Evolution of market shares with repeated purchases and heterogeneous network externalities
Abstract
We investigate how market shares change when a new, superior technology exhibiting network externalities is introduced in a market initially dominated by an old technology. This is done under the assumption that consumers are heterogeneous in their valuation of technology quality and network externalities and that goods are not (perfectly) durable and thus have to be bought repeatedly. When both technologies are unsponsored, the old technology dominates when the quality difference is small, and it disappears when the quality difference is large. When the new technology is sponsored, the relationship between the quality difference and the longrun market share of the new technology is nonmonotonic and the old technology always continues to exist.
Keywords
Technology adoption Network externalities Market share dynamicsJEL Classification
L1 L13 D431 Introduction
The value individuals attach to consuming many technological products (such as telephones, software and hardware) or products that require maintenance depends on how many others are using these goods. This phenomenon is known as network externality. In the literature, it is wellknown that network externalities may create barriers to entry, preventing adoption of new goods, possibly of a higher quality. This can lead to the society being “lockedin” with an inefficient technology. A classical—although, according to some (e.g. Liebowitz and Margolis 1990), mistaken—example is the QWERTY standard commonly used in typewriters and computer keyboards (see David 1986). Nuclear power reactors in Europe are another example—the dominant technology is light water, although many scientists consider it to be inferior to heavy water or gas graphite technology (Cowan 1990).
In many situations, the importance of network externalities for the individual adoption decision will differ between different consumers. An important reason for this differentiation is that people use the same technology in a variety of ways, and some require more coordination than others. In the QWERTY example, a large company with many typists and a high rotation of personnel will care more about the network externality than a freelance journalist, who uses her keyboard herself and for whom typing speed is important. Similarly, a scientist who frequently writes papers together with his colleagues will care more about the possibility of exchanging files than someone who works primarily alone. In the existing literature on technology adoption, however, consumer heterogeneity with respect to their valuation is usually not modeled. Typically, in models with horizontally nondifferentiated goods, consumers are either homogenous (as, e.g., in Farrell and Saloner 1986; Katz and Shapiro 1986a, 1992), or they have identical preferences with respect to network externalities (as, e.g., in Farrell and Saloner 1985; Cabral 1990; Agliardi 1994). Another typical feature of existing models is the assumption that users are “stuck” forever (or at least for a long time) with the technology they choose (e.g. Farrell and Saloner 1985, 1986; Katz and Shapiro 1986a, b, 1992).
In this paper, we study the classical question concerning the possibility of “lockin” in markets where consumers value network externalities. We do this under two novel assumptions. First, we assume that the consumers’ valuation of network externalities and quality is heterogeneous across the population.^{1} Second, we assume that users adopt (buy) a technology in every period, and thus they cannot be stuck with their past purchases and do not incur any switching costs. The motivation for this assumption is twofold. First, it extends the analysis of technology adoption to markets of goods which are not durable. One can think here of goods, such as software, which depreciate fast, e.g. due to technological progress, like software. Another group of products to which our analysis may apply are goods of immediate consumption, such as entertainment goods, where network externalities arise due to social considerations. Second, it shows that inefficiencies in technology adoption can arise even if users can switch between technologies costlessly.
Specific questions that we ask here are: is it possible that the new technology is not adopted by anyone, despite its higher quality? If it attracts some users, under what conditions will the new technology take over the market? Does there exist an equilibrium with both technologies present? We also investigate the market shares when the new technology is sponsored by a provider that chooses prices to maximize longrun profits.
To answer these questions, we study a model with two products, an old, inferior good and a new, superior good. We assume that quality can be objectively measured and that consumers differ in their valuation of quality. This means that if product A is of a higher quality than product B, then everyone regards A to be better than B, but for some people, the quality difference is relatively more important than for others. Users differ also in their valuation of network externalities. One could think here for instance of software: some users care mostly about speed, reliability or userfriendliness, whereas for others the ability to exchange files with colleagues or to move them between computers is more important. The valuation for quality is independent of the valuation for network externalities. Consumers decide in every period which good to buy solely on the basis of the present (net) expected utility.
