Negative effect of price-matching policy on traditional
retailers in a dual-channel supply chain with different content
formats∗
Yuta Kittaka†∗

Noriaki Matsushima‡

Fuyuki Saruta§

We circulated earlier versions of this paper under the title “Competition between physical and electronic content retailers.” We thank the review team for many constructive comments. We also thank
Masayoshi Maruyama, Toshihiro Matsumura, and Yusuke Zennyo, as well as the seminar participants at
Japan Association for Applied Economics 2021 and Nanzan University. In particular, we thank Tomomichi
Mizuno for his invaluable comments on an earlier version of our paper. We gratefully acknowledge the financial support from JSPS KAKENHI grant numbers JP15H05728, JP17H00984, JP18H00847, JP18K01593,
JP19H01483, JP19J10596, JP20H05631, and JP20J11411. Although Matsushima serves as a member of
the Competition Policy Research Center (CPRC) at Japan Fair Trade Commission (JFTC), our paper
does not reflect the views of JFTC. The usual disclaimer applies.
†
Yuta Kittaka, Graduate School of Economics, Kobe University, 2-1, Rokkodai-cho, Nada-ku, Kobe,
Hyogo Prefecture, 657-8501, Japan. E-mail: kittaka@econ-io.jp.
‡
Noriaki Matsushima, Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka,
Ibaraki, Osaka 567-0047, Japan. E-mail: nmatsush@iser.osaka-u.ac.jp, ORCID: 0000-0002-7865-3875.
§
Corresponding author: Fuyuki Saruta, Graduate School of Economics, Osaka University, 1-7
Machikaneyama-cho, Toyonaka-shi, Osaka, 560-0043. E-mail: saruta@iser.osaka-u.ac.jp.

Negative effect of price-matching policy on traditional retailers in adual-channel supply chain with different content formats
Abstract
We investigate a model in which a monopoly supplier distributes two types of its
product, physical and electronic products, through a traditional/brick-and-mortar retailer with a wholesale price contract and an online retailer with an agency contract,
respectively. The supplier and the online retailer negotiate the royalty rate, which isa ratio of the online retailer’s revenue to the entire revenue of the channel. We also
discuss the case in which the supplier imposes the following self-regulation to balanceits dual-channel supply chain: the retail price of the online retailer is not lower than
the wholesale price for the traditional retailer. What will happen if the supplier employs such self-regulation? Is self-regulation beneficial to the two retailers? To answer
these questions, we first derive the equilibrium without such self-regulation. Second,
we investigate what happens if the supplier employs such self-regulation. We obtainthe following results. First, in the baseline model, an increase in the online retailer’sbargaining power over the supplier benefits the two retailers but harms the supplier.
Second, under self-regulation, the wholesale price is strictly higher than that in thebaseline model. Third, the retailers’ equilibrium prices are also strictly higher thanthose in the baseline model. Self-regulation always benefits the online retailer. It
benefits the traditional retailer if the online retailer’s bargaining power is sufficiently
weak. Furthermore, self-regulation benefits the supplier if the substitutability between
the two retailers is not high. However, it reduces consumer and social welfare.
Keywords: Agency contract, Price-quantity competition, Dual-channel supply chain,
Royalty rate.
JEL Classification: L22, L13, C72, C78.

1

Introduction

The recent advancement of online platforms facilitates selling digital content, including
books, games, movies, and music, and purchasing such content without leaving home. The
facilitation of selling and buying digital content is a competitive threat to the traditional
brick-and-mortar retailers handling physical content. For instance, in the US book industry, the sales share of e-books was about 20% in 2015 (Gilbert, 2015), and in 2018, 34% ofyounger consumers, aged 18–29 years, had read e-books in the previous 12 months (Pew
Research Center, 2019a).1 Contrary to this trend, many consumers still chose only paper
books in 2019. According to Pew Research Center (2019b), among consumers who readbooks, the ratios of purchasing only paper books, only e-books including audiobooks, andboth types are, respectively, 51%, 9.8%, and 39%. Thus, consumers still purchase bothtraditional paper books and e-books, depending on preferences, situations, and places (see,for instance, Bergström and Höglund (2020) for the recent trend in e-books in Sweden).
In sum, it is reasonable to investigate a market in which consumers can purchase both
physical and digital content.
In the distribution of such content, online and brick-and-mortar retailers often facedifferent contract types: agency contracts and wholesale price contracts, respectively. For
instance, in the US book industry, some online retailers (e.g., Apple) use agency contracts,
although brick-and-mortar book retailers employ standard wholesale price contracts (Dantas et al., 2014; Gilbert, 2015).2 Publishers directly determine the downstream prices ofsuch online retailers using agency contracts, contrary to wholesale price contracts in whichbrick-and-mortar retailers choose their downstream pricing at their discretion. The online retailers receive profits based on a prenegotiated percentage of revenues (the royaltyrate). Employing agency contracts eliminates the double-marginalization problem andallows the trading pair to be a quasi-integrated firm.3 The quasi-integrated nature of the
trading pair with agency contracts could cause (partial) foreclosure of brick-and-mortar
retailers through offering higher wholesale prices to those brick-and-mortar retailers, whichis reminiscent of encroachment issues in the supply chain problem (Chiang et al., 2003;
1

The penetration of devices for reading e-books, such as e-readers, smartphones, and tablet PCs, isconstantly increasing (Deloitte, 2018). In 2015, smartphones were the most popular devices for reading
e-books (Maloney, 2015). Therefore, the e-books market share is now substantial.
2

In the Appendix (Section 8.3), we briefly describe the practices of the major publishers in the US

before employing agency contracts became a common practice.
3

Another advantage of distributing digital content over physical content is lower production costs.

1

Arya et al., 2007; Zennyo, 2019; Zheng and Yu, 2021) and the discussion of price/marginsqueeze in economics (see, e.g., Bouckaert and Verboven, 2004; Sidak, 2008; Jullien et al.,2014).4 Given the imbalance between brick-and-mortar retailers and online retailers, content suppliers need to consider balancing their dual channels, to keep better threat points(outside profits) in bargaining with online retailers by ensuring brick-and-mortar retailers
are profitable outside options (Shen et al., 2019; Wang and Miller, 2020).5
Given the disadvantage of brick-and-mortar retailers over online retailers, which are
quasi-integrated with suppliers, we can borrow an idea from regulation economics (e.g.,
Bouckaert and Verboven, 2004; Jullien et al., 2014) to balance content suppliers’ dualchannel supply chains: an integrated firm does not set its retail price lower than wholesale
prices for its independent trading downstream firms (henceforth, self-regulation).6 At firstglance, self-regulation seems to give brick-and-mortar retailers competitive power over(quasi-integrated) online retailers in the downstream market because content suppliers
will increase the retail prices of the (quasi-integrated) online retailers and lower the wholesale prices for the brick-and-mortar retailers to meet the requirement of self-regulation.
Besides, the idea seems less likely to cause antitrust concerns intuitively because it is reminiscent of the remedy in the discussion of price/margin squeeze in economics (Bouckaert
and Verboven, 2004; Jullien et al., 2014).
Related to such commitment to pricing, one may think that a retail-level equal pricing
policy is a direct and plausible way to help independent retailers (Zheng and Yu, 2021).
This policy is effective if independent (large-size) e-retailers compete with manufacturers’
direct e-channels because the comparison of those retail prices is easy, and they easily detect the deviation of the price-matching policy through web scraping (Castrillo-Fernández,2015). In our context, gathering price information about brick-and-mortar retailers is noteasy because of the ways they post their prices (posting in their stores offline), making
retail price comparison difficult.
As an alternative to retail price matching, we consider the self-regulation mentioned
earlier. The integrated firm under self-regulation knows completely the price information
required for self-regulation. Furthermore, independent trading downstream firms can de4

See Rey and Tirole (2007) for a review of market foreclosure issues.

5This matter aligns with issues in channel coordination problems (see the seminal work by Jeuland andShugan (1983), review articles (Cachon, 2003), direct channel in Chiang et al. (2003), and revenue sharing
contracts in Cachon and Lariviere (2005)).
6

This idea is related to the price discount contracts investigated in Section 5.1 of Cai et al. (2009).

2

tect the violation only by gathering the online information of the integrated firm’s online
price through web scraping (Castrillo-Fernández, 2015). Therefore, in the competition
between brick-and-mortar retailers and direct e-retailers, the feasibility of implementing
self-regulation is higher than that of using the retail-level equal pricing policy.
Although the abovementioned self-regulation seems useful to balance the dual-channel
supply chain with a brick-and-mortar retailer and an online retailer, what really happens
if the supplier employs such a pricing policy? Is self-regulation beneficial for the supplierand retailers? Does self-regulation benefit consumers and social welfare?
We investigate a model in which a monopoly supplier distributes two types of its product through a traditional retailer with a wholesale price contract and an online retailer withan agency contract.7 We also consider the impact of the abovementioned self-regulationon the three players and the consumer and total surpluses.
In the baseline model, we consider the following three-stage game. First, the supplierand the online retailer negotiate the royalty rate through Nash bargaining, which is a key
element in our model. Second, the supplier unilaterally determines the wholesale price forthe traditional retailer. The sequence of the first and second stages follows those in therelated papers that endogenize royalty rates (e.g., Abhishek et al., 2016; Zennyo, 2019,2020; Tsunoda and Zennyo, 2021). Third, observing the trading terms determined before,
the traditional retailer and the supplier simultaneously set their own strategic variables.
By taking into account the nature of physical and digital content, we assume that thetraditional retailer and the supplier via the online retailer compete in quantity and price,
respectively (e.g., a subgame of the model with endogenous choices of strategic variables
in Singh and Vives (1984)). We call the competition mode “price-quantity competition.”In the self-regulation case, the supplier cannot set a retail price that is lower than thewholesale price for the traditional retailer in the third stage (Bouckaert and Verboven,
2004; Jullien et al., 2014).
Note that we can classify the research method in our paper as a “closed-form analyticaloperational analyses,” which are standard in the supply chain management literature,7

The literature of closed-loop supply chains investigates the distribution of multiple types of products,

as in our paper (Savaskan et al., 2004; Atasu et al., 2008; Taleizadeh et al., 2018; Taleizadeh and Moshtagh,
2019; Taleizadeh and Sadeghi, 2019; Alizadeh-Basban and Taleizadeh, 2020; Xiao et al., 2020; Dou andChoi, 2021; Qiao and Su, 2021; Sheu et al., 2021; Taleizadeh et al., 2021).

