Discussion Paper No. 1039

TECHNOLOGY SPILLOVERSAND OUTSIDE OPTIONSIN A BILATERAL DUOPOLY

Noriaki Matsushima
Laixun Zhao

August 2018

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

Technology spillovers and outside options in a bilateral duopoly
Noriaki Matsushima†∗

Laixun Zhao‡

August 6, 2018
Abstract
This paper examines the role of outside options in a downstream duopoly with
exclusive vertical relations as in the Japanese automobile industry. In our setup,the downstream firms have outside options, and two upstream firms with exclusive relations can engage in cost reducing investments. More interestingly, each
upstream firm can choose whether to voluntarily generate technology spillovers to
its rival. We show that better outside options of the downstream firms can induce
voluntary technology spillovers in the upstream level, increasing the profits of all
firms on the vertical chain.

Keywords: Strategic sourcing, Outside option, Spillovers, Vertical relations
JEL Codes: L13, O32, M11, C72∗

This paper is a part of the paper circulated under the title “Strategic dual sourcing as a driverfor free revealing of innovation” (2015, ISER Discussion Paper No. 936). We divide the discussion
paper into the two papers. This paper consists of Sections 1, 3, 4, and 5 in the discussion paper anda new explanation for the related real-world case. Another one, Matsushima and Zhao (2018), consistsof Section 2 in the discussion paper and a new introduction. We are grateful to Eric Avenel, Carmen
Beviá-Baeza, Yongmin Chen, Jay Pil Choi, Stefano Colombo, Wolfgang Gick, Junichiro Ishida, Atsushi
Kajii, Inés Macho-Stadler, Masaki Nakabayashi, Ryoji Ohdoi, David Pérez-Castrillo, Joel Sandonı́s-Dı́ez,
Tadashi Sekiguchi, Tetsuya Shimane, Takashi Shimizu, Ken-Ichi Shimomura, Zhaohui Wu and seminar
participants in APIOC2016 (Melbourne), Kobe University, MaCCI2015 (Mannheim), Osaka University,
Tokyo Institute of Technology, and Universitat d’Alacant for their helpful comments. We would also
like to thank Cong Pan and Shohei Yoshida for research assistance. The first author thanks the warm
hospitality at MOVE, Universitat Autònoma de Barcelona where part of this paper was written anda financial support from the “Strategic Young Researcher Overseas Visits Program for Accelerating
Brain Circulation” by JSPS. We gratefully acknowledge financial support from JSPS Grant-in-Aid for
Scientific Research (S) Nos. JP26220503, JP15H05728, (A) No. JP17H00984, (B) Nos. JP24330079,
JP15H03349, JP18H00847 and (C) No. JP24530248 and JP18K01593, and the program of the Joint
Usage/Research Center for ‘Behavioral Economics’ at ISER, Osaka University. The usual disclaimer
applies.
†
Corresponding author: Noriaki Matsushima, Institute of Social and Economic Research, Osaka University, Mihogaoka 6-1, Ibaraki, Osaka 567-0047, Japan. Phone: +81-6-6879-8571. E-mail:nmatsush@iser.osaka-u.ac.jp.
‡
Laixun Zhao, Research Institute of Economics and Business, Kobe University, Rokkodai 2-1, Kobe,
Hyogo 657-8501, Japan. Phone: +81-78-803-7006. E-mail: zhao@rieb.kobe-u.ac.jp.

1

Introduction

While downstream manufactures depend on the cooperation of upstream intermediate
suppliers (sometimes even with exclusive bilateral contracts), often the manufacturers
would try to induce some kind of competition among suppliers, such as encouraging
independent investments. For instance, Japanese firms bring rival parts suppliers towork together in some situations, while in other situations they keep the suppliers awayfrom each other (Asanuma, 1985).1 Such practices still exist even today, and they create
not only cooperation but also competition between suppliers within the network (Wilhelm,2012).
Under such coexistence of cooperation and competition in the upstream level, downstream manufacturers tend to have long-term relations with suppliers (see examples ofToyota and Honda in Aoki and Lennerfors (2013)). In addition, they encourage thesuppliers to engage in cost/quality improvements by using threats of their outside options/suppliers (e.g., Wu and Choi, 2005; Wilhelm, 2012). Kamath and Liker (1994)
report that Japanese automakers would invite guest engineers from rival suppliers tocompete side-by-side, to see who could come up with a better design.2 Moreover, even along-term buyer-supplier relationship does not prevent the buyer from finding alternative
suppliers to create competition (Wu and Choi, 2005).3
Given these stylized facts, in this paper, we examine how the threats of outside
options can be used strategically to influence the incentives of suppliers to engage incost-reducing or quality-upgrading investment, in a framework of downstream compe1

Sourcing strategies have been discussed by management researchers since Porter (1980), who view
multiple sourcing as a mechanism for a firm to aﬀect its bargaining power relative to both inside andoutside suppliers, which is what we rigorously model here.
2

For instance, Dyer and Nobeoka (2000) that document the actual practices in Toyota.

3

In another study, Wu et al. (2010) simulate such “co-opetition,” where competing suppliers work
together to meet the needs of the buyer.