We study the questions outlined above in two different environments. First, in Section 3, we address the pure demand side effect by considering technology adoption in a world where technology is competitively provided and firms are passive. Second, in Section 4, we study the situation in which the new technology is introduced by a profitmaximizing firm, while maintaining the assumption of competitively provided old technology. This assumption, also made by, e.g., Katz and Shapiro (1986a) and Farrell and Saloner (1986), can be justified on the grounds that the new technology is protected by a patent, while the patent on the old technology has already expired so that it is provided competitively, with all firms taking prices as given.^{2}
In every period, market shares adjust to their equilibrium values given the price of the new technology in that period and the market share at the beginning of the period. Our basic results are as follows. When both technologies are not sponsored, two equilibrium market shares may emerge: if the difference in qualities is larger than a certain threshold value, the new technology will be the unique technology in the market. If the quality advantage is lower than this threshold value, the two technologies coexist and the entrant will have the smallest market share. It is easy to see why the new technology has to have the whole market if it has a market share larger than half: if both quality and market share of one technology are higher, all consumers derive more utility from this product than from the other and, hence, will switch to this new technology. The possible emergence of two equilibrium market shares and the discontinuous jump of market shares at the threshold value are the main results of this section.
Section 4 examines whether these results continue to hold when the new technology is protected by a patent and provided by a pricesetting firm. Unlike in the basic model, the market share of the new technology is now nonmonotonic in the quality advantage. When the quality difference is small, the longrun market share of the new technology remains small. When the quality difference takes intermediate values, the sponsor of the new technology is able to get a large market share and keep it in the long run. When the quality difference becomes large, the firm can earn a higher profit by raising price to a level at which fewer people with a low valuation of quality adopt the new technology, which allows it to extract more surplus from the qualityloving consumers. Therefore, the fact that the new technology is sponsored results in its having a lower market share.
We also analyze welfare properties of the different equilibrium configurations. Social welfare is maximized when the new technology takes over the whole market. This only happens when the technologies are unsponsored and when the quality difference is high enough. In both the unsponsored and sponsored case, the introduction of a new technology decreases welfare if its quality advantage is small, and it increases welfare when its quality advantage is large.
Since the literature on technology adoption in the presence of network externalities is extensive, we devote a separate section to a brief overview. This is done in Section 2. The rest of the paper is organized as follows. Section 3 on unsponsored technologies considers the pure demand effects due to heterogeneous network externalities. This section allows us to illustrate the main concepts in a relatively simple setting. Section 4 introduces the sponsor of the new technology. Section 5 concludes. Most proofs are given in the Appendix.
2 Related literature
The seminal papers on the adoption of a new, incompatible technology are those by Farrell and Saloner (1985, 1986) and Katz and Shapiro (1986a, b), followed by Katz and Shapiro (1992) and many others. Just as in this paper, two situations are typically studied: in one, technologies are provided by a competitive industry at marginal cost, while in the other, technologies are sponsored. Another way to classify the literature is to distinguish the models where the total number of users remains constant and the network of the new technology arises due to consumers switching from old to new technology (e.g. Farrell and Saloner 1985, 1986; Agliardi 1994; Michihiro 1998), from those where a technology is adopted only once and the new network can arise due to the arrival of new generations (Farrell and Saloner 1986; Katz and Shapiro 1986a, b, 1992; Choi 1994, 1997; De Bijl and Goyal 1995; Regibeau and Rockett 1996). In some of the switching models, the decision to remain in the old network may be changed later, whereas the choice of the new technology is irreversible (Farrell and Saloner 1985).
The literature shows that both too much adoption (excess momentum) and too little (excess inertia) can take place. Consider the case of unsponsored technologies. Excess inertia may arise because users are not sure whether they will be followed if they switch (Farrell and Saloner 1985), because they expect others not to switch (Katz and Shapiro 1992) or because they are not willing to bear a temporary loss of network benefits (Farrell and Saloner 1986). In general, excess inertia follows from the fact that individuals do not take into account the positive effect that their adoption of the new technology would have on others. In models with several generations excess inertia can also take place because the users who arrive first do not wait for the new technology to appear (Choi 1994, 1997; Choi and Thum 1998), or choose the technology that is cheaper now, instead of one that is expected to be cheaper in the future (Katz and Shapiro 1986b). This decreases the welfare of younger users, who must either lose the network of the old users, or use the less efficient technology.
