Discussion Paper No. 1020

SUPPLIER ENCHROACHMENTAND CONSUMER WELFARE:
UPSTREAM FIRM’S OPPORTUNISM
AND MULTICHANNEL DISTRIBUTION

Cong Pan

February 2018

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

Supplier Encroachment and Consumer Welfare:
Upstream Firm’s Opportunism and Multichannel
Distribution*
Cong Pan†
Faculty of Economics, Nagoya University of Commerce & BusinessFebruary 14, 2018

Abstract
I revisit supplier encroachment under the framework of a two-part tariﬀ contract.
When a monopoly manufacturer supplies competing retailers and each retailer’s contracting process is unobservable to the rival, the retailer’s lack of knowledge vis-à-vis its
rival’s contract may undermine the manufacturer’s commitment power, which preventsthe manufacturer from achieving optimal profit. I demonstrate that when the manufacturer directly supplies the resale market, it can use the direct channel as a commitment
tool and thus restore its market power. Even though the manufacturer’s encroachment
creates more competitors in the resale market, the resultant higher wholesale prices aggravate double marginalization, which may reduce consumer welfare. This result holds
even when the manufacturer is very eﬃcient in direct selling.

Keywords: channels of distribution; encroachment; two-part tariﬀ contract; supplier
opportunism; consumer welfare
JEL Classification Numbers: L14, L22, M11* I would like to express my sincere thanks to Noriaki Matsushima for his constructive suggestions on

this paper. Besides, I really thank Hiroshi Kitamura, Toshihiro Matsumura, Tomomichi Mizuno and Shohei
Yoshida, the conference participants at Asia-pacific Industrial Organisation Conference 2017, the 44th Annual
Conference of EARIE 2017 and 2017 Spring Meeting of Japanese Economic Association, the seminar participants at Centre d’économie de la Sorbonne-Cachan, ENS Université paris Saclay and Osaka University for theirkind encouragement and suggestions. The author acknowledges the financial support from the Japan Society
for the promotion of Science (Pan, 15J05223) and Nagoya University of Commerce & Business. Needless tosay, all errors are mine.
†
Address: Nagoya University of Commerce & Business, 4-4 Sagamine Komenoki-cho Nisshin-shi, Aichi,
470-0193, Japan. Tel: (81)-561-73-2111. E-mail: cong pan@nucba.ac.jp.

11Introduction

Direct distribution by upstream suppliers is very commonplace in vertically related industries.1 Over decades, incumbent retailers (or franchisees) in traditional indirect channels
have been complaining about the competitive pressure caused by upstream manufacturers’(or franchisers’) downward entry, known as “supplier encroachment,” and making lobbying
eﬀorts in convincing governments that their territory should be legally protected (Kalnins,2004). On the other hand, the conventional wisdom that any policy against manufacturers’
downward entry is anti-competitive and harmful to consumers’ benefits (Dutta et al., 1999;
Blair and Lafontaine, 2005) has put governments in an unusual position when deciding how
encroachment should be regulated.
Surprisingly, despite of the extensive attention from pioneering theoretical studies on
manufacturers’ encroaching activities (e.g., Arya et al., 2007), the welfare magnitude of thisimportant and controversial economic phenomenon has not been rigorously discussed. The
focus on supply chain profit makes a linear contract more suitable for modeling supplierencroachment than a two-part tariﬀ contract. One technical reason is that under a two-parttariﬀ contract (contract terms comprise a wholesale price and a fixed fee) the manufacturer
can always achieve full downstream surplus through the fixed fee and would thus have no
incentive to encroach.2 Actually, under a linear contract, the welfare eﬀect of encroachment
becomes quite straightforward as well known by the conventional wisdom. Specifically, theencroachment promotes competition and alleviates double marginalization, and thus must
1

In the background of franchising contracts, over several decades, franchisors in traditional industries (e.g.,
auto services, business aids, hotels and motels, and laundries and dry cleaners) have operated some stores
directly, while franchising others to third parties (Lafontaine, 1992). Also in manufacturing, the proliferationof e-commerce enables upstream manufacturers (e.g., Nike, Adidas, Apple, and Samsung) to open online
channels to drop-ship directly to consumers (Randall et al., 2006), while maintaining traditional brick-andmortar wholesale channels.
2
Notice that for analytical simplicity, the main trend of theoretical studies on this topic has focused on
the case wherein the upstream manufacturer has full bargaining power. The arguments in this paper are also
restricted to this case. If instead, the upstream manufacture does not have full bargaining power, analyses undera two-part tariﬀ contract would also be feasible. See concluding remarks for the author’s future work about
supplier encroachment and Nash bargaining under a two-part tariﬀ contract.

2

increase consumer welfare.
However, the empirical literature shows a contrasting reality to the existing theoreticalmodels that in industries wherein supplier encroachment occurs, the two-part tariﬀ contract
is far more customary than the linear contract (Lafontaine, 1992; Kalnins, 2004).3 More surprisingly, recent empirical study on food market shows a quite counterintuitive result: local
farmers’ direct selling leads to lower total sale, implying that encroachment may have a negative eﬀect on consumer welfare (Park, 2015). The above contrasting results demonstrated
respectively by empirical and theoretical studies confirm the importance of reexamining theexisting theoretical models and studying how encroachment aﬀects consumer welfare.
Motivated by the above, I develop a two-part tariﬀ framework to study the welfare eﬀect
on consumers caused by encroachment, using a model wherein a monopoly manufacturer
sells through competing retailers. I show that under quantity competition and when eachretailer’s contracting process is unobservable to the rival throughout the game (interim unobservability), the manufacturer’s encroachment always reduces consumer welfare, even whenthe manufacturer is very eﬃcient in direct selling. This result holds even when the contracting process is unobservable but the contracting results can be observed before quantity
competition (interim observability). The same result also holds under price competition withinterim unobservability.
The above result is somewhat counterintuitive in that increasing the number of resalecompetitors (with the manufacturer’s downward entry) may trigger an anticompetitive effect. The intuition is as follows. Under secret contracting, when there is no encroachment,each retailer only accepts a wholesale price as low as the manufacturer’s production cost.
Otherwise, the retailer would be hurt by the manufacturer who always secretly oﬀers a lower
wholesale price to the other retailer. Because of a lack of commitment, the manufacturer
3

Lafontaine (1992) uses data from 548 franchisors to suggest that manufacturers’ multichannel marketing
exists in most industries (e.g., auto services, business aids, education services, and hotels and motels) and thatover 90% (504 out of 548) of observations employed two-part tariﬀ contracts. In Kalnins (2004), the adoption
rate of the two-part tariﬀ contract is 88% (7 out of 8).

3

has to supply the retailers at equal-to-margin wholesale prices, and thus fails to achieve an
optimal profit. On the other hand, when the manufacturer has committed to encroach, its
incentive to raise wholesale prices so as to shift more share to its direct channel is well understood by each retailer. In other words, the manufacturer can use the encroachment as
a commitment tool to charge above-margin wholesale prices, which solves its commitment
problem. The higher wholesale prices amplify double marginalization, thus curtailing consumers welfare.
My main logic that the invalidation of a two-part tariﬀ contract may trigger a manufacturer’s encroaching incentive is consistent with some empirical studies. For example,
Kaufmann and Lafontaine (1994) use the case of McDonald’s to show that a monopolist
with a two-part tariﬀ contract may enhance its profit by opening direct outlets. Michael andMoore (1994) support this conclusion and provide more generalized results among a wider
set of franchise systems. Moreover, my findings concerning consumer welfare have significant policy implications vis-à-vis the debate over anti-encroachment legislation. Since theearly 1990s, managers and legislators have sought to protect incumbent retailers’ territories
against manufacturers’ (franchisors’) direct sales (e.g., Hadfield, 1990; Blair and Lafontaine
2005; Espinoza, 2008). However, their eﬀorts have been considered anticompetitive andthus harmful to consumers’ benefits (Dutta et al., 1999; Blair and Lafontaine, 2005). Some
who protest anti-encroachment legislation argue that encroachment may even benefit incumbent retailers, inducing pareto improvements (Kaufmann and Rangan, 1990; Kalnins, 2004).
My study, on the contrary, demonstrates that supplier encroachment may be anticompetitive
under some market circumstances and provides support for anti-encroachment legislation.
As noted, studies concerning supplier encroachment generally employ linear contract
frameworks. For example, Chiang et al. (2003) discuss a price-setting game and show that amanufacturer uses the direct channel to threaten retailers into reducing their prices, although
no direct sales actually occur. Arya et al. (2007) consider a quantity-setting game and show4

