GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 309

Who benefits from resale-below-cost laws?

Noriaki Matsushima and Akira Miyaoka

June 2013

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Who benefits from resale-below-cost laws?∗
Noriaki Matsushima†
Institute of Social and Economic Research, Osaka University
Akira Miyaoka‡
Graduate School of Economics, Osaka University
June 9, 2013

Abstract
We investigate the effect of banning resale-below-cost offers. There are two retailers withheterogeneous bargaining positions in relation to a monopolistic manufacturer. Each retailer
sells two goods: one procured from the monopolistic manufacturer and the other, from a competitive fringe. In equilibrium, banning resale-below-cost offers can decrease the retailers’prices. The ban can benefit the weak retailer in terms of bargaining position and increase thetotal consumer surplus, although it harms the dominant retailer and the monopolistic manufacturer. Contrary to the basic scenario, when the weak retailer is horizontally separated, the banbenefits the monopolistic manufacturer.

JEL Classification: L13, L41, M38, D21
Keywords: Loss-leader Pricing, Channel Power, Vertical Relations∗

We especially thank Atsushi Kajii, Tadashi Sekiguchi, and the seminar participants at University of Nagasaki for
helpful discussions and comments. The authors gratefully acknowledge financial support from the Japan Society for the
Promotion of Science. The usual disclaimer applies.
†
Noriaki Matsushima. Address: 6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan. E-mail: nmatsush@iser.osakau.ac.jp.
‡
Akira Miyaoka (Corresponding Author). Address: 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, Japan. E-mail:jge013ma@mail2.econ.osaka-u.ac.jp.

1

Introduction

Retailers use price promotions as both an offensive mechanism to attract competitors’ customers
and a defensive strategy to retain current customers (Gupta, 1988; Inman and McAlister, 1993;
Raghubir et al., 2004; Srinivasan et al., 2004; Talukdar et al., 2010). As reported in Talukdar
et al. (2010, p.336), according to the Promotion Marketing Association, in 2004, US retailers
across all product categories spent about $429 billion in such promotions; a more recent estimate
by ACNielsen (2007) suggests that promotional sales account for as much as 36% of total grocery
sales.
One particularly popular price promotion used by retailers is the “loss-leader (or below-cost)pricing,” which refers to setting retail prices for the selected items at or below retailers’ respective
marginal costs (Walters and MacKenzie, 1988). Price promotions through loss-leader pricing draw
some profitable shoppers who would otherwise shop at competitors’ stores. Those promotions
also draw unprofitable shoppers who only buy promoted items, as confirmed by recent empiricalresearch (Gauri et al., 2008; Talukdar et al., 2010).1
While loss-leader pricing is an important promotion strategy in the retail sector, several countries have adopted resale-below-cost (RBC) laws, which prevent retailers from setting resale prices
below purchase prices.2 In Germany, for instance, RBC laws forbid retailers from setting resale
prices below purchase prices for extended periods and allow for private enforcement in the case ofa defendant with a dominant position.3 Similarly, in the US, several states have sales-below-cost
motor fuel laws that typically outlaw the selling of motor fuel (gasoline) at retail prices below cost
1
The effectiveness of loss-leader promotions is empirically unclear. Walters and Rinne (1986) show that certain
portfolios of products promoted as loss leaders have a greater impact on store traffic, store sales, and deal sales than otherproduct portfolios, with no significant impact on retailer profits. Walters and MacKenzie (1988) also find a significant
impact of loss leaders on store traffic and store sales, but only two (out of eight) of their categories had significant
effects on store profits—one positive and one negative. Recent empirical research does not support the effectiveness ofloss-leader promotions either (Srinivasan et al., 2004; Ailawadi et al., 2009).
2
The OECD Roundtable on Predatory Foreclosure identified that Ireland, France, and Germany have stringent RBC
laws (OECD, 2007). Besides these nations, in the EU, RBC laws exist in Belgium, Hungary, Italy, Luxembourg, Portugal,
Spain, and Greece. In the US, although a federal RBC law has not been adopted, many states have RBC laws either
applying to all retail products or to specific goods such as gasoline or dairy products (Allain and Chambolle, 2011).
3
Walmart’s troubles in Germany are blamed on regulations forbidding Walmart to institute its usual loss-leader strategy for common grocery products such as milk and eggs. The German Supreme Court held that Germany’s RBC prohibition applies regardless of any harm to competition from the RBC prices and that being a large firm relative to small andmedium size competitors is sufficient to show that it has “superior” market power (OECD, 2007).

1

(Skidmore et al., 2005).4
However, in spite of the prevalence of RBC laws, these regulations have been the subject ofa heated debate in the area of competition policy. For example, OECD (2007) argues that RBC
laws are likely to protect inefficient competitors and harm consumers. In fact, the existing empiricalliterature on sales-below-cost laws in gasoline retailing has mixed results. Although most studies
show that these laws are associated with higher gasoline prices (e.g., Anderson and Johnson, 1999),
several recent empirical studies show that the laws actually lower average gasoline prices (Skidmore
et al., 2005; Carranza et al., 2009).5
By considering the practical importance of loss-leader promotions, we need to understand how
such promotions influence firms’ pricing strategies, and how banning RBC influences consumer
welfare and firms’ profitability including that of retailers and manufacturers. Therefore, we investigate a linear city (Hotelling) model with two asymmetric retailers, each carrying two products,
and with heterogenous consumer groups. One of the products is produced by a monopolistic manufacturer and the other, by a competitive fringe whose wholesale price is set at its marginal cost.
Retailers are asymmetric in terms of bargaining positions over the monopolistic manufacturer. The“dominant” retailer can procure the good at the wholesale price equalized to the manufacturer’s
marginal cost, but the “weak” retailer must procure at the wholesale price offered by the manufacturer in a take-it-or-leave-it manner.6 Consumer groups are heterogenous in terms of the number of
products they need and per distance transportation costs. The first heterogeneity means that consumers in one group need two products, although those in the other group need only one of the two
products. These heterogeneities generate below-cost prices as in DeGraba (2006) and Azar (2010).
In this basic setting, we first show that when it is legally permitted, below-cost pricing canappear as an equilibrium pricing strategy of both retailers and that the weak retailer, which must
4
The debate over whether to adopt or overturn sales-below-cost restrictions in gasoline retailing is ongoing (Eckert,2013).
5
We briefly discuss the relationship between their results and ours later in this section.
6
This assumption reflects the dispute in Germany mentioned in the previous paragraph. That is, we capture a situationin which large retailers have strong bargaining positions, unlike small and medium size retailers. This assumption also
captures the US gasoline market in the previous paragraph. That is, we capture a situation in which large retailers areintegrated with gasoline wholesalers, which leads to lower internal transfer prices, although small and medium size
retailers are independent from those gasoline wholesalers.

2

procure one of the goods at a higher wholesale price, is more likely to adopt that strategy. We next
investigate how a ban on below-cost pricing affects the firms’ behaviors and show that the ban candecrease the retailers’ prices in equilibrium. We also show that the ban always improves the total
consumer surplus and can benefit the weak retailer, although it always harms the dominant retailer
and the monopolistic manufacturer.
Our results using the basic model are related to the empirical findings in the studies by Skidmore
et al. (2005) and Carranza et al. (2009), which are based on theoretical predictions in which belowcost laws can encourage market entry and lead to reduced retail prices.7 Skidmore et al. (2005) showthat these laws in the US retail gasoline market lower average gasoline prices, in part by increasing
the number of gasoline outlets. Carranza et al. (2009) show that below-cost regulation imposedin the retail gasoline market in Québec led to more competition, lower prices for consumers, andlower productivity. Our study theoretically shows that a weak retailer benefits from the ban on
below-cost pricing, which implies that the ban improves the survival rate of weak retailers. Our
study result is related to the empirical findings in the two studies and might also imply that suchbans worsen the efficiency in retail markets because the weak retailer’s marginal cost that includes
its wholesale price are higher. Our study also shows that such a ban can decrease retail prices eventhough the number of retailers is exogenously fixed. Therefore, we believe that our study provides
another explanation to the empirical findings in the two studies mentioned above, which is usefulwhen setting competition policy.
Further, we extend our model to a setting in which the weak retailer is horizontally separated,thus eliminating the possibility that the separated retailers employ below-cost pricing. This extended model capturers a case in which a big-box retailer (the dominant retailer in our model)
competes with a shopping mall organized by independent retailers (the separated weak retailers in
our model). We show that the ban increases the price of the loss-leader product over most of theparameter range, although the change in the price of the product competing with the loss-leader
product depends on the exogenous parameters. We also show that the ban improves the total consumer surplus and benefits the monopolistic manufacturer and the weak retailer trading with it,
7

The outcomes derived from their theories are similar, although the theories in the two studies differ from each other.

3

although the ban can harm the other weak retailer and the dominant retailer. That is, the effectof the ban on the profitability of the manufacturer also depends on the retailer-level organizational
structure.
Our results have a managerial implication for manufacturers in the context of vertical channels.
In our study, the (weak) retailer trading with the monopolistic manufacturer benefits from a belowcost law, which implies that the law stabilizes the weak retailer as a distribution channel because
the benefit from the law improves the financial stability of the weak retailer. This stability improvement, in itself, benefits the monopolistic manufacturer as well as the weak retailer. However, therelationship between imposing the law and the profitability of the monopolistic manufacturer depends on whether its product is used as a loss-leader product; in other words, it depends on whether
its product is used as a tool to increase store traffic in its trading retailer. Our results imply thatwhen sales managers in manufacturing companies choose retailers, they need to consider not only
the regulatory environment in the market but also the pricing strategies of retailers.
Among the theoretical studies that explain the mechanisms of loss-leader promotions (Hess andGerstner, 1987; Bliss, 1988; Bagwell and Ramey, 1994; Lal and Matutes, 1994; DeGraba, 2006;
Azar, 2010), Innes and Hamilton (2009) show the market condition in which loss leadership can
appear in the context of vertical restraints. The main concern of their study is to investigate how
resale price maintenance (RPM) imposed by a monopolistic manufacturer on two competing symmetric retailers affects the equilibrium prices of those retailers trading two products including thatof the manufacturer and the welfare property in equilibrium. They do not consider the effect ofbanning below-cost pricing and retailers’ asymmetry.8 Several recent studies discuss the effect ofbanning below-cost pricing. Chen and Rey (2012) analyze the effect of banning below-cost pricing
under asymmetric retail competition with general demand structures, although they ignore the interactions between retailers and their suppliers; von Schlippenbach (2008) constructs a downstream
monopoly model in which an upstream supplier produces a core product and competitive fringe
suppliers produce complements to the core product. The effects of RBC laws in her model are8

Rey and Vergé (2010) investigate how RPM can be used by duopoly manufacturers to control the retail pricing of
duopoly retailers. They show that combining two-part tariffs with RPM can eliminate both interbrand and intrabrandcompetition and generate industry-wide monopoly pricing.

