Competitive Behaviour-Based Price Discrimination among Asymmetric Firms

Elias CARRONI

CERPE – March 2015

Department of Economics

Working Papers

2015/01

Competitive Behaviour-Based Price

Discrimination among Asymmetric Firms∗

Elias Carroni†

This version: March 2015

Abstract

This article studies the effects of Behaviour-Based Price Discrimina-tion (BBPD) in a horizontally and vertically differentiated duopoly. In atwo-period model, firms are allowed to condition their pricing policies onthe past purchase behaviour of consumers. The paper shows two differenttypes of equilibria depending on the strength of vertical differentiation. Ifthe difference in quality is small enough, both firms steal each other’s con-sumers and suffer a situation in which prices and profits are lower and theconsumer surplus increases. When quality differentials are instead substan-tial, asymmetric behaviours arise: the high-quality firm sells its productto few consumers at a high price in the first period and then becomes ag-gressive in the second one. As a consequence, customers only move fromthe low-quality to the high-quality firm. If consumers are myopic, theODS scenario is detrimental for them and beneficial for firms in relationto uniform pricing. If instead consumers are forward-looking, they and thelow-quality firm are better off and the high-quality firm is worse off whenBBPD is viable.

JEL codes: L1, D4Keywords: Competitive BBPD, Vertical Differentiation, Switching

∗I wish to thank Eric Toulemonde, Paul Belleflamme, Jean Marie Baland, Marc Bourreau,Rosa Branca Esteves, Paolo Pin, Stefano Colombo and Gani Aldashev for their suggestions. Ithank the participants to the Doctoral Workshops 2014-2015 at UCLouvain, Doctoral Workshop2014 at DiSea-University of Sassari, 2015 RES PhD Meeting at UCL (London), 2014 ASSETConference (Aix-en-Provence), 2014 Simposio SAEE (Palma de Mallorca). I acknowledge the“Programma Master & Back - Regione Autonoma della Sardegna” for financial support.

†Department of Economics, CERPE, University of Namur, email: elias.carroni@unamur.be.

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1 Introduction

Customer’s recognition represents an increasingly important matter in economics.Indeed, the development of big data and the availability to firms of consumers’sensible information have raised issues concerning consumers’ privacy. Moreover,the improvements in obtaining and processing such information enable firms toinfer preferences of consumers and to discriminate based on their past purchasebehaviour (BBPD). Since this pricing strategy is being used frequently, it hascaptured the attention of many scholars,1 whose main concern has been the un-derstanding of its effect on firms’ profits, consumer surplus and level of prices.

The present paper participates in this debate investigating the effects of BBPDwhen competing firms are assumed to be located at the endpoints of a Hotellingline and to offer goods of different qualities. In particular, in a two-period model,forward-looking firms can observe the purchase behaviour of consumers and thusare allowed to discriminate between old and new buyers. Depending on therelative strength of brands’ vertical differentiation (difference in the quality ofthe good offered), the model exhibits two different equilibria. For weak verticaldifferentiation, unsurprisingly, the paper accords with the previous literature ofBBPD with symmetric competitors: firms set prices in such a way that both stealeach other’s consumers in the second period (Two-Direction Switching, TDS).The model is able to replicate the results of Fudenberg and Tirole (2000) if thedifference in quality is assumed to be zero and gives the essential welfare result ofsymmetric BBPD: consumers are better off and firms both worse off when pricediscrimination is feasible.

As soon as a certain level of vertical differentiation is reached, the paperadds important findings to the current literature. Specifically, firms convergeto asymmetric pricing behaviours in the first period: the strong seller adopts amargin-focusing strategy and the low-quality rival conquers most of the market.As a consequence, buyers move in the second period only from the low-qualityto the high-quality firm (One-Direction Switching, ODS). On top of that, whendifferentiation is very pronounced, ODS causes the exit of the small firm, thatwould have been active under uniform pricing.

This inter-temporally unbalanced equilibrium follows from the fact that ver-tical differentiation creates an important asymmetry in the incentives that eachfirm has in the first period. In particular, the stronger the vertical differentiation,

1Starting from Chen (1997), Villas-Boas (1999) and Fudenberg and Tirole (2000). Esteves(2009) provides an extensive and up-to-date survey on the existing literature in this field.

2

the more the high-quality firm best response is to use extreme pricing strategies,i.e., either to be very aggressive focusing only on current market share (and be-coming monopolist in the first period) or to be benevolent focusing on marginsand letting the rival conquer most of the Hotelling line (but becoming monopolistin the second period). Clearly, the first strategy is preferred when the rival setsa low price, since the fight becomes so hard to induce to lay down arms today,aware of the fact that this brings to cheap ODS tomorrow (as switchers are rela-tively close). On the other side, the low-quality firm anticipates that attractingconsumers tomorrow will be more difficult as differentiation becomes strongerand it prefers to focus on conquering the largest possible market share in the firstperiod. This pushes it to be aggressive and to pursue a market-share focusingstrategy.

With these mechanisms in mind, the implications on profits and consumersurplus of the asymmetric equilibrium are straightforward. When consumers aremyopic, BBPD becomes a very powerful tool for the high-quality seller, which isgiven the possibility to decide the destiny of the rival. At equilibrium, the high-quality firm decides to focus on margins and the low-quality firm enjoys a largemarket share. In this scenario, firms reach endogenously a sort of market sharingagreement, which allocates the surplus over time: the high-quality seller tradestoday’s for tomorrow’s market share and the low-quality firm does exactly theopposite. This turns out to be ex-post preferred by firms to the uniform pricingas it reduces price competition in the first period. Concerning the low-qualityfirm, the positive effect of reduced first-period competition compensates the dis-advantage provoked by its exit (or cornering) in the second period. Consumerswill be worse off as they suffer the reduced competition in terms of prices in thefirst period and in term of number of competitors in the second period.

When consumers are instead forward-looking, they anticipate tomorrow switch-ing and the first period “elasticity of the demand” changes compared to the uni-form pricing.2 This is a standard result of BBPD literature: Fudenberg andTirole (2000) discuss how the sensitivity to price of the market splitting locationis weaker when consumers are forward-looking in relation to uniform pricing.This boosts first-period prices under BBPD. Oppositely, in the present model,when switching is uni-directional the elasticity to price is increased by BBPD.

2Technically speaking, the demand in the model is inelastic. Nevertheless, using the ex-pression elasticity helps capture how market shares respond to marginal changes of prices indifferent ways moving from the uniform pricing to the case of BBPD with forward-lookingconsumers.

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This induces the low-quality firm to be more aggressive than in the myopic case.As a consequence, the same force that lets the high-quality firm exert a strongpower when consumers are myopic, becomes a curse when consumers are forward-looking. Namely, if in the first case differentiation helps the high-quality firm tomake high margins and lets the low-quality firm enjoy a weakened competition,in the second one it imposes to the low-quality firm to be aggressive and thehigh-quality firm suffers the increased competition in the first period. The resultoverall is that the low-quality firm and the consumers are better off under thediscriminatory pricing at the expenses of the high-quality firm.

Related Literature. The paper belongs naturally to the literature studyingprice discrimination in oligopolies, which generally agrees on a negative effect onfirms’ profits compared to uniform pricing. This is because the typical positiveeffect in the monopoly case (the so-called Surplus Extraction effect) is accom-panied and often overturned by an intensification of competition in oligopolisticmarkets (Business Stealing effect). As a matter of fact, the information aboutbrands preferences of consumers can be used in two different ways when marketsare duopolistic. On the one hand, each firm wants to charge consumers belong-ing to its “strong” market (i.e., exhibiting relatively strong brand preference)with a high price, thus exploiting information in order to extract their surplus.On the other hand, a given seller also wants to set a low price in its “weak”market to steal rival’s business. In the jargon used by Corts (1998), the marketexhibits best-response asymmetry, as the“strong” market for a firm is “weak”for the competitor. In these cases, the firms’ dominant strategy is to charge lowprices in the rival’s “strong” market and this, in turn, prevents the latter to fullyextract surplus. In a very influential article, Thisse and Vives (1988) show thatif firms know the precise location of each consumer and can accordingly engagein perfect price discrimination, then all prices might fall in relation to uniformpricing as the more distant firm is very aggressive in each location. For givenprices offered by the rival, both firms find it profitable to discriminate, but thisleads to a reduction in prices in the style of a prisoner’s dilemma situation.

The paper is more specifically linked to the literature on BBPD, in which firmslearn consumers’ preferences by observing their purchase behaviour in the pastrather than have full information about their locations. In Chen (1997),Villas-Boas (1999), Fudenberg and Tirole (2000) and Esteves (2010), the observationof consumers’ identities allows sellers to distinguish between “strong” market(previous buyers) and “weak” market (rival’s inherited consumers), as purchase

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reveals how much a consumer is inclined to buy one or the other product. Theloss of firms and consequent gain of consumers are still there: as the latter canbe identified and price discrimination is permitted, both sellers have incentivesto steal each other’s consumers and prices fall down. More recent articles havedemonstrated how results may slightly or substantially differ under different set-tings. In a very recent paper, Colombo (2015) studies the incentives to pricediscriminate shown by a firm facing a discriminating competitor. He demon-strates that if consumers are myopic enough, the optimal choice is to commit touniform prices even if the access to information about purchases of consumersis completely costless. Furthermore, Esteves and Reggiani (2014) show how in-creasing the demand elasticity reduces the negative impact of BBPD on firms’profits, while Chen and Pearcy (2010) demonstrate that when a weak correlationover time between preferences of consumers is assumed, then BBPD will actuallybe beneficial for firms and detrimental for consumers.

The intuition behind the present paper is that the welfare effect of BBPDdepends crucially on the symmetry of the market: if firms are identical ex-anteand compete fiercely for switchers, they end up poaching the same number ofconsumers with the consequence of a lower level of prices and profits. In theanalysis of their two-period model, for example, Fudenberg and Tirole (2000) needspecifically to eliminate asymmetric subgames in order to provide their SPNE.3

Namely, they do not take into account how an inherited market unbalanced infavour of one of the two firms may imply switching only from the dominant tothe dominated firm.4

Other articles dealing with price discrimination in asymmetric duopolies haveresults directly comparable with the ones of this paper. As pointed out by Chen(2008), the effects of dynamic price discrimination change substantially fromsymmetric to asymmetric markets. In a considerably different approach fromthe present paper with regard to time horizon and consumers’ preferences, hefinds that price discrimination can be a tool for a low-cost firm to eliminate theless efficient competitor and if exit happens consumers are worse off comparedto uniform pricing. Shaffer and Zhang (2002) propose a model where vertically

3From the article at page 639:“We will show that, provided that |θ∗| is not too large, thesecond-period equilibrium has this form: Both firms poach some of their rival’s first-periodcustomers, so that some consumers do switch providers”. In their model |θ∗| represents thelocation of the time 1 indifferent agent in a Hotelling with firms symmetrically located aroundzero.

4See Gehrig et al. (2007) for an analysis of Fudenberg and Tirole (2000) second period withthe past taken as given.

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and horizontally differentiated firms are allowed to (costly) target consumerswith one-to-one promotions (perfect price discrimination). They find that eventhough promotional offers intensify price competition they can result in a benefitin terms of market share and profits for the high-quality firm. In Liu and Serfes(2005), firms can costly acquire information about consumers-specific characteris-tics. They show that when information is not too costly, only the high-quality firmwill buy it and engage in price discrimination, with the low-quality firm optingfor a uniform price strategy at equilibrium. Differently from the last two articles,in the present model information cannot be acquired and price discriminationis only based to past purchase behaviour and, for strong vertical differentiation,price discrimination benefits the low-quality firm, as price competition is relaxedin the early stage. Gehrig et al. (2011, 2012) propose models in which the asym-metry of the firms is given by some inherited market dominance and firms areallowed to discriminate prices according to the (exogenous) purchase history ofconsumers.5 Roughly speaking, their analysis is similar and allows for switchingbehaviours switching similar to the subgames of the model presented hereafter,which endogenises the purchase history of consumers.

The rest of the paper is organised as follows. The next section presents themain ingredients of the model. After, sections 3 and 4 are devoted to the analysisof the two benchmarks of uniform and discriminatory pricing. The two regimesare then compared in order to provide a welfare analysis on the effects of BBPDin Section 5. Finally, Section 6 contains some concluding remarks.

2 Description of the model

Two competing firms i = H,L aim to sell a good to a population of customersassumed to be uniformly distributed along a unit segment. Firms locations arekept fixed at the end-points of this segment: firm H is located at lH = 0 andL at lL = 1. Sellers are vertically differentiated, as the qualities of the productsthey sell are different. For the sake of simplicity, firm H is assumed to sell thehigh-quality good. Formally, it is assumed that qH ≥ qL, where qk denotes thequality of the product offered by firm k ∈ {H,L}.

Consumers face a transportation cost normalised to 1 per unit of distancecovered to reach the location of each firm and valuate linearly the quality of the

5In particular, Gehrig et al. (2011) provides the limit case of an entry model.

