OO

F

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

Table 1Price chart (as of date: October 6 2005)

Category Product Amazon.com ($) BarnesNoble.com ($) BN local store ($)

BOOK The World is Flat 16 16.5 19.25Devinci Code 14.5 14.97 17.47Introduction to Probability Model 89.95 89.95 89.95A Course in Game Theory 35 35 35A Million Little Pieces 8.97 10.46 11.96

DVD Why Harry Met Sally 11.21 11.45 14.99The Matrix 14.97 17.98 19.99The Sound of Music 18.89 21.58 26.99

Amazon.com BELK

HOME Full Ultimate Memory Foam Mattress Topper 129.95 130King Ultimate Memory Foam Mattress Topper 189.95 170Corning French White 12-pc Bake and Serve Set 39.99 49.99Hamilton Beach Iced Tea Maker 24.99 19.99Shaper Image Air Purifier Ionic Breeze 3.0 249.99 249.99Euro-pro Sewing Machine Shark 99.99 99.99

2 W. Huang, J.M. Swaminathan / European Journal of Operational Research xxx (2007) xxx–xxx



UN

CO

RR

EC

TED

PRpure e-tailers for DVD titles is 14% lower than those charged by e-tailers with traditional channels. Ancarani and Shankar

[1] develop hypotheses on how prices and price dispersion compare among pure-play Internet, bricks-and-mortar, andbricks-and-clicks retailers and test them through an empirical analysis. Their results, based on an analysis of 13,270 pricequotes, show that when posted prices are considered, traditional retailers have the highest prices, followed by multichannelretailers, and pure-play e-tailers. (2) In a recent research, Cattani et al. [9] study four prevalent pricing strategies for amonopolist under varying degree of autonomy for the Internet channel and find through a detailed computational studythat an identical pricing strategy may indeed be very close to the optimal. (3) We did our own survey of the pricing of asmall subset of popular products. As shown in Table 1, the actual pricing strategy for these items shows a wide dispersion.For example, some products are priced identical by competitors (like Introduction to Probability Model and A Course inGame Theory) whereas for some others the price on the Internet channel offered by a firm that has both a traditional andInternet channel are higher than those offered by a pure Internet player (like World is Flat or Devinci Code). For productssold by two pure competing players, we find that prices could either be higher or lower (like Corning Set and HamiltonBeach Tea Maker). The above pieces of evidence motivated us to explore the possibility of theoretical bounds and optimalpricing in monopoly and duopoly environments with retailers having pure and mixed channels.

As opposed to the micro-level demand model considered in Cattani et al. [9], in this paper we use a stylized lineardemand function that depends on price of the product in the channels as well as on the degree of substitutability acrossthe two channels. First, we analyze a monopolistic retailer who opens a new Internet channel and consider four pricingstrategies with different degrees of autonomy for the Internet channel. We theoretically show that under mild conditionsweb price to optimize the joint profit (while holding the traditional price) is at most 4% from the optimal. If the costs acrossthe two channels are identical then this bound is at most 3% away from optimal and, the profit under the identical pricing isat most 4% away from optimal. Next, we consider the case where the Internet channel is introduced by a different firm, andshow the existence of a unique Nash equilibrium for the cases where the incumbent has a pure traditional channel or acombination of Internet and traditional channel. We develop conditions under which internet price of the firm with bothchannels is greater or less than the price of the pure e-tailer. Through a limited computational study, we explore the behav-ior (price and profits) of the above models under different parameters and consumer preference for the alternative channels.Finally, we validate our results for a variant of the demand model that has also been well studied in literature. In the pro-cess we provide managerial insights on the following questions (1) how sub-optimal are some of the prevalent pricing pol-icies? (2) What happens to optimal prices and profits when a new channel is introduced under monopoly and competition?(3) When do mixed channel firms price goods above the pure firm?

The rest of paper is organized as follows. In Section 2, we discuss relevant literature. In Section 3, we study the monop-olistic setting under four different pricing strategies. In Section 4, we study duopoly with a new entrant. We study a variantof the demand model in Section 5 and conclude in Section 6.

2. Related literature and our contribution

In the last few years, researchers have begun to study issues related to electronic supply chains [31]. Cattani et al. [8]provide a comprehensive review of models which deal with coordination of traditional and Internet channels under monop-oly. Tsay and Agarwal [33] provide a complete review of the same issues under competition in the multi-channels.


wehuan

Cross-Out

wehuan

Replacement Text

2

swaminaj

Inserted Text

using the

swaminaj

Inserted Text

(see Swaminathan and Tayur [30])

swaminaj

Cross-Out

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

W. Huang, J.M. Swaminathan / European Journal of Operational Research xxx (2007) xxx–xxx 3



UN

CO

RR

EC

TED

PR

OO

F

There is a volume of literature in marketing on channel management. Most of these models consider vertical competi-tion and channel coordination. Choi [11] focuses on the intra- and inter-channel price competition and its implication onthe channel structure and profits. Ingene and Parry [20] focus on the case of a single manufacturer selling an identical prod-uct to two competing retailers. They examine the implications of two simple wholesale pricing strategies: a linear quantitydiscount schedule and a two-part tariff, and show the circumstances where a manufacturer prefer different pricing strate-gies. Coughlan [12], Moorth [26], Coughlan and Wernerfelt [13], and Gupta and Loulou [19] study the possibilities wherethe manufacturers are better off using the intermediaries instead of vertically integrating. More recently, Cattani et al. [10]study price coordination issues when a manufacturer opens a direct internet channel. As opposed to the above models, ourfocus is on horizontal competition.

Horizontal competition has been studied in inventory theory by Parlar [28], Lippman and McCardle,[23] and Bernsteinand Federgruen [4]. Parlar [28] and Lippman and McCardle [23] study a pair of ‘‘newsvendor” firms who become compet-itors because their products are partially substitutable, while Bernstein and Federgruen [4] examine an oligopoly in whichsales are awarded based on the competitors’ relative selling prices and fill rates. As opposed to the above papers that focuson a single channel our focus is on studying such competition in dual channels.

Some of the recent related work in the area of managing dual channels include Reinhardt and Levesque [29], Bernstein et al.[5], Lal and Sarvary [22], Zettelmeyer [35], Balasubramanian [3], Druehl and Porteus [15], Cattani et al. [9] and Pan et al. [27].The first four papers are focussed on the interaction effects across the two channels. So the primarily concern in those papers ishow the introduction of a new channel (such as an internet channel) may provide an opportunity for customers to search forthe product on either of the two channels and that eventually could lead to a purchase on the other channel. Our model doesnot take such interactions into account in that, we do not capture the cases where a customer could use the information aboutthe product on the Internet channel to refine their decisions of purchase on the traditional channel or vice versa. We (and thelast four papers) assume that the presence of another channel only acts as a substitute avenue for purchase.

Balasubramanian [3] models competition between direct marketers and conventional retailers. His model considers inelas-tic demand with customers uniformly distributed on a circle, retailers evenly spaced around the circle, and the direct marketerin the center. All retailers have fixed costs and customers of the direct channel incur a monetary disutility that may be of anyvalue. Among other issues, his focus is on the role of information in multiple-channel markets and he shows that even withzero information costs, providing information to all consumers may not be optimal, especially when the product is not welladapted to the direct channel. In that case, high market coverage (from having a greater number of retailers) may depressprofits. Druehl and Porteus [15] consider price competition between a traditional and an internet retailer. They adopt a utilitybased demand model and assume that only some proportion of the market has access to the Internet and is willing to buyonline. They find the Nash and Stackelberg equilibrium prices and show that a pure Nash equilibrium may not exist in somecases. Cattani et al. [9] study coordination of pricing on Internet and traditional channels by modeling micro-level consumerbehavior for demand generation. In their model, customers are at a random physical distance from traditional retailers, and ata random virtual distance from the direct marketer, independent of the physical distance. The market then is segmentedaccording to the utility each customer attains from either the direct channel or the traditional channel. Customers are notexcluded from a specific market; thus both markets have a chance to compete for all customers. Pan et al. [27] consider pricecompetition between a traditional retailer and an e-tailer, and a dual channel retailer and Internet-only firm, based on theHotelling model. They show that the price at a pure e-tailer is lower than that of a dual channel firm. However, the mainassumption of the above paper is that they assume both the stores have the same marginal cost c, and the demand functionis only differentiated by the transaction costs and prices. Further, when they compare the Nash equilibrium prices for the puree-tailer and the brick-and-clicks retailer, they assume that the bricks-and-clicks retailer sets the same price at its two stores.

In contrast to the four papers above, one of the main aims in this paper is to provide theoretical insights on the nearoptimality of some of the prevalent pricing strategies (non-optimal). In particular, we use the demand model introduced byMcGuire and Staelin [25] where the demand depends linearly on the prices on the different channels and the degree of sub-stitutability across them. For the case when there are three possible channels for purchase (a mixed channel competing witha pure Internet channel), we utilize an extension similar to one developed by Granot and Sosic [18]. Further, we do notrestrict ourselves to identical unit costs and prices, and characterize the optimal price and profits over a range of param-eters. The main contributions of our work include – theoretical bounds on performance of sub-optimal yet prevalent pric-ing strategies; analysis of the implication of new alternative channel on prices and profits; and the effect and benefits ofhaving a mixed channel under competition. It is to be noted that in the rest of the paper we will assume that the incumbentpure player is a traditional retailer and the new channel is an Internet channel but since the model is symmetric all theresults should hold if other alternatives to an Internet channel are considered.

