The Japanese Economic ReviewVol. 56, No. 1, March 2005

– 55 –© Japanese Economic Association 2005.

J E R E 3 0 0 Operator: He Hong Dispatch: 19.01.05 PE: Rosie Scott

Journal Name Manuscript No. Proofreader: Chen Xiaoming No. of Pages: 12 Copy-editor:

THE EXISTENCE OF EQUILIBRIUM IN A DIFFERENTIATED DUOPOLY WITH 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 isinvestigated in a duopoly model of vertical differentiation where quality improvementsrequire a quadratic variable cost and network externalities operate. We show that thereexists a parameter region where the incentive to predate at the quality stage prevents firmsfrom reaching a pure-strategy non-cooperative equilibrium with prices above marginalcosts. If network externalities are sufficiently large, a Bertrand equilibrium with zero profitsmay arise, although the amount of product differentiation is strictly positive.JEL Classification Numbers: D62, L13.

1. Introduction

The issue of the existence of a subgame-perfect equilibrium in pure strategies has drawna considerable amount of attention in the existing literature on endogenous product differ-entiation. When vertical differentiation is considered, technology largely affects the equi-librium market structure. If quality improvements hinge either on fixed costs arising fromR&D activity, or on a variable cost that is not too steep in the quality level, then in equi-librium the market supports a limited number of firms with prices above marginal cost. Thisresult is known as the finiteness property (see Gabszewicz and Thisse, 1979, 1980; Shakedand Sutton, 1982, 1983). When variable costs are sufficiently convex in the quality level,then competition obtains as in spatial models à la Hotelling (1929).

As stressed by Gabszewicz and Thisse (1986), vertical product differentiation is usuallyexpected to generate pure-strategy equilibria with prices strictly above marginal costs. Onthe contrary, under horizontal product differentiation an established result is that a pure-strategy 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, asubgame-perfect equilibrium with positive profits may fail to exist, because firms’ locationchoices drive prices to marginal cost.1

In some markets, individuals may take their consumption decisions conditionally uponthe 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 theUniversity 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 andPitchik (1987).

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captured by introducing network externalities into consumer preferences.2 The presenceof network externalities tends to intensify competition. This, in turn, may put into questionthe existence of a pure-strategy equilibrium with positive profits.

Vanity and network effects are analysed by Grilo et al. (2001) in a duopoly model ofspatial competition with full market coverage, which also permits vertical differentiationto be dealt with. They describe price competition for exogenous product locations, finding,among other things, that (i) an increase in network externalities makes price competitiontougher, and (ii) when product differentiation dominates the externality monopoly equilibriamay arise.

We investigate the role of network externalities in a duopoly model with vertical differ-entiation and quadratic costs of quality improvements, where the finiteness property doesnot hold.3 We introduce a network externality in consumers’ utility function, and investigatethe existence and characterization of symmetric pure-strategy subgame-perfect equilibriaunder full market coverage. We show that

• price undercutting is never profitable, independently of the extent of network externalities.

However, we also prove that

• there exists a subset of the parameters measuring hedonic preferences and networkexternalities, where the incentive to predate at the quality stage prevents firms fromreaching a pure-strategy non-cooperative equilibrium with prices above marginal costs.

Therefore, the existence of a pure-strategy subgame-perfect equilibrium, in a duopolymodel 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 zeroprofits 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 2.Section 3 describes the price behaviour of firms at the market stage. The existence of apure-strategy equilibrium at the quality stage is investigated in Section 4, and concludingremarks are presented in Section 5.

2. 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). Weconsider a vertically differentiated duopoly where each firm produces a single good ofquality q, with qH ≥ qL > 0, and then competes in prices against a rival. There exists a con-tinuum of consumers indexed by their marginal willingness to pay for quality θ ∈[θ0, θ1],

2 Seminal contributions in the theory of network externalities are Katz and Shapiro (1985, 1986); Farrelland Saloner (1985, 1986). For an overview, see Katz and Shapiro (1994) and the special issue of theInternational Journal of Industrial Organization, edited by Economides and Encaoua (1996)

3 An analysis of monopoly with the same variable cost technology and network externalities can be foundin Lambertini and Orsini (2001). A monopoly model with vanity effects is in Lambertini and Orsini(2002).

