ISSN (Print) 0473-453X

Discussion Paper No. 1069

ISSN (Online) 2435-0982

ORGANIZATIONAL STRUCTUREANDTECHNOLOGICAL INVESTMENT

Inés Macho-Stadler
Noriaki Matsushima
Ryusuke Shinohara

Secondly Revised Decvember 2020
Revised April 2020
November 2019

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

Organizational structure and technologicalinvestment∗Inés Macho-Stadler†
Department of Economics, Universitat Autònoma de Barcelona andBarcelona GSE
Noriaki Matsushima‡
Institute of Social and Economic Research, Osaka University
Ryusuke Shinohara§
Department of Economics, Hosei University
December 7, 2020∗
We thank Shingo Ishiguro, Akifumi Ishihara, Atsushi Kajii, David Pérez-Castrillo, Yusuke Zennyo, andthe seminar participants at Japan Association for Applied Economics (Nanzan University), Japanese Economic Association (Rissho University), Kagoshima University, and Waseda University for their valuable
discussions and comments. We gratefully acknowledge the financial support from the Japan Society for
the Promotion of Science (JSPS), KAKENHI Grant Numbers JP15H03349, JP15H05728, JP17H00984,
JP18H00847, JP18K01519, JP18K01593, JP19H01483, and JP20H05631, the Generalitat de Catalunya
grant 2017SGR-711, the Severo Ochoa Programme grant number CEX2019-915-S, the MCIU and FEDER
PGC2018-094348-B-I00, Osaka University for its International Joint Research Promotion Program, and theprogram of the Joint Usage/Research Center for “Behavioral Economics” at the ISER, Osaka University.
The second author thanks the warm hospitality at MOVE, Universitat Autònoma de Barcelona where partof this paper was written and a financial support from the “Strategic Young Researcher Overseas Visits
Program for Accelerating Brain Circulation” by JSPS. The usual disclaimer applies.
†
Inés Macho-Stadler, Department of Economics, Faculty of Economics and Business Economics, Edifici
B, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain. E-mail: ines.macho@uab.cat,
ORCID: 0000-0002-2415-7972‡
Corresponding author: Noriaki Matsushima, Institute of Social and Economic Research, Osaka University, Mihogaoka 6-1, Ibaraki, Osaka 567-0047, Japan. E-mail: nmatsush@iser.osaka-u.ac.jp, ORCID:
0000-0002-7865-3875.
§
Ryusuke Shinohara, Department of Economics, Hosei University, 4342 Aihara-machi, Machida, Tokyo
194-0298, Japan. E-mail: ryusukes@hosei.ac.jp, ORCID: 0000-0003-2916-8352.

Organizational structure and technological investment

Abstract
We analyze firms’ decisions on their vertical organization, taking into account thecharacteristics of both the final good competition and the R&D process. We consider
two vertical chains, where upstream sectors invest in R&D. Such investment determines
the production costs of the downstream sector and has spillovers on the production andthe investment costs of the rival. In a general setting, we show that the equilibriumorganizational structure depends on whether the situation considered belongs to one
of four possible cases. We study how final good market competition, R&D spillover,and R&D incentives interact to determine the equilibrium vertical structure.

JEL Codes: L22, L13, O32, C72.
Keywords: R&D, Vertical separation, Market structure, Spillover

1

Introduction

In this paper, we analyze how final good competition and the characteristics of the firms’
R&D investment aﬀect firms’ decisions about their vertical structure under a general demandsystem. We investigate this question by proposing a downstream duopoly in which each of thedownstream firms has an exclusive relationship with an independent supplier that engages
in a process innovation. This innovation process may induce R&D spillovers on the rival’s
production and R&D costs. We provide a complete formulation of the reasons for adopting
either an integrated or a decentralized structure. We consider how upstream suppliers invest
in R&D depending on the vertical structure, the R&D spillovers on the rival, and how the
R&D eﬀorts interact with market competition.
Our formulation helps us understand firms’ decisions and may explain, for example,
the market structures of the automotive industries in the US and Japan take diﬀerent vertical organizational forms. The automotive industry depends heavily on external suppliers for car interiors (such as overhead lighting and doors paneling) and other components.
Those manufacturers engage in many types of R&D (including design and materials), whichcause technological spillovers between them (Sako, 1996; Aoki and Lennerfors, 2013). While
this is common to the industry, diﬀerent manufacturers choose diﬀerent vertical structures.
Japanese automobile manufacturers often separate the development and manufacturing of
critical elements to independently managed entities, which still have close relations with thosemanufacturers (Ahmadjian and Lincoln, 2001). The degree of vertical integration in thosemanufacturers is smaller than that in their US counterparts (Cusumano and Takeishi, 1991).
Several explanations for the diﬀerent degrees of vertical integration in the US and Japan havebeen proposed (Dyer, 1996). Among these, arguments are based on various preferences overthe vertical structure, such as the independence of outside suppliers (Scherer, 1980; Perry,1989), or the fear of losing managerial control (Emerson, 1962; Pfeﬀer and Salancik, 1978).

1

Another important aspect is the business culture: trust in suppliers is essential in Japanesecultural norms (Sako, 1991; Hill, 1995). We will show how market interaction, the R&D
spillovers, and the idiosyncratic relationship between the upstream and downstream sectors
may also help explain the choice of organizational structure.
We consider two competing vertical chains, each consisting of one upstream and one
downstream sector. Each of the upstream sectors decides on R&D investment, which determines the production costs of the downstream sector in converting inputs into final goods.
The first important element of our model is allowing the existence of cross-firm R&D
spillovers: first, a firm’s R&D investment cost may depend on its rival’s R&D eﬀort; second,
the rival’s research eﬀort can also aﬀect a firm’s production cost. We will refer to the first
type of R&D spillovers as spillovers on investment, and to the second type as spillovers on
production (Section 3 provides definitions of both). Each type of spillover may be positiveor negative.
The second important element of our model is that the R&D decision depends on theorganizational structure. Under vertical integration, there is no coordination cost between
the upstream and the downstream sectors.1 They share objectives and provide all relevant
information and eﬀort to help each other to produce the optimal level of R&D for the verticalstructure.2 Nonintegrated structures suﬀer from some lack of coordination. We model thisproblem by assuming that in the case of adopting a separated structure, the downstream
and the upstream sectors sign a secret contract with nonlinear tariﬀs that depends on the1

This is consistent with Gibbons (2005, p. 203): “the implicit definition of integration in the industrialorganization literature is the unification of control rights.”2

The innovation process depends on the ability to monitor the latest market and technological developments and to integrate various sophisticated technologies. In the car industry, this can include know-how,commercially sensitive information, cooperation to develop new materials, safety-related innovations, or more
complex products or components. The coordination in objectives can also be related to trust or better monitoring, and the diﬀerence of this coordination between internal or external R&D may be substantial. These
concerns also exist in the pharmaceutical and biotechnology industries, the service sector, and all industries
with material suppliers such as beverage and food sellers or toy producers.

2

downstream bargaining power (which we can interpret as imperfect coordination). Less
than full bargaining power leads to an underinvestment in R&D activities. In other words,
as argued by Armour and Teece (1980), more vertically integrated firms make R&D decisions
in more eﬃcient ways.
We show that the equilibrium vertical structure depends on whether the situation belongs
to one of four possible cases, which are defined by the two following criteria: (i) whether afirm’s R&D eﬀort helps or harms the rival (nature of the spillovers); and (ii) whether firms’
R&D eﬀorts are strategic substitutes or strategic complements. Case 1 is the situation
where a firm’s R&D investment helps its rival, and R&D eﬀorts are strategic substitutes.
Case 2 is the situation where a firm’s investment helps its rival, and investment levels are
strategic complements. Case 3 is the situation where a firm’s investment harms its rival, andinvestment levels are strategic substitutes. Finally, Case 4 is the situation where a firm’s
investment harms its rival, and R&D investment levels are strategic complements.
The analyses of Cases 2 and 3 are easy. Recall that under a separated structure, theR&D level is lower than in a centralized one. In both cases, the best organization is an
integrated structure. In Case 2, aggressiveness for investment by both firms mutually fosters
R&D eﬀorts, which is beneficial for the firms. Hence, each firm adopts a vertically integrated
structure as a commitment to engage in more investment to induce its rival to also invest
more. In the same way, in Case 3, adopting a vertically integrated structure leads a firm
to undertake more R&D investment, which induces its rival to invest less. That is, firms
vertically integrate because aggressiveness in R&D investment is essential to maintain astrong competitive position in the downstream market.
In contrast, in Cases 1 and 4, firms would prefer to commit to lower R&D eﬀorts. To
understand the results on the equilibrium vertical structure, we also need to know whether
the decrease in a vertically separated firm’s R&D eﬀort is small or large. Firms may beinterested in adopting a decentralized structure as long as the decrease in the R&D eﬀort
3

