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Coordinating Internet and Traditional Channels

Kyle Cattani Wendell Gilland

Jayashankar M. Swaminathan

The Kenan-Flagler Business School The University of North Carolina at Chapel Hill

Chapel Hill, NC 27599-3490

March, 2002

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Coordinating Internet and Traditional Channels Abstract

The Internet has provided traditional retailers a new avenue to conduct their business. As retailers consider the addition of an Internet channel, they need to decide two important issues – the degree of independence of the new channel and the pricing of goods across the two channels. In this paper, we use consumer utility theory to develop the aggregate demand for the product from both traditional and Internet channels and find the optimal prices under different degrees of autonomy for the Internet operations. We explore the effect of cost, customer affinity for shopping online, and the inherent profitability of the product on the optimal prices across the two channels. We also study a duopoly case where a traditional retailer competes with a pure e-tailer and prove the existence of a Nash equilibrium. Finally, through a detailed computational study we explore the behavior (price and profits) of the above models (monopoly and duopoly) under different parameters.

Keywords: Supply Chain Management; Multiple Channels; Channel Conflict; Pricing.

1. Introduction The Internet has provided traditional retailers a new avenue to conduct their business. On one

hand, utilizing the Internet channel potentially could increase the market for the firm and, due to

synergies involved, reduce the costs of operations. On the other hand, a new channel threatens

existing channel relationships through possible cannibalization. The introduction of an Internet

channel poses two important questions – the degree of independence of the Internet channel and the

pricing of goods across the two channels. Retailers that have faced this issue in the last couple of years

have taken different courses of action with varying and instructive results.

For example, Best Buy, an electronics retailer, was quite slow to adopt the Internet channel.

However, on adoption Best Buy decided to treat the customers of the online and physical stores in an

identical manner (Wieffering, 2000). This meant that every part of the business, from purchasing to

distribution, was carefully analyzed and engineered so that at the conclusion of a sale, a customer

would not be able to tell where the traditional stores’ activity ended and where the online experience

took over. On the other hand, Barnes and Noble, a book retailer, decided to create an independent unit

for Internet operations that essentially competed with the traditional retail store. In fact, many experts

advocated such an independent structure for the Internet business (Christensen and Overdorf, 2000)

since it gave the executive team more flexibility and independence to get the business up and running

quickly without the inevitable debates over issues such as pricing, cannibalization and even dress

codes that an integrated structure may have had to encounter. Some retailers, such as Wal-Mart,

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started out by establishing independent Internet operations (walmart.com), but corporate executives

subsequently decided to roll that business back into the existing company because they felt that they

were unable to benefit from the synergies across the two businesses (Prior, 2001). Another set of

retailers, including many grocery stores, decided to ignore completely the Internet channel and focus

on their traditional channel. Yet other retailers do not offer the product for sale over the web, but use

the Internet as a medium to provide information about their products and to direct the customer to the

nearest store.

Given an Internet channel, the question arises as to how one should price products on the

Internet channel. A recent survey by Ernst & Young reports that nearly two thirds of companies price

products identically for their on-line and off-line operations (Ernst&Young, 2001). Strategies of firms

such as Best Buy focus on pricing identically across the two channels so that the Internet enhances

customer experience by providing more information and greater convenience, but not lower prices.

The Ernst & Young survey also indicates that the majority of customers expect to find lower prices

online. Thus, not surprisingly, several researchers have found that prices for specific goods may be

lower on the Internet than in traditional stores. For example, Brynjolfsson and Smith (2000) indicate

that prices of books and CD’s were 9-16% lower on the Internet than in traditional stores. Further, the

prices charged by pure e-tailers and e-tailers with traditional channels may be different. Tang and

Xing (2001) find that the price charged by pure e-tailers for DVD titles is 14% lower than those

charged by e-tailers with traditional channels. It is apparent that the pricing decision is linked to the

autonomy decision regarding Internet operations.

In this paper, we explore the pricing issue for the traditional and Internet channels under

different degrees of autonomy for the Internet operations. We first focus our attention on the

monopolistic situation where there is one traditional retail outlet selling a single product, and the

retailer decides to introduce an Internet channel. We use consumer utility functions to develop the

aggregate demand for the product across the traditional and Internet channels – thereby capturing

potential cannibalization that may occur. We find the optimal prices for the traditional and the Internet

channel for the following cases: (1) the Internet operations are run independently and the traditional

channel is left unchanged, (2) the Internet operations are run so as to maximize profits for the firm as

a whole, but without making pricing changes on the traditional channel, (3) using the same prices on

both channels, and (4) jointly optimizing prices on both channels. We explore the effect of cost,

customer affinity for shopping online, and the inherent profitability of the product on the optimal

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prices across the two channels. Finally, we consider a situation where the traditional retailer faces

competition from a pure e-tailer and prove the existence of a Nash equilibrium.

Among other results, we show that in the monopoly case with independent Internet operations

there exists a threshold value in terms of the differences in the cost of the traditional and Internet

channels that determines whether the Internet channel is priced higher or lower. If the traditional-

channel price is left unchanged and costs in the two channels are identical, we show that the Internet is

priced lower than the traditional channel when profits from the traditional channel are ignored,

whereas the Internet channel is priced higher than the traditional channel when profits from the

traditional channel are considered. We also show that if both channels are priced identically then the

resulting price is higher than that charged by a monopolist with a single channel, thereby illustrating

that opening up a new channel may not always lead to lower prices. For the case with joint optimal

pricing we show, counter to our intuition, that under identical costs the Internet price is greater than

(less than) the traditional price if the Internet channel is less (more) convenient than the traditional

channel. Through a detailed computational study, we explore the behavior (prices and profits) of the

above models (monopoly and duopoly) under different parameters. We confirm that it is not always a

good strategy to price the Internet channel below the traditional channel (Lal and Sarvary, 1999). We

also find that the popular approach of running an independent Internet operation can be very

detrimental in some environments.

The rest of the paper is organized as follows. In section 2, we discuss related literature. We

introduce the demand generation model and present our analytical model and results in section 3. In

section 4, we discuss the duopoly case. We provide computational insights in section 5 and conclude

in section 6.

2. Related Literature Much work has been done toward understanding the conflicts that arise given the different

objectives of channel members. Jeuland and Shugan (1983) defined channel coordination as the

setting of all manufacturer- and retailer-controlled variables at the levels that maximize channel

profits. Many studies have focused on this vertical coordination of the channel using measures such as

transfer pricing schemes, formal agreements, or information sharing (e.g., Choi, et al. (1990) and Choi

(1991), McGuire and Staelin (1986); Moorthy (1987); Ingene and Parry (1995); Desiraju and Moorthy

(1997); Kim and Staelin (1999); Cachon (1999); Lariviere (1999)). Other works have explored the

issue of prices and offerings with a focus on competition (e.g., Shaked and Sutton (1982); McGuire

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and Staelin (1983); Jeuland and Shugan (1983); Moorthy (1988); Choi (1991); Vandenbosch and

Weinberg (1995)).

