ISSN (Print) 0473-453X

Discussion Paper No. 1155

ISSN (Online) 2435-0982

THE IMPACTS OF SUPPLIERS
AND MUTUAL OUTSOURCINGON ORGANIZATIONAL FORMS

Yasuhiro Arai
Noriaki Matsushima

December 2021

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

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
December 28, 2021

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 delegations 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. The total surplus when both firms delegate
is smaller than that when no firm delegates.

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

∗

We are thankful to Michael Kopel for his invaluable comments and suggestions. We gratefully
acknowledge the financial support from the Japan Society for the Promotion of Science (JSPS),
KAKENHI Grant Numbers JP15H03349, JP15H05728, JP17H00984, JP18H00847, JP18K01593,
JP19H01483, and JP20H05631. Although Matsushima serves as a member of the Competition Policy
Research Center (CPRC) at Japan Fair Trade Commission (JFTC), our paper does not reflect theviews of 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 delegations 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. The total surplus when both firms delegate
is smaller than that when no firm delegates.

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

1

Introduction

Deciding on organizational forms within a firm is critical to achieving better performance in almost all industries (Foss and Klein, 2014). When firms consider theiroptimal organizational forms, they need to consider internal resources (Mosakowski,1998), suppliers (Bolandifar et al., 2016), customers (Foss et al., 2011), competitors(Fershtman and Judd, 1987), technological knowledge (Foss et al., 2013), and so on.
In particular, we need to focus on the eﬀects of supplier relations and technologicalknowledge on firms’ internal organization forms because the evolution of technologicalcomplexity in many industries has augmented firms’ dependency on outside resources(Bolandifar et al., 2016). Therefore, we investigate the organizational forms of firms bytaking into account the key elements in the manufacturing industry: the monopolistic
supplier and the mutual outsourcing. Below, we explain why we identified these twoelements as the key factors in the manufacturing industry, particularly focusing on thesmartphone industry, which provides items essential for daily life, as an example.
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 Samsung produces the Nexus
smartphone for Google as an original equipment manufacturer (OEM). Google manages
the Android operating system, which is a critical component for smartphone operation.
This operating system powers the Nexus smartphone sold by Google to final consumers.
Samsung also embeds the Android operating system into its Galaxy smartphone andsells the smartphone to final consumers. In sum, Google and Samsung mutually depend
1

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).

1

on each other to produce their smartphones.
Along with such mutual outsourcing, in the input market for smartphones, Qualcomm plays an indispensable role and exerts strong monopoly power over the smartphone makers including Apple, Google, and Samsung.3 Qualcomm produces global 4G
modem chips, which are a vital input for smartphones. Qualcomm held more than half
the market share in the modem chip industry in 2017 (Ai and Lu, 2019, p.645).
Qualcomm demanded that Apple’s OEMs pay royalty fees that are higher than thestandard-essential patent (SEP) royalty fees (see Ai and Lu, 2019, p.647), although theroyalty rates of Qualcomm’s SEPs for 4G cellular technologies are nominally inflexible
under the committed terms based on 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.4 Altogether, Qualcomm exerts its monopoly powerin various ways in the modem chip market (see the discussions on antitrust concerns in
Hovenkamp (2017) and Hovenkamp and Simcoe (2020)).
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 delegations into
the duopoly model, as in Fershtman and Judd (1987), Sklivas (1987), and Basu (1995),
and derive the equilibrium managerial structure (see excellent surveys on strategic del3

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

Practitioners often refer to this as the “no license, no chip policy.”

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 oligopolistic
industries, including tires, glass, and so on.

2egation by Sengul et al. (2012), Lambertiti (2017), and Kopel and Pezzino (2018);
empirical analysis of delegation, Colombo et al. (2007) and Bloomfield (2021)).
The downstream firms compete in quantity, which is suitable for investigating competition in the manufacturing industries (see, e.g., Sundaram et al., 1996; PlehnDujowich and Serfes, 2010). In the smartphone industry, production capacity constraints 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). The downstream firms mutually outsource their inputs from therivals under linear input prices, where the per-unit payment is constant with respect
to the procuring input quantity. The monopolistic input supplier sells its key input tothe two downstream firms under a common linear input price.6 Each downstream firm
produces one unit of the final product by combining each one unit of those three inputs.
In the first stage, each downstream firm’s owner determines whether to delegate its
decision regarding the downstream production to a manager.7 If the owner delegates,
the firm needs to incur a fixed operational cost to inspect the manager’s performance,
as in Basu (1995). In the second stage, the monopoly supplier and the owners set thecontract terms in the input market regardless of the managerial forms determined in thefirst 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 ofa linear combination of its profit and revenue (Fershtman and Judd, 1987). In the
fourth stage, the decision-makers in the downstream market simultaneously set theirquantities.
First, we show that the delegating owner places a negative weight on the revenue in
6