(which is a function of the bargaining power of the two sectors in a separated structure) is
not too strong. For example, in Case 1, with positive R&D spillovers and substitutabilityof R&D eﬀorts, each firm has an incentive to vertically separate in order to induce its rival
to engage in more R&D (as long as the decrease in its own R&D is small enough). In
this scenario, vertical separation allows firms to increase their free-riding on their rival’s
investment.
To further clarify the forces at work in Case 1, recall that positive spillovers and strategic
substitutability of R&D eﬀorts make firms willing to commit to marginally lower R&D eﬀort,
with the objective of increasing the rival’s R&D eﬀort. Consider the extreme case where in
the separated structure, the upstream sector’s bargaining power is very high so that it chooses
an R&D eﬀort close to the one implemented by a centralized upstream sector. Then, thecost of vertical separation in terms of the reduction of own R&D is of a negligible second
order. However, this marginal decrease in the firm’s R&D eﬀort induces a strategic eﬀect
on the rival’s decision that is of the first order. Therefore, if the bargaining power of the
upstream sector in the case of decentralization is high, then decentralization is profitable.
In contrast, if the bargaining power of the upstream sector in a separated structure is weak,then the R&D eﬀort falls considerably, and the eﬀect of the fall on profits (direct eﬀect) is nolonger marginal. Hence, vertical separation is no longer a good choice. The analysis of Case
4 is similar. With negative spillovers and strategic complementarity of the R&D eﬀorts, afirm would like to commit to marginally decrease its R&D eﬀort to lead the rival to also
decrease its R&D eﬀort. Whether or not the firm decides to vertically separate depends onwhether, under this organizational structure, the decrease in its R&D eﬀort is not too large.
Using the example of the Japanese and US car industries, we explain the consistencyof our conclusions with the vertical organizations in these countries. We now presume that
this industry belongs to Case 1 (or 4), where firms would prefer to commit to decreasing
their R&D eﬀorts to induce the rival to invest more (or less). Japanese suppliers have more4

bargaining power over automobile assemblers than their US counterparts (Matsushima andMizuno, 2013, Section 5.1). Moreover, the degree of vertical integration in those manufacturers is smaller than that in their US counterparts (Cusumano and Takeishi, 1991).
Under these terms, our paper provides a plausible explanation for the diﬀerence between theJapanese and US automobile assemblers’ organizational structures. Japanese firms, whose
R&D investment will decrease little under decentralization, are more likely to adopt vertical separation. In contrast, US firms, whose R&D eﬀort will decrease more under verticalseparation, will be more likely to adopt vertical integration.
Finally, to further highlight the importance of our model settings, we note that in themodel studied in the seminal paper by d’Aspremont and Jacquemin (1988), only Cases 2
and 3 emerge. As we have explained, both situations lead to vertical integration. Given
this result, one may conclude that technological spillovers do not aﬀect the vertical structure
decision. However, this conclusion is not appropriate in more general settings (i.e., thosethat can generate Cases 1 and 4).
The remainder of the paper is structured as follows. Section 2 summarizes the related
literature. Section 3 describes the model. Section 4 considers the incentives for R&D integrated and separated vertical structures. Section 5 classifies the possible market structures
into four cases that depend on the iso-profit curves and the reaction functions of the firms in
the R&D investment stage. Then, we provide some parametric examples to explain how the
incentives to conduct R&D influence the equilibrium vertical structures. Section 6 concludes.

2

Literature review

Our paper considers market structure, R&D activities, and firms’ vertical organization, andis related to three lines of research (see Table 1 for a summary).
Many papers have provided motives for vertical integration or disintegration by considering the features of the final good market in which firms interact (Bonanno and Vickers,
51988; Ziss 1995; Gal-Or, 1999; Choi and Yi, 2000; Chen, 2001, 2005; Lin, 2006; Matsushima,2009). However, those papers do not consider R&D activities (including process and product innovations), which are indispensable for firms in obtaining competitive advantages overtheir rivals. They ignore that these activities may be vital in the upstream or downstream
sectors. In the case of the car industry, for instance, upstream firms may have a leadingrole in the innovation process, and knowledge spillovers related to those activities are often
inevitable. Then, when the upstream section is critical in the innovation process, whether
there is vertical integration or not may have significant consequences.
Another relevant line of literature has studied several eﬀects of vertical structures on the
R&D process. Bolton and Whinston (1993) is the pioneering study of the eﬀect of vertical integration on downstream investments. Buehler and Schmutzler (2008) investigate theimpact of vertical integration on investment incentives when only downstream firms makecost-reducing investments.3 Chen and Sappington (2010) show that vertical integration enhances upstream research eﬀorts under downstream Cournot competition but may decrease
those research eﬀorts under Bertrand competition. Liu (2016) studies the impact of vertical integration on investment incentives when both upstream and downstream firms makeinnovative investments.
In our paper, we are interested in the eﬀects of the market competition conditions andthe R&D investment characteristics on the firms’ organization decision. To our knowledge,
the closest paper is Gupta (2008), which extends an earlier article (Gupta and Loulou, 1998)
in which the manufacturers engage in process innovation without R&D spillovers in a market
formed by two exclusive competing vertical chains.4 Gupta (2008) adds R&D spillovers on
3

Milliou and Petrakis (2012, 2019) consider the eﬀect of vertical integration by an upstream monopolist
and a downstream firm in a downstream duopoly market with technology spillovers.
4

There are a few papers that are less related to the consequences of market competition and R&D
investment on the vertical organization. Wang et al. (2017) introduce price and quality sensitivities into the
Gupta and Loulou (1998) model by using a linear demand system and considering the degree of consumer
loyalty. Lin et al. (2014) study a downstream duopoly circular-city model with a three-tier exclusive supply

6

the unit cost of production (spillovers on production) in a model with linear demand, linear
marginal costs, and linear tariﬀ contracts between the two chains in the case of vertical separation. He shows that vertical integration by both chains is always an equilibrium outcome.
In addition, he shows that vertical separation by both chains is also an equilibrium outcome,
if and only if the substitutability of their products is suﬃciently high. Moreover, verticalseparation by both chains induces high investment levels if the spillovers on production arelarge enough. In our paper, in addition to general R&D spillovers on production, we also
consider spillovers on the rival’s R&D process. Furthermore, we investigate a general settingthat allows us to focus on the strategic properties of the R&D investment eﬀorts (strategic
substitutes and complements) under a general demand system. Finally, and in contrast to
Gupta (2008), we consider secret contracts with nonlinear tariﬀs or lump-sum transfers in
the case of vertical separation. Our results lead to more general and quite diﬀerent conclusions regarding the firms’ decisions on vertical integration or separation. They also show
the crucial diﬀerence between considering linear contracts between the upstream and downstream sectors (as the previous papers do) and assume secret contracts with nonlinear tariﬀs
or lump-sum transfers (as we do). In our model, vertical separation does not have a strategic
eﬀect on profits through the double marginalization problem, which is a critical factor in theprevious papers.
[Table 1 about here]
Finally, our paper contributes to the connection between organizational economics andindustrial organization (e.g., Legros and Newman, 2013, 2014; for a review of vertical integration and market structure, see Bresnahan and Levin, 2013). In the organizational economics
literature, both transaction costs (Williamson, 1975; Klein et al., 1978) and property rights
chain. However, technological spillovers of any type are beyond the scope of those papers.

7

theories (Grossman and Hart, 1986; Hart and Moore, 1990) assume that contracts are incomplete. Those theories argue whether the market or the organization facilitates the control
and coordination of decisions and provides better incentives for relationship-specific investments. We assume that a vertically integrated organization internalizes the benefit frominvestments within a unified entry, achieving higher R&D eﬀorts. We also assume that ina separated structure, the separated upstream investment sector determines its R&D eﬀort
by solely considering its profit through bargaining over the downstream production profits.
Hence, the separated upstream sector does not fully internalize the total profits of the vertical chain.5 As explained in the Introduction, vertical separation in our model may entail astrategic advantage in R&D competition. Thus, the cost of integration in our model is theloss of this advantage. This diﬀers from the cost of integration explained in the studies oforganizational economics, thereby contributing to the literature on the connection between
organizational economics and industrial organization.

3

Basic model

Consider a situation in which two firms or vertical chains, 1 and 2, produce a final good andinvest in R&D before the final good market competition. Each firm consists of two sectors:
an upstream sector and a downstream sector. The vertical chains (downstream sectors)
compete in the final good market, and they (upstream sector) invest in R&D.
We denote the firms’ R&D eﬀorts by ei (i = 1, 2) and we assume that they are publicly
observable. When firm i integrates the two sectors vertically, it organizes them as a single
entity, which takes into account the joint profits of the sectors. This implies that the R&D
eﬀort is decided to maximize the total profits of the vertical chain. In contrast, if a firm5

Our view that decentralization reduces R&D eﬀorts is consistent with Hart (1995, p. 33): “In summary,the benefit of integration is that the acquiring firm’s incentive to make relationship-specific investments
increases since, given that it has more residual control right, it will receive a greater fraction of the ex post
surplus created by such investments.”

8

separates the two sectors vertically, each of them has its own objective, and they negotiate
the trading terms between them. In particular, in a vertically separated firm, the two sectors
in firm i (i = 1, 2) negotiate a lump-sum transfer Fi from the downstream to the upstream
sector.6 If the negotiation breaks down, the downstream sector procures the final goods fromother competitive upstream firms without any technological advantage. In the third stage,
the downstream sectors compete in the final good market using their available technologicalimprovements. We assume that the two chains choose their R&D eﬀort noncooperatively.
The R&D investment made by the upstream sector of a firm has a positive eﬀect on the
productivity of its downstream sector (e.g., it allows a reduction in production costs), andit may also aﬀect the rival. We consider that a firm’s R&D eﬀorts may aﬀect the rival at
two levels: they may influence the R&D process of the other firm, and they can directly
aﬀect the production process of the rival. We will refer to the first type of spillovers (R&D
spillovers on the rival’s R&D process) as spillovers on investment because they aﬀect thecost of investing in R&D. We will refer to the second type of spillovers (R&D spillovers on
the rival’s production process) as spillovers on production because the R&D generated by
the firm directly aﬀects the rival’s production process by influencing its production cost.
We formalize the spillovers on investment by assuming that if firm i chooses eﬀort ei andfirm j(̸= i) chooses ej , then firm i’s R&D investment cost is I i (ei , ej ), which allows for therival’s R&D eﬀort to aﬀect firm i’s cost of R&D. Spillovers on investment can be negativeor positive. When firms invest in R&D in order to be the first to develop an innovation andthen apply for patent rights, the winner rules out the possibility of the competitor’s success6

We assume that the payment between the two sectors is a fixed amount; that is, there is no royalty as
a function of the production of the final good. This payment scheme is optimal when the contract terms are
private information within the two sectors of the vertical chain (so-called secret contracts) and the upstream
sector is not involved in the production of the final output after transferring the R&D outcome. A related
discussion is available in Section 3.2 of Caillaud and Rey (1995) (see also Hart and Tirole, 1990, and Rey and
Tirole, 2007, for discussions of secret contracts). In addition, Rockett (1990) explains situations in which
licensors of new technology face diﬃculties in collecting per production fees from licensees.