3. Single firm: Introduction of a new channel

In this section we first introduce the demand model and then consider four different pricing strategies and finally providebounds on performance.


wehuan

Cross-Out

wehuan

Replacement Text

19

wehuan

Cross-Out

wehuan

Replacement Text

8

wehuan

Cross-Out

wehuan

Replacement Text

5

wehuan

Cross-Out

wehuan

Replacement Text

7

wehuan

Cross-Out

wehuan

Replacement Text

2

wehuan

Cross-Out

wehuan

Replacement Text

7

wehuan

Cross-Out

wehuan

Replacement Text

2

wehuan

Cross-Out

wehuan

Replacement Text

8

wehuan

Cross-Out

wehuan

Replacement Text

1

wehuan

Cross-Out

wehuan

Replacement Text

4

wehuan

Cross-Out

wehuan

Replacement Text

4

wehuan

Cross-Out

wehuan

Replacement Text

6

wehuan

Cross-Out

wehuan

Replacement Text

4

wehuan

Cross-Out

wehuan

Replacement Text

6

wehuan

Cross-Out

wehuan

Replacement Text

4

wehuan

Cross-Out

wehuan

Replacement Text

7

129

130

131

132133135135

136

137

138

140140

141

142

143

144145

147147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

177177178

179

180

181

182

183




3.1. Demand model

We first assume there is one firm that sells products only through the traditional channel. Let qT denote the demand ofthis firm when its price is at pT, and let cT denote its cost, and gT denote its profit. We assume a linear downward slopingdemand function. The demand and profit are given as below.

Pleaits, E

qT ¼ Sð1� bpTÞ; gT ¼ qTðpT � cTÞ; ð1Þ
where b and S are positive real values. The constant S is a scale factor equal to industry demand or market potential and bis the price elasticity factor. In such a setting, the firm decides the optimal price p�T by maximizing its profit. The optimalprice p�T and the optimal profit g�T are given by
Fp�T ¼1þ cTb

2b; g�T ¼

Sð1� cTbÞ2

4b: ð2Þ

OONow consider the case where the firm begins to sell products through another channel (like the Internet). For example, a

firm such as Wal-Mart decides to open walmart.com. We utilize the demand function given by McGuire and Staelin [25]for two channels. Let qT denote the demand for the traditional channel, and qW denote the demand for the Internet chan-nel. Then

RqT ¼ uS 1� b1� h

pT þbh

1� hpW

� �; qW ¼ ð1� uÞS 1þ bh

1� hpT �

b1� h

pW

� �; ð3Þ

NC

OR

REC

TED

Pwhere 0 6 u 6 1; 0 6 h < 1, and b and S are positive. All the proofs for this paper are in Appendix.The parameters u and h capture two different aspects: the absolute difference in demand and the substitutability of the

end products as reflected by the cross-elasticities. The parameter u captures the market share. For example in the case oftraditional and internet channel, if the population density is very high near a retail outlet, then u (the market share for thetraditional channel) is likely to be higher all other factors being constant. On the other hand, in a remote place with lowerpopulation density the market share u of the traditional channel is expected to be lower. Similarly, if the population is dom-inated by families with high Internet literacy and dual career with high premium for time, then 1� u (the market share forthe Internet channel) can be expected to be higher. However, it is quite possible that a high population density such as acity may also have a higher internet literacy in which case their relative strengths will determine which channel will havemore market share. The parameter h, in contrast, affects the substitutability of the product across the two channels in termsof changes in prices. More specifically, h is the ratio of the rate of change of quantity with respect to the price on the otherchannel to the rate of change of quantity with respect to price on the own channel. In our setting, h captures characteristicsrelated to the type of product that is sold. For example, consumers prefer to buy clothes at retail stores, but are more ame-nable to buy computer or electronics product on the Internet. Therefore, the substitutability factor h can be expected to belower for clothes industry than for electronics. Purchase decisions are also affected by factors such as immediate availabilityand savings in traveling and purchase time. A combination of the h and u can capture these effects. As indicated earlier, thismodel does not capture the cross interactions in purchase decisions like an individual first searching the Internet and thenmaking the purchase on the traditional channel. Despite the above shortcoming we believe this model captures severalinteresting dynamics related to managing dual channels and lends itself to theoretical analysis and characterization suchas worst case bounds on prevalent (yet non-optimal) pricing policies.

In order to utilize the above model, it is necessary to impose additional inequality constraints on these parameters inorder to guarantee that (1) prices exceed marginal costs, (2) quantities are nonnegative, and (3) industry demand doesnot increase with increase in prices for either product. McGuire and Staelin [25] derive the conditions given below toaccomplish the same.

Constraint 1: quantities nonnegative: cTb 6 1 and cWb 6 1Constraint 2: negative coefficients of prices in the quantities functions
U
h 61� u

uand h 6

u1� u

� �$ h

1þ h6 u 6

1

1þ h

� �: ð4Þ

3.2. Alternative pricing strategies

Ernst and Young [17] reported that the need for organizational discipline might prompt retailers to consider setting up aseparate online entity to manage itself. In particular they found that in the United States, 27% retailers fully integrated theirtraditional and Internet operations, 53% of Internet operations are semi-autonomous, and 20% are totally autonomous.They conclude that there are no clear best practices in place. Further, coordinating prices on traditional and Internet chan-

se cite this article in press as: W. Huang, J.M. Swaminathan, Introduction of a second channel: Implications for pricing and prof-uropean Journal of Operational Research (2007), doi:10.1016/j.ejor.2007.11.041

wehuan

Cross-Out

wehuan

Replacement Text

4

wehuan

Cross-Out

wehuan

Replacement Text

4

wehuan

Cross-Out

wehuan

Replacement Text

6

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

210210

211

213213

214

216216217

218

219

220

221

222

224224

225

226

227

228

230230




OO

F

nels has been a challenging task for many firms. The natural inclination for many firms is to price products on the web atthe same price as their traditional retailers [2, 24]. However, evidence exists of companies charging both higher and lowerprices on their Internet channel [21, 30].

We consider four prevalent pricing strategies when a retailer adds an Internet channel to an existing traditional channelbased on Cattani et al. [9]. These strategies differ in the degree of autonomy given to the Internet channel. In the first strategy,the retailer optimizes profit on the Internet channel by choosing an independent price for that channel without adjusting theprice on the traditional channel. This strategy is suitable for retailers who have a very large traditional retail business and thenew internet channel is very small in comparison. The logic behind such a strategy is that the firm may not want to makemajor changes on its traditional channel due to the presence of the Internet channel. In the second strategy, the retailer findsthe price which optimizes the combined profit on traditional and Internet channels without adjusting the price on the tra-ditional channel. In this case, the autonomy to the Internet channel is lower and they are somewhat forced to price the prod-uct for the benefit of the whole firm. In the third strategy, the retailer chooses an identical price across the two channelswhich optimizes the combined profits. Such a strategy has been followed by retailers such as Best Buy who want to providea consistent seamless shopping experience for the consumer. Clearly, the autonomy while choosing prices on the Internetchannel is further limited in the case. In the fourth strategy, the retailer chooses prices across the two channels which opti-mize the combined profits. Strategy one and strategy two minimize disruptions to the traditional channel, however they maybe sub-optimal from the profit standpoint. Strategy three (a popular practice in industry) prices the product identicallyacross the two channels, thereby minimizing cannibalization of demand due to differential pricing and maintaining thebrand. Strategy four is the theoretical optimal solution however it affects pricing on both channels.
R
DP3.2.1. Pricing strategy I

The firm chooses the price on the traditional channel which maximizes its own profit in (1), and decides the optimal pricefor the Internet channel only by maximizing the profit from the Internet channel. The policy of profit maximization overthe web channel without consideration of the traditional channel could arise if the web channel were managed separatelyfrom the traditional operations, as recommended by some experts (e.g., [6]). In this case, the objective profit function forthe Internet channel is given by

Pleaits, E

E

gW ¼ ð1� uÞS 1þ ð1þ cTbÞh2ð1� hÞ �

bð1� hÞ pW

� �ðpW � cWÞ; ð5Þ
T
gW is concave with respect to pW, and the optimal price for the Internet channel and the corresponding profit are given by
C
p�W ¼2ð1þ cWbÞ � hð1� cTbÞ

4b; g�W ¼

Sð1� uÞð2ð1� cWbÞ � hð1� cTbÞÞ2

16bð1� hÞ : ð6Þ
E
Given the Internet channel optimal price, the profit for the retailer earned on the traditional channel is
Rg�T ¼Suð1� cTbÞð2hð1� cWbÞ � ð2� h2Þð1� cTbÞÞ
8bð1� hÞ : ð7Þ

CO

R

3.2.2. Pricing strategy II

The firm does not change the price of the product sold on the traditional channel, but it sets the price for the productsold through the Internet channel by maximizing the profit of the whole firm. For any set of parameters, this policy obvi-ously will result in total profits (across both channels) that are greater than or equal to the total profits achieved underpricing strategy I. In this case, the objective is to maximize the total profit given as follows:

NgW þ gT ¼ ð1� uÞS 1þ ð1þ cTbÞh2ð1� hÞ �

bð1� hÞ pW

� �ðpW � cWÞ þ uS 1� 1þ cTb

2ð1� hÞ þbh

1� hpW

� �1þ cTb

2b� cT

� �: ð8Þ

ULemma 1. The profit of the whole firm ðgW þ gTÞ is concave with respect to price for the Internet channel ðpWÞ.