4

similar to that in our model, although her model cannot investigate asymmetric effects of RBC laws
on heterogenous retailers. Allain and Chambolle (2011) investigate the effect of RBC laws in thecontext of the vertical restraints discussed in Innes and Hamilton (2009).
Our study is also in line with game-theoretical analyses on marketing channels (Jeuland andShugan, 1983; McGuire and Staelin, 1983; Coughlan, 1985; Moorthy, 1988; Choi, 1991; Chu andDesai, 1995; Lal and Narasimhan, 1996; Lee and Staelin, 1997; Purohit, 1997; Kim and Staelin,1999). We incorporate power asymmetry between retailers as in Geylani et al. (2007). Existing
marketing literature also considers power within the channel—for instance, Butaney and Wortzel(1988), Messinger and Narasimhan (1995), Bloom and Perry (2001), Banks et al. (2002), and Iyer
and Villas-Boas (2003). Raju and Zhang (2005) and Dukes et al. (2006) also investigate asymmetricchannel relationships in retail competition. Raju and Zhang (2005) investigate how a manufacturer
can best coordinate a channel in the presence of a dominant retailer. Dukes et al. (2006) focus
on characterizing the impact of asymmetric retailing costs on wholesale bargaining and on profit
distribution.
The rest of this paper is organized as follows. In the following section, we set up the model. In
Section 3, we derive the equilibrium pricing of each firm when below-cost pricing is feasible andwhen a ban on below-cost pricing is introduced. In Section 4, we compare the equilibrium with andwithout the ban and explore the effect of banning below-cost pricing. In Section 5, we analyze theextended model where the weak retailer is horizontally separated. Finally, in Section 6, we present
concluding remarks. Proofs of all propositions are provided in Appendix A.9

2

The Model

There are two downstream retailers, two upstream manufacturers, and two goods. While good 1 is
supplied to both retailers by a monopolistic manufacturer, good 2 is supplied to both retailers by acompetitive fringe. The marginal production cost of each good is constant and denoted by c1 andc2 , respectively. For simplicity, the retailers’ marginal costs of retailing are assumed to be zero.
9

The Mathematica files used to obtain the results in this paper are available from the authors upon request.

5

In the downstream market, we adopt a standard Hotelling framework (Hotelling, 1929), with
two retailers located at the endpoints of the unit interval. Each retailer purchases both goods 1 and 2
from upstream manufacturers and sells them to consumers. In addition, as in Geylani et al. (2007),
we assume that the two retailers are asymmetric with respect to the monopolistic manufacturer’s
control over its wholesale prices. One retailer is a “dominant” retailer (denoted by D), and therefore,
the manufacturer must set the wholesale price to this retailer at its marginal cost level, that is,
w1D = c1 . However, the other retailer is a “weak” retailer (denoted by W), and the manufacturer
can offer a take-it-or-leave-it wholesale price w1W . As for good 2, the competitive fringe sets thewholesale prices to both retailers at its marginal cost level, that is, w2D = w2W = c2 . We assume thatretailer D (W) is located at the left (right) end of the unit interval. The price set by retailer i = D, W
for good k = 1, 2 is given as pki . Figure 1 illustrates the industrial structure of our model explained
here.
[Figure 1 about here.]In the downstream market, there are two types of consumers: type 2 and B. Type 2 consumers
purchase only good 2 and have a mass of λ > 0. Type B consumers purchase both goods, 1 and
2, and have a mass that is normalized to 1. Each type of consumers is assumed to be uniformly
distributed on the unit interval.10 In order to travel to either retailer, consumers must incur a linear
transportation cost for the distance traveled. While the cost per unit of distance for type 2 consumers
is t, that for type B consumers is τ. The transportation cost of a type 2 consumer located at x is tx
if he/she purchases good(s) from retailer D or t(1 − x) if he/she purchases good(s) from retailer W.
Similarly, the transportation cost of a type B consumer located at x is τx if he/she purchases good(s)from retailer D or τ(1 − x) if he/she purchases good(s) from retailer W.
Type 2 consumers purchase one unit of good 2 from one of the retailers. Similarly, type B
consumers purchase one unit of both goods from one of the retailers. We assume that type B
consumers do not split purchases between different retailers.11 Therefore, each consumer purchases
10

We do not consider “type 1” consumers who purchase only good 1. If the population of type 1 consumers is smaller
than that of the other two types, adding type 1 consumers to the model does not alter our main results qualitatively.
11This is because the cost of standing in a checkout line and searching for goods is relatively larger than the traveling6from the retailer for which the sum of the retail prices and the transportation cost is lower. The sumof those costs incurred by a type 2 consumer is given as(tx + p2D ,if buying from retailer D,t(1 − x) + p2W , if buying from retailer W.

(1)

The sum of those costs incurred by a type B consumer is given as(

τx + p1D + p2D ,if buying from retailer D,τ(1 − x) + p1W + p2W , if buying from retailer W.

(2)

The game runs as follows: First, the monopolistic manufacturer sets the wholesale price, w1W , forretailer W. Note that the rest of the wholesale prices are set at the manufacturers’ marginal costlevels. Second, given the wholesale prices, the retailers set the retail prices, pki (k = 1, 2, i = D, W),
in the downstream market. Finally, following his/her preference, each consumer purchases good(s)from either retailer.

3

Equilibrium

In this section, we derive the equilibrium pricing of the monopolistic manufacturer and the retailers.
In Section 3.1, we first explore the case where downstream retailers can set their retail prices freely.
Then, in Section 3.2, we analyze the case when the ban on below-cost pricing is introduced.

3.1

When below-cost pricing is feasible

In this subsection, we derive the equilibrium wholesale and retail prices when downstream retailers
can adopt below-cost pricing. Because each type of consumers purchases one good or two goods
from the retailer that minimizes the total cost, there is a consumer who is indifferent between thetwo retailers. Let x̄2 ∈ [0, 1] represent the location of the indifferent consumer of type 2. From (1),x̄2 is given as follows:
p2D + t x̄2 = p2W + t(1 − x̄2 ) ⇔ x̄2 =
cost.

7

1 p2W − p2D
+
.
2
2t

(3)

Similarly, for type B consumers, from (2), the location of the indifferent consumer x̄B ∈ [0, 1] isgiven by
p1D + p2D + τ x̄B = p1W + p2W + τ(1 − x̄B ) ⇔ x̄B =1 p1W + p2W − p1D − p2D
+
.
2
2τ

(4)

Therefore, given the wholesale prices, the dominant and weak retailers simultaneously choose theirretail prices to maximize the following profits, respectively:
ΠD = (p1D + p2D − w1D − w2D ) x̄B + (p2D − w2D )λ x̄2 ,(5)ΠW = (p1W + p2W − w1W − w2W )(1 − x̄B ) + (p2W − w2W )λ(1 − x̄2 ).

(6)

From (5), we have the following first-order conditions for the dominant retailer:
∂ΠD
=
∂p1D
∂ΠD
=
∂p2D

!
1 p1W + p2W − p1D − p2D
p1D + p2D − w1D − w2D
+
−
= 0,
2
2τ
2τ
!
1 p1W + p2W − p1D − p2D
p1D + p2D − w1D − w2D
+
−
2
2τ
2τ
!
1 p2W − p2D
λ(p2D − w2D )
+λ +
−
= 0.
2
2t
2t

(7)

(8)

From (6), the first-order conditions for the weak retailer are the same as (7) and (8) with the subscripts D and W reversed. Then, by solving these four first-order conditions, we have the following
retail prices for the given wholesale prices:
2w1D + w1W+ τ − t;3
w1D + 2w1W
p1W =+ τ − t;3
p1D =p2D = w2D + t;

(9)p2W = w2W + t.

(10)

The upstream manufacturers decide their wholesale prices anticipating the retailers’ pricing
given by (9) and (10). The competitive fringe prices good 2 at its marginal cost, that is, w2D =w2W = c2 . However, while the monopolistic manufacturer sets the wholesale price of good 1 tothe dominant retailer at the marginal cost, that is, w1D = c1 , it offers a take-it-or-leave-it wholesale
price w1W to the weak retailer in order to maximize the following profit:Π1 = (w1W − c1 )(1 − x̄B ).
8(11)

Let mki denote the price-cost margin of good k for retailer i, that is, mki = pki − wki . In addition,
we denote with an asterisk (∗) the equilibrium values when the ban on below-cost pricing is not
imposed. Then, we obtain the following results that show that below-cost pricing can appear as anequilibrium strategy of downstream retailers.
Proposition 1. When the ban on below-cost pricing is not imposed, both retailers’ price-cost margins for good 2 are always positive, that is, m∗2i > 0 for i = D, W. By contrast, the price-cost
margins of good 1 can be negative when the value of τ/t is sufficiently small. More precisely,(a) for 0 < τ/t < 2/3, both retailers’ price-cost margins of good 1 are negative, that is, m∗1i < 0for i = D, W,(b) for 2/3 < τ/t < 2, only the weak retailer’s price-cost margin of good 1 is negative, that is,m∗1D > 0 and m∗1W < 0, and(c) for τ/t > 2, both retailers’ price-cost margins of good 1 are positive, that is, m∗1i > 0 for
i = D, W.
[Figure 2 about here.]
Figure 2(a) summarizes the results of Proposition 1. These results suggest that the retailers
may set the price of good 1 below the wholesale price in equilibrium when below-cost pricing isfeasible. This is due to the existence of type B consumers who purchase both goods at either retailer.
Since type B consumers care about the sum of both goods’ prices, the retailers can attract them bylowering either the price of good 1 or the price of good 2, as in the models of DeGraba (2006)
and Azar (2010). Lowering the price of good 1 does not affect the profits from type 2 consumers,
although lowering the price of good 2 reduces those profits. Therefore, it is more attractive for theretailers to lower the price of good 1. In particular, when τ/t is sufficiently small (τ/t < 2), type 2
consumers are so price inelastic that the retailers can charge a high markup on good 2, which is toohigh from the perspective of type B consumers. Therefore, it is optimal for the retailers to make theprice-cost margin on good 1 negative to attract type B consumers, while charging a high markup on
good 2.
9In addition, in equilibrium, the monopolistic manufacturer sets the wholesale price for the weakretailer higher than for the dominant retailer (that is, w∗1W > w∗1D = c1 ). Therefore, the weak retailer
is more likely to price good 1 below cost than the dominant one.12 In fact, some survey evidence
suggests that small stores are just as likely as large stores to find it advantageous to employ belowcost pricing as a marketing tool in order to generate store traffic (OECD, 2007).