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good they buy. According to these assumptions, the per-period utility of an agentlocated at x who buys good i will be given by:

U(x, i) = qi − pi − |x− li|. (1)

Firms set prices in order to maximise profits, facing a unitary cost normalisedto 0 in each time period and discounting the future at a factor δ < 1. Each timeperiod is composed of two stages. In stage (1.1) firms simultaneously set pricespH1 and pL1 and in stage (1.2) consumers decide upon purchase. In stage (2.1),firms simultaneously set prices knowing who bought which good in period 1: piH2is defined as the price set by firm i for a consumer who bought good H in period1, while piL2 is charged to L’s inherited clients. In the last stage (2.2) consumersobserve the new prices and buy again.

The following sections provide a complete analysis of the model. In particular,the next section introduces a benchmark case in which customer’s recognitionis not allowed, useful to isolate the effects of BBPD. The subsequent sectiondescribes the possible equilibria when firms are allowed to engage in BBPD.

3 Uniform Pricing

Assume there exists a ban on price discrimination or that customers’ purchasescannot be observed. In this scenario, the utility of an agent buying respectivelygood H and good L will be:

U(x,H) = qH − pH − x, U(x, L) = qL − pL − (1− x).

Accordingly, the indifferent consumer is located at:

x̄ =1

2+

∆+ pL − pH

2, (2)

where ∆ ≡ qH − qL. We assume hereafter that qH and qL are high enough sothat consumers of all locations prefer to buy one of the two products (full marketcoverage) and that the prices chosen by the two firms are not too different inorder to get rid of situations in which one firm corners the market. Accordingly,the cutoff x̄ determines a demand of x̄ for firm H and 1− x̄ for firm L. Moreover,the attention is restricted only to cases in which the difference in quality is nottoo large to eject the low-quality firm out of the market. As can be clearly seenbelow, the necessary and sufficient condition for this to be the case is ∆ < 3,

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which allows firm L to charge an above-marginal-cost price at equilibrium. Thisassumption is maintained hereafter.

Anticipating the reaction of consumers, firms set prices in order to maximisethe following static profits:

πH = pH(1

2+

∆ + pL − pH

2

)

, πL = pL(1

2− ∆+ pL − pH

2

)

.

It is worth noticing that, in comparison with the standard Hotelling with equalqualities, firm H can charge higher prices as ∆ > 0 and the opposite happens tothe low quality firm. Indeed, the equilibrium prices are the following:

pHu = 1 +∆

3, pLu = 1− ∆

3.

They take into account both horizontal (through the transportation cost, 1) andvertical (through the term ∆/3) differentiation. Specifically, 1 represents themarket power that both firms enjoy on consumers, whereas ∆/3 is the resultof the competitive advantage that firm H enjoys because sells a higher qualityproduct. The prices above result in the following static equilibrium profits:

πHu =

(3 + ∆)2

18, πL

u =(3−∆)2

18.

Under uniform price in both periods, subgame perfect Nash equilibrium givesa replication of the static equilibrium, with the following overall profits:

πHu =

(1 + δ)

18(3 + ∆)2, πL

u =(1 + δ)

18(3−∆)2. (3)

4 Observation of Consumer Purchases and BBPD

In this section, first-period prices as well as the behaviour of first-period con-sumers are assumed to be observable to both firms when they choose second-period discriminatory prices. Subgame perfection is used as equilibrium concept.

4.1 Second-Period Subgames

In stage (2.2) consumers observe prices for loyalists and for switchers offered byboth firms. In the inherited turf of firm H, a consumer prefers to buy again good

8

H rather than switch seller when qH − pHH2 −x > qL− pLH2 − (1−x), which gives

the following indifferent location:

x̂H2 =

1

2+

∆+ pLH2 − pHH2

2, (4)

so that x̂H agents buy again good H. Defining x̂1 as the inherited market share offirm H,6 x̂1 − x̂H

2 agents will instead switch towards firm L. Concerning the turfof firm L, consumers compare qH − pHL

2 − x with qL − pLL2 − (1 − x). It meansthat all agents located on the right of

x̂L2 =

1

2+

∆+ pLL2 − pHL2

2(5)

will buy again good L, whereas agents located in the interval[x̂1, x̂

L2

]will switch

to firm H.Firms anticipate this reaction of consumers in term of purchase and set prices

in stage (2.1). The analysis at this stage depends on the market shares (x̂1, 1−x̂1)inherited from the first period, which determine the actual chances to have switch-ing from one firm to the other one and the other way around. Differently fromFudenberg and Tirole (2000), who assume the inherited markets to be symmetricenough, here all possible subgames are analyzed in the backward-induction anal-ysis of the model. In particular, we have subgames with two-direction switching(TDS) and subgames with switching only towards one of the two firms (one-direction switching or ODS).

When firms expect switching to occur in both directions, the thresholds de-scribed by equations (4) and (5) are located in such a way that prices can be foundin both turfs such that x̂H

2 < x̂1 < x̂L2 . When instead firms expect switching to

occur only towards the high-quality firm (H), the thresholds above are locatedin such a way that x̂1 ≤ x̂H

2 and x̂1 < x̂L2 . These two examples are depicted in

the figure below.Considering all the possible subgames, the following proposition contains all

possible scenarios when the inherited base of customers x̂1 is considered exoge-nous.

Proposition 1. When firms are allowed to price discriminate between old andrival’s previous consumers, the second-period equilibrium prices are:

6Notice that under the assumption of fully covered market, 1− x̂1 is the first period marketshare of firm L.

9

0 x̂H2

x̂1 x̂L2

1

loyal to H︷ ︸︸ ︷

︸ ︷︷ ︸

switchers to L

switchers to H︷ ︸︸ ︷

︸ ︷︷ ︸

loyal to L

0 x̂1 x̂L2

1

loyal to H︷︸︸︷

switchers to H︷ ︸︸ ︷

︸ ︷︷ ︸

loyal to L

Figure 1: Different Switching Scenarios

(i)pHH2 = 1 +∆− 2x̂1, pLH2 = 0,

pHL2 = 1 + ∆

3 − 43 x̂1, pLL2 = 1− ∆+2

3 + 23(1− x̂1)

}

when x̂1 ≤ 1+∆4

(ii)pHH2 = 1 + ∆

3 + 23 x̂1, pLH2 = 1− ∆

3 − 43(1− x̂1),

pHL2 = 1 + ∆

3 − 43 x̂1, pLL2 = 1− ∆+2

3 + 23(1− x̂1)

}

when x̂1 ∈(1+∆4 , 3+∆

4

)

(iii)pHH2 = 1 + ∆

3 + 2x̂1

3 , pLH2 = 1− ∆3 − 4

3(1− x̂1),pHL2 = 0, pLL2 = 1−∆− 2(1− x̂1)

}

when x̂1 ≥ 3+∆4

Proof. See mathematical appendix.In order to better grasp the intuition behind Proposition 1 let us consider the

equilibrium prices in point (ii). Unsurprisingly, a stronger vertical differentiationis associated with a competitive advantage in favour of the high-quality firm,whose equilibrium prices for old and new consumers are both increasing in ∆.Exactly the opposite relation exists between the prices of the low-quality firmand the vertical differentiation parameter.

On the other hand, the own inherited market share7 affects positively the pricea given firm charges to the old loyal consumers and negatively the one offered tothe switchers. Intuitively, the relation between prices and market share followsdirectly from the effective power that the size of the first-period market createsin each turf for the “attacking” (else turf) and the “defending” firm (own turf).Clearly, the attack in the rival turf turns out to be more costly as the size of themarket already conquered in the first period becomes higher. In other words,the price offered to the switchers should be lower when a lot of consumers wereattracted in the first period, since the non-conquered portion is very far awayin the Hotelling line. For extreme levels of the market share,8 attracting newconsumers is not profitable as it would require a below-marginal-cost price. These

7The market shares will be x̂1 for the high-quality firm and the remaining 1 − x̂1 for thelow-quality rival given the assumption of full market coverage.

8According to the proposition, this level will be 3+∆

4for firm H and 1− 1+∆

4for firm L.

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cases are presented in points (i) and (iii), where respectively firm H and L preferthe dominating strategy of setting prices equal to the marginal cost (i.e., 0) inthe rival turf.

Therefore, from the point of view of the defending firm, the higher the marketshare inherited from the past the weaker the price competition in its own turf,as the rival becomes less aggressive. For this reason, the equilibrium price forloyalists9 is increasing in the inherited market share. In the extreme cases inwhich the attacking rival sets the price equal to the marginal cost (points (i) and(iii) in the proposition), then the optimal response of the defending firm is tooffer to past consumers a price just sufficient not to lose any of them.

These equilibrium prices will determine peculiar switching behaviours of con-sumers. If the first-period market is balanced enough, then both firms succeed infinding profitable prices to offer to rival’s consumers and both are able to attract(and respectively suffer the loss of) some new (old) consumers. If instead themarket is strongly dominated by a firm in the first period, the dominating firmdoes not attract any rival consumers, even though it charges a price equal tothe marginal cost. For this reason, switching will be one-direction towards thedominated firm. These results are formally presented in the following corollary:

Corollary 1. Given the equilibrium prices in Proposition 1: (i) when x̂1 ≤ ∆+14

,consumers only switch to firm H (ODS); (ii) when x̂1 ∈

(∆+14

, ∆+34

), consumers

switch from H to L and vice-versa (TDS); and (iii) when x̂1 ≥ max{∆+34

, 1},consumers only switch to firm L (ODS).

Proof. Plugging the equilibrium prices in proposition 1, it is easy to find thefollowing cutoffs: (i) when x̂1 ≤ ∆+1

4, x̂H

2 = x̂1 and x̂L2 = ∆+2x̂1+3

6; (ii) when

x̂1 ∈(∆+14

, ∆+34

), x̂H

2 = ∆+2x̂1+16

and x̂L2 = ∆+2x̂1+3

6; (iii) when x̂1 ≥ max{∆+3

4, 1},

x̂H2 = ∆+2x̂1+1

6and x̂L

2 = 1− x̂1.

4.2 First Period

In stage (1.2) consumers observe prices and buy the good giving them the highestutility. In what follows, consumers are assumed to be myopic i.e., they only careabout the utility they get at stage (1.2), without anticipating the second-period(possible) switching.10 In what follows, only the case of myopic consumers is

9See prices pHH2 and pLL

2 in point (ii) of proposition 1.10As discussed in the book of Belleflamme and Peitz (2010), the effect of consumers’ far-

sightedness is essentially that BBPD weakens price competition in the first period compared to

11

presented whereas the main features of the forward-looking consumers case arediscussed in a separate paragraph and formally in the appendix.

Under myopia, the first-period indifferent consumer will be located at:

x̂1 =1

2+

∆+ pL1 − pH12

, (6)

so that all agents to the left of the cutoff above buy the high-quality good andall agents to the right buy the low-quality good. Following a backward inductionreasoning, in the initial stage (1.1) forward-looking firms correctly anticipateboth purchase decisions in stage (1.2) and all possible subgames. Anticipatingfirst-period purchase behaviour of consumers expressed by the cutoff in (6) anddiscounting future profits, firm H and L respectively maximize the followinginter-temporal profits:

πH1 + δπH

2 = pH1 x̂1 + δ[pHH2 min

{x̂H2 , x̂1

}+ pHL

2 max{x̂L2 − x̂1, 0

}],

πL1 + δπL

2 = pL1 (1− x̂1) + δ[pLL2 min

{1− x̂L

2 , 1− x̂1

}+ pLH2 max

{x̂1 − x̂H

2 , 0}]

.

Clearly, the future profits depend on the expectations firms have about tomor-row’s movements of consumers. In particular, plugging the prices in proposition1 and the resulting cutoffs expressed in the proof of Corollary 1, second-periodprofits depend on the inherited market share as follows:

πH2 (x̂1) =

πH2H = ∆2+(9−2x̂1(10x̂1+3))+2∆(5x̂1+3)

18if firms expect x̂1 ≤ ∆+1

4,

πH2TDS =

∆2+5(2x̂2

1−2x̂1+1)−2∆(x̂1−2)

9if x̂1 ∈

(∆+14

, ∆+34

),

πH2L = (∆+2x̂1+1)2

18if x̂1 ≥ max{∆+3

4, 1}.

for firm H. Similarly, firm L anticipates profits:

πL2 (x̂1) =

πL2H = (∆+(2x̂1−3))2

18if firms expect x̂1 ≤ ∆+1

4,

πL2TDS =

∆2+5(2x̂2

1−2x̂1+1)−2∆(x̂1+1)

9if x̂1 ∈

(∆+14

, ∆+34

),

πL2L =

∆2+(46x̂1−20x̂2

1−17)+2∆(5x̂1−8)

18if x̂1 ≥ max{∆+3

4, 1}.

uniform pricing because of a lower first-period elasticity to price. In the context of the presentpaper, which analyses all possible scenarios and not only the symmetric one, one can observein the appendix how the farsightedness of consumers would bring to richer results because theresponsiveness to price of the infra-marginal consumer is very strong in the asymmetric cases.

