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For simplicity we assume that both rms’ production costs are nil. The entrant’s cost ofentering market A, f E , is non-negative.4In our notation, the subscript always identi es the market whereas the superscript identi esthe rm. The variables associated with uniform pricing have an upper bar, and the letters mand d in the subscript denote monopoly and duopoly, respectively.3Discriminatory versus uniform pricingIf entry does not occur the incumbent remains a monopolist. Under discriminatory pricing themonopolist always covers the two markets whereas under uniform pricing one can show that heppcovers both markets if and only if 2 1 k2 1. For very low (high) k values, demand inmarket A (B) is considerably larger than demand in market B (A) and the incumbent prefers tosell in market A (B) only. In addition, it is trivial to show that, when both markets are coveredunder uniform pricing, the monopoly discriminatory price in market A will be higher than theuniform price if and only if market A is larger and less elastic than market B (k 1).In this section we analyze the Nash equilibrium under discriminatory and uniform pricingwhen the entrant decides to enter in market A.3.1Discriminatory pricingSince the incumbent may set a di erent price in each market and he continues to be a monopolistkaA2 .in market B, the optimal solution in this market is pIBm However, market A is now aduopoly and we need to nd the Nash equilibrium of the price-game:Lemma 1 Under discriminatory pricing, the post-entry unique Nash equilibrium in market Ais symmetric and it is given by:pIAd pE aA (bA dA ):2bA dAThe post-entry pro ts in market A are:IAd4 E bA (bAdA ) a2A(bA dA ) (2bAdA )2:(3)We assume that the entrant’s cost of entering in market B is so high that entry in this market would alwaysbe unpro table.6

One can show that pIAd pIAm andandIAd IAmIAdIAm as long as dA 0. Conversely pIAd pIAmwhen dA 0. This result is quite obvious: as long as there exists some productsubstitutability, the incumbent looses with the competition of the entrant. Also notice that thedi erence between pIAd and pIAm increases as products become more substitutable.3.2Uniform pricingThe Nash equilibrium under uniform pricing depends on whether the incumbent is better o selling in both markets or just in one of the markets. Let kA and kB be de ned as follows:qbA 8b2A 5d2A2b2A d2A2b2A d2A 4b2A d2AkA 2 2b2A d2A (bA dA ) (2bA dA )qih2222222 (2bA dA ) 4 bA dA 2bA dA 8bA 5dA2b2A d2A b2A d2AkB :(bA dA ) 32b4A 32b2A d2A 9d4AThe next lemma characterizes the unique Nash equilibrium under uniform pricing when entryoccurs.5Lemma 2 Under uniform pricing, the post-entry equilibrium is as follows:(i) When k kA , the incumbent is better o selling only in market A. The Nash equilibriumprices and post-entry pro ts are:pId pE (ii) When kAkaA (bA dA )2bA dAIdand E bA (bAdA ) a2AdA )2(bA dA ) (2bA:kB , the incumbent is better o selling in both markets. The Nashequilibrium prices and pro ts are given by:pId pE IdE5 aA (bAdA ) [2k (bA dA ) (2bA dA )]8b2A 5d2AaA (bAdA ) bA dA (k 1) 2 2b2AbA 8b2A 5d2Aa2A (bAdA ) 2b2Ad2A [2k (bA dA ) (2bA dA )]2bA (bA dA ) 8b2Aa2A (bAd2A kd2A5d2AdA ) bA dA (k 1) 2 2b2AbA (bA dA ) 8b2A5d2A2d2A kd2A22:When k kA and k kB the incumbent is indi erent between covering both markets or covering just onemarket. In these cases, we assume the incumbent covers the two markets.7