9through a similar technology (patent race models, such as Loury (1979), study this eﬀect). In
this setting, an increase in a firm’s R&D eﬀorts increases the probability of its own success,but lowers that of its rival, generating negative spillovers on investment (e.g., Crépon andDuguet (1997) estimate a negative eﬀect of rival R&D on a firm’s R&D). However, when
the patent rights are not well defined or when the R&D approach followed by the firms isvery diﬀerent, a firm may benefit from the other firm’s knowledge stock accessed throughpublic sources. Then, the rival’s R&D eﬀort produces a positive spillover on investment
because a firm is more likely to create an innovation thanks to another firm’s R&D eﬀort.
For example, in the absorptive capacity literature, it is shown that “knowledge spillovers”
induce complementarities in firms’ R&D eﬀorts (Cohen and Levinthal, 1989).
We consider a general reduced form for the profits obtained in the final output market.
The downstream sector in firm i obtains the gross profit V i (ei , ej ) if the downstream sector
in firm i uses the technological improvements created by the upstream sector in firm i; otherwise, it obtains V̄ i (ej ) ≡ V i (0, ej ), which does not depend on the technological improvements
made by the upstream sector i. Expressions V i (ei , ej ) and V̄ i (ej ) may depend on the rival’s
R&D eﬀort, to allow for firm i to experience spillovers in the production stage. As with thespillovers on investment, spillovers on production can also be positive or negative. Several
papers consider positive spillovers on production by defining knowledge spillovers as the decrease in production costs because of the R&D investment emanating from other firms (e.g.,
Spence, 1984; d’Aspremont and Jacquemin, 1988; Bernstein and Nadiri, 1989). However,there are situations where spillovers on production are negative because an innovation froma rival can increase the production cost of the firm. Consider an innovation that allows therival to improve its product quality significantly at the same production cost. If the firm
wants to stay in the market (or retain its market share), it may be under pressure to produce a similar quality, which would involve a higher marginal cost. Similarly, an innovation
may allow the rival to address some employees’ demands or some consumers’ environmental
10

concerns, which would increase the production cost of the firm to meet the same standard.
Thus, in the final stage of the game, when firm i uses the technology of its upstream
sector, firm i’s net profit isΠi (ei , ej ) ≡ V i (ei , ej ) − I i (ei , ej ).

(1)

Summarizing, we analyze the three-stage game where:- In the first stage, firms simultaneously and noncooperatively determine their organizational structures: vertically separated or integrated.
- In the second stage, the two upstream sectors simultaneously and noncooperatively choose
their own R&D eﬀort ei (i = 1, 2).
- In the third stage, the downstream sectors compete in the final good market.
We assume that values V i (ei , ej ) and I i (ei , ej ) that firm i obtains in the Nash equilibriumof the third stage, for i = 1, 2, satisfy the following assumptions:(A1) V i (ei , ej ) and I i (ei , ej ) are twice continuously diﬀerentiable with respect to their own
R&D eﬀort,(A2) V i (ei , ej ) is increasing in its own eﬀort in R&D: Vii (ei , ej ) ≡ ∂V i (ei , ej )/∂ei > 0,
(A3) I i (ei , ej ) is increasing and convex in its own R&D eﬀort: Iii (ei , ej ) ≡ ∂I i (ei , ej )/∂ei > 0,Iiii (ei , ej ) ≡ ∂ 2 I i (ei , ej )/∂e2i ≥ 0, and(A4) Πi (ei , ej ) is concave in its own R&D eﬀort: Πiii (ei , ej ) ≡ ∂ 2 Πi /∂e2i < 0.
For illustrative purposes, in the examples of Section 5.3, we present particular functional
forms of V i (ei , ej ) and I i (ei , ej ), by assuming that in the final good market, standard Cournot
competition exists, and considering markets with diﬀerent characteristics in terms of demand,
spillovers, and bargaining power in the case of vertical separation.
11

4

Stage 2: R&D investment

We study the second stage decision for the two possible organizational forms that a firm canadopt. We obtain a firm’s best response function when it is vertically integrated, and when it
is vertically separated. Note that firm i’s best response function does not depend on firm j’sorganizational form, only on firm j’s R&D investment. Of course, the equilibrium decisionof firm i in the R&D stage will depend on the organizational structures of both firms.

4.1

Best response function of a vertically integrated firm

When firm i is vertically integrated, the two sectors in firm i are organized as one entity
and they make the R&D decisions together. The first-order condition of the maximizationof Πi (ei , ej ) with respect to ei isΠii (ei , ej ) = Vii (ei , ej ) − Iii (ei , ej ) = 0.

(2)

From equation (2), and assuming that the solution is interior, we can obtain the best response
function of firm i to any R&D eﬀort of the rival that we denote eIi (ej ), where the superscript
refers to the type of organizational structure (here, Integration).

4.2

Best response function of a vertically separated firm

When firm i is vertically separated, the two sectors negotiate a lump-sum transfer Fi fromthe downstream to the upstream sector in firm i. Because the agreement is negotiated after
the R&D eﬀort takes place, the R&D cost is sunk. If the negotiation succeeds, the upstream
sector’s profit is Fi − I i (ei , ej ) and the downstream sector’s profit is V i (ei , ej ) − Fi . If thenegotiation fails, then the upstream sector’s profit is −I i (ei , ej ) and the downstream sector’s
profit is V̄ i (ej ). Thus, if the negotiation succeeds, the net surplus of the negotiation isV i (ei , ej ) − V̄ i (ej ).
We assume that, given (ei , ej ), Fi is the outcome of a (possibly asymmetric) Nash bargaining where the bargaining power of the separated upstream sector in firm i is β i , with12

β i ∈ (0, 1), exogenously given.7 Thus, the fixed fee Fi is characterized by(1 − β i )(Fi − 0) = β i (V i (ei , ej ) − Fi − V̄ i (ej ))orFi = β i (V i (ei , ej ) − V̄ i (ej )).

Anticipating the terms of the agreement, firm i’s upstream sector maximizes the following
expression with respect to ei :Fi − I i (ei , ej ) = β i (V i (ei , ej ) − V̄ i (ej )) − I i (ei , ej ).
The maximization problem is equivalent tomax β i V i (ei , ej ) − I i (ei , ej ).
ei(3)

The functional form diﬀers from that in which firm i is integrated.8 The coeﬃcient of
V i (ei , ej ) is β i in the separation case, while it is 1 in the integration case. Given that β i < 1,
the incentive of a separated upstream sector to invest in R&D is weaker than that of an
integrated firm.
The first-order condition of the previous maximization in (3) is
β i Vii (ei , ej ) − Iii (ei , ej ) = 0.

(4)

If the solution is interior, equation (4) provides the best response function eSi (ej ) of a vertically separated firm i to any investment level of the rival (superscript indicates Separation).
By comparing (2) and (4), we obtain the following lemma.
7

β i is likely to be large if the upstream sector is more important than the downstream sector in termsof size or weight in the production process. β i is also more likely to be large if the upstream sector is lesscompetitive (e.g., there are fewer firms) than the downstream sector.
8What matters for our result is how a firm can credibly commit to underinvest in R&D. Adopting a
vertical separation is a credible option, easy to verify by the rival. Integration with less than fully centralized decision-making and accountability, causing the upstream sector to be less than fully accountable fordownstream benefits from R&D, could also be an option if those features are verifiable for the rival. In thissense, we could also interpret β i as the preestablished share of the final profit. If the upstream sector infirm i gains β i (V i (ei , ej ) − V̄ i (ej )), which is contingent on the R&D outcomes, from the R&D input, β i isinterpreted as the profit-sharing ratio of the upstream sector in firm i.

13

Lemma 1 Given ej , eSi (ej ) is always lower than eIi (ej ). In addition, eIi (ej )−eSi (ej ) decreases
with β i .
Lemma 1 states that vertical separation is a credible commitment to investing less in R&D.

4.3

R&D equilibrium

∗
∗
BR ∗
The equilibrium R&D decisions (e∗1 , e∗2 ) are the ones that satisfy eBR
1 (e2 ) = e1 and e2 (e1 ) =IBRSe∗2 , where eBRi (ej ) = ei (ej ) or ei (ej ) = ei (ej ) depending on whether firm i is verticallyintegrated or separated, for i, j = 1, 2 and j ̸= i.

5Organizational structure decision

In this section, we first investigate the basic properties of the iso-profit curve in the eﬀort
space and the investment reaction function. Then, using the characteristics of these functions, we classify the market structures into four cases. Finally, we further investigate theequilibrium organizational structures using parametric examples.

5.1

Iso-profit curve and reaction function

As we have seen in the previous section, vertical separation induces a distortion in the
incentive to invest in R&D, whenever the bargaining power of the upstream sector is smaller
than 1. To understand whether this distortion is profitable for the firm, that is, whether
vertical separation is preferable, it is useful to analyze the slope of the iso-profit curve andthe slope of the reaction function eIi (ej ) at the equilibrium (e∗1 , e∗2 ).
First, we examine the properties of the firms’ iso-profit curves. For a certain profit levela ≥ 0, the iso-profit curve of firm i is defined as {(e1 , e2 ) ∈ R2+ : Πi (e1 , e2 ) = a}. Totallydiﬀerentiating Πi (e), we obtain
Πii dei + Πij dej = 0.