We obtain a qualitatively similar result even when the supplier can price discriminate against thedownstream firms.
7

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

3

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 the fixed operational
cost in the case of delegation: (i) no owner delegates; (ii) only one owner delegates; (iii)both owners delegate. Because the manager is less aggressive than an owner-managed
firm, the sum of the quantities supplied in case (i) is the largest. This comparison
implies that the consumer surplus in case (i) is the largest.
The logic behind the first result is diﬀerent from those in several papers related to
strategic delegation (e.g., labor union (Szymanski, 1994) and monopoly supplier (Liao,2010)) but similar to that in Macho-Stadler and Verdier (1991), who consider crossshareholdings, although the realized result is similar to those in the former related
papers.
The logic behind the second result is new in the context of strategic delegation.8
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) and that
Google is more likely to employ delegated management (Steiber, 2014). Following oursecond result, we can conclude that the co-existence of asymmetric managerial forms
is a considerable consequence. Note that the interaction of the two essential elements(the monopolistic supplier and mutual outsourcing) is the key to obtaining our result.
We explain the intuition behind the results. In our model, there are two forcesthat influence incentive contracts: (i) the downstream firms’ input prices in mutual
outsourcing and (ii) the supplier’s input price. In the fourth stage, when the decision8

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makers set their quantities, they do not take into account their revenues from the input
market, as in the standard quantity competition with internal market licensing (e.g.,
Faulı́-Oller and Sandonı́s, 2002, p.195). However, when they mutually outsource, eachof the firms gains a larger licensing revenue if it is less aggressive in the downstream
market. The delegating owner can internalize the licensing revenue by oﬀering an
incentive contract that makes its manager less aggressive than an owner-managed firm.
Concretely, the delegating owner sets a negative weight on the revenue in the linear
combination of the incentive contract. The monopolistic supplier does not have any
key role in the establishment of the first main result, which is in contrast to the related
papers whose timing structures diﬀer from ours (Szymanski, 1994; Liao, 2010).
Now, considering the asymmetric managerial structure, we explain the intuition
behind the second main result. The delegating owner enjoys a lower input price oﬀered
by the competing owner because the rival nondelegating owner anticipates that thedelegating owner makes its manager less aggressive than an owner-managed firm. Sucha lower input price oﬀered by the nondelegating owner allows the monopolistic supplierto oﬀer a higher input price because of the strategic substitution in the complementary
input market. The monopoly power of the supplier is strongest when no firm delegatesbecause each firm gives priority to the profit from the downstream market. Because of
the rent extraction by the monopolistic supplier, the asymmetric managerial structure
can be attainable if the cost of delegation is moderate.
The interaction of the two forces mentioned above is important for obtaining theasymmetric managerial structure because the negative weight in the incentive contract
stems from the first force mentioned above. Note that in the asymmetric managerial
structure in which only one owner decides to delegate, the firm with delegation supplies

5the same quantity as that without delegation in equilibrium.9 The equilibrium quantityof the delegating firm implies that the input price reduction through delegation provides
an advantage to the delegating firm. Furthermore, the firm with delegation earns ahigher profit than that without delegation when the asymmetric managerial structure
occurs on the equilibrium path.
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 discusses how the two key elements in our model aﬀect
the analytical results. Section 6 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 andKopel, 2020), network externalities (e.g., Hoernig, 2012; Lee and Choi, 2018)).10 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.11 The diﬀerent
timing structure is the key to deriving such less aggressive behavior.
Although most of those papers with vertical relations do not investigate the equi9

If the supplier can price discriminate, the delegating firm supplies a larger quantity than theowner-managed firm in equilibrium.
10

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.
11

Fanti and Meccheri (2013) and Chatterjee and Saha (2017) further investigate the model framework in Szymanski (1994).