(iii) When k kB , the incumbent is better o selling only in market B and the entrantbecomes a monopolist in market A. The Nash equilibrium prices and pro ts are given by:pI Id kaA;2k 2 a2A4bApE aA;2Eand a2A:4bAProof: The equilibrium prices are the solution of the system of best response functions ofthe two rms, taking into account that the incumbent’s best response function depends on kand dA .Figure 1 illustrates these results. For the set of parameters in dark grey, the incumbent’sresidual demand in market A is considerably larger than demand in market B. Thus, theincumbent prefers to sell only in market A even when there is a competitor in this market. Onthe other hand, when demand in market B is considerably larger than the incumbent’s residualdemand in market A, the set of parameters in white, the incumbent prefers to sell in market Bonly. In this case, the entrant becomes a monopolist in market A. The incumbent prefers to sellin both markets when demand in market B and incumbent’s residual demand in market A aresimilar, i.e, kAkkB (set of parameters in light grey).dAbA0.8 bA0.6 bAkBkA0.4 bA0.2 bA02 10.611.522 1kFigure 1: Uniform pricing under duopoly in market AWhen the entrant decides to enter market A, the incumbent’s decision of serving both orjust one of the markets depends on the relative size of demand in market B and the incumbent’sresidual demand in market A. This comparison depends on the di erence in market demands,k, but it also depends on the degree of product substitutability, dA . The higher dA is, the larger8

is the reduction in the incumbent’s residual demand in market A when entry occurs. Thus, asdA increases, demand in market A has to be much bigger (k has to be lower) in order for theincumbent to prefer to sell only in market A under duopoly. Thus the value of k below whichthe incumbent prefers to sell only in market A, kA , is decreasing with dA . For the same reason,kB is also decreasing with dA . As dA increases, market A becomes more competitive. Thus, theincumbent prefers to abandon this market for smaller values of k.Notice that as products become virtually homogeneous (dA ! bA ), competition in market Ais so erce that even when market A is much larger (very low values of k) the incumbent prefersto sell in market B only. On the other hand, when products are very di erentiated (dA ! 0),competition in market A is soft and consequently market B has to be much larger than marketA for the incumbent to start selling in market B only. When dA 0 the two products arecompletely di erentiated and the incumbent’s demand is the same as under monopoly. Thus hepp2 1.serves both markets as long as 2 1 kAs expected, when the incumbent sells in market A under duopoly, kkB , the incumbent’sprice in market A with competition is lower than under monopoly except when there is completedi erentiation, in which case the price is the same. The incumbent’s pro t under entry is alsolower than under monopoly unless the two products are completely di erentiated (dA 0).3.3Comparison of pricing policiespWhen the incumbent serves both markets under monopoly uniform pricing, i.e, when 2 1pk2 1, one can show that the price decrease in market A due to competition is higherunder discriminatory pricing than under uniform pricing. This happens because uniform pricingmakes the incumbent softer. When E enters, the incumbent has an incentive to decrease priceto maintain a larger market share in market A. However the incentive to decrease price is lowerunder uniform pricing because it implies a reduction in pro ts in market B. Thus the reaction tocompetition is higher under discriminatory pricing.6 This is the e ect described by Armstrongand Vickers (1993).6When kAk p21 and k kE the e ect is reversed. In this case, market B is not served undermonopoly uniform pricing. When entry occurs the incumbent has an incentive to decrease the uniform price andstart serving market B. Since pro ts in market B increase as the price decreases, uniform pricing makes theincumbent tougher.9

However the previous e ect does not mean that the incumbent’s equilibrium price in marketA is always lower under discriminatory than under uniform pricing. If demand in market Ais much larger than demand in market B, the discriminatory price may be higher than theuniform price even when there is a competitor in market A. This is the e ect described byCheung and Wang (1999). When demand in market A is much larger than demand in marketB the e ect of the di erence in market demands may overwhelm the e ect of the di erence inthe reaction to competition. Thus, the price decrease in market A due to competition is higherunder discriminatory pricing than under uniform pricing but the incumbent’s equilibrium pricein market A is higher under discriminatory than under uniform pricing.As described in the next section, the degree of product substitutability a ects the value of ksuch that the e ect of the di erence in market demands overwhelms the e ect of the di erencein the reaction to competition. Thus, we add a third e ect to Armstrong and Vickers (1993)and Cheung and Wang’s (1999) analysis.4Entry decisionThe entrant’s decision to enter market A depends on whether his post-entry pro t is smaller orhigher than his entry costs. When the entrant’s post-entry pro t is higher under discriminatorypricing than under uniform pricing, there is an interval of entry costs values where entry ispro table under discriminatory pricing but unpro table under uniform pricing. Therefore weconsider that discriminatory pricing is more favorable to entry than uniform pricing.Under discriminatory pricing the incumbent always sells in market A and the entrant’s pro tfunction is given by (3). However, under uniform pricing, the entrant’s pro t function dependson whether the incumbent sells in market A only, sells in both markets or sells in market Bonly:E E E bA (bA dA )a2A(bA dA )(2bA dA )22a2A (bA dA )[bA dA (k 1) 2(2b2A d2A ) kd2A ]2bA (bA dA )[8b2A 5d2A ]a2A4bAfor k kAfor kAkkB(4)for k kB :The next proposition describes which of the two pricing policies implies higher entrant’spost-entry pro t, for di erent degrees of market demands di erence and degrees of product10

