GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 370

Strategic dual sourcing as a driver for free revealing ofinnovation

Noriaki Matsushima and Laixun Zhao

May 2015

GCOE Secretariat
Graduate School of Economics
OSAKA UNIVERSITY
1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan∗
Strategic dual sourcing as a driver for free revealing of innovation
Noriaki Matsushima†
Institute of Social and Economic Research, Osaka University
Laixun Zhao‡
Research Institute of Economics and Business, Kobe University
May 11, 2015

Abstract
This paper examines the role of dual sourcing (e.g., outside options) in vertical andhorizontal relations. In a bilateral monopoly market, if either the upstream or downstreamﬁrm has outside options, the other ﬁrm could lose from seemingly positive shocks, e.g., market expansion or technology improvements. We extend this setting to a bilateral duopoly
market in which each downstream ﬁrm has outside options and upstream ﬁrms can engage
in cost reducing investments and generate technological spillovers. We ﬁnd that each upstream ﬁrm has an incentive to voluntarily generate technological spillovers to its upstream
rival if the downstream ﬁrms have better outside options.

Keywords: Dual sourcing, Outside option, Spillover, Vertical relations
JEL Codes: L13, O32, M11, C72∗

We would like to thank Eric Avenel, Carmen Beviá-Baeza, Yongmin Chen, Jay Pil Choi, Stefano Colombo,
Atsushi Kajii, Inés Macho-Stadler, Ryoji Ohdoi, David Pérez-Castrillo, Joel Sandonı́s-Dı́ez, Tadashi Sekiguchi,
Tetsuya Shimane, and the seminar participants in Kobe University, MaCCI (Mannheim), Tokyo Institute ofTechnology, and Universitat d’Alacant for their comments. We would also like to thank Cong Pan and Shohei
Yoshida for their research assistance. The ﬁrst author thanks the warm hospitality at MOVE, Universitat
Autònoma de Barcelona where part of this paper was written. We gratefully acknowledge ﬁnancial support from
JSPS Grant-in-Aid for Scientiﬁc Research (B) Nos. 24330079, 15H03349 and (C) No. 24530248, RISTEX (JST),and the program of the Joint Usage/Research Center for ‘Behavioral Economic’ 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

The management literature documents a long list of “dual sourcing.” Asanuma (1985) showsthat Japanese ﬁrms bring rival parts suppliers to work together in some situations, while inother situations they keep the suppliers away from each other to create competition.1 Kamath and Liker (1994) describe how Japanese automakers would invite guest engineers fromrival suppliers to compete side-by-side, to see who could come up with a better design. Dyer
and Nobeoka (2000) document the practices in Toyota, which constitute of diﬀerent layers
within the supplier network, ranging from bilateral relations (i.e., sending Toyota’s internallean-production experts to advise suppliers), to sub-networks (i.e., establishing supplier learning groups where rival suppliers help each other study and implement the Toyota Production
System), to more encompassing network structures (i.e., regular meetings within Toyota’s supplier association). Subsequently, Wilhelm (2012) ﬁnds that such practices still exist today inJapanese ﬁrms and they create not only cooperation but also competition between suppliers
within the network. Wu and Choi (2005) investigate the cases of eight buyers, and ﬁnd thateven a long-term buyer-supplier relationship does not prevent the buyer from ﬁnding alternative suppliers to create competition.2 In another study, Wu et al. (2010) simulate such“co-opetition,” where competing suppliers work together to meet the needs of the buyer.
The present paper examines the aforementioned phenomenon of dual-sourcing, in an extended setup of both vertical and horizontal competition. We start by considering a simple
bilateral monopoly where a supplier sells its input to a buyer. The payment to the supplieris negotiated between the two parties,3 but aﬀected by the buyer’s eﬀorts to search for fringe
1

Soucing strategies have been discussed by management researchers since Porter (1980), which views multiple

sourcing as a mechanism for a ﬁrm to aﬀect its bargaining power relative to both inside and outside suppliers,
which is exactly what we rigorously model here.
2

In the context of international trade, recently several researchers investigate the eﬀects of bi-sourcing(make-and-buy) from home and foreign markets (Kogut and Kulatilaka, 1994; Rob and Vettas, 2003; Choi andDavidson, 2004; Mukherjee, 2008).
3

Such negotiations are common in the literature (see Davidson, 1988; Horn and Wolinsky, 1988; Dobson

and Waterson, 1997; O’Brien and Shaﬀer, 2005; Björnerstedt and Stennek, 2007; Inderst, 2007; Milliou andPetrakis, 2007). Recently, Iozzi and Valletti (2014) comprehensively discuss vertical relations with bilateral1suppliers. In other words, the buyer has an outside option to procure its input with, i.e., “dualsourcing.”
With this simple setup, we ﬁrst show that dual sourcing can aﬀect proﬁtability in unconventional ways. For instance, the usual sources for potential gains such as a market expansion
may not beneﬁt the supplier, opposite to what one might expect. Intuitively, as the market
size rises, the buyer raises its search eﬀort for alternative sourcing, weakening the bargaining
position of the supplier. As a result, a market expansion while beneﬁting the downstream
ﬁrm, can hurt the upstream ﬁrm. Furthermore, in the Appendix, we reverse the roles of the
upstream and downstream ﬁrms, and ﬁnd our basic mechanism remains robust.
We then generalize the above setup to the case of a bilateral duopoly, again with dual
sourcing by buyers, and incorporating cost-reducing investments (e.g., R&D) by the suppliers.
We ﬁnd surprisingly that each supplier has an incentive to unilaterally generate technology
spillovers to its rival for free, if its own downstream buyer can ﬁnd cheap alternative sourcing.
Such spillovers generate a market-size shrink, which can hurt the downstream buyer but beneﬁt the supplier, via the bargaining mechanism described above. We show that the unilateralspillovers are strategic complements in the sense that it can induce the rival to also generate technology spillovers, and such technology spillovers can beneﬁt both the buyers and the
suppliers.
Our results provide rationale for why the Japanese “suppliers’ associations” still have strong
support (see Sako, 1996) from both suppliers and auto makers, even though these associations
may at times generate outward spillovers to rivals. Sako ﬁnds that a supplier would typicallyjoin in several suppliers’ associations and hold multiple memberships, aiming at obtaining information on production plans and solicitating suggestion for common problems such as standardization and pollution control, etc.4 As such, the suppliers’ association has been eﬀective
Nash bargaining.
4

In 1990s, almost all major Japanese auto manufacturers (except Honda) have suppliers’ associations, andmany suppliers join multiple associations (Sako, 1996, p.651). Part suppliers that join in multiple suppliers’
associations tend to be large in size and play a leading role in association activities (Sako, 1996, p. 656).