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where the notation H,L, and TDS in the subscripts are used to indicate thatswitching is respectively towards firm H only, towards firm L only or towardsboth directions. The following paragraph is devoted to the description of bestresponses, that exhibit peculiar features due to the fact that firms can choosevery different pricing strategies according to the inter-temporal objective theywant to pursue. Subsequently, the equilibria of the model are presented, givingalso some insights on the main characteristics of prices and switching behaviour.Finally, the last paragraph of this section discusses the case of forward-lookingconsumers, highlighting the main difference with the case of myopia.

Best Responses and Vertical Differentiation. In order to build up thebest responses of firms, the following approach is used. For given price of therival pj1, firm i has three different alternatives. Namely, it can optimally choosea price pi1H(p

j1), p

i1TDS(p

j1) or p

i1L(p

j1) leading respectively to the market splitting

cutoff x̂1 in the interval [0, 1+∆4

] or (1+∆4

, 3+∆4

) or [3+∆4

, 1] and giving firm i thecorrespondent second-period profits. The correspondent profits in each of thethree cases are then compared: the best response will be the one leading to thehighest profit.

Albeit the complete construction of the best responses is left to the ap-pendix, it is worth discussing their main features. The best-reply price will inter-temporally trade-off between today’s profits (market share and per-consumermargin) and tomorrow’s cost of poaching consumers. In particular, choosing anaggressive pricing strategy focusing on today’s market share will entail a relativelow per-consumer margin and makes the attraction of new consumers very costly.When a firm is particularly aggressive today, it conquers a large market and to-morrow only the rival succeed in attracting new consumers. The best responseprice in this case will be respectively

pH1L =(9− 2δ)

18− 2δpL1 +

9− 4δ

18− 2δ+

(9− 4δ)∆

18− 2δ

and

pL1H =(9− 2δ)

18− 2δpH1 +

9− 4δ

18− 2δ− (9− 4δ)∆

18− 2δ.

In principle, when a firm is very aggressive this strategy can also lead to the cornercase in which it becomes monopolist, i.e., pH1M = pL1 +∆−1 or pL1M = pH1 −∆−1.11

11As it will be shown afterwards, these prices are never part of an equilibrium but can bebest responses for high levels of the rival’s price.

13

On the other hand, choosing a benevolent pricing strategy that focuses on highmargins on few consumers in the first period will make the attack of the rivalturf less costly in the second period. Being benevolent today let a firm conquerof a small market today and benefit ODS tomorrow. The best response price inthis case will be respectively

pH1H =(10δ + 9)

18 + 10δpL1 +

9 + 13δ

18 + 10δ+

∆

2

and

pL1L =(10δ + 9)

18 + 10δpH1 +

9 + 13δ

18 + 10δ− ∆

2.

Choosing a more inter-temporally balanced strategy will instead lead to to-morrow’s TDS. In this case, the best response prices will be:

pH1TDS =(9− 10δ)

18− 10δpL1 +

9

18− 10δ+

(9− 8δ)∆

18− 10δ

and

pL1TDS =(9− 10δ)

18− 10δpH1 +

9

18− 10δ− (9− 8δ)∆

18− 10δ.

The global best response is thus found by choosing the alternative leading tothe highest profit among the three strategies described above. It turns out thatthe optimal pricing behaviours change dramatically according to the strength ofthe asymmetry present in the market.

In particular, when vertical differentiation is weaker than horizontal differen-tiation (∆ < 1), the best responses have the following forms:

pH1 (pL1 ) =

pH1H if pL1 ≤ p̂,

pH1TDS if pL1 ∈ (p̂, p̂LC) ,

pH1L if pL1 ≥ p̂LC ,

pL1 (pH1 ) =

pL1L if pH1 ≤ p̃,

pL1TDS if pH1 ∈ [p̃, p̃HC ] ,

pL1H if pH1 ≥ p̃HC ,

where

p̂ ≡√

(9∆+9)2−(5δ∆+5δ)2

30+ 65δ−15δ∆

90− 3∆+3

10, p̃LC ≡ 18−3δ∆−5δ

9,

p̃ ≡√

(9−9∆)2−(5δ−5δ∆)2

30+ 65δ+15δ∆

90+ 3∆−3

10and p̃HC ≡ 18+3δ∆−5δ

9

are the cutoff values of rival’s prices that induce each firm to switch from onestrategy (aggressive, balanced, benevolent) to another one.

14

pH1 (pL1 )

pL1

pL1 (pH1 )

pH1H pH1TDS

pH1L

pL1L

pL1TDS

pL1H

Figure 2: Best Responses: ∆ < 1

15

Intuitively, being aggressive (respectively benevolent) today is preferred whenthe rival is benevolent (aggressive). Indeed, if the rival sets a high price, beingaggressive today does not entail a substantial cost in terms of lower margins.Therefore, since a firm is given the possibility to make high margins on a largemarket today, it does not care at all about tomorrow switching. Oppositely, whenthe rival is very aggressive, a seller will let it conquer a large part of the marketenjoying uni-directional switching tomorrow. In other words, the seller lays downarms today when the fight becomes too hard, aware of the fact that this bringsto a cheap conquest of rival territory tomorrow.

Finally, firm i will prefer pi1TDS when the rival chooses an intermediate price.In this segment, the best response is less steep than in the two extreme cases dueto the fact that involves an inter-temporal balance of incentives. Namely, oncea firm prefers to play a ODS equilibrium (either to itself or to the rival), thensecond period turns out to be less important because the change determined by aprice cut (or increase) today does not involve sizeable changes in the movementsof consumers tomorrow. Thus, the best response will be more sensitive too anyprice change of the rival compared to the TDS case, in which future movementsof consumers are more crucial.12

As vertical differentiation is stronger than horizontal differentiation, ODS tofirm L is no more a threat for the high-quality firm, as it is reachable only ifthe latter becomes a monopolist in the first period.13 In particular, as the pricecharged by the low-quality firm reaches a cutoff (i.e., pL1 > p̂M ≡ 27+2δ∆−10δ−9∆

9),

pH1TDS is not optimal any more and firm L cannot enter the market in the firstperiod. Moreover, the cutoff price p̂M is decreasing in the level of vertical differ-entiation, meaning that the stronger the asymmetry the narrower the segment inwhich the high-quality firm finds it optimal the inter-temporally balanced strat-egy leading to TDS. This will determine the following firm H best response forincreasing levels of asymmetry:

12Notice that the higher the discount factor, the less sensitive the best response today. Whenthe discount factor is very high (i.e., δ > 9/10), this tendency brings to situations in whichprices are strategic substitute.This is because, when the future is very important, firms see aprice cut of the rival as an opportunity of conquering a large market tomorrow rather thana threat of losing market share today. Accordingly, a price cut of the rival induces a firm toslightly increase the price today making high per-consumer margins, mainly focusing on theattraction of consumers tomorrow.

13Indeed, according to Corollary 1 ODS to L can be the case only if x̂1 ≥ max{∆+3

4, 1}.

16

if ∆ ∈ [1, 3− 12δ9−2δ

], pH1 (pL1 ) =

pH1H if pL1 ≤ p̂,

pH1TDS if pL1 ∈ (p̂, p̂M) ,

pH1M if pL1 ≥ p̂M ,

if ∆ ∈ [3− 12δ9−2δ

, ∆̂], pH1 (pL1 ) =

pH1H if pL1 ≤ p̂,

pH1TDS if pL1 ∈ (p̂, p̂M) ,

pH1H if pL1 ∈ [p̂M , p̂H ] ,

pH1M if pL1 > p̂H ,

if ∆ ≥ ∆̂, pH1 (pL1 ) =

{

pH1H if pL1 ≤ p̂H ,

pH1M if pL1 > p̂H .

where ∆̂ ≡ 3− 12(√81−25δ2−(9−2δ))

36−29δ.14

On the other hand, firm L entry is prevented in the first-period if rival’sprice is lower than a certain level (i.e., pH1 < p̃M ≡ 10δ−9+(9−2δ)∆

9). Again, since

p̃ increases in ∆, increasing vertical differentiation entails that the segment inwhich TDS is possible is narrower. Formally the best response of firm L when∆ > 1 will be:

pL1 (pH1 ) =

{

pL1TDS if pH1 ∈ [p̃M , p̃HC),

pL1H if pH1 ≥ p̃HC .

Figure 3 gives a graphical representation of the best responses when firmsare more vertically than horizontally differentiated. Two different cases are de-picted.15 The left figure describes situations in which vertical differentiation isnot so strong to eliminate completely the TDS price from the best response of thehigh-quality firm. In particular, TDS is optimally chosen in second tiny segmenton the left the dashed vertical line. For a rival price on the right of the dashedline, the high-quality firm is given the possibility to choose between charging ahigh price today enjoying ODS tomorrow and conquering the entire market to-day. Clearly, the first solution is preferred for relatively low prices charged by the

14It is easy to verify that√81− 25δ2 − (9− 2δ) > 0 for all δ ∈ [0, 1], meaning that ∆̂ < 3.

15The case in which ∆ ∈ [1, 3− 12δ

9−2δ] is not represented in the figures as it is similar to the

one with ∆ < 1. The only difference is that ODS to L can be the case only if the low-qualityfirm does not enter the market in the first period.

17

rival (and depicted in the third segment of the best response) as being aggres-sive would entail a too high loss in terms of per-consumer margins. Oppositely,when the rival is benevolent and sets a high price, the conquest of the entireHotelling line in the first period turns out to be preferred (fourth segment in thebest response). The right figure follows exactly the same reasoning, with the onlydifference that TDS is never chosen by the high quality firm. Graphically, thesecond segment in the left figure disappears completely and thus for low levels offirm L prices, only pH1H is chosen.

∆ ∈ [3− 12δ9−2δ

, ∆̂] ∆ > ∆̂

Figure 3: Best Responses for Increasing ∆

To conclude, it is worth noticing a sudden discontinuity in the best responsewhen switching to the ODS scenario.16 This ”jump” is due to a sharp changeof strategy when we consider a rival’s undercut. Assume a price of the rivaljust above the maximal level inducing a firm to use a benevolent pricing strategyenjoying ODS tomorrow. If the rival slightly lowers the price, the high-margins fo-cused strategy becomes suddenly preferred to the alternative in which the marketshares are inter-temporally balanced (or to the market-share focused solution).The discontinuity in the best response indicates that the focus on margins isparticularly intense, as the responding firm will sudden increase the price.17

16In figure 2, this jump is present in the passage from the first (leading to ODS) to the secondsegment (leading to TDS) of the best responses of both firms. In figure 3, this jump is notpresent in the best response of the low-quality firm as ODS disappears. For the high-qualityfirm, we have three jumps, the first two when we pass from pH1H to pH1TDS

and vice-versa andthe second passing from pH1H to pH1L.

17As it will be clarified when discussing the forward-looking case, this jumps is made possibleand preferred by the fact that myopic consumers are not affected by tomorrows switching, and

18

Existence and Uniqueness of Equilibria. In the analysis of equilibria, oneshould take into account two main aspects. On the one hand, it is importantto see under which conditions asymmetric equilibria in which switching is onedirection can be reached at equilibrium. This is an important novelty of thepaper compared to the symmetric approach leading always to TDS proposed byFudenberg and Tirole (2000), which result can be found in the present paper justsetting vertical differentiation parameter ∆ = 0. On the other hand, due to thediscontinuities in the best-response functions, equilibria might fail to exist or canbe multiple. The following proposition summarise all possible scenarios

Proposition 2. (Equilibria)

1. (TDS) If ∆ < 3− 8δ9−4δ

≡ ∆̄, there exists an equilibrium in which the pricesare:

(pH∗1TDS, p

L∗1TDS) =

(1 + ∆

3− 4δ∆

81−60δ, 1− ∆

3+ 4δ∆

81−60δ

)

resulting in TDS in the second period.

2. (ODS) If ∆ > 3−min{

8δ9−6δ

, 28δ(2δ+9)δ(14δ+27)+162

}

≡ ∆, there exists an equilibrium

in which prices are:

(pH∗1H , p

L∗1H) =

(

1 + ∆3+ 2δ(22+11(1−δ)+∆(1+5δ)

24δ+81, 1− ∆

3+ δ(15(1−δ)+11∆+(10∆−7)δ)

24δ+81

)

resulting in ODS to firm H in the second period.

3. (Existence and Multiplicity) If the discount factor is not too high (i.e., δ <0.93875), then the two equilibria coexist in the interval [∆, ∆̄]. Otherwise,no equilibrium exists in the interval [∆̄,∆].

Proof. See Appendix for a complete proof.The proposition tells that we can observe two different sorts of equilibria.

In the first one, both firm choose an inter-temporally balanced pricing strategythat leads to TDS (point 1.). When quality differentiation is rather substantial,asymmetric behaviours arise. Specifically, the high-quality firm finds it profitableto implement a benevolent pricing strategy, which allows for the obtainmentof high first-period margins associated with second-period ODS. The followingcorollary of Proposition 2 formally explains how market is shared differently inthe two equilibria and how this affects the second-period switching of consumers.

thus their responsiveness to price in the first period do not change passing from TDS to ODSscenarios.