substitutability.Proposition 1 The entrant’s post-entry pro t under discriminatory pricing is:(i) Equal to the entrant’s pro t under uniform pricing when k kA , or when there is completedi erentiation or also when k k E , where k E 2(bA dA )2bA dA .(ii) Higher than the entrant’s pro t under uniform pricing when kAk kE .(iii) Lower than the entrant’s pro t under uniform pricing when k k E where for k kB theentrant is the only rm in market A.Proof: The result is a direct consequence of comparing the entrant’s pro t under discriminatory pricing (3) and under uniform pricing (4).Figure 2 illustrates this proposition where the area marked with circles stands for (i), thearea marked with squares stands for (ii) and the area in white stands for (iii). This propositionshows that discriminatory pricing may be more, less or equally favorable to entry than uniformpricing.dAbA0.8 bA0.6 bAkEkAkB0.4 bA0.2 bA02 10.611.522 1kFigure 2: Entrant’s pro ts comparisonWhen the competitive market is much larger than the captive market, k kA , under uniformpricing the incumbent is better o serving only the competitive market. Thus equilibrium pricesin this market are precisely the same than under discriminatory pricing and the two pricingpolicies are equally favorable to entry. If there is complete di erentiation, the two pricing policies11

also have the same e ect on entry, because the entrant’s pro ts do not depend on the incumbent’sprice.When the captive market is much larger than the competitive market, k kB , the incumbentabandons the competitive market under uniform pricing, thus uniform pricing is more favorableto entry.When both markets are covered under uniform pricing, kAkkB ; there exist alwayssome levels of the di erence in market demands where discriminatory pricing is more favorableto entry than uniform pricing and other levels where the reverse is true. Discriminatory pricingis more favorable to entry than uniform pricing for k 2 kA ; k E . In this region, the impact onentry due to the di erence in market demands overwhelms the impact due to the di erences inthe reaction to competition.7 Notice that k E is decreasing with dA and the interval kA ; k Edepends on the degree of product substitutability: when products are highly substitutable theinterval is small, when products are highly di erentiated the interval is large.The previous result shows that the degree of product substitutability a ects the impact ofpricing policies upon entry. The entrant’s pro t depends on the incumbent’s price in the competitive market. Discriminatory pricing is more favorable to entry if and only if the incumbent’sdiscriminatory price in the competitive market is higher than the uniform price. When the incumbent serves both markets, the discriminatory price in the competitive market is higher thanthe uniform price when the incumbent’s residual demand in the competitive market is largerand less elastic than the captive market’s demand (i.e., when the competitive market is strong).The comparison of these demand elasticities depends on the di erence in market demands, k,but it also depends on the degree of product substitutability, dA . When products are highlysubstitutable, the reduction in the incumbent’s residual demand due to competition is verylarge. Thus, unless the competitive market is much larger, the incumbent’s residual demand inthe competitive market is smaller and more elastic than the captive market’s demand, thus thediscriminatory price in the competitive market will be lower. In this case, there exists only asmall interval of di erences in market demands where discriminatory pricing is more favorableto entry than uniform pricing. On the other hand, when products are very di erentiated thereaction to competition is smaller, thus the di erence in market demands is the most relevantfactor in the elasticities comparisons and there exists a larger interval of di erences in market7For kA k p21 and k kE both e ects point to discriminatory price being higher than uniform price.12

