Vertical Di erentiation With Congestion E ect Khaïreddine Jebsi Rim Lahmandi-Ayed y March 13, 007 Abstract Adding a congestion e ect to a vertical di erentiation model, the qualitythen-price equilibrium is characterized with given capacity levels. With regard to the literature on standard vertical di erentiation without congestion e ects, new features arise. Mainly, on a narrow market where only one rm survives with a standard vertical di erentiation model, the presence of congestion allows to two rms to co-exist. Moreover the maximal di erentiation principle may be violated. Indeed under some conditions on the model s parameters, rms choose at equilibrium not to di erentiate their products yet make positive pro ts. JEL Classi cation: C7, L13, L14 Keywords: vertical di erentiation, congestion, quality choice, prices. 1 Introduction Many goods are characterized by negative consumption externalities: consumers utility decreases with the number of users buying the same good. Indeed when these consumers share a xed amount of capacity, the increase in the aggregate consumption decreases the users utility. This may be linked to the consumption delay caused by the development of queues, ow congestion or crowding. The crowding in a store or tra c and the necessary time to load a le provide appropriate examples. The higher the delay, the greater the stress on the user because he bears a delay cost, interpreted as a congestion cost which depends on the congestion level, i.e. the aggregate consumption. Corresponding author: Faculté de Droit et des Sciences Economiques et Politiques de Sousse, Cité Erriadh 403, Sousse, Tunisia and LEGI-Ecole Polytechnique de Tunisie. Tel. + 16 73 3 666. E-mail: khairy.jebsi@topnet.tn y LEGI, Ecole Polytechnique de Tunisie, BP 743, 078 La Marsa, Tunisia and ESSAI. E-mail: rim_lahmandi@yahoo.com 1
In this paper, we suppose that the consumers utility depends on the price, the congestion level and the inherent quality" of the good. By inherent quality" (which will be further referred to simply by quality we mean a characteristic independent from the congestion dimension. Considering a costless duopoly, we characterize the subgame perfect equilibrium in terms of quality and price choice in a vertical di erentiation model with congestion e ect. The separation between the congestion level which creates in itself vertical differentiation, and another vertical quality, may be illustrated by competition between Internet Service Providers (ISP. Indeed individual connection decisions are based on the ISP s connection prices and on their qualities of services. The service quality depends on four basic parameters which are: delay, jitter, bandwidth and reliability. 1 The rst three parameters are related to the congestion level which depends on the number of subscribers and on the capacity level of the ISP. However, the reliability can be considered as the inherent quality, because it results from transmission and switching systems. Two types of literature deal with di erentiation in presence of network externalities. A rst set of papers identify the quality of each product to the network externality level. Katz and Shapiro (1985, Bental and Spiegel (1995, Crémer et al. (000, Foros and Hansen (001, Lee and Mason (001 and La ont et al. (003 among others, consider positive network externalities. Scotchmer (1985, de Palma and Leruth (1989 and Reitman (1991 consider negative network externalities or what is more commonly referred to as congestion. A second set of papers consider two independent parameters of di erentiation: the network externality level and an inherent quality. Baake and Boom (001 and Ghazzai and Lahmandi-Ayed (006 consider positive network externalities combined with another vertical quality. Navon et al. (1995 consider congestion in a horizontal di erentiation model but deal only with price competition with exogenous xed locations. We consider in this paper congestion e ects in a vertical di erentiation model and we deal with both price and quality choice. In this paper we use a vertical di erentiation model to which we add a congestion component. The choice of each consumer between products is made depending on the prices, the qualities and the congestion levels of the products. Even if the ranking of qualities is unanimously admitted by consumers, the quality is not the only parameter taken into account by a consumer who may prefer to buy a lower quality product if 1 The jitter is the variation in end-to-end transit delay. See Ferguson and Huston (1998. The reliability can be thought of as the average error rate of the medium (Ferguson and Huston, 1998.