2

in diﬀusing innovative practices in the automobile industry. Our model supports her ﬁndings
with a rigorous theory: dual sourcing by automakers can stabilize the suppliers’ associations
for strategic reasons.
The present paper while simple, generates novel results that match the real practices ofmany ﬁrms on vertically and horizontally related production networks, such as the automobile
makers and parts suppliers documented in the literature. We explicitly show conventionallycounterintuitive 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, and even worker training in thelabor market.
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. Their results are obtained based on downstream competition and asymmetric bargaining power in each vertical chain, which are notrequired in our model.
Several other papers investigate cases in which downstream ﬁrms engage in R&D andendogenously determine the degree of R&D spillover.5 Kultti and Takalo (1998) and PoyagoTheotoky (1999) discuss whether downstream duopolists generate R&D spillover after determining their investment levels. Kamien and Zhang (2000), Gil-Moltó et al. (2005), Piga andPoyago-Theotoky (2005) and Milliou (2009) discuss cases in which downstream ﬁrms noncooperatively set the degree of R&D spillover before setting investment levels.6 Milliou (2009)shows that oligopolists prefer generating outward R&D spillover if the degree of product differentiation is high enough, a result that does not hold if the products are homogenous as in5

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

6

In Kamien and Zhang (2000), no ﬁrm has an incentive to generate spillover. In Gil-Moltó et al. (2005) andPiga and Poyago-Theotoky (2005), because spillover is reciprocal, ﬁrms have incentives to generate a positive
degree of spillover.

3

our paper. Subsequently, Milliou and Petrakis (2012) show that a vertically integrated ﬁrmchooses 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 product diﬀerentiation is low. Yoshida (2015) also showsa possibility that a downstream ﬁrm has an incentive to give its superior technology to its
rival in a downstream multi-product duopoly with vertical relations. Product multiplicity is
the key factor to derive his main result. And ﬁnally, De Fraja (1993), Katsoulacos and Ulph(1998), and Pacheco-de-Almeida and Zemsky (2012) investigate market environments where
free revealing of technological knowledge can appear, in the context of continuous-time R&D
competition.7 In contrast, our paper provides another rationale behind such behavior, basedon bargaining and outside options.
The paper proceeds as follows. Section 2 investigates a bilateral monopoly and establishesa benchmark result. Section 3 extends the benchmark model to the case of bilateral duopoly
with cost-reducing activities by upstream suppliers, and examines both one-sided and crossspillovers in technology. Section 4 brieﬂy discusses the relation between the main results andthe real-world phenomena. Section 5 concludes the paper. In the Appendix, we demonstrate
the robustness of our results by reversing the roles played by the downstream and upstreamﬁrms in the benchmark model.

2

Bilateral Monopoly

We start with a benchmark bilateral monopoly model and demonstrate a basic result in thesimplest way—a general positive shock such as a market expansion can hurt the ﬁrm withworse or even ﬁxed outside option in negotiations, even though the shock increases the size ofthe total pie (rents). But, it beneﬁts the ﬁrm that can improve its outside option.
7

Harhoﬀ et al. (2003) is a seminal work of voluntary information spillovers in the context of management.

See also Pénin (2007) for a recent survey on open knowledge transfer.

4

2.1

The basic setting

Consider a downstream ﬁrm and an upstream ﬁrm without any production cost for simplicity,
whose outputs are related in a one-to-one ratio. However, the downstream ﬁrm has an outside
option: by incurring a ﬁxed cost F , it can procure the input from a diﬀerent source at pricew(e) if negotiation with the upstream ﬁrm breaks down, where e is the search eﬀort of thedownstream ﬁrm to improve the value of its outside option, at a cost of S(e).8 We assumew′ (e) < 0, w′′ (e) > 0; and S ′ (e) ≥ 0, S ′′ (e) > 0.
Let us ﬁrst investigate the simplest game structure: in the ﬁrst stage, the downstream ﬁrm
makes a search eﬀort e to improve its outside option; in the second stage, both ﬁrms bargain
over the trading terms (a two-part tariﬀ, mq + T , where m is the wholesale price and T is theﬁxed payment). If bargaining breaks down, the downstream ﬁrm executes its outside option
at the ﬁxed cost F , with a marginal cost of w(e) (similar to Inderst and Valletti, 2009); ﬁnally,in the third stage, the downstream ﬁrm sets the quantity of ﬁnal output. The game is solved
by backward induction.
To keep the model simple and clean, we assume the search cost to be sunk, in a way similar
to R&D investments in innovation models, where actual production and any price or proﬁt
negotiation occurs afterwards.9 Let the inverse demand function in the downstream market bep(q; a), where q is the quantity and a is a positive parameter that can shift up the demand,such as market size, income, etc.
In the last stage, the downstream ﬁrm chooses the ﬁnal quantity q to maximize its proﬁts
8

The assumption concerning outside options follows that in Inderst and Valletti (2009). However, here we

endogenize the price.
9

An alternative is to let the two parties bargain ﬁrst and then the downstream ﬁrm search if bargaining

breaks down. In that case, however, since search begins after bargaining breaks down, delay of production
occurs, which is costly to the downstream ﬁrm. In order to avoid such delay cost, the downstream ﬁrm thuschooses to search before bargaining occurs. Also, Feng and Lu (2012) assume simultaneous bargaining over
output and transfer payment. In contrast, the sequential timing structure in our model clariﬁes what happensin each stage. Furthermore, our model formulation endogenizes the outside opportunity of the downstream ﬁrm
along the lines of Inderst and Valletti (2009).

5

(p(q; a) − m)q − T , resulting in the following ﬁrst-order and second-order conditions:
F.O.C.

p(q; a) − m + pq (q; a)q = 0,

S.O.C.