demands where discriminatory pricing is more favorable to entry than uniform pricing.Our results have some similarities and some di erences relatively to Cheung and Wang(1999). Like in Cheung and Wang’s analysis, when both markets are covered under uniformpricing we may have cases where entry discriminatory pricing is more favorable to entry thanuniform pricing and cases where the reverse is true. In addition, the e ects of the di erence inmarket demands and of the di erence in the reaction to competition are also present. However,if we consider the case of homogenous products entry never occurs under discriminatory pricingwhile it occurs under uniform pricing. Thus the dominant rm model and duopoly model implyvery di erent result in the homogeneous product case. Moreover, when there is duopolisticcompetition under entry, the degree of products substitutability is a key factor to take intoaccount in the comparison of the impact of pricing policies upon entry.5Welfare analysisWe now compare the two pricing policies in terms of social welfare. In this comparison it isimportant to consider whether the pricing policy a ects or does not a ect the entry decisionand to analyze the impact of the pricing policy on the equilibrium prices.When the entry decision is a ected, a very important variable is the degree of product substitutability.8 In our model, entry has two e ects: it increases competition and it increasesproduct variety. The competition e ect bene ts consumers in the competitive market. Thise ect is stronger when product di erentiation is low. On the other hand, the variety e ect ishigher when product di erentiation is high (for example, when dA 0, entry it is equivalentto creating a whole new market). This e ect bene ts both consumers and producers. Consumers always gain with entry, but the industry pro t may either be lower or higher with entry.When products are almost homogeneous the duopoly pro t is smaller than the monopoly pro t(competition decreases industry pro ts). On the contrary, when the entrant product is verydi erentiated the industry pro ts are higher under entry.In the welfare comparison there are four alternative scenarios for consideration, dependingon the level of the entrant’s entry costs. When the entry cost is high, entry will not occur8This discussion is also relevant when entry occurs under the two pricing regimes but the incumbent abandonsmarket A under uniform pricing.13

under both pricing policies and thus we are comparing the two pricing policies under monopoly.Second, when entry cost is low, entry will occur under both pricing regimes, thus we need tocompare social welfare under entry in the competitive market. For intermediate entry costs,entry may occur under one of the pricing regimes but not in the other, which leads to the lasttwo scenarios.95.1High Entry CostsWhen entry cost is high, f E maxE;E, entry does not occur under both pricing policiesand thus the welfare analysis is limited to the well known case of monopoly third-degree pricediscrimination with linear demands.ppWhen 2 1k2 1, the incumbent covers the two markets under both pricingpolicies. For k 6 1, the incumbent’s pro t is higher and the consumer surplus is lower underdiscriminatory pricing. As total quantity does not rise under discriminatory pricing, socialwelfare is lower under discriminatory pricing.10 When k 1, social welfare is the same underboth pricing policies since market A and B are identical and thus uniform pricing leads to thesame results as discriminatory pricing.ppWhen k 2 1 or k 2 1 one of the markets is not covered under uniform pricingand the price charged in the market which is covered is the same under the two pricing policies,thus social welfare is clearly higher under discriminatory pricing.Figure 3 illustrates the welfare comparison under monopoly. Welfare is higher under uniformpricing in the light grey area and smaller in the white area. Notice that consumers and rm’sinterests go in the same direction when one of the markets is not covered, but go in oppositedirections when both markets are covered.5.2Low Entry CostsWhen entry cost is low, f E minE;E, entry occurs under both pricing policies. Conse-quently, the pricing policy does not a ect entry and the welfare comparison is based only on9Notice that these two scenarios only occur for kkA , k 6 kE and dA 0 since the entrant pro t is the sameunder both pricing policies when k kA or dA 0 or k kE .10This is a well known result in the literature of third degree price discrimination (see, for example, Schmalensee(1981)).14

dAbA0.8 bA0.6 bAΠD ΠUΠD ΠUΠD Π UCSD CSUCSD CSUCSD CSU0.4 bA0.2 bA2 102 11kFigure 3: Welfare comparison when entry costs are high.the implications of pricing policies on the equilibrium prices. Figure 4 summarizes the welfarecomparison in this scenario.dAbA0.8 bA0.6 bAkACSD CSUΠD ΠU

and captive markets. We say that discriminatory pricing is more favorable to entry than uniform pricing when there exist levels of entry costs where entry is pro-table under discriminatory pricing but unpro-table under uniform pricing, i.e., when entrant s post-entry pro-t is higher under discriminatory pric-ing than under uniform pricing.