19

Corollary 2. (Market Shares & Switching)

1. When (pH∗1TDS, p

L∗1TDS) are the equilibrium prices, then x̂1 = 1

2+ (9−4δ)∆

54−40δ,

x̂H2 = 1

3+ 2(3−2δ)∆

27−20δand x̂L

2 = 23+ 2(3−2δ)∆

27−20δ. In the second period x̂1 − x̂H

2 =16− (3−4δ)∆

54−40δconsumers switch from H to L and x̂L

2 − x̂1 =16+ (3−4δ)∆

54−40δmove

to the opposite direction.

2. When (pH∗1H , p

L∗1H) are the equilibrium prices, then x̂1 = x̂H

2 = 12− (7δ+9)∆−17δ

16δ+54,

and x̂L2 = 1

2+ 5δ∆−3δ+12∆+9

16δ+54. In the second period, min

{x̂L2 − x̂1, 1− x̂1

}

consumers switch from L to A.

Proof. The results are found by plugging the firs-period equilibrium prices intothe cutoffs expressed in equations (4), (5) and (6).

The corollary above highlights two different scenarios. In the first one firmsshare first-period market in a relatively balanced way and both succeed in stealingrival consumers in the second period. In the second one, reached when verticaldifferentiation is important enough, we observe a sort of market sharing agree-ment, according to which firms allocate market shares and surplus over time inan asymmetric way. In particular, firm H pursues a benevolent pricing strategy,consisting in being inoffensive today in order to induce a favourable response ofthe low-quality rival. This strategy lets firm H make high unitary margins ona small number of consumers, allowing for the opportunity of a large market toconquer cheaply in the second period.

Unsurprisingly, this benevolent pricing strategy will lead to a mitigated pricecompetition for increasing ∆ compared both to the TDS and to the uniformpricing case. This result is formally stated in the following corollary:

Corollary 3. For what concerns equilibrium prices, the following holds:

(a) high-quality firm:∂pH∗

1H

∂∆>

∂pH∗

1TDS

∂∆> 0 and pH∗

1H > pHu ;

(b) low-quality firm:∂pL∗

1TDS

∂∆<

∂pL∗

1H

∂∆< 0 and pL∗1H > pLu .

Corollary 3 highlights how the equilibrium price set by the high-quality firm(respectively by the low-quality firm) is affected positively (negatively) by thedifference in quality both in the TDS and in the ODS-to-H scenario. This simplyfollows from the fact that an increase in the vertical differentiation gives the highquality a stronger power. The difference between the two scenarios is that the

20

positive effect on strong firm equilibrium price is amplified and the negative effecton weak firm’s price is mitigated passing form TDS to ODS. This together withthe fact that eh ODS-to-H equilibrium is associated to higher levels of verticaldifferentiation suggests that price competition is less severe when the asymmetricequilibrium arises.

Moreover, the ODS equilibrium entails a attenuation of competition in relationwith the uniform pricing in two different ways. On the one hand, Corollary 3shows that first-period prices are higher than the uniform price. This is the casebecause of the benevolent pricing strategy implemented by the high-quality firm.On the other hand, the equilibrium ODS to H in general entails the exit of thelow quality firm which cannot find any profitable way to compete in the secondperiod. This result is formally explained in the following corollary.

Corollary 4. (Exit of the low quality firm) If ∆ > max{11δ+185δ+12

,∆}, then ODSto H determines the exit of the low quality firm from the market.

Proof. Take the x̂L2 = 1

2+ 5δ∆−3δ+12∆+9

16δ+54resulting from (pH∗

1H , pL∗1H). It holds that:

12+ 5δ∆−3δ+12∆+9

16δ+54> 1 ⇔ ∆ > 11δ+18

5δ+12. Since ODS-to-H equilibrium exists only

if ∆ > ∆, the result of the corollary is proved.

ODS to H + Exit

TDS

TDS + ODS

ODS to H no ExitNo equilibrium

Figure 4: Equilibria with myopic consumers.

The result in the corollary above is due by the fact that the high-qualityfirm conquers a very small market in the first period, with the consequence thatcompeting in the rival’s territory in the second period becomes very easy. Thisgives the high-quality firm a way to profitably attack all the rival turf and conquerthe entire Hotelling segment. For the sake of completeness, when the discountfactor is very high (δ > 6/7), there exist situations in which the low-quality firm

21

survives, albeit it is relegated to a very tiny corner of the market. As can beseen in Figure 4, these instances can arise only for very specific combinations ofvertical differentiation and discount factor.

Forward-Looking Consumers: Best Responses and Equilibria. Thisparagraph is devoted to a brief description of best responses and equilibria underconsumers’ farsightedness, providing a discussion about the main differences be-tween myopic and forward-looking consumers. For all technical detail, the readeris invited to read the appendix.

In the present setting, a forward-looking consumer should be able to correctlyanticipate tomorrow’s switching scenarios (TDS or ODS). In order to find the in-different first-period consumer, the following approach is used. When consumersobserve prices pH1 , p

L1 offered by firms, they become aware about which game

firms are playing in terms of tomorrow’s switching (TDS or ODS). Accordingly,the indifferent period-one consumer will locate differently according to her expec-tations. Let us assume that TDS is expected. The rational consumer indifferentin period one foresees that if she buys good H in time 1, she will switch to L inthe next period and vice-versa. As proved in the appendix, this consumer will belocated at:

x̂FL1TDS =

1

2+

(3− δ)∆ + 3(pL1 − pH1 )

2δ + 6. (7)

When ODS to H is expected, the rational consumer foresees that if she buysgood H in time 1, she will buy it again in the second period, whereas if shepurchases product L, she will switch to H in the subsequent period. As provedin the appendix, this consumer will be located at:

x̂FL1H =

1

2+

(3− 2δ)∆ + 3(pL1 − pH1 ) + δ

6− 2δ. (8)

The main difference between expecting TDS and ODS-to-H is that the indif-ferent consumer is more sensitive to price in the second case. On top of that,as expressed in the following lemma, the “elasticity of first period demand” inrelation to uniform pricing and myopic consumers is respectively reduced whenTDS occurs and increased when ODS occurs.

Lemma 1. If consumers foresee the future switching scenarios, the location of theindifferent consumer is more sensible to price changes when tomorrow’s switching

22

is expected to be uni-directional, i.e.,

|∂x̂FL1H

∂pi1| = 3

6− 2δ> |∂x̂1

∂pi1| = 1

2= | ∂x̄

∂pi1| > 3

6 + 2δ= |∂x̂

FL1TDS

∂pi1|

with i ∈ {H,L}.

The result presented in Lemma 1 is not surprising. Assume two differentequilibria in which the market splits at given locations x̂FL

1H , x̂FL1TDS. Ceteris

paribus, consider an increase in the price of the low-quality firm. This will havethe effect to change the optimal behaviour of agents located just on the right ofthe cutoffs, who will decide to buy good H rather than good L. This will move thecutoff towards right. Clearly, the responsiveness of consumers to the increase inthe price of firm L is stronger as long as consumers are closely firm H’s location.Thus, since x̂FL

1H < x̂FL1TDS, a higher number of agents will change buying firm H’s

product in the ODS-to-H compared to the TDS case.Compared to the non-discriminatory regime, the “elasticity” changes due to

the fact that forward-looking consumers take into account not only the directimpact of a price variation,18 but also the indirect effect of a variation over thesecond period’s prices. Colombo (2015) provides a very accurate and preciseexplanation of this effect in the TDS case with symmetric firms and points outhow the demand “elasticity” is lower under BBPD.19 Oppositely, when ODS isassumed to be the case, consumers anticipate that tomorrow’s discounted priceswill be less attractive as firm H will not need to lower the price too much toattract switchers. As a result, the first period benefit from switching after a pricedecrease is higher than the uniform case.

This result is very important to understand the discontinuity in the bestresponses that we can observe in figure 5. Differently from the myopic case, dis-continuities emerge when the price of the rival is high rather than when it is low.In particular, when the price of the rival reaches a threshold, we have the pas-sage from TDS to an aggressive best-response leading to uni-directional switchingtowards the rival. When this change happens, a forward-looking consumer an-ticipates it. And, since the responsiveness to price of the indifferent consumer

18Notice that they would consider only this direct effect both in the uniform pricing regimeand in the myopic-consumers case.

19Studying an increase in the price of firm L, he concludes the following:“It follows thatthe first-period benefit from shifting from firm L to firm H is lower when future is taken intoaccount. Hence, the higher δ is, the lower is the benefit from shifting after a first-period pricedecrease.”

23

Figure 5: Best Responses for Increasing ∆: Forward-Looking Consumers.

becomes suddenly stronger, a firm should sudden lower the price in order to at-tract a lot of consumer. For this reason we observe jumps downwards when whenrival’s price reaches a certain level. This also entails that the best response of thehigh-quality firm becomes continuous when the vertical differentiation is strongerthan horizontal differentiation and ODS to L is not a threat (last three plots ofFigure 5).20

Concerning the shape of the best response prices and the relation with thestrength of vertical differentiation, the essential ingredients are still there. In par-ticular, Figure 5 shows how firms trade-off between first-period market share andsecond-period switching by being essentially aggressive in pricing when the rivalis benevolent and benevolent when the rival is aggressive. An inter-temporallybalanced TDS is instead chosen when the rival price is intermediate.

Similarly to the case of myopic consumers, the best response is less steep in the

20As a matter of fact, the continuity of firm H optimal price together with the fact that bothbest responses are always upward sloping is what technically determines the non-emergence ofmultiple equilibria, as it can never happen that the two functions cross each other more thanonce.

24

latter intermediate case than in the former extreme ones. The only difference isthat, no matter the discount factor, prices are always strategic complement. Thisdifference is due to the fact that, since consumers correctly anticipate tomorrow’sswitching, any price cut today would bring to less substantial movements ofconsumers in the second-period compared to the myopic-consumer case. For thisreason, even when the future is very important, a tentative undercut of the rivaldoes never push a firm to increase the price today focusing on the attractionof consumers in the second-period, because the change in the size of this set ofconsumers is not as important as in the myopic consumers case. The equilibriaare summarised in the following proposition.

Proposition 3. (Equilibria with Forward-Looking Consumers)

1. If ∆ < 3 − 20δ9+3δ

≡ ∆̄FL, there exists a unique equilibrium in which pricesare:

(pH∗1TDS, p

L∗1TDS) =

(

1 + ∆3+ δ

3− (13−9δ)δ∆

81−33δ, 1− ∆

3+ δ

3+ (13−9δ)δ∆

81−33δ

)

,

resulting in TDS in the second period.

2. If ∆ > 3−min

{

20δ9+3δ

,20δ

(

9(

3√

(9−2δ)(9−4δ)+227)

−δ(

4δ+√

(9−2δ)(9−4δ)+1029))

3(1296−δ(279−δ(δ+138)))

}

≡∆FL, there exists a unique equilibrium in which prices are:

(pH∗1H , p

L∗1H) =

(

1 + ∆3+ 4δ(12−9δ−(5−δ)∆)

3(27−δ), 1− ∆

3+ δ((δ+29)∆−21(δ+1))

3(27−δ)

)

,

resulting in ODS to firm H in the second period.

3. For δ > 0.6871, no equilibrium exists in the interval [∆̄FL,∆FL].

Similarly to the myopic case, when vertical differentiation is substantial, theinter-temporally balanced strategy leading to TDS gradually disappears from thebest responses of the two firms. Nevertheless, some important differences emerge.First, the region in which equilibria are multiple disappears. In particular, if thediscount factor is sufficiently low ODS to H emerges as a unique equilibrium forstrong vertical differentiation and TDS for weak vertical differentiation. On theother hand, as in the case of myopia, when the discount factor is high and thevertical differentiation not sufficiently strong, the equilibrium fails to exist.

25

Secondly, the elasticity of the demand today changes according to the factconsumers anticipate tomorrow switching. This will lead to an asymmetric equi-librium equivalent to the on of the myopic case, with firm L relatively aggressive,firm H relatively benevolent and ODS-to-H in the second period. But since theelasticity of consumers is stronger, firm L is much more aggressive, making thebenevolent pricing strategy implemented by the high-quality firm less effective interms of reduction of price competition. In other words, vertical differentiationin association with BBPD pushes the low-quality firm to be more aggressive to-day and to let the rival enjoy ODS tomorrow. All these results are formally areexpressed in the following corollary.

Corollary 5. For what concerns equilibrium prices, the following holds:

(a) high-quality firm:∂pH∗

1TDS

∂∆>

∂pH∗

1H

∂∆> 0 and pH∗

1H < pHu ;

(b) low-quality firm:∂pL∗

1TDS

∂∆<

∂pL∗

1H

∂∆< 0 and pL∗1H > pLu .