2pq (q; a) + pqq (q; a)q < 0,where pq (q; a) = ∂p(q; a)/∂q and pqq (q; a) = ∂ 2 p(q; a)/∂q 2 .
Let q(m; a) denote the equilibrium quantity in which the wholesale price is m. Then the
gross proﬁts of the downstream and the upstream ﬁrms (excluding the search costs of thedownstream ﬁrm) in the second stage are given as respectively,
πD = [p(q(m; a); a) − m]q(m; a) − T ; πU = mq(m; a) + T,
The ﬁrms’ outside options are respectively,OπD= [p(q(w(e); a); a) − w(e)]q(w(e); a) − F ; πUO = 0.

During bargaining, they jointly maximize10O
G = {πD − πD}{πU − 0}.

The ﬁrst-order conditions ∂G/∂T = 0 and ∂G/∂m = 0 can be reexpressed as:O
πD − πD− πU = 0,O(πD − πD)[q(m; a) + mqm (m; a)] − πU q(m; a)O
OO= (πD − πD
− πU )q(m; a) + (πD − πD
)mqm (m; a) = (πD − πD
)mqm (m; a) = 0.

From the above, we obtainm = 0, T =p(q(0; a); a)q(0; a) − [p(q(w(e); a); a) − w(e)]q(w(e); a) + F.
2

Since bargaining is eﬃcient, the two ﬁrms set the wholesale price equal to the marginal costof the upstream ﬁrm, i.e., m = 0, to maximize their joint proﬁts ﬁrst, and then, they split the10

Matsushima and Shinohara (2014, pp. 430–1) explain a plausibility of using Nash bargaining to capture

the negotiations between buyers and suppliers in the Japanese automobile industry.

6

overall proﬁts through the lump-sum payment, T . As such, the “double marginalization” problem in a standard bilateral-monopoly situation is avoided. Expecting the bargained outcome,the proﬁts of the two ﬁrms respectively becomeΠD (e) = [p(q(m); a) − m]q(m; a) − T − S(e)[p(q(0; a); a)]q(0; a) + [p(q(w(e); a); a) − w(e)]q(w(e); a) − F − 2S(e),2[p(q(0; a); a)]q(0; a) − [p(q(w(e); a); a) − w(e)]q(w(e); a) + F
ΠU (e) =.
2=

Then in the ﬁrst stage, the downstream ﬁrm chooses search eﬀort, satisfying the followingﬁrst-order condition,∂ΠD (e)w′ (e∗ )q(w(e∗ ); a) + 2S ′ (e∗ )
=0⇔−
= 0.
∂e
2

2.2

(1)

Market size

As promised, we now investigate how a marginal increase in parameter a (e.g., market size or
income) aﬀects the equilibrium outcome. Applying the envelop theorem yields
de∗2∂ΠU (e∗ )= pa (q(0; a); a)q(0; a) − pa (q(w(e∗ ); a); a)q(w(e∗ ); a) + w′ (e∗ )q(w(e∗ ); a)
.
∂a
da

(2)

where pa ≡ ∂p(q; a)/∂a and e∗ is the equilibrium level of search eﬀort. Total diﬀerentiation of(1) givesde∗w′ (e∗ )qa (w(e∗ ); a)=,da−{w′′ (e∗ )q(w(e∗ ); a) + [w′ (e∗ )]2 qm (w(e∗ ); a) + 2S ′′ (e∗ )}where qm ≡ ∂q(m, a)/∂m. It is easy to show de∗ /da > 0, because the denominator is derived
from the second-order condition of (1).
Let us examine (2) in detail. If w(e∗ ) is suﬃciently small (e.g., approaching zero), the ﬁrst
and second terms on the RHS cancel out, leaving only the third term and hence ∂ΠU (e∗ )/∂a <0. Thus,
Proposition 1 An increase in the parameter a decreases ΠU (e∗ ) if w(e∗ ) is suﬃciently small.

7

This Proposition implies that a market expansion actually harms the upstream ﬁrm, opposite to what one might conventionally think. Intuitively, as the market size rises, the downstream ﬁrm raises its search eﬀort, which then weakens the bargaining position of the upstream
ﬁrm, leading to our result.
To summarize, positive shocks such as market size increases enlarge the pie and could bring
potential gains for all players. However, if some players can raise their outside options, they
can take away more than the increase of the pie, leaving others worse oﬀ. Also note that thesame logic applies to a setting with n outside options, where the ﬁrm would use the best oneof them and the rest of the options (n − 1) becomes irrelevant.
ExampleAssume that p = a − q, w(e) = w − e, and S(e) = γe2 /2, with a and γ being

positive constants.11 The second stage net proﬁts of the two ﬁrms are then given asΠD (e) =

)
)
1( 2
1( 2
a + [a − w(e)]2 − 4F − S(e), and ΠU (e) =a + [a − w(e)]2 + 4F
8
8

In the ﬁrst stage, the ﬁrst-order condition of the downstream ﬁrm is
a − w − (4γ − 1)ea−w∂ΠD (e)== 0 → e∗ =.
∂e44γ − 1Since the quantity under which bargaining breaks down, (a − w(e∗ ))/2, is smaller than whenagreement is reached, we require a < 4γw. Substituting e∗ into ΠU (e) givesΠU (e∗ ) =[(8γ − 1)a − 4γw](4γw − a) F
+ .
8(4γ − 1)2
2

The ﬁrst term is positive if
4γw< a < 4γw.
8γ − 1
Diﬀerentiating ΠU (e∗ ) with respect to a yields
16γ 2 w − (8γ − 1)a∂ΠU (e∗ )=,∂a4(4γ − 1)2
that is negative if and only if
16γ 2 w
< a < 4γw,
8γ − 1
under which a rise in market size a decreases the proﬁt of the upstream ﬁrm.
11

We assume w(e) = w − e for mathematical simplicity, although w′′ (e) = 0 here.

8

3

Bilateral Duopoly

In this section we extend the above benchmark model to a case of two pairs of downstream
and upstream ﬁrms, denoted by Di and Ui respectively (i = 1, 2). By doing so, we addcompetition to the upstream and downstream ﬁrms, forming a setup of both vertical andhorizontal competition. We further show that improving one’s outside option has strategic
eﬀects, which surprisingly can hurt oneself under certain conditions. And upstream ﬁrms,
being aware of this mechanism, may give away their own technology to rivals, for free in an
extreme.