14

From this expression, we compute the slope of the iso-profit curve:dejΠi= − ii .
deiΠjNote that Πii > 0 for ei < eIi (ej ) and Πii < 0 for ei > eIi (ej ).
The second derivative of the iso-profit curve is
(
)
dej
d
Πiii Πij − Πii Πijidei.
=−( i )2deiΠj(5)

By (A4) we know that Πiii < 0. Moreover, at the optimal eﬀort eIi (ej ) chosen by an integrated()
firm i, Πii eIi , ej = 0, Therefore, by (5), at this best reply, the iso-profit curve of firm i isconcave with respect to ei if and only if Πij < 0; and it is convex if and only if Πij > 0.
The convexity of firm i’s iso-profit curve is equivalent to the fact that firm j’s investments
benefit firm i. Conversely, the concavity of firm i’s iso-profit is equivalent to having firm j’s
investments harm firm i.
Second, to find the slope of the reaction function eIi (ej ), we substitute ei by eIi (ej ) into(2) and diﬀerentiate the expression with respect to ej to obtainor∂ 2 Πi (ei , ej ) deIi (ej ) ∂ 2 Πi (ei , ej )
+
=0
∂e2i
dej
∂ei ∂ejΠiij (ei , ej )deIi (ej )
=− i
.
dej
Πii (ei , ej )Because the denominator of the latter fraction is negative, we see thatdeIi (ej )⋛ 0 iﬀ Πiij (e) ⋛ 0.
dej

(6)

Thus, the cross-partial derivative of the net profit is the key factor related to the slope of thereaction function. Using a similar argument, we can study the slope of the reaction function
under separation, eSi (ej ):deSi (ej )⋛ 0 iﬀ Πiij (e) ⋛ 0.
dej
15

(7)

The two inequalities in (6) and (7) mean that firms’ investment levels are strategic substitutes
if and only if the cross-partial derivative of the net profit is negative. Furthermore, firms’
investment levels are strategic complements if and only if the cross-partial derivative of the
net profit with respect to the R&D eﬀorts is positive.
To clarify the relationship between spillovers and the slope of the reaction functions,ei (ej ), consider a scenario without spillovers on investment, that is, without spillovers via
I i (ei , ej ). Here, the cross-partial derivative of the profit with respect to the R&D eﬀorts only
depends on their eﬀect on net profits via the market competition and the cost function thatmay present spillovers on production (e.g., d’Aspremont and Jacquemin, 1988). Now, we
add R&D spillovers on investment to this scenario. Positive (resp. negative) R&D spillovers
on investment (adding a positive (resp. negative) term to the eﬀect via net profits) are more
likely to make the slope of the reaction function ei (ej ) positive (resp. negative). However,
the existence of R&D spillovers on investment is not a suﬃcient determinant of the slope ofthe reaction function ei (ej ) when there are spillovers on production.

5.2

Possible cases

From the previous analysis, four cases can arise depending on the signs of Πij and Πiij . To
illustrate the classification, and to see how it may aﬀect the choice of the organizational
structure at stage 1, we present four graphical examples in Figure 1.
[Figure 1 about here]
We discuss the four cases separately.
Case 1. The iso-profit curve of each firm is convex with respect to its R&D eﬀort and thereaction function of each firm is downward sloping.
In this case, the rival’s investment helps the firm (i.e., it increases the convexity of the isoprofit curves), and this eﬀect is stronger if the spillovers on production are high. Furthermore,16

R&D eﬀorts are strategic substitutes. Therefore, at stage 1, each firm faces the following
trade-oﬀ. On the one hand, a high R&D eﬀort directly results in a better position in themarket. Such a high R&D eﬀort can be achieved by adopting vertical integration (Lemma
1). On the other hand, each firm can increase the rival’s R&D investment by committing
to a lower R&D eﬀort (strategic substitution). Such beneficial commitment is attainable byadopting vertical separation.
The trade-oﬀ between the two eﬀects will depend on (i) the size of the spillover eﬀects
and (ii) the bargaining power that aﬀects the shift of the reaction function under verticalseparation. For the first aspect, we should expect vertical integration if those spillovers are
low and vertical separation if they are high. For the second aspect, we should expect vertical
integration if the bargaining power of the upstream sector is weak, and vertical separation
if it is strong.
For the sake of argument, let us consider a scenario where β i = 1. In this case, separationand integration are equivalent. When β i marginally decreases from β i = 1, the cost of verticalseparation in terms of the direct eﬀect is negligible because the loss from lowering firm i’s
R&D level is marginal around the neighborhood of the optimal R&D eﬀort under vertical
integration, β i = 1. Contrary to the marginal loss, the strategic gain from vertical separation
has a first-order eﬀect because vertical separation induces the rival to invest more, helping
firm i because of the convex iso-profit curve. However, as the value of β i decreases, the costof vertical separation increases because the R&D eﬀort under vertical separation diverges
significantly from the optimal R&D level under vertical integration. As a result, the direct
cost of vertical separation outweighs the strategic benefit of vertical separation when β i issmall.
Case 2. The iso-profit curve of each firm is convex with respect to its R&D eﬀort and thereaction function of each firm is upward sloping.

17Here, the rival’s R&D eﬀort helps the firm, and R&D levels are strategic complements.
Each firm directly profits from its R&D eﬀort and also benefits indirectly, because investing
more induces its rival to invest more. In this situation, vertical integration is superior.
Case 3. The iso-profit curve of each firm is concave with respect to its R&D eﬀort and thereaction function of each firm is downward sloping.
If a rival’s R&D investment harms the firm and R&D eﬀorts are strategic substitutes,
then the direct and indirect eﬀects of investment will make aggressive investment attractive
for firms. Such an aggressive investment is attainable by adopting vertical integration atstage 1.
Case 4. The iso-profit curve of each firm is concave with respect to its R&D eﬀort and thereaction function of each firm is upward sloping.
In this case, the rival’s R&D investment harms the firm, and R&D eﬀorts are strategic
complements. Then, a higher investment level has a direct eﬀect on the firm’s competitiveness but induces the rival to invest more. The latter indirect eﬀect harms the firm. As in
Case 1, there are two forces at work, and the intuition follows the same lines. The decision ofeach firm regarding vertical separation or integration at stage 1 depends on (i) whether the
degree of strategic complementarity is high or low, and (ii) whether the upstream sector’s
bargaining power is weak or strong. For high enough values of β i , the optimal decision is tovertically separate, while for low values of β i , the optimal organization is integration.
We summarize the above discussion in the following theorem (the proof is relegated to
the Appendix).
Theorem 1 In Cases 2 and 3, regardless of the rival firm’s vertical structure, a firm prefers
vertical integration to vertical separation. In Case 1 (resp. Case 4), there exists a β̂ki ∈(0, 1) (resp. β̃ki ∈ (0, 1)) such that for firm i, vertical separation is preferable if β i ≥ β̂ki
18

(respectively, β i ≥ β̃ki ) and the organizational structure of firm j is k ∈ {I, S}, wherei, j = 1, 2, j ̸= i, and I and S represent vertical Integration and Separation.
Note that for Case 1 (resp. Case 4), Theorem 1 shows the existence of β 1 < 1 at whichfirm 1 prefers vertical separation to integration under assumptions (A1) to (A4). However,for the existence of the unique threshold value β̂k1 (resp. β̃k1 ), a more demanding conditionon the profit function is needed to ensure the global convexity of firm 1’s iso-profit curves.
This condition is guaranteed under the assumption Πiii Πij < Πii Πiji .
We can compare our taxonomy with the one proposed by Fudenberg and Tirole (1984).
One dimension, the slope of the reaction curves, is the same. For the other dimension, we
can interpret the cases where the R&D investment of a firm helps the rival as a case where
the investment makes the firm that invests “soft,” and the cases where the R&D investmentof a firm harms the rival as a case where the investment makes the firm “tough.”9Remark (Possibilities of nonexclusive vertical chains)

Although we have considered

exclusive vertical chains, let us briefly discuss a scenario in which a separated upstream sector
in a vertical chain can supply to the downstream sector of the rival vertical chain (we call thisthe “dual supply capability”). Assume that each upstream sector incurs adjustment costs,
τ , to supply to the downstream sector of the rival vertical chain, because of the firm-specificrelationship.
The game discussed here also has three stages. In the first stage, firms simultaneously
and noncooperatively determine their organizational structures: vertically separated or integrated. In addition to the decision, if a firm vertically separates, it also determines whether
the relationship between the upstream and downstream sectors is exclusive or not.10 The
9Using Fudenberg and Tirole’s terminology, Case 1 corresponds to “lean and hungry,” Case 2 to “fatcat,” Case 3 to “top dog,” and Case 4 to “puppy dog.” However, these very intuitive names (inspired by asituation where the incumbent prevents or accommodates an entry) are not well suited for our setup.
10

Milliou (2008) considers a downstream monopoly model with an endogenous choice of exclusive vertical19

exclusive relationship is the same as that in the main model. The nonexclusive relationshipmeans that the upstream sector can supply to the competing downstream sector in the rival
vertical chain. In the second stage, each of the two upstream sectors simultaneously andnoncooperatively chooses their own R&D eﬀort ei (i = 1, 2). Then, after the upstream sectors make their R&D investments, if firm i (i = 1, 2) vertically separates, the two sectors infirm i negotiate a lump-sum transfer Fi from the downstream to the upstream sector in firm
i. Note that if both firms vertically separate, the disagreement profit of the downstream
sector in firm i diﬀers from that in the main model because it can potentially procure fromthe upstream sector in the competing vertical chain. The disagreement profit of this downstream sector in firm i is V̄ i (ej ) = V i (ej − τ, ej ) − fi or V̄ i (ej ) = V i (0, ej ), which reflects theeﬀort, ej , the adjustment cost of the upstream sector in firm j, τ , and an oﬀ-path lump-sum
payment for the upstream sector j, fi . We assume that τ is not small enough such thatmonopolization of the upstream market is not profitable for separated upstream sectors.
Finally, each of the downstream sectors competes with its rivals in the final good market byusing their available technological improvements.
Because both V i (ej − τ, ej ) − fi and V i (0, ej ) are independent of ei , the maximization
problem of the upstream sector of firm i (i = 1, 2) in the second stage is the same as that in
the main model (see (3)). Therefore, the possibility of a nonexclusive relationship does notchange the R&D investment incentives of the two upstream sectors.

5.3

Illustrating examples

To illustrate the analysis as a function of the nature of the spillovers on investment andon production, we consider a market where two identical firms compete in quantities in amarket with the following inverse demand function:p=3− q 1 − q2 ,2

relationship.