There are a number of recent papers that study issues related to supply chains in

e-business (see Swaminathan and Tayur (2002) for a detailed review). Balasubramanian (1998) and

Druehl and Porteus (2001) are closely related to this paper. Balasubramanian (1998) models

competition between direct marketers and conventional retailers. His model considers inelastic

demand with customers uniformly distributed on a circle, retailers evenly spaced around the circle,

and the direct marketer in the center. All retailers have fixed costs and customers of the direct channel

incur a monetary disutility that may be of any value. Among other issues, his focus is on the role of

information in multiple-channel markets and he shows that even with zero information costs,

providing information to all consumers may not be optimal, especially when the product is not well

adapted to the direct channel. In this case, high market coverage (from having a greater number of

retailers) may depress profits. Our model assumes that the utility customers receive from each channel

is independent; some value the traditional channel more, while others prefer the direct channel. Our

focus is on the pricing in the competing channels given their relative levels of convenience to the

customer and under different degrees of autonomy for the direct channel.

Druehl and Porteus (2001) consider competition between an Internet firm and a bricks and

mortar firm. Consumers are assumed to have different reservation prices for the two channels and they

solve for the firm’s optimal prices in Nash and Stackelberg games. The competitive outcomes are

interpreted in terms of the degree of innovation. In their model, only a fraction of customer population

has access to the Internet and are willing to buy from there. They show that the Internet firm might

not dominate the market, even if it delivers at a lower cost and is preferred by the best customers.

Price competition between independent retailers and manufacturer-owned stores also is

studied by Ahn, et al. (2000). Their model assumes that manufacturing-owned stores are

geographically isolated from the traditional retailers (i.e., they consider outlet stores on the periphery

of metropolitan areas). The equilibrium pricing strategy is always one of three: monopoly,

competition, or elimination.

Our research follows up on the idea of market segmentation with a somewhat different

approach. In our models, customers are a random physical distance from the traditional retailers, and

a random virtual distance from the direct marketer, independent of the physical distance. The market

then is segmented according to the utility each customer attains from either the direct channel or the

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traditional channel. Customers are not excluded from a specific market; thus both markets have an

equal chance to compete for all customers.

Pricing issues have been examined by Brynjolfsson and Smith (2000) in an empirical study

that concludes that prices for books and CDs on the Internet are 9-16% lower than prices in

conventional outlets in competitive markets. Lal and Sarvary (1999) develop a model that finds cases

where the use of the Internet leads to higher prices and can discourage consumers from engaging in

search. In particular, this scenario arises when the cost of undertaking the shopping trip is higher than

the cost of searching among stores. Our models consider the relative convenience of a virtual trip to

the direct channel versus the physical distance customers must travel to traditional retailers. We

explore market penetration and market shares for each channel for the case of monopolists

considering the addition of a direct channel as well as for the case of a direct channel as competition

to the traditional channel.

3. Model We model each individual customer using a utility function for the product sold by the firm.

A customers’ utility for the product is decreasing in the price of the product and the distance

(physical or virtual) to purchase the product. We assume that customers in the traditional channel

physically are distributed uniformly from the retailer across a distance that is normalized to vary from

zero to one, and that customers in the web channel are distributed uniformly, in what could be thought

of as a virtual distance that corresponds to the perceived inconvenience of the Internet, varying from

zero to γ . The physical and virtual distances of a customer are assumed to be independent. We use a

linear customer utility function as in Vandenbosch and Weinberg (1995) and Srinivasan (1982): the

utility a customer j receives from a product in channel x is j j

x x xU R P Dα= − − (1)

where { },x T W∈ for traditional and web channels, respectively, and where R = the customer’s

reservation price, Px = the price in channel x, α = the customer’s sensitivity to distance, and Dx =

the inconvenience the customer perceives for channel x, which might be thought of as a physical

distance or virtual distance (which may capture the degree of reluctance to buy via the Internet) from

the retailer, where traditional distance ~ [0,1]TD U and web distance ~ [0, ]WD U γ . Different values

of γ model different types of populations. When γ is small, the population is more comfortable in

terms of Internet purchases whereas a very large γ indicates that customers are more spread in terms

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of their preference for the Internet. Over time one might expect that γ will decrease for a particular

product.

In contrast to the models in many of the related works such as Balasubramanian (1998), our

models do not assume that R is sufficiently high that all customers participate. Customers will

purchase the product only if the net utility from the channel is positive. If both channels provide

positive utility, the customer will buy from the higher-utility channel. Heterogenity of the channels is

captured by the parameter γ . We will focus on retail prices in the range

*TR P Rα− ≤ < , and *

WR P Rαγ− ≤ < (2)

although we will indicate interesting conditions where the range is suboptimal. Prices at or above

these upper limits will provide positive utility to 0% of the population while prices at or below these

lower limits will provide positive utility to 100% of the population. We assume that the cost to

produce and stock a unit xC R< ; otherwise the retailer will not participate.

Consider a retailer with only the traditional channel. It costs the retailer CT to stock a unit

which he sells to customers at price PT. Prior to consideration of the web channel, the traditional

channel faces demand based on the customer’s distance from the retailer. The probability that a

randomly selected customer will purchase the product provides a downward sloping demand curve:

{ } { } { } ( ){ } ( )Pr a customer will buy Pr 0 Pr 0 Pr / /jT T T T T TU R P D D R P R Pα α α= > = − − > = < − = −

Next consider the addition of a web channel. We must now account for both the traditional

(physical) distance as well as the web “distance,” which is independent of the traditional distance.

Consumers with a low web distance may be thought of as experienced web users who value the

convenience of e-tailing, whereas those with a high web distance may not have convenient web access

and/or may be concerned about the security of conducting financial transactions over the Internet. If

utility is positive from at least one channel, customers will purchase the product from the channel that

provides the greatest utility. If neither channel provides positive utility, the consumer elects not to

purchase a product.

Expected profit over the random distances is calculated over the appropriate regions of the

joint distribution of DT and DW, the random variables for traditional and web distances, respectively.

The calculation depends on the relative prices of PT and PW, shown below as Case 1 and Case 2.

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Case 1: PW > PT

Figure 1: Customer Channel Preference Based on Distances (PW>PT)

Sales in traditional and web channels: PW>PT

0

1

0 � DW

DT

W2

T1

W1

T2

In Case 1, utility is greater than zero for all customers whose web distance ( )W WD R P α≤ − (regions

W2, W1, and T1 of Figure 1) and for all customers whose traditional distance ( )T TD R P α≤ −

(regions W1, T1, and T2). Customers in regions W1 and T1 receive positive utility from both channels.

However, customers in region W1 , above the diagonal line ( )T W T WD P P Dα= − + , have a

preference for the web channel, while those in region T1, have a preference for the traditional channel.

For a given set of parameter values and prices, the percentage of customers in each region is

calculated as follows:

W2 = Pr{Cust. buys over web but would not buy from trad. channel} 11 WT R PR Pα α γ

−− = −

W1= Pr{Customer would buy from either channel but prefers web}21

2WR P

γ α− =

T = T1+T2 = Pr{Customer buys from traditional channel} 1TR P W

α− = −

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Case 2: PW < PT

Figure 2: Customer Channel Preference Based on Distances (PW<PT)

Sales in traditional and web channels: PW<PT

0 γ 0

DW

DT

1

W2

T2 T1

W1

In Case 2, the line ( ( )T W T WD P P Dα= − + ) that separates regions 1W ′ and 1T ′ crosses the DT axis below

zero. When W TP P< , the percentage of customers in each region is calculated as follows:

2W ′ = Pr{Cust. buys over web but would not buy from trad. channel} 11 WT R PR Pα α γ

−− = −

1W ′ = Pr{Cust. would buy from either channel but prefers web} 1 12

W T WT R P P PR Pα α α γ

− −− = +

1 2T T T′ ′′ = + = Pr{Cust. buys from traditional channel } 1TR P W

α− ′= −

3.1. Base Case: A Traditional Channel We first consider a traditional retailer operating prior to the existence of a web channel. The

traditional retailer chooses a price PT to maximize his expected profit TΠ . Without loss of generality

we assume that the market size is one, and thus ( ) ( )( )( )/T T T T TP R P P CαΠ = − − . In the following

sections, we focus our analysis on the interior point solutions which arise with values of CT such that

2 TR C Rα− < < . It is easy to show that ( )T TPΠ is strictly concave and that prices are optimized at

PT*=(R+CT)/2. (3)

Expected optimal profits at interior points in the domain are ( ) ( )2* / 4T T TP R C αΠ = − . At TC R= ,

PT* equals R, and expected profits are zero. At 2TC R α= − , PT* equals R α− , the entire population

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buys, and expected profits are α per customer. There is no incentive to lower prices further.