The optimal price and profit for the firm on the traditional and Internet channel are computed as

p�W ¼2ð1þ cWbÞð1� uÞ � hð1� cTbÞð1� 2uÞ

4ð1� uÞb ;

g�W ¼Sð2ð1� cWbÞð1� uÞ � hð1� cTbÞÞð2ð1� cWbÞð1� uÞ þ hð1� cTbÞð2u� 1ÞÞ

16ð1� uÞbð1� hÞ ;

g�T ¼Suð1� cTbÞð2ð1� cTbÞð1� uÞ � 2hð1� cWbÞð1� uÞ � h2ð1� cTbÞð1� 2uÞÞ

8ð1� uÞbð1� hÞ :

ð9Þ


wehuan

Cross-Out

wehuan

Replacement Text

3

wehuan

Cross-Out

wehuan

Replacement Text

0

wehuan

Cross-Out

wehuan

Replacement Text

29

231

232

233

234

235

237237

238

239

240

241

243243

244

246246

247

248

249

250

252252

253

254

255

256

258258

259

260261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280




3.2.3. Pricing strategy III

In this case the retailer charges the same price in both channels. Although setting pT ¼ pW adds a constraint to the opti-mization problem (thereby potentially reducing profits), it has the benefit of avoiding channel conflict and perceived con-sumer inequality. In this case, the firm utilizes the same price ðpT ¼ pW ¼ pÞ for the traditional and Internet channel whichoptimizes its total profit given as follows:

Pleaits, E

gT þ gW ¼ Sð1� pbÞðp þ cWð�1þ uÞ � cTuÞ: ð10Þ

Lemma 2. The total profit of the firm is concave with respect to pT ¼ pW ¼ p.

The identical optimal price for the tradition and Internet channels is

Fp�T ¼ p�W ¼1þ cWbþ cTub� cWub

2b: ð11Þ

The corresponding profits of the two channels are given by

OO

g�T ¼Suð1þ cTð�2þ uÞb� cWð�1þ uÞbÞð1þ cWð�1þ uÞb� cTubÞ

4b;

g�W ¼Sð�1þ uÞð1þ cWð�1þ uÞb� cTubÞð�1� cTubþ cWð1þ uÞbÞ

4b:

PR

3.2.4. Pricing strategy IV

In this case, the firm maximizes the total profits without constraining the prices. The total profit for the firm is given asfollows:

DgT þ gW ¼S

�1þ hðbp2

Wð1� uÞ � ðcT � pTÞuð�1þ pTbþ hÞ þ cWð�1þ uÞð�1þ pWbþ h� pTbhÞ þ pWð�1

þ uþ h� uh� pTbhþ cTubhÞÞ: ð12Þ

TE

Lemma 3. When 4u� 4u2 � h2 > 0, the total profit of the firm is jointly concave with respect to pT and pW.

The optimal prices for the traditional and Internet channels are given by

ECp�T ¼

2uð1þ cTbÞðu� 1Þ � ð1� cWbÞhð2u� 1Þðu� 1Þ þ h2ð1� uþ ucTbÞbð�4uþ 4u2 þ h2Þ

;

p�W ¼2uð1þ cWbÞðu� 1Þ þ uhð1� cTbÞð1� 2uÞ þ h2ðcWbþ u� ucWbÞ

bð�4uþ 4u2 þ h2Þ:

ð13Þ

CO

RR3.3. Prices and profits

Proposition 1. The Internet channel optimal price is less than the traditional channel optimal price ðp�W < p�TÞ under thedifferent pricing strategies as follows:

(i) Strategy I: p�W < p�T when cT P cW or cT < cW and cW < bð2�hÞcTþh2b ;

(ii) Strategy II: p�W < p�T when cT P cW and u < 1=2;

(iii) Strategy III: p�WðIIIÞ < p�TðI ; IIÞ when cT P cW;

(iv) Strategy IV: p�W < p�T when cT P cW and u < 1=2.

UN

3.3.1. DiscussionFrom the above proposition it is to be noted that the Internet channel price p�W is less than the traditional price p�T in

most cases when cW < cT. For strategy I, p�W < p�T even when cW > cT as long as it not too large. Note that bð2�hÞcTþh2b is

increasing in h due to constraint 1 ðcTb < 1Þ. Therefore, as h increases, i.e., when consumers feel products sold on the Inter-net channel are more substitutable, p�W is more likely to be less than p�T, even when cW P cT. For strategy II, the aboveresult shows that it is worthwhile to price lower on Internet channel only if it has larger market share and sufficiently lowerproduction cost in order to maximize the total profits for the firm. The above result for strategy III shows that if the cost ofselling products on the traditional channel is higher than the cost of selling the product on the Internet channel, then theoptimal equal price is still lower than the original traditional channel price. Thus, even in a monopolistic situation, con-sumers may benefit from the introduction of a new channel if the Internet channel is less expensive to operate than thetraditional channel. In this case, the presence of an Internet channel helps in subsidizing the price on the traditional


281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

311311

312

313

314

315

316

317

318

319

320

321




channel. For strategy IV it is worthwhile to price lower on Internet channel if it has larger market share and sufficientlylower production cost in order to maximize the total profits for the firm.

Next we provide theoretical bounds for the profit for strategies II and III as compared to the optimal solution undermild assumptions.

Theorem 1. The ratio of the total profit of the firm under pricing strategy II to that under pricing strategy IV is at least 96%.

Theorem 2. When the costs in the two channels are identical ðcW ¼ cTÞ, the ratio of the total profit of two channels under pric-

ing strategy II to that under strategy IV is at least 97%.

Theorem 3. Under the assumption cT ¼ cW ¼ c, the ratio of the total profit of two firms in pricing strategy III to that in pricingstrategy IV is at least 96%.
F
REC

TED

PR

OO3.3.2. Discussion

Theorems 1–3 provide some interesting insights that may be influencing prevalent pricing practices. As indicated earlier,many firms when they introduced their Internet channel, tended to give semi-autonomy to the Internet channel in that theInternet channel price was set assuming minimal changes to the traditional channel. Our results indicate that if a firm keptthe same price on the traditional channel and optimized the firm profit with the Internet channel price (strategy II) ormatched prices on traditional and Internet channels (strategy III) then in the worst case the profits of the firms are no morethan 4% away from the maximum profits. This has an important implication for managers because changing prices on twochannels is onerous and has large implications and influences on customers. Our results show that semi-autonomy to theInternet channel (via strategy II or III) maybe a very pragmatic non-optimal strategy for managers in practice. This may beone of the reasons as why to 53% of retailers (operating under monopoly and competition) in practice have adopted a semi-autonomous structure as indicated in the Ernst and Young report.

4. Two firms: Introduction of a new channel

In this section, we study the duopoly setting where the Internet channel is introduced by a different firm. We first con-sider the case where the incumbent has a pure traditional channel and then the case where the incumbent has both a tra-ditional and Internet channel.

4.1. Incumbent with a traditional channel

In the case where there is an incumbent with a traditional channel and an entrant with an Internet channel, this problemmathematically is identical to the problem studied by McGuire and Staelin [25] who show the existence of a Nash equilib-rium in a different context.

Lemma 4. There exists a Nash equilibrium where optimal prices are given by

Pleaits, E

CO

R

p�NT ¼�2� 2cTbþ h� cWbhþ h2

bð�4þ h2Þ;

p�NW ¼�2� 2cWbþ h� cTbhþ h2

bð�4þ h2Þ:

ð14Þ

UNIn addition, we can also show the following result regarding concavity.

Proposition 2. The optimal prices of the two firms in the Nash equilibrium are concave decreasing functions with respect to h,

the substitutability factor between the two firms’ products.

The equilibrium prices are decreasing concavely in h because the channels become more substitutable as h increases.Therefore, the competition intensifies and prices get pushed down at a faster rate.

We now provide analytical insights on the effect of the new entrant on incumbent profits. In particular, we consider acase where the new entrant brings an additional set of customers to the market. Let S represent the whole market whenboth firms are present and S1 represent the market when only the first firm is present. Further let the market share ofthe incumbent be u after the new e-tailer being introduced into market. Then we assume that S1 ¼ Su, i.e., the introductionof a new player does not affect retailers share.


wehuan

Cross-Out

wehuan

Replacement Text

4

322

323

324

325

326

327

328

330330

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

346

347

349349

350

351

352

353

354

355

356

357

359359

360

361

363363

364

365




Theorem 4.

(i) The ratio of the incumbent firm’s profit before and after the presence of new e-tailer is decreasing with cT and increasing

with cW.