3

following the explanation in footnote 2 of Choi and Guo (2020).8
The assumption of price-quantity competition is consistent with the standard viewon when price and quantity are strategic variables of firms (e.g., see the textbooks byBelleflamme and Peitz (2015, Section 3.3.3, p. 67) and Cabral (2017, Section 8.3, pp.
200–201)). Before we consider this view, we discuss an interpretation of quantity settingin oligopoly models. Setting a quantity in oligopoly models is interpreted as a reduced
form of the following procedure: a firm precommits its production capacity, then sets its
retail price by considering its precommitted capacity. In this procedure (capacity-thenprice decision), the retail price equalizes the capacity (the supply) to the market demandfor its product (the demand). Thus, the market is clear, and there is no inventory problem.
The logic holds even when more than one firm simultaneously conducts such procedures.
The seminal work by Kreps and Scheinkman (1983) provides a micro-foundation for theinterpretation. Since then, many researchers have relied on this interpretation to formulate oligopoly models with capacity-then-price decisions. The operations research andmanagement science (OR/MS) literature is no exception. For instance, Arya et al. (2007),David and Adida (2014), Shulman (2014), and Yang et al. (2018) employ quantity-setting
models in supply chain problems. Furthermore, Farahat et al. (2019) provide a microfoundation of quantity competition in the context of OR/MS. Our modeling approach
follows the stream of quantity competition models in the context of OR/MS.
We now explain the plausibility of our price-quantity competition using a well-cited
textbook on industrial organization by Belleflamme and Peitz (2015), who summarize thediscussion of Kreps and Scheinkman (1983) (Belleflamme and Peitz, 2015, pp. 61–65)
and then explain how to use strategic variables in oligopoly models. Belleflamme andPeitz (2015) mention that when capacity is unlimited, assuming a firm is a price setter isappropriate; when capacity is limited, assuming a firm is a quantity setter is appropriate.
Case 3.4 on page 68 in Belleflamme and Peitz (2015) explains the change in productcharacteristics in the publishing industry: “it seems that the quantity competition model
fits better with the batch printing technology (because prices will adjust to selling theexisting capacity) and the price competition model with the POD technology (because
8

Because the purpose of this methodology is to derive closed-form solutions theoretically, the analyticalframeworks are not complex (Choi and Guo, 2020), which is in contrast to the mathematical programmingbased operational research studies. Even in recent years, many papers in operations research studies employ
closed-form analytical operational analyses (e.g., Taleizadeh et al., 2019; Choi, 2021; Dou and Choi, 2021;
Xu and Choi, 2021; Cao and Choi, 2021; Sheu et al., 2021).

4

quantity can be adjusted immediately at the announced prices)” (POD in the quotation is“publish on demand”). Thus, we think that the market structure discussed in our paper
properly captures the nature of product characteristics in real-world industries, including
books, games, movies, and music.
First, we obtain the following results in the baseline model. The equilibrium royalty
rate positively correlates with the online retailer’s bargaining power over the supplier.
The wholesale price for the traditional retailer is higher than the online retailer’s retailprice in equilibrium if the bargaining power is lower than a threshold value, which islarger than 0.3 but lower than 0.5. An increase in the online retailer’s bargaining powerover the supplier benefits the two retailers but harms the supplier. Furthermore, suchan increase in bargaining power lowers the wholesale price and the online retailer’s retailprice, enhancing consumer and social welfare.
Second, we also obtain the following results in the self-regulation case. The equilibrium royalty rate positively correlates with the online retailer’s bargaining power over the
supplier as in the baseline model. The online retailer’s retail price is always binding to
the wholesale price for the traditional retailer under self-regulation because the suppliercan mitigate the downstream competition by setting a higher wholesale price, leading toa higher retail price of the online retailer. The wholesale price under self-regulation isstrictly higher than that in the baseline model, contrasting to the intuition that the supplier decreases the wholesale price but increases the retail price of the online retailer tomeet the self-regulation. The retail prices under self-regulation are also strictly higher
than those in the baseline model. The royalty rate under self-regulation is strictly higher
than that in the baseline model. In particular, the incremental level of the royalty rateincreases as the product substitutability between the two contents increases.
Furthermore, self-regulation always benefits the online retailer because it weakens thedownstream competition and increases the royalty rate. The self-regulation benefits the
traditional retailer if the online retailer’s bargaining power is sufficiently weak because theself-regulation limits the supplier’s strong dependency on the online retailer with weak bargaining power, which dominates the cost from increasing the wholesale price. In addition,
the self-regulation benefits the supplier if the substitutability between the two retailers
is not sufficiently high because the benefit from mitigating the downstream competition
dominates the cost from augmenting the royalty rate (see the abovementioned relation
between the royalty rate and the product substitutability).
5

The results of our analyses have an important managerial implication for content suppliers. Our model fits the distribution problems in copyrighted content industries, including books, games, movies, and music. A content supplier’s bargaining power over online
retailers can be stronger if the originality/uniqueness of its content is higher. Our theoretical model shows that self-regulation benefits all firms if the online retailer’s bargaining
power is sufficiently weak and the product substitutability between the two contents is not
sufficiently high; in other words, if the royalty rate is sufficiently low and the two contents
are horizontally differentiated. The suppliers/providers of content with high originality
can commit to the self-regulation discussed here without causing complaints by brickand-mortar retailers. However, if content suppliers/providers do not have content withhigh originality, imposing the pricing constraint can cause conflicts with brick-and-mortar
retailers because the constraint harms those retailers. In sum, the bargaining power ofcontent suppliers/providers is one of the key factors in determining whether implementing
the pricing constraint discussed here is feasible. However, we discuss another importantfactor in implementing the pricing constraint below.
Note that implementing the price constraint might cause concern on competition policy
if the competition authority follows our welfare analysis. The pricing constraint intuitively
seems acceptable from the viewpoint of competition policy because it is similar to the remedy in the discussion of price/margin squeeze in economics (e.g., Bouckaert and Verboven,
2004; Jullien et al., 2014). However, our results show that the pricing constraint harms
consumer and social welfare because of the increases in the wholesale price and the online
retailer’s retail price. Those price increases stem from the structure of price-quantity competition, which captures competition modes in the content industries. Therefore, giventhat content suppliers employ the pricing constraint discussed here, if the competition
authority recognizes that capacity constraints of the traditional retailers influence theirpricing policies, it might apply our results to the conduct of the content suppliers. This
concern is the other factor in determining whether implementing the pricing constraint
discussed here is feasible.
The remainder of this paper is organized as follows. Section 2 surveys the related
papers. Section 3 constructs the model. Section 4 analyzes the game and presents theresults. Section 5 analyzes the effect of self-regulation on retail prices. Section 6 discusses
the case in which two retailers compete in price. Section 7 offers concluding remarks.

6

2

Related literature

Because the market structure in our model is the dual-channel supply chain in which
the monopoly supplier distributes two types of its product through a traditional retailer
under linear wholesale pricing and an online retailer under an agency contract, our paper
is related to the numerous studies on dual-channel supply chain management issues in thecontext of OR/MS based on game theory (earlier works, Chiang et al., 2003; Arya et al.,2007; Cai, 2010; Li et al., 2014, 2015b; Matsui, 2016; recent works, Guan et al., 2020; Heet al., 2020; Xu et al., 2020; Xue et al., 2020; Zhen et al., 2020; Zhang et al., 2021; Zhang
and Hezarkhani, 2021; Zheng and Yu, 2021). Except for a few papers discussed in themain text, the focus of our paper differs from those in other papers, although the market
structures in those papers are similar to ours.
In particular, our paper contributes to the literature on dual-channel supply chainswith agency contracts (e.g., Abhishek et al., 2016; Tan et al., 2016; Tan and Carrillo,
2017; Lu et al., 2018; Tian et al., 2018; Yan et al., 2018; Shen et al., 2019; Zennyo, 2019,2020; Zhang and Zhang, 2020; Chen et al., 2021; Fu et al., 2021; Song et al., 2021; Tsunoda
and Zennyo, 2021; Zhen and Xu, 2022).9 We can classify the themes of those papers into
the following: endogenous contract forms (wholesale or agency) (Abhishek et al., 2016;
Tan et al., 2016; Tan and Carrillo, 2017; Lu et al., 2018; Tian et al., 2018; Yan et al., 2018;
Zennyo, 2019, 2020; Fu et al., 2021; Tsunoda and Zennyo, 2021); optimal channel structure(Shen et al., 2019; Zhang and Zhang, 2020; Song et al., 2021; Zhen and Xu, 2022); andmultiperiod sales problem (Chen et al., 2021). We investigate (i) the relation between
the bargaining power of the online retailer and the equilibrium royalty rate, (ii) the effectof the online retailer’s bargaining power on the profits and consumer and social welfare,(iii) the effect of self-regulation imposed by the supplier on the profits and consumer andsocial welfare, and (iv) how the competition mode (price-quantity competition) influences
the effect of self-regulation.10 Although our main focus listed above does not overlap with9

See also Foros et al. (2017); Johnson (2017); Maruyama and Zennyo (2020). Our paper also com-plements the literature on store-within-store formats in traditional retailers (Jerath and Zhang, 2010;
Netemeyer et al., 2012).
10

Among the papers listed in the main text, only Tan and Carrillo (2017) and Shen et al. (2019)

explicitly investigate the relationship between the retailer’s bargaining power and the endogenous royalty
rate in extension sections. The other papers, which endogenize royalty rates, assume that e-retailers
unilaterally set royalty rates (Abhishek et al., 2016; Zennyo, 2019, 2020; Fu et al., 2021; Tsunoda and
Zennyo, 2021).

7

those of the previous papers, the market structures of Shen et al. (2019) and Zennyo (2019)
partially overlap with that of ours. We discuss the difference between Zennyo (2019) andour paper in Section 6 because the demand system in that paper is closer in nature toours than that in Shen et al. (2019).11
Our paper also contributes to the literature on price parity clauses (price-matchingpolicies) with agency contracts by providing a different mechanism behind the welfarereducing price parity,12 although we consider the parity of different tier prices (wholesale
and retail prices).
We review several models dealing with price-matching policies under the dual-channel
supply chain in the context of operations management and marketing. Cattani et al. (2006)
is the seminal work that discusses a retail price-matching policy in a dual-channel supply
chain in which a direct retail channel and an indirect retail channel compete. Such a pricematching policy benefits the indirect channel. Cai et al. (2009) discuss a pricing constraint
related to ours in Stackelberg price-setting games with direct and indirect channels. They
discuss a price discount contract that ensures that the wholesale price for the indirectchannel should be lower than the retail price of the direct channel by some exogenous
ratio.13 When the manufacturer is the leader, such a price discount contract is beneficial
to the retailer if the exogenous ratio is lower than a threshold value.14 Their price-matchingpolicies are different to that in our paper. Furthermore, agency contracts are beyond thescope of their paper. Ding et al. (2016) also consider price-matching constraints related toCattani et al. (2006) and show that such constraints benefit the retailer, contrasting to our
result. Zhou et al. (2018) discuss a price-matching policy in a dual-channel supply chain
with investments for services and show that such a policy benefits the indirect channel.
11

Tsunoda and Zennyo (2021) incorporate uncertainty into Zennyo (2019). Note that Shen et al. (2019)

also consider Nash bargaining to derive the royalty rate in an extension section. One technical advantage
of our analysis is that we analytically obtain the royalty rate in the single-part contract, although Shenet al. (2019) also obtain the equilibrium royalty rate, which is independent of the bargaining power, undera two-part contract form with a royalty rate and a lump-sum payment. The nature of the equilibrium
royalty rate in their paper is reminiscent of the standard two-part tariff contract (setting the royalty rate tomaximize the total channel profits and then splitting them through the lump-sum payment, which depends
on the bargaining power).
12

The related works in economics are, for instance, Johnson (2017), Wang and Wright (2020), andBisceglia et al. (2021).
13

They also consider the case in which the reference retail price is that of the indirect channel.