20

(8)

where qi is the quantity supplied by firm i (i = 1, 2). We allow for diﬀerent combinations of
R&D spillovers, and analyze the equilibrium of the three-stage game.
The R&D process produces a reduction in the marginal cost. As a result of the R&D,
the marginal cost of firm i, ci , is given asci (ei , ej ) = 1 − (ei + θej ),(9)where ei and ej are the R&D eﬀort of firms i and j (i, j = 1, 2, j ̸= i), and θ ∈ [0, 1] is an
exogenous parameter. The first term of ci (ei , ej ) is an ex ante marginal cost level (normalizedto 1); the second term is the degree of marginal cost reduction, which includes the degreeof R&D spillovers on production, measured by θ. Note that ci (ei , ej ) in (9) is essentiallyequivalent to the expression for the marginal cost in d’Aspremont and Jacquemin (1988).
We assume that the R&D investment cost of firm i is given as
I i (ei , ej ) = 2e2i − δei ej ,(10)

where δ ∈ [−1, 1] is an exogenous parameter measuring the spillovers on investment.
In these scenarios defined by (θ, δ), the gross profit of firm i in the second stage is givenas
V i (ei , ej ) =(3/2 + cj (ei , ej ) − 2ci (ei , ej ))2
.
9

(11)

The net profit of firm i in the first stage is given as
Πi (ei , ej ) = V i (ei , ej ) − I i (ei , ej )=(1/2 − (ej + θei ) + 2(ei + θej ))2− (2e2i − δei ej ).
9(12)

Before presenting parametric examples of the cases listed in Section 5.2, let us consider
the situation where there is no spillover in any stage (on production or on investment), thatis, θ = δ = 0. Without spillovers, the iso-profit curve of each firm is concave with respect
to its R&D eﬀort, and the reaction function of each firm is downward sloping, a situation21

of strategic substitutes. This scenario belongs to Case 3, where for any bargaining power of
separated upstream sectors, the firms vertically integrate in equilibrium.

The shapes of iso-profit curves and reaction functions

Following the classificationin Section 5.2, in what follows we discuss the decisions of the two symmetric firms on the
scenarios generated by the parameter values summarized in Table 2.
[Table 2 about here]
In these four scenarios, we will consider symmetric firms, both in terms of the natureand the magnitude of the spillovers θ and δ (which define the case a firm is facing) and withrespect to their bargaining power. After we explain the four cases, we briefly discuss what
happens if the spillovers the firms generate are not the same. Then in Section 5.3.3, we
present in one of the scenarios the equilibrium organizational structure for any (β 1 , β 2 ) toillustrate the eﬀect of the bargaining power.
5.3.1

Parametric examples of Cases 1 and 2 (θ = 1)

We first consider the extreme case where R&D spillovers on production are positive and large,θ = 1, for example, because R&D outcomes stem from the two firms’ open innovation. In
addition, we assume that R&D spillovers on investment are negative, implying that the firm
R&D eﬀort gives an important disadvantage to the rival. As an example of this combinationof R&D spillovers on production and R&D spillovers on investment, one can think of anopen innovation that requires a scarce labor force (e.g., programmers with high IT skills).
In this scenario, innovation has a public good nature, and at the same time, it increases the
R&D costs of the rival through the labor market competition for high-skill programmers.
We illustrate that Case 1 emerges when the negative R&D spillovers on investment areimportant. In contrast, when the negative spillovers on investment are very small, then thesituation belongs to Case 2.
22

From (9) and (10), the marginal cost of production and R&D cost of firm i are given as
ci = 1 − (ei + ej ), I i (ei , ej ) = 2e2i − δei ej ,(13)

where we assume δ ≤ 0. In scenario (13), firm j’s R&D eﬀort improves the eﬃciencies offirms j and i equally. In other words, the cost-reducing eﬀorts of the firms are a public good.
Furthermore, there are negative spillovers on investment, δ ≤ 0, in such a way that the R&D
eﬀort of a firm increases the R&D cost of the rival.
For any δ ≤ 0, the iso-profit curve of firm i is convex with respect to ei . In contrast,
the slope of the reaction function depends on the magnitude of the negative spillovers on
investment, δ. If the magnitude is strong (resp. weak), in particular, if δ < −2/9 (resp.
−2/9 ≤ δ ≤ 0), the reaction function is downward (upward) sloping.11 A substantiallynegative δ means that the substitutability of the R&D eﬀorts is large, implying that the
R&D eﬀorts of the firms are strategic substitutes.
When the degree of negative spillovers on investment is significant, the market structure
belongs to Case 1. As shown in Theorem 1, the equilibrium vertical structure depends on
the bargaining power β (β = β 1 = β 2 ) in case of separation. If β is suﬃciently close to 1, thecombination of the positive spillovers on production and the negative spillovers on investment
induces the firms to adopt vertical separation as a credible commitment to engage in lower
R&D eﬀorts, which cause the rivals to invest more.
When the degree of negative spillovers on investment is small, say, for instance, δ = 0,then the situation with perfect spillovers on production à la d’Aspremont and Jacquemin
11

The net profit of firm i is
Πi =(1 + 2ei + 2ej )2− (2e2i − δei ej ).
36

The first-order and cross-partial derivatives are∂Πi
1 − 34ei + (2 + 9δ)ej∂ 2 Πi2 + 9δ
=
,
=
.
∂ei
9
∂ei ∂ej
9
The cross-partial derivative is negative (resp. positive) if and only if δ < −2/9 (resp. −2/9 < δ ≤ 0).

23

(1988) belongs to Case 2. In this case, for any bargaining power in the case of separation, thefirms adopt vertical integration in equilibrium. Their R&D eﬀorts translate into a universal
reduction in production costs, and R&D investment levels are strategic complements. The
firms employ vertical integration to commit to investing more credibly.
Proposition 1 When the market is defined by equations (8) and (13), and for suﬃciently
large β = β 1 = β 2 , then in equilibrium, the firms adopt vertical separation if δ < −2/9;
otherwise, they select vertical integration.
5.3.2

Parametric examples of Cases 3 and 4 (θ = 0)

Now we consider situations where there are no R&D spillovers on production (a setting withstrong patent protection of the innovation, and firms have diﬀerent production processes) andthe spillovers on investment are positive. For example, imagine that the firms’ production
technologies are considerably diﬀerent, and they use diﬀerent production processes so thata firm cannot benefit from the rival’s innovation. Nevertheless, a firm can profit from therival’s R&D ideas and eﬀort to improve its own R&D process. Then, for low enough positive
spillovers on investment, the situation belongs to Case 3, while Case 4 emerges when thepositive R&D spillovers on investment are strong enough.
From (9) and (10), the marginal cost and investment cost of firm i are given as
ci = 1 − ei , I i (ei , ej ) = 2e2i − δei ej ,(14)

where now we assume δ ≥ 0, in such a way that firm j’s R&D eﬀort aﬀects only its marginal
production cost but does not aﬀect the marginal production cost of firm i. In other words,
the cost-reducing eﬀorts of the firms are private goods. Second, there are positive spillovers
on investment, δ > 0, in such a way that the R&D eﬀort of a firm decreases the R&D
investment cost of the rival.
In all these situations, that is, for all δ ≥ 0, the iso-profit curve of firm i is concavewith respect to ei . As in Section 5.3.1, the slope of the reaction function depends on the24magnitude of the positive spillovers on investment, measured by δ. If the positive spillovers
are weak (resp. strong), in particular 0 ≤ δ ≤ 4/9 (resp. δ > 4/9), the reaction function is
downward (resp. upward) sloping.12 A higher δ indicates that the complementarity of thefirms’ R&D eﬀorts through the positive spillovers is larger. This relation implies that thecomplementarity overrides the substitutability induced by market competition. As a result,the firms’ R&D eﬀorts become strategic complements for high values of δ.
When positive spillovers on investment are small, say, for example, δ = 0 (standard
Cournot competition without any spillovers), the situation belongs to Case 3. Verticalseparation merely diminishes the R&D investment level of the separated firm, which weakens
its competitiveness but strengthens the rival’s competitiveness.
When the positive R&D spillovers on investment are significant, the situation belongs
to Case 4. The considerable spillovers on investment mean that a decrease in ei induces
a reduction in ej . As indicated in Theorem 1, the equilibrium vertical structure depends
on the bargaining power β (β = β 1 = β 2 ) in case of separation. For a suﬃciently large β,
the combination of the positive spillovers on investment and the eﬀect through the Cournot
market interaction induces the firms to adopt vertical separation, because choosing a vertically separated organization works as a credible commitment to engage in lower R&D eﬀorts,
which induces the rival to invest less.
Proposition 2 When the market is defined by equations (8) and (14), and for suﬃciently
large β = β 1 = β 2 , then in equilibrium, the firms adopt vertical separation if δ > 4/9;
12

The net profit of firm i is
Πi =(1 + 4ei − 2ej )2− (2e2i − δei ej ).
36

The first-order and cross-partial derivatives are∂Πi
2 − 28ei − (4 − 9δ)ej∂ 2 Πi4 − 9δ
=
,
=−
.
∂ei
9
∂ei ∂ej
9
The cross-partial derivative is negative (resp. positive) if and only if δ < 4/9 (resp. 4/9 < δ ≤ 1).

25

otherwise, they choose vertical integration.
In sum, comparing the results obtained in Sections 5.3.1 and 5.3.2, we conclude that(i) significantly positive spillovers on production with significantly negative spillovers on
investment (Case 1) and (ii) significantly large positive spillovers on investment without
spillovers on production costs (Case 4) lead to similar outcomes. In those situations, firms
choose vertical separation if the bargaining power parameter in the case of separation is
suﬃciently large.
Additional remarks

To complete the discussion, let us make two remarks.

