THE EXISTENCE OF EQUILIBRIUM IN A DIFFERENTIATED DUOPOLY WITH NETWORK EXTERNALITIES*

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1 Vol. 56, No. 1, March 005 TE EXISTENCE OF EQUILIBRIUM IN A DIFFERENTIATED DUOPOLY WIT NETWORK EXTERNALITIES* By LUCA LAMBERTINI and RAIMONDELLO ORSINI Università degli Studi di Bologna, Italy The existence of a pure-strategy subgame-perfect equilibrium in qualities and prices is investigated in a duopoly model of vertical differentiation where quality improvements require a quadratic variable cost and network externalities operate. We show that there exists a parameter region where the incentive to predate at the quality stage prevents firms from reaching a pure-strategy non-cooperative equilibrium with prices above marginal costs. If network externalities are sufficiently large, a Bertrand equilibrium with zero profits may arise, although the amount of product differentiation is strictly positive. JEL Classification Numbers: D6, L Introduction The issue of the existence of a subgame-perfect equilibrium in pure strategies has drawn a considerable amount of attention in the existing literature on endogenous product differentiation. When vertical differentiation is considered, technology largely affects the equilibrium market structure. If quality improvements hinge either on fixed costs arising from R&D activity, or on a variable cost that is not too steep in the quality level, then in equilibrium the market supports a limited number of firms with prices above marginal cost. This result is known as the finiteness property (see Gabszewicz and Thisse, 1979, 1980; Shaked and Sutton, 198, 1983). When variable costs are sufficiently convex in the quality level, then competition obtains as in spatial models à la otelling (199). As stressed by Gabszewicz and Thisse (1986), vertical product differentiation is usually expected to generate pure-strategy equilibria with prices strictly above marginal costs. On the contrary, under horizontal product differentiation an established result is that a purestrategy equilibrium in prices may fail to exist (see, inter alia, d Aspremont et al., 1979; Gabszewicz and Thisse, 1986; Economides, 1986; Anderson, 1988). More precisely, a subgame-perfect equilibrium with positive profits may fail to exist, because firms location choices drive prices to marginal cost. 1 In some markets, individuals may take their consumption decisions conditionally upon the number of other consumers purchasing the same brand or good. This phenomenon is * We thank the editor of this Journal, Makoto Yano, an anonymous referee and the seminar audience at the University of Copenhagen for useful comments and discussion. The usual disclaimer applies. 1 A price equilibrium in mixed strategies always exists. See Dasgupta and Maskin (1986); Osborne and Pitchik (1987). 55

2 captured by introducing network externalities into consumer preferences. The presence of network externalities tends to intensify competition. This, in turn, may put into question the existence of a pure-strategy equilibrium with positive profits. Vanity and network effects are analysed by Grilo et al. (001) in a duopoly model of spatial competition with full market coverage, which also permits vertical differentiation to be dealt with. They describe price competition for exogenous product locations, finding, among other things, that (i) an increase in network externalities makes price competition tougher, and (ii) when product differentiation dominates the externality monopoly equilibria may arise. We investigate the role of network externalities in a duopoly model with vertical differentiation and quadratic costs of quality improvements, where the finiteness property does not hold. 3 We introduce a network externality in consumers utility function, and investigate the existence and characterization of symmetric pure-strategy subgame-perfect equilibria under full market coverage. We show that price undercutting is never profitable, independently of the extent of network externalities. owever, we also prove that there exists a subset of the parameters measuring hedonic preferences and network externalities, where the incentive to predate at the quality stage prevents firms from reaching a pure-strategy non-cooperative equilibrium with prices above marginal costs. Therefore, the existence of a pure-strategy subgame-perfect equilibrium, in a duopoly model with convex costs, can be threatened by predatory behaviour at the product stage, rather than undercutting behaviour at the market stage. Whenever network externalities are large enough, a Bertrand equilibrium with zero profits may arise, although the amount of product differentiation is strictly positive. The remainder of the paper is structured as follows. The model is laid out in Section. Section 3 describes the price behaviour of firms at the market stage. The existence of a pure-strategy equilibrium at the quality stage is investigated in Section 4, and concluding remarks are presented in Section 5.. The model This setting extends the analysis carried out by several authors (e.g. Champsaur and Rochet, 1989; Cremer and Thisse, 1994; Lambertini, 1996; Ecchia and Lambertini, 1997). We consider a vertically differentiated duopoly where each firm produces a single good of quality q, with q q L > 0, and then competes in prices against a rival. There exists a continuum of consumers indexed by their marginal willingness to pay for quality θ [θ 0, θ 1 ], 3 Seminal contributions in the theory of network externalities are Katz and Shapiro (1985, 1986); Farrell and Saloner (1985, 1986). For an overview, see Katz and Shapiro (1994) and the special issue of the International Journal of Industrial Organization, edited by Economides and Encaoua (1996) An analysis of monopoly with the same variable cost technology and network externalities can be found in Lambertini and Orsini (001). A monopoly model with vanity effects is in Lambertini and Orsini (00). 56