14

If the retailer is the leader, the manufacturer can completely exploit the retailer’s rent by equalizing

its wholesale price to the predetermined retail price when no price discount contract exists. The channel
members can escape the problem by employing the price discount contract.

8

Recently, Zheng and Yu (2021) consider a bilateral monopoly model in which the
manufacturer can open its direct channel, which is less efficient than the indirect channel(the encroachment option). They assume that the manufacturer can employ a pricematching policy requiring the direct channel to equalize its price to the indirect channel’s
price and show that the availability of the policy enhances the incentive to encroach,harming the indirect channel. The key factor of their result is the enhanced incentiveof the manufacturer to encroach through the price-matching policy. This fact implies
that the encroachment option is necessary, contrasting to our model’s fixed dual-channel
supply chain. Zhen and Xu (2022) investigate three channel structures: (i) The monopoly
manufacturer sells its product to a retailer, which resells it in the retailer’s own channel
and through a third-party platform under an agency contract, as in the encroachment
subgame of Zheng and Yu (2021) and our paper; (ii) the manufacturer sells its product tothe retailer that resells it in the retailer’s own channel and uses the third-party platform
under the agency contract; (iii) the manufacturer also sells its product through the thirdparty platform under structure (ii). Under the three structures, they consider two pricing
strategies: offering discriminatory retail prices and offering a uniform price set by theretailer. They show that a higher exogenous royalty rate makes uniform pricing profitable
for the two firms in the third channel structure (Proposition 7), contrasting to our result,which shows that self-regulation is more likely to benefit the supplier and the traditional
retailer when the bargaining power of the online channel is lower (the endogenous royalty
rate becomes lower).
There are several further differences between those two papers and ours. First, we
introduce an online platform employing an endogenous agency contract with a monopoly
supplier. The two players negotiate the contract term based on a percentage of the platform’s revenue. Second, the constraint is looser than those in the related papers in that
we assume that the retail price on the online platform should be greater than or equal
to the wholesale price for the indirect channel. That is, we allow the possibility that theretail price is strictly higher than the wholesale price, although the constraint is always
binding in equilibrium. The looser restriction is technically richer than the restrictions in
the related papers. Third, the tiers of matched prices are different as mentioned earlier.
Finally, the competition mode in our paper is the price-quantity competition, which issuitable for the content markets, including books, games, movies, and videos.
Technically, our modeling approach is related to the literature on price-quantity com-9

petition. Although there are many papers discussing price-quantity competition in various
contexts since the seminal works by Bylka and Komar (1976) and Singh and Vives (1984)(e.g., Askar, 2014; Bian et al., 2018; and the comprehensive survey of Tremblay and Tremblay (2019)), only a few papers discuss price-quantity competition with vertical relations(e.g., Yang et al., 2015; Fanti and Scrimitore, 2019). The main themes in the latter two
papers are the endogenous choices of strategic variables (price and quantity) differing fromours.
The modeling approach in the supplementary appendix in Lei (2019) is the closest
to ours. The market structures in her model and ours use price-quantity competition.
The difference between her model and ours is threefold. First, the main focuses of thetwo models are different. She investigates the optimal channel structure for a monopoly
manufacturer, as in Chiang et al. (2003) and Arya et al. (2007). We investigate theequilibrium royalty rate and the impact of self-regulation by an upstream firm, but fix
the dual-channel supply chain. Second, the properties of the realized royalty rates in thetwo models differ from each other. The endogenous royalty rate in her model binds atthe exogenous upper bound because the retailer unilaterally sets the royalty rate. The
endogenous royalty rate in our model is flexible because we consider Nash bargaining.
Third, self-regulation by the supplier is only discussed in our model. In this sense, weshould regard our model as an extension of price-quantity competition in a direction
different from her extension.
Table 1 summarizes the literature that is closely related to our research.

3

Problem description

We introduce three players below. A monopolistic supplier produces two versions of agood, physical and digital versions, and distributes through two retailers: (i) a traditional
retailer who resells the physical version of the good, and (ii) an e-commerce retailer who
handles the digital version of the good. For the sake of clarity, we focus on a book market
where a monopolistic supplier publishes a book and sells it through a traditional bookstore(henceforth referred to as T -retailer) and an e-commerce platform (E-retailer). T -retailer
sells a traditional paper version of the book (hereafter, we refer to it as paper book), andE-retailer sells an electronic version of the book (e-book). The market structure is similar

10Table 1: Summary of the related literature
ArticleAgency model

Price matching

Cattani et al. (2006)(direct)✓

Cai et al. (2009)(direct)✓

Yang et al. (2015)(direct)

Abhishek et al. (2016)
Ding et al. (2016)
Tan and Carrillo (2017)✓✓(direct)✓✓✓

Bian et al. (2018)
Lu et al. (2018)

price-quantity
competition✓(direct)✓

Lei (2019)✓✓

Shen et al. (2019)✓

Tsunoda and Zennyo (2021)✓

Fanti and Scrimitore (2019)

Zheng and Yu (2021)(direct)✓

Fu et al. (2021)✓

Zhen and Xu (2022)✓✓

This research

✓

✓

✓

to those in the papers discussing e-books (Li et al., 2015a; Lu et al., 2018).15
We assume that the product characteristics of the paper book and the e-book are
different, as explained in Sections 1 and 2. Hereafter, the subscripts t and e denotethe T -retailer and E-retailer, respectively. For the paper book, T -retailer determines its
sales quantity, qt , because it must consider the capacity of the book in the physical store(Kreps and Scheinkman (1983), Farahat et al. (2019), and the discussion in Section 1). By
contrast, for the e-book, the monopolistic supplier and E-retailer sign an agency contractthat allows the supplier to set its retail price, pe , at the E-retailer directly without worrying
about the capacity of the book thanks to the unlimited capacity of the electronic format(Kreps and Scheinkman (1983) and the discussion in Section 1). These assumptions are
consistent with the standard view on when price and quantity are strategic variables offirms (e.g., Kreps and Scheinkman (1983); Belleflamme and Peitz (2015), Section 3.3.3
on p. 67; Cabral (2017), Section 8.3 on pp. 200–201; Yang et al. (2018); Farahat et al.
(2019)). From the assumptions, the competition mode in the retail market is the pricequantity competition (Singh and Vives (1984), Askar (2014), Bian et al. (2018, Section3.3)). T -retailer and the supplier simultaneously determine qt and pe , respectively.
15

The competition mode in their models (price competition) differs from ours (price-quantity competi-tion).

11

To describe the price-quantity competition in the retail market, we use the demandsystem in the related papers (Singh and Vives (1984), Cai (2010), Askar (2014), Matsui(2017), Bian et al. (2018, Section 3.3), Fanti and Scrimitore (2019), Zhang and Zhang(2020), Zheng and Yu (2021)). The inverse demand functions for the paper book and thee-book are:pt = α − qt − γqe ,pe = α − qe − γqt ,where α(> 0) is a positive constant and γ ∈ (0, 1) is the degree of product substitutability
between the two products. The demand system means that the product characteristicsof the paper book and the e-book generate horizontal product differentiation (the firstparagraph in Section 1).16 We convert the demand system to meet the competition mode,the price-quantity competition (e.g., Askar, 2014; Bian et al., 2018, Section 3.3):17
qe = α − pe − γqt ,(1)

pt = α − qt − γ(α − pe − γqt )= (1 − γ)α − (1 − γ 2 )qt + γpe .

(2)For analytical and notational simplicity, we assume that the market potentials for the two
versions of the good (the constant terms α in the inverse demand functions) are symmetricand that their marginal costs are zero. In Section 4.5, we explain that those assumptions
do not qualitatively affect the results of Section 4.
The monopolistic supplier transacts with T -retailer and E-retailer with a wholesale
contract and an agency contract, respectively (e.g., Lu et al., 2018; Zennyo, 2020).18 Witha wholesale contract, the supplier determines its wholesale price, wt , before T -retailer
sets its sales quantity, qt . With an agency contract, the supplier sets the retail price ofthe e-book, pe , directly, and the supplier and E-retailer split sales revenues according
to a royalty rate, r, that is determined in negotiation between them before the supplier16

If γ is close to zero (resp. one), the two versions are almost different (resp. homogeneous).

17

Equation (2) in Askar (2014) contains a typo (the sign of the last term in the first line). Of course,

the mathematical procedure is correct.
18

Those papers consider endogenous choices of contract forms for E-retailers as in Hao and Fan (2014)

and Abhishek et al. (2016).

12

determines wt (and pe ). Concretely, the supplier pays r times its revenue earned via theE-retailer’s channel to the E-retailer. The E-retailer’s bargaining power over the supplieris β ∈ (0, 1/2].19 We formulate the bargaining problems in the subsequent sections. The
timing for the royalty rate follows those in the related papers (e.g., Abhishek et al., 2016;
Zennyo, 2019, 2020; Tsunoda and Zennyo, 2021). We believe that the assumptions oncontract terms fit the reality where most publishers adopt wholesale contracts with brickand-mortar bookstores, and those publishers adopt agency contracts with online retailers
to sell e-books (see Dantas et al. (2014) and Gilbert (2015) who describe the pricing
arrangements between publishers and retailers in book industries).20
From the discussions and equations (1) and (2), the profits of the supplier and theretailers are:πs = wt qt + (1 − r)pe qe = wt qt + (1 − r)pe (α − pe − γqt ),πt = (pt − wt )qt = ((1 − γ)α − (1 − γ 2 )qt + γpe − wt )qt ,(3)πe = r pe qe = r pe (α − pe − γqt ),where the subscript s represents the supplier.
We define consumer surplus as:
CS ≡∑ {∫ qii=t,e}(α − xi − γqj − pi ) dxi ,(4)0where j ̸= i, and social welfare is
SW ≡ CS + πs + πt + πe .

(5)Table 2 and Figure 1 summarize the problem description.
Under the market structure, we discuss the effect of the following self-regulation by
the supplier: the price at E-retailer, pe , must be higher than or equal to the wholesale19

We restrict the parameter range of β within (0, 1/2] because the realized royalty rate under the range(1/2, 1] is too high, which does not match the real-world royalty rates very closely.
20

For instance, Amazon yearly enters Annual Vendor Negotiations with various vendors, including pub-lishers (McLeod, 2018; The U.S. House of Representatives, 2019). In 2014, when the five famous publishers
in the US (so-called the big five) changed the contract form from wholesale contracts to agency contracts
with Amazon, two of the big five, Simon & Schuster and Hachette, initially engaged in negotiations with
Amazon (Gilbert, 2015). Therefore, the bargaining scenario in our model reflects the reality. Jarsulic(2020) provides a useful comprehensive case study on the market power of platform monopolies.