Excess momentum, on the other hand, often arises because, as some users adopt the new technology, those who continue to use the old technology are left with a smaller network (Farrell and Saloner 1986; De Bijl and Goyal 1995; Choi 1994). It can also arise if users have heterogeneous preferences and if it is only possible to commit to the new technology (Farrell and Saloner 1985). In that case, those with a preference for the new technology will switch and the other users may be forced to switch to their less preferred technology. In general, excess momentum arises because users who adopt a new technology do not take into account the negative effect that they have on old users, or on those who will switch later.
New technology sponsorship does not systematically increase or decrease efficiency. Sponsorship can give the new technology an advantage if both technologies are always present, but the old one has a cost (or quality) advantage now, and the new one later (Katz and Shapiro 1986a). In that case, the new producer can price below cost first to build an installed base and raise prices later. On the other hand, if only the new technology is sponsored and is not available in the first period, early users may be unwilling to adopt it because they will expect to be exploited by its sponsor in the future (Choi and Thum 1998). When the timing of the introduction of the new technology is a decision variable, the new technology will tend to be introduced too early, because its sponsor does not take into account the lost network benefits of consumers of the old technology, and the profits of the incumbent (Katz and Shapiro 1992; Regibeau and Rockett 1996).
In the model presented in this paper, the new network arises due to switching, and the cases of both unsponsored technologies and sponsored new technology are studied. Unlike in some models described above, excess momentum is not possible in our model, since the optimal outcome has all users switching to the new technology. That is because, for a given network size, the new technology is preferred by everyone, and because switching is costless. On the other hand, excess inertia can arise when users expect that it will be the case. Sponsorship exacerbates the inertia because a profitmaximizing firm may prefer to extract more surplus from consumers with a high valuation of quality rather than gain as large a market share as possible.
3 Unsponsored technologies
In this section, we describe the demand side in detail and show whether and if so to what extent a new technology will be adopted in markets where the valuations of network size and quality are heterogeneous. We assume here that both technologies are competitively provided, that is, their presence is not connected to any particular seller and their prices are equal to marginal cost (which we assume to be zero). In this section we also define the concept of a stable equilibrium given the initial market share, which we will use in the rest of the paper. We explicitly show how stable equilibrium market shares depend on the quality difference between both technologies.
The technology that consumers buy lasts one period. In each period, therefore, consumers decide whether or not to adopt one of the available technologies. Each consumer chooses the technology that maximizes expected utility in that period.
Definition 1
 1.
Each consumer (α,β) maximizes \( E{\left[ {u_{{\alpha ,\beta }} {\left( t \right)}} \right]} \) given Ex _{ t } = Ex,
 2.
Ex = x.
From the Figure it is clear that x _{1} is the stable equilibrium for x _{0} < x _{2}, and x = 1 is the stable equilibrium if x _{0} > x _{2}. x _{2} is also an equilibrium, but it is not stable: a very small change in market share in either direction would make the system converge to either x _{1}, or x = 1. Using this dynamic analysis, we may define a stable equilibrium given an initial market share of x _{0} as follows:
Definition 2
A stable equilibrium given an initial market share of x _{0} is a pair (Ex,x) which (1) satisfies Definition 1 and (2) emerges as the stable outcome of the dynamic model in which consumers’ expectations are given by \( Ex_{t} = x_{{t  1}} \) and the initial market share of the new technology is x _{0}.
The above analysis can then be summarized as follows.
Proposition 1
If the new technology has only a small market share in equilibrium, the welfare effect is not obvious: the consumers who use the new technology in the equilibrium derive more utility from its quality, but they lose a large part of the network externality. The remaining consumers do not gain anything in terms of quality and they lose a part of the network benefits.