that supplier encroachment triggers the manufacturer’s incentive to reduce the wholesale
price and may thus benefit the incumbent retailer. Tsay and Agrawal (2004) consider thecase wherein both manufacturer and retailer invest in sales promotions. Cattani et al. (2006)
discuss whether a manufacturer would promise an incumbent retailer to maintain a consistent
wholesale or retail price after it encroaches. In sum, under a linear contract framework, thesestudies reveal that supplier encroachment works as an approach to alleviate double marginalization.My study diﬀers from previous literature at the outset by focusing on how supplierencroachment performs under a two-part tariﬀ contract. From this outset, my analysis oﬀers
a diﬀerent standpoint–encroachment provides the manufacturer with a strategic tool to solve
its commitment problem, which aggravates double marginalization.
To the best of my knowledge, Li et al. (2015) is the only study that analyzes supplierencroachment within a framework of nonlinear contracts. It discusses a bilateral monopoly
with asymmetric market information in which the retailer knows the true market size whereas
the manufacturer knows only the distribution. The manufacturer oﬀers a nonlinear contract
comprising a unit wholesale price and a corresponding quantity. The manufacturer faces a
trade-oﬀ between using a direct channel to reduce information rent and causing distortions
in retailer’ order quantity. Essentially, in that study, the nonlinear contract always perfectly
resolves double marginalization regardless of encroachment. Consumers always benefit fromthe encroachment because the downstream competition becomes more intense.4 Since mystudy does not incorporate asymmetric market information, reducing information rent is nota concern. Then, I can focus on investigating how encroaching upon the resale market helps
the manufacturer restore market power and how this aﬀects consumer welfare. Therefore,
despite of new results obtained in this research, I add several new insights to the literature.
My study is also related to the literature discussing upstream monopolist’s commitment
problem. A large theoretical literature shows that when a monopoly manufacturer supplies
4

Their earlier study (Li et al., 2014) discusses supplier encroachment under asymmetric market information
using a framework of linear contract.

5

competing retailers and secretly oﬀers each a two-part tariﬀ contract, each retailer’s lack ofknowledge of its rival’s contracting results may undermine the manufacturer’s commitment
power. Specifically, the lack of commitment triggers each retailer to distrust the manufacturer’s oﬀer, which prevents the manufacturer from achieving optimal profit via the fixedfee (e.g., Hart and Tirole, 1990; O’Brien and Shaﬀer, 1992; McAfee and Schwartz, 1994).
Solutions that help the manufacturer solve the commitment problem include imposing minimum resale price maintenance (O’Brien and Shaﬀer, 1992), allowing for nondiscrimination clauses for retailers (Marx and Shaﬀer, 2004),5 utilizing upstream capacity constraint(Avenel, 2012), and carrying out vertical integration (Reisinger and Tarantino, 2015).6 In
this study, I demonstrate that an encroachment can also help a manufacturer solve this problem by creating a commitment tool (i.e., a direct channel).
The remainder of the paper is organized as follows. Section 2 introduces quantity competition with retailers’ interim unobservability. Section 3 uses a general demand to demonstrate changes in wholesale prices, the manufacturer’s profit and consumer welfare after theencroachment. Section 4 uses some examples to derive the manufacturer’s encroaching decision in equilibrium and how consumer welfare is aﬀected. Section 5 carries out an extension
to analyze diﬀerentiated products. Section 6 discusses quantity competition with retailers’interim observability, and price competition with both interim unobservability and observability. Section 7 concludes the paper and discusses the case of the linear wholesale pricecontract.

5

Marx and Shaﬀer (2004) challenge one of McAfee and Schwartz (1994)’s results that nondiscrimination
clauses is not able to solve the monopoly supplier’s opportunism and thus fails to solve the commitment problem. Marx and Shaﬀer (2004) show that the commitment can be solved when retailers are allowed to invoke
their nondiscrimination clauses.
6
Also as demonstrated by law literature, more solutions to the manufacturer’s commitment problem could
include operating through a sole outlet (Hadfield, 1990) and creating exclusive territories for each downstream
retailer (Gilo, 1999).

6

2

Basic Model

Consider a vertically related industry in which manufacturer M uses an indirect channel tosupply wholesale products to two retailers, R1 and R2 , that compete in quantity in the resalemarket. The manufacturer can also develop a direct channel through which it encroaches
upon the resale market and competes with incumbent retailers. The retailer’s resale cost is
normalized to zero, and the manufacturer’s marginal direct selling cost when it encroaches
is c > 0.7 For simplicity, I assume the manufacturer’s production cost is zero.
Let us first consider a homogeneous product oligopoly. Denote each retailer’s quantity
by qi , q j , i, j = 1 or 2, and the manufacturer’s by q M (if it encroaches). I assume that thedemand for retailers’ products is symmetric and that the inverse demand functions for resale
products, p = P(·), is continuous, nonnegative, strictly decreasing, and twice diﬀerentiable.
To guarantee that profit functions are strictly quasi-concave and that resale competition involves strategic substitutability, I assume P′ (·) + P′′ (·)q M < 0 (if encroachment happens) andP′ (·) + P′′ (·)qi < 0 (Vives, 1999).
In the contracting stage, the manufacturer oﬀers each retailer a take-it-or-leave-it twopart tariﬀ contract T i ≡ (wi , Fi ), where wi is the wholesale price per wholesale product and
Fi is a fixed fee. I assume the contract is oﬀered secretly in the sense that each retailer
cannot observe its rival’s contract. The manufacturer and each retailer’s profit, π M and πi , arerealized as follows:
πi = [P(q M + qi + q j ) − wi ]qi − Fi ,7

(1)

For example, suppose the incumbent retailers are brick-and-mortar ones and the manufacturer’s directchannel is employed through an online store. Then, the brick-and-mortar retailers are more familiar with consumers’ preferences from direct contact than the online store (Arya et al., 2007). Moreover, the manufacturer
pays higher transportation costs to ship directly to consumers, whereas the brick-and-mortar retailers do notneed to do this (Li et al., 2015). When the manufacturer utilizes an online store to sell directly, it must risk
returns and redress because consumers cannot physically inspect products before ordering (Pan, 2016). This
assumption is also standard in the literature in that the manufacturer acts as an entrant and is thus less eﬃcientthan the incumbent retailers.

7π M = (wi qi + Fi ) + (w j q j + F j ) + [P(q M + qi + q j ) − c]q M ,(2)s.t. Fi ≤ [P(q M + qi + q j ) − wi ]qi ,F j ≤ [P(q M + qi + q j ) − w j ]q j .
Note that q M and the third term in Eq. (2) does not exist when the manufacturer chooses not
to encroach.
The game proceeds as follows: In period 1, M decides whether to encroach and commit tothis decision in the following games. In period 2, M secretly oﬀers a two-part tariﬀ contract
T i to each retailer Ri . Retailers simultaneously and independently decide whether to accept
or reject the contract. If one accepts, it immediately pays the fixed fee. In period 3, Retailers
order the quantity of wholesale products (qi ) and pay the wholesale price per product. If the
manufacturer encroaches, it chooses its quantity (q M ) and simultaneously competes with theretailers.
The timing line that the manufacturer decides whether to encroach before the contracting
process reflects the idea that starting a direct channel is relatively irreversible and thus must
be taken prudently. To start direct sale as an market entrant, the manufacturer has to deal withresale issues such as inventory and siting locations, which are always regarded as long-term
decisions. Moreover, the above timing reflects the assumption that a manufacturer’s commitment of encroachment is credible for the incumbent retailers. This assumption follows
from the fact that the manufacturer’s direct sale must be in compliance with the incumbent
retailer’s favor, which is requested by anti-encroachment legislation. Conducting direct salewithout informing the incumbent retailer will possibly cause legal dispute afterwards.8
The above timing also follows the setting with “interim unobservability” in McAfee and
Schwartz (1994) in the sense that each retailer can only observe its own contracting results
before competing on the resale market (except for the encroachment decision by the man8

For example, some manufacturers (or franchisors in the franchising literature) are requested to open direct
stores certain distance out of its incumbent retailers (franchisees)’s locations. A manufacturer will be accused
if its direct sale is justified to have caused share losing of the incumbent retailers (Blair and Lafontaine, 2002;
Espinoza, 2008).

8

ufacturer). In Section 6, I will discuss the other case with “interim observability” whereineach retailer observes its rival’s contracting results before the competition starts.
For simplicity, I also make the following tie-breaking assumptions: 1) if each retailer isindiﬀerent between accepting and rejecting the contract oﬀered by the manufacturer in period
2, the retailer accepts it. 2) If the manufacturer is indiﬀerent between encroaching and not,it encroaches. 3) If the manufacturer is indiﬀerent between supplying both retailers andforeclosing both of them, it supplies both of them. The equilibrium is solved by backward
induction. Throughout the analysis, equilibria are derived by first-order conditions, and thesolutions satisfy the weak perfect Bayesian equilibrium.