6librium 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 also assume that a delegating firm incurs a fixed
operational cost to inspect the manager’s performance, as in Basu (1995) and our paper,an asymmetric managerial structure does not appear.12
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).13 The stream of this research interprets the organizational design as
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. Initially, if the owner of a firm decides to decentralize vertically, the (upstream) production
manager determines the internal transfer price after the decisions regarding the incentive contracts for the marketing managers in the two firms. If not, the internal transfer
price is set equal to the marginal cost of the firm. That is, regardless of organizational
choices, the two owners oﬀer incentive contracts to their marketing managers.
He shows the possibility that one owner chooses decentralization and the otherchooses centralization. In this asymmetric case, the stronger the centralized firm’s
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.
12

13

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

7

aggressiveness, the lower the decentralized firm’s internal transfer price because thedecentralized firm needs to maintain its demand by lowering the internal transfer price.
Anticipating the decentralized firm’s retaliation through lowering its transfer price, thecentralized firm sets the least aggressive incentive contract, mitigating the downstream
competition.
In the 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. That is, internal transfer pricing is the key to obtaining the asymmetric case,although external sourcing is out of scope in Vroom (2006).
Our paper provides a diﬀerent source of balancing the two firms’ competitive advantages. The delegating/decentralized firm acquires an advantage (the lower input prices)
and a disadvantage (the less aggressive behavior in the downstream market). That is,
we provide another plausible scenario leading to an asymmetric managerial structurein oligopoly. That is, external sourcing is the key in our model.
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. They8

show 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
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 proposed by
Ishibashi and Matsushima (2009).14 Under some parameter sets, one of the firms does
not delegate to a manager in order to avoid the worse equilibrium. That is, the demandsystem 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 − q i − 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).
14

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

9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 each one unit of the three inputs, respectively. We assume that the 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 and the supplier, ri and w, that is, ci = ri + w. Here, the supplier oﬀers
a common input price w to firms 1 and 2. Figure 1 summarizes the trading structurein the industry.
S
w
r2
1

2

r1
q1

q2

consumers
Figure 1: The market structure
The profits of firm i and the supplier, Πi and ΠS , areΠi = (Ri − (ri + w)qi ) + rj 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 a10

fixed 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. The reservation utilities ofthe managers are assumed to be zero.15 The linear formulation of gi is consistent withthe previous studies showing the plausibility of linear contracts (Bhattacharyya andLafontaine, 1995; Chu and Sappington, 2007; Bose, Pal, and Sappington, 2011).
We mention the plausibility of employing managerial delegation. Bloomfield (2021)
finds that Cournot firms are more likely to commit to revenue-based executive pay after
the eﬀectuation of the new Compensation Discussion and Analysis (CD&A) section of
the proxy exchanges, which ordered public listing firms on American stock exchanges tofurnish more detailed disclosures concerning the compensation plans for their executives.
He concludes that the behavioral changes of those public listing firms are consistent withthe theoretical prediction by Fershtman and Judd (1987). He also mentions “many firms
add revenue-based pay to their CEO’s pay packages immediately after adopting the new
CD&A requirements” (Bloomfield, 2021, pp. 620-621).16
We consider the following four-stage game:17
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 a15

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.
16

Gipper (2021) shows the changes in incentive pay packages after the revision of CD&A.

17

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.

11fixed 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.
2. The supplier and the owners of firms 1 and 2 simultaneously determine the linear
input prices, w, r2 , and r1 . 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 5).
The time horizon of the contract terms in input markets is longer than that of
the incentive contracts. Manufacturers tend to sign long-term contracts to avoid
opportunistic behaviors (e.g., hold-up problems). The signed trading terms tend
to be 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.

12

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.

The fourth stage The objective of the manager in firm i is
gi = αi Πi + (1 − αi ) Ri
= αi {(a − qi − qj − ri − w)qi + rj 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 (ri + 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 (ri + w) is weaker, the quantity offirm i increases. The best response function does not include the input price imposedby firm i, rj , 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).18 This manager’s ignorance regarding therevenue from selling input influences the choice of firm i’s owner on αi in the third
18

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.

13

stage. Solving the simultaneous equations mentioned in (1), we obtain the quantity offirm i:qi (α, r, w) =a − 2αi (ri + w) + αj (rj + 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 )(ri + w))qi (α, r, w) + rj 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.

