J Ind Compet Trade DOI 10.1007/s10842-013-0170-0

Compatibility Under Differentiated Duopoly with Network Externalities: A Comment Tsuyoshi Toshimitsu

Received: 30 May 2013 / Revised: 13 October 2013 / Accepted: 17 October 2013 # Springer Science+Business Media New York 2013

Abstract Chen and Chen (J Ind Compet Trade 11:43−55, 2011) analyze the effects of compatibility under system product Cournot competition with network externalities. They show that a firm’s optimal strategy is to set an incompatible system standard, even though perfect compatibility is socially optimal. In this case, a social dilemma arises. However, their result depends on a specific assumption about the network size. We use the framework of Shy (1995) to modify this assumption, and hence show that the social dilemma identified by Chen and Chen (J Ind Compet Trade 11:43−55, 2011) does not arise. Keywords Network externality . Compatibility . System product . Cournot competition JEL Classification D21 . D43 . D62

1 Introduction Chen and Chen (2011), hereafter Chen–Chen, analyze the effects of compatibility under system product Cournot competition with network externalities. They show that a firm’s optimal strategy is to set an incompatible system standard, even though perfect compatibility is socially optimal. In this case, a social dilemma arises. This result has important policy implications for administrators in the information and communications industries. However, Chen–Chen’s result depends on a specific assumption about the network size. That is, Chen–Chen assume that the effect of the network size of a given system product is composed of the own effect and the spillover effect. The latter effect is determined by the market share of a rival firm, multiplied by the degree of product compatibility of the rival firm’s system product. Although an increase in the degree of product compatibility of a system product has no effect on the consumer’s willingness to pay for it, greater compatibility increases the consumer’s willingness to buy a competing system product. Thus, if a firm raises the degree of product compatibility of its own system product, its output falls whereas that of the rival firm increases. This is because of the strategic substitutability relationship under Cournot competition. Hence, a firm that increases the degree of product compatibility of T. Toshimitsu (*) School of Economics, Kwansei Gakuin University, 1-155, Nishinomiya, Japan 662-8501 e-mail: [email protected]

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its own system product reduces its own profits but raises those of the rival firm. Therefore, the firm’s optimal strategy is to set an incompatible standard system, even though a perfectly compatible standard is socially optimal. In this paper, we use the framework of Shy (1995) to modify Chen–Chen’s assumption about the network size. To be specific, we assume that the spillover effect is determined by the market share of the rival firm multiplied by the degree of product compatibility of the firm’s own system product. In this case, an increase in the degree of product compatibility of the system product increases the firm’s own output and profits. Therefore, no social dilemma arises.

2 The Model 2.1 Modifications1 Based on the framework of Economides (1996), Chen–Chen assume that the inverse demand function of system product i is: Pi =A−(qi +θqj)+f(Si), where A is the potential market size, qi (qj) is the output of firm i (j), and θ∈(0,1) represents product substitutability.2 The networkexternality function is given by f(Si), where Si represents the network size of system product i. Chen–Chen assume that f(Si)=aSi, where a∈(0,1) is a network-externality parameter for network size. Chen–Chen assume that the network size of system product i is: S i ¼ qi þ α j q j ;

ð2Þ

where αi,αj ∈[0,1]. If αi =αj =1(0), the components of one system product work (do not work) in the other system product. In this case, qi (αjqj) represents the own (spillover) effect. Eq. (2) implies that the spillover effect in the network size of system product i depends on the degree of product compatibility of the rival firm’s system product j. We modify Eq. (2) so that is the same as Eq. (10.11) in Shy (1995), as follows: S i ¼ qi þ αi q j :

ð20 Þ

Equation (2') implies that the spillover effect, αiqj, depends on the degree of product compatibility of the firm’s own system product. This assumption differs significantly different from the one made by Chen–Chen. Given this modification, we derive the following inverse demand function for firm i:     ð30 Þ Pi ¼ A− qi þ θq j þ a qi þ αi q j ¼ A−ð1−aÞqi −ðθ−aαi Þq j : Equation (3') represents the consumer’s willingness to pay for system product i, taking into account network externalities. In this case, we assume that the own-price effect exceeds the dPi dPi cross-price effect; i.e., dq ¼ 1−a > dq ¼ jθ−aαi j: Given Eq. (3'), the effect of an increase i

1

j

Hereafter, we follow the number of equations given by Chen–Chen. Chen–Chen assume that θ∈[0,1]. That is, they allow the two system products to be homogenous, i.e., θ=1. Hence, the products are perfect substitutes. As we show subsequently, even in this case, the two firms can set the degree of product compatibility to zero; i.e., αi =αj =0. This means that these products are not compatible even though they are perfect substitutes. This seems unreasonable. At the other extreme of independent products, θ=0, suppose that αi =αj =1. This implies that these products are perfectly compatible even though they are independent. This is also unreasonable. Hence, we assume that 0<θ<1. 2

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in the degree of product compatibility of system product i on the consumer’s willingness to pay dPi is positive; i.e., dα ¼ aq j > 0: i Chen–Chen’s version of firm i’s inverse demand function, based on Eq. (2), is:    
Pi ¼ A− qi þ θq j þ a qi þ αi q j ¼ A−ð1−aÞqi − θ−aα j q j : ð3Þ dPi Hence, dα ¼ aq j > 0: That is, an increase in the degree of product compatibility of firm j’s j system product increases the consumer’s willingness to pay for firm i’s system product, which increases firm i’s price and output. For simplicity, we assume that ci =cj =0.