3 Equilibrium Analysis
The manufacturer’s decision in period 1 gives rise to two subgames: the manufacturer encroaches or not. I first derive equilibrium outcomes in each subgame, following the process
in the literature on multilateral contracting under the passive belief assumption. The passive belief assumption requires that each retailer’s conjecture about the contract oﬀered to its
retail competitor is unaﬀected by the out-of-equilibrium contract it receives from the manufacturer. Hence, in equilibrium, each retailer, say Ri , always expects that its rival R j isoﬀered an equilibrium wholesale price, w∗j . This implies that each retailer’s quantity is decided by only its own wholesale price, which is denoted by qi (wi ). I will go back to this point
hereafter.

3.1 No-encroachment
First, let us consider the no-encroachment subgame. Under the assumption of passive belief,
each retailer, say Ri , always expects that R j receives an equilibrium wholesale price and thatR j also expects this in the same way. This implies that Ri always expects that R j will choose9an equilibrium quantity, q∗j . The manufacturer solves the following maximizing problem:max

wi ,w j ,Fi ,F j

qi (wi )wi + Fi + q j (w j )w j + F j{}s.t. Fi ≤ P[qi (wi ), q∗j ] − wi qi (wi )}{F j ≤ P[q∗i , q j (w j )] − w j q j (w j ).

(3)

Plugging Fi and F j , the wholesale price wi oﬀered by the manufacturer to Ri only maximizes
their joint profit. The manufacturer acts as if it and the contracting partner are integrated andface a given residual downstream demand. The first-order eﬀect on the retailer’s profit fromchanging Ri ’s quantity qi in response to wi is zero by the envelope theorem. The direct eﬀectof changing wi is an internal transfer and is thus canceled. Therefore, the optimal wi mustsatisfy
wi

∂qi
= 0.
∂wi

(4)

Because ∂qi /∂wi < 0, the wholesale price in equilibrium equals the manufacturer’s marginal
production cost (i.e., wi = 0). Such an equal-to-margin wholesale pricing outcome isstandard in the literature (Hart and Tirole, 1990; O’Brien and Shaﬀer, 1992; McAfee and
Schwartz, 1994; Rey and Vergé, 2004; Reisinger and Tarantino, 2015). As an upstream monopolist, the manufacturer should have optimally set positive wholesale prices such that theresultant total quantity equals the monopoly level. However, because the contracting process
is unobservable, each retailer only accepts a zero wholesale price. If one retailer accepts a
positive wholesale price and pays the corresponding fixed fee, it will be hurt by the manufacturer’s undercutting. More precisely, if one retailer accepts a positive wholesale price,
the manufacturer would always secretly oﬀers a lower wholesale price to the other retailer.
Because of a lack of commitment, a two-part tariﬀ contract does not ensure a monopoly
profit for the monopoly manufacturer. The manufacturer can only commit to zero wholesale
prices and the resultant competition gives rise to the total quantity of Cournot level. Noticethat the manufacturer is better oﬀ when it forecloses either retailer and oﬀers the remaining
retailer wi = 0 and Fi equals to the monopoly retailer’s resale profit. However, the retailer
10must reject this oﬀer because the manufacturer’s commitment of foreclosure can never beguaranteed.9 Such a commitment problem causes retailers to oversupply and prevents the
manufacturer from achieving the downstream monopoly profit.

3.2 Encroachment
Next, I examine the subgame wherein the manufacturer encroaches. The optimal wholesale
price wi is determined as follows.

max

wi ,w j ,Fi ,F j

qi (wi )wi + Fi + q j (w j ) + Fi
{ (
}
)
+
P qi (wi ), q j (w j ), q M [qi (wi ), q j (w j )] − c q M [qi (wi ), q j (w j )],{ (})
s.t. Fi ≤ P qi (wi ), q∗j , q M (wi , q∗j ) − wi qi (wi ),{}F j ≤ P[q∗i , q j (w j ), q M (q∗i , w j )] − w j q j (w j ).

(5)

The first row in Eq. (5) denote the profit from the indirect channels, and the second row
denotes that from the direct channel. Because each retailer only knows its own wholesale
price, even though the quantity competition happens in period 3, qi has already been decided
by Ri in period 2, based on wi and its expectation about q j and q M .10 Thus, the manufacturer acts as a Stackelberg follower who decides q M based on qi (wi ) and q j (w j ), or formallyq M [qi (wi ), q j (w j )] in the second row of Eq. (5). Note that as denoted in the third row of
Eq. (5), when Ri decides whether to accept the highest fixed fee, its expectation about q M inperiod 3 is also formed from its expectation about q∗j . Now, the manufacturer still faces thecommitment problem that it can only maximize the joint profit with Ri when choosing wi .
The optimal wholesale price is decided in the following lemma:

9Given that R1 chooses the monopoly resale quantity q M in period 3, the manufacturer always maintains
R2 a positive resale quantity, q2 = arg maxq [P(q M + q)]q > 0, implying the manufacturer does not actuallyforeclose R2 .
10
Even though c is public information, Ri cannot make a precise expectation about q M unless it knows w j .

11Lemma 1 In the subgame with encroachment, wi satisfies the follow:

wi∂qi∂P ∂q M∂P ∂qi
+ qi
+ qM
= 0.
∂wi
∂q M ∂wi∂qi ∂wi(6)

proof. The maximization problem in Eq. (5) can be derived as

wi

∂qi
∂qi
∂P ∂q M+ qi + (P′ qi + P − wi )
+ qi
− qi
∂wi
∂wi
∂q M ∂wi∂q M∂P ∂qi+ (P′ q M + P − c)
+ qM
.
∂wi
∂qi ∂wi(7)

Using envelope theorem, I obtain Lemma 1.
From Lemma 1, the sign of wi is decided by the relative share of the wholesale channel
and its direct channel, namely qi and q M . Comparing with the subgame without encroachment, because now the manufacturer has committed that it will encroach in period 1, theretailer has to take into account the strategic interaction from the direct channel. Specifically, how wi aﬀects q M is known by Ri . This implies that when encroachment happens, themanufacturer faces the following trade-oﬀ: to levy a positive wholesale price in order to shift
more share to its direct channel, or to subsidize the retailer to promote the retailers’ sales.
This trade-oﬀ makes the current research diﬀerent with those existing ones in the sense thatthe manufacturer may use the direct channel to solve its commitment problem.
In order to formulate a rigorous analysis on the sign of equilibrium wi , I derive the manufacturer’s maximization problem in an alternative way. Because wi decided in period 2 will
ultimately determine qi decided in period 3, the contract can be regarded as a quantity based
two-part tariﬀ, namely T i = (qi , Fi ). Each retailer, say Ri , only knows its own quantity qi
and forms an expectation about the rival’s quantity q∗j . In period 3, the manufacturer chooses
q M to maximize its own profit, ignoring its impact on the profit from the retailer’s channel,

12because it has already levied Fi in period 2:
max[P(qi + q j + q M ) − c]q M .
qM(8)The corresponding FOC will result in a best response, q M (qi +q j ). In period 2, the contracting
between M and Ri requires solving{}
max Fi + wi qi + F j + w j q j + P[qi + q j + q M (qi + q j )] − c q M (qi + q j ),qi}{s.t. Fi ≤ P[qi + q∗j + q M (qi + q∗j )] − wi qi ;{}F j ≤ P[q∗i + q j + q M (q∗i + q j )] − w j q j .

(9)Plugging Fi and F j , the manufacturer chooses qi to maximize the joint profit with Ri . Therefore, the problem in period 2 can be rewritten as{}
max P[q M (qi + q∗j ) + qi + q∗j ]qi + P[q M (qi + q j ) + qi + q j ] − c q M (qi + q j ).
qi

(10)

In this subgame, the manufacturer acts as if it plays a Stackelberg game with itself, but witha diﬀerent objective function in each period–in period 2, it takes into account the impact on
the indirect channel, but in period 3, it ignores such an impact. Following this idea, it isstraightforward to derive the each retailer’s equilibrium share in this subgame:
Lemma 2 In the subgame with encroachment, each retailer’s quantity is given byqi =−c × (P′′ q M + 2P′ )> 0.
(P′ )2(11)

proof. The FOC in period 3 is given by
P′ q M + P − c = 0.