13

Table 2: Notation and explanations
Notation

Description

qtj

The demand of traditional retailer (strategic variable), j = {A, B, C, D, R}

qej

The demand of e-commerce platform/E-retailer

pjt

The retail price of traditional retailer

pje

The retail price of E-retailer (strategic variable)

αMarket size (constant)β

The bargaining power of E-retailerγ

The degree of product substitutability

πij

The profit of each player (i = s, t, e)wtj

Wholesale price for traditional retailer (strategic variable)

weWholesale price for E-retailer (used only in Section 6)

r

j

Os
λ

j

The royalty rate for E-retailer (determined in negotiation)
The value of outside options for the supplier (defined in Section 4)
The Lagrange multiplier (defined in Section 5)

CS

Consumer surplus (CS R represents consumer surplus under supplier’s self-regulation)

SW

Social welfare (SW R represents social welfare under supplier’s self-regulation)

Else

Superscript ∗ denotes the equilibrium solution in the baseline model; superscript j
represents different cases from the baseline model (used in Sections 5 and 6)

price for T -retailer, wt ; mathematically, pe ≥ wt . The formulation of the self-regulation
borrows the ideas of the remedy in the context of price/margin squeeze (see, e.g., Bouckaert
and Verboven, 2004; Sidak, 2008; Jullien et al., 2014) and the price discount contracts
investigated in Section 5.1 of Cai et al. (2009).21
We consider two games: (i) The game of the dual-channel supply chain without any
self-regulation; (ii) the game of the dual-channel supply chain with self-regulation pe ≥ wt .
In both games, the timing is as follows.
Stage 1. The supplier and E-retailer determine the royalty rate, r, through bargaining.
Stage 2. The supplier unilaterally sets a wholesale price, wt , for T -retailer.
Stage 3. Observing the previous outcomes, T -retailer chooses the sales quantity, qt , andthe supplier chooses the price of the e-book, pe , simultaneously.
We solve each game by backward induction. We have a remark on the assumption. Contract observability does not matter (we explain the reason for this in Section 4.1).
21

Cai et al. (2009) also discuss different types of constrained pricing in Section 4.

14

Supplier (s)(πs in eq. (3))
wtpe

T -retailer (t)(πt in eq. (3))

E-retailer (e)(πe in eq. (3))
r: royalty rate

qt
Consumers(Eqs. (1) and (2))
Note: The arrows show the directions of the offers.

Figure 1: The market structure

4

Analysis

In this section, we analyze the game of the dual-channel supply chain without any selfregulation (hereafter, the baseline model).

4.1

Stage 3To begin with, we derive the results of Stage 3. In this stage, T -retailer sets qt and the
supplier sets pe simultaneously. They maximize their profits expressed in (3) given thewholesale price, wt , and the royalty rate, r. The first-order conditions of T -retailer andthe supplier are:∂πt
= (1 − γ)α + γpe − 2(1 − γ 2 )qt − wt = 0,∂qt

∂πs
= (1 − r)(α − γqt − 2pe ) = 0. (6)∂pe∂πs /∂pe in (6) provides three remarks on the pricing of the supplier. First, it does
not internalize the profit through the wholesale price wt , leading to fiercer competition in
the downstream market. The discrepancy between the internalization and the first-ordercondition stems from the price-quantity competition discussed here.22 If the strategic
variable of T -retailer is also price (say pt ), T -retailer’s quantity is a function of pt and pe(say qt (pt , pe )), and then the supplier indirectly takes into account the wholesale profit,
wt qt (pt , pe ), mitigating the discrepancy (we discuss this in Section 6). Second, the supplier
22This effect in itself is not new. Such a discrepancy occurs in the context of licensing under quantity

competition (see, e.g., Faulí-Oller and Sandonís (2002), p. 195).

15

does not care about the royalty rate, r. As the royalty rate increases, the discrepancy
mentioned above becomes serious. Third, the second remark implies that T -retailer need
not know the realized r when it sets its quantity. Because both T -retailer and the supplierknow wt , our model does not have the matter of contract observability.
From (6), we obtain the optimal quantity and price:α(2 − γ) − 2wt
qt (wt ) =,4 − 3γ 2()
α 2 − γ − γ 2 + γwt
pe (wt ) =.
4 − 3γ 2(7)

We can check that ∂qt (wt )/∂wt < 0 and ∂pe (wt )/∂wt > 0; that is, when the wholesale
price of the paper book increases, T -retailer reduces its sales quantity of the paper bookand the supplier raises its retail price for the e-book. Therefore, the supplier can intensify
or weaken the retail competition by adjusting the wholesale price.

4.2

Stage 2

In Stage 2, given the quantity and price in (7), the supplier chooses the optimal wt thatmaximizes its profit. The supplier tries to balance the wholesale revenue of the paper bookand the sales revenue of the e-book.23 The optimal wholesale price is:{}α 8 − 4 rγ − 2(4 − r)γ 2 + (1 + 2 r)γ 3
wt (r) =.
2 {8 − γ 2 (7 − r)}(8)

From (8), it is straightforward that ∂wt (r)/∂r < 0. If, for example, the supplier’s shareof the e-book’s revenue decreases, the supplier places weight on the wholesale revenue ofthe paper book and lowers the wholesale price to strengthen T -retailer’s competitiveness.
Regarding the contract form, although we believe that assuming the nature of thewholesale contract is reasonable given real-world practices in the book industry (see, e.g.,
Dantas et al., 2014; Gilbert, 2015), it is reasonable to question what happens if the supplieremploys a two-part tariff contract. Even under a two-part tariff contract, the suppliercannot set a low wholesale price to internalize the profit of T -retailer because such a low
price unnecessarily intensifies the downstream competition in the third stage. Although
the wholesale price under the two-part tariff contract is lower than that under the wholesale
contract, the former wholesale price can be higher than the equilibrium retail price of E23

This trade-off between the two sources of revenue is similar to that of vertically integrated producers

who consider both wholesale revenue from independent retailers and sales revenue of their integrated
retailers (see, e.g., Arya et al., 2008; Arya and Mittendorf, 2018).

16

retailer under the two-part tariff contract for some parameter range.

4.3

Stage 1In Stage 1, based on their expectations of the results in the subsequent stages, the supplierand E-retailer negotiate over the royalty rate, r. We employ Nash bargaining to derive
the negotiation outcome, following the formulations in the related papers on supply chain
issues (e.g., Choi and Guo, 2020; Choi, 2020; Matsui, 2020, 2021).24 For simplicity, we
assume that E-retailer has no outside options. However, we assume that, if the negotiation
breaks down, the supplier sells the paper book only through T -retailer. At that time, thesupplier sets wto = α/2 and T -retailer chooses qto = α/4, where the superscript o indicatesthe case in which the supplier executes its outside option. Therefore, the disagreement
profit of the supplier is:
Os =

α2
.
8

(9)

The equilibrium royalty rate, r∗ , solves the following problem:max [πs (r) − Os ]1−β [πe (r)]β , where
r

{
(
) }
(
)2
α2 12 − 8γ − 3γ 2 − 4 2 − γ − γ 2 r
α2 8 − 2γ − 5γ 2 r
πs (r) =, πe (r) =.
4 {8 − (7 − r)γ 2 }4 {8 − (7 − r)γ 2 }2

(10)

(11)

Solving the problem in (10), we obtain the equilibrium royalty rate:
Proposition 1 In equilibrium, as a result of their Nash bargaining, the supplier andE-retailer set the royalty rate as:{1r =(8 − 2γ − 5γ 2 )2βγ 2 (16 − 8γ − 7γ 2 )}√− (8 − 2γ − 5γ 2 )4 − β 2 γ 2 (16 − 8γ − 7γ 2 )(16 − 16γ + γ 2 )(8 − 7γ 2 ) , (12)∗where the superscript “∗” represents the equilibrium outcome in the baseline model. Figure
2 shows the royalty rates in the three cases (β = 1/6, 1/3, 1/2). Given the equilibrium
royalty rate r∗ under β, if the degree of product substitutability is moderate (aroundγ = 0.5), β around 1/3 approximates the actual real-world royalty rate for e-books (e.g.,
24

Matsui (2020) explains the theoretical foundation of using Nash bargaining in noncooperative games.

17

30% in Amazon). This might imply that publishers have reasonably strong bargaining
power over online retailers thanks to the copyrights.

r
0.5
β=1/2

0.4
0.3

β=1/3

0.2
β=1/6

0.1

0.0

0.2

0.4

0.6

0.8

1.0

γ

Figure 2: Equilibrium royalty rates (β = 1/6, β = 1/3, β = 1/2)
We obtain the equilibrium profit of the supplier, that of T -retailer, and that of Eretailer, πs∗ , πt∗ , and πe∗ , respectively. The other variables such as qi∗ , p∗i for i = t, e, andwt∗ are also derived and shown in Table 3 in the Appendix.
The simple but tedious partial derivatives of r∗ with respect to β and γ, respectively,
lead to ∂r∗ /∂β > 0 and ∂r∗ /∂γ < 0 for any β ∈ (0, 1/2] and γ ∈ [0, 1). The former
outcome is intuitive in that the stronger bargaining power of E-retailer leads to its higher
profit share. We explain the latter outcome. The royalty rate r is based on the additional
contribution of E-retailer, which is related to the two channels’ relative profitability.
An increase in γ has two contrasting effects: (i) intensifying retail competition, which
diminishes E-retailer’s profitability, and (ii) enhancing the direct channel advantage of
E-retailer because of the nonexistence of double marginalization. The mixture of the two
effects determines the sign of ∂r∗ /∂γ. When β is less than 1/2, the first effect dominates
the latter one, then ∂r∗ /∂γ < 0.
A simple comparison between p∗e and wt∗ leads to Proposition 2:
Proposition 2 In the equilibrium of the baseline model, the supplier sets the retail priceof the e-book below the wholesale price of the paper book depending on β and γ, that is,18

p∗e < wt∗ , if and only if
0 < β < β̄ and 0 < γ < 1, or




12
and 0 < γ < γ1 (β), γ2 (β) < γ < 1, orβ̄ < β ≤

37




 12 < β < 1 and 0 < γ < γ1 (β),
37
2

(13)

where β̄ ≃ 0.306, and γ1 (β) and γ2 (β) are the first and the second root of 128 − 256β −(32 + 64β)γ − (112 − 216β)γ 2 + (8 + 74β)γ 3 + (20 − 7β)γ 4 .

0.5
wt < pe
0.4
0.3
βw > pe

0.2

t

0.1
0.0
0.0

0.2

0.4

0.6

0.8

1.0

Figure 3: Parameter ranges where the supplier uses belowwholesale-price retail pricing
The supplier implements below-wholesale-price retail pricing when the parameters γ
and β are in the shaded area in Figure 3. A small value of β means that the supplier’s
bargaining power over E-retailer is strong; therefore, (1 − r∗ ) is large. At that time, thesupplier prioritizes the sales revenue of the e-book, (1−r∗ )pe qe , over the wholesale revenueof the paper book, wt qt ; the supplier tries to soften the retail competition and earn larger
sales revenue from the e-book by raising wt and pe . To sell the e-book well, the increasein pe is moderate compared with that in wt . Therefore, wt∗ exceeds p∗e when β is small.

