The impacts of suppliers and mutual outsourcing onorganizational forms∗
Yasuhiro Arai†
Faculty of Humanities and Social Sciences, Kochi University
Noriaki Matsushima‡
Institute of Social and Economic Research, Osaka University
May 8, 2022

Abstract
We consider a downstream duopoly model with a monopolistic common supplier and mutual outsourcing between the two symmetric downstream ﬁrms. The
market structure captures the recent procurement environment in the smartphoneindustry. We also incorporate managerial delegation into the duopoly model because deciding on organizational forms within a ﬁrm is critical to achieving better
performance in almost all industries. There is an equilibrium in which only one of
the ﬁrms delegates its downstream production to its sales manager. A delegating
ﬁrm becomes less aggressive. The proﬁts when both ﬁrms delegate can be higher
than those when no ﬁrm delegates. Social welfare when both ﬁrms delegate canbe smaller than that when no ﬁrm delegates.

Keywords: Mutual outsourcing, Supplier, Delegation, Asymmetric managerial forms.
JEL Codes: L13, D43.

∗

We have beneﬁted from many constructive comments from the Co-Editor, Dan Bernhardt, andanonymous referees. We are thankful to Michael Kopel for his invaluable comments and suggestions. We gratefully acknowledge the ﬁnancial support from the Japan Society for the Promotionof Science (JSPS), KAKENHI Grant Numbers JP15H03349, JP15H05728, JP17H00984, JP18H00847,
JP18K01593, JP19H01483, JP20H05631, JP21H00702, JP21K01452, and JP21K01541. Although Matsushima serves as a member of the Competition Policy Research Center (CPRC) at the Japan Fair
Trade Commission (JFTC), the views expressed in this paper are solely ours and should not be attributed to the JFTC. The usual disclaimer applies.
†
Yasuhiro Arai, Faculty of Humanities and Social Sciences, Kochi University, 2-5-1 Akebono-cho,
Kochi-shi, Kochi 780-8520 JAPAN. E-mail: yasuhiro68@gmail.com‡
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

The impacts of suppliers and mutual outsourcing onorganizational forms

Abstract
We consider a downstream duopoly model with a monopolistic common supplier and mutual outsourcing between the two symmetric downstream ﬁrms. The
market structure captures the recent procurement environment in the smartphoneindustry. We also incorporate managerial delegation into the duopoly model because deciding on organizational forms within a ﬁrm is critical to achieving better
performance in almost all industries. There is an equilibrium in which only one of
the ﬁrms delegates its downstream production to its sales manager. A delegating
ﬁrm becomes less aggressive. The proﬁts when both ﬁrms delegate can be higher
than those when no ﬁrm delegates. Social welfare when both ﬁrms delegate canbe smaller than that when no ﬁrm delegates.

Keywords: Mutual outsourcing, supplier, delegation, asymmetric managerial forms.
JEL Codes: L13, D43.

1

Introduction

Mutual outsourcing has become a common business practice in technology-intensive
industries, typically the smartphone industry. 1 For instance, Google and Samsung mutually outsource their key components to each other.2 Google manages the Android
operating system, which is a critical component for smartphone operation. Samsung
embeds the Android operating system into its Galaxy smartphone and sells the smartphone to ﬁnal consumers. On the other hand, Samsung produces the Nexus smartphone
for Google as an original equipment manufacturer (OEM). Along with such mutual
outsourcing, in the input market for smartphones, Qualcomm plays an indispensablerole and exerts strong monopoly power over the smartphone makers including Apple,
Google, and Samsung in various ways in the modem chip market. 3 In fact, Qualcomm
held more than half the market share in the modem chip industry in 2017 (Ai and Lu,
2019, p.645).4
Given the current technological environment mentioned above, we consider a downstream duopoly model with a monopolistic common supplier and mutual outsourcing
between the two downstream ﬁrms.5 We also incorporate managerial delegation into the1

Pun (2015) and Milliou and Serfes (2020) provide some real-world examples of mutual outsourcing
in some industries.
2

We borrowed this example from Pun (2015, p.2092).

3

Although Samsung is also a modem chip maker, it has depended on Qualcomm to produce its
high-end smartphones (see, for instance, Srivastava (2016) and Leswing (2019)).
4

Qualcomm demanded that Apple’s OEMs pay royalty fees that are higher than the standardessential patent (SEP) royalty fees (see Ai and Lu, 2019, p.647), although the royalty rates of Qualcomm’s SEPs for 4G cellular technologies are nominally inﬂexible under the committed terms basedon the FRAND (fair, reasonable, and nondiscriminatory) licenses. Any refusal to this demand would
lead to Qualcomm’s refusal to supply its modem chip to the refusing OEM. See the discussions on
antitrust concerns in Hovenkamp (2017) and Hovenkamp and Simcoe (2020).
5

Our analytical framework would also apply to the automobile industry because mutual outsourcing is a common practice. For instance, competitors Nissan and Daimler mutually traded their carcomponents (Pun, 2015). On the one hand, Nissan produced its luxury car, Inﬁnity, by purchasing thefront-wheel-drive architecture platform used in Daimler’s Mercedes. On the other hand, Nissan supplied diesel and gas engines to Daimler. Automobile manufacturers depend on suppliers in oligopolistic1duopoly model, as in Fershtman and Judd (1987), Sklivas (1987), and Basu (1995), andderive the equilibrium managerial structure because deciding on organizational forms
within a ﬁrm is critical to achieving better performance in almost all industries (Fossand Klein, 2014).6
The details of the model are as follows. The downstream ﬁrms compete in quantity,
which is suitable for investigating competition in the manufacturing industries (see,e.g., Sundaram et al., 1996; Plehn-Dujowich and Serfes, 2010). 7 The downstream ﬁrms
mutually outsource their inputs from the rivals under linear input prices, where theper-unit payment is constant with respect to the quantity demanded for input. The
monopolistic input supplier sells its key input to the two downstream ﬁrms under a common linear input price.8 Each downstream ﬁrm produces one unit of the ﬁnal productby combining each one unit of those three inputs.
We consider the following four stage game.9 In the ﬁrst stage, each downstream
ﬁrm’s owner determines whether to delegate its decision regarding the downstream
production to a manager. If the owner delegates, the ﬁrm needs to incur costs ofdelegation, as in Basu (1995). Those costs come from hiring a manager. In the second
stage, the monopoly supplier and the owners set the contract terms in the input market
regardless of the managerial forms determined in the ﬁrst stage. In the third stage,
each owner oﬀers an incentive contract to the manager if she/he delegates in the ﬁrst
stage. The incentive contract in each ﬁrm consists of a linear combination of its proﬁt
industries, including tires, glass, and so on.
6

See excellent surveys on strategic delegation by Sengul et al. (2012), Lambertiti (2017), and Kopel
and Pezzino (2018); empirical analysis of delegation, Colombo et al. (2007) and Bloomﬁeld (2021).
7

For instance, in the smartphone industry, capacity constraints on production in input markets(e.g., modem chip production) can be a bottleneck to producing enough smartphones (see, for instance,
Sohn, 2021, a recent article about the global chip shortage).
8

We obtain a qualitatively similar result even when the supplier can price discriminate against thedownstream ﬁrms.
9

We explain the plausibility of the timing structure in Section 3.