Thus, if the equilibrium market share of the new technology is small, the loss of network benefits dominates the gain from higher quality and social welfare decreases. The larger the equilibrium market share of the new technology, the larger the loss.
4 Sponsored new technology
In this section, we analyze the case where the new technology is put on the market by a firm that sets prices to maximize profits. The old technology is still unsponsored and available at price zero. To study the changing incentives of the firm as its market share changes, we consider an infinite pricesetting process, in which, in every period T, first the firm chooses the price, and then the equilibrium market share given initial x _{ 0 } is determined by the dynamic adjustment process described in Section 3, where consumers make their decisions by comparing net utilities.
The outcome of the maximalization will be a sequence of prices and market shares, \( {\left[ {p_{T} ,x_{T} } \right]}^{\infty }_{{T = 1}} \). We first derive the perperiod demand function for the new technology, i.e. the stable equilibrium market share which arises in each period given the price of the new technology and initial market shares. Next, we describe the optimal pricing strategy of the firm.
Note that the situation represented in Fig. 6a cannot be an equilibrium: it is clear from the figure that x _{ T } < 1/2, which contradicts \( x_{T} = Ex_{T} > {{\left( {p_{T}  q + 1} \right)}} \mathord{\left/ {\vphantom {{{\left( {p_{T}  q + 1} \right)}} 2}} \right. \kern\nulldelimiterspace} 2 \). That leaves four possible equilibria to be considered. The eventual outcomes depend on the value of q, the quality difference between technologies. Lemmas 1–3 characterize the demand function of the firm for different values of the quality difference q. The proof of Lemma 2, the most difficult one, can be found in the Appendix; other proofs are available upon request.
Lemma 1
Lemma 2

\( \begin{array}{*{20}c} {{x_{T} = 0}} & {{if\;p_{T} > q,}} \\ \end{array} \)

\( x_{T} = \frac{{q  {\sqrt {q{\left( {  4p^{2}_{T} + 8p_{T} q + q  4q^{2} } \right)}} }}} {{4q}} \) \(if\;p_{T} > {{\sqrt q }} \mathord{\left/ {\vphantom {{{\sqrt q }} 2}} \right. \kern\nulldelimiterspace} 2 \) , or \( \begin{array}{*{20}l} {{if\;q \mathord{\left/ {\vphantom {q {{\left( {q + 1} \right)}}}} \right. \kern\nulldelimiterspace} {{\left( {q + 1} \right)}} < p_{T} < {{\sqrt q }} \mathord{\left/ {\vphantom {{{\sqrt q }} 2}} \right. \kern\nulldelimiterspace} 2} \hfill} & {{and} \hfill} & {{x_{{T  1}} < {{\left( {3q  {\sqrt {q{\left( {q  4p^{2}_{T} } \right)}} }} \right)}} \mathord{\left/ {\vphantom {{{\left( {3q  {\sqrt {q{\left( {q  4p^{2}_{T} } \right)}} }} \right)}} {4q}}} \right. \kern\nulldelimiterspace} {4q},\,\quad or} \hfill} \\ {{if\;{q^{2} } \mathord{\left/ {\vphantom {{q^{2} } {{\left( {q + 1} \right)}}}} \right. \kern\nulldelimiterspace} {{\left( {q + 1} \right)}} < p_{T} < q \mathord{\left/ {\vphantom {q {{\left( {q + 1} \right)}}}} \right. \kern\nulldelimiterspace} {{\left( {q + 1} \right)}}} \hfill} & {{and} \hfill} & {{x_{{T  1}} < {{\left( {1  2q + 2p_{T} } \right)}} \mathord{\left/ {\vphantom {{{\left( {1  2q + 2p_{T} } \right)}} {{\left[ {2{\left( {1  q} \right)}} \right]}}}} \right. \kern\nulldelimiterspace} {{\left[ {2{\left( {1  q} \right)}} \right]}},\quad or} \hfill} \\ {{if\;{{\left( {2q  {\sqrt q }} \right)}} \mathord{\left/ {\vphantom {{{\left( {2q  {\sqrt q }} \right)}} 2}} \right. \kern\nulldelimiterspace} 2 < p_{T} < {q^{2} } \mathord{\left/ {\vphantom {{q^{2} } {{\left( {q + 1} \right)}}}} \right. \kern\nulldelimiterspace} {{\left( {q + 1} \right)}}} \hfill} & {{and} \hfill} & {{x_{{T  1}} < {{\left( {q + {\sqrt {q{\left( {  4p^{2}_{T} + 8p_{T} q + q  4q^{2} } \right)}} }} \right)}} \mathord{\left/ {\vphantom {{{\left( {q + {\sqrt {q{\left( {  4p^{2}_{T} + 8p_{T} q + q  4q^{2} } \right)}} }} \right)}} {4q}}} \right. \kern\nulldelimiterspace} {4q},} \hfill} \\ \end{array} \)

\( x_{T} = \frac{{3q + {\sqrt {q{\left( {q  4p^{2}_{{T  1}} } \right)}} }}} {{4q}} \) otherwise.