1tition with vertical relations and endogenous outside options. We consider a bilateral
duopoly in which two downstream buyers can make eﬀorts to enrich their own outside
options (e.g., searching for other suppliers and/or keeping in-house substitutable input
production) and two upstream suppliers can engage in cost-reducing investments (e.g.,
R&D). Moreover, before they engage in their eﬀorts, each upstream firm is able to determine if it wants to generate technological spillovers to its rival upstream firm. This
market structure follows the practices in the Japanese automobile industry, and we wish
to investigate the incentives behind them, instead of considering other schemes such astechnology licensing.
We find surprisingly that each supplier has an incentive to unilaterally generate atechnology spillover to its rival for free, if its own downstream buyer can find cheap
alternative sourcing. Such a strategy causes a market-size shrink within the vertical
chain, which can hurt the downstream buyer but benefit the supplier, via dampening
the buyer’s eﬀort to enrich its outside option. We show that the unilateral spillovers
are strategic complements in the sense that it can induce the rival to also generate atechnology spillover; and the resulted cross spillovers can benefit both the buyers andthe suppliers.
Our results provide rationale for why the Japanese “suppliers’ associations” still havestrong support from both suppliers and auto makers (see Sako, 1996), even though theseassociations may at times generate outward spillovers to rivals.4 In Figure 1, we illustrate
the structure in the Japanese automobile industry. Both exclusiveness within vertical
chains and multiple memberships by very large auto-parts suppliers coexist. One might
think that the parties on an exclusive vertical chain would worry about technologicalspillovers by the very large suppliers in an association, because such spillovers may help
4

See also Aoki and Lennerfors (2013), for an analysis of the recent transformation of vertical keiretsu
in the Japanese automotive industry.

2the rival vertical chains. However, as Sako (1996) and Aoki and Lennerfors (2013)
show, the suppliers’ associations remain stable, as the logic behind our result would
indicate.5 Large suppliers would typically join in several suppliers’ associations andhold multiple memberships, aiming at obtaining information on production plans andsoliciting suggestion for common problems such as standardization and pollution control,etc., examples include Akebono Brake, Bridgestone, Denso, etc.6 Simultaneously, longterm relationships and even exclusive dealings within vertical chains are observed in
Toyota and Honda (Aoki and Lennerfors, 2013).7 In addition, as mentioned earlier, along-term buyer-supplier relationship does not prevent the buyer from finding alternative
suppliers to create competition (Wu and Choi, 2005).
[Figure 1 about here]
Our bilateral duopoly model contributes to the literature of vertical relationships.
Many researchers in the literature investigate the eﬀects of disagreement payoﬀs on
realized outcomes in buyer-supplier relations (e.g., Davidson, 1988; Horn and Wolinsky,
1988; Inderst, 2007; Milliou and Petrakis, 2007; Iozzi and Valletti, 2014, Inderst and5

Sako (1996) finds that the suppliers’ associations are eﬀective in diﬀusing innovative practices
and sharing information in the automobile industry. In the 1990s, almost all major Japanese auto
manufacturers (except Honda) have suppliers’ associations, and many suppliers join multiple associations(Sako, 1996, p.651). Note that Aoki and Lennerfors (2013, p.79) point out that Honda also establishedsuppliers’ associations similar to other automakers, although the word ‘keiretsu’ is not used.
6

Sako (1996. p.656) points out that main suppliers “employ around 3500 workers on average, andproduce brake systems, bearings, springs, spark plugs, tyres, belting and batteries. They tend to play aleading role in association activities, by taking on chairmanship and executive positions in associationcommittees and meetings.” In the main model of the present paper, for analytical simplicity, we do
not explicitly incorporate common suppliers. We expect that such an extension would not qualitatively
aﬀect our main results due to the symmetric impact of common suppliers on each vertical chain (except
the case of one-sided spillover).
7

Aoki and Lennerfors (2013) also find that the number of Toyota keiretsu members who also participate in the Nissan keiretsu is stable but that of Nissan keiretsu members who also participate in the
Toyota keiretsu increased since 2003 (see their Table 2), confirming that the findings in Sako (1996) arestill true in the current Japanese automobile industry.

3

Shaﬀer, 2018). Our model further extends the literature by incorporating upstream
firms’ cost-reducing eﬀorts with endogenous technology spillovers, and shows the abovementioned interesting results.
Note that Feng and Lu (2012) also use a similar market structure as ours to examine
bilateral duopoly, and show that a simultaneous eﬃciency improvement of both suppliers can harm the downstream firm although the total industry profits rise. However,their results are obtained based on asymmetric bargaining powers for the upstream anddownstream firms, which are not required in our model.
Several other papers investigate cases in which downstream firms engage in R&D andendogenously determine the degree of R&D spillovers.8 Kultti and Takalo (1998) andPoyago-Theotoky (1999) discuss whether downstream duopolists generate R&D spillover
after determining their investment levels, while in Kamien and Zang (2000), Gil-Moltó
et al. (2005), and Milliou (2009), downstream firms noncooperatively set the degree of
R&D spillover before setting investment levels.9 Milliou (2009) shows that oligopolists
prefer generating outward R&D spillover if the degree of product diﬀerentiation is highenough, a result that does not hold if the products are homogenous as in our paper.
Elsewhere, the literature has investigated knowledge disclosure among competingfirms.10 Pérez-Castrillo and Sandonı́s (1997) discuss know-how disclosure in a researchjoint venture although the main concern is free-riding problems.11 De Fraja (1993), Katsoulacos and Ulph (1998), and Pacheco-de-Almeida and Zemsky (2012) study environments where free revealing of knowledge can appear, in the context of continuous-time
8

De Bondt (1997) and Rockett (2012) nicely survey the literature of R&D competition.

9

In Kamien and Zang (2000), no firm has an incentive to generate spillover. In Gil-Moltó et al.
(2005), given that spillover is reciprocal, firms have incentives to generate a positive degree of spillovers.
10

In the management literature, see Harhoﬀ et al. (2003) for voluntary information spillovers andPénin (2007) for open knowledge transfer.
11

Subsequent researches further investigate the eﬀectiveness of research joint ventures and cooperative
R&D (e.g., Pastor and Sandonı́s, 2002; Gil-Moltó et al., 2005).