2

and revenue (Fershtman and Judd, 1987). In the fourth stage, the decision-makers in
the downstream market simultaneously set their quantities.
We obatin the following two main results. First, we show that the delegating ownerplaces a negative weight on the revenue in the linear combination of the incentive contract to its manager, inducing the manager to be less aggressive than an owner-managed
ﬁrm. Second, we also show that there are three types of equilibrium managerial structures depending on costs of delegation: (i) no owner delegates; (ii) only one ownerdelegates; (iii) both owners delegate. Moreover, the ﬁrst and the second results imply
that the sum of the quantities supplied in case (i) is the largest. This comparison meansthat the consumer surplus in case (i) is the largest.
We explain the intuition behind the ﬁrst result. The objective of each decisionmaker in the downstream market consists of two parts: (i) a linear combination of
proﬁts and revenues in the downstream market, and (ii) revenue from the input market, which is independent of the decision-maker’s strategic variable (its downstream
quantity). Because the input market revenue and its downstream quantity are independent, the decision-maker sets its quantity without considering the input market
revenue. If an owner delegates, it can internalize the input market revenue by oﬀering
an incentive contract that induces its manager to produce less than an owner-managed
ﬁrm. Speciﬁcally, the owner who delegates sets a negative weight on the revenue inthe linear combination of the incentive contract. Note that the monopolistic supplierdoes not play an important role in establishing the ﬁrst main result, in contrast to therelated papers whose timing structures diﬀer from ours (Szymanski, 1994; Liao, 2010).
We illustrate the intuition behind the second main result. The owner who delegates
enjoys a lower input price oﬀered by the competing owner than if it does not delegate.
This is because the competing owner anticipates that the delegating owner induces
its manager to produce less than if it does not delegate, meaning that the demand3for input produced by the competing owner is smaller. The competing owner’s lower
input price allows the monopolistic supplier to oﬀer a higher input price because of thestrategic substitution in the complementary input market. Thus, delegation strengthens
the monopolistic supplier’s market power, oﬀsetting the delegating owner’s gain fromthe lower input price oﬀered by the competing owner. The eﬀect of such an oﬀset isstrong under an asymmetric managerial structure, making the asymmetric structureattainable.
The logic behind the second result is new in the context of strategic delegation. 10
The second result implies that mimicking the well-performed rival’s organizational form
can be an inconsiderate decision. Related to the result, the well-known facts are that
Samsung is more likely to employ centralized management (Joseph et al., 2016) andthat Google is more likely to employ delegated management (Steiber, 2014). Following
our second result, we can conclude that the coexistence of asymmetric managerial forms
is a considerable consequence. Note that the interaction of the two essential forces (themonopolistic supplier and mutual outsourcing) is the key to obtaining the result because
the negative weight in the incentive contract stems from the ﬁrst force mentioned above.
The remainder of this paper is organized as follows. Section 2 surveys the related
papers. Section 3 constructs the model. Section 4 analyzes the three subgames andshows the main result. Section 5 oﬀers concluding remarks.

2

Related papers

Several papers show the possibility of making managers less aggressive than ownermanaged ﬁrms (e.g., vertical relations (e.g., Szymanski, 1994; Liao, 2010; Habiger and10

Several papers also show asymmetric managerial structures in diﬀerent contexts (Vroom, 2006;
Mujumdar and Pal, 2007; Kopel and Löﬄer, 2008, 2012; Pan et al., 2020). We explain the diﬀerence
between those papers and ours in Section 2.

4

Kopel, 2020), network externalities (e.g., Hoernig, 2012; Lee and Choi, 2018)). 11 Szymanski (1994) incorporates ﬁrm-speciﬁc labor unions into the standard duopoly modelwith delegation. His numerical analysis shows that delegating ﬁrms are less aggressive
than proﬁt-maximizing ﬁrms because these delegating ﬁrms anticipate that the labor
unions extract rents after the determination of incentive contracts. 12 The timing structure in the related papers (e.g., Liao, 2010; Claude, 2018) is diﬀerent from ours. Thoserelated papers consider the following sequence: incentive contracts, input prices, andstrategic variables in the downstream market. The diﬀerent timing structure in thosepapers is the key to deriving such less aggressive behavior.
Although most of those papers with vertical relations do not investigate the equilibrium managerial structure, Choi et al. (2020) is the exception. They investigate
endogenous managerial forms in duopoly models with two exclusive vertical relations.
They show that employing managerial delegation is a dominant strategy, which contrasts with our result. Even if they would also assume that a delegating ﬁrm incurs costsof delegation, as in Basu (1995) and our paper, an asymmetric managerial structure
does not appear.13
Some theoretical studies have investigated how strategic organizational design aﬀects
ﬁrm asymmetry (Vroom, 2006; Mujumdar and Pal, 2007; Kopel and Löﬄer, 2008, 2012;
Pan et al., 2020).14 The stream of this research interprets the organizational design as
11

Liao (2014) and Claude (2018) extend Liao (2010). Fanti and Scrimitore (2019) consider a duopoly
model where a vertically integrated ﬁrm and an independent downstream ﬁrm that procures inputs
from the integrated ﬁrm compete in quantity. Vertical integration is the key in their model to obtaining
the negative weight on the sales volume in equilibrium.
12

Fanti and Meccheri (2013) and Chatterjee and Saha (2017) further investigate the model framework in Szymanski (1994).
We check it by comparing πiddC − πindC (the gross gain from delegation given that the rivaldelegates) and πidnC − πinnC (the gross gain from delegation given that the rival does not delegate) in
their paper, and ﬁnd that (πiddC − πindC ) > (πidnC − πinnC ), implying that an asymmetric managerial
structure does not appear.
13

14

Recently, Macho-Stadler et al. (2021) discuss a downstream duopoly with technological spillovers

5

a commitment device for ﬁrms’ output decisions. Speciﬁcally, when a ﬁrm’s ownerhires a manager with an incentive contract, the owner can make the manager more/lessaggressive through the contract terms than an owner-managed ﬁrm.
Vroom (2006) discusses a duopoly market where a ﬁrm’s owner oﬀers a relativeperformance-based incentive contract to the (downstream) marketing manager. Heshows the possibility that one owner chooses decentralization and the other chooses
centralization. In this asymmetric case, the decentralized ﬁrm acquires an advantage(the relative aggressiveness in the downstream market) and a disadvantage (the double
mark-up problem). The two contrasting eﬀects make the asymmetric organizational
forms attainable. External sourcing is beyond the scope of analysis in Vroom (2006).
Several papers consider duopoly models with sequential choice of production volumes to obtain asymmetric managerial forms.
Mujumdar and Pal (2007) consider a symmetric duopoly with twice the production
opportunities. They show that the ﬁrst mover delegates to a manager with a proﬁtbased incentive contract, while the second mover delegates with a sales-based contract.
The ﬁrst mover does not have an incentive to change the owner’s best response because
she/he chooses her/his quantity based on the second mover’s best response, although
the second mover can beneﬁt from making it more aggressive.
Kopel and Löﬄer (2008) consider the interaction of two commitment devices: processinnovation R&D and delegation to a manager in a Stackelberg duopoly model. Theyshow that both ﬁrms invest in process innovation and that only the follower delegatesto a manager because (i) a Stackelberg leader cannot beneﬁt from the strategic delegation, and (ii) the leader’s cost-reducing investment does not directly aﬀect the follower’s
quantity choice. They also show that the follower can overcome the disadvantage of its

and show an equilibrium asymmetric managerial structure. The key factor is the interaction between
the modes of downstream competition and the strategic nature of R&D investments.