4.4

Effects of countervailing power

To take a more in-depth look at the equilibrium characteristics of the baseline model,
we analyze how a change in β affects the equilibrium results in this subsection. Check19

ing the simple comparative statics for each equilibrium outcome, we derive the following
proposition (we omit the proof).
Proposition 3 An increase in E-retailer’s bargaining power lowers the retail price of thee-book. Moreover, this increases the profits of T -retailer and E-retailer, the consumer
surplus, and the social surplus; however, it reduces the supplier’s profit. Specifically,
∂p∗e
∂wt∗
∂πt∗
∂πe∗
∂πs∗
∂CS ∗
∂SW ∗
< 0,
< 0,
> 0,
> 0,
< 0,
> 0,
> 0.
∂β
∂β
∂β
∂β
∂β
∂β
∂β

(14)

First, we can confirm that E-retailer’s bargaining power leads to a lower retail price inour model. Second, it may be more surprising that T -retailer improves its profit froman increase in E-retailer’s bargaining power. These two effects arise from the supplier’s
reaction to an increase in β. When β increases, the supplier places more weight on thewholesale revenue of the paper book over the retail revenue of the e-book. As a result,the supplier lowers wt , inducing a low pe ; that is, ∂p∗e /∂β < 0 and ∂wt∗ /∂β < 0.25 Even
though the retail price of the paper book, pt , also decreases, both (pt − wt ) and qt increase.
Consequently, T -retailer can earn more profit when E-retailer’s bargaining power becomesstrong. This complementarity between T -retailer and E-retailer might be related to therecent empirical finding in Li (2021), who points out the complementarity between theoffline print channel and the e-channel on the supply side.
As an aside, in addition to the supplier’s profit, total sales of the e-book decrease inβ, specifically, ∂p∗e qe∗ /∂β < 0. In our model, an increase in E-retailer’s bargaining power
does not promote e-book sales but rather suppresses e-book sales because, for its profit,
the supplier comes to depend more on the wholesale revenue of the paper book when β
becomes large.
Decreases in p∗e and wt∗ through an increase in β improve the consumer and social
surplus.

4.5

Heterogeneous market potentials

We discuss how the heterogeneity of market potentials and marginal costs of the twocontent formats influences the results in this section. The demand system is: pt = (1 +25This mechanism is very similar to those revealed in Chen (2003) and Matsushima and Yoshida (2018).

In their papers, a supplier, which deals with a dominant retailer and fringe retailers under wholesale
contracts, responds to an increase in the dominant retailer’s bargaining power by lowering the wholesale
price for fringe retailers to boost its sales through the fringe retailers.

20b)α − qt − γqe and pe = α − qe − γqt , where b(> 0) is the market potential advantage of thepaper book over the e-book. Furthermore, we incorporate the supplier’s marginal cost of
the paper book ct (> 0) into the baseline model with the modified demand system.
In the modified model, we can summarize the additional components into the differenceη ≡ bα − ct , which affects the profitability of the firms.26 On the one hand, given the
consumers’ strong preference for paper books in the real world (see the first paragraph in
Section 1), the market potential for the paper book (the constant term (1 + b)α for thepaper book) is higher than that for the e-book. On the other hand, the marginal costof producing one unit of paper books is higher than that of e-books. If the former effectdominates (is dominated by) the latter one, an equilibrium wholesale price is more (less)likely to be higher than an equilibrium supplier’s retail price. In other words, as the valueof η increases, the price of the e-book is more likely to be lower than the wholesale priceof the paper book.
0.5

0.5

0.4

0.4

0.3

0.3

β

β
0.2

0.2

0.1

0.1

0.0
0.0

0.2

0.4

0.6

0.8

0.0
0.0

1.0

γ

0.2

0.4

0.6

0.8

1.0

γ

η = −0.025

η = 0.025

Figure 4: Parameter ranges where (wt∗ − ct ) − p∗e > 0
We calculate (wt∗ − ct ) − p∗e in the modified model.27 If this difference is positive,p∗e < wt∗ . We can numerically show that (wt∗ − ct ) − p∗e > 0 if η ≥ 0 and β < 0.305.
The shaded areas in Figure 4 show the parameter regions in which (wt∗ − ct ) − p∗e > 0
for η = −0.025 and η = 0.025 under α = 1. Figure 4 shows that this inequality is more
likely to hold as η becomes larger. We can conclude that the result here is qualitatively
similar to that in Proposition 2 when the paper book is advantageous over the e-book(when η ≥ 0).
26

The mathematical procedure is available in the Supplementary Appendix.

27

We include ct to make the numerical calculation work well. If the difference is positive, the inequalitywt∗ − p∗e > 0 also holds because ct > 0.

21

We also obtain outcomes qualitatively similar to those in Proposition 3 when η ≥ 0.
Moreover, except for the sign of ∂πt /∂β when η < 0, the outcomes of comparative statics
in Proposition 3 hold even for −1/10 ≤ η ≤ 0 under α = 1.

5

Self-regulation on below-wholesale-price retail pricing

Proposition 2 in the previous section shows the possibility that the wholesale price forT -retailer is higher than the retail price of E-retailer even if we have assumed that themarginal costs of the two books are zero.
In this section, we consider a simple way to escape the imbalance between the wholesale
price for T -retailer and the retail price of E-retailer by following the insights in the context
of price/margin squeeze (Bouckaert and Verboven, 2004; Jullien et al., 2014) and the pricematching policy in Cai et al. (2009).
We impose the following self-regulation regarding wt and pe on the supplier: wt ≤pe , which prevents the supplier from setting wt that is higher than pe . We derive new
equilibrium results with this self-regulation and compare them with those of the baselinemodel.
In the same vein as the baseline model, we derive a subgame perfect Nash equilibriumof the self-regulation case by backward induction. Therefore, at first, we solve the retailcompetition in Stage 3.

5.1

Stage 3 with self-regulation

The maximization problems of T-retailer and the self-regulated supplier in Stage 3 are:max πt = (pt − wt )qt ,qt

max π̃s = wt qt + (1 − r)pe qe + λ(pe − wt ),
pe

(15)

where λ ≥ 0 is a Lagrange multiplier. The first-order conditions are:∂πt
= (1 − γ)α + γpe − 2(1 − γ 2 )qt − wt = 0,
∂qt
∂ π̃s
= (1 − r)(α − γqt − 2pe ) + λ = 0.
∂pe

(16)

From equation (16), we obtain the optimal retail quantity and price and the condition

22

that the constraint, wt ≤ pe , is binding. The constraint is binding if and only if:
λ=(1 − r)(−α(2 + γ) + (4 + 3γ)wt )> 0,
2(1 + r)in other words, if and only if
wt >

5.2

(2 + γ)α.
4 + 3γ(17)Stage 2 with self-regulation

Following the outcome in (17), we consider two cases in Stage 2: (Case A) wt > (2 +γ)α/(4 + 3γ) (the constraint in Stage 3 is binding); (Case B) wt ≤ (2 + γ)α/(4 + 3γ) (theconstraint in Stage 3 is not binding).
Case A (wt > (2 + γ)α/(4 + 3γ))In this case, the optimal retail quantity and price are:qtA (wt ) =α − wt,2(1 + γ)(18)pA
e (wt ) = wt .
Taking the quantity and the price as given, the supplier sets wt to maximize its profit
subject to the constraint (17). The supplier solves the following maximization problem:(A
max πs (qtA (wt ), pAe (wt )) + λwt(2 + γ)αwt −4 + 3γ),(19)

where λA ≥ 0. The optimal wholesale price and the Lagrange multiplier in this case are:
wtA =

α
,
2

(20)

A

λ = 0.
We find that this wtA is greater than the lower bound of (17), (2 + γ)α/(4 + 3γ).
Case B (wt ≤ (2 + γ)α/(4 + 3γ)) When wt ≤ (2 + γ)α/(4 + 3γ), the constraint inStage 3, wt ≤ pe , is not binding. The optimal retail quantity and price are derived bysubstituting λ = 0 into the first-order conditions in equation (16):(2 − γ)α − 2wt,
4 − 3γ 2(2 − γ − γ 2 )α + γwt
.
pB
(w
)
=
t
e
4 − 3γ 2
qtB (wt ) =

23

(21)

Given the quantity and the price, the supplier solves the following problem:(Bmax πs (qtB (wt ), pBe (wt )) + λwt)(2 + γ)α− wt ,
4 + 3γ(22)

where λB ≥ 0. We can obtain both a corner solution and an interior solution in thiscase. The supplier chooses the corner solution (wt = (2 + γ)α/(4 + 3γ) and λB > 0) when0 ≤ r < (2 − γ)/(4 + 2γ), and the interior solution (wt < (2 + γ)α/(4 + 3γ) and λB = 0)
when (2 − γ)/(4 + 2γ) ≤ r < 1. However, the former corner solution is dominated by theoutcome of Case A from the viewpoint of the supplier’s profit. Therefore, we show only
the optimal wholesale price and the Lagrange multiplier of the latter. They are:
α
wtB =

{
}
8 − 4 rγ − 2(4 − r)γ 2 + (1 + 2 r)γ 3,
2 {8 − γ 2 (7 − r)}(23)

λB = 0.
Note that this wtB is consistent with that in the baseline model without self-regulation.
Comparison of the profits under Cases A and B

We can now compare the sup-

A
B
B
B
B
plier’s profit under Cases A and B: πs (qtA (wtA ), pAe (wt )) and πs (qt (wt ), pe (wt )). We

find that the former is larger (smaller) than the latter if and only if the royalty rate, r, is√smaller (larger) than (1 + 9 − 4γ − 4γ 2 )/2(2 + γ). Moreover, the threshold value under
this condition is higher than (2 − γ)/(4 + 2γ). To sum up, we derive the following subgame
equilibrium:

5.3

√

9 − 4γ − 4γ 2
1
+

A


wt = wt if 0 < r ≤2(2 + γ)√

1 + 9 − 4γ − 4γ 2
wt = wtB if< r.
2(2 + γ)(24)

Stage 1 with self-regulation

We move on to Stage 1, the bargaining stage. We have assumed that β ≤ 1/2 to focus on
realistic equilibrium outcomes. In the Appendix, we show that the Nash product in (10)is maximized at a royalty rate that leads the supplier to set wtA in the following subgame.
The equilibrium is as follows:
Proposition 4 When the supplier imposes self-regulation, wt ≤ pe , and when β ∈ (0, 1/2],
then the supplier and E-retailer choose the royalty rate rR = 2β/(2 + γ). The supplierR
sets wtR = α/2 and the constraint is binding, pR
e = wt = α/2. The quantities are24qeR = (2 + γ)α/(4(1 + γ)) and qtR = α/(4(1 + γ)).
The superscript “R” represents the equilibrium outcome in the self-regulated supplier case.
We also obtain the equilibrium profit of the supplier, that of T -retailer, and that of
E-retailer, πsR , πtR , and πeR , respectively (see Table 3 in the Appendix).
As in Section 4.4, we conduct comparative statics. Except for πsR and πeR , the outcomes
are independent of β (Proposition 4). We find that ∂πsR /∂β < 0 and ∂πeR /∂β > 0 as in
Proposition 3.

5.4

Comparison of the results with and without self-regulation

By comparing the royalty rate and the prices in Section 5 with those of the baseline model,
we derive the following proposition (we omit the proof because of the simple comparisons).
Proposition 5 In the equilibrium of the self-regulated supplier case with β ∈ (0, 1/2], forγ ∈ (0, 1),• the royalty rate with self-regulation, rR , is always higher than that of the baselinemodel, r∗ , and the difference, rR − r∗ , increases as γ increases;• the wholesale price with self-regulation, wtR , is higher than that of the baseline model,wt∗ ;R• the retail prices with self-regulation, pRe and pt , are also higher than those of thebaseline model, p∗e and p∗t .
Specifically, for γ ∈ (0, 1) and β ∈ (0, 1/2],∗
R∗
rR > r∗ , wtR > wt∗ , pR
e > pe , pt > pt .