First, in the previous examples, firms are identical in all respects. It can be the casethat the nature and the magnitude of the parameters representing R&D spillovers, θ and δ,(which define the case a firm is facing) diﬀer for firms 1 and 2.
Let us consider the same spillover structure as before,ci = 1 − (ei + θi ej ) and I i (ei , ej ) = 2e2i − δei ej ,assume that δ = −1/2, and let us comment on three combinations of spillovers on production:
symmetric medium and large spillovers, θ1 = θ2 = 1/2 and θ1 = θ2 = 3/4, and asymmetricspillovers (hence asymmetric firms) θ1 = 3/4 and θ2 = 1/2.13
For θ1 = θ2 = 1/2, both firms are in Case 3, and both are better oﬀ by adopting verticallyintegrated structures. For θ1 = θ2 = 3/4, both firms are in Case 1, and at least one of thefirms has an incentive to vertically separate if β i > 0.942 (i = 1, 2). In the asymmetriccase, θ1 = 3/4 and θ2 = 1/2, firm 1 is in Case 1 and the other firm is in Case 3. Firm 1
has an incentive to vertically separate if β 1 > 0.939, while firm 2 prefers to employ vertical
integration. Hence, we expect that a firm that benefits from a higher spillover from therival’s R&D eﬀort on its production is more likely to choose vertical separation if in addition
there are negative spillovers on investment.
13

The Mathematica file used to derive the results in this remark is available upon request.

26

Second, Cohen et al. (2002) find that R&D spillovers are significant and also thatspillovers are greater in Japan than in the US. They identify the diﬀerences in patent policy
as a key reason for greater R&D spillovers in Japan, where there is an important strategic
use of patents, particularly for negotiations. The finding in Cohen et al. (2002) supports
the diﬀerence between the vertical structures in Japan and the US because R&D eﬀorts are
strategic substitutes. The asymmetric degrees of spillovers lead to diﬀerent organizational
strategies in Japan, with large spillovers (Case 1), and in the US, with lower spillovers (Case3).
Finally, we have framed the example in a market where firms compete in quantities.
Note that one can obtain qualitatively similar results under price competition with thelinear demand system (Dixit, 1979; Singh and Vives, 1984).14
5.3.3

Generalized values of β i in Case 1

Which of the four cases that appears depends only on demand, and the production and
R&D cost parameters (including spillovers). In Cases 2 and 3, in which the two firms
employ vertical integration to invest more, the value of β i does not matter in the decisions ofvertical structure in equilibrium. In contrast, in Cases 1 and 4, in which vertical separation
may work as a commitment device to invest less, the parameter β i is a critical parameterthat influences the decisions of vertical separation.
To illustrate how the bargaining power aﬀects the vertical structure, here we consider amarket with θ = 1, δ = −1, and any β i ∈ [0, 1] and we check how the values of β 1 and β 2
influence the equilibrium vertical structure. Except for the relaxation of the two parameters
14

In our model, we consider a lump-sum transfer within separated sectors in a vertical chain. The
diﬀerence between price and quantity competition is weaker because in a vertical chain the wholesale priceis always equal to the marginal cost (note that the price scheme is related to t = 0 in equation (1) in
Chen and Sappington (2010)). In other words, the wholesale price of each vertical chain does not work as
a strategic commitment to downstream behavior. Therefore, the strategic eﬀect of vertical separation isdiluted compared with the approach of the related works (e.g., Bonnano and Vickers, 1988; Gupta, 2008;
Gupta and Loulou, 1998).

27

β i and the imposition of δ = −1, we use the same parameters as in Section 5.3.1. Using a
numerical analysis, we obtain Figure 2.15
[Figure 2 about here]
Result 1 summarizes Figure 2, where the functions βSj (β i ) for i, j = 1, 2 are increasing
and such that βSj (1) = 0.648, and βSj (0.629) = 0.629.
Result 1 When the market is defined by equations (8) and (13), and for δ = −1,
1. if β i > 0.648 and β j > β̂Sj (β i ) for i, j = 1, 2 (j ̸= i) where β̂Si (β j ) for i, j = 1, 2 (j ̸= i)
is in Figure 2, then the firms vertically separate (area B in Figure 2);
2. if β i > 0.648 and β j < β̂Sj (β i ), then firm i separates and firm j integrates (areas Oi(i = 1, 2) in Figure 2);
3. if β̂Si (β j ) ≤ β i ≤ 0.648 for β i > 0.629 (i, j = 1, 2, j ̸= i), there are two possible
equilibria in which either the two firms separate or the two integrate (area BN in
Figure 2);
4. otherwise, in equilibrium, no firm separates (area N in Figure 2).
Result 1 shows how the equilibrium organizational structure varies as a function of thevalues of bargaining power, (β 1 , β 2 ), in the case of separation. Part 2 of Result 1, in particular, shows that when β 1 and β 2 are very diﬀerent, asymmetric organizational structuresarise. We apply the logic in Result 1 based on Case 1 to a reference scenario based on Case 4:
the diﬀerences between the upstream sectors’ bargaining powers may explain that diﬀerent
firms adopt diﬀerent vertical organizations.
Result 1 is complementary to the explanation in Matsushima and Mizuno (2013) whoshow that vertical separation is more likely to appear in equilibrium if the bargaining power of
15

The Mathematica file used to derive Figure 2 is available upon request.

28

independent complementary input suppliers is high. The existence of multiple suppliers is the
key ingredient that leads to vertical separation in Matsushima and Mizuno (2013). Contrary
to their setting, we do not assume multiple inputs but formulate the interaction between
upstream sectors in two competing vertical chains, and show that technological spillover
can be a driving force of vertical separation. The consideration of technological spillover isplausible, given that there are suppliers’ associations in the Japanese automobile industry(see Sako, 1996; Aoki and Lennerfors, 2013). In addition, some major parts suppliers havejoined multiple suppliers’ associations (Sako 1996, p. 656), which fosters mutual learning
among parts suppliers (Sako, 1996, p. 664).
As summarized in Matsushima and Mizuno (2013, Section 5.1), Japanese suppliers havestronger bargaining power over automobile assemblers than their US counterparts. Furthermore, the degrees of vertical integration in those manufacturers are smaller than those in
their US counterparts (Cusumano and Takeishi, 1991).
In sum, our paper provides another plausible explanation for the diﬀerence between theorganizational structures of Japanese and US automobile assemblers. The diﬀerences of
the two countries between the degrees of R&D spillovers and between the upstream sectors’
bargaining power are the key reasons for the diﬀerence between the organizational structuresof Japanese and US automobile assemblers.
5.3.4

Discussion

In Sections 5.3.1–5.3.3, we consider spillovers on investment to investigate market environments in which vertical separation can occur. We briefly discuss a duopoly competition
without spillovers on investment to explain that spillovers on investment in themselves are
not essential for vertical separation to occur.
Let us consider the inverse demand in (8) and the following marginal cost and investment

29

cost functions of firm i:
ci = 1 − (ei + ej )1/5 and I i (ei , ej ) = 2e2i .

(15)

In this scenario, there is no spillover on investment (as in Cases 2 and 3 in Table 2, δ = 0).
Furthermore, the eﬀorts of the firms have a public good nature (as in Section 5.3.1, θ = 1)
in the sense that ei reduces the marginal cost of firm j as much as that of firm i. However,
the marginal cost is not linear in ei and ej . Amir et al. (2003) use a similar functional form
to discuss a general form of R&D spillovers on production in a standard duopoly model.
For functional forms of the type (e1 + e2 )α with α < 1, the marginal cost is concave withrespect to ei . The concavity substantially influences the incentives of the firms to reduce
their marginal costs. When the value of α is small enough (the concavity of (e1 + e2 )α is
significant), as is the case for α = 1/5, only one of the firms’ eﬀorts is suﬃcient to achievea reasonable level of cost reduction. This implies that the iso-profit curve is convex withrespect to its investment level. In addition, the reaction curve is downward sloping, in sucha way that this scenario belongs to Case 1.16 As a consequence, following Theorem 1 andalong with Proposition 1, firms adopt vertical separation if the upstream sectors’ bargaining
power is large enough.
Result 2 When the market is defined by equations (8) and (15),
1. if β 1 > β̂S1 (β 2 ) and β 2 > β̂S2 (β 1 ), where β̂Si (β j ) is in Figure 3, the firms verticallyseparate (area B in Figure 3);
2. if β i > 0.456 and β j ≤ 0.456 or if β i > β̂Si (β j ) and 0.456 < β j ≤ β̂Sj (β i ), then firm ivertically separates and firm j vertically integrates (area Oi (i = 1, 2) in Figure 3);
16

Recall that in Section 5.3.1, with linear production costs, full spillovers on production (θ = 1), and no
spillovers on investment (δ = 0), the reference case is Case 2. Then, the only equilibrium is where both firms
vertically integrate.

30

3. if 0.456 < β i ≤ β̂Si (β j ) for i, j = 1, 2 (j ̸= i), there are two possible equilibria in which
one of the two firms separates and the other integrates (area O12 in Figure 3);
4. otherwise, in equilibrium, no firm separates (area N in Figure 3).
This illustrates the importance of the functional form of the market and the robustnessof the conclusions obtained with linear functional forms.17[Figure 3 about here]
The relation between the equilibrium organizational structures and the parameters ofbargaining power, β 1 and β 2 , is similar to that in Proposition 1 except for the existence ofasymmetric equilibria (area O12 ) in Result 2.
The conclusions in Results 1 and 2 diﬀer because of the diﬀerence in how investment
eﬀorts reduce the marginal cost under the two cost structures. In both structures, if a firm’s
opponent vertically separates, adopting vertical separation has three direct eﬀects on thefirm: (i) underinvestment from the viewpoint of the vertical chain; (ii) a lower reductionin production cost; and (iii) a reduction of its R&D cost. The first eﬀect is similar in thetwo cost structures. The second eﬀect is milder in the first cost structure because of thelinearity of the marginal cost. The third eﬀect, the only positive one, is larger in the first
cost structure because of the direct spillover on investment. In addition to these three direct
eﬀects, the adoption of vertical separation has an indirect impact on the rival’s R&D eﬀort.
In essence, the indirect impact enhances the rival’s eﬀort, which recovers the loss from thesecond direct eﬀect and enlarges the spillover on investment in the first cost structure. This
positive indirect eﬀect is larger in the first cost structure because of the linearity of ci .
However, the spillover on investment partially oﬀsets the gain from this positive eﬀect. In
total, adopting vertical separation is less costly in the first cost structure in Section 5.3.3,
17

The Mathematica file to derive Figure 3 is available upon request.