3.1

The basic setting of duopoly

Just as in the benchmark, 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 if the negotiation with Ui breaks
down, where ei is the search eﬀort of Di , with a cost of S(ei ). Simultaneously, Ui engages
in cost-reducing activities such as process R&D (which was absent in the benchmark case),through which it can reduce its marginal cost to ci (Ii ) = 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.
We then take into account technological spillovers between upstream ﬁrms, i.e., the costreducing 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 . We assume
that the marginal cost of Ui is ci (Ii , Ij ) = c − Ii − rIj if Uj chooses to give its reduced cost to
Ui , where r ∈ [0, 1] is the degree of knowledge spillover.12
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,
12

This setup of spillovers is quite diﬀerent from the related literature on research joint ventures (see for

instance, Amir et al., 2003; d’Aspremont and Jacquemin, 1988; and Suzumura, 1992), where ﬁrms conduct joint
R&D in the ﬁrst stage and then compete in the product market in the second stage.

9

each of the four ﬁrms simultaneously sinks an investment cost that determines the eﬀort levelto improve its outside option (downstream ﬁrms) or to reduce its marginal cost (upstreamﬁrms). In the third stage, observing the eﬀort levels in the second stage, the upstream andthe downstream ﬁrms on the same vertical chain negotiate over a transfer payment (a two-parttariﬀ). The determined transfer payment is privately known in the vertical chain, but unknown
to outsiders. This assumption simpliﬁes the analysis since the Nash equilibrium wholesale priceis set at the marginal cost of the upstream ﬁrm on each vertical chain.13 If an agreement
is reached, the downstream ﬁrm procures its input from the upstream ﬁrm; otherwise, thedownstream ﬁrm exercises its outside option. For expositional simplicity, we omit the ﬁxed
cost. Finally, in the forth stage, all downstream ﬁrms simultaneously set quantities to maximize
their own proﬁts. The game is again solved by backward induction.
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 thevertical chains i and j, and Ti as the payment from the downstream to the upstream ﬁrm
when bargaining reaches an agreement. Then πi (ci , cj ) − Ti is the gross proﬁt of Di , excluding
investment costs already sunk in the second stage, and Ti becomes the gross proﬁt of Ui . Notethat to obtain the net proﬁt of each ﬁrm, the sunk cost of investment in the second stage mustbe subtracted from the above.
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 on thevertical 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(3)

If Di and Ui reach an agreement, di is ci (Ii ); otherwise, it is w̃i (ei ). Note that the functional
form of ci (Ii ) is replaced by ci (Ii , Ij ) if Uj chooses to give its reduced cost technology to Ui .
13

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). The qualitative nature of our results would
not change if contracts are observable.

10

In the third stage, the pair Di and Ui on the same vertical chain i maximize the following
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 ) =

(4)
(5)

Similar to the discussion in the previous section, a decrease in cj reduces qi , inducing Di tolower 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

Spillover eﬀects

In the second stage, we consider three scenarios: no upstream ﬁrm generates spillover; only
one upstream ﬁrm generates spillover; both upstream ﬁrms generate spillovers.
3.2.1

No spillover

First, we look into the case when no upstream ﬁrm generates spillover to its rival. In theinvestment stage, from (5) 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
ΠU
i =

(6)
(7)

The ﬁrst-order conditions lead to the reaction functions:
Ii (Ij ) =α − Ijα − 2β − Ij
, ei (Ij ) =,9γ − 29γ − 211(8)

where α ≡ a − c and β ≡ w − c. Note that ei (Ij ) in (8) includes only the investment level of therival’s upstream ﬁrm Uj because Di ’s eﬀort is related to its outside option, πi (w̃i (ei ), cj (Ij )).
Solving the simultaneous equations, we have the investment levels:Ii∗ =α(9γ − 2)α − 2(9γ − 1)β, 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)2
9γ 2 α2γ((9γ − 2)α − 2(9γ − 1)β)2ΠD∗+,i (N, N ) =2(9γ − 1)22(9γ − 1)2 (9γ − 2)∗
ΠUi (N, N ) =U∗
where k and l in ΠD∗i (k, l) and Πi (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 theother way around. The cost function of U2 can be rewritten as c2 (I2 , I1 ) = c − I2 − rI1 . From(5) 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 ))ΠU− f (I2 ),2 =
2π2 (c2 (I2 , I1 ), c1 (I1 )) + π2 (w̃2 (e2 ), c1 (I1 ))ΠD− S(e2 ).
2 =
2
ΠU
1 =

(9)
(10)
(11)
(12)

Then, the eﬀort level of U1 , I1 , inﬂuences the outside value of D1 , π1 (w̃1 (e1 ), c2 (I2 , I1 )), as wellas the downstream proﬁt, π1 (c1 (I1 ), c2 (I2 , I1 )). The voluntary spillover generates a strategicinteraction between the endogenous eﬀort levels of U1 and D1 , through π1 (w̃1 (e1 ), c2 (I2 , I1 )).
Speciﬁcally, an increase in I1 decreases c2 , inducing D1 to lower e1 . This eﬀect can beneﬁt U1
especially when w̃1 (e1 ) is small, since the marginal eﬀect of lowering e1 on the outside proﬁtof D1 increases as its “eﬃciency,” w̃1 (e1 ), improves.
12In the investment stage, each ﬁrm’s reaction function is obtained as respectively
I1 (I2 , e1 ) =I2 (I1 ) =e1 (I2 , I1 ) =e2 (I1 ) =α − rβ − I2 + re1,
9γ − 2(1 − r)
α − (1 − 2r)I1,
9γ − 2α − 2β − I2 − rI1,
9γ − 2α − 2β − I1.
9γ − 2

(13)
(14)
(15)
(16)

e2 (I1 ) is the same in both (16) and (8). 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 (13)), 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 (14)), which is similar to that in thecontext of research joint ventures (e.g., d’Aspremont and Jacquemin, 1988). Finally, the eﬀort
by U1 lowers the incentive of D1 to increase its outside value (see (15)), similar to the reason
in the ﬁrst eﬀect. Putting these all together, the spillover generates a positive impact on theeﬃciency of the rival upstream ﬁrm but a negative one on the downstream partner’s eﬀort toimprove its outside option. The former impact hurts the technology giver while the latter one
beneﬁts it, and which eﬀect is bigger can be explained as follows.
Solving the simultaneous equations gives:
I1∗ =
I2∗ =
e∗1 =
e∗2 =