Similarly to the case of myopic consumers, Corollary 5 shows that the equi-librium price set by the high-quality firm (respectively by the low-quality firm)is affected positively (negatively) by the difference in quality both in the TDSand in the ODS-to-H scenario. As in the case of myopic consumers, the negativeeffect on weak firm’s price is mitigated passing form TDS to ODS. Things changeinstead dramatically for the high-quality firm, for which the positive effect of ∆on the equilibrium price is attenuated going from the TDS to the the ODS-to-Hequilibrium. This is due to the fact that the implementation of the strategy ofbenevolent pricing makes the consumer more elastic to price changes comparedto the inter-temporally balanced pricing entailing TDS. Clearly, since the asym-metric inter-temporal strategy is anticipated by consumers, the high-quality firmenjoys less power than in the case of myopic consumers. Another proof of thisreduction of power is that the first-period price of the high-quality firm is lowerthan the uniform price.

Concerning the exit of the low-quality firm, results are very similar to themyopic case, as shown in the following corollary.

Corollary 6. (Market Shares, Switching & Exit)

1. When (pH∗1TDS, p

L∗1TDS) are the equilibrium prices, then x̂1 =

12+ (9−7δ)∆

27−11δ, x̂H

2 =13+ 3(2−δ)∆

27−11δand x̂L

2 = 23+ 3(2−δ)∆

27−11δ. In the second period x̂1− x̂H

2 = 16− (δ+3)∆

54−22δ

consumers switch from H to L and x̂L2 −x̂1 =

16+ (δ+3)∆

54−22δmove to the opposite

direction.

26

2. When (pH∗1H , p

L∗1H) are the equilibrium prices, then x̂1 = x̂H

2 = 12+ δ(∆−14)+9∆

2(27−δ),

and x̂L2 = 1

2+ 12∆+9−5δ

2(27−δ). In the second period, min

{x̂L2 − x̂1, 1− x̂1

}con-

sumers switch from L to A.

3. If ∆ > max{2δ+96

,∆FL}, then ODS to H determines the exit of the lowquality firm from the market.

Proof. The first two results are found by plugging the first-period equilibriumprices into the cutoffs expressed in equations (4), (5), (7) and (8). For the result

in point 3., take the x̂L2 = 3(2(∆+3)−δ)

27−δresulting from (pH∗

1H , pL∗1H). It holds that:

3(2(∆+3)−δ)27−δ

> 1 ⇔ ∆ > 2δ+96

. Since ODS-to-H equilibrium exists only if∆ > ∆FL, the result of the corollary is proved.

5 Welfare Analysis

The current section presents the effects of BBPD on both firms’ and consumers’welfare. In order to provide this analysis, profits and consumer surplus result-ing under customer’s recognition are compared with the benchmark case of noBBPD or no customer’s recognition, which serves to isolate the effect of pricediscrimination.

5.1 Myopic Consumers

Firms’ Profits. First, let us consider the effects of BBPD on firms’ profits.According to the results shown in Proposition (2), firms will enjoy different prof-its at equilibrium according to the difference in quality and the occurrence ofTDS,ODS and exit of the low-quality seller. Leaving all technical details to theappendix, the comparison of profits attained under BBPD with the ones result-ing in the benchmark case of Section 3, it is very easy to verify the followingproposition.

Proposition 4. If consumers are myopic, price discrimination according to pastpurchase behaviour will be:

(i) detrimental for both firms if TDS occurs and ∆ ∈ [0,min{∆̄, ∆̃}),

(ii) beneficial for the low-quality firm and detrimental for the high-quality firmif ∆̄ > ∆̃ and ∆ ∈ [∆̃, ∆̄]

27

(iii) beneficial for both firms if ODS occurs and δ < 0.98.

∆̃ ≡ 60δ−81√

(27−20δ)2(16δ(20δ−61)+765)

2(4δ(20δ−61)+189)represents the minimal level of asymmetry

needed for the low-quality seller to be better off in the TDS scenario.

Proof. See Appendix.Point (i) tells that low levels of vertical differentiation yield the firms-damaging

scenario shown in the traditional literature of BBPD. In particular, assuming ∆to be zero replicates the results of Fudenberg and Tirole (2000), demonstratingde facto that their assumption of symmetry is not needed, as firms reach thesymmetric equilibrium endogenously.

Things change radically when firms are assumed to be sufficiently asymmetric.In particular, as vertical differentiation becomes stronger, the high-quality firmis given the choice to decide the destiny of the low-quality competitor. Theequilibrium price configuration sees the high-quality firm implementing a fat catstrategy, consisting in being inoffensive today in order to induce a favourableresponse of the rival. This scenario will be always profit-enhancing compared tothe uniform pricing case because reduces price competition in the first period andyields the exit (or, at least, the cornering) of the low quality opponent. Therefore,the high-quality firm enjoys high margins on a small market in the first periodand conquers the entire second-period market without excessive effort in termsof prices.

This strategy gives the lower-quality rival the opportunity to obtain a largefirst-period market share, without the need to charge an extremely low price.This is clearly beneficial in the early competition, but it becomes harmful in thesecond period. Indeed, the high-quality firm steals a lot of consumers and thelow-quality firm only loses market share. Moreover, this ODS result leads inmostly of the cases to the exit. The balance between these two opposite effectsis always positive for the low quality seller, no matter if it survives or exits themarket. At a first sight, the fact that firm L is happier out of the market can becounterintuitive. Nevertheless, when the difference in quality is very pronounced,a situation without customer’s recognition is not so appealing for this firm, whichwould conquer a niche of the market in any case. Therefore, the ODS scenariogives to the low-quality firm the possibility to get a level of profits that wouldnot be reached in the benchmark case, not even in two periods.

Consumer Surplus. This paragraph provides the analysis on the effects ofBBPD on consumer surplus. In what follows, U ij(x) refers to the inter-temporal

28

utility of a consumer located at x who buys good j in the first period and good iin the second one, with possibly i 6= j in case of switching in the second period.21

This utility will be equal to:

U ij(x) = qi + qj − pi1 − pij2 − |x− li| − |x− lj| where i, j ∈ {H,L}.Indeed, the consumer above enjoys quality i product in the first period and

quality j in the second one, pays the respective prices and bears the respectivetransportation cost to reach the supplier location. If i = j, the consumer aboveis loyal to firm j in both periods so that the total transportation cost becomes2|x − lj|, lj being the location of firm j. Oppositely, when i 6= j the consumerswitches from firm j to firm i in the second period and the transportation costis faced in the whole segment. The total consumer surplus is thus given by thesum of utilities of all consumers:

CS =

lH∫

0

UHH(x)dx+

1∫

lL

ULL(x)dx+

lL∫

sL

UHL(x)dx+

sH∫

lH

ULH(x)dx,

where lH = min{x̂1, x̂H2 }, lL = max{x̂1, x̂

L2 }, sL = min{x̂1, x̂

L2 } and sH =

max{x̂1, x̂H2 }. The first two terms in the sum represents the surplus for loyal-

ist whereas the second ones are taken by switchers. Comparing the consumersurplus under BBPD with the one obtained under uniform pricing, we find thenet effect of BBPD on surplus, which is presented in the following proposition

Proposition 5. If consumers are myopic, price discrimination according to pastpurchase behaviour:

(i) will increase consumer surplus if TDS happens,

(ii) will decrease consumer surplus if ODS happens.

Proof. See Appendix.The intuition behind this very sharp result is nothing but the other side of the

coin of the discussion made for firms’ profits. The more symmetric firms are, thehigher the odds that BBPD brings to an intensification of competition benefitingconsumers in terms of lower prices. When instead vertical differentiation becomesstrong, consumer surplus is gradually eroded due to the mitigated first-periodprice competition and to the exit of the low-quality firm.

21When consumers are myopic, their discount factor in the first period is implicitly assumedto be 0. Thus the consumer’s utility is computed as a non-weighted sum of per-period utilities.

29

5.2 Forward-Looking Consumers

Again, in order to look at the effects of BBPD on firms’ profits, we compareBBPD profits with the ones resulting in the benchmark case of Section 3 Theresults are contained in the following proposition.

Proposition 6. If consumers are forward-looking, price discrimination accordingto past purchase behaviour will be:

(i) detrimental for both firms if TDS occurs and ∆ ∈ [0,min{∆̄FL, ∆̃FL}),

(ii) beneficial for the low-quality firm and detrimental for the high-quality firmif ODS-to-H occurs or if TDS occurs and ∆ ∈ [∆̃FL, ∆̄FL].

∆̃FL ≡ 3− (27−11δ)(

2√

117−δ(37−8δ)−3(7−δ))

27−δ(23δ−22)represents the minimal level of asymme-

try needed for the low-quality seller to be better off in the TDS scenario.

Proof. See Appendix.Proposition 6 highlights how the high-quality firm suffers the BBPD equilib-

rium when this leads to ODS-to-H. In particular, the same mechanism that givesa very powerful tool in the hands high-quality firm when consumers are myopicturns out to be a condemn when consumers are forward-looking. Here, the low-quality firm sets a very low price in the first period and the high-quality firm is“forced” to postpone the attack to tomorrow. Oppositely,bwhen consumers aremyopic the high-quality firm sets a high price in the first period enjoying ODStomorrow and the rival “enjoys” the reduced price competition in the first pe-riod. This strategy gives the lower-quality rival the opportunity to obtain a largefirst-period market share and this is always positive and more than compensatethe lost of market share suffered in the second period.

On the other hand, the loss suffered by the high-quality firm is captured notonly by the low-quality firm but also by the consumers.

Proposition 7. If consumers are forward-looking consumers, price discrimina-tion according to past purchase behaviour will always increase consumer surplus.

Proof. See Appendix.Indeed, in aggregate terms the enjoy a higher surplus than under the uniform

pricing due to the fact that most of them pay a relative low price in the firstperiod to the low-quality firm and then switch at a relative low price to thehigh-quality supplier.

30

6 Conclusion

Despite the issues in terms of privacy created by the access of firm to consumerspecific information, BBPD literature has been in favour of consumers recog-nition due to the fact that consumers benefit from it in terms of lower pricesand increased competition. In particular, the main message of Fudenberg andTirole (2000) is the same sent by the traditional price discrimination literaturein oligopolistic markets: once firms can discriminate prices, they suffer a moreintense competition, leading to lower prices and a positive effect for consumersurplus.

This paper participates to the debate arguing that the result above does notnecessarily hold anymore if firms sell good of different qualities. As expected,the model gives the same results as Fudenberg and Tirole (2000) when firmsare not (enough) vertically differentiated. However, if firms are different enoughregarding the quality of the good sold then asymmetric pricing behaviours arise.This will be the case when vertical differentiation is sufficiently strong.

Specifically, the high-quality firm optimally chooses to set a high price in thefirst period serving a small number of consumers attaining high per-consumermargins and then to compete fiercely in the second period attracting a lot ofnew and keeping all previous consumers. This outcome turns out to be ex-postpreferred to uniform pricing as the renounce in terms of first-period demandexceeds the gain in margin at that time and the ODS in the second period.Moreover, the high-quality firm becomes a monopolist or , at least, conquers amarket share very close to one in the thanks to BBPD.

The low-quality seller responds to this strategy in a yielding way, as it simplytakes what is left by the high quality competitor. Nonetheless, this accommodat-ing behaviour turns out to be positive, as in the early competition low-qualityfirm can enjoy a large part of demand charging a price relatively high comparedto uniform pricing. The positive effect is not neutralised even when this asym-metric scenario leads to the exit of the low-quality firm from the market. Inany case, the possible exit of a firm caused by price discrimination suggests thenatural policy implication that an authority which main concern is to preservecompetition should block BBPD as it reduces the number of firms active in themarket.

All these considerations give peculiar effects on the consumer surplus, which isclearly enhanced by BBPD only under low enough vertical differentiation. Indeed,when the asymmetric result of ODS to the high-quality firm occurs, BBPD may

31

results in a reduction of consumer surplus compared to the uniform price regime.As a consequence, the second policy implication of the paper is that if consumersurplus is the antitrust authority’s criterion, then BBPD should be banned inmarkets exhibiting strong vertical differentiation.

Appendix

Proof of Proposition 1

Let us analysis second period pricing decisions. Given the cutoffs x̂H2 and x̂L

2 inequation (4) and (5), firms solve the following problem in A’s turf:

maxpHH2

pHH2 x̂H

2 = pHH2

(12+

∆+pLH2

−pHH2

2

)

,

maxpLH2

pLH2(x̂1 − x̂H

2

)= pLH2

(

x̂1 − 12− ∆+pLH

2−pHH

2

2

)

.

Solving the maximization problem, firm H’s best response turns out to be:

pHH2 =

1 +∆+ pLH22

,

and firm L best response is to set a price

pLH2 = x̂1 +pHH2 − 1−∆

2.

which give the following equilibrium prices:

pHH2 =

∆+ 2x̂1 + 1

3and pLH2 =

4x̂1 − 1−∆

3.

Doing the same in L’s turf yields:

pHL2 =

∆+ 3− 4x̂1

3and pLL2 =

3− 2x̂1 −∆

3.

Notice that charging a price pLH2 < 0 is a dominated strategy for firm L.Therefore,whenever the equilibrium price pLH2 < 0 then the best option for thisfirm is to set pLH2 = 0. This will happen when 4x̂1−1−∆

3≤ 0 ⇔ x̂1 ≤ ∆+1

4.