it has a su ciently low congestion level or a su ciently low price. The congestion e ect taken into account in our model concerns the industries for which the aggregate demand is served at the same time such as computer loads, electronic mail systems, etc. 3. For given capacity levels, we characterize the rms choices in terms of quality and price. This analysis may account for mid-run competition, capacities being less easily adjustable than the other strategic variables. In our paper the presence of two vertical dimensions of product di erentiation implies new insights in relation to the standard results of vertical product di erentiation models without network externalities on the one hand (Anderson, de Palma and Thisse, 199; Lahmandi-Ayed, 000; among numerous others, and of models of competition between congested networks where the congestion level is the only dimension of product di erentiation on the other hand (Scotchmer, 1985; de Palma and Leruth, 1989; Reitman, 1991. With regard to the standard results of vertical product di erentiation models three interesting features arise. First, when the market is narrow so as only one rm is viable with a standard vertical di erentiation model, we show that the presence of congestion allows to both rms to be active. Second, we show that the maximal di erentiation principle may be violated. It is indeed possible that rms choose not to di erentiate their products at equilibrium. Third, at price equilibrium the rm which has the highest quality does not always charge the highest price. In the literature on congested networks where the quality of the service is assimilated to the network congestion level, the highest capacity rm xes the highest price. This result is no longer true in our model where the highest capacity rm xes at equilibrium the highest price only when its capacity is su ciently high w.r.t. its competitor s one. Our paper is organized as follows. Section describes the model. In section 3, the rms demands are characterized. In section 4, we determine the price equilibrium for given qualities. Finally, the quality choice is dealt with in section 5. Section 6 concludes. The model We use a vertical di erentiation model to which we add a congestion component. We consider two rms each o ering a congestible network good, with a xed amount of capacity K i > 0 (i = 1; ; for a continuum of consumers who di er by their intensity of preference for the inherent quality of the good. For given network capacity levels, 3 The used model does not allow to study the case where congestion is due to queues. For this see Luski (1976, Levhari and Luski (1978 and Wilson (1989. 3
the choice of the good depends not only on the quality and the price levels but also on the level of congestion involving disutility. Indeed, each consumer bears a delay cost due to congestion when he consumes the good. To simplify the analysis, we assume that this cost is the same for users belonging to the same network. More precisely we consider a linear delay cost given by y i ; where y i is the level of congestion on the network of Firm i and is the unit waiting cost of the user (or the willingness to pay of the consumer in order to avoid congestion. Each consumer is characterized by a parameter which represents his intensity of preference for quality. We assume that the distribution of is uniform over [; ] with a density normalized to 1. When consumer buys one unit of product q i with congestion level y i at price p i, he obtains the utility: U i = q i y i p i (1 Each consumer is supposed to buy one unit of the product that ensures to him the highest utility 4. q i [q; q] is the quality of Firm i, with q > 0 is the minimal quality required to o er the product and q represents the technologically feasible maximal quality, and p i is the price of the product of Firm i. Since the capacity is shared by the rm s customers, the congestion function is given by the ratio : y i = X i K i where X i is the aggregate demand of Firm i and K i is the level of its capacity. This ratio measures the intensity of use of the rm s network. The considered model is thus adequate to account for the industries in which the aggregate demand is served at the same time such as e-mail delivery 5. We suppose in this paper that rms capacity levels are exogenous. Making such an assumption may appear restrictive but it is a rst step to take congestion e ects into account along with quality di erentiation that may account for mid-run competition, capacities being less easily adjustable than the other strategic variables. Moreover 4 In standard vertical di erentiation models, sometimes consumers are supposed to buy one unit if the obtained utility is positive, otherwise they buy nothing. This issue is ignored in this paper for simplicity sake. But the parameters of the model may be adjusted so as the calculated equilibrium is also an equilibrium with the possibility of no purchase. 5 This congestion function is appropriate for processor sharing modeling as rm s capacity is shared by users. See Reitman (1991 for the formulation of congestion functions of di erent forms of congestion in particular discounting and systems saturation. 4
