k, τ , and the proﬁt functions in Eq. (13).

24

k
0.5

0.4

H12,12L
H12,0L

0.3

0.2

H0,0L

H0,0L
H12,12L

0.1

0.1

0.2

0.3

0.4

0.5

Figure 2: The equilibrium outcome.

25Τ∂ΠDi /∂di
0.01

0.005

di
0

0.5

Figure 3: The marginal beneﬁt from ﬁrm 1’s investment (d2 = 0).

26

∂ΠD1 /∂di
0.03

0.015

di
0

0.5

Figure 4: The partial derivative of πD1 (d2 = 1/2).

27

ǸU

ǸU

0.056

ǸD(di = 0.5)

0.12
ǸU (di = xe )

0.11
0.054

ǸD(di = xe )

0.10

0.052
0.050

ǸU(di = 0.5)

0.09
0.08
ǸD (di = 0)ǸU (di = 0)

c

0.045

0.050

0.055

c

0.060

0.045

(downstream ﬁrm)

0.050

0.055

0.060

(upstream ﬁrm)

Figure 5: The proﬁts of the downstream and the upstream ﬁrms.

28

∂ΠDi /∂di , cdi
0.03∂ΠDi /∂di (d2 = 0.5)
cdi

0.015

∂ΠDi /∂di (d2 = 0.1)
0

di

0.5

0.1

Figure 6: Complementarity between the choices of di (c = 15625/351232).

29

di
0.5

c
0

2272
99225

32
729

1
16

Figure 7: Multiple equilibria

30Technical appendix
Proof of Proposition 2

We must verify the equilibrium outcomes in the eight regions inFigure 1.
Region (a): The values of k and τ are in region (a). For any di (i = 1, 2), the ﬁrst proﬁtfunction (1) in equation (13) applies. The payoﬀ matrix is described as follows (note that,γ = 1 − d1 − d2 and we assume that F (di ) = 0 in this section):d2 = 0d2 = 1/2(1 − k)2

d1 = 09(1 − k)29d1 = 1/2
4(3 − 3k − 2τ )2225(6 − 6k + τ )2
225

(6 − 6k + τ )2225

(2 − 2k − τ )216

4(3 − 3k − 2τ )2225(2 − 2k − τ )2
16

Given that d2 = 0, ﬁrm 1 also chooses d1 = 0 if
(1 − k)2 4(3 − 3k − 2τ )2(k − (1 − 4τ ))(11 − 11k − 4τ )
−
=
≥ 0.
9
225
225
We ﬁnd that both ﬁrms choose di = 0 (i = 1, 2) in equilibrium if k ≥ 1 − 4τ . Given thatd2 = 1/2, ﬁrm 1 also chooses d1 = 1/2 if(6 − 19τ − 6k)(54 − 54k − 11τ )(2 − 2k − τ )2 (6 − 6k + τ )2
−
=
≥ 0.
16
225
3600
We ﬁnd that both ﬁrms choose di = 1/2 (i = 1, 2) in equilibrium if k ≤ (6 − 19τ )/6. From thediscussion, we ﬁnd that no case in which only one ﬁrm chooses di = 0 appears in equilibrium
because there is no k that satisﬁes both k ≤ 1 − 4τ and k ≥ (6 − 19τ )/6.
Region (b): The values of k and τ are in region (b). When d1 = d2 = 0 or d1 = d2 = 1/2,
the ﬁrst proﬁt function (1) in equation (13) applies. When di = 0 and dj = 1/2 (i = 1, 2,i ̸= j), the second proﬁt function (2) in equation (13) applies. The payoﬀ matrix is describedas follows (note that, γ = 1 − d1 − d2 and we assume that F (di ) = 0 in this section):31

d2 = 0d2 = 1/2(1 − k)2

d1 = 09(1 − k)29d1 = 1/2(3 + k)2225(3 + k)2
225(27 − 31k)2
3600(27 − 31k)23600(2 − 2k − τ )216

(2 − 2k − τ )2
16

Given that d2 = 0, ﬁrm 1 also chooses d1 = 0 if
8(2 − k)(1 − 3k)(1 − k)2 (3 + k)2
−
=
≥ 0.
9
225
225
We ﬁnd that both ﬁrms choose di = 0 (i = 1, 2) in equilibrium for any k and τ . Given thatd2 = 1/2, ﬁrm 1 also chooses d1 = 1/2 if(2 − 2k − τ )2 (27 − 31k)2(3 − 15τ + k)(57 − 61k − 15τ )
−
=
≥ 0.
16
3600
3600
We ﬁnd that both ﬁrms choose di = 1/2 (i = 1, 2) in equilibrium if k ≥ 15τ − 3. Given thatd2 = 0, ﬁrm 1 chooses d1 = 1/2 if(1 − k)2 (3 + k)28(2 − k)(1 − 3k)
−
=
≤0
9
225
225
(2 − 2k − τ )2 (27 − 31k)2(3 − 15τ + k)(57 − 61k − 15τ )
−
=
≤ 0.
16
3600
3600
We ﬁnd that only one ﬁrm chooses di = 0 in equilibrium if k = 1/3 and τ ≥ 2/9.
Region (c): The values of k and τ are in region (c). When d1 = d2 = 0, the ﬁrst proﬁtfunction (1) in equation (13) applies. When di = 0 and dj = 1/2 (i = 1, 2, i ̸= j), the second
proﬁt function (2) in equation (13) applies. When d1 = d2 = 1/2, the third proﬁt function (3)in equation (13) applies. The payoﬀ matrix is described as follows (note that, γ = 1 − d1 − d2and we assume that F (di ) = 0 in this section):32