3 L. Lambertini and R. Orsini: Equilibrium in a Differentiated Duopoly with θ 0 θ 1 1. The distribution of consumers is uniform, with density f(θ) 1, so that the population of consumers is 1. The net consumer surplus from the purchase of one unit of product i is U θq i + αx i p i, i, L, where p i is the price of good i, x i is the market demand for good i, and α 0 (the same for all agents) is the weight of the network externality in the utility function. This amounts to saying that the network externality is productspecific. 4 We confine our attention to the case where the market is completely covered by firms; that is, the poorest consumer is always willing to buy: 5 max{θ 0 q + αx p, θ 0 q L + αx L p L } 0. (1) Given qualities and prices set by the firms, a generic consumer indexed by θ [θ 0, θ 1 ], will choose to buy the good i that maximizes his net surplus θq i + αx i p i. In order to define market demands, we have to find, along the support [θ 0, θ 1 ], the location # of the consumer who is indifferent between the high and the low-quality good. From the indifference condition #q + αx p #q L + αx L p L, () we obtain # (α αθ 1 + p p L )/(q q L α). 6 The market demand functions are, respectively, x θ 1 #, x L # θ 0 if # (θ 0, θ 1 ); (3) x 1, x L 0 if # θ 0 ; (3 ) x 0, x L 1 if # θ 1. (3 ) Production technology involves variable costs, which are quadratic in the quality level and linear in the output level: Ci qi xi, i, L, (4) where x i indicates the output level of firm i, whose profit function is π i p i q i x i ( ). (5) This can be the case in the telecommunications industry. For a model analysing country-specific networks and their effects on comparative advantage in the telecommunications industry, see Kikuchi (00). See also Shy (001, chapter 5). Of course, there exist goods for which the externality is not product-specific. The inclusion of a weight attached to the network of the other variety would give rise to an external effect α(x i + δx j ), with δ [0, 1]. owever, given the assumption of full market coverage, this translates into a neutral reparametrization of the consumer surplus function we have assumed above. Notice that, unlike Katz and Shapiro (1985, 1986), we assume that consumers have heterogeneous tastes. This entails that any price differentials across firms may not induce all consumers to switch from one good (or network) to the other. As a consequence, we may expect the present model not to generate multiple equilibria. 57

4 For future reference, we can now define the interval of individually preferred qualities (see Cremer and Thisse, 1994). Consider a generic consumer characterized by a marginal willingness to pay equal to θ [θ 0, θ 1 ]. is preferred quality q θ maximizes his net surplus when he is able to purchase such quality at marginal cost, i.e. when q arg max U θq q, θ q which yields q θ θ/. 7 This entails that the interval of individually optimal qualities is [θ 0 /, θ 1 /] [(θ 1 1)/, θ 1 /]. Given the a priori symmetry of the model, we may expect that in equilibrium, if any, firms locate symmetrically above and below the quality q m (θ 1 1)/4 preferred by the median (and average) consumer at (θ 1 1)/. ence, in general, q q m q L > 0. The game is fully non-cooperative and takes place in two stages. In the first stage firms set their respective quality levels; then, in the second, which is the proper market stage, they compete in prices. In both stages moves are simultaneous. Subgame-perfect equilibrium by backward induction is adopted as the solution concept. We confine our attention to the existence and characterization of symmetric equilibria in pure strategies, defining internal solutions in both stages. 8 (6) 3. The price stage Proceeding backwards, we examine first the firms price behaviour at the market stage for a generic quality pair. We prove what follows. Lemma 1: Candidate equilibrium prices {p, p L } decrease as α increases; i.e., p i / α < 0 for all α > 0. Proof. From the first-order conditions (henceforth FOCs) π p pl p + q α + θ1( q ql) q q α L 0, (7) π p L L p pl + ql α + ( θ1 1)( q ql) q q α the following candidate equilibrium prices obtain: L 0, (8) 7 8 Notice that, in the present setting, (6) should be written qθ arg max U θq q + αx. q owever, in the social optimum, x 1/. This can be easily shown by solving the social planner s problem, which consists in maximizing welfare w.r.t. prices and qualities. (The proof is omitted for the sake of brevity.) Therefore, the interval of socially efficient qualities remains the same as in the model without network effects. Corner solutions at the price stage are examined in Shaked and Sutton (198, 1983) and Wauthy (1996), in models without network effects. 58