3.2

When below-cost pricing is banned

Next, we explore the equilibrium pricing of the upstream monopolist and downstream retailers
when below-cost pricing is banned. When a ban on below-cost pricing is introduced, each retailer
cannot price its goods below the wholesale prices at which it procures the goods. Therefore, given
the wholesale prices, the dominant and the weak retailers, respectively, choose their retail prices inorder to maximize (5) and (6) under the constraints that pkD ≥ wkD and pkW ≥ wkW for k = 1, 2.
Anticipating the retail pricing, the upstream manufacturers set their wholesale prices as in theprevious subsection. Since good 2 is supplied competitively, its wholesale prices for both retailers
are set at the marginal cost level. However, for good 1, the upstream monopolist chooses its wholesale price for the weak retailer to maximize its own profit given by (11), while setting the wholesale
price for the dominant retailer at its marginal cost.
As shown in Proposition 1, even though the ban on below-cost pricing is inactive, the retailers
never set the prices below their wholesale prices when τ/t ≥ 2. Therefore, when τ/t ≥ 2, the ban is
not binding, and it affects neither the retail nor wholesale pricing in equilibrium. On the other hand,when τ/t < 2, the retailers have incentives to adopt below-cost pricing. Therefore, the retailers’ andmanufacturers’ equilibrium prices will alter by introducing the ban on below-cost pricing.
Let variables with a tilde (∼) denote equilibrium values when below-cost pricing is banned.
Then, the following proposition summarizes the equilibrium retail pricing under a ban on belowcost pricing.
Proposition 2. When the ban on below-cost pricing is imposed, both retailers’ price-cost margins
12
In Section 5, we consider an extended model where the weak retailer is horizontally separated. In that setting, only
the dominant retailer can adopt below-cost pricing in equilibrium, as in Chen and Rey (2012). See Section 5 for details.

10

of good 2 are always positive, that is, m̃2i > 0 for i = D, W. On the other hand, the price-cost
margins of good 1 can be zero when τ/t is sufficiently small. More precisely,(a) for 0 < τ/t ≤ T , both retailers’ price-cost margins of good 1 are zero, that is, m̃1i = 0 for
i = D, W,(b) for T < τ/t ≤ 2, only the weak retailer’s price-cost margin of good 1 is zero, that is, m̃1D > 0
and m̃1W = 0, and(c) for τ/t > 2, both retailers’ price-cost margins of good 1 are positive, that is, m̃1i > 0 for
i = D, W,
where the exact expression for T is derived in the Appendix.
Figure 2(b) illustrates the results of Proposition 2. Proposition 1 shows that retailer W’s pricecost margin of good 1 is smaller than that of retailer D. The pricing of retailer W is more likely tobind the constraint of the ban.

4

The Effects of Banning Below-cost Pricing

In the previous section, we derived the equilibrium price with and without the ban on below-cost
pricing. In this section, by comparing the results in the previous section, we analyze the effects ofbanning below-cost pricing.
First, we explore the effects of the ban on the wholesale and the retail prices.
Proposition 3. The effects of banning below-cost pricing on wholesale and retail prices are summarized as follows:(a) The monopolistic manufacturer’s wholesale price always decreases.
(b) Both retailers’ prices of good 1 decrease when τ/t is sufficiently large.
(c) Both retailers’ prices of good 2 never increase. In particular, they decrease except when τ/tis too large.
11

[Figure 3 about here.]
When the ban on below-cost pricing is not imposed, as explained in Proposition 1, the weakretailer tries to attract consumers away from the dominant retailer by setting the price of good 1 at
a level below the wholesale price. When the ban is imposed, the weak retailer must set the priceof good 1 at least equal to the wholesale price. Anticipating this higher retail price set by the weakretailer, the monopolistic manufacturer sets a lower wholesale price to mitigate the demand loss ofgood 1 distributed by the weak retailer.
Figure 3(a) summarizes the effect of banning below-cost pricing on the retail prices.13 Because
the products sold by each retailer are complements, an increase in the price of one product is more
likely to decrease the price of the other one. As shown in Figure 3(a), however, when τ/t is large(T p < τ/t < T ), both p1i and p2i simultaneously decrease by banning below-cost pricing.14 We
explain the mechanism below.15
As for the weak retailer’s pricing of good 1, banning below-cost pricing has two opposite effects.
While it can lower the price through the reduction of the wholesale price as explained above, it canalso raise the price by preventing the weak retailer from having a negative price-cost margin on
good 1. This trade-off depends on the pricing for good 2. When the ban is imposed, the price-cost
margin of good 1 increases from a negative level to zero, which encourages the weak retailer toattract consumers of type B through a decrease in p2W . This incentive to decrease p2W is relatively
weaker as τ/t is larger, because an increase in the price-cost margin on good 1 through the ban isrelatively smaller.16 In such a situation, the monopolistic manufacturer must reduce its wholesale
13

The exact expressions for T p and T in the figure are derived in the Appendix.
A similar result is obtained by von Schlippenbach (2008). In her model, the downstream monopolist supplies themonopolistic manufacturer’s product and complementary products procured from competitive suppliers. Although competitive suppliers’ products are complements to the monopolistic supplier’s product, a decrease in the price of the monopolistic manufacturer’s product causes decreases in the prices of those complementary products.
15
For T < τ/t < 2, the mechanism behind a decrease in p1i through the ban is the same as explained below. However,
the reason the ban does not affect p2i is as follows. In this range, it is optimal for the monopolistic manufacturer to set
its wholesale price such that the constraint of the ban for the weak retailer’s price of good 2 is just binding. Therefore,
given the wholesale price set optimally, the constraint of the ban is not effective, and the equilibrium retailers’ responses
are given by (9) and (10).
16This can be also confirmed from the Kuhn-Tucker conditions derived in the Appendix. From (29) and (30), the best
response price of good 2 for retailer W can be derived as p2W = (p2D + t + c2 )/2 − tµW /λ, where µW is the Lagrange
multiplier of retailer W’s maximization problem. The Lagrange multiplier µW is decreasing in τ/t and it is equal to zero
14

12

price significantly in order to maintain the demand for good 1 distributed by the weak retailer.
Therefore, when τ/t is large, the effect of the reduction of the wholesale price dominates the otherone, and then, the ban decreases the weak retailer’s prices of both goods. In addition, becauseof strategic complementarity between both retailers’ pricing, the dominant retailer’s prices of both
goods also decrease through the ban.
Next, we consider the effects of banning below-cost pricing on the profits of the monopolistic
manufacturer and downstream retailers.
Proposition 4. The effects of banning below-cost pricing on firms’ profits are summarized as follows:(a) The monopolistic manufacturer’s profit always decreases.
(b) The dominant retailer’s profit always decreases.
(c) The weak retailer’s profit increases when τ/t is sufficiently large.
These results are illustrated in Figure 3(b).17 The ban prevents the weak retailer from setting
the price of good 1 at a below-cost level, which makes it difficult for the monopolistic manufacturer
to set a higher wholesale price. This reduces the profit of the monopolistic manufacturer. The banlowers the prices of good 2, which lowers the profits of the retailers from type 2 consumers. The
lower τ/t is, the stronger is the negative effect, because the relative profitability of the market for
type 2 consumers becomes larger. Because of the negative effect, the ban always reduces the profitof the dominant retailer. For the weak retailer, however, the negative effect does not always dominate the positive effect of the decrease in the wholesale price set by the monopolistic manufacturer.
When τ/t is large, the positive effect can dominate the negative one.
Finally, we analyze how introducing the ban affects consumer surplus. In this study, since
we assume that all consumers are served in equilibrium, consumer surplus for type 2 and type B
without a ban on below-cost pricing. Therefore, the ban shifts the best response curve downward less significantly when
τ/t is large.
17
In the Appendix, we explain how to derive T Π in the figure.

13

consumers are, respectively, given by
CS 2 = −λZ x̄2(p2D + tm)dm +0

CS B = −

Z x̄B

!

Z 1

(p2W + t(1 − m))dm ,(12)(p1W + p2W + τ(1 − m))dm.

(13)

x̄2

Z 1
(p1D + p2D + τm)dm −0x̄BThe total consumer surplus is given by CS T = CS 2 + CS B . Then, we have the following results on
the effects of banning below-cost pricing on consumer surplus.
Proposition 5. The effects of banning below-cost pricing on consumer surplus are summarized asfollows:(a) Banning below-cost pricing never harms type 2 consumers. In particular, it benefits them
except when τ/t is too large.
(b) Banning below-cost pricing benefits type B consumers when τ/t is sufficiently large.
(c) Banning below-cost pricing always increases the total consumer surplus.
These results are also summarized in Figure 3(c). As shown in Proposition 3, when the ban
is imposed, the prices of good 2 set by both retailers do not increase and, in particular, decrease
when τ/t is not large enough. Therefore, the ban at least does not harm type 2 consumers, whopurchase only good 2, and can benefit them through a decrease in the price of good 2. By contrast,
type B consumers, who buy both goods, can be hurt by the ban, which directly eliminates the
below-cost prices of good 1. When τ is relatively smaller than t (τ/t is smaller than 1), attracting
type B consumers is relatively easy for each retailer because those consumers are relatively priceelastic than type 2 consumers. To do so, controlling the price of good 1 is relatively useful for eachretailer to attract type B consumers. The ban restricts the retailer’s pricing for good 1, which leads
to increases in those prices. As a result, the restriction mitigates competition between the retailers
to attract type B consumers, which increases the total payment of those consumers.