13

(12)

Totally diﬀerentiating Eq. (12) gives rise todq M
P′′ q M + P′
= − ′′.
dqi
P q M + 2P′(13)

Using Eq. (12), the maximization problem in Eq. (10) can be derived as[P′ q M + P − c]

dq M
dq M
+ P′ qi+ P′ q M + P′ qi + p = 0
dqi
dqi

dq M
+ c + P′ qi = 0dqi−c⇒ qi = ′.
P × (1 + dq M /dqi )⇒ P′ qi(14)

Substituting Eq. (13) gives rise to the expression in Lemma 2. Because of strategic substitutability, qi is positive.
Lemma 2 implies that as long as retailers have a cost advantage in resale activities, it isalways optimal for the manufacturer to assign a positive share to each retailer. The valueof each retailer’s share is determined by how eﬃcient its resale is. At extreme, when the
manufacturer is as eﬃcient as the retailers (c = 0), the manufacturer assigns each retailer
zero quantity so that its optimal direct sale becomes a monopoly level which is decided by
Eq. (12). Therefore, due to the continuity, when c is suﬃciently small, the second term in Eq.
(6) is trivial, implying that the manufacturer levies a positive wi . This finding is summarized
in the next proposition:
Proposition 1 ∃c̄ > 0 such that ∀c < c̄, the manufacturer oﬀers each retailer an abovemargin wholesale price when it encroaches.
The intuition is as follows. When the the manufacturer is almost as eﬃcient as the retailers,
it depends quite less on them. But, as the manufacturer still cannot make its commitment offoreclosing either retailer credible, it still faces the commitment problem. Since the manufacturer’s low resale cost is observed by each retailer, its incentive to charge a positive wi to
shift share to the direct channel is also well understood. Each retailer knows that even if it

14

accepts the positive wholesale price, it would not be undercut by the manufacturer as in thecase without encroachment. This is because secretly undercutting either retailer harms the
manufacturer’s benefit from the direct channel. Thus, the encroachment acts as a commitment tool that makes the manufacturer’s oﬀer of positive wholesale prices credible. In otherwords, the encroachment authorizes the manufacturer a strategic option to control the total
quantity sold in the resale market, which may partially solve its commitment problem.

3.3 No-encroachment vs. Encroachment
Now, I compare the manufacturer’s profits and consumer welfare in the above two subgames. Let all notations with superscripts N and E denote equilibrium outcomes in the
no-encroachment and encroachment cases, respectively. Let CW denote consumer welfare.
Following the similar logic in deriving Proposition 1, I derive how the encroachment aﬀects
consumer welfare, which is summarized in the next proposition:
Proposition 2 ∃c̄ > 0 such that ∀c < c̄, the encroachment helps the manufacturer increase
profit, which reduces consumer welfare.
proof. Let us consider the extreme case wherein the manufacturer’s marginal resale cost
becomes zero.
When the manufacturer does not encroach, q1N and q2N are solutions to the optimization
systemq1N = arg max P(q + q2 )q, q2N = arg max P(q1 + q)q.
q

q

(15)

The manufacturer’s Profit and consumer welfare are given byΠNM = Πduopoly ≡ P(q1N + q2N ) × (q1N + q2N ),∫ qN +qN
1
2
N
duopoly
CW = CW≡
P(z)dz − Πduopoly .
0

15

(16)
(17)

When the manufacturer encroaches, the equilibrium quantities are
qiE = 0, qEM = arg max P(q)q.
q(18)

The manufacturer’s profit and consumer welfare are given by
πEM = πmonopoly ≡ P(qEM ) × qEM ,∫ qEM
Emonopoly
CW = CW≡
P(z)dz − πmonopoly .

(19)
(20)

0

Because q1N + q2N > qEM and πduopoly < πmonopoly ,πNM |c=0 < πEM |c=0 , CW N |c=0 > CW E |c=0 .

(21)

Using continuity, I finish proving Proposition 2.
The above proposition is seriously counterintuitive because consumers may be worse oﬀ
with an additional supplier (the manufacturer) joining the resale competition, even thoughsuch an additional supplier is very eﬃcient. The manufacturer’s encroachment turns resalecompetition from a duopoly to a triopoly, generating a pro-competitive eﬀect. Meanwhile,
the above-margin wholesale prices incurred by the encroachment cause eﬃciency losses in
the indirect channels, generating an anti-competitive eﬀect. How consumers are aﬀected isdetermined by the above trade-oﬀs. Specifically, when c is small enough, the manufacturer
is capable to use the direct sale to commit to the wholesale prices which are high enough to
restrain the retailers’ oversupply. In this way, the manufacturer solves its commitment problem and achieves a profit that is close to the monopoly level. On the other hand, consumers
are harmed by the encroachment because the manufacturer restrains the total supply in theresale market. This implies that even if c is very small, the anti-competitive eﬀect caused by
the encroachment can outweigh the pro-competitive eﬀect.
Notice that Proposition 2 holds with more downstream retailers. This is because thegrowing number of retailers will aggravate oversupply, which makes the manufacturer’s
16commitment problem more severe. Then, the manufacturer must have a stronger willingness
to develop a direct channel to restrict the total quantity in the resale market. Only by restricting the total quantity can the manufacturer enhance profit, implying that a profit-enhancing
encroachment and a consumer welfare-improving encroachment must be paradoxical.
Even though comparing the two subgames is indicative of a downward entry that harms
consumers, to derive the manufacturer’s equilibrium decision in period 1, I still need to confirm the deviation cases wherein the manufacturer forecloses one or both retailers in the noencroachment subgame. Note first that the case wherein the manufacturer encroaches while
supplying only one retailer can not occur in equilibrium. This imperative is established in theliterature on supplier encroachment. When only one retailer constitutes the resale market, themanufacturer’s opportunism is no longer a concern. Because the manufacturer can fully extract the monopoly retailer’s surplus with a two-part tariﬀ contract, it never encroaches upon
the territory of the retailer who is more eﬃcient in resale activities. This would give riseto the case wherein the manufacturer supplies only one retailer without encroaching, which
actually never occurs under the assumption of unobservable contracting. In other words, aslong as πEM > πNM is satisfied, the manufacturer never deviates by foreclosing one retailer.
Therefore, I only need to exclude the deviation case wherein the manufacturer forecloses
both retailers. This will be confirmed in the following sections.

4

Examples

4.1 Example 1: Linear Demand
In this subsection, I use a linear demand system to confirm the results in Proposition 1 and 2
and derive another result regarding consumer welfare. Let p = a − b(q1 + q2 ) when there isno encroachment and p = a − b(q1 − q2 − q M ) when encroachment happens.
No-encroachment: In equilibrium, each retailer is oﬀered zero wholesale price, and the
manufacturer obtains a profit comprising retailers’ Cournot profits. Equilibrium outcomes17

are as follows:
w1N = w2N = 0;aq1N = q2N =;3b2a2
πNM = 2 × [a − b(q1N + q2N )] =;9b
∫ qN +qN
1
2
2a2
N
CW =
;
P(z)dz − πNM =
9b
0

(22)
(23)

Encroachment: Notice again that because the manufacturer has levied the fixed fee in period
2, it ignores the retailers’ profits In period 3. Based on the the assumption of passive belief,
Ri believes that R j would choose an equilibrium quantity q∗j which is given by11
q j (w∗i , w∗j )=a + c − 3w∗j + w∗i
4b

.

(24)

passive belief requires that w∗i = w∗j ≡ w∗ , from whichq∗j =a + c − 2w∗
.
4b

(25)

Besides, Ri also believes that the manufacturer would choose q M in response to q∗j . Then, qi
and Ri ’s expectation about q M are decided by simultaneously solving maxqi [a − b(qi + q∗j +q M ) − wi ]qi and maxqM [a − b(qi + q∗j + q M ) − c]q M , which are given as follow:
qi (wi ) =3a + 3c − 8wi + 2w∗
3a − 9c + 4wi + 2w∗; q M (wi , q∗j ) =
.
12b
12b

(26)

This implies that when each retailer is oﬀered a wholesale price in period 2, it can decide
which quantity to sell in period 3. Knowing this, the manufacturer decide its quantity by
maximizing the profit from direct selling, maxqM [P(qi + q j + q M ) − c]q M , from which theq j (wi , w j ) is the solution to the maximization problems maxqi [a − b(qi + q j + q M ) − wi ]qi and maxqM [a −b(qi + q j + q M ) − c]q M , where i = 1, 2.
11

18

manufacturer’s optimal direct share can be derived as follows:q M [q1 (w1 ), q2 (w2 )] =a − c − q1 (w1 ) − q2 (w2 ) 3a − 9c + 4w1 + 4w2 − 2w∗
=
.
2
12b

(27)

Substituting Eq. (25), (26) and (27) to Eq. (5) the equilibrium outcomes can be derived
as follow:
a − 7c;22c E
a − 5c=, qM =;b2b(a − c)2=;4b∫ q1E +q2E +qEM
(a + 3c)2.
=p(z)dz − πEM − cqEM =8b0

w1E = w2E =q1E = q2E
πEM
CW E

(28)
(29)

Deviation: When the manufacturer forecloses both retailers and supplies the resale market
directly, it obtains monopoly output q′M and profit π′M , with marginal resale cost c:a−c;2b(a − c)2
=
;
4b
∫ q′M
(a − c)2.
=[p(z) − c]dz − π′M =
8b
0

q′M =
π′M
CW ′

(30)
(31)