4

R&D competition without vertical relation. And finally, Milliou and Petrakis (2012)
show that a vertically integrated firm chooses to fully disclose its production knowledge
to the downstream rival to expand downstream production, which eventually benefits
the integrated upstream sector. Their result is less likely to hold if the degree of productdiﬀerentiation is low.12 In contrast to the above, our paper provides another rationale
behind such behavior, based on bargaining and outside options.
The present paper while simple, generates novel results that match the real practices
of many firms on vertically related production networks, such as the automobile makers
and parts suppliers documented in the literature. We explicitly demonstrate conventionally counterintuitive situations when the supplier may lose from a market expansion
and when it may choose to give out its own technology to rivals. Our mechanism has
wide applications under diﬀerent circumstances, for instance, suppliers’ incentives in cost
reduction, quality improvement, upstream collaboration and technology spillovers, andeven worker training in the labor market.
The rest of the paper proceeds as follows. Section 2 provides a model of bilateral
duopoly with the downstream firms’ outside options and cost-reducing activities by the
upstream suppliers. Section 3 shows the main results. Section 4 concludes the paper. In
the Appendix, we show several outcomes in Section 3.

2

The model

We examine a bilateral duopoly with two pairs of downstream and upstream firms,
denoted by Di and Ui respectively (i = 1, 2). To produce a unit of the final product, firm
Di needs one unit of the input produced by firm Ui but not Uj . However, Di has outside
options: it can procure the input from a diﬀerent source at the price w̃i (ei ) = w − ei
12

Yoshida (2015) shows that a downstream firm may have incentives to give superior technology to
its rival in a downstream multi-product duopoly, where product multiplicity and the existence of strong
upstream bargaining power are the key factors. The logic is quite diﬀerent from ours.

5

if the negotiation with Ui breaks down, where ei is the search eﬀort of Di , with a costof S(ei ). Simultaneously, Ui engages in cost-reducing activities such as process R&D,
through which it can reduce its marginal cost to c − Ii , where Ii is the investment of Ui
at a cost of f (Ii ). We assume c (< w) to be constant to ensure an interior solution.
Aoki and Lennerfors (2013) point out that long-term relationships and exclusivenesswithin vertical chains are observed in Toyota and Honda. In the present model, we canjustify this relationship as follows: if Ui has a long-term trading relationship with Di , itobtains a cost-advantage relative to Uj , because Ui and Di can often communicate faceto-face to improve on cost and quality. Such communication intensifies and becomes
especially important, as the product characteristic complicates. But maintaining a longterm relationship is very costly so each branch has just one such relationship.
We then take into account technological spillovers between upstream firms, i.e., thecost-reducing eﬀort by Uj spills over to Ui . To obtain clear-cut results, following Milliou(2009), here we explicitly solve the game with linear demand and specific investment
functions; p = a − q1 − q2 , S(ei ) = γe2i , and f (Ii ) = γIi2 , where qi is the quantity supplied
by Di , and a(> c) and γ are positive constants. We assume that the marginal cost of Ui
is c − Ii − rIj if Uj chooses to give its reduced cost to Ui , where r ∈ [0, 1] is the exogenous
degree of knowledge spillover.13 For notational simplicity, we define the marginal cost of
Ui as follows:{ci (Ii ; sj ) =c − Ii≡ ci (Ii )if Uj does not spill over (sj = 0),c − Ii − rIj ≡ ci (Ii , Ij ) if Uj spills over (sj = 1).

(1)

Consider the following game structure. In the first stage, each upstream firm simultaneously determines whether to unilaterally generate spillover to its upstream rival. In
the second stage, each of the four firms simultaneously sinks an investment cost that13

The formulation of spillovers follows those in the related literature on research joint ventures (seefor instance, Amir et al., 2003; Choi, 1993, d’Aspremont and Jacquemin, 1988; and Suzumura, 1992),
although those papers consider scenarios in which firms conduct joint R&D in the first stage and thencompete in the product market in the second stage.

6

determines the eﬀort level to improve its outside option (downstream firms) or to reduce
its marginal cost (upstream firms). In the third stage, observing the eﬀort levels in thesecond stage, the upstream and the downstream firms on the same vertical chain negotiate over a transfer payment (a two-part tariﬀ). The determined transfer payment isprivately known in the vertical chain, but unknown to outsiders. This assumption simplifies the analysis because the Nash equilibrium wholesale price is set at the marginal
cost of the upstream firm on each vertical chain.14 If an agreement is reached, the downstream firm procures its input from the upstream firm; otherwise, the downstream firm
exercises its outside option. For expositional simplicity, we omit the fixed cost. Finally,in the fourth stage, all downstream firms simultaneously set quantities to maximize theirown profits. The game is solved by backward induction.

3

Results

Before we proceed the analysis, we denote the gross final stage profit (excluding investments costs sunk in the second stage) on the vertical chain i as πi (ci , cj ), where ci and cj
are respectively the marginal costs on the vertical chains i and j, and Ti as the payment
from the downstream to the upstream firm when bargaining reaches an agreement. Thenπi (ci , cj ) − Ti is the gross profit of Di , excluding investment costs already sunk in thesecond stage, and Ti becomes the gross profit of Ui . Note that to obtain the net profitof each firm, the sunk cost of investment in the second stage must be subtracted fromthe above.