(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)

Further substitution yields the ﬁrms’ equilibrium proﬁts, which are messy algebraically (see13

Figure 1: Condition for voluntary spillover to beneﬁt U1

Appendix 2):
∗
D∗
U∗
D∗
ΠU
1 (Y, N ), Π1 (Y, N ), Π2 (Y, N ), Π2 (Y, N ),given that the cost reduction of U1 is spilled over to U2 .
Using the above, we can examine whether the voluntary spillover increases the proﬁt of U1 .
A simple comparison leads to:14∗∗

U
Proposition 2 ΠU
1 (Y, N ) > Π1 (N, N ) if and only if β < β(Y N ), where β(Y N ) is β such
∗

∗

∗

∗

U
U
U
that ΠU
1 (Y, N ) = Π1 (N, N ). Similarly, Π2 (N, Y ) > Π2 (N, N ) if and only if β < β(Y N ).

The threshold value of β, β(Y N ), is depicted in Figure 1. U1 beneﬁts by giving its technology to the rival for free, if the above condition is satisﬁed. Similarly, we can numerically showthat giving U1 ’s technology to U2 harms D1 .15
14

In Section 3, we explicitly derive the threshold values in the propositions. The ﬁle is available upon request.

15

The ex ante commitment to generating technological spillovers is crucial in deriving the above results. Given

that the eﬀort levels have already been determined, normally each upstream ﬁrm does not have an incentive togive its technology to its rival upstream. For our results to hold, each upstream ﬁrm must commit to giving out
its technology before choosing eﬀort levels, while the size of spillovers is not so essential.

143.2.3Two-sided spillovers

Finally, we examine the case when both U1 and U2 cross spillover, speciﬁcally, the cost functionof Ui is given by ci (Ii , Ij ) = c − Ii − rIj (i, j = 1, 2, j ̸= i). Except this ex-post cost of the
upstream ﬁrms, the timing structure of the game is similar to the case under one-sided spillover
just examined. From (5) and (4), the objective functions of Ui and Di are given as respectively(i, j = 1, 2 and j ̸= i)
1(πi (ci (Ii , Ij ), cj (Ij , Ii )) − πi (w̃i (ei ), cj (Ij , Ii ))) − f (Ii ),21ΠD(πi (ci (Ii , Ij ), cj (Ij , Ii )) + πi (w̃i (ei ), cj (Ij , Ii ))) − S(ei ).
i =
2
ΠU
i =

(17)
(18)

D
Rearranging the ﬁrst-order conditions, ∂ΠUi /∂Ii = 0 and ∂Πi /∂ei = 0, gives the following

reaction functions
Ii (Ij , ei ) =ei (Ii , Ij ) =α − rβ − (1 − r)2 Ij + rei,
9γ − 2(1 − r)
α − 2β − rIi − Ij.
9γ − 2(19)(20)

Under reciprocal spillovers, the reaction function of Ui in (19) diﬀers from that in (13). However,
the reaction function of Di in (20) is the same with that in (15), because the outside proﬁt
does not depend on the inside transfer price, ci (Ii , Ij ).
Solving the simultaneous equations leads toIi∗∗ =e∗∗=i(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

We can explicitly solve the game and derive the equilibrium proﬁts, which are again messy(see Appendix 2):∗
D∗ΠUi (Y, Y ), Πi (Y, Y ).

Here we only show how the exogenous variables (α and r) aﬀect the equilibrium proﬁts.
In Figure 2, the vertical axis indicates the proﬁt level, Πji (j = U, D); and the horizontal axis
indicates the value of r. The ﬁgures show that the degree of spillovers raises ﬁrm proﬁtability.
15

PU
i

PiD

0.040
0.44040.035

0.44020.030

0.44000.025

0.43980.020

0.43960.015

0.43940.010

0.4392
0.2

0.4

0.6

0.8

1.0

r

0.2

Upstream ﬁrm
Note:

0.4

0.6

0.8

1.0

r

Downstream ﬁrm
We set γ = 3, α = 2, and β = 1/20.

Figure 2: The proﬁts of the upstream and downstream ﬁrms.

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 ﬁrms.

Since α = a − c and β = w − c, an increase in α reﬂects an increase in a, expanding themarket size. As previously shown, an increase in α does not always improve the proﬁtabilityof each upstream ﬁrm in the absence of spillovers (Figure 3).
Note that the left-hand side of 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 thedegree of spillovers. ii). The spillover eﬀect is more signiﬁcant on the upstream ﬁrms than thedownstream ﬁrms, caused by the former ﬁrms’ investment to lower the outside option of the16latter ﬁrms. iii). The spillover eﬀect on the rival’s eﬃciency is magniﬁed, when the downstreamﬁrm also has a stronger incentive to raise its own option value. In turn, an increase in r enhances
the incentive of each 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:
U∗∗
Proposition 3 ΠU
1 (Y, Y ) > Π1 (N, Y ) if and only if β < β(Y Y ), where β(Y Y ) is β such
∗
U∗
U∗
U∗
that ΠU
1 (Y, Y ) = Π1 (N, Y ). Similarly, Π2 (Y, Y ) > Π2 (Y, N ) if and only 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 depend on theexogenous 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 of each upstream ﬁrm fortechnology spillovers.

β(Y Y ) − β(Y N )β(Y Y )

Figure 4: The conditions that upstream ﬁrms generate spillovers.

3.3

Endogenous spillover eﬀects

So far we have examined either one-way or two-way but exogenous giveaway of upstream
technology.