(25)

Figure 5 shows a graphical comparison between the royalty rates in the baseline andself-regulation cases.
Proposition 5 states that self-regulation, which seemingly lowers the wholesale price
wt , has anticompetitive effects on the retail competition. The supplier anticipates the
expected outcome of Stage 3, in which the wholesale price set in Stage 2 is the lower
bound of its retail price set in Stage 3. Setting a higher wholesale price allows the supplier
to commit itself to setting a high retail price of the e-book in Stage 3.

25

r*
0.35
0.30
Self-regulation

0.25

Baseline

0.20

0.0

0.2

0.4

0.6

0.8

1.0

γ

Figure 5: Equilibrium royalty rates in the two cases (β = 1/3)
Furthermore, the royalty rate under self-regulation becomes higher than that in thebaseline model as the degree of product substitution γ increases. The reason is that therelative profitability of T -retailer shrinks because of the higher wholesale prices, giving Eretailer a better bargaining position over the supplier. This profit shrinkage of T -retailer
is higher when the downstream competition is effective; in other words, when γ is higher.
We check the effects of self-regulation on the firms’ profits. Comparing πe∗ , πt∗ , andπs∗ with πeR , πtR , and πsR , respectively, we have (we omit the proof because of the simple
comparisons):
Proposition 6 When β ∈ (0, 1/2], the supplier’s self-regulation, wt ≤ pe ,• always increases the profit of E-retailer;• increases the profit of T -retailer iff 0 < γ < 1 and 0 < β < β1 (γ);• increases the profit of the supplier iff 0 < γ ≤ γ̃, where γ̃ ≃ 0.631, or γ̃ < γ < 1 and
0 < β < β2 (γ);
where

32γ + 16γ 2 − 26γ 3 − 15γ 4,
128 + 96γ − 120γ 2 − 86γ 3 + 7γ 4√
1 16 − 8γ − 21γ 2 + 6γ 3 + 7γ 4.
β2 (γ) =2
8 − 7γ 2β1 (γ) =(26)

From Proposition 6, the impact of self-regulation on E-retailer is clear. The self-regulated
supplier can commit to setting a higher pe when it sets a higher wt . The supplier controls
the competitive environment in the downstream market by setting a higher wt , which26

leads to a weaker competitive position of T -retailer. Furthermore, the weaker competitive
position of T -retailer increases the royalty rate r, which is significant when γ is high. As
a result, self-regulation benefits E-retailer.
However, the impact of self-regulation on T -retailer and the supplier depends on the
degree of product substitution γ. In the baseline model, the higher the degree of productsubstitution γ, the lower the royalty rate r, which increases the supplier’s profitability from
E-retailer. As a result, the supplier’s incentive to weaken T -retailer becomes stronger asγ increases. The negative effect on T -retailer is stronger when the bargaining power ofE-retailer is weaker because the supplier’s profitability from E-retailer is higher. Selfregulation limits such an incentive of the supplier, benefiting T -retailer. The benefit
dominates the cost of a high wt if the bargaining power of E-retailer is sufficiently weak
and the degree of product substitution is high (see the south-east area in the left-hand sideof Figure 6). For the supplier, the increase in the royalty rate diminishes the positive effectof self-regulation on its profit. The higher the value of γ, the larger the increment of theroyalty rate through self-regulation, completely offsetting the benefit of self-regulation (seethe north-east area in the right-hand side of Figure 6). Figure 6 illustrates the ambiguous
impact of self-regulation on the profits of T -retailer and the supplier.
0.5

0.5
π s > πs R

0.4

0.4
π t > πt R

0.3

0.3

β

β
0.2

πs

0.2

0.1
0.0
0.0

0.2

0.4

πt

πt R

0.6

0.8

πs R

0.1

1.0

γ

0.0
0.0

0.2

0.4

0.6

0.8

1.0

γ

Figure 6: Effects of self-regulation on T -retailer’s profit (left panel)and on the supplier’s profit (right panel)
Our results provide an important managerial implication for content suppliers. Ourmodel fits the distribution problems in the copyrighted content industries, including books,
games, movies, and music.28 We expect that a content supplier’s bargaining power over28

For example, Capcom Co., Ltd. (2020, p.69) describes recent market trends and forecasts for the game27

online retailers is stronger as the originality/uniqueness of its content is higher. Our theoretical model shows that self-regulation benefits all firms if the online retailer’s bargaining
power is sufficiently weak and the product substitutability between the two contents is not
sufficiently high; in other words, if the royalty rate is sufficiently low and the two contents
are moderately differentiated. The suppliers/providers of content with high originality
can commit to the self-regulation discussed here without causing complaints by brickand-mortar retailers. However, if content suppliers/providers do not have content withhigh originality, imposing the pricing constraint can cause conflicts with brick-and-mortar
retailers because the constraint harms those retailers. In sum, the bargaining power ofcontent suppliers/providers is a key factor in determining whether the pricing constraint
discussed here is feasible.

5.5

Welfare analysis

Finally, we examine the effect of self-regulation on social welfare and consumer surplus.
Before checking the welfare effect, we examine the effect on industry profits (the producer surplus) and show the following result (we omit the proof because of the simple
comparison).
Proposition 7 When β ∈ (0, 1/2], the supplier’s self-regulation, wt ≤ pe , increases industry profits when the degree of substitution is relatively high and the E-retailer’s bargaining
power over the supplier is relatively low, which is illustrated in Figure 7.
Propositions 6 and 7 jointly demonstrate the profitability of self-regulation for the threefirms and the producers’ surplus. There is a parameter area in which self-regulation
benefits all firms.
Comparing the consumer and social surpluses of the baseline model with those of theself-regulated supplier case, we have (we omit the proof because of the simple comparisons):
Proposition 8 When β ∈ (0, 1/2], the supplier’s self-regulation, wt ≤ pe , decreases bothconsumer surplus and social welfare, that is, for γ ∈ (0, 1),
CS ∗ > CS R ,SW ∗ > SW R .

(27)

industry. The sales volumes of the mobile content and PC online markets globally are $92.0 billion and$38.4 billion (US), respectively, whereas that of the consumer (package + digital) market, which is themarket for traditional home video game consoles, is $35.1 billion (US). Online content dominates offline,
contrary to the book industry.

28

0.5
0.4
0.3

πs πe πt> πsR+πeR+πt R

β0.2

πs πe πt< πsR+πeR+πt R

0.1
0.0
0.0

0.2

0.4

0.6

0.8

1.0

γ
Figure 7: Effect of self-regulation on the industry profits
In addition to the increases in the retail prices, the retail quantities are lower for mostvalues of γ. Therefore, self-regulation reduces the consumer surplus. Moreover, eventhough self-regulation can increase industry profits, it decreases social welfare for all γ.
Proposition 8 cautions the supplier to impose the price constraint, which may cause
concern on competition policy if the competition authority follows our welfare analysis. Proposition 8 stems from the price-quantity competition, which captures competition
modes in the content industries. Therefore, given that content suppliers employ the pricing
constraint discussed here, if the competition authority recognizes that capacity constraintsof the traditional retailers influence their pricing policies, it might apply our results to the
conduct of the content suppliers. This concern is also a factor in determining whether
implementing the pricing constraint discussed here is feasible.
In the Appendix (Section 8.4), we numerically confirm the propositions in the main
text.

6

Retailers compete in priceWe develop a modified model in which the two retailers compete in price instead of theprice-quantity competition to clarify the importance of considering the price-quantity competition. Concretely, self-regulation, in which the supplier does not use below-wholesaleprice retail pricing (wt ≤ pe ), does not have any effect on the outcome in the modified

29

model, when we introduce price competition into the retail market. To examine the result,
we solve the self-regulation case with price competition and show that the constraint is
not binding, wt < pe , in equilibrium.
The maximization problems of T -retailer and the self-regulated supplier in the retail
pricing stage are:max πt = (pt − wt )qt ,pt

max π̃s = wt qt + (1 − r)pe qe + λ(pe − wt ),
pe

(28)

where λ ≥ 0 is a Lagrange multiplier. The first-order conditions are:α(1 − γ) − 2pt + γpe + wt
∂πt
=
= 0,
∂pt
1 − γ2∂ π̃s(1 − r){α(1 − γ) − 2pe + γpt } + γwt=+ λ = 0.
∂pe
1 − γ2(29)

Note that the supplier internalizes the wholesale revenue contrary to the price-quantity
model; the supplier’s first-order condition includes wt , whereas it is not included in equation (6). This consideration for the wholesale revenue in the retail pricing stage differsfrom the supplier’s pricing in the baseline model, and it increases the retail price of theE-retailer.29 From equation (29), we obtain optimal retail prices, and the condition forthe constraint, wt ≤ pe , is binding. The constraint is binding if and only if:λ=−α(2 − γ − γ 2 )(1 − r) + {4 − 3γ − γ 2 − (4 − γ − γ 2 )r}wt> 0;2(1 − γ 2 )(30)

in other words, if and only if
4 − 3γ − γ 2≡ r̄, and4 − γ − γ2α(2 − γ − γ 2 )(1 − r)wt ≥≡ w̄t .
4 − 3γ − γ 2 − (4 − γ − γ 2 )r
0<r≤

(31)

In Stage 2, we consider two cases: (Case C) wt ≥ w̄t and r ≤ r̄ (the constraint in Stage
3 is binding); (Case D) wt ≤ w̄t or r > r̄.
29

From equation (6), T -retailer’s reaction function in the baseline case is qt (pe ) = {(1 − γ)α + γpe −wt }/2(1−γ 2 ), and we can see that ∂wt qt (pe )/∂pe > 0 for all γ ∈ (0, 1). Therefore, the lack of consideration
for the wholesale revenue makes the supplier more aggressive and decreases pe to less than the optimallevel for the supplier who cares about the wholesale revenue.

30

Case C (wt ≥ w̄t and r ≤ r̄) In this case, the optimal retail prices are:α(1 − γ) + (1 + γ)wt,
2pC
e (wt ) = wt .

pC
t (wt ) =(32)

Taking these prices as given, the supplier sets wt to maximize its profit subject to theconstraint (31); the supplier solves the following maximization problem:CC
max πs (pCt (wt ), pe (wt )) + λ (wt − w̄t ) ,
wt

(33)

where λC ≥ 0. Then, the optimal wholesale price and the Lagrange multiplier are:α(2 − γ − γ 2 )(1 − r)
= w̄t ,
4 − 3γ − γ 2 − (4 − γ − γ 2 )rαγ{1 − γ + r(1 + γ)}{3 + γ − r(2 + γ)}
λC =.
2(1 + γ){4 − 3γ − γ 2 − (4 − γ − γ 2 )r}
wtC =

(34)

We can also check that λC is always positive for any r ∈ (0, r̄]. In the range wt ≥ w̄t , ifr ≤ r̄, wt = w̄t is optimal in Stage 2, otherwise (r > r̄), Case C is not sustainable.
Case D (wt ≤ w̄t or r > r̄)When wt ≤ w̄t or r > r̄, the constraint in Stage 3,

wt ≤ pe , is not binding. The optimal retail prices are derived by substituting λ = 0 into
the first-order conditions in equation (29):α(2 − γ − γ 2 )(1 − r) + {2(1 − r) + γ 2 }wt,(4 − γ 2 )(1 − r)α(2 − γ − γ 2 )(1 − r) + γ(3 − r)wt
pD
(w
)
=
> wt .
te(4 − γ 2 )(1 − r)pDt (wt ) =(35)

Given these prices, the supplier solves the following problem:DD
max πs (pDt (wt ), pe (wt )) + λ (w̄t − wt ) ,
wt

(36)

where λD ≥ 0. We can obtain an interior solution in this case; the constraint, wt ≤ w̄t , isslack. The optimal wholesale price and the Lagrange multiplier are:wtD =α(2 − γ − γ 2 )(1 − r){4 + 2γ(1 − r) − γ 2 + γ 3 }< w̄t ,
2{8(1 − r) − γ 2 (7 − 8r − r2 ) − γ 4 }λD = 0.