3.1

The third and the forth stages

From the forth-stage game, given that the marginal cost of Di is di , as in the standard
Cournot duopoly outcome, the quantity supplied by Di and the final-stage gross profit
14

Notice the diﬀerence from a case where contract terms are used as a commitment device to foster
aggressive behavior at the downstream level (e.g., Fershtman and Judd, 1987).

7

on the vertical chain i are given as respectively
qi (di , dj ) =a + dj − 2di, and πi (di , dj ) = (qi (di , dj ))2 , i, j = 1, 2, j ̸= i.
3(2)

If Di and Ui reach an agreement, di is ci (Ii ; sj ) in (1); otherwise, it is w̃i (ei ).
In the third stage, the pair Di and Ui on the same vertical chain i maximize thefollowing Nash product with respect to T :Gi = [πi (ci , cj ) − Ti − πi (w̃i , cj )]Ti ,where πi (w̃i , cj ) and 0 are respectively their gross profits if bargaining breaks down.
Optimization givesTi∗ =πi (ci , cj ) − πi (w̃i , cj ).
2Then substitution yields the net profits as respectively
πi (ci , cj ) − πi (w̃i , cj )− f (Ii ),2πi (ci , cj ) + πi (w̃i , cj )∗ΠD
− S(e).
i = πi (ci , cj ) − Ti − S(e) =2
ΠUi = Ti∗ − f (Ii ) =

(3)
(4)

Remark on equations (3) and (4) A decrease in cj reduces qi , inducing Di to lower
ei , which decreases πi (w̃i , cj ) through an increase in w̃i indirectly. Simultaneously, thedecrease in cj also directly lowers both πi (ci , cj ) and πi (w̃i , cj ).

3.2

The second stage: Spillover eﬀects

In the second stage, we consider three scenarios: (i) no upstream firm generates spillover;(ii) only one upstream firm generates spillover; (iii) both upstream firms generate spillovers.
We then compare them and see which one comes on top.

8

3.2.1

No spillovers

First, we look into the case when no upstream firm generates spillover to its rival. In
the investment stage, from (3) and (4), the objective functions are given asπi (ci (Ii ), cj (Ij )) − πi (w̃i (ei ), cj (Ij ))− f (Ii ),2πi (ci (Ii ), cj (Ij )) + πi (w̃i (ei ), cj (Ij ))ΠD− S(ei ).
i =
2
ΠUi =

(5)
(6)

The first-order conditions lead to the reaction functions:
Ii (Ij ) =α − Ijα − 2β − Ij
, ei (Ij ) =,9γ − 29γ − 2(7)whereα ≡ a − c and β ≡ w − c.
Note that ei (Ij ) in (7) includes only the investment level of the rival’s upstream firm
Uj because Di ’s eﬀort is related to its outside option, πi (w̃i (ei ), cj (Ij )). Solving thesimultaneous equations, we have the investment levels:Ii∗ =(9γ − 2)α − 2(9γ − 1)βα, e∗i =.
9γ − 1(9γ − 2)(9γ − 1)

The net profit of each firm then becomes(9γ − 2)γα2 9γ 2 ((9γ − 2)α − 2(9γ − 1)β)2−,2(9γ − 1)22(9γ − 1)2 (9γ − 2)2γ((9γ − 2)α − 2(9γ − 1)β)2
9γ 2 α2
ΠD∗
(N,
N
)
=
+
,
i
2(9γ − 1)22(9γ − 1)2 (9γ − 2)ΠUi ∗ (N, N ) =where k and l in ΠUi ∗ (k, l) and ΠD∗i (k, l) respectively represent the decisions of upstream
firms 1 and 2 for generating spillover (k, l = Y, N ), with Y and N indicating yes and no.
3.2.2

One-sided spillover

Next, consider the case in which only the reduced cost of U1 is spilled over to U2 but not
the other way around. The cost function of U2 can be rewritten as c2 (I2 , I1 ) = c−I2 −rI1
9

(see (1)). From (3) and (4), the objective functions of the firms are respectively
π1 (c1 (I1 ), c2 (I2 , I1 )) − π1 (w̃1 (e1 ), c2 (I2 , I1 ))− f (I1 ),
2
π1 (c1 (I1 ), c2 (I2 , I1 )) + π1 (w̃1 (e1 ), c2 (I2 , I1 ))ΠD− S(e1 ),1 =
2π2 (c2 (I2 , I1 ), c1 (I1 )) − π2 (w̃2 (e2 ), c1 (I1 ))− f (I2 ),ΠU2 =2
π2 (c2 (I2 , I1 ), c1 (I1 )) + π2 (w̃2 (e2 ), c1 (I1 ))ΠD− S(e2 ).
2 =
2
ΠU1 =

(8)
(9)
(10)
(11)

Then, the eﬀort level of U1 , I1 , influences the outside value of D1 , π1 (w̃1 (e1 ), c2 (I2 , I1 )),as well as the downstream profit, π1 (c1 (I1 ), c2 (I2 , I1 )). The voluntary spillover generates a strategic interaction between the endogenous eﬀort levels of U1 and D1 , throughπ1 (w̃1 (e1 ), c2 (I2 , I1 )). An increase in I1 decreases c2 , inducing D1 to lower e1 . This eﬀect
can benefit U1 especially when w̃1 (e1 ) is small because the marginal eﬀect of lowering e1
on the outside profit of D1 increases as its “eﬃciency,” w̃1 (e1 ), improves.
In the investment stage, each firm’s reaction function is obtained as respectively
α − rβ − I2 + re1,
9γ − 2(1 − r)
α − (1 − 2r)I1
I2 (I1 ) =,9γ − 2α − 2β − I2 − rI1
e1 (I2 , I1 ) =,9γ − 2α − 2β − I1e2 (I1 ) =.
9γ − 2I1 (I2 , e1 ) =