In the following sections, we consider pricing issues under a series of policies representing a

range of degrees of autonomy for the Internet operations. Each of the policies makes use of the two

cases. In Section 3.2 we explore a policy where a retailer adds a web-based channel without adjusting

the pricing in the traditional channel. In this policy, the web-based channel experiences relatively

more autonomy. In Section 3.3 we assume the retailer adds a web-based channel and coordinates

prices in both channels, resulting in relatively less autonomy for the web-based channel. In Section

3.4 we consider a scenario where the two channels act as competitors.

3.2. Adding a Web-based Channel While Holding Prices Fixed in the Trad. Channel In this policy the retailer adds a web-based channel and keeps the traditional price at

( )* 2T TP R C= + . This models a situation where the retailer may be experimenting with the web

channel, or the retailer may decide that the web channel is insignificant compared to the traditional

channel and, as a result, not worry about the effect of opening this new channel. An example of this

situation might be a giant retailer such as Wal-Mart experimenting with a web channel. We consider

two optimization strategies for this new web channel: the web channel profits are optimized without

consideration of the traditional channel profits (Section 3.2.1), or profits are optimized over both

channels (Section 3.3.2).

3.2.1. Optimizing profits from the web channel (ignoring traditional channel profits): WW

The policy of profit maximization over the web channel without consideration of the

traditional channel could arise if the web channel were managed separately from the traditional

operations, as recommended by many management experts (e.g., Bower and Christensen 1995). We

abbreviate the various policies by noting which channel’s prices are modified and which channel’s

profits are optimized. Thus, this first policy is denoted as web web (WW).

We show that the relative costs of the channels, versus W TC C , determine the relevant case,

i.e., whether the price over the web should be greater than or less than the traditional channel price.

Proofs of most propositions and lemmas are in the appendix.

Theorem 1: In Policy WW, * W T W TC C P P< + ∆ ⇔ < where ( )2 / 8TR C α∆ = − .

If CW is sufficiently low compared to CT, but not necessarily less than TC , then the optimal

web entrant price is less than the traditional channel price.

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Corollary 1: In Policy WW, if W TC C= , then *W TP P< .

Proof: This follows directly from Theorem 1.

The above result states that if the costs in the two channels are equivalent, the optimal web

price will be lower than the traditional price for a profit-maximizing entrant that ignores the traditional

channel. In maximizing their own profits (and ignoring the profits of the traditional channel) the web

channel entrant has an incentive to undercut the traditional channel, cannibalizing significant market

share (and profits) from the traditional channel.

Lemma 1a: If W TC C< + ∆ (causing *W TP P< ) and 2WC R αγ≥ − + ∆ then the web retailer chooses a

price PW* to maximize expected profit WΠ such that:

( )* / 2W WP R C= + − ∆ , (4)

with expected profit ( ) ( )2* / 4W W WP R C αγΠ = − − ∆ .

As expected, PW* is increasing in CT, CW, and R. The web entrant’s price will be greater if her

costs or the traditional channel costs are greater and for higher reservation prices. The restriction

2WC R αγ≥ − + ∆ on CW ensures an interior point solution. The interesting result that arises when CW

is very small (i.e., the restriction does not hold) is highlighted in Lemma 1a′.

Lemma 1a′: If W TC C< + ∆ and 2WC R αγ< − + ∆ , then *WP R αγ< − .

Lemma 1a’ is a sub case of Case 2, which we identify as Case 2b: PW < PT and

2WC R αγ< − + ∆ . Figure 3 illustrates Case 2b, where *WP R αγ< − .

Figure 3: Case 2b

Sales in traditional and web channels: PW<PT

0

1

0 � DW

DT

W2

W1

T

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When *WP drops below R αγ− , the vertical line segmenting customers who receive positive versus

negative utility from the web channel lies to the right of γ , meaning that the entire population

receives positive utility from the web channel offering.

If CW is sufficiently low, then the price charged by the web entrant is less than the price

required to ensure 100% of the population buys the product. At this point, the web entrant is pricing

aggressively low with the exclusive intent to steal market share from the traditional channel (as

opposed to also enticing new customers to buy). There is a floor for *WP : the price for the web will

always be such that *W TP P αγ≥ − . Beyond this limit, the web will have 100% market share and

100% customer participation, so further price reductions would only increase consumer surplus.

Lemma 1b: If W TC C≥ + ∆ (causing *W TP P≥ ), then the web retailer chooses a price PW

* to maximize

expected profit WΠ such that:

( )* 1 23W T WP C C R Xα= + + + − (5)

where ( ) ( ) ( )( )( )22 3 2T W W T W WX C C R C R C C R C Rα α= + + + − + + + + ,

and expected profit is:

( ) ( )( )( )( )*2

1 2 2 2 2 2 454W W T W T W T WP R C C X R C C X C C R Xα α α

α γΠ = − − − + + − + − − − + +

Theorem 1 demonstrates that the web channel’s price relative to the traditional channel will be

determined by the relative costs of the channels. The web channel will enter with a lower price unless

the cost in the web channel is sufficiently larger (specifically ∆ more) than the cost in the traditional

channel. Lemmas 1a and 1b provide closed-form solutions for the optimal price and the resulting

expected profit when a web channel is added with no response from the traditional channel.

Corollary 2: The optimal price for the web entrant is independent of γ .

Proof: Observed directly from (4) and (5).

This result states that the optimal price of the web offering need not account for how

convenient customers perceive the web to be relative to the traditional channel. The optimal web

price depends on the relative costs of the two channels as well as the customer’s reservation price and

sensitivity to distance. The traditional retailer’s optimal price (equation (3) in section 3.1) is not

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dependent on the range of customer distances because the marginal benefits (increased margin on

each sale) and marginal costs (reduced market penetration) are equally affected by a change in the

range of distances; in a similar manner, the optimal price for Policy WW does not depend on the

relative convenience of the channels.

In summary, a retailer who adds a web-based channel without adjusting the pricing in the

traditional channel and optimizes profits solely for the web-based channel will choose a web price

based on the relative channel costs: if the web channel costs are comparable to the traditional channel

costs (or lower), the web price will be lower than the traditional channel price. In section 4 we will

show that Policy WW can result in significantly lower profits compared to strategies that provide less

autonomy to the new channel. While Policy WW may be effective in launching a new channel, it

may do so at the expense of the traditional channel and profits of the organization as a whole.