studying quality-then-price equilibrium allows a relevant comparison with standard vertical di erentiation models without network externalities and with congestion models without quality di erentiation that involve only those two stages. Finally, production is supposed to be costless. For given rms capacities levels we consider the subgame perfect Nash equilibrium of the following three stage game with complete information: 1. The rms choose simultaneously their qualities in the interval [q; q]:. The rms set simultaneously their prices p 1 and p in [0; y]. y is supposed to be su ciently large so that it is never constraining. 3. Each consumer chooses a product to buy. The game is solved by backward induction. First the demand of each rm is determined. Then, we determine the price equilibrium and the quality choice of the rms. 3 Demand Characterization Consumers decide which product to purchase, taking into account the prices, the inherent qualities and the congestion levels of the products. But the congestion levels depending on the decisions of the other consumers, a sort of recursivity appears 6. Thus to determine the rms demands, we have to determine the network equilibrium, i.e. a situation where no consumer has interest to change his choice, given the choice of the others. Without loss of generality, Firm is supposed to supply a product with a higher quality than Firm 1, i.e. q q 1 : We rst consider the case q > q 1 and prove that in this case rms demands are necessarily intervals (possibly empty and ordered as in the standard model, i.e. the lowest buy the lowest quality and the highest buy the highest quality. Lemma 1 Suppose q > q 1. At a network equilibrium, if there exists some consumer 0 [; ] preferring Firm 1 or being indi erent between both rms, then all consumers < 0 necessarily prefer Firm 1. 6 As for instance in Baake and Boom (001 and Ghazzai and Lahmandi-Ayed (006 who consider a vertical di erentiation model with positive network externalities. 5
Proof. Suppose that a consumer 0 [; ] prefers Firm 1 or is indi erent between both. This means 0 q 1 y 1 p 1 0 q y p ( 0 (q q 1 (y y 1 + p p 1 Therefore 8 < 0 ; (q q 1 < 0 (q q 1 (y y 1 + p p 1 : Thus, consumer prefers also Firm 1. Lemma 1 allows immediately to identify three possible types of network equilibria. Corollary 1 Three network equilibria are possible : Type I : only Firm 1 is active : X 1 = ; X = 0 Type II : only Firm is active : X = ; X 1 = 0 Type III : Firms 1 and are active : X 1 = ^ ; X = ^; where ^ is some consumer in the interval ]; [, who is indi erent between both qualities. The following lemmas determine the conditions under which each type of network equilibrium prevails. Lemma Suppose q > q 1. A type I network equilibrium prevails if and only if: p p 1 (q q 1 + ( Lemma states a condition on rms strategic variables for a type I network equilibrium to prevail. When p is su ciently high or q su ciently low, no consumer nds his interest in buying from Firm even if the congestion level is null, i.e. no other consumer buys from it. Proof. A type I network equilibrium prevails if all consumers prefer Firm 1. Thus X 1 = and X = 0. According to Lemma 1, this occurs if and only if either prefers good 1 or is indi erent between both. u 1 ( u (; which is equivalent to : q 1 y 1 p 1 q p ( p p 1 ( (q q 1 Lemma 3 Suppose q > q 1. A type II network equilibrium prevails if and only if: p p 1 (q q 1 K ( 6
Proof. A type II network equilibrium prevails when all consumers prefer Firm. Thus X 1 = 0 and X =. According to Lemma 1, it occurs if and only if consumer either prefers good or is indi erent between both goods, i.e. u ( u 1 (; which is equivalent to : q 1 p 1 q y p ( p p 1 + K ( (q q 1 Lemma 4 Suppose q > q 1. A type III network equilibrium prevails if and only if: (q q 1 K ( < p p 1 < (q q 1 + ( Proof. According to Lemma 1, a type III network equilibrium prevails if and only if there exists a consumer ^ indi erent between both rms. Thus ^ = p p 1 + ( + K q q 1 + + K ( This is a network equilibrium if each consumer < ^ prefers Firm 1 and each consumer > ^ prefers Firm. u 1 ( u ( = u 1 ( u 1 (^ u ( + u (^ = (q q 1 (^ Hence u 1 ( u ( has the same sign as ^ : An equilibrium with two active rms exists if and only if < ^ <. From Equation (, we deduce the conditions mentioned in the Lemma. The case of equal qualities must be dealt with apart. This is the object of Lemma 5. Lemma 5 (The case of equal qualities Suppose q 1 = q. We have the following: A type I network equilibrium prevails if and only if p p 1 ( : A type II network equilibrium prevails if and only if p p 1 K ( : A type III network equilibrium prevails if and only if ( : K ( < p p 1 < 7