(12)
(13)
(14)
(15)

e2 (I1 ) is the same in both (7) and (15). Note that the technology spillover generates
three additional eﬀects on these reaction functions. First, the eﬀort by D1 enhances
the incentive of U1 to engage in cost reduction (see (12)), because the spillover allows
U1 to directly decrease the outside profit of D1 through its own investment. Second,the eﬀort by U1 can increase the incentive of U2 if the degree of spillover is large (see(13)), which is similar to that in the context of research joint ventures (e.g., d’Aspremontand Jacquemin, 1988). Finally, the eﬀort by U1 lowers the incentive of D1 to increase10

its outside value (see (14)), similar to the reason in the first eﬀect. Putting these all
together, the spillover generates a positive impact on the eﬃciency of the rival upstream
firm but a negative one on the downstream partner’s eﬀort to improve its outside option.
The former impact hurts the technology giver while the latter one benefits it, and whicheﬀect is bigger can be explained as follows.
Solving the simultaneous equations gives:(3γ − 1)(9γ − 2 + r)α − 3γ(9γ − 2)rβ,(9γ − 2 + r)((9γ − 1)(3γ − 1) + 3γr)
(3γ − 1 + r)(9γ − 2 + r)α + 3γ(1 − 2r)rβ=,(9γ − 2 + r)((9γ − 1)(3γ − 1) + 3γr)
(3γ − 1)(9γ − 2 + r)α − (2(3γ − 1)(9γ − 1) + (12γ − 1)r − 3γr2 )β=,(9γ − 2 + r)((9γ − 1)(3γ − 1) + 3γr)((3γ − 1)(9γ − 2) + 3γr)(9γ − 2 + r)α=(9γ − 2)(9γ − 2 + r)((9γ − 1)(3γ − 1) + 3γr)(2(3γ − 1)(9γ − 2)(9γ − 1) + (2 − 30γ + 81γ 2 )r + 6γr2 )β−.
(9γ − 2)(9γ − 2 + r)((9γ − 1)(3γ − 1) + 3γr)

I1∗ =
I2∗
e∗1
e∗2

Further substitution yields the firms’ equilibrium profits, which are messy algebraicallyU∗D∗(see Appendix): ΠU1 ∗ (Y, N ), ΠD∗
1 (Y, N ), Π2 (Y, N ), and Π2 (Y, N ), given that the cost

reduction of U1 is spilled over to U2 .
Using the above, we can explicitly derive the threshold value of β, denoted as β(Y N ),at which U1 ’s profit when it voluntarily generates spillover equals that when it does not.
A simple comparison leads to:∗∗

Proposition 1 ΠU1 (Y, N ) > ΠU1 (N, N ) if and only if β < β(Y N ), where β(Y N ) is
∗

∗

∗

∗

the threshold β satisfying ΠU1 (Y, N ) = ΠU1 (N, N ). Similarly, ΠU2 (N, Y ) > ΠU2 (N, N )if and only if β < β(Y N ).
The threshold value β(Y N ) is depicted in Figure 2. U1 benefits by giving its technology to the rival for free, if the above condition is satisfied. Similarly, we can numerically

11

show that giving U1 ’s technology to U2 harms D1 .15[Figure 2 about here]
3.2.3

Two-sided spillovers

Finally, we examine the case when both U1 and U2 cross spillover, specifically, the costfunction of Ui is given by ci (Ii , Ij ) = c − Ii − rIj (i, j = 1, 2, j ̸= i, see (1)). Except thisex-post cost of the upstream firms, the timing structure of the game is similar to thecase under one-sided spillover just examined. From (3) and (4), the objective functionsof Ui and Di are given as respectively (i, j = 1, 2 and j ̸= i)(πi (ci (Ii , Ij ), cj (Ij , Ii )) − πi (w̃i (ei ), cj (Ij , Ii )))− f (Ii ),2(πi (ci (Ii , Ij ), cj (Ij , Ii )) + πi (w̃i (ei ), cj (Ij , Ii )))− S(ei ).
ΠD
i =
2
ΠUi =

(16)
(17)

Rearranging the first-order conditions, ∂ΠUi /∂Ii = 0 and ∂ΠDi /∂ei = 0, gives the following reaction functions
α − rβ − (1 − r)2 Ij + rei,
9γ − 2(1 − r)
α − 2β − rIi − Ijei (Ii , Ij ) =.
9γ − 2Ii (Ij , ei ) =

(18)
(19)

Under reciprocal spillovers, the reaction function of Ui in (18) diﬀers from that in (12).
However, the reaction function of Di in (19) is the same with that in (14), because theoutside profit does not depend on the inside transfer price, ci (Ii , Ij ).
Solving the simultaneous equations leads to(9γ − (2 − r))α − 9rγβ,
(9γ − 2)(9γ − 1) + r + (9γ − 1)r2
(9γ − (2 − r)(1 + r))α − (18γ − (2 − r)(1 + r))β=.
(9γ − 2)(9γ − 1) + r + (9γ − 1)r2

Ii∗∗ =e∗∗i
15

Notice the sequence of moves. For our results to hold, each upstream firm must choose whether togive out its technology before choosing eﬀort levels, while the size of spillovers is not so essential.