Lemma 3
 1.If p _{ T } > q

\( \begin{array}{*{20}l} {{x_{T} = 0,} \hfill} & {{if\;p_{T} \geqslant {{\left( {4q + 1} \right)}} \mathord{\left/ {\vphantom {{{\left( {4q + 1} \right)}} 8}} \right. \kern\nulldelimiterspace} 8,\quad or} \hfill} \\ {{} \hfill} & {{if\;q \leqslant p_{T} < {{\left( {4q + 1} \right)}} \mathord{\left/ {\vphantom {{{\left( {4q + 1} \right)}} 8}} \right. \kern\nulldelimiterspace} 8\quad and\quad x_{{T  1}} < {{\left( {3  {\sqrt {4q  8p_{T} + 1} }} \right)}} \mathord{\left/ {\vphantom {{{\left( {3  {\sqrt {4q  8p_{T} + 1} }} \right)}} 4}} \right. \kern\nulldelimiterspace} 4.} \hfill} \\ \end{array} \)

\( x_{T} = \frac{{3 + {\sqrt {4q  8p_{T} + 1} }}} {4} \) otherwise.

 2.If p _{ T } < q

\( x_{T} = \frac{{q  {\sqrt {q{\left( {  4p^{2}_{T} + 8p_{T} q + q  4q^{2} } \right)}} }}} {{4q}} \), \( \begin{array}{*{20}l} {{if\;p_{T} > q \mathord{\left/ {\vphantom {q {{\left( {q + 1} \right)}}}} \right. \kern\nulldelimiterspace} {{\left( {q + 1} \right)}}} \hfill} & {{and} \hfill} & {{x_{{T  1}} < {{\left( {3q  {\sqrt {q{\left( {q  4p^{2}_{{T  1}} } \right)}} }} \right)}} \mathord{\left/ {\vphantom {{{\left( {3q  {\sqrt {q{\left( {q  4p^{2}_{{T  1}} } \right)}} }} \right)}} {4q}}} \right. \kern\nulldelimiterspace} {4q},\quad or} \hfill} \\ {{if\;{q^{2} } \mathord{\left/ {\vphantom {{q^{2} } {{\left( {q + 1} \right)}}}} \right. \kern\nulldelimiterspace} {{\left( {q + 1} \right)}} < p_{T} < q \mathord{\left/ {\vphantom {q {{\left( {q + 1} \right)}}}} \right. \kern\nulldelimiterspace} {{\left( {q + 1} \right)}}} \hfill} & {{and} \hfill} & {{x_{{T  1}} < {{\left( {1  2q + 2p_{T} } \right)}} \mathord{\left/ {\vphantom {{{\left( {1  2q + 2p_{T} } \right)}} {{\left[ {2{\left( {1  q} \right)}} \right]}}}} \right. \kern\nulldelimiterspace} {{\left[ {2{\left( {1  q} \right)}} \right]}},\quad or} \hfill} \\ {{if\;p_{T} < {q^{2} } \mathord{\left/ {\vphantom {{q^{2} } {{\left( {q + 1} \right)}}}} \right. \kern\nulldelimiterspace} {{\left( {q + 1} \right)}}} \hfill} & {{and} \hfill} & {{x_{{T  1}} < {{\left( {q  {\sqrt {q{\left( {  4p^{2}_{T} + 8p_{T} q + q  4q^{2} } \right)}} }} \right)}} \mathord{\left/ {\vphantom {{{\left( {q  {\sqrt {q{\left( {  4p^{2}_{T} + 8p_{T} q + q  4q^{2} } \right)}} }} \right)}} {4q}}} \right. \kern\nulldelimiterspace} {4q},} \hfill} \\ \end{array} \)

\( x_{T} = \frac{{3q + {\sqrt {q{\left( {q  4p^{2}_{T} } \right)}} }}} {{4q}} \) otherwise.

