Identifying critical mass in the global cellular telephony market

International Journal of Industrial Organization 30 (2012) 496–507

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International Journal of Industrial Organization journal homepage: www.elsevier.com/locate/ijio

Identifying critical mass in the global cellular telephony market Michał Grajek a, Tobias Kretschmer b,⁎ a b

ESMT European School of Management and Technology, Schlossplatz 1, 10178 Berlin, Germany Institute for Strategy, Technology and Organization, ifo Institute and Centre for Economic Performance, LMU Munich, Schackstr. 4/III, 80539 Munich, Germany

a r t i c l e

i n f o

Article history: Received 3 November 2011 Received in revised form 25 April 2012 Accepted 14 June 2012 Available online 23 June 2012 JEL classifications: C53 L14 M37

a b s t r a c t Technology diffusion processes are often said to have critical mass phenomena. We apply a model of demand with installed base effects to provide theoretically grounded empirical insights about critical mass. Our model allows us to rigorously identify and quantify critical mass as a function of installed base and price. Using data from the digital cellular telephony market, which is commonly assumed to have installed base effects, we apply our model and find that installed base effects were generally not strong enough to generate critical mass phenomena, except in the first cellular markets to introduce the technology. © 2012 Elsevier B.V. All rights reserved.

Keywords: Critical mass Network effects Technology diffusion Cellular telephony

1. Introduction Successful new technologies and innovations typically diffuse in an S-shape. While the focus of past research has often been on the inflection point of a diffusion curve where diffusion slows down after a period of rapid growth, it is arguably just as important to identify the point on the diffusion curve when a technology starts penetrating the mass market. Different literatures have called this phenomenon “product takeoff” (Agarwal and Bayus, 2002; Golder and Tellis, 1997; Lee et al., 2003), a “catastrophe” (Cabral, 1990, 2006), a “punctuated equilibrium” (Loch and Huberman, 1999) or “critical mass” (Cool et al., 1997; Evans and Schmalensee, 2010; Mahler and Rogers, 1999; Markus, 1987). We focus on the last and derive conditions for the existence of critical mass before identifying it empirically in the global cellular telephony market. What, then, is critical mass? A common definition is that at critical mass “diffusion becomes self-sustaining” (Rogers, 2003: 243). This is qualitatively different to most conventional technology diffusion processes that rely on heterogeneous consumers and price decreases and/or quality increases (Grajek and Kretschmer, 2009; Loch and Huberman, 1999). Critical mass phenomena rely on a rapidly evolving endogenous process over time, e.g. installed base effects driving diffusion even in the absence of price decreases. The link between technology diffusion and installed base effects is well established (Cabral, 1990, ⁎ Corresponding author. Tel.: +498921806270; fax: +49892180995717. E-mail addresses: [email protected] (M. Grajek), [email protected] (T. Kretschmer). 0167-7187/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.ijindorg.2012.06.003

2006; Granovetter, 1978; Kretschmer, 2008; Markus, 1987; Rohlfs, 1974), and research identifying multiple stable equilibria separated by an unstable one (Economides and Himmelberg, 1995; Evans and Schmalensee, 2010; Katz and Shapiro, 1985) characterizes the transition from one equilibrium to the other as critical mass. We adapt the model by Cabral (1990) to develop a simple structural demand model of a non-durable good or service with installed base effects. We use the logic of multiple equilibria and endogenous diffusion outlined above to show that critical mass — a self-sustaining diffusion process — will only emerge if boundary conditions on the strength of installed base effects, the size of the installed base, and the current market price, are met. We find that these three parameters are substitutes in terms of reaching critical mass: for stronger network effects, critical mass is reached for higher prices and lower installed bases; for a higher installed base, network effects can be weaker and prices higher for critical mass to still exist; and so on. We estimate demand in the global cellular telephony industry between 1998 and 2007 and find that demand for cellular services displayed critical mass phenomena only in pioneering markets. Our paper makes two contributions to the economics of technology diffusion: First, we operationalize and test the model by Cabral (1990) empirically, offering a simple but rigorous formal test for identifying critical mass, i.e. whether a technology displays periods of endogenous and rapid diffusion. We show that the critical mass point depends on price given that sufficiently strong installed base effects exist in a market. Our approach is complementary to work focusing on the adoption dynamics of durable goods with indirect network effects (Dubé et al.,

M. Grajek, T. Kretschmer / International Journal of Industrial Organization 30 (2012) 496–507

2010; Gowrisankaran and Stavins, 2004; Ohashi, 2003). Our work also complements simulation models on industry dynamics and transitions (Lee et al., 2003; Loch and Huberman, 1999) as we derive information about demand conditions and especially the strength of installed base effects in real-life markets which can help calibrate simulation models. Our empirical model has two key advantages: (i) it imposes modest data requirements and (ii) it gives a linear (in parameters) diffusion equation with fixed effects, which is convenient to work with empirically. Second, we show for the case of digital cellular telephony that critical mass was a local rather than a global phenomenon. That is, we find critical mass only in markets that pioneered the technology, i.e. that started offering 2G services early. Our empirical results suggest that critical mass is a function of both installed base and price, the latter being more important. Specifically for pioneering markets, we find that cellular telephony would have “taken off” without any installed base at an average price of 36 US cents per minute. However critical mass is reached at a slightly higher price (38 US cents) when the installed base of subscribers is about 24% of the population, suggesting that installed base can only substitute for the “right” price to some extent.

2. Prior work Much of the empirical literature on emerging technologies with installed base effects 1 can be divided in two streams — competition between emerging technologies and diffusion of a new technology. The first stream looks at explaining and documenting the dynamics of competing standards (Cantillon and Yin, 2008; Dranove and Gandal, 2003; Dubé et al., 2010; Jenkins et al., 2004; Ohashi, 2003) and the phenomenon of market tipping (Shapiro and Varian, 1999). The second stream studies the impact of network effects on the diffusion speed or adoption timing of a new technology (Gowrisankaran and Stavins, 2004; Saloner and Shepard, 1995) or a set of complementary technologies (Gandal et al., 2000). Many of these studies find significant installed base effects resulting in faster diffusion, as predicted by Rogers (2003). None of these papers, however, allow for the possibility of an endogenous, almost discontinuous diffusion path. Indeed, they estimate smooth curves that can differ only in the speed of diffusion, not their general shape. By construction, critical mass cannot be identified through these models. Surprisingly, the critical mass literature has developed largely in parallel to work on installed base effects. One stream of the literature derives theoretical or conceptual explanations of why some markets display critical mass phenomena. Markus (1987) uses a set of qualitative indicators on interactive media markets with critical mass behavior and finds the underlying production function and consumer heterogeneity to be especially important in such markets. Loch and Huberman (1999) show in simulations that rapid (i.e., critical mass-like) transition from an old to a new standard can occur if consumers have a high rate of experimentation and the new technology improves rapidly. Finally, Cabral (1990, 2006) finds that the equilibrium adoption path might display “catastrophe points”, that is, critical mass phenomena, only if network effects are sufficiently strong. Evans and Schmalensee (2010) generalize this result for indirect network effects in two-sided markets. All of the above papers outline circumstances (e.g. demand and network parameters, production technologies, market structure) under which critical mass-like phenomena can occur, and most provide anecdotal evidence of market dynamics consistent with critical mass, although none test their findings using an econometric model and real data.2 Empirical work on critical mass

1 We use the term “installed base effects” to describe any effect that results in an increase in consumers' propensity to adopt with increasing installed base. For a detailed discussion on the potential sources of installed base effects, see Section 3.1. 2 Economides and Himmelberg (1995) develop a theoretical model and test it on fax diffusion in the US. Their definition of critical mass differs from ours though — we discuss this in Section 3.3.