12We can explicitly solve the game and derive the equilibrium profits, which are again
messy (see Appendix): ΠUi ∗ (Y, Y ), ΠD∗i (Y, Y ). Thus here we only examine how the exogenous variables (α and r) aﬀect the equilibrium profits. First, an increase in the degreeof spillovers (r) raises firm profitability. Second, because α = a − c and β = w − c, anincrease in α reflects an increase in a, expanding the market size. As Matsushima andZhao (2018) show, an increase in α does not always improve the profitability of each
upstream firm in the absence of spillovers if r is small enough (Figure 3).
[Figure 3 about here]
Note that the left panel in Figure 3 also shows several properties: i). An increase in
the market size monotonically and significantly raises the profit of each upstream firm
when the degree of spillovers is large enough, because the outside option of Di decreases
with the degree of spillovers. ii). The spillover eﬀect is more significant on the upstream
firms than the downstream firms, caused by the former firms’ investment to lower theoutside option of the latter firms. iii). The spillover eﬀect on the rival’s eﬃciency ismagnified, when the downstream firm also has a stronger incentive to raise its own
option value (i.e., when α is large). In turn, an increase in r enhances the incentive ofeach upstream firm to engage in cost-reducing activity, raising their profits.
We can explicitly derive the threshold value of β, denoted as β(Y Y ), at which Ui ’s
profit when both upstream firms generate spillovers equals that when only Uj generates
spillover. A simple comparison gives:
Proposition 2 ΠU1 ∗ (Y, Y ) > ΠU1 ∗ (N, Y ) if and only if β < β(Y Y ), where β(Y Y ) is thethreshold β satisfying ΠU1 ∗ (Y, Y ) = ΠU1 ∗ (N, Y ). Similarly, ΠU2 ∗ (Y, Y ) > ΠU2 ∗ (Y, N ) if andonly if β < β(Y Y ).
The threshold value β(Y Y ), and the diﬀerence between β(Y Y ) and β(Y N ) are summarized in Figure 4. From these threshold values, the decisions of the upstream firms de13

pend on the exogenous values β/α and r. As β/α rises, the incentive of each downstream
firm to increase its search eﬀort becomes weaker, which in turn lowers the incentive ofeach upstream firm for technology spillovers.
[Figure 4 about here]

3.3

The first stage: Endogenous spillover eﬀects

So far we have examined either one-way or two-way but exogenous giveaway of upstream
technology. Next, we examine in the first stage, whether each upstream firm Ui unilaterally decides to generate technological spillover to its rival upstream firm Uj or not.16
Depending on the threshold values of β, we find:
Proposition 3 (i). If β < β(Y Y ), both U1 and U2 voluntarily generate spillovers;(ii). If β > β(Y N ), no upstream firm voluntarily generates spillover; and (iii). Ifβ(Y N ) < β < β(Y Y ), multiple equilibria exist.
Figure 5 shows, under high parameter values of β/α, neither upstream firm is willingto give its technology out; Under intermediate values, both firms can either generatetechnology spillovers to the rival or not at all, i.e., multiple equilibria exist; Under
low values of the same parameter ratio, each upstream firm has incentives to generate
spillover in equilibrium.
[Figure 5 about here]
The logic can be understood as follows. As emphasized in the paragraph right after(11) in Section 3.2.2, a lower β/α enhances the incentives for the downstream firms toincrease search eﬀorts, which induces each upstream firm to generate spillovers to mitigate its downstream partner’s incentive to search. As shown in Figure 6, the investment
16

It can be straightforwardly shown that Uj does not have an incentive to refuse the unilateralspillover.

14

of a spillover-generating upstream firm and that of a downstream firm are negatively
correlated with β/α.
[Figure 6 about here]
Note that investment spillover is a key reason for multiple equilibria under intermediate values of β/α. In Figure 5, the lower line indicates the threshold value ofβ/α for which Ui has incentives to generate spillovers given that U−i does not generate
spillovers. As in (13), generating spillovers increases the rival upstream’s investment because it raises the rival’s quantity, diminishing its own incentive for unilateral spillover.
However, the eﬀect is almost canceled out by the decrease in the downstream partner’s
investment, which is shown by the gentle slope of the lower line in Figure 5.
Next, the threshold value of β/α for which Ui has an incentive to generate spillover
given that U−i also generates spillover, can be reflected by the upper curve in Figure 5,
which is concave and shows the changes in the reactions of the upstream firms. Given thatits rival generates spillovers, a firm’s own spillover causes two eﬀects: (i) the rival’s freeriding on the investment and (ii) the loss of its aggressive investment through unilaterally
receiving spillovers. The free-riding eﬀect is weaker than when only one upstream firm
generates spillover (compare (18) with (13)), which increases its incentive for generating
spillover. As a result, the threshold value of β/α rises above the level when no upstream
firm generates spillover, leading to multiple equilibria. This eﬀect, while positive, ishowever partially canceled out by the latter loss which increases with the degree of
spillovers (see (13)), shifting down the upper curve of β/α for a higher r in Figure 5.
Further, the above results are obtained based on the assumption that the downstream
firms’ eﬀorts are independent from each other (i.e., ei does not influence the outside
option of Dj (i, j = 1, 2, j ̸= i)). If on the contrary, ei also improves the outside optionof Dj , then each upstream firm has a stronger incentive to generate positive spillovers,
just to mitigate the eﬀorts of the downstream firms.
15

Finally, we check whether the voluntary spillover benefits the downstream firms.
Simple calculations lead to:D∗
Proposition 4 ΠD∗i (Y, Y ) > Πi (N, N ) if and only if β < (9γ − 2)α/((1 + r)(9γ − 1)).

Note here the upper bound of β is higher than β(Y Y ), which implies the downstream
firms benefit from such voluntary spillovers from the upstream firms.