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with θ0 = θ1 − 1. The distribution of consumers is uniform, with density f (θ ) = 1, so thatthe population of consumers is 1. The net consumer surplus from the purchase of one unitof product i is U = θqi + αxi − pi, i = H, L, where pi is the price of good i, xi is the marketdemand for good i, and α ≥ 0 (the same for all agents) is the weight of the network exter-nality in the utility function. This amounts to saying that the network externality is product-specific.4 We confine our attention to the case where the market is completely covered byfirms; that is, the poorest consumer is always willing to buy:5

max{θ0qH + αxH − pH, θ0qL + αxL − pL} ≥ 0. (1)

Given qualities and prices set by the firms, a generic consumer indexed by θ ∈ [θ0, θ1],will choose to buy the good i that maximizes his net surplus θqi + αxi − pi. In order todefine market demands, we have to find, along the support [θ0, θ1], the location # of theconsumer who is indifferent between the high and the low-quality good. From the indif-ference condition

#qH + αxH − pH = #qL + αxL − pL, (2)

we obtain # = (α − 2αθ1 + pH − pL)/(qH − qL − 2α).6 The market demand functions are,respectively,

xH = θ1 − #, xL = # − θ0 if # ∈ (θ0, θ1); (3)

xH = 1, xL = 0 if # ≤ θ0; (3′)

xH = 0, xL = 1 if # ≥ θ1. (3″)

Production technology involves variable costs, which are quadratic in the quality leveland linear in the output level:

(4)

where xi indicates the output level of firm i, whose profit function is

(5)

4 This can be the case in the telecommunications industry. For a model analysing country-specific networksand their effects on comparative advantage in the telecommunications industry, see Kikuchi (2002). Seealso Shy (2001, chapter 5).

5 Of course, there exist goods for which the externality is not product-specific. The inclusion of a weightattached to the network of the other variety would give rise to an external effect α (xi + δxj), with δ ∈ [0, 1].However, given the assumption of full market coverage, this translates into a neutral reparametrization ofthe consumer surplus function we have assumed above.

6 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 onegood (or network) to the other. As a consequence, we may expect the present model not to generate mul-tiple equilibria.

C q x i H Li i i , , ,= =2

π i i i ip q x ( ) .= − 2

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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 marginalwillingness to pay equal to θ ∈ [θ0, θ1]. His preferred quality qθ maximizes his net sur-plus when he is able to purchase such quality at marginal cost, i.e. when

(6)

which yields qθ = θ/2.7 This entails that the interval of individually optimal qualities is[θ0/2, θ1/2] = [(θ1 − 1)/2, θ1/2]. Given the a priori symmetry of the model, we may expectthat in equilibrium, if any, firms locate symmetrically above and below the quality qm =(2θ1 − 1)/4 preferred by the median (and average) consumer at (2θ1 − 1)/2. Hence, ingeneral, qH ≥ qm ≥ qL > 0.

The game is fully non-cooperative and takes place in two stages. In the first stage firmsset 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 equilibriumby backward induction is adopted as the solution concept. We confine our attention to theexistence and characterization of symmetric equilibria in pure strategies, defining internalsolutions in both stages.8

3. The price stage

Proceeding backwards, we examine first the firms’ price behaviour at the market stage fora generic quality pair. We prove what follows.

Lemma 1: Candidate equilibrium prices {pH, pL} decrease as α increases; i.e., ∂ pi/∂α< 0 for all α > 0.