497

focuses on identifying a percentage — typically varying between 10% (Mahler and Rogers, 1999) and 25% (Cool et al., 1997) — of market potential as critical mass, assumed to be the penetration level at which diffusion speed picks up significantly. Relatedly, work on sales takeoff (Agarwal and Bayus, 2002; Golder and Tellis, 1997; Tellis et al., 2003) develops heuristics on the link between early sales growth and long-term product success. Finally, the global cellular telephony industry has been studied in some detail. Existing work finds that the technology displays installed base effects in adoption (Gruber and Verboven, 2001; Koski and Kretschmer, 2005) and that cellular diffusion varies across countries and groups of countries, suggesting that different technological, socioeconomic, and regulatory factors affect the diffusion process. However, existing models cannot empirically distinguish the periods of rapid diffusion common to most industrialized countries from critical mass, which does not rest on price decreases or exogenously changing technologies. In contrast, our model and estimation can do this. 3 One exception is Grajek (2010), who uses a similar model to ours, but focuses on compatibility across competing networks in a single market. Since we estimate our model using data from multiple geographic markets, we provide a more extensive analysis of critical mass and identify the role of market-specific factors like competition and demand characteristics. 3. Theoretical model 3.1. Willingness to pay and installed base effects We adapt the model developed by Cabral (1990) in which at each time, t, consumers decide whether or not to subscribe to a service with installed base effects depending on their net benefit. Examples include subscription to a payment system such as a credit card or to a communication service like e-mail or cellular telephony. Installed base effects imply that the installed base of adopters (subscribers) increases consumer willingness to pay. Installed base effects could have a number of origins. First, there could be network effects—direct ones stemming from direct mobile-to-mobile calling and texting among users and indirect ones from the provision of complementary goods, e.g. handsets, ringtones, etc. However, installed base effects could also be due to other social contagion effects including social learning under uncertainty and social-normative pressures. It is difficult to differentiate between them using aggregated data (Hartmann et al., 2008), while conceptually and empirically they all have a similar effect of increasing a user's willingness to pay (or decreasing a user's cost of adoption). We refer to installed base effects throughout the paper to acknowledge the fact that we cannot discriminate between the different effects, but they all can lead to critical mass phenomena. Suppose there is a measure one of infinitely lived consumers with unit demand for a service. Consumer v's preferences are represented by the willingness-to-pay function u(v, xt − δ), where v is the individual preference parameter, xt − δ is lagged network size at time t, and the perception lag δ is a non-negative number. Further, we assume v to be distributed according to a CDFF(v), and u(v, xt − δ) to be strictly increasing and continuous in v. Thus, v establishes a rank ordering of consumers by willingness to pay assumed to be invariant with respect to changes in xt − δ.

3 Grajek and Kretschmer (2009) seek to rule out another potential reason for endogenous diffusion, namely epidemic effects. There, epidemic effects would imply constant usage intensity with increasing penetration. As they find strongly decreasing usage intensity, this effect appears largely absent or at least overshadowed by other factors, most notably consumer heterogeneity, which implies constantly falling prices and/or increasing quality as drivers of diffusion. Here, we take this further and ask if exogenous changes are responsible for all diffusion (which implies no critical mass) or if there was an element of endogenous diffusion (which we find in a subset of countries).

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Including lagged network size xt−δ in the willingness-to-pay function captures network effects in demand for the good, and the perception lag δ works as an equilibrium selection device that yields a unique diffusion path for the network.4 Further, a perception lag is realistic empirically since consumers will not have access to the current numbers of subscribers, but rather to previously published figures, resembling a perception lag.5 Such a lag therefore seems both realistic and is necessary for developing an empirically tractable strategy for identifying critical mass. Clearly however, the use of perception lag renders consumers myopic if switching costs or long-term contracts are binding. Future adoption rates and prices may affect current adoption decisions in this case, but expectations are difficult to capture empirically. 3.2. Short-run and long-run subscription demand At time t, consumer v decides whether to subscribe by considering the net utility from joining: uðv; xt−δ Þ−pt :

ð1Þ

There is one market price, pt. If Eq. (1) is non-negative, the consumer will join, otherwise not. The consumer indifferent between joining or staying out at time t (vt*) is given by the following equation:
  u vt ; xt−δ ¼ pt :

ð2Þ

All consumers with v ≥ vt* will join. Define

 H vt ≡1−F vt ;

ð3Þ

such that H(⋅) equals the number of consumers in the network at time t. The state equation describing network size at time t, that is, short-run demand, is given by: xt ¼ H




 vt :

by Eq. (4) (Cabral, 1990). Because H(⋅) maps the change in network size from time t− δ to t, it is convenient to think of it as of a function of lagged network size xt−δ. We calculate the derivatives of H(⋅) with respect to lagged network size xt−δ and price p in Appendix A.1. The slope of H(⋅) increases in the strength of installed base effects measured by 

vt ; xt−δ Þ η≡ ∂uð∂x , as shown in Lemma 1. t−δ

Fig. 1 illustrates the diffusion dynamics. In the top panel we show H(⋅) as a function of lagged network size xt − δ. Given the steady-state condition (Eq. (5)), the long-run equilibrium network sizes coincide with the fixed points of H(⋅). Without installed base effects, H(⋅) is a horizontal line with a single fixed point. A combination of positive installed base effects and a bell-shaped distribution of types v can result in a function H(⋅) with multiple long-run equilibria as in Fig. 1. The dynamics of the model let us discriminate among these multiple steady states. Suppose market price is p* in Fig. 1. According to state Eq. (4), network size will evolve as indicated in the top panel. If it starts at some size x b x', it will eventually reach x 0; if x > x', it will end up in x". If x = x', it will stay there, but any arbitrarily small shock will lead to an equilibrium at x 0 or x''. Therefore, x 0 and x" are stable steady states, whereas x' is unstable. We can apply the same logic to any price p. Lemma 2 in the Appendix states that lowering price shifts H(⋅) upwards (although not necessarily in parallel). Drawing the steady states for each price gives long-run demand D(p) in the lower panel of Fig. 1, which lets us define a continuum of critical mass points (all proofs are in Appendix A.2): Proposition 1. Downward-sloping parts of long-run demand D(p) consist of stable equilibria, whereas upward-sloping parts are unstable, that is, consist of critical-mass points. Installed base effects must be sufficiently strong for unstable equilibria to exist.

xt

ð4Þ

1

H(v *(x t- δ ,p*))

In steady state, no consumer can increase utility by joining or leaving; the network stays constant over time, which gives the following long-run demand condition: xt ¼ xt−δ :

ð5Þ

Long-run demand is reached when the market is saturated and there are no more consumers to fuel further diffusion. However, long-run demand can also fall short of full saturation depending on prices and consumer preferences. This feature differentiates the model from Bass (1969), in which full saturation is always reached. In other words, the model can accommodate failed products. Note that the steady-state equilibrium coincides with the static fulfilled-expectations equilibrium in the literature (Katz and Shapiro, 1985; Rohlfs, 1974).

x t-δ p

3.3. Network dynamics: critical mass and diffusion takeoff It is useful to consider the dynamics implied by the model to define critical mass and point out differences of our definition to alternative ones. Assume the CDF of F(v) and the willingness-to-pay function u(v, xt−δ) to be continuously differentiable in all arguments. For sufficiently strong installed base effects and lag length δ approaching zero, the equilibrium adoption path is unique and discontinuous as described

ph

p*

pl 4

Cabral (1990) shows that for δ = 0 there are infinitely many equilibrium diffusion paths. A positive δ implies that consumers cannot coordinate their subscription decisions, leading to a unique equilibrium diffusion path. As an alternative, Economides and Himmelberg (1995) allow consumers to coordinate to reach critical mass. 5 Cabral (1990) shows that for infinitely small δ, consumers are rational as their subscription decisions resemble those by forward-looking consumers.