Note that in the case of equal qualities the equivalent of Lemma 1 cannot be proved. The demand of each rm is not necessarily an interval as in the case of di erent qualities. In fact the demand of each rm is even not identi ed, i.e. we do not know who buys from which rm. Lemma 5 allows however to calculate the global demand of each rm at a network equilibrium. Someway we can say that quality di erentiation allows to identify consumers (to guess from his purchases to which interval the consumer belongs not the congestion level alone. Proof. We proceed as in the case q > q 1. The only di erence is that the choice of a consumer does not depend on his type. A type I network equilibrium prevails when each consumer prefer Firm 1 even when all consumers buy from Firm 1: X 1 = and X = 0. This is the case if the resulting utility of each consumer when buying from Firm 1 is higher than his utility when buying from Firm : thus the stated inequality. p 1 ( p ; A type II network equilibrium prevails if all consumers prefer Firm : p 1 p K ( ; thus the stated inequality. A type III network equilibrium. The inherent qualities being equal, the choice of a consumer if he buys from Firm i depends only on price p i and the congestion of the product. For a given choice of the other consumers, the consumer compares: X 1 p 1 and X K p, which does not depend on. A network equilibrium prevails with two active rms (a type III only if the obtained utilities are equal. Otherwise a type I or a type II equilibrium prevails. This can be written as follows: X 1 p 1 = X K p : Writing X 1 + X =, we obtain: X 1 = p p 1 + K ( + K 8
The obtained demands satisfy the supposed hypotheses if 0 < X 1 < (, which are equivalent to the inequalities stated in the lemma on prices and congestion levels. As announced and shown in the proof, in the case q 1 = q, the demand of each rm is not identi ed. The global demand of each rm is however identi ed at a network equilibrium for each couple of prices. Moreover the obtained demands correspond to the limit of the case q > q 1. Now, we are ready to characterize the demands in all cases. Denote by : A = (q q 1 + ( (3 B = (q q 1 K ( (4 Proposition 1 For q 1 q, the rms demands are given by the following: 8 >< if p 1 p A X 1 = ^ if p A < p 1 < p B >: 0 if p 1 p B 8 >< if p p 1 + B X = ^ if p1 + B < p < p 1 + A >: 0 if p p 1 + A Note that only one type of network equilibrium prevails in the same time for each couple of prices. This is not the case with positive network e ects. In Baake and Boom (001 and Ghazzai and Lahmandi-Ayed (006, under some conditions on prices and qualities, the three types of network equilibria may coexist making necessary the adoption of a selection rule. Proof. It consists simply in re-writing Lemmas, 3 and 4 in terms of rms demands. 4 Price equilibrium Given the qualities and the capacity levels, we determine in this section the price equilibrium. Lemma 6 provides the reaction functions of both rms. Lemma 6 The best reply correspondences of rms are: 9
8 [0; y] if p >< B p ' 1 (p = B if B < p < A B >: p A if p A B 8 < p 1 + A if p 1 A B ' (p 1 = : p 1 + B if p 1 A B Proof. Best reply of Firm 1: Using the expression of pro t 1 = p 1 (^ and the expression of the marginal consumer given by Equation (, we obtain (It is easy to check the second order condition: ~p 1 = p B (5 The obtained price is the best reply of Firm 1 if it satis es the following conditions: 8 >< >: ~p 1 0 and These conditions are equivalent to: p A ~p 1 p B B < p < A B with A B = ( ( + K + ( (q q 1 When p > A B; we have ~p 1 < p A. The pro t is thus increasing up to p A and is always decreasing afterwards. Hence 1 reaches its maximal value at p 1 = p A: 7 Finally when p < B, according to Proposition (1 the demand is always null thus 1 = 0 for all prices p 1, so that Firm 1 is indi erent between all its strategies. Hence the pro t reaches its maximal value at every price p 1 [0; y]: Best reply of Firm : We proceed similarly for Firm. Using the expression = p ( ^, we write the rst order condition, which yields: ~p = p 1 + A (6 7 Note that since p > A B; we have p A > A B > 0. 10
The obtained price is the best reply of Firm if it satis es: p 1 + B ~p p 1 + A, which is equivalent to p 1 A B: 8 When p 1 > A B, we have ~p < p 1 + B. The pro t is increasing up to ~p and is always decreasing afterwards. It thus reaches its maximal value at p = p 1 + B: The intersection between the best replies in the plane (p 1 ; p gives the price equilibrium that depends on the sign of A B: This is the object of the following proposition. 9 1 Proposition If A > B, both rms are active. The equilibrium prices and the corresponding pro ts are given by: 8 >< >: p 1 = A B = 1[( (q 3 3 q 1 + ( ( + K ] p = A B = 1[( (q 3 3 q 1 + ( ( + K ] 8 >< >: 1 = [( (q q 1 + ( ( + K ] 9(q q 1 + + K = [( (q q 1 + ( ( + K ] 9(q q 1 + + K If 1 A B: only Firm is active. The equilibrium prices and the pro ts are given by: ( ( 1 = 0 p 1 = 0 p = (q q 1 K ( = ( [(q q 1 K ( 1 Proof. First case: A > B: The unique possible intersection between the reaction functions calculated in Lemma 6 is the price equilibrium given by Equation (7. Replacing these prices in Equation (, we obtain: ] (7 (8 (9 (10 ^ = 1 3 [( ( K + (q q 1 ( + ] + ( + K q q 1 + + K (11 Substituting the prices given by Equations (7 and (11 in the pro t expressions 1 = p 1 (^ and = p ( ^, we obtain the pro ts given by Equations (8. 8 ~p < p 1 + A is equivalent to A < p 1, which is always true as A > 0. 9 By active rm we mean a rm that has a positive demand. 11