31

in which the marginal cost is linear in eﬀorts. As a result, in the first cost structure, thereis the area BN in which a firm follows its opponent’s adoption of vertical separation, andmultiple symmetric equilibria can emerge.

6Conclusion

In oligopolistic industries, understanding why firms adopt vertical integration or separation
has attracted a great deal of attention. However, we are still far from understanding thewhole picture, and there is no apparent and straightforward reason for firms’ decisions tointegrate or not. Various authors explain the gains that firms will derive from meeting
their input requirements internally or externally (see Section 2). In addition to the market
structure, some studies include process R&D among the determinants, although very few
consider the possible R&D spillovers in the analysis. To our knowledge, no studies investigate
the eﬀects of positive or negative spillovers on the rival’s R&D investment. Our paper isthus a first step toward clarifying the importance of the technological environment when we
consider the equilibrium organizational structures of firms in oligopolistic markets.
We stress the interaction between incentives for R&D, R&D spillovers, and firms’ decisions regarding their organizational structures. In this paper, we consider a general andflexible framework for market competition environments, which includes R&D spillovers on
production and on investment. This framework allows us to present the conditions that determine the firms’ organization decisions, and to understand how the combination of the characteristics of the final good market and those of the R&D process (incentives and spillovers)
makes the strategy of integration superior or inferior to the strategy of adopting verticalseparation.
In our model, the forces at work come from the market conditions (as a function of thedemand and the market competition), how a firm’s R&D decisions influence its production
cost, and how the rival’s R&D eﬀorts aﬀect the firm through spillovers on production and32those on investment. Those factors interact with the fact that the incentives to perform
R&D in a vertical chain are greater under vertical integration.
We show that the equilibrium organizational structure depends on whether the situation
belongs to one of four cases. They depend on whether R&D investments are mutually beneficial or not, and whether investment eﬀorts are strategic substitutes or complements. When
R&D investments are mutually helpful, and investment eﬀorts are strategic substitutes, orwhen R&D investments are not mutually beneficial, and investment eﬀorts are strategic
complements, vertical separation can emerge. In the remaining two cases, however, verticalseparation never occurs in equilibrium. In the former two cases, the degrees of upstream
sectors’ bargaining power in the two vertical chains are critical for a vertical separation to
emerge or not. If the degree of an upstream sector’s bargaining power is suﬃciently high,the vertical chain decides to separate vertically. Conversely, if the degree of an upstream
sector’s bargaining power in a vertical chain is not high, this vertical chain is more likely
to integrate vertically. Interestingly, even in entirely symmetric environments, it can be thecase that two vertical chains adopt diﬀerent organizational structures in equilibrium.
The eﬀects analyzed in our paper may help us to understand other contexts. Instead of
considering firms competing in the production market and making decisions regarding R&D
eﬀorts, one can consider, for instance, advertising expenditures that could have (positive or
negative) eﬀects on the rival’s demand or sales and its cost of advertising. There, verticalseparation could be a way to commit to underinvest in advertising.
Our analysis clarifies that we cannot reach conclusions on which vertical integration strategy is better without considering both technological and competitive perspectives together.
For example, as we have illustrated in Section 5.3, we cannot conclude that the level ofintegration depends negatively on the degree of spillovers in the industry: the technologicalenvironment in the industry matters. Therefore, we need to be cautious regarding the importance of knowledge and R&D spillovers that previous studies on vertical organizations have33not considered. This caveat can be useful for empirical studies on firms’ behavior because
bypassing the R&D decisions and its spillovers may induce researchers to assign to market
competition the realized vertical structure for which R&D investments are accountable.
In this paper, we have omitted several important issues that also determine the firms’
optimal organization decisions. Further research will resolve these diﬃcult questions about
why some firms decide to integrate their sectors, and others separate them.

Appendix
Proof of Theorem 1 In Case 1, irrespective of firm 2’s vertical structure, vertical separation in firm 1 causes a leftward shift of firm 1’s reaction function, inducing a left-upward shiftof the Nash equilibrium because of the downward sloping of firm 2’s reaction function.18 Because the slope of firm 1’s iso-profit curve is zero and the slope of firm 2’s best response curveis negative at the original Nash equilibrium, there is an ϵ > 0 such that (e∗1 − ϵ, ē2 ) is on firm
1’s iso-profit curve passing through the original Nash equilibrium (e∗1 , e∗2 ) and ē2 < ek2 (e∗1 − ϵ),where k ∈ {I, S} represents the organizational structure of firm 2, Integration or Separation.
We obtain Π1 (e∗1 , e∗2 ) = Π1 (e∗1 − ϵ, ē2 ) < Π1 (e∗1 − ϵ, ek2 (e∗1 − ϵ)) because Π12 > 0. Because firm
1’s best response eS1 (e2 ) is continuous and increasing in β 1 for any e2 (see (4)), the above
inequalities imply that there exists β̂k1 < 1, such that firm 1 prefers vertical separation to
integration for any β 1 ∈ [β̂k1 , 1).
In Case 2, irrespective of firm 2’s vertical structure, vertical separation in firm 1 causes
a leftward shift of firm 1’s reaction function, inducing a left-downward shift of the Nash
18

The shift of the Nash equilibrium referred to in this proof is guaranteed by the condition1
dek2 (e1 )<de1deI1 (e2 )/de2(k = I, S),

which implies that irrespective of firm 2’s organizational structure, the slope of firm 1’s best response is
steeper than that of firm 2’s best response. This condition seems innocuous because it guarantees thestability of the Nash equilibrium. The numerical examples presented later satisfy this condition.

34

equilibrium because of the upward sloping of firm 2’s reaction function. We check the eﬀectof the left-downward shift of the Nash equilibrium on firm 1 in two steps. A leftward shiftof the equilibrium point from the original Nash equilibrium decreases the profit of firm 1.
Moreover, a downward shift of e2 further decreases the profit of firm 1 (Π12 > 0). Therefore,a left-downward shift of the Nash equilibrium through vertical separation harms firm 1. We
can apply the same logic to the eﬀect of vertical separation on firm 2.
In Case 3, irrespective of firm 2’s vertical structure, vertical separation in firm 1 causes
a leftward shift of firm 1’s reaction function, inducing a left-upward shift of the Nash equilibrium because of the downward sloping of firm 2’s reaction function. We check the eﬀectof the left-upward shift of the Nash equilibrium on firm 1 in two steps. A leftward shiftof the equilibrium point from the original Nash equilibrium decreases the profit of firm 1.
Moreover, an upward shift of e2 also decreases the profit of firm 1 (Π12 < 0). Therefore, a
left-upward shift of the Nash equilibrium through vertical separation harms firm 1. We canapply the same logic to the eﬀect of vertical separation on firm 2.
In Case 4, irrespective of firm 2’s vertical structure, vertical separation in firm 1 causes
a leftward shift of firm 1’s reaction function, inducing a left-downward shift of the Nashequilibrium because of the upward sloping of firm 2’s reaction function. Because the slope offirm 1’s iso-profit curve is zero and the slope of firm 2’s best response is positive at the original
Nash equilibrium, then there is an ϵ > 0, such that (e∗1 − ϵ, ē2 ) is on firm 1’s iso-profit curve
passing through the original Nash equilibrium (e∗1 , e∗2 ) and ē2 > ek2 (e∗1 − ϵ), where k ∈ {I, S}
represents the organizational structure of firm 2. We obtain Π1 (e∗1 , e∗2 ) = Π1 (e∗1 − ϵ, ē2 ) <Π1 (e∗1 − ϵ, ek2 (e∗1 − ϵ)) because Π12 < 0. Because firm 1’s best response eS1 (e2 ) is continuous andincreasing in β 1 for any e2 (see (4)), the above inequalities imply that there exists β̃k1 < 1,such that firm 1 prefers vertical separation to integration for any β 1 ∈ [β̃k1 , 1).

35

Q.E.D.

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41

Question

PapersMarket structure⇒ Vertical integration

Bonanno and Vickers (1988), Ziss (1995),
Gal-Or (1999), Choi and Yi (2000), Chen
(2001,2005), Lin (2006), Matsushima (2009),
and so on.

Vertical integration⇒ R&D incentives

Bolton and Whinston (1993), Buehler and
Schmutzler (2008), Chen and Sappington(2010), Liu (2016).

Market structure and R&D Gupta and Loulou (1998), Gupta (2008), Lin
technology
et al. (2014), Wang et al. (2017), Our paper.
⇒ Vertical integration

Table 1: Related literature

42

Example ofCase 1Case 2
Case 3
Case 4

θ
1
1
0
0

δ
δ < −2/9−2/9 ≤ δ ≤ 0
0 ≤ δ ≤ 4/9
4/9 < δ

Table 2: Parametric examples for the four cases

43

0.06

0.06

e1 (e2 )0.05

e1 (e2 )

0.05

0.04

0.04

π1 (e1 ,e2 )

0.03

e2 (e1 )π1 (e1 ,e2 )
0.03

e2 (e1 )

0.02

0.02

0.01

0.01

0.00

0.00
0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.00

1. Convex and downward

0.01

0.02

0.03

0.04

0.05

0.06

2. Convex and upward

0.12
0.14

0.10
0.12

e1 (e2 )e1 (e2 )0.08

e2 (e1 )

0.10

0.08

e2 (e1 )

0.06

0.06

π1 (e1 ,e2 )π1 (e1 ,e2 )

0.04
0.04

0.02
0.02

0.00

0.00

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.00

3. Concave and downward
Note

0.02

0.04

0.06

0.08

0.10

0.12

0.14

4. Concave and upward

Horizontal axis: e1 ; Vertical axis: e2 ;The arrow indicates the direction of increasing profit for firm 1.