0%

x0

x’

x’’

100%

Fig. 1. Stable vs. unstable equilibria.

x

M. Grajek, T. Kretschmer / International Journal of Industrial Organization 30 (2012) 496–507

The intuition of Proposition 1 is that downward-sloping parts of the demand correspondence are locally “well-behaved,” that is, every price p has a single corresponding long-run network size given by D(p). Conversely, critical mass points are unstable in the sense that they divide regions of attraction towards the stable equilibria. When the installed base reaches critical mass, there is a qualitative change in the diffusion process; a switch from low-adoption to high-adoption equilibrium occurs and diffusion takes off without a change in prices or qualities. Consider a case with no initial installed base and falling prices over time to compare our definition of critical mass to Cabral's (1990) catastrophe point. 6 That is, let pt be a continuous and decreasing function of time, and let pt = 0 > p h (as in Fig. 1) and x(pt = o) be the unique steady-state network size givenpt = 0. As price falls, network size initially follows the low-adoption steady state. Eventually price reaches p l and network size jumps to the high-adoption steady state and progresses along it. Thus, the low-adoption steady-state network size at price p l corresponds to the catastrophe point. Economides and Himmelberg (1995) use a similar model but include expected rather than lagged network size in the benefit function. Since they allow the consumers to expect efficient network size to be realized, the catastrophe point already occurs at p h and thus the critical mass of subscribers needed for self-sustaining diffusion is smaller. Our definition of critical mass encompasses both and all the price-installed base combinations between p h and p l. Basically, we define critical mass as all combinations of prices and installed base constituting unstable equilibria which are all points that separate low-adoption from high-adoption equilibria. By upholding the lag structure of the willingness to pay we do not rely on coordination among consumers to obtain a high-adoption equilibrium. However, for takeoff to occur at a price higher than p l, the suppliers need to grow the installed base to critical mass point (x' in Fig. 1) at that price, for example through temporary discounts or free sampling. We can now formulate some predictions about the comparative static behavior of the critical mass points. Proposition 2. If sufficiently strong installed base effects exist to generate multiple steady-state equilibria, critical mass is reached at a lower (higher) installed base for lower (higher) price. Ceteris paribus, stronger installed base effects imply critical mass at a lower installed base and/or higher price. The first part of Proposition 2 follows immediately from Proposition 1. Since critical mass points are on the upward sloping part of the long-run demand function, a higher price implies higher critical mass and vice versa. The second part of Proposition 2 is also intuitive. With stronger installed base effects, it takes a smaller installed base to make a consumer with a given intrinsic valuation adopt if price and distribution of types are held constant.

499

4. Empirical implementation 4.1. Data We use country-level quarterly data from the Merrill Lynch Global Wireless Matrix on the global cellular telephony market in the early stages of the first digital generation (2G), up to the third quarter of 2007 and covering 36 countries and 36 quarters. 7 The decade from 1998 onwards is one of the most dynamic episodes in the global cellular phone market with global penetration rates increasing from 6% to more than 50%. Since we define critical mass to be a function of both price and installed base, we require a sufficiently long period of price and diffusion figures per country to capture critical mass adequately, and quarterly data affords the necessary degrees of freedom for testing the robustness of our results with respect to the specification of the perception lag, which we cannot observe empirically. Table 1 shows descriptive statistics of our variables. Cellular penetration and cellular penetration squared are calculated as ratios of the total number of cellular subscribers to the population in a given country and GDP per capita measures the average wealth of the population. The price variable measures the average price of a 1-min call in a given country and is defined as an average price across operators weighted by their respective penetration rates. 8 Table 1 also reports the instrumental variable used to account for the potential endogeneity of price in our demand equation. We define it as the average price in other countries of the region (Grajek and Kretschmer, 2009). 9 Finally, we construct three dummy variables, WEALTH, PIONEER, and COMPETITION, which we use to assess whether the estimates of network effects and critical mass differ across various country groups. WEALTH indicates countries with above-median GDP per capita in our sample, PIONEER captures the earliest countries to introduce 2G cellular telephony and COMPETITION indicates countries with three or more active cellular telephony providers on average in the sample.10

4.2. Functional specification and identification issues We now specify functional forms for the underlying demand model. The specification in this section has been chosen for three reasons, (1) it gives a simple linear (in parameters) diffusion equation with fixed effects, which is convenient to work with empirically, (2) it facilitates analysis based on multiple markets, and (3) it generates the Bass (1969) diffusion equation for the single market case. 11 It bears emphasizing that the fixed effects in our diffusion equation also help us correctly identify the installed base effects. This issue is challenging because if a country has an unobserved preference for mobile phones, it will exhibit both high installed based and high future adoption, but the relationship between the two will not be causal. Fixed effects account for such preference-driven unobserved heterogeneity among countries.

3.4. Firm strategies We do not model the supply side because we focus on identifying conditions under which demand for a good displays critical mass phenomena. That is, given the demand conditions in the model, decisionmakers can subsequently implement appropriate supply-side strategies to reach and utilize critical mass. Moreover, econometrically we do not need to impose any structure for the supply relation to correctly estimate network effects and identify critical mass, as we can resolve endogeneity issues regarding the price variable via instrumental variable techniques.

6 Cabral (1990) does not include price in his model but rather exogenous benefit that increases with time. This exogenous benefit is analogous to price falling over time.

7 These data have been used in Grajek and Kretschmer (2009) and Genakos and Valletti (2011) and covers the following countries: Australia, Austria, Belgium, Brazil, Canada, China, Czech Republic, Denmark, Finland, France, Germany, Greece, Hong Kong, Hungary, Ireland, Israel, Italy, Japan, Korea, Malaysia, Mexico, Netherlands, New Zealand, Norway, Poland, Portugal, Russia, South Africa, Singapore, Spain, Sweden, Switzerland, Thailand, Turkey, UK, and the US. 8 The price for an individual operator, as obtained from the ML Global Wireless Matrix, is defined as the revenue from services divided by the total number of minutes on the operator's network. 9 The regions are classified as follows: USA/Canada, Western Europe, Eastern Europe, Asia/Pacific, Africa, and Americas. The identification of price in our demand equation is further explained in Section 4.3. 10 2G was first introduced in Finland in 1992. We define PIONEER as a country which introduced 2G by 1993. 11 In the Appendix we show how the model simplifies to the Bass diffusion equation.