6second-mover position.
Kopel and Löﬄer (2012) consider a duopoly model in which ﬁrms can choose theirorganizational governance and leader–follower roles. They consider the delegation costand show that a ﬁrm chooses to be the market leader and does not delegate, whereas
the other ﬁrm hires a manager. The key factor in the three papers is sequential choicesof quantities.
Recent work by Pan et al. (2020) also shows the possibility that an asymmetricmanagerial structure emerges in a duopoly with quantity competition. The key factor
in their paper is equilibrium multiplicity under the convex inverse demand function
proposed by Ishibashi and Matsushima (2009).15 Under some parameter sets, one of
the ﬁrms does not delegate to a manager in order to avoid the worse equilibrium.
That is, the demand system is the key factor in obtaining an asymmetric managerial
structure.

3

Model

We consider a Cournot duopoly market with a monopolistic supplier. The inverse
demand in the downstream market isp = a − qi − qj ,where p is the price, a is a positive constant, and qi is the quantity supplied by ﬁrm i(i, j = 1, 2; j ̸= i).
Each of the proﬁt-maximizing downstream ﬁrms needs three types of inputs produced by the downstream ﬁrms and the monopolistic supplier (hereinafter we refer tothe latter as “the supplier”). To produce one unit of the ﬁnal product, each downstream ﬁrm uses one unit of each of the three inputs, respectively. We assume that the15

Pan (2018, 2020) are applications of the inverse demand function in Ishibashi and Matsushima(2009) to the problems of endogenous timing and ﬁrm entry, respectively.

7three ﬁrms do not incur any production cost in producing their inputs for notational
simplicity. The total marginal cost of ﬁrm i, ci (i = 1, 2), is the sum of the input
prices imposed by the rival ﬁrm j (j = 1, 2, j ̸= i) and the supplier, rj and w; that is,ci = rj + w. Here, the supplier oﬀers a common input price w to ﬁrms 1 and 2. Figure
1 summarizes the trading structure in the industry.
S
w
r1
1

2

r2
q1

q2

consumers
Figure 1: The market structure
The proﬁts of ﬁrm i and the supplier, Πi and ΠS , areΠi = (Ri − (rj + w)qi ) + ri qj , where Ri = (a − qi − qj )qi , (i, j = 1, 2, j ̸= i),ΠS = w(q1 + q2 ).
Πi (i = 1, 2) is the sum of the proﬁts from the downstream and upstream/input markets.
In the duopoly market, each downstream ﬁrm (owner) determines whether to delegate the downstream production to its manager. Following the tradition in the context
of strategic delegation (Fershtman and Judd, 1987; Basu, 1995), we assume that the
managerial compensation contracts are of the form Ai + Bi gi , where Ai and Bi are aﬁxed wage and a weight on the performance measure gi :gi = αi Πi + (1 − αi )Ri ,where αi is the weight imposed by the owner of ﬁrm i.16 The reservation utilities of the16

Several articles explored cases where a manager’s compensation is determined by the combinationof various parameters (proﬁts and sales quantities (Vickers, 1985), own proﬁts and rival’s proﬁts (SalasFumás, 1992; Miller and Pazgal 2002), proﬁts and market share (Ritz, 2008)). In this paper, we use thetraditional compensation scheme to highlight the new sources of asymmetric managerial structures.

8

managers are assumed to be Y . We assume that the owner can earn proﬁts X by hiring
a manager, where X < Y . Examples of X include money from time saved by the ownerand proﬁt generated from the manager’s expertise.17 We deﬁne F as the diﬀerence
between Y and X, that is, F ≡ Y − X, which becomes a ﬁxed cost of delegation tohire a manager.18
The linear formulation of gi is consistent with the real-world practices summarizedin Murphy (1999) and Kelly and Matthews (2020). Murphy (1999, p.2500) points
out that companies generally use multiple performance measures, which are essentially
additive and separable, in their incentive plans. Recently, in the Willis Towers Watson
2020 Annual Incentive Design Survey, Kelly and Matthews (2020) also mention thatthe multiple measures used by about 80% of companies are additive and independentof each other.
In reality, for instance, the executive oﬃcer compensation in Apple consists of three
primary components (Apple Inc., 2021): annual base salary (ﬁxed element); annual
cash incentive, which depends on an equal weight for net sales and operating income;and long-term equity awards (stock prices).19 Because those compensation components
are calculated independently, we can say that the remuneration system in Apple is alinear combination of proﬁts and other indicators. 20
17Given the inherent nature of delegation, the gain assumed here is plausible (see, Kopel and Löﬄer,2012).
The compensation Ei = Ai + Bi gi should be larger than or equal to Y ; that is, Ai + Bi gi ≥ Y .
When the owner oﬀers the contract term, it needs to consider the maximization problem: max Π i − Eisuch that Ei = Ai + Bi gi ≥ Y . As a result, the owner’s net payment to the manager is Ei = Y andthe owner gains X. In total, the owner’s net proﬁt is Πi − Y + X = Πi − F when it hires a manager.
We thank an anonymous referee for providing us with this formulation.
18

19

It is diﬃcult to determine precisely how equity awards inﬂuence managers’ incentives. If managers
think that collusive conduct contributes to long-term proﬁts, they behave less aggressively. This
scenario seems plausible if they face equal footing rivals and the market is stable. However, if managers
think that excluding rivals contributes to long-term proﬁts, they behave aggressively. This scenario
seems plausible if they are in progressive or growing markets.
20

We also pick up Siemens as another example. Siemens uses linear managerial contracts as well.

9The assumption on strategic delegation means that the owners of ﬁrms in our model
can delegate managerial decisions to their managers for saving time and strategic motives. Such a strategic motive of delegation is consistent with the empirical ﬁnding inBloomﬁeld (2021) that shows that many ﬁrms add revenue-based pay to their CEO’s
pay packages immediately after the requirement to disclose compensation plans for their
executives, which facilitates the strategic commitment to executives’ objectives as in
the scenario of Fershtman and Judd (1987).21
We consider the following four-stage game:22
1. Each downstream ﬁrm’s owner determines whether to delegate overall downstream
production to its manager. If the owner delegates, the ﬁrm needs to incur aﬁxed cost, F , to delegate. This assumption follows Basu (1995). We denote thedecisions of delegation and no delegation as D and N , respectively.
Firms are often reluctant to change their organizational forms, so-called “organizational inertia,” because of routines (e.g., Yi et al., 2016). Changing the forms
can be a long-term decision because of such inertia. Therefore, we assume thatthe ﬁrms decide whether to delegate in the ﬁrst stage.
The compensation of Managing Boardmembers in Siemens also consists of three components: Fixedcompensation (approx. 40%); Short-term variable compensation (bonus) (approx. 25%); and Longterm variable compensation (stock award) (approx. 35%) (note that the report (Siemens, 2021, p.9)
explicitly mentions those percentages). The company’s proﬁts and proﬁtability are reﬂected in the
short-term variable. The long-term variable depends on the total shareholder return and the Environmental, Social, and Governance (ESG) index. Because those components are independent, we cansay that the remuneration system in Siemens is a linear combination of proﬁts and another indicator(long-term incentive).
21

Gipper (2021) shows the changes in incentive pay packages after the revision of the Compensation
Discussion and Analysis statement.
22

The timing structure is diﬀerent from those in the related papers (e.g., Liao, 2010; Claude,2018) and suitable for the motivating example in the introduction. Those related papers consider thefollowing sequence: incentive contracts, input prices, and strategic variables in the downstream market.
The timing structure in those related papers appears to be plausible in other contexts. We believethat the timing structure in a game-theoretical model should depend on the context that researchersconsider.