The third stage The owner of firm i sets the parameter αi to maximize profit
Πi (α, r, w) . Using (3), we solve the maximization problem and then obtain the owner’s
best response function:
αi (αj , r, w) =3rj
6 (ri + w) − a αj (rj + w)−+
, (i, j = 1, 2, j ̸= i).
4 (ri + w)
4 (ri + w)
4 (ri + 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).19 Solving the19

Macho-Stadler and Verdier (1991) also show such delegation incentives to lower quantities in a14

simultaneous equations in (4), we obtain αi (r, w)
αi (r, w) =4rj − ri−a + 8(ri + w) − 2(rj + w)+.
5 (ri + w)
5 (ri + 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(ri + w) + 2(rj + w)) 2ri − 3rj
+
.
5
5

(6)

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, ri , 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, rj , induces a smaller quantity.
The second stage The supplier and firms 1 and 2 set w, r2 , and r1 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 ri (resp., rj ) comes from the strategic substitution between the prices of firm i’s (resp. firm j’s) complementary inputs, w and ri (resp., rj ),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).

15

which is a key factor for deriving the asymmetric outcome. The strategic substitution
resembles that in the standard Cournot complementary input market. Therefore, as ri
or rj 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 )(ri + w))qi (r, w) + rj qj (r, w).

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

rj =

(10)

The first fraction in the first line comes from the gain from the downstream competition.
The marginal gain from weakening the rival through rj is larger as firm i’s eﬃciency
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 (i = 1, 2) are
w∗ =

35a
34a
14a
79
, rj∗ =, qi∗ =
, αi∗ = .
121
121
121
69

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

980a2
812a2
− 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:

16

Result 1 We hypothetically suppose that the supplier does not have any bargaining
power, that is, w = 0. If both firms delegate, the equilibrium input prices, output levels,and αi (i = 1, 2) arew∗ = 0, rj∗ =17a ∗ 7a22
, qi = , αi∗ = .
43
43
17

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

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, andαi (i = 1, 2) arew∗ =17
4a ∗ 3a ∗ 9a ∗ 3a ∗ 3a
, rj = , ri = , qi = , qj = , αi∗ = 1, αj∗ = .
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.

17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 αi (i = 1, 2) are
a
a
a
a
3
w∗ = 0, rj∗ = , ri∗ = , qi∗ = , qj∗ = , αi∗ = 1, αj∗ = .
3
2
6
6
2
The profits under (N, D) are
πi =

a2
5a2
≃ 0.0833a2 , πj =− F ≃ 0.139a2 − F, πS = 0.
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 rj ) oﬀsets the less aggressive behavior caused by the higher αj . Then, the quantities of the two firms are the same.

18

4.3

No firm delegates, case (N, N )Next, we consider the case in which no firm delegates to managers, that is, αi = αj = 1.
As in the previous subsection, because the mathematical procedures are similar to thosein Section 4.1, we directly mention the proposition in this case:
Proposition 3 If no firm delegates, the equilibrium input prices and output levels (i =1, 2) are
w∗ =

6a
5a
2a
, rj∗ = , qi∗ = .
17
17
17

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

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, j = 1, 2, j ̸= i).
To clarify the role of the strategic interaction in the input market, we also solve thesimultaneous equations in (16) and w = 0. Then, we obtain the result:
Result 3 We hypothetically suppose that the supplier does not have any bargaining
power, that is, w = 0. If no firm delegates, the equilibrium input prices, output levels,and αi (i = 1, 2) arew∗ = 0, rj∗ =5a ∗ 2a, q = .
11 i
11

19

The profits under (N, N ) are
πi = π j =14a2≃ 0.1157a2 , πS = 0.
121

The quantities in Result 3 are the largest among the three hypothetical cases.

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
0.0555a2 − F

0.0399a2

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.20 We discuss the welfare properties of the subgames in
Section 4.5.
20

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

20

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.21
From the payoﬀ matrix, we obtain the following proposition:
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 the
fixed cost for delegation F increases, delegation becomes less profitable. One of thefirms stops employing delegation at the threshold value of F (F = 0.0156a2 ). At the
fixed delegation cost 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.
Proposition 5 implies that mimicking the well-performed rival’s organizational form
can be an inconsiderate decision. Actually, Samsung is more likely to employ centralized
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-existence
21

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

21F(N,D),(D,N)(N,N)

F=0.0182a 2(D,D)

F=0.0156a 2
aFigure 2: The parameter range in Proposition 5of asymmetric managerial forms is a considerable consequence.
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, the total surplus in our model is the sum of the firms’ profits and theconsumer surplus. From the assumption that no firm incurs production costs except
for input prices, the total surplus minus the delegation costs monotonically increases

22

in the total quantity supplied by the downstream firms. In addition to this argument,a delegating firm incurs the fixed cost F . The total delegation cost in the case of
Proposition 3 is the lowest (no delegation cost), and that in the case of Proposition 1 isthe highest (two delegations). From the discussions, 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 total surplus in case (N, N ) is the highest. The total surplus
in case (D, D) is the lowest if F > 2419a2 /4948658 ≃ 4.889 × 10−4 ; otherwise, that incase (N, D) is the lowest.