Second case: 1 A B: From lemma (6, we have 8p 1; p = p 1 + B. The only Nash equilibrium is p 1 = 0 and p = B. In the case 1 A > B where both rms are active, the prices xed by the rms can be rewritten (inserting Equation (11 in X 1 = ^ and in X = ^ as the following expression: p i = X i (q j q i + X i ( 1 K i + 1 K j i; j = 1; i 6= j Then, we get the traditional principle of the textbook congestion pricing which allows the e cient use of networks: each user should pay a price that covers the marginal cost of congestion that he imposes on the other users consuming the same good. In our setting this cost is equal to: X i ( 1 K i + 1 K j It corresponds to the product of the number of users X i by the marginal cost of congestion that a user imposes on another, ( 1 K i + 1 K j. The latter is the sum of two marginal congestion costs: the marginal congestion cost that a user imposes on another belonging to network i, K i, and the marginal congestion cost that he imposes on a user belonging to network j, K j : According to this pricing rule, the price xed by a rm to a consumer who wishes to join its network, covers not only the marginal cost of congestion that a user imposes on another user of his network but also covers the marginal cost of congestion that he imposes on a user belonging to the other network 10. This result is explained as follows. Since all users have the same unit waiting cost, when a user chooses network i, he limits the choice of the other users because he makes this network less attractive for them, and therefore he obliges some users to join network j. He thus indirectly imposes a congestion cost to a user joining network j. In the case 1 A B where only one rm is active (Firm, we note that the rm which has the highest quality has the ability to undercut the price in order to induce the exit of its competitor from the market. the users a compensation equal to their congestion cost which equals: To do so, Firm attributes to K (, (see price p in Equation (9. This possibility does not exist in the models dealing with competition between congested networks where the quality of the service is assimilated to the network congestion level, as in Scotchmer (1985, De Palma and Leruth (1989 10 In the case of a monopoly o ering a congestible network good from a capacity K, the marginal cost of congestion that a user imposes on another is K (see for instance Mackie-Mason and Varian, 1995, Jebsi and Thomas, 006. 1
and Reitman (1991. Indeed in this literature, when a rm cuts its price, it cannot attract all users from its competitor for, as its demand increases, its congestion level also increases. Hence the number of users who switch to this rm is limited by the accompanying deterioration in its quality level. The existence of two independent characteristics of the good makes entry deterrence possible with xed qualities. In the case where there is room for both rms ( 1 A > B, the comparison between the prices charged by both rms is not obvious and depends on capacities. This is stated precisely in the following corollary. Corollary When 1 A > B; we have: p p 1 ( (q q 1 ( + ( ( 1 1 K When K, Firm which o ers both the highest quality and the highest capacity xes the highest price. This is natural as Firm has the best product in all respects. When > K, the last comparison is not always valid. Firm 1 may x a price higher than Firm if the capacity level of Firm 1 is su ciently high with respect to Firm s one, although the latter o ers the highest quality level. This result mitigates the traditional result of the standard vertical di erentiation models where the highest quality rm always xes the highest price. On the other hand, the highest capacity rm does not necessarily x the highest price, which mitigates the traditional result of the models of competition between congested networks with congestion as the only dimension of di erentiation. 5 Quality choice We determine in this section the quality choice of the rms anticipating price competition. To do so, we study each rm s pro t at price equilibrium w.r.t. its quality. The pro t of a rm does not have the same expression depending on whether it produces the highest quality or not. For a good readability of the paper, we choose to denote by q 1 and q the rms qualities when q q 1 and by q i and q j when the ranking of the rms qualities is not known. Note that 1 A > B is equivalent to ( (q q 1 + ( ( + K > 0; and that this inequality is always true when, which means that in this case there is always room for two rms as in the classical case without congestion. 13
Lemma 7 The pro t function of the rm o ering the highest quality is always increasing on its own quality: 8q q 1 @ @q > 0 Proof. If 1 A B, from Equation (10 we have: @ @q = ( > 0 8 [; ] If 1A > B, by using the second expression in Equation (8, we show that @ @q same sign as the following expression: has the ( (q q 1 + + K [( (q q 1 + ( ( + K ]; which may be re-written as: which is always positive. ( (q q 1 + + K (3 ; The analysis of the pro t of the lowest quality rm is di erent depending on whether < or. We begin with the case <. Lemma 8 Suppose < : Denote by C = ( + K > 0. The pro t of the rm with the lowest quality at price equilibrium may be written as: 8q 1 q, 8 0 if q 1 q C >< 1 = [( (q q 1 + ( ( + K ] >: 9(q q 1 + + if q C < q 1 q K Moreover, Proof. Suppose < : when q C < q 1 q @ 1 @q 1 > 0 Inequality 1A > B may be written as q 1 > q C: By deriving 1 given by Equation (8 with respect to q 1 ; we show that @ 1 @q 1 has the same sign as the following expression: ( (q q 1 + + K + A B (1 which is positive. 14