102. The supplier and the owners of ﬁrms 1 and 2 simultaneously determine the linear
input prices, w, r1 , and r2 . As explained in the introduction, even if we assume
that the supplier can oﬀer diﬀerent input prices for the downstream ﬁrms, thequalitative nature of the main results does not change (see Section 4.5).
The duration of alliances in input markets is longer than that of the incentive
contracts. For instance, Qualcomm announced a multi-year strategic relationshipagreement with Samsung in 2018 (e.g., Qualcomm Technologies Inc., 2018). We
could expect that the signed trading terms are valid for longer periods.
3. Each downstream ﬁrm’s owner determines the incentive scheme for its manager,
αi , if it delegates the production in the ﬁrst stage.
We treat the proposal of an incentive scheme for managers as a yearly based
decision, which is shorter than those in the input market.
4. The agents with the right to control the downstream production determine thequantities in the downstream market.
We derive the subgame perfect Nash equilibrium by backward induction.

4

Analysis

We consider three subgames: (i) both ﬁrms delegate, case (D, D); (ii) one of the ﬁrms
delegates, case (N, D); (iii) no ﬁrm delegates, case (N, N ). Finally, we use the outcomesin the three subgames and obtain the main result.

4.1

Both ﬁrms delegate, case (D, D)

We consider the case in which both ﬁrms delegate.

11

The fourth stage The objective of the manager in ﬁrm i is
gi = αi Πi + (1 − αi ) Ri
= αi {(a − qi − qj − rj − w)qi + ri qj } + (1 − αi ) (a − qi − qj )qi .
In this stage, both ﬁrms’ managers simultaneously set the quantities. We can obtain
the best response function of ﬁrm i’s manager as follows:qi =a − qj − αi (rj + w)
, (i, j = 1, 2, j ̸= i).
2(1)This best response function is a decreasing function of αi . As αi becomes smaller, inother words, as the manager’s attention to the cost (rj + w) is weaker, the quantity ofﬁrm i increases. The best response function does not include the input price imposedby ﬁrm i, ri , that is, the manager does not take into account the revenue from the
input market, as in the standard quantity competition with internal market licensing(e.g., Faulı́-Oller and Sandonı́s, 2002, p.195). 23 This manager’s ignorance regarding therevenue from selling input inﬂuences the choice of ﬁrm i’s owner on αi in the third
stage. Solving the simultaneous equations mentioned in (1), we obtain the quantity ofﬁrm i:qi (α, r, w) =a − 2αi (rj + w) + αj (ri + w)
.
3

(2)

When αj is small, manager i knows that the rival ﬁrm commits to set a higher qj , inducing manager i to set a small output because of the strategic substitution of downstream
outputs. Using the result in (2), we have the proﬁt in the fourth stage:Πi (α, r, w) = (qi (α, r, w) − (1 − αi )(rj + w))qi (α, r, w) + ri qj (α, r, w).

(3)

The ﬁrst term is the proﬁt from the downstream market and the second term is the
proﬁt from the input market.
23

Faulı́-Oller and Sandonı́s (2002) discuss the Cournot duopoly with a licenser (ﬁrm 1) and a licensee(ﬁrm 2). The licenser (ﬁrm 1)’s maximization problem maxx1 p1 (x1 , x2 )x1 +rx2 leads to the ﬁrst-ordercondition: p1 + x1 (∂p1 /∂x1 ) = 0, where p1 (x1 , x2 ) is the inverse demand for the licenser, xi is thequantity of ﬁrm i (i = 1, 2), and r is the per unit fee on the licensee. This condition does not include
r, as in our paper. This property is common in the context of licensing under quantity competition.

12

The third stage The owner of ﬁrm i sets the parameter αi to maximize proﬁtΠi (α, r, w) . Using (3), we solve the maximization problem and then obtain the owner’s
best response function:
αi (αj , r, w) =6 (rj + w) − a αj (ri + w)
3ri−+
, (i, j = 1, 2, j ̸= i).
4 (rj + w)
4 (rj + w)
4 (rj + w)

(4)

The ﬁrst and second terms in (4) are the standard best response in the delegation game,and the last term in (4) is the element, which reﬂects ﬁrm i’s revenue from the input
market.
To maximize proﬁt, ﬁrm i must consider the trade-oﬀ between the downstream proﬁt
and the proﬁt from ﬁrm j. However, the Cournot best response function (1) shows thatthe manager of ﬁrm i does not consider the revenue in the input market.
In the third stage, each ﬁrm can use the degree of delegation αi as a commitment device to decrease the quantity in the fourth stage to balance the proﬁts fromthe downstream and input markets (Macho-Stadler and Verdier, 1991). 24 Solving thesimultaneous equations in (4), we obtain αi (r, w)
αi (r, w) =4ri − rj−a + 8(rj + w) − 2(ri + w)+.
5 (rj + w)
5 (rj + w)

(5)

The ﬁrst term in (5) captures the degree of delegation in the standard delegation gamewithout the last term in (4), and the second term in (5) captures the impact of theirrevenues from inputs on αi . As explained before, the input revenue makes the ﬁrm lessaggressive (a high αi ) in the fourth-stage competition.
By substituting (5) into (2), we can rewrite the output as a function of (r, w):
qi (r, w) =2(a − 3(rj + w) + 2(ri + w)) 2rj − 3ri
+
.
5
5

24

(6)

Macho-Stadler and Verdier (1991) also show such delegation incentives to lower quantities in a
duopoly market with cross-shareholding. Because of the cross-shareholding by one of the ﬁrms, theholding ﬁrm weighs heavily on the proﬁt in the delegation stage to mitigate competition (Proposition
1 in their paper).

13

The second term in (6) reﬂects the impact of their revenues from inputs on the quantity
through the delegation levels. An increase in its input price, rj , leads to a larger quantity
because the rival becomes less aggressive by oﬀering a higher αj . The converse holds,
that is, an increase in its rival’s input price, ri , induces a smaller quantity.
The second stage The supplier and ﬁrms 1 and 2 set w, r1 , and r2 to maximize
their own proﬁts. The supplier’s proﬁt function isΠS (r, w) = wq1 ((r, w) + q2 (r, w)).

(7)

Solving the ﬁrst-order condition, we obtain the best response function of the supplier:w=

4a − 3(ri + rj )
.
8

(8)

The negative coeﬃcient of rj (resp., ri ) comes from the strategic substitution between the prices of ﬁrm i’s (resp. ﬁrm j’s) complementary inputs, w and rj (resp., ri ),which is a key factor for deriving the asymmetric outcome. The strategic substitution
resembles that in the standard Cournot complementary input market. Therefore, as rj
or ri increases, the monopoly power of the supplier weakens.
Firm i’s proﬁt function is (i, j = 1, 2, j ̸= i):
Πi (r, w) = (qi (r, w) − (1 − αi )(rj + w))qi (r, w) + ri qj (r, w).