Comparing the outcomes under the “no-encroachment” and “encroachment,” I can confirm the arguments that the encroachment helps the manufacturer enhance profit and always
reduces consumer welfare. Taking into account the deviation case, πEM = π′M implies that
foreclosing both retailers cannot bring a better profit than encroachment for the manufacturer. By the tie-breaking assumption 3), a consumer welfare-reducing downward entry can
actually happen in equilibrium, which is summarized as follows:
Corollary 1 In a linear demand system,

19i) the manufacturer encroaches in equilibrium if and only if√
c 3−2 2
0< <
;
a
3

(32)

ii) the encroachment always reduces consumer welfare.
Moreover, CW E > CW ′ implies that even though the manufacturer achieves the same profit
when employing multichannel distribution as when running the direct channel only, the multichannel distribution results in a larger total quantity than that in the case when both retailers
are foreclosed, implying a better consumer welfare.
It is noteworthy that CW E increases in c. This finding is also counterintuitive because in
the linear demand system, one firm’s lower marginal cost always gives rise to a smaller total
quantity and a higher market price, which is summarized in the following proposition:
Proposition 3 In a linear demand system, when encroachment happens, the manufacturer’s
cost reduction always harms consumer welfare.
Now I discuss the intuition. In a quantity competition with homogeneous products, according
to the theorem of Bergstrom and Varian (1985), price, total quantity and consumer welfare
depend only on the total marginal costs of all firms, namely w1E + w2E + c. When running
multichannel distribution, the manufacturer’s main task becomes managing its direct andindirect channels such that it achieves an optimal aggregate profit. A lower c motivates the
manufacturer to become more active in the direct channel and to rely less on the retailers who
cause a commitment problem for the manufacturer. Therefore, the manufacturer’s incentive
to charge a higher wholesale price becomes credible for each retailer who then accepts sucha wholesale price. Because there are two indirect channels in this market, the eﬃciency
loss due to higher wholesale prices is doubled nad outweighs the gain due to a lower c.
Consequently, under multichannel distribution, a lower c enables the manufacturer to better
solve its commitment problem, meanwhile causing loss for consumers.

20

c × 103
πEM × 103
πNM × 103π′M × 103
CW E × 103
CW N × 103

10379.1503
353.5534
379.1411
137.7128235.7023

8.5
380.0089
353.5534
380.0031
136.3260
235.70237
380.8691
353.5534
380.8658
134.9308
235.70235.5
4
2.5
381.7307 382.5937 383.4579
353.5534 353.5534 353.5534
381.7291 382.5931 383.4577
133.5269 132.1140 130.6918
235.7023 235.7023 235.7023

Table 1: β = 2

4.2 Example 2: Non-linear Demand
In this subsection, I use a nonlinear demand system to confirm my results. Let p = a −(q1 + q2 )β when there is no encroachment and p = a − (q1 + q2 + q M )β when encroachment
happens. Due to mathematical complexity involved, I cannot explicitly derive the close-form
solutions. Therefore, I specify a to 1, and provide numerical results as denoted Tables 1. The
following information can be obtained: 1) even though with a non-linear demand system, an
encroachment in equilibrium may still reduce consumer welfare. 2) the manufacturer’s cost
reduction harms the consumers.

5 Extension: Diﬀerentiated Products
In this section, I consider a case wherein the manufacturer and retailers sell heterogeneous
products and products sold by diﬀerent retailers are perfect substitutes.12 Let π M denote the
manufacturer’s profit including the fixed fee. A representative consumer’s utility is quadraticwith the form1u(q1 , q2 , q M ) = a(q1 + q2 + q M ) − [(q1 + q2 )2 + 2γ(q1 + q2 )q M + q2M ] + I,
2(33)

where γ ∈ (0, 1] denotes the substitutability between the two varieties and I denotes consumer income. Notice that all results here are consistent with the case of homogeneous

12

In Section 6, I will consider the case wherein products sold by diﬀerent retailers are diﬀerentiated.

21

products when γ = 1. The inverse demand functions for products R and M are denoted by
pR = a − (q1 + q2 ) − γq M , p M = a − q M − γ(q1 + q2 ).

(34)

Consumer welfare when the encroachment happens can be denoted as
CW = u(q1 , q2 , q M ) − cq M − π M .

(35)

I omit the detailed calculations because they are quite similar with those in our basicmodel. Details can be provided upon request. The equilibrium outcomes in each subgame
and in the deviation case are given as follow:
No-encroachment:a2awiN = 0; qiN = ; πNM =392; CW =∫ qN +qN
1

N

0

2

pR (z)dz − πNM =

2a2
.
9

(36)

Encroachment:γ[(6 − 6γ + γ2 )a − (6 + γ2 )c];2(6 − 5γ2 )2[(1 − γ)a + γc] E(6 − 4γ − γ2 )a + (6 − γ2 )c
=
,
q
;
=
M
6 − 5γ2
6 − 5γ2(68 − 64γ − 60γ2 + 64γ3 − 7γ4 )a2 − 2(36 − 32γ − 28γ2 + 32γ3 − 7γ4 )ac=4(6 − 5γ2 )22
4 2(36 − 28γ − 7γ )c+;(37)4(6 − 5γ2 )2(100 − 80γ − 92γ2 + 56γ3 + 17γ4 )a2 − 2(36 − 40γ − 44γ2 + 28γ3 + 17γ4 )ac=8(6 − 5γ2 )2(36 − 44γ2 + 17γ4 )c2+.
(38)
8(6 − 5γ2 )2

wiE =
qiE
πEM

CW E

22

Deviation:a−c;2(a − c)2;=4(a − c)2
=
.
8

q′M =
π′M
CW ′

(39)
(40)

Comparing πEM , πNM and π′M , I can confirm that when πEM > πNM is satisfied, πEM > π′M mustalso be satisfied. The next proposition summarizes the conditions under which our argumentsin the case of homogeneous products hold.
Proposition 4 When products in the direct and indirect channels are diﬀerentiated,i) the manufacturer encroaches in equilibrium if and only if 0 < c/a < f1 (γ), where f1 (γ)
is the threshold value satisfying πEM = πNM ;ii) the encroachment reduces consumer welfare if and only if f2 (γ) < c/a < f3 (γ), where
f2 (γ) and f3 (γ) are threshold values satisfying CW ME = CW MN ;iii) the manufacturer’s cost reduction harms consumer welfare if and only if c/a > f4 (γ),
where f4 (γ) is the threshold value satisfying ∂CW ME /∂c = 0.

f1 (γ) ≡f2 (γ) ≡f3 (γ) ≡f4 (γ) ≡√36 − 32γ − 28γ2 + 32γ3 − 7γ4 2γ(6 − 5γ2 ) 16 − 14γ2−,
36 − 28γ2 − 7γ43(36 − 28γ2 − 7γ4 )√36 − 40γ − 44γ2 + 28γ3 + 17γ4 4γ(6 − 5γ2 ) 8 − 17γ2−,
36 − 44γ2 + 17γ43(36 − 44γ2 + 17γ4 )√36 − 40γ − 44γ2 + 28γ3 + 17γ4 4γ(6 − 5γ2 ) 8 − 17γ2+,
36 − 44γ2 + 17γ43(36 − 44γ2 + 17γ4 )36 − 40γ − 44γ2 + 28γ3 + 17γ4.
36 − 44γ2 + 17γ4

(41)

Figure 1 depicts the ranges in Proposition 4. In the range wherein the encroachment
happens (c/a < f1 (γ)), the encroachment reduces consumer welfare when γ is relatively
23

large, which is denoted by the shaded area. When products sold in the direct and indirect

Figure 1: The Change in CW
channels are closely substitutable, the manufacturer’s commitment problem is solved to thebest, which enables the manufacturer to greatly restrict the total quantity. The resultant highprice implies that the procompetitive eﬀect due to the downward entry is outweighed by the
anticompetitive eﬀect due to the eﬃciency loss.

6 Discussion
In this section, following McAfee and Schwartz (1994), I discuss quantity competition with“interim observability” and price competition with both “interim observability” and “interim
unobservability.”13 Now, the quantity eventually sold by each retailer, whether in quantity
or price competition, depends on the contracting results of its rival retailer, which triggers
the manufacturer’s incentive for multilateral deviation on both retailers. As noted in Rey andVergé (2004), such a deviation occurs when products sold by retailers are homogeneous (as13

In the game with “interim observability,” each retailer observes its rival’s contracting results before thecompetition starts. In the game with “interim unobservability,” each retailer never observes the rival’s contracting results.

24

in my basic setting), resulting in the non-existence of equilibrium. I then consider the casewherein retailers compete in diﬀerentiated products. To construct a symmetric marketing
position for both retailers and to better track the main mechanism, I assume the products
sold by the manufacturer are substitutable with those by each retailer to the same extent.
Such substitutability is denoted by γ. The inverse demand functions are
pi = a − qi − γ(q j + q M ), p M = a − q M − γ(qi + q j ), i, j = 1, 2.