The shape of the demand function depends on the value of the quality difference between technologies. Most striking is the difference between the cases q > 1 and 0.25 < q < 1. For large q, the demand function is continuous and each price can only lead to one value of demand, which is not the case if q < 1. The reason is that, for large q, network externalities, which are the cause of the discontinuity, are relatively less important. The total value of network externalities, βx or β(1 − x _{ T }), cannot be larger than one, while the total value of increased quality, αq, is larger than one for at least some consumers. Therefore, the demand function for the case q > 1 resembles the demand function for the case where no network externalities are present (thus β = 0 for all consumers). For a comparison, we depict this demand function, described by the expression \( x_{T} = 1  p \mathord{\left/ {\vphantom {p q}} \right. \kern\nulldelimiterspace} q \), in Fig. 7a as a dashed line. Compared to the case with β = 0, the presence of network externalities increases the elasticity of demand, except for the case when p is very low or very high. In these cases, the consumers of the new technology benefit from large network externalities and therefore are more reluctant to switch away from it when its price goes up.
When q < 1, network externalities are relatively more important in determining the consumers’ choice, leading to a multiplicity of equilibrium market shares for a given price. Whenever two equilibria are possible, the upper curve of the demand correspondence represents the case where high quality has a high initial market share. For a certain critical price, \( \widetilde{p} \), the demand function will jump from the upper segment of the demand function to the lower one. The discontinuity point depends positively on the initial market share.
We now proceed to describing the optimal price path for the sponsor of the new technology. As the firm sets its prices to maximize total profits, \( \Pi = {\sum\nolimits_{T = 1}^\infty {\delta ^{{T  1}} \Pi _{T} } } \), and as profits in period T are equal to \( \Pi _{T} = p_{T} x_{T} {\left( {p_{T} ,x_{{T  1}} } \right)} \), past prices influence the current profits through the impact on today’s market share. Clearly, a large initial market share is an advantage: at given prices in the twoequilibrium range, a larger initial market share results in a larger demand.
Proposition 2 characterizes the optimum prices of the sponsor of the new technology for different values of q.
Proposition 2
 1.
\( \begin{array}{*{20}c} {{If\;q \geqslant q\prime ,\;\;then}} & {{p^{ * }_{T} = p^{m} }} & {{for\;T = 1,2 \ldots }} \\ \end{array} \)
 2.
\( \begin{array}{*{20}c} {{If\;q \leqslant q < q\prime ,\;\;then}} & {{p^{ * }_{T} = p^{l} }} & {{for\;T = 1,2 \ldots }} \\ \end{array} \)
 3.
\( \begin{array}{*{20}c} {If\;q\prime < q < q,\;\;then}{p^{ * }_{T} = p^{u} }{for\;T = 1} \\ {and}{p^{ * }_{T} = p^{l} }{for\;T = 2,3, \ldots } \\ \end{array} \)
 4.
\(\begin{array}{*{20}c} {If\;\;1 \mathord{\left/ {\vphantom {1 4}} \right. \kern\nulldelimiterspace} 4 < q < q\prime \prime \prime ,\;\;then}{p^{ * }_{1} = p^{u} }{for\;T = 1} \\ {and}{p^{ * }_{1} = p^{l} }{for\;T = 2,3, \ldots if\;\delta > \delta ^{ * } ,^{4} } \\ {}{p^{ * }_{1} = p^{s} }{for\;all\;T = 1,2,3, \ldots if\;\delta < \delta ^{ * } .} \\ \end{array} \) ^{4}
 5.