(9)Solving the ﬁrst-order conditions, we have the best response functions:7(a − (2rj + 3w) + 2w) 5(2a + (rj + 4w) − 6w)+
34
34
17a − 17w − 9rj=, (i, j = 1, 2, j ̸= i).
34

ri =

(10)

The ﬁrst fraction in the ﬁrst line comes from the gain from the downstream competition.
The marginal gain from weakening the rival through ri is larger as ﬁrm i’s eﬃciency

14

increases. The second fraction in the ﬁrst line comes from the direct gain from the
input market.
Solving the simultaneous equations in (8) and (10) and substituting the derived
outcome into (7) and (9), we obtain the proposition:
Proposition 1 If both ﬁrms delegate, the equilibrium input prices, output levels, andαi are
w∗ =

35a
34a
14a
79
, ri∗ =, qi∗ =
, αi∗ = , (i, j = 1, 2, j ̸= i).
121
121
121
69

The resulting overall proﬁts under (D, D) are
πi =

812a2
980a2
− F ≃ 0.0555a2 − F, πS =≃ 0.0669a2 .
14641
14641

To clarify the role of mutual outsourcing, we also solve the simultaneous equationsin (10) and w = 0. Then, we obtain the result:
Result 1 We hypothetically suppose that the supplier does not have any bargaining
power, that is, w = 0, which equals to its marginal cost. If both ﬁrms delegate, theequilibrium input prices, output levels, and αi areri∗ =17a ∗ 7a22
, qi = , αi∗ =(i, j = 1, 2, j ̸= i).
43
43
17

The proﬁts under (D, D) are
πi =

203a2
− F ≃ 0.1098a2 − F.
1849

Note that αi (i = 1, 2) is strictly larger than 1 in Proposition 1 and Result 1. Contrary
to the standard delegation game in Cournot competition, each owner places weights on
its proﬁt rather than its revenue, leading to less aggressive behavior in the downstream
market. Proposition 1 and Result 1 clarify that the existence of mutual outsourcing is
suﬃcient to induce less aggressive behavior in the downstream market.
15

4.2

Only one ﬁrm delegates, case (N, D)

Now suppose that only ﬁrm j delegates. Because the mathematical procedures are
similar to those in Section 4.1, we directly mention the proposition in this case.
Proposition 2 If only ﬁrm j delegates, the equilibrium input prices, output levels, andαj arew∗ =4a ∗ 3a ∗ 9a ∗ 3a ∗ 3a17
, ri = , rj = , qi = , qj = , αj∗ =(i, j = 1, 2, j ̸= i).
13
13
26
26
26
14

The resulting overall proﬁts under (N, D) are
πi =

45a2
12a2
27a2
≃ 0.0399a2 , πj =− F ≃ 0.0666a2 − F, πS =≃ 0.0710a2 .
676
676
169

We have a remark on the input price w.
Remark 1 The input price w in case (N, D), w = 4a/13 ≃ 0.308a, is higher than thatin case (D, D), w = 35a/121 ≃ 0.289a.
We explain the intuition behind Remark 1. Firm i without delegation anticipates alower quantity supplied by ﬁrm j, which delegates, and sets an input price lower thanthat in case (D, D). Because of the mechanism of the complementary input pricing,the supplier in case (N, D) sets a higher input price than in case (D, D). The supplier’s
higher input price implies that abolishing delegation causes a reduction of the actual
market size for the downstream ﬁrms.
To clarify the role of the strategic interaction in the input market, we also solve thesimultaneous equations in (12) and (13) and w = 0. Then, we obtain the result:
Result 2 We hypothetically suppose that the supplier does not have any bargaining
power, that is, w = 0. If only ﬁrm j delegates, the equilibrium input prices, output
levels, and αj are
a
a
a
a
3
ri∗ = , rj∗ = , qi∗ = , qj∗ = , αj∗ =(i, j = 1, 2, j ̸= i).
3
2
6
6
2
16

The proﬁts under (N, D) are
a2
5a2
2
πi =
≃ 0.0833a , πj =− F ≃ 0.139a2 − F.
1236
As in Proposition 1 and Result 1, the delegating owner places a negative weight on
its revenue in the manager’s objective function in Proposition 2 and Result 2. Thisdelegation decision balances the proﬁts from the downstream and input markets.
Firm j’s better procurement condition (the lower ri ) oﬀsets the less aggressive behavior caused by the higher αj . Then, the quantities of the two ﬁrms are the same.

4.3

No ﬁrm delegates, case (N, N )Next, we consider the case in which no ﬁrm delegates to managers. As in the previous
subsection, because the mathematical procedures are similar to those in Section 4.1, we
directly mention the proposition in this case:
Proposition 3 If no ﬁrm delegates, the equilibrium input prices and output levels are
w∗ =

6a
5a
2a
, ri∗ = , qi∗ =(i, j = 1, 2, j ̸= i).
17
17
17

The resulting proﬁts under (N, N ) are
πi =

14a2
24a2
≃ 0.0484a2 , πS =≃ 0.0830a2 .
289
289

We have a remark on the supplier’s input price w.
Remark 2 Similar to Remark 1, the supplier’s input price in case (N, N ), w = 6a/17 ≃0.353a, is higher than that in case (N, D), w = 4a/13 ≃ 0.308a. The incremental
amount of the supplier’s input price in Remark 2, 0.045a, is signiﬁcantly higher thanthat in Remark 1, 0.018a.
The quantities in case (N, N ) are larger than those in case (D, D). The increase in downstream competition through no delegation dominates the supplier’s monopoly power in
response to the reduction of ri (i = 1, 2).
17

4.4

The ﬁrst stage

From Sections 4.1, 4.2, and 4.3, we obtain the payoﬀ matrix in the ﬁrst stage (see Table
1). Using the outcomes, we obtain Proposition 4.
1/2

D

N
0.0555a − F
20.0399a2

D
0.0555a2 − F

0.0666a2 − F

0.0666a2 − F
0.0484a2

N
0.0399a2

0.0484a2
D: Delegation; N : No delegation.

Table 1: The ﬁrst-stage decision
Proposition 4 The proﬁts under (D, D) are higher than those under (N, N ) if F <0.0071a2 .
As we check in Remarks 1 and 2, the larger the number of delegating ﬁrms, the lower
the supplier’s input price. Furthermore, as we show in Propositions 1 and 2, employing
delegation induces the ﬁrm to be less aggressive than an owner-managed ﬁrm, which
mitigates downstream competition. From the two eﬀects, the proﬁts under (D, D) canbe higher than those under (N, N ). Proposition 4 implies that mutual outsourcing
can facilitate collusive conduct.25 We discuss the welfare properties of the subgames in
Section 4.5.
The proﬁt ranking in Proposition 4 is not common but resembles those in Liao(2010) and Meccheri and Fanti (2014) whose mechanisms diﬀer from ours. 26
From the payoﬀ matrix, we obtain the following proposition (see also Figure 2):25

Of course, mutual outsourcing saves ﬁxed costs to produce additional complementary inputs,
although such costs are beyond the scope of this paper.
26

Bhattacharjee and Pal (2014) show a similar result in a duopoly model with network externality.

18

Proposition 5 Both owners delegate if and only if F ≤ 0.0156a2 ; Only one of theowners delegates if and only if 0.0156a2 ≤ F ≤ 0.0182a2 ; No owner delegates if andonly if 0.0182a2 ≤ F .
In the symmetric downstream duopoly, there is a possibility that asymmetry of thedelegation decisions occurs. In many previous related papers, decisions on strategicdelegation are strategic complements, that is, the outcomes of (D, D) and (N, N ) can
coexist (e.g., Colombo and Scrimitore, 2018), which is in contrast to Proposition 5.
Here, we explain the mechanism behind Proposition 5. First, if delegation is costless,both ﬁrms employ delegation. Setting the costless situation as a starting point, as theﬁxed cost of delegation F increases, delegation becomes less proﬁtable. One of theﬁrms stops employing delegation at the threshold value of F (F = 0.0156a2 ). At theﬁxed cost of delegation slightly higher than the threshold value of F , F = 0.0156a2 , therival downstream ﬁrm does not change its managerial form, anticipating the substantial
increase of the supplier’s input price (Remark 2). The substantial increase comes fromthe strategic interaction in the input market. Therefore, when F is moderate, theasymmetric managerial forms occur in equilibrium.
F(N,D),(D,N)(N,N)

F=0.0182a 2(D,D)

F=0.0156a 2
aFigure 2: The parameter range in Proposition 5
Proposition 5 implies that mimicking the well-performed rival’s organizational form
can be an inconsiderate decision. Actually, Samsung is more likely to employ centralized19management (Joseph et al., 2016), and Google is more likely to employ delegated management (Steiber, 2014). Following Proposition 5, we can conclude that the co-existenceof asymmetric managerial forms is a considerable consequence. 27
Finally, we restate the logic behind the asymmetric outcome in Vroom (2006) andcompare it with ours. In his asymmetric outcome, the decentralized ﬁrm acquires
an advantage from aggressiveness in the downstream market and the disadvantage ofthe double mark-up problem. We provide a diﬀerent source to balance the two ﬁrms’competitive advantages. The delegating/decentralized ﬁrm acquires the advantage ofthe lower input price and the disadvantage of less aggressive behavior in the downstream
market.