14

5

Extension: A Big-box Retailer vs. a Shopping Mall

In the basic model, we assumed that each retailer sells both goods, and this assumption enables both
retailers to engage in below-cost pricing. How does the equilibrium property change if we assume
that the weak retailer is separated? In this section, we extend the basic model and investigate how
the structural change of the retail sector affects the effect of banning below-cost pricing.

5.1

The extended model

We consider the case where the dominant retailer competes with two weak retailers, W1 and W2,
who carry good 1 and 2, respectively (see Figure 4). We interpret the group of the separated weakretailers as a shopping mall, which competes with a big-box retailer.18 In this setting, since eachweak retailer carries only one brand, it cannot set below-cost prices, which always lead to negative
profits. However, the dominant retailer sells both brands and can engage in below-cost pricing, as
in the basic model. As shown below, this structural change of the retail sector drastically alters theeffect of banning below-cost pricing.
[Figure 4 about here.]
The profits of the separated weak retailers are, respectively, given as follows:ΠW1 = (p1W − w1W )(1 − x̄B ),(14)

ΠW2 = (p2W − w2W )((1 − x̄B ) + λ(1 − x̄2 )).

(15)

Note that each weak retailer does not internalize the effect of its own pricing on the other weakretailer through the behavior of type B consumers. The profits of the dominant retailer and themonopolistic manufacturer are the same as in the basic model and represented by (5) and (11),respectively. The timing structure of the game is also the same as in the basic model.
18

The case discussed here is also related to a case in which one manufacturer sets up a store-within-a-store, and theretailer buys the other’s product at a wholesale price and sells it at a marked-up retail price (Jerath and Zhang, 2010).
The separated retailer trading with a competitive fringe is related to a manufacturer within a store and that trading withthe monopolistic manufacturer is related to the retailer buying the other manufacturer’s product at a wholesale price.

15

As mentioned above, in this extended model, only the dominant retailer can set below-costprices in equilibrium. Figure 5 illustrates the equilibrium price-cost margin of good 1 set by thedominant retailer with and without the ban on below-cost pricing.19 When below-cost pricing islegally permitted, the dominant retailer sets the price of good 1 below the wholesale price if τ/t issufficiently small. The intuition behind this result is the same as in the basic model. By contrast,
when the ban is imposed, it is binding in the parameter range where the dominant retailer engaged
in below-cost pricing without the ban.20 In the relevant range of parameters, the dominant retailer
sets the price of good 1 equal to the wholesale price, and the ban affects the prices, profits, andconsumer surplus in equilibrium, as summarized in Figure 6.21 We examine each of these effects indetail below.
[Figure 5 about here.][Figure 6 about here.]

The effect of the ban on prices

The ban increases p1D for most of the cases where the ban isbinding for the dominant retailer in equilibrium. Obviously, this is because the ban prevents thedominant retailer from setting the price of good 1 below the wholesale price, c1 .22 The constraint
on p1D has a direct negative impact on p2D .
Next, we consider the effect of the ban on the monopolistic manufacturer’s wholesale pricing.
We call the weak retailer trading with the monopolistic manufacturer and the other weak retailer “thetrading weak retailer” and “the outside weak retailer”, respectively. From (11), the monopolistic
19

Proofs of all results in this section are provided in Appendix B.
According to Figure 5, the ban is binding even in the parameter range where the dominant retailer does not engage
in below-cost pricing without the ban (i.e., T e < τ/t < T̂ e ). In this range, under the ban, the monopolistic manufacturer
sets a lower wholesale price in order to induce the weak retailer to decrease p1W . Since the dominant retailer cannot
reduce p1D below c1 in response to the decrease in p1W owing to the ban, the monopolistic manufacturer can significantly
attract the demand for its product through the weak retailer.
21
In Figure 6, we do not illustrate the effects on the monopolistic manufacturer’s profit and total consumer surplus,
because the direction of those effects is independent of the parameter values, as explained below.
22
Note that for τ/t around the threshold value of τ/t in which the ban is binding for the dominant retailer in equilibrium,
p1D decreases by the ban. The reason for this phenomenon is explained in footnote 20.
20

16manufacturer’s first-order condition is given as∂Π1∂(1 − x̄B )= (1 − x̄B ) + (w1W − c1 )
.
∂w1W
∂w1W

(16)

The ban on below-cost pricing affects both terms in the right-hand side of this condition.
First, the ban works as a commitment of the dominant retailer to set p1D = c1 , which implies
that it becomes the first-mover in the price competition with the trading weak retailer. This also
implies that the trading weak retailer has the second-mover advantage, which increases the demand
from type B consumers for the weak retailers, 1 − xB . This induces the monopolistic manufacturer
to increase its wholesale price for the trading weak retailer.
Second, the price commitment has another effect on the competition between the dominantretailer and the trading weak retailer. Under the ban, the dominant retailer cannot reduce its price ofgood 1 in response to a decrease in the trading weak retailer’s price. Therefore, the ban changes thestrategic interaction between them from strategic complements to strategic independence, whichmeans that a decrease in the trading weak retailer’s price attracts the demand for it more than thecase without the ban. In other words, the elasticity of demand for this retailer, which is correlated
to ∂(1 − x̄B )/∂w1W , becomes higher. This effect gives the monopolistic manufacturer an incentive
to decrease its wholesale price.23
The relative scale of these two effects depends on τ/t. When τ/t is small, the first effect is large
enough to dominate the second effect. However, when τ/t is large, while the second effect remains
effective, the first effect becomes sufficiently small because an increase in p1D through the ban isvery small. Therefore, as illustrated in Figure 6(a), the ban increases (decreases) w1W when τ/t issmall (large).

The effect of the ban on profits

The ban always benefits the monopolistic manufacturer and theweak retailer trading with it because the dominant retailer becomes less aggressive in the pricing
23
To be precise, there is another effect of the ban that counteracts the price-commitment effect explained in this paragraph. Under the ban, since the dominant retailer cannot adjust p1D flexibly, it responds to a decrease in p1W by decreasing
p2D more aggressively than the case without the ban. This effect makes the demand for the trading weak retailer less elastic and gives the monopolistic manufacturer an incentive to increase w1W . However, since this effect is relatively weaker
than the price-commitment effect explained in this paragraph, the overall effect of the ban on the demand elasticity tends
to reduce w1W .

17

of p1D . The other weak retailer also benefits from the ban if and only if its retail price increases inequilibrium.
As regards the dominant retailer’s profit, Figure 6(c) shows that the effect of the ban is nonmonotonic in τ/t. Behind this result, there are three effects of the ban on the dominant retailer’s
profit. First, the ban restricts the scope of the dominant retailer to control demand for the goods
through a change in p1D because it cannot set p1D at a below-cost level. This loss of flexibility hurts
the dominant retailer, and this effect is stronger as τ/t is smaller because the below-cost pricing ismore valuable.
Second, as explained above, the ban affects the wholesale price of good 1 for the weak retailer.
In particular, when τ/t is large enough, the ban reduces the wholesale price, which gives the weakretailers a competitive advantage vis-a-vis the dominant retailer.
Third, since the ban works as a commitment of the dominant retailer to set p1D = c1 , this canmitigate the price competition and benefit the dominant retailer. The strength of this effect is nonmonotonic in τ/t. When τ/t is nearly equal to the threshold value, this effect does not work because
p1D = c1 is nearly equal to the equilibrium price in the case without the ban. On the other hand,when the value of τ/t is small enough, this effect does not work, because a small τ/t implies highelasticity of type B consumers’ demand. Therefore, this effect is strong for an intermediate rangeof τ/t.
These three effects produce the result in Figure 6(c). When τ/t is sufficiently small or large,
since the first or second effects become dominant, the ban hurts the dominant retailer. However,when τ/t falls within an intermediate range, the third effect dominates the other effects and the banbenefits the dominant retailer.

The effect of the ban on consumer surplus

The ban always increases the total consumer surplus,

which seems similar to the result in the basic model. The main factor behind the welfare property
is decreases in the prices of good 2.

18

5.2

Discussion

When compared to that of the basic model, the results in this section imply that the effects ofbanning below-cost pricing on the prices and the profits of each firm crucially depend on the downstream market structure.24 As discussed in the Introduction, the existing empirical literature has
mixed results on the effects of (banning) below-cost pricing on the prices and retailers’ profits. In
addition, there are also mixed empirical or anecdotal evidences on whether (banning) below-cost
pricing benefits or hurts the upstream manufacturers. On one hand, Srinivasan et al. (2004) show
empirical findings that price promotions have a predominantly positive impact on manufacturer
revenues. On the other hand, it is also observed that manufacturers sometimes object to loss-leader
pricing on the ground that competition between retailers can bring pressure on the manufacturers tolower the wholesale prices.25 Our findings could be one possible explanation for these seemingly
contradictory observations.

6

Conclusion

This study analyzed below-cost pricing by retailers and the effects of resale-below-cost laws thatprohibit retailers from pricing their goods below wholesale prices. The key feature of our modelis the power asymmetry between retailers. One retailer (weak retailer) does not have bargaining
power over a monopolistic manufacturer; the other retailer (dominant retailer) has full bargaining
power over manufacturers, including the monopolistic one.
Under a basic model in which each retailer sells two products including that of the monopolistic
manufacturer, we showed that below-cost pricing could appear as an equilibrium pricing strategy ofboth retailers and that the weak retailer is more likely to adopt that strategy. We also showed thatthe ban could decrease the prices of the retailers in equilibrium. This result is related to empirical24

In the basic model of previous sections, we assumed that resale-below-cost laws are equally imposed on both dominant and weak retailers. If the laws are imposed only on the dominant retailer because of its strong market power, theeffect of the ban on the monopolistic manufacturer is similar to that in the model of this section. That is, the ban benefitsit.
25
For example, recently in Germany, the German Farmer Association (Deutscher Bauernverband) and the German
Association of Brand Manufacturers (Deutscher Markenverband) complained against below-cost prices by large retailers(von Schlippenbach, 2008). In addition, Marvel (1994) also reports that manufacturers often object to loss-leader statusbecause it makes retailers reluctant to order inventories in the presence of demand uncertainty.