Proof. From the first-order conditions (henceforth FOCs)

(7)

(8)

the following candidate equilibrium prices obtain:

7 Notice that, in the present setting, (6) should be written

However, in the social optimum, x = 1/2. This can be easily shown by solving the social planner’s prob-lem, which consists in maximizing welfare w.r.t. prices and qualities. (The proof is omitted for the sakeof brevity.) Therefore, the interval of socially efficient qualities remains the same as in the model withoutnetwork effects.

8 Corner solutions at the price stage are examined in Shaked and Sutton (1982, 1983) and Wauthy (1996),in models without network effects.

q U q qq

θ θ arg max ,= = − 2

q U q q xq

θ θ α arg max .= = − +2

∂π∂

α θα

H

H

L H H H L

H Lp

p p q q q

q q

( )

,=

− + − + −− −

=2

20

21

∂π∂

α θα

L

L

H L L H L

H Lp

p p q q q

q q

( )( )

,=

− + − + − −− −

=2 1

20

21

L. Lambertini and R. Orsini: Equilibrium in a Differentiated Duopoly

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(9)

From (9), it can be immediately verified that ∂ pi/∂α = −1. �

The above result can be given the following interpretation. Although the presence of anetwork externality component in the utility function raises the reservation price for allconsumers compared with the standard case, where α = 0, the ultimate effect of networkexternalities is pro-competitive, in that firms are led to decrease prices in order to enlargetheir 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 pricecauses demand to rise first as a result of the price effect, and second because an increasein 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. Pro-vided that firm j charges the candidate equilibrium price pj, firm i may charge a price sufficiently lower than pi, so as to steal firm j’s market share.

Given the symmetry of the model, in evaluating the undercutting incentives we mayconfine ourselves to firm H ’s viewpoint. Here we consider the situation that is mostfavourable to the undercutting firm, namely, the case where the poorest consumer evalu-ates the opportunity of switching to qH given that any other richer consumer does switch.This amounts to taking the solution of the coordination problem for granted. Given pL =pL, the undercutting price solves xH = 1, or, equivalently, xL = 0 or # = θ1 − 1:9

(10)

Substituting (9) into firms’ objective functions and rearranging, we get the profit functionsdefined exclusively in terms of qualities:

(11)

(12)

Notice that candidate equilibrium profits #i are positive for all α ∈ [0, (qH − qL)/2). Weprove the following results.

9 In a model with linear costs, the undercutting price would be the price at which the consumer whose pre-ferred quality is qL switches to qH. Since the vertically differentiated duopoly with quadratic variable costsof quality is equivalent to the quadratic transportation cost version of Hotelling’s spatial model (see Cremerand Thisse, 1991), the undercutting price must ensure that the poorest consumer switches from qL to qH.

p

p

HH L H L

LH L L H

q q q q

q q q q

( )( )

;

( )( )

.

=− + + +

−

=− − + +

−

θ α

θ α

12 2

12 2

1 2

32 2

3

piu

pq q q q

Hu H H L L

( ) ( ).=

+ − − − −2 1 2 2 1

31 1θ θ

#H H LH L H L

H L

q qq q q q

q q( , )

[( )( ) ]

( );=

+ − − + −− −

θ αα

12 2 21 3

9 2

#L H LH L H L

H L

q qq q q q

q q( , )

[( ) ( ) ]

( ).=

− − + − −− −

2 3

9 21

2 2 2θ αα

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Lemma 2: Consider the standpoint of the high-quality firm. For all qL < qH, undercuttingis never profitable.

Proof. Assume that pL = pL and pH as in (10). If so, xH = 1. The resulting undercuttingprofits accruing to firm H are10

(13)

and

(14)

The above expression is positive for all α ∈ [0, (qH − qL)/2). Hence undercutting is neverprofitable 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 toundercut.