4.5

Welfare analysis

We brieﬂy check the welfare ranking between the three cases.
The consumer surplus in our model with the linear demand is (q1 + q2 )2 /2. Thatis, the consumer surplus monotonically increases in the total quantity supplied by thedownstream ﬁrms. From Propositions 1, 2, and 3, we ﬁnd that the total quantity in
Proposition 3 is the highest and that in Proposition 2 is the lowest.
Furthermore, social welfare in our model is the sum of the ﬁrms’ proﬁts including
beneﬁts from delegation, X, and the consumer surplus. We denote it in each caseby SWk (k ∈ {(D, D), (N, D), (N, N )}).28 From the assumption that no ﬁrm incurs
production costs except for input prices, social welfare minus the delegation beneﬁts
monotonically increases in the total quantity supplied by the downstream ﬁrms. In
addition to this argument, a delegating ﬁrm obtains the delegation beneﬁt X. The
total delegation beneﬁt in the case of Proposition 3 is the lowest (no ﬁrm delegating),
27

We implicitly assume that the ﬁrms properly coordinate the delegation choices in multiple equilibria. We can not predict which ﬁrm delegates from our theoretical result.
28

The payments for the managers are cancelled out when we evaluate social welfare.

20and that in the case of Proposition 1 is the highest (two ﬁrms delegating). From thediscussions, we have the following proposition:
Proposition 6 The consumer surplus in case (N, N ) is the highest and that in case(N, D) is the lowest. The ranking of social welfare in the three cases is in the following
order:

6308a2



SW
>
SW
>
SW
if
0
≤
X
<
≃ 0.00149a2 ,
(N,N
)
(D,D)
(N,D)


4231249

339a2
2
SW
≥
SW
>
SW
if
0.00149a≤X<≃ 0.00347a2 ,

(D,D)
(N,N
)
(N,D)


97682


 SW
2
(D,D) > SW(N,D) ≥ SW(N,N ) if 0.00347a ≤ X.

4.6

Robustness check

We have incorporated the two key elements in the smartphone industry: the supplierand the mutual outsourcing. We further investigate several scenarios to clarify thatthe key elements are crucial. First, we derive the two results with only one of thetwo elements in the model, respectively, to clarify that the existence of multiple inputs
is essential for obtaining the asymmetric organizational forms. Second, we show that
input price discrimination by the supplier does not bear on the main result (Proposition
5). The detail is available in an earlier version (Arai and Matsushima, 2021).

5

Conclusion

We consider a downstream duopoly model with a monopolistic common supplier andmutual outsourcing between the two downstream ﬁrms. The recent procurement environment in the smartphone industry motivates us to formulate the market structure.
We also incorporate managerial delegation into the duopoly model, as in Fershtman
and Judd (1987), because deciding on organizational forms within a ﬁrm is critical toachieving better performance in almost all industries (Foss and Klein, 2014).

21

Section 4 shows that depending on the ﬁxed cost of delegating to a manager, we
obtain three types of managerial forms in equilibrium: (i) both owners delegate if andonly if the ﬁxed cost is low; (ii) only one of the owners delegates if and only if theﬁxed cost is medium; (iii) no owner delegates if and only if the ﬁxed cost is high.
Section 4.6 shows that the two key elements in our model are necessary for obtaining
the asymmetric managerial forms in the main results.
Following the literature of strategic delegation, we assume the standard linear combination of proﬁt and revenue as an incentive contract to a manager. The linear formulation is consistent with the real-world practices (Murphy, 1999). 29 Contrary to theformulation, recently, Fanti et al. (2017) and Dickson et al. (2021) propose interesting
incentive structures. They show nonstandard eﬀects of the incentive structure.
We brieﬂy discuss nonlinear incentive contracts in our main model, as in Dickson etal. (2021). Following their formulation, we set the nonlinear performance measure as
follows: gi = αi Πi + (1 − αi )v(Ri ), where v(Ri ) is a concave function of the revenue Ri .
Incorporating the concave function diﬀers from the performance measure in the mainmodel. Because of the concavity, each ﬁrm is less aggressive than when v(Ri ) is linear
in Ri , v(Ri ) = Ri . Even in this modiﬁed formulation, each ﬁrm needs to internalize
the licensing revenue by oﬀering an incentive contract that makes its manager less aggressive. We guess that each delegating ﬁrm sets αi to be greater than 1 unless theconcavity of v(Ri ) is strong. If this logic holds, the relationship between the decisions toemploy delegation and the supplier’s monopoly power remains in the modiﬁed formulation. Therefore, we guess that the main results hold even in the modiﬁed formulation.
However, because the modiﬁed model requires us to solve complex problems, we leavethese problems to future research.

29

This is also consistent with the previous studies ﬁnding the plausibility of linear contracts (Bhattacharyya and Lafontaine, 1995; Chu and Sappington, 2007; Bose, Pal, and Sappington, 2011).

22

Besides, it may be interesting to investigate oligopoly competition with nonproﬁt
maximizing ﬁrms in our model framework (for instance, socially concerned ﬁrms (e.g.,
Kopel and Putz, 2021, and the references therein)). This possibility is also left for
future research. Furthermore, considering the endogenous choice of outsourcing in ourmodel framework is an interesting topic (e.g., Colombo and Scrimitore, 2018) for future
research.

Appendix
The stage games in Section 4.2
Suppose that only ﬁrm j delegates. We can use the derived outcome on the quantities
in the fourth stage in the previous subsection. Thus, ﬁrst, we consider the third stage.

The third stage In this subgame, ﬁrm i does not delegate, which is equivalent tothe case in which ﬁrm i sets αi = 1 in this stage. From (4), we can obtain αj (1, r, w)
as
αj = (1, r, w) =3rj
−a + 6 (ri + w) − (rj + w)+.
4 (ri + w)
4 (ri + w)

By using this equation and (2), we can rewrite the outputs as a function of (r, w):
a + 2 (ri + w) − 3(rj + w) rj+ ,44
a + (rj + w) − 2 (ri + w) rj
qj (r, w) =− .
22
qi (r, w) =The second stage Using the quantities derived immediately prior to equation (7),we can obtain the supplier’s best response function as:w=

3a − 2(ri + rj )
6

23

(11)

The best response functions of the downstream ﬁrms are:a − (2rj + 3w) + 2w 2(a + w − 2w)
3a − 3w − 2rj
+
=
,
6
6
6
a − 2(ri + w) + w a + 2ri − 3w
a−w
rj =
+
=
.
4
4
2
ri =

(12)
(13)

Solving the simultaneous equations in (11), (12), and (13), we obtain Proposition 2.