19

findings (that is, in Skidmore et al., 2005 and Carranza et al., 2009) that are based on theoreticalpredictions in which below-cost laws can encourage market entry leading to reduced retail prices.
Our study also showed that the ban could decrease retail prices even though the number of retailers is
exogenously fixed. We therefore believe that our study provides another explanation to the empiricalfindings mentioned above.
In addition, we analyzed an extended model in which the weak retailer is horizontally separated.
This extended model captures a case in which a big-box retailer competes with a shopping mallorganized by independent retailers. We found that when a ban on below-cost pricing is binding, theban benefits the monopolistic manufacturer and the weak retailer trading with it, although the banharms the other weak retailer and the dominant retailer. The result implies that the effect of the banon the manufacturer’s profitability crucially depends on the retailer’s pricing strategy.
In this study, we employed the Hotelling framework with two multi-product retailers and withheterogenous consumer groups, as in DeGraba (2006) and Azar (2010). However, some previous
studies use other demand structures to analyze loss-leader pricing.26 We leave it for future work
to analyze the retailers’ pricing behavior and the effects of banning loss-leader pricing under suchdemand structures.

Appendix A: Proofs
Proof of Proposition 1
Substituting wholesale prices (other than w1W ) and retailers’ responses given by (9) and (10) into
the manufacturer’s profits (11), we have!
1 w1W − c1
Π1 = (w1W − c1 ) −
.
2
6τ

(17)

Then, the monopolistic manufacturer optimally sets the wholesale price atw∗1W = c1 +
26

3τ
.
2

(18)

For instance, Innes and Hamilton (2009) use the Hotelling-type spatial competition model combined with the demand
functions derived from a quasi-linear utility function. Allain and Chambolle (2011) use a linear demand framework
developed by Dobson and Waterson (1996), which captures both interbrand and intrabrand competition.

20

From (18), the equilibrium retail prices are obtained as follows:p∗1D = c1 +

3τ
− t;
2

p∗1W = c1 + 2τ − t;andp∗2D = p∗2W = c2 + t.

(19)

For good 2, it is easy to see that m∗2i = p∗2i − c2 = t and that both retailers’ price-cost margin is
always positive. On the other hand, for good 1, each retailer’s price-cost margin becomes3τ
τ 2−t R0⇔ R ,
2
t
3
τ
τ
m∗1W = p∗1W − w∗1W = − t R 0 ⇔ R 2.
2tm∗1D = p∗1D − c1 =Therefore, we obtain the results in Proposition 1.

(20)
(21)


Proof of Proposition 2
In the following proof, we first derive an equilibrium under a ban, temporarily ignoring the constraints for good 2 (p2i ≥ w2i for i = D, W), and later we check that those constraints are certainly
not binding in that equilibrium. Then, the Lagrange functions of each retailer’s profit maximization
problem are as follows:
LD = (p1D + p2D − c1 − c2 ) x̄B + (p2D − c2 )λ x̄2 + µD (p1D − c1 ),(22)

LW = (p1W + p2W − w1W − c2 )(1 − x̄B ) + (p2W − c2 )λ(1 − x̄2 ) + µW (p1W − w1W ),(23)

where x̄2 and x̄B are respectively given by (3) and (4), and µD and µW are Lagrange multipliers.
Note that we have w1D = c1 and w2D = w2W = c2 by assumption. Then, we have the following
Kuhn-Tucker conditions for the dominant retailer:
∂LD
=
∂p1D
∂LD
=
∂p2D

!
1 p1W + p2W − p1D − p2D
p1D + p2D − c1 − c2+−
+ µD = 0,22τ
2τ
!
1 p1W + p2W − p1D − p2D
p1D + p2D − c1 − c2
+
−
2
2τ
2τ
!
1 p2W − p2D
λ(p2D − c2 )
+λ +
−
= 0,
2
2t
2t

∂LD
= p1D − c1 ≥ 0,
∂µD

(24)

(25)
(26)

µD ≥ 0,(27)µD (p1D − c1 ) = 0.

(28)
21

Similarly, we obtain the Kuhn-Tucker conditions for the weak retailer as follows:
∂LW
=
∂p1W
∂LW
=
∂p2W

!
1 p1D + p2D − p1W − p2W
p1W + p2W − w1W − c2+−
+ µW = 0,22τ
2τ
!
1 p1D + p2D − p1W − p2W
p1W + p2W − w1W − c2
+
−
2
2τ
2τ
!
1 p2D − p2W
λ(p2W − c2 )
+λ +
−
= 0,
2
2t
2t

∂LW
= p1W − w1W ≥ 0,
∂µW

(29)

(30)
(31)

µW ≥ 0,(32)µW (p1W − w1W ) = 0.

(33)

In the analysis below, we deal with the following three cases separately: (i) both (26) and (31) arebinding; (ii) only (26) is binding; (iii) only (31) is binding. Note that the case where neither (26)nor (31) are binding corresponds to the analysis without a ban on below-cost pricing in Section 3.1.
Case I: Both (26) and (31) are binding
First, we explore the case where both retailers’ prices of good 1 are bound by the wholesale prices
in equilibrium. From (24), (25), (29), (30), and binding conditions, we haveλ(t − τ + c1 − w1W );2(t + λτ)(34)

λ(t − τ + w1W − c1 ).
2(t + λτ)(35)t(w1W − c1 + 3(1 + λ)τ);3(t + λτ)µD =t(c1 − w1W + 3(1 + λ)τ);3(t + λτ)µW =p1D = c1 ;p2D = c2 +p1W = w1W ;p2W = c2 +Substituting the above equations into (11), the monopolistic manufacturer’s profit becomes"Π1I = (w1W − c1 )#
1 (w1W − c1 )(t + 3λτ)−.
2
6τ(t + λτ)(36)

The monopolistic manufacturer maximizes this profit under the constraints (27) and (32), which
can be rewritten as c1 + τ − t ≤ w1W ≤ c1 + t − τ. Note that, in this case, there exists an equilibrium
as long as t − τ ≥ 0. Assuming an interior solution, we obtain the optimal wholesale price and the
manufacturer’s profit as follows:Iw̃1W= c1 +

3τ(t + λτ),2(t + 3λτ)

22

(37)

e I = 3τ(t + λτ) .
Π18(t + 3λτ)(38)

I ≤ c + t − τ, which implies that
This is valid as long as w̃1W1√
τ 6λ − 5 + 36λ2 + 12λ + 25
0< ≤
.
t
18λ

(39)

I = c + t − τ, and the manufacturer’s profit becomes
Otherwise, we have w̃1W
1#"
1 (t − τ)(t + 3λτ)Ie.
Π1 = (t − τ) −2
6τ(t + λτ)(40)Case II: Only (31) is binding
Second, we focus on the case where only the weak retailer’s equilibrium price of good 1 is boundby the wholesale price. Note that, since (26) holds with strict inequality, we have µD = 0 from (28).
Then, in the same way as in the previous case, from (24), (25), (29), (30), and binding conditions,
we have
p1D = c1 +

τ − t + w1W − c1;2p1W = w1W ;p2W = c2 +p2D = c2 +t(c1 − w1W + 3(t + τ + 2λτ));6(t + λτ)t(c1 − w1W + 3(1 + λ)τ);3(t + λτ)µW =µD = 0;(41)

λ(3(t − τ) + w1W − c1 ).
4(t + λτ)(42)

Substituting these equations into (11), the profit of the upstream monopolist can be rewritten as
follows:"
Π1II = (w1W − c1 )#
1 (w1W − c1 )(2t + 3λτ) + 3λτ(t − τ)−.
212τ(t + λτ)(43)

The monopolistic manufacturer maximizes this profit under the constraints (26) with strict inequality and (32), which can be rewritten as w1W ≥ c1 +max{t−τ, 3(τ−t)}. Assuming an interior solution,
the optimal wholesale price and the manufacturer’s profit are
IIw̃1W= c1 +

3τ(t(2 − λ) + 3λτ),2(2t + 3λτ)

(44)

2

e II = 3τ(t(2 − λ) + 3λτ) .
Π116(t + λτ)(2t + 3λτ)II ≥ c + max{t − τ, 3(τ − t)}, which implies that
This is valid as long as w̃1W1√√
9λ − 10 + 81λ2 + 60λ + 100 τ 5λ − 2 + 25λ2 + 28λ + 4
< ≤
.
30λ
t
6λ

23

(45)

(46)

√
For τ/t > (5λ − 2 + 25λ2 + 28λ + 4)/6λ, since Π1II is decreasing in w1W ≥ c1 + 3(τ − t), the optimal
wholesale price and the associated profit are
IIw̃1W= c1 + 3(τ − t),(47)e II = 3t(τ − t) .
Π
(48)
1
2τ
√
√
For (3λ − 4 + 9λ2 + 16)/12λ ≤ τ/t ≤ (9λ − 10 + 81λ2 + 60λ + 100)/30λ, since Π1II is decreasing
II = c + t − τ andin w1W ≥ c1 + t − τ, we have w̃1W
1#"
1 (t − τ)(t + 3λτ)
II
e
.
(49)
Π1 = (t − τ) −2
6τ(t + λτ)√
Finally, for 0 < τ/t < (3λ − 4 + 9λ2 + 16)/12λ, the monopolist cannot earn positive profit for any
w1W ≥ c1 + t − τ.
Case III: Only (26) is binding
Finally, we turn to the case where only the dominant retailer’s equilibrium price of good 1 is boundby the wholesale price. Note that, since (31) holds with strict inequality, we have µW = 0 from (33).
Then, from (24), (25), (29), (30), and binding conditions, we have
p1D = c1 ;p1W = w1W +p2D = c2 +t(w1W − c1 + 3(1 + λ)τ);3(t + λτ)τ − t + c1 − w1W;2

p2W = c2 +µD =λ(3(t − τ) + c1 − w1W );4(t + λτ)t(w1W − c1 + 3(t + τ + 2λτ);6(t + λτ)µW = 0.