(ii) The incumbent firm’s profit before the introduction of new e-tailer is greater than that after introduction when cT < cW

and

Pleaits, E

1� cWb < ð1� cTbÞ 4� 2h2 � ð4� h2Þffiffiffiffiffiffiffiffiffiffiffi1� hp

2h

!: ð15Þ

OO

FCounter to normal intuition, the above proposition shows that the competition from the new e-tailer can in fact be prof-itable for the incumbent traditional-only firm when the Internet channel costs are substantially higher than the traditional

channel costs since the expression 4�2h2�ð4�h2Þffiffiffiffiffiffi1�hp

2h

� �6 1. On a closer look, we find that indeed the new Internet channel

helps the incumbent because it brings along with it a new set of customers (since S1 ¼ Su) thereby increasing the potentialmarket. However, once those customers are in the market they switch to the traditional channel because of channel choiceand lower prices on the traditional channel (resulting from substantially lower cost). If the customers are assumed to becompletely loyal to the Internet channel then one would not observe this result.
R
TED

P4.2. Incumbent with a mixed channel

In this section, we study the case where the incumbent firm has both traditional and Internet channels and faces com-petition from a new Internet firm. The demand functions obtained are natural extensions of the demand functions usedearlier. The incumbent firm sells products through two channels, the traditional and Internet channels. The new entrantsells products only through its own Internet channel. We consider the three channels among the two firms as three distinc-tive players. Assume that there is a central decision maker for the two channels in the incumbent firm which aims to max-imize its total profit. There is a separate decision maker for the new entrant firm which aims to maximize its own profit. Let1, 2 denote the Internet and traditional channel of the incumbent firm, respectively. Let 3 denote the Internet channel of thenew entrant. In such a setting, the deterministic and linear demand functions, introduced by Granot and Sosic [18] aregiven by
C
qi ¼ uiS 1� b1

ð1� hijÞð1� hikÞpi þ b

hijð2� hikÞ2ð1� hikÞð1� hijÞ

pj þ bhikð2� hijÞ

2ð1� hikÞð1� hijÞpk

ð16Þ
E
OR

Rwith i; j; k 2 f1; 2; 3g such that i 6¼ j 6¼ k; 0 6 u1; u2; u3 6 1; u1 þ u2 þ u3 ¼ 1; b > 0; S > 0, and 0 6 hij < 1. Here let S repre-sent a scale factor corresponding to the industry demand when the prices of all products are set to zero, while the u’s and h’srepresent two aspects of product differentiation – the absolute difference in demands and the substitutability of two prod-ucts, respectively. When hij ¼ 0, product i and j have independent demands; product substitutability increases with hij, andthey become highly substitutable as hij ! 1.

It is necessary to impose additional inequality constraints on these parameters in order to guarantee (1) prices exceedmarginal costs; (ii) quantities are nonnegative; (iii) industry demand must not increase with increases in prices.

Constraint 1: quantities nonnegative:
C
2ð1� hijÞð1� hikÞ � 2bpi þ bhijð2� hikÞpj þ bhikð2� hijÞpk P 0 ð17Þ
N
when i; j; k 2 f1; 2; 3g; i 6¼ j 6¼ k.Constraint 2: negative coefficients of prices in the quantities functions
U
u1ð2þ ð�1þ h12Þh13ð�2þ h23Þ � 2h23Þ þ ðu2ðh12 � h13Þ þ h13 � h12h13Þð�2þ h23Þ > 0;

ð�2þ h13Þðu1ðh12 � h23Þ þ h23 � h12h23Þþ u2ð2þ 2h23 � 2h12h23 þ h13ð�2þ ð�1þ h12Þh23ÞÞ > 0;

2� 2h12 þ u1ð�2þ 2h13ð�1þ h23Þ þ h12ð2þ h13 � h13h23ÞÞþ u2ð�2þ 2ð�1þ h13Þh23 þ h12ð2þ h23 � h13h23ÞÞ > 0:

ð18Þ

We must guarantee that the whole profit function of the incumbent firm is jointly concave with respect to p1 and p2 in orderto analyze the Nash equilibrium.


wehuan

Cross-Out

wehuan

Replacement Text

7

366

367

368

369

371371

372

373

374

375

376

377

378

379

380

381

382

383

384

385

386

387

388

389

390

391

392

393

394

396396

397

399399

400

402402



Constraint 3: condition to preserve concavity
2 2

Pleaits, E

�h12ðu2ð�1þ h13Þð�2þ h23Þ þ u1ð�2þ h13Þð�1þ h23ÞÞ þ 16u1u2ð�1þ h13Þð�1þ h23Þ > 0: ð19Þ

ED

PR

OO

F

In the following passages, we will provide theoretical insights on how changes in costs and substitutability factors affectoptimal pricing and profits. We will consider symmetric cases where most parameters are identical and asymmetric caseswhere firms may have differential costs, substitutability and markets shares across channels.

4.2.1. Effect of cost and substitutability on prices

Under the above conditions (constraints 1–3), we consider the game beginning with both the incumbent and entrantfirms simultaneously choosing prices, where pN1; pN2 are the prices of the two channels of the incumbent firm, and pN3

is the price of the Internet-only entrant firm. We can show that there exists a unique Nash equilibrium in this game underthe above assumptions. In this section, we first characterize the equilibrium prices for some special settings where we canprove a definite ordering of prices based on cost and substitutability. Then we use numerical results to demonstrate theireffect under general settings.

Identical substitutability factors: Let h12 ¼ h13 ¼ h23 ¼ h in this case.

Proposition 3. If the costs of the products from the two firms are identical ðc1 ¼ c2 ¼ c3 ¼ cÞ and the substitutability factor

across the three players are identical ðh12 ¼ h23 ¼ h13 ¼ hÞ, the Nash equilibrium price for the Internet channel for the

incumbent firm is always greater than the price for the Internet-only new entrant firm ðpN3 < pN1Þ.

Proposition 4. If we assume that c1 ¼ c2 ¼ c; h12 ¼ h13 ¼ h23 ¼ h; u1 ¼ u2 ¼ u, there exists a c3 ¼ cbð2�2hþh2Þ2þð2�hÞð�1þhÞ2hbð4�6hþ3h2Þ such

that if c3 > c3, then the Nash equilibrium price of the Internet-only entrant firm is greater than that of the incumbent firm

ðpN3 > pN1Þ.

Under identical substitution across the three channels, it appears that unless the cost for the entrant is significantly high(greater than c3), the incumbent is likely to have higher prices on the Internet channel. This may provide some theoreticalvalidation as to why Tang and Xing [32] empirically found that the price charged by pure e-tailers for DVD titles is 14%lower than those charged by e-tailers with traditional channels. Although it is important to note that not all retailers intheir sample were operating in a duopoly market. In cases where the costs across the three channel are non-identical thenthe characterization is more complex. The difference between the prices pN1 � pN3 is given as follows.
T
pN1 � pN3 ¼ �A

4bð�8þ 8h� 4h3 þ h4Þ; ð20Þ
Cwhere
A ¼ ð�4c3bð4� 6hþ 3h2Þ þ c1bð16� 24hþ 16h2 � 4h3 þ h4Þ þ ð�2þ hÞhð�4ð�1þ hÞ2 þ c2bð4� 6hþ 3h2ÞÞÞ ð21Þ
Eand ð�8þ 8h� 4h3 þ h4Þ < 0 since 0 < h < 1.
UN

CO

RR

ð22Þ


wehuan

Cross-Out

wehuan

Replacement Text

1

403

404

405

406

407

408

409

410

411

412

413

414

415

416

417

418

419

420

421

422

423

425425

426

427

428

429

430

431

432

433

434

435

436

437

438

439

440




PR

OO

F

If A > 0 then pN1 > pN3. Clearly, when c1 > c3, in cases 1, 2 and 3 above where the incumbent’s internet costs are greaterthan the Internet-only firms costs, it is beneficial for the incumbent to price higher on the Internet channel. In some cases 5and 6, even when ðc1 < c3 and c1 > c2Þ but not by a huge margin, pN1 > pN3. Finally, there are other cases 4 and 7, whereðc1 < c3 and c1 < c2Þ in which case, whether the Internet price for the incumbent is higher (or lower) depends in addition onthe difference between c2 and c3.

Identical costs: In this case, we will assume that c1 ¼ c2 ¼ c3 ¼ c and u1 ¼ u2 ¼ u; u3 ¼ 1� 2u.There are two effects that could determine consumers purchase in a major way – firm loyalty and channel loyalty. If

consumers choose products more based on firm than channel, then h12 P h13 P h23. On the other hand, under channel loy-alty consumers choose products more based on the channel rather than the retail firm, i.e., h13 P h12 P h23.

Proposition 5. If the costs of the products from the two firms are identical ðc1 ¼ c2 ¼ c3 ¼ cÞ and the substitutability factorsamong the channels are h12 > h13 ¼ h23, the Nash equilibrium price for the Internet channel for the incumbent firm is greater

than the price for the Internet-only new entrant firm ðpN3 < pN1Þ when h12 <2

3�h13.

It is to be expected that both pN1 and pN2 should decrease with respect to h12. The above result indicates that there existsa threshold value for h12 (that is dependent on both h13 and h23Þ such that the incumbent firm charges a higher price onlywhen h12 is below the threshold. An intuitive explanation of this phenomenon is that when h12 becomes high then the priceincrease on the Internet channel leads to internal cannibalization of demand and hence makes this strategy less attractivefor the firm. Thus, when customers choose products based on the firm then h12 should be low enough for the incumbentfirm to charge higher price on the Internet channel.