(42)

6.1 Quantity Competition with Interim Observability
I first consider the case of no-encroachment. The results are consistent with Rey and Vergé
(2004) except that the manufacturer’s production cost is assumed to be zero. In the last stage,
each retailer decides its quantity, observing the contracting results of the rival. By symmetry,the equilibrium quantities and profits are given by
2 − γ − 2wi + γw j;
4 − γ2)(2 − γ − 2wi + γw j 2.
πi (wi , w j ) =4 − γ2qi (wi , w j ) =

(43)
(44)

The manufacturer chooses wi and w j to solve
max wi qi (wi , w j ) + πi (wi , w∗j ) + w j q j (wi , w j ) + π j (w∗i , w j ),wi ,w j(45)

from which I have
wiN = wNj =−aγ2.
2(2 − γ2 )(46)

Because wiN < 0, the interim observability aggravates the manufacturer’s commitment problem: it has to commit to a subsidy to each retailer, otherwise its oﬀer would be rejected. To

25

guarantee that the manufacturer does not unilaterally deviate, I need that
∂π2M (wi , w j ) ∂π2M (wi , w j )+≤ 0 ⇒ γ ≤ 0.806.
∂qi ∂q j
∂q2i

(47)

The equilibrium outcomes are(2 − γ)a;2(2 − γ2 )(1 − γ)(4 − γ2 )a2=;2(2 − γ2 )2(1 + γ)(2 − γ)2 a2=.
4(2 − γ2 )2qiN = qNj =
πNM
CW N

(48)
(49)

Next, I consider the case where the manufacture encroaches. The equilibrium quantities
and profits are given by(2 − γ)a + γc − (2 + γ)wi + γw j;
2(2 − γ)(1 + γ)
(2 − γ)a − (2 + γ)c + γwi + γw j;
q M (wi , w j ) =2(2 − γ)(1 + γ)][
(2 − γ)a + γc − (2 + γ)wi + γw j 2.
πi (wi , w j ) =2(2 − γ)(1 + γ)
qi (wi , w j ) =

(50)
(51)
(52)

The manufacturer chooses wi and w j to solve
max π M (wi , w j ) = wi qi (wi , w j ) + πi (wi , w∗j ) + w j q j (wi , w j ) + π j (w∗i , w j )wi ,w j+ [P M (wi , w j ) − c]q M (wi , w j ),(53)

from which I have
wiE = wEj =γ[(2 − 3γ + γ2 )a − (2 + γ + γ2 )c].
2(1 − γ)(2 + 3γ)(54)

Notice that for ranges wherein all equilibrium quantities are interior solutions, the equi-26librium wholesale price under the encroachment case is always higher than that in the noencroachment case. This is because the manufacturer now obtains additional profit from thedirect channel and thus relies less on the incumbent retailers. To guarantee that the manufacturer does not unilaterally deviate, I need that
∂π2M (wi , w j ) ∂π2M (wi , w j )+≤ 0 ⇒ γ ≤ 0.756.
∂wi ∂w j
∂w2i

(55)

The equilibrium outcomes are
qiE = qEj =
qEM =
πEM =
+
CW E =+(1 − γ)a + γc;
(1 − γ)(2 + 3γ)(2 − γ − γ2 )a − (2 + γ − γ2 )c;2(1 − γ)(2 + 3γ)
3(1 − γ)2 (4 + 4γ − γ2 )a2 − 2(1 − γ)(4 − 3γ2 + 3γ3 )ac4(1 − γ)2 (2 + 3γ)2
(1 + γ)(4 − 3γ2 − 3γ3 )c2;
4(1 − γ)2 (2 + 3γ)2
(1 − γ)2 (4 + 32γ + 13γ2 )a2 − 2γ(10 − 21γ − 2γ2 + 13γ3 )ac8(2 + γ − 3γ2 )2(1 + γ)(4 + 8γ − 23γ2 + 13γ3 )c2.
8(2 + γ − 3γ2 )2

(56)

(57)

To rule out the manufacturer foreclosing both retailers, I need πEM > π′M or q′M ≤ 0.14 Moreover, I need all equilibrium outcomes to be interior solutions.
The next proposition summarizes how the manufacturer’s encroachment aﬀects consumer welfare in quantity competition with interim observability.
Proposition 5 In a quantity competition with interim observability,i) the manufacturer encroaches in equilibrium if and only if 0 < c/a < f5 (γ), where f5 (γ)
is the threshold value satisfying πEM = πNM ;ii) the encroachment reduces consumer welfare if and only if f6 (γ) < c/a < f5 (γ), where
f6 (γ) are threshold values satisfying CW ME = CW MN .
14

The equilibrium outcomes when the manufacturer forecloses both retailers and monopolizes the resalemarket itself are equivalent to Eq. (31) in Section 4.

27√(1 − γ)(4 − 3γ2 + 3γ3 ) γ(2 + γ − 3γ2 ) 2(8 − 12γ − 9γ2 + 15γ3 + 3γ4 − 3γ5 )f5 (γ) ≡−,(1 + γ)(4 − 3γ2 − 3γ3 )(1 + γ)(2 − γ2 )(4 − 3γ2 − 3γ3 )4 + 8γ − 13γ2 − 22γ3 + 20γ4 + 16γ5 − 13γ6
f6 (γ) ≡4 + 12γ + γ2 − 16γ3 + 6γ5 − 3γ6√γ(2 + γ − 5γ2 + 2γ4 ) 2(8 + 16γ − 29γ2 − 55γ3 + 25γ4 + 31γ5 − 14γ6 )−.
(2 − γ2 )(4 + 12γ + γ2 − 16γ3 + 6γ5 − 3γ6 )The encroachment drives the wholesale price up to a positive value, which enables the manufacturer to restrict the retailers’ total outputs and thus to partially solve its commitment
problem.
Notice that the assumption that the fixed fee must be paid before the competition starts
plays an important role in the setting of interim observability. As shown by Fontenay andGans (2005), with interim observability, the manufacturer’s opportunism is no longer a concern if the manufacturer is allowed to oﬀer a fixed fee contingent on the retailer’s quantity.

6.2 Price Competition
6.2.1 Price Competition with Interim Unobservability
I first consider the game with interim unobservability. In the last stage, each retailer decides its price by solving a Bayesian game. The no-encroachment case is identical to thatin O’Brien and Shaﬀer (1992) and Rey and Vergé (2004). If there exists an equilibrium inprice competition with interim unobservability, I need γ to be smaller than 1/2. The manufacturer’s maximization problem is as follows:max
wi ,w jwi qi [pi (wi ), p j (w j )] + [pi (wi ) − wi ]qi [pi (wi ), p∗j ]{}+ w j q j pi (wi ), p j (w j )] + [p j (w j ) − w j ]q j [p∗i , p j (w j )].

28

(58)

The equilibrium outcomes are summarized as follow:
wiN = wNj = 0;
(1 − γ)apiN = pNj =;2−γ2(1 − γ)a2πNM =;(2 − γ)2 (1 + γ)
a2
N
CW =
.
(2 − γ)2 (1 + γ)

(59)
(60)

In the encroachment case, the manufacturer’s maximization problem is as follows:max
wi ,w j{}wi qi pi (wi ), p j (w j ), p M [pi (wi ), p j (w j )] + [pi (wi ) − wi ]qi [pi (wi ), p∗j , p M (wi , p∗j )]{}+ w j q j pi (wi ), p j (w j ), p M [pi (wi ), p(w j )] + [p j (w j ) − w j ]q j [p∗i , p j (w j ), p M (p∗i , w j )]{} {}+ p M [pi (wi ), p j (w j )] − c q M pi (wi ), p j (w j ), p M [pi (wi ), p j (w j )] .
(61)

The equilibrium outcomes in the encroachment case are summarized as follows:
wiE = wEj =
πEM =
−