(50)(51)

Substituting the above equations into (27) and (31) and combining them lead to
w1W − c1 < τ − t ≤ −w1W − c1.
3(52)

However, in equilibrium, since the upstream monopolist must set its wholesale price such that
w1W ≥ c1 , this inequality cannot hold. Therefore, this case can never happen in equilibrium.
Now, we find the optimal wholesale prices for given τ/t by comparing the optimal manufacturer’s profit derived above in each case.

24

(i) 0 < τ/t < (3λ − 4 +√9λ2 + 16)/12λIn this case, the upstream monopolist can earn positive profit only when it chooses the wholesale price at which both retailers’ prices of good 1 are bound. Therefore, the equilibrium
wholesale price and monopolist’s profit are given by (37) and (38).
(ii) (3λ − 4 +√9λ2 + 16)/12λ ≤ τ/t ≤ (9λ − 10 +√81λ2 + 60λ + 100)/30λ

In this case, a comparison between (38) and (49) leads to2

2 2

eI − Πe II = (2t + (6λ − 5)tτ − 9λτ ) > 0.
Π1
124τ(t + λτ)(t + 3λτ)

(53)

Therefore, the equilibrium wholesale price and monopolist’s profit are given by (37) and (38).
(iii) (9λ − 10 +√81λ2 + 60λ + 100)/30λ < τ/t ≤ (6λ − 5 +√36λ2 + 12λ + 25)/18λIn this case, interior solutions (37) and (44) are both valid. Then, by comparing (38) and (45),
we haveeI − Πe II R 0 if and only if τ/t Q T ,
Π
1
1

(54)

135λ2 + 90λ + 211 + (18λ − 29)X 1/3 + X 2/3,
63λX 1/3

(55)

where
T=

√
√
and X = −63 3(1 + λ) −162λ4 + 486λ3 + 153λ2 + 504λ − 25 + 729λ3 + 4698λ2 + 2889λ +3016. In addition, it can be checked that T satisfies√√
9λ − 10 + 81λ2 + 60λ + 100
6λ − 5 + 36λ2 + 12λ + 25
<T <
30λ
18λ

(56)

for all λ > 0. Therefore, the equilibrium wholesale price and monopolist’s profit are given√by (37) and (38) if (9λ − 10 + 81λ2 + 60λ + 100)/30λ < τ/t ≤ T , and by (44) and (45) if√
T < τ/t ≤ (6λ − 5 + 36λ2 + 12λ + 25)/18λ.
(iv) (6λ − 5 +√36λ2 + 12λ + 25)/18λ < τ/t ≤ 1

By comparing (40) with (45), we have2

2 2

eI − Πe II = − (4t + (9λ − 10)tτ − 15λτ ) < 0.
Π1
1
48τ(t + λτ)(2t + 3λτ)

(57)

Therefore, the equilibrium wholesale price and monopolist’s profit are given by (44) and (45).
25

(v) 1 < τ/t ≤ 2
In this case, the monopolist can obtain positive profit only when it chooses the wholesale
price at which only weak retailers’ price of good 1 is bound. Therefore, as explained in CaseII, the equilibrium wholesale price and monopolist’s profit are respectively given by (44) and(45) if 1 < τ/t ≤ T , and by (47) and (48) if T < τ/t ≤ 2, where T is given by√
5λ − 2 + 25λ2 + 28λ + 4
T=
.
6λ

(58)

Then, the equilibrium under a ban on below-cost pricing can be summarized as follows. If 0 < τ/t ≤T , both retailers’ prices of good 1 are bound (that is, m̃1i = 0 for i = D, W), and the equilibrium
wholesale and retail prices are calculated as follows:3τ(t + λτ);2(t + 3λτ)tτ((3 + 2λ)t + λ(7 + 6λ)τ);p̃1D = c1 ; p̃2D = c2 +2(t + λτ)(t + 3λτ)
3τ(t + λτ)tτ((1 + 2λ)t + λ(5 + 6λ)τ)p̃1W = c1 +; p̃2W = c2 +.
2(t + 3λτ)2(t + λτ)(t + 3λτ)w̃1W = c1 +

(59)
(60)
(61)

If T < τ/t ≤ T , only the weak retailer’s price of good 1 is bound (that is, m̃1D > 0 and m̃1W = 0),
and the equilibrium wholesale and retail prices are obtained as follows:w̃1W = c1 +3τ(t(2 − λ) + 3λτ);2(2t + 3λτ)(62)

−4t2 + (10 − 9λ)tτ + 15λτ2
t(4t2 + (2 + 15λ)tτ + 3λ(1 + 4λ)τ2 ); p̃2D = c2 +; (63)
4(2t + 3λτ)4(t + λτ)(2t + 3λτ)
3τ(t(2 − λ) + 3λτ)tτ((2 + 5λ)t + 3λ(1 + 2λ)τ)p̃1W = c1 +; p̃2W = c2 +.
(64)2(2t + 3λτ)2(t + λτ)(2t + 3λτ)p̃1D = c1 +Finally, if T < τ/t ≤ 2, only the weak retailer’s price of good 1 is bound (that is, m̃1D > 0 andm̃1W = 0), and the equilibrium wholesale and retail prices are given as follows:w̃1W = c1 + 3(τ − t);(65)p̃1D = c1 + 2(τ − t);p̃2D = c2 + t;

(66)

p̃1W = c1 + 3(τ − t);p̃2W = c2 + t.

(67)

It is easy to see that the prices of good 2 set by both retailers are always above their wholesale
prices, that is, m̃2i > 0 for i = D, W. This completes the proof of Proposition 2.
26

Proof of Proposition 3
In the following proofs, we define ∆w1W ≡ w̃1W − w∗1W and ∆pki ≡ p̃ki − p∗ki for k = 1, 2, i = D, W.
(a) From Propositions 1 and 2, we have


3λτ2



−



t + 3λτ







3tλτ
∆w1W = 
−



2(2t
+ 3λτ)
3(2t − τ)



−
2

if 0 <

τ≤ T,
tif T <

τ≤ T,
tif T <

τ
< 2.
t

(68)

It is easy to see that we have ∆w1W < 0 for 0 < τ/t < 2.
(b) For the price of good 1 set by the dominant retailer, we have
2t − 3ττ

if 0 < ≤ T ,



2
t







 4t2 + (3λ − 2)tτ − 3λτ2
τ∆p1D = if T < ≤ T ,



4(2t
+
3λτ)
t






2t − τ
τ
if T < < 2.
−
2
t

(69)

As shown in Figure 3(a), it can be verified that ∆p1D > 0 for 0 < τ/t < T p and ∆p1D < 0 for
T p < τ/t < 2, whereTp =3λ − 2 +√9λ2 + 36λ + 4.
6λSimilarly, for the price of good 1 set by the weak retailer, we have 2
2t + (6λ − 1)tτ − 9λτ2τif 0 < ≤ T ,
2(t + 3λτ)
t






 4t2 + (3λ − 2)tτ − 3λτ2
τ∆p1W = 
if T < ≤ T ,



2(2t
+
3λτ)
t





τ



if T < < 2.
−(2t − τ)
t

(70)

(71)

In the same way as for the dominant retailer’s price of good 1, we can show that we have∆p1W > 0 for 0 < τ/t < T p and ∆p1W < 0 for T p < τ/t < 2.

27

(c) For the price of good 2 set by the dominant retailer, we havet(2t2 + 3(2λ − 1)tτ − 7λτ2 )
τ



−
if 0 < ≤ T ,



2(t
+
λτ)(t
+
3λτ)
t






 t(4t2 + (5λ − 2)tτ − 3λτ2 )τ∆p2D = −
if T < ≤ T ,
4(t + λτ)(2t + 3λτ)
t





τ



if T < < 2.
0
t

(72)

As shown in Figure 3(a), we can show that ∆p2D < 0 for 0 < τ/t ≤ T .
Similarly, for the weak retailer’s price of good 2, we have
τt(2t2 + (6λ − 1)tτ − 5λτ2 )
if 0 < ≤ T ,−
2(t + λτ)(t + 3λτ)
t






2
2
 t(4t + (5λ − 2)tτ − 3λτ )τ∆p2W = 

if
T
<
≤ T,
−



2(t + λτ)(2t + 3λτ)
t





τ



if T < < 2.
0
t

(73)

In the same way as above, it can be verified that ∆p2W < 0 for 0 < τ/t ≤ T .



Proof of Proposition 4
e s − Π∗s for s = 1, D, W.
In the following proofs, we define ∆Π s ≡ Π(a) When below-cost pricing is feasible, the monopolistic manufacturer’s equilibrium profit is
Π∗1 =

3τ
.
8

(74)

On the other hand, when a ban on below-cost pricing is introduced, the monopolistic manufacturer’s equilibrium profit is given by3τ(t + λτ)






8(t + 3λτ)







3τ(t(2 − λ) + 3λτ)2e1 = Π
16(t + λτ)(2t + 3λτ)






3t(τ − t)




2τ

28

if 0 <

τ≤ T,
tif T <

τ≤ T,
tif T <

τ
< 2.
t

(75)

Then, the effect of banning below-cost pricing on the monopolistic manufacturer’s profit canbe calculated as follows:

3λτ2



−



4(t + 3λτ)







 3λτ((4 − λ)t2 − 2(1 − 3λ)tτ − 3λτ2 )∆Π1 = −
16(t + λτ)(2t + 3λτ)







3(2t − τ)2


−
8τ

if 0 <

τ≤ T,
tif T <

τ≤ T,
tif T <

τ
< 2.
t

(76)

For 0 < τ/t ≤ T and T < τ/t < 2, it is easy to see that we have ∆Π1 < 0. For T < τ/t ≤ T ,although the expression is a bit complicated, we can also show that ∆Π1 < 0.
(b) Without a ban, the dominant retailer’s equilibrium profit is given byΠ∗D =4tλ + 9τ.
8