If instead of considering only the web-based channel as it determines the web price, the

retailer considers profits over both channels, we will show in the next section that the relative prices

reverse when costs are comparable: if the web channel costs are comparable to the traditional channel

costs (or higher), the web price will be higher than traditional channel price.

3.2.2. Optimizing profits across both channels: WJ

In this policy, the retailer adds a web-based channel without adjusting the pricing in the

traditional channel, but prices the web channel offering to optimize profits across both channels. For

any set of parameters, this policy obviously will result in total profits (across both channels) that are

greater than or equal to the total profits achieved under section 3.2.1. This policy is denoted as WJ as

the web price is changed to optimize both channels jointly.

The retailer decides the price for the product offered over the web, PW, keeping his traditional

price at ( ) / 2T TP R C= + . Again, we show that there are two cases: the optimal price over the web is

either greater than or less than the traditional channel price, depending on the relative costs of the

channel, versus W TC C . Proofs are in the appendix.

( )2We again let / 8TR C α∆ = − .

Theorem 2: * W T W TC C P P< − ∆ ⇔ < .

Note that as opposed to Theorem 1 where we compare CW with CT + ∆ , here we compare CW

with CT – ∆ . Optimal prices are given in Lemma 2a and 2b.

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Lemma 2a: If W TC C< − ∆ (causing *W TP P< ), then the web retailer chooses a price PW

* to maximize

expected profit Π such that:

( )* / 2W WP R C= + + ∆ . (6)

A closed-formed expression for expected profit is attainable by substituting (6) into the

expected profit equation (equation (10) in the appendix), but is not elegant.

Lemma 2b: If W TC C≥ − ∆ (causing *W TP P≥ ), then the web retailer chooses a price PW

* to maximize

expected profit WΠ such that:

( )* 1 2 3 46W T WP C C R Yα= + + + − (7)

where ( ) ( )2 22 23 8 6 8 4 2 4T T W W WY C R C C R C Cα α α= − + − + + + − + ,

Again, a closed-formed expression for expected profit is attainable, but not elegant, by

substituting (7) into the expected profit equation (equation (10) in the appendix).

Lemmas 2a and 2b provide closed-form solutions for the optimal web prices when a web

channel is added with no response from the traditional channel, but considering profits over both

channels. Proposition 2 demonstrates that the web channel’s price will be determined by the relative

costs of the channels. In contrast to the previous section, the web channel will enter with a higher

price unless the cost in the web channel is sufficiently lower (specifically ∆ less) than the cost in the

traditional channel.

Corollary 3: If W TC C= , then *W TP P> .

Proof: This follows directly from Theorem 2.

The above result states that if the costs in the two channels are equivalent, the optimal web

price will be higher than the traditional price. By adding a web channel with a higher price, additional

customers will be induced to buy (those with a high physical, but low web distance – W2 in Figure 1)

and any cannibalization from the traditional channel (W1 in Figure 1) will be at a higher price, also

providing additional margin to the supply chain.

Corollary 4: The optimal price is independent of γ

Proof: Observed directly from (6) and (7).

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As in section 3.2.1, the optimal price of the web offering need not account for how convenient

customers perceive the web to be relative to the traditional channel, but only on the relative costs of

the two channels as well as the customer’s reservation price and sensitivity to distance.

If WC is sufficiently small ( W TC C≤ − ∆ ), then *W TP P≤ regardless of whether web profits are

maximized independently or if profits across both channels are jointly maximized (while holding TP

fixed). Similarly, if W TC C≥ + ∆ , then *W TP P> for either policy. When WC and TC are relatively

close ( T W TC C C− ∆ < < + ∆), then the resulting relative prices in the two channels depend on

whether web profits are maximized independently, or whether profits in both channels are being

considered simultaneously. In section 4 we perform a numerical study for further insights into the

problem.

3.3. Coordinated Web and Traditional Channel In this section we consider the web and traditional channel to be coordinated such that the

expected profit is optimized jointly over PT and PW. This situation arises when a retailer adds the web

channel and wants to ensure that the combined profits of the web channel and the traditional channel

are maximized.

3.3.1. Same price in both channels (EJ)

We begin our analysis of the coordinated channel policy by assuming that the retailer charges

the same price in both channels. Many firms, such as Best Buy, follow such a strategy for many of

their products. Although setting T WP P= adds a constraint to the optimization problem (thereby

potentially reducing profits), it has the benefit of avoiding channel conflict and perceived consumer

inequity. We denote this as Policy EJ as equal prices are set in both channels and the channels are

optimized jointly.

Lemma 3: If the same price is used in both channels, then the price, *P , that maximizes expected

profits over the two channels is:

( ) ( )( )( )( ) ( )( ) ( ) ( ) ( )( )

*

2

1 2 26

where 2 2 12

T W

T W W T

P C C R R x

x C C R R R C R R C R

α αγ

α αγ α αγ αγ

= + + − + − +

= + + − + − − − − − + −

Corollary 5a: If the costs, C, are identical in both channels, then the price, *P , that maximizes

expected profits over the two channels is:

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( ) ( ) ( ) ( ) ( )( )2* 1 13

P C R R R C R Cα αγ α αγ γ α= + − + − + − − − + − +

Corollary 5b: If the same price is used in both channels, then the optimal price, *P , is greater than

the price charged when only considering the traditional channel ( ( )* 2T TP R C= +

from section 3.1).

Lemma 3 provides a closed-form expression for the optimal price in the coordinated channel

case when the same price is desired in both channels. A closed-formed expression for expected profit

is also attainable, but not elegant.

Corollary 5a provides a closed form expression for the price when costs are identical in the

two channels. Corollary 5b demonstrates that market segmentation places upward pressure on prices

if both segments are priced identically. A firm that opens a new channel under Policy EJ will increase

prices. The existing channel will lose demand because of cannibalization and because of the increased

price. However, the firm’s overall profits will increase through the incremental sales in the new

channel and the higher price for all sales.

3.3.2. Different prices across channels (BJ)

When the prices charged in the two channels are allowed to differ, optimal prices still can be

determined analytically. However, due to their complexity, little insight is gained. This policy is

denoted as BJ as the prices in both channels are changed to optimize both channels jointly. An

interesting result from analyzing this policy is articulated in Theorem 3.

Theorem 3: Assume T WC C= , and profits are quasi-concave in PT and PW.

(i) if 1γ > , then * *W TP P> ;

(ii) if 1γ = , then * *W TP P= ;

(iii) if 1γ < , then * *W TP P< .

Theorem 3 states that if the web is relatively less convenient than the traditional channel (i.e.,

1γ > ), then the optimal web price will be greater than the optimal traditional price. Rather than

having to compensate the customers for the perceived inconvenience, the web channel concedes

market share to the traditional channel but captures the consumer surplus from those customers who

do consider the web to be more convenient. Because the firm is acting as a monopolist that is

17

segmenting its market, it does not have to be concerned with pricing the channel in a competitive way

that compensates the customer for her inconvenience. The objective is to capture consumer surplus

rather than market share.

3.4. Competition between web and traditional channel In this section we assume that the web entrant is distinct from the company operating in the

traditional channel, and that both firms view each other as direct competitors. In addition to the

parameters that characterize the market (R, α, and γ), each firm’s profit function will depend on its

own cost as well as the price charged by each competitor. For a given set of parameters, we seek to

identify a Nash Equilibrium (NE), whereby neither participant has an incentive to alter their price,

given the price being charged by their competitor. In this scenario, both the web price and the

traditional price are adjusted to maximize profits for their respective firm.