17

In the ﬁrst stage, each upstream ﬁrm Ui unilaterally determines whether to generate technological spillover to its rival upstream ﬁrm Uj .16 Depending on the threshold values of β, we
ﬁnd:Proposition 4 (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). If β(Y N ) < β < β(Y Y ),multiple equilibria exist.
Figure 5 shows, under high parameter values of β/α, neither upstream ﬁrm is willing togive its technology out; Under intermediate values, both ﬁrms can either generate technology
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.
ΒΑ
0.050
No firm
0.045
ΒHYYL
0.040

No firm, Two firms

0.035
ΒHYNL
0.030

0.0

Two firms

0.2

0.4

0.6

0.8

1.0

r

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

The logic can be understood as follows. As in the monopoly case, a lower β/α enhances the
incentives of the downstream ﬁrms to increase their search eﬀorts, which induces each upstreamﬁrm to generate spillovers to mitigate its downstream partner’s incentive to search. Figure 6
shows that the investment of a spillover-generating upstream ﬁrm and that of a downstreamﬁrm are negatively correlated with β/α.
16It can be straightforwardly shown that Uj does not have an incentive to refuse the unilateral spillover.

18

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

0.10

ΒΑ

(Investment levels: downstream)

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

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 )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 of β/α for which Ui has
incentives to generate spillover given that U−i does not generate spillover. As in (14), generating
unilateral spillovers increases the rival upstream’s investment since it raises the rival’s quantity(see also the diﬀerence between I1 (Y, N ) and I2 (Y, N ) in Figure 7), diminishing the incentive
for unilateral spillover. The negative eﬀect is stronger as the degree of spillover increases.
However, the negative eﬀect is almost canceled out by the decrease in the downstream partner’s
investment (see e1 (Y, N ) in Figure 7), 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 giventhat U−i generates spillover too, can be reﬂected by the upper curve in Figure 5, which isconcave and shows the changes in the reactions of the upstream ﬁrms through spillovers. Given
that its 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 (19) with (14)), which increases its incentive for generating spillover. As a

19

Ii

ei
0.03600.041

I2 HY,NL

e2 HY,NL

0.0355

0.040
0.039

ei HN,NL

0.0350Ii HN,NL

0.038
0.0345Ii HY,YL

0.037
0.036
0.0

ei HY,YL
0.0340I1 HY,NL
0.2

0.4

0.6

0.8

1.0

r

e1 HY,NL

0.0(Investment levels: upstream)

0.2

0.4

0.6

0.8

1.0

r

(Investment levels: downstream)

Figure 7: The equilibrium investment levels (γ = 3, β/α = 1/25)(Y, Y ):(Y, N ):
(N, N ):

Both upstream ﬁrms generate spillovers.
Only U1 generates spillover.
No upstream ﬁrm generates spillover.

result, the threshold value of β/α rises above than when no upstream ﬁrm generates spillover,leading to multiple equilibria. This eﬀect, while positive, is however partially canceled out by
the latter loss which increases with the degree of spillovers (see (14)), 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 option of Dj , theneach upstream ﬁrm has a stronger incentive to generate positive spillovers, just to mitigate theeﬀorts of the downstream ﬁrms.
Finally, we check whether the voluntary spillover beneﬁts the downstream ﬁrms. Simple
calculations lead to:
Proposition 5 ΠiD∗ (Y, Y ) > ΠD∗i (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.

20

4

Discussion

The above results may explain the following stylized facts. “Mr. Toyoda was instrumental
in forging a technology tie-up agreement between Denso and Bosch” (Anderson, 2010, p.94).
The relationships among autoparts suppliers Denso and Bosch, and major automakers Ford,
Mitsubishi and Toyota are perhaps good examples. Denso originated as a parts division of
Toyota, spun oﬀ from Toyota in 1949 and initially supplied parts only to Toyota, graduallydeveloped other sales channels and now provides parts to all major automakers in the world(e.g., Ford, General Motors, Honda, Mazda, Mitsubishi, except Nissan). Interestingly, Denso
also has licensing and joint venture relationships with its direct rival, Bosch, in Europe andthe U.S., the aim of which was to share technology and gain economies of scale and scope.
Similarly, Suzuki (1993) ﬁnds that there are two types of R&D spillovers in the Japaneseelectrical machinery industry, one among members within a keiretsu group such as NEC andthe other between members of diﬀerent keiretsu groups (e.g., between members of NEC and
Fujitsu), although the former is more signiﬁcant than the latter. These horizontal and vertical
relationships match very well the settings we have examined above.
Furthermore, our results suggest that giving up patents can spur innovation and increase
proﬁts in the industry, which might explain the recent surprise moves by Tesla and Toyota togive up patents. On June 13, 2014, electric carmaker Tesla announced it was giving up its
patents “to the open source movement.” Perhaps surprisingly, its stock price has skyrocketedsince then. And even more surprisingly, on January 5, 2015, Toyota too announced it would
make 5,680 patents related to fuel cell drive systems available, as a means to help other automakers build fuel cell cars. While standardization competition for fuel systems is rumoredto be one reason for them to open their innovation outcomes to the public, our paper showsthat the more fundamental reason lies in the possibility that they could grab the biggest shareof the enlarged pie through diﬀerent bargaining arrangements.

21

5Conclusion

This paper has investigated how dual sourcing inﬂuences proﬁtability in buyer-suppler relationships, and especially as an alternative to overcoming trading frictions. Under a bilateralmonopoly, if the buyer easily ﬁnds alternate suppliers, we ﬁnd that a market size expansion
actually harms the supplier, opposite to what one might conventionally think.
We then apply the mechanism behind this result to bilateral duopoly with cost-reducing
investments by the suppliers, and ﬁnd that each supplier has an incentive to unilaterally generate technology spillovers to its rival if its buyer’s cost to lower the outside wholesale price issmall. Such a free spillover generates a market size shrink, which can beneﬁt the supplier via
the mechanism aforementioned.
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.
We have abstracted from commitment issues and strategic competition among suppliers,
by modelling the second source as a purely competitive ‘fringe’. The wholesale price set byoutside 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 andsuppliers. In their terminology, a lower wholesale price in the present model (set by outside
suppliers) would represent a lower friction level from the viewpoint of buyers.
Emons (1996), Shy and Stenbacka (2003), Beladi and Mukherjee (2012) and Stenbacka andTombak (2012) investigate buyers’ decisions on bi-sourcing.17 In our model, since buyers have
options to procure inputs from potential outside suppliers, they continue dual sourcing andﬁnd alternative suppliers.