4

Conclusion

We have investigated how strategic sourcing influences profitability in buyer-suppler relationships, and especially as an alternative to overcoming trading frictions. Specifically,
we consider a bilateral duopoly with cost-reducing investments by the suppliers, and findthat each supplier has an incentive to voluntarily generate technology spillovers to its
rival if the buyer’s cost to lower the outside wholesale price is small. Such an action
causes a market size shrink within the vertical chain, which can benefit the supplier.
Our mechanism can have wide applications in various situations, such as in suppliers’
incentives in cost reduction, quality improvement, upstream collaboration, technology
spillovers and labor training, which remain as interesting topics for future research. Also,
strategic sourcing in the context of our model is theoretically similar to improving the
production technology at the downstream level and vertical integration with outside(second-best) suppliers, where the implication of our results can be applied.
Finally, we note some caveats. First, we have abstracted from commitment issues
and strategic competition among suppliers, by modelling the second source as a purelycompetitive ‘fringe’. The wholesale price set by outside fringe suppliers is negatively
correlated to the degree of frictions. Chatain and Zemsky (2011) formulate frictions as
a result of probabilistic randomness in the matching of buyers and suppliers. A lower
wholesale price in the present model (set by outside suppliers) would represent a lower

16

friction level in their terminology. Second, we have examined how buyers invest to
improve procurement conditions given that they have options to procure their inputs
from potential outside suppliers. As a future topic, it would be interesting to explicitly
investigate the buyer decisions on bi-sourcing, as discussed in Emons (1996), Shy andStenbacka (2003), Beladi and Mukherjee (2012) and Stenbacka and Tombak (2012).17

17

In the management literature, it is well established that manufacturers are willing to outsourcein the absence of suppliers’ cost advantage because outsourcing mitigates market competition (Cachonand Harker, 2002; Arya et al., 2008; Liu and Tyagi, 2011).

17

5

Appendix: The Equilibrium Profits in Section 3

The equilibrium profits under the two scenarios of spillovers in Section 3 are as follows.

One-sided spillover

The equilibrium profits of the firms are respectively,

ΠU1 ∗ (Y, N )=9γ 2 [(3γ − 1)(9γ − (2 − r))α − r(6γ − 1 − 3γr)β]22(9γ − (2 − r))2 ((3γ − 1)(9γ − 1) + 3γr)2
9γ 2 [(3γ − 1)(9γ − (2 − r))α − (2(3γ − 1)(9γ − 1) + (12γ − 1)r − 3γr2 )β]2−2(9γ − (2 − r))2 ((3γ − 1)(9γ − 1) + 3γr)2γ[(3γ − 1)(9γ − (2 − r))α − 3r(9γ − 2)γβ]2,−(9γ − (2 − r))2 ((3γ − 1)(9γ − 1) + 3γr)2ΠD∗
1 (Y, N )=γ(9γ − 2)[(3γ − 1)(9γ − (2 − r))α − (2(3γ − 1)(9γ − 1) + (12γ − 1)r − 3γr2 )β]22(9γ − (2 − r))2 ((3γ − 1)(9γ − 1) + 3γr)2
9γ 2 [(3γ − 1)(9γ − (2 − r))α − r(6γ − 1 − 3γr)β]2+,2(9γ − (2 − r))2 ((3γ − 1)(9γ − 1) + 3γr)2ΠU2 ∗ (Y, N )=γ(9γ − 2)[((3γ − 1)(9γ − 2) + 3(4γ − 1)r + r2 )α + 3(1 − 2r)rγβ]22(9γ − (2 − r))2 ((3γ − 1)(9γ − 1) + 3γr)2
9γ 2−
×2(9γ − 2)2 (9γ − (2 − r))2 ((3γ − 1)(9γ − 1) + 3γr)2{((9γ − 2)2 (3γ − 1) + (9γ − 2)(6γ − 1)r + 3γr2 )α−(2(3γ − 1)(9γ − 1)(9γ − 2) + (81γ 2 − 30γ + 2)r + 6γr2 )β}2ΠD∗
2 (Y, N )=γ
2(9γ − 2)(9γ − (2 − r))2 ((3γ − 1)(9γ − 1) + 3γr)2×{((9γ − 2)2 (3γ − 1) + (9γ − 2)(6γ − 1)r + 3γr2 )α−(2(3γ − 1)(9γ − 1)(9γ − 2) + (81γ 2 − 30γ + 2)r + 6γr2 )β+9γ 2 [((3γ − 1)(9γ − 2) + 3(4γ − 1)r + r2 )α + 3(1 − 2r)rγβ]2.
2(9γ − (2 − r))2 ((3γ − 1)(9γ − 1) + 3γr)2