As with many of our earlier scenarios, the exact profit function for each competitor depends

on the relative price in the two channels. If W TP P> (see Figure 1), the two profit functions are:

( ) ( )21 1,

2WT

T W T T TR PR PP P P C

α α γ −− Π = − −

( ) ( )21 1, 1

2W WT

W W T W WR P R PR PP P P C

α α α γ − −− Π = − − +

Alternatively, when W TP P< (see Figure 2), the two profit functions are:

( ) ( )

( ) ( )

2

2

1 1,2

1 1,2

W T TT W T T T

W TW W T W W

R P R P R PP P P C

R P R PP P P C

γα α α γ

α α γ

− − − Π = − − +

− − Π = − −

In either case, the first partial derivatives, T TP∂Π ∂ and W WP∂Π ∂ , can be set to zero in order

to determine the two firms’ response functions (i.e., the optimal price they will charge given the price

charged by the competitor). Any pair, WP and TP , that simultaneously satisfies both response

functions is a pure strategy NE, provided the relative prices are consistent with the assumptions

underlying the set of profit equations used to obtain the pair.

18

Theorem 4: Within the price ranges WR P Rαγ− ≤ ≤ and TR P Rα− ≤ ≤ , a pure strategy Nash

Equilibrium exists.

We will focus our analysis of the competitive equilibrium on the case where W TP P> . (The

analysis of the other case is identical, and details of this case are presented in the Appendix.) By

setting 0T

TP∂Π =∂

, we obtain the following response function for the competitor using the traditional

channel: ( )212 4

TT W

R CP R Pαγ

+= − −

Similarly, the web-based competitor adheres to the following response function:

The intersection of these two lines (i.e., the joint solution to the two equations) will determine

the NE pair of prices (provided, of course, that W TP P> in order to remain consistent with the

assumptions underlying the two response functions). In some instances (depending on the specific

parameter values), the two response functions will not intersect within the prescribed price ranges and

the Nash Equilibrium will occur at the boundary (with WP R αγ= − and/or TP R α= − ).

As an example of the Nash Equilibrium, consider the case when R = 12, α = 4, CT = 8, CW = 8,

γ = 0.8. Figure 4 displays the response curves for the two competitors. The NE occurs when PW =

9.65 and PT = 9.57.

Figure 4: Response Curves for Traditional and Web-based Competitors

9

9.2

9.4

9.6

9.8

10

9 9.2 9.4 9.6 9.8 10

Web Price

Trad

ition

al P

rice

WebTraditional

( )2 2 2 22 2 4 4 3 2 2 8 6 6 3W W T W T W T W T TP C P C P R C P C P R RPα α α α α= + + − + + + − − + − −

19

4. Computational Study In sections 3.2 - 3.4, we have discussed four different policies that could arise when a

company begins using the Internet to sell its products, and a fifth scenario representing a competitive

entry (summarized in Table 1).

Table 1: Summary of Scenarios

Scenario How prices are determined

3.2.1 (WW) Web price chosen to maximize web profits (holding TP fixed)

3.2.2 (WJ) Web price chosen to maximize joint profits (holding TP fixed)

3.3.1 (EJ) Prices chosen to maximize joint profits (prices held to be equal)

3.3.2 (BJ) Prices chosen to maximize joint profits (no constraint on prices)

3.4 (NE) Nash Equilibrium prices when the web entrant is a competitor.

In this section we perform a numerical study for further insights into the problem. We analyze

the performance of the Policies WW, WJ, EJ, and BJ using a full-factorial experimental design

varying R, CT, CW, α , and γ , as shown in Table 2. We follow up the experimental design with

additional experiments to test the sensitivity of profits under the various policies to the various

parameters.

Table 2: Parameters for Experimental Design

Parameter Parameter values R 8, 12, 16 CT 4, 6, 8 CW 4, 6, 8 α 4, 6, 8 γ 0.5, 1, 1.5

Table 3 summarizes the results of the experimental design for the monopolist cases in Sections

3.1 through 3.3, showing the average profits of each policy as a percentage of the profits under the

optimal prices of Policy BJ. The base case of Policy 3.1 (the profits from the traditional channel prior

to web entry) provides only 66.8% of the profits on average of Policy BJ. Adding a web-based

channel using the original price from the traditional channel in both channels (see the last column)

increases profits compared to the base case but provides on average only 78.5% of profits under

Policy BJ. Adding a web-based channel without consideration of the existing channel (Policy WW,

see the first column) lowers profits on average from the base case and provides only 50.1% of the

20

profits compared to Policy BJ. Better performance on average arises under Policies WJ and EJ at

93.7% and 92.2% of Policy BJ results, respectively. However, as can be seen from Table 3, while the

average performance under Policies WJ and EJ are above 90% of the optimal policy, for some values

of the parameters these policies perform poorly.

Table 3: Summary of Experimental Design Results for Monopolist Total Profits for Both Channels Relative to Optimal Prices (BJ)

Scenario 3.2.1 (WW) Scenario 3.2.2 (WJ) Scenario 3.3.1 (EJ) Scenario 3.1 P T * in bothWeb-optimal (P T fixed) Joint-optimum (P T fixed) Joint-optimum (equal prices) (1 channel) channels

HIGH 16 49.2% 91.2% 95.4% 73.2% 82.3%R MED 12 51.0% 95.3% 92.3% 61.0% 81.5%

LOW 8 55.2% 99.4% 76.2% 49.6% 49.8%HIGH 8 67.5% 90.7% 90.1% 44.6% 81.2%

C T MED 6 52.3% 92.3% 96.0% 67.3% 84.7%LOW 4 36.2% 95.1% 93.2% 85.1% 75.4%HIGH 8 30.7% 97.1% 90.3% 85.7% 71.3%

C W MED 6 48.8% 92.8% 96.1% 70.9% 83.4%LOW 4 63.5% 90.4% 92.9% 54.6% 83.0%HIGH 8 52.1% 96.2% 94.5% 59.9% 84.7%

Alpha MED 6 48.6% 92.6% 93.7% 65.9% 79.0%LOW 4 50.1% 91.1% 92.0% 75.1% 77.8%HIGH 1.5 38.1% 97.7% 94.0% 81.1% 89.3%

Gamma MED 1 49.4% 95.1% 92.7% 72.6% 83.6%LOW 0.5 59.0% 88.0% 93.1% 55.6% 70.9%

AVERAGE: 50.1% 93.7% 92.2% 66.8% 78.5%

Notes: * 18 cases in scenarios 2a and 2b have infeasible P T *; then: P T *=R -alpha* In about 50% of the cases, "extreme" pricing is infeasible (owing to large C W )

Basecases

We next examine the sensitivity of profits from the various policies to different values of CW,

CT, R, and γ. Figures 5-8 compare combined profits (across both channels) for Policies WW, WJ, EJ,

and BJ, and also include the base case for a one-channel monopoly discussed in Section 3.1 and the

case where a web channel is introduced with both channels using the price from the base case.

Although we illustrate our insight using a particular example, we found these insights to be consistent

across all our experiments.