When below-cost pricing is banned, the profit becomestλ(3t + 2tλ + 7λτ + 6λ2 τ)2






8(t + λτ)(t + 3λτ)2






 tλ(4t2 + (2 + 15λ)tτ + 3λ(1 + 4λ)τ2 )2 τ(t(6 + λ) + 5λτ)2eD = Π+32(t + λτ)2 (2t + 3λτ)232(t + λτ)2







t2 − 4tτ + tλτ + 4τ2



2τ

(77)

if 0 <

τ≤ T,
tif T <

τ≤ T , (78)t

if T <

τ< 2.
tAlthough we do not report the full expression of ∆ΠD here, we can show that we have ∆ΠD <0 for 0 < τ/t < 2.
(c) Without a ban on below-cost pricing, the weak retailer’s equilibrium profit isΠ∗W =4tλ + τ.
8

When a ban on below-cost pricing is put into force, the equilibrium profit becomesτtλ(t + 2tλ + 5λτ + 6λ2 τ)2
if 0 < ≤ T ,


2

t
8(t
+
λτ)(t
+
3λτ)





 tλ(2t + 5tλ + 3λτ + 6λ2 τ)2τeW = 
Π
if T < ≤ T ,


2

t
8(t + λτ)(2t + 3λτ)







t(t + λτ)τif T < < 2.
2τ
t
29

(79)

(80)

Because it is a bit messy, we do not provide the full expression of ∆ΠW here. From (79) and(80), as shown in Figure 3(c), we can find a threshold T Π , which satisfies
4tλ + τ tλ(2t + 5tλ + 3λτ + 6λ2 τ)2=,88(t + λτ)(2t + 3λτ)2(81)

and show that ∆ΠW R 0 if and only if τ/t R T Π .

Proof of Proposition 5
gh − CS ∗ for h = 2, B, T .
In the following proofs, we define ∆CS h ≡ CSh(a) From Proposition 3, we have ∆p2i < 0 if 0 < τ/t ≤ T and ∆p2i = 0 if T < τ/t < 2 for
i = D, W. Therefore, it is straightforward to see that we have ∆CS 2 ≥ 0 for 0 < τ/t < 2 and,in particular, ∆CS 2 > 0 if 0 < τ/t ≤ T .
(b) For T < τ/t < 2, Proposition 3 shows that we have ∆p1i < 0 and ∆p2i = 0 for i = D, W.
Therefore, it is easy to see that we have ∆CS B > 0 for this range. For 0 < τ/t ≤ T , when theban on below-cost pricing is not imposed, the type B consumers’ surplus is
CS ∗B = −(c1 + c2 ) −

31τ
.
16

(82)

On the other hand, when the ban is imposed, we have22 2

g B = −(c1 + c2 ) − τ((16λ + 31)t + 12(4λ + 7)λtτ + 21λ τ ) ,CS
16(t + λτ)(t + 3λτ)(83)

if 0 < τ/t ≤ T , and
2

2

2 2

g B = −(c1 + c2 ) − τ((−λ + 20λ + 124)t + 2(11λ + 114)λtτ + 103λ τ ) .
CS
64(t + λτ)2
if T < τ/t ≤ T . Using (82)–(84), we haveλτ(2t2 + (6λ − 5)tτ − 9λτ2 )



−



2(t + λτ)(t + 3λτ)∆CS B = λτ(τ − t)((20 − λ)t + 21λτ)






64(t + λτ)2

if 0 <

τ≤ T,
tif T <

τ
≤ T.
t

(84)

(85)

As shown in Figure 3(b), we can find a threshold value of τ/t such that ∆CS B Q 0 if and only
if τ/t Q 1.
30(c) From the results in (a) and (b), it is easy to see that we have ∆CS T > 0 for T < τ/t < 2. For
0 < τ/t ≤ T , when below-cost pricing is feasible, type 2 consumers’ surplus is"#
5tλ
CS 2∗ = −λ c2 +4
On the other hand, when below-cost pricing is banned, it becomeshg2 = − λ c2 + t(t3 + (4 + 11λ)t2 τ + (39λ2 + 24λ − 1)tτ2
CSi+λ(45λ2 + 36λ − 1)τ3 )(4(t + λτ)(t + 3λτ)2 )−1

(86)

(87)

if 0 < τ/t ≤ T , andhg2 = − λ c2 + t(112t4 + 56(15λ + 2)t3 τ + (2047λ2 + 428λ − 4)t2 τ2CS+ 6λ(340λ2 + 89λ − 2)tτ3 + 9λ2 (80λ2 + 24λ − 1)τ4 )i
×(64(t + λτ)2 (2t + 3λτ)2 )−1(88)

if T < τ/t ≤ T . Using (85)–(88), we can obtain ∆CS T for 0 < τ/t ≤ T . Although we do notreport the full expression of ∆CS T here for the sake of brevity, we can show that ∆CS T > 0


for 0 < τ/t < T .

Appendix B: Extension
B.1

Equilibrium

When below-cost pricing is feasible
Since the dominant retailer’s profit function is given by (5), which is same as in the basic model,
the first-order conditions for the dominant retailer are given by (7) and (8). On the other hand, from
(14) and (15), the first-order conditions for the weak retailers are given as follows:
∂ΠW1
=
∂p1W
∂ΠW2
=
∂p2W

!
1 p1D + p2D − p1W − p2W
p1W − w1W
+
−
= 0,
2
2τ
2τ
!
1 p1D + p2D − p1W − p2W
p2W − w2W
+
−
2
2τ
2τ
!
λ(p2W − w2W )1 p2D − p2W
−
= 0.
+λ +
2
2t
2t

31

(89)

(90)

By solving these four first-order conditions and substituting w1D = c1 and w2D = w2W = c2 , wehave the following retail prices for given w1W :3(w1W − c1 )(t + λτ) − 4t2 + 9λτ2 + tτ(7 − 6λ),8t + 9λτt(4t + 3τ + 9λτ − (w1W − c1 ))p2D = c2 +,8t + 9λτ(91)
6(w1W − c1 )(t + λτ) + 3τ(t(2 − λ) + 3λτ)p1W = c1 +,8t + 9λτt(3τ(2 + 3λ) − 2(w1W − c1 )).
p2W = c2 +8t + 9λτThe monopolistic manufacturer chooses w1W to maximize its own profit, taking into consideration
p1D = c1 +the above retailers’ responses. Substituting the above retail prices into the manufacturer’s profits(11), we have!
1 (2t + 3λτ)(w1W − c1 ) + tτ(2 + 3λ)Π1 = (w1W − c1 ) −.
216tτ + 18λτ2(92)Then, the monopolistic manufacturer optimally sets the wholesale price at
3τ(2t − tλ + 3λτ),
4t + 6λτ(93)

9τ(t(2 − λ) + 3λτ)2.
8(8t + 9λτ)(2t + 3λτ)

(94)w∗1W = c1 +and its equilibrium profit becomesΠ∗1 =In addition, by substituting (93) into (91) and (92), we obtain the equilibrium retail prices as follows:−16t3 + t2 (46 − 57λ)τ + 3t(41 − 15λ)λτ2 + 81λ2 τ3,2(2t + 3λτ)(8t + 9λτ)t(16t2 + 3t(2 + 21λ)τ + 9λ(1 + 6λ)τ2 )p∗2D = c2 +,2(2t + 3λτ)(8t + 9λτ)
3τ(5t2 (2 − λ) + 3t(9 − 2λ)λτ + 18λ2 τ2 )p∗1W = c1 +,(2t + 3λτ)(8t + 9λτ)
3tτ(t(2 + 7λ) + 3(1 + 3λ)λτ)p∗2W = c2 +.
(2t + 3λτ)(8t + 9λτ)p∗1D = c1 +(95)

For good 2, it is easy to see that both retailers’ price-cost margin is always positive, m∗2i = p∗2i − c2 >0. On the other hand, for good 1, the dominant retailer’s price-cost margin becomesm∗1D = p∗1D − c1 =−16t3 + t2 (46 − 57λ)τ + 3t(41 − 15λ)λτ2 + 81λ2 τ3.
2(2t + 3λτ)(8t + 9λτ)(96)

Let T e denote the threshold value of τ/t at which we have m∗1D = 0. As drawn in Figure 5, we havem∗1D < 0 when τ/t < T e .
32

When below-cost pricing is banned
In the extended model, since each weak retailer carries one good, it cannot set the price of its good
below the wholesale price. Therefore, only the dominant retailer can engage in below-cost pricing
in the equilibrium. In the following proof, we first derive an equilibrium under a ban, temporarily
ignoring the constraints for good 2 (p2D ≥ w2D ), and later we check that this constraint is certainly
not binding in that equilibrium. Then, the Lagrange function and Kuhn-Tucker conditions for thedominant retailer are given by (22) and (24)–(28), respectively. On the other hand, the first-orderconditions for the weak retailers are given by (89) and (90).
When the dominant retailer’s prices of good 1 is bound by the wholesale price in equilibrium,from (24), (25), (29), (30), and binding conditions, we have
p1D = c1 ;p2D = c2 +p1W =p2W = c2 +µD =t((w1W − c1 )(t + λτ) + (t(5 + 4λ) + (7 + 6λ)λτ)τ);2(t + λτ)(2t + 3λτ)(97)3(w1W + τ)(t + λτ) + c1 (t + 3λτ);2(2t + 3λτ)(98)

t((t(3 + 4λ) + (5 + 6λ)λτ)τ − (w1W − c1 )(t + λτ));2(t + λτ)(2t + 3λτ)λ(4t2 − (7 − 6λ)tτ − 9λτ2 − 3(w1W − c1 )(t + λτ)).
4(t + λτ)(2t + 3λτ)

(99)
(100)

Substituting the above equations into (11), the monopolistic manufacturer’s profit becomes#
1 (w1W − c1 + τ)(t + 3λτ)
Π1 = (w1W − c1 ) −.
2
4τ(2t + 3λτ)
"

(101)

The monopolistic manufacturer maximizes this profit under the constraint (27), which can be rewritten as
w1W ≤ c1 +4t2 − t(7 − 6λ)τ − 9λτ2.
3(t + λτ)(102)