4. The quality stage

Given the symmetry of the model, a quality pair that could be candidates for the equilibriumof the first stage must be symmetric around (2θ1 − 1)/4, which is the variety preferred bythe average (and median) consumer (see Cremer and Thisse, 1991, 1994). Hence qH + qL =(2θ1 − 1)/2. Consider firm H. Differentiating #H(qH, qL) w.r.t. qH and imposing the symmetrycondition qL = (2θ1 − 2qH − 1)/2, its first-order condition can be written as:11

(15)

Solving (15) and using again qL = (2θ1 − 2qH − 1)/2, we obtain a unique quality pair:

(16)

which entail the general constraint θ1 > 5/4 in order for qL to be positive. Notice that qual-ities 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 pi

decrease linearly in the weight attached to network externalities.The corresponding candidate equilibrium profits are , which are

non-negative for all α ≤ 3/8. Market demand is equally split between firms, with xH = xL

= 1/2. Observe that the socially optimal qualities would be the first and third quartiles ofthe interval [θ0/2, θ1/2], which obtains from the calculation of the preferred varieties for

10 Undercutting profits , as well as candidate equilibrium profits, are obviously nil when products arehomogeneous.

11 The other critical points of the FOCs w.r.t. qH and qL (which would obtain without imposing the symmetrycondition) can be ruled out by checking second-order conditions. For the sake of brevity, this procedureis omitted.

π Hu

π θHu H L H Lq q q q

( ) ( )

=− − − −2 2 2 1

31

#H Hu H H L L

H L

q q q q

q q

[ ( ) ( ) ]

( ).− =

− − − − − +− −

π θ θ αα

1 122 2 3

9 2

∂∂

θ#H H L

H

Hq q

q

q( , )

.=

− +=

4 8 1

1201

q qH L*

; *

,=+

=−4 1

8

4 5

81 1θ θ

π π αH L* * ( )/= = −3 8 16

L. Lambertini and R. Orsini: Equilibrium in a Differentiated Duopoly

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the richest and the poorest consumer in the market, if such varieties were sold at marginalcost.12 This implies that (i) qualities are set, respectively, too low and too high comparedwith the social optimum; and (ii) this model shares its general features with the model ofspatial 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 requirementsare met. First, profit maximization requires the second-order conditions (SOCs)

(17)

which are met for all α ∈ [0, 9/32). The range α > 3/8 is clearly inadmissible. Second, thepoorest consumer, located at θ0, must be able to buy; i.e., θ0qL + αxL − pL ≥ 0. The necessaryand sufficient condition for this to obtain is

(18)

The following issues arise:

(a) whether there exists an incentive for firms to modify their respective quality levels soas to steal the market share of the rival, for α ∈ [0, 9/32); and

(b) what the individually rational behaviour of firms looks like when α ≥ 9/32.

In the range α ∈ [0, 9/32), the following holds:

Proposition 1:

(i) For all α ∈ [0, 5/24], there exists a subgame-perfect equilibrium where qualities are and .

(ii) For all α ∈ (5/24, 9/32), both firms have an incentive to monopolize the market, andno 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. Weconsider first the case where , and investigate the incentive for the high-qualityproducer to manipulate its own quality level in order to predate the rival’s market share.Predatory quality is

(19)

at which xH = 1 and xL = 0. Observe that

(20)

12 This can be shown by proving that welfare maximization with respect to quality levels is unaffected bythe network externality. Again, the proof is omitted for brevity.

∂∂

∂∂

αα

2

2

2

2

9 32

6 8 30

# #H

H

L

Lq q

( ) ,= =

−−

<

αθ θ

( )

.≥− +16 2 9

961 1

α θ θ [ ( ) ]/ .≥ − +16 2 9 961 1

qH* ( )/= +4 1 81θ q L* ( )/= −4 5 81θ

q qL L *=

q qHP

L( *)

[ ( )]

,=−

++θ α1 2

2

3 3 64

8

√

q q qHP

L L( *) * [ ( )]

,− =+ −√ 3 3 64 3

8

α

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which reveals that, in the absence of network externalities, predation obtains when thehigh quality, in the limit, coincides with the low quality. Overall, both (19) and (20) areincreasing and concave in α. Intuitively, this is due to the fact that, as the weight of net-work externalities increases, predation is obtained with a comparatively smaller decreasein the high quality, with respect to . Indeed, we have that