5 L. Lambertini and R. Orsini: Equilibrium in a Differentiated Duopoly p p L ( q q )( θ + 1) + q + q 3 ( q q )( θ ) + q + q 3 L 1 L L 1 L From (9), it can be immediately verified that p i / α 1. α; α. (9) The above result can be given the following interpretation. Although the presence of a network externality component in the utility function raises the reservation price for all consumers compared with the standard case, where α 0, the ultimate effect of network externalities is pro-competitive, in that firms are led to decrease prices in order to enlarge their respective market shares. As network externalities increase, each firm s demand curve (given the price of the other firm) becomes more elastic. A decrease in a firm s price causes demand to rise first as a result of the price effect, and second because an increase in demand generates a further increase through the network effect. 3.1 Existence of equilibrium in prices In order to check whether (9) are indeed equilibrium prices, we have to investigate firms undercutting incentives. In general, the undercutting argument unravels as follows. Provided that firm j charges the candidate equilibrium price p j, firm i may charge a price p i u sufficiently lower than p i, so as to steal firm j s market share. Given the symmetry of the model, in evaluating the undercutting incentives we may confine ourselves to firm s viewpoint. ere we consider the situation that is most favourable to the undercutting firm, namely, the case where the poorest consumer evaluates the opportunity of switching to q given that any other richer consumer does switch. This amounts to taking the solution of the coordination problem for granted. Given p L p L, the undercutting price solves x 1, or, equivalently, x L 0 or # θ 1 1: 9 q q q q p u ( θ ) L( L ) θ1 1. (10) 3 Substituting (9) into firms objective functions and rearranging, we get the profit functions defined exclusively in terms of qualities: [( θ + 1)( q q ) q + q 3α] 1 L L # ( q, q L ) 9( q ql α ) [( θ1)( q ql) + q ql 3α] # L ( q, q L ). (1) 9( q ql α ) Notice that candidate equilibrium profits # i are positive for all α [0, (q q L )/). We prove the following results. ; (11) 9 In a model with linear costs, the undercutting price would be the price at which the consumer whose preferred quality is q L switches to q. Since the vertically differentiated duopoly with quadratic variable costs of quality is equivalent to the quadratic transportation cost version of otelling s spatial model (see Cremer and Thisse, 1991), the undercutting price must ensure that the poorest consumer switches from q L to q. 59

6 Lemma : Consider the standpoint of the high-quality firm. For all q L < q, undercutting is never profitable. Proof. Assume that p L p L and p as in (10). If so, x 1. The resulting undercutting profits accruing to firm are 10 ( q θ π u ql)( 1 q ql 1) (13) 3 and [ q u ( θ1 q ) ql( θ1 ql ) + 3α] # π. (14) 9( q ql α ) The above expression is positive for all α [0, (q q L )/). ence undercutting is never profitable in the parameter range where the profit function # i is positive. By symmetry, analogous conclusions can be drawn if one evaluates firm L s incentive to undercut. 4. The quality stage Given the symmetry of the model, a quality pair that could be candidates for the equilibrium of the first stage must be symmetric around (θ 1 1)/4, which is the variety preferred by the average (and median) consumer (see Cremer and Thisse, 1991, 1994). ence q + q L (θ 1 1)/. Consider firm. Differentiating # (q, q L ) w.r.t. q and imposing the symmetry condition q L (θ 1 q 1)/, its first-order condition can be written as: 11 # L ( q, q ) q 4θ 1 8q Solving (15) and using again q L (θ 1 q 1)/, we obtain a unique quality pair: 4θ θ1 5 q* ; q* L, (16) 8 8 which entail the general constraint θ 1 > 5/4 in order for q L to be positive. Notice that qualities in (16) coincide with those obtained in the model without network externalities (cf. Cremer and Thisse, 1994). This is due to the fact that candidate equilibrium prices p i decrease linearly in the weight attached to network externalities. The corresponding candidate equilibrium profits are π* π* L ( 3 8α)/ 16, which are non-negative for all α 3/8. Market demand is equally split between firms, with x x L 1/. Observe that the socially optimal qualities would be the first and third quartiles of the interval [θ 0 /, θ 1 /], which obtains from the calculation of the preferred varieties for (15) Undercutting profits homogeneous. π u, as well as candidate equilibrium profits, are obviously nil when products are The other critical points of the FOCs w.r.t. q and q L (which would obtain without imposing the symmetry condition) can be ruled out by checking second-order conditions. For the sake of brevity, this procedure is omitted. 60