Assuming an interior solution, we obtain the optimal wholesale price and the manufacturer’s profit
as follows:w̃1W = c1 +e1 =
Π3τ(t + λτ),2(t + 3λτ)

9τ(t + λτ)2.
16(t + 3λτ)(2t + 3λτ)
33

(103)
(104)

e

This is valid as long as w̃1W satisfies (102). Let T denote the value of τ/t at which w̃1W satisfiese(102) with equality. Then, (103) is valid when τ/t ≤ T .
ee

By comparing T e with T , we can find that T e < T for λ > 0. This implies that for T e < τ/t <ee1 .
T , the monopolistic manufacturer chooses w∗1W or w̃1W depending on relative size of Π∗1 and Πe1 −Π∗ directly, we can find a threshold T̂ e ∈ (T e , T e ) such that we have Π
e1 > (<)Π∗
By computing Π11
when τ/t < (>)T̂ e .
Then, the equilibrium under a ban on below-cost pricing can be summarized as follows. If
0 < τ/t ≤ T̂ e , the dominant retailer’s price of good 1 is bound (that is, m̃1D = 0), and the equilibrium
wholesale and retail prices are calculated as follows:w̃1W = c1 +p̃1D = c1 ;p̃2D = c2 +tτ(t2 (13 + 8λ) + 2tλ(25 + 18λ)τ + 9λ2 (5 + 4λ)τ2 );4(t + 3λτ)(t + λτ)(2t + 3λτ)

(105)

(106)

3τ(t + λτ)(5t + 9λτ);4(t + 3λτ)(2t + 3λτ)(107)

tτ(t2 (3 + 8λ) + 2tλ(11 + 18λ)τ + 9λ2 (3 + 4λ)τ2 ).
4(t + 3λτ)(t + λτ)(2t + 3λτ)(108)p̃1W = c1 +p̃2W = c2 +3τ(t + λτ);2(t + 3λτ)It is easy to see that the price of good 2 set by the dominant retailer is always above their wholesale
price, that is, m̃2D > 0. On the other hand, if τ/t > T̂ e , the dominant retailer’s price of good 1 is notbound and the equilibrium under the ban corresponds to that without the ban.

B.2

Effects of banning below-cost pricing

Wholesale and retail prices
From the above analysis, we can derive the changes of the wholesale and retail prices through theban on below-cost pricing as follows:
3λτ(t2 − (4 − 3λ)tτ − 6λτ2 ),2(t + 3λτ)(2t + 3λτ)(109)

16t3 + t2 (57λ − 46)τ + 3tλ(15λ − 41)τ2 − 81λ2 τ3,2(2t + 3λτ)(8t + 9λτ)(110)

3λτ(20t3 + t2 (84λ − 71)τ + 18t(4λ − 11)λτ2 − 135λ2 τ3 ),4(t + 3λτ)(2t + 3λτ)(8t + 9λτ)(111)

∆w1W =∆p1D =∆p1W =

34

∆p2D = −t(32t4 + 2t3 (95λ − 46)τ + t2 λ(348λ − 451)τ2 + 18tλ2 (11λ − 39)τ3 − 351λ3 τ4 )
, (112)4(t + λτ)(t + 3λτ)(2t + 3λτ)(8t + 9λτ)∆p2W = −tλτ(20t3 + t2 (84λ − 71)τ + 18tλ(4λ − 11)τ2 − 135λ2 τ3 ).
4(t + λτ)(t + 3λτ)(2t + 3λτ)(8t + 9λτ)

(113)Then, we can obtain Figure 6(a) and (b).
Profits of firms
For the monopolistic manufacturer, from (94) and (104), we have∆Π1 =9τ((t + λτ)2 (8t + 9λτ) − 2(t(2 − λ) + 3λτ)2 (t + 3λτ)).
16(t + 3λτ)(2t + 3λτ)(8t + 9λτ)(114)

For the relevant parameter range where the ban is binding, it can be found that we have ∆Π1 > 0.
When below-cost pricing is not banned, the equilibrium profits of retailers are obtained as follows:Π∗D =(256t5 λ + 4t4 (676 + 204λ + 513λ2 )τ + 3t3 λ(4588 + 1188λ + 1935λ2 )τ2+ 9t2 λ2 (2884 + 570λ + 765λ2 )τ3 + 81tλ3 (265 + 30λ + 36λ2 )τ4+ 6561λ4 τ5 )(8(2t + 3λτ)2 (8t + 9λτ)2 )−1 ,(115)
9τ(t(2 − λ) + 3λτ)2,Π∗W1 =
8(8t + 9λτ)2Π∗W2 =

9tτ(t + λτ)(t(2 + 7λ) + 3(1 + 3λ)λτ)2.
2(2t + 3λτ)2 (8t + 9λτ)2On the other hand, under the ban on below-cost pricing, we have
2

2

2 2

e D = tτ(t (13 + 8λ) + 2tλ(25 + 18λ)τ + 9λ (5 + 4λ)τ ) ,Π32(t + λτ)(t + 3λτ)2 (2t + 3λτ)22eW1 = 9τ(t + λτ) ,Π32(2t + 3λτ)2

(116)

2

2

2 2

eW2 = tτ(t (3 + 8λ) + 2tλ(11 + 18λ)τ + 9λ (3 + 4λ)τ ) .
Π32(t + λτ)(t + 3λτ)2 (2t + 3λτ)2Although we do not provide the full expression of ∆ΠD , ∆ΠW1 , and ∆ΠW2 , we can obtain Figure6(c) and (d).
35Consumer surplus
When the ban on below-cost pricing is not imposed, type 2 and type B consumers’ surplus arerespectively calculated as follows:h
CS 2∗ = − λ c2 + t(1792t4 + 672t3 (2 + 21λ)τ + 9t2 (4039λ2 + 588λ − 4)τ2
+ 54tλ(714λ2 + 127λ − 2)τ3 + 81λ2 (180λ2 + 36λ − 1)τ4 )i
×(16(2t + 3λτ)2 (8t + 9λτ)2 )−1 ,(117)

τ(t2 (2140 + 228λ − 9λ2 ) + 54tλ(86 + 5λ)τ + 2511λ2 τ2 ).
16(8t + 9λτ)2On the other hand, when the ban is imposed, we havehg2 = − λ c2 + t(16t5 + 32t4 (2 + 7λ)τ + t3 (1188λ2 + 576λ − 25)τ2
CS
CS ∗B = − (c1 + c2 ) −+ t2 λ(2988λ2 + 1872λ − 115)τ3 + 9tλ2 (396λ2 + 288λ − 19)τ4
i+81λ3 (20λ2 + 16λ − 1)τ5 )(16(t + λτ)(t + 3λτ)2 (2t + 3λτ)2 )−1 ,(118)
g B = − (c1 + c2 ) − τ(t (535 + 256λ) + 6t λ(511 + 256λ)τ
CS
43+ 180t2 λ2 (33 + 16λ)τ2 + 54tλ3 (81 + 32λ)τ3 + 837λ4 τ4 )× (64(t + λτ)(t + 3λτ)(2t + 3λτ)2 )−1 .
Although we do not report the full expression of ∆CS 2 and ∆CS B , we can obtain Figure 6(e) and6(f).

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40

Monopolistic
manufacturer

Competitive
fringeDominant
retailerWeakretailer

Consumers

Figure 1: Industrial structure of the model

41

l
1.0

l
1.0

0.8

0.8
∗

0.6

∗

<0

∗

<0

∗

>0

∗

>0

<0

∗

>0

0.4

0.2

0.2

0.5

1.0

1.5

2.0

>0

=0

=0

0.6

0.4

0.0
0.0

=0

t êt
2.5

0.0
0.0

(a) When the ban is not imposed

0.5

1.0

1.5

(b) When the ban is imposed

Figure 2: Equilibrium price-cost margin of good 1

42

t êt
2.0

l
1.0

0.8

0.6

↑

↓

↓

↓

↓

→

0.4

0.2

0.0
0.0

0.5

1.0

1.5

t êt
2.0

(a) Retail prices
l
1.0

l
1.0

0.8

0.8

Π ↓

Π ↓

Π ↓

Π ↓

Π ↓

Π ↑

0.6

0.4

0.6

↑

→

↓

↑

↑

0.4

0.2

0.0
0.0

↑

0.2

0.5

1.0

1.5

t êt
2.0

0.0
0.0

(b) Profits

0.5

1.0

1.5

(c) Consumer surplus

Figure 3: The effect of banning below-cost pricing

43

t êt
2.0

Monopolistic
manufacturer

Competitive
fringeWeakretailer 1

Dominant
retailerWeakretailer 2

Consumers

Figure 4: Industrial structure of the extended model

44

l
2.0

∗

>0

1.5

=0
∗

<0
=0

1.0

0.5
∗

0.0
0.0

0.1

0.2

0.3

0.4

=

>0

t êt
0.5

Figure 5: Equilibrium price-cost margin of good 1 sold by the dominant retailer

45

l
2.0

l
2.0

↓
↓
1.5

1.5

↑

1.0

1.0

↑

↓

↓

↑

↑

↑

↓

↓

↓
0.5

0.5

↓
↑

0.0
0.0

0.1

0.2

0.3

0.4

t êt
0.5

0.0
0.0

0.1

(a) Wholesale price
l
2.0

1.5

1.5

Π ↑

Π ↓

0.3

t êt
0.5

0.4

(b) Retail prices

l
2.0

1.0

0.2

1.0

Π

↑

Π

↓

Π ↓
0.5

0.5

0.0
0.0

0.1

0.2

0.3

0.4

t êt
0.5

0.0
0.0

(c) Dominant retailer’s profit

0.2

0.3

↑

Π

↑

0.4

l
2.0

1.5

1.5

↑

↓

1.0

↓

↑
0.5

0.5

0.0
0.0

t êt
0.5

(d) Weak retailers’ profits

l
2.0

1.0

0.1

Π

0.1

0.2

0.3

0.4

t êt
0.5

0.0
0.0

(e) Type 2 consumers’ surplus

0.1

0.2

0.3

0.4

(f) Type B consumers’ surplus

Figure 6: The effect of banning below-cost pricing in the extended model
46

t êt
0.5