(21)

which is positive and decreasing for all α ∈ [0, 3/8); hence, a fortiori, for all α ∈ [0, 9/23).The predatory monopoly profits at amount to /8> 0 for all α ∈ [0, 3/8). The incentive towards predation is measured by , whichis positive for all α ∈ (5/24, 3/8). This holds a fortiori in the range α ∈ (5/24, 9/32), whereprofits #H are concave. �

The analysis carried out in the above proof is illustrated in Figure 1 for the parameterspace {θ1, α}. Within the region ABCD, the market is not affluent enough to allow for fullmarket coverage to obtain; to the right of , producing the Nashequilibrium quality dominates predating, for all α ∈ [0, 5/24), and conversely for all α∈(5/24, 9/32). If α = 5/24, firms are indifferent between producing or , and itcan be assumed that they play the Nash equilibrium.

What happens when α ≥ 9/32 remains to be investigated? In this respect, we confineourselves to quality pairs that are symmetric around the quality level preferred by the average(and median) consumer, i.e. qH + qL = (2θ1 − 1) /2. This amounts to considering quality pairssuch that qL = (2θ1 − 1) /4 − k and qH = (2θ1 − 1)/4 + k, where k > 0. We prove the following.

Theorem 1: Let α ≥ max{9/32, [4θ1(1 − θ1) + 16k 2 + 16k − 1]/24}, in order to obtainfull market coverage. (i) Firms can gain profits by producing qH = (2θ1 − 1)/4 + k

and qL = (2θ1 − 1)/4 − k, provided k > max{α, 3/8}.(ii) Given any such quality pair, the incentive to predate exists; i.e., for all α

∈ (9/32, k). For all α ≥ k, #i, and firms stop producing.

qH*

q q qH HP

L* ( *) [ ( )]

,− =− +9 √ 3 3 64

8

α

q qHP

L( *) π α αHP { [ ( )] ( )}= + − +√ 3 3 64 3 16

π πHP

H *−

FIGURE 1. Equilibrium analysis

α θ θ ( )/= − + +16 32 9 9612

1

qi* q qiP

j( *)

# i ik= − > = −( )/ * ( )/α π α2 3 8 16

π iP

i > #π i

P ≤ 0

L. Lambertini and R. Orsini: Equilibrium in a Differentiated Duopoly

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Proof. First of all, if α ≥ 9/32, qualities (16) identify a minimum equal to .Therefore, firms find it preferable to supply an alternative symmetric pair of products.

Considering qL = (2θ1 − 1)/2 − qH, the profit function of the high-quality firm simplifiesto πH = (1 − 4α − 2θ1 + 4qH)/8, which is larger than for all qH > (4θ1 + 1)/8. Thelatter condition conveys the relevant information that qH 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 > 9/32 firms adopt a symmetric solution in qualities with , and qL = (2θ1 − 1)/2 − qH. It is possible to derive a predatory quality for a generic

level of qL:

(22)

Suppose, at the candidate equilibrium, qL = (2θ1 − 1)/4 − k and qH = (2θ1 − 1)/4 + k. Thecondition (and, equivalently, ) implies that k > 3 /8. The consumerlocated at θ0 = θ1 − 1 is able to buy if and only if α ≥ [4θ1(1 − θ1) + 16k2 + 16k − 1]/24.

Moreover, becomes

(23)

Plugging this expression into the high-quality firm’s profit function, we get the corre-sponding predatory profits:

(24)

The above predatory profits have to be evaluated against the candidate equilibrium profitsthat the high-quality firm can get at qH = (2θ1 − 1) /4 + k, amounting to #H = (k − α) /2 ≥ 0for all k ≥ α. The incentive to predate is measured by

(25)

which is positive for all α ∈ (min{(1 − k)/3, k}, max{(1 − k)/3, k}). Since α > 9/32 andk > 3/8, expression (25) is positive for all

(26)

For all α ≥ k, both and #H are non-positive, so that the market is inoperative. Bysymmetry, 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 withprices above marginal production costs not to exist is α > 9/32.