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M. Grajek, T. Kretschmer / International Journal of Industrial Organization 30 (2012) 496–507

Table 1 Descriptive statistics. Variable

Variable name Mean

Cellular penetration Cellular penetration squared GDP per capita Population (in millions) Average price/min Others' average price Above median GDP per capita First to adopt 2 G Average no. of competitors ≥ 3

Xi,t X2i,t (GDP/POP)i,t POPi,t pi,t pj,t WEALTH PIONEER COMPETITION

.676 .557

Std. Dev. Min .317 .424

21.469 13.916 80.090 209.500 .200 .089 .217 .085 .504 .500 .231 .694

.422 .461

.009 .00008 .870 3.813 .019 .070 0 0 0

Max 1.484 2.202 77.589 1329.48 .870 .770 1 1 1

2

X i;t ¼ a0i b þ a1 bðGDP=POPÞi;t −bpi;t þ bcX t−1 þ bdX t−1 :

2

ð6Þ

where c and d are parameters that determine the extent of installed base effects, with the square term capturing possible nonlinearities, for example congestion effects (Swann, 2002), and installed base Xi,t−1 is defined as the number of subscribers normalized by population size in a given geographic market (i.e. country) in period t− 1 (Xi,t−1 = xi,t−1/ POPi,t−1).12 The installed base effects as a function of relative rather than absolute number of subscribers facilitate analysis of multiple differently sized markets. We further assume the preference parameter v to be uniformly distributed over (−∞, ai,t] with density bi,t >0. This distribution assumption, which implies that the absolute population of potential subscribers is infinite, lets us avoid corner solutions.13 We also assume the distribution parameters to depend on demographics as follows: ai;t ¼ a0i þ a1 ðGDP=POPÞi;t

ð7Þ

bi;t ¼ bPOPi;t :

ð8Þ

The highest consumer type in the population depends on a country's GDP per head and the unobserved heterogeneity across countries; the extent to which demand reacts to price changes depends on the country's overall population. We choose market shares in our specification of the willingnessto-pay function (6) and our functional assumptions about the distribution of types (7) and (8) to facilitate cross-market comparability. In particular, the density of the distribution of types depends on the size of population to allow for a plausible representation of the price effect across markets; e.g. a given price change in Greece will in absolute terms have a much smaller effect on subscriptions than in the US because Greece is a much smaller country. In the same vein, a given change in the installed base (measured in market shares) will have a much smaller effect on subscriptions (in absolute terms) in Greece than in the US in our model. This seems plausible. Moreover, it is straightforward to recast the model using absolute rather than relative installed base measure for a single market case, because the problem of capturing price effects across differentlysized markets disappears. In contrast, if we assume that the willingness-to-pay function (6) depends on the absolute level of subscribers (and keep all other functional form assumptions), a given change in the installed base (measured in absolute terms) will 12 Note that in the empirical model, δ is determined by data frequency. Consequently, we replace δ with 1 meaning “one period” from now. 13 Alternatively, the distribution support could be bounded from below to limit the population of consumers and the bound assumed to be low enough to avoid corner solutions with all consumers subscribing. Note also that the population size is defined here in absolute terms and not in relative terms as in the theoretical model in Section 3.

ð9Þ

The structural parameters of this model can be recovered from the following estimation equation: X i;t ¼ α 0i þ α 1 ðGDP=POPÞi;t þ βpi;t þ γ1 X i;t−1 þ γ2 X i;t−1 þ εi;t ;

We specify consumer v's willingness-to-pay function as follows:   2 u v; X i;t−1 ¼ v þ cX i;t−1 þ dX i;t−1 ;

still have a much smaller effect on the subscriptions (in absolute terms) in Greece than in the US in the model. This restriction does not accord with our intuition, because it implies that the installed base effects are weaker in Greece than in the US.14 As shown in Appendix A.4, given these functional forms, diffusion Eq. (4) becomes

ð10Þ

where εi,t denotes the error term, which we allow to be heteroscedastic and correlated across time t, but not across markets i. The error term captures the effects of variables that affect subscriptions, but are not observed in our data set, e.g. marketing effort of operators, or the degree of non-price competition more generally, in each geographic market. The coefficient estimates in Eq. (10) identify our structural model parameters as follows: the highest consumer type in each market is identified via country fixed effects and the parameters on price and GDP, a0i = –α0i/β and a1 = −α1/β, and the density of the distribution of types is given by the price parameter, b = −β. The installed base effect parameters c and d are identified via −γ1/β and −γ2/β, respectively. Installed base effects in our model are thus identified by separating the impact of installed base on current subscriptions from the impact of price. The identification of structural parameters of our model in the data depends critically on the ability to consistently estimate the coefficients on the installed base variables in (10). One problem mentioned above is the unobserved heterogeneity across countries, which we address by fixed effects. Another problem is the appropriate choice of the perception lag δ. Unless we have more detailed information on how frequently the consumers update their installed base estimate, the choice of perception lag will be ad hoc and in practice determined by data frequency. Yet another problem, pointed out by Hartmann et al. (2008) in relation to the Bass (1969) model is that the relationship between the installed base and current network might be driven by serial correlation in sales-related unobservables over time. We address these concerns by comparing the estimated model using various perception lags. We also test for serial correlation of residuals in the model to see if omitted unobservable variables are a potential concern. The identification of price coefficient β is subject to the usual endogeneity concerns, as prices may be set in direct response to a change in subscriber base. Utilizing the panel nature of the data, we construct instrumental variables based on the geographical proximity between countries (Hausman, 1997). To the extent that there are some common cost elements in the cellular service provision across regions (e.g., costs of equipment and materials), we can instrument for prices in a given country by average prices in all other countries of the region (Grajek and Kretschmer, 2009). For instance, prices in Germany can be instrumented with a cellular price index for the rest of Western Europe. The identification assumption we make is that while unobserved cost shocks are correlated across countries in a given region unobserved demand shocks are not. We believe that this assumption is reasonable given language and cultural differences across our sample countries. In particular, advertising campaigns—a common example of correlated demand shocks across states in the US—will typically be designed and run at the national level, so they are uncorrelated across countries. The strength of these geographical 14 We estimated such a model and the installed base parameters turned out statistically insignificant.

M. Grajek, T. Kretschmer / International Journal of Industrial Organization 30 (2012) 496–507 Table 2 Baseline estimation results.

Table 3 Identified structural parameters from the baseline estimation results. (1)

(2)

(3)

Estimation:

OLS

IV

IV FE

GDP per capita

0.001⁎⁎⁎ (0.000) −0.259⁎⁎⁎ (0.076) 1.091⁎⁎⁎

0.002⁎⁎⁎ (0.001) −0.623⁎⁎⁎ (0.151) 1.042⁎⁎⁎

0.023⁎⁎⁎ (0.005) −1.845⁎⁎⁎ (0.222) 0.770⁎⁎⁎

(0.068) −0.147⁎⁎ (0.065) 0.049⁎⁎⁎ (0.016) 0.055⁎⁎ (0.026) 1134

(0.076) −0.118⁎ (0.065) 0.129⁎⁎⁎ (0.035) 0.223⁎⁎⁎ (0.072) 1112

(0.203) −0.370⁎⁎ (0.151) 0.246a (0.255) 1.20

Average price Lagged penetration Lagged penetration squared Constant Autocorrelationb N

501

1057

⁎p b 0.1, ⁎⁎p b 0.05, ⁎⁎⁎p b 0.01. Robust standard errors in parentheses. a Mean and standard deviation of the country-specific fixed effects. b In column (3) we report the Arellano–Bond test statistic for AR(2) in first differences. A regression-based test for the autocorrelation in the error term is reported in columns (1) and (2).