The stage games in Section 4.3
We consider the case in which no ﬁrm delegates to managers. This case is equivalent
to the case in which αi = αj = 1.
The fourth stage From equation (2), the equilibrium output of ﬁrm i is equal toqi (r, w) =a + (ri + w) − 2 (rj + w)
.
3

(14)

The second stage Anticipating the quantities in (14), the supplier maximizes the
proﬁt in (7). Solving the ﬁrst-order conditions of the maximization problem, the supplier’s best response function is:w=

2a − ri − rj
, (i, j = 1, 2, j ̸= i).
4(15)

Anticipating the quantities in (14), each downstream ﬁrm maximizes the proﬁt in (9).
Solving the ﬁrst-order condition of the maximization problems, we obtain the best
response function of each downstream ﬁrm:
ri =5a − 5w − rj
, (i, j = 1, 2, j ̸= i).
10

Using the best response functions in (15) and (16), we can obtain Proposition 3.

24

(16)

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32

The impacts of suppliers and mutual outsourcing onorganizational forms
Yasuhiro Arai and Noriaki Matsushima

Supplementary Appendix
We explain the mathematical procedures to derive the results in Section 5.

A

Without mutual outsourcing

We remove mutual outsourcing from the model. In other words, we consider the casein which ri = rj = 0.

A.1

Both ﬁrms delegate

The objective of the manager in ﬁrm i is
gi = αi Πi + (1 − αi ) Ri
= αi {(a − qi − qj − w)qi } + (1 − αi ) (a − qi − qj )qi
We can obtain the best response function of ﬁrm i’s manager as follows:qi =a − qj − α i w, (i, j = 1, 2, j ̸= i).
2Solving the simultaneous equations mentioned above, we obtain the quantity of ﬁrm i:qi (α, w) =a − 2αi w + αj w.
3

The third stage The owner of downstream ﬁrm i sets the parameter αi to maximize
proﬁt Πi (α, w) . We solve the maximization problem and then obtain the owner’s best
response function:33

αi (αj , w) =6w − a αj w−(i, j = 1, 2, j ̸= i).
4w
4w

Solving the simultaneous equations, we obtain αi (w)αi (w) =−a + 8w − 2w.
5w

We can rewrite output as a function of w:

qi (w) =2(a − 3w + 2w).
5

The second stage The supplier sets w to maximize its own proﬁts. From the supplier’s proﬁt function, we obtain the equilibrium parameters:a4aw∗ = , ri∗ = 0, qi∗ = , αi∗ = .
2
5
5The proﬁts under (D, D) are
πi = πj =

A.2

a2
a2
− F, πS = .
50
5Only one ﬁrm delegates

Now suppose that only ﬁrm j delegates. Thus, ﬁrst, we consider the third stage.

The third stage In this subgame, ﬁrm i does not delegate; in other words, ﬁrm i
sets αi = 1 in this stage. We can obtain αj (1, w) as
αj = (1, w) =−a + 6w − w.
4w

By using this equation, we can rewrite the outputs as a function of w:

qi (w) =a + w − 2w
a + 2w − 3w
, qj (w) =
.
4
2
34

The second stage Solving the ﬁrst-order condition of the supplier, we obtainaaa3w∗ = , ri∗ = rj∗ = 0, qi∗ = , qj∗ = , αi∗ = 1, αj∗ = .
2
8
4
4The proﬁts under (N, D) are
a2
a2
3a2
πi = , π j =− F, πS =
.
64
32
16

A.3

No ﬁrm delegates

Next, we consider the case in which no ﬁrm delegates to managers, that is, αi = αj = 1.
The fourth stage The quantity of ﬁrm i is equal toqi (w) =a + w − 2w.
3

The second stage Anticipating the quantities, the supplier maximizes its proﬁt.
Solving the ﬁrst-order conditions of the maximization problem, we can obtain the equilibrium outcome under the case in which no ﬁrm delegates as follows.
aaw∗ = , ri∗ = 0, qi∗ = .
2
6
The proﬁts under (N, N ) are

πi = πj =

A.4

a2
a2
, πS = .
36
6First stage

We can derive the payoﬀ matrix in the ﬁrst stage (see Table 2).
The equilibrium organizational forms are as follows: The outcome that both owners
delegate is sustainable in equilibrium if and only if F ≤ 7a2 /1600. The outcome thatno owner delegates is sustainable in equilibrium if and only if a2 /288 ≤ F . The two
35

1/2

D

N
a /50 − F2

a2 /64D
a2 /50 − F

a2 /32 − F

a2 /32 − F
a2 /36N
a2 /64

a2 /36

D: Delegation; N : No delegation.

Table 2: The ﬁrst-stage decision: without mutual outsourcing
outcomes can be simultaneously sustainable in equilibrium if and only if a2 /288 ≤ F ≤7a2 /1600.

B

Without the monopolistic supplier

We remove the monopolistic supplier from the main model. In other words, we consider
the case in which wi = wj = 0.

B.1

Both ﬁrms delegate

The fourth stage The objective of the manager in ﬁrm i is
gi = αi Πi + (1 − αi ) Ri
= αi {(a − qi − qj − rj )qi + ri qj } + (1 − αi ) (a − qi − qj )qi
We can obtain the best response function of ﬁrm i’s manager as follows:qi =a − qj − α i r j, (i, j = 1, 2, j ̸= i).
2Solving the simultaneous equations, we obtain the quantity of ﬁrm i:qi (α, r) =a − 2αi rj + αj ri.
3Using qi (α, r), we can obtain the proﬁt in the fourth stage.

36

The third stage The owner of downstream ﬁrm i sets the parameter αi to maximize
its proﬁt. From the proﬁt in the fourth stage, we solve the maximization problem andthen obtain the owner’s best response function:
αi (αj , r) =6rj − a αj ri 3ri−+, (i, j = 1, 2, j ̸= i).
4rj
4rj
4rj

Solving the simultaneous equations above, we obtain αi (r)αi (r) =−a + 7rj + 2ri.
5rj

By using αi (r), we can rewrite the output as a function of r:
qi (r) =2a − 4rj + ri.
5

The second stage Firms 1 and 2 set r1 , and r2 to maximize their own proﬁts. Solving
the ﬁrst-order conditions, we have the best response functions:ri =17a − 9rj
, (i, j = 1, 2, j ̸= i).
34

From the best response functions, the equilibrium wholesale prices, output levels, andαi (i = 1, 2) arewi∗ = wj∗ = 0, ri∗ =17a ∗ 7a ∗ 22
, qi = , α i = .
43
43
17

The proﬁts under (D, D) are
203a2
πi = πj =− F ≃ 0.1098a2 − F, πS = 0.
1849

B.2

Only one ﬁrm delegates

Now suppose that only ﬁrm j delegates. We can use the derived outcome on thequantities in the fourth stage in the previous subsection. Thus, ﬁrst, we consider thethird stage.
37

The third stage In this subgame, ﬁrm i does not delegate, which is equivalent tothe case in which ﬁrm i sets αi = 1 in this stage. We can obtain αj (1, r) as
αj = (1, r) =−a + 6ri − rj 3rj
+
.
4ri
4ri

We can rewrite the outputs as a function of r:
qi (r) =a − 2ria + 2ri − 2rj, qj (r) =.
4
2

The second stage The best response functions of the downstream ﬁrms are:ri =3a − 2rja, rj = .
6
2

From these equations, and wi = wj = 0, we can obtain the equilibrium output levels,and αi (i = 1, 2) as follows.
a
a
a
3
a
wi∗ = wj∗ = 0, ri∗ = , rj∗ = , qi∗ = , qj∗ = , αi∗ = 1, αj∗ = .
3
2
6
6
2
The proﬁts under (N, D) are