When x̂1 ≤ ∆+14

it follows that pLH2 = 0 and thus the best response of firm His to charge the maximal possible price compatible with not to lose the marginal

32

consumer located at x̂1, i.e., a price pHH2 such that qH−pHH

2 −x̂1 = qL−0−(1−x̂1),which gives pHH

2 = ∆+1−2x̂1. This will give equilibrium prices when x̂1 ≤ ∆+14

:

pHH2 = ∆+ 1− 2x̂1 and pLH2 = 0;

pHL2 = ∆+(3−4x̂1)

3and pLL2 = 3−2x̂1−∆

3.

In this case switching is one direction towards firm H. Similarly, firm H willset pHL

1 = 0 when ∆+3−4x̂1

3≤ 0 ⇔ x̂1 ≥ ∆+3

4.

When x̂1 ≥ ∆+34

it follows that pLH2 = 0 and thus the best response of firm Lis to charge the maximal possible price compatible with not to lose the marginalconsumer located at x̂1, i.e., a price p

LL1 such that qH−0−x̂1 = qL−pLL2 −(1−x̂1),

which gives pLL2 = 2x̂1− 1−∆. This will give equilibrium prices when x̂1 ≤ ∆+34

:

pHH2 = ∆+2x̂1+1

3and pLH2 = 4x̂1−1−∆

3;

pHL2 = 0 and pLL2 = 2x̂1 − 1−∆.

Notice that if ∆ > 1, this scenario with ODS to firm L cannot be reachedunless. Summarizing, ODS to L can be the case only if ∆ < 1 or x̂1 = 1.

Construction of the Best Replies.

Firm H best response.

(i) If x̂1 = 12+

∆+pL1−pH

1

2∈

(∆+14

, ∆+34

), TDS occurs and firm H enjoys the

following second period profits: πH2TDS =

∆2+5(2x̂2

1−2x̂1+1)−2∆(x̂1−2)

9. Accord-

ingly, firm H solves maxpH1

πHTDS = maxpH

1

pH1 x̂1 + δπH2TDS under the con-

straints ∆+t4t

< x̂1 <∆+3t4t

. The first order condition of this problem gives:

pH1TDS =(9− 10δ)

18− 10δpL1 +

9

18− 10δ+

(9− 8δ)∆

18− 10δ,

with correspondent x̂1 =9pL

1−2δ(∆+5)+9(∆+1)

36−20δ. If ∆ ≤ 1, then x̂1 ∈ (0, 1)

and constraints are met if 5δ−3δ∆9

≡ p̂HC < pL1 < p̂LC ≡ 18−3δ∆−5δ9

. Thecorrespondent profit will be:

πHTDS =

(9pL1−9(∆+1))2−18pL

1(2δ(∆+5))−4δ2(3∆+5)2+36δ(∆(∆+2)+5)

72(9−5δ).

We have three more cases to consider.

33

• When ∆ > 1, the constraint x̂1 < ∆+34

is non-binding. In thiscase, whenever pL1 is such that x̂1(p

HTDS, p

L1 ) ≥ 1, i.e. pL1 ≥ p̂M ≡

2δ∆−10δ−9∆+279

, TDS cannot occur tomorrow and firm H becomes amonopolist setting the price pH1 such that x̂ = 1 or pH1M = ∆+ pL1 − 1and resulting profit of πH

M = 118δ(∆ + 3)2 + ∆ + pL1 − 1, where the

subscript M stays for monopoly of firm H.

• If ∆ ≤ 1, the constraint x̂1 < ∆+34

turns out to be binding when

pL1 ≥ p̂LC , pH1 is such that x̂1 = ∆+3

4, or pH1LC =

∆+2pL1−1

2, leading to

ODS to L. The profit overall will be:

πHLC =

1

72

(18(∆ + 3)pL1 + δ(3∆ + 5)2 + 9(∆− 1)(∆ + 3)

).

• Finally, if pL1 ≤ p̂HC , then x̂1 ≤ ∆+14

. It means that pH1 is such that

the constraint is binding, i.e., pH1HC =∆+2pL

1+1

2, , leading to ODS to L.

The correspondent profit will be:

πHHC =

1

72

(δ(9∆(∆ + 2) + 25) + 9(∆ + 1)2 + 18(∆ + 1)pL1

).

(ii) If x̂1 =12+

∆+pL1−pH

1

2≤ ∆+1

4, ODS occurs only towards firmH, which receives

profit πH2H = ∆2+(9−2x̂1(10x̂1+3))+2∆(5x̂1+3)

18. The maximisation problem will be

the following maxpH1

πHH = maxpH

1

pH1 x̂1 + δπH2H under the constraint x̂1 ≤

∆+14

. The first order condition of this problem gives:

pH1H =(10δ + 9)

18 + 10δpL1 +

9 + 13δ

18 + 10δ+

∆

2,

with correspondent x̂1 =9pL

1+5δ∆−3δ+9∆+9

20δ+36. Constraint is met and x̂1 ∈ (0, 1)

if 0 < pL1 < p̂H ≡ 8δ/9. The correspondent profit will be:

πHH = 1

8

(6δ(5−pL

1 )+9(pL1 +1)2+21δ2

5δ+9+ 2∆

(pL1 + δ + 1

)+ (δ + 1)∆2

)

.

If pL1 < p̂H , x̂1 ≥ ∆+14

and thus firm H sets a price such that x̂1 =∆+14

, i.e.,pH1HC . Moreover, since πH

2H(x̂1 =∆+14

) = πH2TDS(x̂1 =

∆+14

), this profit turnsout to be πHC .

34

(iii) If x̂1 = 12+

∆+pL1−pH

1

2≥ ∆+3

4, ODS occurs only towards firm L. This case

can exist only if ∆ ≤ t or, when ∆ > t, if x̂1 = 1. ODS to L would give

firm H a second period profit of πH2L = (∆+2tx̂1+1)2

18t.

• If ∆ ≤ 1, then firm H solves maxpH1

πHL = maxpH

1

pH1 x̂1 + δπH2L under

the constraint x̂1 ≥ ∆+34

. The first order condition of this problemgives:

pH1L =(9− 2δ)

18− 2δpL1 +

9− 4δ

18− 2δ+

(9− 4δ)∆

18− 2δ,

with correspondent x̂1 =9pL

1+(2δ+9)(∆+1)

4(9−δ). The constraint is met if

pL1 ≥ 18−3δ∆−5δ9

, for other prices ODL cannot occur. The resultingprofit will be:

πHL =

(8δ + 9)(∆ + 1)2 + 9(pL1 )2 + 2(2δ + 9)(∆ + 1)pL1

8(9− δ).

If pL1 ≤ p̂LC , x̂1 ≤ ∆+34

and thus firmH sets a price such that x̂1 =∆+14

,i.e., pL1LC . Moreover, since πH

2L(x̂1 = ∆+34

) = πH2TDS(x̂1 = ∆+3

4), this

profit turns out to be πHLC . The case in which x̂1 = 1 has been already

discussed in the TDS case.

Up to now, we obtained all possible best responses of firm H within eachregime. In order to build up the global best response, we must compare profitsacross regimes in each segment. We have 3 possible cases:

1. If ∆ < 1, then we have the following segments:

(a) pL1 ≤ p̂HC . =⇒ best response pH1H .

(b) pL1 ∈ (p̂HC , p̂H) If pL1 >

√(9∆+9)2−(5δ∆+5δ)2

30+ 65δ−15δ∆

90− 3∆−3

10≡ p̂, then

πHTDS > πH

H . Otherwise, πHH > πH

TDS.

(c) pL1 ∈ [p̂H , p̂LC). =⇒ best response pH1TDS.

(d) pL1 ≥ p̂LC . =⇒ best response pH1L.

2. ∆ ∈ [1, 3− 12δ9−2δ

], then p̂H < p̂M . In segments (a) and (b) nothing changes.Above p̂H we have:

(c.i) pL1 ∈ [p̂H , p̂M ]. =⇒ best response pH1TDS.

35

(d.i) pL1 ≥ p̂M . =⇒ best response pH1M .

3. If ∆ > 3 − 12δ9−2δ

then p̂M > p̂HIn segments (a) nothing changes comparedto point 1. Above p̂HC we have:

(b.ii) pL1 ∈ (p̂HC , p̂M). If ∆ < 3 − 12(√81−25δ2−(9−2δ))

36−29δ≡ ∆̂, then πH

TDS > πHH

for pL1 > p̂ =⇒ best response pH1TDS. Otherwise, the best response isalways pH1H .

(c.ii) pL1 ∈ [p̂M , p̂H ]. πHH > πH

M =⇒ best response pH1H .

(d.ii) pL1 ≥ p̂H . πHM > πH

HC =⇒ best response pH1M .

Putting together all the results above, the best response will depend on thesize of ∆. Indeed, the best response of firm H will be the following:

pH1 (pL1 ) =

pH1H if pL1 ≤ p̂,

pH1TDS if pL1 ∈ (p̂, p̂LC) ,

pH1L if pL1 ≥ p̂LC ,

pH1 (pL1 ) =

pH1H if pL1 ≤ p̂,

pH1TDS if pL1 ∈ [p̂, p̂M ] ,

pH1M if pL1 > p̂M ,

when ∆ < 1, when ∆ ∈ [1, 3− 12δ9−2δ

],

pH1 (pL1 ) =

pH1H if pL1 ≤ p̂,

pH1TDS if pL1 ∈ [p̂, p̂M ] ,

pH1H if pL1 ∈ [p̂M , p̂H ] ,

pH1M if pL1 > p̂H ,

pH1 (pL1 ) =

{

pH1H if pL1 ≤ p̂H ,

pH1M if pL1 > p̂H ,

when ∆ ∈ [3− 12δ9−2δ

, ∆̂], when ∆ > ∆̂.

Notice that 3− 12δ9−2δ

< ∆̂ for any discount factor.

Firm L best response.

(i) If x̂1 =12+

∆+pL1−pH

1

2∈(∆+14

, ∆+34

), TDS occurs and firm L enjoys a second

period profit of πL2TDS =

∆2+5(2x̂2

1−2x̂1+1)−2∆(x̂1+1)

9. Accordingly, firm L solves

maxpL1

πLTDS = maxpL

1

pL1 (1− x̂1)+δπL2TDS under the constraints ∆+1

4< x̂1 <

∆+34

. The first order condition of this problem gives:

pL1TDS =(9− 10δ)

18− 10δpH1 +

9

18− 10δ− (9− 8δ)∆

18− 10δ,

36

with resulting x̂1 =9(∆+3)−9pH

1−2δ(∆+5)

4(9−5δ). If ∆ ≤ 1, then constraints are met

if 3δ∆+5δ9

≡ p̃LC < pH1 < p̃HC ≡ 3δ∆−5δ+189

. The correspondent profit willbe:

πLTDS =

(9pH1−9(∆−1))2+18pH

1(2δ(∆−5)+4δ2(5−3∆)2+36δ((∆−2)∆−5)

72(9−5δ).

We have three more cases to consider.

• If ∆ > 1 the constraint x̂1 <∆+34

is non-binding. Whenever pH1 < p̃M ,x̂1 ≥ 1 firm L cannot enter the market.

• If pH1 ≥ p̃HC , then first constraint is not satisfied and thus pL1 will be

such that x̂1 =∆+14

or equivalently pL1HC =2pH

1−1−∆

2. In this case the

profit will be:

πLHC =

1

72

(δ(5− 3∆)2 + 9(∆− 3)(∆ + 1)− 18(∆− 3)pH1

).

• When ∆ ≤ 1 and pH1 ≤ p̃LC , then the second constraint is notsatisfied and thus pL1 will be such that x̂1 = ∆+3

4or equivalently

pL1LC =2pH

1+1−∆

2. The correspondent profit will be:

πLLC =

1

72

(9(δ + 1)∆2 + 25δ − 18∆(δ + pH1 + 1) + 18pH1 + 9

).

(ii) If x̂1 = 12+

∆+pL1−pH

1

2≤ ∆+1

4, ODS occurs only towards firm H and firm

L gets πL2H = (∆+(2x̂1−3))2

18. The maximisation problem will be maxpL

1

πLH =

maxpL1

pL1 (1 − x̂1) + δπL2H under the constraint x̂1 ≤ ∆+1

4. The first order

condition of this problem gives:

pL1H =(9− 2δ)

18− 2δpH1 +

9− 4δ

18− 2δ− (9− 4δ)∆

18− 2δ,

with correspondent x̂1 =2δ∆−6δ+9∆−9pH

1+27

36−4δ. Constraint is met if pH1 ≥ p̂HC

and the correspondent profit will be:

πLH =

(8δ + 9)(∆− 1)2 + 9(pH1 )2 − 2(2δ + 9)(∆− 1)pH1

8(9− δ).

If pH1 ≤ p̂HC , then x̂1 ≥ ∆+14

and thus firm L sets a price such that x̂1 =∆+14

,i.e., pH1HC . Moreover, since πL

2H(x̂1 = ∆+14

) = πL2TDS(x̂1 = ∆+1

4), the profit

will be πLHC .