18

}2

,

Two-sided spillovers

The equilibrium profits of the firms are respectively,

9γ 2 [(9γ − 2 + r + r2 )α − r(1 + r)β]22((9γ − 1)(9γ − 2) + r + (9γ − 1)r2 )2
9γ 2 [(9γ − 2 − r + r2 )α − (2(9γ − 1) − r + r2 )β]2−2((9γ − 1)(9γ − 2) + r + (9γ − 1)r2 )2γ[(9γ − 2 + r)α − 9rγβ]2−,((9γ − 1)(9γ − 2) + r + (9γ − 1)r2 )2γ(9γ − 2)[(9γ − 2 − r + r2 )α − (2(9γ − 1) − r + r2 )β]2
(Y,
Y
)
=
ΠD∗
i
2((9γ − 1)(9γ − 2) + r + (9γ − 1)r2 )2
9γ 2 [(9γ − 2 + r + r2 )α − r(1 + r)β]2+.
2((9γ − 1)(9γ − 2) + r + (9γ − 1)r2 )2ΠUi ∗ (Y, Y ) =The threshold values in Propositions 1 and 2 The threshold values in Propositions 2 and 3 are given as√α(9γ − 2)(9γ − 2 + r){((9γ − 1)(3γ − 1) + 3γr) AY N − BY N }β(Y N ) =,CY Nα(9γ − 2 + r)β(Y Y ) =CY Y√
×{((9γ − 1)(3γ − 1) + 3γr)(9γ − 2)((9γ − 1)(9γ − 2 + r2 ) + r) AY Y − BY Y },where AY N ≡ (9γ − 1)4 + 6(3γ − 1)(9γ − 1)(9γ − 2)r + 9γ(9γ − 2)r2 , BY N ≡ (3γ − 1)(9γ −1)(81γ 2 −18γ −1)+9γ(81γ 2 −30γ +2)r+27γ 2 r2 , CY N ≡ 3(9γ −1){6γ(3γ −1)(9γ −1)(9γ −2) − (6561γ 5 − 1458γ 4 − 1620γ 3 + 612γ 2 − 64γ + 2)r − 6γ(234γ 3 − 24γ + 2)r2 − 18γ 2 r3 },AY Y ≡ γ{(9γ −1)4 (9γ −2)2 γ +2γ(9γ −2)(9γ −1)(729γ 3 −81γ 2 −45γ +7)r+(−124659γ 5 +121014γ 4 − 43092γ 3 + 7092γ 2 − 543γ + 16)r2 + 4(19683γ 5 − 21141γ 4 + 8343γ 3 − 1485γ 2 +121γ −4)r3 +(13122γ 5 −14580γ 4 +5751γ 3 −1134γ 2 +118γ −4)r4 −2(13122γ 5 −14580γ 4 +6156γ 3 − 1215γ 2 + 114γ − 4)r5 + (9γ − 2)(27γ 2 − 12γ + 1)2 r6 }, BY Y ≡ γ{(9γ − 1)2 (9γ −2)3 (243γ 3 −135γ 2 +15γ +1)+(9γ −2)2 (59049γ 5 −34992γ 4 +5103γ 3 +243γ 2 −87γ +4)r +(3188646γ 7 −4782969γ 6 +3326427γ 5 −1303452γ 4 +296946γ 3 −38421γ 2 +2589γ −70)r2 −3(531441γ 7 − 1417176γ 6 + 1200663γ 5 − 494262γ 4 + 110970γ 3 − 13662γ 2 + 847γ − 20)r3 +(−531441γ 6 + 708588γ 5 − 336798γ 4 + 73143γ 3 − 6966γ 2 + 147γ + 10)r4 + (−177147γ 6 +295245γ 5 −174960γ 4 +52974γ 3 −8883γ 2 +789γ −28)r5 −(9γ −1)2 (729γ 3 −513γ 2 +108γ −198)r6 }, CY Y ≡ 3γ{6γ(3γ − 1)(81γ 2 − 27γ − 2)3 − (81γ 2 − 27γ + 2)2 (6561γ 5 − 2916γ 4 −810γ 3 + 486γ 2 − 56γ + 2)r − 2(9γ − 1)2 (118098γ 6 − 98415γ 5 + 29160γ 4 − 4212γ 3 + 531γ 2 −72γ + 4)r2 − 6(177147γ 7 − 98415γ 6 + 15039γ 5 − 2916γ 4 + 1593γ 3 − 357γ 2 + 32γ − 1)r3 −(5314410γ 7 − 5904900γ 6 + 2742498γ 5 − 676512γ 4 + 93150γ 3 − 6966γ 2 + 260γ − 4)r4 −(−2125764γ 7 + 2834352γ 6 − 1430298γ 5 + 358668γ 4 − 47547γ 3 + 3222γ 2 − 108γ + 2)r5 −18γ(6561γ 5 −5103γ 2 +1944γ 3 −423γ 2 +48γ−2)r6 −2γ(9γ−1)2 (729γ 3 −486γ 2 +99γ−8)r7 }.

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25Supplier
Association(Toyota)U U U U

Supplier
Association(Nissan)U

U

ToyotaU U U U
NissanFigure 1: Supplier Associations in the Japanese automobile industry

Figure 2: Condition for voluntary spillover to benefit U1

Upstream firm

Downstream firm

Note: We set γ = 3 and β = 1/20 to draw the figures.

Figure 3: The profits of the upstream and downstream firms26

β(Y Y ) − β(Y N )β(Y Y )

Figure 4: The conditions that upstream firms generate spillovers
ΒΑ
0.050
No firm
0.045
ΒHYYL
0.040

No firm, Two firms

0.035
ΒHYNL
0.030

Two firms

0.0

0.2

0.4

0.6

0.8

1.0

r

Figure 5: Endogenous determination of spillover eﬀects (γ = 3)
Ii

ei

0.040

I2 HY,NL

0.039

0.038

Ii HN,NL

0.036

ei HN,NL

0.038
Ii HY,YL

0.034

I1 HY,NL

0.032

0.037
0.036
0.00

0.02

0.04

0.06

0.08

0.10

ΒΑ

ei HY,YL
e1 HY,NL

0.02(Investment levels: upstream)e2 HY,NL

0.04

0.06

0.08

(Investment levels: downstream)

Figure 6: The equilibrium investment levels (γ = 3, r = 1/2)
(Y, Y ):(Y, N ):
(N, N ):
Note:

0.10

Both upstream firms generate spillovers.
Only U1 generates spillover.
No upstream firm generates spillover.
e2 (Y, N ) > ei (N, N ) > ei (Y, Y ) > e1 (Y, N )27ΒΑ