d2 = 0d2 = 1/2(1 − k)2

d1 = 09(1 − k)29d1 = 1/2(3 + k)2
225(27 − 31k)2
3600(27 − 31k)23600(3 + k)2
225

1
16

1
16

Given that d2 = 0, ﬁrm 1 also chooses d1 = 0 if
8(2 − k)(1 − 3k)(1 − k)2 (3 + k)2
−
=
≥ 0.
9
225
225
We ﬁnd that both ﬁrms choose di = 0 (i = 1, 2) in equilibrium for any k and τ . Given thatd2 = 1/2, ﬁrm 1 also chooses d1 = 1/2 if
1(27 − 31k)2(42 − 31k)(12 − 31k)
−
=−
≥ 0.
16
3600
3600
We ﬁnd that no case in which both ﬁrms choose di = 1/2 (i = 1, 2) appears in equilibrium.
Given that d2 = 0, ﬁrm 1 chooses d1 = 1/2 if(1 − k)2 (3 + k)28(2 − k)(1 − 3k)
−
=
≤ 0,
9
225
225
1
(27 − 31k)2(42 − 31k)(12 − 31k)
−
=−
≤ 0.
16
3600
3600
We ﬁnd that only one ﬁrm chooses di = 0 in equilibrium if k = 1/3.
Region (d): The values of k and τ are in region (d). When d1 = d2 = 0, the third proﬁtfunction (3) in equation (13) applies. In the remaining cases, the ﬁrst proﬁt function (1) inequation (13) applies. The payoﬀ matrix is described as follows (note that, γ = 1 − d1 − d2and we assume that F (di ) = 0 in this section):33

d2 = 0d1 = 04
81
d1 = 1/2
4(3 − 3k − 2τ )2225

d2 = 1/2
481(6 − 6k + τ )2
225

(6 − 6k + τ )2225

(2 − 2k − τ )216

4(3 − 3k − 2τ )2225(2 − 2k − τ )2
16

Given that d2 = 0, ﬁrm 1 also chooses d1 = 0 if
4(3 − 3k − 2τ )24(14 − 9k − 6τ )(4 − 9k − 6τ )
4
−
=−
≥ 0.
81
225
2025
Because k ≤ (3 − 4τ )/7 in this region, the above inequality does not hold. We ﬁnd that nocase in which both ﬁrms choose di = 0 (i = 1, 2) appears in equilibrium. Given that d2 = 1/2,
ﬁrm 1 also chooses d1 = 1/2 if(2 − 2k − τ )2 (6 − 6k + τ )2
(6 − 19τ − 6k)(54 − 54k − 11τ )
−
=
≥ 0.
16
225
3600
We ﬁnd that both ﬁrms choose di = 1/2 (i = 1, 2) in equilibrium for any k and τ . Given thatd2 = 0, ﬁrm 1 chooses d1 = 1/2 if
4(3 − 3k − 2τ )24(14 − 9k − 6τ )(4 − 9k − 6τ )
4
−
=−
≤ 0,
81
225
2025
(2 − 2k − τ )2 (6 − 6k + τ )2
(6 − 19τ − 6k)(54 − 54k − 11τ )
−
=
≤ 0.
16
225
3600
We ﬁnd that no case in which only one ﬁrm chooses di = 0 appears in equilibrium.
Region (e): The values of k and τ are in region (e). When d1 = d2 = 0, the third proﬁtfunction (3) in equation (13) applies. When di = 0 and dj = 1/2 (i = 1, 2, i ̸= j), the second
proﬁt function (2) in equation (13) applies. When d1 = d2 = 1/2, the ﬁrst proﬁt function (1)in equation (13) applies. The payoﬀ matrix is described as follows (note that, γ = 1 − d1 − d2and we assume that F (di ) = 0 in this section):34