Proposition 6. If the costs of the products from the two firms are identical ðc1 ¼ c2 ¼ c3 ¼ cÞ and ðu1 ¼ u2 ¼ uÞ and the

substitutability factor h13 > h12 ¼ h23, the Nash equilibrium price for the Internet channel for the incumbent firm is greater

than the price for the Internet-only new entrant firm ðpN3 < pN1Þ when

Pleaits, E

D0 < h12 < 0:747405 and h12 < h13 < T 1ðh12Þ;0:747405 < h12 < 1 and h12 < h13 < T 2ðh12Þ:

OR

REC

TEBased on Figs. 1 and 2, it is clear that T 1ðh12Þ and T 2ðh12Þ are very close to one. Therefore, in almost all cases under the

above assumptions, the incumbent firm with two channels has an incentive to price higher on the Internet channel. Intu-itively, the incumbent gains more by charging a higher price on the traditional channel since h13 > h23. For the same reason,the Internet-only firm has a greater incentive to price lower than the incumbent on the Internet channel. Further, for theincumbent to go below the Internet-only firm’s price would hurt profits due to internal cannibalization. Thus, in almost allcases, it seems that the incumbent will price the product higher on the Internet channel as compared to the entrant.

Clearly, a change in substitutability across a pair of channels has a significant and complex effect on resulting equilib-rium prices. Under firm loyalty settings when h12 P h13 P h23 we numerically study how changes in h12 affect the prices andprofits. In order to illustrate those effects in the following example, we consider two possibilities – (1) the substitutabilityfactor between the two firms (i.e., h13; h23) remains unchanged, (2) the substitutability factor between the two firms changesin the same direction as h12, (i.e., h13; h23 also increase with h12Þ. In the first case, a price drop for the incumbent in one ofher two channels could potentially bring new customers from the third channel but at the same time could lead to canni-balization across its two channels. We find (as shown in Fig. 3) that the prices across all channels decrease with h12. Theprices of the incumbent firm’s two channels decrease slowly at first but once the price is lower than the entrant’s price, thereis a rapid drop. One could explain this effect by noting that when the incumbent’s prices are higher than the entrant’s price

UN

C

0.1 0.2 0.3 0.4 0.5 0.6 0.7Theta12

0.985

0.9875

0.99

0.9925

0.995

0.9975

Theta13

Fig. 1. Constraint for h13 and when 0 < h12 < 0:747405.


CTED

PR

OO

F441

442

443

444

445

446

447

0.75 0.8 0.85 0.9 0.95Theta12

0.985

0.99

0.995

1.005

1.01

Theta13

Fig. 2. Constraint for h13 and when 0:747405 < h12 < 1.

0.3 0.4 0.5 0.6 0.7 0.8 0.9Theta12

5.2

5.4

5.6

5.8

6.2

Price

PN3

PN2

PN1

Fig. 3. Price across the two firms where h13 ¼ 0:2; h23 ¼ 0:1 and u1 ¼ u2 ¼ 0:2; c1 ¼ c2 ¼ c3 ¼ 3;b ¼ 0:1.




RR

Ethe best response to an increase in h12 is a drop in the incumbent’s prices so that she could benefit from a larger pool ofcustomers in channel one and two. The price drop is not substantial because there is cannibalization. However when theincumbent’s prices get lower than the entrant’s price, any price drop in addition attracts the entrant’s customer therebyproviding greater incentive to decrease the prices. At that point, the profit of the incumbent increase with h12 as shownin Fig. 4. Note that the profit of the entrant is decreasing in h12. However, if the initial market share of the incumbentis high (as in Figs. 5 and 6), then the profit of the incumbent does not increase even after the price drops below the entrant’sprice because the entrant’s market share is not significant. In the case where h13 and h23 increase with h12, a decrease in the

UN

CO

0.3 0.4 0.5 0.6 0.7 0.8 0.9Theta12

595

600

605

610

615

Profit

Profit3

Profit12

Fig. 4. Profit across the two firms where h13 ¼ 0:2; h23 ¼ 0:1 and u1 ¼ u2 ¼ 0:2; c1 ¼ c2 ¼ c3 ¼ 3;b ¼ 0:1.


UN

CO

RR

EC

TED

PR

OO

F

448

449

450 Q3

0.55 0.6 0.65 0.7 0.75Theta12

6.3

6.4

6.5

Price

PN3

PN2

PN1


0.55 0.6 0.65 0.7 0.75Theta12

200

220

240

260

280

Profit

Profit3

Profit12


0.25 0.3 0.35 0.4 0.45 0.5 0.55Theta12

5.2

5.4

5.6

5.8

6

6.2

6.4

Price

PN3

PN2

PN1

Fig. 7. Price across the two firms where h12 ¼ h13 þ 0:1 ¼ h23 þ 0:2 and u1 ¼ u2 ¼ 0:2; c1 ¼ c2 ¼ c3 ¼ 3; b ¼ 0:1.




incumbent’s prices enable her to capture the entrant’s market more easily therefore we observe a sharp drop in the priceswhen h12 increases, as shown in Fig. 7. In this case, since increase in h12 leads to higher substitutability across all channelswe find that the profits of the two firms decrease with h12, shown in Fig. 8 (Figs. 9 and 10).


wehuan

Cross-Out

swaminaj

Inserted Text

and Fig 9.

swaminaj

Inserted Text

and Fig. 10

UN

CO

RR

EC

TED

PR

OO

F

451

452

453

454

0.3 0.4 0.5Theta12

500

520

540

560

580

600

Profit

Profit3

Profit12

Fig. 8. Profit across the two firms where h12 ¼ h13 þ 0:1 ¼ h23 þ 0:2 and u1 ¼ u2 ¼ 0:2; c1 ¼ c2 ¼ c3 ¼ 3;b ¼ 0:1.

0.3 0.4 0.5 0.6 0.7Theta12

5.75

6.25

6.5

6.75

7

7.25

7.5

Price

PN3

PN2

PN1


0.3 0.4 0.5 0.6 0.7Theta12

200

250

300

350

Profit

Profit3

Profit12





Under channel loyalty, where h13 P h12 P h23 we study how changes in h13 may affect the prices and profits. As above,we consider two possibilities – (1) the other two substitutability factors remain unchanged (i.e., h12; h23), (2) the other twosubstitutability factors change in the same direction as h13 (i.e., h12; h23 also increase with h13). In the first case, we find (asshown in Fig. 11) that the prices across all channels decrease with h13. Further, as h13 increases the incumbent drops the


swaminaj

Inserted Text

and Fig. 13

OF

455

456

457

458

0.25 0.3 0.35 0.4 0.45 0.5Theta13

5.2

5.4

5.6

5.8

6.2

Price

PN3

PN2

PN1





RO

price on the Internet channel in a much sharper fashion than on its traditional channel. An intuitive explanation is that aprice drop for the incumbent on the Internet channel could potentially bring more customers from the third channel withan increase in h13. However, a price drop on the traditional channel does not have a similar effect since h23 is held constant.We also find that the profits of the incumbent and the entrant decrease with h13 as shown in Fig. 12 (Fig. 13).

UN

CO

RR

EC

TED

P

0.25 0.3 0.35 0.4 0.45 0.5Theta13

560

570

580

590

600

Profit

Profit3

Profit12


0.3 0.4 0.5 0.6 0.7 0.8Theta13

4.5

5

5.5

6

Price

PN3

PN2

PN1



wehuan

Cross-Out

swaminaj

Inserted Text

and Fig. 14

459

460

461

462

463

464

465

466

467

468

469

470

471

472

473

474

475

476

477




RO

OF

In the case where h12 and h23 increase with h13, the incumbent firm can capture the entrant’s market more easily bydecreasing the price on both her channels. Therefore we observe a sharp drop in the prices on both the channels whenh13 increases (as shown in Fig. 15). Increase in h13 leads to higher substitutability across all channels, and as expectedwe find that the profits of the two firms decrease with h13, shown in Fig. 16. We also find that the resulting profit ofthe two firms depends on their market share. For example, when the incumbent’s market share is low (Figs. 16 and12), then beyond a threshold value for h13 the incumbent ends up making more profits than the entrant. On the other hand,when the incumbent’s market share is high to begin with (Figs. 14 and 18), the incumbent ends up with a higher profitdespite increases in h13 (Fig. 17).

4.2.2. Change in incumbent profits due to new entrant

In this section, we provide analytical insights on the effect of the new entrant on incumbent profits. In particular, weconsider a case where the new entrant brings an additional set of customers to the market. Let S represent the whole mar-ket when both firms are present and S1 represent the market when only the first firm is present. Further let the market shareof the Internet channel of the incumbent be u1 and that of the traditional channel be u2 ¼ hu1 and h > 0. Then we assumethat S1 ¼ Sðu1 þ hu1Þ. Further we assume that costs across the two channels in the incumbent are the same ðc1 ¼ c2Þ andthe substitutability of the new Internet channel is identical with respect to the existing channels ðh13 ¼ h23Þ.

Proposition 7. If the costs across the two channels in the incumbent are the same ðc1 ¼ c2Þ and the substitutability of the new

Internet channel is identical with respect to the existing channels ðh13 ¼ h23Þ, the sign of the difference between the profits ofthe incumbent before and after the entrance of the new entrant is independent of the market share of the Internet channel of the

incumbent (u1Þ.