2.2 Cournot–Nash Equilibrium and Fixed Product Compatibility Given the modifications described in Section 2.1, we can derive the following reaction function for firm i: qi ¼

A θ−aαi  q: 2ð1−aÞ 2ð1−aÞ j

ð60 Þ

Given Eq. (6'), under Cournot competition, the strategic relationship between the firms depends on the degree of product substitutability and the degree of product compatibility with a ∂qi < ð>Þ0⇔θ > ð< Þaαi : 3 This condition implies that, under network externality; i.e., ∂q j

Cournot competition, there is a relationship of strategic substitutability (complementarity) between the firms if the degree of product substitutability is higher (lower) than the degree of product compatibility with a network externality. In particular, even though the two products are substitutable, there is a relationship of strategic complementarity under Cournot competition if the degree of product compatibility with a network externality is sufficiently high; i.e., θ
Af2ð1−aÞ−ðθ−aαi Þg ; D

ð70 Þ

where D≡4(1−a)2 −(θ−aαi)(θ−aαj)>0 and 2(1−a)−(θ−aαi)>0. Both of these conditions are satisfied because the own-price effect exceeds the cross-price effect. Given Eq. (6'), we can categorize the Cournot–Nash equilibrium into three cases. (i and ii) Strong (weak) product compatibility with a network externality: θ<(>)aαi. When there is strong (weak) product compatibility with a network externality, because of the strategic complementarity (substitutability) relationship under Cournot competition, the reaction curves are upward (downward) sloping. (iii) Asymmetric product compatibility: aαj >θ>aαi. In this case, the reaction curve of firm j (i) is upward (downward) sloping. From Eq. (7'), the effects of an increase in the degree of product compatibility on output levels are: 
2að1−aÞA 2ð1−aÞ− θ−aα j ∂qi 2að1−aÞ ¼ ¼ ð110 Þ q j > 0; D ∂αi D2

3

Chen–Chen assume that θ>aαi. Hence, they only consider strategic substitutability.

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∂qi aðθ−aαi ÞAf2ð1−aÞ−ðθ−aαi Þg aðθ−aαi Þ qi > ð< Þ0 ¼− ¼− 2 D ∂α j D ⇔θ < ð>Þaαi :

ð120 Þ

2.3 Endogenous Decision Making About Product Compatibility, and the Social Dilemma Given Eq. (7'), the profit function is given by πi =Piqi =(1−a)qi2. Hence, based on Eqs. (11') and (12'), we can derive the following: ∂πi ∂q ¼ 2ð1−aÞqi i > 0; ð130 Þ ∂αi ∂αi ∂πi ∂q ¼ 2ð1−aÞqi i > ð< Þ0⇔θ < ð>Þaαi : ∂α j ∂α j

ð140 Þ

The proposition below, based on Eqs. (11'), (12'), (13'), and (14'), differs significantly from Proposition 1 of Chen–Chen. Proposition 1' An increase in the degree of product compatibility of system product i increases firm i’s output and profits. However, the effect on the rival firm’s output and profits depends on both the degree of product substitutability and the degree of product compatibility with a network externality. To clarify the latter point, if the degree of product compatibility with a network externality is higher (lower) than that the degree of product substitutability, then there is a relationship of strategic complementarity (substitutability) under Cournot competition. In this case, an increase in the degree of product compatibility increases (decreases) the rival firm’s output, and thus increases (decreases) its profits. When the degree of product compatibility of system product i is variable, based on Eq. (13'), we can derive the following important result. Proposition 4' Under system product Cournot competition, given Eq. (2'), the firm’s optimal strategy is to set a perfectly compatible standard, i.e., αi =αj =1. When firms endogenously choose the degree of product compatibility, depending on whether θ<(>)a, the Cournot–Nash equilibrium exists. The effect of an increase in the degree of product compatibility on social welfare is given by:   ∂q j 3að1−aÞAq j ∂SW ∂qi ¼ 3ð1−aÞ qi þ qj H i; ð210 Þ ¼ ∂αi ∂αi ∂αi D2 where Hi =2(1−a){[(1−a)−(θ−aαi)]+[(1−a)−(θ−aαj)]}+(θ−aαj)2 >0, because 1−a>|θ− ∂SW aαi|,|θ−aαj|. Therefore, ∂SW ∂αi > 0: Similarly, for system product j, ∂α j > 0 . These results confirm Chen–Chen’s Proposition 5: the perfectly compatible standard is socially optimal; i.e., αi =αj =1. Therefore, from Propositions 4' and 5, given Eq. (2'), there is no social dilemma.

3 Conclusion In this paper, we make a simple comment on Chen and Chen (2011). If the assumption about the network size is based on the framework of Shy (1995), no social dilemma arises. That is, the perfectly compatible standard prevails, and this solution is socially and privately optimal.

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Like Chen and Chen (2011), we deal with the case of one-way compatibility. Moreover, Chen and Chen (2011) and ourselves adopt the opposite extreme approaches to the direction of compatibility in the setting of network size. These extremes are conveyed, respectively, by Eqs. (2) and (2'). As suggested by a referee, one possible extension of this comment would be to integrate Chen and Chen’s (2011) approach into the approach described in this comment. The resulting generalized model would enable analysis of two-way compatibility.

References Chen H-C, Chen C-C (2011) Compatibility under differentiated duopoly with network externalities. J Ind Compet Trade 11:43–55 Economides N (1996) Network externalities, complementarities, and invitations to enter. Eur J Polit Econ 12: 211–233 Shy O (1995) Industrial organization: theory and application. MIT, Cambridge