d2 = 0d1 = 04
81
d1 = 1/2(3 + k)2
225

d2 = 1/2
4
81(27 − 31k)2
3600(27 − 31k)23600(2 − 2k − τ )216

(3 + k)2225(2 − 2k − τ )2
16

Given that d2 = 0, ﬁrm 1 also chooses d1 = 0 if
(3 + k)2(3k − 1)(19 + 3k)
4
−
=−
≥ 0.
81
225
2025
We ﬁnd that both ﬁrms choose di = 0 (i = 1, 2) in equilibrium if k = 1/3. Given thatd2 = 1/2, ﬁrm 1 also chooses d1 = 1/2 if(2 − 2k − τ )2 (27 − 31k)2(3 − 15τ + k)(57 − 61k − 15τ )
−
=
≥ 0.
16
3600
3600
We ﬁnd that both ﬁrms choose di = 1/2 (i = 1, 2) in equilibrium if k ≥ 15τ − 3. Given thatd2 = 0, ﬁrm 1 chooses d1 = 1/2 if
4(3 + k)2(3k − 1)(19 + 3k)
−
=−
≤ 0,
81
225
2025
(2 − 2k − τ )2 (27 − 31k)2(3 − 15τ + k)(57 − 61k − 15τ )
−
=
≤ 0.
16
3600
3600
We ﬁnd that only one ﬁrm chooses di = 0 in equilibrium if k ≤ 15τ − 3.
Region (f ): The values of k and τ are on region (f). When d1 = d2 = 0 or d1 = d2 = 1/2,
the third proﬁt function (3) in equation (13) applies. When di = 0 and dj = 1/2 (i = 1, 2,i ̸= j), the second proﬁt function (2) in equation (13) applies. The payoﬀ matrix is describedas follows (note that, γ = 1 − d1 − d2 and we assume that F (di ) = 0 in this section):35

d2 = 04
81

d1 = 04
81
d1 = 1/2(3 + k)2
225

d2 = 1/2(27 − 31k)2
3600(27 − 31k)2
3600

1
16

(3 + k)2
225

1
16

Given that d2 = 0, ﬁrm 1 also chooses d1 = 0 if
(3 + k)2(3k − 1)(19 + 3k)
4
−
=−
≥ 0.
81
225
2025
We ﬁnd that both ﬁrms choose di = 0 (i = 1, 2) in equilibrium if k = 1/3. Given thatd2 = 1/2, ﬁrm 1 also chooses d1 = 1/2 if
1(27 − 31k)2(42 − 31k)(31k − 12)
−
=
≥ 0.
16
3600
3600
We ﬁnd that both ﬁrms choose di = 1/2 (i = 1, 2) in equilibrium if k ≥ 12/31. Given thatd2 = 0, ﬁrm 1 chooses d1 = 1/2 if
4(3 + k)2(3k − 1)(19 + 3k)
−
=−
≤ 0,
81
225
2025
1
(27 − 31k)2(42 − 31k)(31k − 12)
−
=
≤ 0.
16
3600
3600
We ﬁnd that only one ﬁrm chooses di = 0 in equilibrium if 1/3 ≤ k ≤ 12/31.
Region (g): The values of k and τ are in region (g). When d1 = d2 = 0 or di = 0 and
dj = 1/2 (i = 1, 2, i ̸= j), the third proﬁt function (3) in equation (13) applies. Whend1 = d2 = 1/2, the ﬁrst proﬁt function (1) in equation (13) applies. The payoﬀ matrix is
described as follows (note that, γ = 1 − d1 − d2 and we assume that F (di ) = 0 in this section):36

d2 = 0d1 = 04
81
d1 = 1/2641225

d2 = 1/2

4
81

64
1225

64
1225

(2 − 2k − τ )2
16

64
1225
(2 − 2k − τ )2
16

Given that d2 = 0, ﬁrm 1 also chooses d1 = 0 if
4
64
284
−
=−
≥ 0.
81 122599225
We ﬁnd that no case where both ﬁrms choose di = 0 (i = 1, 2) appears in equilibrium. Given
that d2 = 1/2, ﬁrm 1 also chooses d1 = 1/2 if(2 − 2k − τ )2
64
(102 − 70k − 35τ )(38 − 70k − 35τ )
−
=
≥ 0.
16
1225
19600
Because k ≤ (1 − τ )/2 in this region, the above inequality always holds. We ﬁnd that bothﬁrms choose di = 1/2 (i = 1, 2) in equilibrium for any τ and k. Given that d2 = 0, ﬁrm 1chooses d1 = 1/2 if
4
64
284
−
=−
≤ 0,
81 122599225(2 − 2k − τ )2
64
(102 − 70k − 35τ )(38 − 70k − 35τ )
−
=
≤ 0.
16
1225
19600
We ﬁnd that no case where only one ﬁrm chooses di = 0 appears in equilibrium.
Region (h): The values of k and τ are in region (h). For any di , the third proﬁt function (3)in equation (13) applies. The payoﬀ matrix is described as follows (note that, γ = 1 − d1 − d2and we assume that F (di ) = 0 in this section):
d1 = 0d1 = 1/2

d2 = 04/81
4/81
64/1225
64/122537

d2 = 1/2
64/1225
64/1225
1/16
1/16

We ﬁnd that both ﬁrms choose di = 1/2 (i = 1, 2) in equilibrium for any k and τ . The othertwo cases do not appear in equilibrium.

38