Figure 5: Sensitivity to CW

Profits under different policies(C T =6, alpha=4, R =12, gamma 1.5)

1.75

2.25

2.75

3.25

3.75

3.50 4.50 5.50 6.50 7.50C W

$Pro

fit

BJEJWJWWboth PT*Base

21

As expected, profits decrease with the cost in the web channel. If costs in the web channel are

lower than costs in the traditional channel, then Policy BJ is significantly more profitable than all

other scenarios and there is a tremendous benefit possible from adding the additional channel (Figure

5). For higher web channel costs, Policy WJ provides comparable profits to Policy BJ.

Figure 6: Sensitivity to CT

Profits under different policies(C W =6, alpha=4, R =12, gamma 1.5)

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

3.50 4.50 5.50 6.50 7.50C T

$Pro

fit

BJEJWJWWboth PT*Base

c

From Figure 6, we observe that if costs in the traditional channel are relatively low, Policy

WW performs especially poorly. In this example, Policy WJ performed well across all values of CT;

for these parameters it is reasonable to keep PT fixed at its original value, as long as profits are

considered across both channels. Furthermore, if the costs of the traditional channel are high (in

magnitude and in comparison to the web channel costs) then all the strategies (except base case)

perform very well. On the other hand, when costs of the traditional channel are substantially lower, it

is more important to consider both channels while setting prices (the difference in profits of the

alternative strategies is much bigger).

Figure 7: Sensitivity to R

Profits under different policies(C T =6, C W =6, alpha=4, gamma 1.5)

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

6.00 8.00 10.00 12.00 14.00 16.00R

$Pro

fit

BJEJWJWWboth PT*Base

22

Figure 7 shows the effect of R on the performance of the different policies. Higher values of R

lead to higher margins and a greater relative improvement in profits of Policy BJ, especially versus

Policy WW.

Figure 8: Sensitivity to Gamma

Profits under different policies(C T =6, C W =6, alpha=4, R=12)

2.00

2.50

3.00

3.50

4.00

4.50

5.00

5.50

6.00

0.00 0.50 1.00 1.50 2.00gamma

$Pro

fit

BJEJWJWWboth PT*Base

Figure 8 shows the effect of γ on the performance of the various policies. As expected, profits

decrease with γ. Policy WW performs poorly across all values of γ. For lower values of γ, only Policy

EJ remains approximately competitive with BJ, while for higher values of γ, Scenarios WJ, BJ, and

EJ each perform similarly to BJ. The performance of EJ is particularly good when the costs of the two

channels are identical, as in this figure.

Table 4: Prices and Profits Under Competition (Nash Equilibrium)

P*T P*W ProfitT ProfitW Total ProfitsHIGH 12 8.044 7.992 1.275 1.107 2.382

R LOW 8 6.510 6.501 0.309 0.273 0.582HIGH 8 9.787 8.959 0.412 1.095 1.507

C T MED 6 8.661 8.800 0.990 0.925 1.915LOW 4 7.571 8.640 1.837 0.733 2.570HIGH 8 8.911 9.762 1.286 0.343 1.629

C W MED 6 8.758 8.605 1.103 0.834 1.937LOW 4 8.621 7.508 0.899 1.566 2.465HIGH 8 7.830 7.812 0.469 0.396 0.865

Alpha MED 6 7.782 7.758 0.590 0.487 1.077LOW 4 7.658 7.617 0.756 0.611 1.367HIGH 1.5 7.567 7.528 0.456 0.290 0.746

Gamma MED 1 7.523 7.523 0.431 0.431 0.862LOW 0.5 7.387 7.515 0.361 0.838 1.199

Average Prices and Profits Under Competition

23

Table 4 summarizes the results of the competition scenario. As we would expect, the average

PT* and PW* increase with R. We also note that a decrease in costs in a single channel causes prices

in both channels to drop. Somewhat less intuitive is that prices also both increase with α and γ,

although it is interesting to note that neither of the prices is particularly sensitive to α and γ.

Surprisingly, the price in the traditional market, PT*, is more sensitive to changes in γ than is PW*.

Under competition, we note that the average price in the traditional channel is lower than the price in

the web channel if the web channel is more convenient (γ<1). This is consistent with the general

intuition that prices should be lower in the traditional channel in order to entice customers who may

otherwise be inclined to buy from the web channel. It is to be noted that this result is the reverse of the

result we noted when total channel profits are maximized by a monopolist (Policy BJ).

In every scenario that we ran, the highest prices occurred when total channel profits are

maximized in a coordinated manner (Policy BJ). This result is consistent with economic theory,

which suggests that increased competition (arising from greater autonomy) will trigger lower prices

and greater consumer surplus. It is also consistent with empirical studies indicating that prices

charged by pure dot-com retailers are lower than the web prices charged by multi-channel retailers.

Our research suggests that a multi-channel retailer trying to maximize profits across both channels in

a coordinated fashion would naturally set prices higher than a pure dot-com company that views the

traditional channel strictly as a competitor.

As retailers consider adding a web channel, we find that it is important not to ignore profits

from the existing channel in the determination of the web-channel price. In particular, Policy WW,

which encourages the web channel to act entrepreneurially and maximize its own profits, does so at

the expense of the existing channel and often leaves total profits (across both channels) far from

optimal. In general, Policy WJ, which holds traditional prices steady but chooses web prices to

maximize profits over both channels, seems to be a reasonable strategy based on our computational

experiments. This might be comforting to managers who wish to experiment with a web channel

while not altering the existing channel. Policy EJ, which prices both channels equivalently, performs

reasonably well, especially when the costs in each channel are nearly identical. As the discrepancy

widens between costs in different channels, the level of profitability obtainable by keeping both prices

equal decreases relative to the optimal profit level. In practice, many companies have opted to match

prices in the web channel to the traditional channel prices, but when costs differ in the two channels,

the overall profits from such a strategy can be significantly lower than the optimal profits that would

arise from choosing the optimal prices for each channel.

24

5. Conclusions As firms complement traditional channels with the addition of web channels, they must

determine the level of autonomy for the new channel, and they are faced with determining optimal

pricing across the channels. The conclusions summarized in Table 5 highlight the basic relationship

between prices under various scenarios given comparable costs.

We find that in a monopoly case, the web entrant need not worry about perceived convenience

of the web in setting web prices if the traditional price will not be adjusted based on a web entry. The

web price is determined by the relative cost of the web channel. If only the web channel is being

optimized, the web price will be lower under identical costs for the two channels while if both

channels are being optimized, the web price will be higher under identical costs.

Table 5: Summary of Relative Pricing Section Scenario PW PT γγγγ Result

WW

Decision

(R+CT)/2

Doesn’t matter

CW=CT ⇒ PW*<PT

3.2: Web entrant, no

response from traditional

channel

WJ

Decision

(R+CT)/2

Doesn’t matter

CW=CT ⇒ PW*>PT

EJ

Decision (PW=PT=P)

CW=CT ⇒ P*>PT

BJ

Decision

Decision

γ > 1

CW=CT ⇒ PW*>PT*

3.3: Web

entrant, prices jointly

optimized for supply chain

profits BJ

Decision

Decision

γ < 1

CW=CT ⇒ PW*<PT*

NE

Decision

Decision

γ > 1

CW=CT ⇒ PW*<PT*

3.4: Web

entrant, with response from

competitor

NE

Decision

Decision

γ < 1

CW=CT ⇒ PW*>PT*

When choosing to keep prices in both channels equal (Policy EJ), the optimal price is higher

than the original price in the traditional channel. The relative prices under BJ are influenced by the

magnitude of γ; if the web is relatively inconvenient (γ>1), the web price is higher than the traditional

price and visa versa. In the scenarios with competition (NE), the relative prices again are determined

25

by the magnitude of γ; but in this case if the web is relatively inconvenient (γ>1), the web price is

lower than the traditional price and visa versa.