\( \begin{array}{*{20}c} {{q < 0.25,\;\;then}} & {{p^{ * }_{1} = p^{s} }} & {{for\;all\;T = 1,2,3, \ldots }} \\ \end{array} \)
Figure 8 shows the intuitive notion that the larger the quality difference, the larger the price the provider of the superior technology charges. It is interesting to note that, apart from an intermediate range of quality difference [0.25, q″], the optimal price is constant over time and varies only with the exogenous quality difference. When the quality difference is very small, this price is very small and equal to p ^{ s }, represented in the Figure by the bold curvature. When the quality difference is somewhat larger, the optimal pricing structure depends on the discount factor. If the future is valued enough, i.e., if δ is large enough, the optimal pricing path is such that, in the first period, a low undercutting price p ^{ u } is asked, represented in the Figure by the dashed line, to gain a substantial market share and a large price p ^{ l }, represented in the Figure by the thin curvature, ever after. If the discount factor is relatively low, it is better for the provider of the new technology to set one constant, somewhat moderate, price p ^{ s }. When the quality difference is larger, i.e., q > q″′, it is optimal, for all discount factors, to start with a low price in period 1 and continue with a large price from period 2 onwards. When the quality difference grows, the undercutting price p ^{ u } and the large price p ^{ l } converge towards one another when the quality difference increases, and from q″ they coincide.
An interesting outcome, illustrated in Fig. 9, is that the market share is nonmonotonic in q. The longrun market share (all segments drawn in continuous line) first increases, then stays constant, and then decreases in q. A higher q has a twofold effect on the firm’s market share. The direct effect is positive: a higher q leads to a higher x for a given price. The indirect effect is negative: a higher q leads to a higher optimal price, which impacts market share negatively. When there are no network externalities (see the dotted line), the two effects exactly cancel each other so that the market share stays constant. The firm chooses its price so that it serves only people with a relatively high valuation of quality.
With network externalities, the relationship between the two effects becomes more complicated. For small q and a small market share, the direct positive effect dominates. For intermediate q, both effects cancel out, but at a much higher market share than without network externalities. The firm keeps a large market share in order to attract not only consumers with a high valuation of quality, but also those with a low valuation of quality but high willingness to pay for network externalities. When the quality difference becomes very large, it becomes more profitable to give up this last group of consumers in order to be able to extract more surplus from qualityloving consumers.
Consider now the longrun welfare change when the new technology is sponsored. The welfare change is equal to the change in gross consumers surplus of consumers who choose the new technology, ΔGCS _{1}, and in the consumer surplus of consumers of old technology ΔCS _{0}. Just as in the case of unsponsored technologies, the social welfare is maximized when everyone adopts the new technology. However, unlike in that case, this never happens when the new technology is sponsored. Proposition 3 describes the welfare change as a consequence of the introduction of the new technology.
Proposition 3
The introduction of the new technology increases (decreases) social welfare in the long run if \( q > q\prime {\left( {q < 0.25} \right)} \). If 0.25 < q < q″′, social welfare increases if, and only if the new technology has a dominant market share.
5 Conclusions
We have examined the adoption of technology in a market where consumers have to make a purchasing decision in every period and have different valuations of both network size and quality. In Section 3, we considered the case of unsponsored technologies, and in Section 4, we analyzed the implication of the new technology being supplied by a profitmaximizing seller. We showed that, even though consumers are not stuck with past purchases, the better technology may not gain the whole market in equilibrium, due to lagging expectations. We also found that market outcomes depend on the quality advantage of the new technology. In the case of unsponsored new technology, the higher the quality difference, the higher the market share of the new technology. There is a critical value such that, for larger quality differences, the new superior technology will be used in the whole market, and for lower quality differences, the old technology remains dominant in the market.