5

Remarks

We have incorporated the two key elements in the smartphone industry: the supplierand the mutual outsourcing. Below, 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 upon request.

Without mutual outsourcing

We remove mutual outsourcing from the mainmodel. In other words, we consider the case in which rj = ri = 0.
We can show that the profits under (N, N ) are higher than those under (D, D), as
in the standard delegation game with quantity competition, in contrast to Proposition
4. Furthermore, the supplier oﬀers the same input price w = a/2 regardless of the
managerial structures, which is in contrast to the propositions in Section 4.
The parameters of the incentive contracts in the two cases, (i) one delegation by
firm 2 and (ii) two delegations, are α2 = 3/4 and αi = 4/5 (i = 1, 2), respectively. The
23

equilibrium weights in the delegation cases also diﬀer from those in the outcomes in
Section 4.
In this case, an asymmetric organizational structure does not appear as an equilibrium outcome.

Without the supplier

We remove the supplier from the main model. In otherwords, we consider the case where w = 0.
We can show that the profits under (N, N ) are higher than those under (D, D) in
contrast to Proposition 4. From the two scenarios, we find that the two key elements
are necessary for obtaining Proposition 4 in which the profits under (D, D) are higher
than those under (N, N ).
The parameters of the incentive contracts in the two cases, (i) one delegation by
firm 2 and (ii) two delegations, are α2 = 3/2 and αi = 22/17 (i = 1, 2), respectively.
The equilibrium weights in the delegation cases are similar to those in the outcomes in
Section 4.
In this case, an asymmetric organizational structure does not appear as an equilibrium outcome. From the two scenarios, the two key elements in our model are necessary
for obtaining the asymmetric outcome in Proposition 5.

Diﬀerent input prices by the supplierThis part is the robustness check of

the main model. Now, we suppose that the supplier can price discriminate to thedownstream firms, that is, wi and wj can be diﬀerent.
We can show that the profits under (D, D) are higher than those under (N, N ), as
in Proposition 4.
The parameters of the incentive contracts in the two cases, (i) one delegation by
firm 2 and (ii) two delegations, are α2 = 19/16 and αi = 79/69 (i = 1, 2), respectively.
24

The equilibrium weights in the delegation cases are similar to those in the outcomes in
Section 4.
In this case, an asymmetric organizational structure appears as an equilibrium outcome when F is moderate, as in Proposition 4.

6

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 delegations 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).
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 the
fixed cost is medium; (iii) no owner delegates if and only if the fixed cost is high.
Section 5 shows that the two key elements in our model are necessary for obtaining theasymmetric 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 previous studies finding the plausibility of linear contracts(Bhattacharyya and Lafontaine, 1995; Chu and Sappington, 2007; Bose, Pal, and Sappington, 2011). Contrary to the formulation, recently, Fanti et al. (2017) and Dickson
et al. (2021) propose interesting incentive structures. They show nonstandard eﬀects
of the incentive structure.

25

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.
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.

26

The third stage In this subgame, firm i does not delegate; in other words, firm isets αi = 1 in this stage. From (4), we can obtain αj (1, r, w) as
αj = (1, r, w) =−a + 6 (rj + w) − (ri + w)
3ri+.
4 (rj + w)
4 (rj + w)

By using this equation and (2), we can rewrite the outputs as a function of (r, w):
a + 2 (rj + w) − 3(ri + w) ri+ ,44
a + (ri + w) − 2 (rj + w) ri
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

(11)

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

rj =

(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, that is, αi = αj = 1.
The fourth stage From equation (2), the equilibrium output of firm i is equal toqi (r, w) =a + (rj + w) − 2 (ri + w)
.
3

27

(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 − rj − ri
, (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:rj =5a − 5w − ri
, (i, j = 1, 2, j ̸= i).
10

(16)

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

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hidden source of organizational variation. Organization Science, 27(3), 782–800.