7 L. Lambertini and R. Orsini: Equilibrium in a Differentiated Duopoly the richest and the poorest consumer in the market, if such varieties were sold at marginal cost. 1 This implies that (i) qualities are set, respectively, too low and too high compared with the social optimum; and (ii) this model shares its general features with the model of spatial competition with quadratic transportation costs (see Cremer and Thisse, 1991). 4.1 Existence of equilibrium in qualities The quality pair (16) are candidates for the equilibrium of the first stage if two requirements are met. First, profit maximization requires the second-order conditions (SOCs) # # L 9 3α < 0, (17) q q L 68 ( α 3) which are met for all α [0, 9/3). The range α > 3/8 is clearly inadmissible. Second, the poorest consumer, located at θ 0, must be able to buy; i.e., θ 0 q L + αx L p L 0. The necessary and sufficient condition for this to obtain is The following issues arise: (a) whether there exists an incentive for firms to modify their respective quality levels so as to steal the market share of the rival, for α [0, 9/3); and (b) what the individually rational behaviour of firms looks like when α 9/3. In the range α [0, 9/3), the following holds: Proposition 1: α [ 16θ ( θ ) + 9]/ (i) For all α [0, 5/4], there exists a subgame-perfect equilibrium where qualities are q* ( 4θ 1 + 1)/ 8 and q* L ( 4θ 1 5)/ 8. (ii) For all α (5/4, 9/3), both firms have an incentive to monopolize the market, and no equilibrium exists in pure strategies, where prices are above marginal costs. Proof. Given the symmetry of the model, we take the high-quality firm s viewpoint. We consider first the case where ql q * L, and investigate the incentive for the high-quality producer to manipulate its own quality level in order to predate the rival s market share. Predatory quality is θ q P ( q*) L at which x 1 and x L 0. Observe that 16θ1( θ1) + 9 α [( α)] +, 8 (18) (19) q ( q*) q* P L L [( α)] 3, 8 (0) 1 This can be shown by proving that welfare maximization with respect to quality levels is unaffected by the network externality. Again, the proof is omitted for brevity. 61

8 which reveals that, in the absence of network externalities, predation obtains when the high quality, in the limit, coincides with the low quality. Overall, both (19) and (0) are increasing and concave in α. Intuitively, this is due to the fact that, as the weight of network externalities increases, predation is obtained with a comparatively smaller decrease in the high quality, with respect to q*. Indeed, we have that 9 [( α)] q* q P ( q*) L, (1) 8 which is positive and decreasing for all α [0, 3/8); hence, a fortiori, for all α [0, 9/3). The predatory monopoly profits at q P ( q*) L amount to π P { [( α)] ( α)} /8 > 0 for all α [0, 3/8). The incentive towards predation is measured by π P π *, which is positive for all α (5/4, 3/8). This holds a fortiori in the range α (5/4, 9/3), where profits # are concave. The analysis carried out in the above proof is illustrated in Figure 1 for the parameter space {θ 1, α}. Within the region ABCD, the market is not affluent enough to allow for full market coverage to obtain; to the right of α ( 16θ1 + 3θ1 + 9)/ 96, producing the Nash equilibrium quality dominates predating, for all α [0, 5/4), and conversely for all α (5/4, 9/3). If α 5/4, firms are indifferent between producing q* i or qi P ( q*) j, and it can be assumed that they play the Nash equilibrium. What happens when α 9/3 remains to be investigated? In this respect, we confine ourselves to quality pairs that are symmetric around the quality level preferred by the average (and median) consumer, i.e. q + q L (θ 1 1)/. This amounts to considering quality pairs such that q L (θ 1 1)/4 k and q (θ 1 1)/4 + k, where k > 0. We prove the following. Theorem 1: Let α max{9/3, [4θ 1 (1 θ 1 ) + 16k + 16k 1]/4}, in order to obtain full market coverage. (i) Firms can gain profits # i ( k α)/ > π* i ( 3 8α)/ 16 by producing q (θ 1 1)/4 + k and q L (θ 1 1)/4 k, provided k > max{α, 3/8}. P (ii) Given any such quality pair, the incentive to predate exists; i.e., π i > # i for all α P (9/3, k). For all α k, # i, π i 0 and firms stop producing. FIGURE 1. Equilibrium analysis 6