We state the rst result concerning the quality choice in the case of a narrow market. Proposition 3 Suppose <. qualities and prices are given by: At the unique subgame perfect Nash equilibrium, 8 >< p 1 = q 1 = q = q ( 3 + K >: 8 >< p = 1 = [( ( 3 + K ( + K ] 9( + K >: = [( ( + K ] 9( + K In the case of a narrow market, i.e. < both rms are active at equilibrium selling the highest quality. Three remarks may be noted regarding this case. First our result di ers from the one obtained in the standard vertical di erentiation models where variants di er only by their inherent qualities. In those models, when < only one rm is active, selling the highest quality. Proposition 3 shows that adding a congestion component allows to both rms to be active. The existence of a negative network externality allows to two rms to co-exist even on a narrow market. Second, although the inherent qualities are the same, the price levels are di erent as they depend on the rms capacity levels. The rm with the highest capacity xes the highest price, thus obtaining the same result as De Palma and Leruth (1989. This is a natural result for, when the inherent qualities are the same, the best variant for consumers is the one with the lowest congestion level thus with the highest capacity, which allows a higher price. Finally, although both rms produce the same quality, they make positive pro ts, contrarily to Bertrand competition between homogenous products. This is so even if both capacities are equal making identical the two products. Hence the congestion level does not simply amount to an anodyne additional characteristic. When consumers su er from congestion, it is useless for rms to launch in pro t destructive competition as the attraction of new consumers makes the product less interesting. 15
Proof. Suppose <. Consider Firm i s pro t for a xed q j [q; q] of its competitor. According to Lemmas 7 and 8, For q i q j (The case dealt with in Lemma 8, i = 0, thus constant for q i q j C, i is increasing for q i [q j C; q j ]. For q i > q j (The case dealt with in Lemma 7, the pro t i is increasing w.r.t. q i. It may happen that q j C < q, in which case, the pro t function is always increasing, which does not change the analysis. Hence the pro t of Firm i achieves its maximal value at q i = q, whatever q j. We now consider the case. Which has to be studied is the pro t of the lowest quality rm, the analysis of the pro t of the highest quality one being the same independently of the width of the market. Lemma 9 Suppose : The pro t of the lowest quality rm satis es the following where D = 1 [ K + (3 ] ( if q D q 1 q then if q 1 q D q then @ 1 @q 1 0 @ 1 @q 1 0 Proof. Suppose : Two rms are always active in this case at price equilibrium. When deriving the pro t (given by Equation 8 of the lowest quality rm (selling product q 1 with q 1 < q, after calculations, we prove that @ 1 @q 1 has the same sign as ( + which is positive if and only if q 1 > q D. + K q q 1 + + K ; Lemma 10 Suppose. To maximize its pro t, each rm necessarily chooses either q or q. Proof. Consider Firm i. For a given strategy of its competitor q j, the pro t of Firm i satis es the following: 16
when q i < q j (case dealt with in Lemma 9 for q i < q j D, i is decreasing. for q j D < q i < q j, i is increasing. for q i > q j (case dealt with in Lemma 7, i is increasing. Hence on [q; q], i is always decreasing (if q j = q and D < 0, always increasing (when q j D < q, or decreasing then increasing. In all cases, it reaches its maximal value at q or at q, which are thus the only relevant strategies for each rm. Denote by ^ i = i (q i = q; q j = q = i (q i = q; q j = q = [( ( K i + K j ] 9( K i + K j i = i (q i = q; q j = q = [( (q q + ( ( K i + K j ] 9(q q + K i + K j i = i (q i = q; q j = q = [( (q q + ( ( K i + K j ] 9(q q + K i + K j the pro ts of the rms at the relevant qualities. The game may be more clearly visualized as follows: 1n q q q (^ 1 ; ^ ( 1 ; q ( 1 ; (^ 1 ; ^ Lemma 11 We always have for i = 1;, ^ i < i : Thus the best reply of each rm to the quality q of its competitor is q. This comparison implies that the situation in which both rms produce quality q never emerges at equilibrium. Proof. Note rst that ^ i is obtained when one makes q i. q = 0 in the expression of 17
Considering now i as a function of q q, it is an increasing function 11 thus reaches its minimal value for when q q = 0. Therefore, the inequality ^ i < i always holds. Lemma 1 Equalities i = ^ i hold for i = 1; simultaneously if and only if = K = ( (15 3 (q q( This is to say that the special case that requires a special treatment where both equalities hold together is a very special one. Proof. Suppose indeed that we have in the same time i = ^ i for i = 1;. Replacing the pro ts by their equilibrium values, this implies for i = 1; and j 6= i: [( ( K i + K j ] 9( K i + K j = [( (q q + ( ( K i + K j ] 9(q q + K i + K j Thus Note that the denominator of each pro t does not change as i and j are inverted. [( ( + K ] [( (q q + ( ( + K ] = [( ( K + ] [( (q q + ( ( K + ] This leads after simple calculations to = K. Consider now the case = K = K. Note that the two equalities amount to the same one equivalent after calculations to K = ( (15 3 (q q( Proposition 4 Suppose. In terms of quality choice, at equilibrium, exactly one, two or three outcomes are simultaneously possible among the three candidates: (q; q, (q; q, and (q; q. More precisely, 1. In the special case K = (q; q and (q; q. ( (15 3 (q q(, the game admits three equilibria: (q; q, 11 Its derivative w.r.t. this variable is positive, using only that <. 18