UN

CO

RR

EC

TED

P

0.25 0.3 0.35 0.4 0.45 0.5Theta13

5.25

5.5

5.75

6

6.25

6.5

Price

PN3

PN2

PN1

Fig. 15. Price across the two firms where h13 ¼ h12 þ 0:1 ¼ h23 þ 0:2 and u1 ¼ u2 ¼ 0:2; c1 ¼ c2 ¼ c3 ¼ 3;b ¼ 0:1.

0.3 0.4 0.5 0.6 0.7 0.8Theta13

400

500

600

700

Profit

Profit3

Profit12

Fig. 14. Profit across the two firms where h12 ¼ 0:2; h23 ¼ 0:1 and u1 ¼ u2 ¼ 0:3; c1 ¼ c2 ¼ c3 ¼ 3; b ¼ 0:1.


wehuan

Cross-Out

swaminaj

Inserted Text

and Fig. 17

swaminaj

Inserted Text

and Fig. 18

UN

CO

RR

EC

TED

PR

OO

F

478

479

480

481

482

483

0.25 0.3 0.35 0.4 0.45 0.5Theta13

520

540

560

580

600

Profit

Profit3

Profit12


0.3 0.4 0.5 0.6Theta13

4.5

5.5

6

6.5

Price

PN3

PN2

PN1


0.3 0.4 0.5 0.6Theta13

450

500

550

600

650

700

Profit

Profit3

Profit12





Proposition 8. When the Internet channel and traditional channel in the incumbent firm has the same market share ðu1 ¼ u2Þ,the same substitutability factor ðh13 ¼ h23Þ and the same costðc1 ¼ c2 ¼ cÞ, the gap between the profit of the incumbent firm

after and before the entrance of the new entrant ðprofitafter � profitbeforeÞ is increasing with the cost of the new entrant when

c < c3.

The above results point out that changes in the existing firm’s profits are not necessarily dependent on the existing mar-ket share. Given that a new set of customers are getting added to the market with the entrant, it depends on the cost dif-


484

485

486

487

488

489

490

491

492

493

495495

496

497

498

499

500

501

502

504504

505

506

507

508

509

510

511

512

513

515515

516

518518

519

520




ferentials. This illustrates a possibility for the incumbent to have the higher profit with the new entrant when the cost of thenew entrant is sufficiently higher than the cost of the incumbent firm. In that case the incumbent may benefit from the newmarket that the new entrant brings in.

5. A variant: Model with different demand functions

One of the shortcomings of the above general demand function is that when u > 1=2 and pW > pT then oðqTþqWÞoh > 0 as

indicated in Choi ([11]). Clearly this is not an intuitive behavior that as substitutability increases the demand should go up.In order to overcome the above shortcoming they consider a modified demand function that we study in this section.

OF5.1. Single firm: Introduction of a new channel

We adopt the deterministic and linear demand functions similar to one introduced by Choi ([11]) which are givenby

Pleaits, E

OqT ¼ 1� pT þ cðpW � pTÞ;qW ¼ 1� pW þ cðpT � pWÞ;

Rwhere c > 0 represents the channel differentiation between the traditional and Internet channels. It can be easily shown thatthe above is a special case of the demand function in Eq. (3) when l ¼ 1=2; lS ¼ 1; b ¼ 1, and h ¼ c

1þc. Hence, all the resultsin Section 3 hold correctly for this modified demand function.
P
ED

5.2. Two firms: Introduction of a new channel

In this setting there are two firms who are competing. When the incumbent firm has a traditional channel (like Section4.1), the modified demand function is still a special case of the general function as a result, all results hold. The case wherethe incumbent firm has a mixed channel (as in Section 4.2) is analyzed here. In this case, we can represent the demand as

Tq1 ¼ 1� p1 þ aðp2 � p1Þ þ bðp3 � p1Þ;q2 ¼ 1� p2 þ aðp1 � p2Þ þ bðp3 � p2Þ;q3 ¼ 1� p3 þ aðp2 � p3Þ þ bðp1 � p3Þ;

OR

REC

where a > 0 represents channel differentiation, and b > 0 represents firm differentiation. It is necessary to impose addi-tional inequality constraints on these parameters in order to guarantee (i) prices exceed marginal costs; (ii) quantitiesare nonnegative, (iii) industry demand does not increase with increase in price.

In the following passages, we will provide theoretical insights on how changes in costs and substitutability factors affectoptimal prices and profits. Under the above conditions, we consider the game beginning with both the incumbent andentrant firms simultaneously choosing prices, where pN1; pN2 are the prices of the two channels of the incumbent firms,and pN3 is the price of the Internet-only entrant firm.

Proposition 9. There exists a unique Nash equilibrium in this game under the above assumptions and the resulting prices are

given as follows:

UN

CpN1 ¼að4þ c2bþ 3c3bÞ þ 2ð2þ ð3þ c3Þbþ c3b

2Þ þ c1ð4ð1þ bÞ2 þ að4þ 3bÞÞ2ð4þ 8bþ 3b2 þ að4þ 3bÞÞ

;

pN2 ¼2ð2þ c3bÞð1þ aÞ þ c2ð4þ 8bþ 3b2 þ 4að1þ bÞÞ þ 6bþ c1b

2 þ 2c3b2

2ð4þ 8bþ 3b2 þ að4þ 3bÞÞ;

pN3 ¼2þ 3bþ c1bþ c1b

2 þ 2c3ð1þ bÞð1þ aþ bÞ þ að1þ c2 þ c2bÞ4þ 8bþ 3b2 þ að4þ 3bÞ

:

Hence, we get

pN1 � pN3 ¼�2c3ð2þ bÞð1þ aþ bÞ þ c1ð4þ 4aþ 6bþ 3abþ 2b2Þ � að�2þ c2ð2þ bÞÞ

2ð4þ 8bþ 3b2 þ að4þ 3bÞÞ¼ ðc1 � c3Þð4þ 4aþ 6bþ 2abþ 2b2Þ þ ðc1 � c2Þabþ 2að1� c2Þ: ð23Þ


521

522

524524

525

526

527

528

529

530

531

532

533

534

535

536

537

538

539

540

541

542

543

544

545

546

547

548

549

550

551 Q1

552

553

554

555

556557

558

559

561561




Proposition 10. Let c1; c2 be the cost of product on the two channels of the incumbent firm, and c3 be the cost of the Internet-

only entrant firm. pN3 < pN1 when

Pleaits, E

ðiÞ c1 < c3; c1 > c2; c2 < 1 and c3 <4c1ð1þ aÞ þ 2að1� c2Þ þ 2c1bð3þ bÞ þ abð3c1 � c2Þ

4þ 4aþ 6bþ 2abþ 2b2

or

ðiiÞ c1 < c3; c1 < c2; c2 < 1 and c2 <2þ c1b2þ b

; c3 <4c1ð1þ aÞ þ 2að1� c2Þ þ 2c1bð3þ bÞ þ abð3c1 � c2Þ

4þ 4aþ 6bþ 2abþ 2b2

or

ðiiiÞ c1 > c3; c1 > c2; c2 < 1:
F
UN

CO

RR

EC

TED

PR

OO6. Conclusions

In this paper we study the optimal pricing strategies when a product is sold on two channels. We assume a stylized deter-ministic demand model where the demand on a channel depends on prices, degree of substitution across channels and theoverall market potential. We first study four pricing strategies for the monopolistic environment which differ in the degreeof autonomy for the Internet channel. We provide theoretical bounds for the pricing strategies. Next, we consider the casewhere the Internet channel is introduced by a different firm, and show the existence of a unique Nash equilibrium. Wederive conditions under which the profits of the incumbent may indeed increase. Next, we study the case where the firstfirm has both traditional and Internet channels, and faces the competition from a new Internet firm. We show the existenceof a unique Nash equilibrium for this situation. We develop conditions under which Internet price of the firm with bothchannels is greater than the price of the pure e-tailer. Finally, through a limited computational study, we explore the behav-ior (price and profits) of the above models under different parameters.

Our research has several important managerial insights. Firstly we show that prevalent practices related to providinglimited autonomy in terms of pricing on the new Internet channel by firms is not necessarily such a bad idea. In fact,we theoretically prove that such polices restrict the profits by at most 4% in many cases. Secondly, we show that havinga new competing channel is not necessarily bad for the incumbent firm particularly in cases where the new channel brings inmore customers to the market and the incumbent has a significant cost advantage. In those case, competition becomes asource of opportunity. Finally, through our model we demonstrate that mixed channel firms have a greater likelihood ofpricing their products higher on the competing channel while facing competition from a pure player. This validates some ofthe empirical findings of earlier researchers. The degree of dispersion of prices across the channels depends on the cost dif-ferences, relative substitutability and market share of the different channels.

There are several possible extensions of our work. Firstly, we adopt a simple linear demand model for analytical trac-tability. It would be interesting to test if similar insights could be obtained for more detailed demand models, even com-putationally. Secondly, extending the analysis to multiple channels and retailers would be another interesting avenue forthis research. Finally, analyzing the pricing issue from the manufacturer’s perspective could potentially lead to otherinsights. These will be the focus of our future research.

7. Uncited reference

[14].

Acknowledgements

The authors wish to thank seminar participants at the Stern School of Business, Fuqua School of Business, EcclesSchool of Business, Harvard Business School, Workshop on Optimization in Supply Chain Management at Universityof Minnesota and POMS Conference in Savannah for their comments.