9 L. Lambertini and R. Orsini: Equilibrium in a Differentiated Duopoly Proof. First of all, if α 9/3, qualities (16) identify a minimum equal to πi* ( 3 8α)/ 16. Therefore, firms find it preferable to supply an alternative symmetric pair of products. Considering q L (θ 1 1)/ q, the profit function of the high-quality firm simplifies to π (1 4α θ 1 + 4q )/8, which is larger than π* i for all q > (4θ 1 + 1)/8. The latter condition conveys the relevant information that q must increase as α increases, and conversely. The intuition behind this fact is that an increase in α entails, as we know from Lemma 1, a decrease in price, which must be compensated by an increase in product differentiation. Therefore, when α > 9/3 firms adopt a symmetric solution in qualities with q > q *, q q * P L < L and q L (θ 1 1)/ q. It is possible to derive a predatory quality q for a generic level of q L : q P θ + [ ( + ) + + ql ql + ql] ( ql) 1 α θ1 θ θ1 4. () Suppose, at the candidate equilibrium, q L (θ 1 1)/4 k and q (θ 1 1)/4 + k. The condition ql < q * L (and, equivalently, q > q * ) implies that k > 3/8. The consumer located at θ 0 θ 1 1 is able to buy if and only if α [4θ 1 (1 θ 1 ) + 16k + 16k 1]/4. Moreover, q ( q ) becomes P L q P θ + ( k k + α + / ) ( ql) (3) Plugging this expression into the high-quality firm s profit function, we get the corresponding predatory profits: ( 16k 4k + 48α + 9) 3 π P + k α > 0 α [, 0 k). 4 (4) The above predatory profits have to be evaluated against the candidate equilibrium profits that the high-quality firm can get at q (θ 1 1)/4 + k, amounting to # (k α)/ 0 for all k α. The incentive to predate is measured by π P # ( 16k 4k + 48α + 9) 3 6α + k, 4 (5) which is positive for all α (min{(1 k)/3, k}, max{(1 k)/3, k}). Since α > 9/3 and k > 3/8, expression (5) is positive for all α,. (6) 9 3 k P For all α k, both π and # are non-positive, so that the market is inoperative. By symmetry, the foregoing discussion applies to firm L as well. Theorem 1 produces the following relevant corollaries. Corollary 1: A sufficient condition for a pure-strategy Nash equilibrium in qualities with prices above marginal production costs not to exist is α > 9/3. 63