instruments depends on the extent to which the cost structure of 2 G operators is correlated across countries. The existence of a global input market for the telecommunications industry suggests that cost structures will be significantly correlated. Finally, while critical mass manifests itself in upward-sloping demand, it is helpful to devise a more formal test in order to identify critical mass. The test we propose rests on the intuition that in the presence of critical mass, the maximum (choke) price at which consumers are willing to buy occurs at a positive installed base rather than zero. We can then test whether this positive installed base, Xmax, is statistically different from zero. If so, this implies that there is an upward sloping part of the demand curve and hence critical mass. Details can be found in Appendix A.5. 4.3. Baseline estimation results We estimate Eq. (10) using OLS, Instrumental Variables (IV), and panel data techniques to accommodate the endogeneity of price and the unobserved heterogeneity across markets. Results are reported in Table 2. Column (1) reports the OLS results, column (2) the IV results, and column (3) the results of the IV with country-specific fixed effects (IV FE). For the identification of the price coefficient in the IV estimations (columns 2 and 3) we use the average price in other countries in the region as an instrument as explained above., 15, 16 The instrument is very strong as evidenced by the first-stage statistics in the FE regression (reported in Appendix A.6); the coefficient on price in other countries in the region is positive and highly significant, as expected. In the IV FE regression (column 3) we additionally use GMM-type instruments for identification of the coefficients on the lagged dependent variable and the squared lagged dependent variable (Arellano and Bond, 1991); here, the identifying assumption is the lack of serial correlation in the estimated demand equation, which can be empirically tested. Consequently the IV 15 We do not report the overidentification test of the instrument because the price variable is exactly identified in our model, i.e. we have one instrument for one endogenous variable. 16 Our results are robust to the inclusion of landline prices, the price of a close substitute. The fixed line price is positive and significant in the OLS and the IV regressions as expected. In the IV FE regression the coefficient is positive but not significant. Thus, the impact of landlines seems to be controlled by the country-specific effects and does not affect our results much. Given that fixed-line price is not significant in our most ambitious specification (IV FE) and it significantly limits our sample size, we decided not to include this specification.

Estimation:

(1)

(2)

(3)

OLS

IV

IV FE

b

(0.056) 0.006⁎⁎⁎ (0.001) 0.259⁎⁎⁎

0.207⁎⁎⁎ (0.030) 0.003⁎⁎⁎ (0.000) 0.623⁎⁎⁎

0.133a (0.138) 0.012⁎⁎⁎ (0.003) 1.845⁎⁎⁎

c

(0.076) 4.220⁎⁎⁎

(0.151) 1.674⁎⁎⁎

(0.222) 0.417⁎⁎⁎

a0 a1

d Xmax

0.188⁎⁎⁎

(1.184) −0.570⁎⁎ (0.221) 0.310⁎⁎⁎ (0.107)

(0.435) −0.190⁎ (0.104) 0.180 (0.231)

(0.144) −0.201⁎⁎ (0.097) −0.311 (0.397)

⁎p b 0.1, ⁎⁎p b 0.05, ⁎⁎⁎p b 0.01. Standard errors calculated with the delta method based on the coefficients in Table 2. a Mean and standard deviation of the country-specific fixed effects normalized by the price coefficient.

FE regression is estimated by GMM. Note that only the IV FE regression allows for country-specific unobserved heterogeneity; the OLS and the IV regressions implicitly assume a0i = a0 for all i. We first discuss the regression results and their robustness across specifications, and then assess their implications for critical mass. Comparing coefficients across the different specifications in Table 2, we see that all coefficients are statistically significant and have the expected signs. Wealth measured by GDP per capita positively affects cellular penetration and price has a negative effect in all three regressions. Moreover, the lagged installed base of subscribers has a positive and diminishing effect on current subscriptions, indicating diminishing marginal installed base effects (Swann, 2002). As expected, we also find the price effect to be larger in our IV regressions (columns 2 and 3) because they correct for reverse causality and omitted variable bias. Failing to control for reverse causality would bias the price effect downwards because operators might have an incentive to increase prices (or at least decrease at a slower pace) as the installed base grows. Our preferred specification is the IV FE not only because it is the most flexible, but also because it does not suffer from serial correlation in the error term, thereby yielding consistent estimates of the lagged dependent variables' coefficients. The test statistics reported at the bottom of Table 2 show that only in the IV FE regression the null hypothesis of no serial correlation cannot be rejected. Our estimations let us recover the structural parameters of our model as outlined above. We can use the parameters reported in Table 3 to identify combinations of installed base (Xt − 1) and prices (pt) that give an upwardsloping long-run demand curve—points at which critical mass occurs. Fig. 2 gives steady-state demand functions based on the parameters in Table 3. Fig. 2 shows that the existence of critical mass in the global cellular telephony market is supported by our OLS and IV regression results. In the OLS regression, critical mass exists if average price per minute is between 31 and 37 US cents. Below this range, the market does not exhibit critical mass, and only the high-adoption equilibrium exists; above it, demand is zero. Within this range, critical mass ranges from 0% to approximately 31% installed base depending on price. For example, for prices slightly below 37 US cents, an installed base of roughly 31% of the market would be needed to facilitate the jump from the low-adoption to the high-adoption equilibrium. 17 Conversely, if price was around 31 US cents, the diffusion process 17 Specifically, we obtain 31% of the market at the price of 36.8 US cents as the maximum of the estimated demand curve (Xmax in Table 3). It is statistically different from zero, hence according to our test, we cannot reject the hypothesis that critical mass exists in the model estimated by OLS.

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M. Grajek, T. Kretschmer / International Journal of Industrial Organization 30 (2012) 496–507 Table 4 Estimation results using varying data frequency.

Frequency: GDP per capita Average price Lagged penetration Lagged penetration squared Constanta Autocorrelationb N

Fig. 2. Simulation of long-run demand: Baseline model.

would take off immediately at zero installed base converging to approximately 60% penetration in the long run. In the IV regression, in which the estimated price effect is stronger, there are still significant installed base effects, although the price range for critical mass to exist is much smaller, oscillating around 28 US cents. Moreover, the critical mass is not statistically significant in the IV regression, as evidenced by the insignificant Xmax value in Table 3. Once controlling for country fixed-effects in our preferred IV FE regression however, critical mass does not occur; the long-run demand function is downward sloping for the entire range of the installed base. We now explore whether this is robust across subsamples and for alternative data frequencies. 4.4. Alternative data frequencies and varying network effects Our relatively long panel allows us to test alternative frequencies with which the installed base affects consumer demand for cellular services, thereby determining subscription dynamics. 18 Quarterly frequency is the default choice given by the data availability. We also experiment with semiannual and annual frequencies. The results of the IV FE regressions using these three frequencies (Table 4) show significant differences. In particular, installed base shows a stronger effect in the semiannual and annual regressions than in the quarterly regressions: the installed base coefficient is higher and/or the coefficient on the squared installed base is lower (i.e. less negative). This is intuitive as stronger network effects empirically offset the effects of lower updating frequency on diffusion speed. We also observe large changes in the estimated price coefficient, because price codetermines diffusion speed. The direction of these changes is less intuitive. Simulating long-run demand based on the estimates in Table 4 demonstrates, however, that the alternative frequencies hardly matter for critical mass. As evident from panel a in Fig. 3, the longrun demand functions are all downward-sloping. Another line of investigation is to allow for more flexible cross-country heterogeneity. Our baseline model in the previous section pools the entire sample of countries to identify a global range of installed base/price combinations that generate critical mass phenomena in each country. Although we take into account between-country heterogeneity in our estimation of ai,t and bi,t, the degree of installed base effects c and d may also realistically vary across countries or groups of countries. This

Cabral (1990) shows that when the perception lag δ → 0 in his model, i.e. the installed base is updated instantaneously in the consumer demand, the network good's diffusion around the critical mass becomes discontinuous. The lower the frequency of updating, the slower the diffusion around the critical mass becomes.