It is important for a monopolist adding a web channel to take into account the impact on the

existing channel. A skunk works approach to developing a web channel may provide a means to

develop expertise with the new channel, but may also lead to a reduction in profits for the firm as a

whole. In pricing the web channel, the firm is better off by accounting for profits from the existing

channel as well, especially if costs in the channels differ. In many cases, it is effective to use a

simplified strategy of keeping the traditional price unchanged, while setting the web-channel price to

optimize profits over both channels, unless the web channel is perceived to be significantly more

convenient on average (i.e., low γ ). If costs in the two channels are comparable, it is also reasonable

to price the two channels identically, although not at the original price of the traditional channel.

Within the assumptions of this simple model, future work could be done to explore other

channel management challenges such as the interactions in a dealer network with exclusive territories

that face competition from an alternate channel such as the Internet. The framework might also be

modified to explore the channel management problem from the perspective of the manufacturer.

While the restrictive assumption about the randomness of distances and the simplified structure of the

supply chain in this model limit our ability to generalize fully our results, maintaining tractability

likely would become an issue when the assumptions are relaxed.

Acknowledgements The authors thank David Rubin for his suggestions on section 3.2.2 and Sebastian Heese for

his extensive computational assistance in this research. In addition, the helpful comments of seminar

participants at the Kenan-Flagler Business School, University of North Carolina; The McDonough

School of Business, Georgetown University; OR Department at North Carolina State University and

the First East Coast Round Table on Operations Research and E-Commerce at MIT are gratefully

acknowledged. The authors also acknowledge the research support provided by the Center for

Technology and Advanced Commerce at University of North Carolina.

26

Appendix

Proofs of Section 3.2

In section 3.2 we assume that the traditional channel does not respond to the web entry and thus the

web entry can price assuming that PT remains constant at ( ) / 2T TP R C= + . We must prove Theorems

1 and 2 for both Case 1 and Case 2: for Theorem 1, we prove that * W T W TC C P P≥ + ∆ ⇔ ≥ from

Case 1, and * W T W TC C P P< + ∆ ⇔ < from Case 2. The proof for Theorem 2 is exactly parallel.

Lemmas 1b and 2b follow from the proofs from Case 1, while Lemmas 1a and 2a follow from the

proofs from Case 2.

Section 3.2.1:

In this scenario, expected profit is maximized over only the web channel. Expected profit is

( ) ( ) ( )1 2W W W WP W W P CΠ = + − . (8)

Proof of Theorem 1 and Lemmas 1b and 1a

Using Case 1,

( ) ( ) ( )

( ) ( )

1 2

22

1 112

W W W WW W

WTW W W

W

P W W P CP P

R PR P R P P CP α α α γ

∂ ∂Π = + −∂ ∂

− �∂ − � �= − + − − � � �∂ �

First order conditions lead to the result for Lemma 1b:

( ) ( )( ) ( ) ( )( )2* 1 2 2 3 2 2 23W W T W T W T TP C P C P C P R P Rα α α α = + + ± + + − + + − +

Given 2

TT

R CP += , then

( ) ( ) ( )( )( )( )2* 1 2 2 3 23W T W T W W T W WP C C R C C R C R C C R C Rα α α= + + + ± + + + − + + + + (9)

It can be shown that the negative root is a local maximum and the negative root a local minimum of

the cubic profit function. The positive root can be shown to occur only in regions where Pw>R, which

is outside our feasible range. The negative root from (9) is either viable ( *WR P Rαγ− ≤ ≤ ) or below

our considered domain ( R αγ− ), in which case the boundary is optimal: *WP R αγ= − .

We use the results above to prove Theorem 1 by noting the following equivalent statements:

*W TP P≥ is equivalent to

27

( ) ( ) ( )( )( )( )( ) ( ) ( )( )( )

2

2

1 2 2 3 23 2

2 3 2 22

TT W T W W T W W

TT W W T W W W

R CC C R C C R C R C C R C R

R CC C R C R C C R C R C

α α α

α α α

++ + + − + + + − + + + + ≥

++ + + − + + + + ≤ + −

Both sides can be squared and,

( ) ( ) ( )( )( )2

22 3 2 22

TT W W T W W W

R CC C R C R C C R C R Cα α α + + + + − + + + + ≤ + −

which leads to (after much algebra): T WC C+ ∆ ≤ . Therefore, *W T W TP P C C≥ ⇔ ≥ + ∆ .

We now use Case 2 to prove *W T W TP P C C< ⇔ < + ∆ .

Using Case 2, first order conditions lead to the result for Lemma 1a:

( ) ( )( )

( )( ) ( )

1 2

2

1 11 22

W W W WW W

WTT W T W W

W

P W W P CP P

R PR P R P R P P P CP α α γ α γ

∂ ∂ ′ ′Π = + −∂ ∂

−∂ − = − + − − + − ∂

Using 2

TT

R CP += we obtain

( )2

*2 16

2

TWW

W

R CR CP

R Cα

−+= −

+ − ∆=

Expected profit ( )2

* 1, 22

WW W T

R CP Pαγ

− Π = − ∆

( )W WPΠ is concave in PW since:

( ) ( )( ), , 1 2 2 / 0W W W WP P W W P P W WP W W P C αγ∂ Π = ∂ + − = − ≤

We use the results above to prove Theorem 1 by noting the following equivalent statements:

*W TP P< is equivalent to

( ) ( )/ 2 / 2W T

W T

R C R CC C

+ − ∆ < +< + ∆

Therefore, *W T W TC C P P< + ∆ ⇔ < .

Proof of Lemma 1a’: Case 2b (see Figure 3) arises when 2WC R αγ< − + ∆ , shown as follows: The

derivative of PW* with respect to CW can be shown to be strictly positive in Case 2 (seen directly from

28

equation (4)) and in Case 2b, where

( ) ( )( )2 2 21* 2 2 2 33W W T T W T WP C P P C P Cαγ αγ α α γ= + − + − − − − + and

( )( )( )2

* 1 1 03 6

T WW

WT W

P CPC P C

αγ

αγ αγ

− −∂ = − > ∂ − − +

. Thus, since 2WC R αγ= − + ∆ when *WP R αγ= − , if

2WC R αγ< − + ∆ , then *WP R αγ< − .

Proofs of Section 3.2.2:

In WJ, the scenario represented in Lemmas 2a and 2b and in Proposition 2, we again assume

that the traditional channel does not respond to the web entry and determine PW assuming that PT

remains constant at ( ) / 2T TP R C= + . In this scenario however, expected profit is maximized over

both channels: ( ) ( )( ) ( ) ( )1 2 1 2W W W T TP W W P C T T P CΠ = + − + + − . (10)

Proof of Theorem 2 and Lemmas 2b and 2a:

The proof of Proposition 2 is exactly parallel to the proof for Proposition 1 and is not repeated

here. First order conditions, along with ( ) / 2T TP R C= + and Case 1 lead to Lemma 2b:

( ) ( ) ( ) ( )( )1 2 1 2W W W T TW W

P W W P C T T P CP P∂ ∂Π = + − + + − ∂ ∂

( ) ( )( )* 2 22 21 2 3 4 3 8 6 8 4 2 46W T W T T W W WP C C R C R C C R C Cα α α α= + + + ± − + − + + + − + (11)

Next we show that only the second of these roots is viable through a contradiction.