. When i > ^ i, for i = 1;, or when i = ^ i and j > ^ j for i = 1; and j 6= i, the game admits two equilibria: (q; q and (q; q. This is the case when the parameters are = 10; = 4; = 1; q = 6; q = 1; = K = 100: Thus the pro ts are: ^ i = 0:18, i = 14:9 and i = :9, for i = 1;. This is also the case when the parameters are = 10; = 4; = 1; q = 6; q = 1; = 100; K = 4:75: Thus the pro ts are: ^ 1 = 1 = 3:37; ^ = 0:96; = :75; 1 = 145:16 and = 140:97: 3. When i < ^ i, for i = 1;, the game admits exactly one equilibrium: (q; q. This is the case when the parameters are = 10; = 4; = 1; q = 1; q = 0:8; = K = 100: Thus the pro ts are: ^ i = 0:18, i = 5:77 and i = 0:17. 4. When i = ^ i and j < ^ j for i = 1; and j 6= i, the game admits two equilibria: (q i ; q j and (q; q where q i = q and q j = q: This is the case for the parameters = 10; = 4; = 1; q = 1; q = 0:8; = 5; K = 00; where the pro ts are: ^ 1 = 1 = 0:; ^ = 0:64; = 0:38; 1 = 5:56 and = 6:4: Thus, the game admits two equilibria: (q; q and (q; q: If rms are inverted, we obtain (q; q and (q; q: 5. Finally, when i > ^ i and j < ^ j, at equilibrium rm i chooses q and rm j chooses q. This is the case when the parameters are = 10; = 4; = 1; q = 1; q = 0:7; = 50; K = 100; for which the pro ts are: ^ 1 = 0:1, ^ = 0:33, 1 = 0:3, = 0:7, 1 = 8:55 and = 8:75: Thus the game admits exactly one equilibrium: (q; q. If rms are inverted, we obtain (q; q as the unique equilibrium. Proof. The proof is immediate. The veri cation that a numerical case exists for each pointed case was necessary as the expression of the pro ts does not allow to write the above conditions function of the parameters. Therefore it was necessary to check that each enumerated case is a non-empty one. Considering both propositions 3 and 4, three outcomes may arise at equilibrium in terms of quality choice: either both rms choose q or the two rms di erentiate maximally their products: one rm or the other produces the highest quality while the other produces the lowest one, the two situations not necessarily being simultaneously obtained at equilibrium. In standard vertical di erentiation without congestion e ect, only maximal di erentiation obtains at equilibrium. The presence of congestion e ects 19
makes possible at equilibrium the situation where both rms produce the highest quality. The presence of congestion e ects may have a positive e ect in quality terms. This is so because the production of the same quality, when there are congestion e ects, does not destroy pro ts as in the standard case. When consumers su er from congestion, it may be optimal to produce the same quality as the competitor as congestion makes useless a erce price competition. In terms of prices, to make sense the comparison of our results with the standard vertical di erentiation model without congestion must be done carefully. Two cases must be distinguished. When maximal di erentiation obtains at equilibrium with congestion (Proposition 4, cases 1,, 4 and 5, the equilibrium prices of the qualities q and q are given by (Firm 1 being the rm producing q, p C 1 = 1 3 [( (q q + ( ( + K ] p C = 1 [( (q q + ( ( + ]: 3 K In the standard vertical di erentiation model, maximal di erentiation with two active rms always obtains when, and the equilibrium prices for the qualities q and q are given by: p 1 = 1 [( (q q] 3 p = 1 [( (q q]; 3 The equilibrium prices obtained with congestion are higher relative to the case without congestion as we already mentioned. When in both cases (with and without congestion e ect rms maximally di erentiate their qualities at equilibrium, the presence of congestion lessens price competition allowing higher prices, which is the same conclusion as Navon et al. (1995. For the same quality the di erence between the prices in the two cases, and p C 1 p 1 = 1 3 [( ( + K ] p C p = 1 [( ( + ]; 3 K 0
decreases with each capacity. This is perfectly intuitive as the increase in each capacity releases the congestion e ect. When both capacities go to in nity, the congestion e ect disappears and the situation amounts to the standard vertical di erentiation. When both rms choose the same quality q at equilibrium with congestion (case < dealt with in Proposition 3 and Case of Proposition 4, the prices are given by and ~p C 1 = 1 3 ( ( + K ~p C = 1 ( ( + : 3 K Obviously when compared with Bertrand competition between homogenous goods, the presence of congestion increases equilibrium prices. When capacities increase, the congestion e ects decrease and the prices converge to the Bertrand prices values. But this situation never emerges at equilibrium in a standard vertical di erentiation model. What would be more relevant is the comparison between the prices of a quality emerging at equilibrium