(2)

(3)

Quarterly

Semiannual

Annual

0.023⁎⁎⁎ (0.005) −1.845⁎⁎⁎ (0.222) 0.770⁎⁎⁎ (0.203) −0.370⁎⁎ (0.151) 0.246 (0.255) 1.20 1057

0.003⁎⁎ (0.001) −0.312⁎⁎⁎ (0.097) 0.893⁎⁎⁎

0.003⁎⁎ (0.001) −0.534⁎⁎⁎ (0.160) 0.766⁎⁎⁎

(0.073) −0.044 (0.055) 0.136 (0.048) 1.56 479

(0.102) −0.027 (0.085) 0.288 (0.082) −0.86 191

⁎p b 0.1, ⁎⁎p b 0.05, ⁎⁎⁎p b 0.01. Robust standard errors in parentheses. a Mean and standard deviation of the country-specific fixed effects. b Arellano–Bond test statistic for AR(2) in first differences.

could imply that some countries do not display critical mass while others do, or that critical mass exists over different price ranges in different economic areas. Three important distinctions among countries are if they are “rich” or “poor,” if they are “pioneers” or “followers” in cellular telephony, and if the cellular market is highly competitive or not. We assess the differences in coefficients by creating three corresponding dummy variables—WEALTH, PIONEER and COMPETITION—and interacting them with the (linear and squared) installed base. Results for the one-quarter lag IV FE regressions are in Table 5.19 All interaction terms in Table 5 are significant, suggesting that the extent of installed base effects substantially differs across subsamples.20 Installed base effects in rich countries (WEALTH) and countries pioneering cellular telephony technology (PIONEER) seem stronger when penetration is small as indicated by the positive coefficient on the linear cellular penetration interaction, but weaker when penetration is high, as shown by the negative coefficient on the squared interaction term. Installed base effects in competitive markets (COMPETITION) see the opposite effect, possibly due to the reduced network effects due to splintering among different competitors (Kretschmer, 2008). Interestingly however, we find that only pioneering countries face critical mass phenomena. As can be seen in panel c of Fig. 3, critical mass exists in pioneering countries in the price range between 36 and 38 US cents. This critical mass is not rejected by our proposed statistical test, as the value of Xmax is equal to 23.9% market penetration and is significant at the 10% level. The stronger installed base effect in pioneering markets driving critical mass suggests that consumers value local installed base more when little is known about the technology overall. A speculative interpretation of this is that once cellular telephony became a global phenomenon domestic peers mattered less for adoption because its use and benefits were widely known and roaming, which implies active networks in other countries, was important in late-adopting countries. Further, in pioneering markets there was little “external knowledge” and experience to draw from so that adopters knew about 2G telephony mainly through their peers in the same country. Moreover, 2G adoption became widespread especially after new users were attracted by text messaging, which was an experience good especially in early-adopting countries. Conversely, users in late-adopting countries probably knew about texting from experiences and reports from other countries—providing a possible source of cross-country spillovers.

19

18

(1)

Our results are robust to including all three interaction terms simultaneously. All but one interaction terms are individually significant. The interaction terms are also jointly significant in each of the three regressions in Table 5. Note that only the indicators interacted with the installed base are present because the individual indicators are not identified in the IV FE regressions. 20

M. Grajek, T. Kretschmer / International Journal of Industrial Organization 30 (2012) 496–507

503

Fig. 3. Long-run demand with varying network effects and data frequency.

Table 5 Varying network effects: estimation results with interaction terms. (1)

(2)

(3)

INDICATOR:

WEALTH

PIONEER

COMPETITION

GDP per capita

0.021⁎⁎⁎ (0.003) −1.761⁎⁎⁎

0.022⁎⁎⁎ (0.004) −1.848⁎⁎⁎

0.022⁎⁎⁎ (0.004) −1.838⁎⁎⁎

(0.228) 0.523⁎⁎ (0.221) −0.101 (0.151) 0.465 (0.322) −0.448⁎

(0.220) 0.617⁎⁎ (0.242) −0.275 (0.177) 0.731⁎⁎ (0.341) −0.453⁎

(0.223) 0.758⁎⁎ (0.370) −0.356 (0.311) -0.002 (0.420) 0.012

(0.229) 0.084 (0.218) 0.429 (0.165) 1.02 1057

(0.261) 0.026 (0.202) 0.331 (0.258) 1.11 1057

(0.337) 0.266 (0.244) 0.266 (0.247) 1.11 1057

Average price Lagged penetration Lagged penetration squared Lagged penetration ⁎INDICATOR Lagged penetration squared ⁎INDICATOR Constant (INDICATOR = 1)a Constant (INDICATOR = 0)a Autocorrelationb N

⁎p b 0.1, ⁎⁎p b 0.05, ⁎⁎⁎p b 0.01. Robust standard errors in parentheses. a Mean and standard deviation of the country-specific fixed effects in the subsample defined by the INDICATOR's value. b Arellano–Bond test statistic for AR(2) in first differences.

Our estimates of critical mass depart from the traditional estimates in one important way: they explicitly recognize the impact of price. Whereas most prior work estimates the market penetration needed for takeoff of a new product (or critical mass) to range from some 2.5% (Golder and Tellis, 1997; Rogers, 2003) to 10% (Mahler and Rogers, 1999) to 25% (Cool et al., 1997) our estimates suggest that the primary driver of takeoff is price. In pioneer countries for example, subscriptions do not take off and market penetration remains small unless the price reaches the threshold of 36 to 38 US cents on average. 21 Once price passes this threshold, takeoff occurs, leading to adoption by the mass market.

4.5. Sensitivity analyses In this section, we study how sensitive long-run demand and critical mass are to changes in the strength of installed base effects and the country-specific demand heterogeneity parameters in our model. This is useful because although we found the observation lag not to affect critical mass excessively, we cannot rule out that ill-specified observation lags 21 To be precise, the model predicts that for prices higher than the threshold the number of subscribers will be exactly zero, a property driven by our chosen uniform type distribution. More generally, we think of zero market penetration as a small penetration of specialized users who are qualitatively different from mass-market users.

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M. Grajek, T. Kretschmer / International Journal of Industrial Organization 30 (2012) 496–507

Fig. 4. Sensitivity of long-run demand to changes in the network effects and the taste distribution parameters.

affect the point estimates of our parameters. Changing them one by one then helps us assess if critical mass is likely to depend disproportionately on individual parameters to be estimated. We first change the estimated installed base effects by one standard deviation from the estimated parameter in our preferred (IV FE on quarterly data) regression in Section 4.5.1, and change the demand heterogeneity parameters in 4.5.2.