Figure 1: Iso-profit curves and reaction functions in the eﬀort space defining the four possible
cases

44

β2

O2

1

β̂S1 (β 2 )BO2

β̂S2 (β 1 )β̂I2 = 0.648

BN

0.629
0.5

O1

N

β1
0

0.5 0.6291
β̂I1 = 0.648

Note

B: Both separate; O1 : Only firm 1 separates; O2 : Only firm 2 separates;
BN : Both separate or No firm separates; N : No firm separates.
(The threshold curves β̂Si (β j ) are not straight lines).

Figure 2: Equilibrium vertical structures under Case 1 (θ = 1, δ = −1)

45

β2
1
O2
0.617
0.570
β̂I2 = 0.456

β̂S1 (β 2 )BO2O12 O1N

β̂S2 (β 1 )

O1
β1

0
Note

0.456 0.617
β̂I1 0.570

1

B: Both separate; O1 : Only firm 1 separates; O2 : Only firm 2 separates;
O12 : Only one of the firms separates; N : No firm separates.
(The threshold curves β̂Si (β j ) are not straight lines).

Figure 3: Equilibrium vertical structures under Case 1 (α = 1/5, θ = 1)

46

Online Appendix
Price and quantity competition in the linear demand system
We show the last remark in Section 5.3.2 and footnote 14 hold true. We consider thefollowing inverse demand functions:pi =3a− qi − γqj , i, j = 1, 2, j ̸= i, and γ ∈ [0, 1].
2

The demand functions of the firms areqi =3(1 − γ)a − 2pi + 2γpj
, i, j = 1, 2, j ̸= i.
2(1 − γ 2 )In the following, we show the signs of the key partial diﬀerentials in price competition are
similar to those in quantity competition. Those key partial diﬀerentials are as follows:∂ 2 (πi − I i ) ∂ 2 (πi − I i ),,∂e2i∂ej ∂ei∂(πi − I i ),∂eje∗ ,e∗
1

2

in which πi and I i represent the profit and R&D investment cost of firm i under pricecompetition in Section A and under quantity competition in Section B.

A

Price competition

Suppose that they compete in price. Under the demand system, if γ is not large enoughsuch that both firms are active, the prices and the resulting profits under the marginal costci (i = 1, 2) are
pi =

A.1

3(2 + γ)(1 − γ)a + 2(2ci + γcj )(3(2 + γ)(1 − γ)a − 2((2 − γ 2 )ci − γcj ))2
,
π
=
.
i
2(4 − γ 2 )4(4 − γ 2 )2 (1 − γ 2 )Price competition: Parametric examples of Cases 1 and 2

We set the marginal cost and investment cost of firm i:
ci = a − (ei + ej ), I i (ei , ej ) = 2e2i − δei ej .
1

Those are equation (13) in the Cournot example if a = 1.
Under the functional form, we obtain the following:(1 − γ)a − 2(7 + γ − 6γ 2 + 2γ 3 )ei + (2(1 − γ) + (2 − γ)2 (1 + γ)δ)ej∂(πi − I i )=,∂ei(2 − γ)2 (1 + γ)
(1 − γ)a⇒ e∗i =,
2
4(3 + γ − 3γ + γ 3 ) − (2 − γ)2 (1 + γ)δ
∂(πi − I i )(1 − γ)a + 2(1 − γ)(ei + ej ) + (2 − γ)2 (1 + γ)δei=,∂ej(2 − γ)2 (1 + γ)−2(7 + γ − 6γ 2 + 2γ 3 )∂ 2 (πi − I i )=(< 0),∂e2i(2 − γ)2 (1 + γ)
∂ 2 (πi − I i )2(1 − γ) + (2 − γ)2 (1 + γ)δ
2(1 − γ)=
> 0 if and only if δ > −.
2∂ej ∂ei(2 − γ) (1 + γ)
(2 − γ)2 (1 + γ)
At the equilibrium investment levels, ∂(πi − I i )/∂ej is∂(πi − I i )4(1 − γ)a
=
> 0.
∂ej
4(3 + γ − 3γ 2 + γ 3 ) − (2 − γ)2 (1 + γ)δ
e∗ ,e∗
1

A.2

2

Price competition: Parametric examples of Cases 3 and 4

We set the marginal cost and investment cost of firm i:
ci = a − ei , I i (ei , ej ) = 2e2i − δei ej .
Those are equation (14) in the Cournot example if a = 1.
Under the functional form, we obtain the following:(1 − γ)(2 + γ)(2 − γ 2 )a − 2(28 − 44γ 2 + 17γ 4 − 2γ 6 )ei∂(πi − I i )=∂ei(4 − γ 2 )2 (1 − γ 2 )(2γ(2 − γ 2 ) − (4 − γ 2 )2 (1 − γ 2 )δ)ej−(4 − γ 2 )2 (1 − γ 2 )(2 − γ 2 )a⇒ e∗i =,
2(14 + 8γ − 11γ 2 − 2γ 3 + 2γ 4 ) − (2 − γ)2 (1 + γ)(2 + γ)δ
(1 − γ)γ(2 + γ)a + 2γ((2 − γ 2 )ei − γej ) − (4 − γ 2 )2 (1 − γ 2 )δei∂(πi − I i )=−,∂ej(4 − γ 2 )2 (1 − γ 2 )∂ 2 (πi − I i )2(28 − 44γ 2 + 17γ 4 − 2γ 6 )=−< 0 if and only if γ < 0.9695,∂e2i(4 − γ 2 )2 (1 − γ 2 )∂ 2 (πi − I i )(2γ(2 − γ 2 ) − (4 − γ 2 )2 (1 − γ 2 )δ)2γ(2 − γ 2 )
=−
>
0
if
and
only
if
δ
>
.
∂ej ∂ei(4 − γ 2 )2 (1 − γ 2 )(4 − γ 2 )2 (1 − γ 2 )2At the equilibrium investment levels, ∂(πi − I i )/∂ej is∂(πi − I i )(4γ − (2 + γ − γ 2 )δ)a
=−
<0
∂ej
2(14 + 8γ − 11γ 2 − 2γ 3 + 2γ 4 ) − (2 − γ)2 (1 + γ)(2 + γ)δ
e∗ ,e∗
1

2

if and only if δ < 4γ/(2 + γ − γ 2 ).

B

Quantity competition

Suppose that they compete in quantity. Under the demand system, both firms are active,
and the quantities and the resulting profits under the marginal cost ci (i = 1, 2) are3(2 − γ)a − 2(2ci − γcj )(3(2 − γ)a − 2(2ci − γcj ))2qi =, πi =.
2(4 − γ 2 )4(4 − γ 2 )2

B.1

Quantity competition: Parametric examples of Cases 1 and 2

We set the marginal cost and investment cost of firm i:
ci = a − (ei + ej ), I i (ei , ej ) = 2e2i − δei ej .
Those are equation (13) in the Cournot example if a = 1.
Under the functional form, we obtain the following:∂(πi − I i )a − 2(7 + 8γ + 2γ 2 )ei + (2 + (2 + γ)2 δ)ej,=
∂ei
(2 + γ)2a⇒ e∗i =,
4(1 + γ)(3 + γ) − (2 + γ)2 δ∂(πi − I i )a + 2ej + (2 + (2 + γ)2 δ)ei=,
∂ej
(2 + γ)2∂ 2 (πi − I i )−2(7 + 8γ + 2γ 2 )=(< 0),∂e2i(2 + γ)2∂ 2 (πi − I i )2 + (2 + γ)2 δ
2=> 0 if and only if δ > −.
2∂ej ∂ei(2 + γ)(2 + γ)2
At the equilibrium investment levels, ∂(πi − I i )/∂ej is∂(πi − I i )
4a
> 0.
=
∂ej
4(1 + γ)(3 + γ) − (2 + γ)2 δe∗ ,e∗
1

2

3

The signs of ∂ 2 (πi − I i )/∂e2i and ∂(πi − I i )/∂ej |e∗ ,e∗ in quantity competition under Cases
12

1 and 2 are the same as those in price competition under Cases 1 and 2. Besides, the signof ∂ 2 (πi − I i )/∂ej ∂ei in quantity competition under Cases 1 and 2 is similar to that in pricecompetition under Cases 1 and 2 in the sense that it is positive if δ is large.

B.2

Quantity competition: Parametric examples of Cases 3 and 4

We set the marginal cost and investment cost of firm i:
ci = a − ei , I i (ei , ej ) = 2e2i − δei ej .
Those are equation (14) in the Cournot example if a = 1.
Under the functional form, we obtain the following:∂(πi − I i )2(2 − γ)a − 4(14 − 8γ 2 + γ 4 )ei − (4γ − (4 − γ 2 )2 δ)ej=∂ei
(4 − γ 2 )22a,⇒ e∗i =
2
4(7 + 4γ − 2γ − γ 3 ) − (2 − γ)(2 + γ)2 δ∂(πi − I i )γ(2 − γ)a − 2γ 2 ej + (4γ − (4 − γ 2 )2 δ)ei
= −
,
∂ej
(4 − γ 2 )24(14 − 8γ 2 + 4γ 4 )∂ 2 (πi − I i )
=
−
< 0,
∂e2i
(4 − γ 2 )24
∂ 2 (πi − I i )−4γ + (4 − γ 2 )2 δ
> 0 if and only if δ >.
=22∂ej ∂ei(4 − γ )(4 − γ 2 )2At the equilibrium investment levels, ∂(πi − I i )/∂ej is∂(πi − I i )(4γ − (2 + γ)δ)a
=−
<0
∂ej
4(7 + 4γ − 2γ 2 − γ 3 ) − (2 − γ)(2 + γ)2 δe∗ ,e∗
1

2

if and only if δ < 4γ/(2 + γ).
The sign of ∂ 2 (πi − I i )/∂e2i in quantity competition under Cases 3 and 4 is almost thesame as that in price competition under Cases 3 and 4 except for suﬃciently high γ. Besides,
the signs of ∂ 2 (πi − I i )/∂ej ∂ei and ∂(πi − I i )/∂ej |e∗ ,e∗ in quantity competition under Cases
12

3 and 4 are similar to those in price competition under Cases 3 and 4 in the sense that eachof them is positive if δ is large.
4