10 FIGURE. The high-quality firm s profit function Proof. To prove the above corollary, it suffices to observe that, on the basis of Theorem 1, we know that k > max{α, 3/8}, and predation is profitable for all α (9/3, k). ence, for all α > 9/3, either predation obtains, or the market is inoperative. Corollary : For any positive α, there exists α Nash equilibrium at the quality stage where the Bertrand paradox with zero profits obtains, even if firms are supplying strictly differentiated goods. P Proof. When q L (θ 1 1)/4 α and π i # i 0. ence the incentive to predate disappears, and such a quality pair is indeed admissible as an equilibrium. owever, the equilibrium profits are nil for both firms, notwithstanding the fact that the market is supplied with a strictly positive degree of product differentiation. The shape of the profit function of the high-quality firm is illustrated in Figure. A symmetric picture would describe the behaviour of the low-quality firm s profits. It is known that, under full market coverage, the spatial duopoly model with quadratic transportation costs (d Aspremont et al., 1979) is a special case of a vertical differentiation model with quadratic costs of quality improvement (Cremer and Thisse, 1991). As a consequence, the above results must hold also in the horizontal setting with convex disutility of transportation. Therefore the non-existence result also applies to the model in Grilo et al. (001), if a linear externality is considered. (This result does not emerge in their contribution, since locations are exogenously given in their model.) Should we consider negative values of parameter α in the present model, i.e. a positional externality in the utility function of consumers, the issue could be dealt with in two alternative ways. First, from Figure 1, one should note that the region wherein predatory behaviour is observed cannot intersect the negative ortant of α. This is due to the fact that, positional externalities being operative for both varieties, the demand basin for each one must be as small as possible, because every consumer patronizing a particular good prefers to belong to a small group purchasing the same good. This sharply contrasts with the idea of a firm trying to steal the rival s market share. Consequently, the presence of positional externalities tends to discourage predatory behaviour. Second, strictly speaking, the positional externality should be interpreted as an asymmetric phenomenon affecting the consumption of the high-quality product but 64

11 L. Lambertini and R. Orsini: Equilibrium in a Differentiated Duopoly not the low-quality one. 13 A duopoly model with vertical differentiation where positional effects operates only in the high-quality segment is presented in Lambertini and Orsini (1998), where a unique duopoly equilibrium is shown to exist. 5. Concluding remarks In the foregoing analysis, we have investigated the existence of a pure-strategy subgameperfect equilibrium in a vertically differentiated duopoly with convex variable costs of quality and a network externality component in the consumers utility function. We have proved that price undercutting is never a profitable strategy. owever, at the quality stage there exists an incentive towards predation, which undermines the existence of a non-cooperative equilibrium in pure strategies with prices above marginal costs. If network externalities are above a critical threshold, equilibrium profits are nil notwithstanding that products are differentiated. The same results hold in the spatial differentiation model with quadratic transportation costs. The above findings shed some new light on the longstanding issue of existence and stability of equilibrium in market models with endogenous differentiation. It appears that assuming convex disutility of transportation or, as in the model we use here, convex production costs is not sufficient to guarantee the existence of equilibrium, in that a relatively small amount of network externality suffices to destroy it. The more specifically we try to describe market interaction, the more delicate the issues of existence and stability seem to become. Final version accepted 7 February 004. REFERENCES Anderson, S. P. (1988) Equilibrium Existence in the Linear Model of Spatial Competition, Economica, Vol. 55, pp Champsaur, P. and J.-C. Rochet (1989) Multiproduct Duopolists, Econometrica, Vol. 57, pp Cremer,. and J.-F. Thisse (1991) Location Models of orizontal Differentiation: A Special Case of Vertical Differentiation Models, Journal of Industrial Economics, Vol. 39, and (1994) Commodity Taxation in a Differentiated Oligopoly, International Economic Review, Vol. 35, Dasgupta, P. and E. Maskin (1986) The Existence of Equilibrium in Discontinuous Economic Games, II: Applications, Review of Economic Studies, Vol. 53, pp d Aspremont, C., J. J. Gabszewicz and J.-F. Thisse (1979) On otelling s Stability in Competition, Econometrica, Vol. 47, pp Ecchia, G. and L. Lambertini (1997) Minimum Quality Standards and Collusion, Journal of Industrial Economics, Vol. 45, pp Economides, N. (1986) Minimal and Maximal Differentiation in otelling s Duopoly, Economics Letters, Vol. 1, pp and D. Encaoua (eds.) (1996) Special Issue on Network Economics: Business Conduct and Market Structure, International Journal of Industrial Organization, Vol. 14. Farrell, J. and G. Saloner (1985) Standardization, Compatibility and Innovation, RAND Journal of Economics, Vol. 16, pp and (1986) Standardization and Variety, Economics Letters, Vol. 0, pp For instance, take the hi-fi market. An esoteric handmade tube amplifier such as Audio Note Ongaku is a status good and, accordingly, entails a positional externality. The same does not hold for a standard solid-state amplifier thought to be for mass consumption, with the same power output. 65

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