in the two cases, with and without congestion. In the standard vertical di erentiation model, either only one rm is viable (case < producing quality q at price p = (q or the two rms maximally di erentiate their products (case and the highest quality equilibrium price is given by q; p 0 = 1 ( (q q; 3 The last two prices of quality q obtained at equilibrium in a standard model without congestion, may be lower or higher than ~p C 1 and ~p C each rm s equilibrium price of quality q in the model with congestion e ect. Thus in price terms, the overall e ect of congestion is ambiguous. More precisely, the equilibrium price of the highest quality is higher with congestion relative to the case without, when capacities are su ciently low or the quality segment is su ciently narrow. Indeed on the one hand, a low capacity resulting in a high congestion level increases the highest quality price. On the other hand, when the quality segment is su ciently narrow, competition is ercer in a standard vertical di erentiation model as both products tend to be homogenous; while the presence of congestion allows to two rms producing the same quality to x prices that may be higher as they depend only on capacities. 1
6 Conclusion We have shown in this paper that new features stem from competition between rms selling goods characterized by a vertical quality and congestion e ects, with regard to the standard results of vertical di erentiation models without network externalities and related to the models of competition between congested networks where products are di erentiated only by the congestion levels. Several directions may be given for future research. The most natural one is to calculate endogenously capacity levels, adding a stage in the game where rms choose the capacity levels prior to qualities and prices. It is not sure that all the cases pointed out in Proposition 4 will necessarily obtain at the subgame perfect equilibrium of the three-stage game. Second the purchase obligation may be relaxed supposing that a consumer buys a product only if it improves his utility relative to no purchase. This may be especially important in the three stage game (capacity, quality, price if rms pro ts at the quality subgame are decreasing w.r.t. capacity 1, which means that rms will choose very low capacities. Indeed in this case, the congestion level will be very high and the utility of consumers very low making purchase obligation unreasonable. Finally we may consider that consumers di er w.r.t. their waiting cost. 7 References Anderson S, De Palma A,Thisse J. Discrete choice theory of product di erentiation. The MIT Press: Cambridge; 199. Baake P, Boom A. Vertical product di erentiation, network externalities and compatibility decisions. International Journal of Industrial Organization 199;19; 67-84. Bental B, Spiegel M. Network competition, product quality and market coverage in the presence of network externalities. Journal of Industrial Economics 1995;43; 197-08. Crémer J, Rey P, Tirole J. Connectivity in the commercial Internet. Journal of Industrial Economics 000;48; 433-47. De palma A, Leruth L. Congestion and game in capacity. Annales d Economie et de Statistique 1989;15/16; 389-407. Ferguson P, Huston G. Quality of service: delivering QoS on the internet and in corporate networks. John Wiley & Sons, Inc.: New York; 1998. Foros O, Hansen B. Competition and compatibility among internet service providers. 1 which is easy to check for instance when <.
Information Economics and Policy 001;13; 411-45. Ghazzai H, Lahmandi-Ayed R. Vertical di erentiation, network externalities and compatibility decisions: an alternative approach. Cahiers de la MSE, Série Bleue, Université Paris 1-Panthéon Sorbonne 006; Ref: 006-13. Jebsi K, Thomas L. Optimal pricing of a congestible good with random participation. Economics Letters 006;9; 19-197. Katz M, Shapiro C. Network externalities, competition and compatibility. American Economic Review 1985;75; 44-440. La ont J, Marcus S, Rey P, Tirole J. Internet interconnection and the o -net-cost principle. Rand Journal of Economics 003;34; 370-390. Lahmandi-Ayed R. Natural oligopolies: a vertical di erentiation model. International Economic Review 000;41; 971-987. Lee I, Mason R. Market Structure in congestible markets. European Economic Review 001;45; 809-818. Levhari D, Luski I. Duopoly pricing and waiting lines. European Economic Review 1978;11; 17-35. Luski I. On partial equilibrium in a queuing system with two servers. The Review of Economic Studies 1976;43; 519-55. MacKie-Mason J, Varian H. Pricing congestible network resources. IEEE Journal of Selected Areas in Communications 1995;13; 1141-1149. Navon A, Shy O, Thisse J. Product Di erentiation in the Presence of Positive and Negative Network E ects. CEPR Discussion Paper 1995;1306. London, Centre for Economic Policy Research. Web site: www.cepr.org/pubs/dps/dp1306.asp Reitman D. Endogenous quality di erentiation in congested markets. The Journal of Industrial Economics 1991;39; 61-647. Scotchmer S. Two-tier pricing of shared facilities in a free-entry equilibrium. Rand Journal of Economics 1985;16; 456-47. Wilson R. E cient and competitive rationing. Econometrica 1989;57; 1-40. 3