Assume the positive root from (9) is viable. Then, from (2),

( ) ( )( )2 22 21 2 3 4 3 8 6 8 4 2 46 T W T T W W WC C R C R C C R C C Rα α α α+ + + ± − + − + + + − + ≤ . Then,

( ) ( )2 22 23 8 6 8 4 2 4 3 2 4T T W W W T WC R C C R C C R C Cα α α α− + − + + + − + ≤ − − − . If the right hand

side is negative, then a contradiction arises. If it is positive, then

( ) ( ) ( )( ) ( )( )

22 22 23 8 6 8 4 2 4 3 2 4

2 0

T T W W W T W

W T

C R C C R C C R C C

R C C R

α α α α

α

− + − + + + − + ≤ − − −

− − − <

which is a contradiction since both terms are positive.

First order conditions, along with ( ) / 2T TP R C= + and Case 2 lead to Lemma 2a:

29

( ) ( ) ( ) ( ) ( )( )1 2 1 2W W W T TW W

P W W P C T T P CP P∂ ∂ ′ ′ ′ ′Π = + − + + −

∂ ∂

( )* 2T WP R C= + + ∆

Proofs of Section 3.3

Proof of Lemma 3 and Corollary 5a: Given identical prices, either Case 1 or Case 2 can be used

for determining probabilities W1, W2, T1, and T2. Expected profit is

( ) ( )( ) ( )( )1 2 1 2W TP W W P C T T P CΠ = + − + + − .

( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )

1 2 1 2

2 23 4 2 2

W T

T W T W W T

P W W P C T T P CP P

P C C P R P R C C P R C R C Pα α α γ α γ

∂ ∂ Π = + − + + − ∂ ∂

= + − + − − − − + + + + −

First order conditions lead to Lemma 3:

( ) ( )( )( ) ( )( ) ( )( ) ( )( )

*2

2 216 2 2 12

T W

T W W T

C C R RP

C C R R R C R R C R

α αγ

α αγ α αγ αγ

+ + − + − + = + + − + − − − − − + −

.

( ) ( ) ( )2

2 2

2 3 2* T WC C P R

PP

α αγα γ

+ − − + +∂ Π =∂

( ) ( ) ( ) ( )2

2 2 W TR P R P P C P Cα αγα γ

− − + − − − − − −= ,

which is less than zero, since and P R P Rα αγ> − > − .

Substituting C for CW and CT leads directly to Corollary 5a.

Proof of Corollary 5b:

Consider a retailer that is currently charging the price chosen in section 3.1 in both the

traditional and web channels. By increasing the price, the retailer gains increased margin on those

customers that still purchase the product, but loses the margin on customers that no longer purchase

the product.

11R P R P R PGain dPα α α γ

− − − = + −

30

( ) 11dP R P R PLoss P C γα α α γ

− − � � �= − − + − � � � �

( ) 11 1d dP R P R P R P R PGain Loss R P P CdPΠ γ

α α γ α α γ − − − − = − = − + − − − − + − � � � � � �

� �

( ) ( )1 1dP R P R PR P P C R P P C P Cα γ α αγ

− − = − − + + − − − + + −

We wish to evaluate ddPΠ at

2R CP += , so 0R P P C− − + = .

( ) ( )2 0d dP P C R PdPΠ

α γ= − − >

Since an increase in price increases profits, the optimal price is greater than 2

R C+ .

Proof of Theorem 3:

Consider the optimal price P* for Policy EJ. In relaxing this constraint for Policy BJ the optimal

prices will change if 1≠γ . As we start from the optimal solution of the constrained problem, the

change in prices can only take the form of lowering one price while increasing the other (otherwise

P* would not have been optimal). Let CPm −= * be the original margin for both products.

Figure 9

α*PR −

α*PR −

γα

)*( ∆−− PR

T

W ∆W

α)*( ∆−− PR

α∆

α∆

0

1

W

T

∆W

∆T

31

We compare the effects of a price reduction of magnitude ∆ in either channel on overall profits. Let

W∆Π and T∆Π be the changes in profits owing to a price reduction for the web-channel and the

traditional channel, respectively. We look at the difference TW ∆Π−∆Π and use Figure 9 as

illustration. As the original case is symmetric, the effects of the price-reductions on the part of the

market that is represented by the lower left corner are identical and cancel each other out.

Decreasing the web-price by ∆ leads to new sales at margin ∆−m in area

( ) ( )( )1 *W R P∆ ∆ αγ α= − − and to losses owing to reduced prices in area

( )( ) ( )( )1 * *W R P R Pα αγ= − − − . Similarly, lowering the traditional price leads to new sales in

( ) ( )( )1 *T R P∆ ∆ α αγ= − − and to losses in ( )( ) ( )( )1 * *T R P R Pαγ α= − − − .

Then: ( )( )( ) ( ) ... (1 )[2 * ]W T W T m W T P C R∆Π ∆Π ∆ ∆ ∆ ∆ ∆ αγ γ ∆− = − − − − = = − − − − . As the

potential for market segmentation causes P* to be greater than the monopolist’s optimal price

2CRPM

+= , we find that, for sufficiently small ∆, the profit increase from lowering the web-price

is greater than the gain from reducing the traditional price if 1<γ and it is smaller if 1>γ . We are

indifferent if 1=γ , i.e. the optimal price remains P* for both channels.

Figure 10

PW

PT

P*

P*

0

PT=PW

PT<PW

(γ>1)

PT>PW

(γ<1)

32

Comparing the changes at P* suffices to determine which price is greater, as the profit function is

jointly quasi concave in both prices and as no further improvement of the objective function can

lead to the region presenting the opposite order of the prices without crossing the line of equality (as

indicated in Figure 10). However, any crossing of this line would contradict the assumption that P*

was the optimal solution for the constrained problem.

Proof of Theorem 4:

Theorem 1.2 in Fudenberg and Tirole (1991) guarantees the existence of a pure strategy Nash

Equilibrium provided that (1) the strategy space for each channel is a nonempty, compact convex

subset, and (2) the profit function for each channel is continuous and quasi-concave with respect to

the channel’s price. For any T TP C< and W WP C< , the profits will clearly be negative. Profits are

zero when T TP C= or W WP C= . Consider the case when W TP P< . WΠ is a quadratic function that is

concave with respect to WP . Setting 0T TP∂Π ∂ = identifies two extreme points (the positive and

negative roots of the quadratic equation). Under our condition that WP R αγ≥ − , the positive root can

easily be shown to be greater than or equal to R , and therefore outside our range of consideration.

Since TΠ is non-decreasing below the negative root and non-increasing beyond this root, TΠ is

quasi-concave when *WR P Rαγ− ≤ < . Since a similar argument can be made for the case when

W TP P> , the two profit functions meet the conditions presented in Fudenberg and Tirole (1991).

Response functions when W TP P< :

By setting 0W WP∂Π ∂ = , we obtain the following response function for the competitor using

the web channel: ( ) ( )22 4W W TP R C R P α= + − − .

Similarly, the traditional competitor adheres to the following response function:

( )2 2 2 2 22 2 4 4 3 2 2 8 6 6 3T T W T W T W T W WP C P C P R C P C P R RPαγ α γ αγ αγ αγ= + + − + + + − − + − −

33

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