17

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

22

6

Appendix 1: Reversing the Roles of the Up- and Downstream Firms

In Appendix 1, we switch the roles of the upstream and the downstream ﬁrms in the basicmodel, and show a similar result when the upstream (instead of the downstream) ﬁrm has
better outside options.
Again assume the two ﬁrms’ outputs are related by a one-to-one ratio, with the upstream
ﬁrm’s marginal cost being constant at c. As its outside option, the upstream ﬁrm can supply
a ﬁnal product directly, of quality v(e), if negotiation with the downstream ﬁrm breaks down,
where e is its eﬀort to improve the value (v ′ (e) > 0 and v ′′ (e) < 0), at a cost of SU (e) withSU′ (e) ≥ 0 and SU′′ (e) > 0.
The game structure is as follows: in the ﬁrst stage, the upstream ﬁrm makes an eﬀort e
to improve its outside options; in the second stage, both ﬁrms bargain over the trading terms;
ﬁnally, in the third stage, the downstream ﬁrm sets the quantity of ﬁnal output. The game issolved by backward induction as before.
As in the benchmark, when the upstream ﬁrm uses a two-part tariﬀ contract, it sets thewholesale price at its marginal cost, c. The ﬁxed payment from the downstream ﬁrm to the
upstream ﬁrm is Td . In an abstract form, the gross proﬁt of the downstream ﬁrm is πd (c),where πd′ (c) < 0 and πd′′ (c) > 0.
On the other hand, its outside option is when the upstream ﬁrm directly enters thedownstream market and supplies the ﬁnal product, with a gross proﬁt of πo (c, v), where∂πo (c, v)/∂c < 0 and ∂ 2 πo (c, v)/∂c2 > 0; that is, an increase in c diminishes the gross proﬁt, at
a decreasing rate. But the converse holds for quality, ∂πo (c, v)/∂v > 0 and ∂ 2 πo (c, v)/∂v 2 > 0.
The cross partial derivative is ∂ 2 πo (c, v)/∂c∂v < 0.
Thus, Td is chosen to satisfy
πd (c) − Td = Td − πo (c, v).

23

Then the ﬁrms’ net proﬁts in the second stage are respectively
ΠD =
ΠU

=

πd (c) − πo (c, v(e)),
2πd (c) + πo (c, v(e))− SU (e).
2

And in the ﬁrst stage, the upstream ﬁrm’s maximization problem leads to:∂πo (c, v(e)) ′· v (e) − 2SU′ (e) = 0.
∂v
Total diﬀerential of the above yields
[ 2
]
[
]
∂ πo (c, v(e))∂πo (c, v(e)) ′′∂πo (c, v(e)) ′′2′′· (v (e)) +· v (e) − 2SU (e) de +· v (e) dc = 0,
∂v 2
∂v
∂c∂v
which gives de/dc < 0, since the terms in the ﬁrst brackets are the second-order conditions andnegative, and the last term in the second brackets is also negative.
Finally, we examine how the upstream ﬁrm’s marginal cost aﬀects the downstream ﬁrm’s
proﬁt.
2dΠDdc∂πo (c, v(e)) ∂πo (c, v(e)) ′ de−· v (e)
∂c
∂v
dc
∂π
(c,
v(e))
de
o
= πd′ (c) −−2SU′ (e) .
| {z } |∂c{z}|{z dc}= πd′ (c) −

(−)

(+)

(+)

If the third term is strong enough, the sign of dΠD /dc becomes positive. That is, the eﬃciency
improvement of the upstream ﬁrm can harm the downstream ﬁrm, even though it generates abigger pie, analogous to the result in the basic model.

7

Appendix 2: The Equilibrium Proﬁts in Section 3

The equilibrium proﬁts under the two scenarios of spillovers in Section 3 can be explicitly
written as follows.

24One-sided spillover

The equilibrium proﬁts of the ﬁrms are respectively,∗
ΠU
1 (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)2∗ΠU
2 (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

25

}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]2ΠD∗
i (Y, Y ) =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 )2∗ΠU
i (Y, Y ) =References
Amir, R., Evstigneev, I., Wooders, J., 2003. Noncooperative versus cooperative R&D withendogenous spillover rates. Games and Economic Behavior 42(2), pp. 183–207.
Anderson, E., 2010. Toyota’s competitive advantage: Path dependency, dynamic capabilities,
and sources of inimitability — a contrastive study with Nissan, in Sanchez R. and A.
Heene (eds), Advances in Applied Business Strategy Vol.12, Enhancing competences forcompetitive advantage, pp. 87–120.
Asanuma, B., 1985. The contractual framework for parts supply in the Japanese automotive
industry. Japanese Economy 13(4), pp. 54–78.
Arya, A., Mittendorf, B., Sappington, D.E.M., 2008. The make-or-buy decision in the presenceof a rival: Strategic outsourcing to a common supplier. Management Science 54(10), pp.
1747–1758.
Beladi, H., Mukherjee, A., 2012. Market structure and strategic bi-sourcing. Journal of
Economic Behavior and Organization 82(1), pp. 210–219.
Björnerstedt, J., Stennek, J., 2007. Bilateral oligopoly — The eﬃciency of intermediate goods
markets. International Journal of Industrial Organization 25(5), pp. 884–907.

26

Cachon, G.P., Harker, P.T., 2002. Competition and outsourcing with scale economies. Management Science 48(10), pp. 1314–1333.
Chatain, O., Zemsky, P., 2011. Value creation and value capture with frictions. Strategic
Management Journal 32(11), pp. 1206–1231.
Choi, J.P., Davidson, C., 2004. Strategic second sourcing by multinationals. International
Economic Review 45(2), pp. 579–600.
d’Aspremont, C., Jacquemin, A. 1992. Cooperative and noncooperative R&D in duopoly withspillovers. American Economic Review 78(5), pp. 1133–1137.
Davidson, C., 1988. Multiunit bargaining in oligopolistic industries. Journal of Labor Economics 6(3), pp. 397–422.
De Bondt, R., 1997. Spillovers and innovative activities. International Journal of Industrial
Organization 15(1), pp. 1–28.
De Fraja, G., 1993. Strategic spillovers in patent races. International Journal of Industrial
Organization 11(1), pp. 139–146.
Dobson, P.W., Waterson, M., 1997. Countervailing power and consumer prices. Economic
Journal 107(441), pp. 418–430.
Dyer, J.H., Nobeoka, K., 2000. Creating and managing a high-performance knowledge-sharing
network: The Toyota case. Strategic Management Journal 21(3), pp. 345–367.
Emons, W., 1996. Good times, bad times, and vertical upstream integration. International
Journal of Industrial Organization 14(4), pp. 465–484.
Feng, Q., Lu, L.X., 2012. The strategic perils of low cost outsourcing. Management Science
58(6), pp. 1196–1210.
Fershtman, C., Judd, K., 1987. Equilibrium incentives in oligopoly. American Economic
Review 77(5), pp. 927–940.
27