4.5.1. Increase in installed base effects The effect of a change in installed base effects is shown in panels c and d of Fig. 4. The other structural parameters, a, the maximum willingness to pay for a subscription when network size is zero, and b, the density of the distribution of consumer types, are left at the values implied by our preferred regression (3) in Table 3 (i.e., a = 0.401, b = 1.845). 22 We increase and decrease the value of each parameter in question by one standard deviation. In panel c, we see that as installed base effects become stronger (i.e., increasing c), the demand function becomes more concave and is close to having an upwardsloping portion (critical mass). In panel d, we find that a change in d (i.e., congestion effects setting in more or less rapidly) does not have much impact on the occurrence of critical mass. We could potentially observe critical mass (for a given price p) earlier when network effects become stronger (i.e., increasing c or decreasing d). This effect is more

22 The parameter a is evaluated at the mean income in the sample: 0.401≈ 0.1334+0.0124⁎21.5.

pronounced in panel c of Fig. 4, as c affects the extent of installed base effects more than d when installed base is small.

4.5.2. Changes in price sensitivity and consumer stand-alone valuation We now show the effects of changes in the distribution of consumer types, again using the values of the other parameters implied by regression (3) in Table 3. Panels a and b of Fig. 4 show the effects of changes in these parameters for installed base effects set at c = 0.417 and d = − 0.201. In panel a, we simulate the long-run demand for the highest and lowest parameter ai,t, as defined in Eq. (7), given the country-specific fixed effect and income in each country. We can see that for larger values of ai,t there is an upward shift as expected. That is, although the average country in our sample does not display critical mass (as seen in Section 4.3), it would be reached at higher prices for higher-income countries. In panel b, we see that the impact of increased density of consumer types' distribution, b, which determines short-run price sensitivity of demand, mirrors the impact of stronger network effects. With higher b, more consumers are willing to subscribe at each price. If b is sufficiently high, installed base effects “kick in” early, generating critical mass at relatively high prices. This is intuitively appealing as countries have different “densities” of high-value consumers that determine when critical mass is reached. Many consumers willing to subscribe at high prices may be enough to generate critical mass and penetrate the mass market. Our counterfactuals indicate that in otherwise identical markets (i.e., with the same structural parameters and price sensitivity),

M. Grajek, T. Kretschmer / International Journal of Industrial Organization 30 (2012) 496–507

markets with more pronounced installed base effects are more likely to display critical mass phenomena while others with more moderate installed base effects (or strong congestion effects) are not. Similarly, more price-sensitive goods are more likely to display critical mass phenomena, suggesting that small changes in price might generate more extreme changes in demand for a good than might be anticipated from a static demand curve. It is important to emphasize that the possible shape of steady-state demand as simulated in Figs. 2 to 4 is influenced by our functional form assumptions. With the assumptions of uniform and bounded valuations, we do not obtain two downward-sloping parts in our demand correspondence as in Fig. 1.23 However, the simulated demand functions in Figs. 2 to 4 approximate the more general function in Fig. 1 because the vertical axis above the minimum critical mass point (i.e., X = 0 for sufficiently high p) is also part of long-run demand. Thus, the intuition behind critical mass dividing high and low (zero in our model) demand regions holds even for our simplifying functional assumptions.

505

(Grajek, 2010), as well as integrate richer information about the distribution of tastes in a population (Economides and Himmelberg, 1995). As our empirical implementation shows, the model yields new insights about critical mass phenomena without excessive data requirements. We therefore believe our approach offers an effective alternative to existing models of critical mass, and highlights novel aspects of technologically dynamic markets. Acknowledgments We thank conference participants at WIEM 2007, ITS 2007, EARIE 2007, AOM 2009, Luís Cabral, Manfred Schwaiger, Luc Wathieu, seminar audiences at ESMT and LMU Munich and an Associate Editor and two referees for helpful comments, and Jan Krancke for data access. Appendix A A.1. Derivatives of the function H(.) with respect to xt − δ and p

5. Conclusion We develop a simple structural econometric model of demand for a new network technology to identify critical mass that can easily be implemented empirically. Most existing papers either propose a rigorous theoretical model or provide a simple empirical heuristic. We define critical mass points as combinations of price, installed base effects and current installed base that lead to multiple equilibria. The parameters recovered from the empirical implementation of the model can be used to identify and analyze critical mass, the point at which “further diffusion is self-sustaining” (Rogers, 2003), which implies that (rapid) diffusion occurs without any further changes in price. In our empirical setting, we observe critical mass for digital cellular telephony in some countries and find that timing of the technology introduction has an important effect on the existence of critical mass. Specifically, pioneering markets exhibit stronger network effects (and higher likelihood of critical mass) perhaps because the experience with the technology is obtained primarily from local markets early on in the global diffusion process. Later on, the installed base in pioneering countries may put other countries “over the hump” towards widespread adoption due to spillover effects between countries, which are not explicitly picked up by the model. Of course, countries being stuck in a low-adoption equilibrium may seem unrealistic today as mobile phones are ubiquitous in every country, but an important finding of our model is that with critical mass the diffusion path to the eventual (high-adoption) equilibrium contains unstable and self-sustaining periods of diffusion. Our model extends the empirical literature on critical mass and sales takeoff in several ways. First, and most importantly, we can estimate the range of prices for which takeoff of sales occurs. Second, our method can accommodate in the same estimation procedure different markets in terms of size and income heterogeneity, which is important for cross-country or even cross-technology comparison. Third, we identify two “diffusion regimes” over time, one in which diffusion is driven by price changes (along the downward-sloping parts of the demand curve), and one in which diffusion occurs endogenously without further price changes (in the critical mass region of demand). Fourth, we show that the timing of technology introduction matters for the existence of critical mass in new technologies. Despite the limitations inevitably imposed by our simple model, we believe it to be sufficiently flexible to be applied to a variety of empirical settings. In particular, it can be modified to accommodate between-network competition at the firm rather than market level

23 Fig. 1 implies a long tail in the distribution of types that captures consumers with very high willingness to pay even if no one else subscribes (the innovators).

For simplicity, we slightly abuse the notation in this section by treating price as a constant parameter p. Recall that v⁎t is an implicit function of xt − δ and p is defined by
  u vt ; xt–δ ¼ p:

ðA:1Þ

To calculate the derivative of H with respect to the lagged network size xt-δ, we first apply the chain rule to the definition of H(.) given in (3). We obtain

 ∂F vt  ∂vt  ∂ H vt ¼ − ⋅ ∂xt−δ ∂vt  ∂xt−δ

ðA:2Þ

The first term on the RHS of (A.2) is just the density of v at v⁎t . To calculate the second term, note that the total derivative of u(v⁎t , xt − δ) with respect to xt − δ must stay constant in order to satisfy Eq. (A.1). This holds for
 ∂u vt ; xt−δ ∂uðvt ; xt−δ ∂vt  ¼− ⋅ ∂xt−δ ∂vt  ∂xt−δ

ðA:3Þ



∂vt and substituting that into (A.2) yields Solving (A.3) for ∂x t−δ



 ∂uðvt ; xt−δ ∂ H vt  ¼ f vt  ⋅ ∂xt−δ ∂vt 

!−1 ⋅


 ∂u vt ; xt−δ ; ∂xt−δ

ðA:4Þ

where f is the density function of v. Examination of (A.4) gives the following lemma. Lemma 1. Whenever the solution to Eq. (2) exists and is unique so that v⁎t is well defined, the extent of network externalities measured vt  ;xt−δ Þ by η≡ ∂uð∂x determines the slope of the function H in the xt-δ dot−δ main, such that (i) H is non-decreasing if and only if network effects are non-negative, (ii) the slope of H equals zero if there are no network effects, and (iii) the slope of H increases with network effects whenever the density of types is strictly positive. Proof of Lemma 1. According to (A.4), the slope of function H in the xt-δ domain is determined by a product of the three components: density of consumer types, inverse of the partial derivative of the willingness-to-pay function with respect to consumer type, and partial derivative of the willingness-to-pay function with respect to the installed base, all evaluated at the indifferent type v⁎t . The first component of this product is non-negative (density function), the second is