35

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 rj = ri = 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 − αi 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:36

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∗ = , rj∗ = 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 isets 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
37

The second stage Solving the first-order condition of the supplier, we obtainaaa3w∗ = , rj∗ = ri∗ = 0, qi∗ = , qj∗ = , αi∗ = 1, αj∗ = .
2
8
4
4The profits under (N, D) are
πi =

A.3

a2
a2
3a2
, πj =− F, πS =
.
64
32
16

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∗ = , rj∗ = 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 that
38

1/2

D

N
a /50 − F2

a2 /64D
a2 /50 − F

a /32 − F

a2 /32 − F2

a2 /36N
a2 /64

a2 /36

D: Delegation; N : No delegation.

Table 2: The first-stage decision: without mutual outsourcing
no owner delegates is sustainable in equilibrium if and only if a2 /288 ≤ F . The twooutcomes 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 − ri )qi + rj 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 ri, (i, j = 1, 2, j ̸= i).
2Solving the simultaneous equations, we obtain the quantity of firm i:qi (α, r) =a − 2αi ri + αj rj.
3Using qi (α, r), we can obtain the profit in the fourth stage.
39

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) =6ri − a αj rj 3rj−+, (i, j = 1, 2, j ̸= i).
4ri
4ri
4ri

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

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

The second stage Firms 1 and 2 set r2 , and r1 to maximize their own profits. Solving
the first-order conditions, we have the best response functions:rj =17a − 9ri
, (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, rj∗ =17a ∗ 7a ∗ 22
, q i = , αi = .
43
43
17

The profits under (D, D) are

πi = πj =

B.2

203a2
− F ≃ 0.1098a2 − F, πS = 0.
1849

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.
40

The third stage In this subgame, firm i does not delegate; in other words, firm isets αi = 1 in this stage. We can obtain αj (1, r) as
αj = (1, r) =−a + 6rj − ri 3ri
+
.
4rj
4rj

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

The second stage The best response functions of the downstream firms are:rj =3a − 2ria, ri = .
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
a
3
wi∗ = wj∗ = 0, rj∗ = , ri∗ = , 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, that is, αi = αj = 1.
The fourth stage In this case, the quantity of firm i is equal toqi (r) =a + rj − 2ri.
341The second stage We obtain the best response function of each downstream firm:rj =5a − ri, (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, rj∗ =5a ∗ 2a, q = .
11 i
11The profits under (N, N ) are
14a2≃ 0.1157a2 , πS = 0.
πi = πj =121

B.4

First stage

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

D

N
203a /1849 − F2

a2 /12D
203a2 /1849 − F

5a2 /36 − F

5a2 /36 − F
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.
42

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 − ri − wi )qi + rj 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 (ri + 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 (ri + wi ) + αj (rj + wj ).
3

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

43

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

wi =

rj

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

The profits under (D, D) are
980a2
812a2
2
− F ≃ 0.0555a − F, πS =≃ 0.0669a2 .
πi = πj =
14641
14641

C.2

Only one firm delegates

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

The third stage In this subgame, firm i sets αi = 1 in this stage. We can obtain
αj (1, r, w) as
αj = (1, r, w) =3ri
−a + 6 (rj + wj ) − (ri + wi )+.
4 (rj + wj )4 (rj + wj )We can rewrite the outputs as a function of
a + 2 (rj + wj ) − 3(ri + wi ) ri
+ ,44
a + (ri + wi ) − 2 (rj + wj ) ri
qj (r, w) =− .
22
qi (r, w) =44

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

wi =

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

rj =

We can obtain

a
3a
7a
a
a
2a
19
wi∗ = , wj∗ = , rj∗ = , ri∗ = , 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, that is, αi = αj = 1.
The fourth stage The quantity of firm i is equal toqi (r, w) =a + (rj + wj ) − 2 (ri + 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 + (rj + wj ) − 2ri wj+ , (i, j = 1, 2, j ̸= i).
4445

Similarly, we obtain the best response function of each downstream firm:rj =5a − 4wj − wi − ri
, (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 j
17
17

The profits under (N, N ) are
24a2
14a2
2
≃ 0.0484a , πS =≃ 0.0830a2 .
π i = πj =
289
289

C.4

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

0.0644a2 − F

0.0644a2 − F
0.0484a2

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 .

46