31(37)

From the discussions in Cases C and D, the supplier prefers the interior solution in Case
D to the corner solution in Case C, which is the same as the corner in Case D.
This result implies that the supplier does not set wt that induces the retail price at theE-retailer, pe , to be binding to wt . Therefore, the self-regulation, wt ≤ pe , does not affectthe equilibrium prices and profits in price competition. Furthermore, the retail price at theE-retailer is not lower than the wholesale price of the T -retailer, wt , in equilibrium. We canconclude that the price-quantity competition is crucial in obtaining the counterintuitiveeffect of self-regulation.
Here, we briefly describe the model and results in Zennyo (2019), as well as the differences with our paper. Zennyo (2019) focuses on the equilibrium contractual form between
the supplier and E-retailer in price–price competition under a similar market structure.
The channel structure is the same as in our paper: there is a monopoly supplier,a traditional retailer (T -retailer), and an e-commerce retailer (E-retailer). The gameproceeds as follows. In Stage 1, E-retailer unilaterally offers the royalty rate r to themonopoly supplier. In Stage 2, if the supplier accepts it, the two firms employ the agency
contract, in which the supplier sets the retail price for E-retailer in Stage 3, and the supplierunilaterally sets a wholesale price wt for T -retailer. Otherwise, the supplier unilaterally
sets wholesale prices wt and we for both retailers under the wholesale contract, in which
T -retailer sets its retail price in Stage 3. In Stage 3, T -retailer and the designated firm
under the contract offer their retail prices simultaneously.30
In his study, because of the advantage of the agency contract, the supplier and Eretailer always employ the agency contract in equilibrium, and thus the realized market
situation becomes the same as ours.31 The differences between his study and ours are as
follows. First, T -retailer’s strategic variable is price in Zennyo (2019) and quantity in ourpaper; second, the procedure for determining the royalty rate is a unilateral offer in Zennyo(2019) and Nash bargaining in our paper; and third, the main focus is the equilibrium
contract form in Zennyo (2019) and the market outcomes under price-quantity competitionand self-regulation in our paper. Despite these similar situations, we emphasize that ourdifferent (but natural) assumptions yield very different outcomes and rich implications.
30

He also endogenizes the choice of whether the supplier uses E-retailer and shows that the supplieralways uses E-retailer in addition to T -retailer in equilibrium.
31

In addition to the advantage of the agency contract discussed above, E-retailer prefers the agency

contract to the wholesale contract because he can obtain a first-mover advantage. See Johnson (2017) for
more details.

32

In this sense, we believe that his study and our own complement each other.

7

Conclusion

Following the recent advancement of online platforms facilitating digital content sales, weinvestigate a model in which a monopoly supplier distributes two types of its productthrough a traditional retailer with a wholesale price contract and an online retailer withan agency contract. The supplier and the online retailer negotiate the royalty rate throughNash bargaining. A notable feature of our model is assuming that the traditional retailer
and the supplier, via the online retailer, compete in quantity and price, respectively. We
believe that the assumption of price-quantity competition is consistent with the standard
view on when price and quantity are strategic variables of firms (e.g., the textbook by
Belleflamme and Peitz (2015), Section 3.3.3 on p. 67). We also discuss self-regulation by
the supplier such that the retail price of the online retailer is not lower than the wholesale
price for the traditional retailer, seemingly helping the traditional retailer.
The additional discussion on the abovementioned self-regulation by the supplier leads
to the following results. The wholesale price under self-regulation is strictly higher thanthat in the baseline model. The retailers’ prices under self-regulation are also strictly
higher than those in the baseline model. The retail price of the online retailer is always
binding by the constraint. The royalty rate under self-regulation is strictly higher than thatin the baseline model because the negotiating party anticipates lower profitability of the
traditional retailer, giving the online retailer a strong bargaining position over the supplier.
The self-regulation benefits the online retailer. However, it benefits the traditional retailer
if the bargaining power of the online retailer is sufficiently weak. Furthermore, the selfregulation benefits the supplier if the product substitutability between the two contents
is not sufficiently high. The consumer and total surpluses under self-regulation are lower
than those in the baseline model. In sum, the self-regulation can benefit all firms in somesituations but always harms consumers.
We have shown that incorporating the heterogeneity of market potentials and marginal
costs of the two content formats does not qualitatively change the results derived in Section 4. However, incorporating the heterogeneity makes the first stage analysis in the
self-regulation case (Section 5) unsolvable.32 Solving the self-regulation case with the het32

The first-order condition in the bargaining problem becomes the fourth-order equation of r.

33

erogeneity of market potentials and marginal costs could be challenging future research.
We consider the cases where the monopoly supplier uses two different channels to
distribute two different content formats: brick-and-mortar and online retailers. We canalso consider a case in which an online retailer handles both content formats in addition
to the existence of brick-and-mortar retailers. Considering the additional element is also
important for future research.

34

8

Appendix

8.1

Proof of Proposition 4

In Stage 1, the supplier and E-retailer negotiate the royalty rate. Considering (24), we
derive the optimal r that maximizes the Nash product in (10). The Nash product under
the self-regulation case is:

where




N P A (r)


if 0 < r ≤
N P B (r)



if

√

1+

9−4γ−4γ 2,2(2+γ)

√

1+

(38)

9−4γ−4γ 2< r < 1,2(2+γ)

]1−β [ 2]β[α (2+γ)rN P A (r) ≡ α2 (2−(2+γ)r)
8(1+γ)
8(1+γ)
,


[
]
[ 2
]β

2 )r) 1−βα (8−2γ−5γ 2 )2 r
N P B (r) ≡ α2 (16−16γ+γ 2 −(16−8γ−7γ
8(8−(7−r)γ 2 )4(8−(7−r)γ 2 )

(39)

.

√
For 0 < r ≤ (1 + 9 − 4γ − 4γ 2 )/2(2 + γ), r = rA ≡ 2β/(2 + γ) is optimal, which
is characterized by ∂N P A (rA )/∂r = 0. The second-order derivative of N P A with r is√
negative; that is, ∂ 2 N P A (r)/∂r2 < 0. However, for (1 + 9 − 4γ − 4γ 2 )/2(2 + γ) <r < 1, ∂N P B (r)/∂r < 0 for any r. Therefore, the corner solution, r = rB ≡ (1 +√9 − 4γ − 4γ 2 )/2(2 + γ), is optimal in this case.
Finally, we compare N P A (rA ) and N P B (rB ) to derive the solution r to the Nashbargaining. As Figure 8 shows, N P A (rA ) > N P B (rB ) for all β ∈ (0, 1/2] and γ ∈ (0, 1);
rA is the optimal royalty rate for the self-regulation case.

NPA (rA )-NPB (rB )
0.15

0.10
β=1/6

β=1/3

0.05

β=1/2

0.0

0.2

0.4

0.6

0.8

1.0

γ

Figure 8: The difference between the equilibrium Nash products
for Cases A and B in the self-regulation case (α = 1)

35

8.2

Equilibrium results
Table 3: Equilibrium outcome for the baseline case and the selfregulation case
Baseline caseSelf-regulation case√γ (16−8γ−7γ )(16−16γ+γ )(8−7γ )r∗ = (8−2γ−5γ ) − (8−2γ−5γβγ) 2−β(16−8γ−7γ 2 )2βrR = 2+γwt

α{8−4γr ∗ −2γ 2 (4−r∗ )+γ 3 (1+2r∗ )}2{8−γ 2 (7−r∗ )}

α
2

pe

α(8−2γ−5γ 2 )2{8−γ 2 (7−r∗ )}

α
2

pt

α{12−2γ(2+r∗ )−2γ 2 (6−r)+5γ 3 }2{8−γ 2 (7−r∗ )}α(3−γ)4qeα(8−2γ−5γ 2 )2{8−γ 2 (7−r∗ )}α(2+γ)
4(1+γ)qt

α{2−γ(2−r∗ )}8−γ 2 (7−r∗ )α4(1+γ)πsα2 {12−8r∗ −4γ(2−r∗ )−γ 2 (3−4r∗ )}4{8−γ 2 (7−r∗ )}α2 {3−2rR +γ(1−rR )}8(1+γ)

r

2 2

2 4

2 2

2

2

2

2

πe
πt
CS
SW

α2 (8−2γ−5γ 2 ) r∗
4{8−γ 2 (7−r∗ )}2

α2 (2+γ)rR8(1+γ)α2 (1−γ 2 ){2−γ(2−r∗ )}2{8−γ 2 (7−r∗ )}2

α2 (1−γ)16(1+γ)α2 {80+16γr∗ −4γ 2 (35−4r∗ −(r∗ )2 )−4γ 3 (1+2r∗ )+5γ 4 (13−4r∗ )}8{8−γ 2 (7−r∗ )}2

α2 {304−48γ(4−r∗ )−4γ 2 (89−8r∗ +(r∗ )2 )+4γ 3 (43−18r∗ +2(r∗ )2 )+75γ 4 }8{8−γ 2 (7−r∗ )}236

α2 (5+3γ)
32(1+γ)α2 (19+5γ)
32(1+γ)8.3

A brief summary of the five major publishers’ conduct

We briefly describe the actions of the five major publishers in the US when e-books entered
the market.
We believe that the self-regulation discussed in our study is a realistic and reasonable strategy, taking into account the history of the US publishing industry, where thecurrent five major publishers (Penguin Random House, HarperCollins, Simon & Schuster, Hachette, and Macmillan) have continued to attempt to raise the prices of e-books(Sang, 2017) (Penguin and Random House merged in 2013 and became Penguin Random House). We believe their conduct reflects concern for the sales of paper books andbookstores’ profits.
Before e-books became widely accepted around 2010, Amazon adopted a strategy ofselling e-books of best sellers and new releases at $9.99, which was lower than the wholesale
prices of those e-books. This strategy aimed to promote sales of Kindle, an e-book reader
released by Amazon.
Those publishers argued that selling e-books at such a low price would hurt the salesof paper books, damaging the profitability of bookstores and distributors. As a result,some publishers raised the wholesale price of e-books, which had originally been lower
than that of paper books; others delayed the release of e-versions for weeks or months
after the release of paper books. Furthermore, they considered strategies such as retailprice maintenance, mandatory minimum advertised pricing, and a joint venture to selle-books (Cote, 2013). However, these strategies were not enough to force Amazon to raise
the prices of e-books.
The five publishers except for Random House in the US adopted agency contracts todirectly set the prices of e-books when Apple entered the e-book market with the iPadlaunch in 2010.33 These arrangements included most favored nations (price parity) clauses,
which put pressure on Amazon to move to agency contracts. However, Apple and the five
publishers were sued by the US Department of Justice in 2012 for conspiring to raise theprices of e-books. As a result, the agency contracts were prohibited for two years. In fact,
De los Santos and Wildenbeest (2017) empirically show that the return to the wholesale
contracts from the agency contracts lowered e-book prices by 18% at Amazon.