Appendix. Proofs

Proof of Theorem 1. In this case, the ratio of the total profit of the firm under pricing strategy II to that under pricingstrategy IV is shown to be

R ¼ ð�4uþ 4u2 þ h2ÞA16ð1� uÞ2uð1� hÞð1þ hÞB

; ð24Þ


562

564564

565

567567

568

569

570

572572

573

575575

576

577

579579

580

581

583583

584

585

587587

588

590590

591

593593

594

596596




where

Pleaits, E

A ¼ ðð�2þ 2cWbþ h� cTbhÞ2 � 4uð1þ 2c2Wb2 þ cWbð�4þ hÞ � hþ h2

þ cTbð2þ h� cWbh� 2h2Þ þ c2Tb2ð�1þ h2ÞÞ

þ 4u2ð�2cwbþ c2Wb2 þ h2 � 2cTbð�1þ h2Þ þ c2

Tb2ð�1þ h2ÞÞÞ;B ¼ ð�1þ c2

Wð�1þ uÞb2 � c2Tub2 þ cTbð2u� hÞ þ hþ cWbð2� 2u� hþ cTbhÞÞ:

The first order derivatives of R with respect to cT and cW are given by

F

dRdcT

¼ �ðbð�1þ cTbÞð�1þ cWbÞðh� 2uhÞ2ð2� 2cWbþ 2uð�1þ cWbÞ � hþ cTbhÞð�4uþ 4u2 þ h2ÞÞð16ð1� uÞ2uð1� hÞð1þ hÞBÞ2

;

dRdcW

¼ ðð1� 2uÞ2bð�1þ cTbÞ2h2ð2� 2cWbþ 2uð�1þ cWbÞ � hþ cTbhÞð�4uþ 4u2 þ h2ÞÞð16ð1� uÞ2uð1� hÞð1þ hÞBÞ2

: ð25Þ

OO

In order to guarantee that the quantity of the Internet channel positive, i.e., ð2� 2cWbþ 2uð�1þ cWbÞ � hþ cTbhÞ > 0;Rdecreases with cT and increases with cW, we know that cTb < 1 as well. Therefore, ratio is greater than RðcT ¼ 1

bÞ which isshown to be

RR� ¼ R cT ¼1

b

� �¼ 4u� 4u2 � h2

4ð1� uÞuð1� hÞð1þ hÞ : ð26Þ

The first order derivative of R� with respect to h is given by
P dR�
dh¼ � ð1� 2uÞ2h

2ð1� uÞuð�1þ h2Þ2< 0; ð27Þ
Dthen R� is decreasing with h. From the constraints, we know that 0 < u < 1=2; 0 6 h 6 u
1�u or 1=2 < u < 1; 0 6 h 6 1�uu , it

shows that

TE

R� P4� 5uþ 2u2

4� 4ufor 0 < u < 1=2;

R� P1þ uþ 2u2

4ufor 1=2 < u < 1:
C
We can show that 4�5uþ2u2

4�4u is convex with respect to u when 0 < u < 1=2 and 1þuþ2u2

4u is convex with respect to u when1=2 < u < 1. The minimum points are shown to be
E
R4� 5uþ 2u2

4� 4u6 0:957107 when u ¼ 0:292893;

1þ uþ 2u2

4u6 0:957107 when u ¼ 0:707107: �
R
OProof of Theorem 2. Assuming cT ¼ cW, it is easy to compute the ratio of the total profit of two firms in Cases II and IV. Itis computed to be

CR ¼ �ð�4uþ 4u2 þ h2Þðð�2þ hÞ2 þ 4u2h2 � 4uð1� hþ h2ÞÞ16ð1� uÞ2uð1� hÞ2ð1þ hÞ

: ð28Þ

NThe first order derivative of this ratio with respect to h is given as follows:

UdRdh¼ ð1� 2uÞ2h 8� 8hþ 4h2 þ h3 � h4 � 4uð4� hþ h2Þ þ 4u2ð2þ hþ h2Þ

16ð�1þ uÞ2uð�1þ hÞ3ð�1þ hÞ2

!: ð29Þ

The above derivative is negative, i.e., the ratio decreases with respect to h, when

0 < u 61

2and 0 6 h 6

u1� u

ð30Þ

or

1

26 u 6 1 and 0 < h < Root½4� 4u� 2x� 2uxþ x2 � 2ux2 þ x3; 2�; ð31Þ


597

598

600600

601

603603

604

606606

607

608

610610

611

612

613

615615

616

617

619619

620

622622

623

624

626626




where Root [f, k] represents the kth root of the polynomial equation f ðxÞ ¼ 0. Let h ¼ Root½4� 4u� 2x� 2uxþx2 � 2ux2 þ x3; 2�. The derivative dR

dh becomes positive when

Pleaits, E

1=2 6 u 6 1 and h < h < 1: ð32Þ

Therefore,

R Pð4� 5uþ 2u2Þð�4þ 8u� 5u2 þ 2u3Þ

16ð�1þ uÞ3¼ R1 when 0 6 u 6

1

2;

R P Rðh ¼ hÞ ¼ R2 when1

2< u 6 1:

FThe first order derivative of R1 with respect to u is given by

OdR1

du¼ �4þ 32u� 79u2 þ 80u3 � 40u4 þ 8u5

16ð�1þ uÞ4ð33Þ

Owhich is negative when 0 < u < 0:22299 and positive when 0:22299 < u < 1=2. Therefore the minimal point in the first caseis

Rmin ffi 0:97 when u� ffi 0:22299: ð34Þ
R PThe graph of R2 for 1=2 < u < 1 is shown as in Fig. 19. From the graph, we could easily derive that Rmin ffi 0:99795 when
u� ffi 0:685.The above quantities respectively are convex with respect to u. The minimal points in two cases are given as follows:

DRmin ffi 0:97 when u� ffi 0:22299;

Rmin ffi 0:99795 when u� ffi 0:685: �
EProof of Theorem 3. Under the assumption of cT ¼ cW, the ratio of the total profit of the firm in pricing strategy III andpricing strategy IV is computed as T
CR ¼ 4u� 4u2 � h2

4ð1� uÞuð1� hÞð1þ hÞ : ð35Þ

EThe first order derivative of the ratio with respect to h is given as follows:

RdRdh¼ � ð1� 2uÞ2h

2ð1� uÞuð�1þ h2Þ2ð36Þ

Rwhich is negative when 0 < u < 1.The second order derivative of the ratio with respect to u is given as follows:
OdR2
d2u¼ � ð1� 3uþ 3u2Þh2

2ð1� uÞ3u3ð1� h2Þ: ð37Þ

UN

C

0.5 0.6 0.7 0.8 0.9u

0.998

0.9985

0.999

0.9995

R

Fig. 19. Value of R2 when 1=2 < u < 1.


627

628

629

631631

632

634634

635

636

637

639639

640

642642

643

644

646646

647

648

649

650

651

653653

654

655

656

658658




which is negative when 0 < u < 1 and 0 < h < 1. From the derivatives of this ratio with respect to h and u, the ratio de-creases with h and is concave with respect to u. Because the necessary conditions, i.e., h 6 u

1�u when 0 6 u 6 12

and h 6 1�uu

when 126 u 6 1,

Pleaits, E

R P4� 5uþ 2u2

4� 4uwhen 0 6 u 6

1

2;

R P4u3 � 4u4 � 1� u2 þ 2u

4uð�1þ 3u� 2u2Þ when1

26 u 6 1:

ð38Þ

In both cases, the above quantities are convex with respect to u. We may compute the minimal points in the two cases as

OFRmin ¼

1

4þ 1ffiffiffi

2p ffi 0:96 when u� ¼ 2�

ffiffiffi2p

2ffi 0:2929;

Rmin ¼5� 4

ffiffiffi2p

�12þ 8ffiffiffi2p ffi 0:96 when u� ¼ 1ffiffiffi

2p ffi 0:7071: �

ROProof of Proposition 9.

ProofThe total profits of the two channels of the incumbent firm and of the Internet-only entrant firm are given by

g1 þ g2 ¼ q1ðp1 � c1Þ þ q2ðp2 � c2Þ and g3 ¼ q3ðp3 � c3Þ: ð39Þ
PThe second order derivative of the profit with respect to the prices are as follows.
TEDo

2ðg1 þ g2Þop2

1

¼ o2ðg1 þ g2Þ

op22

¼ �2ð1þ aþ bÞ < 0;

o2ðg1 þ g2Þop2

1

o2ðg1 þ g2Þop2

2

� o2ðg1 þ g2Þop1op2

� �2

¼ 4ð1þ aþ bÞ2 � 4a2 > 0;

o2g3

op23

¼ �2ð1þ aþ bÞ < 0:

CHence, the profit functions are concave with respect to the prices. We derive the unique Nash equilibrium prices from thefirst order conditions, and the optimal prices are given by

RR

E

pN1 ¼að4þ c2bþ 3c3bÞ þ 2ð2þ ð3þ c3Þbþ c3b

2Þ þ c1ð4ð1þ bÞ2 þ að4þ 3bÞÞ2ð4þ 8bþ 3b2 þ að4þ 3bÞÞ

;

pN2 ¼2ð2þ c3bÞð1þ aÞ þ c2ð4þ 8bþ 3b2 þ 4að1þ bÞÞ þ 6bþ c1b

2 þ 2c3b2

2ð4þ 8bþ 3b2 þ að4þ 3bÞÞ;

pN3 ¼2þ 3bþ c1bþ c1b

2 þ 2c3ð1þ bÞð1þ aþ bÞ þ að1þ c2 þ c2bÞ4þ 8bþ 3b2 þ að4þ 3bÞ

:

NC

O

h

Proof of Proposition 10.