37

(iii) If x̂1 = 12+

∆+pL1−pH

1

2≥ ∆+3

4, ODS occurs only towards firm L. This case

can exist only if ∆ < 1 or, when ∆ > 1, if x̂1 = 1. ODS to L would give

firm L a second period profit of πL2L =

∆2+(46x̂1−20x̂2

1−17)+2∆(5x̂1−8)

18.

If ∆ < 1, then firm L maximizes maxpL1

πLL = maxpL

1

pL1 (1− x̂1)+δπL2L under

the constraint x̂1 ≥ ∆+34

. The first order condition of this problem gives:

pL1L =(10δ + 9)

18 + 10δpH1 +

9 + 13δ

18 + 10δ− ∆

2,

with correspondent x̂1 =27−9pH

1+5δ∆+23δ+9∆

20δ+36. Constraint is met if pH1 ≤

p̃L ≡ 8δ9and the correspondent profit will be:

πLL =

1

8

(21δ2 + 6δ(5− pH1 ) + 9(pH1 + 1)2

5δ + 9+ (δ + 1)∆2 − 2∆(δ + pH1 + 1)

)

.

If the constraint is not satisfied (i.e., pH1 ≥ p̃L ), then we are back to thecase with price pL1LC and profit πL

LC . If instead x̂1 = 1, firm L entry isprevented.

Up to now, we obtained all possible best responses of firm L within eachregime. In order to build up the global best response, we must compare profitsacross regimes in each segment. We have two cases:

1. If ∆ ≤ 1, then p̃L ≥ p̃LC . We will have four segments:

(a) pH1 < p̃LC πLL > πL

L =⇒ pL1L

(b) pH1 ∈ (p̃LC , p̃L). If pH1 <

√−25δ2∆2+50δ2∆−25δ2+81∆2−162∆+81

30+15δ∆+65δ+27∆−27

90≡

p̃, then πLL > πL

TDS. When pH1 > p̂ =⇒ πLTDS > πL

1L.

(c) pH1 ∈ (p̃L, p̃HC). πLTDS > πL

L =⇒ pL1TDS.

(d) pH1 ≥ p̃HC . πLH > πL

HC =⇒ pL1H .

2. ∆ ≥ 1, then in the last segment nothing changes compared to the case with∆ < 1. For pH1 ≤ p̃HC

(a.i) pH1 < p̃M . Firm L is out of the market.

(b.i) pH1 ∈ (p̃M , p̃HC). πLTDS > πL

L =⇒ pL1TDS.

38

Putting together all the results above, the best response of firm L is

pL1 (pH1 ) =

pL1L if pH1 ≤ p̃,

pL1TDS if pH1 ∈ [p̂, p̃HC ] ,

pL1H if pH1 > p̃HC ,

pL1 (pH1 ) =

{

pL1TDS if pH1 ∈ (p̃M , p̃HC) ,

pL1H if pH1 > p̃HC ,

∆ < 1, ∆ ≥ 1.

Proof of Proposition 2

Existence and uniqueness of the equilibria. We have three cases.

• TDS scenario. The only couple of prices generating this scenario is

(pH∗

1TDS, pL∗

1TDS) =

(

1 + ∆3− 4δ∆

81−60δ,1− ∆

3+ 4δ∆

81−60δ

)

. This is an equi-

librium whenever ∆ < 3 − 8δ9−4δ

≡ ∆̄. If ∆ ≥ ∆̄, then the marketsplitting cut-off will be located above 1, so that TDS cannot be the case inthe second period.

• ODS to H scenario. The only couple of prices is:

(pH∗1H , p

L∗1H) =

(

1 + ∆3+ 2δ(22+11(1−δ)+∆(1+5δ)

24δ+81, 1− ∆

3+ δ(15(1−δ)+11∆+(10∆−7)δ)

24δ+81

)

.

From this situation, firm L would never deviate provided that ∆ > 3− 8δ9−6δ

since pH∗1H > p̃HC . On the other hand, firm H does not deviate ( i.e., pL1H ∈

[p̂M , p̂A]) whenever ∆ > 3 − min{

8δ9−6δ

, 28δ(2δ+9)δ(14δ+27)+162

}

≡ ∆. Summarizing,

if ∆ ≥ ∆, this is always an equilibrium.

• ODS to L scenario. Two cases:

1. When ∆ < 1, only one couple of prices can lead to this scenario, i.e.,

(pH1L, pL1L) =

(27(∆+3)−δ(10δ∆+22δ+3∆−39)

24δ+81, −2δ(5δ∆+11δ+3∆−45)+27(3−∆)

24δ+81

)

.

This cannot be an equilibrium because the best response of firm H topL1L is pH1TDS.

2. If ∆ ≥ 1, the only possibility is to have a monopoly of firm H inthe first period, choosing price pH1M = pL1 + ∆ − 1. This strategy iseffective (i.e., firm L cannot enter the market) only if pL1 + ∆ − 1 <p̃M = 10δ+9∆−2δ∆−9

9⇔ pL1 < 10δ−2δ∆

9. Firm H always deviates to ODS

when ∆ > 97because 10δ−2δ∆

9< 8δ

9= p̂H and when ∆ < 9

7because

10δ−2δ∆9

< 27+2δ∆−10δ−9∆9

= p̂M .

39

Case of forward-looking consumers

When consumers are forward-looking, they take into account the possibility oftomorrow’s switching. H forward looking consumer buys good i today and po-tentially switches to firm j enjoying a discount price. This will give him utilityU i(x) = qi − pi1 − |x− li|+ δ(qj − pij2 − |x− lj|).

Firm H best response.

(i) If x̂1 ∈(∆+14

, ∆+34

). Compared to the case of myopia, the rational consumer

who is indifferent in period 1 anticipates that if she buys product H in period1, she will switch to product L in period 2, whereas if she chooses productL in period 1 she will switch to product H in period 2. Thus, the indifferentconsumer is located in the x̂1 such that

qH−pH1 −x̂1+δ[qL − pLH2 − (1− x̂1)

]= qL−pL1 −(1−x̂1)+δ(qH−pHL

2 −x̂1)

where pLH2 and pHL2 are the ones in point (ii) of proposition 1. Rewriting:

x̂1 =1

2+

(3− δ)∆ + 3(pL1 − pH1 )− δ∆

2δ + 6. (9)

Following the same notation used in the construction of best replies of themyopic case, we find the following:

Prices

pH1TDS =(9−7δ)pL

1+(δ(3δ−8)+9)∆+(δ+3)2

18−4δ, pH1M = ∆+ pL1 − 1− δ(∆+1)

3,

pH1LC =−3δ∆−δ+3∆+6pL

1−3

6, pH1HC =

−3δ∆+δ+3∆+6pL1+3

6.

Profits

πHTDS =

9(3pL1 −δ∆+δ+3∆+3)2

+12δ(3∆(δ(∆+4)+∆+2)+5(δ+3)−3(∆+5)pL1)−4δ2(3∆+5)2

72(9−2δ),

πHLC =

18(∆+1)pL1+4δ(3∆+7)+9(∆+1)2

72, πH

HC =18(∆+3)pL

1+16δ+9(∆−1)(∆+3)

72,

πHM = pL1 + (∆− 1) + (∆−1)∆δ+2δ

9.

Cutoffs

p̂HC = δ(3∆+5)9

, p̂LC ≡ 3δ∆+δ+189

.

40

(ii) If x̂1 ≤ ∆+14

, ODS occurs only towards firm H. Here, the rational consumerwho is indifferent in period 1 anticipates that if she buys product H in period1, she will buy it again in period 2, whereas if she chooses product L inperiod 1 she will switch to product H in period 2. Thus, the indifferentconsumer is located in the x̂1 such that

qH − pH1 − x̂1 + δ[qH − pHH

2 − x̂1

]= qL − pL1 − (1− x̂1) + δ(qH − pHL

2 − x̂1)

where pHH2 and pHL

2 are the ones in point (ii) of proposition 1. Rearranging:

x̂1 = ∆+3(1 + pL1 −∆− pH1 )

2(3− δ).

Using the same notation of the myopia case, we will have:

pH1H =(7δ+9)pL

1−δ(3δ∆+δ+4∆−10)+9(∆+1)

18+4δ, p̂H = δ(3∆+5)

9,

πHH =

9(pL1 )2

−2pL1(δ(∆+3)−9(∆+1))+δ2(∆+3)2+2δ(∆+3)(∆+5)+9(∆+1)2

8(9+2δ).

(iii) If x̂1 ≥ ∆+34

, ODS occurs only towards firm L. The rational consumer whois indifferent in period 1 anticipates that if she buys product L in period 1,she will buy it again in period 2, whereas if she chooses product H in period1 she will switch to product L in period 2. Thus, the indifferent consumeris located in the x̂1 such that

qH−pH1 −x̂1+δ[qL − pLH2 − (1− x̂1)

]= qL−pL1−(1−x̂1)+δ

[qL − pLL2 − (1− x̂1)

],

where pLH2 and pLL2 are the ones in point (ii) of Proposition 1. Rearranging:

x̂1 = ∆+ 1− 3(pH1 − pL1 +∆+ 1

)

2 (3− δ).

Using the same notation of the myopia case, we will have:

pH1L =(9−5δ)pL

1−(1−δ)(9−4δ)(∆+1)

18−8δ, πH

L =2(9−4δ)(∆+1)pL

1+9(pL1 )2+(9−4δ)(∆+1)2

8(9−4δ),

p̂L = 2− 8δ9.

41

Doing the same analysis done for the myopic consumers’ case, we can distin-guish four possible cases.

1. If ∆ < 1 , then we have the following segments:

(i) pL1 ≤ p̂H . πHH > πH

LC , πHHC =⇒ best response pH1H .

(ii) pL1 ∈ (p̂H , p̂L). πHTDS > πH

HC , πHLC =⇒ pH1TDS.

(iii) pL1 ∈ [p̂L, p̂LC ]. πHTDS > πH

L if

pL1 < p̂ ≡ 118

(

4δ(3∆ + 5) + 3(√

(2δ − 9)(4δ − 9)(∆ + 3)2 − 9∆− 15))

,

the opposite is true otherwise.

(iv) pL1 > p̂L. πHL > πH

HC , πHLC =⇒ pH1L.

2. If 3 > ∆ > 1 , then the best response remains unchanged in segment (i).For pL1 ≥ p̂H , we have the following segments:

(ii) pL1 ∈ (p̂H , p̂M). πHTDS > πH

HC =⇒ pH1TDS.

(iii) pL1 > p̂M . πHM > πH

HC =⇒ pH1MH .

Putting together all the results above, the best response will depend on thesize of ∆. Namely, the best response of firm H will be:

pH1 (pL1 ) =

pH1H if pL1 ≤ p̂H ,

pH1TDS if pL1 ∈ (p̂H , p̂),

pH1L if pL1 ≥ p̂,

pH1 (pL1 ) =

pH1H if pL1 ≤ p̂H ,

pH1TDS if pL1 ∈ [p̂H , p̂M ] ,

pH1M if pL1 > p̂M ,

when ∆ < 1, when ∆ ∈ [1, 3].

Firm L best response.

(i) If x̂1 ∈(∆+14

, ∆+34

). Doing the same as done for the high-quality firm and

using the same notation of the myopic case, we find:

Prices

pL1TDS =(9−7δ)pH

1−(9−δ(8−3δ))∆+(δ+3)2

18−4δ, pL1M = ∆+ pL1 − 1− δ(∆+1)

3,

pL1LC =−3δ∆−δ+3∆+6pL

1−3

6, pL1HC =

−3δ∆+δ+3∆+6pL1+3

6.

42

Profits

πLTDS =

9(3pH1 +(δ−3)∆+δ+3)2

+12δ(3(∆−5)pH1+3∆(δ(∆−4)+∆−2)+5(δ+3))−4δ2(5−3∆)2

72(9−2δ),

πLHC =

18(3−∆)pH1+16δ−9(3−∆)(∆+1)

72, πL

LC =18(1−∆)pH

1+4δ(7−3∆)+9(1−∆)2

72.

Cutoffs

p̃LC = δ(5−3∆)9

, p̃HC = 18−3δ∆+δ9

, p̃M = ∆+ δ(7−5∆)9

− 1.

(ii) If x̂1 ≤ ∆+14

, ODS occurs only towards firm H. Doing the same as done forthe high-quality firm and using the same notation of the myopic case, wefind:

pL1H =(9−5δ)pH

1+(1−δ)(9−4δ)(1−∆)

18−8δ, p̃H = 2− 8δ

9, πL

H =2(9−4δ)(1−∆)pH

1+9(pH1 )2+(9−4δ)(∆−1)2

8(9−4δ).

(iii) If x̂1 ≥ ∆+34

, ODS occurs only towards firm L. Doing the same as done forthe high-quality firm and using the same notation of the myopic case, wefind:

pL1L =(7δ+9)pH

1+δ(δ(3∆−1)+4∆+10)−9∆+9

4δ+18, p̃L = δ(5−3∆)

9,

πLL =

2pH1(δ(∆−3)−9∆+9)+9(pH1 )2+δ2(∆−3)2+2δ(∆−5)(∆−3)+9(∆−1)2

8(2δ+9).