33

Random House employed agency contracts in 2011 after the five publishers did.

37

8.4

Numerical examples

We numerically show how γ influences the equilibrium outcomes in Tables 4 and 5, confirming the propositions.
First, we confirm the propositions in Section 4. r∗ in each table shows that r∗ decreases
in γ (Proposition 1 and Figure 2). Table 4 shows that the inequality wt∗ > p∗e holds forγ ≤ 0.7; similarly, Table 5 shows that the inequality wt∗ > p∗e always holds (Proposition 2
and Figure 3). Comparing the values in Tables 4 and 5, we find that the values of wt∗ , p∗e ,and πs∗ in Table 4 are smaller than those in Table 5 and that the values of πe∗ , πt∗ , CS ∗ ,and SW ∗ in Table 4 are higher than those in Table 5 (Proposition 3).
Second, we confirm the propositions in Section 5. rR in each table shows that rR
decreases in γ, but the values of wtR and pR
e remain the same irrespective of γ (Proposition
4). The difference rR − r∗ is always positive and increases with γ in each table, and theRdifferences, wtR − wt∗ and pRe − pe , are always positive in each table (Proposition 5 andFigure 5). For profits, πs∗ < πsR holds for γ ≤ 0.8 in Table 4, although πs∗ < πsR always
holds in Table 5; πe∗ < πeR always holds in each table; πt∗ > πtR always holds in Table
4, although πt∗ < πtR hold for γ ≥ 0.8 in Table 5 (Proposition 6 and Figure 6). The
industry profit πs∗ + πe∗ + πt∗ in the baseline model is higher than πsR + πeR + πtR in the
self-regulation case if γ ≤ 0.6 in Table 4; and if γ ≤ 0.3 in Table 5 (Proposition 7 andFigure 7). CS ∗ > CS R and SW ∗ > SW R always hold in both tables (Proposition 8).

38

39

γ

r∗

rR

wt∗

wtR

p∗e

pR
e

p∗t

pR
t

qe∗

qeR

qt∗

qtR

0.1

0.317132

0.31746

0.491678

0.5

0.488455

0.50.720262

0.725

0.488455

0.477273

0.230893

0.227273

0.2

0.301851

0.30303

0.483408

0.5

0.478526

0.5

0.689557

0.7

0.478526

0.458333

0.214738

0.208333

0.3

0.287438

0.289855

0.475220.5

0.469857

0.5

0.658089

0.675

0.469857

0.442308

0.200954

0.192308

0.4

0.273799

0.277778

0.467126

0.5

0.462173

0.5

0.625998

0.65

0.462173

0.428571

0.1891330.178571

0.50.260803

0.266667

0.459130.5

0.455251

0.5

0.593378

0.625

0.455251

0.416667

0.178997

0.166667

0.6

0.248249

0.256410.451235

0.5

0.448884

0.5

0.560282

0.6

0.448884

0.406250.170387

0.156250.7

0.235788

0.246914

0.443451

0.5

0.442852

0.5

0.526724

0.575

0.442852

0.397059

0.163279

0.147059

0.8

0.222644

0.238095

0.435839

0.5

0.436861

0.5

0.492664

0.55

0.436861

0.388889

0.157847

0.138889

0.9

0.206355

0.229885

0.4287460.50.430491

0.5

0.458094

0.525

0.430491

0.381579

0.154464

0.131579

γ

πs∗

πsR

πe∗

πeR

πt∗

πtR

CS ∗

CS R

πs∗ + πe∗ + πt∗πsR + πeR + πtR

SW ∗

SW R

0.1

0.276449

0.276515

0.0756641

0.0757576

0.0527784

0.0511364

0.157228

0.150568

0.404892

0.403409

0.562120.553977

0.2

0.263673

0.263889

0.0691201

0.0694444

0.0442679

0.0416667

0.1581010.145833

0.377061

0.375

0.5351630.5208330.30.252807

0.253205

0.0634563

0.0641026

0.0367482

0.0336538

0.15890.141827

0.353011

0.350962

0.511911

0.492788

0.40.243469

0.244048

0.0584847

0.0595238

0.0300478

0.0267857

0.159653

0.138393

0.332001

0.330357

0.491654

0.468750.50.235384

0.236111

0.0540523

0.0555556

0.0240299

0.0208333

0.160391

0.135417

0.313466

0.31250.473857

0.447917

0.6

0.228360.229167

0.0500214

0.0520833

0.0185804

0.015625

0.161155

0.132813

0.296962

0.296875

0.458116

0.429688

0.7

0.222282

0.223039

0.0462423

0.0490196

0.0135967

0.0110294

0.162005

0.130515

0.282121

0.283088

0.444126

0.4136030.80.217153

0.217593

0.042491

0.0462963

0.00896969

0.00694444

0.163048

0.128472

0.268613

0.270833

0.431661

0.3993060.9

0.213306

0.212719

0.0382423

0.0438596

0.00453324

0.00328947

0.164437

0.126645

0.256082

0.259868

0.420518

0.386513

Table 4: Numerical examples (α = 1, β = 1/3)

40

γ

r∗

rR

wt∗

wtR

p∗e

pR
e

p∗t

pR
t

qe∗

qeR

qt∗

qtR

0.1

0.15855

0.15873

0.495555

0.5

0.488553

0.5

0.722205

0.725

0.488553

0.477273

0.22894

0.227273

0.2

0.150867

0.151515

0.490665

0.5

0.47890.5

0.693223

0.7

0.47890.458333

0.210997

0.208333

0.3

0.143594

0.144928

0.485464

0.50.470681

0.5

0.663334

0.675

0.470681

0.442308

0.195462

0.192308

0.4

0.136686

0.138889

0.480054

0.5

0.463642

0.5

0.632755

0.65

0.463642

0.428571

0.181788

0.178571

0.5

0.130071

0.133333

0.474524

0.5

0.457619

0.5

0.601667

0.625

0.457619

0.416667

0.169524

0.166667

0.6

0.123642

0.128205

0.468973

0.5

0.452529

0.5

0.570245

0.6

0.452529

0.40625

0.158238

0.156250.7

0.117202

0.123457

0.463550.5

0.448413

0.5

0.53872

0.575

0.448413

0.397059

0.147391

0.147059

0.8

0.110325

0.119048

0.458578

0.5

0.445607

0.5

0.507532

0.55

0.445607

0.388889

0.135982

0.138889

0.9

0.101704

0.114943

0.4551340.5

0.445618

0.5

0.478095

0.525

0.445618

0.381579

0.120849

0.131579

γ

πs∗

πsR

πe∗

πeR

πt∗

πtR

CS ∗

CS R

πs∗ + πe∗ + πt∗πsR + πeR + πtR

SW ∗

SW R

0.1

0.314293

0.314394

0.0378434

0.0378788

0.0518893

0.0511364

0.156734

0.150568

0.404025

0.403409

0.560759

0.553977

0.2

0.298274

0.298611

0.0346006

0.0347222

0.0427391

0.0416667

0.157142

0.145833

0.375614

0.375

0.532756

0.520833

0.3

0.284618

0.285256

0.0318119

0.0320513

0.0347668

0.0336538

0.157473

0.141827

0.351197

0.350962

0.50867

0.492788

0.4

0.27285

0.273810.0293825

0.0297619

0.0277593

0.0267857

0.157719

0.138393

0.329991

0.330357

0.487711

0.46875

0.5

0.262619

0.263889

0.0272389

0.0277778

0.0215537

0.0208333

0.157865

0.135417

0.311412

0.31250.469277

0.447917

0.6

0.253672

0.255208

0.0253197

0.0260417

0.016025

0.015625

0.157875

0.132813

0.295016

0.296875

0.452891

0.429688

0.7

0.245831

0.247549

0.0235664

0.0245098

0.0110793

0.0110294

0.157664

0.130515

0.280477

0.283088

0.438140.413603

0.8

0.2390180.240741

0.0219067

0.0231481

0.00665684

0.00694444

0.157004

0.128472

0.267581

0.270833

0.424585

0.399306

0.90.233382

0.234649

0.0201959

0.0219298

0.00277485

0.00328947

0.155057

0.126645

0.256353

0.259868

0.411410.386513

Table 5: Numerical examples (α = 1, β = 1/6)

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Supplementary Appendix
We explain the mathematical procedure in Section 4.5.
The profits of the supplier and the retailers are:πs = (wt − ct )qt + (1 − r)pe qe = (wt − ct )qt + (1 − r)pe (α − pe − γqt ),πt = (pt − wt )qt = ((1 + b − γ)α − (1 − γ 2 )qt + γpe − wt )qt ,(40)

πe = rpe qe = rpe (α − pe − γqt ).
The first-order conditions in Stage 3 are:∂πs
= (1 − r)(α − 2pe − γqt ) = 0,∂pe∂πt
= (pt − wt )qt = (1 + b − γ)α − 2(1 − γ 2 )qt + γpe − wt = 0.
∂qt

(41)

Solving the simultaneous equation, we have:pe (wt ) =(2 − (1 + b)γ − γ 2 )α + γwt(2(1 + b) − γ)α − 2wt, qt (wt ) =.
2
4 − 3γ
4 − 3γ 2(42)Solving the first-order condition in Stage 2, we have:wt (r) =(8(1 + b)(1 − γ 2 ) + γ 3 − 2γ(2 − (1 + b)γ − γ 2 )r)α + 2(4 − 3γ 2 )ct.
2(8 − (7 − r)γ 2 )(43)Using the outcomes in Stages 2 and 3, we have πs (r), πt (r), and πe (r).
Solving the maximization problems in the bilateral monopoly with the supplier andthe E-retailer, we have the disagreement profit of the supplier:Os =((1 + b)α − ct )2.
8

We need to solve the following bargaining problem to obtain r:
max [πs (r) − Os ]1−β [πe (r)]β .
r

To simplify the exposition, we define η ≡ bα − ct . The royalty rate r is:(8 − 2γ − 5γ 2 )2 α2 − 4(8 − 2γ − 5γ 2 )γηα − 4γ 2 η 2 +r=βγ 2 ((16 − 8γ − 7γ 2 )α2 − 2(4 − γ)γηα + γ 2 η 2 )

50

√

R

,

(44)

where R ≡ ((8 − 2γ − 5γ 2 )α − 2γη)4 − β 2 γ 2 (8 − 7γ 2 )((16 − 8γ − 7γ 2 )α2 − 2(4 − γ)γηα +γ 2 η 2 )((16 − 16γ + γ 2 )α2 − 2(8 − 7γ)γηα + 7γ 2 η 2 ). If η = 0, the derived r is the same asthat in (12).

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