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M. Grajek, T. Kretschmer / International Journal of Industrial Organization 30 (2012) 496–507

positive (due to the assumed rank ordering), and the third is the extent of network effects. ■ Analogously, to calculate the derivative of H with respect to the price p we first apply the chain rule to obtain
 ∂F vt  ∂vt  ∂
 : H vt ¼ − ∂vt  ⋅ ∂p ∂p

ðA:5Þ

Then we note that from (A.1) we have
 ∂u vt ; xt−δ ∂vt  ¼ 1; ⋅ ∂p ∂v 

is a critical mass point, that is, there exists a price p⁎ for which H(.) crosses the 45-degree line hfrom ibelow at x'. Therefore, there must exist a neighborhood of x',  x0 ; x0 , such that  x0 bx1 bx0 and H(.) has a h i 0 0 positive slope over  x ; x . Note that the density of types that correh i 0 0 sponds to  x ; x and p⁎ must be strictly positive for H(.) to have a h i positive slope over  x0 ; x0 (Lemma 1). An increase in network effects h i in the neighborhood  x0 ; x0 thus increases the slope of H(.) over the entire neighborhood, shifting the critical mass point x' to the left.

ðA:6Þ

t

■

A.3. Relation to the Bass (1969) model

and substitute to get
!−1
 ∂u vt ; xt−δ ∂ : Hðvt Þ ¼ −f vt ⋅ ∂p ∂v 

ðA:7Þ

t

Lemma 2 follows directly from examination of (A.7). Lemma 2. Whenever the solution to Eq. (2) exists and is unique so that v⁎t is well defined, changes in price p determine the shifts of the function H in the xt–δ domain, such that H(v⁎(xt-δ,p') ≥ H(v⁎(xt-δ,p”) for every xt-δ and H(v⁎(xt-δ,p') > H(v⁎(xt-δ,p”) for at least some xt-δ if p' b p''.

In this Appendix, we illustrate that our model applied to a single market simplifies to the original Bass (1969) diffusion equation. First, we specify consumer v's willingness-to-pay function in a single market as follows: 2

uðv; xt−1 Þ ¼ v þ cxt−1 þ dxt−1 ;

ðA:8Þ

where c and d are parameters that determine the extent of network effects, with the square term capturing possible nonlinearities, as before. Further, assume v to be uniformly distributed over (-∞, a] with density b > 0. Given these functional forms, diffusion Eq. (4) becomes 2

Proof of Lemma 2. Because the density of types is by definition non-negative (and strictly positive over some range), and the derivative of the willingness-to-pay function with respect to consumer type is positive due to the assumed rank ordering, (A.7) is always non-negative and strictly positive for at least some values of the installed base xt−δ. ■ A.2. Proofs of Propositions

xt ¼ ab−bpt þ bcxt−1 þ bdxt−1 :

ðA:9Þ

The structural parameters of this model can be recovered from the coefficients of the following estimation equation: 2

xt ¼ α þ βpt þ γ 1 xt−1 þ γ2 xt−1 þ εt ;

ðA:9Þ

where εt denotes the error term. Eq. (A.9) simplifies to the original Bass model if β = 0. To see this, rearrange the terms to obtain: 2

xt −xt−1 ¼ α þ ðγ 1 −1Þxt−1 þ γ2 xt−1 þ εt : Proof of Proposition 1. First, we prove that the downward-sloping parts of the long-run demand consist of stable equilibria and the upward-sloping parts are unstable. The long-run demand condition (5) implies that the long-run equlibria in our model correspond to the fixed points of the function H(.). The stable fixed points of the function H(.) are the long-run attractors of the dynamic process described by Eq. (4) (illustrated by the arrows in the upper panel of Fig. 1). This means that for a fixed point to be stable, the function H(.) must cross the 45-degree line from above. The reverse is true at unstable fixed points (critical mass points): the function H(.) must cross the 45-degree line from below. It follows that a price decrease that shifts the function H(.) upwards (Lemma 2) moves the stable fixed points to the right and the unstable ones to the left. Hence, downward-sloping parts of long-run demand must consist of stable, and upward-sloping parts of unstable, equlibria. Second, we prove that network effects must be strong enough for the unstable equilibria to exist. For an unstable equilibrium to exist, the function H(.) must cross the 45-degree line from below, which is possible if and only if the network effects are strong enough. This follows from Lemma 1, which shows that the slope of function H(.) increases with, and is zero without, network effects. ■ Proof of Proposition 2. The first part of Proposition 2 follows immediately from Proposition 1, which says that unstable equilibria are located on the upward-sloping part of the long-run demand function. Whenever critical mass exists, an increased (decreased) price leads to a higher (lower) critical mass. The second part of Proposition 2 follows from Lemma 1. It states that the slope of H(.) increases with network effects whenever the density of types is positive. Suppose that x'

ðA:10Þ

The left-hand side of (A.10) corresponds to sales at t and the righthand side is a square function of cumulative sales through period t-1, which matches exactly the discrete analog of the Bass (1969) diffusion equation. A.4. Derivation of Eq. (9) Given the willingness-to-pay function (6), an indifferent consumer in market i and time t, v*i,t, is given by 

2

pi;t ¼ vi;t þ cX i;t−1 þ dX i;t−1 ; hence 

ðA:11Þ

2

vi;t ¼ pi;t −cX i;t−1 −dX i;t−1 :

ðA:12Þ

All types higher than v*i,t subscribe to the service. The number of subscribers in market i and time t under the uniform distribution assumption is then given by (ai,t − v*i,t )bi,t, which equals 2

xi;t ¼ ai;t bi;t −bi;t pi;t þ bi;t cX i;t−1 þ bi;t dX i;t−1 :

ðA:13Þ

Eq. (9) results from substituting (7) and (8) for the parameters ai,t and bi,t in (A.13). A.5. A formal test of critical mass Taking advantage of the simple functional forms in our model we observe that if multiple equilibria are to exist, the steady-state demand function must exhibit an upward sloping part and hence the

M. Grajek, T. Kretschmer / International Journal of Industrial Organization 30 (2012) 496–507

maximum must be achieved at a positive level of the subscriber base. Thus, one intuitive yet formal test of multiple equlibria consists of computing the level at which the maximum is achieved and testing whether this level is significantly different from zero. Given the functional form assumptions the steady-state (inverse) demand is pi ¼ −

α 0i α 1 γ −1 γ 2 Xi − 2 Xi : − ðGDP=POPÞi − 1 β β β β

ðA:14Þ

And the level at which the maximum price is achieved is 1 X max ¼ 1−γ 2γ2 . We report Xmax along with the other identified structural parameters of the model in Table 3. A.6. First-stage results of the baseline IV estimation Table A1 Baseline estimation results: first stage. Dependent variable:

Average price

Estimation:

IV

GDP per capita

0.002*** (0.000) 0.559*** (0.053) −0.015 (0.061) −0.005 (0.041) 0.057** (0.024) 1112

Others' average price Lagged penetration Lagged penetration squared Constant N ⁎p b 0.1, ⁎⁎p b 0.05, ⁎⁎⁎pb 0.01. Robust standard errors in parentheses.

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