Proof

(i) c1 < c3; c1 > c2; c2 < 1,We show that pN1 � pN3 > 0, when
U
c3 <4c1ð1þ aÞ þ 2að1� c2Þ þ 2c1bð3þ bÞ þ abð3c1 � c2Þ

4þ 4aþ 6bþ 2abþ 2b2ð40Þ

and pNi � ci > 0; i ¼ 1; 2; 3.(ii) c1 < c3; c1 < c2; c2 < 1,We show that pN1 � pN3 > 0, when

c2 <2þ c1b2þ b

and c3 <4c1ð1þ aÞ þ 2að1� c2Þ þ 2c1bð3þ bÞ þ abð3c1 � c2Þ

4þ 4aþ 6bþ 2abþ 2b2: ð41Þ


659

660

661

662

663664665666667668669670671672673674675676677678679 Q2

680681682683684685686687688689690691692693694695696697698699700701702703704705706707708709710711712713714715

716




Under the above conditions, pN1 > c1; pN2 > c2, and pN3 > c3.(iii) c1 > c3; c1 > c2; c2 < 1,Under this condition, pN1 > pN3. h

UN

CO

RR

EC

TED

PR

OO

F

References

[1] Fabio Ancarani, Venkatesh Shankar, Price levels and price dispersion within and across multiple retailer types: Further evidence and extension,Academy of Marketing Science 32 (2) (2004).

[2] W. Baker, M. Marn, C. Zawada, Price smarter on the net, Harvard Business Review 79 (2001).[3] S. Balasubramanian, Mail versus mall: A strategic analysis of competition between direct marketers and conventional retailers, Marketing Science 17

(1998).[4] F. Bernstein, A. Federgruen, A general equilibrium model for decentralized supply chains with price and service competition, Working Paper, The

Fuqua School of Business, Duke University, Durham, NC, 2001.[5] Fernando Bernstein, Jing-Sheng Song, Xiaona Zheng, Free riding in a multi-channel supply chain, Working Paper, The Fuqua School of Business,

Duke University, Durham, NC, 2004.[6] Joseph L. Bower, Clayton M. Christensen, Disruptive technologies: Catching the wave, Harvard Business Review (1995), January–February.[7] E. Brynjolfsson, M.D. Smith, Frictionless commerce? A comparison of internet and conventional retailers, Management Science 46 (4) (2000).[8] K. Cattani, W. Gilland, J.M. Swaminathan, Coordinating internet and traditional supply chains, in: David Simchi-Levi, David Wu, Max Shen

(Eds.), Handbook of Quantitative Supply Chain Analysis: Modeling in the E-Business Era, Elsevier Publishers, 2004.[9] K. Cattani, W. Gilland, J.M. Swaminathan, Adding a direct channel? how autonomy of the direct channel affects prices and profits, Working Paper,

The Kenan-Flagler Business School, UNC-CH, Chapel Hill, NC, 2002 (revised 2004).[10] K. Cattani, W. Gilland, S. Heese, J.M. Swaminathan, Boiling frogs: Pricing strategies for a manufacturer adding a direct channel that competes with

the traditional channel, Production and Operations Management, in press.[11] S.C. Choi, Price competition in a duopoly common retailer channel, Journal of Retailing 72 (2) (1996).[12] A.T. Coughlan, Competition and cooperation in marketing channel choice: Theory and application, Marketing Science 4 (2) (1985).[13] A.T. Coughlan, B. Wernerfelt, On credible delegation by oligopolists: A discussion of distribution channel management, Management Science 35 (2)

(1989).[14] Dholakia Ruby Roy, Miao Zhao, Nikhilesh Dholakia, Multichannel retailing: A case study of early experiences, Journal of Interactive Marketing 19

(2) (2005).[15] C. Druehl, E. Porteus, Price competition between an internet firm and a bricks and mortar firm, Working Paper, Graduate School of Business,

Stanford University, Stanford, CA, 2002.[16] Ernst and Young. The Global Face of E-Tailing. <http://www.ey.com/GLOBAL/content.nsf/International/Industries_-_RCP_-_Global_Sup-

ply_Chain_Survey/>, 2000.[17] Ernst and Young, The Multi-Channel Imperative. <http://www.ey.com/global/download.nsf/International/RCP_-_The_Multi-Channel_Imperative/

file/rcp_multichannel.pdf/>, 2001.[18] D. Granot, G. Sosic, Formation of alliances in internet-based supply exchanges, Management Science 51 (1) (2005).[19] S. Gupta, R. Loulou, Process innovation, product differentiation, and channel structure: Strategic incentives in a duopoly, Marketing Science 17 (4)

(1998).[20] C.A. Ingene, M.E. Parry, Channel coordination when retailers compete, Marketing Science 14 (4) (1995).[21] L. Johannes, Competing online, drugstore chains virtually undersell themselves-their web sites charge less than their own stores, with some strings

attached, Wall Street Journal B1 (2000).[22] R. Lal, M. Sarvary, When and how is the internet likely to decrease price competition, Marketing Science 18 (4) (1999).[23] S.A. Lippman, K.F. McCardle, The competitive newsboy, Operations Research 45 (1997).[24] K. McIntyre, E. Perlman, Nike-Channel Conflict Stanford GSB Case, EC-9B, 2000.[25] T.W. McGuire, R. Staelin, An industry equilibrium analysis of downstream vertical integration, Marketing Science 2 (1983).[26] K.S. Moorth, Product and price competition in a duopoly, Marketing Science 7 (2) (1988).[27] X. Pan, V. Shankar, B.T. Ratchford, Price competition between pure play vs. bricks-and-clicks e-tailers: Analytical model and empirical analysis,

Advances in Applied Microeconomics 11 (2002).[28] M. Parlar, Game theoretic analysis of the substitutable product inventory problem with random demands, Naval Research Logistics 35 (1988).[29] G. Reinhardt, M. Levesque, Virtual versus bricks-and-mortar retailing, Working Paper, Department of Management, Depaul University, 2001.[30] J.M. Schlesinger, New economy: If E-Commerce helps kill inflation, why did prices just spike?, Wall Street Journal A1 (1999)[31] J.M. Swaminathan, S. Tayur, Models for supply chains in E-Business, Management Science 49 (10) (2003).[32] F. Tang, X. Xing, Will the growth of multi-channel retailing diminish the pricing efficiency of the web?, Journal of Retailing 77 (2001)[33] A.A. Tsay, N. Agrawal, Manufacturer and reseller perspectives on channel conflict and coordination in multiple-channel distribution, in: David

Simchi-Levi, David Wu, Max Shen (Eds.), Handbook of Quantitative Supply Chain Analysis: Modeling in the E-Business Era, Elsevier Publishers,2004.

[34] Wieffering Eric, Net Shopping Becomes Way of Life, But Buyers Demand Better Service, The Patriot Ledger, March, 14, 2000.[35] F. Zettelmeyer, Expanding to the internet: Pricing and communications strategies when firms compete on multiple channels, Journal of Marketing

Research 37 (3) (2000).


http://www.ey.com/GLOBAL/content.nsf/International/Industries_-_RCP_-_Global_Supply_Chain_Survey

http://www.ey.com/GLOBAL/content.nsf/International/Industries_-_RCP_-_Global_Supply_Chain_Survey

http://www.ey.com/global/download.nsf/International/RCP_-_The_Multi-Channel_Imperative/file/rcp_multichannel.pdf

http://www.ey.com/global/download.nsf/International/RCP_-_The_Multi-Channel_Imperative/file/rcp_multichannel.pdf

wehuan

Cross-Out

wehuan

Replacement Text

15 (1) (2006).

wehuan

Cross-Out

wehuan

Cross-Out

wehuan

Replacement Text

4

wehuan

Cross-Out

wehuan

Replacement Text

5

wehuan

Cross-Out

wehuan

Replacement Text

6

wehuan

Cross-Out

wehuan

Replacement Text

7

wehuan

Cross-Out

wehuan

Replacement Text

8

wehuan

Cross-Out

wehuan

Replacement Text

19

wehuan

Cross-Out

wehuan

Replacement Text

0

wehuan

Cross-Out

wehuan

Replacement Text

1

wehuan

Cross-Out

wehuan

Replacement Text

2

wehuan

Cross-Out

wehuan

Replacement Text

3

wehuan

Cross-Out

wehuan

Replacement Text

4

wehuan

Cross-Out

wehuan

Replacement Text

5

wehuan

Cross-Out

wehuan

Replacement Text

6

wehuan

Cross-Out

wehuan

Replacement Text

7

wehuan

Cross-Out

wehuan

Replacement Text

8

wehuan

Cross-Out

wehuan

Replacement Text

29

wehuan

Cross-Out

wehuan

Replacement Text

0

wehuan

Cross-Out

wehuan

Replacement Text

1

wehuan

Cross-Out

wehuan

Replacement Text

2

wehuan

Cross-Out

wehuan

Replacement Text

3

wehuan

Cross-Out

wehuan

Replacement Text

4

introduction of a second channel: implications for pricing

Documents