πi =

B.3

a2
5a2
, πj =− F, πS = 0.
1236

No ﬁrm delegates

Next, we consider the case in which no ﬁrm delegates to managers, which is equivalent
to the case in which αi = αj = 1.
The fourth stage In this case, the quantity of ﬁrm i is equal toqi (r) =a + ri − 2rj.
338

The second stage We obtain the best response function of each downstream ﬁrm:
ri =5a − rj, (i, j = 1, 2, j ̸= i).
10

From these best response functions, we can obtain the equilibrium outcomes under thecase that no ﬁrm delegates:wi∗ = wj∗ = 0, ri∗ =5a ∗ 2a, q = .
11 i
11The proﬁts under (N, N ) are

πi = πj =

B.4

14a2
≃ 0.1157a2 , πS = 0.
121

First stage

We can derive the payoﬀ matrix in the ﬁrst stage (see Table 3).
1/2

D

N
203a2 /1849 − F

a2 /12D
203a2 /1849 − F

5a /36 − F

5a2 /36 − F2

14a2 /121

N
a2 /12

14a2 /121
D: Delegation; N : No delegation.

Table 3: The ﬁrst-stage decision: without the supplier
The equilibrium organizational forms are as follows: The outcome that both owners
delegate is sustainable in equilibrium if and only if F ≤ 587a2 /22188. The outcome thatno owner delegates is sustainable in equilibrium if and only if 101a2 /4356 ≤ F . The twooutcomes can be simultaneously sustainable in equilibrium if and only if 101a2 /4356 ≤F ≤ 587a2 /22188.

39

C

Wholesale price discrimination by the monopolistic supplierWe suppose that the monopolistic supplier can price discriminate to the downstream
ﬁrms.

C.1

Both ﬁrms delegate

The fourth stage The objective of the manager in ﬁrm i is
gi = αi Πi + (1 − αi ) Ri
= αi {(a − qi − qj − rj − wi )qi + ri qj } + (1 − αi ) (a − qi − qj )qi
We can obtain the best response function of ﬁrm i’s manager as follows:qi =a − qj − αi (rj + wi ), (i, j = 1, 2, j ̸= i).
2Solving the simultaneous equations mentioned above, we obtain the quantity of ﬁrm i:qi (α, r, w) =a − 2αi (rj + wi ) + αj (ri + wj ).
3

The third stage We solve the owner’s maximization problem and then obtain its
best response function:
αi (αj , r, w) =3ri
6 (rj + wi ) − a αj (ri + wj )−+, (i, j = 1, 2, j ̸= i).
4 (rj + wi )4 (rj + wi )4 (rj + wi )Solving the simultaneous equations above, we obtain αi (r, w)
αi (r, w) =4ri − rj−a + 8(rj + wi ) − 2(ri + wj )+.
5 (rj + wi )5 (rj + wi )By substituting αi (r, w) into qi (α, r, w), we can rewrite output as a function of (r, w):
qi (r, w) =2(a − 3(rj + wi ) + 2(ri + wj )) 2rj − 3ri
+
.
5
5

40

The second stage The supplier and ﬁrms 1 and 2 set w, r1 , and r2 to maximize theirown proﬁts. Solving the ﬁrst-order conditions, we have the best response functions:
2a − 4rj + (ri + 4wj ) 4wj+
12
12
2a − 4rj + ri + 8wj, (i, j = 1, 2, j ̸= i),=127(a − (2rj + 3wi ) + 2wj ) 5(2a + (rj + 4wi ) − 6wj )
=
+
34
34
17a − (wi + 16wj + 9rj )=, (i, j = 1, 2, j ̸= i).
34

wi =

ri

In this case, the equilibrium parameters arewi∗ =35a ∗ 34a ∗ 14a ∗ 79
, r =
, q =
, αi = .
121 i
121 i
121
69

The proﬁts under (D, D) are

πi = πj =

C.2

812a2
980a2
− F ≃ 0.0555a2 − F, πS =≃ 0.0669a2 .
14641
14641

Only one ﬁrm delegates

Now suppose that only ﬁrm j delegates. We consider the third stage.

The third stage In this subgame, ﬁrm i does not degelate, which is equivalent tothe case in which ﬁrm i sets αi = 1. We can obtain αj (1, r, w) as
αj = (1, r, w) =3rj
−a + 6 (ri + wj ) − (rj + wi )+.
4 (ri + wj )4 (ri + wj )We can rewrite the outputs as a function of
a + 2 (ri + wj ) − 3(rj + wi ) rj
+ ,44
a + (rj + wi ) − 2 (ri + wj ) rj
qj (r, w) =− .
22
qi (r, w) =41

The second stage Using the quantities derived immediately above, we can obtainthe supplier’s best response functions as:a − 2rj + 2 (ri + wj ) 2wj
a − 2rj + 2ri + 4wj
+
=
,
6
6
6
a − 2ri + 2wi
a − 2ri + wi wi
+
=
.
wj =
4
4
4
wi =

The best response functions of the downstream ﬁrms are:a − (2rj + 3wi ) + 2wj 2(a + wi − 2wj )3a − 2wj − wi − 2rj
+
=
,
6
6
6
a − 2(ri + wj ) + wi a + 2(ri + wj ) − 3wi
a − wi
rj =
+
=
.
4
4
2
ri =

We can obtain
a
3a
7a
a
a
2a
19
wi∗ = , wj∗ = , ri∗ = , rj∗ = , qi∗ = , qj∗ = , αi∗ = 1, αj∗ = .
3
10
30
3
10
15
16
The proﬁts under (N, D) are
πi =

C.3

37a2
29a2
11a2
≃ 0.0411a2 , πj =− F ≃ 0.0644a2 − F, πS =≃ 0.0733a2 .
900
450
150

No ﬁrm delegates

Next, we consider the case in which no ﬁrm delegates to managers, which is equivalent
to the case in which αi = αj = 1.
The fourth stage The quantity of ﬁrm i is equal toqi (r, w) =a + (ri + wj ) − 2 (rj + wi ).
3

The second stage Solving the ﬁrst-order conditions of the maximization problem,
we can obtain the supplier’s best response functions as:wi =a + (ri + wj ) − 2rj wj+ , (i, j = 1, 2, j ̸= i).
4442Similarly, we obtain the best response function of each downstream ﬁrm:
ri =5a − 4wj − wi − rj
, (i, j = 1, 2, j ̸= i).
10

Using those best response functions, we can obtain the equilibrium wholesale prices andoutput levels as follows.
wi∗ =6a ∗ 5a ∗ 2a
, r = , qi = .
17 i
17
17

The proﬁts under (N, N ) are
πi = πj =

C.4

14a2
24a2
≃ 0.0484a2 , πS =≃ 0.0830a2 .
289
289

First stage

We can derive the payoﬀ matrix in the ﬁrst stage (see Table 4).
1/2

D

N
0.0555a − F
20.0411a2

D
0.0555a2 − F

0.0644a − F

0.0644a2 − F

20.0484a2

N
0.0411a2

0.0484a2
D: Delegation; N : No delegation.

Table 4: The ﬁrst-stage decision: diﬀerent input prices by the supplier
The equilibrium organizational forms are as follows: The outcome that both owners
delegate if and only if F < 0.0144a2 ; Only one of the owners delegates if and only if
0.0144a2 < F < 0.016a2 ; No owner delegates if and only if 0.016a2 < F .

43