+
CW E =−+

γ[(1 − γ)(2 + 3γ)2 a − (4 + 10γ + 3γ2 − 3γ3 )c];2(4 + 10γ + γ2 − 9γ3 − 3γ4 )(1 + γ)(1 − γ)2 (48 + 240γ + 348γ2 − 28γ3 − 391γ4 − 176γ5 + 36γ6 )a2
4(1 − γ)(1 + 2γ)(4 + 10γ + γ2 − 9γ3 − 3γ4 )22(1 − γ2 )(16 + 64γ + 44γ2 − 88γ3 − 75γ4 + 55γ5 + 12γ6 − 36γ7 )ac4(1 − γ)(1 + 2γ)(4 + 10γ + γ2 − 9γ3 − 3γ4 )216 + 80γ + 108γ2 − 52γ3 − 211γ4  2(1 + γ) c − 110γ5 + 45γ6 + 72γ7 + 36γ8 ;(62)
4(1 − γ)(1 + 2γ)(4 + 10γ + γ2 − 9γ3 − 3γ4 )2(1 + γ)2 (1 − γ)(48 + 208γ + 188γ2 − 268γ3 − 399γ4 + 70γ5 + 180γ6 )a2
8(1 − γ)(1 + 2γ)(4 + 10γ + γ2 − 9γ3 − 3γ4 )22(1 + γ)2 (1 − γ)(16 + 64γ + 36γ2 − 128γ3 − 131γ4 + 66γ5 + 84γ6 )ac8(1 − γ)(1 + 2γ)(4 + 10γ + γ2 − 9γ3 − 3γ4 )2(1 + γ)2 (16 + 64γ + 44γ2 − 80γ3 − 51γ4 + 69γ5 + 6γ6 − 36γ7 )c2.
(63)
8(1 − γ)(1 + 2γ)(4 + 10γ + γ2 − 9γ3 − 3γ4 )229

Given γ < 1/2, the manufacturer never forecloses both retailers, and the manufacturer’s
multilateral deviation never happens in this case.
The next proposition summarizes how the manufacturer’s encroachment aﬀects consumer welfare in price competition with interim unobservability.
Proposition 6 Under the two-part tariﬀ contract, in price competition with interim unobservability,i) the manufacturer encroaches in equilibrium if γ < 1/2 and c/a < f7 (γ), where f7 (γ)
is the threshold value satisfying πEM = πNM ;ii) the encroachment always reduces consumer welfare.

16 + 48γ − 20γ2 − 132γ3 + 13γ4 + 130γ5 − 43γ6 − 48γ7 + 36γ8
16 + 80γ + 108γ2 − 52γ3 − 211γ4 − 110γ5 + 45γ6 + 72γ7 + 36γ8
2γ(1 − γ)(4 + 10γ + γ2 − 9γ3 − 3γ4 )−16 + 80γ + 108γ2 − 52γ3 − 211γ4 (2 − γ)(1 + γ) 
 + −110γ5 + 45γ6 + 72γ7 + 36γ8 √
2(1 + 2γ)(8 + 28γ + 5γ2 − 76γ3 − 63γ4 + 50γ5 + 56γ6 )× ,16 + 80γ + 108γ2 − 52γ3 − 211γ4 (2 − γ)(1 + γ) 
 + −110γ5 + 45γ6 + 72γ7 + 36γ8 

f7 (γ) ≡

6.2.2

(64)

Price Competition with Interim Observability

I now consider the game with interim observability. When the manufacturer does not encroach, in the last stage, each retailer’s equilibrium resale prices and profits are given by(2 − γ − γ2 )a + 2wi + γw j;
4 − γ2[(2 − γ − γ2 )a − (2 − γ2 )wi + γw j ]2πi (wi , w j ) =.
(4 − γ2 )2 (1 − γ2 )pi (wi , w j ) =

30

(65)
(66)

The manufacturer chooses wi and w j to solve[]
max π M (wi , w j ) = wi qi pi (wi , w j ), p j (wi , w j ) + πi (wi , w∗j )wi ,w j[]
+ w j q j pi (wi , w j ), p j (wi , w j ) + π j (w∗i , w j ),(67)

from which I have
wiN = wNj =

aγ2
.
4

(68)

In price competition with interim observability, even if the manufacturer does not encroach,it can charge each retailer a positive wholesale price, implying that the commitment problem
is not as serious as the cases I discussed above. To guarantee that the manufacturer does notunilaterally deviate, I need that
∂π2M (wi , w j ) ∂π2M (wi , w j )≤ 0 ⇒ γ ≤ 0.806.
+∂wi ∂w j
∂w2i

(69)

The equilibrium outcomes are(2 − γ)a;4(4 − γ2 )a2=;
8(1 + γ)
(2 + γ)2 a2=.
16(1 + γ)

piN = pNj =
πNM
CW N

(70)
(71)

Next, I consider the case when the manufacture encroaches. The equilibrium quantities31

and profits are given by(2 + γ − 3γ2 )a + (γ + γ2 )c + (2 + 3γ + γ2 )wi + (γ + γ2 )w j;
2(2 + 3γ)
(2 + γ − 3γ2 )a + (2 + 3γ + γ2 )c + (γ + γ2 )wi + (γ + γ2 )w j
p M (wi , w j ) =2(2 − γ)(1 + γ)

2

2
2 
(2 + γ − 3γ )a + (γ + γ )c(1 + γ) 
 + −(2 + 3γ − γ2 )w + (γ + γ2 )w ij.
πi (wi , w j ) =4(1 − γ)(1 + 2γ)(2 + 3γ)2pi (wi , w j ) =

(72)
(73)

(74)

The manufacturer chooses wi and w j to solve
max π M (wi , w j ) = wi qi [pi (wi , w j ), p j (wi , w j ), p M (wi , w j )] + πi (wi , w∗j )wi ,w j

+ w j q j [pi (wi , w j ), p j (wi , w j ), p M (wi , w j )] + π j (w∗i , w j )+ [p M (wi , w j ) − c]q M [pi (wi , w j ), p j (wi , w j ), p M (wi , w j )],(75)

from which I have
wiE = wEj =γ(1 + 2γ)[(2 + γ − 3γ2 )a − (2 + γ − γ2 )c].
2(1 + γ)(2 + 3γ − 2γ2 − 2γ3 )(76)

To guarantee that the manufacturer does not unilaterally deviate, I need that
∂π2M (wi , w j ) ∂π2M (wi , w j )+≤ 0 ⇒ γ ≤ 0.884.
∂wi ∂w j
∂w2i

32

(77)

The equilibrium outcomes are
πEM =+CW E =−+(1 − γ)(12 + 48γ + 45γ2 − 29γ3 − 49γ4 − 9γ5 )a2 − 2(1 − γ2 )(2 + 3γ − γ2 )ac4(1 + 2γ)(2 + 3γ − 2γ2 − 2γ3 )2(1 + γ)(4 + 16γ + 17γ2 − 3γ3 − 13γ4 − 7γ5 )c2;(78)
4(1 + 2γ)(2 + 3γ − 2γ2 − 2γ3 )2(1 + γ)(4 + 4γ − 5γ2 )(3 + 7γ − γ2 − 7γ3 )a2
8(1 − γ)(1 + 2γ)(2 + 3γ − 2γ2 − 2γ3 )2(1 + γ)(4 + 12γ − γ2 − 23γ3 − 3γ4 + 13γ5 )ac8(1 − γ)(1 + 2γ)(2 + 3γ − 2γ2 − 2γ3 )2(1 + γ)(4 + 12γ + γ2 − 16γ3 + 6γ5 − 3γ6 )c2.
(79)
8(1 − γ)(1 + 2γ)(2 + 3γ − 2γ2 − 2γ3 )2To rule out the manufacturer foreclosing both retailers, I need πEM > π′M or q′M ≤ 0. Moreover,I need all equilibrium outcomes to be interior solutions.
The next proposition summarizes how the manufacturer’s encroachment aﬀects consumer welfare in quantity competition with interim observability.
Proposition 7 In a price competition with interim observability,i) the manufacturer encroaches in equilibrium if and only if γ ≤ 0.594 and 0 < c/a <f8 (γ), where f8 (γ) is the threshold value satisfying πEM = πNM ;ii) the encroachment always enhances consumer welfare in equilibrium.