When the new technology is sponsored, the relationship between the quality difference and the longrun market share becomes nonmonotonic. The market share initially increases, but then decreases as the quality difference increases. The reason is that, for intermediate q, the firm finds it profitable to serve not only the consumers with a high valuation for quality, but also those with a low valuation of quality and high valuation of network externalities. When the quality difference is large, the firm prefers to give up the consumers who care mostly about network externalities in order to extract more surplus from those who care much about quality.
The introduction of the new technology does not necessarily increase social welfare. When the quality difference is small and the equilibrium market share of the new technology small, the gain of consumers who switch to the new technology does not outweigh the loss in network externalities for consumers who do not switch. When the new technology has a much higher quality, its introduction increases social welfare. However, the social optimum, in which everyone uses the new technology, can only be achieved when the new technology is unsponsored.
Footnotes
 1.
De Palma and Leruth (1996) also model heterogeneous network externalities, but they do not allow for heterogeneity in the valuation of quality, nor do they study the introduction of a new technology.
 2.
In a discussion paper (Janssen and Mendys 2000), we analyze the situation where both technologies are sponsored in the case when consumers’ valuation for quality and network externalities are inversely correlated. With the independent valuations assumed in this article, the analysis becomes analytically intractable.
 3.
The exact values are \( q\prime = {{\left( {{\sqrt {17} } + 3} \right)}} \mathord{\left/ {\vphantom {{{\left( {{\sqrt {17} } + 3} \right)}} 4}} \right. \kern\nulldelimiterspace} 4 \), \( q = {{\left( {3{\sqrt {17} } + 3 + 4{\sqrt 2 }{\sqrt {3{\sqrt {17} }  5} }} \right)}} \mathord{\left/ {\vphantom {{{\left( {3{\sqrt {17} } + 3 + 4{\sqrt 2 }{\sqrt {3{\sqrt {17} }  5} }} \right)}} {32}}} \right. \kern\nulldelimiterspace} {32} \), \( q\prime = {{\left( {6{\sqrt 3 }  9} \right)}} \mathord{\left/ {\vphantom {{{\left( {6{\sqrt 3 }  9} \right)}} 4}} \right. \kern\nulldelimiterspace} 4 \) and θ is implicitly defined by \( \cos \theta = {2q{\left( {4q + 27} \right)}} \mathord{\left/ {\vphantom {{2q{\left( {4q + 27} \right)}} {{\left( {{\left( {4q + 9} \right)}{\sqrt {q{\left( {4q + 9} \right)}} }} \right)}}}} \right. \kern\nulldelimiterspace} {{\left( {{\left( {4q + 9} \right)}{\sqrt {q{\left( {4q + 9} \right)}} }} \right)}} \).
 4.
Where δ* is such that \( \Pi ^{u} + {\delta * } \mathord{\left/ {\vphantom {{\delta * } {}}} \right. \kern\nulldelimiterspace} {}{\left( {1  \delta * } \right)}\Pi ^{l} = 1 \mathord{\left/ {\vphantom {1 {}}} \right. \kern\nulldelimiterspace} {}{\left( {1  \delta * } \right)}\Pi ^{s} \)with \( \Pi ^{u} = p^{u} x{\left( {p^{u} ,x_{{T  1}} < 0.25} \right)} \), \( \Pi ^{l} = p^{l} x{\left( {p^{l} ,x_{{T  1}} > 0.75} \right)} \) and \( \Pi ^{s} = p^{s} x{\left( {p^{s} ,x_{{T  1}} < 0.25} \right)} \).
 5.
If \( q = {3{\sqrt 3 }} \mathord{\left/ {\vphantom {{3{\sqrt 3 }} 2}} \right. \kern\nulldelimiterspace} 2  9 \mathord{\left/ {\vphantom {9 4}} \right. \kern\nulldelimiterspace} 4 \), then both p′ and p″ are real, but only p″ is a maximum.
 6.
Calculations available by request.
Notes
Acknowledgment
We thank the editor and an anonymous referee for their valuable comments and audiences in Oxford, Lausanne, Amsterdam and Rotterdam.
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