We have two cases:

1. If ∆ ≤ 1, we will have four segments:

(a) pH1 < p̃LC . πLL > πL

HC , πLLC =⇒ pL1L

(b) pH1 ∈ (p̃LC , p̃H). πLTDS > πL

LC , πLHC =⇒ pL1TDS.

(c) pH1 ∈ (p̃H , p̃HC). πLTDS > πL

H if

pH1 < p̃ ≡ 118

(

3(√

(2δ − 9)(4δ − 9)(∆− 3)2 + 9∆− 15)

+ 4δ(5− 3∆))

,

the opposite is true otherwise.

(d) pH1 > p̃HC . πLH > πL

HC , πLLC =⇒ pL1H

2. ∆ ≥ 1, then

43

(a) pH1 < p̃M . Firm L is out of the market.

(b) pH1 ∈ (p̃M , p̃H). πLTDS > πL

HC =⇒ pL1TDS.

(c) pH1 ∈ (p̃H , p̃HC). πLTDS > πL

H if pH1 < p̃.

(d) pH1 > p̃HC . πLH > πL

HC =⇒ pL1H .

Putting together all the results above, the best response of firm L is

pL1 (pH1 ) =

pL1L if pH1 ≤ p̃LC ,

pL1TDS if pH1 ∈ (p̃LC , p̃),

pL1H if pH1 ≥ p̃,

pL1 (pH1 ) =

{

pL1TDS if pH1 ∈ (p̃M , p̃),

pL1H = if pH1 > p̃,

when ∆ ≤ 1, when ∆ > 1.

Existence and uniqueness of the equilibria. We have three cases.

• TDS scenario. The only couple of prices generating this scenario is

(pH∗

1TDS, pL∗

1TDS) =

(

1 + ∆3+ δ

3−

(13−9δ)δ∆

81−33δ,1− ∆

3+ δ

3+ (13−9δ)δ∆

81−33δ

)

.

When ∆ < 1, both firms are on their best responses. If ∆ ≥ 1, then firmH always deviates if ∆ > 3 − 20δ

9+3δ≡ ∆̄FL, since pL∗1TDS < p̂H . For what

concern firm L, ∆ > ∆̄FL implies pH∗1TDS ≤ p̃M . Summarizing, This is an

equilibrium iff ∆ < ∆̄FL.

• ODS to H scenario. Given the best responses, the candidate equilibriumprices will be

(pH∗

1H, pL∗

1H) =(

1 + ∆3+ 4δ(12−9δ−(5−δ)∆)

3(27−δ),1− ∆

3+ δ((δ+29)∆−21(δ+1))

3(27−δ)

)

.

When ∆ < 1, this is not an equilibrium. When ∆ > 1, the monotonicityand continuity of firm H best response implies that whenever TDS can bean equilibrium, ODS to H cannot. Therefore, a necessary condition forODS to H to be an equilibrium is that ∆ > ∆̄FL. From this situation, firm

L would never deviate when δ ≤ 9(2√103−19)17

≈ 0.6871. Otherwise, it isneeded the stricter condition that

∆ > 3− 20δ(

9(

3√

(9−2δ)(9−4δ)+227)

−δ(

4δ+√

(9−2δ)(9−4δ)+1029))

3(1296−δ(279−δ(δ+138))),

otherwise pH∗1H > p̃. On the other hand, firm H does not deviate ( i.e.,

pL∗1H < p̂H) whenever ∆ > ∆̄FL. Therefore, if

44

∆ ≥ 3−min

{

20δ9+3δ

,20δ

(

9(

3√

(9−2δ)(9−4δ)+227)

−δ(

4δ+√

(9−2δ)(9−4δ)+1029))

3(1296−δ(279−δ(δ+138)))

}

≡ ∆FL,

this is always an equilibrium.

• ODS to L scenario. Two cases:

1. When ∆ < 1, only one couple of prices can lead to this scenario, i.e.,

(pH1L, pL1L) =

(27(∆+3−δ2(∆+21)−6δ(5∆+4))

3(27−δ), (9−4δ)(δ(∆+9)−3∆+9)

3(27−δ)

)

. These prices

cannot be an equilibrium because pH∗1L < p̂.

2. If ∆ ≥ 1, the only possibility is to have a monopoly of firm H inthe first period, choosing price pH1MH = pL1 + ∆ − 1. This strategy iseffective (i.e., firm L cannot enter the market) only if pL1 + ∆ − 1 <−2δ∆+10δ+9∆−9

9⇔ pL1 < 10δ−2δ∆

9. Firm H always deviates when ∆ > 9

7

because 10δ−2δ∆9

< 8δ9(the minimal price charged by the rival to find

profitable the monopoly case) and when ∆ < 97because 10δ−2δ∆

9<

2δ∆−10δ−9∆+279

.

Proof of Proposition 4

Under BBPD, three different cases may arise:

(a) Exit of the low-quality firm (∆ > 11δ+185δ+12

). In this case, firm H and firm Lrespectively get:

πHE = δ3(∆(133∆−358)+789)+66δ2(∆(8∆+3)+27)+162δ(∆+3)(4∆+5)+243(∆+3)2

6(8δ+27)2,

πLE =

(81++39δ−22δ2+(2δ−3)(5δ+9)∆)(δ(25−7∆)+9(3−∆))

6(8δ+27)2.

It is easy to verify that πHE −πH

u and πLE−πL

u are both positive if ∆ > 11δ+185δ+12

.

(b) ODS with the low-quality firm active. Since ∆ < 11δ+185δ+12

only if δ > 6/7, this

case can exist only if δ > 6/7. Here firm H gets πHH =

δ3(∆(103∆−82)+327)+δ2(501∆2+846∆+729)+108δ(∆+3)(7∆+6)+243(∆+3)

6(8δ+27)2

45

which is higher than πHu if δ <≈ 0.978601. When the discount factor is very

close to 1, it will be higher only if

∆ >9

7− 30(251δ + 540)

7δ(245δ + 1007) + 7749+ 30

√

(5δ + 9)(8δ + 27)2

(δ(245δ + 1007) + 1107)2,

lower otherwise. Since this case is very specific, we assume a discount factorreasonably lower than 0.978601.

On the other hand, firm L is always better off under the discriminatoryregime, as the profit it gets, i.e.,

πLH = δ3(∆+17)(5∆−11)+3δ2(∆(83∆−354)+523)+54δ(∆(11∆−52)+75)+243(∆−3)2

6(8δ+27)2

is always higher that πLu .

(c) TDS. In this case, both firms are strictly worse off under the discriminatoryregime. Indeed, the profits they get:

πHTDS = 80δ3(3∆+5)2−72δ2(∆(23∆+60)+25)+81δ(∆(5∆−22)−75)+729(∆+3)2

18(27−20δ)2,

πLTDS = 80δ3(5−3∆)2−72δ2(∆(23∆−60)+25)+81δ(∆(5∆+22)−75)+729(∆−3)2

18(27−20δ)2,

are both lower than the respective profits resulting under the uniform pric-

ing if ∆ <60δ−81

√(27−20δ)2(16δ(20δ−61)+765)

2(4δ(20δ−61)+189)≡ ∆̃. Otherwise, the low-quality

firm is better off and the high-quality worse off under the discriminatoryregime.

Proof of Proposition 5

The benchmark case. If BBPD is not viable, prices are equal in both periodsand there is not switching. By simple computation, the total surplus will be:

CSu =

x̄∫

0

UHH(x)dx+

1∫

x̄

ULL(x)dx = qH + qL − 45−∆2

18,

where U iiu (x) = 2 (qi − piu − |x− li|) represents the utility of buying in the two

period good i paying the non-discriminatory price.

46

Discriminatory Price. Under BBPD, three different cases may arise:

(a) Exit of the low-quality firm .

CSE =x̂1∫

0

UHHH (x)dx+

1∫

x̂1

UHLH (x)dx =

(24qH+30qL−2∆2)27

− 2,

where U ijH is simply the utility of buying good j in the first period and good

i in the second when prices are the one leading to a scenario in which onlythe high-quality firm poaches rival’s consumers. Compared with CSu, thisis always lower.

(b) ODS with the low-quality firm active. In this case, the total surplus willbe:

CSH =x̂1∫

0

UHHH (x)dx+

x̂L2∫

x̂1

UHLH (x)dx+

1∫

x̂L2

ULLH (x)dx

= 1486

(582qL + 147qH − (16 + 81qH)∆2 − 954

).

This is always lower than the benchmark case of uniform pricing.

(c) TDS. In this case, the total surplus will be:

CSTDS =x̂H2∫

0

UHHTDS(x)dx+

x̂1∫

x̂H2

ULHTDS(x)dx+

x̂L2∫

x̂1

UHLTDS(x)dx+

1∫

x̂L2

ULLTDS(x)dx

= 1225

(243∆2

27−20δ− 1053∆2

(27−20δ)2−∆2 + 225(qH + qL)− 475

)

,

where U ij2TDS is simply the utilities of buying good j in the first period and

good i in the second when prices are the one leading to a two-directionswitching. Compared with CSu, this is always higher.

Proof of Proposition 6

Under BBPD, three different cases may arise:

47

(a) Exit of the low-quality firm (∆ >). In this case, firm H and firm L respec-tively get:

πHE = δ3(13−∆)(87−7∆)−3δ2(603+∆(240−23∆))+81δ(∆+3)(∆+9)+243(∆+3)2

6(27−δ)2

πLE = (δ(13−∆)+9(3−∆))(δ(δ+30)∆−3δ(7δ+8)+27(3−∆))

6(27−δ)2.

It is easy to verify that πHE − πH

u < 0 and πLE − πL

u > 0.

(b) ODS with the low-quality firm active. Here firm H gets

πHH = δ3(∆(7∆−166)+1023)+3δ2(∆(11∆−126)−801)+189δ(∆+3)2+243(∆+3)2

6(δ−27)2,

which is always lowerer that πHu .

On the other hand, firm L is always better off under the discriminatoryregime, as the profit it gets, i.e.,

πLH =

2δ(−(δ(δ+16)+27)∆2+3δ(9δ+59)∆−9δ(19δ+24)+486(∆−1))9(δ−27)2

is always higher that πLu .

(c) TDS. In this case, both firms are strictly worse off under the discriminatoryregime. Indeed, the profits they get:

πHTDS = 4δ3(3∆(12∆+55)+242)−9δ2(∆(55∆+246)+407)+162δ((∆−2)∆+3)+729(∆+3)2

18(27−11δ)2,

πLTDS = 4δ3(3∆(12∆−55)+242)−9δ2(∆(55∆−246)+407)+162δ(∆(∆+2)+3)+729(∆−3)2

18(27−11δ)2,

are both lower than the respective profits resulting under the uniform pric-

ing if ∆ < 3 − (27−11δ)(

2√

117−δ(37−8δ)−3(7−δ))

27−δ(23δ−22)≡ ∆̃FL. Otherwise, the low-

quality firm is better off and the high-quality worse off under the discrimi-natory regime.

Proof of Proposition 5.2

The benchmark case. If BBPD is not viable, prices are equal in both periodsand there is not switching. By simple computation, the total surplus will be:

CSFLu =

x̄∫

0

UHH(x)dx+

1∫

x̄

ULL(x)dx = (1 + δ)

(

qH + qL − 45−∆2

36

)

,

48

where U iiu (x) = (1 + δ) (qi − piu − |x− li|) represents the utility of buying in the

two period good i paying the non-discriminatory price.

Discriminatory Price. Under BBPD, three different cases may arise:

(a) Exit of the low-quality firm .

CSFLE =

x̂FL

1H∫

0

UHHH (x)dx+

1∫

x̂FL

1H

UHLH (x)dx =

81(18(qH+qL)+∆2−45)−δ2(98qH+114qL+23∆2−1703)4(27−δ)2

+18δ(70qH+7∆2+80qB−174)−2δ3(287−2qH−48qL+2∆2)

4(27−δ)2

where U ijH is simply the utility of buying good j in the first period and good

i in the second when prices are the one leading to a scenario in which onlythe high-quality firm poaches rival’s consumers. Compared with CSu, thisis always higher.

(b) ODS with the low-quality firm active. In this case, the total surplus willbe:

CSFLH =

x̂FL

1H∫

0

UHHH (x)dx+

x̂L2∫

x̂FL

1H

UHLH (x)dx+

1∫

x̂L2

ULLH (x)dx

=δ2(1847−194qH−18qL−23∆2)+18δ(46qH+104qL+15∆2−156)−2δ3(279−2qH+2∆2−48∆)+81(∆2−18∆−45)

4(27−δ)2.

This is always higher than the benchmark case of uniform pricing.

(c) TDS. In this case, the total surplus will be:

CSTDS =729((∆−18)∆−45)+δ3(243∆2+2046qH+2310qL−5203)−9δ2(89∆2+954qH+938qL−2233)

36(27−11δ)2,

+81δ(13∆2+42qH+18qL−57)

36(27−11δ)2

where U ij2TDS is simply the utilities of buying good j in the first period and

good i in the second when prices are the one leading to a two-directionswitching. Compared with CSu, this is always higher.

49

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