2(1 + γ)(2 + 3γ − γ2 )22(1 + γ)(4 + 16γ + 17γ2 − 3γ3 − 13γ4 − 7γ5 )√γ(2 + 3γ − 2γ2 − 2γ3 ) 2(1 + 2γ)(8 + 20γ + 3γ2 − 9γ3 + 13γ4 + 7γ5 )−. (80)
2(1 + γ)(4 + 16γ + 17γ2 − 3γ3 − 13γ4 − 7γ5 )f8 (γ) ≡7

Concluding Remarks

In this study, I discuss supplier encroachment within a framework of two-part tariﬀ contracts.
I consider a multilateral contracting case wherein a monopoly manufacturer supplies competing retailers. Given the assumption of unobservable contracting, the monopoly manufacturer
33

loses market power even when oﬀering two-part tariﬀ contract because its opportunism triggers each retailer to form a belief about the rival’s contracts that is independent from whatitself is oﬀered; this prevents the manufacturer from optimizing wholesale profit through afixed fee. In a quantity-setting game with interim unobservability, I demonstrate that the
manufacturer restores its market power by encroaching upon the resale market and that theencroachment may result in higher wholesale prices; this aggravates double marginalization
and always harms consumer welfare. When the quantity-setting game is with interim observability, the encroachment may still reduce consumer welfare. On the other hand, I show thatin a price competition with interim unobservability, the encroachment always reduces consumer welfare; in a price competition with interim observability, the encroachment always
enhances consumer welfare.
This paper mainly studies the eﬀect of encroachment on consumer welfare instead ofsocial welfare. This is because I want to focus on the price change caused by the manufacturer’s downward entry. If we take producer surplus into account as well, whether theencroachment increases or reduces social welfare is decided by the trade-oﬀ between the
manufacturer’s gain and consumers’ loss. Actually, the encroachment may also reduce social
welfare in equilibrium, whether under a quantity or price competition. Details are provided
upon request.
Following most existing literature, the manufacturer’s opportunism is captured by thetwo-part tariﬀ contract. Recently, this issue under a linear wholesale price contract has been
discussed by several researchers (e.g., Aghadadashli et al., 2016; Gaudin, 2016). As noticed by Gaudin (2016), when a monopoly manufacturer supplies competing retailers witha linear wholesale price contract, the equilibrium wholesale price under secret contracts islower than that under public contracts. Following the main mechanism in the current study,
the encroachment triggers the manufacturer’s incentive to raise the wholesale price, which
aggravates the double marginalization. However, because the double marginalization before
encroachment is not as severe as the case with two-part tariﬀ contract (in the case with linear
wholesale price contract, the equilibrium wholesale price is always above the manufacturer’s

34

production cost, whether with or without the encroachment), whether the encroachment enhances consumer welfare or not is also worth discussing.
As already noted in the introduction, theoretical analysis on supplier encroachment withthe base of two-part tariﬀ contract is an important issue worth further elaboration. This
paper presents one standpoint wherein the manufacturer fails to obtain the optimal supply
chain profit via the two-part tariﬀ contract because of the opportunism behavior. Anotherpossible attempt is to consider a bargaining problem wherein the manufacturer can only
obtain a part of the supply chain profit based on its bargaining power. To summarize, under a
two-part tariﬀ contract, the analysis of supplier encroachment is quite diﬃcult with the caseof a linear contract which is discussed by most existing literature, and is an interesting topic
for future research.

35

References[1] Aghadadashlia, H.; Dertwinkel-Kalt, M. and Wey, C., 2016, ‘The Nash Bargaining
Solution in Vertical Relations with Linear Input Prices,’ Economics Letters, 145, pp.
291-294.
[2] Arya, A.; Mittendorf, B. and Sappington, D. E. M., 2007, ‘The Bright Side of Supplier
Encroachment,’ Marketing Science, 26(5), pp. 651-659.
[3] Avenel, E., 2012, ‘Upstream Capacity Constraint and the Preservation of Monopoly
Power in Private Bilateral Contrasting,’ Journal of Industrial Economics, 60(4), pp.
578-598.
[4] Bergstrom, C. T. and Varian, H. R., 1985, ‘When Are Nash Equilibria Independent of
the Distribution of Agents’ Characteristics,’ Review of Economic Studies 52, 715-718.
[5] Blair, R. D. and Lafontaine, F., 2005, The Economics of Franchising, Cambridge, UK:
Cambridge University Press.
[6] Cattani, K.; Gilland, W.; Heese, S. and Swaminathan, J., 2006, ‘Boiling Frogs: Pricing
Strategies for a Manufacturer Adding an Internet Channel,’ Production and Operations
Management, 15(1), pp. 40-56.
[7] Chiang, W. K.; Chhajed, D. and Hess, J. D., 2003, ‘Direct Marketing, Indirect Profits: a Strategic Analysis of Dual-Channel Supply-Chain Design,’ Management Science.
49(1), pp, 1-20.
[8] Dutta, S.; Heide, J. and Bergen, M., 1999, ‘Vertical Territory Restrictions and Public
Policy: Theories and Industry Evidence,’ Journal of Marketing, 63, pp. 121-134.
[9] Espinoza, G., 2008, ‘Franchisors’ Encroachment in the Eyes of the Beholder: How
to Protect a Franchisee?’ Available at SSRN: http://ssrn.com/abstract=1137945 or
http://dx.doi.org/10.2139/ssrn.1137945.

36

[10] Fontenay C. C. and Gans J. S., 2005, ‘Optional Fixed Fees in Multilateral Vertical
Relations,’ Economic Letters. 88(2005), pp, 184-189.
[11] Gaudin, G., 2016, ‘Vertical Relations, Opportunism, and Welfare,’ Available at SSRN:
https://ssrn.com/abstract=2819489.
[12] Gilo, D., 1999, ‘Does a Supplier Have the Market Power We Thought it Had? The Useof Vertical Integration and Vertical Restraints to Restore the Supplier’s Market Power,’
Harvard Law School John M. Olin Center for Law, Economics and Business Discussion
Paper Series, paper 262.
[13] Hart, O. and Tirole, J., 1990, ‘Vertical Integration and Market Foreclosure,’ Brookings
Papers on Economic Activity: Microeconomics, 1990, pp. 205-286.
[14] Hadfield, G. K., 1990, ‘Problematic Relations: Franchising and the Law of Incomplete
Contracts,’ Stanford Law Review. 42(4), pp. 927-992.
[15] Kalnins, A., 2004, ‘An Empirical Analysis of Territorial Encroachment within Franchised and Company-owned Branded Chains,’ Management Science, 23(4), pp. 1-20.
[16] Kaufmann, P. J. and Lafontaine, F., 1994, ‘Costs of Control: The Source of Economic
Rents for McDonald’s Franchisees,’ The Journal of Law and Economics, 37(2), pp.
417-453.
[17] Kaufmann, P. J. and Rangan, V. K., 1990, ‘A Model for Managing System Conflict
During Franchise Expansion,’ Journal of Retailing, 66(2), pp. 155-173.
[18] Lafontaine, F., 1992, ‘Agency Theory and Franchising: Some Empirical Results,’
RAND Journal of Economics, 23(2), pp. 263-283.
[19] Li, Z.; Gilbert, S. M. and Lai, G., 2014, ‘Supplier Encroachment under Asymmetric
Information,’ Management Science, 60(2), pp. 449-462.

37

[20] Li, Z.; Gilbert, S. M. and Lai, G., 2015, ‘Supplier Encroachment as an Enhancement
or a Hindrance to Nonlinear Pricing,’ Production and Operations Management, 24(1),pp. 89-109.
[21] Marx, L. M. and Shaﬀer, G., 2004, ‘Opportunism in Multilateral Vertical Contracting: Nondiscrimination, Exclusivity, and Uniformity: Comment,’ American Economic
Review, 94(3), pp. 796-801.
[22] McAfee, R. P. and Schwartz, M., 1994, ‘Opportunism in Multilateral Vertical Contracting: Nondiscrimination, Exclusivity and Uniformity,’ American Economic Review,
84(1), pp. 210-230.
[23] Michael, S. C. and Moore, H. J., 1995, ‘Returns to Franchising,’ Journal of Corporate
Finance, 2(1995), pp. 133-155.
[24] O’Brien, D. P. and Shaﬀer, G., 1992, ‘Vertical Control with Bilateral Contracts,’ RAND
Journal of Economics, 23(3), pp. 299-308.
[25] Pan, C., 2016, ‘Retailer’s Product Line Choice with Manufacturer’s Multichannel Marketing,’ ISER Discussion Paper No. 938. Available at SSRN:
http://ssrn.com/abstract=2808837.
[26] Park, T., 2015, ‘Direct Marketing and the Structure of Farm Sales: An Unconditional
Quantile Regression Approach,’ Journal of Agricultural and Resource Economics,
40(2), pp. 266-284.
[27] Randall, T.; Netessine, S. and Rudi, N., 2006, ‘An Empirical Examination of the Decision to Invest in Fulfillment Capabilities: A Study of Internet Retailers,’ Management
Science, 52(4), pp. 567-580.
[28] Reisinger, M. and Tarantino, E., 2015, ‘Vertical Integration, Foreclosure, and Productive Eﬃciency,’ RAND Journal of Economics, 46(3), pp. 461-479.

38

[29] Rey, P. and Vergé, T., 2004, ‘Bilateral Control with Vertical Contracts,’ RAND Journalof Economics, 35(4), pp. 728-746.
[30] Tannenbaum, J., 1995, ‘Franchisees Resist Poaching via Software, Kiosks, Internet,’
Wall Street Journal, October 30.
[31] Tsay, A. A. and Agrawal, N., 2004, ‘Channel Conflict and Coordination in the ECommerce Age,’ Production and Operations Management, 13(1), pp. 93-110.
[32] Villas-Boas, J. M., 1998, ‘Product Line Design for a Distribution Channel,’ Marketing
Science, 17(2), pp. 156-169.
[33] Vives, X. 1999. Oligopoly Pricing: Old Ideas and New Tools, Cambridge, Mass. MIT
Press.

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