π αi* ( )/= −3 8 16

π i*

q qH H *>q qL L *< qH

P

q qq q q

HP

LL L L( )

[ ( ) ].=

− + + − + + − +θ α θ θ θ1 1 12

122 4 1 3 8 4 4

2

√

q qL L *< q qH H *>

q qHP

L( )

q qk k

HP

L( ) ( / )

.=− + − + +θ α1

22 4 6 12 9 4

2

√

π α α αHP k k

k k ( )

[ , ).=− + + −

+ − > ∀ ∈√ 16 24 48 9 3

42 0 0

2

π α αHP

Hk k k

( )

,− =− + + − − +

#√ 16 24 48 9 3 6 2

4

2

α , .∈

9

32k

πHP

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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 ∈ (9/32, k). Hence,for all α > 9/32, either predation obtains, or the market is inoperative. �

Corollary 2: For any positive α, there exists α Nash equilibrium at the quality stagewhere the Bertrand paradox with zero profits obtains, even if firms are supplying strictlydifferentiated goods.

Proof. When qL = (2θ1 − 1)/4 − α and . Hence the incentive to predate dis-appears, and such a quality pair is indeed admissible as an equilibrium. However, theequilibrium profits are nil for both firms, notwithstanding the fact that the market is suppliedwith a strictly positive degree of product differentiation. �

The shape of the profit function of the high-quality firm is illustrated in Figure 2. Asymmetric 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 quadratictransportation costs (d’Aspremont et al., 1979) is a special case of a vertical differentiationmodel with quadratic costs of quality improvement (Cremer and Thisse, 1991). As a con-sequence, the above results must hold also in the horizontal setting with convex disutilityof transportation. Therefore the non-existence result also applies to the model in Grilo et al.(2001), 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 valuesof parameter α in the present model, i.e. a positional externality in the utility function ofconsumers, the issue could be dealt with in two alternative ways. First, from Figure 1, oneshould note that the region wherein predatory behaviour is observed cannot intersect thenegative ortant of α. This is due to the fact that, positional externalities being operativefor both varieties, the demand basin for each one must be as small as possible, becauseevery consumer patronizing a particular good prefers to belong to a small group purchasingthe same good. This sharply contrasts with the idea of a firm trying to steal the rival’s marketshare. Consequently, the presence of positional externalities tends to discourage predatorybehaviour. Second, strictly speaking, the positional externality should be interpreted asan asymmetric phenomenon affecting the consumption of the high-quality product but

π iP

i= = # 0

FIGURE 2. The high-quality firm’s profit function

L. Lambertini and R. Orsini: Equilibrium in a Differentiated Duopoly

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not the low-quality one.13 A duopoly model with vertical differentiation where positionaleffects 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 subgame-perfect equilibrium in a vertically differentiated duopoly with convex variable costs of qualityand a network externality component in the consumers’ utility function. We have provedthat price undercutting is never a profitable strategy. However, at the quality stage thereexists an incentive towards predation, which undermines the existence of a non-cooperativeequilibrium in pure strategies with prices above marginal costs. If network externalitiesare above a critical threshold, equilibrium profits are nil notwithstanding that products aredifferentiated. The same results hold in the spatial differentiation model with quadratictransportation costs.

The above findings shed some new light on the longstanding issue of existence and stab-ility of equilibrium in market models with endogenous differentiation. It appears that assumingconvex disutility of transportation—or, as in the model we use here, convex productioncosts—is not sufficient to guarantee the existence of equilibrium, in that a relatively smallamount of network externality suffices to destroy it. The more specifically we try to describemarket interaction, the more delicate the issues of existence and stability seem to become.

Final version accepted 27 February 2004.

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