Gil-Moltó, M.J., Georgantzı́s, N., Orts, V., 2005. Cooperative R&D with endogenous technology diﬀerentiation. Journal of Economics & Management Strategy 14(2), pp. 461–476.
Harhoﬀ, D., Henkel, J., von Hippel, E., 2003. Proﬁting from voluntary information spillovers:
How users beneﬁt by freely revealing their innovations. Research Policy 32(10), pp.
1753–1769.
Horn, H., Wolinsky, A., 1988. Bilateral monopolies and incentives for merger. RAND Journalof Economics 19(3), pp. 408–419.
Inderst, R., 2007. Leveraging buyer power. International Journal of Industrial Organization
25(5), pp. 908–924.
Inderst, R., Valletti, T., 2009. Price discrimination in input markets. RAND Journal of
Economics 40(1), pp. 1–19.
Iozzi, A., Valletti, T., 2014. Vertical bargaining and countervailing power. American Economic Journal: Microeconomics 6(3), pp. 106–135.
Kamath, R.R., Liker, J.K., 1994. A second look at Japanese product development. Harvard
Business Review 72(6), pp. 154–165.
Kamien, M.I., Zang, I., 2000. Meet me halfway: Research joint ventures and absorptive
capacity. International Journal of Industrial Organization 18(7), pp. 995–1012.
Katsoulacos, Y., Ulph, D., 1998. Endogenous spillovers and the performance of research joint
ventures. Journal of Industrial Economics 46(3), pp. 333–357.
Kogut, B., Kulatilaka, N., 1994. Operating ﬂexibility, global manufacturing, and the option
value of a multinational network. Management Science 40(1), pp. 123–139.
Kultti, K., Takalo, T., 1998. R&D spillovers and information exchange. Economics Letters
61(1), pp. 21–23.

28

Liu, Y., Tyagi, R., 2011. The beneﬁts of competitive upward channel decentralization. Management Science 57(4), pp. 741–751.
Matsushima, N., Shinohara, R., 2014. What factors determine the number of trading partners? Journal of Economic Behavior & Organization 106(1), pp. 428–441.
Milliou, C., 2009. Endogenous protection of R&D investments. Canadian Journal of Economics 41(1), pp. 184–205.
Milliou, C., Petrakis, E., 2007. Upstream horizontal mergers, vertical contracts, and bargaining. International Journal of Industrial Organization 25(5), pp. 963–987.
Milliou, C., Petrakis, E., 2012. Vertical integration, knowledge disclosure and decreasing
rival’s cost. Departamento de Economı́a, Universidad Carlos III de Madrid, Economics
Series, Working Paper 12-13.
Mukherjee, A., 2008. Unionised labour market and strategic production decision of a multinational. Economic Journal 118(532), pp. 1621–1639.
O’Brien, D.P., Shaﬀer, G., 2005. Bargaining, bundling, and clout: The portfolio eﬀects ofhorizontal mergers. RAND Journal of Economics 36(3), pp. 573–595.
Pacheco-de-Almeida, G., Zemsky, P., 2012. Some like it free: Innovators’ strategic use of
disclosure to slow down competition. Strategic Management Journal 33(7), pp. 773–793.
Pénin, J., 2007. Open knowledge disclosure: An overview of the evidence and economic
motivations. Journal of Economic Surveys 21(2), pp. 326–348.
Piga, C., Poyago-Theotoky, J., 2005. Endogenous R&D spillovers and location choice, Regional Science and Urban Economics 35(2), pp. 127–139.
Porter, M., 1980. Competitive Strategy: Techniques for Analyzing Industries and Competitors.
Free Press, New York, NY.

29

Poyago-Theotoky, J., 1999. A note on endogenous spillovers in a non-tournament R&D
duopoly. Review of Industrial Organization 15(3), pp. 253–262.
Rob, R., Vettas, N., 2003. Foreign direct investment and exports with growing demand.
Review of Economic Studies 70(3), pp. 629–648.
Rockett, K., 2012. Perspectives on the knowledge-based society: An introduction to thespecial issue, Economics: The Open-Access, Open-Assessment E-Journal, 6(2012-35):
pp. 1–26. Doi: 10.5018/economics-ejournal.ja.2012-35
Sako, M., 1996, Suppliers’ association in the Japanese automobile industry: Collective action
for technology diﬀusion. Cambridge Journal of Economics 20(6), pp. 657–671.
Shy, O., Stenbacka, R., 2003. Strategic outsourcing. Journal of Economic Behavior andOrganization 50(2), pp. 203–224.
Stenbacka, R., Tombak, M., 2012. Make and buy: Balancing bargaining power. Journal of
Economic Behavior and Organization 81(2), pp. 391–402.
Suzuki, K. 1993. R&D spillovers and technology transfer among and within vertical keiretsu
groups–evidence from the Japanese electrical machery industry, International Journal ofIndustrial Organization 11(4), pp. 573–591.
Suzumura, K. 1992. Cooperative and noncooperative R&D in an oligopoly with spillovers,
American Economic Review 82(5), pp. 1307–1320.
Wilhelm, M.M., 2012. Managing coopetition through horizontal supply chain relations: Linking dyadic and network levels of analysis. Journal of Operations Management 29(7-8),
pp. 663–676.
Wu, Z., Choi, T.Y., 2005. Supplier-supplier relationships in the buyer-supplier triad: Building
theories from eight case studies. Journal of Operations Management 24(1), pp. 27–52.

30

Wu, Z., Choi, T.Y., Rungtusanatham, M.J., 2010. Supplier–supplier relationships in buyer–
supplier–supplier triads: Implications for supplier performance. Journal of Operations
Management 28(2), pp. 115–123.
Yoshida, S., 2015. Multiproduct competition in vertically related industries, ISER Discussion
Paper No. 935, Osaka University.

31